Lecture 7: Live cell microscopy & Stochastic models
Part II: Cellular Dynamics
week
lecture
lab
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Fluorescence microscopy / Stochastic models / Random walks
Markov model of a two-state ion channel, (anomalous) diffusion
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Law of mass action / Deterministic models
Cellular clock ODE model
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Molecule Tracking
Template matching / Optimum assignment problem / Particle Tracking
Table 6: Timetable
Stochastic description of molecular processes
Example 13. A two state model of a single ion channel
• Remark: Depolarization of Xenopus oocytes expressing voltage-gated channels to
potentials > -20 mV results in Calcium influx indicated by a fluorescent dye
• Model: Stochastic gating as a continuos time Markov process
• Open state (Calcium influx) / Closed state k+
C (closed) O (open) k−
• k+, k− are rate constants with units [1/sec]
• Random variable s ∈ {C, O}
• Pi(t) = Prob{s = i,t} Probability for channel to be in state i at time t .
• Probability of all states must add up to 1: PC (t) + PO (t) = 1
• Probability for a channel closed at time t to open in interval △t P rob{s = O, t + △t | s = C, t} = k+△t
• The probability for a C → O transition depends on PC (t)
k+PC(t)△t
• Analogously, the probability for a O → C transition is:
k−PO(t)△t
PC(t + △t) = PC(t) + [k−PO(t) − k+PC(t)]△t = PC + △PC(t)△t
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• Together, we obtain:
• Taking the limit △t → 0 we obtain the ODEs (Chemical Master Equation) dPC =k−PO(t)−k+PC(t)
dt
dPO =k+PC(t)−k−PO(t) dt
• Because PC (t) = 1 − PO (t) we can eliminate one of the equations, and write dPO =k+(1−PO(t))−k−PO(t)
dt
• Remark: Later we will arrive at similar expressions where Pi(t) is replaced by
concentrations of a large number of molecules
Transition probability matrix
Prob{C,t+△t | C,t} Prob{C,t+△t | O,t} Q = P rob{O, t + △t | C, t} P rob{O, t + △t | O, t}
• Qi,j Probability of transition from state j to i . • Columns of Q add up to 1:
Qi,j =1 i
1−k+△t = k+△t
k+△t 1 − k+△t
• Using vector notation for the states
→− Prob(C,t)
we can write
and generally
Dwell times
P (t) =
P rob(O, t) →− →−
P (t + △t) = QP (t) →− n →−
P(t+n△t)=Q P(t)
• Dwell time: Average amount of time channel stays open or closed
• Channelclosedduring[t,t+n△t]: Prob{C,[t,t+n△t]|C,t}=(1−k+△t)n
• Letτ =n△tandrewrite: Prob{C,[t,t+τ]|C,t}=(1−k+τ)n n
• Inthelimitofn→∞and△t→0,andusinglimn→∞1−αn =e−α weobtain: n
Prob{C,[t,t+τ] | C,t} = e−k+τ
Remark: The probability that a channel stays closed decreases exponentially
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• Closed dwell time: channel closed during [t,t+τ], opens during [t+τ,t+τ +△t] Prob{C,[t,t+τ] | C,t}Prob{O,t+τ +△t | C,t+τ} = e−k+τk+△t
• The averaged closed time can be computed as ⟨τC⟩= 1
Similarly:
k+ ⟨τO⟩= 1
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k−
Lecture 8: Simulating stochastic reactions and molecular movement
Monte Carlo simulation of a single channel
• Recall that in the transition matrix Q columns add up to 1: Qi,j =1
i
1 − k+△t [remains closed] k+△t [open to closed] Q = k+△t [closed to open] 1 − k+△t [remains open]
• If current state is open then draw a uniformly distributed random number Y ∈ [0, 1].
• If0≤Y
Nerve axon (r = 1m): 109 sec = 30 years !
Anomalous diffusion
• Plotting msd(t) against t gives a straight line 2nDt
• Generally, we can state msd(t) = 2nDtα , and have – normal diffusion for α = 1
– superdiffusion for α > 1
– subdiffusion for α < 1
• Example for superdiffusion: motor molecules moving cargo (vesicles) along micro- tubule tracks
• Example for subdiffusion: Molecules are confined to a specific area (see lab problem)
Diffusion as a random walk
Because msd(t)/t = 2nD = constant, diffusion can be approximated by a (fully dis- crete) random walk. For a large number of N = t/△t steps (N > 10), with D :=
1 (△x)2/△t 2
n 4Dt P(x,t)≈ 1 e− x2
(4πDt) 2
is the probability distribution for a single random walker in n dimensions.
Comparing to a Gaussian,
f(x,μ=0,σ)= 2 ne 2σ2 ,
i=1
(ri,t − ri,0)2
2 1−x2 (2πσ )2
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we see that σ = √2Dt = msd(t). In terms of the “speed of diffusion” we can therefore state: If N molecules is released at a single point at t = 0, we can expect to find ∼ 68%
√
of molecules within distance ± 2Dt.
Smoluchovski equation and diffusion
• Generate a normally distributed random number ξ with zero mean and unit vari- ance for each spatial dimension (in the discrete version this would have been steps of ±1)
• Compute the position at time t + △t using
X(t + △t) = X(t) + 2D△tξx Y(t+△t) = Y(t)+2D△tξy Z(t + △t) = Z(t) + 2D△tξz
noting that diffusion is independent in each spatial direction. For no-flux boundary conditions (particles in a box) we reflect the particle on the boundary should the predicted position fall outside the box.
• Note how this is directly related to the Euler-Maruyama Integration scheme we used earlier where X(t) was considered the concentration of a transcription factor.
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