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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 .

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robustness analysis and tuning of synthetic gene networks

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
  • Synthetic biology: design and construct biological systems with desired behaviors
synthetic biology3
Synthetic biology
  • Synthetic biology: design and construct biological systems with desired behaviors

banana-smelling bacteria

synthetic biology4
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 biology5
Synthetic biology
  • Synthetic biology: design and construct biological systems with desired behaviors
    • engineering and medical applications
    • study biological system properties in controlled environment
synthetic biology6
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 biology7
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 biology8
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
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 tuning10
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

robustness analysis and tuning11

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 ?

robustness analysis and tuning12
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
Overview
  • Introduction: rational design of synthetic gene networks
  • Modeling and specification
  • Robustness analysis
  • Tuning
  • Application: tuning a synthetic transcriptional cascade
  • Discussion and conclusions
overview14
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
gene network models

A

B

b

a

Gene network models
  • Genetic networks modeled by class of differential equations using ramp functions to describe regulatory interactions
gene network models16

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

gene network models17

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

gene network models18

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
gene network models19
Gene network models
  • Differential equation models
gene network models20
Gene network models
  • Differential equation models
gene network models21
Gene network models
  • Differential equation models
gene network models22
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 models23
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
Specifications of dynamical properties
  • Dynamical properties expressed in temporal logic (LTL)
specifications of dynamical properties25
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 properties26

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:

specifications of dynamical properties27
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 properties28
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

...

...

...

overview29
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
overview30
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
discrete abstraction definition

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
discrete abstraction definition32
Discrete abstraction: definition
  • Discrete transition system,, where
discrete abstraction definition33

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
discrete abstraction definition34

R11

R6

R1

representation of the flow for some

Discrete abstraction: definition
  • Discrete transition system,, where
    • finite set of rectangles
    • transition relation
discrete abstraction definition35

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
discrete abstraction definition36

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
discrete abstraction computation
Discrete abstraction: computation
  • Transition between rectangles iff for some parameter, the flow at a common vertex agrees with relative position of rectangles
discrete abstraction computation38
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 computation39

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

discrete abstraction computation40

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

discrete abstraction model checking
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 checking42
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 checking43
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 checking44
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 checking45
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 checking46
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

overview47
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
overview48
Overview
  • Introduction: rational design of synthetic gene networks
  • Modeling and specification
  • Robustness analysis
  • Tuning
  • Application: tuning a synthetic transcriptional cascade
  • Discussion and conclusions
tuning
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

tuning50

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

tuning51
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:

tuning52
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
tuning53
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

overview54
Overview
  • Introduction: rational design of synthetic gene networks
  • Modeling and specification
  • Robustness analysis
  • Tuning
  • Application: tuning a synthetic transcriptional cascade
  • Discussion and conclusions
summary
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
overview56
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
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
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
Transcriptional cascade: specification

Expected input/output behaviorof cascade

Temporal logic specification

transcriptional cascade tuning
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
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:

overview62
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
overview63
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
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
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
Acknowledgements

Thank you for your attention!

  • Calin Belta(Boston University, USA)
  • Ron Weiss(Princeton University, USA)
  • Boyan Yordanov (Boston University, USA)
discrete abstraction definition67
Discrete abstraction: definition
  • Discrete transition system,, where
    • finite set of rectangles
    • transition relation

P1

X0

P2

discrete abstraction definition68
Discrete abstraction: definition
  • Discrete transition system,, where
    • finite set of rectangles
    • transition relation

P1

X0

P2

discrete abstraction model checking69
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:

P1

X0

P2