synthesizing units for modelling cell physiology bas kooijman dept of theoretical biology vrije...
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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