Abstract
In this work we propose a model that can be used to study the dynamics of mass action systems, systems consisting of a large number of individuals whose behavior is influenced by other individuals that they encounter. Our approach is rather synthetic and abstract, viewing each individual as a probabilistic automaton that can be in one of finitely many discrete states. We demonstrate the type of investigations that can be carried out on such a model using the Populus toolkit. In particular, we illustrate how sensitivity to initial spatial distribution can be observed in simulation.
This work was partially supported by the French ANR project Cadmidia and the NIH Grants K25 CA131558 and R01 GM104973. Part of the work done while the second author was visiting CNRS-VERIMAG. This paper is an extended version of [14].
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Notes
- 1.
Actually bilinear if one assumes the probability of triple encounters to be zero, as is often done in Chemistry.
- 2.
A probabilistic automaton [15] is a Markov chain with an input alphabet where each input symbol induces a different transition matrix. It is also known as a Markov Decision Process (MDP) in some circles.
- 3.
We export the primed variable notation from program verification where x stands for x[t] and \(x'\) denotes \(x[t+1]\).
- 4.
We write the algorithm using the normalized state notation x but the combinatorial calculation underlying the derivation of probabilities will be based on the particle count X.
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Acknowledgment
We thank Eric Fanchon for many useful comments.
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Maler, O., Halász, Á.M., Lebeltel, O., Maler, O. (2015). Exploring Synthetic Mass Action Models. In: Maler, O., Halász, Á., Dang, T., Piazza, C. (eds) Hybrid Systems Biology. HSB 2014. Lecture Notes in Computer Science(), vol 7699. Springer, Cham. https://doi.org/10.1007/978-3-319-27656-4_6
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