Computational Methods for Systems and Synthetic Biology

Computational Methods for Systems and Synthetic Biology

François Fages

Inria Saclay – Ile de France

Lifeware project-team

http://lifeware.inria.fr/

Biochemical Kinetics

Study the concentration of chemical substances in a biological system as a function of time.

Molecules: A

,…, A

|A|=Number of molecules A

[A]=Concentration of A in the solution: [A] = |A| / Volume ML

Molecular solutions: S, S’, …

multiset of molecules

Mass Action Law Kinetics

Assumption: each molecule moves independently of other molecules in a random walk (diffusion, dilute solutions, low concentration)

Law: The number of collisions (and reactions) between A and B is proportional to the number of A and B molecules present in the solution

The rate of a reaction A + B => C is k*[A]*[B] for some reaction constant k The time evolution of concentrations obeys the ordinary differential equation

dC/dt = k A B

dA/dt = -k A B

dB/dt = -k A B

Interpretation of Mass Action Constants

Diffusion speeds (small molecules > substrates > enzyme)

Average travel in one random walk: 1 μm in 1s, 2μm in 4s, 10μm in 100s (cells are 10-100μm long) For one enzyme:

500000 random collisions per second with a substrate concentration of 10

50000 random collisions per second with a substrate concentration of 10

Complexation rate constant: probability of reaction upon collision

specificity, affinity, position of matching surfaces and energy of bonds

Reaction Rate Functions

Mass action law kinetics k*[A] for A => B

k*[A]*[B] for A+B => C

k*[A]^m*[B]^n for m*A + n*B => R Michaelis-Menten kinetics

Vm*[A]/(Km+[A]) for A => B Hill kinetics

Vm*[A]^n/(Km+[A]^n) for A => B

Subject of this lecture: where do those different kinetics come from ?

Guldberg and Waage, 1864

Henry 1903, Michaelis Menten 1913

Archibald Hill 1910

Continuous Semantics of Reaction Rules

To a set of reaction rules with kinetic expressions e

{ f

for S

=> S’

}

one associates the Ordinary Differential Equation (ODE) over {A

,…, A

} (first moment closure approximation of CME)

dA

/dt = Σ

( r

(A

) - l

(A

) ) * f

= Σ

v

(A

) * f

Numerical Integration Methods

dX/dt = f(X) with initial conditions X

Idea: discretize time t

, t

=t

+Δt, t

=t

+Δt, … and compute a trace (t

,X

), (t

,X

), …, (t

,X

)…

• Forward Euler’s method: t

=t

+ Δt X

=X

+f(X

)*Δt

estimation error(X

)=|f(X

)-f(X

)|*Δt

• Midpoint method (Runge-Kutta): intermediate computation at Δt/2

• Adaptive step size: Δt

= Δt

/2 while error>error

, otherwise Δt

= 2*Δt

• Implicit method (Rosenbrock): solve X

=X

+f(X

)*Δt by root finding

In Biocham 4, use of GSL library, the default is an implicit method.

Single Enzymatic Reaction

• An enzyme E binds to a substrate S to catalyze the formation of product P:

E+S à

C à

E+P E+S ß

C

• Mass action law kinetics ODE:

dE/dt =

Single Enzymatic Reaction

• An enzyme E binds to a substrate S to catalyze the formation of product P:

E+S à

C à

E+P E+S ß

C

• Mass action law kinetics ODE:

dE/dt = -c

ES+(c

+c

)C

dS/dt =

Single Enzymatic Reaction

• An enzyme E binds to a substrate S to catalyze the formation of product P:

E+S à

C à

E+P E+S ß

C

• Mass action law kinetics ODE:

dE/dt = -c

ES+(c

+c

)C dS/dt = -c

ES+c

C

dC/dt =

Single Enzymatic Reaction

• An enzyme E binds to a substrate S to catalyze the formation of product P:

E+S à

C à

E+P E+S ß

C

• Mass action law kinetics ODE:

dE/dt = -c

ES+(c

+c

)C dS/dt = -c

ES+c

C

dC/dt = c

ES-(c

+c

)C

dP/dt =

Single Enzymatic Reaction

• An enzyme E binds to a substrate S to catalyze the formation of product P:

E+S à

C à

E+P E+S ß

C

• Mass action law kinetics ODE:

dE/dt = -c

ES+(c

+c

)C dS/dt = -c

ES+c

C

dC/dt = c

ES-(c

+c

)C dP/dt = c

C

• Two conservation laws (i.e. Σ

dMi/dt = 0 so that Σ

Mi = constant):

Single Enzymatic Reaction

• An enzyme E binds to a substrate S to catalyze the formation of product P:

E+S à

C à

E+P E+S ß

C

• Mass action law kinetics ODE:

dE/dt = -c

ES+(c

+c

)C dS/dt = -c

ES+c

C

dC/dt = c

ES-(c

+c

)C dP/dt = c

C

• Two conservation laws (i.e. Σ

dMi/dt = 0 so that Σ

Mi = constant):

E+C=constant, S+C+P=constant,

• Assuming C

=P

=0, we get E=E

-C and P=S

-S-C

dS/dt = -c

(E

-C)S+c

C dC/dt = c

E

S-(c

S+c

+c

)C

Slow/Fast Time Scales

Hydrolysis of benzoyl-L-arginine ethyl ester by trypsin

present(E,1e-8). present(S,1e-5). absent(C). absent(P). E << S parameter(c1,4e6). parameter(c2,25). parameter(c3,15). c1 >> c2, c3

c1*[E]*[S] for E+S => C. c2*[C] for C => E+S . c3*[C] for C => E+P.

Complex formation 5e-9 in 0.1s Product formation 1e-5 in 400s

Michaelis Menten Reduction

parameter(E0,1e-8). macro(Vm, c2*E0). macro(Km, (c2+c3)/c1).

Vm*[S]/(Km+[S]) for S ==> P.

Same product formation in 400 s

Slightly different early trajectory in 0.1s Same stable state

TD 2: Michaelis-Menten Reduction

1. Connect to http://lifeware.inria.fr/biocham4 2. Open examples/MPRI

3. Run the notebook TD2_michaelis_menten.ipynb 4. Run simulation at different time scales

5. Compare the results with QSSA and QE reductions

Assume quasi-steady state dC/dt=0=c

E

S-(c

S+c

+c

)C Then C = c

E

S/(c

S+c

+c

) = E

S/(K

+S)

where K

=(c

+c

)/c

Quasi-Steady State Approximation (QSSA)

Assume quasi-steady state dC/dt=0=c

E

S-(c

S+c

+c

)C Then C = c

E

S/(c

S+c

+c

) = E

S/(K

+S)

where K

=(c

+c

)/c

We get dP/dt = -dS/dt = V

S / (K

+S)

where V

= c

E

maximum velocity at saturing substrate concentration

Michaelis-Menten quasi-steady state kinetics: V

S /(K

+S) for S => P

Quasi-Steady State Approximation (QSSA)

Assume quasi-steady state dC/dt=0=c

E

S-(c

S+c

+c

)C Then C = c

E

S/(c

S+c

+c

) = E

S/(K

+S)

where K

=(c

+c

)/c

substrate concentration with half maximum velocity We get dP/dt = -dS/dt = V

S / (K

+S)

where V

= c

E

maximum velocity at saturing substrate concentration

Michaelis-Menten quasi-steady state kinetics: V

S /(K

+S) for S => P

Quasi-Steady State Approximation (QSSA)

Assume quasi-steady state dC/dt=0=c

E

S-(c

S+c

+c

)C Then C = c

E

S/(c

S+c

+c

) = E

S/(K

+S)

where K

=(c

+c

)/c

substrate concentration with half maximum velocity We get dP/dt = -dS/dt = V

S / (K

+S)

where V

= c

E

maximum velocity at saturing substrate concentration

Michaelis-Menten quasi-steady state kinetics: V

S /(K

+S) for S => P

Quasi-Steady State Approximation (QSSA)

Time Scales ?

“Time taken for a significant change”

Time scale of f(t) ≈

|^*+_*,|"#$

Time scale of C(t) ≈ 5e−8

4e−8/0.01 =

Formally, supposing S(t)=S

we get dC/dt = c

(E

+C

)S

- (c

S

+c

+c

) C

C(t) = (C

-C) e

+ C where k= c

S

+c

+c

and C = (E

+C

)S

/(K

+S)

Taking k

as time scale of e

(i.e. decrease of e

≈1/3 in k

time)

gives in the Trypsin example 1/(10

.4.10

+25+15)=1/80 =0.0125s

Validity Condition of QSSA ?

When the time scale of S is much longer than the time scale of C…

S varies from S

to 0

| dS/dt = -c

(E

-C)S+ c

C | is maximal when S=S

, C=C

Time scale of S ≈ 1/(c

E

)

(in the Trypsin example 1/(4.10

. 10

)≈25s)

Validity condition: c

E

<< c

S

+c

+c

(e.g. Trypsin: 4.10

<<80)

Assume reaction equilibrium c

ES=c

C (fast complexation/decomplexation cycle) E+S à

C à

E+P

E+S ß

C

Quasi-Equilibrium Approximation (QE)

Assume reaction equilibrium c

ES=c

C (fast complexation/decomplexation cycle) From E=E

-C we get c

E

S-c

CS = c

C

C = c

E

S/(c

+c

S) C = E

S/(K

+S) where K

=c

/c

Hence dP/dt = -dS/dt = V

S / (K

+S) where V

= c

E

Quasi-Equilibrium Approximation (QE)

Assume reaction equilibrium c

ES=c

C (fast complexation/decomplexation cycle) From E=E

-C we get c

E

S-c

CS = c

C

C = c

E

S/(c

+c

S) C = E

S/(K

+S) where K

=c

/c

substrate concentration with half maximum velocity Hence dP/dt = -dS/dt = V

S / (K

+S)

where V

= c

E

maximum velocity at saturing substrate concentration

Quasi-Equilibrium Approximation (QE)

Assume reaction equilibrium c

ES=c

C (fast complexation/decomplexation cycle) From E=E

-C we get c

E

S-c

CS = c

C

C = c

E

S/(c

+c

S) C = E

S/(K

+S) where K

=c

/c

substrate concentration with half maximum velocity Hence dP/dt = -dS/dt = V

S / (K

+S)

where V

= c

E

maximum velocity at saturing substrate concentration

Quasi-Equilibrium Approximation (QE)

Enzymatic Competitive Inhibition

present(En,1e-8). present(S,1e-5). present(I,1e-5). parameter(k3,5e5).

parameter(k1,4e6). parameter(km1,25). parameter(k2,15).

(k1*[E]*[S],km1*[C]) for E+S <=> C. k2*[C] for C => E+P.

k3*[C]*[I] for C+I => CI.

Complex formation 4e-9 in 0.04s Product formation 3e-8 in 3s

Isosteric Inhibition

present(En,1e-8). present(S,1e-5). present(I,1e-5). parameter(k3,5e5).

parameter(k1,4e6). parameter(km1,25). parameter(k2,15).

(k1*[E]*[S],km1*[C]) for E+S <=> C. k2*[C] for C => E+P.

k3*[E]*[I] for E+I => EI.

Complex formation 2.5e-9 in 0.4s Product formation 2.5e-9 in 1000s

Allosteric Inhibition/Activation

(i*[E]*[I],im*[EI]) for E+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

TD 3: Hill kinetics for Allosteric Reactions

1. Connect to http://lifeware.inria.fr/biocham4 2. Open examples/MPRI

3. Run the notebook TD3_hill.ipynb

4. Run simulations and plot C (as reaction velocity) against S

5. Compare different parameter values and observe cooperativity/non

Hill Kinetics (several binding sites for same ligands)

E.g. Dimer enzyme with two promoters:

E+S à

C

à

E+P C

+S à

C2 à

C

+P E+S ß

C

C

+S ß

C2

Let K

=(c

+c

)/c

and K’

=(c’

+c’

)/c’

Non-cooperative if K’

=K

Michaelis-Menten rate: V

S / (K

+S) where V

=2*c

*E

. hyperbolic velocity vs substrate concentration

Cooperative if K’

>K

Hill equation rate: V

S

/ (K

*K’

+S

) order 2

where V

=2*c

*E

“sigmoid” velocity vs substrate concentration

Conclusion

Michaelis-Menten kinetics, Hill kinetics of order n and more general kinetics come from reductions of elementary reaction systems with mass action law kinetics

and more precisely from projections on the slow dynamics variables

(eliminate fast dynamics species E, C acting as slaves of slow species S)

The slow/fast separation of species may be different in different regimes of

the system. This leads to a hybrid automaton of piece-wise reduced systems.