Lecture 3

Implications of theory

Mass & energy balance

The standard DEB model specifies fluxes of 4 organic compounds food, faeces, stucture (growth), reserve (including reproduction)

The fluxes of 4 mineral compouds (CO2, H2O, O2, NH3) follow from conservation of chemical elements C, H, O, N and strong homeostasis

The standard DEB model assumes that only food is limiting

Dissipating heat follows from conservation of energy and strong homeostasis (constant chemical potentials)

Method of indirect calorimetry

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

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

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

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 compositionData Esener et al 1982, 1983; Kleibsiella on glycerol at 35°C

nHW

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

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 extensionsanimal heterotroph phototroph

symbiosis plant

Interactions of substrates

Photosynthesis

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

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

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 foodGrowth 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

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

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

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

Life span

10log aVolume

10log m3earth

whale

bacterium

water molecule

life on earth

whale

bacteriumATP

Inter-species body size scaling• 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

assimilation {JEAm} max surface-specific assim rate Lm

feeding {b} surface- specific searching rate

digestion yEX yield of reserve on food

growth yVE yield of structure on reserve

mobilization v energy conductance

heating,osmosis {JET} surface-specific somatic maint. costs

turnover,activity [JEM] volume-specific somatic maint. costs

regulation,defence kJ maturity maintenance rate coefficient

allocation partitioning fraction

egg formation R reproduction efficiency

life cycle [MHb] volume-specific maturity at birth

life cycle [MHp] volume-specific maturity at puberty

aging ha aging acceleration

maximum length Lm = {JEAm} / [JEM] Kooijman 1986J. Theor. Biol. 121: 269-282

Follows from:1. maturity at birth equals a given value2. reserve density at birth equals that of mother

State variables:

Parameters:

Problem: Given parameter values, find

Initial reserve of an egg

Theory in Kooy2008

Effects of nutrition

scaled res density at birth

scaled res density at birth

scaled res density at birth

scal

ed le

ngth

at b

irth

scal

ed in

itial

res

erve

scal

ed a

ge a

t birt

h

Reduction of initial reserve

1

0.8

0.5scaled age

scaled age

scaled age

scal

ed m

atur

itysc

aled

str

uct v

olum

e

scal

ed r

eser

ve

Scaling relationships

log zoom factor, z

log zoom factor, z

log zoom factor, z

log

scal

ed in

itial

res

erve

log

scal

ed a

ge a

t birt

h

log

scal

ed le

ngth

at b

irth

approximate slope at large zoom factor

Length at puberty

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

Body weight has contributions from structure and reserveIf reserve allocated to reproduction hardly contributes:

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 rateslope = 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

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

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)

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

a↑0

Von Bertalanffy growth rate