Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta-Notch protein signalling

Abstract

Hybrid automata are an eminently suitable modelling framework for biological protein regulatory networks, as the protein concentration dynamics inside each biological cell are modelled using linear differential equations; inputs activate or deactivate these continuous dynamics through discrete switches, which themselves are controlled by protein concentrations reaching given thresholds. This paper proposes an iterative refinement algorithm for computing discrete abstractions of a class of hybrid automata with piecewise affine continuous dynamics and forced discrete transitions, defined completely in terms of symbolic variables and parameters. Furthermore, these discrete abstractions are utilised to compute symbolic parametric backward reachable sets from the equilibria of the hybrid automata, that are guaranteed to be exact or conservative under-approximations. The algorithm is then implemented using MATLAB and QEPCAD, to compute reachable sets for the biologically observed equilibria of the multiple cell Delta-Notch protein signalling automaton with symbolic parameters. The results are analysed to show that novel, non-intuitive, and biologically interesting properties can be deduced from the reachability computation, thus demonstrating the utility of the algorithm.

Mathematical modelling in developmental biology

“…Non-diffusive patterning mechanisms can be provided by direct contacts between cells, of which the classical example is Delta-Notch signalling associated with the binding of non-diffusive Delta to the Notch receptor of neighbouring cells (Collier et al 1996). Cell determination patterns due to contact signalling can be studied using the cellular automaton models, represented by sets of rules defining the interactions and states of cells or using so-called hybrid models combining cellular automata with differential equation (Ghosh & Tomlin 2004).…”

Hybrid Control Models and Tools for Biological Regulatory Networks

“…The algorithm was implemented using MATLAB and QEPCAD, to compute reachable sets for the biologically observed equilibria of the multiple cell Delta Notch protein signaling automaton with symbolic parameters. The results were analyzed to show that novel, non-intuitive, and biologically interesting properties can be deduced from the reachability computation, thus demonstrating the utility of the algorithm [3,7,9].…”

Stochastic hybrid systems for studying biochemical processes

“…Deterministic hybrid systems that integrate continuous dynamics with discrete events have been used to model a wide array of biological processes that exhibit switching behaviour (Belta et al 2004;Ghosh & Tomlin 2004;Lincoln & Tiwari 2004;Tanaka et al 2008). However, deterministic frameworks often fail to capture biochemical processes within living cells where low population counts of mRNAs and proteins can set the stage for significant stochastic effects (Raj & van Oudenaarden 2008).…”