Computational Methods in Systems Biology and Synthetic Biology Fran

1
Computational Methods inSystems Biology and
Synthetic BiologyFrançois Fages, Constraint
Programming Group, INRIA Rocquencourt
mailtoFrancois.Fages_at_inria.frhttp//contraintes.
inria.fr/
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Overview of the Lectures

• Formal molecules and reaction models in BIOCHAM
• Kinetics
• Qualitative properties formalized in temporal
  logic CTL
• Quantitative properties formalized in LTL(R) and
  pLTL(R)
• LTL(R) formulae with constraints over the reals
• Trace-based forward model-checking algorithm
• Quantitative models of the cell cycle
• Parameter search from LTL(R) specifications
• Constraint-based backward model-checking
  algorithm
• pLTL(R) formulae for the stochastic semantics

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Linear Time Logic with Constraints LTL(R)

• Constraints over concentrations and derivatives
  as formulae over the reals
• M gt 0.2
• MP gt Q
• d(M)/dt lt 0

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Linear Time Logic with Constraints LTL(R)

• Constraints over concentrations and derivatives
  as formulae over the reals
• M gt 0.2
• MP gt Q
• d(M)/dt lt 0
• LTL(R) formulae
• F(Mgt0.2)
• FG(Mgt0.2)

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Linear Time Logic with Constraints LTL(R)

• Constraints over concentrations and derivatives
  as formulae over the reals
• M gt 0.2
• MP gt Q
• d(M)/dt lt 0
• LTL(R) formulae
• 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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Linear Time Logic with Constraints LTL(R)

• Constraints over concentrations and derivatives
  as formulae over the reals
• M gt 0.2
• MP gt Q
• d(M)/dt lt 0
• LTL(R) formulae
• F(Mgt0.2)
• FG(Mgt0.2)
• F (Mgt2 F (d(M)/dtlt0 F (Mlt2
  d(M)/dtgt0 F(d(M)/dtlt0))))
• oscil(M,n) defined as at least n alternances of
  sign of the derivative

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Real-time LTL(R)

• Real-time variable T
• Reachability of a threshold value with a minimum
  delay
• F(Mgt0.2) G(Tlt5 ?Mlt0.2)
• Formalization of numerical data time series
  (experimental curve)
• F(T1 M0.05 F(T2 M0.12 F(T3
  M0.25)))
• Constraint of period
• Period(A,75) ? 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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How to Evaluate a LTL(R) Formula ?

• Consider the ODEs of the concentration semantics
  
• dX/dt f(X)
• Initial conditions X0

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How to Evaluate a LTL(R) Formula ?

• Consider the ODEs of the concentration semantics
  
• dX/dt f(X)
• Initial conditions X0
• Numerical integration methods produce a (clever)
  discretization of time
• (adaptive step size Runge-Kutta or Rosenbrock
  method for stiff syst.)

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How to Evaluate a LTL(R) Formula ?

• Consider the ODEs of the concentration semantics
  
• dX/dt f(X)
• Initial conditions X0
• Numerical integration methods produce a (clever)
  discretization of time
• (adaptive step size Runge-Kutta or Rosenbrock
  method for stiff syst.)
• The trace is a linear Kripke structure over
  concentrations and their derivatives at discrete
  time points
• (t0,X0,dX0/dt), (t1,X1,dX1/dt), ,
  (tn,Xn,dXn/dt),
• Apply a model-checking algorithm on a finite
  prefix of that Kripke structure.

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LTL(R) Trace-based Model Checking Algo

• Input An ODE system M given with initial
  conditions, an LTL(R) formula f
• Hypothesis the formula can be checked over a
  finite period of time 0,T
• Output whether f is true in M

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LTL(R) Trace-based Model Checking Algo

• Input An ODE system M given with initial
  conditions, an LTL(R) formula f
• Hypothesis the formula can be checked over a
  finite period of time 0,T
• Output whether f is true in M
• Compute a trace by numerical integration from 0
  to T

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LTL(R) Trace-based Model Checking Algo

• Input An ODE system M given with initial
  conditions, an LTL(R) formula f
• Hypothesis the formula can be checked over a
  finite period of time 0,T
• Output whether f is true in M
• Compute a trace by numerical integration from 0
  to T
• Label each state of the trace with the formulas
  constraints that are true,

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LTL(R) Trace-based Model Checking Algo

• Input An ODE system M given with initial
  conditions, an LTL(R) formula f
• Hypothesis the formula can be checked over a
  finite period of time 0,T
• Output whether f is true in M
• Compute a trace by numerical integration from 0
  to T
• Label each state of the trace with the formulas
  constraints that are true,
• Iteratively label the states with the
  sub-formulae that are true
• Add X f1 to the immediate predecessors of states
  labeled by f1,

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LTL(R) Trace-based Model Checking Algo

• Input An ODE system M given with initial
  conditions, an LTL(R) formula f
• Hypothesis the formula can be checked over a
  finite period of time 0,T
• Output whether f is true in M
• Compute a trace by numerical integration from 0
  to T
• Label each state of the trace with the formulas
  constraints that are true,
• Iteratively label the states with the
  sub-formulae that are true
• Add X f1 to the immediate predecessors of states
  labeled by f1,
• Add f1 U f2 to the predecessors of states
  labelled by f2 while they satisfy f1,

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LTL(R) Trace-based Model Checking Algo

• Input An ODE system M given with initial
  conditions, an LTL(R) formula f
• Hypothesis the formula can be checked over a
  finite period of time 0,T
• Output whether f is true in M
• Compute a trace by numerical integration from 0
  to T
• Label each state of the trace with the formulas
  constraints that are true,
• Iteratively label the states with the
  sub-formulae that are true
• Add X f1 to the immediate predecessors of states
  labeled by f1,
• Add f1 U f2 to the predecessors of states
  labelled by f2 while they satisfy f1,
• Add f1 W f2 to the states labelled by f1? f2, to
  the last state if it is labelled by f1, and to
  the predecessors of states labelled by f1 W f2
  while they satisfy f1,

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LTL(R) Trace-based Model Checking Algo

• Input An ODE system M given with initial
  conditions, an LTL(R) formula f
• Hypothesis the formula can be checked over a
  finite period of time 0,T
• Output whether f is true in M
• Compute a trace by numerical integration from 0
  to T
• Label each state of the trace with the formulas
  constraints that are true,
• Iteratively label the states with the
  sub-formulae that are true
• Add X f1 to the immediate predecessors of states
  labeled by f1,
• Add f1 U f2 to the predecessors of states
  labelled by f2 while they satisfy f1,
• Add f1 W f2 to the states labelled by f1? f2, to
  the last state if it is labelled by f1, and to
  the predecessors of states labelled by f1 W f2
  while they satisfy f1,
• Return true if the initial state is labelled by
  f, and false otherwise

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Naïve Parameter Search Algorithm

• Input an ODE model M(p) with n parameters p in
  range pmin,pmax,
• an LTL(R) specification ?
• Output parameter values v such that M(v) ?
• or fail if no such values

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Naïve Parameter Search Algorithm

• Input an ODE model M(p) with n parameters p in
  range pmin,pmax,
• an LTL(R) specification ?
• Output parameter values v such that M(v) ?
• or fail if no such values
• Scan the parameter value space pmin,pmaxn with
  a fixed step

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Naïve Parameter Search Algorithm

• Input an ODE model M(p) with n parameters p in
  range pmin,pmax,
• an LTL(R) specification ?
• Output parameter values v such that M(v) ?
• or fail if no such values
• Scan the parameter value space pmin,pmaxn with
  a fixed step
• Test whether M(v) ? by trace-based model
  checking

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Naïve Parameter Search Algorithm

• Input an ODE model M(p) with n parameters p in
  range pmin,pmax,
• an LTL(R) specification ?
• Output parameter values v such that M(v) ?
• or fail if no such values
• Scan the parameter value space pmin,pmaxn with
  a fixed step
• Test whether M(v) ? by trace-based model
  checking
• Return the first value set v which satisfies ?

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Naïve Parameter Search Algorithm

• Input an ODE model M(p) with n parameters p in
  range pmin,pmax,
• an LTL(R) specification ?
• Output parameter values v such that M(v) ?
• or fail if no such values
• Scan the parameter value space pmin,pmaxn with
  a fixed step
• Test whether M(v) ? by trace-based model
  checking
• Return the first value set v which satisfies ?
• Exponential complexity in O(sn) where s is the
  maximum number of tried values in a range
• Gradient-based methods need a satisfaction degree
  for LTL(R) formulae

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Cell Cycle Control Qu et al. 2003
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Quantitative Model of 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).

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Learning Parameters from Temporal Properties

• biocham learn_parameter(k3,k4,(0,200),(0,200)
  ,20,
• 
  oscil(Cdc2-Cyclinp1,3),150).

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Learning Parameters from Temporal Properties

• biocham learn_parameter(k3,k4,(0,200),(0,200)
  ,20,
• 
  oscil(Cdc2-Cyclinp1,3),150).
• First values found
• parameter(k3,10).
• parameter(k4,70).

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Learning Parameters from Temporal Properties

• biocham learn_parameter(k3,k4,(0,200),(0,200)
  ,20,
• oscil(Cdc2-Cyclinp1,3)
  F(Cdc2-Cyclinp1gt0.15), 150).
• First values found
• parameter(k3,10).
• parameter(k4,120).

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Learning Parameters from Temporal Properties

• biocham learn_parameter(k3,k4,(0,200),(0,200)
  ,20,
• 
  period(Cdc2-Cyclinp1,35), 150).
• First values found
• parameter(k3,10).
• parameter(k4,280).

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LTL(R) Satisfaction Degree and Bifurcation
Diagram

• Satisfaction degree of LTL(R) formulas
  Bifurcation diagram on k4, k6
• for oscillation with amplitude constraint
  Tyson 91
• Rizk Batt Fages Soliman 08

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Linking the Cell and Circadian Cycles through Wee1
Cell cycle
Leloup and Goldbeter (1999)
Wee1P
Wee1
.. ..
preMPF
MPF
APC
APC
Cdc25P
.. ..
.. ..
Cdc25
.. ..
L
L. Calzone, S. Soliman 2006
31
PCN
Wee1m
BN
Wee1
MPF
Cdc25
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Condition of Entrainment in Periodon Wee1/Cdc25
Entrainment in period constraint expressed in LTL
with the period formula
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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.
• Represent sets of real-valued states with
  constraints polyhedrons defined by linear
  inequalities.

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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 a Constraint Logic Program

• 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

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Translation to a Constraint Logic Program

• 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
• 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.

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Translation to a Constraint Logic Program

• 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
• 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 to a Constraint Logic Program over
  the reals P (here dt1)
• Init - X0, Y0, p(X,Y).
• p(X,Y)- Xgt0, Ygt0, Ylt0.8, X1X-02X01,
  Y1Y0.1X, p(X1,Y1).
• p(X,Y)- Xgt0, Ygt0, Ygt0.8, X1X-02X,
  Y1Y0.1X, p(X1,Y1).

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CTL(R) Backward Model Checking
Theorem Delzanno Podelski 99 EF(f)lfp(TP?p(x)
-f) the set of states satisfying EF(f) can be
computed by bottom-up
execution of the CLP program P with p(x)-
f. EG(f)gfp(TP?f) greatest fixed point of the
CLP program P with f added to clauses
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CTL(R) Backward Model Checking
Theorem Delzanno Podelski 99 EF(f)lfp(TP?p(x)
-f) the set of states satisfying EF(f) can be
computed by bottom-up
execution of the CLP program P with p(x)-
f. EG(f)gfp(TP?f) greatest fixed point of the
CLP program P with f added to clauses Safety
property AG(?f) iff ?EF(f) iff init?lfp(TP?f) Li
veness property AG(f1?AF(f2)) iff
init?lfp(TP?f1?gfp(T P?f2 ) )
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CTL(R) Backward Model Checking
Theorem Delzanno Podelski 99 EF(f)lfp(TP?p(x)
-f) the set of states satisfying EF(f) can be
computed by bottom-up
execution of the CLP program P with p(x)-
f. EG(f)gfp(TP?f) greatest fixed point of the
CLP program P with f added to clauses Safety
property AG(?f) iff ?EF(f) iff init?lfp(TP?f) Li
veness 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,Bo
olean) Fourier-Motzkin elimination and
Simplex algorithm.
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CTL(R) Backward Model-Checking

• 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).
  
• ? Unreachable(xgt0.6).

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CTL(R) Backward Model-Checking

• 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).
  
• ? Unreachable(xgt0.6).
• True. Execution time 0
• ? ls.
• s(0, p(s_s,A,_),Agt0.6,1,(0,0)).

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CTL(R) Backward Model-Checking

• ? Unreachable(xgt0.2).
• False. 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,Bgt0,Agt0.1938775510204081
  6,2,(2,1)).
• s(26, p(s_s,A,B), Bgt0,Agt0,
• B0.1982676351105516Alt0.7741338175552753,
  27, (2,26)).
• s(27, init, , 28, (1,27)).
• Amounts to execute symbolically the CLP program
  with linear constraints

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pLTL(R) for the Stochastic Semantics

• Probabilistic Linear Time Logic over the reals
  pLTL(R) generalizes LTL(R) by replacing the path
  quantifier A? by a probability operator Pp ?
  which represents a constraint on the probability
  of realization of ?.
• A(?) is Pgt1(?)
• E(?) is P?0(?)

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pLTL(R) for the Stochastic Semantics

• Probabilistic Linear Time Logic over the reals
  pLTL(R) generalizes LTL(R) by replacing the path
  quantifier A? by a probability operator Pp ?
  which represents a constraint on the probability
  of realization of ?.
• A(?) is Pgt1(?)
• E(?) is P?0(?)
• Monte Carlo pLTL(R) model-checking algorithm
• Check the validity of the formula on traces of
  the stochastic semantics using the LTL(R)
  model-checker
• Evaluate the probability of realization of the
  formula by sampling stochastic traces
• Not practical issues of performance and of
  low-pass filters