Dynamic Energy Budget theoryfor metabolic organization of life

Bas KooijmanDept of Theoretical Biology

Vrije Universiteit, Amsterdamhttp://www.bio.vu.nl/thb/deb/

Oldenburg, 2004/05/05

adul

t

embryo

juvenile

Dynamic Energy Budget theoryFirst 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

Applications in• ecotoxicology• biotechnologyDirect links with empiry

Space-time scales

molecule

cell

individual

population

ecosystem

system earth

time

spac

e

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 model

1780 Lavoisier multiple regression of heat against mineral fluxes

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

1920 Pütter von Bertalanffy growth of individuals 1959 Holling hyperbolic functional response

1927 Pearl logistic population growth 1962 Marr & Pirt maintenance in yields of biomass

1928 Fisher & Tippitt

Weibull aging 1973 Droop reserve (cell quota) dynamics

1932 Kleiber respiration scales with body weight3/ 4 1974 Rahn & Ar water loss in bird eggs

1932 Mayneord cube root growth of tumours 1975 Hungate digestion

1950 Emerson cube root growth of bacterial colonies 1977 Beer & Anderson development of salmonid embryos

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 & 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

Surface area/volume interactions• nutrient supply to ecosystems (erosion) surface area production (nutrient concentration) volume

• food availability for cows: grass weight/ surface area food availability for daphnids: algal weight/ volume

• feeding rate surface area; maintenance rate volume isomorphs: surface area volume2/3

V0-morphs: surface area volume0

V1-morphs: surface area volume1

• many active enzyme linked to membranes (surfaces) substrate and product concentrations linked to volumes

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(s): unequal turnover times, maintenance costs

Reasons to delineate reserve, distinct from structure• metabolic memory• 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 respiration patterns (freshly laid eggs don’t respire) • fate of metabolites (e.g. conversion into energy vs buiding blocks)

reserve structuresubstrate(s)

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

nHW

nOW

nNWO2

CO2

Spec 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

yVE 0.904 yXE 1.35rm 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

General assumptions• 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• 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 (weak homeostasis)• 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, reproduction. or change reserve dynamics; cease maturation, reprod.; do or do not shrink in structure

-rule for allocation

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

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 johnstoni,Data from Whitehead 1987

yolk

embryo

Synthesizing unitsGeneralized enzymes that follow classic enzyme kinetics E + S ES EP E + Pwith two modifications:• back flux is negligibly small E + S ES EP E + P• specification of transformation is on the basis of arrival fluxes of substrates rather than concentrations

Concentration: problematic (intracellular) environments: spatially heterogeneous state variables in dynamic systems In spatially homogeneous environments: arrival fluxes concentrations

Mitochondria

Transformations:1 Oxaloacetate + Acetyl CoA + H2O = Citrate + HSCoA2 Citrate = cis-Aconitrate + H2O3 cis-Aconitrate + H2O = Isocitrate4 Isocitrate + NAD+ = α-Ketoglutarate + CO2 + NADH + H+

5 α-Ketoglutarate + NAD+ + HSCoA = Succinyl CoA + CO2 + NADH + H+

6 Succinyl CoA + GDP 3- + Pi 2- + H+ = Succinate + GTP 4- + HSCoA

7 Succinate + FAD = Fumarate + FADH2

8 Fumarate + H2O = Malate9 Malate + NAD+ = Oxaloacetate + NADH + H+

TriCarboxylic Acid cycle

Enzymes pass metabolites directly to other enzymes enzymes catalizing transformations 5 & 7: bound to inner membrane (and FAD/FADH2)Net transformation: Acetyl-CoA + 3 NAD+ + FAD + GDP 3- + Pi

2- + 2 H2O = 2 CO2 + 3 NADH + FADH2 + GTP 4- + 2 H+ + HS-CoA

Dual function of intermediary metabolites building blocks energy substrate

all eukaryotes once possessed mitochondria,most still do

Pathways & allocation

reserve

reservereserve

maintenance

maintenance

maintenance

structure structure

structure

Mixture of products &intermediary metabolites

that is allocated tomaintenance (or growth)has constant composition

Kooijman & Segel, 2004

Numerical matching for n=4P

rodu

ct f

lux

Rej

ecte

d fl

ux

Unb

ound

ed f

ract

ion

= 0.73, 0.67, 0.001, 0.27 handshaking = 0.67, 0.91, 0.96, 0.97 binding probk = 0.12, 0.19, 0.54, 0.19 dissociation nSE = 0.032,0.032,0.032,0.032 # in reservenSV = 0.045,0.045,0.045,0.045 # in structureyEV = 1.2 res/struct kE = 0.4 res turnover jEM = 0.02 maint flux n0E = 0.05 sub in res

0

0

1

1

1

2

2

23

3

3

4

4

Spec growth rate

Spec growth rate

Matching pathway whole cellNo exact match possible between production of products and intermediary metabolites by pathway and requirements by the cell

But very close approximation is possible by tuning abundance parameters and/or binding and handshaking parameters

Best approximation requires all four tuning parameters per node growth-dependent reserve abundance plays a key role in tuning

VSES iinn ,

ii αρ ,

Kooijman, S. A. L. M. and Segel, L. A. (2004) How growth affects the fate of cellular substrates.Bull. Math. Biol. (to appear)

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

Applies to all products, heat & non-limiting substrates

Indirect calorimetry (Lavoisier, 1780): heat = wO JO + wC JC + wN JN

No reserve: 2-dim basis for product formation

Symbiosis

product

substrate

Product formation is basic to symbioses

Symbiosis

substrate substrate

Product formation is basic to symbioses

Internalization

Structures merge Reserves merge

Free-living, clusteringFree-living, homogeneous

Steps in symbiogenesis

Symbiogenesis

• symbioses: fundamental organization of life based on syntrophy ranges from weak to strong interactions; basis of biodiversity• symbiogenesis: evolution of eukaryotes (mitochondria, plastids)• DEB model is closed under symbiogenesis: it is possible to model symbiogenesis of two initially independently living populations that follow the DEB rules by incremental changes of parameter values such that a single population emerges that again follows the DEB rules• essential property for models that apply to all organisms

Kooijman, Auger, Poggiale, Kooi 2003 Quantitative steps in symbiogenesis and the evolution of homeostasisBiological Reviews 78: 435 - 463

Central Metabolism

polymers

monomers

waste/source

source

• Pentose Phosphate (PP) cycle glucose-6-P ribulose-6-P, NADP NADPH• Glycolysis glucose-6-P pyruvate ADP + P ATP • TriCarboxcyl Acid (TCA) cycle pyruvate CO2

NADP NADPH• Respiratory chain NADPH + O2 NADP + H2O ADP + P ATP

Modules of central metabolism

Evolution of central metabolism

i = inverseACS = acetyl-CoA Synthase pathway PP = Pentose Phosphate cycleTCA = TriCarboxylic Acid cycle

RC = Respiratory Chain Gly = Glycolysis

Kooijman, Hengeveld 2003 The symbiontic nature of metabolic evolution Acta Biotheoretica (to appear)

in prokaryotes (= bacteria)3.8 Ga 2.7 Ga

Prokaryotic metabolic evolution

Chemolithotrophy • acetyl-CoA pathway• inverse TCA cycle• inverse glycolysis

Phototrophy:• el. transport chain• PS I & PS II• Calvin cycle

Heterotrophy:• pentose phosph cycle• glycolysis• respiration chain

Symbiogenesis1.5-2 Ga 1.2 Ga

Bacillariophyceae(diatoms)

(brown algae)Phaeophyceae

Prymnesiophyceae

RaphidophyceaeXanthophyceae

EustigmatophyceaeDictyochophyceae

Pelagophyceae

ChrysophyceaeSynurophyceae

Cryptophyceae

(plants)Cormophyta

(green algae)Chlorophyceae

(red algae)Rhodophyceae

Glaucophyceae

animals

Euglenozoa

Dinozoa

Rhizopoda

Bicosoecia

Actinopoda

Pseudofungi

Labyrinthulomycota

MyxomycotaProtostelida Ciliophora

Sporozoa

Bacteria

Zygomycota

BasidiomycotaAscomycota

Archamoeba

Microsporidia

Chytridiomycota

Percolozoa

Bigyromonada

Metamonada

Choanozoa

GranuloreticulataXenophyophora

Loukozoa

PlasmodiophoromycotaChlorarachnida

Cercomonada

Apusozoa

Pedinellophyceae

Bolidophyceae

Composed byBas Kooijman

Opalinata

Glomeromycota

Survey of organisms

mitochondria

secondarychloroplast

primary chloroplast

tertiarychloroplast

Sizes of blobsdo not reflect

number of species

Bacteria

Opi

stho

kont

s

Chromista

Amoebozoa

Alveo-lates

Plantae

Excavates

Ret

aria

Cercozoa

fungi

animals

forams

cort

ical

alv

eoli

Bik

ont

DH

FR

-TS

gen

e fu

sion

chlo

ropl

asts

mem

br. d

ynun

ikon

t

loss phagoc.gap junctions tissues (nervous)

bicentriolarmainly chitin

EF1 insertion

trip

le r

oots

mai

nly

cell

lose

photosymbionts

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 pκpL

}{ Ap][ Mp

mA Lp }{

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

Scaling of metabolic rate

comparison intra-species inter-species

maintenance

growth

weight

nrespiratio3

32

dl

llls

43

32

ldld

lll

EV

h

structure

reserve

32 lll

l0l

0

3lllh

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

3lllh

0hl

Scaling of metabolic rate

Log weight, g

Log m

etabolic 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)

Von Bertalanffy growth

trb

BeLLLtL )()( rategrowh Bert von length; BrL

Len

gth,

mm

Age, d

Arrhenius

1T

BrlogK6400AT

Data from Greve, 1972

Von Bertalanffy growth rate

11 ][])[]([3

)()(

MmGB

trb

pEκfEr

eLLLtL B

costsmaint spec][fractioncapacity reserve spec][resp funccostsgrowth spec][length

m

m

G

pκEfEL