clark_2010_a mechanistic model of the enzymatic hydrolysis cellulose

8/6/2019 Clark_2010_A Mechanistic Model of the Enzymatic Hydrolysis Cellulose

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ARTICLE

A Mechanistic Model of the Enzymatic Hydrolysisof Cellulose

Seth E. Levine, Jerome M. Fox, Harvey W. Blanch, Douglas S. ClarkEnergy Biosciences Institute, Department of Chemical Engineering,

University of California-Berkeley, Berkeley, California 94720;

telephone: 510-642-2408; fax: 510-643-1228; e-mail: [email protected]

Received 19 January 2010; revision received 16 April 2010; accepted 26 April 2010

Published online 18 May 2010 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/bit.22789

ABSTRACT: A detailed mechanistic model of enzymaticcellulose hydrolysis has been developed. The behavior ofindividual cellulase enzymes and parameters describing the

cellulose surface properties are included. Results obtainedfor individual enzymes (T. reesei endoglucanase 2 andcellobiohydrolase I) and systems with both enzymes presentare compared with experimental literature data. The modelwas sensitive to cellulase-accessible surface area; the EG2CBHI synergy observed experimentally was only predicted ata sufficiently high cellulose surface area. Enzyme crowding,which is more apparent at low surface areas, resulted indifferences between predicted and experimental rates ofhydrolysis. Model predictions also indicated that theobserved decrease in hydrolysis rates following the initialrate of rapid hydrolysis is not solely caused by productinhibition and/or thermal deactivation. Surface heterogene-ities, which are not accounted for in this work, may play arole in decreasing the hydrolysis rate. The importance of

separating the enzyme adsorption and complexation steps isillustrated by the models sensitivity to the rate of formationof enzymesubstrate complexes on the cellulose surface.

Biotechnol. Bioeng. 2010;xxx: xxxxxx.

2010 Wiley Periodicals, Inc.

KEYWORDS: cellulase activity; cellulose; enzymatic hydro-lysis; kinetic models; mechanistic models; enzyme crowd-ing; cellulase synergy

Introduction

The mechanism of enzymatic cellulose hydrolysis is notwell understood. The complexity of the system, whicharises from the concerted action of several enzymes actingon a heterogeneous solid substrate, makes experimentalkinetic and mechanistic studies difficult. As a result, the rate-determining features of the cellulasecellulose system have

yet to be identified, and the physical and chemical details ofthe enzymesubstrate interaction remain to be elucidated.

Improved understanding of the overall system is crucial forthe design of optimal enzymatic hydrolysis processes.

During enzymatic cellulose hydrolysis, saccharificationrates typically decline dramatically after an initial burstphase (Huang, 1975; Jeoh et al., 2006; Jeoh et al., 2007; Leeand Fan, 1983; Medve et al., 1998). This decrease inhydrolysis rates makes high enzyme loadings and longreaction times necessary to achieve desired celluloseconversions. While many hypotheses for the cause or causesof this decline have been proposed (Bommarius et al., 2008;Desai and Converse, 1997; Erikkson et al., 2002; Nidetzkyand Steiner, 1993; Ohmine et al., 1983; Valjamae et al., 1998;Yang et al., 2006; Zhang et al., 1999), the origin of thisphenomenon is still unresolved.

Detailed mechanistic models provide tools for studyingthe kinetics and mechanisms of complex reaction systems.

Their importance in efforts to understand and improveenzymatic cellulose hydrolysis processes has been describedearlier (Bansal et al., 2009; Okazaki and Moo-Young, 1978).Mechanistic models allow investigators to rapidly probe aproposed reaction network through the use of computa-tional predictions and sensitivity analysis. By understandingthe underlying kinetics and mechanism of the cellulasereaction system more directed and rational approaches canbe used for cellulase engineering and process optimization.

Many models of cellulase-catalyzed cellulose hydrolysishave been developed over the past 30 years. Most of thesemodels contained simplified representations of the cellulases

and/or the substrate. The activities of the different cellulaseenzymes that work together during cellulose hydrolysis arecommonly lumped together as a single enzyme concentra-tion. The cellulose substrate has been represented using avariety of simplified representations: (1) considering thesolid cellulose phase to be equivalent to a soluble polymericsubstrate (Okazaki and Moo-Young, 1978), (2) describingthe cellulose by a single bulk concentration (Holtzappleet al., 1984; Kadam et al., 2004; Zheng et al., 2009), (3)treating the cellulose as a mixture of digestible and inertsubstrate concentrations (Fan and Lee, 1983; Gan et al.,

Correspondence to: H.W. Blanch or D.S. Clark

Contract grant sponsor: Energy Biosciences Institute JMF is an NSF Fellow

Additional Supporting Information may be found in the online version of this article.

2010 Wiley Periodicals, Inc. Biotechnology and Bioengineering, Vol. xxx, No. xxx, 2010 1


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2003), (4) using one or two adsorption sites (e.g., crystallineand amorphous) on a surface area that is then related to thebulk cellulose concentration (Converse and Optekar, 1993;Converse et al., 1988; Movagharnejad and Sohrabi, 2003;Movagarnejad et al., 2000; Wald et al., 1984), and (5) using ahypothetical periodic surface composed of random poly-meric chains (Fenske et al., 1999). These simplificationslimit the ability of these models to explain enzymeenzymecooperativity or to predict evolution of the cellulose

substrate surface morphology as hydrolysis occurs.Most models that include an enzyme adsorption step rely

on simple assumptions about the enzymesubstrate inter-action. Cellulase adsorption onto cellulose has typically beendescribed through the use of a Langmuir-type equilibriumequation (Holtzapple et al., 1984; Peri et al., 2007; Waldet al., 1984; Zhou et al., 2009a,b). The Langmuir isotherm,while often able to be fit to experimental data, is based onassumptions that are not representative of the cellulasecellulose system. In addition, inconsistencies betweenexperimental adsorption data and the Langmuir equationpredictions have been reported in the literature (Medveet al., 1997). Several models have avoided using anequilibrium adsorption model (Converse and Optekar,1993; Gan et al., 2003), but their reliance on lumped enzymeand/or substrate concentrations has remained, rendering thephysical meaning of the adsorption step unclear.

Representing adsorption in a mechanistic model is furthercomplicated by the formation of the enzymesubstratecomplex on the cellulose surface. Early models assumed thatan enzyme adsorbs to the cellulose surface and complexeswith a cellulose chain in one concerted reaction step(Okazaki and Moo-Young, 1978; Suga et al., 1975a,b); latermodels derived from these early models have also made this

assumption (Zhang and Lynd, 2006; Zhou et al., 2009a,b).Most cellulases, however, have a structural architecture thatsuggests a two-step process. Cellulases contain a carbohy-drate-binding module (CBM) that is separated from thecatalytic domain by a short linker; adsorption of the CBMdomain to the surface and complexation of the catalyticdomain to a chain may occur as two distinct events. A recenthigh-speed AFM study of cellulase enzymes on cellulosesupports this two-step interaction (Igarashi et al., 2009).

Mathematical representations must capture this complexa-tion event. The utility of previous models in predictingenzymatic cellulose hydrolysis behavior has been severelylimited by their reliance on lumped terms and inadequaterepresentations of cellulasecellulose interactions.

In this work a model has been developed which avoidssome of the unrealistic assumptions that have limited earlierefforts. It is based on a mechanistic description that includesdistinct enzyme adsorption and complexation steps. Theequilibrium assumption is abandoned, thus permittingdynamic interactions between cellulases and the cellulosesurface to be incorporated. Individual cellulases of a well-defined enzyme mixture are explicitly tracked; substrateconcentration and the degree of cellulose polymerization aremonitored; and surface concentrations of each cellulosechain length are individually described. Representation ofthe substrate is focused on capturing the time course ofcellulose surface area; as the cellulose particles shrink, newchains are exposed, and the total cellulose surface area isreduced. This approach provides useful insights into theimpact of substrate surface area on the hydrolysis rate, theroles of various mechanisms in the slowdown in the rate ofhydrolysis, and the importance of an adequate representa-tion of the enzymesubstrate interaction.

BOX 1. Mapping Arbitrary Particle Shapes to Collections of Spheres

Monodisperse distributions of spheres can be used to model properties for static shapes. When the arbitrary shapes ofinterest change with time, a polydisperse distribution of spheres must be used to properly map the properties. Figure A1shows the requirements and capabilities for matching static and dynamic properties of a given shape.

The polydisperse distribution of spheres required to map a given shape is difficult to determine. It requires knowledgeof how the surface area changes with time. Thus a reverse approach may be used, where a given polydisperse distributionis used to model the dynamic change of a particles properties. The form of these results is compared to measuredresults. Thus, instead of directly finding the sphere distribution to match an arbitrary shape, an arbitrary shape is foundthat matches a given sphere distribution.

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Methods

Cellulose Substrate

The solid cellulose substrate is represented as an assembly ofspherical particles. This representation reflects an estab-lished mapping of arbitrary shapes into an assembly ofmonodisperse spheres that has the same total surface areaand volume as the original shape (Hansen and Travis, 1974).

This approach has been used to describe the radiativescattering and absorption of atmospheric ice crystals(Hansen and Travis, 1974; Warren and Thomas, 1999).The rate of cellulose hydrolysis is directly dependent to theamount of substrate surface area; the manner in whichsurface area changes depends on the initial shape. In orderto describe the change of surface area with time, we positthat the original shape can be represented by apolydisperse assembly of spheres (Box 1 provides thedetails of arbitrary shape mapping to spheres). In thepresent work, both monodisperse and bimodal distributionsare considered.

Particles are composed of cellulose chains of varying

length described by a Poisson distribution based on theinitial degree of polymerization. The cellulose particlesshrink as soluble cello-oligosaccharides are released from thesurface. A material balance describes the rate at which theradius changes with time (Eq. 1). The surface area is relatedto the density and the radius of a spherical particle by asimple relationship (Eq. 2).

dR

dt R

SA

3MW1

X6i1

ir0i (1)

SA 3Rrcellulose

(2)

As soluble cello-oligosaccharides are released from thesolid cellulose surface, fresh cellulose chains become partof the surface. A term is required in the equations forcellulose chain surface concentrations to account for thisnew chain exposure. Particles are assumed to be composedof a continuum of cellulose chains rather than discretelayers, and their physical characteristics (e.g., sites per areaand density) are assumed to remain constant as the

particles shrinks. The rate at which new chains are exposedis equal to the rate at which old chains are lost, as thenumber of sites per area is assumed to be constant.Individual chains have an exposure term that dependson the rate of loss of old chains and the initial degreeof polymerization probability distribution function F(Eq. 3).

dC0i

dt FDP0; i

1

DP0

X6j1

jr0i (3)

The cellulose chains on the surface are represented as alattice of glucose units. In the case of lignocellulosicsubstrate, the representation of the surface can be modifiedto include non-hydrolyzable sites. Each cellulase enzymethat adsorbs onto this lattice occupies a certain number ofsites, described by its footprint. Footprint estimates arebased on small angle X-ray scattering measurements of theTrichoderma reesei cellobiohydrolase I (CBHI) (Abuja et al.,1988a,b). If the size of each glucose unit is assumed to be

0.25 nm2, CBHI occupies approximately 84 glucose sites. Ifall cellulases are assumed to have this footprint, regardless ofshape, a monolayer of perfectly packed enzymes would coverall lattice sites. A more physically realistic description ofadsorption can be developed by considering randomadsorption by cellulases of a defined shape to the surface.Consequently, random sequential adsorption (RSA) simu-lations of a cellulase-shaped adsorbate consisting of a largecatalytic domain, a short linker, and a small CBM, adsorbingonto a periodic square lattice were used to determine aneffective footprint. An adsorbed enzyme represented thisway occupies 156 glucose units, nearly double the physical

footprint. The details of the RSA simulation are provided inAppendix B of the Supplemental Material.

Hydrolysis Mechanism

The mechanism by which cellulases catalyze the hydrolysisof cellulose can be considered in three steps: (1) adsorption,(2) complexation, and (3) reaction. Adsorption andcomplexation were treated as reversible steps, while thereaction step was treated as irreversible. A schematic of thismechanism is given in Figure 1, which illustrates these steps

for an endoglucanase (EG) and a cellobiohydrolase (CBH).Adsorption and desorption of the cellulases weredescribed using site and enzyme balances, which provideconcentrations of cellulose surface sites and solution-phaseenzymes. All cellulases are allowed to adsorb to identical free

Figure 1. Schematic of the general mechanistic steps of cellulase-catalyzedhydrolysis of cellulose by an endoglucanase and an exoglucanase.

Levine et al.: Enzymatic Cellulose Hydrolysis 3

Biotechnology and Bioengineering


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sites on the cellulose surface. Adsorption and desorptionreactions are treated as elementary reactions, leading tobalance equations for adsorbed, uncomplexed enzymeconcentrations (Eq. 4).

dE0xads

dt kxadsExfree

0 kxdesE0

xads (4)

The terms resulting from adsorbed enzyme complexing

and decomplexing with a cellulose chain and catalysis arenot included in Equation (4). The catalysis step is assumedto be slow in comparison with the other reactions, allowingthe complexation/decomplexation step to be considered atequilibrium.

The equilibrium assumption for complexation leads to asimple relationship between the surface concentrationsof adsorbed complexed enzyme, adsorbed uncomplexedenzymes, and surface cellulose chains (Eq. 5). The differencein the equilibrium relationship for endoglucanase andcellobiohydrolases enzymes stems from the ability ofendoglucanase to complex with any glycosidic bond on

the cellulose chain while cellobiohydrolases can onlycomplex with a specific chain end (reducing or non-reducing).

E0EGads E0EGadsuEGi 1C

0

i

KMEGcel

E0CBHads E0CBHadsuCBHC

0

i

KMCBHcel

(5)

Using this equilibrium relationship and the mechanismdepicted in Figure 1, balance equations for the surfaceconcentration of solid cellulose chains of length i can be

derived (Eq. 6).

i > 6

dC0i

dt

kcatEGcel

KMEGcelE0EGadsuEG 2

X1ji1

C0j i 1C0

i

!

kcatCBHcel

KMCBHcelE0CBHadsuCBHC

0

i2 C0

i

(6)

Additional endoglucanase or cellobiohydrolases enzymescan be incorporated into the model; this will add terms toEquation (6) that are similar to those above.

Soluble cello-oligosaccharides are formed by enzymesacting on both soluble, short-chain sugars and on insolublechains within the solid substrate. The equations describingthe generation of soluble short-chain sugars (DP 6)assume a MichaelisMenten mechanism for the solublephase reaction terms, and rely on assumptions similar to

those in Equation (6) for the solid phase reaction terms. Theequation for cellobiose concentration is shown as anexample in Equation (7). The full set of material balanceequations can be found in Appendix A with detailedderivations in Appendix C of the Supplemental Material.While it is not shown here, the action ofb-glucosidase canbe integrated into the model equations with an additionalMichaelisMenten type term in the cellobiose and glucoseequations. Both free and adsorbed cellulases are competi-tively inhibited by glucose and cellobiose. Terms for thisinhibition are present in the enzyme and site balances. Thefull set of model equations is shown in Appendix A (withdetailed derivations in Appendix C of the SupplementalMaterial).

Model Parameters

The model requires a variety of adsorption, kinetic, andphysical parameters. Sets of experimental values weredetermined in the present work, or values from theliterature were used. In the absence of reliable experimen-tally determined parameters, estimates were made by fittingthe model to experimental data.

Kinetic parameters for the activity of cellulase enzymes on

a variety of substrates have been reported in the literature.The work detailed here relies on accurate determination ofMichaelisMenten constants for cellulase activity on solublecello-oligosaccharides. For T. reesei parameters are availablefor CBH1 (Nidetzky et al., 1994), CBH2 (Koivula et al.,2002) and a range of endoglucanases (Karlsson et al., 2002).Recent results for the rate at which a CBH1 enzyme moves asit degrades the end of a cellulose chain (Igarashi et al., 2009)reveal that the turnover number of a cellulase complexedwith a solid cellulose chain ($7 s1) is roughly equivalent tothat of a cellulase acting on a soluble cello-oligosaccharidesin solution ($9 s1). Accordingly, kcat values determined on

dC2

dt 2

kcatEGcel

KMEGcelE0EGads

uEG

X1i7

C0i kcatCBHcel

KMCBHcelE0CBHadsuCBH

X1i7

C0i

2kcatEGsol

KMEGsolE0EGads

X6i3

CikcatCBHsol

KMCBHsolE0CBHads

X6i3

Ci C4

!!Acel

Vliq

2kcatEGsol

KMEGsolEEGfree

X6i3

Ci kcatCBHsol

KMCBHsolECBHfree

X6i3

Ci C4

!(7)



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substrates like cellohexaose and cellopentaose can be usednot only for solution-phase reactions but also for solidcellulose substrates. There are no comparable experimentalvalues for the complexation constants of adsorbed enzymewith chains on the surface of a solid substrate. Theseparameters were thus optimized to best fit the experimentaldata.

Product inhibition equilibrium constants have beenreported for cellulase hydrolysis of soluble substrates.

Reported values range over several orders of magnitude(Hsu et al., 1980; Koivula et al., 1996; Nidetzky et al., 1994;von Ossowski et al., 2003), making the determination ofreliable values difficult. Literature values were used only toset reasonable bounds for the optimization of the productinhibition parameters.

The majority of cellulase adsorption studies haveemployed solution depletion measurements. A Langmuirmodel has typically been fit to such data to determineadsorption parameters. Because the model proposed hererequires values of kadsorption and kdesorption for each enzymein the system (Eq. 4), fits to the Langmuir isotherm have notbeen employed. Values for these parameters have beenestimated for a commercial cellulase mixture (Celluclast1.5 L) using data obtained from ellipsometry (S. Mauer andC.J. Radke, 2009, unpublished results).

The parameters associated with the cellulose substrate arethe initial surface area, the sites per area, and the averagedegree of polymerization of the cellulose chains.The substrate surface is assumed to consist of a homo-geneous lattice of glucose units. While an experimental valuefor sites per area is not available an upper bound for thisvalue can be calculated based on the size of a single glucoseunit (0.25 nm2); accordingly a glucose lattice structure has asurface concentration of 6.64 105 mmol dm2. The

degree of polymerization of several of cellulosic substrates(including Avicel, cotton, bacterial cellulose) has beendetermined using a variety of techniques (Kleman-Leyeret al., 1992; Ng and Zelkus, 1980; Ryu et al., 1982; Valjamaeet al., 1999; Wood, 1988). These values provide an estimateof a typical degree of polymerization for a given cellulosicsubstrate. The literature provides only a limited set ofsurface area measurements. Nitrogen BET measurementsare available; however, nitrogen molecules are far smallerthan proteins, and this method requires the use of drysubstrates; therefore, these measurements will give inaccu-rate values for the surface area accessible to cellulases.

Attempts to use surface depletion of fluorescent cellulase-like constructs have also been made (Hong et al., 2007), butthese use only one type of CBM and thus likely under-estimate the accessible surface area. Appendix D in theSupplemental Material provides more detail about theparameter values used in this study.

Model Assembly and Solution

The model consists of a system of differential and algebraicequations. Balance equations for the surface concentrations

for chains of DP 7 to DPmax (set to 1.5 DP0); theconcentrations of soluble cello-saccharides from glucose tocellohexaose; the adsorbed, uncomplexed enzyme surfaceconcentrations; and the radius of the cellulose particles areincluded. Algebraic equations for enzyme and site balancesare also part of the system of equations that comprise themodel. The model was solved in Matlab using the ode23tfunction. For parameter optimization the fmincon functionwas used, which utilizes a sequential quadratic program-

ming optimization algorithm. A normalized sum of squareerror between experimental values and model results wasused as the objective function.

Results

Surface Area Requirements

The inadequacy of currently available surface area measure-ments is made clear by a comparison between measuredvalues and the cellulose surface area required based onadsorbed enzyme footprint estimates. A typical enzyme

loading of 0.15mmolg1

(approximately 9 mg enzyme pergram substrate) requires a surface area of 1.9 m2 g1

assuming an enzyme footprint based on the crystal structureenzyme dimensions, and a surface area of 3.5m2 g1 isrequired based on the effective (RSA) enzyme footprint. The0.15mmol g1 loading is usually considered to be indust-rially relevant. The accessible surface area of Avicel hasrecently been reported as 2.38 m2 g1 (Hong et al., 2007).However, the 2.38m2 g1 estimate for the cellulase-accessible surface area for Avicel is insufficient toaccommodate a loading of 0.15mmolg1. Despite beingexperimentally elusive, reliable measurements for enzymeaccessible surface area are critical for understanding and

Table I. Kinetic parameters used in model simulations for comparison to

experimental data from Medve et al. (1998).

Kinetic parameter Valuea Units

kEG2-ads 8640b L mmol1 h1

kCBH1-ads 8640b L mmol1 h1

kEG2-des 19.3b hr1

kCBH1-des 164b hr1

kcat-EG2 65c s1

kcat-CBH1 9.4c s1

KM-EG2-cel 0.0067d mmol dm2

KM-CBH1-cel 0.000014d mmol dm2

KM-EG2-sol 0.053c mmolL1

KM-CBH1-sol 0.0032c mmolL1

Ki-EG2-cellobiose 0.01d mmolL1

Ki-CBH1-cellobiose 0.093d mmolL1

Ki-EG2-glucose 16.9d mmolL1

Ki-CBH1-glucose 31c mmolL1

aFor expanded details about the model parameters and their sources seeAppendix C in the Supplemental Material.

bParameters based on values from elipsiometry experiments (unpub-lished data).

cExperimental parameters from various literature sources.dParameters optimized to early time data (less than 3 h) for single

enzyme hydrolysis results from Medve et al. (1998).




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modeling the enzymesubstrate interactions duringhydrolysis.

Enzyme Activity With Low Surface Area(Monodisperse Spheres)

The model was tested using T. reesei CBHI and endoglu-canase 2 (EG2) hydrolysis time-course data reported byMedve et al. (1998). The hydrolysis conditions were asfollows: 10 g L1 Avicel; 0.16mmol g1 of either EG2 orCBHI (individual pure enzymes) (approximately 10 mgenzyme per g cellulose); 0.32 mmolg1 total enzyme (1:1EG2/CBHI mixed enzymes) (approximately 20 mg enzymeper g cellulose); pH 4.8; 50 mM sodium acetate buffer; 408C.The experiments and simulations conducted with a singlepure enzyme (EG2 or CBHI) will be referred to as singleenzyme hydrolysis in this work. The summation of the total

effective glucose release rates of the EG2 and CBHI singleenzyme hydrolysis experiments and simulations will bereferred to as theoretical mixed enzyme hydrolysis. Theexperiments and simulations that use a 1:1 EG2/CBHImixture will be referred to as mixed enzyme hydrolysis.The parameters used in the model are shown in Table I(more details can be found in Appendix C of theSupplemental Material).

An Avicel surface area of 8m2 g

1 was used as a lowsurface area condition. This value was chosen because itcorresponds to a case where an enzyme loading of0.16mmolg1 represents 50% surface coverage based onthe effective (RSA) area occupied by an enzyme. The modelresults are compared to the experimental values in Figure 2.The activities for single enzyme hydrolysis are shown inFigure 2a. The model exhibits excellent quantitativeagreement at early times (up to 3 h) and qualitativeagreement for extended times for single enzyme hydrolysis.The model fails to capture the decrease in the rate ofhydrolysis. The model does not capture the experimental

data for the hydrolysis rates with both enzymes present, asshown in Figure 2b. While the experimental results show asynergistic relationship between EG2 and CBH1, the modelpredicts a competitive relationship.

The model allows the cause of this inconsistency to beexamined. Figure 3 shows the fraction of available sites onthe substrate surface for each enzyme during single enzymeand mixed enzyme hydrolysis. The fraction of sites availableto EG2 for hydrolysis is much lower in mixed cellulasehydrolysis compared to hydrolysis with EG2 alone. Thisdecrease in substrate accessibility severely hinders the

Figure 2. Model results using monodisperse spheres with the low initial surfacearea value (8 m2 g1) compared to experimental data for (a) single enzyme hydrolysis

with CBH1 ( , experiment; - - -, model) and EG2 ( , experiment; , model) and

(b) theoretical mixed enzyme hydrolysis based on single enzyme experiments ( ,experiment; , model) and the actual mixture (*, experiment; - - -, model). The

experimental data are from Medve et al. (1998). The insets show an expanded view of

the early time points (up to 3 h).

Figure 3. Fraction of sites on the cellulose surface available for catalysis byeach enzyme type during the model simulation using monodisperse spheres with the

low initial surface area value (8m2 g1). Only the first half hour is shown because the

values have reached steady state at that point.



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effectiveness of EG2 in hydrolyzing cellulose within amixture. The CBH1 enzyme, on the other hand, suffersalmost no change in the fraction of available sites (chainends) on the surface. The difference between how a crowdedsurface affects EG2 and CBHI is expected. EG2 has far moresites to react with than CBHI because there are many moreglycosidic bonds than chain ends. An enzyme on the surfacetherefore obstructs more EG2 sites than CBHI sites. Inaddition, EG2 activity converts glycosidic bonds into chain

ends, thus adding to the number of CBHI sites. A low surfacearea under mixed enzyme hydrolysis conditions causes thesynergistic relationship between EG2 and CBH1 to breakdown; EG2 is unable to effectively cleave cellulose chains to

increase the number of chain ends available for CBH1 tohydrolyze to cellobiose.

High Surface Area (Monodisperse Spheres)

The effect of surface area on hydrolysis was furtherexamined by employing an initial Avicel surface area of47.6 m2 g1 (all other conditions unchanged). This value

was chosen to minimize any effect a lack of available surfacearea would have on the cellulose and still be within the rangeof surface areas reported for other types of cellulose (e.g.,PASC and bacterial cellulose) (Hong et al., 2007). Forsingle enzyme hydrolysis (Fig. 4a), the model results andexperimental data show good quantitative agreement atearly times (up to 3 h) and qualitative agreement at extendedtimes. The mixed enzyme model results (Fig. 4b) are nowalso in good quantitative agreement with the experimentaldata at early times (up to 3 h) and qualitative agreement atextended time. It is clear that the increased cellulose surfacearea is required to predict the observed synergisticinteraction between the cellulase enzymes. However, the

model fails to capture the full extent of the observed decreasein the rate of hydrolysis.

The role of increased surface area in allowing thesynergistic interactions to occur can be explored byexamining the change in the fraction of available sites forcomplexation for each enzyme in the model (Fig. 5). WhileEG2 still demonstrates a decreased fraction of accessible

Figure 4. Model results using monodisperse spheres with the high initial sur-face area value (47.6 m2 g1) compared to experimental data for (a) single enzyme

hydrolysis with CBH1 (&, experiment; -- -, model) and EG2 ( , experiment; , model)

and (b) theoretical mixed enzyme hydrolysis based on single enzyme experiments ( ,

experiment; , model) and the actual mixture ( , experiment; - - -, model). The

experimental data are from Medve et al. (1998). The insets show an expanded view of

the early time points (up to 3 h).

Figure 5. Fraction of sites on the cellulose surface available for catalysis byeach enzyme type during the model simulation using monodisperse spheres with the

high initial surface area value (47.6 m2 g1). Only the first half hour is shown because

the values have reached steady state at that point.




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surface sites in the mixed enzyme hydrolysis case comparedto the single enzyme hydrolysis case, this decline in availablesurface sites is much lower than that seen in the low surfacearea case. CBH1 has a large increase in the fraction ofavailable sites (chain ends) on the substrate surface duringmixed enzyme hydrolysis compared to the single enzymehydrolysis case. This result is expected because EG2 cleavescellulose chains on the surface into shorter chains, therebyincreasing the number of chain ends available. The results of

the high and low surface area scenarios illustrate that inorder for synergy to occur, the endoglucanase must beable to act almost as efficiently in the mixture as it doesalone.

Inclusion of Thermal Deactivation(Monodisperse Spheres)

In the model cases above only changes in the availablesurface area and product inhibition are capable of causing adecrease in the rate of hydrolysis. Clearly, even with strongproduct inhibition from cellobiose, the model is unable tocapture the full extent of the observed decrease in the rate ofhydrolysis. The extent to which thermal deactivation couldcontribute to the slower rate of hydrolysis was includedthrough addition of first-order thermal decay of enzyme.The entire enzyme population was assumed to be identicallysusceptible to the thermal deactivation, and for simplicitythermal deactivation was assumed to occur only in solution.

When the enzyme half-lives were reduced significantly,the model results achieved excellent agreement with theexperimental data at all hydrolysis times (Fig. 6). The half-life of EG2 was set to 4.3 h and the half-life of CBH1 was set

to 10.6 h. These are much shorter thermal half-life valuesthan the value of the half-life for a commercial T. reesei

cellulase mixture reported in the literature, 42.5 h (Drissenet al., 2007).

Effect of a Bimodal Particle Size Distribution

All of the simulations discussed above used a monodispersespherical particle distribution. As a demonstration of the useof polydisperse spherical particle distributions to capture the

hydrolysis behavior of non-spherical cellulose particles, arange of bimodal distributions was employed under thesame conditions as the high surface area case withoutthermal enzyme deactivation. The time course of cellulosesurface area in the reactor with different bimodal distribu-tions is shown in Figure 7. In the cases shown in Figure 7, theproportion and size of small spheres in the distributionhas been varied; the fraction of initial surface areaarising from small spheres was held constant at 83%. Thedecrease in cellulose area can also be altered by changing thefraction of the initial area resulting from each of he two

Figure 6. Comparison of model results with experimental data using monodis-perse spheres with the high initial surface area (47.6 m2 g1) case with the inclusion of

first-order thermal enzyme deactivation in the model. Results are shown for hydrolysis

with EG2 (^, experiment; , model), CBH1 ( , experiment; - - -, model), theoretical

mixture from single enzyme hydrolysis results ( , experiment; , model), and the

actual EG2 and CBH1 mixture ( , experiment; , model). The experimental data are

from Medve et al. (1998). The insets show an expanded view of the early time points

(up to 3h).

Figure 7. Model results for the accessible surface area per reactor volumeunder various sphere size distributions. The monodisperse cases are for high initial

surface area (47.6 m2 g1) ( ) and low initial surface area (8 m2 g1) ( ). The bimodal

cases all have a total initial surface area that matches the high initial surface area

value. The distribution is initially set to split the surface area with 83% on the small

spheres and 17% large sphere. The cases presented vary weight % of the smallspheres in the distribution (and the size to maintain the initial surface area ratios): 5%

1.5mm spheres:95% 142.5mm spheres ( ), 7.5% 2.25mm spheres:92.5% 139mm

spheres (-- -), 10% 3mm spheres:90% 135mm spheres (), 15% 4.5mm spheres:85%

127.5mm spheres ( ).



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spherical particle in the bimodal distribution (data notshown).

The model results for the bimodal particle distribution2.25mM particles at a concentration of 0.75 g L1 (83% ofthe initial surface area) and 139mM particles at aconcentration of 9.25 g L1 (17% of the initial surface area)are shown in Figure 8. This bimodal distribution was able tofit the experimental data for a mixture of EG2 and CBHI atall times but failed to match the experimental data for the

single enzyme hydrolysis cases at longer times (Fig. 8). Thedecline in the hydrolysis rate is captured in the enzymemixture but not in the single enzyme hydrolysis, resulting inthe inability of the model to capture the synergy exhibited inthe experimental data. By examining the predictedfraction of available sites (Fig. 9), it can be seen that thesites available to EG2 initially exhibit a slight decrease,similar to that seen in the high surface area case (Fig. 5).However, the fraction of available sites for EG2 forcomplexation rapidly decreases as the particle size distribu-tion transitions between being similar to the high surfacearea case to that of the low surface area case. The predictedfraction of available sites for CBHI exhibits a similar trend,where the fraction matches the behavior seen in the highsurface area case (Fig. 5) but rapidly looses available sites assurface area declines.

Discussion

The Role of Surface Area in Hydrolysis

The total cellulase-accessible cellulose surface area is agoverning parameter for enzymatic hydrolysis. The modelsability to capture EG2CBHI synergy depends strongly on

surface area. The activities of enzymes acting alone orpresent in a mixture depend on the amount of surface that isaccessible for cellulase complexation. The enzyme crowdingeffect is exacerbated as the amount of enzyme on the surfaceincreases.

The extent to which enzyme crowding occurs may beinfluenced by the existence of enzyme-specific interactionsbetween different cellulases and heterogeneities within thecellulose surface. When exposed to a low surface areacellulosic substrate, the adsorption of CBHI and EG2 resultsin high enzyme surface concentrations of both enzymeslimiting the activity of EG2. Because the two enzymes are

allowed to adsorb anywhere on the homogeneous surface,CBHI and EG2 compete for surface area. Lignocellulosesubstrates, however, are heterogeneous, containing crystal-line and amorphous subsites. Furthermore, there is evidencethat different cellulose binding domains exhibit specificityfor particular morphological regions (Boraston et al., 2003;Carrard et al., 2000; Creagh et al., 1996; Voutilainene et al.,2008). By altering the manner in which cellulases competefor sites on the surface, the distribution of subsites mayimpact the competitive behavior that leads to crowdingeffects.

Figure 9. Fraction of sites on the cellulose surface available for catalysis byeach enzyme type during the model simulation using the bimodal sphere distribution

with 7.5wt%2.25mm spheres and 92.5 wt%139mm spheres withan initial total surface

area of 47.6 m2 g1.

Figure 8. Comparison of model results with experimental data using a bimodalsphere distribution with 7.5 wt% 2.25mm spheres and 92.5wt% 139mm spheres with

an initial total surface area of 47.6 m2 g1. Results are shown for hydrolysis with EG2

(^, experiment; , model), CBH1 ( , experiment; - - -, model), theoretical mixture

from single enzyme hydrolysis results ( , experiment; , model), and the actual EG2and CBH1 mixture (*, experiment; , model). The experimental data are from Medve

et al. (1998). The insets show an expanded view of the early time points (up to 3 h).




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The Decrease in the Rate of Hydrolysis

With significant enhancement in cellobiose inhibition andthermal inactivation parameters, the model was able tocapture the slowdown in the hydrolysis rate exhibited by theexperimental data (Fig. 6). Strong cellobiose inhibition wasrequired: Ki 0.01mM for EG2 and Ki 0.093 mM forCBHI. While cellobiose is known to be a strong inhibitor ofcellulases, these values are low relative to reported valuesreported. Product inhibition alone did not enable the modelto quantitatively capture the experimental data (Figs. 4and 5). The sensitivity of the model to these parametersunderscores the need to obtain reliable experimental valuesfor the inhibition constants.

The inclusion of short thermal inactivation half-livesallowed the model to achieve quantitative agreement withthe experimental data. These half-lives, however, were up toan order of magnitude smaller than those reported for aT. reesei cellulase mixture. The interaction of the cellulases atthe solidliquid interface may enhance their rate of thermaldenaturation; alternatively, another rate-limiting mechan-ism may be at play. Regardless, the discrepancy between

experimentally determined inactivation parameters andthose required by the model indicates that the molecularpicture is not complete. As with inhibition parameters,accurate half-life measurements for individual cellulasesneed to be obtained. Additional mechanisms that limitenzyme activity also need to be explored.

The model also indicates that enzyme crowding is not anexclusive explanation for the decrease in the hydrolysis rate.The kinetic slowdown has been observed under a wide rangeof hydrolysis conditions; however, enzyme crowding onlyarises with high enzyme loadings or low surface areas. Theexperimental data referenced in this work show the decline

in hydrolysis rate both with single enzymes acting alone andwith enzymes acting within a mixture. Model results showcrowding only with mixed enzymes and low substratesurface area. Again the need to explore additional rate-limiting mechanisms is illustrated by the incidence of ahydrolysis rate decrease.

Parameter Sensitivity

A sensitivity analysis was used to examine the importance ofspecific mechanistic steps in the performance of the model.The adsorption, catalytic, complexation, and cellobiose

inhibition constants were increased and decreased by 20%under the high surface area, mixed enzyme, with thermaldeactivation scenarios. These results were used to determinethe fractional change in the amount of cellulose solubilizedwith respect to variations in each parameter. This is astandard method for sensitivity analysis of a kineticmodel (Turanyi, 1990). The sensitivity analysis indicatedthat the model was highly responsive to changes infour parameters: KM-CBH1-cel, KM-EG2-cel, Ki-EG2-cellobiose,and Ki-CBH1-cellobiose, which are listed from most to leastsensitive. These parameters relate to the complexation of the

enzymes either with the solid cellulose surface for catalysis orwith cellobiose to inhibit the enzymes. The model wasrelatively insensitive to changes in the adsorption, deso-rption, and catalytic parameters for both EG2 and CBHI.The sensitivity results suggest that complexation of adsorbedenzyme with the substrate is the governing kinetic step incellulose hydrolysis. The sensitivity of the model to theinhibition parameters reinforces the need for reliableexperimentally determined values.

Capturing Particle Shape Effects

The bimodal distribution serves as a demonstration thatmore complicated rates of change in the total surface area ofsubstrate can be represented using a distribution of sphereschosen to match the initial total area and volume of theparticles. More complex particle size distributions would berequired to match the rate of change in surface area for aspecific particle shape. To determine these particle sizedistributions, detailed information about how a particleshape decreases during hydrolysis would be necessary. Box 1

details an approach for developing sphere size distributionsfor different particle shapes.

The model results for the bimodal case (Figs. 8 and 9)demonstrate the large effect more complex rates of surfacearea change can have on the predicted hydrolysis behavior.The inability of the model to capture synergy in the bimodalcase reinforces the point that by itself, enzyme crowdingrelated to a lack of accessible surface area cannot explain thedecline in the rate of hydrolysis. If the decline in the rate ofhydrolysis was caused simply by surface area and crowdingthere would be evidence available that lowering the enzymeloading can reduce, delay, or alleviate the effect; this has not

been reported in the literature. In fact, the single enzymehydrolysis results of Medve et al. (1998) show a decline inthe rate of hydrolysis at a lower conversion than the mixedenzyme hydrolysis results, even though the single enzymeloading was only half the mixed enzyme loading.

Broader Implications of the Model Results

The model used kcat values for soluble oligosaccharide(cellohexaose or cellopentaose) in representing catalysissteps performed by enzymes bound to soluble chains as wellas to the solid substrate. The use of soluble-substrate kcat

values relies on the assumption that once the substrate isbound the rate of chemical catalysis does not change.Recently published experimental evidence supports thisassumption: high-speed AFM results (Igarashi et al., 2009)showed the CBHI enzyme moving along a cellulose surfaceat 3.5nms1, which corresponds to a kcat of 7 s

1, similar tothe soluble cello-oligosaccharide values.

If parameters describing the catalytic activity of enzymeson solid and soluble cellulose chains are identical,improvements in intrinsic cellulase kinetics (kcat) on solublesubstrates should improve the enzymes activity on solid



11/15

substrates. However, the model shows that the conversion ofsolid substrates to soluble sugars is not sensitive to the kcatvalues of the enzymes. As discussed above, the complexationof adsorbed enzyme with solid substrate is the mostsignificant parameter that influences the model predictions.Therefore, improvements in kcat may not translate intoimprovements in solid-substrate cellulolytic activity.

The importance of accessible surface area is alsohighlighted by the model results. The discrepancy in activity

and synergy behaviors between the low and high surface areacases indicates the broadly beneficial role of maximizing theenzyme accessible surface area before hydrolysis starts. Apretreatment process that enhances substrate surface areacan reduce the likelihood that enzyme crowding will becomerate limiting.

Conclusions

A detailed mechanistic model for cellulase-catalyzedhydrolysis of cellulose has been developed. The modelexplicitly tracks individual cellulases and key cellulosesurface properties. Independent enzyme adsorption andcomplexation steps have been incorporated in an attempt tocapture the most important details of the enzymesubstrateinteraction.

Individual enzyme hydrolysis (EG2 or CBHI) and mixedenzyme hydrolysis scenarios were used to compare modelresults with experimental data from the literature. Themodel results were not consistent with all of the experi-mental data in the case of relatively low surface area. Whenthe surface area was increased, limiting the effect of enzymecrowding during mixed enzyme hydrolysis, the modelachieved good agreement with the experimental data,including EG2CBHI synergy.

The model was not capable of capturing the full extent ofthe decrease in the rate of cellulose hydrolysis often reportedin the literature. Strong product inhibition and shortenzyme half-lives were required to match the slowdownapparent in experimental data. Neither of these effects is theprimary cause of the observed slowdown. The ability tocapture the rate of change in available surface area for shapesmore complex than spheres, using a distribution of sphericalparticle sizes was demonstrated using a bimodal distribu-tion. The available substrate surface area and how this areachanges during hydrolysis was shown to be important, butthese changes are unlikely to explain the decline in the rate of

hydrolysis alone. The differential manner in which cellulasesinteract with structural heterogeneities within the substratemay play an important role in causing the rapid reduction inhydrolysis rate. These physical subtleties are not currentlyincluded in the model. The changes to the cellulosemorphology that occur during hydrolysis may alter thecharacter of the subsites on the surface leading to areduction in the observed hydrolysis rate. The model resultspresented in this article illustrate the importance ofunderstanding the effect of relevant surface areas to enzymehydrolysis activity. This work also highlights the utility of

future investigations attempting to elucidate further detailsof cellulasecellulose interaction.

Nomenclature

Acel total surface area of cellulose in the system (dm2)

C0i

solid cellulose chain of length i (i> 6)

Ci soluble cello-oligosaccharide chain of length i (i< 7)

D P degree of polymerization

DP0

initial degree of polymerization

DPmax maximum degree of polymerization tracked in the model

E0CBHads uncomplexed cellobiohydrolase adsorbed to the surface

E0CBHadsCi cellobiohydrolase adsorbed to the surface and complexed

with an i length cellulose chain

ECBH-free uncomplexed cellobiohydrolase in solution

E0EGads uncomplexed endoglucanase adsorbed to the surface

E0

CBHadsCi endoglucanase adsorbed to the surface and complexed with

an i length cellulose chain

ECBH-free uncomplexed endoglucanase in solution

E0

xads uncomplexed cellulase of type x adsorbed to the surface

Ex-free uncomplexed cellulase of type x in solution

kcat-CBH-cel catalytic constant for cellobiohydrolase acting on a solid

cellulose chain (s

1

)kcat-CBH-sol catalytic constant for cellobiohydrolase acting on a soluble

cello-oligosaccharide (s1)

kcat-EG-cel catalytic constant for endoglucanase acting on a solid

cellulose chain (s1)

kcat-EG-sol catalytic constant for endoglucanase acting on a soluble

cello-oligosaccharide (s1)

KM-CBH-cel complexation equilibrium constant for cellobiohydrolase

with a solid cellulose chain (mmoldm2)

KM-CBH-sol complexation equilibrium constant for cellobiohydrolase

with a soluble cello-oligosaccharide (mmol L1)

KM-EG-cel complexation equilibrium constant for endoglucanase with a

solid cellulose chain (mmol dm2)

KM-EG-sol complexation equilibrium constant for endoglucanase with a

soluble cello-oligosaccharide (mmol L1)

kx-ads adsorption constant for a cellulase of type x onto a cellulose

surface (L mmol1 s1)

kx-des desorption constant for a cellulase of type x from a cellulose

surface (s1)

MW1 cellulose monomer molecular weight (g mmol1)

n concentration of cellulose particles (mmol L1)

R cellulose particle radius (dm)

r0i

rate of formation of a length i soluble sugar from solid

substrate (mmol dm2 s1)

SA cellulose surface area (dm2 g1)

t time (s)

Vliq volume of reactor (L)

uCBH fraction of free sites on the surface available to acellobiohydrolase for complexation

uEG fraction of free sites on the surface available to an

endoglucanase for complexation

rcellulose cellulose density during hydrolysis (g L1)

free sites

[ ] concentration (mmol L1) if unprimed symbol or surface

concentration (mmoldm2) if primed symbol

The authors wish to thank Clayton Radke and Sam Mauer for helpful

discussions.




12/15

Figure A1. Hexagonal columns, cylinders, and shape combinations represented as spheres: a monodisperse distribution of spheres has the same totalvolume and total surface area as the original shape, a polydisperse distribution of spheres has the same total volume, total surface area, and overall hydrolysis rate as the

original shape.

Appendix A: Model EquationsIn the following equations a prime (0) indicates that the quantity is using a per area basis as opposed to a per volume basis.

Cellulose Particle Size Determination from a Material Balance:

dR

dt R

SA

3MW1

X6i1

ir0i (1)

SA 3

Rrcellulose(2)

Cellulose Area per Volume Equation:

Acel

Vliq 4pR2n (A1)



13/15

Soluble Saccharide Mass Balances:

dC1

dt 2

kcatEGcel

KMEGcelE0EGads

uEG

X1i7

C0i 2kcatEGsol

KMEGsolE0EGads

X6i3

CikcatCBHsol

KMCBHsolE0CBHadsC3

Acel

Vliq

2kcatEGsol

KMEGsolEEGfree

X6i3

Ci kcatCBHsol

KMCBHsolECBHfreeC3 (A2)

dC2

dt 2

kcatEGcel

KMEGcelE0EGads uEGX

1

i7

C0i kcatCBHcel

KMCBHcelE0CBHadsuCBHX

1

i7

C0i

2kcatEGsol

KMEGsolE0EGads

X6i3

CikcatCBHsol

KMCBHsolE0CBHads

X6i3

Ci C4

!!Acel

Vliq

2kcatEGsol

KMEGsolEEGfree

X6i3

Ci kcatCBHsol

KMCBHsolECBHfree

X6i3

Ci C4

!(7)

i 3; 4

dCi

dt 2

kcatEGcel

KMEGcelE0EGads

uEGX

1

j7

C0j 2kcatEGsol

KMEGsolE0EGads X

6

j3

Cj i 1Ci

!

kcatCBHsol

KMCBHsolE0CBHadsCi2 Ci

Acel

Vliq 2

kcatEGsol

KMEGsolEEGfree

X6j3

Cj i 1Ci

!

kcatCBHsol

KMCBHsolECBHfreeCi1 Ci A3

i 5; 6dCi

dt 2

kcatEGcel

KMEGcelE0EGads

uEG

X1j7

C0j kcatCBHcel


0

i2

2kcatEGsol

KMEGsolE0EGads

X6j3

Cj i 1Ci

!

kcatCBHsol

KMCBHsolE0CBHadsCi

Acel

Vliq

2

kcatEGsol

KMEGsol EEGfreeX6j3

Cj i 1Ci !

kcatCBHsol

KMCBHsol ECBHfreeCi

(A4)

Solid Cellulose Mass Balances:

i > 6dC0i

dt

kcatEGcel

KMEGcelE0EGadsuEG 2

X1ji1

C0j i 1C0

i

!

kcatCBHcel


0

i2 C0

i

(6)

Cellulose Site Balance Equations:

0

0max

u

1 sEGE

0

EGads

0max1 X1i7

uEGi 1C0

i

KMEGcelX6i3

i 1Ci

KMEGsol

C1

KiEGglucose

C2

KiEGcellobiose

!

sCBHE

0

CBHads

0max1 X1i7

uCBHC0

i

KMCBHcelX6i3

Ci

KMCBHsol

C1

KiCBHglucose

C2

KiCBHcellobiose

!(A5)




14/15

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(A9)



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