DEB theory for metabolic organisation

Bas KooijmanDept theoretical biology

Vrije Universiteit AmsterdamBas@bio.vu.nl

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