Dynamic Stability of Signal Transduction Networks Depending on Downstream and Upstream Specificity of Protein Kinases Bernd Binder and Reinhart Heinrich Theoretical Biophysics, Institute of Biology, Math.-Nat. Dept. I, Humboldt University Berlin, Invalidenstr. 43, D 5 Berlin (Germany) Abstract. Mathematical methods are used for explaining the structural design of signal transduction networks, e.g. MAP kinase cascades, which control cell proliferation, differentation or apoptosis. Taking into account protein kinases and phosphatases the interrelation between the topology of signaling networks and the stability of their ground state are analysed. It is shown that the stability is closely related to the system s dimension and to the number of cycles within the network. Systems with a higher number of kinases and/or cycles tend to be more unstable. In contrast to that increasing phosphatase activity stabilises the ground state. Keywords: signaling network, dynamical stability, ground state, MAP kinases, specificity, digraphs, cycles Abbreviations: digraph directed graph;. Introduction There are several different MAPK cascades in mammalian cells acting in parallel and exhibiting crosstalks among each other and with other pathways like PI3K/Akt. These signaling pathways start with a receptor tyrosine kinase located at the membrane, from where the signal is passed first to adaptor or docking proteins (e.g. Grb). These form a complex with enzymes (e.g. Sos), acting as guanine nucleotide exchange factors for G-proteins like the Ras/Rac proteins. These G-proteins interact with a wide spectrum of downstream effectors, for instance with the MAPKKK (e.g. Raf/MEKK), constituting the first step in MAP kinase modules. The final step of these cascades, the MAP kinases (e.g. ERK,JNK/SAPK) or RSK kinases, act as regulators for various enzymes and activators of transcription factors (Cobb). A topological picture arises, in which some of the kinases are highly specific to downstream molecules (i.e. MEK), whereas others are quite unspecific. Moreover, unspecific action of kinases towards upstream components result in positive feedback loops. As for metabolic networks, the design of signal transduction networks is thought to be the result of an evolutionary optimization process. For glycolysis, TCA-cycle or oxidative respiration, optimization criteria, like high ATP production rate and long-term homeostasis, have been sucessfully applied to explain structural characteristics of metabolic networks (Stephani et al. and Meléndez-Hevia et al.). It has been argued that signal transduction pathways Kluwer Academic Publishers. Printed in the Netherlands. abstract.tex; /7/; :5; p.
B. Binder, R. Heinrich need a special design permitting both stability of the unstimulated ground state and high signal output upon stimulation (Heinrich et al.). In the present mathematical treatment we consider dynamical stability, in terms of the real part of the maximum eigenvalue of the Jacobi matrix, and signal amplification as important characteristics of signal transduction networks. Using simple mass action kinetics for characterising the activity of kinases and phosphatases the dynamics of a protein kinase/phosphatase network is described by a differential equation system (Heinrich et al.): dx i dt j α i j x j x i c i β i x i () where x i is the concentration of the activated kinase i, α i j the second order rate constant for the activation of kinase i by kinase j, and β i the rate constant for inactivation of kinase i by phosphatase i. The concentration of active and inactive kinase i sum up to the total amount of kinase i: c i x i x i. Stability of ground states can be easily determined by considering the Jacobian of the linearized system which can be written as follows: J i j α i j α β i δ i j i j () if kinase i activates kinase j α i j otherwise To further simplify the analysis here we assumed all rate constants of the kinases as well as the rate constants of the phosphatases to be equal: α if kinase i activates kinase j α i j otherwise (3) and for the phosphatases β i β (4) Due to this simplification, the interaction of kinases can be mapped to a directed graph, with the adjacency matrix A: A T J βi where β β α (5) abstract.tex; /7/; :5; p.
Dynamic Stability of Signal Transduction Networks 3. results We generated all possible non-isomorphic digraphs of 4,5 and 6 nodes. According to Pólya Enumeration Theorem (Harray et al.) there are 8, 968, and 54944 digraphs, respectively. The histograms in Figure show the distributions of maximum eigenvalue of Jacobians (with β=) for three different system dimensions (n 4 5 6). The abscissa represents the real part of maximum eigenvalue and the ordinate the number of networks within small intervals Real(λ max ). The percentage of stable networks (Real λ max ) decreases towards higher system dimension. For the given parameter values there are about 4%, 3% and 4% stable systems, respectively. 6 4 node graphs 5 node graphs 4 x 4 6 node graphs 5 8 counts 4 3 6 4 8 6 4 4 λ max 4 4 λ λ max max Figure. Distribution of λ max (spectral radius) of non-isomorphic digraphs with 4,5 and 6 nodes. The rate constants of phosphatases are β=. On the other hand the percentage of systems with stable ground state becomes higher with increasing phosphatase activity. Increasing β results in a shift of the histograms to the left, as becomes clear by the definition of J i j (see Eq. ). In Figure the dependence of the percentage of stable networks (N st N st N inst ) is depicted as a function of β. The plateaus for β are due to the fact, that there arent t any networks for Real λ max (when β ), which can be seen in in Figure. Changing β from to keeps the number of stable networks constant. We also investigated the influence of cycles within digraphs on the stability of the ground state. In Figure networks with λ max are exclusively those with no cycles at all; they are Acyclic Digraphs or trees. For details of relations between spectral and structural properties of digraphs (see Cvetkovic et al.). One would expect cycles to make a dynamical system more unstable, in particular with the assumed parameters values (α ). Figure 3 shows the abstract.tex; /7/; :5; p.3
4 B. Binder, R. Heinrich.8 4 nodes 5 nodes 6 nodes #stable network increases with phosphatase rate constant stable networks [%].6.4..5.5.5 3 3.5 4 β Figure. Correlation between the number of stable networks and β. maximum eigenvalue distribution depending on the amount of cycles within a graph. It is seen that networks with a higher number of cycles have generally higher values for the maximum eigenvalues, meaning that cycles have indeed an destabilising effect. counts (x 4 ) max. eigenvalue distribution of 6 node digraphs depending on number of cycles cycles cycle cycles 3 cycles 4 cycles 5 cycles.5.5.5.5.5 λ max Figure 3. Max. eigenvalue distribution of digraphs with 6 nodes containing different numbers of cycles (β ). abstract.tex; /7/; :5; p.4
Dynamic Stability of Signal Transduction Networks 5 The results presented in this paper for the case of simple mass action kinetics are representive also for less simplified kinetic equations. This is due to the fact that pathway stability is determined by the properties of the Jacobian of the ground state which become independent of saturation phenomena. Beside the number of cycles a characteristic topolocical feature of directed networks is the distribution of degrees (indegrees/outdegrees) among the nodes. The aim of the present work is to investigate correlations between stability of signal transduction networks and these topolocical features resulting from more or less unspecific protein kinases. Because the number of possible non-isomorphic digraphs increases strongly with the number nodes, systematic generation of all digraphs could only be performed up to 6 nodes. References Cobb, M.H., Progr. Biophys. Mol. Bio., 7 (999) 479. Stephani, A., Nuno, J. C. and Heinrich, R., J. Theor. Biol., 99 (999) 45. Meléndez-Hevia, E., Wadell, T.G., and Cascante, M., J. Mol. Evol., 43 (996) 93. Heinrich, R., Neel, B.G., Rapoport, T.A., Molecular Cell, 9 () 957. Harary, F. and Palmer, E.M. (Ed.) Graphical Enumeration, Academic Press, NY, 973, p. 4 and p. 4. Cvetkovic, D.M., Doob M. and Sachs, H. (Ed.) Spectra of Graphs, Academic Press, NY, 98, p. 8. Address for Offprints: e-mail: bernd.binder@rz.hu-berlin.de abstract.tex; /7/; :5; p.5
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