Synthesizing Unitsfor modelling cell physiology

Bas KooijmanDept of Theoretical Biology

Vrije Universiteit, Amsterdamhttp://www.bio.vu.nl/thb/deb/

Leiden, 2004/06/24

adul

t

embryo

juvenile

Research program:Dynamic Energy Budget

theory

Weird world at small scaleAlmost all transformations in cells are enzyme mediatedClassic enzyme kinetics: based on chemical kinetics (industrial enzymes)• diffusion/convection• larger number of molecules• constant reactor volume• law of mass action: transformation rate product of conc. of substrates

Problematic application in cellular metabolism:• definition of concentration (compartments, moving organelles) • transport mechanisms (proteins with address labels, targetting, allocation) • crowding (presence of many macro-molecules that do not partake in transformation)• intrinsic stochasticity due to small numbers of molecules• liquid crystalline properties • surface area - volume relationships: membrane-cytoplasm; polymer-liquid• connectivity (many metabolites are energy substrate & building block; dilution by growth)

Alternative approach: reconstruction of transformation kinetics on the basis of cellular input/output kinetics

Self-ionization of water in cells

A cell of volume 0.25 m3

and pH 7 at 25°C hasn = 15 protons N = 8 109 water molecules

confidence intervals of pH 95, 90, 80, 60 %

pH

cell volume, m3

modified Bessel function

7

Diffusion cannot occur in cells

Crowding affects transport

cytoskeletal polymers

ribosomes

nucleic acids

proteins

ATP generation & use5 106 ATP molecules in bacterial cell enough for 2 s of biosynthetic work

Only used if energy generating & energy demanding transformations are at different site/time

If ADP/ATP ratio varies, then rates of generation & use varies, but not necessarily the rates of transformations they drive

Processes that are not much faster than cell cycle, should be linked to large slow pools of metabolites, not to small fast pools

DEB theory uses reserve as large slow pool for driving metabolism

Yield vs growth

1/spec growth rate, 1/h

1/yi

eld,

mm

ol g

luco

se/

mg

cells

Streptococcus bovis, Russell & Baldwin (1979)

Marr-Pirt (no reserve)DEB

spec growth rate

yield

Russell & Cook (1995): this is evidence for down-regulation of maintenance at low growth ratesDEB theory: high reserve density gives high growth rates structure requires maintenance, reserves not

Methanotrophy

NWOWHW nnnWX3NX2OX2CX4 NOCH Y NH Y O Y CO Y CH

AC Assim (catabolic) -1 1 2 -2 0 0 0

AA Assim (anabolic) -1 0 1 0

M Maintenance 0 1 -1 0

GC Growth (catabolic) 0 1 -1 0

GA Growth (anabolic) 0 0 -1 1

C Carbon 1 1 0 0 0 1 1

H Hydrogen 4 0 2 0 3

O Oxygen 0 2 1 2 0

N Nitrogen 0 0 0 0 1 2/2/2/

2/32/2/

2/2/1

2/2/3

2/2/

2/2/32

From

GHEOVOE

GOE

GNEHVHE

GHE

NVNEG

NE

MHEOE

MOE

HENEM

HE

OEA

HXA

OX

HEA

NXA

HX

NEA

NX

YnnY

YnnY

nnY

YnY

nnY

nYY

nYY

nY

nY0A

HXY AOXY A

NXYM

HEY

GHEY

MHEY

MOEYM

OEYG

OEY GNEY

NEn

NEn

HEn

OEn

NEn

HVn

OVn

NVn

sym

bol

proc

ess

X: m

etha

ne

C: c

arbo

n di

oxid

e

H: w

ater

O: d

ioxy

gen

N: a

mm

onia

E: r

eser

ve

V: s

truc

ture

EAXE jy )1(

EAj

EGVE jy )1(

EGVE jy

EMj

EVE

EMEEVV

EVEG

MEVEM

EAmEA

ym

jkmM

dt

dMr

ryj

kyjXK

Xjj

1

reserve density mE = ME/MV

rate

Yie

ld c

oeff

icie

ntsT

Che

mic

al in

dice

s

Macroscopic transformation (variable yield coefficients and indices):

Microscopic transformations (constant coefficients and indices):

Methanotrophy

spec growth rate, h-1 spec growth rate, h-1

X/O

N/O

C/O

flux

rat

io, m

ol.m

ol-1

spec

flu

x, m

ol.m

ol-1.h

-1

CE

N

X

O

X: methaneC: carbon dioxideO: dioxygenN: ammoniaE: reserve

jEAm = 1.2 mol.mol-1.h-1

yEX = 0.8yVE = 0.8kM = 0.01 h-1

kE = 2 h-1

nHE = 1.8nOE = 0.3nNE = 0.3

nHV = 1.8nOV = 0.3nNV = 0.3

chemical indices

Kooijman et al, 2004Ecology, 85, 1230-1243

Enzyme kinetics A+BC

ABCBAABBBAAAB

BAABABBABAB

ABBABAAABBA

BBAABBAAABC

θkkkθXbθXbθdt

d

θXbkθXbθkθdt

d

θXbkθXbθkθdt

d

θXbXbθkθkθkθdt

d

)(

)(

)(

)(

..

...

...

......

0

0

0

1

10

10

10

11111

.

.

..

CAAAB

AAAB

ABAA

AB

B

A

ρρρxx

ρρxx

ρxρx

θ

θ

θ

θ

;/;/

/;/

BCCBAA

BBBBAAAA

kkρkkρ

kbXxkbXx

ABCmCC θJJXdt

d

CBA XXX ,, : conc of compounds A,B,C

ABBA θθθθ ,,, .... : fractions of bounded enzymes ABBA θθθθ ....1

Cm

C

BAAB

BABA

BA

AB

B

A

BABA

CmC

C

BBB

C

AAABA

J

J

xxxx

xxxx

xx

θ

θ

θ

θ

xxxx

JJ

k

bXx

k

bXxkk

1

1

;;0,

11

11

1

.

.

..

111

Cm

C

A

B

BA

AB

B

A

BA

CmC

B

BBB

A

AAA

B

B

A

ABABA

J

J

x

x

xx

θ

θ

θ

θ

xx

JJ

k

bXx

k

bXx

k

b

k

bbbkk

1

11;;

,;,,,

1

1

11

.

.

..

11

constant

Syn

thes

izin

g U

nit

Rej

ecti

on U

nit

CBA kkk ,, : dissociation rates : association ratesBA bb ,

0,

forRU,

SU,

BA

C

C

XX

J

J

Isoclines for rate A+BC

.2 .2.4 .4

.6 .6.8

Conc A Conc A

Con

c B

Synthesizing Unit Rejection Unit

almost singlesubstr limitationat low conc’s

Synthesizing units

Generalized enzymes that process generalized substrates and 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 In spatially homogeneous environments: arrival fluxes concentrations

Simultaneous Substrate Processing

Chemical reaction: 1A + 1B 1CPoisson arrival events for molecules A and B

blocked time intervals

• acceptation event¤ rejection event

11111 BABACmC JJJJJJFlux of C:

production

production

Kooijman, 1998Biophys Chem73: 179-188

SU kinetics: n1X1+n2X2X

0 tb tc

time

productrelease

productrelease

binding prod.

cycle

112

1

2

1

1

1

1

0

1

01

21

21

2

2

1

1

0

1

0

110

1

1

1 2

)(!!

)!()(

fn gamma incomplete !/}exp{1),(

),()(pdf)(1)(1

:substrates For .,/1

i i

i

i i

iXmX

n

i

n

jji

ji

tb

n

j

j

n

iii

n

i

t

t

n

itt

bXmccX

n

J

n

JJJ

JJ

JJ

ji

ji

J

n

J

ndttSt

jsssnP

tJnPdsstStS

ntJttJ

b

bibib

E

EEE

)(tSt

JJ

bt

b

Xm

X

E

flux of Xmax flux of XExpected value of tb

Survivor function of tb

Period between subsequent arrivals is exponentially distributedSum of exponentially distributed vars is gamma distributed

Production flux not very sensitive for details of stoichiometryStoichiometry mainly affects arrival rates

Kooijman, 1998Biophys Chem73: 179-188

Simultaneous Nutrient Limitation

Specific growth rate of Pavlova lutheri as function of intracellular phosphorus and vitamin B12 at 20 ºC

Data from Droop 1974Note the absence of high contents for both compounds

due to damming up of reserves, andlow contents in structure (at zero growth)

P content, fmol/cell

B12 content,

10 -21 mol/cell

Kooijman, 1998Biophys Chem73: 179-188

Reserve interactions 5.2.4

Spec growth rate, d-1 Spec growth rate, d-1 Spec growth rate, d-1

P-c

onte

nt, f

mol

.cel

l-1P

-con

c, μ

M

B12

-con

c, p

M

B12

-con

t., 1

0-21 .m

ol.c

ell-1

P Vitamin B12

kE 1.19 1.22 d-1

yXV 0.39 10-

15

2.35 mol.cell-1

jEAm 4.91 10-

21

76.6 10-15 mol.cell-1. d-1

κE 0.69 0.96

kM 0.0079 0.135 d-1

K 0.017 0.12 pM, μM

Data from Droop 1974 on Pavlova lutheri

P(μM) B12(pM)

1.44 68

14.4 6.8

1.44 20.4

1.44 6.8

C,N,P-limitation

Nannochloropsis gaditana (Eugstimatophyta) in sea waterData from Carmen Garrido PerezReductions by factor 1/3 starting from 24.7 mM NO3, 1.99 mM PO4

N,P reductions

N reductions

P reductions

C,N,P-limitationNannochloropsis gaditana in sea water

CkJXjrnmrkCdt

dCCCCVCE )(

XjrnmrkPdt

dPPVPE )(

rXXdt

d

iEii mkjmdt

d

PNCPNPCNCPNC rrrrrrrrrrrrr

11111111

iK

jj

i

imi /1

iiEV

Ei m

y

rkr

For PNCi ,, XjrnmrkN

dt

dNNVNE )(

DIC

nitrate

phosphate

res. dens.

structure

uptake rate

spec growth rate

spec growth

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 dynamics

Consumer nutrient limited

Consumer notnutrient limited

Hopf bifurcation

Hopf bifurcation

tangent bifurcation

transcritical bifurcation

homoclinicbifurcation

Interactions of substrates

Kooijman, 2001Phil Trans R Soc B356: 331-349

Photosynthesis2 H2O + 4 h O2 + 4 H+ + 4 e-

CO2 + 4 H+ + 4 e- CH2O + H2O

CO2 + H2O + light CH2O + O2

3222

32

NHOOHCONOCH

NOOCH

ENOEHECEnnn

ENEC

HNEOEHE

OH

yyyy

yy

no synthesis ofhydrocarbons at

compensation point

PhotorespirationRuP2 ribulose 1,5-biphosphate

(C5 + C 2 C3)

(C5 C3 + C2 )

Transformations are catalized by Rubisco, which evolved in anaerobic environmentsO2 competes with CO2

which gives an oxidation, rather than a reduction

Co-metabolismConsider coupled transformations A C and B DBinding probability of B to free SU differs from that to SU-A complex

Co-metabolismCo-metabolic degradation of 3-chloroaniline by Rhodococcus with glucose as primary substrateData from Schukat et al, 1983

Brandt et al, 2003Water Research37, 4843-4854

Co-metabolismCo-metabolic anearobic degradation of citrate by E. coli with glucose as primary substrateData from Lütgens and Gottschalk, 1980

Brandt et al, 2003Water Research37, 4843-4854

Adaptation

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

Inhibition

))/1/()((

)(

1

)(

'''

'

**.

....

...'

.

.'

...'

.

..''

....

kjjkkj

kkjyj

θθkyj

θθθθ

θkθkθjθdt

d

θjθkθkθjθdt

d

θjjθkθkθdt

d

BAABB

BAACAC

ABAACAC

ABBA

BBABABB

ABAAABBAA

BABBAA

A does not affect B in yACAC; B inhibits binding of A

CA

A

A

A

ykjρθ..

BA

AAA

kkkjρj

'

unbounded fractionbinding prob of Aarrival rate of Adissociation rate of Ayield of C on A

A inhibits binding of B in yACAC; B inhibits binding of A

BAABBA

BAACAC

AACAC

ABBA

BAABABB

ABABBAA

BABBAA

kkkjkj

kkjyj

θkyj

θθθθ

θjθkθjθdt

d

θjθkθjθdt

d

θjjθkθkθdt

d

''

'

*.

....

.'

..'

.

.'

..'

.

..''

....

1

)(

Adaptation

)(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

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

Diauxic growthbi

omas

s co

nc.,

OD

590

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

Social inhibition of x esequential parallel

dilution rate

subs

trat

e co

nc.

biom

ass

conc

.

No

soci

aliz

atio

n

Implications: stable co-existence of competing species “survival of the fittest”? absence of paradox of enrichment

x substratee reservey species 1z species 2

Collaboration:Van Voorn, Gross, Feudel, Kooi, Kooijman

Aggressive competitionV structure; E reserve; M maintenance substrate priority E M; posteriority V MJE flux mobilized from reserve specified by DEB theoryJV flux mobilized from structure amount of structure (part of maint.) excess returns to structurekV dissociation rate SU-V complex kE dissociation rate SU-E complex kV kE depend on such that kM = yMEkE(E. + EV)+yMVkV .V is constant

J EM,

J VM

J EM,

J VM

JE

kV = kE

kV < kE

Collaboration:Tolla, Poggiale, Auger, Kooi, Kooijman