Model Checking Genetic Regulatory Networks with Parameter Uncertainty Grégory Batt, Calin Belta, Ron Weiss HSCC 2007 Presented by Spring Berman ESE : Systems Biology

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)

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

Approach 1) Model genetic network with piecewise- multiaffine (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

Approach 1) Model genetic network with piecewise- multiaffine (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

PMA models of genetic networks State vector: n genes; x i = concentration of protein encoded by gene i Parameter vector: Network dynamics: Production rate possibly uncertain Degradation rate parameters Regulation function (products of ramp functions)

Example: Toggle Switch gene repressor protein

Approach 1) Model genetic network with piecewise- multiaffine (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

Dynamical Property as LTL Formula  T emporal L ogic : System for describing how the truth of assertions changes over time  L inear: 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 MIT Press, 1990.

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 Example: Toggle Switch

Approach 1) Model genetic network with piecewise- multiaffine (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

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 π

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

Example: Toggle Switch Continuous dynamics Discrete abstraction

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…

Approach 1) Model genetic network with piecewise- multiaffine (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

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)

Example: Toggle Switch 32 affine expressions, only 4 non-constant ones Parameter space

Example: Toggle Switch Hierarchy among parameter sets Parameter equivalence classes Valid for bistability property

Approach 1) Model genetic network with piecewise- multiaffine (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

System over- and under-approximations Contains all transitions present in at least one Contains only transitions present in all or, inspect subsets of P  Algorithm: Recursively explore the tree of parameter sets, starting from ; stop search at condition (A) or (B) (A) (B) (C)

Computation of, 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

Implementation in RoVerGene (Robust Verification of Gene Networks)  Multi-Parametric Toolbox for polyhedral operations  Library matlabBGL for graph operations  CTL/LTL model checker NuSMV Grégory Batt, Calin Belta

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

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, >200,000 equivalence classes, 350 parameter sets analyzed, < 2 hours runtime