The application of DEB theory to fish energetics

Bas KooijmanDept theoretical biology

Vrije Universiteit AmsterdamBas@bio.vu.nl

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

Sète, 2005/01/12

Contents• DEB theory introduction

• Allocation & growth

• Body parts

• Scaling

• Schooling

Sète, 2005/01/12

Dynamic Energy Budget theoryfor metabolic organisation

Uptake 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 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

: 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

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

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

1- maturitymaintenance

maturityoffspring

maturationreproduction

Basic DEB scheme

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

-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, dwei

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

Carettochelys insculptaData from Web et al 1986

yolk

embryo

Embryonic development

time, d

wei

ght,

g

Salmo truttaData from Gray 1926

yolkembryo

Growth at constant food

time, d ultimate length, mm

leng

th, m

m

M

M

δfVfLLvLδkr

trLLLtL

mm

MB

Bb

//33

)exp()()(

3/1

11

LLLt

b

Mδkvr

M

BtimeLengthL. at birthultimate L.

von Bert growth rateenergy conductancemaint. rate coefficientshape coefficient

vδ /3 M

Von

Ber

t gro

wth

rat

e -1, d

13 Mk

Von Bertalanffy growth curve:

Von Bertalanffy growth

trb

BeLLLtL )()( rategrowh Bert von length; BrL

Len

gth,

mm

Age, d

Arrhenius

1T

BrlogK6400AT

Data from Greve, 1972

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

Competitive 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

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

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

Reproduction Definition:Conversion of adult reserve(s) into embryonic reserve(s)Energy to fuel conversion is extracted from reserve(s)Implies: products associated with reproduction (e.g. CO2, NH3)

Allocation to reproduction in adults:

Allocation per time increment is infinitesimally smallWe therefore need a buffer with buffer-handling rules for egg prod (no buffer required in case of placental mode)

Strong homeostasis: Fixed conversion efficiencyWeak homeostasis: Reserve density at birth equals that of motherReproduction rate: follows from maintenance + growth costs, given amounts of structure and reserve at birth

constantwith)1( ,,,, JEJECERE JJJκJ

eggpercostswith/ 00, EEJκR RER0E

Reproduction at constant food

length, mm length, mm

103 e

ggs

103 e

ggs

Gobius paganellusData Miller, 1961

Rana esculentaData Günther, 1990

)(foodconstantat

)()1(),(

332

3/2

0

feLδv

k

f

fgLδ

v

kL

VgkVkvVeg

geκ

Ve

κVeR

pMM

pMMm

R

ΜΜ

Application to flatfishname (english) plaice flounder dab sole

name (latin) Pleuronectes platessa

Platichthys flesus Limanda limanda Solea solea

habitat cold warm, euryhaline cold warm

max life span (a) 30 10-15 10-12 20

max length (cm) 78 56 51 75

max weight (kg) 5 2 1.3 3

reprod/ body wght 0.2 0.45 0.2 0.15

length at pub m,f (cm) 15,22 11,13 10,11 12.15

Arrhenius temp (K) 5878, 7963 6957, 11134 3958, 4931 7301, 9708

partitioning fraction 0.85 0.65 0.85 0.9

{pXm} (W m-2, 283 K) 57 55 36 45

E0 (J/egg) 5 0.7 0.4 1.5

{pAm}/{pXm} = 0.2[pM] = 225 W m-3

[EG] = 7 kJ cm-3 [Em] 2.5 kJ cm-3

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

Body weight

Body weight has contribution from structure and reserveIf reserves allocated to reproduction hardly contribute:

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

Usually quantified in three different ways• consumption of dioxygen• production of carbon dioxide• dissipation of heat

DEB theory: These fluxes are weighted sums of• assimilation• maintenance• growth

Weight coefficients might differ

Not constant, depends on size & feeding conditions

Metabolic rate

nconsumptiodioxygen

production dioxidecarbon Quotient n Respiratio

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)

Von Bertalanffy growth rate

11 ][])[]([3

)()(

MmGB

trb

pEκfEr

eLLLtL B

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

m

m

G

pκEfEL

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

Spatial structure: schooling

Isomorphic schools: Number of feeding individuals N 2/3

Feeding rate per individual N -1/3

Population models require rules for birth and death of schools; shools are just “super individuals”

Scomber scombrus

DEB tele-course 2005Feb – April 2005, 10 weeks, 200 h no financial costs

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

Download slides of Sète lecture by Bas Kooijmanhttp://www.bio.vu.nl/thb/users/bas/lectures/

Vacancies at Dept Theor Biol VUA EU-projects Modelkey (1PhD+1PD), Nomiracle (1PhD)

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