# Lecture 10

# Dynamics: Filament Polymerization

## Polymerization at Both Ends (cont.)

### Actin

Unlike microtubules, actin does not have the same *critical concentration* for both ends. The “” end is the side where the filament grows and the “” end is the side where the filament shrinks.

Plotting as a function of will illustrate a different phase space than that of microtubules. Assume that , this gives us the following relationships between and .

The three relationships above yield 3 different phases for actin.

: Both ends shrink : The end grows, end shrinks : Both ends grow

For some value of $c$ within the regime of the rate at which the end grows equals the rate the end shrinks. This concentration is called the *treadmilling concentration* .

Simple treadmilling animation

For actin, . Both ends grows and shrink at the same rate resulting in the length of the actin remaining fixed. The system is in a steady state, however it is not in thermodynamic equilibrium because the filaments are constantly shifting. We can derive by noting that the total number of filaments remains constant.

## Problems with the Model

Our model, given by the rate equation has a couple of problems.

- Nothing in this model says that cannot be negative.
- Physically, the number of filaments must be zero or positive.

- At small values of a continuous differential equation is no longer realistic. We can no longer approximate changes in as being continuous.

In order to try to solve these problems in our model we take a probabilistic approach. Let’s examine what may happen to a filament during small interval of time and their respective probabilities.

- Add a filament:
- Subtract a filament:
- Nothing happens:

Using these relations we can formulate the probability that at some time the actin has filaments, . There are three different ways to get to a state with filaments: add 1 filament, subtract 1 filament, and the actin had filaments and nothing changed. Quantitatively this can be expressed as the following:

This can apply to all values of except for edge cases. For example, at the probability of having zero filaments is

We can re-write the two equations above to show how the probabilities vary as a function of time.

For

For

The two above equations dictating the behavior of are called the (stochastic) *Master Equation*.

## Expectation Value

We can calculate the expectation value (mean) of using the following formula:

The expression for can be simplified by rewriting the different summation terms. By changing the summation index and by noting that for we can make a few simplifications.

For the first summation term:

For the second summation term:

For the third dummation term:

This simplifies to the following:

## If , what happens at long times?

Let’s assume that as , the change in probabilities goes to zero, . Using this assumption and the *master equation* we can find a recursion relation between the probabilities of different numbers of filaments.

For

For

A pattern emerges, and we can therefore say that the probability of being in a state with filaments is given by

Let so we can express the above statement as

In order for our probabilities to be **normalized** we enforce that all the probabilities must sum to one.

This is known as a geometric distribution, it is like a discrete version of an exponential distribution.

Now that we have a general form of we can calculate the mean of .

This result is good because it shows that does not become negative in this model.

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