Stochastic modeling of molecular reaction networks Daniel Forger University of Michigan.
Stochastic modeling of molecular reaction networks
description
Transcript of Stochastic modeling of molecular reaction networks
We can list the basic reaction rates and stochiometry
numsites = total # of sites on a gene, G = # sites bound M = mRNA, Po = unmodified protein, Pt = modified protein
Transcription trans or 0 +MTranslation tl*M +PoProtein Modification conv*Po -Po, +PtM degradation degM*M -MPo degradation degPo*Po -PoPt degradation degPt*Pt -PtBinding to DNA bin(numsites - G)*Pt -Pt, +GUnbinding to DNA unbin*G -G
We normally track concentrationLet’s track # molecules instead
• Let M, Po, Pt be # molecules• First order rate constants (tl, unbin, conv,
degM, degPo and degPt) have units 1/time and stay constant
• Zero order rate constant (trans) has units conc/time, so multiply it by volume
• 2nd order rate constant (bin) has units 1/(conc*time), so divide it by volume
numsites = total # of sites on a gene, G = # sites bound M = mRNA, Po = unmodified protein, Pt = modified proteinV = Volume
Transcription trans*V or 0 +MTranslation tl*M +PoProtein Modification conv*Po -Po, +PtM degradation degM*M -MPo degradation degPo*Po -PoPt degradation degPt*Pt -PtBinding to DNA bin/V(numsites - G)*Pt -Pt, +GUnbinding to DNA unbin*G -G
How would you simulate this?
• Choose which reaction happens next– Find next reaction– Update species by stochiometry of next
reaction– Find time to this next reaction
How to find the next reaction
• Choose randomly based on their reaction rates
trans*V tl*M conv*Po degM*M degPo*Po degPt*Pt bin/V(numsites - G)*Ptunbin*G
Random #
Now that we know the next reaction modifies the protein
• Po = Po - 1
• Pt = Pt + 1
• How much time has elapsed– a0 = sum of reaction rates
– r0 = random # between 0 and 1
⎟⎟⎠
⎞⎜⎜⎝
⎛=
00
1ln
1
raτ
This method goes by many names
• Computational Biologists typically call this the Gillespie Method– Gillespie also has another method
• Material Scientists typically call this Kinetic Monte Carlo
• Here a protein can be in 3 states, A, B or C
• We start the system with 100 molecules of A
• Assume all rates are 1, and that reactions occur without randomness (it takes one time unit to go from A to B, etc.)
A B
C
Mass Action represents a limiting case of Stochastics
• Mass action and stochastic simulations should agree when certain “limits” are obtained
• Mass action typically represents the expected concentrations of chemical species (more later)
What matters is the number of reactions
• This is particularly important for reversible reactions
• By the central limit theorem, fluctuations dissapear like n-1/2
• There are almost always a very limited number of genes, – Ok if fast binding and unbinding
There are several representations in between Mass Action and Gillespie
• Chemical Langevin Equations
• Master Equations
• Fokker-Planck
• Moment descriptions
We will illustrate this with an exampleKepler and Elston Biophysical Journal 81:3116
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Master Equations describe how the probability of being in
each state
€
dpm0
dt= −Kk0 + δm + α 0( ) pm
0 + Kk1pm1 + δ(m +1)pm +1
0 + α 0 pm−10
dpm1
dt= −Kk1 + δm + α 1( ) pm
1 + Kk0 pm0 + δ(m +1)pm +1
1 + α 1pm−11
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Sometimes we can solve for the mean and variance
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moments = m j
s≡ m j pm
s
m
∑
at steady state
mean =α 0k1 + α 1k0
δ
var iance = mean + k0k1
α 0 −α 1
δ
⎛
⎝ ⎜
⎞
⎠ ⎟2
δ
δ + K
Moment Descriptions
• Gaussian Random Variables are fully characterized by their mean and standard deviation
• We can write down odes for the mean and standard deviation of each variable
• However, for bimolecular reactions, we need to know the correlations between variables (potentially N2)
Towards Fokker Planck
• Let’s divide the master equation by the mean m*.
• Although this equation described many states, we can smooth the states to make a probability distribution function
€
pms (t) ≡ dxps(x, t)
(m−1/ 2)/ m*
(m +1/ 2)/ m*
∫
Note
€
ps x +1
m*
⎛
⎝ ⎜
⎞
⎠ ⎟=
1
j!∂x( )
jps(x)
1
m*
⎛
⎝ ⎜
⎞
⎠ ⎟j
= e1
m*∂ x
ps(x)j
∑
If 1/m* is small, we can then derive a simplifedVersion of the Master equations
€
∂t ps(x) = −∂x
α s
m*−δx
⎛
⎝ ⎜
⎞
⎠ ⎟ps(x)
⎡
⎣ ⎢
⎤
⎦ ⎥+
1
2m*∂x
2 α s
m*+ δx
⎛
⎝ ⎜
⎞
⎠ ⎟ps(x)
⎡
⎣ ⎢
⎤
⎦ ⎥+ K[k ˆ s pˆ s (x) − ks ps(x)]
Chemical Langevin Equations
• If we don’t want the whole probability distribution, we can sometimes derive a stochastic differential equation to generate a sample
€
dX
dt= A(X) + B(X)ξ (t)
Adalsteinsson et al. BMC Bioinformatics 5:24
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Rossi et al. Molecular Cell
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Ozbudak et al. Nature 427:737
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Guantes and Poyatos PLoS Computational Biology 2:e30
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SNIC BifurcationSNIC Bifurcation
Invariant Circle
Limit Cycle
x2
p1
node
saddle
Saddle-Node on anInvariant Circle
max
min
max
SNIC
Liu et al. Cell 129:605
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