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 applicationsdetection 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

Ultrasensitive input/output responseat steady-state

Transcriptional cascade in E. coli

Synthetic biology

Synthetic biology: design and construct biological systems with desired behaviors

engineering and medical applications

study biological system properties in controlled environment

Network design is difficult

Most newly-created networks need tuning

Ultrasensitive input/output responseat steady-state

Transcriptional cascade in E. coli

Synthetic biology

Synthetic biology: design and construct biological systems with desired behaviors

engineering and medical applications

study biological system properties in controlled environment

Network design is difficult

Most newly-created networks need tuning

How can the network be tuned ?

Robustness analysis and tuning

Problem for network design: parameter uncertainties current limitations in experimental techniques

fluctuating extra and intracellular environments

Need for designing or tuning networks having robust behavior

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: parameter uncertainties current limitations in experimental techniques

fluctuating extra and intracellular environments

Need for designing or tuning networks having robust behavior

Robust behavior if system presents expected property despite parameter

variations

Two problems of interest:

1) find parameters such that system satisfies property2) check robustness of proposed modifications

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

fixed initial conditionfixed parameter

x0

p1

p2

X0

P1

P2

set of initial conditionsset of parameters

How to define the expected dynamical properties ?

How to reason with infinite number of parameters and initial conditions ?

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

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

Genetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions

b

B

a

A

Gene network models

Genetic 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

Genetic 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

Genetic 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

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

PMA models are related to piecewise affine modelsGlass and Kauffman, J. Theor. Biol., 73 de Jong et al., Bull. Math. Biol., 04

Belta et al., CDC, 02

Gene network models

Differential equation models

is piecewise-multiaffine (PMA) function of state variables

is piecewise-affine function of rate parameters (’s and ’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 ,

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:

b

B

a

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 ,

Semantics of LTL formulas defined over executions of transition systems

...

...

...

q q q qq

qq q q q

qqqp , qp , qp ,

:Fq

:Gq

:Uqp

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

...

...

...

q q q qq

qq q q q

qqqp , qp , qp ,

:Fq

:Gq

:Uqp

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

Threshold hyperplanes partition state space: set of rectangles

R1 R2 R3 R4 R5

R6 R7 R8 R9 R10

R15R14R13R12R11

Discrete abstraction: definition

Discrete transition system, , where

Discrete abstraction: definition

Discrete transition system, , where finite set of rectangles

R1 R2 R3 R4 R5

R6 R7 R8 R9 R10

R15R14R13R12R11

Discrete abstraction: definition

Discrete transition system, , where finite set of rectangles

transition relation

representation of the flow for some

R1

R6

R11

Discrete abstraction: definition

Discrete transition system, , where finite set of rectangles

transition relation

R1 R2 R3 R4 R5

R6 R7 R8 R9 R10

R15R14R13R12R11

Discrete abstraction: definition

Discrete transition system, , where finite set of rectangles

transition relation

satisfaction relation

R1 R2 R3 R4 R5

R6 R7 R8 R9 R10

R15R14R13R12R11

How can we compute ?

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

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)In every rectangular region, the flow is a convex combination of its values at the vertices

Belta and Habets, Trans. Autom. Contr., 06

R1 R2

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)

In every rectangular region, the flow is a convex combination of its values at the vertices

Belta and Habets, Trans. Autom. Contr., 06

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

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

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

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

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

bistability property:

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

bistability property:

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

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

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

Robustness analysis

provides finite description of the dynamics of original system in

state space for parameter sets

can be computed by polyhedral operations

is a conservative 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

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

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

Hooshangi et al., PNAS, 05

Input/output response

Transcriptional cascade in E. coli

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

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

Robustness analysis and tuning of genetic regulatory networks dynamical properties expressed in temporal logic

unknown parameters, initial conditions and inputs given by intervals

piecewise-multiaffine differential 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

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

Antoniotti et al., Theor. Comput. Sci., 04Calzone et al., Trans. Comput. Syst. Biol, 06

de Jong et al., Bull. Math. Biol., 04Ghosh 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

X0

P1

P2

Discrete abstraction: definition

Discrete transition system, , where finite set of rectangles

transition relation

X0

P1

P2

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:

X0

P1

P2