Pathway Logic: Symbolic Analysis of Biological Signaling

Formal Methods for Computational Systems Biology

Abstract. Biologists use diagrams to represent interactions between molecular species, and on the computer, diagrammatic notations are also employed in interactive maps. These diagrams are fundamentally of two types: reaction graphs and activation/inhibition graphs. In this tutorial, we study these graphs with formal methods originating from programming theory. We consider systems of biochemical reactions with kinetic expressions, as written in the Systems Biology Markup Language (SBML), and interpreted in the Biochemical Abstract Machine (Biocham) at different levels of abstraction, by either an asynchronous boolean transition system, a continuous time Markov chain, or a system of Ordinary Differential Equations over molecular concentrations. We show that under general conditions satisfied in practice, the activation/inhibition graph is independent of the precise kinetic expressions, and is computable in linear time in the number of reactions. Then we consider the formalization of the biological properties of systems, as observed in experiments, in temporal logics. We show that these logics are expressive enough to capture semi-qualitative semi-quantitative properties of the boolean and differential semantics of reaction models, and that model-checking techniques can be used to validate a model w.r.t. its temporal specification, complete it, and search for kinetic parameter values. We illustrate this modelling method with examples on the MAPK signalling cascade, and on Kohn's map of the mammalian cell cycle.

Section: Theorem 1 ([13]) For Any Reaction Modelmentioning

confidence: 99%

Section: Corollary 3 the Differential Influence Graph Of A Reaction mentioning

confidence: 99%

Formal Cell Biology in Biocham

Formal Methods for Computational Systems Biology

“…This approach has rapidly developed in Systems Biology for reasoning on large interaction networks, with for instance, the analysis of qualitative attractors in a logical dynamics of gene networks à la Thomas [3,4,5], reachability and temporal logic properties in reaction networks [6,7,8,9,10], structural invariants in the Petri net representation of the reactions [11,12,13,14,15,16], or model reductions using graph theory concepts [17,18]. These qualitative analysis tools do not rely on kinetic information, but on the structure of the reaction network which has thus to be correctly written as a set of formal reactions, with well-identified reactants, products and modifiers (and in certain cases their stoichiometry) for each reaction.…”

Section: Introductionmentioning

confidence: 99%

Inferring reaction systems from ordinary differential equations

Gay

Theoretical Computer Science

2015

In Mathematical Biology, many dynamical models of biochemical reaction systems are presented with Ordinary Differential Equations (ODE). Once kinetic parameter values are fixed, this simple mathematical formalism completely defines the dynamical behavior of a system of biochemical reactions and provides powerful tools for deterministic simulations, parameter sensitivity analysis, bifurcation analysis, etc. However, without requiring any information on the reaction kinetics and parameter values, various qualitative analyses can be performed using the structure of the reactions, provided the reactants, products and modifiers of each reaction are precisely defined. In order to apply these structural methods to parametric ODE models, we study a mathematical condition for expressing the consistency between the structure and the kinetics of a reaction, without restricting to Mass Action law kinetics. This condition, satisfied in particular by standard kinetic laws, entails a remarkable property of independence of the influence graph from the kinetics of the reactions. We derive from this study a heuristic algorithm which, given a system of ODEs as input, computes a system of reactions with the same ODE semantics, by inferring well-formed reactions whenever possible. We show how this strategy is capable of automatically curating the writing of ODE models in SBML, and present some statistics obtained on the model repository biomodels.net. 1 .

“…In the early days of systems biology, propositional temporal logic was proposed by computer scientists to formalize the Boolean properties of the behavior of biochemical reaction systems [11,5] or gene regulatory networks [4,3]. Generalizing these techniques to quantitative models can be done in two ways: either by discretizing the different regimes of the dynamics in piece-wise linear or affine models [8,2], or by relying on numerical simulations and taking a first-order version of temporal logic with constraints on concentrations, as query language for the numerical traces [1,13,14].…”

Section: Introductionmentioning

confidence: 99%

Trace Simplifications Preserving Temporal Logic Formulae with Case Study in a Coupled Model of the Cell Cycle and the Circadian Clock

Traynard

Computational Methods in Systems Biology

2014

Abstract. Calibrating dynamical models on experimental data time series is a central task in computational systems biology. When numerical values for model parameters can be found to fit the data, the model can be used to make predictions, whereas the absence of any good fit may suggest to revisit the structure of the model and gain new insights in the biology of the system. Temporal logic provides a formal framework to deal with imprecise data and specify a wide variety of dynamical behaviors. It can be used to extract information from numerical traces coming from either experimental data or model simulations, and to specify the expected behaviors for model calibration. The computation time of the different methods depends on the number of points in the trace so the question of trace simplification is important to improve their performance. In this paper we study this problem and provide a series of trace simplifications which are correct to perform for some common temporal logic formulae. We give some general soundness theorems, and apply this approach to period and phase constraints on the circadian clock and the cell cycle. In this application, temporal logic patterns are used to compute the relevant characteristics of the experimental traces, and to measure the adequacy of the model to its specification on simulation traces. Speed-ups by several orders of magnitude are obtained by trace simplification even when produced by smart numerical integration methods.