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