Approximating Probabilistic Behaviors of Biological Systems Using Abstract Interpretation

Abstract

In this paper we apply the Abstract Interpretation approach [8,9] for approximating the behavior of biological systems, modeled specifically using the Chemical Ground Form calculus [4], a new stochastic calculus rich enough to model the dynamics of biochemical reactions. Our analysis computes an Interval Markov Chain (IMC) that safely approximates the Discrete-Time Markov Chain, describing the probabilistic behavior of the system, and reports both lower and upper bounds for probabilistic temporal properties. Our analysis has several advantages: (i) the method is effective (even for infinite state systems) and allows us to systematically derive an IMC from an abstract labeled transition system; (ii) using intervals for abstracting the multiplicity of reagents allows us to achieve conservative bounds for the concrete probabilities of a set of concrete experiments which differs only for initial concentrations.

Probabilistic Abstract Interpretation

“…S p P ¨ Γ or equivalently ∀s ∈ S p P : ∃s ∈ Γ : s˙ s that is ∀s ∈ S p P : ∃s ∈ Γ : ∀ω ∈ Ω : s(ω) s (ω) which holds since [3,3] …”

“…1). This definition covers most probabilistic models of computation found in the literature such as program semantics [17], Markov decision processes [2,3,10,11,22,29], etc. Example 1.…”

“…There is a lot of work on probabilistic program construction and verification methods [13,15,19,23], probabilistic model-checking [11], probabilistic abstract modelchecking [2,27,29], probabilistic abstract interpretation [21,25,28], with, in the case of model-checking and abstract interpretation, existing applications to biological pathways [1,3,18]. One of our objectives is to unify and generalize these frameworks.…”

Probabilistic Model Checking of Biological Systems with Uncertain Kinetic Rates

“…The closest approach to ours is presented in [12], where a systematic technique for abstracting a set of DTMC, each representing a concrete experiment, is proposed. Their abstraction approximates the information about the multiplicity of reagents present in a solution by means of intervals of integers.…”

Best Probabilistic Transformers

“…While in [10][11][12][13] ideas from abstract interpretation have been applied to probabilistic models, to the best of our knowledge, there is no preceding framework for MDP abstraction with Galois connections and best transformers. Papers [11][12][13] target deterministic models and not MDPs.…”