A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

master equation of our pdf we need to represent the transitional probability functions
that describe the probability that our walker takes a step. These functions should
depend on W since our walker is under the influence of the control species. Define
and
T +
i (W ) = transitional probability per unit time of a jump from i to i + 1,
T −
i (W ) = transitional probability per unit time of a jump from i to i − 1.
We can write the master equation [19]
∂pi
∂t
= T −
i+1 (W )pi+1(t) + T +
In plain English the above can be stated (roughly) as:
24
i−1 (W )pi−1(t) − (T +
−
i (W ) + Ti (W ))pi(t) (3.1)
The change in probability of finding our walker at i
�
(Probability the walker moves from i + 1 to i) × (the probability the walker was at i + 1)
+
(Probability the walker moves from i − 1 to i) × (the probability the walker was at i − 1)
|
(Probability the walker moves from i) × (the probability the walker was at i).
Moreover, we can say that the quantity
(T +
i
(W ) + T −
i (W ))−1