Robustness Analysis and Tuning of Synthetic Gene Networks

1
Robustness Analysis and Tuning of Synthetic Gene
Networks

• Calin Belta
• (with Grégory Batt)
• Center for Information and Systems Engineering
• and Center for BioDynamics
• Boston University

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
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
• Network design is difficult
• Most newly-created networks are non-functioning
  and need tuning

How can the network be tuned ?
8
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

9
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
• property
• 2) check robustness of proposed modifications

10
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

11
Overview

• Introduction rational design of synthetic gene
  networks
• Problem definition
• Robustness analysis
• Tuning
• Application tuning a synthetic transcriptional
  cascade
• Discussion and conclusions

12
Overview

• Introduction rational design of synthetic gene
  networks
• Problem definition
• Models piecewise-multiaffine differential
  equations
• Dynamical property specification LTL formulas
• Meaning of a system satisfies a property
• Robustness analysis
• Tuning
• Application tuning a synthetic transcriptional
  cascade
• Discussion and conclusions

13
Gene network models

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

14
Gene network models

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

A
B
b
15
Gene network models

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

A
B
a
16
Gene network models

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

17
Gene network models

• Differential equation models

18
Gene network models

• Differential equation models

19
Gene network models

• Differential equation models
• is piecewise-multiaffine (PMA) function of
  state variables

Belta et al., CDC, 02
20
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
21
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
22
Specifications of dynamical properties

• Dynamical properties expressed in temporal logic
  (LTL)

23
Specifications of dynamical properties

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

24
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
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 ,
• Semantics of LTL formulas defined over executions
  of transition systems

...
...
...
26
State space partition

• PMA system
• Threshold hyperplanes partition state space set
  of rectangles

27
Embedding transition system

• PMA system, , associated with
  embedding transition system,
  , where

28
Embedding transition system

• PMA system, , associated with
  embedding transition system,
  , where
• continuous state space

R12
R15
R14
R13
R11
R6
R7
R8
R9
R10
R3
R4
R5
R1
R2
29
Embedding transition system

• PMA system, , associated with
  embedding transition system,
  , where
• continuous state space
• transition relation

x4
x3
x2
x1
30
Embedding transition system

• PMA system, , associated with
  embedding transition system,
  , where
• continuous state space
• transition relation
• satisfaction relation

x4
x3
x2
x1
31
Embedding transition system

• PMA system, , associated with
  embedding transition system,
  
• captures almost all solution
  trajectories of

32
Embedding transition system

• PMA system, , associated with
  embedding transition system,
  
• captures almost all solution
  trajectories of
• satisfies property for parameter p if
• Then, p is a valid parameter

33
Embedding transition system

• PMA system, , associated with
  embedding transition system,
  
• captures almost all solution
  trajectories of
• satisfies property for parameter p if
• Then, p is a valid parameter
• Problem definitions
• Robustness
• Synthesis
• How can we test whether for all parameters
  in set P ?

34
Overview

• Introduction rational design of synthetic gene
  networks
• Problem definition
• 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

35
Discrete abstraction definition

• Discrete transition system,
  , where

36
Discrete abstraction definition

• Discrete transition system,
  , where
• finite set of rectangles

37
Discrete abstraction definition

• Discrete transition system,
  , where
• finite set of rectangles
• quotient transition relation

38
Discrete abstraction definition

• Discrete transition system,
  , where
• finite set of rectangles
• quotient transition relation

39
Discrete abstraction definition

• Discrete transition system,
  , where
• finite set of rectangles
• quotient transition relation
• quotient satisfaction relation

x4
R11
x3
R6
x2
...
x1
R1
40
Discrete abstraction definition

• Discrete transition system,
  , where

41
Discrete abstraction definition

• Discrete transition system,
  , where
• transition relation

42
Discrete abstraction computation

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

R1
R2
43
Discrete abstraction computation

• Transition between rectangles iff for some
  parameter, the flow at a common vertex agrees
  with relative position of rectangles
• 
  , with
• is a union of polytopes in parameter
  space
• Because in every rectangle, is an affine
  function of p
• can be computed by polyhedral
  operations

44
Discrete abstraction model checking

• Model checking is automated technique for
  verifying that finite transition systems satisfy
  temporal logic properties
• Efficient computer tools are available to perform
  model checking

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

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
• is conservative approximation of original
  system
• (simulation relations between
  transition systems)
• Issue verification of liveness properties and
  progress of time

Alur et al., Proc. IEEE, 00
Batt et al., TACAS, 07
47
Overview

• Rational design of synthetic gene networks
• Problem definition
• Robustness analysis
• Tuning
• Synthesis of parameter constraints
• Iterative parameter space exploration
• Hierarchical parameter space exploration
• Application tuning a synthetic transcriptional
  cascade
• Discussion and conclusions

48
Synthesis of parameter constraints

• are affine constraints defining
  existence of transitions between rectangles
• Set of constraints define polyhedral partition of
  parameter space

49
Synthesis of parameter constraints

• are affine constraints defining
  existence of transitions between rectangles
• Set of constraints define polyhedral partition of
  parameter space

50
Synthesis of parameter constraints

• are affine constraints defining
  existence of transitions between rectangles
• Set of constraints define polyhedral partition of
  parameter space

R1
R2
51
Synthesis of parameter constraints

• are affine constraints defining
  existence of transitions between rectangles
• Set of constraints define polyhedral partition of
  parameter space

52
Synthesis of parameter constraints

• are affine constraints defining
  existence of transitions between rectangles
• Set of constraints define polyhedral partition of
  parameter space

53
Synthesis of parameter constraints

• are affine constraints defining
  existence of transitions between rectangles
• Set of constraints define polyhedral partition of
  parameter space
• All parameters in a same region are equivalent
• Equivalent parameters correspond to a same
  discrete abstraction

54
Iterative parameter space exploration

• Collect all constraints by inspecting all
  vertices
• Construct the partition of parameter space

55
Iterative parameter space exploration

• Collect all constraints by inspecting all
  vertices
• Construct the partition of parameter space
• Iteratively test the validity of each region in
  parameter space
• Approach not efficient large number of regions
  in parameter space

bistability property
56
Hierarchical parameter space exploration

• Collect all constraints by inspecting all
  vertices
• Model check while constructing the partition
• Additional transition system used to
  detect that refinement in parameter space will
  not improve results

bistability property
57
Hierarchical parameter space exploration

• Collect all constraints by inspecting all
  vertices
• Model check while constructing the partition
• Additional transition system used to
  detect that refinement in parameter space will
  not improve results
• Approach implemented in publicly-available tool
  RoVerGeNe
• Exploits tools for polyhedra operations (MPT),
  graph operations (MatlabBGL), and model checker
  (NuSMV)

bistability property
Batt et al., HSCC, 07
58
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 polyhedral partition of
  parameter space
• Efficiently explore parameter space using
  hierarchical approach

59
Overview

• Introduction rational design of synthetic gene
  networks
• Problem definition
• 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

60
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
61
Transcriptional cascade modeling

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

Computation of I/O behavior of cascade
62
Transcriptional cascade specification
Expected input/output behaviorof cascade
Temporal logic specification
63
Transcriptional cascade tuning

• Tuning search for valid parameter sets
• Let 3 production rates unconstrained
• Answer 1 set found (lt2 h.)

Computation of I/O behavior of cascade for some
parameters in the set
64
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
  (lt4 h.) No for 20 parameter variations
  

11 uncertain parameters
65
Overview

• Introduction rational design of synthetic gene
  networks
• Problem definition
• Analysis for fixed parameters
• Analysis for sets of parameters
• Tuning of a synthetic transcriptional cascade
• Discussion and conclusions

66
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

67
Discussion

• Related work formal analysis of uncertain
  biological networks (deterministic dynamics)
• 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
• Automatic state-space partition refinement
• Verification of properties involving timing
  constraints
• Compositional verification to exploit network
  modularity

Antoniotti et al., Theor. Comput. Sci.,
04Calzone 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
68
Acknowledgements

• Thank you for your attention!

• Grégory Batt (Boston University, USA)
• Ron Weiss (Princeton University, USA)
• Boyan Yordanov (Boston University, USA)