Innovations by DEB theory to understand metabolic organisation

Bas KooijmanDept theoretical biology

Vrije Universiteit AmsterdamBas@bio.vu.nl

http://www.bio.vu.nl/thb/

Nantes, 2005/04/22

Contents

• introduction

• static & production models

• principles

• surface-volume interactions

• respiration patterns

• biomass composition

• product formation

• body size scaling

Nantes, 2005/04/22

Dynamic Energy Budget theoryfor metabolic organisationUptake of substrates (nutrients, light, food) by organisms and their use (maintenance, growth, development, reproduction)

First principles, quantitative, axiomatic set upAim: Biological equivalent of Theoretical Physics

Primary target: the individual with consequences for• sub-organismal organization• supra-organismal organizationRelationships between levels of organisation

Many popular empirical models are special cases of DEB

molecule

cell

individual

population

ecosystem

system earth

time

spac

e

Space-time scales

When changing the space-time scale, new processes will become important other will become less importantIndividuals are special because of straightforward energy/mass balances

Each process has its characteristic domain of space-time scales

Empirical special cases of DEB year author model year author model1780 Lavoisier multiple regression of heat

against mineral fluxes1950 Emerson cube root growth of bacterial

colonies

1825 Gompertz Survival probability for aging 1951 Huggett & Widdas foetal growth

1889 Arrhenius temperature dependence of physiological rates

1951 Weibull survival probability for aging

1891 Huxley allometric growth of body parts 1955 Best diffusion limitation of uptake

1902 Henri Michaelis--Menten kinetics 1957 Smith embryonic respiration

1905 Blackman bilinear functional response 1959 Leudeking & Piret microbial product formation

1910 Hill Cooperative binding 1959 Holling hyperbolic functional response

1920 Pütter von Bertalanffy growth of individuals

1962 Marr & Pirt maintenance in yields of biomass

1927 Pearl logistic population growth 1973 Droop reserve (cell quota) dynamics

1928 Fisher & Tippitt

Weibull aging 1974 Rahn & Ar water loss in bird eggs

1932 Kleiber respiration scales with body weight3/ 4

1975 Hungate digestion

1932 Mayneord cube root growth of tumours 1977 Beer & Anderson development of salmonid embryos

DEB theory is axiomatic, based on mechanisms not meant to glue empirical models

Since many empirical models turn out to be special cases of DEB theory the data behind these models support DEB theory

This makes DEB theory very well tested against data

Classic energetics

Anabolism: synthetic pathwaysCatabolism: degradation pathwaysDuality: compounds as source for energy & building blocks

From: Mader, S. S. 1993 Biology, WCB

This decomposition occursat several places in DEBs

Static Energy Budgets Scope for Growth

From: Brafield, A. E. and Llewellyn, M. J. 1982 Animal energetics, Blackie, Glasgow

C energy from foodP production (growth)F energy in faecesU energy in urineR heat

Numbers: kJ in 28 d

Basic difference with dynamic budgets:Production is quantified as energy fixed in new tissue, not as energy allocated to growth: SEBs exclude overheadsHeat includes overheads of growth, reproduction and other processes it does not quantify maintenance costs

Production model

food faecesassimilation

feeding defecation

maintenance

offspring

reproductionreserve

structurestructure

growth

1- maturitymaintenance

maturityoffspring

maturationreproduction

Basic DEB scheme

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

Production models• no accommodation for embryonic stage; require additional state variables (no food intake, still maintenance costs and growth)

• no metabolic memory, no growth during starvation

• require switches in case of food shortage (reserves allocated to reproduction used for maintenance)

• no natural dynamics for reserve; descriptive rules for growth vs reprod.

• no explanation for body size scaling of metabolic rates, changes in composition of biomass, metabolic memory

• require complex regulation modelling for fate of metabolites (ATP vs building blocks; consistency problem with lower levels of org.)

• dividing organisms (with reserve) cannot be included

• typically have descriptive set points for allocation, no mechanisms (weight-for-age rules quantify allocation to reproduction)

Some DEB pillars

• life cycle perspective of individual as primary target embryo, juvenile, adult (levels in metabolic organization)

• life as coupled chemical transformations (reserve & structure)

• time, energy, entropy & mass balances

• surface area/ volume relationships (spatial structure & transport)

• homeostasis (stoichiometric constraints via Synthesizing Units)

• syntrophy (basis for symbioses, evolutionary perspective)

• intensive/extensive parameters: body size scaling

Biomass: reserve(s) + structure(s)

Reserve(s), structure(s): generalized compounds, mixtures of proteins, lipids, carbohydrates: fixed compositionCompounds in reserve(s): equal turnover times, no maintenance costs structure: unequal turnover times, maintenance costs

Reasons to delineate reserve, distinct from structure• metabolic memory• explanation of respiration patterns (freshly laid eggs don’t respire) • biomass composition depends on growth rate• fluxes are linear sums of assimilation, dissipation and growth basis of method of indirect calorimetry• explanation of inter-species body size scaling relationships

Change in body shape

Isomorph: surface area volume2/3

volumetric length = volume1/3

V0-morph: surface area volume0

V1-morph: surface area volume1

Ceratium

Mucor

Merismopedia

Mixtures of V0 & V1 morphs

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

growing in length only

Respiration ontogeny in birds

age, d age, d

ml C

O2

d-1

ml O

2 d-1

altricialTroglodytes aëdon

precocialGallus domesticus

Observations: just prior to hatching • respiration shows a plateau in precocial, not in altricial birds • pore size and frequency in egg shell is such that O2 flux has constant resistance

Conclusion: ontogeny is constrained by diffusion limitation in precocial birds (Rahn et al 1990)

DEB theory: reserve dynamics controls ontogeny (same pattern in species without shells) Minimization of water loss causes observed constant flux resistance

Embryonic development

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 johnstoniData from Whitehead 1987

yolk

embryo

Heat increment of feeding

O2 consumption increases sharply with feeding

Present situation: little understood expected increase due to chemistry of protein processing

only explains 10 % of observed increase

Explanation by DEB theory: elemental balance for assimilation process

fluxes of CO2, H2O, O2, N-waste

follow from fluxes of substrate, reserve, structure, products

Indirect calorimetry

Empirical finding (Lavoisier, 1780): heat dissipation = weighted sum of O2, CO2, N-waste fluxes

Explanation by DEB theory: Mass & energy balances show that dissipating heat, and all mineral and organic fluxes are weighted sums of 3 basic energy fluxes (assimilation, maintenance, growth)

Empirical finding for many micro-organisms: heat dissipation O2 flux

DEB theory offers: • constraints on elemental composition of reserve relative to structure involving chemical potentials of minerals and organic (generalized) compounds• gives expression for proportionality factor

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

DEB theory:

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

For pyruvate: wG<0

Leudeking & Piret (1959):

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

Cannot explain observed pattern

Yield vs growth

1/spec growth rate, 1/h

1/yi

eld,

mm

ol g

luco

se/

mg

cells

Streptococcus bovis, Russell & Baldwin (1979)

Marr-Pirt (no reserve)DEB

spec growth rate

yield

Russell & Cook (1995): this is evidence for down-regulation of maintenance at high growth ratesDEB theory: high reserve density gives high growth rates structure requires maintenance, reserves do not

Interactions of substrates

Substrate interactions in DEB theory are based on Synthesizing Units (SUs): generalized enzymes that follow the rules of classic enzyme kinetics but• working depends in fluxes of substrates, rather than concentrations “concentration” only has meaning in homogeneous environments• backward fluxes are small in S + E SE EP E + P

Basic classification• substrates: substitutable vs complementary• processing: sequential vs parellel

Mixture between substitutable & complementary substrates: grass cow; sheep brains cow; grass + sheep brains cow

Dynamics of SU on the basis of time budgetting offers framework for foraging theory example: feeding in Sparus larvae (Lika, Can J Fish & Aquat Sci, 2005): food searching sequential to mechanic food handling food processing (digestion) parellel to searching & handling gives deviations from Holling type II

low low high

• parameter values tend to co-vary across species• parameters are either intensive or extensive• ratio’s 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

Inter-species body size scaling

][/}{ MAmm pκpL

}{ Amp][ Mp

mAm Lp }{

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

Primary scaling relationships

Extensive parameters

K saturation constant

Lb length at birth

Lp length at puberty

{JXm} max spec feeding rate

{pAm} max spec assim rate

[Em] max reserve capacity

Intensive parameters

[pM] volume-spec maint. costs

{pT} surface-spec maint. costs

[EG] spec structure costs

ha aging acceleration

partitionning fraction

R reproduction efficiency

maximum length Lm = {pAm} / [pM]

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, gLo

g m

etab

olic

rat

e,

w

endotherms

ectotherms

unicellulars

slope = 1

slope = 2/3

Length, cm

O2 c

onsu

mpt

ion,

l

/h

Inter-speciesIntra-species

0.0226 L2 + 0.0185 L3

0.0516 L2.44

2 curves fitted:

(Daphnia pulex)

West-Brown: scaling of respiration

Explanation: Minimizing of transportation costs in space-filling fractally branching tube systems results in ¾ - “law”

Problems:• Protostomes have open circulation system, no tube system

scaling of respiration also applies to protostomes

• Flux in capillaries is much less than in big tubes, not equal

• Transport rate must match peak metabolic requirements

rather than standard

• No differentiation between inter- and intra-specific scaling

• Transport costs are tiny fraction of maintenance costs

minimum argument is not convincing (nor demonstrated)

• Scaling of respiration does not explain all other scaling “laws”

nor “the growth curve” of demand systems

These gouramis are from the same nest, These gouramis are from the same nest, they have the same age and lived in the same tankthey have the same age and lived in the same tankSocial interaction during feeding caused the huge size differenceSocial interaction during feeding caused the huge size differenceAge-based models for growth are bound to fail;growth depends on food intake Age-based models for growth are bound to fail;growth depends on food intake Growth depends on food availability; no so according to Brown 2001, Growth depends on food availability; no so according to Brown 2001, NatureNature

Trichopsis vittatus

Size, not ageSize, not age

Banavar: scaling of respiration

Explanation: Dilution of biomass with transport material between maintenance-requiring nodes in efficient networks results in ¾ -”law”

Problems:• Transport rate must match peak metabolic requirements

rather than standard• No differentiation between inter- and intra-specific scaling• Efficiency criterion

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

Incubation time

10log egg weight, g 10log egg weight, g

10lo

g in

cuba

tion

tim

e, d

10lo

g in

cuba

tion

tim

e, d

lb equal° tube noses

slope = 0.25

Data from Harrison 1975

European birds

4/104

0

EaLE

Lab

m

mb

Incubation timeEgg weight

Conclusion: tube noses are special,not because of their long incubationbut because of their large eggsIntra-species scaling: large eggs short time to fledging

Von Bertalanffy growth rate

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

Some innovations by DEB theory

• Unifies all life on earth (bacteria, protoctists, fungi/animals, plants)

• Links levels of organisation• Explains body size scaling relationships• Deals with energetic and stoichiometric constraints• Individuals that follow DEB rules can merge smoothly

into a symbiosis that again follows DEB rules• Method for determining entropy of living biomass • Biomass composition depends on growth rate• Product formation has 3 degrees of freedom• Explains indirect calorimetry• Explains how yield of biomass depends on growth rate• Quantitative predictions have many practical applications