Biochemical Reactions: how types of molecules combine.

Playing by the Rules

+ +2a b c

Biochemical Reactions

9

6

7

cellspecies count

+

8

5

9

Discrete chemical kinetics; spatial homogeneity.

Biochemical Reactions

+

+

+

slow

medium

fast

Relative rates or (reaction propensities):

Discrete chemical kinetics; spatial homogeneity.

Biochemical ReactionsLingua Franca of computational biology.

1 molecule of type A combines with2 molecules of type B to produce2 molecules of type C.

Reaction

CBAk

2 21

Reaction is annotated with a rate constant and physical constraints (localization, gradients, etc.)

Biochemical ReactionsLingua Franca of computational biology.

• Elementary molecules (e.g., hydrogen, phosphorous, ...)• Complex molecules (e.g., proteins, enzymes, RNA ...)

Species:

Reaction:

H2HHk

O1

O • Elementary step (e.g., ) • Conglomeration of steps (e.g., transcription of gene

product)

CBAk

2 21

Reaction

Coupled Set Reactions

BCA

ACB

CBA

k

k

k

2

2

2 2

3

2

1

R1

R2

R3

Goal: given initial conditions, analyze (predict) the evolution of such a system.

Lingua Franca of computational biology.

Biochemical Reactions

Biochemical Reactions

• Assumes that molecular quantities are continuous values that vary deterministically over time.

Convential Approach: numerical calculations based on coupled ordinary differential equations.

d[A]/dt = 2k_2 [B][C] -- k_1 [A][B]^2 -- k_3 [A][C]

d[B]/dt = 2k_3 [A][C] -- k_1 [A][B]^2 -- k_2 [B][C]

d[C]/dt = 2k_1 [A][B]^2 -- k_2 [B][C]^-- k_3 [A][C]

BCA

ACB

CBA

2

3

2 3

2

k1

k2

k3

R1

R2

R3

See D. Gillespie, “Stochastic Chemical Kinetics”, 2006.

The probability that a given reaction is the next to fire is proportional to:

• Its rate.• The number of ways that the

reactants can combine.

Discrete Stochastic Kinetics

Choose the next reaction according to:

jj

iiR

)Pr(

Ri ...... kXnXn 2211

let

...

2

2

1

1

n

X

n

Xki

For each reaction

Stochastic Kinetics

Track precise (integer) quantities of molecular species.

“States”

A B C

4 7 5

2 6 8

22 0 997

S1

S2S3

A reaction transforms one state into another:

21 1SS

Re.g.,

Gillespie’s Framework

Reactions

BCA

ACB

CBA

2

3

2 3

2

k1

k2

k3

R1

R2

R3

S1 = [5, 5, 5] 0

Choose the next reaction according to:

StochasticSimulation

Ri ikiiii XnXn 2,2,1,1,

jj

iiR

)Pr(

where

R1 R2 R3

2

2

1

1i n

X

n

Xki

StochasticSimulation

Ri ikiiii XnXn 2,2,1,1,R1 R2 R3

Choose the time of the next reaction according to:

S1 = [5, 5, 5] 0

dettt

jj

jj

0

00 )Pr(

StochasticSimulation

R1 R2 R3

See D. Gillespie, “Exact Stochastic Simulation of Coupled Chemical Reactions”,J. Phys. Chem. 1977

S1 = [5, 5, 5] 0

StochasticSimulationS1 = [5, 5, 5] 0

S2 = [4, 7, 4]

Choose R3 and t = 3 seconds.

R1 R2 R3

S3 = [2, 6, 7] 4

Choose R1 and t = 1 seconds.

S4 = [1, 8, 6] 6

Choose R3 and t = 2 seconds.

3

Choose R2 and t = 1 seconds.

StochasticSimulationS1 = [5, 5, 5] 0

S2 = [4, 7, 4]

Choose R3 and t = 3 seconds.

S3 = [2, 6, 7] 4

Choose R1 and t = 1 seconds.

S4 = [1, 8, 6] 6

Choose R3 and t = 2 seconds.

Choose R2 and t = 1 seconds.

37