Korunumlu döngülerin stokastik simülasyon algoritmalarında kullanımı

Year 2016, Volume: 20 Issue: 3, 617 - 625, 02.09.2016

Derya Altintan

https://doi.org/10.16984/saufenbilder.22901

Abstract

Biyokimyasal reaksiyon sistemleri farklı reaksiyonlar  aracılığıyla etkileşime giren birçok farklı  türü içerir. Sistem içerisinde yer alan türlerin sayıları ve miktarları çok yüksek olduğunda, diferansiyel denklemlere dayanan saf modelleme yaklaşımları çok boyutluluktan muzdarip olurlar. Eğer bir sistem korunumlu döngüler içerirse,  bazı türlerin miktarları cebirsel bağlantılar yoluyla elde edilebilir bu da sistemin dinamiklerini  temsil eden diferansiyel denklemlerin boyutunu düşürür. Bu çalışmada, biyokimyasal reaksiyon sistemlerinde yer alan korunumlu döngüleri elde etmek için Gauss-Jordan metodunu kullanan bir nümerik algoritma öneriyoruz. Algoritmayı stokastik modelleme yaklaşımında konum vektörünün tam realizasyonlarını elde eden Direk  Metod (DM), İlk Reaksiyon Metodu (FRM) ve Sonraki Reaksiyon Metodu (FRM) içerisinde verdik. Bu üç algoritmayı korunum bağıntılarını içerecek/içermecek şekilde farklı boyutlardaki biyokimyasal sistemlere uyguladık ve   her tam lagoritmanın farklı iki versiyonunun hesaplama miktarları kıyasladık.

Keywords

korunum bağıntıları, gauss-jordan metodu, direk metod, ilk reaksiyon metodu, sonraki reaksiyon metodu.

Using conserved cycles in exact stochastic simulation algorithms

Abstract

Biochemical reaction systems involve many different species interacting via many different reaction channels. When the number of species and the abundance of species are so high, pure modeling approaches based on differential equations suffer from curse of dimensionality. If a system involves conserved cycles, abundances of some species can be obtained via algebraic relations which in turn will reduce the dimension of differential equations representing the dynamics of the system. In the present paper, we propose a numerical algorithm that uses Gauss-Jordan method to obtain conserved cycles in biochemical systems. We give this algorithm in Direct Method (DM), First Reaction Method (FRM) and Next Reaction Method (NRM) which obtain exact realizations of the state vector in stochastic modeling approach. We apply these three algorithms with/without using conservation relations to biochemical systems in different sizes and compare the computational costs of two different versions of each exact algorithm.

 

Keywords

conservation relations, gauss-jordan method, direct method, first reaction method, next reaction method

References

• A. Kremling, J. Saez-Rodriguez, Systems biologyan engineering perspective, Journal
• of Biotechnology 129 (2007) 329-351.
• D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem 81 (1977) 2340-2361.
• N. G. v. Kampen, Stochastic processes in physics and chemistry, Amsterdam ; New York : North-Holland ; New York : sole distributors for the USA and Canada,
• Elsevier North-Holland, 1981.
• D. Wilkinson, Stochastic modelling for systems biology, Chapman & Hall/CRC mathematical and computational biology series, Boca Raton, FL : Taylor & Francis,
• D. Anderson, T. Kurtz, Continuous time markov chain models for chemical reaction networks, in: H. Koeppl, G. Setti, M. d. Bernardo, D. Densmore (Eds.), Design and
• Analysis of Biomolecular Circuits, Springer-Verlag, 2011.
• D. T. Gillespie, A rigorous derivation of the chemical master equation, Physica A 188 (1992) 404-425.
• F. Cordero, A. Horvath, D. Manini, L. Napione, M. De Pierro, S. Pavan, A. Picco, A. Veglio, M. Sereno, F. Bussolino, G. Balbo, Simplication of a complex signal
• transduction model using invariants and flow equivalent servers, Theoretical Computer Science 412 (2011) 6036-6057.
• T. R. Maarleveld, R. A. Khandelwal, B. G. Olivier, B. Teusink, F. J. Bruggeman, Basic concepts and principles of stoichiometric modeling of metabolic networks, Biotechnol.
• J. 8 (2013) 997-1008.
• D. W. Schryer, M. Vendelin, P. Peterson, Symbolic
• flux analysis for genome-scalemetabolic networks, BMC Systems Biology 5 (81).
• R. Winkler, Stochastic dierential algebraic equations in transient noise analysis, in: Scientic Computing in Electrical Engineering, Mathematics in Industry, Springer,
• , pp. 151-158.
• H. Sauro, B. Ingalls, Conservation analysis in biochemical networks computational issues for software writers, Biophysical Chemistry 109 (2004) 1-15.
• E. Klipp, W. Liebermeister, C. Wierling, A. Kowald, H. Lehrach, R. Herwig, Systems Biology: A Textbook, WILEY - VCH Verlag GmbH & Co. KGaA, 2009.
• R. R. Vallabhajosyula, V. Chickarmane, H. M. Sauro, Conservation analysis of large biochemical networks, Bioinformatics 22 (3) (2006) 346-353.
• H. S. Haraldsdottir, R. M. T. Fleming, Identication of conserved moieties in metabolic networks by graph theoretical analysis of atom transition networks, arXiv
• (2016). arXiv:arXiv:1605.05371.
• S. Schuster, T. Hofer, Determining all extreme semi-positive conservation relations in chemical reaction systems : A test criterion for conservativity, J. Chem. Soc. Faraday
• Trans. 87 (16) (1991) 2561-2566.
• A. Cornish-Bowden, J.-H. Hofmeyr, The role of stoichiometric analysis in studies of metabolism : An example, Journal of Theoretical Biology 216 (2002) 179-191.
• R. Winkler, Stochastic dierential algebraic equations of index 1 and applications in circuit simulation, Journal of Computational and Applied Mathematics 157 (2003)
• -505.
• A. Ganguly, D. Altntan, H. Koeppl, Jump-diusion approximation of stochastic reaction dynamics: Error bounds and algorithms, Multiscale Model. Simul. 13 (4) (2015) 1390-1419.
• D. Gillespie, The chemical langevin equation, Journal of Chemical Physics 113 (1) (2000) 297-306.
• T. Jahnke, W. Huisinga, Solving the chemical master equation for monomolecular reaction systems analytically, J. Math. Biol. 54 (2007) 1{26.
• D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys. 22 (1976) 403-434.
• M. Gibson, J. Bruck, Ecient exact stochastic simulation of chemical systems with
• many species and many channels, The Journal of Physical Chemistry A 104 (9)
• (2000) 1876-1889.
• Y. Cao, H. Li, L. Petzold, Ecient formulation of the stochastic simulation algorithm
• for chemically reacting systems, J. Chem. Phys 121 (9) (2004) 4059-4067.