Example Matrix From Previous Page..
| 6/16 | 3/20 | 7/24 |
so, for example, the probability that if we start with C we'll
end up with B after one step is 3/24 = 1/8.
|
| 3/16 | 14/20 | 3/24 |
| 7/16 | 3/20 | 14/24 |
Ok, now we get a little technical. If you don't know matrix multiplication, don't worry.
If you do, then you'll have no trouble.
The matrix above contains the mutation probabilities for one step, we assume. Question: if we
start with a gene containing 64 A's, 40 B's, and 48 C's, after one step how many A's do we expect
we'll have given this mutation probability matrix?
Answer: well, we expect (6/16)x64 = 24 of the A's to stay A's; and (3/20)x40 = 6 of the B's to
change to A's; and (7/24)x48 = 14 of the C's to turn into A's. So we expect around 24+6+14=44 A's
after one step.
In a similar manner we expect 12+28+6=46 B's, and 28+6+28=62 C's. Well, so we take that one step
into the future, and here we are in the present with 44 A's, 46 B's, and 62 C's.
What if we now want to predict what will happen after another step, which is one step
from where we are. Again we use the one step mutation probability matrix, now applied
to our three new numbers
{44, 46, 62}. In this way we can predict what will happen after
two steps from where we were. And obviously, continuing on in this manner we can
predict what will happen after 3 steps, 4 steps, ..., and 250 steps.
The matrix above is a one step probability matrix. If you're familiar with matrix multiplication
then you probably recognize that we can find a single matrix to take us n steps into the future
by multiplying the one step matrix times itself n times. To see how this works I've written a
little JavaScript program below that will take our initial one-step matrix and with the aid of
a little button produce successive matrices for higher steps.
A couple points to make here: notice that initially the matrix changes a fair amount with each step,
but the more steps you take the less it changes; in fact after about 15 steps it becomes clear that
the numbers across each row are getting closer and closer to the same value, ie., they're converging;
which is way interesting, but equally important to us, since the change becomes less and less, with
each step the initial accuracy of our one-step probabilities becomes more important. We'll discuss
these issues again when we look at the PAM-250 scoring matrix. Right now, let's finish making a
one-step scoring matrix.
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