Model Checking Genetic Regulatory Networks with Parameter Uncertainty

1
Model Checking Genetic Regulatory Networks with
Parameter Uncertainty

• Grégory Batt, Calin Belta, Ron Weiss
• HSCC 2007
• Presented by Spring Berman
• ESE 680-003 Systems Biology

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Motivation

• Uncertainty in biological parameters limits the
  development and analysis of models of genetic
  regulatory networks
• - Sources gene expression noise, mutation,
  cell death, changing intra- and extra-cellular
  environments
• - Direct determination of rate constants in
  vivo is still inaccurate and nontrivial 1
• Network tuning is a central problem in synthetic
  biology
• - Most initial attempts at building gene
  networks fail to produce the desired behavior 1
• 1 E. Andrianantoandro, S. Basu, D. Karig, R.
  Weiss Synthetic biology New engineering rules
  for an emerging discipline. Mol. Syst. Biol.
  (2006)

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Objective

• Problem 1 Robustness analysis
• Check whether a dynamical property is
  satisfied by every parameter in a given set and
  for every initial state in a given region.
• Problem 2 Parameter constraint synthesis
• Find a subset of a given parameter set that
  satisfies a certain dynamical property.
• Assume no sliding modes

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Approach

• 1) Model genetic network with piecewise-multiaffi
  ne (PMA) differential equations
• 2) Formulate the property to be checked in
  Linear Temporal Logic (LTL)
• 3) Define an embedding transition system for the
  PMA model and its discrete abstraction
• 4) Define a hierarchy of parameter equivalence
  classes
• 5) Explore the parameter space efficiently

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Approach

• 1) Model genetic network with piecewise-multiaffi
  ne (PMA) differential equations
• 2) Formulate the property to be checked in
  Linear Temporal Logic (LTL)
• 3) Define an embedding transition system for the
  PMA model and its discrete abstraction
• 4) Define a hierarchy of parameter equivalence
  classes
• 5) Explore the parameter space efficiently

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PMA models of genetic networks

• State vector
• n genes xi concentration of protein
  encoded by gene i
• Parameter vector
• Network dynamics
• Production rate
  possibly uncertain
• Degradation rate
  parameters
• Regulation
  function (products of ramp functions)

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Example Toggle Switch
repressor protein
repressor protein
gene
gene
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Approach

• 1) Model genetic network with piecewise-multiaffi
  ne (PMA) differential equations
• 2) Formulate the property to be checked in
  Linear Temporal Logic (LTL)
• 3) Define an embedding transition system for the
  PMA model and its discrete abstraction
• 4) Define a hierarchy of parameter equivalence
  classes
• 5) Explore the parameter space efficiently

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Dynamical Property as LTL Formula

• Temporal Logic System for describing how the
  truth of assertions changes over time
• Linear Events occur along a single timeline
• Dynamical property of a gene network can be
  expressed as an LTL formula, which is built from
• - Atomic propositions in this case
• , ,
• - Boolean operators
• not ( ), and (
  ), or ( )
• - Temporal operators 2
• Fp eventually p, Gp always p,
  p U q p until q
• 2 E. A. Emerson. Temporal and modal logic.
  In J. van Leeuwen, ed., Handbook of Theoretical
  Computer Science, vol B, pp. 995-1072. MIT
  Press, 1990.

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Example Toggle Switch

• Bistability property expressed in LTL
• If concentration of A is low
  and B is high, then the system always
  remains in this state
• If concentration of A is high and B is low,
  then the system always remains in
  this state

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Approach

• 1) Model genetic network with piecewise-multiaffi
  ne (PMA) differential equations
• 2) Formulate the property to be checked in
  Linear Temporal Logic (LTL)
• 3) Define an embedding transition system for the
  PMA model and its discrete abstraction
• 4) Define a hierarchy of parameter equivalence
  classes
• 5) Explore the parameter space efficiently

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Embedding Transition System

• PMA system
• PMA function set of all
  atomic propositions
• Partition into rectangles
• 
  
  is a
  threshold constant or atomic proposition constant
• Embedding Transition System associated with
  
• Parameter vector
• Union of all rectangles in
• Transition relation
  iff a path from x to x ,
  where x and x are in the same or adjacent
  rectangles
• Satisfaction relation
  iff x satisfies proposition p

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Discrete abstraction

• Finite transition system preserving dynamical
  properties of
• Discrete abstraction of
• Set of rectangles (equivalence classes)
• Transition relation
  iff R R , or R is adjacent to R
  and there is a vertex v on the shared facet such
  that
• (exploits convexity property of MA
  functions on rectangles)
• Satisfaction relation
  iff for every

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Example Toggle Switch
Continuous dynamics
Discrete abstraction
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Property Verification

• A parameter set P is valid for an LTL formula ?
  iff
  for almost all
• ?
• Can compute and use model checking
  to test whether
• If , no conclusion on
  validity of p

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Approach

• 1) Model genetic network with piecewise-multiaffi
  ne (PMA) differential equations
• 2) Formulate the property to be checked in
  Linear Temporal Logic (LTL)
• 3) Define an embedding transition system for the
  PMA model and its discrete abstraction
• 4) Define a hierarchy of parameter equivalence
  classes
• 5) Explore the parameter space efficiently

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Parameter equivalence classes

• f is a piecewise-affine, continuous function of
  p
• partition the
  parameter space into polyhedra represent these
  regions by Boolean numbers
• Define parameter sets
• If for some
  then for all
• just
  need to test a random p per
• (but exponential increase with number of
  predicates)

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Example Toggle Switch
32 affine expressions, only 4 non-constant ones
Parameter space
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Example Toggle Switch
Hierarchy among parameter sets
Parameter equivalence classes
Valid for bistability property
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Approach

• 1) Model genetic network with piecewise-multiaffi
  ne (PMA) differential equations
• 2) Formulate the property to be checked in
  Linear Temporal Logic (LTL)
• 3) Define an embedding transition system for the
  PMA model and its discrete abstraction
• 4) Define a hierarchy of parameter equivalence
  classes
• 5) Explore the parameter space efficiently

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System over- and under-approximations

• Contains all transitions present in at least one
• Contains only transitions present in all
• or
  , inspect subsets of P

(A)
(B)
(C)

• Algorithm Recursively explore the tree of
  parameter sets, starting from stop search
  at condition (A) or (B)

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Computation of ,

• iff R R , or R is adjacent to
  R and
• iff R R , or R is adjacent
  to R and

• f is affine in p ? are unions
  of polytopes

• Computation of ,
  intersections and inclusions of polytopes

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Implementation in RoVerGene (Robust
Verification of Gene Networks)
Grégory Batt, Calin Belta
http//iasi.bu.edu/batt/rovergene/rovergene.htm

• Multi-Parametric Toolbox for polyhedral
  operations
• Library matlabBGL for graph operations
• CTL/LTL model checker NuSMV

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Tuning of a transcriptional cascade

• Analysis of steady-state input/output behavior of
  synthetic transcriptional cascade made of 4 genes
• PMA model of system 5-D state space (4 states, 1
  input)
• EYFP should increase at least 1000x for a 2x
  increase in aTc

input
output
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Results

• Actual network does not meet specifications
  used RoVerGeNe to find a
  valid parameter set by tuning 3
  production rates
• 1500 rectangles, 18 affine predicates, gt200,000
  equivalence classes, 350 parameter sets analyzed,
  lt 2 hours runtime