Nicolas Le Novère, Babraham Institute, EMBL-EBI n.lenovere ... · Modelling chemical kinetics Nicolas Le Novère, Babraham Institute, EMBL-EBI [email protected]. Bioinformatics

Bioinformatics for the neuroscientist, 28 September 2015

Introduction to modelling in biology, Babraham Institute, 24 November 2016

Modelling chemical kinetics

Nicolas Le Novère, Babraham Institute, EMBL-EBI

[email protected]



Systems Biology models ODE models

→ Reconstruction of state variable evolution from process descriptions:

Processes can be combined in ODEs (for deterministic simulations); transformed in propensities (for stochastic simulations)

Systems can be reconfigured quickly by adding or removing a process

A

B

P

Q

R

a

b

p

q

substancesA and B

areconsumed

by

reaction R thatproduces

substancesP and Q



ATP is consumed by processes 1 and 3, and produced by processes 7 and 10(for 1 reactions 1 and 3, there are 2 reactions 7 and 10)

1 2 3 45

6

78910



Chemical kinetics and fluxes

S1

S2

E

P



Statistical physics and chemical reaction




Probability to find anobject in a containerwithin an interval of time




Probability to find anobject in a containerwithin an interval of time



Law of Mass Action

Waage and Guldberg (1864)

rate-constant

velocity

stoichiometry

activity



Law of Mass Action

Waage and Guldberg (1864)

activity

rate-constant

velocity

stoichiometry

gas

solution



Evolution of a reactant

Velocity multiplied by stoichiometry

negative if consumption, positive if production

Example of a unimolecular reaction













[x]0

[x]0/e

t1/kln2/k

[x]0/2



Reversible reaction

is equivalent to



Reversible reaction

is equivalent to



Example of an enzymatic reaction









t

[x]

Not feasible in general

Numerical integration




Euler method:

Numerical integration (only for info. Not needed)

t

[x]

Dt

t

[x][x]

t+Dt – [x]

t



Euler method:


t

[x]

Dt

t

[x][x]

t+Dt – [x]

t



Euler method:


t

[x]

Dt

t

[x][x]

t+Dt – [x]

t

t

[x]

Dt

4th order Runge-Kutta:

with



Choose the right formalism




irreversible catalysis





product escapes before rebinding





product escapes before rebinding

quasi-steady-state



Enzyme kinetics

Victor Henri (1903) Lois Générales de l'Action des Diastases. Paris, Hermann.

Leonor Michaelis, Maud Menten (1913). Die Kinetik der Invertinwirkung, Biochem. Z. 49:333-369

George Edward Briggs and John Burdon Sanderson Haldane (1925) A note on the kinetics of enzyme action, Biochem. J., 19: 338-339



Briggs-Haldane on Henri-Michaelis-Menten (only for info. Not needed)

[E]=[E0]-[ES]



[E]=[E0]-[ES]

steady-state!!!

Briggs-Haldane on Henri-Michaelis-Menten (only for info. Not needed)



Generalisation: activators

x y




a

x y

x y

d[y]/dt

[a]

v

50%v

Ka




d[y]/dt

[a]

v

50%v

Ka

d[y]/dt

log[a]

v

50%v

Ka

a

x y

x y




d[y]/dt

[a]

v

50%v

Ka

d[y]/dt

log[a]

v

50%v

Ka

a

x y

x y

(NB: You can derive that as the fraction of target bound to the activator)



Phenomenological ultrasensitivity

d[y]/dt

log[a]

v

50%v

Kad[y]/dt

log[a]

v

50%v

Kad[y]/dt

log[a]

v

50%v

Ka



The Hill function

Hill (1910) J Physiol 40: iv-vii.



Hill (1910) J Physiol 40: iv-vii.

1

0

Y

[x]1/K

The Hill function



Generalisation: inhibitors

d[y]/dt

log[i]

v

50%v

Ki

x y

i

x y

(NB: You can derive that as the fraction of target not bound to the inhibitor)



Mathematics are beautiful



Generalisation: activators and inhibitors

log [a]log [i]

x y

a

x y

i



absolute Vs relative activators

d[x]/dt

log[a]

v

50%v

Ka

a

x y



absolute Vs relative activators

d[y]/dt

log[a]

v

50%v

Ka

d[y]/dt

log[a]

v(1+

v

Ka

a

x y

a

x y



1 compartment



2 compartments



2 compartments

A B

Per unit of time



2 compartments … with different volumes

AB




AB

Per unit of time




AB

Per unit of time




AB

Per unit of time




AB

Per unit of time

Stoichiometries (concentration change per Reaction events) are in factscaling with volumes:



Homeostasis

How can-we maintain a stable level with a dynamic system?

Ø Øx



Homeostasis


Ø Øx



Homeostasis


Ø Ø

[x]

time

x



Homeostasis


Ø Ø

[x]

time

0

x



Homeostasis


Ø Ø

[x]

time

0

1

x



Homeostasis


[x]

time

0

1

Ø Øx



Questions?



Conformational equilibrium



Binding equilibrium



How does a ligand activate its target?







hint: K1>1



Add energies

Multiply constants

+1 quantum energy = constant divided by 10

Explore constants exponentially:

Parameter space

-2.3 -4.6 -6.9 -9.2 -11.5 -13.8 -16.1

...

10-1 10-2 10-3 10-4 10-5 10-6 10-7

0.10.2

0.30.4

0.50.6

0.7

Documents

Nicolas Le Novère, Babraham Institute, EMBL-EBI n.lenovere ... · Modelling chemical kinetics Nicolas Le Novère, Babraham Institute, EMBL-EBI [email protected]. Bioinformatics