1- maturitymaintenance

maturityoffspring

maturationreproduction

Basic DEB scheme

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

Feeding 3.1

Feeding has two aspects• disappearance of food (for food dynamics): JX,F

• appearance of substrate for metabolic processing: JX,A= JX,F

Faeces • cannot come out of an animal, because it was never in it• is treated as a product that is linked to assimilation: JP,F= yPX JX,F

Feeding 3.1

time

time

bind

ing

prob

.bi

ndin

g pr

ob.

fast SU

slow SU

arrival events of food items

raten associatio:rate;on dissociati:density; food:

:response functionalwith ;/

:ratefeeding

:fraction mequilibriu;)1(:SUoffractionunbounded

,*

,

*

bkXXK

XfJf

Xbk

kXbXθJ

bXk

kθbXθθkθ

dt

d

FmXFX

0

0

Busy periods not only include handling but also digestion and other metabolic processing

Assimilation 3.3

Definition:Conversion of substrate(s) (food, nutrients, light) into reserve(s)Energy to fuel conversion is extracted from substratesImplies: products associated with assimilation (e.g. faeces, CO2)

Depends on:• substrate availability• structural (fixed part of) surface area (e.g. surface area of gut)

Consequence of strong homeostasis:Fixed conversion efficiency for fixed composition of substrate

However, biomass composition is not fixed many species feed on biomass

EXAXEXAE yJyJ fixedfor,,

Assimilation 3.3

KX

Xf

JyJJfJ

VJJ

AmXEXAmEAmEAE

AEAE

responsefunctionalscaledand

}{}{fixedfor}{}{with

}{onassimilatitolinkedfluxreserve

,,,,

3/2,,

EXyEVKX food density

saturation constantstructural volumereserveyield of E on X

Reserve dynamics 3.4

Increase: assimilation surface areaDecrease: catabolism reserve density (= reserve/structure)

First order process on the basis of densities follows from• weak homeostasis of biomass = structure + reserve• partitionability of reserve dynamics (essential for symbioses)

Mechanism: structural & local homeostasis

-rule for allocation to growth + somatic maintenance: constant fraction of catabolic rate

AA

AA

pκpκ )1(

Reserve partitioning 3.4

EκA )1(reserve

EκA

reserve stru

ctur

e, V

GA pκ )1(

GA pκ

MA pκ )1(

MA pκIf reserves are partitioned e.g. into lipids and non-lipids maintenance and growth are partitioned as wellPartitioning requirement for catabolic power ( use of reserves, [pM] = pM/V and [EG] constant)

for some function [pC]= pC/V of state variables [E],V

)],[],[|],[]([)],[],[|],]([[

θEκpκVEκpθEpVEpκ

GAMAAC

GMCA

vectorparameter factorarbitrary

costgrowth spec][powergrowth power maint.power assim. volumestructuraldensity reserve][

θκEpppVE

A

G

G

M

A

]1,0[somefor Aκ

Cp

Reserve dynamics 3.4

• Relationship assimilation, growth and maintenance

• Weak homeostasis

• Partionability

• Conclusions Function H is first degree homogeneous:

Function is zero-th degree homogeneous in [E]: : So may depend on V, but not on [E]• Result

]/[]][[])/[][1]([][][

][)],([

/][with,ln][][][][

GMGCA

GMC

AACA

EpEEEκppEdt

d

Vdt

dEppVEκ

VppVdt

dEppE

dt

d

3/1][with0][at0][ VpEdt

dE

dV

dA

]/[])[|],([1

]/[)](][[)|]([][

3/1

G

GMC EEθVEκ

EVpEθEHVp

)],[],[|],[]([)],[],[|],]([[ θEκpκVEκpθEpVEpκ GAMAACGMCA

][

}{,

][

][for)(or

][

][}{][

3/13/1m

Am

mm

Am

E

pv

E

Eeef

V

ve

dt

d

E

Ef

V

pE

dt

d

vEvEHθEκHθEHκ EE constantfor][])([so),|][()|]([

)],([)],[( VEκVEκκ E

θfvκpppp

VEEE

C

M

A

A

G

m

}{

][][

][ reserve densitymax reserve densityspec growth coststructural volumespec assim powerassim powermaint. powercatabolic powerfraction catabol.energy conductancescaled funct. resp.parameter vector

Reserve dynamics 3.4

XK

XffVppppE

dt

dAmACA

;}{; 3/2

V

EE

E

pv

E

Eeef

V

ve

dt

d

E

Ef

V

pE

dt

d

m

Am

mm

Am

][;

][

}{,

][

][for)(or

][

][}{][

3/13/1

Isomorphs

V1-morphs

)]([ 3/2 Vdt

dvVEpC

3/1)/()(with{multiply dAm VVVp M}

][

][for)(;][

m

AmEEAmA E

pkefke

dt

dfVpp

C

A

ppVEX

][][},{

m

AmAm

Epp

Kf

Ekvfood density

reserve energystructural volumeassimilation powercatabolic power

scaled functional responsesaturation constantmax spec assimilation powermax reserve capacity

energy conductancereserve turnover rate

Reserve dynamics

Reserve dynamics

• reserve & structure: spatially segregated

• reserve mobilized at rate surface area of reserve-structure interface

• rejected reserve flux returns to reserve

• SU-reserve complex dissociates to demand-driven maintenance supply-driven growth (synthesis of structure)

• abundance of SUs such that local homeostasis is achieved

Reserve dynamics

SU abundance, relative to DEB value

sd s

peci

fic

use

of r

eser

ve

for assimilation being an alternating Poisson process

10 h-1

50 h-1

2 h-1

assim = 0 assim = 1

0

1

time

assimilation

10 h-1

10 h-1

10 h-1

hazard rates

Reserve dynamics

time, h

PH

B d

ensi

ty, m

ol/m

ol

in starving active sludge

Data fromBeun, 2001

Yield of biomass on substrate

1/spec growth rate, h-1

cusStreptococ mg

glucose mg

Data fromRussel & Cook, 1995

maintenance

reserve

-rule for allocation 3.5

Age, d Age, d

Length, mm Length, mm

Cum

# of young

Length,

mm

Ingestion rate, 105

cells/h

O2 consum

ption,

g/h

• 80% of adult budget to reproduction in daphnids• puberty at 2.5 mm• No change in ingest., resp., or growth • Where do resources for reprod. come from? Or:• What is fate of resources in juveniles?

Respiration Ingestion

Reproduction

Growth:

32 LkvL M2fL

332 )/1( pMM LkfgLkvL

)( LLrLdt

dB

Von Bertalanffy

Somatic maintenance 3.6

Definition of maintenance (somatic and maturity):Collection of processes not associated with net productionOverall effect: reserve excreted products (e.g. CO2, NH3)

Somatic maintenance comprises:• protein turnover (synthesis, but no net synthesis)• maintaining conc gradients across membranes (proton leak)• maintaining defence systems (immune system)• (some) product formation (leaves, hairs, skin flakes, moults)• movement (usually less than 10% of maintenance costs)

Somatic maintenance costs paid from flux JE,C: • structural volume (mosts costs), pM

• surface area (specific costs: heating, osmo-regulation), pT

Maturity maintenance 3.6

Definition of maturity maintenance:Collection of processes required to maintain current state of maturity

Main reason for consideration:making total investment into maturation independent of food intake

Maturity maintenance costs paid from flux (1-)JE,C: • structural volume in embryos and juveniles, pJ

• constant in adults (even if they grow)

Else: size at transition depends on history of food intake

p

MEpJE

Vκ

κjVVj

sizefixedatoccurstransitionstage

1)/,1min(If ,,

0

num

ber

of d

aphn

ids

Maintenance first 3.6

106 cells.day-1

300

200

100

01206030126

max

num

ber

of d

aphn

ids

30 35

400

300

200

100

8 11 15 18 21 24 28 32 37time, d

30106 cells.day-1

Chlorella-fed batch cultures of Daphnia magna, 20°Cneonates at 0 d: 10winter eggs at 37 d: 0, 0, 1, 3, 1, 38

Kooijman, 1985 Toxicity at population level. In: Cairns, J. (ed) Multispecies toxicity testing. Pergamon Press, New York, pp 143 - 164

Maitenance requirements:6 cells.sec-1.daphnid-1

Growth 3.7

Definition:Conversion of reserve(s) into structure(s)Energy to fuel conversion is extracted from reserve(s)Implies: products associated with growth (e.g. CO2, NH3)

Allocation to growth:

Consequence of strong homeostasis:Fixed conversion efficiency

][fixedfor][

fixedfor,,,

VVV

EVGEEVVVGVVGV

MVMM

yJyMrMjMdt

dJ

constantandwith ,,,,,, MEVMEMEMECEGE jMjJJJκJ

Mixtures of V0 & V1 morphs 3.7.2

volu

me,

m

3vo

lum

e,

m3

volu

me,

m

3

hyph

al le

ngth

, mm

time, h time, min

time, mintime, min

Fusarium = 0Trinci 1990

Bacillus = 0.2Collins & Richmond 1962

Escherichia = 0.28Kubitschek 1990

Streptococcus = 0.6Mitchison 1961

structural volume

reserve density max res densityspec assim powerspec heating costsspec som maint costsspec growth costsfraction catabolic p

Growth 3.7

ge

VVVVVeVvVeV

dt

d mmh

3/13/13/23/2 /)/(

),(

κEppp

E

Ee

V

G

M

T

A

m

][][}{}{

][

][

][

}{][

][][

][][

}{][

}{

3/1

3/1

m

Am

m

G

G

MM

MM

Amm

M

Th

E

pv

Eκ

Eg

E

pk

gk

v

p

pκV

p

pV

heating length

max length

maint rate coeff

en investment ratio

energy conductance

Growth at constant food 3.7

time, d ultimate length, mm

leng

th, m

m

M

M

δfVfLLvLδkr

trLLLtL

mm

MB

Bb

//33

)exp()()(

3/1

11

LLLt

b

Mδkvr

M

BtimeLengthL. at birthultimate L.

von Bert growth rateenergy conductancemaint. rate coefficientshape coefficient

vδ /3 M

Von

Ber

t gro

wth

rat

e -1, d

13 Mk

Von Bertalanffy growth curve:

Embryonic development 3.7.1

time, d time, d

wei

ght,

g

O2 c

onsu

mpt

ion,

ml/

h

l

ege

dτ

d

ge

legl

dτ

d

3

3,

3, l

dτ

dJlJJ GOMOO

; : scaled timel : scaled lengthe: scaled reserve densityg: energy investment ratio

Crocodylus johnstoni,Data from Whitehead 1987

yolk

embryo

Foetal development 3.7.1

wei

ght,

g

time, d

Mus musculus

Foetes develop like eggs, but rate not restricted by reserve (because supply during development)Reserve of embryo “added” at birth Initiation of development can be delayed by implantation egg cellNutritional condition of mother only affects foetus in extreme situations

33/20 )3/()(;0)0(;:For vttVVvVV

dt

dE

Data: MacDowell et al 1927

Maturation 3.8

Definition:Use of reserve(s) to increase the state of maturityThis, however, does not increase structural massImplies: products associated with maturation (e.g. CO2, NH3)

Allocation to maturation in embryos and juveniles:

This flux is allocated to reproduction in adults

Dissipating power: with R = 0 in embryos and juvenilesNotice that power also dissipates in association with

constantandwith)1( ,,,,,, JEVJEJEJECERE jMjJJJκJ

RRJTMD pκpppp )1(

MD

GA

pppp

e,maintenanc n,dissipatio :morphs-V1For ,growth,,onassimilati

Reproduction 3.9.1

Definition:Conversion of adult reserve(s) into embryonic reserve(s)Energy to fuel conversion is extracted from reserve(s)Implies: products associated with reproduction (e.g. CO2, NH3)

Allocation to reproduction in adults:

Allocation per time increment is infinitesimally smallWe therefore need a buffer with buffer-handling rules for egg prod (no buffer required in case of placental mode)

Strong homeostasis: Fixed conversion efficiencyWeak homeostasis: Reserve density at birth equals that of motherReproduction rate: follows from maintenance + growth costs, given amounts of structure and reserve at birth

constantwith)1( ,,,, JEJECERE JJJκJ

eggpercostswith/ 00, EEJκR RER0E

Reproduction at constant food 3.9.1

length, mm length, mm

103 e

ggs

103 e

ggs

Gobius paganellusData Miller, 1961

Rana esculentaData Günther, 1990

)(foodconstantat

)()1(),(

332

3/2

0

feLδv

k

f

fgLδ

v

kL

VgkVkvVeg

geκ

Ve

κVeR

pMM

pMMm

R

ΜΜ

General assumptions 3.10

• State variables: structural body mass & reserves they do not change in composition• Food is converted into faeces Assimilates derived from food are added to reserves, which fuel all other metabolic processes Three categories of processes: Assimilation: synthesis of (embryonic) reserves Dissipation: no synthesis of biomass Growth: synthesis of structural body mass Product formation: included in these processes (overheads)• Basic life stage patterns dividers (correspond with juvenile stage) reproducers embryo (no feeding initial structural body mass is negligibly small initial amount of reserves is substantial) juvenile (feeding, but no reproduction) adult (feeding & male/female reproduction)

Specific assumptions 3.10

• Reserve density hatchling = mother at egg formation foetuses: embryos unrestricted by energy reserves• Stage transitions: cumulated investment in maturation > threshold embryo juvenile initiates feeding juvenile adult initiates reproduction & ceases maturation• Somatic & maturity maintenance structure volume (but some maintenance costs surface area) maturity maintenance does not increase after a given cumulated investment in maturation• Feeding rate surface area; fixed food handling time• Partitioning of reserves should not affect dynamics comp. body mass does not change at steady state• Fixed fraction of catabolic energy is spent on somatic maintenance + growth (-rule)• Starving individuals: priority to somatic maintenance do not change reserve dynamics; continue maturation, reprod. or change reserve dynamics; cease maturation, reprod.; do or do not shrink in structure