Theoretical Analysis of Molecular Transport Across Membrane Channels and Nanopores - Computational Modeling of Biological Systems - page 304

Fig. 4 A general schematic
picture for channel with
N D 2 binding sites, multiple
occupancy, and with
intermolecular interactions.
Open circles describe empty
sites, while filled circles
denote occupied sites. Two
molecules sitting next to each
other interact with energy "

u

u 2

u 0

u

u u

u 1

u 2

u u 2

with x D exp."=k B T/ . The case " D 0 corresponds to the situation analyzed in Zilman
[ 30 , 31 ]. This observation allows us to write explicit expressions,

u 0 x ; u x 1 ;

u 1 D u 2 D (11)

where the coefficient 0 1 again specifies how the interparticle interaction
modifies these entrance and exit rates. We can define P.i;j I t/ as the probability of
finding channel in the state .i; j / at time t , and temporal evolution of the system
dynamics can be found by analyzing corresponding master equations:

dP.0;0 I t/
dt

D u P.0;1 I t/ C u P.1;0 I t/ u 0 P.0;0 I t/;

dP.1;0 I t/
dt

D u 0 P.0;0 I t/ C u 2 P.1;1 I t/ C u P.0;1 I t/ 2 u P.1;0 I t/;

dP.0;1 I t/
dt

D u P.1;0 I t/ C u 2 P.1;1 I t/ .2 u C u 1 /P .0; 1 I t/;

dP.1;1 I t/
dt

D u 1 P.0;1 I t/ 2 u 2 P.1;1 I t/: (12)

Solving these equations at large times ( dP.i;j I t/
dt

0 ) yields the expression for the !

molecular flux,

uu 0 u

2 x

u 0

C

J 2 D

uu 0 3 C

: (13)

u 0
2

x
2

3 u 2 C .x C x / C