Formal Biology of the Cell Modeling, Computing and Reasoning with Constraints Fran

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Formal Biology of the CellModeling, Computing
and Reasoning with ConstraintsFrançois Fages,
Constraint Programming Group, INRIA
Rocquencourt mailtoFrancois.Fages_at_inria.frhttp
//contraintes.inria.fr/
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Overview of the Lectures

• Introduction. Formal molecules and reactions in
  BIOCHAM.
• Formal biological properties in temporal logic.
  Symbolic model-checking.
• Continuous dynamics. Kinetics models.
• Computational models of the cell cycle control
  L. Calzone.
• Mixed models of the cell cycle and the circadian
  cycle L. Calzone.
• Machine learning reaction rules from temporal
  properties.
• Learning kinetic parameter values.
  Constraint-based model checking.
• Constraint Logic Programming approach to protein
  structure prediction.

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Cell Cycle Control Qu et al. 03

• k1 for _gtCyclin.
• k2Cyclin for Cyclingt_.
• k3CyclinCdc2p1 for CyclinCdc2p1gtCdc2
  p1-Cyclinp1.
• k4pCdc2p1-Cyclinp1 for
  Cdc2p1-Cyclinp1gtCdc2-Cyclinp1.
• k4Cdc2-Cyclinp12Cdc2p1-Cyclinp1
  for
• Cdc2p1-Cyclinp1Cdc2-Cyclinp1gtCd
  c2-Cyclinp1.
• k5Cdc2-Cyclinp1 for Cdc2-Cyclinp1gtCdc2
  p1-Cyclinp1.
• k6Cdc2-Cyclinp1 for Cdc2-Cyclinp1gtCdc2C
  yclinp1.
• k7Cyclinp1 for Cyclinp1gt_.
• k8Cdc2 for Cdc2gtCdc2p1.
• k9Cdc2p1 for Cdc2p1gtCdc2.
• parameter(k1,0.015). parameter(k2,0.015).
  parameter(k3,200).
• parameter(k4p,0.018). parameter(k4,180).
  parameter(k5,0).
• parameter(k6,1). parameter(k7,0.6).
  parameter(k8,100).
• parameter(k9,100).
• present(Cdc2,1). make_absent_not_present.

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Cell Cycle Control Qu et al. 2003
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Constraint-Based Linear Time Logic

• Constraints over concentrations and derivatives
  as FOL formulae over the reals
• M gt 0.2
• MP gt Q
• d(M)/dt lt 0
• LTL operators for time X, F, G, U (no
  non-determinism).
• F(Mgt0.2)
• FG(Mgt0.2)
• F (Mgt2 F (d(M)/dtlt0 F (Mlt2
  d(M)/dtgt0 F(d(M)/dtlt0))))

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Traces from Numerical Simulation

• From a system of Ordinary Differential Equations
• dX/dt f(X)
• Numerical integration (by Euler, Runge-Kutta,
  adaptive step size Runge-Kutta, Rosenbrock
  methods) produces a discretization of time
• The trace is a linear Kripke structure
• (t0,X0), (t1,X1), , (tn,Xn).
• the derivatives can be added to the trace
• (t0,X0,dX0/dt), (t1,X1,dX1/dt), ,
  (tn,Xn,dXn/dt).
• Equality xv true if xiv xi1v or if xiv
  xi1v (Rolles theorem!)

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Constraint-Based LTL (Forward) Model Checking

• Hypothesis 1 the initial state is completely
  known
• Hypothesis 2 the formula can be checked over a
  finite period of time 0,T
• Simple algorithm based on the trace of the
  numerical simulation
• Run the numerical simulation from 0 to T
  producing
  values at a finite sequence of time points
• Iteratively label the time points with the
  sub-formulae of f that are true
• Add f to the time points where a FOL formula
  f is true,
• Add F f (X f) to the (immediate) previous
  time points labeled by f,
• Add f1 U f2 to the predecessor time points
  of f2 while they satisfy f1,
• Add G f to the states satisfying f until T
  (optimistic abstraction)

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Model-Checking Specific First-Order LTL Formulae

• Let us introduce the time variable t
• We can model-check a First-Order Logic LTL
  formula such as
• period(A,75) defined as
• T ? v F(T t A v d(A)/dt gt 0
  X(d(A)/dt lt 0)
• F(T t 75 A v d(A)/dt gt
  0 X(d(A)/dt lt 0)))

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Example in Qus Model of the Cell Cycle

• K10, K5u0

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Learning Parameter Values from LTL Specification

• ? learn_parameter(k5u,k1,(0,10),(0,500), 20,
  oscil(CycB-CDKp1,2,10.0),300).
• parameter(k5u,0.5). parameter(k1,350).

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Learning Parameter Values from LTL Specification

• ? learn_parameter(k5u,k1,(0,10),(0,500), 20,
  period(CycB-CDKp1,75), 300).
• parameter(k5u,2). parameter(k1,200).

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Backward Constraint-based Model Checking

• Reason backward from the set of states satisfying
  a formula
• to the set of initial states for which the
  formula is true.
• Makes it possible to reason with a partially know
  initial state.
• Approximate set of states with constraints
  polyhedrons defined by linear constraints.

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Hybrid (Continuous-Discrete) Dynamics

• Gene X activates gene Y but above some threshold
  gene Y inhibits X.
• 0.1X for
• _ Xgt Y.
• if Ylt0.8 then 0.1 for
• _ gt X.
• 0.2X for
• X gt _.
• absent(X). absent(Y).

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Translation to Constraint Logic Programs over
Reals

• Hybrid Differential Equation System
• dx/dt 0.1 0.2x if y lt 0.8 dx/dt
  0.2x if y 0.8
• dy/dt 0.1x
• (Concurrent) transition system of the trace using
  Eulers method
• y lt 0.8 ? x x dt(0.1-0.2x) , y y
  dt0.1x
• y 0.8 ? x x dt(0.1-0.2x) , y y
  dt0.1x
• Initial condition x0, y0.
• Translation into a Constraint Logic Program over
  the reals (dt1)
• Init - X0, Y0, p(X,Y).
• p(X,Y)- Xgt0, Ygt0, Ylt0.8, X1X-02X01,
  Y1Y01X, p(X1,Y1).
• p(X,Y)- Xgt0, Ygt0, Ygt0.8, X1X-02X,
  Y1Y01X, p(X1,Y1).

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Constraint-based CTL Backward Model Checking
Theorem Delzanno Podelski 99
EF(f)lfp(TP?p(x)-f),
EG(f)gfp(TP?f). Safety property AG(?f) iff
?EF(f) iff init?lfp(TP?f) Liveness property
AG(f1?AF(f2)) iff init?lfp(TP?f1?gfp(T P?f2 )
) Deductive Model Checking DMC system Delzanno
00 Implemented in Sicstus-Prolog
CLP(Herbrand,Real,Boolean)
Fourier-Motzkin elimination and Simplex algorithm.
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Constraint-based Backward Reasoning in DMC

• r(init, p(s_s,A,B), A0,B0).
• r(p(s_s,A,B), p(s_s,C,D), Agt0,Bgt0.8,CA-02A,D
  B01A).
• r(p(s_s,A,B), p(s_s,C,D), Agt0,Bgt0,Blt0.8,
• 
  CA-02A01,DB01A).
• ? prop(P,S).
• P unsafe, S ps(xgt0.6)
• ? ti.
• Property satisfied. Execution time 0
• ? ls.
• s(0, p(s_s,A,_), Agt0.6, 1, (0,0)).

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Constraint-based Backward Simulation in DMC

• ? prop(P,S).
• P unsafe, S ps(xgt0.2) ?
• ? ti.
• Property NOT satisfied. Execution time 1.5
• ? ls.
• s(0, p(s_s,A,_), Agt0.2, 1, (0,0)).
• s(1, p(s_s,A,B), Blt0.8,Bgt-0,Agt0.193877551020408
  16, 2, (2,1)).
• s(26, p(s_s,A,B), Bgt0,Agt0,
• B0.1982676351105516Alt0.7741338175552753,
  27, (2,26)).
• s(27, init, , 28, (1,27)).