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Robustness Analysis and Tuning of Synthetic Gene Networks

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### 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

Synthetic biology

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

Synthetic biology

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

banana-smelling bacteria

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....

Synthetic biology

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

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

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

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 ?

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

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

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 ?

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

Overview

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

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

B

b

a

Gene network models- Genetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions

: 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

: 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

x : protein concentration

: threshold concentration

B

, : rate parameters

b

a

Gene network modelsGene network models

- Differential equation models

Gene network models

- Differential equation models

Gene network models

- Differential equation models

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

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

Specifications of dynamical properties

- Dynamical properties expressed in temporal logic (LTL)

Specifications of dynamical properties

- Dynamical properties expressed in temporal logic (LTL)
- Syntax of LTL formulas
- set of atomic proposition
- usual logical operators
- temporal operators ,

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:

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

...

...

...

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

...

...

...

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

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

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

Discrete abstraction: definition

- Discrete transition system,, where

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

R6

R1

representation of the flow for some

Discrete abstraction: definition- Discrete transition system,, where
- finite set of rectangles
- transition relation

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

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

Discrete abstraction: computation

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

Discrete abstraction: computation

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

R1

R2

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

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(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

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

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

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

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:

Discrete abstraction: model checking

- is a finite transition system and can be model-checked
- can be used for proving properties of the original system

bistability property:

Discrete abstraction: model checking

- 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

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

Overview

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

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

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

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

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

- More efficient approach: model check while constructing the partition

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

- More efficient approach: model check while constructing the partition
- Approach implemented in publicly-available tool RoVerGeNe

Exploits tools for polyhedra operations (MPT) and model checker (NuSMV)

Batt et al., HSCC07

Overview

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

Summary

- Robustness analysis
- provides finite description of the dynamics of original system in state space for parameter sets
- can be computed by polyhedral operations
- is aconservative approximation of original system
- Tuning
- Use affine constraints appearing in transition computation to define partition of parameter space
- Explore every region in parameter space

Overview

- Introduction: rational design of synthetic gene networks
- Modeling and specification
- Analysis for fixed parameters
- Analysis for sets of parameters
- Application: tuning a synthetic transcriptional cascade
- Modeling the actual network
- Tuning the network
- Verifying robustness of tuned network
- Discussion and conclusions

Transcriptional cascade: problem

- Approach for robust tuning of the cascade:
- develop a model of the actual cascade
- tune network by modifying 3 key parameters
- check that property still true when all parameters vary in ±10% intervals

Transcriptional cascade in E. coli

Input/output response

Hooshangi et al., PNAS, 05

Transcriptional cascade: modeling

- PMA differential equation model (1 input and 4 state variables)
- Parameter identification

Computation of I/O behavior of cascade

Transcriptional cascade: specification

Expected input/output behaviorof cascade

Temporal logic specification

Transcriptional cascade: tuning

- Tuning: search for valid parameter sets
- Let 3 production rates unconstrained
- Answer: 1 set found (<2 h.)

Computation of I/O behavior of cascade for some parameters in the set

Transcriptional cascade: analysis

- Robustness: test robustness of proposed modification
- Assume
- Is property true if all rate parameters vary in a ±10% interval? or ±20%?
- Answer: ‘Yes’ for ±10% parameter variations (<4 h.) ‘No’ for ±20% parameter variations

11 uncertain parameters:

Overview

- Introduction: rational design of synthetic gene networks
- Modeling and specification
- Analysis for fixed parameters
- Analysis for sets of parameters
- Tuning of a synthetic transcriptional cascade
- Modeling the actual network
- Tuning the network
- Verifying robustness of tuned network
- Discussion and conclusions

Overview

- Introduction: rational design of synthetic gene networks
- Modeling and specification
- Analysis for fixed parameters
- Analysis for sets of parameters
- Tuning of a synthetic transcriptional cascade
- Discussion and conclusions

Conclusion

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

Discussion

- Related work: formal analysis of uncertain biological networks
- Iterative search in dense parameter space of ODE models using model checking
- Exhaustive exploration of finite parameter space of logical models using model checking
- Analysis of qualitative PA models by reachability analysis or model checking
- Robust stability and model validation of ODE models using SOSTOOLS
- Further work
- Verification of properties involving timing constraints
- Compositional verification to exploit network modularity

Antoniotti et al., Theor. Comput. Sci., 04

Calzone et al., Trans. Comput. Syst. Biol, 06

Bernot et al., J. Theor. Biol., 04

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!

- Calin Belta(Boston University, USA)
- Ron Weiss(Princeton University, USA)
- Boyan Yordanov (Boston University, USA)

Discrete abstraction: definition

- Discrete transition system,, where
- finite set of rectangles
- transition relation

P1

X0

P2

Discrete abstraction: definition

- Discrete transition system,, where
- finite set of rectangles
- transition relation

P1

X0

P2

Discrete abstraction: model checking

- is a finite transition system and can be model-checked
- can be used for proving properties of the original system

bistability property:

P1

X0

P2

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