Continuous time boolean modeling for biological signaling: application of Gillespie algorithm

BMC Systems Biology

Laurence Calzone¹  Emmanuel Barillot¹  Eric Viara²  Gautier Stoll¹ 

[1] Mines ParisTech, Fontainebleau, F-77300 France;Sysra, Yerres, F-91330 France

关键词: Gillespie algorithm;    Markov process;    Continuous time;    Boolean modeling;   

Others  :  1143658
DOI  :  10.1186/1752-0509-6-116

 received in 2012-04-04, accepted in 2012-08-15,  发布年份 2012

【 摘 要 】

Mathematical modeling is used as a Systems Biology tool to answer biological questions, and more precisely, to validate a network that describes biological observations and predict the effect of perturbations. This article presents an algorithm for modeling biological networks in a discrete framework with continuous time.

Background

There exist two major types of mathematical modeling approaches: (1) quantitative modeling, representing various chemical species concentrations by real numbers, mainly based on differential equations and chemical kinetics formalism; (2) and qualitative modeling, representing chemical species concentrations or activities by a finite set of discrete values. Both approaches answer particular (and often different) biological questions. Qualitative modeling approach permits a simple and less detailed description of the biological systems, efficiently describes stable state identification but remains inconvenient in describing the transient kinetics leading to these states. In this context, time is represented by discrete steps. Quantitative modeling, on the other hand, can describe more accurately the dynamical behavior of biological processes as it follows the evolution of concentration or activities of chemical species as a function of time, but requires an important amount of information on the parameters difficult to find in the literature.

Results

Here, we propose a modeling framework based on a qualitative approach that is intrinsically continuous in time. The algorithm presented in this article fills the gap between qualitative and quantitative modeling. It is based on continuous time Markov process applied on a Boolean state space. In order to describe the temporal evolution of the biological process we wish to model, we explicitly specify the transition rates for each node. For that purpose, we built a language that can be seen as a generalization of Boolean equations. Mathematically, this approach can be translated in a set of ordinary differential equations on probability distributions. We developed a C++ software, MaBoSS, that is able to simulate such a system by applying Kinetic Monte-Carlo (or Gillespie algorithm) on the Boolean state space. This software, parallelized and optimized, computes the temporal evolution of probability distributions and estimates stationary distributions.

Conclusions

Applications of the Boolean Kinetic Monte-Carlo are demonstrated for three qualitative models: a toy model, a published model of p53/Mdm2 interaction and a published model of the mammalian cell cycle. Our approach allows to describe kinetic phenomena which were difficult to handle in the original models. In particular, transient effects are represented by time dependent probability distributions, interpretable in terms of cell populations.

【 授权许可】

   
2012 Stoll et al.; licensee BioMed Central Ltd.

【 图 表 】

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

【 参考文献 】

• [1]Kauffman S: Homeostasis and differentiation in random genetic control networks. Nature 1969, 224:177-178.
• [2]Thomas R: Regulatory networks seen as asynchronous automata: a logical description. J Theor Biol 1991, 153:1-23.
• [3]Stoll G, Rougemont J, Naef F: Few crucial links assure checkpoint efficiency in the yeast cell-cycle network. Bioinformatics 2006, 22(20):2539-2546.
• [4]Stoll G, Bischofberger M, Rougemont J, Naef F: Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico. Biosystems 2010, 102:3-10.
• [5]Garg A, Mohanram K, Di Cara A, De Micheli G, Xenarios I: Modeling stochasticity and robustness in gene regulatory networks. Bioinformatics 2009, 25(12):i101—i109.
• [6]Fauré A, Chaouiya C, Thieffry D, Naldi A: Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle. Bioinformatics 2006, 22(14):e124—e131.
• [7]Shmulevich I, Dougherty E, Kim S, Zhang W: Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics 2002, 18(2):261.
• [8]Gonzalez A, Naldi A, Sanchez L, Thieffry D, Chaouiya C: GINsim: a software suite for the qualitative modelling, simulation and analysis of regulatory networks. Biosystems 2006, 84(2):91-100.
• [9]Wunderlich Z, DePace A: Modeling transcriptional networks in Drosophila development at multiple scales. Curr Opin Genet Dev 2011, 21(6):711-718.
• [10]MacArthur B, Ma’ayan A, Lemischka I: Systems biology of stem cell fate and cellular reprogramming. Nat Rev Mol Cell Biol 2009, 10(10):672-681.
• [11]Saez-Rodriguez J, Alexopoulos L, Zhang M, Morris M, Lauffenburger D, Sorger P: Comparing signaling networks between normal and transformed hepatocytes using discrete logical models. Cancer Res 2011, 71(16):5400.
• [12]Siebert H, Bockmayr A: Temporal constraints in the logical analysis of regulatory networks. Theor Comput Sci 2008, 391(3):258-275.
• [13]Öktem H, Pearson R, Yli-Harja O, Nicorici D, Egiazarian K, Astola J, et al.: A computational model for simulating continuous time Boolean networks. In Proceedings of IEEE International Workshop on Genomic Signal Processing and Statistics (GENSIPS’02). NC, USA: Raleigh; October 2002.
• [14]Vahedi G, Faryabi B, Chamberland J, Datta A, Dougherty E: Sampling-rate-dependent probabilistic Boolean networks. J Theor Biol 2009, 261(4):540-547.
• [15]Sevim V, Gong X, Socolar J: Reliability of transcriptional cycles and the yeast cell-cycle oscillator. PLoS Comput Biol 2010, 6(7):e1000842.
• [16]Teraguchi S, Kumagai Y, Vandenbon A, Akira S, Standley D: Stochastic binary modeling of cells in continuous time as an alternative to biochemical reaction equations. Phys Rev E 2011., 062903(6)
• [17]Bauer A, Jackson T, Jiang Y, Rohlf T: Receptor cross-talk in angiogenesis: mapping environmental cues to cell phenotype using a stochastic, Boolean signaling network model. J Theor Biol 2010, 264(3):838-846.
• [18]Abou-Jaoudé W, Ouattara D, Kaufman M: From structure to dynamics: frequency tuning in the p53-Mdm2 network: I. Logical approach. J Theor Biol 2009, 258(4):561-577.
• [19]Van Kampen N: Stochastic Processes in Physics and Chemistry. Amsterdam, Netherlands: Elsevier; 2004.
• [20]Shiryaev A: Probability, volume 95 of Graduate texts in mathematics. Springer-Verlag: New York, USA; 1996.
• [21]Chaves M, Albert R, Sontag E: Robustness and fragility of Boolean models for genetic regulatory networks. J Theor Biol 2005, 235(3):431-449.
• [22]Chaouiya C: Petri net modelling of biological networks. Briefings in Bioinformatics 2007, 8(4):210-219.
• [23]Young W, Elcock E: Monte Carlo studies of vacancy migration in binary ordered alloys: I. Proceedings of the Physical Society 1966, 89:735.
• [24]Gillespie D: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 1976, 22(4):403-434.
• [25]Klamt S, Saez-Rodriguez J, Gilles E: Structural and functional analysis of cellular networks with CellNetAnalyzer. BMC Syst Biol 2007, 1:2. BioMed Central Full Text
• [26]Müssel C, Hopfensitz M, Kestler H: BoolNet–an R package for generation, reconstruction and analysis of Boolean networks. Bioinformatics 2010, 26(10):1378-1380.
• [27]De Jong H, Geiselmann J, Hernandez C, Page M: Genetic Network Analyzer: qualitative simulation of genetic regulatory networks. Bioinformatics 2003, 19(3):336-344.
• [28]Di Cara A, Garg A, De Micheli G, Xenarios I, Mendoza L: Dynamic simulation of regulatory networks using SQUAD. BMC Bioinf 2007, 8:462. BioMed Central Full Text
• [29]Novak B, Tyson J: A model for restriction point control of the mammalian cell cycle. J Theor Biol 2004, 230(4):563-579.
• [30]Sahin Ö, Fröhlich H, Löbke C, Korf U, Burmester S, Majety M, Mattern J, Schupp I, Chaouiya C, Thieffry D, et al.: Modeling ERBB receptor-regulated G1/S transition to find novel targets for de novo trastuzumab resistance. BMC Syst Biol 2009, 3:1. BioMed Central Full Text
• [31]Schlatter R, Schmich K, Vizcarra I, Scheurich P, Sauter T, Borner C, Ederer M, Merfort I, Sawodny O: ON/OFF and beyond-a boolean model of apoptosis. PLoS Comput Biol 2009, 5(12):e1000595.