Transcript The application of DEB theory to fish energetics

The application of DEB theory to
fish energetics
Bas Kooijman
Dept theoretical biology
Vrije Universiteit Amsterdam
[email protected]
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 theory
for metabolic organisation
Uptake of substrates (nutrients, light, food)
by organisms and their use (maintenance, growth,
development, reproduction)
First principles, quantitative, axiomatic set up
Aim: Biological equivalent of Theoretical Physics
Primary target: the individual with consequences for
• sub-organismal organization
• supra-organismal organization
Relationships between levels of organisation
Many popular empirical models are special cases of DEB
Space-time scales
space
Each process has its characteristic domain of space-time scales
system earth
ecosystem
population
individual
cell
molecule
When changing the space-time scale,
new processes will become important
other will become less important
Individuals are special because of
straightforward energy/mass balances
time
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
1902
temperature dependence of
DEB theory
is rates
axiomatic, 1951 Weibull
physiological
allometric
of body parts
Huxleybased
1955
Best
on growth
mechanisms
Michaelis--Menten
kinetics empirical
Henri not meant
1957
Smith
to glue
models
1905
Blackman
1910
Hill
1891
1920
1927
bilinear functional response
1959
Leudeking & Piret
survival probability for aging
diffusion limitation of uptake
embryonic respiration
microbial product formation
Cooperative binding
hyperbolic functional response
1959
Holling
Since many
empirical models
von Bertalanffy growth of
maintenance in yields of biomass
Pütter
1962
Marr & Pirt
individuals
turn out
to be special cases of DEB theory
logistic population growth
reserve (cell quota) dynamics
Pearl the data
Droopsupport DEB
behind these 1973
models
theory
1928
Fisher &
Tippitt
1932
Kleiber
Weibull aging
1974
Rahn & Ar
water loss in bird eggs
This makes
theory
very
tested against
data
respirationDEB
scales with
body
digestion
1975 well
Hungate
weight3/ 4
1932
Mayneord
cube root growth of tumours
1977
Beer & Anderson
development of salmonid embryos
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
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 composition
Compounds 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
Basic DEB scheme
food
feeding
defecation
faeces
assimilation
reserve
somatic
maintenance
growth
structure

1-
maturity
maintenance
maturation
reproduction
maturity
offspring
-rule for allocation
vL2  kM L3
Ingestion rate, 105 cells/h
O2 consumption, g/h
Respiration 
Length, mm
• 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
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
Embryonic development
embryo
weight, g
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
Carettochelys insculpta
Data from Web et al 1986
time, d
: scaled time
l : scaled length
e: scaled reserve density
g: energy investment ratio
Embryonic development
Salmo trutta
Data from Gray 1926
yolk
weight, g
embryo
time, d
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
3kM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
Length, mm
Von Bertalanffy growth
Data from Greve, 1972
Age, d
log rB
Arrhenius
TA  6400K
T 1
L(t )  L  (L  Lb ) erB t
L length; rB
von Bert growh rate
Competitive tumour growth
food
defecation
feeding
faeces
assimilation
reserve
somatic
maintenance
growth
structure

maint
1-
u 1-u
tumour
maturity
maintenance
Allocation to tumour
 relative maint workload
[ pMu ] Vu (t )
κu (t ) 
[ pM ] V (t )  [ pMu ] Vu (t )
Isomorphy:
κu is constant
Tumour tissue:
low spec growth costs
low spec maint costs
maturation
reproduction
maturity
offspring
Van Leeuwen et al., 2003
The embedded tumour: host physiology
is important for the evaluation of
tumour growth.
British J Cancer 89, 2254-2268
Competitive organ growth
Allocation to velum vs gut
 relative workload
κvelum (t )  κ κ assim (1  f (t ))
κ gut (t )  κ κ assim f (t )
fraction of
catabolic flux
κu
Macoma
low food
Macoma
high food
Collaboration:
Katja Philipart (NIOZ)
Change in body shape
Isomorph:
surface area  volume2/3
volumetric length = volume1/3
Mucor
Ceratium
V0-morph:
surface area  volume0
Merismopedia
V1-morph:
surface area  volume1
volume, m3
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
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:
J E ,R  (1  κ) J E ,C  J E , J
with J E , J constant
Allocation per time increment is infinitesimally small
We therefore need a buffer with buffer-handling rules for egg prod
(no buffer required in case of placental mode)
Strong homeostasis: Fixed conversion efficiency
Weak homeostasis: Reserve density at birth equals that of mother
Reproduction rate: R  κR J E , R / E0 with E0 costsper egg
E0 follows from maintenance + growth costs,
given amounts of structure and reserve at birth
103 eggs
103 eggs
Reproduction at constant food
Gobius paganellus
Data Miller, 1961
Rana esculenta
Data Günther, 1990
length, mm
length, mm
 ge

κR
(1  κ )
(vV 2 / 3  k M V )  gkM V p 
e0Vm
 g e

kM
g  f kM
2
3
L 
δΜ L 
δΜ L3p at constantfood (e  f )
v
f
v
R(e,V ) 
Application to flatfish
{pAm}/{pXm} = 0.2
[pM] = 225 W m-3
[EG] = 7 kJ cm-3
[Em]  2.5 kJ cm-3
name (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
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 Lm  { pA}κ / [ pM ]
 allocation fraction to growth + maint. (intensive)
[ pM ] volume-specific maintenance power (intensive)
{ p A} surface area-specific assimilation power (extensive)
• conclusion : { pA}  Lm (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
Kooijman 1986
Energy budgets can explain body size scaling relations
J. Theor. Biol. 121: 269-282
Body weight
Body weight has contribution from structure and reserve
If reserves allocated to reproduction hardly contribute:
f 1

W  dV V  E wE μ  V dV  [ Em ] wE μE1
W  V dV  [ Em 0 ] (V/V0 )1/ 3 wE μE1
 V  V 4 / 3 LW1

W
W
V
V
E
LW
1
E
intra-spec body weight
inter-spec body weight
intra-spec structural volume
Inter-spec structural volume
reserve energy
compound length-parameter
dV
wE
μE
[ Em ]


specific density for structure
molecular weight for reserve
chemical potential of reserve
maximum reserve energy density
Metabolic rate
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
Respiration Quotient
carbon dioxide production
dioxygenconsumption
Not constant, depends on size & feeding conditions
Scaling of metabolic rate
Respiration: contributions from growth and maintenance
Weight: contributions from structure and reserve
3
Structure  l ; l = length; endotherms lh  0
comparison
intra-species
inter-species
maintenance
 lh l   l 3
 lh l   l 3
growth
 l l 2  l 3
0
 l0
l
ls l 2  l 3

dl3
lh l 2  l 3

dV l 3  d E l 4
reserve
structure
respiration
weight
Metabolic rate
slope = 1
0.0226 L2 + 0.0185 L3
0.0516 L2.44
Log metabolic rate, w
O2 consumption, l/h
2 curves fitted:
endotherms
ectotherms
slope = 2/3
unicellulars
Length, cm
Intra-species
(Daphnia pulex)
Log weight, g
Inter-species
Von Bertalanffy growth rate
L(t )  L  ( L  Lb ) e  rB t
rB1  3 ([EG ]  f κ[ Em ])[ pM ]1
L length
[ EG ] spec growth costs
f func resp [ Em ] spec reservecapacity
κ fraction [ pm ] spec maintcosts
Lp, cm
Length at puberty
Clupoid fishes
 Clupea
• Brevoortia
° Sprattus
 Sardinops
Sardina
 Sardinella
+ Engraulis
* Centengraulis
 Stolephorus
Data from Blaxter & Hunter 1982
L, cm
Length at first reproduction Lp  ultimate length L
Spatial structure: schooling
Scomber scombrus
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”
DEB tele-course 2005
Feb – April 2005, 10 weeks, 200 h
no financial costs
http://www.bio.vu.nl/thb/deb/course/deb/
Vacancies at Dept Theor Biol VUA
EU-projects Modelkey (1PhD+1PD), Nomiracle (1PhD)
see http://www.bio.vu.nl/thb/
Download slides of Sète lecture by Bas Kooijman
http://www.bio.vu.nl/thb/users/bas/lectures/