Robustness Analysis and Tuning of Synthetic Gene Networks Grégory Batt Center for Information and Systems Engineering and Center for BioDynamics Boston University

Synthetic biology vSynthetic biology: design and construct biological systems with desired behaviors

Synthetic biology vSynthetic biology: design and construct biological systems with desired behaviors banana-smelling bacteria

Synthetic biology vSynthetic biology: design and construct biological systems with desired behaviors l engineering and medical applications detection of toxic chemicals, depollution, energy production destruction of cancer cells, gene therapy....

Synthetic biology vSynthetic biology: design and construct biological systems with desired behaviors l engineering and medical applications l study biological system properties in controlled environment

Synthetic biology vSynthetic biology: design and construct biological systems with desired behaviors l engineering and medical applications l study biological system properties in controlled environment Ultrasensitive input/output response at steady-state Transcriptional cascade in E. coli

Synthetic biology vSynthetic biology: design and construct biological systems with desired behaviors l engineering and medical applications l study biological system properties in controlled environment vNetwork design is difficult Most newly-created networks need tuning Ultrasensitive input/output response at steady-state Transcriptional cascade in E. coli

Synthetic biology vSynthetic biology: design and construct biological systems with desired behaviors l engineering and medical applications l study biological system properties in controlled environment vNetwork design is difficult Most newly-created networks need tuning How can the network be tuned ?

Robustness analysis and tuning vProblem for network design: parameter uncertainties l current limitations in experimental techniques l fluctuating extra and intracellular environments vNeed for designing or tuning networks having robust behavior Robust behavior if system presents expected property despite parameter variations vTwo problems of interest: l Robustness analysis: check whether properties are satisfied for all parameters in a set l Tuning: find parameter sets such that properties are satisfied for all parameters in the sets

Robustness analysis and tuning vProblem for network design: parameter uncertainties l current limitations in experimental techniques l fluctuating extra and intracellular environments vNeed for designing or tuning networks having robust behavior Robust behavior if system presents expected property despite parameter variations vTwo problems of interest: 1) find parameters such that system satisfies property 2) check robustness of proposed modifications

Robustness analysis and tuning vConstraints on robustness analysis and tuning of networks l genetic regulations are non-linear phenomena l size of the networks l reasoning for sets of parameters, initial conditions and inputs fixed initial condition fixed parameter x0x0 p1p1 p2p2 X0X0 P1P1 P2P2 set of initial conditions set of parameters How to define the expected dynamical properties ? How to reason with infinite number of parameters and initial conditions ?

Robustness analysis and tuning vConstraints on robustness analysis and tuning of networks l genetic regulations are non-linear phenomena l size of the networks l reasoning for sets of parameters, initial conditions and inputs vApproach: l dynamical properties specified in temporal logic (LTL) l unknown parameters, initial conditions and inputs given by intervals l piecewise-multiaffine differential equations models of gene networks l use of tailored combination of discrete abstraction, parameter constraint synthesis and model checking

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification III.Robustness analysis IV.Tuning V.Application: tuning a synthetic transcriptional cascade VI.Discussion and conclusions

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification I. Models: piecewise-multiaffine differential equations II. Dynamical property specifications: LTL formulas III.Robustness analysis IV.Tuning V.Application: tuning a synthetic transcriptional cascade VI.Discussion and conclusions

Gene network models vGenetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions b B a A

Gene network models vGenetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions x : protein concentration, : rate parameters : threshold concentration b B A

Gene network models vGenetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions x : protein concentration, : rate parameters : threshold concentration B a A

Gene network models vGenetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions b B a A x : protein concentration, : rate parameters : threshold concentration

Gene network models vDifferential equation models

Gene network models vDifferential equation models

Gene network models vDifferential equation models

Gene network models vDifferential equation models v is piecewise-multiaffine (PMA) function of state variables vPMA models are related to piecewise affine models Glass and Kauffman, J. Theor. Biol., 73de Jong et al., Bull. Math. Biol., 04 Belta et al., CDC, 02

Gene network models vDifferential equation models v is piecewise-multiaffine (PMA) function of state variables v is piecewise-affine function of rate parameters ( s and s) Belta et al., CDC, 02

Specifications of dynamical properties vDynamical properties expressed in temporal logic (LTL)

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

Specifications of dynamical properties vDynamical properties expressed in temporal logic (LTL) vSyntax of LTL formulas l set of atomic proposition usual logical operators temporal operators, bistability property: b B a A

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

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification I. Models: piecewise-multiaffine differential equations II. Dynamical property specifications: LTL formulas III.Robustness analysis IV.Tuning V.Application: tuning a synthetic transcriptional cascade VI.Discussion and conclusions

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification III.Robustness analysis I. Definition of discrete abstraction II. Computation of discrete abstraction III. Model checking the discrete abstraction IV.Tuning V.Application: tuning a synthetic transcriptional cascade VI.Discussion and conclusions

Discrete abstraction: definition vThreshold hyperplanes partition state space: set of rectangles R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 15 R 14 R 13 R 12 R 11

Discrete abstraction: definition Discrete transition system,, where

Discrete abstraction: definition Discrete transition system,, where l finite set of rectangles R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 15 R 14 R 13 R 12 R 11

Discrete abstraction: definition Discrete transition system,, where l finite set of rectangles transition relation representation of the flow for some R1R1 R6R6 R 11

Discrete abstraction: definition Discrete transition system,, where l finite set of rectangles transition relation R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 15 R 14 R 13 R 12 R 11

Discrete abstraction: definition Discrete transition system,, where l finite set of rectangles transition relation satisfaction relation R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 15 R 14 R 13 R 12 R 11 How can we compute ?

Discrete abstraction: computation vTransition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles

Discrete abstraction: computation vTransition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles R1R1 R2R2

Discrete abstraction: computation vTransition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles ( Because is a piecewise-multiaffine function of x ) In every rectangular region, the flow is a convex combination of its values at the vertices Belta and Habets, Trans. Autom. Contr., 06 R1R1 R2R2

Discrete abstraction: computation vTransition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles ( Because is a piecewise-multiaffine function of x ) vTransitions can be computed by polyhedral operations (Because is a piecewise-affine function of p ) In every rectangular region, the flow is a convex combination of its values at the vertices Belta and Habets, Trans. Autom. Contr., 06 R1R1 R2R2

Discrete abstraction: model checking vModel checking is automated technique for verifying that finite transition system satisfy temporal logic property Efficient computer tools are available to perform model checking

Discrete abstraction: model checking vModel checking is automated technique for verifying that finite transition systems satisfy temporal logic properties v is a finite transition system and can be model-checked

Discrete abstraction: model checking vModel checking is automated technique for verifying that finite transition systems satisfy temporal logic properties v is a finite transition system and can be model-checked v can be used for proving properties of the original system is conservative approximation of original system (simulation relation between transition systems) Alur et al., Proc. IEEE, 00

Discrete abstraction: model checking vModel checking is automated technique for verifying that finite transition systems satisfy temporal logic properties v is a finite transition system and can be model-checked v can be used for proving properties of the original system bistability property:

Discrete abstraction: model checking vModel checking is automated technique for verifying that finite transition systems satisfy temporal logic properties v is a finite transition system and can be model-checked v can be used for proving properties of the original system bistability property:

Discrete abstraction: model checking vModel checking is automated technique for verifying that finite transition systems satisfy temporal logic properties v is a finite transition system and can be model-checked v can be used for proving properties of the original system bistability property: Property robustly satisfied for parameter set P

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification III.Robustness analysis I. Definition of discrete abstraction II. Computation of discrete abstraction III. Model checking the discrete abstraction IV.Tuning V.Application: tuning a synthetic transcriptional cascade VI.Discussion and conclusions

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification III.Robustness analysis IV.Tuning V.Application: tuning a synthetic transcriptional cascade VI.Discussion and conclusions

Tuning vSynthesis of parameter constraints Collect affine constraints defining existence of transitions between rectangles: vParameter space exploration Construct partition of parameter space using parameter constraints

Tuning bistability property: vSynthesis of parameter constraints Collect affine constraints defining existence of transitions between rectangles: vParameter space exploration Construct partition of parameter space using parameter constraints Test the validity of each region in parameter space

Tuning bistability property: vSynthesis of parameter constraints Collect affine constraints defining existence of transitions between rectangles: vParameter space exploration Construct partition of parameter space using parameter constraints Test the validity of each region in parameter space

Tuning vSynthesis of parameter constraints Collect affine constraints defining existence of transitions between rectangles: vParameter space exploration Construct partition of parameter space using parameter constraints Test the validity of each region in parameter space vMore efficient approach: model check while constructing the partition

Tuning vSynthesis of parameter constraints Collect affine constraints defining existence of transitions between rectangles: vParameter space exploration Construct partition of parameter space using parameter constraints Test the validity of each region in parameter space vMore efficient approach: model check while constructing the partition vApproach implemented in publicly-available tool RoVerGeNe Exploits tools for polyhedra operations (MPT) and model checker (NuSMV) Batt et al., HSCC07

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification III.Robustness analysis IV.Tuning V.Application: tuning a synthetic transcriptional cascade VI.Discussion and conclusions

Summary vRobustness analysis l provides finite description of the dynamics of original system in state space for parameter sets l can be computed by polyhedral operations is a conservative approximation of original system vTuning l Use affine constraints appearing in transition computation to define partition of parameter space l Explore every region in parameter space

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification III.Analysis for fixed parameters IV.Analysis for sets of parameters V.Application: tuning a synthetic transcriptional cascade I. Modeling the actual network II. Tuning the network III. Verifying robustness of tuned network VI.Discussion and conclusions

Transcriptional cascade: problem vApproach for robust tuning of the cascade: l develop a model of the actual cascade l tune network by modifying 3 key parameters l check that property still true when all parameters vary in ±10% intervals Hooshangi et al., PNAS, 05 Input/output response Transcriptional cascade in E. coli

Transcriptional cascade: modeling vPMA differential equation model (1 input and 4 state variables) vParameter identification Computation of I/O behavior of cascade

Transcriptional cascade: specification Expected input/output behavior of cascade Temporal logic specification

Transcriptional cascade: tuning vTuning: search for valid parameter sets l Let 3 production rates unconstrained l Answer: 1 set found (<2 h.) Computation of I/O behavior of cascade for some parameters in the set

Transcriptional cascade: analysis vRobustness: test robustness of proposed modification l Assume l Is property true if all rate parameters vary in a ±10% interval? or ±20%? l Answer: Yes for ±10% parameter variations (<4 h.) No for ±20% parameter variations 11 uncertain parameters:

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification III.Analysis for fixed parameters IV.Analysis for sets of parameters V.Tuning of a synthetic transcriptional cascade I. Modeling the actual network II. Tuning the network III. Verifying robustness of tuned network VI.Discussion and conclusions

Overview I.Introduction: rational design of synthetic gene networks II.Modeling and specification III.Analysis for fixed parameters IV.Analysis for sets of parameters V.Tuning of a synthetic transcriptional cascade VI.Discussion and conclusions

Conclusion vRobustness analysis and tuning of genetic regulatory networks l dynamical properties expressed in temporal logic l unknown parameters, initial conditions and inputs given by intervals l piecewise-multiaffine differential equations models of gene networks vTailored combination of discrete abstraction, parameter constraint synthesis and model checking used for proving properties of uncertain PMA systems vMethod implemented in publicly-available tool RoVerGeNe vApproach can answer efficiently non-trivial questions on networks of biological interest

Discussion vRelated work: formal analysis of uncertain biological networks l Iterative search in dense parameter space of ODE models using model checking l Exhaustive exploration of finite parameter space of logical models using model checking l Analysis of qualitative PA models by reachability analysis or model checking l Robust stability and model validation of ODE models using SOSTOOLS vFurther work l Verification of properties involving timing constraints l Compositional verification to exploit network modularity Bernot et al., J. Theor. Biol., 04 Antoniotti et al., Theor. Comput. Sci., 04 Calzone et al., Trans. Comput. Syst. Biol, 06 de Jong et al., Bull. Math. Biol., 04 Ghosh and Tomlin, Systems Biology, 04; Batt et al., HSCC, 05 El-Samad et al., Proc. IEEE, 06

Acknowledgements Thank you for your attention! l Calin Belta (Boston University, USA) l Ron Weiss (Princeton University, USA) l Boyan Yordanov (Boston University, USA)

Discrete abstraction: definition Discrete transition system,, where l finite set of rectangles transition relation X0X0 P1P1 P2P2

Discrete abstraction: definition Discrete transition system,, where l finite set of rectangles transition relation X0X0 P1P1 P2P2

Discrete abstraction: model checking vModel checking is automated technique for verifying that finite transition systems satisfy temporal logic properties v is a finite transition system and can be model-checked v can be used for proving properties of the original system bistability property: X0X0 P1P1 P2P2