Upload
reginald-pierce
View
220
Download
0
Tags:
Embed Size (px)
Citation preview
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