1
Inferring Dynamic Properties of Biochemical
Reaction Networks
from Structural Knowledge
Edda Klipp1
Wolfram Liebermeister2
Christoph Wierling3
klipp@molgen.mpg.de
lieber@math.fu-berlin.de
wierling@molgen.mpg.de
1
Berlin Center for Genome Based Bioinformatics (BCB), Max-Planck Institute for Molecular Genetics,
Dept. Vertebrate Genomics, Ihnestr. 73, 14195 Berlin, Germany
www.molgen.mpg.de/~ag_klipp
2
Free University Berlin, Biocomputing Group, Arnimallee 2-6, 14195 Berlin, Germany
http://page.inf.fu-berlin.de/~lieber
3
Max-Planck Institute for Molecular Genetics, Dept. Vertebrate Genomics, Ihnestr. 73, 14195 Berlin,
Germany
Abstract
An important aspect of Systems Biology is the prediction of (hitherto) immeasurable properties of
biological objects from measurable features, like protein function from sequence similarity or genetic
networks from gene expression data. The other way round it is of interest to characterize the dynamic
behavior of cellular networks given the network structure.
It is possible to construct a complete metabolic network or a sub-network of interest form information
in publicly available form like the KEGG database. The problem is to use this topological knowledge
in order to produce a description of the dynamic features of the network under given circumstances.
This concerns fluxes or at least the net direction of a flux through a pathway and the pattern of control
within the network. In order to investigate the potential dynamics of biochemical networks we study
systematically the possible distributions of fluxes, concentrations and of control pattern in typical
biochemical networks (linear and branched networks, signaling and gene expression cascades).
Keywords: biochemical network, steady state behavior, large-scale simulation
1 Introduction
Systems biology [Kitano, 2002] aims at investigating cellular networks by combining
experiments, mathematical modelling, and computer simulations. Knowledge of the network
elements and their interactions is a fundamental prerequisite for forward modelling of
biochemical reaction networks.
Currently, networks are constructed with enormous speed and with various methods.
Examples: protein interaction networks [ ], gene regulatory networks, genome-scale
reconstructed metabolic networks [Famili, 2003; Förster, 2003]. Networks are stored in
databases: KEGG [Kanehisa, 2002], more, which makes them easily available for further
analysis.
But the quantitative knowledge about the interactions and their individual kinetics is rather
incomplete. Detailed measurements of concentrations, fluxes, and time courses of individual
compounds are rare, at least compared to the number of compounds and the complexity of the
2
networks. Even if kinetics is measured, the models are incomplete since we have to deal with
measurement errors, dependence on experimental conditions, and individual variations in cell
composition and state.
Question arises: what can we learn about the dynamics of these networks without knowing
the detailed kinetics of their elements, i.e. the individual reactions? Is the dynamics, at least
partially, determined by the topology of the network instead of the individual kinetics?
Different methods have been developed for the case that one has at least a limited set of
experimental data at hand, for example estimation of fluxes from metabolite isotopomer
distribution [ ]. In this case one can make a mathematical model of the dynamics and may try
to estimate parameters of that model. A major problem with this approach is the
identifyability of the system given the set of experiments.
Here, we employ an inverse approach. We assume a known network topology. We chose the
kinetic constants of all reactions from a distribution and observe all possible steady states.
This way we can learn (a) what range of states are possible at all, (b) which structures like
feedback or coupling of fluxes limit the range of dynamics
2 Materials and Methods
We study the biochemical networks given in Scheme/Table 1: a linear chain of five reactions,
a small branching network, a cascade with two states on every level, a cascade with three
states on every level, a small glycolysis model, and a glycolysis model downloaded form
KEGG.
The dynamics of a biochemical reaction system is described by a set of ordinary differential
equations (ODEs)
dci dt 
r
 nij v j
( i  1,.., m )
j 1
where m is the number of biochemical species Mi with the concentrations ci and r the
number of reactions with the rates v j , and the quantities nij denote the stoichiometric
coefficients. All reactions v j are populated with kinetics, either simple mass action kinetics
( v j  k j  ci1  k  j  ci 2 )
or Michaelis-Menten kinetics (denoted in the following by MM)

Vmax,
j
vj 
K m,i1
1
 ci1 
ci1
K m,i1

Vmax,
j
 ci 2
K m,i 2
.
ci 2

K m,i 2
The respective kinetic constants ( k  j , Vmax , K m ) are taken from one of the following
distributions: (1) the uniform distribution (UD) or (2) the logarithmic normal distribution
(LND). The choice of the distribution LND has the following reason: We collected measured
and published kinetic constants for various networks from [Hynne, etal., Rizzi et al., Brenda].
3
The distribution of their values in [µM/min] could be best approximated by a LND with
  1,   2.5 .
Flux or concentration control coefficients are calculated as
X
Cv i 
j
d ln X i
d ln v j
where X i denotes the flux through reaction i or the concentration of metabolite i,
respectively. The control coefficients provide a quantitative measure for changes of X i at
perturbation of reaction vj.
We will compare mean values (Mean) and the coefficients of correlation (standard deviation
divided by mean value, CV) of fluxes and concentrations as well as frequency profiles
(relative frequencies of values being in the range 10 y  10 y 1 with y  7,6,...,2 .
Calculations have been performed with the respective routines offered by Mathematica,
Wolfram Research or, in the case of the KEGG glycolysis model, with the routine of the
modelling and simulation environment PyBioS [Wierling, 2004].
3 Results
Fluxes in the linear reaction chain
We consider a linear pathway consisting of 5 successive reactions as depicted in Figure 1. In
the basic version all reactions are reversible (i). The kinetic constants are chosen either from
the UD or LND distribution. Furthermore, we investigate the cases that (ii) all equilibrium
constants are known and have a fixed value equal to 5 (k(Eq)), (iii) negative feedback from
metabolite 4 to reaction 2 (Feedb.), (iv) irreversibility of reaction 2, (v) a combination of (iii)
and (iv), (vi) coupling of reaction 2 with another reaction like coupling with ATP
consumption (Coupl.). In order to get positive fluxes, the concentration of the external
metabolites are set to m0  1 and m5  0 , respectively.
M0
v1
M1
v2
M2
v3
M3
v4
M4
(i)
(iii)

(iv)

(v)
(vi)
Figure 1: Schematic representation of pathway examples
v5
Mm
4
Table 1: Reaction rates (Mean and CV) for the linear reaction chain
Kinetics Reaction
UD
UD LND LND
LND
LND
LND
rates
k(Eq)
k(Eq)
k(Eq)
Feedb. Feedb. Feedb.
Irrev.
Linear
Mean.
CV
MM
Mean
CV
115.34 325.42 19.40
1.27
0.58 4.55
0.76
5.97
3.55
4.82
2.61 0.11
1.44 10.96
LND
LND
Irrev.
11.92
5.37
2.16
5.35
17.33 35.69
4.41 3.33
Coup.
0.19
1.76
0.11
11.79
0.13
23.94
0.10 0.12
14.29 10.09
0.18
1.30
In Table 1 the mean reaction rates and coefficients of variation for in each case 10 5 simulation
runs are given for all considered pathway examples for population with linear and MichaelisMenten kinetics. Figure 2 shows the relative frequencies of flux values in the intervals
( 10 y  10 y 1 ) with y  7,6,...,2 .
0.4
0.3
0.2
0.1
0
Michaelis-Menten Kinetics
Rel. Frequency
Rel. Frequency
Linear Kinetics
- 7 - 6 - 5 - 4- 3 - 2 1 0 1 2
Interval y
0.4
0.3
0.2
0.1
0
- 7 - 6 - 5 - 4- 3 - 2 1 0 1 2
Interval y
Figure 2: Relative frequencies of flux values for a linear chain with linear or MichaelisMenten kinetics. From left to right: red bar: UD (i), Orange bar: LND (i), Yellow-green bar:
LND + Feedback (iii), Green bar: LND + Irrev. (iv), Light blue bar: LND + Irrev. + Feedback
(v), Blue bar: LND + Coupl. (vi). The value for the basic version (i) and the case that all
kinetic constants are equal to 1 is 0.56 for linear kinetics and 0.11 for Michaelis-Menten
kinetics.
In all cases all considered intervals of flux values are covered. The application of the uniform
distribution to the values of the kinetic constants gives a trend to higher flux values compared
to the log normal distribution and a trend to lower values of the coefficient of variance.
Applying the log normal distribution leads to a smeared distribution of fluxes over all
considered order of magnitude. This effect is almost independent of features like
irreversibility or feedback. For the linear kinetics with uniform distribution
For the case of linear kinetics with LND the only sharp distribution results from a coupling
with another reaction (and fixed total concentration of coupling compounds).
The flux values for a reaction chain with Michaelis-Menten kinetics are also smeared, but
they give sharper peaks for both distributions of kinetic constants. Nevertheless, they also
tend to higher values of the coefficients of correlation.
Flux control coefficients
In order to assess the possible effect of perturbations on the flux values we calculated the flux
control coefficients. Figure 4 shows the mean flux control coefficients assigned to the number
of a reaction in the chain together
5
Basic reaction chain (i)
With feedback (iii)
0.4
C vJ
i
0.3
0.3
0.2
0.2
0.1
1
2
3
4
5
0.1
1
Reaction number
2
3
4
5
Irrev. + Feedb. (v)
0.6
0.5
0.4
0.3
0.2
0.1
0
1
Reaction number
2
3
4
5
Reaction number
Figure 4: Flux control coefficients (mean values and error bars) presented depending on the
number of the reaction. The mean CV are 2.21 (i), 1.70 (iii), and 0.65 (v).
It worth to mention that here the direction of net flux is fixed (positive) due to the choice of
external metabolite concentrations ( c Mm  0 ). Hence all flux control coefficients are nonnegative. For the basic chain (i) we find higher flux control coefficients at the beginning and
at the end of the chain compared to the middle reactions. With feedback (iii) the control is
enhanced for the reaction 5 degrading the metabolite 4 which exerts the feedback on reaction
2. With feedback the error bars and the mean CV are smaller than without feedback. In case
of an irreversible reaction, the subsequent reactions may not exert flux control (reaction 3 and
4), except the reaction 5 which degrades the inhibitor. This pattern is very fixed for the
investigated parameter regions.
Concentration control coefficients
The concentration control coefficients allow an estimation of the direction of change of a
metabolite concentration after perturbation of a reaction. Tendetially, producing reactions
have a positive control and degrading reactions exert a negative control over the concentration
of a metabolite. This can be confirmed by our high-throughput simulations as shown in Figure
5.
2
3
4
1 2 3 4 5
Reaction
1
Irrev. + Feedb. (v)
Metabolite
1
With feedback (iii)
Metabolite
Metabolite
Basic reaction chain (i)
2
3
4
1 2 3 4 5
Reaction
1
2
3
4
1 2 3 4 5
Reaction
- 3 - 2 - 1
0
1
2
3
Figure 5: Concentration control coefficients, gray level coded: light or dark squares indicate
positive or negative values of control exerted by a reaction on the concentration of the
respective metabolite.
In the basic chain with or without feedback ((i) and (iii)) producing reactions yield light
squares and degrading reactions yield dark squares. This patterns is slightly changed in the
case of irreversibility and feedback (v): The negative control of the directly degrading
reactions is a bit stronger, and reactions 3 and 4 have on average no are very low positive
control over the concentration of metabolite 1.
6
Branched reaction networks
We have analyzed a series of flat reaction networks of different complexity containing
various numbers of features like feedback inhibition or feedback activation, irreversibility of
individual reactions or coupling of reactions by common metabolites (not graphically
presented). The results are qualitatively the same as for linear reaction chains: the distribution
of steady state flux values is smeared over all considered ordered of magnitude. Also
metabolite concentrations are rarely determined. However, stoichiometric restrictions lead to
fixed ratios of fluxes between respective branches of the network. The flux and concentration
control coefficients tend to fixed pattern. These patterns are increasingly robust, with an
increasing number of feedback or feedforward loops or irreversible reactions.
0.5
0.51
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.6
0.5
0.61
1.
0.63
0.5
0
0.38
0.62
0.45
0.48
0.48
0.53
0.71
0.57
0.5
0.5
0.5
0.54
0.6
0.5
0.6
0.53
0.5
0.52 0.5
0.52
0.48
0.65
0.76
0.45
0.49
0.58
0.53
0.52
0.46
0.53
0.51
Figure 6: Network example. Left: basic version, middle: with one irreversible reaction, right:
with feedbacks from metabolites to forming reactions. All external metabolite concentrations
set to 1. Numbers at arrows indicate the relative frequency of positive compared to negative
flux values at 105 simulation runs.
In the network given in Figure 6, the fluxes may assume positive and negative values with a
probability of 0.5. The absolute values of the CV are in the order of magnitude of 100.
Imposing restrictions to the direction of single reactions (making one reaction irreversible)
has mainly local effects: the signs of the fluxes through neighbouring reactions (involving the
same reactant as substrate or product) are affected. If an input or output flux is irreversible,
then the signs of the other input/output fluxes are shifted (not shown).
Interestingly, this network tends to an accumulation of metabolite. If the values of all kinetic
constants are chosen as 1, then the concentration of all metabolites is 1. At random choice of
constants, the metabolite concentrations reach an order of magnitude of 103 to 105 with CV
between 10 and 100. Thus, in real systems mechanisms should be implemented, that prevent
metabolite accumulation. We tested feedback mechanisms as indicated in Figure 6, right. This
imposed a strong restriction on the absolute values of the fluxes (though not on their sign), but
no significant decrease of the metabolite concentrations. Only the CVs of metabolite
concentrations decreased to values between 4 and 8.
Glycolysis network form KEGG
In order to apply the approach to a system relevant in cellular metabolism we downloaded the
glycerol network for Saccharomyces cerevisiae from the KEGG database. The reaction
network has been populated with linear kinetics and the kinetic constants have been chosen
from LND. We performed 1000 simulation runs with the PyBioS simulation environment
yielding the results presented in Table 2.
7
Table 2. Simulation results for the glycolysis network of Saccharomyces cerevisiae.
Mean
Highest
Highest
Lowest
Lowest
(value)
(instance)
(value)
(instance)
Mean
28.55
496.16 -D-Fructose-1,60.77 Pyruvate
Concentration
CV
Concentration
Mean Flux
bisphosphate
18.68 -D-Fructose-1,6-
3.87
4 602.86
315 268.48
0.91
18.94
CV Flux
bisphosphate
fructose-bisphosphate
aldolase
fructose-bisphosphate
aldolase
0.64 2-(-Hydroxyethyl)
thiamine diphosphate
330.59 Pyruvate-outflow
0.59 dihydrolipoamide
dehydrogenase
Cascades and hierarchical structures
Structures with hierarchy, i.e. structures where the object of one reaction serves as catalyst for
another reaction, are important parts of regulatory pathways in cells.
Signal
1
M1 M2
2
3
M3 M4
4
5
M 5 M6
6
Signal
M1 1 M2 3 M 3
2
4
7
M4 5 M5 M6
8
6
9
1
M7 M8 M9
11
1
0
2
Figure 7: Hierachical structures with one (left) or two (right) loops on every level.
The structures shown in Figure 7 may represent in a simplified form the gene expression
cascade (left structure) comprising DNA activation by transcription factors (upper level),
mRNA formation (middle), and protein production (lower level). The right structure
corresponds to the structure of a MAP kinase cascade.
Mean values and coefficients of variance are calculated for fluxes and concentrations using
105 simulation runs for low, intermediate and high input signal strength (Tables 3 to 6). The
metabolite concentrations on each level must fulfill moiety conservation (set arbitrarily to 1).
Table 3: Fluxes at low, intermediate, high activation for the one-loop cascade
Stimulus Reaction
1 and 2
3 and 4
Strength Rates
5 and 6
Low
Mean
CV
0.29
0.30
1.88
0.14
2.10
0.14
Medium
Mean
CV
6.69
0.23
4.25
0.19
3.00
0.17
High
Mean
CV
30.68
0.30
6.23
0.21
3.76
0.18
Table 4: Metabolite concentrations at low, intermediate, high activation (one loop cascade)
Stimulus Metabolite
1
2
3
4
5
6
strength
Concentration
Low
Mean.
CV
0.77
2.12
0.23
0.65
0.75
1.98
0.25
0.65
0.76
2.03
0.24
0.63
8
Medium
Mean
CV
0.50
1.15
0.50
1.17
0.63
1.50
0.37
0.89
0.69
1.70
0.31
0.76
High
Mean
CV
0.23
0.65
0.77
2.12
0.53
1.24
0.47
1.09
0.64
1.53
0.36
0.86
Table 5: Fluxes at low, intermediate, high activation for the double-loop cascade
Stimulus Reaction
1, 2
3, 4
5, 6
7, 8
9, 10
strength
Rates
11, 12
Low
Mean.
CV
0.21
0.25
2.41
0.13
0.79
0.09
0.55
0.08
0.65
0.08
0.51
0.07
Medium
Mean
CV
4.09
0.19
3.69
0.19
1.59
0.13
1.41
0.12
0.95
0.10
0.84
0.08
High
Mean
CV
15.48
0.22
5.98
0.22
2.08
0.14
1.99
0.14
1.24
0.11
0.81
0.09
Table 6: Metabolite concentrations at low, intermediate, high activation (double-loop
cascade)
Stimulus Metab.
1
2
3
4
5
6
7
8
strength
Conc.
9
Low
Mean.
CV
0.62
1.46
0.14
0.50
0.24
0.66
0.74
1.88
0.14
0.50
0.12
0.41
0.85
2.54
0.08
0.36
0.07
0.32
Medium
Mean
CV
0.36
0.87
0.27
0.75
0.37
0.88
0.63
1.47
0.19
0.59
0.18
0.53
0.78
2.05
0.11
0.43
0.11
0.39
High
Mean
CV
0.15
0.50
0.40
0.98
0.45
1.06
0.55
1.25
0.21
0.63
0.24
0.63
0.73
1.80
0.14
0.48
0.13
0.44
For both types of cascade (one and double-loop) we find clear pattern of fluxes and
concentrations with comparably low CVs.
Interestingly, the coefficients of variance are even lower in the case of double-loop cascade.
Furthermore the signal amplification comparing the concentration of the metabolite with the
highest number (6 in the one loop case, 9 in the double-loop case) at high or low activation is
clearly higher in double-looped hierarchies.
Concentration control coefficients
1
2
Metabolite
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 9 10 11 12
Reaction
- 3 - 2 - 1
0
Figure 8: Concentration control coefficients in the double-loop hierarchy.
1
2
3
9
The concentration control coefficients for both types of cascade (see Figure 8 for the double
loop cascade) exhibit a very robust pattern with an extremely low CV. Although it is clear
from detailed analysis of signaling pathways that effective signal transduction demands
appropriate parameters of kinases and phosphatases our results support the idea, that the
structure of signaling cascade already ensures and stabilizes the transfer of the signal for a
wide range of parameters. The individual dynamics is, of course, determined by the precise
kinetic values.
4 Discussion
Inspired by two recent developments – first, biochemical networks can be reconstructed from
databases; second, attempts are made to use this knowledge for large-scale or whole cell
modeling without caring about individual kinetics – we wanted to test, whether one can
extract information about systems dynamic from the structure of biochemical networks
without applying no or very restricted knowledge about the kinetics of the individual reaction.
To this end we investigated the possible range of steady state fluxes and concentrations and
the respective control coefficients of different artificial and real biochemical networks using
linear and Michaelis-Menten kinetics and chosing the kinetic constants from different
distributions.
We suggest relating the robustness of network properties like steady state fluxes and
concentrations as well as control coefficients with respect to the choice of the individual
kinetics to the respective coefficients of variants.
For networks (without intrinsic hierarchy) like the metabolic network we find, that flux values
are rarely determined by the structure. Even consideration of irreversibility or feedback does
not contribute much to robustness of fluxes. This kind of network is designed to allow various
fluxes and concentration and an adaptation to the actual need of the cell. Prediction of
dynamics is not possible without detailed knowledge of kinetics (kinetic types of the
reactions, respective kinetic parameters, concentrations of external metabolites, equilibrium
constants) as well as concentrations of external metabolites. This is in accordance with many
observations. E.g. Ihmels et al. [Ihmels et al., 2004] argue, that although the structure of
metabolic networks is far from linear reflecting the need for flexibility and diversity of
metabolic flow, only the transcriptional regulation leads the metabolic flow toward
linearity.
In networks with structural hierarchy, i.e. structures where the object of one reaction serves as
catalyst for another reaction, flux and concentration values are well determined, and the
coefficients of variance are quite low. This corresponds to their function of transferring a
signal. The ability of signal amplification is already implemented in this structure, even more
in the cascade with two loops. For these structures, the prediction of fluxes and concentrations
is more promising – although also here, better predictions can be attained with more prior
knowledge. The sign and the value of flux and concentration control coefficients are much
more restricted than in networks of the metabolic type.
Conclusion: in systems biology, the investigation of networks structures must be
accompanied or accomplished by studying the kinetics of the individual interactions.
References