Abstract
This chapter introduces the reader to the concept of stochastic systems. It motivates the importance of noise and stochastic fluctuations in biological modeling and introduces some of the basic concepts of stochastic systems, including Markov chains and partition functions. The main objective of this theoretical part is to provide the reader with sufficient theoretical background to be able to understand original research papers in the field. Strong emphasis is placed on conveying a conceptual understanding of the topics, while avoiding burdening the reader with unnecessary mathematical detail. The second part of this chapter describes PRISM, which is a powerful computational tool for formulating, analyzing and simulating Markov-chain models. Throughout the chapter, concepts are illustrated using biologically-motivated case studies.
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Notes
- 1.
The binomial coefficient \(\left( {\begin{array}{c}n\\ k\end{array}}\right) \) is defined as the number of ways to choose k elements from a set of n. It can be calculated as follows:
$$\begin{aligned} \left( {\begin{array}{c}n\\ k\end{array}}\right) \doteq \frac{n!}{k! (n-k)!}. \end{aligned}$$ - 2.
Here we truncate the values after two decimal places, which is why the state vectors do not sum to exactly 1.
- 3.
Depending on the exact details of the reader’s working environment, it may be necessary to precede the prism command by a full specification of the path to the prism executable. Exact details of this will depend on the operating system and the installation.
- 4.
From here, we omit the full command line unless it introduces a new feature.
- 5.
We have truncated the time values to 3 decimal places.
- 6.
The astute reader will notice that our sample path is only 16 lines long even though we specified a length of 20. The reason for this is that, after 16 transitions the chain has reached its absorbing state and no more transitions are possible. Hence it halted.
- 7.
We are referring here to PRISM version 4.2.1.
References
Gardiner, C.: Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences. Springer, Berlin (2008)
Chu, D., Zabet, N., Mitavskiy, B.: Models of transcription factor binding: sensitivity of activation functions to model assumptions. J. Theor. Biol. 257(3), 419–429 (2009). doi:10.1016/j.jtbi.2008.11.026
Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) Proceedings of the 23rd International Conference on Computer Aided Verification (CAV’11). LNCS, vol. 6806, pp. 585–591. Springer, Berlin (2011)
PRISM. http://www.prismmodelchecker.org/. Accessed 27 June 2015
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Barnes, D.J., Chu, D. (2015). Other Stochastic Methods and Prism. In: Guide to Simulation and Modeling for Biosciences. Simulation Foundations, Methods and Applications. Springer, London. https://doi.org/10.1007/978-1-4471-6762-4_6
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DOI: https://doi.org/10.1007/978-1-4471-6762-4_6
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