Robustness Analysis and Tuning of Synthetic Gene Networks Grégory Batt 1 Boyan Yordanov 1 Calin Belta 1 Ron Weiss 2 1 Centers for Information and Systems Engineering and for BioDynamics Boston University ( now at ) 2 Departments of Molecular Biology and of Electrical Engineering Princeton University Towards Systems Biology 2007

Synthetic biology vSynthetic biology: application of engineering approaches to produce novel artificial devices using biological building blocks

Synthetic biology vSynthetic biology: application of engineering approaches to produce novel artificial devices using biological building blocks banana-smelling bacteria

Synthetic biology vSynthetic biology: application of engineering approaches to produce novel artificial devices using biological building blocks banana-smelling bacteria

Synthetic biology vSynthetic biology: application of engineering approaches to produce novel artificial devices using biological building blocks vNumerous potential engineering and medical applications l biofuel production, environment depollution,... l biochemical synthesis, tumor cell destruction,... banana-smelling bacteria

Synthetic gene networks vGene networks are networks of genes, proteins, small molecules and their regulatory interactions Ultrasensitive I/O response at steady-state Transcriptional cascade [Hooshangi et al, PNAS, 05]

Need for rational design vGene networks are networks of genes, proteins, small molecules and their regulatory interactions vNetwork design: analysis of non-linear dynamical system with parameter uncertainties l current limitations in experimental techniques l fluctuating extra and intracellular environments Problem: most newly-created networks are non-functioning and need tuning

Robustness analysis and tuning vTwo problems of interest: l robustness analysis: check whether dynamical properties are satisfied for all parameters in a set l tuning: find parameter sets such that dynamical properties are satisfied for all parameters in the sets vApproach: l unknown parameters, initial conditions and inputs given by intervals l piecewise-multiaffine differential equations models of gene networks l dynamical properties specified in temporal logic (LTL) l adapt techniques from hybrid systems theory and model checking

Hybrid systems approach vAnalysis of dynamical systems l Traditional view: fixed initial condition and fixed parameter l More interesting: set of initial conditions and set of parameters x0x0 p1p1 p2p2 X0X0 P1P1 P2P2

Hybrid systems approach vAnalysis of dynamical systems l Traditional view: fixed initial condition and fixed parameter l More interesting: set of initial conditions and set of parameters vHow to reason with infinite number of parameters and initial conditions ? x0x0 p1p1 p2p2 X0X0 P1P1 P2P2

Hybrid systems approach vAnalysis of dynamical systems l Traditional view: fixed initial condition and fixed parameter l More interesting: set of initial conditions and set of parameters vHow to reason with infinite number of parameters and initial conditions ? direct vs indirect approaches x0x0 p1p1 p2p2 X0X0 P1P1 P2P2 X0X0 P1P1 P2P2 X0X0 P1P1 P2P2

Hybrid systems approach vAnalysis of dynamical systems l Traditional view: fixed initial condition and fixed parameter l More interesting: set of initial conditions and set of parameters vHow to reason with infinite number of parameters and initial conditions ? direct vs indirect approaches x0x0 p1p1 p2p2 X0X0 P1P1 P2P2 X0X0 P1P1 P2P2 X0X0 P1P1 P2P2 model checking possible

Overview I.Introduction II.Problem definition III.Robust design of gene networks IV.Application: tuning a synthetic transcriptional cascade V.Discussion and conclusions

Overview I.Introduction II.Problem definition III.Robust design of gene networks IV.Application: tuning a synthetic transcriptional cascade V.Discussion and conclusions

Gene network models cross-inhibition network

Gene network models cross-inhibition network x : protein concentration, : rate parameters : threshold concentration

Gene network models cross-inhibition network x : protein concentration, : rate parameters : threshold concentration x 0 1 Hill function x 0 1 step function x 0 1 ramp function Hill-type models PMA models PA models regulation functions:

Gene network models cross-inhibition network x : protein concentration, : rate parameters : threshold concentration

Gene network models vFind parameters such that network is bistable cross-inhibition network x : protein concentration, : rate parameters : threshold concentration

Gene network models vPartition of the state space: rectangles

Gene network models vPartition of the state space: rectangles vDifferential equation models, with l is piecewise-multiaffine (PMA) function of state variables l is affine function of rate parameters ( s and s) (multiaffine functions: products of different state variables allowed)

Specifications of dynamical properties vDynamical properties expressed in temporal logic (LTL) l set of atomic proposition usual logical operators temporal operators,

Specifications of dynamical properties vDynamical properties expressed in temporal logic (LTL) l set of atomic proposition usual logical operators temporal operators, vSemantics of LTL formulas defined over executions of transition systems...

Specifications of dynamical properties vDynamical properties expressed in temporal logic (LTL) l set of atomic proposition usual logical operators temporal operators, vSemantics of LTL formulas defined over executions of transition systems Solution trajectories of PMA models are associated with executions of embedding transition system...

Specifications of dynamical properties vDynamical properties expressed in temporal logic (LTL) l set of atomic proposition usual logical operators l temporal operators, vSemantics of LTL formulas defined over executions of transition systems bistability property:

Overview I.Introduction II.Problem definition III.Robust design of gene networks IV.Application: tuning a synthetic transcriptional cascade V.Discussion and conclusions

PMA model specifications gene network intervals for uncertain parameters Robust design of gene networks

model checking PMA model specifications synthesis of parameter constraints gene network discrete abstractions convexity properties intervals for uncertain parameters Robust design of gene networks Valid parameter setNo conclusion [Batt et al., HSCC07]

Computation of discrete abstraction Multiaffine function: in every rectangular region, the flow is a convex combination of its values at the vertices [Belta and Habets, Trans. Autom. Contr., 06]

Computation of discrete abstraction vTransition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles Multiaffine function: in every rectangular region, the flow is a convex combination of its values at the vertices [Belta and Habets, Trans. Autom. Contr., 06]

Computation of discrete abstraction vTransition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles vTransitions can be computed by polyhedral operations where (Because is a piecewise-multiaffine function of x and an affine function of p ) Multiaffine function: in every rectangular region, the flow is a convex combination of its values at the vertices [Belta and Habets, Trans. Autom. Contr., 06]

RoVerGeNe vApproach implemented in publicly-available tool RoVerGeNe Written in Matlab, exploits polyhedral operation toolbox MPT and model checker NuSMV

Overview I.Introduction II.Problem definition III.Robustness design of gene networks IV.Application: tuning a synthetic transcriptional cascade V.Discussion and conclusions

Transcriptional cascade: approach vApproach for robust tuning of the cascade: l develop a model of the actual cascade l specify expected behavior l tune network by searching for valid parameter sets l verify robustness of tuned network Transcriptional cascade [Hooshangi et al, PNAS, 05]

Transcriptional cascade: modeling vPMA differential equation model (1 input and 4 state variables) vParameter identification

Transcriptional cascade: specification vExpected input/output behavior of cascade at steady state and for all initial states vTemporal logic specifications vLiveness property: additional fairness constraints needed [Batt et al., TACAS07]

Transcriptional cascade: tuning vTuning: search for valid parameter sets l Let 3 production rate parameters unconstrained l Answer: 15 sets found (<4 h., 1500 rectangles, 18 parameter constraints) comparison with numerical simulation results in parameter space and for input/output behavior [Batt et al., Bioinfo, 07]

Transcriptional cascade: robustness vRobustness: check that tuned network behaves robustly l Let all production and degradation rate parameters range in intervals centered at their reference values (with ±10% or ±20% variations) l Answer for ±10% parameter variations: Yes (< 4hrs) proves that specification holds despite ±10% parameter variations l Answer for ±20% parameter variations: No (< 4hrs) suggests that specification does not hold for some parameters in the ±20% set (confirmed by manual analysis of counter-example) 11 uncertain parameters:

Overview I.Introduction II.Problem definition III.Analysis for fixed parameters IV.Analysis for sets of parameters V.Tuning of a synthetic transcriptional cascade VI.Discussion and conclusions

Summary vGene networks modeled as uncertain PMA systems l piecewise-multiaffine differential equations models l unknown parameters, initial conditions and inputs given by intervals l dynamical properties expressed in temporal logic vUse of tailored combination of parameter constraint synthesis, discrete abstractions, and model checking vMethod implemented in publicly-available tool RoVerGeNe vApproach can answer non-trivial questions on networks of biological interest

Discussion vFirst computational approach for tuning synthetic gene networks vRelated work: l qualitative/discrete approaches (reachability or model checking) l quantitative approaches with fixed parameter values (reachability or MC) l quantitative approaches with uncertain parameters (optimisation-based) vFurther work: l verification of properties involving timing constraints (post doc, Verimag) l deal with uncertain threshold parameters too l use of compositional verification for design of large modular networks [de Jong et al., Bull. Math. Biol. 04; Ghosh and Tomlin, Syst.Biol. 04; Batt et al., Bioinfo. 05] [Bernot et al., J.Theor.Biol. 04; Gonzalez et al., Biosystems 06, Calzone et al., Trans.Comput.Syst.Biol 06] [Belta et al., CDC02; Berman et al., HSCC07; Fages and Rizk, CMSB07] [Kuepfer et al., BMC Bioinfo. 07]

Acknowledgements Thanks to Calin Belta, Boyan Yordanov, Ron Weiss… … and to Ramzi Ben Salah and Oded Maler References: G. Batt, B. Yordanov, C. Belta and R. Weiss (2007) Robustness analysis and tuning of synthetic gene networks. In Bioinformatics, 23(18): G. Batt, C. Belta and R. Weiss (2007) Temporal logic analysis of gene networks under parameter uncertainty. Accepted to Joint Special Issue on Systems Biology of IEEE Trans. Circuits and Systems and IEEE Trans. Automatic Control Center for BioDynamicsCenter for Information and Systems EngineeringBoston University Verimag LabGrenoble Polytechnic Institute