Transcript Het gebruik van modellen in de biologie

The use of models
in DEB research
Bas Kooijman
Dept theoretical biology
Vrije Universiteit Amsterdam
[email protected]
http://www.bio.vu.nl/thb/
Nijmegen, 2005/02/23
Contents
• DEB theory
introduction
• Scales in space & time
• Empirical cycle
• Dimensions
• Plasticity in parameters
• Stochastic vs deteriministic
• Dynamical systems
Nijmegen, 2005/02/23
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
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
DEB
theory
very
tested against
data
respiration
scales with
body
digestion
1975 well
Hungate
weight3/ 4
1932
Mayneord
cube root growth of tumours
1977
Beer & Anderson
development of salmonid embryos
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
Basic DEB scheme
food
feeding
defecation
faeces
assimilation
reserve
somatic
maintenance
growth
structure

1-
maturity
maintenance
maturation
reproduction
maturity
offspring
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
Modelling 1
• model:
scientific statement in mathematical language
“all models are wrong, some are useful”
• aims:
structuring thought;
the single most useful property of models:
“a model is not more than you put into it”
how do factors interact? (machanisms/consequences)
design of experiments, interpretation of results
inter-, extra-polation (prediction)
decision/management (risk analysis)
Empirical cycle
Modelling 2
• language errors:
mathematical, dimensions, conservation laws
• properties:
generic (with respect to application)
realistic (precision)
simple (math. analysis, aid in thinking)
plasticity in parameters (support, testability)
• ideals:
assumptions for mechanisms (coherence, consistency)
distinction action variables/meausered quantities
core/auxiliary theory
Dimension rules
• quantities left and right of = must have equal dimensions
• + and – only defined for quantities with same dimension
• ratio’s of variables with similar dimensions are only dimensionless if
addition of these variables has a meaning within the model context
• never apply transcendental functions to quantities with a dimension
log, exp, sin, … What about pH, and pH1 – pH2?
• don’t replace parameters by their values in model representations
y(x) = a x + b, with a = 0.2 M-1, b = 5  y(x) = 0.2 x + 5
What dimensions have y and x? Distinguish dimensions and units!
Models with dimension problems
• Allometric model: y = a W b
y: some quantity
a: proportionality constant
W: body weight
b: allometric parameter in (2/3, 1)
Usual form ln y = ln a + b ln W
Alternative form: y = y0 (W/W0 )b, with y0 = a W0b
Alternative model: y = a L2 + b L3, where L  W1/3
• Freundlich’s model: C = k c1/n
C: density of compound in soil k: proportionality constant
c: concentration in liquid
n: parameter in (1.4, 5)
Alternative form: C = C0 (c/c0 )1/n, with C0 = kc01/n
Alternative model: C = 2C0 c(c0+c)-1 (Langmuir’s model)
Problem: No natural reference values W0 , c0
Values of y0 , C0 depend on the arbitrary choice
Allometric functions
O2 consumption, μl/h
Two curves fitted:
a L2 + b L3
with a = 0.0336 μl h-1 mm-2
b = 0.01845 μl h-1 mm-3
a Lb
with a = 0.0156 μl h-1 mm-2.437
b = 2.437
Length, mm
Model without dimension problem
Arrhenius model: ln k = a – T0 /T
k: some rate
T: absolute temperature
a: parameter
T0: Arrhenius temperature
Alternative form:
k = k0 exp{1 – T0 /T}, with k0 = exp{a – 1}
Difference with allometric model:
no reference value required to solve dimension problem
Arrhenius relationship
ln pop growth rate, h-1
T A TA
r1 exp{  }
T1 T
r (T ) 
TAH TAH
TAL TAL
}

}  exp{

1  exp{
T
TH
TL
T
103/T, K-1
103/TH
103/TL
r1 =
1.94 h-1
T1 =
TH =
TL =
310 K
318 K
293 K
TA = 4370 K
TAL = 20110 K
TAH = 69490 K
Biodegradation of compounds
n-th order model
d
X  kX n
dt
(1n ) 1
1 n
X (t )  X 0  (1  n)kt


n 0
X (t )  X 0  kt ; t  X 0 / k
n 1
X (t )  X 0 exp{ kt}
1 n
1

a
t (aX 0 )  X 01n k 1
1 n
Monod model
d
X
X  k
dt
KX
0  X (t )  X 0  K ln{ X (t ) / X 0 }  kt
K  X 0
X (t )  X 0  kt ; t  X 0 exp{kt / K}
K  X 0
X (t )  X 0 exp{kt / K}
1
1
t (aX 0 )  X 0 k (a  1)  Kk ln a
X : conc. of compound,
t : time
n : order
X0 : X at time 0
k : degradation rate
K : saturation constant
Biodegradation of compounds
Monod model
scaled conc.
scaled conc.
n-th order model
scaled time
scaled time
Plasticity in parameters
If plasticity of shapes of y(x|a) is large as function of a:
• little problems in estimating value of a from {xi,yi}i
(small confidence intervals)
• little support from data for underlying assumptions
(if data were different: other parameter value results,
but still a good fit, so no rejection of assumption)
Stochastic vs deterministic models
Only stochastic models can be tested against experimental data
Standard way to extend deterministic model to stochastic one:
regression model: y(x| a,b,..) = f(x|a,b,..) + e, with e N(0,2)
Originates from physics, where e stands for measurement error
Problem:
deviations from model are frequently not measurement errors
Alternatives:
• deterministic systems with stochastic inputs
• differences in parameter values between individuals
Problem:
parameter estimation methods become very complex
Statistics
Deals with
• estimation of parameter values, and confidence in these values
• tests of hypothesis about parameter values
differs a parameter value from a known value?
differ parameter values between two samples?
Deals NOT with
• does model 1 fit better than model 2
if model 1 is not a special case of model 2
Statistical methods assume that the model is given
(Non-parametric methods only use some properties of the given
model, rather than its full specification)
Dynamic systems
Defined by simultaneous behaviour of
input, state variable, output
Supply systems: input + state variables  output
Demand systems input  state variables + output
Real systems: mixtures between supply & demand systems
Constraints: mass, energy balance equations
State variables: span a state space
behaviour: usually set of ode’s with parameters
Trajectory: map of behaviour state vars in state space
Parameters:
constant, functions of time, functions of modifying variables
compound parameters: functions of parameters
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 3  J O ,G
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
C,N,P-limitation
N,P reductions
N reductions
P reductions
Nannochloropsis gaditana
(Eugstimatophyta) in sea water
Data from Carmen Garrido Perez
Reductions by factor 1/3
starting from 24.7 mM NO3, 1.99 mM PO4
79.5 h-1
CO2
0.73
h-1
HCO3-
CO2 ingestion only
No maintenance, full excretion
C,N,P-limitation
Nannochloropsis gaditana in sea water
half-saturation parameters
KC = 1.810 mM for uptake of CO2
KN = 3.186 mM for uptake of NO3
KP = 0.905 mM for uptake of PO4
max. specific uptake rate parameters
jCm = 0.046 mM/OD.h, spec uptake of CO2
jNm = 0.080 mM/OD.h, spec uptake of NO3
jPm = 0.025 mM/OD.h, spec uptake of PO4
reserve turnover rate
kE = 0.034 h-1
yield coefficients
yCV = 0.218 mM/OD, from C-res. to structure
yNV = 2.261 mM/OD, from N-res. to structure
yPV = 0.159 mM/OD, from P-res. to structure
carbon species exchange rate (fixed)
kBC = 0.729 h-1 from HCO3- to CO2
kCB = 79.5 h-1 from CO2 to HCO3-
initial conditions (fixed)
HCO3- (0) = 1.89534 mM, initial HCO3- concentration
CO2(0) = 0.02038 mM, initial CO2 concentration
mC(0) = jCm/ kE mM/OD, initial C-reserve density
mN(0) = jNm/ kE mM/OD, initial N-reserve density
mP(0) = jPm/ kE mM/OD, initial P-reserve density
OD(0) = 0.210 initial biomass (free)
Vacancies at VUA-TB
• PhD 4 yr: 2005/02 – 2009/02 EU-project Modelkey
Effects of toxicants on canonical communities
• Postdoc 2 yr: 2006/02 – 2008/02 EU-project Modelkey
Effects of toxicant in food chains
• PhD 4 yr: 205/06/01 – 2009/06/01 EU-project Nomiracle
Toxicity of mixtures of compounds
Further reading
Basic methods of theoretical biology
freely downloadable document on methods
http://www.bio.vu.nl/thb/course/tb/
Data-base with examples, exercises under construction
Dynamic Energy Budget theory
http://www.bio.vu.nl/thb/deb/