# Symbolic Dynamics of Biochemical Pathways as Finite States Machines

## Abstract

We discuss the symbolic dynamics of biochemical networks with separate timescales. We show that symbolic dynamics of monomolecular reaction networks with separated rate constants can be described by deterministic, acyclic automata with a number of states that is inferior to the number of biochemical species. For nonlinear pathways, we propose a general approach to approximate their dynamics by finite state machines working on the metastable states of the network (long life states where the system has slow dynamics). For networks with polynomial rate functions we propose to compute metastable states as solutions of the tropical equilibration problem. Tropical equilibrations are defined by the equality of at least two dominant monomials of opposite signs in the differential equations of each dynamic variable. In algebraic geometry, tropical equilibrations are tantamount to tropical prevarieties, that are finite intersections of tropical hypersurfaces.

## Notes

### Acknowledgements

O.R and A.N are supported by INCa/Plan Cancer grant N\(^\circ \)ASC14021FSA.

### References

- 1.Andrieux, G., Fattet, L., Le Borgne, M., Rimokh, R., Théret, N.: Dynamic regulation of Tgf-B signaling by Tif1\(\gamma \): a computational approach. PloS One
**7**(3), e33761 (2012)CrossRefGoogle Scholar - 2.Chiavazzo, E., Karlin, I.: Adaptive simplification of complex multiscale systems. Phys. Rev. E
**83**(3), 036706 (2011)CrossRefGoogle Scholar - 3.Gorban, A., Karlin, I.: Invariant Manifolds for Physical and Chemical Kinetics. Lecture Notes in Physics, vol. 660. Springer, Heidelberg (2005)MATHGoogle Scholar
- 4.Gorban, A., Radulescu, O.: Dynamic and static limitation in reaction networks, revisited. In: Guy B. Marin, D.W., Yablonsky, G.S. (eds.) Advances in Chemical Engineering - Mathematics in Chemical Kinetics and Engineering. Advances in Chemical Engineering, vol. 34, pp. 103–173. Elsevier, Amsterdam (2008)Google Scholar
- 5.Grigoriev, D.: Complexity of solving tropical linear systems. Comput. Complex.
**22**(1), 71–88 (2013)MathSciNetCrossRefMATHGoogle Scholar - 6.Grigoriev, D., Podolskii, V.V.: Complexity of tropical and min-plus linear prevarieties. Comput. Complex.
**24**, 31–64 (2015)MathSciNetCrossRefGoogle Scholar - 7.Haller, G., Sapsis, T.: Localized instability and attraction along invariant manifolds. SIAM J. Appl. Dyn. Syst.
**9**(2), 611–633 (2010)MathSciNetCrossRefMATHGoogle Scholar - 8.Maas, U., Pope, S.B.: Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combust. Flame
**88**(3), 239–264 (1992)CrossRefGoogle Scholar - 9.Noel, V., Grigoriev, D., Vakulenko, S., Radulescu, O.: Tropical geometries and dynamics of biochemical networks application to hybrid cell cycle models. In: Feret, J., Levchenko, A. (eds.) Proceedings of the 2nd International Workshop on Static Analysis and Systems Biology (SASB 2011). Electronic Notes in Theoretical Computer Science, vol. 284, pp. 75–91. Elsevier (2012)Google Scholar
- 10.Noel, V., Grigoriev, D., Vakulenko, S., Radulescu, O.: Tropicalization and tropical equilibration of chemical reactions. In: Topical and Idempotent Mathematics and Applications, vol. 616. American Mathematical Society (2014)Google Scholar
- 11.Palis, J.: A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque
**261**, 339–351 (2000)Google Scholar - 12.Radulescu, O., Gorban, A.N., Zinovyev, A., Lilienbaum, A.: Robust simplifications of multiscale biochemical networks. BMC Syst. Biol.
**2**(1), 86 (2008)CrossRefGoogle Scholar - 13.Radulescu, O., Gorban, A.N., Zinovyev, A., Noel, V.: Reduction of dynamical biochemical reactions networks in computational biology. Front. Genet.
**3**(131) (2012)Google Scholar - 14.Radulescu, O., Vakulenko, S., Grigoriev, D.: Model reduction of biochemical reactions networks by tropical analysis methods. Mathematical Model of Natural Phenomena (2015, in press)Google Scholar
- 15.Samal, S.S., Radulescu, O., Grigoriev, D., Fröhlich, H., Weber, A.: A tropical method based on newton polygon approach for algebraic analysis of biochemical reaction networks. In: 9th European Conference on Mathematical and Theoretical Biology (2014)Google Scholar
- 16.Soliman, S., Fages, F., Radulescu, O.: A constraint solving approach to model reduction by tropical equilibration. Algorithms Mol. Biol.
**9**(1), 24 (2014)CrossRefGoogle Scholar - 17.Theobald, T.: On the frontiers of polynomial computations in tropical geometry. J. Symbolic Comput.
**41**(12), 1360–1375 (2006)MathSciNetCrossRefMATHGoogle Scholar - 18.Tsuda, I.: Chaotic itinerancy as a dynamical basis of hermeneutics in brain and mind. World Futures: J. Gen. Evol.
**32**(2–3), 167–184 (1991)CrossRefGoogle Scholar