Robustness Analysis and Tuning of Synthetic Gene Networks

1
Robustness Analysis and Tuning of Synthetic
Gene Networks

• Grégory Batt1 Boyan Yordanov1 Calin Belta1
  Ron Weiss2
• 1 Centers for Information and Systems Engineering
  and for BioDynamics
• Boston University
  ( now at )
• 2 Departments of Molecular Biology and of
  Electrical EngineeringPrinceton University
• Towards Systems Biology 2007

2
Synthetic biology

• Synthetic biology application of engineering
  approaches to produce novel artificial devices
  using biological building blocks

3
Synthetic biology

• Synthetic biology application of engineering
  approaches to produce novel artificial devices
  using biological building blocks

banana-smelling bacteria
4
Synthetic biology

• Synthetic biology application of engineering
  approaches to produce novel artificial devices
  using biological building blocks

banana-smelling bacteria
5
Synthetic biology

• Synthetic biology application of engineering
  approaches to produce novel artificial devices
  using biological building blocks
• Numerous potential engineering and medical
  applications
• biofuel production, environment depollution, . .
  .
• biochemical synthesis, tumor cell destruction, .
  . .

banana-smelling bacteria
6
Synthetic gene networks

• Gene networks are networks of genes, proteins,
  small molecules and their regulatory interactions

Transcriptional cascade Hooshangi et al, PNAS,
05
Ultrasensitive I/O response at steady-state
7
Need for rational design

• Gene networks are networks of genes, proteins,
  small molecules and their regulatory interactions
• Network design analysis of non-linear dynamical
  system with parameter uncertainties
• current limitations in experimental techniques
• fluctuating extra and intracellular environments

Problem most newly-created networks are
non-functioning and need tuning
8
Robustness analysis and tuning

• Two problems of interest
• robustness analysis check whether dynamical
  properties are satisfied for all parameters in a
  set
• tuning find parameter sets such that dynamical
  properties are satisfied for all parameters in
  the sets
• Approach
• unknown parameters, initial conditions and
  inputs given by intervals
• piecewise-multiaffine differential equations
  models of gene networks
• dynamical properties specified in temporal logic
  (LTL)
• adapt techniques from hybrid systems theory and
  model checking

9
Hybrid systems approach

• Analysis of dynamical systems
• Traditional view fixed initial condition and
  fixed parameter
• More interesting set of initial conditions and
  set of parameters

10
Hybrid systems approach

• Analysis of dynamical systems
• Traditional view fixed initial condition and
  fixed parameter
• More interesting set of initial conditions and
  set of parameters
• How to reason with infinite number of parameters
  and initial conditions ?

11
Hybrid systems approach

• Analysis of dynamical systems
• Traditional view fixed initial condition and
  fixed parameter
• More interesting set of initial conditions and
  set of parameters
• How to reason with infinite number of parameters
  and initial conditions ? direct vs indirect
  approaches

P1
X0
P2
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Hybrid systems approach

• Analysis of dynamical systems
• Traditional view fixed initial condition and
  fixed parameter
• More interesting set of initial conditions and
  set of parameters
• How to reason with infinite number of parameters
  and initial conditions ? direct vs indirect
  approaches

P1
X0
P2
model checking possible
13
Overview

• Introduction
• Problem definition
• Robust design of gene networks
• Application tuning a synthetic transcriptional
  cascade
• Discussion and conclusions

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Overview

• Introduction
• Problem definition
• Robust design of gene networks
• Application tuning a synthetic transcriptional
  cascade
• Discussion and conclusions

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Gene network models
cross-inhibition network
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Gene network models
cross-inhibition network
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Gene network models
cross-inhibition network
regulation functions
1
1
1
0
0
0
x
x
x
Hill function
step function
ramp function
?Hill-type models
?PMA models
?PA models
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Gene network models
cross-inhibition network
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Gene network models

• Find parameters such that network is bistable

cross-inhibition network
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Gene network models

• Partition of the state space rectangles

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Gene network models

• Partition of the state space rectangles
• Differential equation models ,
  with
• is piecewise-multiaffine (PMA) function of
  state variables
• is affine function of rate parameters (?s
  and ?s)
• (multiaffine functions products of different
  state variables allowed)

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Specifications of dynamical properties

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

23
Specifications of dynamical properties

• Dynamical properties expressed in temporal logic
  (LTL)
• set of atomic proposition
• usual logical operators
• temporal operators ,
• Semantics of LTL formulas defined over executions
  of transition systems

...
...
...
24
Specifications of dynamical properties

• Dynamical properties expressed in temporal logic
  (LTL)
• 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

...
...
...
25
Specifications of dynamical properties

• Dynamical properties expressed in temporal logic
  (LTL)
• set of atomic proposition
• usual logical operators
• temporal operators ,
• Semantics of LTL formulas defined over executions
  of transition systems

bistability property
26
Overview

• Introduction
• Problem definition
• Robust design of gene networks
• Application tuning a synthetic transcriptional
  cascade
• Discussion and conclusions

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Robust design of gene networks
gene network
PMA model
intervals for uncertain parameters
specifications
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Robust design of gene networks
gene network
PMA model
synthesis of parameter constraints
intervals for uncertain parameters
discrete abstractions convexity properties
specifications
model checking
Valid parameter set
No conclusion
Batt et al., HSCC07
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Computation of discrete abstraction
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Computation of discrete abstraction

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

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Computation of discrete abstraction

• Transition between rectangles iff for some
  parameter, the flow at a common vertex agrees
  with relative position of rectangles
• Transitions can be computed by polyhedral
  operations
• where
• (Because is a piecewise-multiaffine function
  of x and an affine function of p)

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RoVerGeNe

• Approach implemented in publicly-available tool
  RoVerGeNe

Written in Matlab, exploits polyhedral
operation toolbox MPT and model checker NuSMV
http//iasi.bu.edu/batt
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Overview

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

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Transcriptional cascade approach

• Approach for robust tuning of the cascade
• develop a model of the actual cascade
• specify expected behavior
• tune network by searching for valid parameter
  sets
• verify robustness of tuned network

Transcriptional cascade Hooshangi et al, PNAS,
05
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Transcriptional cascade modeling

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

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Transcriptional cascade specification

• Expected input/output behavior of cascade at
  steady state and for all initial states
• Temporal logic specifications
• Liveness property additional fairness
  constraints needed

Batt et al., TACAS07
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Transcriptional cascade tuning

• Tuning search for valid parameter sets
• Let 3 production rate parameters unconstrained
• Answer 15 sets found (lt4 h., 1500 rectangles,
  18 parameter constraints)

Batt et al., Bioinfo, 07
comparison with numerical simulation results in
parameter space and for input/output behavior
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Transcriptional cascade robustness

• Robustness check that tuned network behaves
  robustly
• Let all production and degradation rate
  parameters range in intervals centered at their
  reference values (with 10 or 20 variations)
• Answer for 10 parameter variations Yes (lt
  4hrs)
• ? proves that specification holds despite 10
  parameter variations
• Answer for 20 parameter variations No (lt
  4hrs)
• ? suggests that specification does not hold for
  some parameters in the 20 set (confirmed
  by manual analysis of counter-example)

11 uncertain parameters
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Overview

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

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Summary

• Gene networks modeled as uncertain PMA systems
• piecewise-multiaffine differential equations
  models
• unknown parameters, initial conditions and inputs
  given by intervals
• dynamical properties expressed in temporal logic
• Use of tailored combination of parameter
  constraint synthesis, discrete abstractions, and
  model checking
• Method implemented in publicly-available tool
  RoVerGeNe
• Approach can answer non-trivial questions on
  networks of biological interest

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Discussion

• First computational approach for tuning synthetic
  gene networks
• Related work
• qualitative/discrete approaches (reachability or
  model checking)
• quantitative approaches with fixed parameter
  values (reachability or MC)
• quantitative approaches with uncertain parameters
  (optimisation-based)
• Further work
• verification of properties involving timing
  constraints (post doc, Verimag)
• deal with uncertain threshold parameters too
• use of compositional verification for design of
  large modular networks

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Acknowledgements

• Thanks to Calin Belta, Boyan Yordanov, Ron Weiss
• and to Ramzi Ben Salah and Oded Maler
• References
• G. Batt, B. Yordanov, C. Belta and R. Weiss
  (2007) Robustness analysis and tuning of
  synthetic gene networks. In Bioinformatics,
  23(18)2415-1422
• G. Batt, C. Belta and R. Weiss (2007) Temporal
  logic analysis of gene networks under parameter
  uncertainty. Accepted to Joint Special Issue on
  Systems Biology of IEEE Trans. Circuits and
  Systems and IEEE Trans. Automatic Control

Center for BioDynamics
Center for Information and Systems Engineering
Boston University
Verimag Lab
Grenoble Polytechnic Institute