Heijnen_2010.

Heijnen_2010.
Adv Biochem Engin/Biotechnol
DOI: 10.1007/10_2009_63
Impact of Thermodynamic Principles
in Systems Biology
J.J. Heijnen
Abstract It is shown that properties of biological systems which are relevant for
systems biology motivated mathematical modelling are strongly shaped by general
thermodynamic principles such as osmotic limit, Gibbs energy dissipation, near
equilibria and thermodynamic driving force. Each of these aspects will be demonstrated both theoretically and experimentally.
Keywords Black box kinetics, Gene regulation, Lin-log kinetics, Metabolic networks, Model reduction, Pseudo-steady state, Stoichiometry, Systems biology,
Thermodynamics
Contents
1
2
3
4
5
6
7
Introduction
Thermodynamic Principles in Mathematical Models of Biological Systems
The Osmotic Limit Dictates Low Concentrations of Intracellular Metabolites
Consequences of Low Metabolite Concentrations from a Systems Biology Point of View
Thermodynamic Approach to Obtain Network Stoichiometry and Fluxes
5.1 Thermodynamic Approach to Stoichiometry
5.2 Thermodynamic Approach for Maximal Growth Rate, mmax
Prediction of Gene Regulation of Enzymes Using Energy Optimality
Thermodynamics Based Model Reduction in System Biology
7.1 Black Box Models for Design of Biotechnological Processes: From Complexity to
Simplicity Due to Pseudo-Steady State Coupling
7.2 Metabolic Reaction Network Models to Redesign Organisms: Reducing Complexity
J.J. Heijnen
Department Biotechnology, Bioprocess technology, Delft University of Technology, Julianalaan
67, 2628 BC Delft, The Netherlands
e-mail: [email protected]
# Springer‐Verlag Berlin Heidelberg 2010
J.J. Heijnen
8
Thermodynamics Inspired Kinetics: Lin-Log Kinetics
8.1 Introduction
8.2 In Silico Studies with Lin-Log Kinetics
8.3 Application of Lin-Log Kinetics to Experimental Data
9 Conclusion
References
1 Introduction
Thermodynamic principles apply to all physical and chemical systems, including
biological systems. In this chapter it will be shown how these principles shape
their properties, especially from a quantitative model based, systems biology, point
of view.
2 Thermodynamic Principles in Mathematical Models
of Biological Systems
The key aspect of a living cell is the formation of new cells, called growth.
Growth requires that a cell produce each of the molecules present in the newly
formed cells. This occurs in a large and complex metabolic (reaction) network. This
network is composed of many reactions, which consume and produce small molecules, called metabolites. Each reaction is catalysed by a specific enzyme, which is
under genetic/environmental control. Prediction of growth requires a mathematical
model of this network for which the fundamental equations are the mass balances of
intracellular metabolites. In vector notation:
dX
¼ SV ðe; X; pÞ mX;
dt
(1)
where X is the vector containing the individual intracellular metabolites Xj .
V is the vector of the rates of enzyme catalyse reactions, with vi the rate of
reaction catalysed by enzyme present at an amount ei . The rate vi depends on the
amount of enzyme present, ei, on the kinetic effect of metabolites Xj involved (e.g.
substrate, product and possible allosteric effectors) and the parameters p (e.g.
V max , affinities, Hill coefficient etc.). S is the so-called stoichiometric matrix
which represents the structure of the reaction network. Its rows represent the
metabolites, its columns the reactions [44]. The term mX is the so-called dilution
term.
Equation (1) requires information on the dynamic behaviour and values of
metabolite concentrations, on the values of stoichiometric coefficients, on enzyme
levels resulting from genetic regulation and on the shape/algebraic nature of the
Impact of Thermodynamic Principles in Systems Biology
Van’t Hoff
Osmotic limit
Energy optimality
for gene regulation
dx
= S · v(e,x,p) –μ · x
dt
Model
reduction
Gibbs energy
dissipation in
the network
Thermodynamic inspired,
lin-log kinetics
Fig. 1 Impact of thermodynamic principles in systems biology
enzyme kinetic relations, and (1) is the basis of parameter estimation from experimental data and the associated need for model reduction.
In the following sections we will show that thermodynamic principles can be
used to shed light on this information (Fig. 1):
l
l
l
l
l
Metabolite concentration levels ðXj Þ and their control mechanisms
The stoichiometry of growth (S)
The genetic regulation of enzyme levels (ei )
Principles of model reduction
The kinetics of enzyme catalysed reactions based on thermodynamic driving
force
3 The Osmotic Limit Dictates Low Concentrations
of Intracellular Metabolites
Cells have a genome which contains about 5,000 genes. These genes code for about
5,000 proteins, of which about 2,000 are enzymes. Therefore, in a cell, one can
expect about 2,000 different metabolites which are small molecules (e.g. metabolites in central metabolism and in pathways for amino acid, nucleotide, lipid and
carbohydrate/cell wall synthesis). Many of these metabolites are negatively charged
(having phosphate and carboxylate groups) and therefore there are also considerable concentrations of counter cations ðKþ ; Mg2þ Þ. The sum concentration of all
these small molecules is limited by a thermodynamic property called osmotic
pressure [1]. Because cells contain a cell membrane that is water permeable, the
presence of a high intracellular sum concentration of membrane impermeable
anionic/cationic small molecules leads to a water activity inside cells that is lower
than outside. This creates a flow of water into the cell, leading to increase of
intracellular pressure. The water inflow stops when the pressure has reached the
J.J. Heijnen
osmotic pressure. According to van ’t Hoff the osmotic pressure is linear in the sum
concentration C of all intracellular metabolites ðPosm ¼ CRTÞ. For example, for
C = 1 mol/L it follows that Posm ¼ 25 bar (C ¼ 1 mol/L ¼ 1,000 mol/m3, R
¼ 8.314 J/mol K, T ¼ 298 K gives Posm ¼ 24:8 105 N=m2 ¼ 25bar). From a
mechanical point of view, this pressure is counteracted by the mechanical strength
of the cell membrane/cell wall, which is obviously bound to physical limits.
Therefore there must exist a maximal sum concentration of small intracellular
molecules. Assuming a limit of 25 bar gives for this maximal sum C 1.0 mol/L,
which gives a sum concentration of organic (anionic) metabolites of order 0.5 mol/L.
Assuming the presence of about 1,000 different metabolites in cells gives, for the
average intracellular metabolite concentration Xj ; a value of about 103 mol/L.
Of course, there will be a wide distribution of concentrations, so we can expect
an intracellular concentration range of 102 to 104 mol/L, which is equivalent
to 20–0.20 mmol/g dry biomass. These values are indeed found as shown in
Table 1.
4 Consequences of Low Metabolite Concentrations
from a Systems Biology Point of View
The general property of low intracellular metabolite concentrations has very important consequences at system level.
A first consequence is the near absence of spontaneous reactions. Usually the
metabolic network is considered to be totally enzyme catalysed and one assumes
implicitly that non-enzymatic reactions (which occur spontaneously) are absent.
Given the multitude of reactive molecules inside cells, one would expect much
more spontaneous reactions. Such reactions would be disadvantageous because
they are not under genetic control and they cause loss of material. The key to
suppress such reactions, in favour of enzyme catalysed reactions, is to have a
very low metabolite concentration (which kinetically “kills” the rate of a spontaneous reaction) in combination with matching high affinities of enzymes. This
is indeed found. So one could state that the osmotic limit enforces high affinity
enzymes.
A second general consequence is the need of active export.
Many biological systems are used in industrial processes (antibiotics, fuels,
amino acids, organic acids etc.). From an economic point of view, one aims at
high ( 1 M) extracellular concentrations of product. This implies that the final
step of product metabolism, export, has to deal with an unfavourable concentration
gradient of about 103 M inside and 1 M outside. Clearly, this requires active
export [6].
Another general aspect of metabolites is a fast, order of seconds, turnover time
X
(t.o.t.) of each metabolite. The t.o.t. of a metabolite Xj is defined as ðt:o:t:Þ X j ¼ Vsumj ,
with Xj the metabolite concentration and Vsum the sum of all production rates of this
Impact of Thermodynamic Principles in Systems Biology
Table 1 Intracellular metabolite concentrations and turnover time in glucose limited aerobic
cultures of several organisms (Saccharomyces cerevisiae from [2], Penicillium chrysogenum
from [3, 4] E. coli from [5])
Metabolites
Intracellular level (mmol/gDW)
Turnover time (s)
P. chrysogenum S. cerevisiae E. coli P. chrysogenum S. cerevisiae E. coli
Central metabolites
G6P
4.64
5.2
1.42
23.3
17
3.6
F6P
0.71
1.4
0.38
5.7
7.3
1.2
T6P
0.55
0.13
47.8
NA
M6P
1.95
0.48
NA
6PG
0.25
0.48
0.10
3.7
4.5
1.1
Mannitol-1P
0.99
NA
G3P
0.13
0.17
57
13.1
FBP
0.9
0.64
0.82
7.2
3.2
2.5
F2,6bP
0.01
0.35
NA
2PG+3PG
0.59
2.8
1.65
2.3
6.6
2.5
PEP
0.24
2.3
1.61
0.9
5.7
2.7
Pyruvate
0.22
1.1
0.75
0.9
1.7
1.5
a-Ketoglutarate 2.05
0.31
22.1
0.6
Succinate
0.23
4.0
2.65
3.3
20
8.9
Fumarate
0.65
0.85
0.22
13.0
4.1
0.7
Malate
3.33
7.3
0.94
19.0
30
2.8
Amino acids
Alanine
Asparagine
Aspartate
Glutamate
Glutamine
Glycine
Histidine
Isoleucine
Leucine
Lysine
Methionine
Phenylalenine
Proline
Serine
Threonine
Tryptophan
Tyrosine
Valine
Ornithine
21.7
1.5
16.3
53.0
28.7
2.1
0.72
0.33
0.73
1.2
0.14
0.19
0.95
5.7
5.9
0.11
0.26
2.1
Adenine nucleotides
ATP
7.39
ADP
1.03
AMP
0.27
32
4.7
21
170
64
2.9
6.0
1.6
1.0
4.1
0.20
1.6
3.9
269
459
717
658
1,243
244
432
111
131
356
58.8
61.2
206
453
758
130
145
243
3,268
1,142
577
1,112
2,401
247
3,141
140
125
619
66
430
925
4.0
0.51
1.6
10
4.1
1.34
0.58
2.57
74.69
6.14
1.51
0.15
0.11
0.36
1.21
0.05
0.13
0.66
0.53
0.47
0.02
0.18
0.51
0.49
220
788
832
490
502
76.7
81.7
35.0
229.0
80.0
31.0
53.8
12.9
27.1
119.7
10.5
23.8
101.4
8.0
29.3
11.9
44.3
40.9
49.1
7.0
1.3
0.6
5.95
2.31
0.91
NA
NA
NA
1.4
0.25
3.1
2.0
0.8
9.4
J.J. Heijnen
metabolite. Because Xj is low and Vsum can be high, one indeed finds t.o.t. of the
order of seconds (Table 1).
These fast t.o.t. have several important consequences:
l
l
Considering product formation, where a substrate molecule is processed along a
multistep pathway to the secreted product, it follows that the time between
substrate entrance and product leaving the cell is only of order minutes. Clearly
cell factories follow the just in time principle.
Considering the metabolite mass balances (1), we can safely neglect:
– The dilution term mXi , which is orders of magnitude smaller than the synthesis term SV:
i
– The dynamic term dX
dt for time scales larger than minutes (which follows
from t.o.t. of order seconds). This leads to pseudo-steady state.
l
This pseudo-steady state property, which is a direct consequence of the low
metabolite levels due to an osmotic limit, is one of the most important network
properties. It allows one to write for the metabolite mass balances:
SV ¼ 0
l
(2)
These balance equations are the basis of the well-known stoichiometric analysis
of metabolic networks. We should also realise that, due to the pseudo-steady
state property of the metabolic reaction network, these balances also apply to
dynamic situations which allow one to formulate so-called black box stoichiometric/kinetic models which are reliable in a wide range of conditions (see also
“model reduction”).
A final consequence is the need for control mechanisms on the production/
consumption of each intracellular metabolite. Cells are, in their natural environment, continuously exposed to perturbations which change the rate of synthesis/
consumption of metabolites. Given the low concentration of a metabolite, such a
perturbation leads to very quick (second time scale) and drastic (up or down)
changes in metabolite concentrations, which propagate through the network
leading to potential damaging system responses. Control of metabolite levels
is needed to limit these effects, and indeed such control mechanisms are widely
found in biological systems and, most interesting, they operate at proper time
scales. Most well known are allosteric feed back inhibition (e.g. in amino acid
synthesis pathways), and allosteric feed forward activation (e.g. in glycolysis)
mechanisms which operate within seconds. This is exactly the time scale
expected from the t.o.t. for metabolite levels. The other mechanisms are slower.
The post translational modification mechanisms (adenylation, (de)phosphatation, . . . .) take in the order of minutes, consume ATP and interconvert active/
inactive enzyme, but do not change the total enzyme amount. The genetic
mechanisms (induction, repression) take in the order of tens of minutes and
change the amount of enzyme.
Impact of Thermodynamic Principles in Systems Biology
5 Thermodynamic Approach to Obtain Network
Stoichiometry and Fluxes
To obtain the network fluxes (and therewith the stoichiometry of the network)
requires one to solve (2). These balances put linear constraints on the reaction/
uptake/secretion rates. The number of rates in a realistic network is typically several
hundred; however the number of metabolite mass balances in (2) (¼ constraints) is
also large. A general problem is the ATP-balance which contains uncertain ATP
stoichiometric parameters (P/O ratio, unknown growth related ATP ð¼ KX Þ and the
unknown growth unrelated ATP ðmATP Þ. Van Gulik and Heijnen [7] and Van Gulik
et al. [8] have shown how in vivo values for these ATP-parameters can be obtained
using extensive experiments. For many organisms this ATP-information is not
available. This means that the number of equations in (2) is always at least two
lower than the number of rates (underdetermined). This means that solving all rates
needs the specification of at least two experimental rates. Without this experimental
information one cannot predict stoichiometry. Another approach which has
received considerable attention in the past decade, and which aims to predict both
rates and stoichiometry of networks, is constraint based modelling [43]. This
approach uses an optimality criterion (e.g. maximal biomass yield) to obtain a
solution of the underdetermined (2). However, close inspection reveals that this
method still requires the above-mentioned experimentally based information:
l
l
Specification of the uncertain ATP stoichiometric coefficients (P/O, growth
related and unrelated maintenance values). This information is needed to make
stoichiometry predictions!!
Kinetic information, such as an experimentally determined substrate uptake rate
or a maximum O2 -uptake rate. This is needed to calculate fluxes and, e.g. mmax .
Together with this experimental information the optimality criterion forces ATP
requiring processes such as futile cycles to zero and therewith one obtains a unique
flux solution and therewith stoichiometry. When the above-mentioned (ATP and
kinetic) experimental information is not available, constraint based modelling does
not lead to a unique flux solution.
Thermodynamics offers an alternative, more widely applicable, approach to
solve the network stoichiometry and fluxes for arbitrary organisms.
5.1
Thermodynamic Approach to Stoichiometry
Thermodynamics allows one, for a given specific growth rate m (under substrate
limited growth in absence of a non-catabolic product, hence only growth), to
calculate all uptake/secretion rates. Herewith, all yields are also available (yield
is ratio of rates). Heijnen and Van Dijken [9] and Heijnen [10] apply this approach
J.J. Heijnen
to any heterotrophic growth system under substrate limited condition. The only
information needed is the nature of carbon source, electron donor, electron acceptor, N-source and temperature.
In this method, the ATP-balance in the network equation (2) is replaced by
a Gibbs energy balance on all uptake and secretion rates qi. We can define qi as a
biomass specific rate in mol of i/h per Cmol biomass. The quantity 1 Cmol of
biomass (which has the average composition C1 H1:8 O0:5 N0:2 ) is the amount (24.6 g
organic dry matter) of biomass which contains 12 g of carbon (¼ 1 mol C-atom).
Compounds taken up have negative qi values, secreted compounds have positive
qi-values. The Gibbs energy balance follows then as
X
qi DG01
fi þ qG ¼ 0:
(3a)
P
Here
qi DG01
fi is a negative quantity (second law of thermodynamics) and is the
total biomass specific rate of Gibbs energy of conversion. qG is the Gibbs energy
produced, which follows from (3a). The second law requires qG > 0.
DG01
fi is the Gibbs energy of formation of compound i at standard condition (1 M,
298 K and at pH ¼ 7.0). In principle, one needs to take actual concentrations into
account, but this leads only in special cases to significant changes in qG [10].
The key to the use of (3a) is to obtain a relation for qG . Because cells require
Gibbs energy for growth and maintenance, we can write a Herbert–Pirt type of
relation for qG , which expresses that (in absence of non-catabolic product) the cell
needs Gibbs energy for growth and maintenance:
qG ¼
For mG and
5.1.1
1
max
YGX
1
max m þ mG :
YGX
(3b)
; correlations have been established [9, 11].
Gibbs Energy for Maintenance
All living systems need to generate Gibbs energy for their maintenance (which
represents all processes where energy is needed for example to repair degradation
and export compounds that entered due to membrane leakage, etc.). Because living
cells have similar membranes and composition, it can be assumed that different
cells require a similar amount of energy expenditure for maintenance. It has indeed
been found that Gibbs energy needed for maintenance is very similar for a large
range of microorganisms and only depends on absolute
temperature (T)
[11].
kJ Gibbs energy=h
The following correlation has been found for mG in
:
CmolX
mG ¼ 4:5 exp
69;000 1
1
:
R
298 T
(4)
Impact of Thermodynamic Principles in Systems Biology
This correlation shows that at 25 C (298 K) 1 Cmol of cells ( 25 g dry matter)
spends, and therefore needs to generate, 4.5 kJ of Gibbs energy per hour to cover the
energy for maintenance. Also there is a steep temperature effect: for each 8 C
temperature increase mG doubles!!
This correlation applies to aerobic/anaerobic systems and arbitrary electron
donors/acceptors. The Gibbs energy is generated from the catabolic reaction, so
the maintenance reaction equals the catabolic reaction. For specific cases, one can
always set up the catabolic reaction for 1 mol donor and obtain the catabolic Gibbs
energy of reaction for 1 mol donor, called DG01
cat;D ; which is negative and is in kJ of
Gibbs energy per 1 mol donor consumed in the catabolic reaction.
It is thenclear
substrate=h
that we can write for the substrate (or donor) consumption ms in molCmolX
that
must be catabolised for maintenance:
ms ¼
mG
:
DG01
cat;D
(5)
Table 2 shows examples of catabolic reactions consuming
1 mol
donor, and
the Gibbs energy of catabolism per mol consumed donor DG01
cat;D . It is obvious
can
be
two
orders
of
magnitude
different,
dependent
on the specific
that DG01
cat;D
catabolism.
For example, consider Saccharomyces cerevisiae at 30 C. It follows from (4) that
mG ¼
7:1 kJ=h
CmolX
Under aerobic conditions using glucose as substrate, catabolism DG01
cat;D ¼
2,843.1 kJ per mol glucose. This gives
ms ¼
7:1
mol glucose=h
¼ 0:0025
ð2;843:1Þ
CmolX
O2 =h
Also mO2 ¼ 6 ms ¼ 0:015 mol
CmolX . Under anaerobic conditions the catabolic ethanol forming reaction shows DG01
cat;D ¼ 225:4 kJ per mol glucose. This
glucose=h
ethanol=h
7:1
gives ms ¼ 225:4
¼ 0:0315 molCmolX
and methanol ¼ 0:063 molCmolX
:
Table 2 Catabolic reactions and their Gibbs energy of reaction, DG01
cat;D
Catabolic reactions and their Gibbs energy DG01
cat;D (kJ/mol donor)
Donor
Glucose
Ethanol
Glucose
Methanol
Iron (2þ)
Acetate
DG01
cat;D
Catabolic reaction
þ
C6 H12 O6 þ 6O2 ! 6HCO3 þ 6H
C2 H6 O þ 3O2 ! 2HCO3 þ 2Hþ þ 1H2 O
C6 H12 O6 þ 2H2 O ! 2C2 H6 O þ 2HCO3 þ 2Hþ
CH4 O þ 1:20NO3 þ 0:20Hþ ! 0:60N2 þ HCO3 þ 1:60H2 O
Fe2þ þ 14 O2 ! Fe3þ þ 12 H2 O ðpH ¼ 1:85Þ
C2 H3 O2 þ H2 O ! HCO3 þ CH4
2,843.1
1,308.9
225.4
649.4
33.9
31.0
J.J. Heijnen
So, the same organism has widely different ms values due to different catabolism, but still has the same Gibbs energy need for maintenance!!
5.1.2
1
max
YGX
Gibbs Energy for Growth
is the amount of Gibbs energy needed to synthesise 1 CmolX (in kJ/CmolX).
This amount has been found to depend only on two factors [9, 10]: first, the nature
of the carbon source for heterotrophic growth and, second, the nature of electron
donor for autotrophic growth.
Regarding the first factor, the nature of the carbon source for heterotrophic
growth, more Gibbs energy is needed when the carbon source has a smaller number
(C) of C-atoms and when its degree of reduction per C-atom (g) is different from the
degree of reduction of biomass ( 4.2). The explanation is straightforward that
synthesis of biomass monomer molecules (which contain order 6 carbon atoms with
g 4) requires more C–C-coupling and redox reactions for C-sources with a
low number of C-atoms and which need reduction or oxidation because g of the
C-source differs from 4.2. These extra reactions lead to a higher Gibbs energy need.
For heterotrophic growth this intuition is quantified in (6a), which is a correlation:
1
max
YGX
h
i
¼ 200 þ 18ð6 CÞ1:8 þ exp j3:8 gj0:32 ð3:6 þ 0:4CÞ :
(6a)
This correlation shows that to synthesise 1 Cmol biomass one needs between
236 and 1,087 kJ Gibbs energy (dependent on C-source, e.g. 236 for glucose
(g ¼ 4, C ¼ 6) and 1,087 for CH4 (g ¼ 8, C ¼ 1).
Regarding the second factor, the nature of electron donor for autotrophic
growth, in autotrophic growth CO2 is the C-source which must be reduced to
biomass using electrons from the electron donor. The Gibbs energy needed follows
from (6b), which is also a correlation [9]:
1
max ¼ 1; 000ðRETÞ¼ 3; 500ðþRETÞ:
YGX
(6b)
The nature of the electron donor determines the absence (RET) or need
(þRET) of Reverse Electron Transport. For several electron donors (e.g.
NH4þ ; NO2 ; Fe2þ ) the reduction of CO2 to biomass is not feasible thermodynamically. Therefore cells spend extra Gibbs energy to make the redox potential of
electrons obtained from the available donor more negative (e.g. in the production of
NADPH from the donor electrons, in a process called RET, NADPH is then used
to reduce CO2 to biomass). This extra Gibbs energy is very considerable (compare
in (6b) 3,500 and 1,000). For H2 as electron donor, (RET), this problem does
not exist (sufficient negative redox potential) and the Gibbs energy need is 1,000 kJ/
CmolX (6b).
Impact of Thermodynamic Principles in Systems Biology
Summarising, the Gibbs energy needed to make 1 Cmol biomass ranges, dependent on C-source and electron donor, is between 200 and 3,500 kJ and does not
depend on the type of electron acceptor. When the carbon source, electron donor
1
and temperature are known, the correlations (4, 6a/6b) give the coefficients Y max
and
GX
mG in the Gibbs energy Herbert Pirt relation (3b), which completes the Gibbs
energy balance (3a). This linear relation can be combined with all metabolite
mass balances specified in (2), using the stoichiometric matrix S where the ATPbalance is also absent (due to unknown P/O, Kx, etc.). A constraint of minimal
Gibbs energy dissipation will put futile cycles to zero and parallel pathways are also
resolved. This set of linear balances gives, for any selected m, all rates in the
network (reaction/uptake/secretion)!! With these known rates all yields are
known such as the biomass yield YSX ¼ qms or yield of catabolic products.
This thermodynamic approach has been shown to predict biomass yield with
10–15% error for a wide variety (aerobic/anaerobic/hetero/autotrophic) of microbial systems where YSX spans a range of near two orders of magnitude [9].
5.2
Thermodynamic Approach for Maximal Growth Rate, mmax
Microorganisms show a very large range (0.001–1 h1) in mmax -values and it is
relevant to understand why this is so!! A simple approach was proposed [10] which
reproduces most of this range. The concept is that cells during evolution ultimately
become limited in their energy production capacity. Most organisms generate
energy by electron transport phosphorylation. This occurs by electron transport
proteins embedded in membranes. Because membranes are space limited for
protein embedding it is logicalto propose that cells have evolved to a maximal
electrons=h
electron transport capacity qmax
in mol CmolX
which depends mainly on temperel
ature. Also it is known that smaller organisms have higher maximal growth rates,
e.g. E. coli mmax ¼ 2 h1, Saccharomyces cerevisiae mmax ¼ 0.4 h1 and tissue cell
cultures mmax ¼ 0.04 h1. This is in line with the smaller surface/volume ratio
( 6/d, with cell diameter d) which leads to a membrane surface area, hence
maximal electron capacity and mmax which is inverse to the cell diameter and
which is indeed largely observed (e.g. S. cerevisiae has a 5 larger cell diameter
compared to E. coli). This concept has in addition been inspired by the observation
that in E. coli, for different substrates which lead to different mmax , the qmax
O2 -value is
nearly constant [12].
The following correlation was proposed:
qmax
el
69; 000 1
1
¼ 3 exp
:
R
298 T
(7a)
This electron capacity determines the maximal production rate of Gibbs energy
by catabolism (qmax
G ):
J.J. Heijnen
Table 3 Estimated mmax values for different microbial catabolic classes at 25 C based on limiting
Gibbs energy production
mmax (h1)
gD (mol electron/
Catabolic classes
DG01
cat; D
mol donor)
(kJ/mol donor)
Aerobic/glucose
Aerobic/acetate
Anaerobic/(acetate!CH4)
Aerobic/Fe2+ oxidation
Aerobic/nitrification
2,843.1
844.2
31.0
38.6
274.8
24
8
8
1
6
max
qmax
G ¼ qel
DG01
cat;D
gD
1.5
0.70
0.015
0.030
0.040
:
(7b)
Here, gD is the number of electrons released in catabolism of 1 mol donor and
DG01
cat;D is the catabolic Gibbs energy per mol donor. This maximal Gibbs energy
sets mmax according to (3b).
Combination of (3b) and (7b), (2) and using the correlations (4), (6a, b) and (7a)
max max
for mG ; YGX
; qel and the available value for DG01
cat;D and gD immediately allows
one to calculate mmax -values for any growth system. These values agree reasonably
with known values. Table 3 shows that this simple approach can explain a 100-fold
difference in mmax .
Some final remarks:
l
l
l
This thermodynamic approach uses only three correlations (4, 6a/6b, 7), is
simple and general and gives maximal qi-rates and stoichiometry.
Effect of temperature is included (in maintenance and mmax ).
The three correlations are based on a wide range of experimental microbial
growth systems and reflect that similar biochemical pathways are used (unity of
biochemistry). When the predicted mmax or/and stoichiometry are very different
from experimental values, this indicates unusual anabolic and/or catabolic routes
which might be novel. So this method can act as a filter for unusual behavior of
biological systems.
6 Prediction of Gene Regulation of Enzymes Using Energy
Optimality
Gene regulation seems at first glance highly complicated. For example, enzyme
induction upon exposure to a new catabolic substrate involves many mechanisms
between signal transduction, gene expression and production of enzyme for the new
catabolic pathway. Model based prediction of enzyme induction therefore seems
hopeless. However one could expect that evolution has fine tuned the available
Impact of Thermodynamic Principles in Systems Biology
regulation mechanisms such that growth yield is optimal. This optimality principle
was tested using mixed substrates with Saccharomyces cerevisiae, which was
chemostat (aerobic, substrate limited) cultivated at a dilution rate D ¼ 0.1 h1.
Different feed mixtures of glucose and ethanol as substrates were applied between
100% glucose and 100% ethanol) [7, 13, 14].
Growth on glucose differs from that on ethanol. On 100% ethanol the cell has:
l
Fully induced:
– Glyoxylic acid pathway enzymes (isocitrate lyase (ICL) and malate synthase
(MS))
– Gluconeogenic enzymes PEP-carboxykinase (PEPCK) and F16 Bisphosphate (F16BP-ase)
l
Fully repressed (or inactivated):
– Pyruvate kinase/pyruvate carboxylase (PK/PYC)
– F16 bis P-kinase (PFK)
It is obvious that, under substrate limited condition for mixed substrates, the
induction/repression pattern of these enzymes is determined by the residual
ethanol and glucose concentration, which would be the basis of a complicated
gene regulation mechanism and model to predict the occurrence and concentration of these enzymes as function of the residual ethanol and glucose
concentration.
A different approach, based on energy optimality of gene expression, was
followed. Using a metabolic stoichiometric model endowed with experimentally
obtained stoichiometric values for the ATP-balance (P/O, Kx, mATP), linear programming was applied with maximal biomass production (or the equivalent minimal energy consumption) as optimality criterion. It was possible to calculate, for
each glucose/ethanol supply ratio (which is virtually equal to the ratio of their
consumption rates due to the low residual ethanol and glucose concentration) the
optimal rates of all reactions in central metabolism. It was observed that, for the
above inducible enzymes, clear predictions were made on their need as function of
increasing ethanol fraction. This approach predicted:
l
l
The ethanol/glucose feed ratio where a particular enzyme started to be induced
The enzyme amount then increased linear with increasing ethanol fraction
These predictions were qualitatively, but more surprising also quantitatively,
validated using the wild type yeast [13, 14]. Later additional validation was
performed with null-mutants in the above enzymes, leading to a predicted maximal
ethanol uptake rate of each mutant.
This prediction was again quantitatively confirmed [15]. This example clearly
indicates that gene regulation mechanisms might have evolved to provide maximal
biomass yield (giving a competitive edge). This maximal biomass yield is the same
as Gibbs energy optimality because enzyme induction is such that futile cycles are
avoided (e.g. simultaneous activity of FBP-ase/FPK or pyruvate carboxylase/PEP
J.J. Heijnen
carboxy kinase). To the authors knowledge this is one of the earliest and most
successful examples of experimentally demonstrated thermodynamic optimality of
living cells.
7 Thermodynamics Based Model Reduction in System Biology
Mathematical models of biological systems are useful to design processes and/or to
redesign organisms using the rec-DNA tool box. Model reduction is an important
issue, given the complexity of biological systems. Model reduction aspects will be
discussed for two categories of mathematical models.
7.1
Black Box Models for Design of Biotechnological
Processes: From Complexity to Simplicity Due to
Pseudo-Steady State Coupling
We have observed that the thermodynamically based osmotic limit leads to very
low metabolite concentrations in intracellular metabolism. The immediate consequence is that at process time scales larger than about 10 min (as occurs in bath, fed
batch processes) the pseudo-steady state condition for all intracellular metabolites
holds. The consequence of this condition is that all uptake and secretion rates are
directly coupled. This pseudo-steady state coupling can be evaluated by linear
rearranging the metabolite mass balances (2). Usually, in the absence of byproducts, there are only a few (two to three) degrees of freedom, meaning that all
uptake/secretion rates can be written as linear combination of only two to three rates
(usually growth rate, product formation rate, maintenance). A prime example of
such a linear relation is the Herbert–Pirt relation for substrate distribution:
qs ¼
1
1
max m þ Y max qp þ ms :
YSX
SP
(8a)
Furthermore, the substrate uptake relation is usually a hyperbolic relation in the
limiting substrate concentration:
qs ¼ qmax
s
Cs
:
Ks þ Cs
(8b)
A final relation, which holds generally under single substrate limited condition is
that there is a unique relation between qp and m:
qp ¼ fp ðmÞ:
(8c)
Impact of Thermodynamic Principles in Systems Biology
The three equations (8a, b, c) have only one degree of freedom (e.g. Cs or m). All
the other uptake/secretion rates ðO2 ; CO2 ; NH4þ ; water; Hþ ; heat; etc:Þ can be
obtained from qs ; m and qp and the conservation relations (elements and charge).
This black box approach for kinetic modeling of biological processes is only
possible due to the general biological system property of low intracellular metabolite concentrations, which has its origin in a thermodynamic property (osmotic
limit).
Examples of this black box approach can be widely found. A very nice example
is the model for penicillin production [8, 16] and the model developed for
biological P-removal using mixed cultures in a cyclic process [17]. These black
box models show that highly complex biological systems, comprising thousands of
reactions, can be effectively modeled with a reduced model containing only about
6–12 parameters. These models are the basis of process design. This simple
behavior of complex biological systems (from complexity to simplicity!!) is the
direct result of the metabolite pseudo-steady state property which results from an
osmotic limit!
7.2
Metabolic Reaction Network Models to Redesign
Organisms: Reducing Complexity
Genetic intervention in metabolic reaction networks is possible in many ways by,
e.g. changing enzyme levels, changing enzyme kinetics (e.g. abolish feed back
inhibition), introducing different reactions, etc. The problem is that, following such
interventions, the prediction of changes in secretion or uptake rates is very difficult
due to the complex nature of the network structure and of the highly non-linear
kinetics of the enzymes and gene regulation mechanisms. It is generally recommended that a mathematical model of a metabolic (reaction) network can help to
select rationally genetic engineering targets in the redesign of organisms.
Building such a model is essentially straightforward and based on (1). Usually
matrix S is known with high confidence. Uncertainties in S (e.g. cycles, parallel
reactions, etc.) can be addressed using 13C approaches [18].
A much bigger problem is to obtain kinetic relations for each enzyme, meaning
the function v (e, x, p) in (1) for each enzyme.
Traditionally, in vitro obtained kinetic functions and parameters have been used.
Here two problems arise. First, it was found that in vitro kinetics do not reflect
in vivo kinetics [19]. Second, for many enzymes in vitro kinetics are not available.
The only solution therefore is to perform experiments with whole cells to obtain
in vivo kinetic behavior of all enzymes simultaneously. Here rapid pulse experiments as pioneered by the groups of Reuss [20, 21] and Heijnen [22] offer
advantages, but the challenge is the parameter identifiability problem which calls
for model reduction as shown by Nikerel et al. [23–25].
Model reduction in biological systems has a thermodynamic basis as will be
outlined below.
J.J. Heijnen
7.2.1
Pseudo-Equilibrium Reactions
A first approach for model reduction is to replace a kinetic function by an equilibrium relation. This is possible when the enzyme catalysed reaction is so rapid
that the reaction remains very close to equilibrium, even when the reaction rates
increase.
A strong indication of this situation is that the so-called mass action ratio of a
reaction remains nearly constant (independent from flux) and close to the reported
equilibrium constant. Many examples have been found which show this behaviour
in Penicillium chrysogenum [3, 4] and in Saccharomyces cerevisiae [26] such as for
phospho glucose isomerase, enolase, phosphoglycerate mutase, phosphoglucomutase, fumarase.
Recently a near equilibrium reaction ðF6P þ NADH þ H þ ! mannitol 1
P þ NADþ Þwas used as a heterologous sensor reaction to obtain the cytosolic
NAD/NADH ratio in Saccharomyces cerevisiae [2], even under dynamic conditions. It was found that the cytosolic NAD/NADH ratio was about 100. In contrast,
a ratio of 4 was obtained by analysing the total amounts of NADH and NAD in
whole yeast cells. The large difference is due to the fact that nearly all NADH is
present in the mitochondria. Even more important is that, using the cell average
NAD/NADH ratio, the DGR of glycolysis between F16BP and (2þ3 PG) was > 0,
which is impossible. Use of the cytosolic value of 100 leads to DGR which is slightly
negative, as expected. It can be expected that metabolism shows much more near
equilibrium reactions. However here we need much more accurate data on in vivo
equilibrium constants on a genome wide scale!!
7.2.2
Pseudo-Steady State Lumping
We have seen that many metabolites have low concentration levels (due to an
osmotic upper limit) which leads to very low turnover times (<1 s) of metabolite
pools. This allows, even in pulse experiment at a 300-s time scale, lumping of the
synthesis and consumption reactions of a fast metabolite pool into 1 “lumped
reaction”. This model reduction based on pseudo-steady state lumping leads to
much less parameters and much less parameter identifiability problems (see [23–25]).
7.2.3
Thermodynamic Inspired Kinetics
Replacing kinetics of individual reactions by equilibrium constants (pseudo-equilibrium) and by lumped kinetics (pseudo-steady state) significantly reduces the
number of required kinetic functions and the number of parameters which need to
be indentified, with no loss of model performance!! (see [24, 25]).
Nevertheless, there remain a significant number of far from equilibrium reactions into and from metabolite pools with slower turnover times for which we need
Impact of Thermodynamic Principles in Systems Biology
to specify a kinetic function. Unfortunately, mechanism based enzyme kinetics
provide highly non-linear rate functions which contain many parameters. Examples
are bi–bi kinetics, Hill functions, etc.
Although the number of parameters has been significantly reduced (nearequilibrium, pseudo-steady state based reduction) the identification of the remaining parameters still poses enormous problems, due to the non-linear parameter
characteristics. Non-linear parameter estimation algorithms need a decent initial set
of parameter values, which is not available, and in addition they do not guarantee an
optimal result; they often end in a local minimum of the error criterion. Finally,
these identification problems scale very unfavourably with increasing network size
(e.g. genome scale metabolic models contain in the order of 1,000 reactions!!).
A possible solution is the use of proper approximative kinetic functions [27].
Here lin-log kinetics has been developed recently [28], which has its roots in
the concept that the rate of a process is related to the thermodynamic driving
force!! [29].
8 Thermodynamics Inspired Kinetics: Lin-Log Kinetics
8.1
Introduction
Lin-log kinetics is an approximative kinetic format which is a generalisation of the
driving force concept and has been compared recently to other approaches such as
linear, powerlaw, loglin (for review see [27]) and it was concluded that the lin-log
format has significant advantages. Therefore we will focus here on use of lin-log
kinetics.
Consider (Fig. 2) an enzyme e1 which is kinetically affected by its substrate
x1 and product x2 , and is also allosterically affected by a metabolite x8. Moreover,
we consider a reference steady state (superscript 0).
Allosteric
effector
x1
v1
0
J1
=
e1
e 01
Fig. 2 Lin-log kinetics
e1
x8
x2
. 1+ ε 011 . ln x1
0
x1
0
+ ε 12
. ln
x2
0
x2
+ ε 18 . ln
0
x8
0
x8
J.J. Heijnen
2
1.5
v/J0
Fig. 3 Comparison between
lin-log approximative and
hyperbolic mechanistic
kinetics
1
Vlin-log
0.5
0
VMM
0
1
2
3
4
5
x/x0
We can then formulate lin-log kinetics:
v1 e1
x1
x2
x8
o
o
o
¼
1 þ e11 ln þ e12 ln o þ e18 ln o :
J1o eo1
xi
x2
x8
(9)
We can compare approximative lin-log kinetics with traditional mechanistic
hyperbolic kinetics and we see (Fig. 3) a very good approximation with respect to
large changes (factor 5 up and down) in metabolite concentration.
Other approximative formats, such as power law as S- or GMA-systems and
loglin, show similar quality of approximation for such changes in metabolite
levels [27].
However metabolic reaction systems not only show large changes in metabolite
concentrations but especially one faces (in the light of metabolic engineering
ambitions) large changes in enzyme levels. For example, one can easily achieve
enzyme concentration changes of factor 10 up or down due to genetic interventions
in gene regulation (promoter libraries) or in gene dosis. Here, lin-log kinetics
shows distinct advantages compared to the other approximative kinetic formats,
as explained before [27]. A final favourable property of lin-log kinetics is that its
parameters (elasticities) are linear in (1) which is significant in the light of parameter identification analysis and parameter estimation algorithms [23–25].
Lin-log kinetics has, after its conception, been successfully applied to kinetic
modelling of metabolic reaction networks using in silico studies but has also been
applied to experimental systems as will be discussed below.
General important aspects of lin-log kinetics are:
l
l
l
l
The format is non-linear in metabolite concentrations.
The format has a minimum number of kinetic parameters which helps minimise
the identification problem.
The parameters (elasticities) are linear in (1) which has significant advantages
with respect to the parameter identifiability and parameter estimation [23–25].
The elasticity parameters are bounded, e.g. Michaels–Menten kinetics jej < 1,
for Hill kinetics jej < n (with n subunits in the protein and signs (þ or ) are
known. This constrains the e-estimation problem.
Impact of Thermodynamic Principles in Systems Biology
l
Lin-log kinetics cannot be applied for datasets where concentrations become
zero [28], [30]. An analogous problem also occurs in powerlaw format where an
inhibitor concentration cannot become zero.
8.2
In Silico Studies with Lin-Log Kinetics
Although Fig. 3 suggests good performance for lin-log kinetics for an individual
reaction, a basic question remains how good does the lin-log approximation work in
networks. This has been studied in several simulation studies. In a first study, where
the lin-log concept was introduced [28], a small branched reaction network, including cofactors, was used as a test case. The strongly non-linear kinetic model
was approximated with a lin-log model using the theoretical elasticity parameters.
A rapid perturbation experiment, where metabolite concentrations did change
several fold, was successfully reproduced. Moreover, use of lin-log kinetics leads
to the so-called “design equation”, which allows one to specify new fluxes/metabolite levels and to calculate analytically the required large changes in enzyme levels.
In a follow up study [31] a non-linear model of glycolysis in E. coli was successfully approximated using lin-log kinetics and a connected product pathway was
successfully redesigned in silico with respect to the required large changes in
enzyme activities. Even more interesting was a recent study of Smallbone et al.
[32] in which they showed that a lin-log kinetic model of glycolysis in Saccharomyces cerevisiae, in which elasticities were assumed equal to their reaction stoichiometric coefficients, gave surprising agreement with the mechanistic Teusink
model!! More recently [30] it was shown that lin-log kinetics could also successfully simulate a genetic network with strongly non-linear kinetics. These simulation
results show that lin-log kinetics provide a convenient and satisfying approximation
of mechanistic kinetic functions for small and large models (metabolic, genetic),
provided that metabolite concentrations do not become zero.
Having ascertained that lin-log kinetics provides a decent approximation to nonlinear kinetics of networks, the next important problem is to identify the lin-log
parameters (elasticities) from experimental data. The obvious experimental protocol is to perform rapid pulse experiments. The identification of lin-log parameters
(elasticities) from such experiments was studied first in silico. Kresnowati et al. [33]
showed, for a simple linear pathway, that elasticities can easily be obtained by using
the integrated equation (1). The important aspect is that the resulting equations are
linear in elasticities, so that linear regression can be used to obtain the elasticities.
Subsequently a more complex glycolysis model was used for an in silico study of
parameter identification aspects to obtain elasticities from dynamic pulse experiments [23–25]. It was found that, in such rapid pulse experiments, fundamental
identification problems occur due to occurrence of pseudo-steady state metabolite
pools (with turnover time <1 s) in the rapid pulse experiment. The identification
problem could only be resolved by providing additional information (e.g. combined
J.J. Heijnen
steady state and dynamic perturbations. Also, a novel non-linear algorithm for
elasticity parameter estimation was introduced where, due to properties of the linlog format, one can obtain a reliable initial estimate of their values using linear
regression of the integrated equation (1).
Finally, a completely different approach to the parameter identifiability problem
in pulse experiments was shown to be possible due to the unique properties of linlog kinetics [34]. It was shown that use of lin-log kinetics allows a priori model
reduction by lumping pseudo-steady state pool coupled reactions. The reduced
model reproduced the dynamic pulse experiment (S. cer, anaerobic glycolysis)
accurately and allowed calculation of, e.g. flux control coefficients [34].
An interesting application of a lin-log kinetic model is to identify the function of
so-called silent genes [35, 36] which shows how lin-log kinetic models could be
used to resolve gene-annotation problems.
8.3
Application of Lin-Log Kinetics to Experimental Data
Lin-log kinetics allows a general steady state analytical solution of networks which
gives fluxes as a non-linear function of enzyme activities with flux control coefficients CJ as parameters [28]. This equation was successfully used to obtain
CJ-values from experimental data:
l
l
l
A linear product pathway in Penicillin synthesis using fed batch data on flux
and enzymes in the penicillin pathway [37]
The lower glycolysis in a reconstituted linear three enzyme pathway [38]
A branch point for lysine or glutamate synthesis [39]
Estimation of elasticities from experimental data using lin-log format requires
experimental information of fluxes, enzyme levels and metabolite levels. Using an
extensive steady state dataset for lower glycolysis [38] showed that elasticity values
are easily obtained using linear regression. Also lin-log kinetics was used to set-up a
kinetic model for leucine/valine synthesis [40], glycolysis in Lactococcus lactis
[30] and a batch fermentation [41]. In the last two studies it was shown that lin-log
kinetics should not be applied to datasets where metabolite concentrations become
zero (which is obvious). Even more impressive was a recent study [42] in which a
lin-log model was parameterised on a rapid pulse experiment. The model represents
E. coli central metabolism and anabolism to all cell compounds, comprising 126
reactions and 130 metabolites (7 conserved moieties). The presence of allosteric
mechanisms was taken from the MetaCyc database. In total, 921 elasticities were
estimated using evolutionary algorithms and high performance computing. This
work is a first, genome like scale, whole cell dynamic model and shows the power
of the canonical lin-log format approach.
Parameter identification from experiments, however, remains an important
issue. Here recently Nikerel et al. [34] showed the successful application of their
Impact of Thermodynamic Principles in Systems Biology
a priori model reduction approach (made possible due to lin-log format) on anaerobic glucose pulse experiments in Saccharomyces cerevisiae.
This short overview shows that the recently introduced lin-log kinetics, inspired
by thermodynamic principles, shows considerable promise in achieving genome
scale kinetic modelling of metabolic networks.
9 Conclusion
Thermodynamic principles do shape the properties of biological systems, with
considerable and highly interesting consequences for their mathematical models
needed in systems biology. Especially noteworthy are the far reaching consequences of the osmotic limit, such as pseudo-steady state, black box kinetics, just
in time, control mechanisms, model reduction etc. Also of future importance is the
principle of energy optimality for modelling of genetic mechanisms and of thermodynamic driving force (lin-log kinetics) for kinetic modelling. Lin-log kinetics
seems to hold considerable promise to obtain realistic genome scale kinetic models
(especially due to the lin-log based possibilities towards model reduction, a priori
identifiability analysis and an initial estimate of the elasticity parameters).
Abbreviations and Symbols
X
S
e
p
m
qi
Posm
v
DG0fi
max
YGX
max
YSX
max
YSP
mG
C
g
R
T
DG01
cat;D
Ks
Cs
t.o.t.
J
eij
mi
Intracellular metabolite level
Stoichiometry matrix
Enzyme amount per cell mass
Parameter
Specific growth rate
Specific uptake/secretion rate
Osmotic pressure
Rate of intracellular reaction
Standard Gibbs energy of formation
Maximal yield of biomass on Gibbs energy
Maximal yield of biomass on substrate
Maximal yield of product on substrate
Maintenance Gibbs energy requirement
Number of C-atoms in carbon source
Degree of reduction of C-source (per C-atom)
Gas constant 8.314
Absolute temperature
Gibbs energy of the catabolic reaction per mol donor
Affinity for substrate
Substrate concentration
Turn over time
Flux
x
i
Elasticity coefficient vji @v
@xj of metabolite j on enzyme i
Maintenance coefficient for i (O2, substrate, ethanol, etc.)
m mol/g DM
h1
mol (or kJ/h)/CmolX
N/m2
mol i/h/CmolX
kJ/mol
CmolX/kJ
CmolX/molS
molP/molS
kJ/h/CmolX
J/molK
K
kJ/mold
molS/m3
molS/m3
s
mol/h/C-molX
mol of i=h
CmolX
J.J. Heijnen
Subscripts
j
i
p
S
x
G
o
el
D
Cat
Metabolite j
Reaction i
Product
Substrate
Biomass
Gibbs energy
O2
Electrons
Donor
Catabolic
Superscripts
1
0
Biochemical standard (pH ¼ 7)
Standard condition (1 mol/L, 1 bar, 298 K) or reference condition, also reference steady state
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