DEB theory for metabolic organisation
Bas KooijmanDept theoretical biology
Vrije Universiteit [email protected]
http://www.bio.vu.nl/thb
Helsinki, 2010/03/24
Contents
• Preliminary concepts methodology
• Outline of basic theory for a 1-reserve, 1-structure isomorph
• Implications of theory body size scaling relationships
• Evolutionary aspects syntrophy, symbiogenesis
• Population consequences interactions between individuals
>
Dynamic Energy Budget theory
• links levels of organization molecules, cells, individuals, populations, ecosystems scales in space and time: scale separation• interplay between biology, mathematics, physics, chemistry, earth system sciences• framework of general systems theory• quantitative; first principles only equivalent of theoretical physics• fundamental to biology; many practical applications (bio)production, medicine, (eco)toxicity, climate change
for metabolic organization
molecule
cell
individual
population
ecosystem
system earth
time
spac
eSpace-time scales
When changing the space-time scale, new processes will become important other will become less importantIndividuals are special because of straightforward energy/mass balances
Each process has its characteristic domain of space-time scales
Some DEB principles
• life as coupled chemical transformations• life cycle perspective of individual as primary target• energy & mass & time balances• homeostasis• stoichiometric constraints via Synthesizing Units• surface area/ volume relationships• spatial structure & transport• synthrophy (basis for symbioses)• intensive/extensive parameters: scaling• evolutionary perspective
: These gouramis are from the same nest, These gouramis are from the same nest, they have the same age and lived in the same tank they have the same age and lived in the same tankSocial interaction during feeding caused the huge size differenceSocial interaction during feeding caused the huge size differenceAge-based models for growth are bound to fail;Age-based models for growth are bound to fail; growth depends on food intake growth depends on food intake
Not age, but size:Not age, but size:
Trichopsis vittatus
Homeostasis
strong constant composition of pools (reserves/structures) generalized compounds, stoichiometric contraints on synthesis
weak constant composition of biomass during growth in constant environments determines reserve dynamics (in combination with strong homeostasis)
structural
constant relative proportions during growth in constant environments isomorphy .work load allocation
thermal ectothermy homeothermy endothermy
acquisition supply demand systems; development of sensors, behavioural adaptations
the ability to run metabolism independent of the (fluctuating) environment
Flux vs Concentration• concept “concentration” implies spatial homogeneity (at least locally) biomass of constant composition for intracellular compounds• concept “flux” allows spatial heterogeneity• classic enzyme kinetics relate production flux to substrate concentration• Synthesizing Unit kinetics relate production flux to substrate flux• in homogeneous systems: flux conc. (diffusion, convection)• concept “density” resembles “concentration” but no homogeneous mixing at the molecular level density = ratio between two amounts
Synthesizing units
Are enzymes that follow classic enzyme kinetics E + S ES EP E + PWith 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
The concept concentration is problematic in spatially heterogeneous environments, such as inside cellsIn spatially homogeneous environments, arrival fluxes are proportional to concentrations
Interactions of substrates
Surface area/volume interactions• biosphere: thin skin wrapping the earth light from outside, nutrient exchange from inside is across surfaces production (nutrient concentration) volume of environment
• food availability for cows: amount of grass per surface area environ food availability for daphnids: amount of algae per volume environ
• feeding rate surface area; maintenance rate volume (Wallace, 1865)
• many enzymes are only active if linked to membranes (surfaces) substrate and product concentrations linked to volumes change in their concentrations gives local info about cell size ratio of volume and surface area gives a length
Change in body shapeIsomorph: surface area volume2/3
volumetric length = volume1/3
V0-morph: surface area volume0
V1-morph: surface area volume1
Ceratium
Mucor
Merismopedia
Shape correction functionShape correction function
at volume Vactual surface area at volume V
isomorphic surface area at volume V=
1)( VΜ for dVV
V0-morphV1-morph isomorph 0
3/1
3/2
)/()(
)/()(
)/()(
d
d
d
VVV
VVV
VVV
Μ
Μ
Μ
3/13/2
3/13/2
)/(2
2)/(
2)(
)/(3
3)/(
3)(
dd
dd
VVδ
VVδ
δV
VVδ
VVδ
V
Μ
Μ
Static mixtures between V0- and V1-morphs for aspect ratio
V1-morphs are special because• surfaces do not play an explicit role• their population dynamics reduce to an unstructured dynamics; reserve densities of all individuals converge to the same value in homogeneous environments
Mixtures of V0 & V1 morphs
volu
me,
m
3vo
lum
e,
m3
volu
me,
m
3
hyph
al le
ngth
, mm
time, h time, min
time, mintime, min
Fusarium = 0Trinci 1990
Bacillus = 0.2Collins & Richmond 1962
Escherichia = 0.28Kubitschek 1990
Streptococcus = 0.6Mitchison 1961
Biofilms Mixtures of iso- & V0-morphs
Isomorph: V1 = 0
V0-morph: V1 =
mixture between iso- & V0-morph
biomass grows, butsurface area that is involvedin nutrient exchange does not
solid substratebiomass
3/2
1
1)(
d
d
VV
VV
V
VVΜ
Mixtures of changes in shape
Dynamic mixtures between morphs
Lichen Rhizocarpon
V1- V0-morph
V1- iso- V0-morph
outer annulus behaves as a V1-morph, inner part as a V0-morph. Result: diameter increases time
Biomass: reserve(s) + structure(s)Reserve(s), structure(s): generalized compounds, mixtures of proteins, lipids, carbohydrates: fixed composition
Reasons to delineate reserve, distinct from structure• metabolic memory• biomass composition depends on growth rate• explanation of respiration patterns (freshly laid eggs don’t respire) method of indirect calorimetry fluxes are linear sums of assimilation, dissipation and growth fate of metabolites (e.g. conversion into energy vs buiding blocks) inter-species body size scaling relationships
Reserve vs structure
Reserve does not mean: “set apart for later use” compounds in reserve can have active functions
Life span of compounds in• reserve: limited due to turnover of reserve all reserve compounds have the same mean life span
• structure: controlled by somatic maintenance structure compounds can differ in mean life span
Important difference between reserve and structure: no maintenance costs for reserveEmpirical evidence: freshly laid eggs consist of reserve and do not respire
Body size
• length: depends on shape and choice (shape coefficient) volumetric length: cubic root of volume; does not depend on shape contribution of reserve in lengths is usually small use of lengths unavoidable because of role of surfaces and volumes
• weight: wet, dry, ash-free dry contribution of reserve in weights can be substantial easy to measure, but difficult to interpret
• C-moles (number of C-atoms as multiple of number of Avogadro) 1 mol glucose = 6 Cmol glucose useful for mass balances, but destructive measurement
Problem: with reserve and structure, body size becomes bivariateWe have only indirect access to these quantities
StoragePlants store water and carbohydrates,
Animals frequently store lipids
Many reserve materials are less visible
specialized Myrmecocystus
serve as adipose tissue
of the ant colony
Arrhenius relationship
ln r
ate
104 T-1, K-1
reproductionyoung/d
ingestion106 cells/h
growth, d-1
aging, d-1
K 293K; 6400
}exp{)(
1
11
TTT
T
T
TkTk
A
AA
Daphnia magna
Arrhenius relationship
103/T, K-1
ln p
op g
row
th r
ate,
h-1
103/TH 103/TL
r1 = 1.94 h-1
T1 = 310 KTH = 318 KTL = 293 K
TA = 4370 KTAL = 20110 KTAH = 69490 K
}exp{}exp{1
}exp{
)( 11
TT
TT
TT
TT
TT
TT
r
TrAH
H
AH
L
ALAL
AA
Life stages
embryo juvenile adult
fertilization birth puberty deathweaning
baby infant
Essential: switch points, not periods birth: start of feeding puberty: start of allocation to reproductionSwitch points sometimes in reversed order (aphids)
Isomorph with 1 reserve & 1 structure feeds on 1 type of food has 3 life stages (embryo, juvenile, adult)
Processes:
Balances: mass, energy , entropy, time
Standard DEB model
Extensions:• more types of food and food qualities• more types of reserve (autotrophs)• more types of structure (organs, plants)• changes in morphology• different number of life stages
feeding digestionmaintenance
storageproduct formation maturation
growthreproductionaging
1- maturitymaintenance
maturityoffspring
maturationreproduction
Standard DEB model
food faecesassimilation
reserve
feeding defecation
structurestructure
somaticmaintenance
growth
-rule for allocation
Age, d Age, d
Length, mm Length, mm
Cum
# of young
Length, m
mIngestion rate, 105
cells/h
O2 consum
ption,
g/h
• 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 in juveniles?
Respiration Ingestion
Reproduction
Growth:
32 LkvL M2fL
332 )/1( pMM LkfgLkvL
)( LLrLdt
dB
Von Bertalanffy
Embryonic development
time, d time, d
wei
ght,
g
O2 c
onsu
mpt
ion,
ml/h
l
ege
dτ
d
ge
legl
dτ
d
3
3,
3, l
dτ
dJlJJ GOMOO
; : scaled timel : scaled lengthe: scaled reserve densityg: energy investment ratio
Crocodylus johnstoni,Data from Whitehead 1987
yolk
embryo
Growth at constant food
time, dultimate length, mm
leng
th, m
m
Von
Ber
t gro
wth
rat
e -1, d
Von Bertalanffy growth curve:
Reproduction at constant food
length, mm length, mm
103
eggs
103
eggs
Gobius paganellusData Miller, 1961
Rana esculentaData Günther, 1990
Concept overview
• DEB principles • not age, but size
• 5 types of homeostasis
• flux vs concentration Synthesizing Units
• surface area/volume iso-, V0-, V1-morphs shape correction function
• reserve & structure• body size: weight, Cmol, ..
• effects of temperature
• life stages • standard DEB model
Scales of life
Life span
10log aVolume
10log m3earth
whale
bacterium
water molecule
life on earth
whale
bacteriumATP
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 allocation fraction to growth + maint. (intensive) volume-specific maintenance power (intensive) surface area-specific assimilation power (extensive)• conclusion : (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
]/[}{ MAm ppL
}{ Ap
][ Mp
mA Lp }{
Kooijman 1986 Energy budgets can explain body size scaling relationsJ. Theor. Biol. 121: 269-282
Primary scaling relationshipsassimilation {JEAm} max surface-specific assim rate Lm
feeding {b} surface-specific searching rate
digestion yEX yield of reserve on food
growth yVE yield of structure on reserve
mobilization v energy conductance
heating,osmosis {JET} surface-specific somatic maint. costs
turnover,activity [JEM] volume-specific somatic maint. costs
regulation,defence kJ maturity maintenance rate coefficient
allocation partitioning fraction to soma
egg formation R reproduction efficiency
life cycle [MHb] volume-specific maturity at birth
life cycle [MHp] volume-specific maturity at puberty aging
aging ha Weibull aging acceleration Lm
aging sG Gompertz stress coefficientmaximum length Lm = {JEAm} / [JEM]
Kooijman 1986J. Theor. Biol. 121: 269-282
Scaling of metabolic rate
intra-species inter-species
maintenance
growth
weight
nrespiratio3
32
dl
llls
43
32
ldld
lll
EV
h
structure
reserve
32 vll
l0l
0
3lllh
Respiration: contributions from growth and maintenanceWeight: contributions from structure and reserveStructure ; = length; endotherms 3l l
3lllh
0hl
Metabolic rate
Log weight, g
Log metabolic rate,
w
endotherms
ectotherms
unicellulars
slope = 1
slope = 2/3
Length, cm
O2 consum
ption,
l/h
Inter-speciesIntra-species
0.0226 L2 + 0.0185 L3
0.0516 L2.44
2 curves fitted:
(Daphnia pulex)
13/113/1 /3/3/3/3
vkvVkr MMB V
At 25 °C : maint rate coeff kM = 400 a-1
energy conductance v = 0.3 m a-1
25 °CTA = 7 kK
10log ultimate length, mm 10log ultimate length, mm
10lo
g vo
n B
ert
grow
th r
ate
, a-1
)exp()()( 3/13/13/13/1 arVVVaV Bb
3/1V
a
3/1V
3/1bV
1Br
↑
↑0
Von Bertalanffy growth rate
Evolution of DEB systemsvariable structure
composition
strong homeostasisfor structure
delay of use ofinternal substrates
increase ofmaintenance costs
inernalization of maintenance
installation ofmaturation program
strong homeostasisfor reserve
reproductionjuvenile embryo + adult
Kooijman & Troost 2007 Biol Rev, 82, 1-30
54321
specialization of structure
7
8
an
ima
ls
6
pro
ka
ryo
tes
9plants
Evolution of central metabolism
i = inverseACS = acetyl-CoA Synthase pathway PP = Pentose Phosphate cycleTCA = TriCarboxylic Acid cycle
RC = Respiratory Chain Gly = Glycolysis
Kooijman & Hengeveld 2005 In: Reydon & Hemerik (eds): Current Themes in theor. biol.. Springer
in prokaryotes (= bacteria)3.8 Ga 2.7 Ga
Prokaryotic metabolic evolution
Chemolithotrophy • acetyl-CoA pathway• inverse TCA cycle• inverse glycolysis
Phototrophy:• el. transport chain• PS I & PS II• Calvin cycle
Heterotrophy:• pentose phosph cycle• glycolysis• respiration chain
Symbiogenesis2.7 Ga 2.1 Ga 1.27 Ga
phagocytosis
Resource dynamicsTypical approach
Resource dynamics
Nutrient
Resource dynamics
Nutrient
Producer/consumer dynamics
PnCnNPm
ChrCdt
d
CjPrPdt
d
NPNCN
C
PAP
)(
PK
jj
my
kr PAm
PANNP
NP /1
;1
CNCPCNCPC rrrrr
1111
MNPANCNCNMPPACPCP kjmyrkjyr ;
producer
consumer
nutr reserveof producer
: total nutrient in closed system
N
h: hazard rate
CPCCN rry special case: consumer is not nutrient limited
spec growthof consumer
Kooijman et al 2004 Ecology, 85, 1230-1243
Producer/consumer dynamicsConsumer nutrient limited
Consumer notnutrient limited
Hopf bifurcation
Hopf bifurcation
tangent bifurcation
transcritical bifurcation
homoclinicbifurcation
Producer/Consumer DynamicsDeterministic model
Stochastic model
in closed homogeneous system
Producer/Consumer Dynamics
0 2 4 6 8
0
10
20
con
sum
ers
nutrient
1.75 2.3 2.4
2.5
2.7
3.0
1.23
1.15
1.0
2.81.231.53
tang
ent
focu
s
Hop
f
Bifurcation diagram
isoclines
Food chains n=2
time, h time, h
glucose
Escherichia coli
Dictyostelium
mg/
ml
mm
3 /m
lm
m3 /
ml
cell
vol
, m
3ce
ll v
ol,
m3
X0(0) 0.433 mg. ml-1
X1(0) 0.361 X2(0) 0.084 mm3.ml-1
e1(0) 1 e2(0) 1 -
XK1 0.40 XK2 0.18
g1 0.86 g2 4.43 -
kM1 0.008 kM2 0.16 h-1
kE1 0.67 kE2 2.05 h-1
jXm1 0.65 jXm2 0.26
ml
mm,
ml
g 3μ
13
h,hmm
mg
Data from Dent et al 1976h = 0.064 h-1, Xr = 1mg ml-1, 25 °C
Kooijman & Kooi,1996 Nonlin. World 3: 77 - 83
Canonical communityShort time scale:Mass recycling in a community closed for mass open for energy
Long time scale:Nutrients leaks and influxes
Memory is controlled by life span (links to body size)Spatial coherence is controlled by transport (links to body size)
DEB tele course 2011http://www.bio.vu.nl/thb/deb/
Free of financial costs; some 250 h effort investment
Program for 2011: Feb/Mar general theory (5w) April course + symposium in Lisbon (2w + 3 d) Target audience: PhD students
We encourage participation in groups who 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/2010/03/30: 10 h DEB video-course by Roger Nisbet
Cambridge Univ Press 2009
Audience: thank you for your attention
Top Related