Page 10 - Cardiac Electrophysiology | A Modeling and Imaging Approach

P. 10
). A formalism is needed that
inactivation occurs preferentially from the channel open state 12, 13
represents specific kinetic states of the ion channel and transitions between them that depend
on their present occupancy. Such models are required for simulating state-specific processes, for
example, mutations that affect specific transitions (e.g. failure of inactivation from the open state)
or state-specific binding of drugs (e.g. open channel blockade).

Markovian models meet these requirements and can replace equations (2.2)-(2.5) in
computing I of equation (2.1). In a Markov model, occupancy of the channel in its various kinetic
ion
states is computed as a function of voltage and time (and possibly other factors, e.g. ligand

binding). The channel conducts ions when it occupies its open state (or multiple open states). The
macroscopic current density through an ensemble of channels (equivalent to equation (2.2) of the
Hodgkin-Huxley formalism) is given by:

Ι i = g g sc sc,i i , • N • P ( ) ( Vo • m − E i ) (2.6)

In this equation, g is the single channel conductance, P(o) is the (voltage- and time-
sc,i
dependent) probability that the channel occupies the open state, N is the number of channels per
unit area of membrane, and (V -E) is the driving force.
m i

The simplest Markov scheme describes a (hypothetical) channel with a single open (O) and a
single closed (C) state. The following first-order differential equations describe the rate of change of
occupancy in these states:

dC dC = −α • C + β • O

dt
dt
(2.7)
dO dO = α • C − β • O

dt dt

C and O are the probabilities that the channel resides in the open or closed state; α and ß
are voltage dependent transition rates between these states. Following this scheme, additional
closed, open and inactivation states can be included to represent the kinetic properties of a given
ion channel. 10