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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The binomial coefficient \({N\choose k}\) is defined as the number of ways to choose k elements from a set of N. It can be calculated as follows:
$${N\choose k}\doteq{N!\over k!(N-k)!}.\vspace*{-5mm}$$ - 2.
Here we truncate the values after two decimal places, which is why the state vectors do not sum to exactly 1.
- 3.
- 4.
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.
- 5.
From here, we omit the full command line unless it introduces a new feature.
- 6.
Actually PRISM produces 4 more columns, corresponding to the transition and state rewards in the two reward constructs. We have omitted these here for the sake of clarity.
- 7.
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.
- 8.
We are referring here to PRISM version 3.3.
- 9.
References
Cherry, J., Adler, F.: How to make a biological switch. Journal of Theoretical Biology 203(2), 117–133 (2000). doi:10.1006/jtbi.2000.1068
Chu, D., Zabet, N., Mitavskiy, B.: Models of transcription factor binding: sensitivity of activation functions to model assumptions. Journal of Theoretical Biology 257(3), 419–429 (2009). doi:10.1016/j.jtbi.2008.11.026
Gardiner, C.: Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Springer, Berlin (2008)
Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Kemper, P. (ed.) Proc. Tools Session of Aachen 2001 International Multiconference on Measurement, Moddeling and Evaluation of Computer-Communication Systems, pp. 7–12, September 2001
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag London Limited
About this chapter
Cite this chapter
Barnes, D.J., Chu, D. (2010). Other Stochastic Methods and Prism. In: Introduction to Modeling for Biosciences. Springer, London. https://doi.org/10.1007/978-1-84996-326-8_6
Download citation
DOI: https://doi.org/10.1007/978-1-84996-326-8_6
Publisher Name: Springer, London
Print ISBN: 978-1-84996-325-1
Online ISBN: 978-1-84996-326-8
eBook Packages: Computer ScienceComputer Science (R0)