biogeosciences a total quasi-steady-state formulation of

Biogeosciences 10 8329ndash8351 2013wwwbiogeosciencesnet1083292013doi105194bg-10-8329-2013copy Author(s) 2013 CC Attribution 30 License

Biogeosciences

Open A

ccess

A total quasi-steady-state formulation of substrate uptake kineticsin complex networks and an example application to microbial litterdecomposition

J Y Tang and W J Riley

Earth Science Division Lawrence Berkeley National Laboratory (LBL) Berkeley CA USA

Correspondence toJ Y Tang (jinyuntanglblgov)

Received 21 May 2013 ndash Published in Biogeosciences Discuss 28 June 2013Revised 27 September 2013 ndash Accepted 17 November 2013 ndash Published 16 December 2013

Abstract We demonstrate that substrate uptake kinetics inany consumerndashsubstrate network subject to the total quasi-steady-state assumption can be formulated as an equilib-rium chemistry (EC) problem If the consumer-substratecomplexes equilibrate much faster than other metabolicprocesses then the relationships between consumers sub-strates and consumer-substrate complexes are in quasi-equilibrium and the change of a given total substrate (freeplus consumer-bounded) is determined by the degradationof all its consumer-substrate complexes In this EC formu-lation the corresponding equilibrium reaction constants arethe conventional MichaelisndashMenten (MM) substrate affinityconstants When all of the elements in a given network are ei-ther consumer or substrate (but not both) we derived a first-order accurate EC approximation (ECA) The ECA kineticsis compatible with almost every existing extension of MMkinetics In particular for microbial organic matter decom-position modeling ECA kinetics explicitly predicts a specificmicrobersquos uptake for a specific substrate as a function of themicrobersquos affinity for the substrate other microbesrsquo affinityfor the substrate and the shielding effect on substrate uptakeby environmental factors such as mineral surface adsorption

By taking the EC solution as a reference we evaluatedMM and ECA kinetics for their abilities to represent severaldifferently configured enzyme-substrate reaction networksIn applying the ECA and MM kinetics to microbial modelsof different complexities we found (i) both the ECA and MMkinetics accurately reproduced the EC solution when multi-ple microbes are competing for a single substrate (ii) ECAoutperformed MM kinetics in reproducing the EC solutionwhen a single microbe is feeding on multiple substrates (iii)

the MM kinetics failed while the ECA kinetics succeeded inreproducing the EC solution when multiple consumers (iemicrobes and mineral surfaces) were competing for multi-ple substrates We then applied the EC and ECA kinetics to aguild based C-only microbial litter decomposition model andfound that both approaches successfully simulated the com-monly observed (i) two-phase temporal evolution of the de-composition dynamics (ii) final asymptotic convergence ofthe lignocellulose index to a constant that depends on initiallitter chemistry and microbial community structure and (iii)microbial biomass proportion of total organic biomass (litterplus microbes) In contrast the MM kinetics failed to realis-tically predict these metrics We therefore conclude that theECA kinetics are more robust than the MM kinetics in repre-senting complex microbial C substrate and mineral surfaceinteractions Finally we discuss how these concepts can beapplied to other consumerndashsubstrate networks

1 Introduction

Many natural systems involve processes that can be modeledas consumerndashsubstrate (or consumerndashresource in a broadercontext) interactions These interactions include but arenot limited to (i) multicomponent adsorption in aqueouschemistry (eg Jennings et al 1982 Choy et al 2000)(ii) aerosol and cloud droplet interactions in atmosphericchemistry (eg Pilinis et al 1987 Jacobson et al 1996)(iii) protein interaction networks in molecular biology (egChilds and Bardsley 1974 Ciliberto et al 2007) and (iv)many more in natural ecosystems such as plantndashmicrobe

Published by Copernicus Publications on behalf of the European Geosciences Union

8330 J Y Tang and W J Riley Total quasi-steady-state formulation of substrate uptake kinetics

competition for inorganic nitrogen and phosphorus (egReynolds and Pacala 1993 Lambers et al 2009) plantcompetition for light (eg Dybzinski et al 2011) mi-crobial competition for carbon substrates and mineral nu-trients (eg Caperon 1967 Moorhead and Sinsabaugh2006 Allison 2012 Bouskill et al 2012) algae compe-tition for mineral nutrients (eg Tilman 1977 Follows etal 2007) and predator competition for prey (eg Holling1959a Arditi and Ginzburg 1989 Ginzburg and Akcakaya1992 Vayenas and Pavlou 1999 Abrams and Ginzburg2000 Koen-Alonso 2007) Because of this prevalence ofconsumerndashsubstrate interactions in natural systems particu-larly in ecosystem dynamics many mathematical develop-ments have been proposed to interpret and predict ecosystembehavior under a wide range of environmental and biolog-ical conditions (eg Lotka 1923 Volterra 1926 Holling1959b Campbell 1961 Murdoch 1973 Williams 1973Tilman 1977 Pasciak and Gavis 1974 Persson et al 1998Maggi et al 2008 Bonachela et al 2011 Bouskill et al2012) In this study we present developments focusing onthe consumerndashsubstrate network that regulates organic matterdecomposition However our results should be applicable toany problem that can be similarly formulated as a consumerndashsubstrate network

In general the growth of any biological organism mini-mally involves two steps (i) substrate uptake and (ii) sub-strate assimilation Once a substrate is captured it is as-similated to produce energy and biomass for a series ofmetabolic processes including but not limited to cell main-tenance enzyme production cell division and reproduc-tion Therefore explicit modeling of the interactions be-tween many consumers substrates and their habitats re-quires a consistent mathematical representation of substrateuptake under a wide range of biotic and abiotic conditionsAmong the many existing substrate uptake kinetics (Hill1910 Michaelis and Menten 1913 Burnett 1954 Holling1959b Cleland 1963) the MichaelisndashMenten (MM) kinetics(or equivalently Monod (Monod 1949) or Hollingrsquos type II(Holling 1959b) kinetics) is the most widely applied becauseof its simple form solid theoretical foundation (eg Liu2007) and successes under a wide range of conditions (egHolling 1959b Tilman 1977 Reynolds and Pacala 1993Legovic and Cruzado 1997 Hall 2004 Kou 2005 Rileyand Matson 2000 Maggi et al 2008 Allison 2012)

In their seminal paper Michaelis and Menten (1913) as-sumed that enzymes and substrates adsorb to each otherto form enzyme-substrate complexes By assuming theenzyme-substrate complex is of a much lower concentrationthan that of the substrate they obtained by law of mass ac-tions the so-called MM kinetics which states

v =VmaxS

KS+ S(1)

wherev (mol sminus1) is the substrate uptake rateVmax (mol sminus1)

is the maximum substrate uptake rateKS (mol mminus3) is

the half saturation (or substrate affinity) constant andS(mol mminus3) is the free substrate concentration (a full list ofsymbols is given at the end of the text) Later Briggs andHaldane (1925) derived Eq (1) from the enzyme-catalyzedreaction

S+Ek+1harrkminus1

Ck+2rarrP +E (2)

where E (mol mminus3) is (free) enzymeC (mol mminus3) isenzyme-substrate complex from binding (free) substrateS

to enzymeE P is the product (mol mminus3) resulting from theirreversible part of reaction (2)k+

1 (m3 molminus1 sminus1) andk+

2(sminus1) are forward reaction coefficientskminus

1 (sminus1) is the back-ward reaction coefficient andKS =

(kminus

1 + k+

2

)k+

1 Laterstudies (eg Segel and Slemrod 1989 Schnell and Maini2000) indicated that Eq (1) was obtained with the stan-dard quasi-steady-state assumption (sQSSA) which statesthat dC

dt asymp 0 and dSdt = minusk+

1 SE+ kminus

1 C (note that dSdt is the

changing rate of the free substrate which is different fromthe total substrate being used in the total quasi-steady-stateassumption (tQSSA) to be introduced later) Equation (1) isvalid only whenS+KS ET whereET is the total enzymeconcentration including both free and substrate-bound

The MM kinetics has been successful in many applica-tions but there are also many studies demonstrating thatmodifications must be made to account for discrepancies be-tween predictions from applying Eq (1) and observations(eg Cha and Cha 1965 Williams 1973 Suzuki et al1989 Maggi and Riley 2009 Druhan et al 2012) For in-stance Cha and Cha (1965) in studying cyclic enzyme sys-tems noticed that the substrate uptake kinetics when approx-imated with first order accuracy should be

v =VmaxST

KS+ET + ST (3)

Others have obtained Eq (3) or a similar form for var-ious problems (eg Reiner 1969 Segel 1975 Schulz1994 Borghans and De Boer 1995 Borghans et al 1996Schnell and Maini 2000 Wang and Post 2013) In particu-lar Borghans et al (1996) using the total quasi-steady-stateapproximation (tQSSA which also assumesdC

dt asymp 0 but de-

fines a total substrateST = S+C and usesdSTdt = minusk+

2 C)showed Eq (3) is valid ifk+

2 ET ltlt k+

1 (KS+ET + ST)2

Equation (3) is of good accuracy for a much wider range ofsubstrate and enzyme concentrations than Eq (1) It also alle-viates the problem thatv rarr infin asET rarr infin if Eq (1) is used(note Vmax prop ET) In addition when applied to predatorndashprey systems (ie predator =E prey =S) Eq (3) predictspredation depends on both (i) the ratio between predator andprey density and (ii) prey density While we have not foundan example in the literature of Eq (3) being evaluated withpredation data a few studies (eg Vucetich et al 2002Schenk et al 2005) indicated the predation rate is not only

Biogeosciences 10 8329ndash8351 2013 wwwbiogeosciencesnet1083292013

J Y Tang and W J Riley Total quasi-steady-state formulation of substrate uptake kinetics 8331

ratio dependent as proposed in Arditi and Ginzburg (1989)nor only density dependent as implied by MM kinetics

Extension of the MM kinetics to more general cases suchas (i) one enzyme (henceforth without loss of generalitywe use enzymes as the consumers) competing for multiplesubstrates (Schnell and Mendoza 2000 Koen-Alonso 2007Maggi and Riley 2009) (ii) multiple enzymes competing forone substrate (Suzuki et al 1989 De Boer and Perelson1995 Grant et al 1993) and (iii) many enzymes interactingwith many substrates (De Boer and Perelson 1994 Cilib-erto et al 2007) Though the most general case (iii) has beenattempted in various contexts we are not aware of any ana-lytical representation presented in the literature

An analytically and computationally tractable formulationfor case (iii) mentioned above is practically important tosolve many problems such as trait-based modeling of micro-bial ecosystems (Follows et al 2007 Allison 2012 Bouskillet al 2012) and complicated trophic networks (Lindeman1942) Since in a trophic network a predatorrsquos predation ona prey can be practically considered as a random pairing pro-cess between the predator and prey and the feeding processis just the conversion of a prey into internal biomass of thepredator (eg Caperon 1967) the uptake and assimilationof a substrate in a predatorndashprey system can thus be analog-ically described by Eq (2) with the predatorrsquos rates of preyforaging prey escape and prey handling (ie activities likekilling and eating) described respectively by parametersk+

1 kminus

1 andk+

2 Trait-based modeling of a general microbial ecosystem is

different than that of a trophic network due to the unavoid-able interactions between substrates and the aqueous chem-ical environment Particularly the soil microbial ecosystemis further complicated by substrate interactions with vari-ous adsorption surfaces (eg mineral surfaces and biochar)Existing approaches often model the interactions betweenmicrobial substrate uptake and aqueous chemistry and min-eral surface interactions in separate steps while ignoring themathematical similarities between microbial substrate up-take aqueous chemistry and mineral surface interactions(eg Maggi et al 2008 Gu et al 2009) InterestinglyMichaelis and Menten (1913) recognized that Eq (1) couldbe derived from the law of mass action by assuming equilib-rium between the formation and degradation of the enzyme-substrate complexes (though a formal mathematical treat-ment was done by Briggs and Haldane 1925) In their studyMichaelis and Menten also considered a single enzyme thatcould bind with three different substrates and obtained amodified substrate uptake function under the assumptionof negligible enzyme-substrate complex concentration com-pared to substrates Therefore the apparent mathematicalequivalence between the enzymendashsubstrate binding processand that of the chemical interaction between mineral (or or-ganic) surfaces and aqueous chemical species where the lat-ter can usually be described as being in equilibrium (eg

Jennings 1982 Wang et al 2013) should provide a frame-work to model the biotic substrate uptake kinetics and abioticchemistry simultaneously If such a framework can be iden-tified it will consistently describe the substrate uptake by amicrobe (or a consumer in a broader definition) as a func-tion of the microbersquos traits traits of other microbes and theimpacts from different abiotic environmental factors Such aframework will fit well with the idea of game theory (thatis often used to describe biological evolutionary systems)which states ldquothe fitness of an individual is simultaneouslyinfluenced by its own strategy the strategies of others andother features of the abiotic and biotic environmentrdquo (McGilland Brown 2007)

In this study we propose a general approach to model-ing a consumerndashsubstrate network that has an arbitrary butfinite number of consumers and substrates and present itsanalytical approximations under some simplified conditionsWe organize the paper as follows Sect 2 presents the theo-retical aspects of our approach and the design of illustrativenumerical experiments to evaluate our approach and an ap-plication to the modeling of microbial litter decompositionSect 3 presents relevant results and discusses the limitationsand potential applications of our developments and finallySect 4 summarizes the major findings of this study

2 Methods

In this section we first derive the full equilibrium chem-istry (EC) formulation of the consumerndashsubstrate networkand its analytical approximation (ECA) that is at best first or-der accurate We then describe illustrative numerical experi-ments that are used to evaluate the classical MM kinetics andthe ECA kinetics in modeling complex transient consumerndashsubstrate networks including a simple model exercise of themicrobial litter decomposition problem

21 An equilibrium chemistry based formulation ofconsumerndashsubstrate networks

We consider here enzymes as the consumers in ourconsumerndashsubstrate network so that our derivation is basedon enzyme kinetics However the substrate uptake kineticsin other systems can be represented analogously as long asthe following assumptions hold (i) consumers and substratesare well mixed in their environment (ii) consumers and sub-strates only exist in free and complexed states (which couldbroadly include organic and inorganic chemical adsorptionand even engagement in social activities for predatorndashpreysystems) and (iii) the equilibration between formation anddegradation of consumer-substrate complexes is much fasterthan the change of total substrates (free plus complexed) andtotal enzymes (free plus complexed) due to all possible bioticand abiotic sinks and sources Assumption (i) is commonlymade in environmental biogeochemistry although it can be

wwwbiogeosciencesnet1083292013 Biogeosciences 10 8329ndash8351 2013


violated at small scales (eg Molins et al 2012) or underwater-stressed conditions where substrates can become dis-connected from consumers (eg Schimel et al 2011) Allconsumerndashsubstrate networks typically satisfy assumption(ii) A rigorous proof is still lacking for assumption (iii) butKumar and Josie (2011) showed with mathematical rigor thatit holds well for some special consumerndashsubstrate networksCiliberto et al (2007) showed it worked well for proteinndashprotein interactive networks and the MM kinetics (whichalso applies assumption iii) has demonstrated its success innumerous cases (but MM kinetics fails for some cases suchas isotopic fractionation Maggi and Riley 2009 Druhan etal 2012) However Maggi and Riley (2009) concluded thatif assumption (iii) was paired with the sQSSA the resultantsubstrate kinetics failed to describe the isotopic fractionationat high enzyme concentrations

With these three assumptions we consider an enzyme(Ej j = 1 middot middot middot J ) catalyzed reaction that converts a sub-strate (Si i = 1 I ) into a final productPij

Si +Ejk+ij1harrkminusij1

Cijk+ij2rarr Ej +Pij (4)

wherek+

ij1 (m3 molminus1 sminus1) andk+

ij2 (sminus1) are reaction co-

efficients for the forward reactionskminus

ij1 (sminus1) is the reaction

coefficient for the reverse reactions andCij (mol mminus3) is theenzyme-substrate complex formed by bindingSi with Ej

Under the sQSSA (also the tQSSA)Cij is constant dur-ing a modeling (or measurement) time step (Michaelis andMenten 1913) which leads to

SiEjk+

ij1 =

(kminus

ij1 + k+

ij2

)Cij (5)

and can be rewritten as

KSij =kminus

ij1 + k+

ij2

k+

ij1

=SiEj

Cij (6)

Therefore Eq (5) describes the following chemical equilib-rium

Si +EjKSijharr Cij (7)

By taking the remaining procedures to obtain the MM kinet-ics it can be shown thatKSij (mol mminus3) is just the substrateaffinity (or half saturation) constant (see Eq 1 in Michaelisand Menten 1913) Note asKSij rarr infin the complexationbetween substrateSi and enzymeEj becomes increasinglydifficult

For a reaction network that involves many substrates andenzymes one can write a chemical equilibrium for each re-action in the form of Eq (7) Therefore the reaction networkcan be viewed as an equilibrium chemistry (EC) problemwhich have been intensively studied in atmospheric aerosol

chemistry (Pilinis et al 1987 Jacobson et al 1996 Ja-cobson 1999) and reactive transport modeling (eg Jen-nings et al 1982) This EC formulation enables one to useexisting software such as MINTEQ (Felmy et al 1984)SOILCHEM (Sposito and Coves 1988) and EQUISOLV(Jacobson 1999) to solve for all the substrate-enzyme com-plexes and then apply the equation

dPijdt

= k+

ij2Cij (8)

to compute the production rate ofPij from processing of sub-strateSi by enzymeEj

Under the sQSSA the change of a free substrateSi dueto the degradation of all its relevant enzyme-substrate com-plexes is

dSidt

= minus

k=Jsumk=1

(k+

ik1SiEk minus kminus

ik1Cik

) (9)

Under the tQSSA (Borghans et al 1996) one defines

SiT = Si +

k=Jsumk=1

Cik (10)

and then by combing Eqs (6) (9) and (10) one obtains

dSiTdt

= minus

k=Jsumk=1

k+

ik2Cik (11)

We then obtain the full EC formulation by combiningEqs (6) (11) and the enzyme mass balance

EjT = Ej +

k=Isumk=1

Ckj (12)

We note that ifk+

ij2 = 0 then the complex formed with en-zymeEj effectively becomes a shelter for any substrate itcan bind to This constraint allows us to quantify the im-pact of different adsorption surfaces (eg mineral surfacesand biochar) on microbial substrate uptake in a consumerndashsubstrate network Further with the great flexibility pro-vided by the EC formulation (Jennings et al 1982 Jacobson1999) one could simultaneously simulate biotic and abioticinteractions for arbitrarily complex networks subjected tocomputational resource constraints In addition we note thatthe development by Cleland (1963) and the binding strategyin the synthesizing unit approach by Kooijman (1998) arejust special cases of the EC formulation

22 An at-best first-order accurate analyticalapproximation to the equilibrium chemistry basedformulation for some special consumerndashsubstratenetworks

If a consumerndashsubstrate network satisfies two conditions ndash (i)binding does not occur between substrates or between con-sumers and (ii) a consumer-substrate complex once formed



does not bind with another substrate or consumer to formnew complexes ndash we find an at-best first-order accurate equi-librium chemistry approximation (ECA) (see Appendix A forderivation details)

Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkTKSkj

+

k=Jsumk=1

EkTKSik

) (13)

where we have assumed the reaction network includesI sub-strates andJ enzymes (a visualizing way to write Eq 13is shown in Fig 1) By combining Eq (11) with Eq (13)this ECA kinetics states that the uptake of substrateSi byconsumerEj depends on (i) the characteristics of the con-sumer and substrate of interest (throughKSij ) and (ii) thecharacteristics of abiotic and biotic interactions with othersubstrates and consumers (throughKSkj andKSik) In par-ticular when applied to predatorndashprey systems the ECA ki-netics indicates that predation rate is neither ratio nor densitydependent a problem that is yet still under debate (Arditiand Ginzburg 1989 Abrams 2000 Vucetich et al 2002Schenk et al 2005 Kratina et al 2009) Next we derive afew interesting results from Eq (13)

First for a reaction that has only one enzyme interactingwith one substrate we have

C11 =S1TE1T

KS11+ S1T +E1T (14)

which is equivalent to Eq (3) When the substrate concentra-tion is much higher than the enzyme concentration such thatthe microbial process barely changes the total substrate con-centration in the temporal window of interestKS11+ S1Tis almost constant and Eq (14) becomes the reverse MM ki-netics (Schimel and Wintraub 2003) When the substrate ischanging significantly while the overall enzyme concentra-tion is much lower than the substrate so thatKS11+E1Tis almost constant Eq (14) is reduced to the classical MMkinetics (Michaelis and Menten 1913)

Second when enzyme concentrations are very high moreinactive enzymes (eg transporters of dead cells) will com-pete with the active enzymes for substrate adsorption conse-quently introducing an inhibition By treating the active andinactive fractions of an enzyme as two different enzymesEq (14) can be reformulated as

C11 =S1TE1T

S1T +KS11

(1+

E1TKS11

+E2TKS12

) (15)

whereE1T (mol mminus3) andE2T (mol mminus3) are the total con-centrations of the active and inactive enzymes respectivelyBy takingα1 as the transient partitioning coefficient betweenactive and inactive enzyme concentrations (ieE1T = α1ETandE2T = (1minusα1)ET with ET = E1T +E2T) Eq (14)can be rewritten as

13 E1 13 13 Ej1 13 Ej 13 Ej+1 13 13 EJ 13

S1 13 KS11 13 13 KS1 j1 13 KS1 j 13 KS1 j+1 13 13 KS1J 13

13 13 13 13 13 13 13 13

Si1 13 KSi11 13 13 KSi1 j1 13 KSi1 j 13 KSi1 j+1 13 13 KSi1J 13

Si 13 KSi1 13 13 KSi j1 13 KSij 13 KSi j+1 13 13 KSiJ 13

Si+1 13 KSi+11 13 13 KSi+1 j1 13 KSi+1 j 13 KSi+1 j+1 13 13 KSi+1J 13

13 13 13 13 13 13 13 13

SI 13 KSI1 13 13 KSI j1 13 KSIj 13 KSI j+1 13 13 KSIJ 13

13

Fig 1 A matrix-based representation of the parameter configura-tion for the ECA substrate kinetics (Eq 13)

C11 =α1S1TET

S1T +KS11

[1+

(α1KS11

+1minusα1KS12

)ET

] (16)

=α1S1TET

S1T +KS11

(1+

ETKI

) whereKI =

(α1KS11

+1minusα1KS12

)minus1and the term after the sec-

ond equal sign is equivalent to Eq (2) derived in Suzuki etal (1989) where they used it to explain the inhibition effectfrom ineffective binding between substrate and inactive cellsWe point out that Eq (16) could be used to represent the inhi-bition effect from soil minerals which can compete for sub-strates analogously as an ineffective enzyme (that does notresult in a new chemical product but may protect the sub-strates from microbial attack)

Third for the case of many enzymes competing for a sin-gle substrate Eq (13) can be reduced to

C1j =S1TEjT

KS1j

(1+

k=Jsumk=1

EkTKS1k

)+ S1T

(17)

Grant et al (1993) used a variant of Eq (17) to represent thecompetitive uptake of a substrate in the presence of many mi-crobes (see their Eqs 3 and 4) However Grant et al (1993)directly generalized the results by Suzuki et al (1989) (with-out explicit derivation) and also implicitly assumed that thereare ineffective enzymes competing for substrates With thislatter assumption Eq (17) can rewritten as

C1j =αjS1TEjT

KS1j

(1+

k=Jsumk=1

EkTKI1k

)+ S1T

(18)



whereαj is the transient active fraction of enzymeEj andthe inhibition constants are

KI1k =

(αk

KS1k+

1minusαk

KS1kd

)minus1

(19)

whereKS1k andKS1kd are affinity constants of the activeand inactive enzymeEk respectively Note that the value ofJ in Eq (18) is half of that in Eq (17) since Eq (18) groupsthe active and inactive fractions of an enzyme into one

Fourth in the case of a single enzyme interacting withmany substrates Eq (13) is reduced to

Ci1 =SiTE1T

KSi1

(1+

k=Isumk=1

SkTKSk1

)+E1T

(20)

WhenE1T is constant Eq (20) can be equivalently rewrittenas

Ci1 =SiTE1T

KSi1

(1+

KSi1

KSi1

k=Isumk=1

SkTKSk1

) (21)

where KSi1 =KSi1 +E1T If further assumingKSi1

E1T such thatKSi1 =KSi1 then Eq (21) is just the multi-component Langmuir isotherm for multicomponent adsorp-tion in aqueous chemistry (eg Choy et al 2000) and hasbeen used for multi-prey predation in predatorndashprey mod-els (eg Murdoch 1973) We also note the multicomponentLangmuir isotherm is based on sQSSA

If there are only two substrates (ieI = 2) Eq (20) canbe rewritten as

C11 =S1TE1TKS11

1+S1TKS11

+S2TKS21

+E1TKS11

(22a)

C21 =S2TE1TKS21

1+S1TKS11

+S2TKS21

+E1TKS21

(22b)

Then by further assumingE1TKS11 andE1TKS21 aremuch smaller than the other terms one obtains the Eq (20)in Maggi and Riley (2009) Druhan et al (2012) have usedEq (20) by Maggi and Riley (2009) to explain sulfur isotopefractionation in a field subsurface acetate amendment exper-iment Our Eq (22) is based on the tQSSA which makesit valid for a wider range of substrate and enzyme concen-trations This contrasts our Eq (22) with Maggi and RileyrsquosEq (20) which is based on the sQSSA and was found toincorrectly predict isotopic fractionations when enzyme con-centrations were comparable or higher than substrate concen-trations

23 Extension to other inhibitory mechanisms

The EC and ECA kinetics inherently account for competi-tive inhibition (inhibition mechanism (i)) including product

competitive inhibition For enzyme kinetics or more broadlymicrobendashsubstrate networks there are three additional maininhibitory mechanisms often considered (Cornish-Bowden1995) (ii) uncompetitive inhibition (inhibitor binds to theenzyme-substrate complex to make the binding ineffective)(iii) noncompetitive inhibition (inhibitor binds equally wellto both free enzyme and enzyme-substrate complexes andreduces the number of effective bindings but does not af-fect the enzymersquos substrate affinity) and (iv) mixed inhibi-tion (a mixture of competitive and noncompetitive inhibitionbut the inhibitor has different affinity for free enzyme and theenzyme-substrate complex)

The EC kinetics is compatible with all these four in-hibitory mechanisms as long as the reaction coefficients canbe properly defined for all the inhibitor binding equationsHowever the simplified ECA kinetics is only able to repre-sent competitive and noncompetitive inhibition (with somemodifications discussed below) Including mixed and non-competitive inhibition is only possible when many substratesare competing for a single enzyme or vice versa (and therelevant mathematics is much more complicated than wehave presented for competitive and noncompetitive inhibi-tions here) In addition as will be demonstrated later (see thenumerical experiments) even the ECA kinetics without in-hibitory mechanisms (ii) (iii) and (iv) are not always highlyaccurate (compared to EC kinetics) nor can they be cali-brated robustly due to parameterization equifinality (ie dif-ferent combinations of parameters can result in very simi-lar model predictions (eg Beven 2006 Tang and Zhuang2008)) Since including these other inhibitory mechanisms(beside competitive inhibition) will generally introduce moreparameters making the simulations more uncertain the gainin mechanistic representation is thus smaller than the loss ofpredictive capability

Nevertheless a first order approximation for the noncom-petitive inhibition can be achieved by first modifying thesubstrate affinity coefficients (used in Eq 13) that are subjectto the inhibitorsIkk = 1 middot middot middot L as

KSij =KSij

k=Lsumk=1

IkTKIijk

+ 1

(23)

whereKIijkk = 1 L are the inhibition coefficients ofeach inhibitor on enzyme-substrate complexCij In derivingEq (24) we assume that any two inhibitors cannot bind si-multaneously to an enzyme-substrate complex

Second substituting the modified substrate affinity coeffi-cientsKSij into Eq (13) one obtains the enzyme-substratecomplex concentration (under the influence of noncompeti-tive inhibition)

Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkT

KSkj+

k=Jsumk=1

EkT

KSik

) (24)



24 Linking with microbial traits

An appealing application of the EC and ECA kinetics isto trait-based modeling of marine and soil microorganisms(Follows et al 2007 Litchman et al 2007 Allison 2012Bouskill et al 2012) In the trait-based modeling approachparameters of the substrate uptake kinetics are determinedby the microorganismsrsquo traits such as cell size and trans-porter density (eg Follows et al 2007 Armstrong 2008Bonachela et al 2011) Both the EC and ECA kineticsare compatible with such concepts and the incorporation ofthese traits can be accomplished efficiently through the dy-namic update of relevant microbial state variables in the nu-merical model For instance to consider the effect of cell size(which affects substrate diffusion between the environmentand the cell) and transporter density (which affects process-ing rate and affinity to the substrate) on substrate uptakeone has the updated substrate uptake (see diagram shownin Fig 2) With the stationary flux assumption (Pasciak andGavis 1974 1975) one obtains the diffusive flux to a spher-

ical cell as8D = 4πDircjnj(Si minus Si

) whereDi (m2 sminus1)

is the diffusivity of the substrateSi in water (when in soilthis diffusivity depends on soil matric potential soil struc-ture and temperature)rcj (m) is the average size of cellj(by assuming a spherical cell shape in the first order approxi-mation) andnj (number of cells mminus3) is the number densityof cell j The impact of advection on the flux8D can also beincluded using the dimensionless Sherwood number (Karp-Boss et al 1996) but that will not change our derivationessentially Further assuming the internal substrateSi con-centration (that is close to the cell) is also stationary (thus8D is equal to the net enzyme-substrate complex formationrate betweenSi and the cellrsquos transporter) one obtains (as afirst order approximation)

Si =4πDircnjSi

k+

ij1Ej + 4πDircnjasymp

4πDircnjSik+

ij1EjT + 4πDircnj (25)

where we have assumed the reverse dissociation of theenzyme-substrate complex (kminus

ij1) is negligibleEj asymp EjTand the changing rate of the enzyme (or transporter) abun-dance due to new growth is much slower than the enzyme-substrate complex equilibration rate Therefore one can rep-resent the enzyme-substrate complex with Eq (7) and a mod-ified equilibrium coefficient

KSij =kminus

ij1 + k+

ij2

k+

ij1

(1+

k+

ij1EjT

4πDircjnj

)(26)

which leads to a new representation of the substrate affinityparameter for Eq (13) as

KSij =KSij

(1+

k+

ij1EjT

4πDircjnj

) (27)

Si +Ejkij 1

k+ij 1SiEj Pij +Ej

Si

E

D

Fig 2 Diagram of the updated substrate uptake process which in-cludes diffusive substrate flux (between the external environmentand near cell environment) and new enzyme production8E HereSi is any environmental substrate abundanceSi is the correspond-ing local (close to the transporter) substrate abundance andPij isthe assimilated product from the processing ofSi by enzymeEj

Under the assumption thatkminus

ij1 k+

ij2 and defining

Vmaxij = k+

ij2EjT one has

KSij =KSij

(1+

Vmaxij

4πDircjnjKSij

) (28)

which extends the modified MM kinetics derived for a singleenzyme single substrate system (Bonachela et al 2011) toan enzymendashsubstrate network of arbitrary size

Equation (29) implies that if a cell increases its volumet-ric transporter density (EjTnj transporters per cell) it de-creases its substrate affinity However consideringEjT =

njψj4πr2cj if a cell j decreases its volumetric size while

keeping the same area-based transporter densityψj (trans-porters mminus2) it can increase its substrate affinity Further bysubstitution of Eq (28) or Eq (29) into Eq (13) one obtainsa new representation of the moisture effect on organic matterdecomposition (through diffusivityDj and aqueous substrateconcentrationSi) that is more mechanistic than the usuallyapplied simple multiplier factors (eg Andren and Paustian1987 Rodrigo et al 1997 Bauer et al 2008 Parton et al1988)

25 Evaluation of the ECA kinetics and the classicalMM kinetics

We focus our evaluation on the efficacy of the ECA (Eq 13)and the MM kinetics (Appendix C) in approximating predic-tions by the EC kinetics and leave the analysis of the impactof the EC and ECA kinetics on trait-based modeling (Eq 28)for future studies In these comparisons we used the ECkinetics as a baseline to predict the enzyme-substrate com-plexes involved in a reaction network with arbitrary numberof enzymes and substrates We implemented the EC kinet-ics with the analytical equilibrium iteration (AEI Jacobson



1999) and then compared its predictions to those from theclassical MM kinetics and the ECA kinetics for different net-work configurations We conducted the evaluation with threegroups of experiments (E1) random sampling (E2) appli-cations to simple microbial models of different complexi-ties and (E3) simulating litter decomposition with a differentcarbon-only model We remark that in all our evaluations allthe substrate kinetics used the same number of parameterstherefore when one formulation is found to perform worsethan others it is inferior in our evaluation framework

In the first group of experiments (E1 random sampling)we tested the hypothesis that our ECA kinetics is more accu-rate than the MM kinetics for arbitrary consumerndashsubstratenetworks Specifically we randomly generated substrateaffinity parameters using an exponential distribution over therelative range 1ndash104 (mol mminus3) which is sufficiently wideto represent the range of microorganisms in the natural envi-ronment (eg Wang et al 2012) The enzymes and substrateconcentrations were then generated from the least informa-tive uniform distributionU [01] We performed 9 scenariosusing combinations of three substrate enzyme (SE) ratios(10 1 and 001) and three network sizes (60 substrates andone enzyme 10 substrates and 60 enzymes and 20 substratesand 20 enzymes) Each scenario has 10 random replicates re-sulting in a total of 9times 10= 90 evaluations We normalizedthe variance of the EC solution for each replicate of the 9 sce-narios and summarized the results using the Taylor diagram(Taylor 2001) which simultaneously presents the correlationcoefficient and root mean square error between the baselineEC solution and solutions using MM or ECA kinetics Sincethe equilibrium reaction Eq (7) is symmetric to representedsubstrates and enzymes the 9 scenarios effectively represent18 different enzymendashsubstrate networks (eg 10 substratesand 60 enzymes is equivalent to 60 substrates and 10 en-zymes) We remark that the high enzyme substrate ratio maynot be ecologically significant for modeling litter decompo-sition such as E3 but it is important to be investigated givenour EC and ECA approaches are also applicable for problemssuch as predatorndashprey systems where moderate to high ratioof predator to prey (analogously to that between enzyme andsubstrate) can easily occur

In the second group of experiments (E2 simple micro-bial models of different complexities) we used the followinggeneric model structure to illustratively evaluate the impactof different substrate kinetics on microbial system dynamics

dSidt

= minus

j=Jsumj=1

k+

ij2Cij i = 1 I (29)

dBjdt

=

i=Isumi=1

microijk+

ij2Cij minus γjBj j = 1 J (30)

wheremicroij (unitless) is the biomass yield rate of microbeBj(mg C dmminus3) from feeding on substrateSi andγj (dayminus1) is

respiration rate which is defined accordingly (in the captionsof the relevant parameter tables) for different models WecomputedCij using EC kinetics MM kinetics (Appendix C)and ECA kinetics (Eq 13) and evaluated the ability of thetwo analytical approximations to reproduce the temporal dy-namics simulated by the EC kinetics We considered four mi-crobial models of different complexities (Tables 1 2 and 3note the fourth model assigned different units to the variablescompared to the other three models in order to use the param-eters from Moorhead and Sinsabaugh (2006) We note thatusing the parameters from Moorhead and Sinsabaugh (2006)is simply a choice of convenience but it is sufficient for aqualitative assessment of the predicted differences betweenour ECA or EC-based models and the MM model) (i) threesubstrates and one microbe (S3B1) (ii) three substrates onemicrobe and one mineral surface (S3B1M1) (iii) one sub-strate and five microbes (S1B5) and (iv) three microbes andthree substrates (S3B3) For the three models with three sub-strates (S3B1 S3B1M1 and S3B3) we related the substratesto water-soluble carbon cellulose and lignin respectivelySince the results from (E2) are applicable to other similarproblems we labeled the three substrates asS1 S2 andS3For model S1B5 we ran the model with different kineticsusing 20 randomly generated parameter sets and evaluatedtheir performance by the relative model error

err(t)=3

N

i=Nsumi=1

∣∣∣∣ yiEC(t)minus yiapp(t)

yiEC(t)+ yiMM (t)+ yiECA (t)

∣∣∣∣ (31)

whereN = 6 the number of model state variables and apprefers to MM or ECA The above metric avoids division byzero as long as the model difference is non-zero

For all models (including S1B5) we specified the relevantparameter values randomly but kept their nominal values inthe ranges documented in the literature (for microbial param-eters see Li et al 1992 Allison et al 2010 Wang et al2012 for mineral adsorption parameters see Mayes et al2012) As an extra comparison we also included the multi-component Langmuir isotherm (ie Eq 21 which we no-tate as ECA-ML) to compute the substrate uptake in modelsS3B1 S3B1M1 and S3B3 Since ECA-ML can be derivedbased on sQSSA and it assumes enzyme concentrations aremuch lower than the substrate concentrations comparing itsperformance with that of ECA and EC will reveal the advan-tage of tQSSA in representing networks with high enzymeconcentrations

We assumed all microbial transporters are generic (so thatthey can capture all substrates that the microbe can process)and are uniformly distributed over the microbersquos cell surfaceThe total transporter abundance of a given microbe is scaledto the microbial biomass with a constantz which is set to005 a number that falls in the middle of values appliedin other studies (Berg and Purcell 1977 Maggi and Riley2009) We used these experiments to test two hypotheses(i) the ECA kinetics is more robust than the MM kinetics in



Table 1Parameter values for microbial models S3B1 and S3B1M1The parameter vectors are presented in the form

(KSij k

+

ij2microij

)

whose units are respectively mg C dmminus3 dminus1 and none The ini-tial microbial biomass is defined in the parentheses afterB1 whoseunit is mg C dmminus3 The mineral surface is characterized with theLangmuir dissociation parameter (equivalentlyKSij ) and the max-imum adsorption capacity (in the parentheses followingM1) whoseunits are respectively mg C dmminus3 and mg C dmminus3 For both mod-els we used a microbial respiration rate 003 dminus1 The microbialparameters were randomly specified based on prior knowledge fromWang et al (2012) and the mineral surface parameters were speci-fied for Alfisol based on Mayes et al (2012)

S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)

Table 2 Parameter ranges for microbial model S1B5 The param-

eter vectors are presented in the form(KSij k

+

ij2microij

) whose

units are respectively mg C dmminus3 dminus1 and none Numbers in theparentheses following the state variables are their initial valueswhose units are mg C dmminus3 All five microbes used a respirationrate 0005 dminus1 The maximum and minimum parameter values werespecified based on Wang et al (2012)

S1 (300)

Minimum Maximumvalues values

B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)

approximating the EC solution and (ii) only the ECA kinet-ics is analytically tractable and sufficiently accurate to modelmicrobialndashmineral surface interactions

For the third set of experiments (E3 simulating litter de-composition) we tested whether the S3B3 model with dif-ferent substrate kinetics can be calibrated to simulate the77-month red pine litter decomposition data of Melillo etal (1989) We first calibrated model S3B3 with both theECA and MM kinetics and analyzed if the calibrated modelscan reproduce the (i) two-phase evolution of remaining or-ganic matter (ii) increase of lignocellulose index (LCI) dur-ing decomposition and (iii) reasonable fraction of microbialbiomass with respect to the remaining organic matter Wethen ran the models with 9 different initial litter chemistries(Table 4) for a qualitative assessment of the extrapolated pre-dictability (based on observational data if available) of thecalibration We were able to obtain some time series data forthe Massachusetts (MA) site (Magill et al 1998) but failedto extract any useful time series data for the Wisconsin (WI)

Table 3Prior parameters for microbial model S3B3 The parameter

vectors are presented in the form(KSij k

+

ij2microij

) whose units

are respectively g C dminus1 and none All values are adapted fromMoorhead and Sinsabaugh (2006) All three respiratory coefficients(ieγj j = 123 as defined in Eq (30) are set to 003 dminus1 Num-bers in the parentheses following the state variables are their initialvalues whose units are g C

S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)

Table 4 Characteristics of initial litter chemistry for the data inlitterbag decomposition field studies in Wisconsin (WI) and Mas-sachusetts (MA) The table is organized based on Table 3 in Moor-head and Sinsabaugh (2006) who obtained data from Aber etal (1984) and Magill et al (1998) The final LCI is model predicted(see Sect 332 for details) We have also extracted time series datafrom the Magill et al (1998) study for model assessment (Figs 15and S1)

Litter type Labile Holocellu- Lignin Initial Finalby site () lose () () LCI LCI

Wisconsin (WI)Sugar maple 448 431 121 022 055Aspen 311 475 214 031 056White oak 324 474 202 030 056White pine 328 447 225 033 059Red oak 300 452 248 035 059

Massachusetts (MA)Red pine 359 386 255 040 067Red maple 477 354 169 032 068Black oak 350 396 254 039 066Yellow birch 434 403 163 029 062

site (Aber et al 1984) from the original literature or by con-tacting the authors In addition we noticed the original datain Magill et al (1998) indicated a rise of lignin during the de-composition for some unexplained reasons (see their Fig 4)We corrected this by replacing the unreasonable lignin data(ie those higher than the initial lignin mass) with the initiallignin mass (see Fig S1 for details)

We solved all microbial models with the mass positive firstorder ordinary differential equation integrator (Broekhuizenet al 2008) This numerical solver deals with stiff and dis-continuous differential equations well and always ensuresmass balance as long as the elemental stoichiometry is prop-erly formulated in the model We ran all models half hourlya time step that was selected through trial and error by evalu-ating the differences of the predictions when using differentmodel time steps For experiment E2 the total runtime formodel S3B1 S3B1M1 and S1B5 were set to 50 days while



Table 5 Best-fit parameters for model S3B3-ECA by optimizing the simulation outputs to the 77-month red pine litter decomposition

experiment data in Melillo et al (1989) The parameter vectors are presented in the form(KSij k

+

ij2microij

) whose units are respectively

g C dminus1 and none The respiratory coefficients (ieγj j = 123 as defined in Eq (31) of the three microbes are respectively set to

001 005 and 0001 dminus1 Numbers in the parentheses following the state variables are their initial values whose units are g C In doing thecalibration we assumed (i)KS1j j = 123 are same for all three microbes (ii)KS22 =KS23 (iii) for microbeBj k+

ij2 i = 123 aresame for all three substrates By further fixingmicroij to the values in the parentheses we effectively had a total 12 parameters in the calibration

S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)

that for model S3B3 runtime was 1500 days All models inexperiment E3 were run for the length of the observations(80 months)

Bayesian inference based calibrations for experiment E3were performed to invert the relevant parameters (see cap-tion of Table 5 for descriptions) of the substrate uptake ki-netics from fitting the model (eg S3B3-ECA) output tothe time series data of the remaining litter mass and ligno-cellulose index (LCI) from Melillo et al (1989) We imple-mented the Bayesian inference using the MCMC algorithmDREAM (Vrugt et al 2008) A uniform prior was used forall the parameters with the cost function (or the negative log-likelihood function) defined by

Jcost= (8σLCI)minus1

k=8sumk=1

∣∣LCIk minus LCIECAk∣∣ (32)

+ (17σMass)minus1

k=17sumk=1

∣∣rMassk minus rMassECAk∣∣

where LCIk is thekth observation of lignocellulose index (ofwhich there are 8 data points) andrMassk is thekth observa-tion of relative remaining organic matter biomass (microbeplus litter of which there are 17 data points) BothσLCI andσMass are set to 001 For the posterior parameters the setthat minimizesJcost is defined as the modal (ie best fitting)parameter

3 Results and Discussion

31 E1 computing enzyme-substrate complexes forlarge networks

For the first set of experiments we found that ECA kineticsperformed better or as well as MM kinetics in approximat-ing the baseline EC solutions When the substrate to enzymeratio was high (ie enzyme availability is limiting decompo-sition green symbols in Fig 3) the ECA solutions agreedwith the EC solutions with correlation coefficients higherthan 095 and root mean square errors less than 05 standarddeviations (σ ) except for 2 (out of 10) random replicates

for caseS(60)E(1)r(10) (ie 60 substrates 1 enzyme anda substrate to enzyme abundance ratio of 10 similar nomen-clature is used henceforth) and 3 (out of 10) replicates forcaseS(20)E(20)r(10) whose correlation coefficients werestill good (sim 080) and the corresponding root mean squareerrors were between 05σ and 15σ

MM kinetics also achieved good accuracy in approximat-ing the EC solution with correlation coefficients between075 and 097 but in general higher root mean square er-rors For caseS(10)E(60)r(10) MM kinetics only achieveda correlation coefficient of 080 and root mean square er-rors greater than 05σ whereas ECA kinetics achieved cor-relation coefficients ofsim 099 and root mean square errorssmaller than 03σ However the worst approximations (interms of root mean square error) by the MM kinetics (ietwo replicates green diamond symbols forS(60)E(1)r(10))were better than those from the ECA kinetics (for these twopoorly simulated replicates)

Similarly contrasting results were found for the caseswhen the substrate to enzyme ratio was one (purple symbolsin Fig 3) (i) the best approximation by the ECA kineticswas better than that using the MM kinetics and (ii) the MMkinetics resulted in 2 outliers for the caseS(20)E(20)r(1)with root mean square errors greater than 2σ and correla-tion coefficients less than 090 ECA kinetics also produced4 random replicates (2 for caseS(10)E(60)r(1) and 2 forcaseS(20)E(20)r(1)) that had correlation coefficients closeto 08 but the root mean square error was less than 15σ

When substrate was limiting (blue symbols) both the ECAand MM kinetics produced poor approximations (with moreoutliers) compared to the EC solutions Both approaches pro-duced 4 outliers (2 for caseS(10)E(60)r(001) and 2 for caseS(20)E(20)r(001)) with correlation coefficients between070 and 080 and root mean square errors greater than 1σ The worst results (the 2 replicates forS(20)E(20)r(001)) byMM kinetics were again worse than those by ECA kinetics

When all sampling experiments were normalized together(cyan circles) we found the ECA kinetics better approx-imated the baseline EC solution (with similar coefficientsof correlation but smaller root mean square errors) than theMM kinetics did Therefore we summarize that the ECA



Correlat ion CoefficientCo

rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400

ECA MMECS(60)E(1)r(10)S(60)E(1)r(1)S(60)E(1)r(001)S(10)E(60)r(10)S(10)E(60)r(1)S(10)E(60)r(001)S(20)E(20)r(10)S(20)E(20)r(1)S(20)E(20)r(001)Overall

Fig 3A Taylor diagram based summary of the random sampling experiment (E1) that compared the ability of the ECA and the MM kineticsto approximate different enzymendashsubstrate networks simulated by the EC kinetics Each symbol has 10 random replicates The values in theparentheses indicate the number of substrates or enzymes The nomenclatureS(x)E(y)r(z) indicate a network ofx substratesy enzymesand a substrate to enzyme abundance ratio ofz

kinetics is superior to the MM kinetics in representing large-size consumerndashsubstrate networks

32 E2 application to simple microbial models

We found three (EC ECA ECA-ML) of the four differentsubstrate kinetics led to almost identical model predictionsfor the S3B1 scenario over the 50-day time period (Figs 4and 5) The MM predictions deviated from the others slightlyHowever the good agreement between the MM kinetics andthe other kinetic formulations is serendipitous The MM ki-netics is poor in describing enzyme competition in the pres-ence of multi-substrates which has been identified in severalstudies (eg Maggi and Riley 2009 Druhan et al 2012)We also replicated this behavior with an isotope-modelingexample (see Supplement) where it was shown the MM ki-netics has very poor predictability for multi-isotopic fraction-ations (Fig S3) even though it predicted the bulk substrateand microbial dynamics with acceptable accuracy (Fig S2)

When mineral surface interactions were further included(in the S3B1 model) to form the S3B1M1 model we foundthat the ECA kinetics again predicted very similar time se-ries compared to that from EC kinetics (Figs 6 and 7) be-cause both ECA and EC are able to consistently representthe substrate competition by microbe and mineral surfacesHowever both the ECA-ML and MM kinetics resulted inpredictions substantially different from the EC solution TheECA-ML predicted a much faster turnover rate of all threesubstrates because it did not include the inhibitory termdue to the presence of consumers (which can be confirmedby comparing Eq 13 to Eq 21) and thence resulted in a

0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML

Fig 4 Time series of the relevant state variables simulated fromthe applications of the four different substrates uptake kinetics tomicrobial model S3B1

weaker substrate adsorption to the mineral surface Further-more throughout the 50-day period the microbe grew inits biomass and consequently increased its quota to capturesubstrate whereas the mineral surface had a fixed quotawhich together with the growing microbe resulted in a fasterturnover of the three substrates In contrast MM kineticsfavored more substrate adsorption to the mineral surface



04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML

Fig 5Time series of state variable ratios simulated from the appli-cation of the four different substrates uptake kinetics to microbialmodel S3B1

because (by comparing Eq C1 to Eq 13) it did not in-clude the nonlinear competitive inhibition on substrate up-take or the inhibition due to the presence of the consumerIn addition the mineral adsorption sites (counted as adsorp-tion capacity by the state variableM1 in Table 1) is muchmore abundant than microbial transporters which resulted ina strong limitation on microbial substrate uptake This sub-strate limitation (due to mineral surface adsorption) led to amuch lower microbial growth which then led to a greatlyunderestimated substrate turnover rate (by the MM kinetics)Therefore the reduction in turnover of the three substratesin presence of mineral surface adsorption leads us to con-jecture that mineral adsorption (and consequently protectionwhich is not considered here but can be incorporated by us-ing approaches such as in Wang et al 2013) is an impor-tant mechanism impacting organic matter degradation withdepth in the soil profile Implementing the ECA or EC ki-netics could thus potentially avoid the ad hoc parameteriza-tion of soil organic matter (SOM) decomposition rate slow-down with depth as has been implemented in some verti-cally resolved SOM models (eg Jenkinson and Coleman2008 Koven et al 2013) In accordance with the predictedsubstrate dynamics we note that MM kinetics predicted theslowest increase in LCI and fractional microbial C (with re-spect to total organic C including both substrates and mi-crobial biomass) while the ECA-ML kinetics predicted thefastest increase (Fig 7) These findings lead us once againto state that the MM kinetics is qualitatively not appropriatewhen the problem involves multiple substrates and multiple

0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML

Fig 6 Time series of the relevant state variables simulated fromthe applications of the four different substrate uptake kinetics tomicrobial model S3B1M1

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)

(c)MMECAECECAminusML

Fig 7Time series of state variable ratios simulated from the appli-cation of the four different substrates uptake kinetics to microbialmodel S3B1M1

consumers In addition as we will show in experiment E3such deficiencies cannot be remedied through calibration

When five microbes are competing for a single substrate(S1B5) we found the three different substrate kinetics (nowECA-ML kinetics has the same functional form as MM ki-netics when there is only one substrate) made equally goodpredictions (Fig 8) For the worst case (according to the



0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC

Fig 8Comparison of simulations using different substrate kineticsin scenario S1B5(andashf) are from the simulation that showed themost distinctive differences(g) and(h) are summaries of the modeldifferences (with respect to the EC results) from the ensemble of 20random simulations

metrics defined in Eq 32) among the 20 runs with randomlygenerated parameters (see Table 2 for parameters being sam-pled) the prediction by the MM kinetics fit the EC predic-tions better than did the ECA kinetics (Fig 8andashf) When sum-marized over the 20 runs (Fig 8gndashh) we found that the MMkinetics is slightly superior for problems that are in the formof many microbes competing for a single substrate This re-sult is consistent with model result such as that in Bouskillet al (2012) where the ammonia and nitrite oxidizers have avery weak overlap in substrates However if one tries to useisotopic data to improve the parameterization of such modelsthe MM kinetics should be replaced with the ECA kinetics orthe EC kinetics

For model S3B3 three of the four substrate kinetics(ECA-ML ECA and EC) made very similar predictions(see Figs 9 and 10) The predictions from the MM kinet-ics were completely different both qualitatively and quanti-tatively The MM kinetics predicted a gradual reduction inLCI (which stabilized at a constant value smaller than theinitial value see Fig 10b) whereas the other kinetic mod-els predicted a gradual increase in LCI which stabilized ata greater (than the initial) value In addition the MM ki-netics predicted a much higher peak fractional total micro-bial biomass (compared to the total biomass accounting forboth litter and microbes) than did the other substrate kinetics(Fig 10c) We note that ECA-ML ECA and EC all predictedsimilar temporal evolutions of the remaining litter and lit-ter LCI that qualitatively agreed with findings from litterbag

0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML

Fig 9 Time series of relevant state variables simulated from theapplication of different substrate kinetics to microbial model S3B3

0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML

Fig 10Time series of state variable ratios simulated from the ap-plication of the four different substrates uptake kinetics to microbialmodel S3B3

experiments (Fig 10) (i) the litter decomposition has twodistinct phases where the fist phase is fast and the secondphase is much slower and (ii) the LCI increases along withthe decomposition and finally stabilizes at a higher valuethan the initial state (eg Melillo et al 1989 Aber et al1990 Magill et al 1998) This finding leads us to assert thatthe explicit modeling of nonlinear substrate competition (asformulated in EC ECA and ECA-ML) in microbial litter



decomposition is important to represent measured litter dy-namics Once this nonlinear competition is accounted forthe observed temporal evolution of LCI (and consequentlylignin degradation) emerges from the proposed model (ECand ECA) On the other hand MM kinetics is not structuredto account for such nonlinear competition thus one has toenforce an otherwise unconstrained lignin shielding effect oncellulose degradation (though we do not rule out its possibleexistence) to make the model well behaved (eg Moorheadand Sinsabaugh 2006 Allison 2012)

33 E3 simulating litter carbon decomposition

331 Calibrating model S3B3 with different substratekinetics

After calibrating the S3B3-ECA model (model S3B3 imple-mented with ECA same nomenclature are used henceforth)to the 77-month red pine litterbag experiment data we foundthe posterior best-fit parameters led to predictions in goodagreement with the measured time series of remaining lit-ter and LCI (Fig 11) The posterior microbial biomass alsoseemed qualitatively reasonable which stayed below 15 of total organic carbon (including both litter and microbialbiomass) Observational data indicate the fractional micro-bial biomass is relatively low usually within 10 of the totalorganic carbon (Ladd et al 1994 Dilly and Munch 1996)Therefore considering the parameterization equifinality dueto insufficient observational data to constrain the relevant pa-rameters (eg Tang and Zhuang 2008) and the qualitativelygood agreement between posterior simulations and the avail-able data we conclude that ECA kinetics is a better choicethan MM kinetics in our parsimonious framework to repre-sent litter decomposition dynamics

We also ran the S3B3 model with the ECA-ML and EC ki-netics using the same parameters obtained from S3B3-ECAmodel calibration and obtained almost identical predictions(see red and cyan lines in Fig 11) As a sensitivity test wefurther introduced the temperature effect on substrate up-take (labeled as ECA-T in Figs 11 and 13) by applyingthree different Q10 values (whose values are respectively27 15 and 17 based on Bayesian inversion on top of thedefault S3B3-ECA model calibration) to the three biomassyield rates We found the predictions (blue lines in Fig 11)changed slightly compared to the simulations without ac-counting for temperature effects Though the Q10 values arequite uncertain because of data limitations the result indi-cates that temperature was not the single mechanism that ledto the differences between measurement and posterior modelprediction Other mechanisms such as leaching nutrient dy-namics and moisture effects should be investigated in futurestudies to improve the EC and ECA litter decomposition ki-netics

Calibrating the model with the MichaelisndashMenten kinet-ics (S3B3-MM) to the 77-month litterbag data failed to ob-

04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)

ECAminusEnsDataECAECAminusMLECECAminusT

Fig 11 Posterior simulations from calibrating model S3B3-ECAto the red pine litter decomposition experimental data in Melillo etal (1989) ECA-Ens indicates the posterior ensemble simulationsand ECA-T indicates the additional temperature impact on top ofECA (ie the best fitting posterior simulation) The best-fit kineticparameters for ECA EC ECA-ML and ECA-T are in Table 5 Seetext for further details

tain reasonable posterior predictions of the litter decomposi-tion dynamics (Fig 12) A few parameter combinations ledto qualitatively reasonable predictions of the two-phase evo-lution of remaining biomass and the increasing then stabi-lizing behavior of LCI Yet the fractional microbial biomassvaried wildly Many parameter combinations predicted thetotal biomass as microbial-C dominated (almost 100 ) dur-ing the second phase of litter decomposition We also foundthe model S3B3-MM is much more sensitive to the parame-ters than the models implementing ECA-ML ECA and ECkinetics Therefore we conclude that MM kinetics is not suit-able for modeling microbial litter decomposition and SOMdynamics in our more parsimonious framework (than otherexisting models) since these problems always involve multi-ple substrates and multiple microbes

332 The interaction between litter chemistry andmicrobial diversity

Distinct shifts in microbial community structure were ob-served in the posterior model predictions for the 77-monthlitter decomposition experiment (Figs 13 and S4) While wehad no measurements from this experiment to assess whethersuch predictions are realistic some other studies (eg Keeleret al 2009 Wickings et al 2012) indicate such micro-bial community structure shifts often occur in long-term in-cubation experiments For instance Wickings et al (2012)observed significant changes in exoenzyme activities and



02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM

Fig 12Posterior S3B3-MM simulations by calibrating the modelto the 77-month red pine litterbag experimental data in Melillo etal (1989) MM-Ens indicates the posterior ensemble simulationsThe best-fit posterior simulation is in red whose corresponding pa-rameters are in Table S2

fungal bacterial ratio in their long-term (730 days) litter de-composition experiment Considering that fungi often domi-nate lignin decomposition (Osono 2007) our predicted dom-inance of the fungi-like microbe in the second phase of the77-month decomposition is qualitatively reasonable Never-theless a comprehensive assessment should use a model thathas a complete representation of the relevant nutrient dy-namics (eg N and phosphorus) and such a model shouldbe compared to detailed observational characterization of lit-ter chemistry and microbial community structure Howeverdetailed observational characterization of both substrate andmicrobial community structure is lacking in long-term exper-iments that cover temporal scales varying from diurnal cyclesto multiple years These types of observations are critical tothe development of the types of models discussed here

Considering each member of the posterior ensemble sim-ulation as a single red pine litter decomposition exper-iment with a different microbial community our results(Figs 11 and 13) indicate that the evolution of litter chem-istry is strongly regulated by microbial community structureIn addition parameterization equifinality (see gray lines inFigs 11 and 13) indicate different microbial communitieswill sometimes lead to similar litter chemistry after a rel-atively long time The latter is manifested as a weak con-vergence of litter chemistry in terms of LCI throughout the77-month period (Fig 11b also see the review about mea-surements in Melillo et al 1989) Yet we found that the fi-

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3

ECAminusEnsECAECAminusMLECECAminusT

Fig 13Simulated time series of microbial abundances from usingthe best-fit parameters (calibrated with S3B3-ECA) in Table 5 Thefour models ECA ECA-ML EC and ECA-T used the same kineticparameters The ECA-Ens simulations correspond to the ensemblesimulations in Fig 11

nal seemingly constant LCI is not a single value but rather arange between 06 and 08 for the red pine litter being mod-eled here

When we applied the model S3B3-ECA using the best-fitparameters (Table 5) from the Bayesian calibration to 9 dif-ferent litter types (Table 4) the results (Fig 14) indicateda clear dependence of litter decomposition on initial litterchemistry The predictions indicate all 9 litters were degradedin two phases and their LCIs rose asymptotically to differentfinal constant values Furthermore the final constant LCI is afunction of both its initial value and the microbial communitydiversity and dynamics For instance the red maple startedwith a medium initial LCI (032) but reached a final valueof 068 the highest among the 9 litters (Table 4) While wefailed to obtain sufficient data to evaluate the 9 predictionsthe evaluation of the 4 litter types in the study by Magill etal (1998) indicated our model predictions were qualitativelyreasonable (Fig 15) We also applied the MichaelisndashMentenkinetics model (S3B3-MM) with its best-fit parameter (Ta-ble S2) to the 9 litter types its prediction was again poor (seeFig S5)

Therefore we summarize that litter decomposition iscoregulated by both the initial litter chemistry and micro-bial community structure and dynamics Our prediction sup-ports the conclusion drawn in Wickings et al (2012) andchallenges the assumptions of constant final LCI and con-stant microbial community structure in many existing bio-geochemical models eg the GDM model (Moorhead andSinsabaugh 2006) which used a constant final LCI and



02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

Sugar mapleAspenWhite oakWhite pineRed oakRed pineRed mapleBlack oakYellow birch

Fig 14Model (S3B3-ECA) predicted temporal evolution of litterdecomposition dynamics for the 9 different litters in Table 4 Theused parameters are in Table 5

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)

Predicted remaining litter ()

(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)

Red pineRed mapleBlack oakYellow birch

Fig 15Evaluation of model (S3B3-ECA) prediction using Magillet al (1998) litterbag experiment data(a) remaining litter biomassand(b) litter lignocellulose index (LCI) The original and correctedlignin data are in Fig S1

models such as TEM (McGuire et al 1997) CENTURY(Parton et al 1988) and Roth-C (Jenkinson and Coleman2008) which implicitly assumed the relevant microbial com-munity structures are constant

333 The emergent lignin decomposition dynamics

Lignin dynamics play a critical role in litter decomposition(Berg et al 1982 Melillo et al 1982 Machinet et al 2011)The physically reasonable prediction by model S3B3-ECAprovided us with some new insights on lignin decomposi-tion We found (Fig 16) that lignin decomposition does notfollow the conceptual model proposed by Berg and Staaf(1980) which states that no lignin will be degraded until itreaches a threshold concentration (with respect to the totallitter) Rather our predictions support the conceptual modelof Klotzbucher et al (2011) which states that lignin decom-position depends on the availability of easily degradable la-bile carbon However our results add further insights that

0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)

Sugar maple litterAspen litterWhite oak litterWhite pine litterRed oak literRed pine litterRed maple litterBlack oak litterYellow birch litterSugar maple ligninAspen ligninWhite oak ligninWhite pine ligninRed oak ligninRed pine ligninRed maple ligninBlack oak ligninYellow birch lignin

Fig 16 Model (S3B3-ECA) predicted temporal patterns of totallitter and lignin degradation for the 9 different litter types

besides litter chemistry degradation is also regulated by mi-crobial community structure When different groups of mi-crobes are degrading the same type of litter the litter chem-istry could evolve differently (eg Fig 11) We note thatlignin decomposition is also regulated by nutrient dynamicssuch as the nitrogen availability (Herman et al 2008) whichwill be explored in our follow up studies

34 Potential improvements to the EC and ECAsubstrate kinetics for modeling microbial systems

Substrate uptake is a process regulated by many biotic andabiotic factors For soil microbial systems relevant abioticfactors are soil moisture temperature mineralogy aggre-gation and redox potentials (eg Davidson and Janssens2006) As we explained in Sect 21 EC kinetics allow adirect and consistent description of these abiotic processesusing the existing knowledge of reactive transport modeling(Jennings et al 1982 Jin and Bethke 2007) Incorporatingthese factors within the ECA kinetics is more difficult How-ever besides the diffusion limitation (which partly accountsfor the soil moisture effect as we discussed in Sect 21)accounting for the temperature effect in ECA kinetics isstraightforward by recognizing that all parameters in Eq (4)are temperature dependent For instance by using Eyringrsquostransition state theory (Eyring 1935a b) it can be shownthatKSij prop exp(minus1HRT ) where1H (J molminus1) is an ac-tivation energy that should be deducible from measurements(as was done in Davidson et al (2012) though there theyassumedKSij was a linear function of temperature) Deter-mining the activation energies ofk+

ij1 kminus

ij1 andk+

ij2 couldbe challenging but we note that it has been done for in-organic chemistry kinetics (eg Bonner et al 1935) By



combining these ideas with the theory of half reactions (egMcCarty 2007) and assuming other abiotic factors such assoil aggregation and thermal degradation can be representedby chemical kinetics one could (and we hope to in futurework) construct a thermodynamically-based model of micro-bial organic matter decomposition

Other biological factors such as exoenzyme abundanceand microbial transporters affect the substrate uptake pro-cess indirectly by changing the abundance of consumersin the consumerndashsubstrate network Developing mechanisticrepresentations of these factors is an important area of study(Allison 2012 Kooijman and Troost 2007) that we will alsoaddress in follow-on studies with the EC and ECA kineticsbased model

35 Potential applications to different network systems

Because EC kinetics only relies on the premise that theequilibration of the consumer-substrate complexes (betweentheir formation and degradation) is much faster than othermetabolic processes it can in principle be applied to arbitraryfood web structures (eg Lindeman 1942) and proteinndashprotein interaction networks (Ciliberto et al 2007) We notehowever that a model implemented with the EC kinetics maybecome computationally expensive as the problem size in-creases For such cases proper numerical preconditioningbecomes necessary

Compared to EC kinetics the approximate ECA kineticsis applicable to a smaller scope of problems constrained bythe condition that any element in the network be either asubstrate or consumer but not both Still the ECA kineticsis much more general than other existing formulations usedfor predatorndashprey systems (Murdoch 1973 Koen-Alonso2007) and is computationally very efficient For proteinndashprotein interaction networks one feasible application exam-ple is the phosphorylationndashdephosphorylation cycle analyzedby Goldbeter and Koshland (1981) which lends itself to besolved by ECA kinetics under the tQSSA (compare theirEqs 1 and 2 with the form of Eq 4)

4 Conclusions

In this study we proposed that an equilibrium chemistry(EC) formulation could be used to predict the dynamicsof consumer-substrate complexes involved in an arbitraryconsumerndashsubstrate network When the given consumerndashsubstrate network satisfies the condition that any element ofthe network is either consumer or substrate but not both weobtained a first-order accurate approximation to EC (termedECA) Both the EC and ECA kinetics allow a simultaneousand consistent treatment of biotic and abiotic interactions inmicrobial systems (though the ECA kinetics is more limited)which cannot be achieved with the classical MM kinetics orwith other existing MM kinetics based extensions With a

few examples we demonstrated that if a network involvesmultiple substrates and consumers direct application of theclassical MM kinetics is inaccurate We further showed acarbon-only model implemented with the ECA kinetics pre-dicted litter decomposition dynamics reasonably These pre-dictions indicated that litter decomposition is coregulated bylitter chemistry and microbial community structure and dy-namics We hope our results can help develop a benchmarkmodel for microbially-mediated organic matter decomposi-tion in terrestrial and other ecosystems and stimulate applica-tions in other fields involving consumerndashsubstrate networks

Appendix A

Derivation of Eq (13)

In this section we present the derivation of Eq (13) Fromthe mass balance constraint to substrateSi (Eq 10) one has

Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)

Similarly (from Eq 12) one has

Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)

Substituting Eqs (A1) and (A2) into Eq (6) one obtains

Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)

We then apply the perturbation theory (eg Bender andOrzag 1999 Tang et al 2007) to Eq (A3) in which weassume

Cij = εCij1 + ε2Cij2 + (A4a)

Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)

whereε is a very small numberExpanding Eq (5) and keeping the first two orders ofε

gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)



Table A1 Symbols used in paper their definitions and corresponding units

Symbol Definition Unit

Bj Biomass of microbej mol C mminus3 or g CCCij Enzyme-substrate complex mol C mminus3

Cij1Cij2 Scaled first and second order accurate terms of complexCij mol C mminus3

Di Diffusivity of substrateSi in water m2 sminus1

EEj Free enzyme abundance mol mminus3

Ej0Ej1Ej2 Zero first and second order accurate terms of enzymeEj mol mminus3

1H Activation energy J molminus1

IkT Total abundance of inhibitork mol mminus3

Jcost Cost function unitlessk+1 k

+

ij1 Forward reaction coefficients m3 molminus1 sminus1

kminus1 kminus

ij1 Backward reaction coefficients sminus1

k+2 k+

ij2 Forward reaction coefficients sminus1

KSKSij KSi1 KSij Substrate affinity coefficients mol C mminus3

KIij KIijk Inhibitory coefficients mol C mminus3

nj Cell number density of microbej cells mminus3

Pij Product from degradation of complexCij mol C mminus3 or g Crcj Mean cell size of microbej mR Universal gas constant J Kminus1 molminus1

SSi Si Free substrate abundance mol C mminus3 or g CSi0Si1 Zero first and second order accurate terms of substrateSi mol mminus3

SiT Total substrate abundance mol mminus3

T Temperature Kv Substrate uptake rate mol mminus3 sminus1

Vmax Maximum substrate uptake rate mol mminus3 sminus1

αk Active fraction of the enzymek unitlessε Small number unitlessmicroij Biomass yield when microbeBj feeds onSi unitlessψj Area-based transporter density of cellj mol mminus2

σLIC Standard deviation of lignocellulose index unitlessσMass Standard deviation of the remaining organic biomass unitlessγj Respiration rate of microbej hminus1 or dminus1

8D Substrate flux mol mminus3 sminus1

8E Changing rate of new enzymes (transporters) mol mminus3 sminus1

ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0

where the third subscript onCij indicates the associated or-der ofε Substituting Eq (A5) into Eq (A3) gives

Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij

(KSil +KSnj minusKSnl

KSilKSnj

)

Therefore by usingEjT = Ej0 andSiT = Si0 at the zeroorder approximation Eq (A6a) is equivalent to Eq (13) inthe main text The derivation of Eq (A6b) is given in theSupplement

Because of the many unspecified parameters (prior to aspecific application) we were not able to identify a best es-timate of the condition when Eq (A6a) is exactly first orderaccurate However from Eq (A6b) we obtain a crude con-dition



(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1

Condition Eq (A7) holds ifSi0 Ej0 orSi0 Ej0 Evenif Si0 asymp Ej0 Eq (A7) may still hold because of the manypossible parameter combinations in a complicated reactionnetwork

While it is cumbersome to verify Eq (A6) for all combi-nations of substrate and consumer under the single-substrateand single-consumer condition we have

C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+

S1TE1T(KS11+ S1T +E1T

)2]

which is equivalent to Eq (3) in Cha and Cha (1965) whentruncated to second order accuracy

Appendix B

An alternate analytic approximation

Substituting the mass balance relationships Eqs (A1) and(A2) into Eq (6) one has(SiT minus

k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)

Then by expanding Eq (B1) and ignoring quadratic termsone obtains a set of linear equations

KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)

where we again used the zero order approximationEjT =

Ej0 andSiT = Si0Since we have not been able to find an analytical solution

to Eq (B2) we attempted to solve it using existing linearalgebra packages This effort turned out to be numericallyvery difficult and often resulted in unrealistic and negativecomplex concentrations However using results derived byDe Boer and Perelson (1995) we developed an approximatesolution

Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)

which satisfies Eq (B2) exactly whenI = 1 or J = 1 Weevaluated Eq (B3) with random sampling tests and found itwas generally inferior to Eq (13) (results not shown)

Appendix C

The MM kinetics based approximation to the EC solution

The MM kinetics based representation of the enzyme-substrate complexCij is

Cij =SiTEjT

KSij + SiT(C1)

The solution Eq (C1) is then scaled linearly to satisfy themass constraint

k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)

We point out that Eqs (C2) have been implemented dif-ferently from other studies eg Moorhead and Sinsabaugh(2006 GDM) Riley et al (2011 CLM4Me) and Allison(2012 DEMENT) Those studies imposed the constraint ontotal substrate flux within a single time step rather than on theoverall enzyme-substrate complexes It is only with Eqs (C2)that MM and ECA-ML kinetics were able to model the ad-sorption surface effect on substrate dynamics but they wereless accurate than the ECA kinetics as we have demonstratedin the main text (see discussions on scenario S3B1M1 inSect 32)

Supplementary material related to this article isavailable online athttpwwwbiogeosciencesnet1083292013bg-10-8329-2013-supplementpdf

AcknowledgementsThis research was supported by the DirectorOffice of Science Office of Biological and Environmental Researchof the US Department of Energy under contract no DE-AC02-05CH11231 as part of their Regional and Global Climate ModelingProgram and by the Next-Generation Ecosystem Experiments(NGEE Arctic) project supported by the Office of Biologicaland Environmental Research in the DOE Office of Science undercontract no DE-AC02-05CH11231

Edited by G Herndl



References

Aber J D Melillo J M and Mcclaugherty C A Predictinglong-term patterns of mass-loss nitrogen dynamics and soilorganic-matter formation from initial fine litter chemistry in tem-perate forest ecosystems Can J Bot 68 2201ndash2208 1990

Abrams P A and Ginzburg L R The nature of predation preydependent ratio dependent or neither Trends Ecol Evol 15337ndash341 doi101016S0169-5347(00)01908-X 2000

Allison S D A trait-based approach for modelling microbial litterdecomposition Ecol Lett 15 1058ndash1070 doi101111j1461-0248201201807x 2012

Allison S D Wallenstein M D and Bradford M A Soil-carbonresponse to warming dependent on microbial physiology NatGeosci 3 336ndash340 doi101038Ngeo846 2010

Andren O and Paustian K Barley straw decomposition inthe field ndash a comparison of models Ecology 68 1190ndash1200doi1023071939203 1987

Arditi R and Ginzburg L R Coupling in predator prey dy-namics ndash ratio-dependence J Theor Biol 139 311ndash326doi101016S0022-5193(89)80211-5 1989

Armstrong R A Nutrient uptake rate as a function of cell size andsurface transporter density A Michaelis-like approximation tothe model of Pasciak and Gavis Deep-Sea Res Pt I 55 1311ndash1317 doi101016JDsr200805004 2008

Balser T C and Wixon D L Investigating biological controlover soil carbon temperature sensitivity Glob Change Biol 152935ndash2949 doi101111J1365-2486200901946X 2009

Bauer J Herbst M Huisman J A Weihermuller L andVereecken H Sensitivity of simulated soil heterotrophic res-piration to temperature and moisture reduction functions Geo-derma 145 17ndash27 doi101016JGeoderma200801026 2008

Bender C M and Orzag S A Advanced Mathematical Methodsfor Scientists and Engineers Springer-Verlag New York 1999

Berg H C and Purcell E M Physics of Chemoreception Bio-phys J 20 193ndash219 1977

Berg B and Staaf H Decomposition rate and chemical changesof Scots pine litter II Influence of chemical composition EcolBull 32 373ndash390 1980

Berg B Hannus K Popoff T and Theander O Changes inorganic-chemical components of needle litter during decompo-sition ndash long-term decomposition in a scots pine forest 1 CanJ Bot 60 1310ndash1319 1982

Beven K A manifesto for the equifinality thesis Journal of Hy-drology 320 18ndash36 doi101016JJhydrol200507007 2006

Bonachela J A Raghib M and Levin S A Dynamic model offlexible phytoplankton nutrient uptake P Natl Acad Sci USA108 20633ndash20638 doi101073Pnas1118012108 2011

Bonner W D Gore W L and Yost D M The thermal reac-tion between gaseous iodine monochloride and hydrogen J AmChem Soc 57 2723ndash2724 doi101021Ja01315a502 1935

Borghans J A M and Deboer R J A minimal model forT-cell vaccination P Roy Soc B-Biol Sci 259 173ndash178doi101098Rspb19950025 1995

Borghans J A M DeBoer R J and Segel L A Extendingthe quasi-steady state approximation by changing variables BMath Biol 58 43ndash63 doi101007Bf02458281 1996

Bouskill N J Tang J Y Riley W J and Brodie E L Trait-based representation of biological nitrification model develop-

ment testing and predicted community composition Front Mi-crobiol 3 364 doi103389fmicb201200364 2012

Broekhuizen N Rickard G J Bruggeman J and Meis-ter A An improved and generalized second order un-conditionally positive mass conserving integration schemefor biochemical systems Appl Numer Math 58 319ndash340doi101016JApnum200612002 2008

Burnett T Influences of natural temperatures and controlled hostdensities on oviposition of an insect parasite Physiol Zool 27239ndash258 1954

Campbell A Conditions for existence of bacteriophage Evolution15 153ndash165 doi1023072406076 1961

Caperon J Population growth in micro-organisms limited by foodsupply Ecology 48 715ndash722 doi1023071933728 1967

Cha S and Cha C J M Kinetics of cyclic enzyme systems MolPharmacol 1 178ndash189 1965

Childs R E and Bardsley W G Steady-state kinetics of per-oxidase with 22rsquo-Azino-Di-(3-Ethylbenzthiazoline-6-SulphonicAcid) as chromogen Biochem J 145 93ndash103 1975

Choy K K H Porter J F and McKay G Langmuir isothermmodels applied to the multicomponent sorption of acid dyes fromeffluent onto activated carbon J Chem Eng Data 45 575ndash584doi101021Je9902894 2000

Ciliberto A Capuani F and Tyson J J Modeling net-works of coupled enzymatic reactions using the total quasi-steady state approximation Plos Comput Biol 3 463-472doi101371JournalPcbi0030045 2007

Cornish-Bowden A Fundamentals of Enzyme Kinetics rev edPortland Press London 1995

Davidson E A and Janssens I A Temperature sensitivity of soilcarbon decomposition and feedbacks to climate change Nature440 165ndash173 doi101038Nature04514 2006

Davidson E A Samanta S Caramori S S and Savage KThe dual Arrhenius and Michaelis-Menten kinetics model fordecomposition of soil organic matter at hourly to seasonal timescales Glob Change Biol 18 371ndash384 doi101111J1365-2486201102546X 2012

De Boer R J and Perelson A S T-cell repertoires andcompetitive-exclusion J Theor Biol 169 375ndash390doi101006Jtbi19941160 1994

De Boer R J and Perelson A S Towards a general functiondescribing T-cell proliferation J Theor Biol 175 567ndash576doi101006Jtbi19950165 1995

Dilly O and Munch J C Microbial biomass content basal res-piration and enzyme activities during the course of decomposi-tion of leaf litter in a black alder (Alnus glutinosa (L) Gaertn)forest Soil Biol Biochem 28 1073ndash1081 doi1010160038-0717(96)00075-2 1996

Druhan J L Steefel C I Molins S Williams K H ConradM E and DePaolo D J Timing the onset of sulfate reductionover multiple subsurface acetate amendments by measurementand modeling of sulfur isotope fractionation Environ Sci Tech-nol 46 8895ndash8902 doi101021Es302016p 2012

Dybzinski R Farrior C Wolf A Reich P B and Pacala S WEvolutionarily stable strategy carbon allocation to foliage woodand fine roots in trees competing for light and nitrogen an ana-lytically tractable individual-based model and quantitative com-parisons to data Am Nat 177 153ndash166 doi1010866579922011



Eyring H The activated complex and the absolute rate of chemi-cal reactions Chem Rev 17 65ndash77 doi101021Cr60056a0061935

Eyring H The activated complex in chemical reactions J ChemPhys 3 107ndash115 doi10106311749604 1935

Felmy A R Girvin D C and Jenne E A MINTEQ Acomputer program for calculating aqueous geochemical equilib-ria EPA-6003-84-032 Office Res Dev USEPA Athens GA1984

Follows M J Dutkiewicz S Grant S and Chisholm S WEmergent biogeography of microbial communities in a modelocean Science 315 1843ndash1846 doi101126Science11385442007

Ginzburg L R and Akcakaya H R Consequences of ratio-dependent predation for steady-state properties of ecosystemsEcology 73 1536ndash1543 doi1023071940006 1992

Goldbeter A and Koshland D E An amplified sensitivity arisingfrom covalent modification in biological-systems P Natl AcadSci-Biol 78 6840ndash6844 doi101073Pnas78116840 1981

Grant R F Juma N G and Mcgill W B Simulation of carbonand nitrogen transformations in soil ndash mineralization Soil BiolBiochem 25 1317ndash1329 doi1010160038-0717(93)90046-E1993

Gu C H Maggi F Riley W J Hornberger G M Xu T Old-enburg C M Spycher N Miller N L Venterea R T andSteefel C Aqueous and gaseous nitrogen losses induced byfertilizer application J Geophys Res-Biogeo 114 G01006doi1010292008jg000788 2009

Hall S R Stoichiometrically explicit competition between graz-ers Species replacement coexistence and priority effectsalong resource supply gradients Am Nat 164 157ndash172doi101086422201 2004

Herman J Moorhead D and Berg B The relationshipbetween rates of lignin and cellulose decay in above-ground forest litter Soil Biol Biochem 40 2620ndash2626doi101016JSoilbio200807003 2008

Holling C S The components of predation as revealed by a studyof small-mammal predation of the european pine sawfly CanEntomol 91 293ndash320 doi104039Ent9129-7 1959

Jacobson M Z Studying the effects of calcium and magne-sium on size-distributed nitrate and ammonium with EQUI-SOLV II Atmos Environ 33 3635ndash3649 doi101016S1352-2310(99)00105-3 1999

Jacobson M Z Tabazadeh A and Turco R P Simulat-ing equilibrium within aerosols and nonequilibrium betweengases and aerosols J Geophys Res-Atmos 101 9079ndash9091doi10102996jd00348 1996

Jenkinson D S and Coleman K The turnover of organic carbonin subsoils Part 2 Modelling carbon turnover Eur J Soil Sci59 400ndash413 doi101111J1365-2389200801026X 2008

Jennings A A Kirkner D J and Theis T L Multicomponentequilibrium chemistry in groundwater quality models WaterResour Res 18 1089ndash1096 doi101029Wr018i004p010891982

Jin Q and Bethke C M The thermodynamics and kinet-ics of microbial metabolism Am J Sci 307 643ndash677doi10247504200701 2007

Karp-Boss L Boss E and Jumars P A Nutrient fluxes to plank-tonic osmotrophs in the presence of fluid motion Oceanogr MarBiol 34 71ndash107 1996

Keeler B L Hobbie S E and Kellogg L E Effects of long-termnitrogen addition on microbial enzyme activity in eight forestedand grassland sites implications for litter and soil organic matterdecomposition Ecosystems 12 1ndash15 doi101007S10021-008-9199-Z 2009

Klotzbucher T Kaiser K Guggenberger G Gatzek C andKalbitz K A new conceptual model for the fate of lignin indecomposing plant litter Ecology 92 1052ndash1062 2011

Koen-Alonso M A process-oriented approach to the multispeciesfunctional response in From Energetics to Ecosystems The Dy-namics and Structure of Ecological Systems edited by RooneyN McCann M S and Noakes D L G Springer DordrechtThe Netherlands 1ndash36 2007

Kooijman S A L M The Synthesizing Unit as model for thestoichiometric fusion and branching of metabolic fluxes Bio-phys Chem 73 179ndash188 doi101016S0301-4622(98)00162-8 1998

Kooijman S A L M and Troost T A Quantitative stepsin the evolution of metabolic organisation as specified bythe Dynamic Energy Budget theory Biol Rev 82 113ndash142doi101111J1469-185x200600006X 2007

Koven C D Riley W J Subin Z M Tang J Y Torn M SCollins W D Bonan G B Lawrence D M and SwensonS C The effect of vertically resolved soil biogeochemistry andalternate soil C and N models on C dynamics of CLM4 Biogeo-sciences 10 7109ndash7131 doi105194bg-10-7109-2013 2013

Kratina P Vos M Bateman A and Anholt B R Functionalresponses modified by predator density Oecologia 159 425ndash433 doi101007S00442-008-1225-5 2009

Kumar A and Josic K Reduced models of networks ofcoupled enzymatic reactions J Theor Biol 278 87ndash106doi101016JJtbi201102025 2011

Ladd J N Amato M Zhou L K and Schultz J E Differential-effects of rotation plant residue and nitrogen-fertilizer on mi-crobial biomass and organic-matter in an Australian Alfisol SoilBiol Biochem 26 821ndash831 doi1010160038-0717(94)90298-4 1994

Lambers H Mougel C Jaillard B and Hinsinger P Plant-microbe-soil interactions in the rhizosphere an evolutionaryperspective Plant Soil 321 83ndash115 doi101007S11104-009-0042-X 2009

Legovic T and Cruzado A A model of phytoplankton growthon multiple nutrients based on the Michaelis-Menten-Monod up-take Drooprsquos growth and Liebigrsquos law Ecol Model 99 19ndash31doi101016S0304-3800(96)01919-9 1997

Li C S Frolking S and Frolking T A A model of nitrous-oxideevolution from soil driven by rainfall events 1 model structureand sensitivity J Geophys Res-Atmos 97 9759ndash9776 1992

Lindeman R L The trophic-dynamic aspect of ecology Ecology23 399ndash418 doi1023071930126 1942

Litchman E Klausmeier C A Schofield O M and FalkowskiP G The role of functional traits and trade-offs in struc-turing phytoplankton communities scaling from cellular toecosystem level Ecol Lett 10 1170ndash1181 doi101111J1461-0248200701117X 2007



Liu Y Overview of some theoretical approaches for derivation ofthe Monod equation Appl Microbiol Biotech 73 1241ndash12502007

Lotka A J Contribution to quantitative parasitology J WashAcad Sci 13 152ndash158 1923

Machinet G E Bertrand I and Chabbert B Assessment ofLignin-Related Compounds in Soils and Maize Roots by Alka-line Oxidations and Thioacidolysis Soil Sci Soc Am J 75542ndash552 doi102136Sssaj20100222 2011

Maggi F and Riley W J Transient competitive complexa-tion in biological kinetic isotope fractionation explains non-steady isotopic effects Theory and application to denitri-fication in soils J Geophys Res-Biogeo 114 G04012doi1010292008jg000878 2009

Maggi F Gu C Riley W J Hornberger G M VentereaR T Xu T Spycher N Steefel C Miller N L andOldenburg C M A mechanistic treatment of the domi-nant soil nitrogen cycling processes Model development test-ing and application J Geophys Res-Biogeo 113 G02016doi1010292007jg000578 2008

Magill A H and Aber J D Long-term effects of experi-mental nitrogen additions on foliar litter decay and humusformation in forest ecosystems Plant Soil 203 301ndash311doi101023A1004367000041 1998

Mayes M A Heal K R Brandt C C Phillips J R and Jar-dine P M Relation between soil order and sorption of dissolvedorganic carbon in temperate subsoils Soil Sci Soc Am J 761027ndash1037 doi102136Sssaj20110340 2012

McCarty P L Thermodynamic electron equivalents model for bac-terial yield prediction Modifications and comparative evalua-tions Biotechnol Bioeng 97 377ndash388 doi101002Bit212502007

McGill B J and Brown J S Evolutionary game theory and adap-tive dynamics of continuous traits Annu Rev Ecol Evol S38 403ndash435 doi101146AnnurevEcolsys360917041755172007

McGuire A D Melillo J M Kicklighter D W Pan Y D XiaoX M Helfrich J Moore B Vorosmarty C J and SchlossA L Equilibrium responses of global net primary productionand carbon storage to doubled atmospheric carbon dioxide Sen-sitivity to changes in vegetation nitrogen concentration GlobalBiogeochem Cy 11 173ndash189 doi10102997gb00059 1997

Melillo J M Aber J D and Muratore J F Nitrogen and lignincontrol of hardwood leaf litter decomposition dynamics Ecol-ogy 63 621ndash626 doi1023071936780 1982

Melillo J M Aber J D Linkins A E Ricca A Fry B andNadelhoffer K J Carbon and nitrogen dynamics along the de-cay continuum ndash plant litter to soil organic-matter Plant Soil115 189ndash198 doi101007Bf02202587 1989

Michaelis L and Menten M L The kenetics of the inversion ef-fect Biochem Z 49 333ndash369 1913

Machinet G E Bertrand I Barriere Y Chabbert B and Re-cous S Impact of plant cell wall network on biodegradationin soil Role of lignin composition and phenolic acids in rootsfrom 16 maize genotypes Soil Biol Biochem 43 1544ndash1552doi101016JSoilbio201104002 2011

Molins S Trebotich D Steefel C I and Shen C P An inves-tigation of the effect of pore scale flow on average geochemical

reaction rates using direct numerical simulation Water ResourRes 48 W03527 doi1010292011wr011404 2012

Moorhead D L and Sinsabaugh R L A theoretical model of lit-ter decay and microbial interaction Ecol Monogr 76 151ndash174doi1018900012-9615(2006)076[0151Atmold]20Co2 2006

Murdoch W W Functional Response of Predators J Appl Ecol10 335ndash342 1973

Osono T Ecology of ligninolytic fungi associated with leaf litterdecomposition Ecol Res 22 955ndash974 doi101007S11284-007-0390-Z 2007

Parton W J Stewart J W B and Cole C V Dynamics of C NP and S in grassland soils ndash a model Biogeochemistry 5 109ndash131 doi101007Bf02180320 1988

Pasciak W J and Gavis J Transport limitation of nutrient uptakein phytoplankton Limnol Oceanogr 19 881ndash898 1974

Pasciak W J and Gavis J Transport limited nutrient uptake ratesin Ditylum-Brightwellii Limnol Oceanogr 20 604ndash617 1975

Persson L Leonardsson K de Roos A M Gyllenberg M andChristensen B Ontogenetic scaling of foraging rates and thedynamics of a size-structured consumer-resource model TheorPopul Biol 54 270ndash293 doi101006Tpbi19981380 1998

Pilinis C Seinfeld J H and Seigneur C Mathematical modelingof the dynamics of multicomponent atmospheric aerosols At-mos Environ 21 943ndash955 doi1010160004-6981(87)90090-4 1987

Reynolds H L and Pacala S W An analytical treatment of root-to-shoot ratio and plant competition for soil nutrient and lightAm Nat 141 51ndash70 doi101086285460 1993

Riley W J and Matson P A NLOSS A mechanistic model ofdenitrified N2O and N-2 evolution from soil Soil Sci 165 237ndash249 doi10109700010694-200003000-00006 2000

Riley W J Subin Z M Lawrence D M Swenson S C TornM S Meng L Mahowald N M and Hess P Barriers to pre-dicting changes in global terrestrial methane fluxes analyses us-ing CLM4Me a methane biogeochemistry model integrated inCESM Biogeosciences 8 1925ndash1953 doi105194bg-8-1925-2011 2011

Rodrigo A Recous S Neel C and Mary B Modellingtemperature and moisture effects on C-N transformations insoils comparison of nine models Ecol Model 102 325ndash339doi101016S0304-3800(97)00067-7 1997

Saggar S Parshotam A Sparling G P Feltham C W andHart P B S C-14-labelled ryegrass turnover and residencetimes in soils varying in clay content and mineralogy Soil BiolBiochem 28 1677ndash1686 doi101016S0038-0717(96)00250-7 1996

Schenk D Bersier L F and Bacher S An experimen-tal test of the nature of predation neither prey- nor ratio-dependent J Anim Ecol 74 86ndash91 doi101111J1365-2656200400900X 2005

Schimel J P Wetterstedt J A M Holden P A and Trumbore SE Dryingrewetting cycles mobilize old C from deep soils froma California annual grassland Soil Biol Biochem 43 1101ndash1103 doi101016JSoilbio201101008 2011

Schnell S and Maini P K Enzyme kinetics at highenzyme concentration B Math Biol 62 483ndash499doi101006Bulm19990163 2000



Schnell S and Mendoza C Enzyme kinetics of multi-ple alternative substrates J Math Chem 27 155ndash170doi101023A1019139423811 2000

Segel L A and Slemrod M The quasi-steady-state assump-tion - a case-study in perturbation Siam Rev 31 446ndash477doi1011371031091 1989

Sposito G and Coves J SOILCHEM A computer program forthe calculation of chemical spcciation in soils Keamey FoundSoil Sci Univ California Riverside 1988

Suzuki I Lizama H M and Tackaberry P D Competitive-inhibition of ferrous iron oxidation by thiobacillus-ferrooxidansby increasing concentrations of cells Appl Environ Microbiol55 1117ndash1121 1989

Taylor K E Summarizing multiple aspects of model performancein a single diagram J Geophys Res-Atmos 106 7183ndash7192doi1010292000jd900719 2001

Tang J Y and Zhuang Q L Equifinality in parameter-ization of process-based biogeochemistry models A sig-nificant uncertainty source to the estimation of regionalcarbon dynamics J Geophys Res-Biogeo 113 G04010doi1010292008jg000757 2008

Tang J Y Tang J and Wang Y Analytical investigationon 3D non-Boussinesq mountain wave drag for wind profileswith vertical variations Appl Math Mech-Engl 28 317ndash325doi101007S10483-007-0305-Z 2007

Thornton P E Lamarque J F Rosenbloom N A andMahowald N M Influence of carbon-nitrogen cycle cou-pling on land model response to CO2 fertilization andclimate variability Global Biogeochem Cy 21 Gb4018doi1010292006gb002868 2007

Tilman D Resource competition between planktonic algae ndashexperimental and theoretical approach Ecology 58 338ndash348doi1023071935608 1977

Vayenas D V and Pavlou S Chaotic dynamics of a food webin a chemostat Math Biosci 162 69ndash84 doi101016S0025-5564(99)00044-9 1999

Volterra V Variazioni e fluttuazioni del numero drsquoindividui inspecie animali conviventi Mew Acad Lincei 6 31ndash113 1926

Vrugt J A ter Braak C J F Clark M P Hyman J Mand Robinson B A Treatment of input uncertainty in hy-drologic modeling Doing hydrology backward with Markovchain Monte Carlo simulation Water Resour Res 44 W00b09doi1010292007wr006720 2008

Vucetich J A Peterson R O and Schaefer C L The effectof prey and predator densities on wolf predation Ecology 833003ndash3013 doi1023073071837 2002

Wang G S and Post W M A note on the reverseMichaelisndashMenten kinetics Soil Biol Biochem 57 946ndash949doi101016jsoilbio201208028 2013

Wang G S Post W M Mayes M A Frerichs J T andSindhu J Parameter estimation for models of ligninolytic andcellulolytic enzyme kinetics Soil Biol Biochem 48 28ndash38doi101016JSoilbio201201011 2012

Wang G S Post W M and Mayes M A Developmentof microbial-enzyme-mediated decomposition model parametersthrough steady-state and dynamic analyses Ecol Appl 23 255ndash272 2013

Wickings K Grandy A S Reed S C and Cleveland C C Theorigin of litter chemical complexity during decomposition EcolLett 15 1180ndash1188 doi101111J1461-0248201201837X2012

Williams P J Validity of Application of Simple Kinetic Analysisto Heterogeneous Microbial Populations Limnol Oceanogr 18159ndash164 1973



competition for inorganic nitrogen and phosphorus (egReynolds and Pacala 1993 Lambers et al 2009) plantcompetition for light (eg Dybzinski et al 2011) mi-crobial competition for carbon substrates and mineral nu-trients (eg Caperon 1967 Moorhead and Sinsabaugh2006 Allison 2012 Bouskill et al 2012) algae compe-tition for mineral nutrients (eg Tilman 1977 Follows etal 2007) and predator competition for prey (eg Holling1959a Arditi and Ginzburg 1989 Ginzburg and Akcakaya1992 Vayenas and Pavlou 1999 Abrams and Ginzburg2000 Koen-Alonso 2007) Because of this prevalence ofconsumerndashsubstrate interactions in natural systems particu-larly in ecosystem dynamics many mathematical develop-ments have been proposed to interpret and predict ecosystembehavior under a wide range of environmental and biolog-ical conditions (eg Lotka 1923 Volterra 1926 Holling1959b Campbell 1961 Murdoch 1973 Williams 1973Tilman 1977 Pasciak and Gavis 1974 Persson et al 1998Maggi et al 2008 Bonachela et al 2011 Bouskill et al2012) In this study we present developments focusing onthe consumerndashsubstrate network that regulates organic matterdecomposition However our results should be applicable toany problem that can be similarly formulated as a consumerndashsubstrate network

In general the growth of any biological organism mini-mally involves two steps (i) substrate uptake and (ii) sub-strate assimilation Once a substrate is captured it is as-similated to produce energy and biomass for a series ofmetabolic processes including but not limited to cell main-tenance enzyme production cell division and reproduc-tion Therefore explicit modeling of the interactions be-tween many consumers substrates and their habitats re-quires a consistent mathematical representation of substrateuptake under a wide range of biotic and abiotic conditionsAmong the many existing substrate uptake kinetics (Hill1910 Michaelis and Menten 1913 Burnett 1954 Holling1959b Cleland 1963) the MichaelisndashMenten (MM) kinetics(or equivalently Monod (Monod 1949) or Hollingrsquos type II(Holling 1959b) kinetics) is the most widely applied becauseof its simple form solid theoretical foundation (eg Liu2007) and successes under a wide range of conditions (egHolling 1959b Tilman 1977 Reynolds and Pacala 1993Legovic and Cruzado 1997 Hall 2004 Kou 2005 Rileyand Matson 2000 Maggi et al 2008 Allison 2012)

In their seminal paper Michaelis and Menten (1913) as-sumed that enzymes and substrates adsorb to each otherto form enzyme-substrate complexes By assuming theenzyme-substrate complex is of a much lower concentrationthan that of the substrate they obtained by law of mass ac-tions the so-called MM kinetics which states

v =VmaxS

KS+ S(1)

wherev (mol sminus1) is the substrate uptake rateVmax (mol sminus1)

is the maximum substrate uptake rateKS (mol mminus3) is

the half saturation (or substrate affinity) constant andS(mol mminus3) is the free substrate concentration (a full list ofsymbols is given at the end of the text) Later Briggs andHaldane (1925) derived Eq (1) from the enzyme-catalyzedreaction

S+Ek+1harrkminus1

Ck+2rarrP +E (2)

where E (mol mminus3) is (free) enzymeC (mol mminus3) isenzyme-substrate complex from binding (free) substrateS

to enzymeE P is the product (mol mminus3) resulting from theirreversible part of reaction (2)k+

1 (m3 molminus1 sminus1) andk+

2(sminus1) are forward reaction coefficientskminus

1 (sminus1) is the back-ward reaction coefficient andKS =

(kminus

1 + k+

2

)k+

1 Laterstudies (eg Segel and Slemrod 1989 Schnell and Maini2000) indicated that Eq (1) was obtained with the stan-dard quasi-steady-state assumption (sQSSA) which statesthat dC

dt asymp 0 and dSdt = minusk+

1 SE+ kminus

1 C (note that dSdt is the

changing rate of the free substrate which is different fromthe total substrate being used in the total quasi-steady-stateassumption (tQSSA) to be introduced later) Equation (1) isvalid only whenS+KS ET whereET is the total enzymeconcentration including both free and substrate-bound

The MM kinetics has been successful in many applica-tions but there are also many studies demonstrating thatmodifications must be made to account for discrepancies be-tween predictions from applying Eq (1) and observations(eg Cha and Cha 1965 Williams 1973 Suzuki et al1989 Maggi and Riley 2009 Druhan et al 2012) For in-stance Cha and Cha (1965) in studying cyclic enzyme sys-tems noticed that the substrate uptake kinetics when approx-imated with first order accuracy should be

v =VmaxST

KS+ET + ST (3)

Others have obtained Eq (3) or a similar form for var-ious problems (eg Reiner 1969 Segel 1975 Schulz1994 Borghans and De Boer 1995 Borghans et al 1996Schnell and Maini 2000 Wang and Post 2013) In particu-lar Borghans et al (1996) using the total quasi-steady-stateapproximation (tQSSA which also assumesdC

dt asymp 0 but de-

fines a total substrateST = S+C and usesdSTdt = minusk+

2 C)showed Eq (3) is valid ifk+

2 ET ltlt k+

1 (KS+ET + ST)2

Equation (3) is of good accuracy for a much wider range ofsubstrate and enzyme concentrations than Eq (1) It also alle-viates the problem thatv rarr infin asET rarr infin if Eq (1) is used(note Vmax prop ET) In addition when applied to predatorndashprey systems (ie predator =E prey =S) Eq (3) predictspredation depends on both (i) the ratio between predator andprey density and (ii) prey density While we have not foundan example in the literature of Eq (3) being evaluated withpredation data a few studies (eg Vucetich et al 2002Schenk et al 2005) indicated the predation rate is not only






1 kminus

1 andk+





2 Methods










wherek+







SiEjk+

ij1 =

(kminus

ij1 + k+

ij2

)Cij (5)


KSij =kminus

ij1 + k+

ij2

k+

ij1

=SiEj

Cij (6)






dPijdt

= k+

ij2Cij (8)



dSidt

= minus

k=Jsumk=1

(k+


ik1Cik

) (9)


SiT = Si +

k=Jsumk=1

Cik (10)


dSiTdt

= minus

k=Jsumk=1

k+

ik2Cik (11)


EjT = Ej +

k=Isumk=1

Ckj (12)

We note that ifk+







Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkTKSkj

+

k=Jsumk=1

EkTKSik

) (13)



C11 =S1TE1T

KS11+ S1T +E1T (14)



C11 =S1TE1T

S1T +KS11

(1+

E1TKS11

+E2TKS12

) (15)


13 E1 13 13 Ej1 13 Ej 13 Ej+1 13 13 EJ 13


13 13 13 13 13 13 13 13




13 13 13 13 13 13 13 13


13


C11 =α1S1TET

S1T +KS11

[1+

(α1KS11

+1minusα1KS12

)ET

] (16)

=α1S1TET

S1T +KS11

(1+

ETKI

) whereKI =

(α1KS11

+1minusα1KS12




C1j =S1TEjT

KS1j

(1+

k=Jsumk=1

EkTKS1k

)+ S1T

(17)


C1j =αjS1TEjT

KS1j

(1+

k=Jsumk=1

EkTKI1k

)+ S1T

(18)




KI1k =

(αk

KS1k+

1minusαk

KS1kd

)minus1

(19)



Ci1 =SiTE1T

KSi1

(1+

k=Isumk=1

SkTKSk1

)+E1T

(20)


Ci1 =SiTE1T

KSi1

(1+

KSi1

KSi1

k=Isumk=1

SkTKSk1

) (21)




C11 =S1TE1TKS11

1+S1TKS11

+S2TKS21

+E1TKS11

(22a)

C21 =S2TE1TKS21

1+S1TKS11

+S2TKS21

+E1TKS21

(22b)







KSij =KSij

k=Lsumk=1

IkTKIijk

+ 1

(23)



Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkT

KSkj+

k=Jsumk=1

EkT

KSik

) (24)








Si =4πDircnjSi

k+


4πDircnjSik+




KSij =kminus

ij1 + k+

ij2

k+

ij1

(1+

k+

ij1EjT

4πDircjnj

)(26)


KSij =KSij

(1+

k+

ij1EjT

4πDircjnj

) (27)

Si +Ejkij 1

k+ij 1SiEj Pij +Ej

Si

E

D



ij1 k+

ij2 and defining

Vmaxij = k+

ij2EjT one has

KSij =KSij

(1+

Vmaxij

4πDircjnjKSij

) (28)












dSidt

= minus

j=Jsumj=1

k+

ij2Cij i = 1 I (29)

dBjdt

=

i=Isumi=1

microijk+




err(t)=3

N

i=Nsumi=1



∣∣∣∣ (31)







(KSij k

+

ij2microij

)


S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)



+

ij2microij

) whose


S1 (300)


B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)





+

ij2microij

) whose units


S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)











+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References

































































































































1 kminus

1 andk+





2 Methods










wherek+







SiEjk+

ij1 =

(kminus

ij1 + k+

ij2

)Cij (5)


KSij =kminus

ij1 + k+

ij2

k+

ij1

=SiEj

Cij (6)






dPijdt

= k+

ij2Cij (8)



dSidt

= minus

k=Jsumk=1

(k+


ik1Cik

) (9)


SiT = Si +

k=Jsumk=1

Cik (10)


dSiTdt

= minus

k=Jsumk=1

k+

ik2Cik (11)


EjT = Ej +

k=Isumk=1

Ckj (12)

We note that ifk+







Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkTKSkj

+

k=Jsumk=1

EkTKSik

) (13)



C11 =S1TE1T

KS11+ S1T +E1T (14)



C11 =S1TE1T

S1T +KS11

(1+

E1TKS11

+E2TKS12

) (15)


13 E1 13 13 Ej1 13 Ej 13 Ej+1 13 13 EJ 13


13 13 13 13 13 13 13 13




13 13 13 13 13 13 13 13


13


C11 =α1S1TET

S1T +KS11

[1+

(α1KS11

+1minusα1KS12

)ET

] (16)

=α1S1TET

S1T +KS11

(1+

ETKI

) whereKI =

(α1KS11

+1minusα1KS12




C1j =S1TEjT

KS1j

(1+

k=Jsumk=1

EkTKS1k

)+ S1T

(17)


C1j =αjS1TEjT

KS1j

(1+

k=Jsumk=1

EkTKI1k

)+ S1T

(18)




KI1k =

(αk

KS1k+

1minusαk

KS1kd

)minus1

(19)



Ci1 =SiTE1T

KSi1

(1+

k=Isumk=1

SkTKSk1

)+E1T

(20)


Ci1 =SiTE1T

KSi1

(1+

KSi1

KSi1

k=Isumk=1

SkTKSk1

) (21)




C11 =S1TE1TKS11

1+S1TKS11

+S2TKS21

+E1TKS11

(22a)

C21 =S2TE1TKS21

1+S1TKS11

+S2TKS21

+E1TKS21

(22b)







KSij =KSij

k=Lsumk=1

IkTKIijk

+ 1

(23)



Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkT

KSkj+

k=Jsumk=1

EkT

KSik

) (24)








Si =4πDircnjSi

k+


4πDircnjSik+




KSij =kminus

ij1 + k+

ij2

k+

ij1

(1+

k+

ij1EjT

4πDircjnj

)(26)


KSij =KSij

(1+

k+

ij1EjT

4πDircjnj

) (27)

Si +Ejkij 1

k+ij 1SiEj Pij +Ej

Si

E

D



ij1 k+

ij2 and defining

Vmaxij = k+

ij2EjT one has

KSij =KSij

(1+

Vmaxij

4πDircjnjKSij

) (28)












dSidt

= minus

j=Jsumj=1

k+

ij2Cij i = 1 I (29)

dBjdt

=

i=Isumi=1

microijk+




err(t)=3

N

i=Nsumi=1



∣∣∣∣ (31)







(KSij k

+

ij2microij

)


S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)



+

ij2microij

) whose


S1 (300)


B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)





+

ij2microij

) whose units


S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)











+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References


































































































































wherek+







SiEjk+

ij1 =

(kminus

ij1 + k+

ij2

)Cij (5)


KSij =kminus

ij1 + k+

ij2

k+

ij1

=SiEj

Cij (6)






dPijdt

= k+

ij2Cij (8)



dSidt

= minus

k=Jsumk=1

(k+


ik1Cik

) (9)


SiT = Si +

k=Jsumk=1

Cik (10)


dSiTdt

= minus

k=Jsumk=1

k+

ik2Cik (11)


EjT = Ej +

k=Isumk=1

Ckj (12)

We note that ifk+







Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkTKSkj

+

k=Jsumk=1

EkTKSik

) (13)



C11 =S1TE1T

KS11+ S1T +E1T (14)



C11 =S1TE1T

S1T +KS11

(1+

E1TKS11

+E2TKS12

) (15)


13 E1 13 13 Ej1 13 Ej 13 Ej+1 13 13 EJ 13


13 13 13 13 13 13 13 13




13 13 13 13 13 13 13 13


13


C11 =α1S1TET

S1T +KS11

[1+

(α1KS11

+1minusα1KS12

)ET

] (16)

=α1S1TET

S1T +KS11

(1+

ETKI

) whereKI =

(α1KS11

+1minusα1KS12




C1j =S1TEjT

KS1j

(1+

k=Jsumk=1

EkTKS1k

)+ S1T

(17)


C1j =αjS1TEjT

KS1j

(1+

k=Jsumk=1

EkTKI1k

)+ S1T

(18)




KI1k =

(αk

KS1k+

1minusαk

KS1kd

)minus1

(19)



Ci1 =SiTE1T

KSi1

(1+

k=Isumk=1

SkTKSk1

)+E1T

(20)


Ci1 =SiTE1T

KSi1

(1+

KSi1

KSi1

k=Isumk=1

SkTKSk1

) (21)




C11 =S1TE1TKS11

1+S1TKS11

+S2TKS21

+E1TKS11

(22a)

C21 =S2TE1TKS21

1+S1TKS11

+S2TKS21

+E1TKS21

(22b)







KSij =KSij

k=Lsumk=1

IkTKIijk

+ 1

(23)



Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkT

KSkj+

k=Jsumk=1

EkT

KSik

) (24)








Si =4πDircnjSi

k+


4πDircnjSik+




KSij =kminus

ij1 + k+

ij2

k+

ij1

(1+

k+

ij1EjT

4πDircjnj

)(26)


KSij =KSij

(1+

k+

ij1EjT

4πDircjnj

) (27)

Si +Ejkij 1

k+ij 1SiEj Pij +Ej

Si

E

D



ij1 k+

ij2 and defining

Vmaxij = k+

ij2EjT one has

KSij =KSij

(1+

Vmaxij

4πDircjnjKSij

) (28)












dSidt

= minus

j=Jsumj=1

k+

ij2Cij i = 1 I (29)

dBjdt

=

i=Isumi=1

microijk+




err(t)=3

N

i=Nsumi=1



∣∣∣∣ (31)







(KSij k

+

ij2microij

)


S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)



+

ij2microij

) whose


S1 (300)


B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)





+

ij2microij

) whose units


S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)











+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References































































































































Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkTKSkj

+

k=Jsumk=1

EkTKSik

) (13)



C11 =S1TE1T

KS11+ S1T +E1T (14)



C11 =S1TE1T

S1T +KS11

(1+

E1TKS11

+E2TKS12

) (15)


13 E1 13 13 Ej1 13 Ej 13 Ej+1 13 13 EJ 13


13 13 13 13 13 13 13 13




13 13 13 13 13 13 13 13


13


C11 =α1S1TET

S1T +KS11

[1+

(α1KS11

+1minusα1KS12

)ET

] (16)

=α1S1TET

S1T +KS11

(1+

ETKI

) whereKI =

(α1KS11

+1minusα1KS12




C1j =S1TEjT

KS1j

(1+

k=Jsumk=1

EkTKS1k

)+ S1T

(17)


C1j =αjS1TEjT

KS1j

(1+

k=Jsumk=1

EkTKI1k

)+ S1T

(18)




KI1k =

(αk

KS1k+

1minusαk

KS1kd

)minus1

(19)



Ci1 =SiTE1T

KSi1

(1+

k=Isumk=1

SkTKSk1

)+E1T

(20)


Ci1 =SiTE1T

KSi1

(1+

KSi1

KSi1

k=Isumk=1

SkTKSk1

) (21)




C11 =S1TE1TKS11

1+S1TKS11

+S2TKS21

+E1TKS11

(22a)

C21 =S2TE1TKS21

1+S1TKS11

+S2TKS21

+E1TKS21

(22b)







KSij =KSij

k=Lsumk=1

IkTKIijk

+ 1

(23)



Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkT

KSkj+

k=Jsumk=1

EkT

KSik

) (24)








Si =4πDircnjSi

k+


4πDircnjSik+




KSij =kminus

ij1 + k+

ij2

k+

ij1

(1+

k+

ij1EjT

4πDircjnj

)(26)


KSij =KSij

(1+

k+

ij1EjT

4πDircjnj

) (27)

Si +Ejkij 1

k+ij 1SiEj Pij +Ej

Si

E

D



ij1 k+

ij2 and defining

Vmaxij = k+

ij2EjT one has

KSij =KSij

(1+

Vmaxij

4πDircjnjKSij

) (28)












dSidt

= minus

j=Jsumj=1

k+

ij2Cij i = 1 I (29)

dBjdt

=

i=Isumi=1

microijk+




err(t)=3

N

i=Nsumi=1



∣∣∣∣ (31)







(KSij k

+

ij2microij

)


S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)



+

ij2microij

) whose


S1 (300)


B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)





+

ij2microij

) whose units


S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)











+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References































































































































KI1k =

(αk

KS1k+

1minusαk

KS1kd

)minus1

(19)



Ci1 =SiTE1T

KSi1

(1+

k=Isumk=1

SkTKSk1

)+E1T

(20)


Ci1 =SiTE1T

KSi1

(1+

KSi1

KSi1

k=Isumk=1

SkTKSk1

) (21)




C11 =S1TE1TKS11

1+S1TKS11

+S2TKS21

+E1TKS11

(22a)

C21 =S2TE1TKS21

1+S1TKS11

+S2TKS21

+E1TKS21

(22b)







KSij =KSij

k=Lsumk=1

IkTKIijk

+ 1

(23)



Cij =SiTEjT

KSij

(1+

k=Isumk=1

SkT

KSkj+

k=Jsumk=1

EkT

KSik

) (24)








Si =4πDircnjSi

k+


4πDircnjSik+




KSij =kminus

ij1 + k+

ij2

k+

ij1

(1+

k+

ij1EjT

4πDircjnj

)(26)


KSij =KSij

(1+

k+

ij1EjT

4πDircjnj

) (27)

Si +Ejkij 1

k+ij 1SiEj Pij +Ej

Si

E

D



ij1 k+

ij2 and defining

Vmaxij = k+

ij2EjT one has

KSij =KSij

(1+

Vmaxij

4πDircjnjKSij

) (28)












dSidt

= minus

j=Jsumj=1

k+

ij2Cij i = 1 I (29)

dBjdt

=

i=Isumi=1

microijk+




err(t)=3

N

i=Nsumi=1



∣∣∣∣ (31)







(KSij k

+

ij2microij

)


S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)



+

ij2microij

) whose


S1 (300)


B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)





+

ij2microij

) whose units


S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)











+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References



































































































































Si =4πDircnjSi

k+


4πDircnjSik+




KSij =kminus

ij1 + k+

ij2

k+

ij1

(1+

k+

ij1EjT

4πDircjnj

)(26)


KSij =KSij

(1+

k+

ij1EjT

4πDircjnj

) (27)

Si +Ejkij 1

k+ij 1SiEj Pij +Ej

Si

E

D



ij1 k+

ij2 and defining

Vmaxij = k+

ij2EjT one has

KSij =KSij

(1+

Vmaxij

4πDircjnjKSij

) (28)












dSidt

= minus

j=Jsumj=1

k+

ij2Cij i = 1 I (29)

dBjdt

=

i=Isumi=1

microijk+




err(t)=3

N

i=Nsumi=1



∣∣∣∣ (31)







(KSij k

+

ij2microij

)


S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)



+

ij2microij

) whose


S1 (300)


B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)





+

ij2microij

) whose units


S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)











+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References

































































































































dSidt

= minus

j=Jsumj=1

k+

ij2Cij i = 1 I (29)

dBjdt

=

i=Isumi=1

microijk+




err(t)=3

N

i=Nsumi=1



∣∣∣∣ (31)







(KSij k

+

ij2microij

)


S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)



+

ij2microij

) whose


S1 (300)


B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)





+

ij2microij

) whose units


S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)











+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References































































































































(KSij k

+

ij2microij

)


S1 (30) S2 (100) S3 (90)

B1 (01) (1 48 05) (10 48 03) (50 48 01)M1 (1094) (212 0 0) (212 0 0) (212 0 0)



+

ij2microij

) whose


S1 (300)


B1 (1) (1 1 04) (100 10 04)B2 (1) (1 1 04) (100 10 04)B3 (1) (1 1 04) (100 10 04)B4 (1) (1 1 04) (100 10 04)B5 (1) (1 1 04) (100 10 04)





+

ij2microij

) whose units


S1 (448) S2 (431) S3 (121)

B1 (033) (1 1 05) (100 1 03) (5000 1 01)B2 (033) (10 08 05) (10 08 03) (1000 08 01)B3 (033) (10 04 05) (10 04 03) (100 04 01)











+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References
































































































































+

ij2microij





S1 (359) S2 (386) S3 (255)

B1 (494) (222 06027 05) (964 06027 03) (2838 06027 01)B2 (422) (222 03605 05) (1852 03605 03) (52161 03605 01)B3 (242) (222 02061 05) (1852 02061 03) (2195 02061 01)




k=8sumk=1


+ (17σMass)minus1

k=17sumk=1














rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References































































































































rrel

ation

Coe ff

icien

t 010

020

030

040

050

060

070

080

090

095

099

010

020

030

040050060

070

080

090

095

099

00

0Standard devision

100100 200200 300300 400400







0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML





04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References






























































































































04

06

08

1

LCI

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

nal o

f mic

robi

al C

Time (day)

(c)

MMECAECECAminusML



0

50

100

S3 (

mg

C d

mminus3

)

(c)

0

50

100

S2 (

mg

C d

mminus3

)

(b)

0

10

20

30

S1 (

mg

C d

mminus3

)

(a)

0 10 20 30 40 500

20

40

60

Time (day)

B1 (

mg

C d

mminus3

)

(d)

MMECAECECAminusML


04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

() (a)

0 10 20 30 40 500

02

04

06

08

1

Fra

ctio

n of

mic

robi

al C

Time (day)







0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References






























































































































0

20

40

60

80(e) B

4

(mg

C d

mminus3

)

0

20

40

60(f) B

5

09

1

11(c) B

2

(mg

C d

mminus3

)

0

5

10(d) B

3

0

100

200

300(a) S

1

(mg

C d

mminus3

)

1

12

14(b) B

1

0 10 20 30 40 500

02

04

06(g) ErrminusMM

Time (day)0 10 20 30 40 50

0

02

04

06(h) ErrminusECA

Time (day)

MMECAEC




0

2

4

B2 (

g C

) (e)

0

50

100

B1 (

g C

) (d)

0

200

400

600

S3 (

g C

) (c)

0

200

400

600

S2 (

g C

) (b)

0

200

400

600

S1 (

g C

) (a)

0 200 400 600 800 1000 1200 14000

02

04

Time (day)

B3 (

g C

) (f)

MMECAECECAminusML


0

01

02

03

04

Litte

r LC

I

(b)

0

20

40

60

80

100R

emai

ning

litte

r (

) (a)

0 200 400 600 800 1000 1200 14000

01

02

03

04

05

Time (day)

Fra

ctio

n of

mic

robi

al C (c)

MMECAECECAminusML











04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References




































































































































04

05

06

07

08

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c)








02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References






























































































































02

04

06

08

1

Litte

r LC

I

(b)

0

20

40

60

80

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

02

04

06

08

1

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)

MMminusEnsDataMM




0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (b) B

2

0

005

01

015

02

Fra

ctio

n of

mic

robi

al C (a) B

1

0 10 20 30 40 50 60 70 800

005

01

015

02

Time (month)

Fra

ctio

n of

mic

robi

al C (c) B

3








02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References






























































































































02

04

06

08

Litte

r LC

I

(b)

0

50

100

Rem

aini

ng li

tter

()

(a)

0 10 20 30 40 50 60 70 800

01

02

03

04

Time (month)

Fra

ctio

n of

mic

robi

al C

(c)



0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

y=069x+2133 R2=087Mea

sure

d re

mai

ning

litte

r (

)


(a)

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

08

09

1

y=058x+022 R2=078

Mea

sure

d LC

I

Predicted LCI

(b)






0 20 40 60 800

10

20

30

40

50

60

70

80

90

100

Time (month)

Rem

aini

ng fr

actio

n (

)






ij1 kminus

ij1 andk+









4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References



































































































































4 Conclusions



Appendix A



Si =SiT

1+

k=Jsumk=1

EkKSik

(A1)


Ej =EiT

1+

k=Isumk=1

SkKSkj

(A2)


Cij =SiTEjT

KSij

(1+

k=Jsumk=1

EkKSik

)(1+

k=Isumk=1

SkKSkj

) (A3)



Ej = Ej0 + εEj1 + ε2Ej2 + (A4b)

Si = Si0 + εSi1 + ε2Si2 + (A4c)


gives

ε Cij1

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)=SiTEjT

εKSij(A5a)













+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References








































































































































+


kminus1 kminus


k+2 k+














ε2 Cij2

(1+

k=Jsumk=1

Ek0

KSik+

k=Isumk=1

Sk0

KSkj

)(A5b)

+Cij1

(k=Jsumk=1

Ek1

KSik+

k=Isumk=1

Sk1

KSkj+

m=Il=Jsumm=1l=1

KSnlCml1

KSilKSmj

)= 0


Cij1 asympSiTEjT

εKSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6a)

Cij2 =Cij1

KSij

(1+

k=Jsumk=1

Ek0KSik

+

k=Isumk=1

Sk0KSkj

) (A6b)

n=Il=Jsumn=1l=1

Cnl1KSij


KSilKSnj

)





(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References






























































































































(KSij

Si0+

k=Jsumk=1

KSij

KSik

Ek0

Si0+

k=Isumk=1

KSij

KSkj

Sk0

Si0

)(A7)(

KSij

Ej0+

k=Jsumk=1

KSij

KSik

Ek0

Ej0+

k=Isumk=1

KSij

KSkj

Sk0

Ej0

) 1



C11 = εC111 + ε2C112 (A8)

=S1TE1T

KS11+ S1T +E1T

[1+


)2]


Appendix B



k=Jsumk=1

Cik

)(EjT minus

k=Isumk=1

Ckj

)=KSijCij (B1)


KSijCij + SiT

k=Jsumk=1

Cik +EjT

k=Isumk=1

Ckj = SiTEjT (B2)




Cij =SiTEjT

KSij +

k=Isumk=1

SkTEjT+KSijEjT+KSkj

+

k=Jsumk=1

EkTSiT+KSijSiT+KSik

(B3)


Appendix C



Cij =SiTEjT

KSij + SiT(C1)


k=Isumk=1

Ckj le EjT (C2a)

k=Jsumk=1

Cik le SiT (C2b)




Edited by G Herndl



References






























































































































References





























































































































biogeosciences a total quasi-steady-state formulation of

Documents