Robustness Analysis and Tuning of Synthetic Gene Networks

1. Robustness Analysis and Tuning of Synthetic Gene Networks Grégory Batt1‡ Boyan Yordanov1 Calin Belta1Ron Weiss2 1 Centers for Information and Systems Engineering and for BioDynamics Boston University (‡ now at ) 2 Departments of Molecular Biology and of Electrical EngineeringPrinceton University Towards Systems Biology 2007

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

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

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

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

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

7. Need for rational design • Gene networks are networks of genes, proteins, small molecules and their regulatory interactions • Network design: analysis of non-linear dynamical system with parameteruncertainties • current limitations in experimental techniques • fluctuating extra and intracellular environments Problem: most newly-created networks are non-functioning and need tuning

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

9. P1 X0 P2 p1 x0 p2 Hybrid systems approach • Analysis of dynamical systems • Traditional view: fixed initial condition and fixed parameter • More interesting: set of initial conditions and set of parameters

10. P1 X0 P2 p1 x0 p2 Hybrid systems approach • Analysis of dynamical systems • Traditional view: fixed initial condition and fixed parameter • More interesting: set of initial conditions and set of parameters • How to reason with infinite number of parameters and initial conditions ?

11. P1 X0 P2 p1 x0 p2 P1 X0 P2 Hybrid systems approach • Analysis of dynamical systems • Traditional view: fixed initial condition and fixed parameter • More interesting: set of initial conditions and set of parameters • How to reason with infinite number of parameters and initial conditions ? direct vs indirect approaches P1 X0 P2

12. P1 X0 P2 p1 x0 p2 P1 X0 P2 Hybrid systems approach • Analysis of dynamical systems • Traditional view: fixed initial condition and fixed parameter • More interesting: set of initial conditions and set of parameters • How to reason with infinite number of parameters and initial conditions ? direct vs indirect approaches P1 X0 P2 model checking possible

13. Overview • Introduction • Problem definition • Robust design of gene networks • Application: tuning a synthetic transcriptional cascade • Discussion and conclusions

14. Overview • Introduction • Problem definition • Robust design of gene networks • Application: tuning a synthetic transcriptional cascade • Discussion and conclusions

15. Gene network models cross-inhibition network

16. x: protein concentration : threshold concentration  , : rate parameters Gene network models cross-inhibition network

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

18. x: protein concentration : threshold concentration  , : rate parameters Gene network models cross-inhibition network

19. x: protein concentration : threshold concentration  , : rate parameters Gene network models • Find parameters such that network is bistable cross-inhibition network

20. Gene network models • Partition of the state space: rectangles

21. Gene network models • Partition of the state space: rectangles • Differential equation models , with • is piecewise-multiaffine (PMA)function of state variables • is affine function of rate parameters (’sand ’s) (multiaffine functions: products of different state variables allowed)

22. Specifications of dynamical properties • Dynamical properties expressed in temporal logic (LTL) • set of atomic proposition • usual logical operators • temporal operators ,

23. Specifications of dynamical properties • Dynamical properties expressed in temporal logic (LTL) • set of atomic proposition • usual logical operators • temporal operators , • Semantics of LTL formulas defined over executions of transition systems ... ... ...

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

25. Specifications of dynamical properties • Dynamical properties expressed in temporal logic (LTL) • set of atomic proposition • usual logical operators • temporal operators , • Semantics of LTL formulas defined over executions of transition systems bistability property:

26. Overview • Introduction • Problem definition • Robust design of gene networks • Application: tuning a synthetic transcriptional cascade • Discussion and conclusions

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

28. Robust design of gene networks gene network PMA model synthesis of parameter constraints intervals for uncertain parameters discrete abstractions convexity properties specifications model checking No conclusion Valid parameter set [Batt et al., HSCC07]

29. 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

30. 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 • Transition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles

31. 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 • Transition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles • Transitions can be computed by polyhedral operations where (Because is a piecewise-multiaffine function of x and an affine function of p)

32. RoVerGeNe • Approach implemented in publicly-available tool RoVerGeNe Written in Matlab, exploits polyhedral operation toolbox MPT and model checker NuSMV http://iasi.bu.edu/~batt

33. Overview • Introduction • Problem definition • Robustness design of gene networks • Application: tuning a synthetic transcriptional cascade • Discussion and conclusions

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

35. Transcriptional cascade: modeling • PMA differential equation model (1 input and 4 state variables) • Parameter identification

36. Transcriptional cascade: specification • Expected input/output behavior of cascade at steady state and for all initial states • Temporal logic specifications • Liveness property: additional fairness constraints needed [Batt et al., TACAS’07]

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

38. Transcriptional cascade: robustness • Robustness: check that tuned network behaves robustly • Let all production and degradation rate parameters range in intervals centered at their reference values (with ±10% or ±20% variations) • Answer for ±10% parameter variations: Yes (< 4hrs)  proves that specification holds despite ±10% parameter variations • 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:

39. Overview • Introduction • Problem definition • Analysis for fixed parameters • Analysis for sets of parameters • Tuning of a synthetic transcriptional cascade • Discussion and conclusions

40. Summary • Gene networks modeled as uncertain PMA systems • piecewise-multiaffinedifferential equations models • unknown parameters, initial conditions and inputs given by intervals • dynamical properties expressed in temporal logic • Use of tailored combination of parameter constraint synthesis, discrete abstractions, and model checking • Method implemented in publicly-available tool RoVerGeNe • Approach can answer non-trivial questions on networks of biological interest

41. [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., CDC’02; Berman et al., HSCC’07; Fages and Rizk, CMSB’07] [Kuepfer et al., BMC Bioinfo. 07] Discussion • First computational approach for tuning synthetic gene networks • Related work: • qualitative/discrete approaches (reachability or model checking) • quantitative approaches with fixed parameter values (reachability or MC) • quantitative approaches with uncertain parameters (optimisation-based) • Further work: • verification of properties involving timing constraints (post doc, Verimag) • deal with uncertain threshold parameters too • use of compositional verification for design of large modular networks

42. 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. InBioinformatics, 23(18):2415-1422 • 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 BioDynamics Center for Information and Systems Engineering Boston University Verimag Lab Grenoble Polytechnic Institute