Exploring Parameter Space of Stochastic Biochemical Systems Using Quantitative Model Checking

Abstract

We propose an automated method for exploring kinetic parameters of stochastic biochemical systems. The main question addressed is how the validity of an a priori given hypothesis expressed as a temporal logic property depends on kinetic parameters. Our aim is to compute a landscape function that, for each parameter point from the inspected parameter space, returns the quantitative model checking result for the respective continuous time Markov chain. Since the parameter space is in principle dense, it is infeasible to compute the landscape function directly. Hence, we design an effective method that iteratively approximates the lower and upper bounds of the landscape function with respect to a given accuracy. To this end, we modify the standard uniformization technique and introduce an iterative parameter space decomposition. We also demonstrate our approach on two biologically motivated case studies.

Precise parameter synthesis for stochastic biochemical systems

“…This section presents a generalization of the parameter exploration procedure originally introduced in [11]. The procedure takes a pCTMC C P and CSL path formula φ, and provides safe under-and over-approximations for the minimal and maximal probability that C P satisfies φ, that is, lower and upper bounds satisfying, for all s ∈ S,…”

“…We also demonstrate a significant speed-up of the max synthesis algorithm through the use of a sampling-based heuristic. The method is demonstrated using three case studies: the SIR epidemic model [27], where we synthesize infection and recovery rates that maximize the probability of disease extinction; the DNA walker circuit [17], where we derive stepping rates that ensure a predefined level of reliability; and a gene regulation model of the mammalian cell cycle [11], where we investigate degradation rates that lead to bi-stability.…”

“…The algorithms exploit a recently published technique that computes safe approximations to the lower and upper bounds for the probability to satisfy a CSL property over a fixed parameter space [11]. Our algorithms automatically derive the satisfying parameter regions through iterative decomposition of the parameter space up to a given tolerance value.…”

Probabilistic Programming Process Algebra

“…-G [3,5] (I > 0): There are still ignorants at all time points between time 3 and 5, i.e. the rumour has not reached everyone in the population.…”

“…Brim et al [5] propose a method of approximating quantitative model checking results over an entire parameter space as an alternative to parameter estimation. Although not directly relevant to inference, the work in [16] is another example of combining machine learning with formal modelling, presenting a Bayesian approach to statistical model checking.…”