Computing the projected reachable set of switched affine systems: an application to systems biology

A fundamental question in systems biology is what combinations of mean and variance of the species present in a stochastic biochemical reaction network are attainable by perturbing the system with an external signal. To address this question, we show that the moments evolution in any generic network can be either approximated or, under suitable assumptions, computed exactly as the solution of a switched affine system. Motivated by this application, we propose a new method to approximate the reachable set of switched affine systems. A remarkable feature of our approach is that it allows one to easily compute projections of the reachable set for pairs of moments of interest, without requiring the computation of the full reachable set, which can be prohibitive for large networks. As a second contribution, we also show how to select the external signal in order to maximize the probability of reaching a target set. To illustrate the method we study a renown model of controlled gene expression and we derive estimates of the reachable set, for the protein mean and variance, that are more accurate than those available in the literature and consistent with experimental data.

[1]  Jorge M. Gonçalves,et al.  Reachability analysis of continuous-time piecewise affine systems , 2008, Autom..

[2]  Jan H. van Schuppen,et al.  Reachability and control synthesis for piecewise-affine hybrid systems on simplices , 2006, IEEE Transactions on Automatic Control.

[3]  John Lygeros,et al.  Designing experiments to understand the variability in biochemical reaction networks , 2013, Journal of The Royal Society Interface.

[4]  Michael P. H. Stumpf,et al.  Control mechanisms for stochastic biochemical systems via computation of reachable sets , 2016 .

[5]  John Lygeros,et al.  Iterative experiment design guides the characterization of a light-inducible gene expression circuit , 2015, Proceedings of the National Academy of Sciences.

[6]  Jeffrey J. Tabor,et al.  Characterizing bacterial gene circuit dynamics with optically programmed gene expression signals , 2014, Nature Methods.

[7]  Panos J. Antsaklis,et al.  Safety and Reachability of Piecewise Linear Hybrid Dynamical Systems Based on Discrete Abstractions , 2003, Discret. Event Dyn. Syst..

[8]  Corentin Briat,et al.  Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[9]  J. Hespanha.  Moment closure for biochemical networks , 2008, 2008 3rd International Symposium on Communications, Control and Signal Processing.

[10]  François Fages,et al.  Symbolic Model Checking of Biochemical Networks , 2003, CMSB.

[11]  A. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Control , 1996 .

[12]  Alberto Bemporad,et al.  Control of systems integrating logic, dynamics, and constraints , 1999, Autom..

[13]  C. Tomlin,et al.  Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta-Notch protein signalling. , 2004, Systems biology.

[14]  G. Vinnicombe,et al.  Fundamental limits on the suppression of molecular fluctuations , 2010, Nature.

[15]  Antonis Papachristodoulou,et al.  Advanced Methods and Algorithms for Biological Networks Analysis , 2006, Proceedings of the IEEE.

[16]  Andreas Beyer,et al.  Posttranscriptional Expression Regulation: What Determines Translation Rates? , 2007, PLoS Comput. Biol..

[17]  B. Krogh,et al.  Hyperplane method for reachable state estimation for linear time-invariant systems , 1991 .

[18]  Hai Lin,et al.  Switched Linear Systems: Control and Design , 2006, IEEE Transactions on Automatic Control.

[19]  Francesca Parise,et al.  On the reachable set of the controlled gene expression system , 2014, 53rd IEEE Conference on Decision and Control.

[20]  D. Gillespie.  A rigorous derivation of the chemical master equation , 1992 .

[21]  Olaf Wolkenhauer,et al.  Stochastic Modeling for Systems Biology edited by D. J. Wilkinson , 2007 .

[22]  Chang Hyeong Lee,et al.  A moment closure method for stochastic reaction networks. , 2009, The Journal of chemical physics.

[23]  George J. Pappas,et al.  Discrete abstractions of hybrid systems , 2000, Proceedings of the IEEE.

[24]  D. Pincus,et al.  In silico feedback for in vivo regulation of a gene expression circuit , 2011, Nature Biotechnology.

[25]  Joost-Pieter Katoen,et al.  Approximate Model Checking of Stochastic Hybrid Systems , 2010, Eur. J. Control.

[26]  Alberto Bemporad,et al.  Optimal control of continuous-time switched affine systems , 2006, IEEE Transactions on Automatic Control.

[27]  F. Fages,et al.  Long-term model predictive control of gene expression at the population and single-cell levels , 2012, Proceedings of the National Academy of Sciences.

[28]  Thomas A. Henzinger,et al.  Hybrid Automata: An Algorithmic Approach to the Specification and Verification of Hybrid Systems , 1992, Hybrid Systems.

[29]  Mario di Bernardo,et al.  Analysis, design and implementation of a novel scheme for in-vivo control of synthetic gene regulatory networks , 2011, Autom..

[30]  Ian M. Mitchell,et al.  A Toolbox of Hamilton-Jacobi Solvers for Analysis of Nondeterministic Continuous and Hybrid Systems , 2005, HSCC.

[31]  Francesca Parise,et al.  Reachability analysis for switched affine systems and its application to controlled stochastic biochemical reaction networks , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[32]  Oded Maler,et al.  Systematic Simulation Using Sensitivity Analysis , 2007, HSCC.

[33]  J. Goutsias,et al.  Markovian dynamics on complex reaction networks , 2012, 1205.5524.

[34]  Claudio Altafini,et al.  The reachable set of a linear endogenous switching system , 2002, Syst. Control. Lett..

[35]  Christel Baier,et al.  Model-Checking Algorithms for , 2003 .

[36]  Daniel Liberzon,et al.  Calculus of Variations and Optimal Control Theory: A Concise Introduction , 2012 .

[37]  M. Khammash,et al.  The finite state projection algorithm for the solution of the chemical master equation. , 2006, The Journal of chemical physics.

[38]  Hidde de Jong,et al.  Symbolic reachability analysis of genetic regulatory networks using discrete abstractions , 2008, Autom..

[39]  Francesca Parise,et al.  On the use of hyperplane methods to compute the reachable set of controlled stochastic biochemical reaction networks , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).