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Permeation through long narrow pores
J. Theoret. BioZ. (1963) 5, 102-107
Permeation Through Long Narrow Pores E. J. A. LEA Department
of Human Anatomy,
University
of Oxford
(Received 20 October 1962 and, in revisedform, 10 February 1963) The passage of molecules through long narrow pores is discussed. An expression is derived relating the ratio between tracer flux and total flux to the number of “sites” or “places” which can accommodate the molecules in the pore. On this basis values are calculated for the number of “sites” in the pore and compared with those calculated using previous treatments of the same effect.
1. Introduction When molecules pass from one side of a membrane to the other through a pore so narrow as to permit only a single file to pass, and on one side a small proportion of molecules is labelled, then the labelled flux is not directly proportional to the total flux but is dependent on the number of molecules or “sites” in the pore. This is known as the long pore effect. This paper will examine previous theoretical treatments of the effect and present a more satisfactory account. 2. Ratio of Unidirectional Fluxes According to Harris (1960), some collisions with one end of a pore will result in a labelled molecule occupying the first “place” in the tie. In the steady state, the probability of advance will be one half per collision, counting collisions on both sides. Thus tracer movement will be c/C x (4)” of the total flux in the steady state, where c/C is the proportion of labelled to unlabelled molecules and n the number of “sites”. This treatment, however, implies that a given molecule may traverse the pore in consecutive forward movements only. In fact, transfer may occur in any one of a finite number of ways, e.g. as a result of collisions on either side of the pore, the molecule may pass from the first place to the second, back to the first, on to the second and so on. The problem is essentially that of a one dimensional random walk with absorbing barriers at 0, n + 1 with initial position 1. Random walk problems are discussed in detail elsewhere, e.g. Feller (1957), Chandrasekhar (1943). 102
PERMEATION
THROUGH
LONG
NARROW
PORES
103
The following is a brief statement of the theory as applied to this problem. It is convenient to begin by considering the movement of a molecule which has reached the xth place in the file. Let U, o be respectively the probabilities of a molecule advancing or retreating one place in the file, and px, (0 < x < n+ 1) the probability of a molecule at x ultimately reaching position IZ+ 1. Since the molecule may reach x by retreating from (x-t 1) with probability V, or by advancing from (x- 1) with probability ZJ PX =
v,+l+uPx-l
(1)
The boundary conditions are: PO = 0, i.e. the probability of a molecule in the absorbing barrier at 0 ultimately reaching the absorbing barrier at n+ 1 is zero. This condition refers only to molecules reaching the barriers from the file. P n+ i= 1. If a molecule is already in the absorbing barrier at n + 1 it is certain to reach it, i.e. probability 1. P z = 1 is a solution (verified by substitution). P, =
: x is a solution (verified by substitution) 0 so more generally x
P, = A-i-B
!f 0 v
(2)
where A, B are arbitrary constants. Applying the boundary conditions, O=A+B
?I+1
l=A+B
!f
0
l=A-A
0
V
n+1
:. :.
ff V
1 u n+1 0-
A =
l-
V x l-
:.
41 0 V
P, = l-
u 0V
II+1
(3)
104
E. J. A. LEA
i.e. of the molecules which reach position 1 from x = 0, a fraction P,, will reach the boundary at n+ 1 and 1 -PI, will return to the boundary at x = 0. Hence flux as measured by tracer total unidirectional flux =
1
u - i n+1
l-
u
0V The case for U, u + 4 may be evaluated from (3) by the use of L’H6pital’s Rule or Taylor’s Theorem giving
Hence
p, = x n+l 1 .*. p,=--n+l flux as measured by tracer 1 total unidirectional flux = -n+l
(5)
TABLE 1 Values for n (water permeability) -_ .m Material
Osmotic permeability? coefficient E lsion permeability coefficient
Beef red cells Dog red cells Human red cells Gall bladder Frog egg
2.98 6.30 240 l-73
Calculated values of n (Harris) l-57 2.65 1 a26 5.78 0.79
l-98 530 140 54 o-73
Source of data
Villegas, Barton & Solomon (1958) Diamond (1962)$ Prescott & Zeuthen (1953)
Osmotic flux t Osmotic permeability coefficient Diffusion permeability c&cient = Flux as measured by tracer’ 1 Diamond calculated a value for n of 56 but the method of his calculation clear.
was not
Table 1 shows values for n, the number of “places” in the Gle calculated from available data using first the theory of Harris (1960) and then equation (5).
PERMEATION
THROUGH
LONG
NARROW
PORES
105
Insofar as the accuracy of the data used permits, the departure of the values of n (col. 4) from integral values could be attributed to a system of pores with different but integral numbers of “sites” if the pores were ideal. It is likely, however, that these non-integral values for n are due, at least in part, to departures from the ideal case, e.g. non-uniformity of pores may occasionally allow two molecules to pass side by side. The other two cases of interest are for fl 4 1 and !! 4 1 u
V
3. Ratio of Fluxes in Opposite Directions A related problem was examined by Hodgkin & Keynes (1955). The result obtained was
where rM2 represents the flux in direction l-2 of balls initially on side 1, similarly for 2M1. a represents the total flux in direction l-2 irrespective of labelling, similarly for j3. The theory of the problem was based on a model consisting of one species of balls on one side of a pore and a different species on the other side, balls passing from one side to the other along the pore. Since the result was to be applied to the study of tracer fluxes, a more realistic model would be one which had a species S of balls common to both sides of the pore, together with a small concentration (X) of labelled balls X on one side and a small concentration (Y) of labelled balls Y on the other side (X, Y representing tracers)
Side1
Side2
106
E.
Flux Jz
J.
A.
LEA
= Flux ,S, x
l--a/S
cx,
1 -(a//I)“+’
from (4) above assuming E = f. v P Flux zYI = Flux J,
x
l~~~;+l
PI
from (4) above, where -! = f since the flux is from side 2 to side 1. 24 P’ Flux ,X, represents the flux, in direction 1-2 of balls X similarly for Flux 2 YI and Flux ,S, represents the flux in direction 1 - 2 of balls S initially at side 1 as opposed to the total flux a in direction 1-2 irrespective of initial position, similarly for Flux &
= cs,lPI+1 (L-a > (Hodgkin & Keynes, 1955), assuming that flux is proportional centration. l-a/B “+l x l-(a/s>“+’
l--P/a -a)“”
to con-
(9)
For a//I 4 1, see equation (6), (9) reduces to Flux iX, Flux zYI
(10)
For j?/a 4 1, see equation (7), (9) reduces to Flux ,X, Flux zYI For a//3 = 1, see equation (5)
(11)
It+1 -[X]
I31 Only for this special case a//3 = 1 does the result (8), (Hodgkin 1955) apply to the ratio of tracer fluxes.
(12)
& Keynes,
PERMEATION
THROUGH
LONG
NARROW
107
PORES
The value of 2.5 estimated for n, the number of “sites” by Hodgkin Keynes is correct, However,
PI1 __
since the estimation was based on the case [s,l
the departure
WI
&
= 1.
of their results from their theory for values of
other than of the order of 1 can be accounted for, at least in part, by
P21
their use of theory correct
only for u
[&I
-
1 (Hodgkin
& Keynes,
1955,
Fig. 7). A better fit is obtained by the use of equations (IO), (1 I), (12). I am indebted to Dr. D. A. T. Dick for helpful discussion and criticism in the preparation of this manuscript. This work was carried out during the tenure of a Wellcome Research Scholarship. REFERENCES CHANDRASRKHAR, S. (1943). Rev. Mod. Phys. 15, 1. DLUOND.J. M. (1962)..r. Phvsiol. 161. 503. FELLER, W. (1957).1; “An VIntroductionto Probability Theory and its Applications”, p. 313.J. Wiley & Sons,Inc., New York. HARRIS, E. J. (1960).“Transport andAccumulationin Biologica Systems”.Butterworths ScientificPublications,London. Honoxo~,A. L. & KEYNES, R. D. (1955).J. Physiol. 128, 61. Pmsaxr, D. M. & ZRUTHEN, E. (1953).Actu physiof. scand. 28, 77. Vnuxus, R., BARTON, T. C. & SOLOMON, A. K. (1958).J. gen. Physiol. 42,355.
Permeation Through Long Narrow Pores E. J. A. LEA Department
of Human Anatomy,
University
of Oxford
(Received 20 October 1962 and, in revisedform, 10 February 1963) The passage of molecules through long narrow pores is discussed. An expression is derived relating the ratio between tracer flux and total flux to the number of “sites” or “places” which can accommodate the molecules in the pore. On this basis values are calculated for the number of “sites” in the pore and compared with those calculated using previous treatments of the same effect.
1. Introduction When molecules pass from one side of a membrane to the other through a pore so narrow as to permit only a single file to pass, and on one side a small proportion of molecules is labelled, then the labelled flux is not directly proportional to the total flux but is dependent on the number of molecules or “sites” in the pore. This is known as the long pore effect. This paper will examine previous theoretical treatments of the effect and present a more satisfactory account. 2. Ratio of Unidirectional Fluxes According to Harris (1960), some collisions with one end of a pore will result in a labelled molecule occupying the first “place” in the tie. In the steady state, the probability of advance will be one half per collision, counting collisions on both sides. Thus tracer movement will be c/C x (4)” of the total flux in the steady state, where c/C is the proportion of labelled to unlabelled molecules and n the number of “sites”. This treatment, however, implies that a given molecule may traverse the pore in consecutive forward movements only. In fact, transfer may occur in any one of a finite number of ways, e.g. as a result of collisions on either side of the pore, the molecule may pass from the first place to the second, back to the first, on to the second and so on. The problem is essentially that of a one dimensional random walk with absorbing barriers at 0, n + 1 with initial position 1. Random walk problems are discussed in detail elsewhere, e.g. Feller (1957), Chandrasekhar (1943). 102
PERMEATION
THROUGH
LONG
NARROW
PORES
103
The following is a brief statement of the theory as applied to this problem. It is convenient to begin by considering the movement of a molecule which has reached the xth place in the file. Let U, o be respectively the probabilities of a molecule advancing or retreating one place in the file, and px, (0 < x < n+ 1) the probability of a molecule at x ultimately reaching position IZ+ 1. Since the molecule may reach x by retreating from (x-t 1) with probability V, or by advancing from (x- 1) with probability ZJ PX =
v,+l+uPx-l
(1)
The boundary conditions are: PO = 0, i.e. the probability of a molecule in the absorbing barrier at 0 ultimately reaching the absorbing barrier at n+ 1 is zero. This condition refers only to molecules reaching the barriers from the file. P n+ i= 1. If a molecule is already in the absorbing barrier at n + 1 it is certain to reach it, i.e. probability 1. P z = 1 is a solution (verified by substitution). P, =
: x is a solution (verified by substitution) 0 so more generally x
P, = A-i-B
!f 0 v
(2)
where A, B are arbitrary constants. Applying the boundary conditions, O=A+B
?I+1
l=A+B
!f
0
l=A-A
0
V
n+1
:. :.
ff V
1 u n+1 0-
A =
l-
V x l-
:.
41 0 V
P, = l-
u 0V
II+1
(3)
104
E. J. A. LEA
i.e. of the molecules which reach position 1 from x = 0, a fraction P,, will reach the boundary at n+ 1 and 1 -PI, will return to the boundary at x = 0. Hence flux as measured by tracer total unidirectional flux =
1
u - i n+1
l-
u
0V The case for U, u + 4 may be evaluated from (3) by the use of L’H6pital’s Rule or Taylor’s Theorem giving
Hence
p, = x n+l 1 .*. p,=--n+l flux as measured by tracer 1 total unidirectional flux = -n+l
(5)
TABLE 1 Values for n (water permeability) -_ .m Material
Osmotic permeability? coefficient E lsion permeability coefficient
Beef red cells Dog red cells Human red cells Gall bladder Frog egg
2.98 6.30 240 l-73
Calculated values of n (Harris) l-57 2.65 1 a26 5.78 0.79
l-98 530 140 54 o-73
Source of data
Villegas, Barton & Solomon (1958) Diamond (1962)$ Prescott & Zeuthen (1953)
Osmotic flux t Osmotic permeability coefficient Diffusion permeability c&cient = Flux as measured by tracer’ 1 Diamond calculated a value for n of 56 but the method of his calculation clear.
was not
Table 1 shows values for n, the number of “places” in the Gle calculated from available data using first the theory of Harris (1960) and then equation (5).
PERMEATION
THROUGH
LONG
NARROW
PORES
105
Insofar as the accuracy of the data used permits, the departure of the values of n (col. 4) from integral values could be attributed to a system of pores with different but integral numbers of “sites” if the pores were ideal. It is likely, however, that these non-integral values for n are due, at least in part, to departures from the ideal case, e.g. non-uniformity of pores may occasionally allow two molecules to pass side by side. The other two cases of interest are for fl 4 1 and !! 4 1 u
V
3. Ratio of Fluxes in Opposite Directions A related problem was examined by Hodgkin & Keynes (1955). The result obtained was
where rM2 represents the flux in direction l-2 of balls initially on side 1, similarly for 2M1. a represents the total flux in direction l-2 irrespective of labelling, similarly for j3. The theory of the problem was based on a model consisting of one species of balls on one side of a pore and a different species on the other side, balls passing from one side to the other along the pore. Since the result was to be applied to the study of tracer fluxes, a more realistic model would be one which had a species S of balls common to both sides of the pore, together with a small concentration (X) of labelled balls X on one side and a small concentration (Y) of labelled balls Y on the other side (X, Y representing tracers)
Side1
Side2
106
E.
Flux Jz
J.
A.
LEA
= Flux ,S, x
l--a/S
cx,
1 -(a//I)“+’
from (4) above assuming E = f. v P Flux zYI = Flux J,
x
l~~~;+l
PI
from (4) above, where -! = f since the flux is from side 2 to side 1. 24 P’ Flux ,X, represents the flux, in direction 1-2 of balls X similarly for Flux 2 YI and Flux ,S, represents the flux in direction 1 - 2 of balls S initially at side 1 as opposed to the total flux a in direction 1-2 irrespective of initial position, similarly for Flux &
= cs,lPI+1 (L-a > (Hodgkin & Keynes, 1955), assuming that flux is proportional centration. l-a/B “+l x l-(a/s>“+’
l--P/a -a)“”
to con-
(9)
For a//I 4 1, see equation (6), (9) reduces to Flux iX, Flux zYI
(10)
For j?/a 4 1, see equation (7), (9) reduces to Flux ,X, Flux zYI For a//3 = 1, see equation (5)
(11)
It+1 -[X]
I31 Only for this special case a//3 = 1 does the result (8), (Hodgkin 1955) apply to the ratio of tracer fluxes.
(12)
& Keynes,
PERMEATION
THROUGH
LONG
NARROW
107
PORES
The value of 2.5 estimated for n, the number of “sites” by Hodgkin Keynes is correct, However,
PI1 __
since the estimation was based on the case [s,l
the departure
WI
&
= 1.
of their results from their theory for values of
other than of the order of 1 can be accounted for, at least in part, by
P21
their use of theory correct
only for u
[&I
-
1 (Hodgkin
& Keynes,
1955,
Fig. 7). A better fit is obtained by the use of equations (IO), (1 I), (12). I am indebted to Dr. D. A. T. Dick for helpful discussion and criticism in the preparation of this manuscript. This work was carried out during the tenure of a Wellcome Research Scholarship. REFERENCES CHANDRASRKHAR, S. (1943). Rev. Mod. Phys. 15, 1. DLUOND.J. M. (1962)..r. Phvsiol. 161. 503. FELLER, W. (1957).1; “An VIntroductionto Probability Theory and its Applications”, p. 313.J. Wiley & Sons,Inc., New York. HARRIS, E. J. (1960).“Transport andAccumulationin Biologica Systems”.Butterworths ScientificPublications,London. Honoxo~,A. L. & KEYNES, R. D. (1955).J. Physiol. 128, 61. Pmsaxr, D. M. & ZRUTHEN, E. (1953).Actu physiof. scand. 28, 77. Vnuxus, R., BARTON, T. C. & SOLOMON, A. K. (1958).J. gen. Physiol. 42,355.