Overview of the Lectures
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Transcript of Overview of the Lectures
François Fages MPRI Bio-info 2006
Formal Biology of the Cell
Modeling, Computing and Reasoning with Constraints
François Fages, Constraint Programming Group,
INRIA Rocquencourt mailto:[email protected]://contraintes.inria.fr/
François Fages MPRI Bio-info 2006
Overview of the Lectures
1. Introduction. Formal molecules and reactions in BIOCHAM.
2. Formal biological properties in temporal logic. Symbolic model-checking.
3. Continuous dynamics. Kinetics models.
4. Learning kinetic parameter values. Constraint-based model checking.
5. …
François Fages MPRI Bio-info 2006
Biochemical Kinetics
Study the concentration of chemical substances in a biological system as a function of time.
BIOCHAM concentration semantics:
Molecules: A1 ,…, Am
|A|=Number of molecules A
[A]=Concentration of A in the solution: [A] = |A| / Volume ML-1
Solutions with stoichiometric coefficients: c1 *A1 +…+ cn * An
François Fages MPRI Bio-info 2006
Law of Mass Action
The number of A+B interactions is proportional to the number of A and B molecules, the proportionality factor k is the rate constant of the reaction
A + B k C
the rate of the reaction is k*[A]*[B].
dC/dt = k A B
dA/dt = -k A B
dB/dt = -k A B
Assumption: each molecule moves independently of other molecules in a random walk (diffusion, dilute solutions, low concentration).
François Fages MPRI Bio-info 2006
Interpretation of Rate Constants k’s
• Complexation: probabilities of reaction
upon collision (specificity, affinity)
Position of matching surfaces
• Decomplexation: energy of bonds
(giving dissociation rates)
Different diffusion speeds (small molecules>substrates>enzymes…)
Average travel in a random walk: 1 μm in 1s, 2μm in 4s, 10μm in 100s
For an enzyme:
500000 random collisions per second for a substrate concentration of 10-5
50000 random collisions per second for a substrate concentration of 10-6
François Fages MPRI Bio-info 2006
Signal Reception on the Membrane
present(L,0.5). present(RTK,0.01).
absent(L-RTK). absent(S).
parameter(k1,1). parameter(k2,0.1).
parameter(k3,1). parameter(k4,0.3).
(k1*[L]*[RTK], k2*[L-RTK]) for L+RTK <=> L-RTK.
(k3*[L-RTK], k4*[S]) for 2*(L-RTK) <=> S.
François Fages MPRI Bio-info 2006
Michaelis-Menten Enzymatic Reaction
An enzyme E binds to a substrate S to catalyze the formation of product P:
E+S k1 C k2 E+P
E+S km1 C
compiles into a system of non-linear Ordinary Differential Equations
dE/dt = -k1ES+(k2+km1)C
dS/dt = -k1ES+km1C
dC/dt = k1ES-(k2+km1)C
dP/dt = k2C
François Fages MPRI Bio-info 2006
Michaelis-Menten Enzymatic Reaction
An enzyme E binds to a substrate S to catalyze the formation of product P:
E+S k1 C k2 E+P
E+S km1 C
compiles into a system of non-linear Ordinary Differential Equations
dE/dt = -k1ES+(k2+km1)C
dS/dt = -k1ES+km1C
dC/dt = k1ES-(k2+km1)C
dP/dt = k2C
After simplification, supposing C0=P0=0, we get E=E0-C,
François Fages MPRI Bio-info 2006
Michaelis-Menten Enzymatic Reaction
An enzyme E binds to a substrate S to catalyze the formation of product P:
E+S k1 C k2 E+P
E+S km1 C
compiles into a system of non-linear Ordinary Differential Equations
dE/dt = -k1ES+(k2+km1)C
dS/dt = -k1ES+km1C
dC/dt = k1ES-(k2+km1)C
dP/dt = k2C
After simplification, supposing C0=P0=0, we get E=E0-C, S0=S+C+P,
François Fages MPRI Bio-info 2006
Michaelis-Menten Enzymatic Reaction
An enzyme E binds to a substrate S to catalyze the formation of product P:
E+S k1 C k2 E+P
E+S km1 C
compiles into a system of non-linear Ordinary Differential Equations
dE/dt = -k1ES+(k2+km1)C
dS/dt = -k1ES+km1C
dC/dt = k1ES-(k2+km1)C
dP/dt = k2C
After simplification, supposing C0=P0=0, we get E=E0-C, S0=S+C+P,
dS/dt = -k1(E0-C)S+km1C
dC/dt = k1(E0-C)S-(k2+km1)C
François Fages MPRI Bio-info 2006
BIOCHAM Concentration Semantics
To a set of BIOCHAM rules with kinetic expressions e i
{ei for Si=>S’i}i=1,…,n
one associates the system of ODEs over variables {A1,, Ak}
dAk/dt=Σni=1ri(Ak)*ei - Σn
j=1lj(Ak)*ej
where ri(A) (resp. li(A)) is the stoichiometric coefficient of A in Si (resp. S’i).
François Fages MPRI Bio-info 2006
BIOCHAM Concentration Semantics
To a set of BIOCHAM rules with kinetic expressions ei
{ei for Si=>S’i}i=1,…,n
one associates the system of ODEs over variables {A1,, Ak}
dAk/dt=Σni=1ri(Ak)*ei - Σn
j=1lj(Ak)*ej
where ri(A) (resp. li(A)) is the stoichiometric coefficient of A in Si (resp. S’i).
Note on compositionality:
The union of two sets of reaction rules is a set of reaction rules…
So BIOCHAM models can be composed to form complex reaction models by set union.
François Fages MPRI Bio-info 2006
Compositionality of Reaction Rules
Towards open decomposed modular models:
• Sufficiently decomposed reaction rules,
E+S<=>C =>E+P, not S<=[E]=>P if competition on C
• Sufficiently general kinetics expression,
parameters as possibly functions of temperature, pH, pressure, light,…
different pH=-log[H+] in intracellular and extracellular solvents (water)
Ex. pH(cytosol)=7.2, pH(lysosomes)=4.5, pH(cytoplasm) in [6.6,7.2]
• Interface variables,
controlled either by other modules (endogeneous variables)
or by fixed laws (exogeneous variables).
François Fages MPRI Bio-info 2006
Numerical Integration Methods
System dX/dt = f(X). Initial conditions X0
Idea: discretize time t0, t1=t0+Δt, t2=t1+Δt, … and compute a trace
(t0,X0,dX0/dt), (t1,X1,dX1/dt), …, (tn,Xn,dXn/dt)…
(providing a linear Kripke structure for model-checking…)
François Fages MPRI Bio-info 2006
Numerical Integration Methods
System dX/dt = f(X). Initial conditions X0
Idea: discretize time t0, t1=t0+Δt, t2=t1+Δt, … and compute a trace
(t0,X0,dX0/dt), (t1,X1,dX1/dt), …, (tn,Xn,dXn/dt)…
(providing a linear Kripke structure for model-checking…)
Euler’s method: ti+1=ti+ Δt
Xi+1=Xi+f(Xi)*Δt
error estimation E(Xi+1)=|f(Xi)-f(Xi+1)|*Δt
François Fages MPRI Bio-info 2006
Numerical Integration Methods
System dX/dt = f(X). Initial conditions X0
Idea: discretize time t0, t1=t0+Δt, t2=t1+Δt, … and compute a trace
(t0,X0,dX0/dt), (t1,X1,dX1/dt), …, (tn,Xn,dXn/dt)…
(providing a linear Kripke structure for model-checking…)
Euler’s method: ti+1=ti+ Δt
Xi+1=Xi+f(Xi)*Δt
error estimation E(Xi+1)=|f(Xi)-f(Xi+1)|*Δt
Runge-Kutta’s method: intermediate computations at Δt/2
François Fages MPRI Bio-info 2006
Numerical Integration Methods
System dX/dt = f(X). Initial conditions X0
Idea: discretize time t0, t1=t0+Δt, t2=t1+Δt, … and compute a trace
(t0,X0,dX0/dt), (t1,X1,dX1/dt), …, (tn,Xn,dXn/dt)…
(providing a linear Kripke structure for model-checking…)
Euler’s method: ti+1=ti+ Δt
Xi+1=Xi+f(Xi)*Δt
error estimation E(Xi+1)=|f(Xi)-f(Xi+1)|*Δt
Runge-Kutta’s method: intermediate computations at Δt/2
Adaptive step method: Δti+1= Δti/2 while E>Emax, otherwise Δti+1= 2*Δti
François Fages MPRI Bio-info 2006
Numerical Integration Methods
System dX/dt = f(X). Initial conditions X0
Idea: discretize time t0, t1=t0+Δt, t2=t1+Δt, … and compute a trace
(t0,X0,dX0/dt), (t1,X1,dX1/dt), …, (tn,Xn,dXn/dt)…
(providing a linear Kripke structure for model-checking…)
Euler’s method: ti+1=ti+ Δt
Xi+1=Xi+f(Xi)*Δt
error estimation E(Xi+1)=|f(Xi)-f(Xi+1)|*Δt
Runge-Kutta’s method: intermediate computations at Δt/2
Adaptive step method: Δti+1= Δti/2 while E>Emax, otherwise Δti+1= 2*Δti
Rosenbrock’s stiff method: solve Xi+1=Xi+f(Xi+1)*Δt by formal differentiation
François Fages MPRI Bio-info 2006
Multi-Scale Phenomena
Hydrolysis of benzoyl-L-arginine ethyl ester by trypsin
present(En,1e-8). present(S,1e-5). absent(C). absent(P).
(k1*[En]*[S],km1*[C]) for En+S <=> C. k2*[C] for C => En+P.
parameter(k1,4e6). parameter(km1,25). parameter(k2,15).
Complex formation 5e-9 in 0.1s Product formation 1e-5 in 1000s
François Fages MPRI Bio-info 2006
Quasi-Steady State Approximation
After short initial period (0.1s), the complex concentration reaches its limit.
Assume dC/dt=0
François Fages MPRI Bio-info 2006
Quasi-Steady State Approximation
After short initial period, the complex concentration reaches its limit.
Assume dC/dt=0
From dC/dt = k1S(E0-C)-(k2+km1)C
we get C = k1E0S/(k2+km1+k1S)
François Fages MPRI Bio-info 2006
Quasi-Steady State Approximation
After short initial period, the complex concentration reaches its limit.
Assume dC/dt=0
From dC/dt = k1S(E0-C)-(k2+km1)C
we get C = k1E0S/(k2+km1+k1S)
= E0S/(((k2+km1)/k1)+S)
= E0S/(Km+S)
where Km=(k2+km1)/k1
François Fages MPRI Bio-info 2006
Quasi-Steady State Approximation
After short initial period, the complex concentration reaches its limit.
Assume dC/dt=0
From dC/dt = k1S(E0-C)-(k2+km1)C
we get C = k1E0S/(k2+km1+k1S)
= E0S/(((k2+km1)/k1)+S)
= E0S/(Km+S)
where Km=(k2+km1)/k1
dS/dt = -dP/dt
= -k2C
= -VmS / (Km+S)
where Vm= k2E0.
François Fages MPRI Bio-info 2006
Quasi-Steady State Approximation
Assuming dC/dt=0, we have dE/dt=0 and C= E0 S / (Km+S).
Michaelis-Menten rate: dP/dt = -dS/dt = VmS / (Km+S) (reaction velocity)
Vm=k2*E0
Km=(km1+k2)/k1
François Fages MPRI Bio-info 2006
Quasi-Steady State Approximation
Assuming dC/dt=0, we have dE/dt=0 and C= E0 S / (Km+S).
Michaelis-Menten rate: dP/dt = -dS/dt = VmS / (Km+S) (reaction velocity)
Vm=k2*E0 (maximum velocity at saturating substrate concentration)
Km=(km1+k2)/k1
François Fages MPRI Bio-info 2006
Quasi-Steady State Approximation
Assuming dC/dt=0, we have dE/dt=0 and C= E0 S / (Km+S).
Michaelis-Menten rate: dP/dt = -dS/dt = VmS / (Km+S) (reaction velocity)
Vm=k2*E0 (maximum initial velocity)
Km=(km1+k2)/k1 (substrate concentration with half maximum velocity)
Experimental measurement:
The initial velocity is
linear in E0
hyperbolic in S0
François Fages MPRI Bio-info 2006
Quasi-Steady State Approximation
Assuming dC/dt=0, hence dE/dt=0 and C= E0 S / (Km+S).
Michaelis-Menten rate: dP/dt = -dS/dt = VmS / (Km+S) (reaction velocity)
Vm=k2*E0
Km=(km1+k2)/k1
BIOCHAM syntaxmacro(Vm, k2*[En]).macro(Km, (km1+k2)/k1).MM(Vm,Km) for S =[En]=> P.
macro(Kf, Vm*[S]/(Km+[S])).Kf for S =[En]=> P.
François Fages MPRI Bio-info 2006
Competitive Inhibition
present(En,1e-8). present(S,1e-5).
(k1*[En]*[S],km1*[C]) for En+S <=> C. k2*[C] for C => En+P.
parameter(k1,4e6). parameter(km1,25). parameter(k2,15).
present(I,1e-5). k3*[C]*[I] for C+I => CI. parameter(k3,5e5).
Complex formation 4e-9 in 0.04s Product formation 3e-8 in 3s
François Fages MPRI Bio-info 2006
Competitive Inhibition (isosteric)
present(En,1e-8). present(S,1e-5).
(k1*[En]*[S],km1*[C]) for En+S <=> C. k2*[C] for C => En+P.
parameter(k1,4e6). parameter(km1,25). parameter(k2,15).
present(I,1e-5). k3*[En]*[I] for En+I => EI. parameter(k3,5e5).
Complex formation 2.5e-9 in 0.4s Product formation 2.5e-9 in 1000s
François Fages MPRI Bio-info 2006
Allosteric Inhibition (or Activation)
(i*[En]*[I],im*[EI]) for En+I <=> EI. parameter(i,1e7). parameter(im,10).
(i1*[EI]*[S],im1*[CI]) for EI+S <=> CI. parameter(i1,5e6). parameter(im1,5).
i2*[CI] for CI => EI+P. parameter(i2,2).
Complex formation 2e-9 in 0.4s Product formation 1e-5 in 1000s
François Fages MPRI Bio-info 2006
Cooperative Enzymes and Hill Equation
Dimer enzyme with two promoters:
E+S 2*k1 C1 k2 E+P C1+S k’1 C2 2*k’2 C1+P
E+S k-1 C1 C1+S 2*k’-1 C2
Let Km=(k-1+k2)/k1 and K’m=(k’-1+k’2)/k’1
Non-cooperative if Km =K’m Michaelis-Menten rate: VmS / (Km+S) where Vm=2*k2*E0.
(hyperbolic velocity vs substrate concentration)
Cooperative if k’1>k1
Hill equation rate: VmS2 / (Km*K’m+S2) where Vm=2*k2*E0.
(sigmoid velocity vs substrate concentration)
François Fages MPRI Bio-info 2006
MAPK kinetics model
François Fages MPRI Bio-info 2006
Cell Cycle Control [Qu et al. 2003]
François Fages MPRI Bio-info 2006
Lotka-Voltera Autocatalysis
0.3*[RA] for RA => 2*RA. 0.3*[RA]*[RB] for RA + RB => 2*RB.
0.15*[RB] for RB => RP. present(RA,0.5). present(RB,0.5). absent(RP).
François Fages MPRI Bio-info 2006
Hybrid (Continuous-Discrete) Dynamics
Gene X activates gene Y but above some threshold gene Y inhibits X.
0.01*[X] for X => X + Y.
if [Y] lt 0.8 then 0.01
for _ => X.
0.02*[X] for X => _.
absent(X).
absent(Y).