François Fages MPRI Bio-info 2006 Formal Biology of the Cell Modeling, Computing and Reasoning with Constraints François Fages, Constraint Programming

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/


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. …


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


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).


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


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.


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




E+S k1 C k2 E+P

E+S km1 C



dS/dt = -k1ES+km1C


dP/dt = k2C

After simplification, supposing C0=P0=0, we get E=E0-C,




E+S k1 C k2 E+P

E+S km1 C



dS/dt = -k1ES+km1C


dP/dt = k2C

After simplification, supposing C0=P0=0, we get E=E0-C, S0=S+C+P,




E+S k1 C k2 E+P

E+S km1 C



dS/dt = -k1ES+km1C


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


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).


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.


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).


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








Xi+1=Xi+f(Xi)*Δt


Runge-Kutta’s method: intermediate computations at Δt/2








Xi+1=Xi+f(Xi)*Δt



Adaptive step method: Δti+1= Δti/2 while E>Emax, otherwise Δti+1= 2*Δti








Xi+1=Xi+f(Xi)*Δt



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


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


Quasi-Steady State Approximation

After short initial period (0.1s), the complex concentration reaches its limit.

Assume dC/dt=0



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)




Assume dC/dt=0



= E0S/(((k2+km1)/k1)+S)

= E0S/(Km+S)

where Km=(k2+km1)/k1




Assume dC/dt=0



= E0S/(((k2+km1)/k1)+S)

= E0S/(Km+S)

where Km=(k2+km1)/k1

dS/dt = -dP/dt

= -k2C

= -VmS / (Km+S)

where Vm= k2E0.



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





Vm=k2*E0 (maximum velocity at saturating substrate concentration)

Km=(km1+k2)/k1





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



Assuming dC/dt=0, hence dE/dt=0 and C= E0 S / (Km+S).


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.


Competitive Inhibition

present(En,1e-8). present(S,1e-5).



present(I,1e-5). k3*[C]*[I] for C+I => CI. parameter(k3,5e5).



Competitive Inhibition (isosteric)

present(En,1e-8). present(S,1e-5).



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


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).



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)


MAPK kinetics model


Cell Cycle Control [Qu et al. 2003]


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).


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).

Documents

François Fages MPRI Bio-info 2006 Formal Biology of the Cell Modeling, Computing and Reasoning with Constraints François Fages, Constraint Programming