Modelling chemical kinetics

(1)

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]

(2)

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

substances A and B

are consumed

reaction R that produces

substances

P and Q

(3)

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 4

5

6 7 8

9

10

(4)

Chemical kinetics and fluxes

S1

S2

E

P

(5)

Statistical physics and chemical reaction

(6)

Statistical physics and chemical reaction

Probability to find an object in a container

within an interval of time

(7)

Statistical physics and chemical reaction

Probability to find an object in a container

within an interval of time

(8)

Law of Mass Action

Waage and Guldberg (1864)

rate-constant

velocity

stoichiometry

activity

(9)

Law of Mass Action

Waage and Guldberg (1864)

activity rate-constant

velocity

stoichiometry

gas

solution

(10)

Evolution of a reactant



Velocity multiplied by stoichiometry



negative if consumption, positive if production



Example of a unimolecular reaction

(11)

Evolution of a reactant



Velocity multiplied by stoichiometry



negative if consumption, positive if production



Example of a unimolecular reaction

(12)

Evolution of a reactant



Velocity multiplied by stoichiometry



negative if consumption, positive if production



Example of a unimolecular reaction

_[x]

0

[x]

₀

/e

1/k t ln2/k

[x]

₀

/2

(13)

Reversible reaction

is equivalent to

(14)

Reversible reaction

is equivalent to

(15)

Example of an enzymatic reaction

(16)

Example of an enzymatic reaction

(17)

Example of an enzymatic reaction

(18)

t [x]

Not feasible in general

Numerical integration

Example of an enzymatic reaction

(19)

Euler method:

Numerical integration (only for info. Not needed)

t [x]

Dt

t [x]

[x]_t+Dt – [x]_t

(20)

Euler method:

Numerical integration (only for info. Not needed)

t [x]

Dt

t [x]

[x]_t+Dt – [x]_t

(21)

Euler method:

Numerical integration (only for info. Not needed)

t [x]

Dt

t [x]

[x]_t+Dt – [x]_t

t [x]

Dt 4

^th

order Runge-Kutta:

with

(22)

Choose the right formalism

(23)

Choose the right formalism

irreversible catalysis

(24)

Choose the right formalism

irreversible catalysis

product escapes before rebinding

(25)

Choose the right formalism

irreversible catalysis

product escapes before rebinding

quasi-steady-state

(26)

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

(27)

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

[E]=[E

₀

]-[ES]

(28)

[E]=[E

₀

]-[ES]

steady-state!!!

Briggs-Haldane on Henri-Michaelis-Menten

(only for info. Not needed)

(29)

Generalisation: activators

x y

(30)

Generalisation: activators

a

x y

d[y]/dt

[a]

v 50%v

Ka

(31)

Generalisation: activators

d[y]/dt

[a]

v 50%v

Ka d[y]/dt

log[a]

v 50%v

Ka

a

x y

(32)

Generalisation: activators

d[y]/dt

[a]

v 50%v

Ka d[y]/dt

log[a]

v 50%v

Ka

a

x y

(NB: You can derive that as the fraction

(33)

Phenomenological ultrasensitivity

d[y]/dt

log[a]

v 50%v

d[y]/dt Ka

log[a]

v 50%v

d[y]/dt Ka

log[a]

v 50%v

Ka

(34)

The Hill function

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

(35)

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

1 0 Y

1/K [x]

The Hill function

(36)

Generalisation: inhibitors

d[y]/dt

log[i]

v 50%v

Ki

x y

i

x y

(NB: You can derive that as the fraction

(37)

Mathematics are beautiful

(38)

Generalisation: activators and inhibitors

log [a]

log [i]

x y

a

x y

i

(39)

absolute Vs relative activators

d[x]/dt

log[a]

v 50%v

Ka

a

x y

(40)

absolute Vs relative activators

d[y]/dt

log[a]

v 50%v

Ka

d[y]/dt v(1+

v

a

x y

a

x y

(41)

1 compartment

(42)

2 compartments

(43)

2 compartments

A B

Per unit of time

(44)

2 compartments … with different volumes

A

B

(45)

2 compartments … with different volumes

A

B

Per unit of time

(46)

2 compartments … with different volumes

A

B

Per unit of time

(47)

2 compartments … with different volumes

A

B

Per unit of time

(48)

2 compartments … with different volumes

A

B

Per unit of time

Stoichiometries

(concentration change per

Reaction events) are in fact

scaling with volumes:

(49)

Homeostasis

How can-we maintain a stable level with a

dynamic system? Ø x Ø

(50)

Homeostasis

How can-we maintain a stable level with a

dynamic system? Ø x Ø

(51)

Homeostasis

How can-we maintain a stable level with a

dynamic system? Ø Ø

[x]

time

x

(52)

Homeostasis

How can-we maintain a stable level with a

dynamic system? Ø Ø

[x]

0 x

(53)

Homeostasis

How can-we maintain a stable level with a

dynamic system? Ø Ø

[x]

time

0

1 x

(54)

Homeostasis

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

[x]

0

1 Ø x Ø

(55)

Questions?

(56)

Conformational equilibrium

(57)

Binding equilibrium

(58)

How does a ligand activate its target?

(59)

How does a ligand activate its target?

(60)

How does a ligand activate its target?

hint: K

₁

>1

(61)

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

Modelling chemical kinetics

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

substances A and B

are consumed

reaction R that produces

substances

P 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 4

5

6

7 8

9

10

Chemical kinetics and fluxes

S1

S2

E

P

Statistical physics and chemical reaction

Statistical physics and chemical reaction

Probability to find an object in a container

within an interval of time

Statistical physics and chemical reaction

Probability to find an object in a container

within 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

Evolution of a reactant

Velocity multiplied by stoichiometry

negative if consumption, positive if production

Example of a unimolecular reaction

Evolution of a reactant

Velocity multiplied by stoichiometry

negative if consumption, positive if production

Example of a unimolecular reaction

[x]

[x]

/e

1/k t ln2/k

[x]

/2

Reversible reaction

is equivalent to

Reversible reaction

is equivalent to

Example of an enzymatic reaction

Example of an enzymatic reaction

Example of an enzymatic reaction

t [x]

Not feasible in general

Systems Biology models  ^{ODE models}

_[x]