Transcript Dynamic Energy Budget theory

Dynamic Energy Budget theory
for metabolic organization of life
Bas Kooijman
adult
Dept of Theoretical Biology
Vrije Universiteit, Amsterdam
http://www.bio.vu.nl/thb/deb/
Oldenburg, 2004/05/05
Dynamic Energy Budget theory
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
Applications in
• ecotoxicology
• biotechnology
Direct links with empiry
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
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
1920
Pütter
von Bertalanffy growth of individuals
1959
Holling
hyperbolic functional response
1927
Pearl
logistic population growth
1962
Marr & Pirt
maintenance in yields of biomass
1928
Fisher &
Tippitt
Weibull aging
1973
Droop
reserve (cell quota) dynamics
1932
Kleiber
respiration scales with body weight3/ 4
1974
Rahn & Ar
water loss in bird eggs
1932
Mayneord
cube root growth of tumours
1975
Hungate
digestion
1950
Emerson
cube root growth of bacterial colonies
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
Surface area/volume interactions
• nutrient supply to ecosystems (erosion)  surface area
production (nutrient concentration)  volume
• food availability for cows: grass weight/ surface area
food availability for daphnids: algal weight/ volume
• feeding rate  surface area; maintenance rate  volume
isomorphs: surface area  volume2/3
V0-morphs: surface area  volume0
V1-morphs: surface area  volume1
• many active enzyme linked to membranes (surfaces)
substrate and product concentrations linked to volumes
Biomass: reserve(s) + structure(s)
substrate(s)
reserve
structure
Reserve(s), structure(s): generalized compounds,
mixtures of proteins, lipids, carbohydrates: fixed composition
Compounds in
reserve(s): equal turnover times, no maintenance costs
structure(s): unequal turnover times, maintenance costs
Reasons to delineate reserve, distinct from structure
• metabolic memory
• 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
respiration patterns (freshly laid eggs don’t respire)
• fate of metabolites (e.g. conversion into energy vs buiding blocks)
nOW
nNW
Spec growth rate, h-1
kE 2.11 h-1 kM 0.021 h-1
yVE 0.904 yXE 1.35
rm 1.05 h-1 g = 1
Spec prod, mol.mol-1.h-1
Data Esener et al 1982, 1983; Kleibsiella on glycerol at 35°C
-1
nHE 1.66 nOE 0.422 nNE 0.312 •μE
nHW
nHV 1.64 nOV 0.379 nNV 0.189 J
C
Weight yield, mol.mol-1
Relative abundance
Biomass composition
pA
pM
pG
0.14
1.00
-0.49
JH
1.15
0.36
-0.42
JO
-0.35
-0.97
0.63
JN
-0.31
0.31
0.02
O2
CO2
Spec growth rate
Spec growth rate, h-1
General assumptions
• State variables: structural body mass & reserves
they do not change in composition
• Food is converted into faeces
Assimilates derived from food are added to reserves,
which fuel all other metabolic processes
Three categories of processes:
Assimilation: synthesis of (embryonic) reserves
Dissipation: no synthesis of biomass
Growth: synthesis of structural body mass
Product formation: included in these processes (overheads)
• Basic life stage patterns
dividers (correspond with juvenile stage)
reproducers
embryo (no feeding
initial structural body mass is negligibly small
initial amount of reserves is substantial)
juvenile (feeding, but no reproduction)
adult (feeding & male/female reproduction)
Specific assumptions
• Reserve density hatchling = mother at egg formation
foetuses: embryos unrestricted by energy reserves
• Stage transitions: cumulated investment in maturation > threshold
embryo  juvenile initiates feeding
juvenile  adult initiates reproduction & ceases maturation
• Somatic & maturity maintenance  structure volume
(but some maintenance costs  surface area)
maturity maintenance does not increase
after a given cumulated investment in maturation
• Feeding rate  surface area; fixed food handling time
• Partitioning of reserves should not affect dynamics
comp. body mass does not change at steady state (weak homeostasis)
• Fixed fraction of catabolic energy is spent on
somatic maintenance + growth (-rule)
• Starving individuals: priority to somatic maintenance
do not change reserve dynamics; continue maturation, reproduction.
or change reserve dynamics; cease maturation, reprod.; do or do not shrink in structure
-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
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
Synthesizing units
Generalized enzymes that follow classic enzyme kinetics
E + S  ES  EP  E + P
with two modifications:
• back flux is negligibly small
E + S  ES  EP  E + P
• specification of transformation is on the basis of
arrival fluxes of substrates rather than concentrations
Concentration: problematic
(intracellular) environments: spatially heterogeneous
state variables in dynamic systems
In spatially homogeneous environments:
arrival fluxes  concentrations
Mitochondria
TriCarboxylic Acid cycle
Enzymes pass metabolites directly to other enzymes
enzymes catalizing transformations 5 & 7:
bound to inner membrane (and FAD/FADH2)
Net transformation:
Acetyl-CoA + 3 NAD+ + FAD + GDP 3- + Pi2- + 2 H2O =
2 CO2 + 3 NADH + FADH2 + GTP 4- + 2 H+ + HS-CoA
Dual function of intermediary metabolites
building blocks  energy substrate
Transformations:
1 Oxaloacetate + Acetyl CoA + H2O = Citrate + HSCoA
2 Citrate = cis-Aconitrate + H2O
3 cis-Aconitrate + H2O = Isocitrate
4 Isocitrate + NAD+ = α-Ketoglutarate + CO2 + NADH + H+
5 α-Ketoglutarate + NAD+ + HSCoA = Succinyl CoA + CO2 + NADH + H+
6 Succinyl CoA + GDP 3- + Pi 2- + H+ = Succinate + GTP 4- + HSCoA
7 Succinate + FAD = Fumarate + FADH2
8 Fumarate + H2O = Malate
9 Malate + NAD+ = Oxaloacetate + NADH + H+
all eukaryotes
once possessed
mitochondria,
most still do
Pathways & allocation
structure
structure
maintenance
reserve
maintenance
reserve
structure
maintenance
reserve
Mixture of products &
intermediary metabolites
that is allocated to
maintenance (or growth)
has constant composition
Kooijman & Segel, 2004
Numerical matching for n=4
Product flux
1
2
3
4
Unbounded fraction
0
4
3
2
1
Spec growth rate
Rejected flux
0
1
2
3
Spec growth rate
 = 0.73, 0.67, 0.001, 0.27 handshaking
 = 0.67, 0.91, 0.96, 0.97 binding prob
k = 0.12, 0.19, 0.54, 0.19 dissociation
nSE = 0.032,0.032,0.032,0.032 # in reserve
nSV = 0.045,0.045,0.045,0.045 # in structure
yEV = 1.2 res/struct kE = 0.4 res turnover
jEM = 0.02 maint flux n0E = 0.05 sub in res
Matching pathway  whole cell
No exact match possible between
production of products and intermediary metabolites by pathway
and requirements by the cell
But very close approximation is possible by tuning
abundance parameters nSi E , nSiV
and/or
binding and handshaking parameters ρi , αi
Best approximation requires all four tuning parameters per node
growth-dependent reserve abundance plays a key role in tuning
Kooijman, S. A. L. M. and Segel, L. A. (2004)
How growth affects the fate of cellular substrates.
Bull. Math. Biol. (to appear)
Product Formation
According to
Dynamic Energy Budget theory:
pyruvate, mg/l
Product formation rate =
wA . Assimilation rate +
wM . Maintenance rate +
wG . Growth rate
For pyruvate: wG<0
Applies to all products, heat
& non-limiting substrates
Indirect calorimetry (Lavoisier, 1780):
heat = wO JO + wC JC + wN JN
No reserve:
2-dim basis for product formation
throughput rate, h-1
Glucose-limited growth of Saccharomyces
Data from Schatzmann, 1975
Symbiosis
substrate
product
Product formation is basic to symbioses
Symbiosis
substrate
substrate
Product formation is basic to symbioses
Steps in symbiogenesis
Free-living, homogeneous
Structures merge
Free-living, clustering
Internalization
Reserves merge
Symbiogenesis
• symbioses: fundamental organization of life based on syntrophy
ranges from weak to strong interactions; basis of biodiversity
• symbiogenesis: evolution of eukaryotes (mitochondria, plastids)
• DEB model is closed under symbiogenesis:
it is possible to model symbiogenesis of two initially independently
living populations that follow the DEB rules by incremental changes
of parameter values such that a single population emerges that
again follows the DEB rules
• essential property for models that apply to all organisms
Kooijman, Auger, Poggiale, Kooi 2003
Quantitative steps in symbiogenesis and the evolution of homeostasis
Biological Reviews 78: 435 - 463
Central Metabolism
source
polymers
monomers
waste/source
Modules of central metabolism
• Pentose Phosphate (PP) cycle
glucose-6-P
ribulose-6-P,
NADP
NADPH
• Glycolysis
glucose-6-P
pyruvate
ADP + P
ATP
• TriCarboxcyl Acid (TCA) cycle
pyruvate
CO2
NADP
NADPH
• Respiratory chain
NADPH + O2
NADP + H2O
ADP + P
ATP
Evolution of central metabolism
in prokaryotes (= bacteria)
3.8 Ga
2.7 Ga
i = inverse
ACS = acetyl-CoA Synthase pathwayRC = Respiratory Chain Kooijman, Hengeveld 2003
The symbiontic nature of
PP = Pentose Phosphate cycle
Gly = Glycolysis
metabolic evolution
TCA = TriCarboxylic Acid cycle
Acta Biotheoretica (to appear)
Prokaryotic metabolic evolution
Heterotrophy:
• pentose phosph cycle
• glycolysis
• respiration chain
Phototrophy:
• el. transport chain
• PS I & PS II
• Calvin cycle
Chemolithotrophy
• acetyl-CoA pathway
• inverse TCA cycle
• inverse glycolysis
Symbiogenesis
1.5-2 Ga
1.2 Ga
Sizes of blobs
do not reflect
number of species
Survey of organisms
Myxomycota
Protostelida
Bikont
DHFR-TS gene fusion
loss phagoc.Apusozoa
membr. dyn
unikont
mainly celllose
gap junctions
tissues (nervous)
mitochondria
bicentriolar
primary
mainly chitin
chloroplast
EF1 insertion
secondary
Plasmodiophoromycota
Chlorarachnida
Cercozoa
Cercomonada
chloroplast Amoebozoa
Archamoeba
tertiary
chloroplast
photo
symbionts
Bacteria
Bacteria
Rhizopoda
Sporozoa
Percolozoa
Excavates
Euglenozoa
Loukozoa
AlveoDinozoa
lates
Ciliophora
chloroplasts
Chytridiomycota
cortical alveoli
Actinopoda
(brown algae)
Phaeophyceae
Xanthophyceae
Raphidophyceae
Chrysophyceae
Synurophyceae
Eustigmatophyceae
Labyrinthulomycota
Dictyochophyceae
Bicosoecia
Pedinellophyceae
Pelagophyceae
Bigyromonada
Bacillariophyceae
Pseudofungi
(diatoms)
Bolidophyceae
Opalinata
Prymnesiophyceae
Metamonada
Cryptophyceae
triple roots
Granuloreticulata
forams
Xenophyophora
Basidiomycota
Ascomycota
fungi
Glomeromycota
Zygomycota
Microsporidia
animals
animals
Choanozoa
Composed by
Bas Kooijman
(plants)
Cormophyta
(green algae)
Chlorophyceae
Plantae
(red algae)
Rhodophyceae
Glaucophyceae
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
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
Scaling of 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
slope = 2/3
ectotherms
unicellulars
Length, cm
Intra-species
(Daphnia pulex)
Log weight, g
Inter-species
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
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