Life history events in DEB theory
for metabolic organisation
Bas Kooijman
Dept theoretical biology
Vrije Universiteit Amsterdam
Bas@bio.vu.nl
http://www.bio.vu.nl/thb/
Oslo, 2008/03/13
Life history events in DEB theory
for metabolic organisation
Contents:
• What is DEB theory?
• Homeostasis
• Standard model & calorimetry
• Allocation
• Embryonic development
• Unexpected links
• Body size scaling relationships
• Parameter estimation
Bas Kooijman
Dept theoretical biology
Vrije Universiteit Amsterdam
Bas@bio.vu.nl
http://www.bio.vu.nl/thb/
Oslo, 2008/03/13
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
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
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
volume, m3
3.7.2
Bacillus  = 0.2
Collins & Richmond 1962
time, min
Fusarium  = 0
Trinci 1990
time, h
volume, m3
volume, m3
hyphal length, mm
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
Ingestion 
fL2
Length, mm
Length, mm
Cum # of young
Reproduction 
3.5
Growth:
d
L  rB ( L  L)
dt
Von Bertalanffy
Age, d
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
3.7.1
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
3.7.1
ml O2 d-1
ml CO2 d-1
Respiration ontogeny in birds
altricial
Troglodytes aëdon
age, d
precocial
Gallus domesticus
age, d
Observations: just prior to hatching
• respiration shows a plateau in precocial, not in altricial birds
• pore size and frequency in egg shell is such that O2 flux has constant resistance
Conclusion: ontogeny is constrained by diffusion limitation in precocial birds (Rahn et al 1990)
DEB theory: reserve dynamics controls ontogeny (same pattern in species without shells)
Minimization of water loss causes observed constant flux resistance
weight, g
Foetal development
Mus
musculus
time, d
3.7.1
Foetes develop like eggs,
but rate not restricted by reserve
(because supply during development)
Reserve of embryo “added” at birth
Initiation of development
can be delayed by implantation egg cell
Nutritional condition of mother only
affects foetus in extreme situations
Data: MacDowell et al 1927
d
For E0   : V  vV 2 / 3 ; V (0)  0; V (t )  (vt / 3)3
dt
Pupal development
pupal weight, mg
17 °C
time, d
green-veined white butterfly, Pieris napi
Data from Forsberg & Wiklund 1988
pupa = embryo in DEB theory
• no uptake of resources
• start of development with
very small amount of structure
• initiation & termination linked
to maturity
Metamorphosis
The larval malphigian tubes are clearly visible in this emerging cicada
They resemble a fractally-branching space-filling tubing system,
according to Jim Brown, but judge yourself ….
Java, Nov 2007
scaled reserve
scaled maturity
Reduction of initial reserve
scaled age
3.7.1
1
0.8
0.5
scaled struct volume
scaled age
scaled age
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
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
3.7
Von Bert growth rate -1, d
length, mm
Growth at constant food
time, d
Von Bertalanffy growth curve:
L(t )  L  ( L  Lb ) exp( rB t )
rB1  3k M1  3δM L / v
L  fLm  fVm1/ 3 / δM
3δM / v
3k M1
ultimate length, mm
t
L
Lb
L
time
Length
L. at birth
ultimate L.
rB
v
kM
δM
von Bert growth rate
energy conductance
maint. rate coefficient
shape coefficient
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
Length at puberty
8.2.2
Clupoid fishes
 Clupea
• Brevoortia
° Sprattus
 Sardinops
Sardina
 Sardinella
+ Engraulis
* Centengraulis
 Stolephorus
Data from Blaxter & Hunter 1982
Length at first reproduction Lp  ultimate length L
Feeding rate
8.2.2
Filtration rate, l/h
slope = 1
Mytilus edulis
poikilothermic tetrapods
Data: Winter 1973
Data: Farlow 1976
Length, cm
Intra-species: JXm  V2/3
Inter-species: JXm  V
8.2.2
log scaled age at birth
log scaled initial reserve
Scaling relationships
log scaled length at birth
log zoom factor, z
log zoom factor, z
approximate slope at large zoom factor
log zoom factor, z
Two-sample case: D. magna 20°C
Optimality of life history parameters?
measured quantities  primary pars
Standard DEB model (isomorph, 1 reserve, 1 structure)
reserve & maturity: hidden variables
measured
for 2 food levels
primary parameters
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 18-22 symposium in Brest
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/
Audience:
thank you for your attention
Stig Omholt:
thank you for the invitation