Dynamic Energy Budget theory

1 Basic Concepts 2 Standard DEB model 3 Metabolism 4 Univariate DEB models 5 Multivariate DEB models 6 Effects of compounds 7 Extensions of DEB models 8 Co-variation of par values 9 Living together10 Evolution11 Evaluation

Linear pathway 7.1.1

SUi SUi+1

FX ij ,1 PX i

j , FX ij , PX i

j ,1

RX ij ,RX i

j ,1

iXPiXXi PyXyXiiii 111

1

11

0 1100with

i

j XXXPXP

n

iiXP jjiiii

yyyPyX

: Product i : Intermediary metabolite iiP iX

=

If all metabolites would follow the full pathway:

Chain of length 1 7.1.1a

111

111

11111

11111

10

0

11

10

10

ρ1

ρ)θ1(

ρ1θ

/ρθ)θ1(θ

kmj

jkmj

kmj

mjkdt

d

SFX

FXSPX

SFX

SFXChange inUnbounded fractionSteady stateUnbounded fraction

Production flux

Closed handshaking at all nodes 7.1.1b

j

kFXiiiSPX

n

inj

j

kjSjFXi

SFXnnn

SFXiiii

SFX

jkmj

kmj

mjkdt

d

mjkdt

d

mjkdt

d

ii

nn

ii

1k1

11

k1

n

i1

11121

ρ)θθ(

ρ1θ

/ρ)θ1(θ

/ρ)θθ(θ

/ρ)θθ(θ

0

0

1

1

10

Change inUnbounded fractions

Steady stateUnbounded fractions

Production fluxes

1.,,2for ni

1.,,1; ni

Open handshaking at all nodes 7.1.1c

1i

i

11i

i

1

1

1

1

ρ1

ρ)θ1(

ρ1θ

/ρθ)θ1(θ

iSFX

FXiiSPX

iSFXi

SFXiiii

kmj

jkmj

kmj

mjkdt

d

ii

i

ii

ii

ii

Change inUnbounded fractions

Steady stateUnbounded fractions

Production fluxes

ni .,,1For

General handshaking 7.1.1d

PXRX

iiiiPXRX

SXXnnFXPX

SXXiiiiFXPX

SFXnn-n

SFXnn-nn

SFXi

SFXiii

SFXnn-nnnn

SFXiiiiiiii

nn

ii

nnnnn

iiiii

nn

nn

ii

ii

nn

ii

jjjj

mykjjmykjj

mjk

mjkmjk

mjk

mjkdt

d

mjkdt

d

,,

111,,

,,

1,,

,1

,1

i1-i

i1-i1ii

,n1

,1-1i

)ρ))θ1(αθ(1(

)θ1()θ)θ1(α1(

/ρ)α1(

/ραθ

/ρ)α1(

/ρα)θαα1(θ

/ρ))θ1(αθ()θ1(θ

/ρ))θ1(αθ()θ)θ1(α1(θ

1

1

1

1

1

1

1

1Change inUnbounded fractions

Steady stateUnbounded fractions

Production fluxes

Rejection fluxes

1.,,1For ni

Trans-stage feeding strategies 7.2

Juvenile mayflies (Emphemeroptera) cover all adult needs for food; Adult’s digestive system is filled with air

Adult Anoplius viaticus collects all food (here Trochosa terricola) for the juvenile stage of the next generation; the adult feeds on nectar

Food deposits 7.2.1

Melanerpes formicivorus (acorn woodpecker)stores acorns & beech-mast in crevices (upto 60000)from hundreds of km around its residence

Farming: external storage 7.2.1a

Ambrosia beetles garden fungi under tree bark, and some have special structures in their head,

mycangia, to transport the fungus

The gardener-nurse caste of leafcutter ants garden fungi under ground, transplanting them onto fresh substrate and weeding out wrong species of fungus

The social amoeba Dictyostelium carries

bacteria in their multi-cellularslug-stage and inoculate fresh stubstrate with them

Fast/slow substrate uptake 7.2.2

DEB-consistent variant of Morel 1987• uptake depends on substrate concentration and reserve density• reserve mobilization independent of uptake

Not yet tested against experimental data

Short vs long-term nutrient uptake 7.2.2a

Morel 1987 on nutrient uptake in algae

Variant that is consistent with DEB/Droop

Satiation-driven feeding 7.2.3

Suppose that satiation s is a system variable which• increases instantaneously with an amount sX

upon feeding on a food particle • decays exponentially at rate kS during starvationFeeding occurs when a food particle arrives, while satiation s(t) < 1The resulting feeding rate is numerically well described by

},min{5/1;

)1ln(;

5/44)(

2

5/4

,,,

,,

,,2

,,,

FmXXFXXFmXX

FmXXFXX

X

SFmX

X

XFmXFmXXFmXX

XFX

JJJsJJ

JJJs

s

kJ

s

JJJJJJ

sJ

Doucet, 2003 The H-response:A satiation-driven functional response (not published)

Extendable to more food types, preferences are set by type-specific satiation thresholds

Functional surface area 7.2.5

Heliozoans and foraminiferans have threat-like extensions of protoplasm on which they take up food particles

functional surface area depends on feeding rate relative to ratio of moving rate and thickness of stagnant water mantle

Diffusion limitation 7.2.5b

distance from membrane

subs

trat

e co

ncen

trat

ion

0

20

0

0,,

42

1

KXX

XKXXX

XK

XJJ

c

cc

FmXFX

Uptake rate:

0,KK saturation constants

0, XX substrate conc., - at membr

FmXJ , max uptake rate

XXKK 00

0

XXKK 00 ,generally:

mantle thickness

External digestion 7.2.6

Yie

ld m

etab

on

enzy

me

time, h

conc

enz

yme,

met

ab

distance from cell, mm distance from cell, mm

solitary feedingsocial feeding

social

feedingstrategy

solitary

intracellular

metabprofilet = ∞

Moving gut 7.3

From: Mader, S. S. 1993 Biology, WCBSleigh, M. 1989 Protozoa, E. Arnold, London

Feeding vacuoles of ciliates travel from the cell mouth to the cell anus, where the feaces is excreted. These vacuoles can be considered as a moving gut.

Paramecium, Tetrahymena (Ciliophora)

Digestive system 7.3a

time

inpu

t, ou

tput

time

inpu

t, ou

tput

time

inpu

t, ou

tput

completely stirred reactor

plugflow reactor

both reactors in series

stomach model

gut model

Stomach• good in buffering• residence times exponentially distributed many short times, few large ones

Gut• bad in buffering• residence time constant digestion requires some time

Mitochondria 7.6

Transformations:1 Oxaloacetate + Acetyl CoA + H2O = Citrate + HSCoA2 Citrate = cis-Aconitrate + H2O3 cis-Aconitrate + H2O = Isocitrate4 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 = Malate9 Malate + NAD+ = Oxaloacetate + NADH + H+

TriCarboxylic Acid cycleEnzymes pass metabolites directly to other enzymesEnzymes catalizing transformations 5 and 7 are linked to the inner membrane (and FAD/FADH2)All enzymes are linked into a metabolonNet transformation: Acetyl CoA + 3 NAD+ + FAD + GDP 3- + Pi

2- + 2 H2O = 2 CO2 + 3 NADH + FADH2 + GTP 4- + 2 H+ + HSCoA

Dilemma of pathway enzymes 7.6b

Many metabolites have a dual function: • building block for synthesis of functional units• substrate to generation of energy (ATP) or reducing power (NADH, NADPH)

Cell’s need for building blocks depends on variable growth rate

Problem:How is cell’s need “known” by enzyme molecules of pathway?

Example of competing needs: growth versus maintenance

Kooijman & Segel 2005 How growth affects the fate of cellular substrates. Bull Math Biol 67: 57-77

Pathway whole cell 7.6c

GiXP

n

iiXX

n

i

MiXP

n

iiXX

n

i

XPyXy

XPyXy

GiGi

MiMi

111

111

1

1

Fixed stoichiometries for maintenance and growth

Variable overall stoichiometry

GXXMXX XYXYXGM 000

)/(1;

/10

0

0

0 ryj

yY

jry

yY

EVEM

XXXX

EMEV

XXXX

G

G

M

M

Can a model for pathway kinetics give this result?

Constraints on pathway dynamics 7.6d

ryjyjryjyj

PVXME

PEXPX

VXMEEXRX

iii

iii

,,

,,

Amount of SU i per mol of structure:

)( ,, 00ryjnj EVMEEXFX

Supply flux to pathway:

VSEESS iiinmnm

VSES

EX

E

VXEX

ME

PXRX

i

i

ii

ii

ii

nnnm

yyrj

jjSX

,

,

,,

0

,

,

metabolite ienzyme irejection, production fluxspec maintenance fluxspec growth rateyield coefficients (fixed)reserve density ME/MV

abundance of X0 in Eabundance of Si in E, V

Requirements by cell for maintenance and growth:

Pathways & allocation 7.6e

rese

rve

rese

rvere

serve

ma

inten

ance

ma

inten

ance

ma

inten

ance

structure structure

structure

Mixture of products &intermediary metabolites

that is allocated tomaintenance (or growth)has constant composition

Kooijman & Segel 2005

Diauxic growth 7.9.4a

time, h

biom

ass

conc

., O

D43

3

acetate

oxalate

Sub

stra

te c

onc.

, mM

Growth of acetate-adapted Pseudomonas oxalaticus OX1data from Dijkhuizen et al 1980

SU-based DEB curves fitted by Bernd Brandt

Adaptation todifferent substratesis controlled by:

enzyme turnover 0.15 h-1

preference ratio 0.5

cells

Brandt, 2002PhD thesisVU, Amsterdam

Numerical matching for n=4 7.6f

Pro

duct

flux

Re

ject

ed fl

ux

Un

bou

nd

fra

ctio

n = 0.73, 0.67, 0.001, 0.27 handshaking = 0.67, 0.91, 0.96, 0.97 binding probk = 0.12, 0.19, 0.54, 0.19 dissociation nSE = 0.032,0.032,0.032,0.032 # in reservenSV = 0.045,0.045,0.045,0.045 # in structureyEV = 1.2 res/struct kE = 0.4 res turnover jEM = 0.02 maint flux n0E = 0.05 sub in res

0

0

1

1

1

2

2

23

3

3

4

4

Spec growth rate

Spec growth rate

Matching pathway whole cell 7.6g

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 and/or binding and handshaking parameters

Good approximation requires all four tuning parameters per node growth-dependent reserve abundance plays a key role in tuning

VSES iinn ,

ii αρ ,

Kooijman and Segel 2005

Aging in adult insects 7.8.1

age after eclosion, d age after eclosion, d age after eclosion, d

surv

ivin

g nu

mbe

r

surv

ivin

g nu

mbe

r

# of

egg

s/be

etle

, d-1

Drosophila melanogaster Notiophilus biguttatus

Data: Rose 1984Data: Ernsting & Isaaks, 1991

High food, 20/10 °C 0.63 a-2

High food, 10 °C 0.547 a-2

Low food, 20/10 °C 0.374 a-2

:

)(2

)(

30

0 0 12230

21

glκ

eh

dtdttRglκ

ehkh

tth

R

a

t t

R

aMa

survival based onobserved reproductionNo growth

R

glκ

ehkh

tth

R

aMa 3

02

2)(

initialrandom

mort

WeibullModel=3

Anchovy Engraulis encrasicolus 7.8.2

time, d

leng

th,

cm 0.16 cm 0.22 cm

0.4 cm

0.9 cm

1.2 cm

>4 cm

embryo

Pecquerie 2008PhD thesis VU A’dam

Adaptation 7.9.3

glucose, mg/l glucose, mg/l

spec

ific

grow

th r

ate,

h-1

“wild type”Schulze & Lipe, 1964

glucose-adaptedSenn, 1989

Glucose-limited growth of Escherichia coli

70 mg/l 0.06 mg/l

max

.5 max

many types of carriers only carriers for glucose

Adaptation 7.9.4

)(1)()0(1)0(

)(

)(

;

;

;

tκtκκκ

κfwκfκ

fwκhrκ

dt

d

κfwκfκ

fκhrκ

dt

d

KS

SfXfκYrS

dt

dKS

SfXfκYrS

dt

d

rfκrfκrrXXdt

d

ABAB

BBBAA

BBB

ABBAA

AAA

BB

BBBBBXBmB

AA

AAAAAXAmA

BmBBAmAA

Batch culture, Monod special case Model elements:• uptake of substrate by specific carriers• carrier densities nA and nB

• metabolic signals from uptake fini

• relative signal sA = pA fAnA/i pi fini

• carrier production by SUs that are fed by relative signals that inhibit reciprocally• carriers have a common turnover rate

Result:Expression fraction 0 asymptotically in absence of substrate

Xi

i

i

i

YKfSX

whκrr

i

im

biomass densitysubstrate i concscaled func responsesaturation coeff for iyield of biom on substr

spec growth ratemax spec growth rate on iexpression fraction for icarrier turnover ratepreference ratio

Brandt et al, 2004Water Research,38, 1003 - 1013

Diauxic growth 7.9.4b

biom

ass

conc

., O

D59

0

Growth of succinate-adapted Azospirillum brasilenseintracellular amounts followed with radio labels

data from Mukherjee & Ghosh 1987

Adaptation todifferent substratesis controlled by:

enzyme turnover 0.7 h-1

preference ratio 0.8

time, h

fruc

tose

con

c, m

M

succ

inat

e co

nc, m

M

succinate

fructose cells

suc in cells

fruc in cells

Brandt, 2002PhD thesisVU, Amsterdam

Adaptation in degradation 7.9.4.c

time, h time, h

Kooijman & Troost 2007Biol Rev 82: 1-30

conc

entr

atio

n, g

/l

conc

entr

atio

n, g

/l

-- E. coli-- fumarate-- pyruvate

-- E. coli-- fumarate-- glucose

Active pyruvate carriers suppress expression of fumarate carriers

Active glucose carriers don’t suppress expression of fumarate carriers

Dynamic Energy Budget theory

1 Basic Concepts 2 Standard DEB model 3 Metabolism 4 Univariate DEB models 5 Multivariate DEB models 6 Effects of compounds 7 Extensions of DEB models 8 Co-variation of par values 9 Living together10 Evolution11 Evaluation