Microsatellites tends to have a relatively high mutation rate, with mutations frequently involving a gain or a loss of a single repeat during DNA replication. This is thought to occur due to slippage of the DNA as the DNA replication machinery passes over the microsatellite.

The probability of slippage is assumed to increase with the number of repeats, n[t], in the microsatellite at time t.

Part 1:

Let n[t] be the probability that the microsatellite increases by one repeat in a generation.

Let n[t] be the probability that the microsatellite decreases by one repeat in a generation.

Write down an equation for the expected value of n[t+1] as a function of n[t]. Simplify your result and answer the following questions:

• Which model of population growth does this equation correspond to?

• What does that tell you about how the number of repeats in a microsatellite will change over time?

• What condition must hold for the number of repeats to remain constant over time?

Assume instead that deletions become more common as the length of the microsatellite increases (perhaps loops form more readily during DNA replication causing more deletions). Let the probability that the microsatellite decreases by one repeat in a generation now equal n[t]².

Write down the expected value of n[t+1] as a function of n[t]. Simplify your result and answer the following questions:

• Which model of population growth does this equation correspond to? (Relate the variables in each model to one another.)

• What does that tell you about how the number of repeats in a microsatellite will change over time?

• When will the number of repeats remain constant over time?

[NOTE: Throughout this question, assume that and are very small positive quantities such that n[t] and n[t]² are always less than one.]