Bas KooijmanDept theoretical biology

Vrije Universiteit AmsterdamBas@bio.vu.nl

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

Introduction to DEB theory & applications in fishery sciences

Trondheim, 2007/11/01

Contents:

• What is DEB theory?• Evolution & homeostasis• Standard model & calorimetry• Product formation• Allocation• Unexpected links • Social behaviour• Reconstruction• Body size scaling

Bas KooijmanDept theoretical biology

Vrije Universiteit AmsterdamBas@bio.vu.nl

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

Introduction to DEB theory & applications in fishery sciences

Trondheim, 2007/11/01

Dynamic Energy Budget theory

• consists of a set of consistent and coherent assumptions • uses framework of general systems theory• links levels of organization scales in space and time: scale separation• quantitative; first principles only equivalent of theoretical physics• interplay between biology, mathematics, physics, chemistry, earth system sciences• fundamental to biology; many practical applications

for metabolic organization

Research strategy1) use general physical-chemical principles to develop an educated quantitative expectation for the eco-physiological behaviour of a generalized species

2) estimate parameters for any specific case compare the values with expectations from scaling relationships deviations reveal specific evolutionary adaptations

3) study deviations from model expectations learn about the physical-chemical details that matter in this case but had to be ignored because they not always apply

Deviations from a detailed generalized expectation provideaccess to species-specific (or case-specific) modifications

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

DEB theory reveals when to expect deviations from these empirical models

Individual Ecosystem

• population dynamics is derived from properties of individuals + interactions between them

• evolution according to Darwin: variation between individuals + selection

• material and energy balances: most easy for individuals

• individuals are the survival machines of life

Evolution of DEB systemsvariable structure

composition

strong homeostasisfor structure

delay of use ofinternal substrates

increase ofmaintenance costs

inernalization of maintenance

installation ofmaturation program

strong homeostasisfor reserve

reproductionjuvenile embryo + adult

Kooijman & Troost 2007 Biol Rev, 82, 1-30

54321

specialization of structure

7

8

an

ima

ls

6

pro

ka

ryo

tes

9plants

Homeostasisstrong homeostasis constant composition of pools (reserves/structures) generalized compounds, stoichiometric contraints on synthesis

weak homeostasis constant composition of biomass during growth in constant environments determines reserve dynamics (in combination with strong homeostasis)

structural homeostasis

constant relative proportions during growth in constant environments isomorphy .work load allocation

ectothermy homeothermy endothermysupply demand systems development of sensors, behavioural adaptations

1- maturitymaintenance

maturityoffspring

maturationreproduction

Standard DEB model

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

Definition of standard model: Isomorph with 1 reserve & 1 structure feeds on 1 type of food has 3 life stages (embryo, juvenile, adult)

Extensions of standard model:• more types of food and food qualities reserve (autotrophs) structure (organs, plants)• changes in morphology• different number of life stages

Three basic fluxes

• assimilation: substrate reserve + products

linked to surface area

• dissipation: reserve products

somatic maintenance: linked to surface area & structural volume

maturity maintenance: linked to maturity

maturation or reproduction overheads

• growth: reserve structure + products

Product formation = A assimilation + B dissipation + C growth

Examples: heat, CO2, H2O, O2, NH3

Indirect calorimetry: heat = D O2-flux + E CO2-flux + F NH3-flux

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

Static 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

-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

• large part 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

Size of body parts

Data: Gille & Salomon 1994 on mallard

whole body heart

Static generalization of -rule

time, dtime, d

wei

ght,

g

1-

1-u

Tumour growth

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

maturitymaintenance

maturityoffspring

maturationreproduction

tumourtumour

u

Allocation to tumour relative maint workload

Van Leeuwen et al., 2003 British J Cancer 89, 2254-2268

maint

Dynamic generalization of -rule

Isomorphy: [pMU] = [pM]Tumour tissue: low spec growth & maint costsGrowth curve of tumour depends on pars no maximum size is assumed a prioriModel explains dramatic tumour-mediated weight loss If tumour induction occurs late, tumours grow slowerCaloric restriction reduces tumour growth but the effect fades

Organ growth

)()())(1()(

tfκκtκtfκκtκ

assimgut

assimvelum

Allocation to velum vs gut

relative workload

uκ

Macomahigh food

Macomalow food

Collaboration:Katja Philipart (NIOZ)

fraction ofcatabolic flux

Relative organ size is weakly homeostatic

Organ size & function

Kidney removes N-waste from bodyAt constant food availability JN = aL2 + bL3

Strict isomorphy: kidney size L3

If kidney function kidney size: work load reduces with sizeIf kidney function L2 + cL3 for length L of kidney or body work load can be constant for appropriate weight coefficients

This translates into a morphological design constraint for kidneys

Initial amount of reserve

Initial amount of reserve E0 follows from

• initial structural volume is negligibly small

• initial maturity is negligibly small

• maturity at birth is given

• reserve density at birth equals that of mother at egg formation

Accounts for

• maturity maintenance costs

• somatic maintenance costs

• cost for structure

• allocation fraction to somatic maintenance + growth

Mean reproduction rate (number of offspring per time):

R = (1-R) JER/E0

Reproduction buffer: buffer handling rules; clutch size

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

DEB theory reveals unexpected links

Length, mm

O2

cons

umpt

ion,

μl/h

1/yi

eld,

mm

ol g

luco

se/

mg

cells

1/spec growth rate, 1/h

Daphnia

Streptococcus

respiration length in individual animals & yield growth in pop of prokaryotes have a lot in common, as revealed by DEB theory

Reserve plays an important role in both relationships, but you need DEB theory to see why and how

: These gouramis are from the same nest, These gouramis are from the same nest, they have the same age and lived in the same tank they 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;Age-based models for growth are bound to fail; growth depends on food intake growth depends on food intake

Not age, but size:Not age, but size:

Trichopsis vittatus

Rules for feedingR1 a new food particle appears at a random site within the cube at the moment one of the resident particles disappears. The particle stays on this site till it disappears; the particle density X remains constant. R2 a food particle disappears at a constant probability rate, or because it is eaten by the individual(s). R3 the individual of length L travels in a straight line to the nearest visible food particle at speed X2/3 L2, eats the particle upon arrival and waits at this site for a time th = {JXm}-1 L-2. Direction changes if the aimed food particle disappears or a nearer new one appears. Speed changes because of changes in length.

R4 If an individual of length L feeds: scaled reserve density jumps: e e + (LX/ L)3

Change of scaled reserve density e: d/dt e = - e {JXm} LX

3/ L; Change of length L: 3 d/dt L = ({JXm} LX

3 e - L kM g) (e + g)-1

At time t = 0: length L = Lb,; reserve density e = f.

R5 a food particle becomes invisible for an individual of length L1, if an individual of length L1 is within a distance Ls (L2/ L1)2 from the food particle, irrespective of being aimed at.

time

time time

rese

rve

dens

ityre

serv

e de

nsity

leng

thle

ngth

time 1 ind

2 ind

determinexpectation

Social interaction Feeding 2.1.2

Otolith growth & opacity• standard DEB model: otolith is a product

• otolith growth has contributions from growth & dissipation (= maintenance + maturation + reprod overheads)

• opacity relative contribution from growth

DEB theory allows reconstruction of functional response from opacity data as long as reserve supports growthReconstruction is robust for deviations from correct temperature trajectory

Laure Pecquerie 2007: reading the otolith

Otolith opacity Functional response

time, d

time, d

time, d

time, d time, d

otolith length, m

otol

ith

leng

th,

mop

acit

y

tem

p co

rrec

tion

rese

rve

dens

ity

func

tion

al r

espo

nse

Laure Pecquerie 2007: reading the otolith

body

leng

th, c

m

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

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

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)

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

DEB tele course 2009http://www.bio.vu.nl/thb/deb/

Free of financial costs; some 250 h effort investment

Program for 2009: Feb/Mar general theory April symposium in Brest (2-3 d) Sept/Oct case studies & applications

Target audience: PhD students

We encourage participation in groups that organize local meetings weekly

Software package DEBtool for Octave/ Matlab freely downloadable

Slides of this presentation are downloadable from http://www.bio.vu.nl/thb/users/bas/lectures/

Cambridge Univ Press 2000

Audience: thank you for your attention

Organizers: thank you for the invitation

Vacancy PhD-position on DEB theory

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