Model checking the evolution of gene regulatory networks

Combining formal methods and Bayesian approach for inferring discrete-state stochastic models from steady-state data

Stochastic population models are widely used to model phenomena in different areas such as cyber-physical systems, chemical kinetics, collective animal behaviour, and beyond. Quantitative analysis of stochastic population models easily becomes challenging due to the combinatorial number of possible states of the population. Moreover, while the modeller easily hypothesises the mechanistic aspects of the model, the quantitative parameters associated to these mechanistic transitions are difficult or impossible to measure directly. In this paper, we investigate how formal verification methods can aid parameter inference for population discrete-time Markov chains in a scenario where only a limited sample of population-level data measurements-sample distributions among terminal states-are available. We first discuss the parameter identifiability and uncertainty quantification in this setup, as well as how the existing techniques of formal parameter synthesis and Bayesian inference apply. Then, we propose and implement four different methods, three of which incorporate formal parameter synthesis as a pre-computation step. We empirically evaluate the performance of the proposed methods over four representative case studies. We find that our proposed methods incorporating formal parameter synthesis as a pre-computation step allow us to significantly enhance the accuracy, precision, and scalability of inference. Specifically, in the case of unidentifiable parameters, we accurately capture the subspace of parameters which is data-compliant at a desired confidence level.

Symbolic Coloured SCC Decomposition

Problems arising in many scientific disciplines are often modelled using edge-coloured directed graphs. These can be enormous in the number of both vertices and colours. Given such a graph, the original problem frequently translates to the detection of the graph’s strongly connected components, which is challenging at this scale.

We propose a new, symbolic algorithm that computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(p\cdot n\cdot \log n)$$\end{document}O(p·n·logn) symbolic steps, where p is the number of colours and n the number of vertices. We evaluate the algorithm using an experimental implementation based on Binary Decision Diagrams (BDDs) and large (up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{48}$$\end{document}248) coloured graphs produced by models appearing in systems biology.

AEON: Attractor Bifurcation Analysis of Parametrised Boolean Networks

Boolean networks (BNs) provide an effective modelling tool for various phenomena from science and engineering. Any long-term behaviour of a BN eventually converges to a so-called attractor. Depending on various logical parameters, the structure and quality of attractors can undergo a significant change, known as a bifurcation. We present a tool for analysing bifurcations in asynchronous parametrised Boolean networks. To fight the state-space and parameter-space explosion problem the tool uses a parallel semi-symbolic algorithm.

Bio-ModelChecker: Using Bounded Constraint Satisfaction to Seamlessly Integrate Observed Behavior With Prior Knowledge of Biological Networks

The in silico study and reverse engineering of regulatory networks has gained in recognition as an insightful tool for the qualitative study of biological mechanisms that underlie a broad range of complex illness. In the creation of reliable network models, the integration of prior mechanistic knowledge with experimentally observed behavior is hampered by the disparate nature and widespread sparsity of such measurements. The former challenges conventional regression-based parameter fitting while the latter leads to large sets of highly variable network models that are equally compliant with the data. In this paper, we propose a bounded Constraint Satisfaction (CS) based model checking framework for parameter set identification that readily accommodates partial records and the exponential complexity of this problem. We introduce specific criteria to describe the biological plausibility of competing multi-valued regulatory networks that satisfy all the constraints and formulate model identification as a multi-objective optimization problem. Optimization is directed at maximizing structural parsimony of the regulatory network by mitigating excessive control action selectivity while also favoring increased state transition efficiency and robustness of the network's dynamic response. The framework's scalability, computational time and validity is demonstrated on several well-established and well-studied biological networks.