Robustness Analysis and Tuning of Synthetic Gene Networks

1. Robustness Analysis and Tuning of Synthetic Gene Networks Grégory Batt Center for Information and Systems Engineering and Center for BioDynamics Boston University Email: batt@bu.edu

2. Synthetic biology • Synthetic biology: design and construct biological systems with desired behaviors

3. Synthetic biology • Synthetic biology: design and construct biological systems with desired behaviors banana-smelling bacteria

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

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

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

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

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

9. Robustness analysis and tuning • Problem for network design: parameteruncertainties • current limitations in experimental techniques • fluctuating extra and intracellular environments • Need for designing or tuning networks having robustbehavior Robust behavior if system presents expected property despite parameter variations • Two problems of interest: • Robustness analysis: check whether properties are satisfied for all parameters in a set • Tuning: find parameter sets such that properties are satisfied for all parameters in the sets

10. Robustness analysis and tuning • Problem for network design: parameteruncertainties • current limitations in experimental techniques • fluctuating extra and intracellular environments • Need for designing or tuning networks having robustbehavior Robust behavior if system presents expected property despite parameter variations • Two problems of interest: 1) find parameters such that system satisfies property 2) check robustness of proposed modifications

11. P1 p1 X0 x0 p2 P2 set of initial conditions set of parameters fixed initial condition fixed parameter Robustness analysis and tuning • Constraints on robustness analysis and tuning of networks • genetic regulations are non-linear phenomena • size of the networks • reasoning for sets of parameters, initial conditions and inputs How to reason with infinite number of parameters and initial conditions ? How to define the expected dynamical properties ?

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

13. Overview • Introduction: rational design of synthetic gene networks • Modeling and specification • Robustness analysis • Tuning • Application: tuning a synthetic transcriptional cascade • Discussion and conclusions

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

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

16. x : protein concentration  : threshold concentration  ,  : rate parameters Gene network models • Genetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions A B b

17. x : protein concentration  : threshold concentration  ,  : rate parameters Gene network models • Genetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions A B a

18. A x : protein concentration  : threshold concentration B  ,  : rate parameters b a Gene network models • Genetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions

19. Gene network models • Differential equation models

20. Gene network models • Differential equation models

21. Gene network models • Differential equation models

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

23. Gene network models • Differential equation models • is piecewise-multiaffine (PMA)function of state variables • is piecewise-affine function of rate parameters (’sand ’s) Belta et al., CDC, 02

24. Specifications of dynamical properties • Dynamical properties expressed in temporal logic (LTL)

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

26. A B b a Specifications of dynamical properties • Dynamical properties expressed in temporal logic (LTL) • Syntax of LTL formulas • set of atomic proposition • usual logical operators • temporal operators , bistability property:

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

28. Specifications of dynamical properties • Dynamical properties expressed in temporal logic (LTL) • Syntax of LTL formulas • 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 ... ... ...

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

30. Overview • Introduction: rational design of synthetic gene networks • Modeling and specification • Robustness analysis • Definition of discrete abstraction • Computation of discrete abstraction • Model checking the discrete abstraction • Tuning • Application: tuning a synthetic transcriptional cascade • Discussion and conclusions

31. R12 R11 R13 R14 R15 R6 R7 R8 R9 R10 R3 R4 R5 R1 R2 Discrete abstraction: definition • Threshold hyperplanes partition state space: set of rectangles

32. Discrete abstraction: definition • Discrete transition system,, where

33. R12 R11 R13 R14 R15 R6 R7 R8 R9 R10 R3 R4 R5 R1 R2 Discrete abstraction: definition • Discrete transition system,, where • finite set of rectangles

34. R11 R6 R1 representation of the flow for some Discrete abstraction: definition • Discrete transition system,, where • finite set of rectangles • transition relation

35. R12 R11 R13 R14 R15 R6 R7 R8 R9 R10 R3 R4 R5 R1 R2 Discrete abstraction: definition • Discrete transition system,, where • finite set of rectangles • transition relation

36. R12 R11 R13 R14 R15 R6 R7 R8 R9 R10 R3 R4 R5 R1 R2 How can we compute ? Discrete abstraction: definition • Discrete transition system,, where • finite set of rectangles • transition relation • satisfaction relation

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

38. Discrete abstraction: computation • Transition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles R1 R2

39. In every rectangular region, the flow is a convex combination of its values at the vertices Belta and Habets, Trans. Autom. Contr., 06 Discrete abstraction: computation • Transition 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) R1 R2

40. In every rectangular region, the flow is a convex combination of its values at the vertices Belta and Habets, Trans. Autom. Contr., 06 Discrete abstraction: computation • Transition 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) • Transitions can be computed by polyhedral operations (Because is a piecewise-affine function of p) R1 R2

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

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

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

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

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

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

47. Overview • Introduction: rational design of synthetic gene networks • Modeling and specification • Robustness analysis • Definition of discrete abstraction • Computation of discrete abstraction • Model checking the discrete abstraction • Tuning • Application: tuning a synthetic transcriptional cascade • Discussion and conclusions

48. Overview • Introduction: rational design of synthetic gene networks • Modeling and specification • Robustness analysis • Tuning • Application: tuning a synthetic transcriptional cascade • Discussion and conclusions

49. Tuning • Synthesis of parameter constraints Collect affine constraints defining existence of transitions between rectangles: • Parameter space exploration Construct partition of parameter space using parameter constraints

50. bistability property: Tuning • Synthesis of parameter constraints Collect affine constraints defining existence of transitions between rectangles: • Parameter space exploration Construct partition of parameter space using parameter constraints Test the validity of each region in parameter space