Lecture 11

Dynamic Instability of the Microtubule

What is the timescale of fluctuations? ($\tau$)

A good guess would be that it depends only on the two rates which we are aware of - the attach and detach rates.

$$\tau \approx \frac{1}{\alpha}+\frac{1}{\mu} \approx \frac{2}{\mu}$$

Dynamic Instability in microtubule-like filaments

• Microtubules have dynamic instability
  
  • One end of the microtubule is composed of stable (GTP) monomers while the rest of the tubule is made up of unstable (GDP) monomers.
    
    • The GTP end comprises a cap of stable monomers.
  • Random fluctuations either increase or decrease the size of the cap.
    
    • This results in 2 different dynamic states for the microtubule.
      
      • Growing: cap is present
      • Shrinking: cap is gone

Lets define some parameters to describe how the microtubule switches between these to states and also how it behaves once in a state.

$$g \equiv \text{switch from shrinking to growing rate} \\ s \equiv \text{switch from growing to shrinking rate} \\ $$

If we are in the growth state then monomers may be added to the filament.

$$\alpha \equiv \text{attach rate}$$

If we are in the shrinkage state then monomers may be released from the filament.

$$\mu \equiv \text{detach rate}$$

The microtubule switches between these two states.

If we were to plot the length of the microtubule as a function of time it would look something like the following:

Lets now try to write a model that can describe how this occurs.

Dynamic Model

Lets define $f$ as being the fraction of time spent in the growth state. Hence the fraction of time spent in shrinkage state is then $1-f$.

Similar to what we have done in previous lectures, our dynamic model is expressed as the following:

$$\bar{n}(t)=\alpha(ft)-\mu[(1-f)t]+n_0$$

The fractions $f$ and $1-f$ can expressed in terms of the rates we defined above.

$$f = \frac{g}{g+s} \\ 1-f=\frac{s}{g+s} \\ \Rightarrow \bar{n}(t)=\left[\frac{\alpha g}{g+s}-\frac{\mu s}{g+s}\right]t + n_0$$

• Growth is bounded if $\alpha g<\mu s$.
• Growth is unbounded if $\alpha g>\mu s$.

Stochastic Model

Now lets think in terms of probabilities. We need to assign probabilities to being in either a growth or shrinkage state with a certain number of links $n$.

$p(n,+)\equiv q_n = \text{probability of being in growth state}$
$p(n,-)\equiv r_n = \text{probability of being in shrinkage state}$

The probability of being a microtubule with $n$ links is $p_n$.
If $n=0$, $p_n = q_0$.
If $n\geq1$, $p_n=q_n+r_n$.

Growth Rate

$$\frac{dp(n,+)}{dt}=\frac{dq_n}{dt}$$

For $n\geq 1$

$$\frac{dq_n}{dt} = \alpha \ q_{n-1}-\alpha \ q_n - s \ q_n + g \ r_n \\ \frac{dr_n}{dt} = \mu \ r_{n+1}-\mu \ r_n+s \ q_n -g \ r_n$$

For $n=0$

$$\frac{dq_0}{dt}=\mu \ r_1-\alpha \ q_0$$

In a bounded state we can solve for the stationary distribution of $p_n$ by setting the time derivatives equal to zero.

For $n=0$

$$\Rightarrow r_1=\frac{\alpha}{\mu}q_0=\lambda q_0 \\ \lambda \equiv \frac{\alpha}{\mu}$$

For $n=1$

$$0 = \alpha \ q_0-\alpha \ q_1 - s \ q_1 + g \ r_1 \\ 0 = \mu \ r_2-\mu \ r_1+s \ q_1 -g \ r_1 \\ x \equiv \lambda \frac{\mu+g}{\alpha+s} \\ \Rightarrow q_1=x \ q_0, \ \ r_2 = \lambda \ x \ q_0$$

If we repeat this process for $n=2,3,4,...$ we notice a pattern emerge.

$$q_n = x^nq_0 \\ r_n=x^{n-1} \lambda \ q_0$$

We can solve for $q_0$ by normalizing the probability distribution.

$$1=\sum_{n=0}^\infty p_n = q_0 + \sum_{n=1}^\infty (q_n+r_n) \\ 1=q_0\left[\frac{1-x+x+\lambda}{1-x}\right] \\ q_0 = \frac{1-x}{1+\lambda}$$

$$p_n = \begin{cases} q_0, & n = 0 \\ q_0 x^n \left(1+\frac{\lambda}{x}\right), & n\geq1 \end{cases}$$

This is not quite a (exponential) geometric distribution. A geometric distribution looks like the following:

$$p_n=(1-\lambda)\lambda^n \\ \frac{p_{n+1}}{p_n}=\lambda$$

Whereas in our case we have the following relationships:

$$\frac{p_1}{p_0}=x+\lambda, \ \ \ \frac{p_{n+1}}{p_n}=x \ \ \text{for} \ n\geq1$$

Notice that when we solved for the series we assumed $x<1$ in order for the series to converge, and so this is in fact a bounded state.

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