Biophysical Models of the Cardiovascular System - Systems Biology: Networks, Models, and Applications - page 270

Figure 9.3 The 11-state LCC model developed by Jafri et al. [11].

where k is the Boltzmann constant, T is the absolute temperature,
h is the Planck constant, R is the gas constant, F is Faraday's constant,
∆ S l is the change in entropy, z l is the

effective valence (i.e., the charge moved times the fractional distance
the charge is moved through the membrane), and V is the membrane
potential in volts.
The probability of occupying any particular channel state, including
the open state, is described by a set of ordinary differential equations,
written in matrix notation as

H l is the change in enthalpy, ∆

dt
dt

P ()

(2)

= W P () t

where P ( t ) is a vector of state occupancy probabilities and W is the state
transition matrix. W is in general a function of voltage and time, since
state transition rates are also in general functions of voltage and time.
However, for the voltage-clamp conditions generally used to constrain
ion current models, W is piecewise time-independent, thus eq. (2) has
the analytic solution

P ( t ) = exp( W t ) P (0) (3)

Current through an ensemble of Ca 2+ channels, denoted I Ca,L , is

calculated as

I Ca,L ( t ) = NG Ca P open ( t )( V ( t ) − E Ca ( t )) (4)

where N is the number of Ca 2+ channels, G Ca is single-channel conduc-
tance, P open ( t ) is the probability of occupying the open state O, V(t) is
membrane potential, and E Ca ( t ) is the reversal potential for Ca 2+ given
by the Nernst equation.
The number of coupled differential equations, and hence the
number of parameters that need to be constrained for the model, may
be reduced through application of the fundamental principle that the
state occupancy probabilities for a Markov chain model must sum to
one. Thus, an N -state Markov model may be reduced to an N − 1 state