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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: [email protected]
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