We introduce a statistical mechanical model of the selectivity filter that accounts for the interaction between ions within the channel and derive Eisenman equation of the filter selectivity directly from the condition of barrier-less conduction

Eisenberg, Robert S.

Gibby, Will A. T.

Kaufman, Igor Kh.

Luchinsky, Dmitry G.

McClintock, Peter V. E.

NASA Technical Reports Server

Theory of alike selectivity in biological channels
Dmitry G. Luchinsky1;2, Will A.T. Gibby1, Igor Kh. Kaufman1, Robert
S. Eisenberg3, Peter V.E. McClintock1
1Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
2SGT, Inc., Greenbelt, MD, 20770, USA
We introduce a statistical mechanical model of the selectivity lter that accounts
for the interaction between ions within the channel and derive Eisenman equation
of the lter selectivity directly from the condition of barrier-less conduction.
Selectivity of the canonical K+ lter, formed by a narrow 15A-long tunnel holding
four K+-binding sites, is usually analyzed [1-6] in terms of the dierence in the excess
chemical potential in the bulk i in and the channel 
c
i at one binding site
K;Na = (Na   cNa)  (K   cK): (1)
The question of how to introduce barrier-less conduction within this formalism
elicited a many-voiced debate over decades.
Let nsi be a number ions of the i-th type in the solution and ni their number within
the lter. We introduce the free energy of the solution plus the lter in the form
G =
X
i
(nsi   ni)kT lnxsi +
X
i
(nsi   ni)~i +
X
i
ni~
c
i + kT ln
 Y
i
ni!
!
+ " (fnjg; nf ) :
In treating the lter lled with ions as excited states of the system, we derive the
following grand canonical ensemble for the lter (cf [?])
P (fnjg) = Z 1
mY
i=1
(xsi )
ni
ni!
e
P
i
nii "(fnjg;nf)
kT : (2)
The current with a constant voltage drop  = L   R through the channel in
linear response is proportional to the rate of charge uctuations passing through the
channel [7] Jsteady =
hQ2(t)i
2kTt
.
For discrete conducting ions, charge uctuations are proportional to uctuations
in the number of ions in the channel. The variance 2ni =  kT

@2

@2i

T;V
of the
number of ions in the lter can be found using the grand potential 
 =  kT lnZ.
To apply obtained results to the KcsA lter, let us consider two types of conducting
ions K+ and Na+ that can bind to any binding site in the lter, but have to have at
https://ntrs.nasa.gov/search.jsp?R=20170001628 2019-08-31T01:55:35+00:00Z
least one water molecule between them. The free energy of the lter lled with nK
and nNa ions (ignoring a prefactor) is G =  nK~K   nNa~Na + "(nK ; nNa; nf ):
The energy of interaction between the ions is "(nK ; nNa; nf ) =
q2
2C
(nK + nNa + nf ),
where nf is the number of xed charges on the channel wall, i.e. the main mutation
parameter. The channel capacitance is C  40wR2
L
.
The lowest conducting free energy level corresponds to nK = 2. The conditions for
fast conduction are: (i) G(nK ; n

f ) = G(nK + 1; n

f ) and (ii) G(nK ; n

f )  0kT . To
tune the selectivity and conductivity of the lter one has to tune: (i) geometry, (ii)
ci , and (iii) the nf . Tuning the lter for fast conduction of K
+ ions gives
nf =  (nK + 1=2) +
C
q2
K : (3)
Substituting this value into the free energy gives the following barrier for Na+ ion to
enter the channel GNa = K  Na, i.e. Eisenman's selectivity relation follows
directly from the condition for fast conduction of K+ ions.
Analysis of the mean and variance for K+ and Na+ ions in the lter shows that
tuning it for fast conduction of K+ ions results in eective block of Na+ conduction
if the width of the dispersion peaks is smaller than the separation of the peaks
determined by the geometry of the channel and G.
In conclusion, we have introduced a statistical mechanical model for the distribution
of alike-charged ions in the K+ selectivity lter and have shown that the Eisenmann
relation for lter selectivity follows directly from the condition for fast barrier-less
conduction of K+ ions in the lter. The results can be extended beyond linear
response using a solution of the Nernst-Planck equation. Models of lters with
distinguishable binding sites will be considered in future work.
References
1. Zhou Y. at al, Nature 414, 43 (2001)
2. Eisenman G. , in Symposium on Membrane Transport and Metabolism, edt A.
Kleinzeller and A. Kotyk (Academic Press, NY) p. 163 (1961)
3. Bezanilla F. and Armstrong C. M., The J of Gen Physiology 60, 588 (1972)
4. Noskov S. Y. and Roux B., The J of Gen Physiology 129, 135 (2007)
5. Dixit P. D.,. Merchant S, and Asthagiri D., Biophysical Journal 96, 2138 (2009)
6. Roux B., Biophysical Journal 77, 139 (1999)
7. Liu Y. and Zhu F., Biophysical Journal 104, 368 (2013)


Theory of Alike Selectivity in Biological Channels

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