Mass balance 4.3

NPNENVNX

OPOEOVOX

HPHEHVHX

CPCECVCX

NN

ON

HN

CN

P

EE

V

X

N

O

H

C

nnnnnnnnnnnnnnnn

nnnn

JJJ

JJ

JJJJ

R

OMOM

OOMMOOMM

nnJJ

JnnJJnJn0

;

000212020001

;;

1

NOHCM

PEVXOminerals

carbon dioxidewaterdioxygennitrogen-waste

organicsfoodstructurereserveproduct

1Cj

ij

i

n

nJ flux of compound i

chemical index for element i in compound j for all compounds j

DEB model specifies organic fluxesMineral fluxes follow from mass balanceExtendable to more elements/compounds

com

poun

ds

Mass-energy coupling 4.3

PGPDPA

EEE

VG

XA

G

D

A

P

EE

V

X

EVEVGVGVGEXEAXAXXA

ηηημμμ

ηη

ppp

JJJ

JJ

μyμμημyμμη

R

111

11

0000

;;

;;;;

OO

OO

ηpJ

pηJ

com

poun

ds

PEVXO organics

foodstructurereserveproduct

pow

ers

GDA assimilation

dissipationgrowth ij

ij

E

ηyμ chemical potential of E

yield of compound i on jcoupler of compound i to power j for faeces:

OOMMMOOMOOMM ηnnηpηpηnnJnnJ 111 for Decomposition of mineral fluxes into contributions from 3 basic energy fluxes:

0 PGPD ηη

Organic fluxesare linear combinations

of 3 energy fluxes

Energy balance 4.9.1

pηnnμμJμJμ OOMTM

TOO

TOM

TM )(0

:heat gdissipatinFor 1

TT

T

ppp

Dissipating heat can be decomposed into contributions from 3 basic energy fluxes

npJημ chemical potentials (energy-mass couplers)

mass-energy couplers fluxes of compounds3 basic energy fluxes (powers)chemical indices

OM minerals

organics

Method of indirect calorimetry 4.9.2

Empirical origin (multiple regression): Lavoisier 1780

Heat production = wC CO2-production + wO O2-consumption + wN N-waste production

DEB-explanation:Mass and heat fluxes = wA assimilation + wD dissipation + wG growthApplies to CO2, O2, N-waste, heat, food, faeces, …

For V1-morphs: dissipation maintenance

Mass fluxes 4.1

dioxidecarbon 2 CJ

water2 HJ

dioxygen2 OJ

ammonia10 NJ

foodXJ

structure40 VJ

reserve)(10

REE JJ faeces

PJ

llength scaled

f

lux

f

lux

bl pl

notice small dent due to transition

maturation reproductionAt abundant food: growth ceases at l = 1

allocation toreproduction

use of reservenot balanced by

feeding in embryo

bl pl

0 1

10

Methanotrophy 4.3.1

Yield coefficients Y and chemical indices n depend on (variable) specific growth rate rNWOWHW nnnWX3NX2OX2CX4 NOCH Y NH Y O Y CO Y CH

AC Assim (catabolic) -1 1 2 -2 0 0 0

AA Assim (anabolic) -1 0 1 0

M Maintenance 0 1 -1 0

GC Growth (catabolic) 0 1 -1 0

GA Growth (anabolic) 0 0 -1 1

C Carbon 1 1 0 0 0 1 1

H Hydrogen 4 0 2 0 3

O Oxygen 0 2 1 2 0

N Nitrogen 0 0 0 0 1

2/2/2/

2/32/2/

2/2/1

2/2/3

2/2/

2/2/32

From

GHEOVOE

GOE

GNEHVHE

GHE

NVNEG

NE

MHEOE

MOE

HENEM

HE

OEA

HXA

OX

HEA

NXA

HX

NEA

NX

YnnY

YnnY

nnY

YnY

nnY

nYY

nYY

nY

nY0

AHXY A

OXY ANXY

MHEY

GHEY

MHEY

MOEYM

OEYG

OEY GNEY

NEn

NEn

HEn

OEn

NEn

HVn

OVn

NVn

sym

bol

proc

ess

X: m

etha

ne

C: c

arbo

n di

oxid

e

H: w

ater

O: d

ioxy

gen

N: a

mm

onia

E: r

eser

ve

V: s

truc

ture

EAXE jy )1(

EAj

EGVE jy )1(

EGVE jy

EMj

EVE

EMEEVV

EVEG

MEVEM

EAmEA

ym

jkmM

dt

dMr

ryj

kyjXK

Xjj

1

For reserve density mE = ME/MV (ratio of amounts of reserve and structure), the macroscopic transformation can be decomposed into 5 microscopic ones with fixed coefficients

rate

Yie

ld c

oeff

icie

ntsT

Che

mic

al in

dice

s

Methanotrophy 4.3.1

spec growth rate, h-1 spec growth rate, h-1

X/O

N/O

C/O

flux

rat

io, m

ol.m

ol-1

spec

flu

x, m

ol.m

ol-1.h

-1

CE

N

X

O

X: methaneC: carbon dioxideO: dioxygenN: ammoniaE: reserve

jEAm = 1.2 mol.mol-1.h-1

yEX = 0.8yVE = 0.8kM = 0.01 h-1

kE = 2 h-1

nHE = 1.8nOE = 0.3nNE = 0.3

nHV = 1.8nOV = 0.3nNV = 0.3

chemical indices

Kooijman, Andersen &Kooi 2004. Ecology, to appear

Biomass composition 4.3.4

Data Esener et al 1982, 1983; Kleibsiella on glycerol at 35°CnHW

nOW

nNW

O2

CO2Spec growth rate, h-1

Spec growth rate

Spec growth rate, h-1

Rel

ativ

e ab

unda

nce

Spe

c pr

od, m

ol.m

ol-1.h

-1

Wei

ght y

ield

, mol

.mol

-1

nHE 1.66 nOE 0.422 nNE 0.312nHV 1.64 nOV 0.379 nNV 0.189

kE 2.11 h-1 kM 0.021 h-1

yEV 1.135 yXE 1.490rm 1.05 h-1 g = 1

•μE-1 pA pM pG

JC 0.14 1.00 -0.49

JH 1.15 0.36 -0.42

JO -0.35 -0.97 0.63

JN -0.31 0.31 0.02

Entropy J/C-mol.K Glycerol 69.7 Reserve 74.9 Structure 52.0

Sousa et al 2004Interface, subm

Product Formation 4.7

throughput rate, h-1

glyc

erol

, eth

anol

, g/l

pyru

vate

, mg/

l

glycerol

ethanol

pyru

vate

Glucose-limited growth of SaccharomycesData from Schatzmann, 1975

According to Dynamic Energy Budget theory:

Product formation rate = wA . Assimilation rate + wM . Maintenance rate + wG . Growth rate

For pyruvate: wG<0

1 Reserve – 1 Structure

2 Reserves – 1 Structure

Reserve Capacity & Growth

low turnover rate: large reserve capacity

high turnover rate: small reserve capacity

Multivariate extensions 5

animal heterotroph phototroph

symbiosis plant

Interactions of substrates 5.1

Photosynthesis 5.1.3

2 H2O + 4 h O2 + 4 H+ + 4 e-

CO2 + 4 H+ + 4 e- CH2O + H2O

CO2 + H2O + light CH2O + O2

3222

32

NHOOHCONOCH

NOOCH

ENOEHECEnnn

ENEC

HNEOEHE

OH

yyyy

yy

Simultaneous nutrient limitation 5.2.3

Specific growth rate of Pavlova lutheri as function of intracellular phosphorus and vitamine B12 at 20 ºC

Data from Droop 1974Note the absence of high contents for both compounds

due to damming up of reserves, andlow contents in structure (at zero growth)

Reserve interactions 5.2.4

Spec growth rate, d-1 Spec growth rate, d-1 Spec growth rate, d-1

P-c

onte

nt, f

mol

.cel

l-1P

-con

c, μ

M

B12

-con

c, p

M

B12

-con

t., 1

0-21 .m

ol.c

ell-1

P Vitamin B12

kE 1.19 1.22 d-1

yXV 0.39 10-15 2.35 mol.cell-1

jEAm 4.91 10-21 76.6 10-15 mol.cell-1. d-1

κE 0.69 0.96

kM 0.0079 0.135 d-1

K 0.017 0.12 pM, μM

Data from Droop 1974 on Pavlova lutheri

P(μM) B12(pM)

1.44 68

14.4 6.8

1.44 20.4

1.44 6.8

Steps in food 7.1.2

Growth of Daphnia magna at 2 constant food levels

time, d time, d time, d time, d

0 d 7 d 14 d 21 dle

ngth

, mm

leng

th, m

m

Only curves at 0 d are fittedNotice • slow response• gut content in down steps

Steps up

Steps down

Growth on reserve 7.1.3

Opt

ical

Den

sity

at 5

40 n

m

Con

c. p

otas

sium

, mM

Potassium limited growth of E. coli at 30 °CData Mulder 1988; DEB model fitted

OD increases by factor 4 during nutrient starvationinternal reserve fuels 9 hours of growth

time, h

Growth on reserve 7.1.3

Growth in starved Mytilus edulis at 21.8 °CData Strömgren & Cary 1984; DEB model fitted

internal reserve fuels 5 days of growth

time, d

grow

th r

ate,

mm

.d-1

Protein synthesis 7.5

spec growth rate, h-1 scaled spec growth rate

RN

A/d

ry w

eigh

t, μg

.μg-1

scal

ed e

long

atio

n ra

te

Data from Koch 1970Data from Bremer & Dennis 1987

RNA = wRV MV + wRE ME

dry weight = wdV MV + wdE ME

Scales of life 8.0

Life span

10log aVolume

10log m3earth

whale

bacterium

water molecule

life on earth

whale

bacteriumATP

Invariance property 8.1

The parameters of two individuals can differ in a very special way such that both individuals behave identically at constant food density if they start with the same values for the state variables (reserve, structure, damage)

At varying food density, two individuals only behave identically if all their parameters are equal

Inter-species body size scaling 8.2

• parameter values tend to co-vary across species• parameters are either intensive or extensive• ratios of extensive parameters are intensive• maximum body length is allocation fraction to growth + maint. (intensive) volume-specific maintenance power (intensive) surface area-specific assimilation power (extensive)• conclusion : (so are all extensive parameters)• write physiological property as function of parameters (including maximum body weight)• evaluate this property as function of max body weight

]/[}{ MAm ppL

}{ Ap

][ Mp

mA Lp }{

Kooijman 1986 Energy budgets can explain body size scaling relationsJ. Theor. Biol. 121: 269-282

Primary scaling relationships 8.1

K2=K1z+X(z-1) {JXm}2={JXm}1z [pM]2=[pM]1 {pT}2={pT}1

Lb2 = Lb1 {pAm}2={pAm}1z [EG]2=[EG]1 ha2 = ha1

Lp2 = Lp1 [Em]2=[Em]1z 2= 1 R2= R1

K2=K1z {JXm}2={JXm}1z [pM]2=[pM]1 {pT}2={pT}1

Lb2 = Lb1z {pAm}2={pAm}1z [EG]2=[EG]1 ha2 = ha1

Lp2 = Lp1z [Em]2=[Em]1z 2= 1 R2= R1

K saturation constant

{JXm} max spec feeding rate

[pM] spec maint. costs

{pT} spec heating costs

Lb length at birth {pAm} max spec assim rate

[EG] spec growth costs

ha aging acceleration

Lp length at puberty

[Em] max reserve capacity

partitionning fraction

R reprod. efficiency

z: arbitrary zoom factor for species 2 relative to species 1: z = Lm2/Lm1

invarianceproperty

(at food density X)

primaryscaling

parameters

Length at puberty 8.2.1

L, cm

Lp,

cm

Clupea• Brevoortia° Sprattus Sardinops Sardina

Sardinella+ Engraulis* Centengraulis Stolephorus

Data from Blaxter & Hunter 1982

Clupoid fishes

Length at first reproduction Lp ultimate length L

Body weight 8.2.2

Body weight has contribution from structure and reserveIf reserves allocated to reproduction hardly contribute:

13/4

13/100

11

1

)(][

][

W

EEmV

EEmV

f

EEV

L

μwEd

μwEdVμwEVdW

VVV/VVW

][ m

E

E

V

Eμwd

WLE

V

W

V

Wintra-spec body weightinter-spec body weightintra-spec structural volumeInter-spec structural volumereserve energycompound length-parameter

specific density for structuremolecular weight for reservechemical potential of reservemaximum reserve energy density

Feeding rate 8.2.2

slope = 1

poikilothermic tetrapodsData: Farlow 1976

Inter-species: JXm VIntra-species: JXm V2/3

Mytilus edulisData: Winter 1973

Length, cm

Filt

ratio

n ra

te, l

/h

Scaling of metabolic rate 8.2.2

intra-species inter-species

maintenance

growth

weight

nrespiratio3

32

dl

llls

43

32

ldld

lll

EV

h

structure

reserve

32 vll

l0l

0

3lllh

Respiration: contributions from growth and maintenanceWeight: contributions from structure and reserveStructure ; = length; endotherms 3l l

3lllh

0hl

Metabolic rate 8.2.2

Log weight, g

Log metabolic rate,

w

endotherms

ectotherms

unicellulars

slope = 1

slope = 2/3

Length, cm

O2 consum

ption,

l/h

Inter-speciesIntra-species

0.0226 L2 + 0.0185 L3

0.0516 L2.44

2 curves fitted:

(Daphnia pulex)

13/113/1 /3/3/3/3

vkvVkr MMB V

At 25 °C : maint rate coeff kM = 400 a-1

energy conductance v = 0.3 m a-1

25 °CTA = 7 kK

10log ultimate length, mm 10log ultimate length, mm

10lo

g vo

n B

ert

grow

th r

ate

, a-1

)exp()()( 3/13/13/13/1 arVVVaV Bb

3/1V

a

3/1V

3/1bV

1Br

↑

↑0

Von Bertalanffy growth rate 8.2.2