Stochastic Modelling of Single Ion Channels - Computational Neuroscience: A Comprehensive Approach - page 134

Figure 5.4
CK mechanism with four states.

Thus, for small t , the conditional probability is given approximately by

Prob ( channel in state j at time t + ∆ t | channel in state i at time t )/ t q ij ∆ t

It is convenient to define q ii as minus the sum of the transition rates away from state
i : that is q ii = − ∑

i q ik . Then the probability of remaining in the same state is k =

Prob ( channel in state i at time t + ∆ t | channel in state i at time t )

= 1 − Prob ( moving to some other state )

− ∑ k = i q ik ∆ 1 t = 1 + q ii ∆ t

m square matrix Q , whose elements are q ij , is called
the transition rate matrix or Q -matrix. The elements in each of its rows sum to zero.
For example, the Q f-matrix for the CK model is

If there are m states, the m ×

− α 0

Q = β − ( k − 1 + β ) k − 1 (5.1)

0 k + 1 x A − k + 1 x A

Note that, as discussed above, the transition rate for binding is the product of the
rate constant k + 1 and the ligand concentration x A . Each of the other transition rates
is given simply by the appropriate rate constant.
When the channel leaves state i it moves into state j with probability

− /

q ii .
Thus, for example, when the channel leaves state 2 (shut with a bound molecule) it
opens, with probability β / ( k − 1 + β ) , or the bound molecule dissociates, with prob-
ability k − 1 / (

q ij

k − 1 + β ) .

5.3.2 A simple channel-block mechanism

Suppose that the open channel in the CK model can be blocked by a molecule of a
blocker substance, whose concentration is x B . The model that had two shut states
and one open state now has an additional shut state, so it can be represented as in
Figure 5.4