Introduction to DEB theory Bas Kooijman

Dept theoretical biologyVrije Universiteit Amsterdam

Bas@bio.vu.nlhttp://www.bio.vu.nl/thb/

Marseille, 2005/12/15

DEB – ontogeny - IBM1980

1990

2000

Daphniaecotox

application

NECs

ISO/OECD

embryos

body sizescaling

morphdynamicsindirect

calorimetry

food chains

SynthesizingUnits

multivarplants

adaptationtumour

induction

von Foerster

epidemiolapplications

bifurcationanalysis

Globalbif-analysis

integralformulations

adaptive dynamics

ecosystem Self-orginazation

numericalmethods

symbioses

ecosystemdynamics

molecularorganisation

DEB 1

DEB 2

DEBtox 1

organfunction

aging

micro’s

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

Empirical special cases of DEB year author model year author model

1780 Lavoisier multiple regression of heat against mineral fluxes

1950 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

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

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 2.2

• biosphere: thin skin wrapping the earth light from outside, nutrient exchange from inside is across surfaces production (nutrient concentration) volume of environment• food availability for cows: amount of grass per surface area environ food availability for daphnids: amount of algae per volume environ• feeding rate surface area; maintenance rate volume (Wallace, 1865)

• many enzymes are only active if linked to membranes (surfaces) substrate and product concentrations linked to volumes change in their concentrations gives local info about cell size; ratio of volume and surface area gives a length

Change in body shapeIsomorph: surface area volume2/3

volumetric length = volume1/3

V0-morph: surface area volume0

V1-morph: surface area volume1

Ceratium

Mucor

Merismopedia

Shape correction functionShape correction function

at volume Vactual surface area at volume V

isomorphic surface area at volume V=

1)( VΜ for dVV

V0-morphV1-morph isomorph 0

3/1

3/2

)/()(

)/()(

)/()(

d

d

d

VVV

VVV

VVV

Μ

Μ

Μ

3/13/2

3/13/2

)/(2

2)/(

2)(

)/(3

3)/(

3)(

dd

dd

VVδ

VVδ

δV

VVδ

VVδ

V

Μ

Μ

Static mixtures between V0- and V1-morphs for aspect ratio δ

Mixtures of changes in shapeDynamic mixtures between morphs

Lichen Rhizocarpon

V1- V0-morph

V1- iso- V0-morph

outer annulus behaves as a V1-morph, inner part as a V0-morph. Result: diameter increases time

Biofilms

Isomorph: V1 = 0

V0-morph: V1 =

mixture between iso- & V0-morph

biomass grows, butsurface area that is involvedin nutrient exchange does not

solid substratebiomass

3/2

1

1)(

d

d

VV

VV

V

VVΜ

Arrhenius relationship 2.6

103/T, K-1

ln p

op g

row

th r

ate,

h-1

103/TH 103/TL

r1 = 1.94 h-1

T1 = 310 KTH = 318 KTL = 293 K

TA = 4370 KTAL = 20110 KTAH = 69490 K

}exp{}exp{1

}exp{

)( 11

TT

TT

TT

TT

TT

TT

r

TrAH

H

AH

L

ALAL

AA

Von Bertalanffy growth

trb

BeLLLtL )()( rategrowh Bert von length; BrL

Len

gth,

mm

Age, d

Arrhenius

1T

BrlogK6400AT

Data from Greve, 1972

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

1- maturitymaintenance

maturityoffspring

maturationreproduction

Basic DEB scheme

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

1-

1-u

Competitive tumour growth

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

maturitymaintenance

maturityoffspring

maturationreproduction

tumourtumour

u

)(][)(][

)(][)(

tVptVp

tVptκ

uMuM

uMuu

Allocation to tumour relative maint workload

Isomorphy: is constantTumour tissue: low spec growth costs low spec maint costs

uκ

Van Leeuwen et al., 2003 The embedded tumour: host physiology is important for the evaluation of tumour growth.British J Cancer 89, 2254-2268

maint

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

-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

Simultaneous Substrate Processing

Chemical reaction: 1A + 1B 1CPoisson arrival events for molecules A and B

blocked time intervals

• acceptation event¤ rejection event

11111 BABACmC JJJJJJFlux of C:

production

production

Simultaneous Nutrient Limitation

Specific growth rate of Pavlova lutheri as function of intracellular phosphorus and vitamin 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)

P content, fmol/cell

B12 content,

10 -21 mol/cell

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

Von Bertalanffy growth rate

11 ][])[]([3

)()(

MmGB

trb

pEκfEr

eLLLtL B

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

m

m

G

pκEfEL

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

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 low growth ratesDEB theory: high reserve density gives high growth rates structure requires maintenance, reserves not

Synthesizing Unit dynamicsSU: Generalized enzyme that operates on fluxes of metabolites

Typical form for changes in bounded fractions

Typical flux of metabolites for

Mixing of types:

Example of mixture between sequential & complementary substrates:

Interactions of substrates

Kooijman, 2001Phil Trans R Soc B356: 331-349

Co-metabolismCo-metabolic degradation of 3-chloroaniline by Rhodococcus with glucose as primary substrateData from Schukat et al, 1983

Brandt et al, 2003Water Research37, 4843-4854

Size-structured Unstructured Population Dynamics

Isomorphs: individual-based or pde formulationV1-morphs: unstructured (ode) formulation

Effect of individuality becomes small if ratio between largest and smallest body size reduces

This suggest a perturbation method to approximate a pde with an ode formulation

Need for simplification of ecosystem dynamics

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 1986Energy 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

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)

1-species mixotroph community

Mixotrophs areproducers, which live off light and nutrientsas well asdecomposers, which live off organic compounds which they produce by aging

Simplest community with full material cycling

Kooijman, Dijkstra, Kooi 2002 J. Theor. Biol. 214: 233-254

Canonical communityShort time scale:Mass recycling in a community closed for mass open for energy

Long time scale:Nutrients leaks and influxes

Memory is controlled by life span (links to body size)Spatial coherence is controlled by transport (links to body size)

Kooijman, Nisbet 2000 How light and nutrients affect life in a closed bottle. In: Jørgensen, S. E (ed) Thermodynamics and ecological modelling. CRC, 19-60

Self organisation of ecosystems• homogeneous environment, closed for mass • start from mono-species community of mixotrophs• parameters constant for each individual• allow incremental deviations across generations link extensive parameters (body size segregation) • study speciation using adaptive dynamics• allow cannibalism/carnivory• study trophic food web/piramid: coupling of structure & function• study co-evolution of life, geochemical dynamics , climate• adaptive dynamics applied to multi-character DEB models

Troost et al 2004 Math Biosci, to appear; Troost et al 2004 Am Nat, submittedCollaboration: Metz, Troost, Kooi, Kooijman