Model Decomposition and Stochastic Fragments

Tatjana Petrov, Arnab Ganguly, Heinz Koeppl
2012 Electronical Notes in Theoretical Computer Science  
In this paper, we discuss a method for decomposition, abstraction and reconstruction of the stochastic semantics of rule-based systems with conserved number of agents. Abstraction is induced by counting fragments instead of the species, which are the standard entities of information in molecular signaling. The rule-set can be decomposed to smaller rule-sets, so that the fragment-based dynamics of the whole rule-set is exactly a composition of species-based dynamics of smaller rule-sets. The
more » ... rule-sets. The reconstruction of the transient species-based dynamics is possible for certain initial distributions. We show that, if all the rules in a rule set are reversible, the reconstruction of the species-based dynamics is always possible at the stationary distribution. We use a case study of colloidal aggregation to demonstrate that the method can reduce the state space exponentially with respect to the standard, species-based description. views. This is because the language Kappa for specifying reactions allows for symbolic encoding of reactants by using site-graphs instead of structureless variables. The decomposition is performed by detecting that, for example, modifications over one site of the agents' interface never condition the state of another site on the interface of the same agent. The equivalent observation is exploited in the framework of stochastic fragments [8], [9] -where we directly, without performing decomposition, observe entities of information that are more abstract than the standard species and are called stochastic fragments. To illustrate the idea behind the decomposition and stochastic fragments, consider a programming module A that contains two Boolean variables x and y. The values of variables change as a discrete-time stochastic process, so that the next value of each variable is conditioned only on the current value of that same variable. Assume that the module can be instantiated more times, and that all instances are running in parallel. For example, consider two instances of module A: A 1 and A 2 . Let Z n ∈ {(i 1 , i 2 , i 3 , i 4 )|i 1 +i 2 +i 3 +i 4 = 2} represent the state (i 1 , i 2 , i 3 , i 4 ) at time n, with i 1 instances of A setting (x, y) to (0, 0), and i 2 , i 3 , i 4 instances of A setting (x, y) to (0, 1), (1, 0) and (1, 1) respectively. Due to the independent updates of variables x and y, we can decompose module A to two smaller modules-A x , that contains only the updates of variable x, and A y , that contains the updates of variable y. Let the random variables X n ∈ {0, 1, 2} and Y n ∈ {0, 1, 2} represent the number of x and y variables that are set to 1 at time n ∈ N. The independence of x and y allows us to compute the correct joint probability of, for example, states with one variable x set to 1, and one variable y set to 1: P ((X, Y ) n = (1, 1)) = P (X n = 1)P (Y n = 1). The sites x and y that are taking value 1 may belong to the same instance of A, that is, Z n = (0, 1, 1, 0), or to different instances of A, that is, Z n = (1, 0, 0, 1). Hence, P ((X, Y ) n = (1, 1)) = P (Z n ∈ {(0, 1, 1, 0), (1, 0, 0, 1)}). Finally, knowing that there is one variable x set to 1, and one variable y set to 1, the conditional probability that they belong to the same instance of A is P (Z n = (1, 0, 0, 1)|X n = 1 and Y n = 1) = 0.5. Along these lines, we show that the decompositions of rule-based systems [14] give rise to counting fragments [9] instead of species, and that we can effectively reconstruct information about the concrete system by only analyzing the abstract one. We start by encoding the rule-based models and assigning the stochastic semantics to it. In Section 2, we detail how to encode the rule-based models. In Section 3, we define the fragment-based abstraction, and how to decompose the rule set into smaller independent units. In Theorem 3.6, we demonstrate how these two frameworks relate. In Section 4, we use a model of colloidal aggregation, to demonstrate that the method can exponentially reduce the state space. Finally, in Section 5, we review the practical aspects of using the fragment-based abstraction. In particular, the probability distributions over the species-based system can be reconstructed from the fragment-based abstraction for certain initial distributions. We show in Theorem 5.2 that the reconstruction is applicable on a set of reversible rules, regardless of the initial distribution, because the underlying process is a non-explosive, irreducible CTMC with a stationary distribution. T. Petrov et al. /
doi:10.1016/j.entcs.2012.05.018 fatcat:irrptfrypvh3vgqgzexqlnxwge