# Expected return time to the initial state for biochemical systems with linear cyclic chains: unidirectional and bidirectional reactions

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## Abstract

Biochemical systems are robust in nature. We define robustness of a biochemical system as the property where during time evolution, a closed system returns to its initial state. In this study, we propose some mathematical formulations to analyse the robustness of a closed biochemical system. We have provided a tentative guideline towards applying the theory to a non-closed system. We know that a biochemical system evolves with time as a continuous-time Markov process. When this Markov chain is irreducible, it can be proved theoretically that the system will always return to its initial state, and also the expected time of return can be determined. This return time depends upon the stationary probability distribution, which is determined as the solution of an eigenvalue equation \(xQ=0\) where *Q* is the transition rate matrix. We calculate this expected return time for five different closed systems: unidirectional cyclic linear chains, bidirectional cyclic linear chains and three real biological systems, and verify the theoretical results against the average return time obtained by stochastic simulation.

## Keywords

Stochastic simulation TCA cycle stationary probability Markov chain Chapman–Kolmogorov Equation## Notes

### Acknowledgements

Mr Debraj Ghosh, one of the authors, gratefully acknowledges CSIR, India, for providing him a Senior Research Fellowship (9/93 (0150)/2013, EMR-I).

## References

- 1.McQuarrie D A 1967 Stochastic approach to chemical kinetics.
*J. Appl. Probab.*4(3): 413–478MathSciNetCrossRefGoogle Scholar - 2.Gillespie D T 1992 A rigorous derivation of the chemical master equation.
*Phys. A Stat. Mech. Appl.*188(1): 404–425CrossRefGoogle Scholar - 3.Goutsias J 2007 Classical versus stochastic kinetics modeling of biochemical reaction systems.
*Biophys. J.*92(7): 2350–2365CrossRefGoogle Scholar - 4.Gillespie D T 1976 A general method for numerically simulating the stochastic time evolution of coupled chemical reactions.
*J. Comput. Phys.*22(4): 403–434MathSciNetCrossRefGoogle Scholar - 5.Gillespie D T 1977 Exact stochastic simulation of coupled chemical reactions.
*J. Phys. Chem.*81(25): 2340–2361CrossRefGoogle Scholar - 6.Cao Y, Li H and Petzold L 2004 Efficient formulation of the stochastic simulation algorithm for chemically reacting systems.
*J. Chem. Phys*121(9): 4059–4067CrossRefGoogle Scholar - 7.McCollum J M, Peterson G D, Cox C D, Simpson M L and Samatova N F 2006 The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behavior.
*Comput. Biol. Chem.*30(1): 39–49CrossRefGoogle Scholar - 8.Ramaswamy R, González-Segredo N and Sbalzarini I F 2009 A new class of highly efficient exact stochastic simulation algorithms for chemical reaction networks.
*J. Chem. Phys.*130(24): 244104CrossRefGoogle Scholar - 9.Ramaswamy R and Sbalzarini I F 2010 A partial-propensity variant of the composition-rejection stochastic simulation algorithm for chemical reaction networks.
*J. Chem. Phys.*132(4): 044102CrossRefGoogle Scholar - 10.Thanh V H, Priami C and Zunino R 2014 Efficient rejection-based simulation of biochemical reactions with stochastic noise and delays.
*J. Chem. Phys.*141(13): 134116CrossRefGoogle Scholar - 11.Thanh V H, Zunino R and Priami C 2016 Efficient constant-time complexity algorithm for stochastic simulation of large reaction networks.
*IEEE/ACM Trans. Comput. Biol. Bioinf.*14(3): 657–667CrossRefGoogle Scholar - 12.Ghosh D and De R K 2017 Slow update stochastic simulation algorithms for modeling complex biochemical networks.
*Biosystems*162: 135–146CrossRefGoogle Scholar - 13.Gillespie D T 2001 Approximate accelerated stochastic simulation of chemically reacting systems.
*J. Chem. Phys.*115(4): 1716–1733CrossRefGoogle Scholar - 14.Chatterjee A, Vlachos D G and Katsoulakis M A 2005 Binomial distribution based \(\tau \)-leap accelerated stochastic simulation.
*J. Chem. Phys.*122(2): 024112CrossRefGoogle Scholar - 15.Tian T and Burrage K 2004 Binomial leap methods for simulating stochastic chemical kinetics.
*J. Chem. Phys.*121(21): 10356–10364CrossRefGoogle Scholar - 16.Cao Y and Petzold L 2005 Trapezoidal tau-leaping formula for the stochastic simulation of biochemical systems. In:
*Proceedings of Foundations of Systems Biology in Engineering (FOSBE 2005)*, pp. 149–152Google Scholar - 17.Marchetti L, Priami C and Thanh V H 2017
*Simulation algorithms for computational systems biology*. Springer International Publishing, BerlinGoogle Scholar - 18.Norris J R 1998 Markov chains. In:
*Cambridge Series in Statistical and Probabilistic Mathematics*. Cambridge University Press, CambridgeGoogle Scholar - 19.Sigman K 2009 Continuous-time Markov chains. In: Notes for 1 IEOR 6711 Stochastic Processes I, Columbia UniversityGoogle Scholar
- 20.Singh V K and Ghosh I 2006 Kinetic modeling of tricarboxylic acid cycle and glyoxylate bypass in
*mycobacterium tuberculosis*, and its application to assessment of drug targets.*Theor. Biol. Med. Model.*3(1): 1CrossRefGoogle Scholar - 21.Chapman S P, dos Santos M T, Johnson G N, Kritz M V and Schwartz J M 2017 Cyclic decomposition explains a photosynthetic down regulation for
*Chlamydomonas reinhardtii*.*Biosystems*162: 119–127Google Scholar