Introduction to DEB theory &
applications in fishery sciences
Bas Kooijman
Dept theoretical biology
Vrije Universiteit Amsterdam
Bas@bio.vu.nl
http://www.bio.vu.nl/thb/
Trondheim, 2007/11/01
Introduction to DEB theory &
applications in fishery sciences
Contents:
• What is DEB theory?
• Evolution & homeostasis
• Standard model & calorimetry
• Product formation
• Allocation
• Unexpected links
• Social behaviour
• Reconstruction
• Body size scaling
Bas Kooijman
Dept theoretical biology
Vrije Universiteit Amsterdam
Bas@bio.vu.nl
http://www.bio.vu.nl/thb/
Trondheim, 2007/11/01
Dynamic Energy Budget theory
for metabolic organization
• 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
Research strategy
1) 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 provide
access 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
1889
DEB theory
is axiomatic, 1951 Huggett & Widdas
Survival probability for aging
based
on mechanisms
temperature
dependence of
Arrhenius
1951
Weibull
physiological rates
not meant to glue empirical models
foetal growth
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
1973
Droop
reserve (cell quota) dynamics
1910
1920
Since many
empirical models
bilinear functional response
microbial product formation
1959
Leudeking & Piret
to binding
be special cases
of DEB theory
Cooperative
hyperbolic functional response
Hill turn out
1959
Holling
von Bertalanffy
growth ofthese 1962
maintenance
in yields of biomass
behind
models
support
DEB
theory
Pütter the data
Marr &
Pirt
individuals
1927
Pearl
1928
Fisher &
Tippitt
1932
Kleiber
logistic population growth
This makes
DEB theory very
tested
against
data
Weibull aging
water loss in bird
eggs
1974 well
Rahn &
Ar
respiration scales with body
3/ 4
1932
digestion
1975
Hungate
DEB theory
weight reveals when to expect deviations
root growth
of tumours
development of salmonid embryos
Mayneord
1977
Beer & Anderson
from cube
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 systems
1
strong
homeostasis
for structure
2
delay of use of
internal substrates
3
increase of
maintenance costs
4
inernalization of
maintenance
5
7
Kooijman & Troost 2007
Biol Rev, 82, 1-30
reproduction
juvenile  embryo + adult
animals
8
strong homeostasis
for reserve
installation of
maturation program
prokaryotes
variable
structure
composition
6
plants
9
specialization
of structure
Homeostasis
strong 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  endothermy
supply  demand systems
development of sensors, behavioural adaptations
Standard DEB model
food
feeding
defecation
faeces
assimilation
reserve
somatic
maintenance
growth
structure

1-
maturity
maintenance
maturation
reproduction
maturity
offspring
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
pyruvate, mg/l
According to
Dynamic Energy Budget theory:
Product formation rate =
wA . Assimilation rate +
wM . Maintenance rate +
wG . Growth rate
For pyruvate: wG<0
throughput rate, h-1
Glucose-limited growth of Saccharomyces
Data from Schatzmann, 1975
volume, m3
Bacillus  = 0.2
Collins & Richmond 1962
time, min
Fusarium  = 0
Trinci 1990
time, h
volume, m3
volume, m3
hyphal length, mm
Static Mixtures of V0 & V1 morphs
Escherichia  = 0.28
Kubitschek 1990
time, min
Streptococcus  = 0.6
Mitchison 1961
time, min
-rule for allocation
vL2  k M L3
Ingestion rate, 105 cells/h
O2 consumption, g/h
Respiration 
Length, mm
• 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
Age, d in juveniles?
vL2  kM L3  (1  g / f )kM L3p
fL2
Length, mm
Length, mm
Cum # of young
Reproduction 
Ingestion 
Growth:
d
L  rB ( L  L)
dt
Von Bertalanffy
Age, d
Size of body parts
Static generalization of -rule
heart
weight, g
whole body
time, d
Data: Gille & Salomon 1994 on mallard
time, d
Tumour growth
Dynamic generalization of -rule
food
Allocation to tumour
 relative maint workload
defecation
feeding
faeces
assimilation
reserve
somatic
maintenance
growth
structure

maint
1-
u 1-u
tumour
maturity
maintenance
maturation
reproduction
maturity
offspring
Isomorphy: [pMU] = [pM]
Tumour tissue:
low spec growth & maint costs
Growth curve of tumour depends on pars
no maximum size is assumed a priori
Model explains dramatic
tumour-mediated weight loss
If tumour induction occurs late,
tumours grow slower
Caloric restriction reduces tumour growth
but the effect fades
Van Leeuwen et al., 2003
British J Cancer 89, 2254-2268
Organ growth
Allocation to velum vs gut
 relative workload
κ velum (t )  κ κ assim (1  f (t ))
κ gut (t )  κ κ assim f (t )
fraction of
catabolic flux
κu
Macoma
low food
Relative organ size is weakly homeostatic
Macoma
high food
Collaboration:
Katja Philipart (NIOZ)
Organ size & function
Kidney removes N-waste from body
At constant food availability JN = aL2 + bL3
Strict isomorphy: kidney size  L3
If kidney function  kidney size: work load reduces with size
If 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
weight, g
embryo
yolk
time, d
d
e
e  g ; d l  g e  l
dτ
l
dτ
3 e g
J O  J O , M l  J O ,G
3
d 3
l
dτ
O2 consumption, ml/h
Crocodylus johnstoni,
Data from Whitehead 1987
time, d
: scaled time
l : scaled length
e: scaled reserve density
g: energy investment ratio
Daphnia
Length, mm
1/yield, mmol glucose/ mg cells
O2 consumption, μl/h
DEB theory reveals unexpected links
Streptococcus
1/spec growth rate, 1/h
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
Not
: age, but size:These gouramis are from the same nest,
they have the same age and lived in the same tank
Social interaction during feeding caused the huge size difference
Age-based models for growth are bound to fail;
growth depends on food intake
Trichopsis vittatus
Rules for feeding
R1 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} LX3/ L;
Change of length L:
3 d/dt L = ({JXm} LX3 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.
2.1.2
determin
expectation
length
reserve density
Social interaction  Feeding
time
time
1 ind
2 ind
length
reserve density
time
time
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 growth
Reconstruction is robust for deviations from
correct temperature trajectory
Laure Pecquerie 2007: reading the otolith
time, d
time, d
opacity
functional response
temp correction
Otolith opacity  Functional response
reserve density
body length, cm
otolith length, m
otolith length, m
time, d
time, d
time, d
Laure Pecquerie 2007: reading the otolith
Primary scaling relationships
assimilation
feeding
digestion
growth
mobilization
heating,osmosis
turnover,activity
regulation,defence
allocation
egg formation
life cycle
life cycle
aging
{JEAm}
{b}
yEX
yVE
v
{JET}
[JEM]
kJ

R
[MHb]
[MHp]
ha
max surface-specific assim rate  Lm
surface- specific searching rate
yield of reserve on food
yield of structure on reserve
energy conductance
surface-specific somatic maint. costs
volume-specific somatic maint. costs
maturity maintenance rate coefficient
partitioning fraction
reproduction efficiency
volume-specific maturity at birth
volume-specific maturity at puberty
aging acceleration
maximum length Lm =  {JEAm} / [JEM]
Kooijman 1986
J. Theor. Biol.
121: 269-282
Scaling of metabolic rate
8.2.2
Respiration: contributions from growth and maintenance
Weight: contributions from structure and reserve
3
Structure  l ; l = length; endotherms lh  0
intra-species
inter-species
maintenance
 lh l   l 3
 lh l   l 3
growth
 l 2  vl3
0
 l0
l
ls l 2  l 3

dl 3
lh l 2  l 3

dV l 3  d E l 4
reserve
structure
respiratio n
weight
Metabolic rate
2 curves fitted:
0.0226 L2 + 0.0185 L3
0.0516 L2.44
Log metabolic rate, w
O2 consumption, l/h
slope = 1
endotherms
ectotherms
slope = 2/3
unicellulars
Length, cm
Intra-species
(Daphnia pulex)
Log weight, g
Inter-species
Von Bertalanffy growth rate
von Bert growth rate, a-1
8.2.2
10log
25 °C
TA = 7 kK
10log

rB  3 / kM  3 V
1/ 3

ultimate length, mm
/v

1

10log
 3 / kM  3V
1/ 3
/v
ultimate length, mm

1
V1/ 3
At 25 °C :
maint rate coeff kM = 400 a-1
energy conductance v = 0.3 m a-1
V 1/ 3
↑
V 1/ 3 (a)  V1/ 3  (V1/ 3  Vb1/ 3 ) exp( rB a)
Vb1/ 3
rB1
↑
0
a
DEB tele course 2009
http://www.bio.vu.nl/thb/deb/
Cambridge Univ Press 2000
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/
Vacancy PhD-position on DEB theory
http://www.bio.vu.nl/thb/
Audience:
thank you for your attention
Organizers:
thank you for the invitation