Content uploaded by John R Clarke
Author content
All content in this area was uploaded by John R Clarke on Jun 02, 2019
Content may be subject to copyright.
J. theor. Biol. (1977) 68, 293-298
Fluidity-induced Changes in Diffusion through Membranes:
a Predictive Model?
JOHN R. CLARKE~
Department of Biology, Florida State University,
Tallahassee, Florida 32306, U.S.A.
(Received 4 August 1976, and in revisedform 24 February 1977)
Diffusion of uncharged solutes through a neutral monolayer is limited by
the probability of sufficiently sized vacancies forming randomly between
membrane barriers. If the one-dimensional distribution of barriers is
described by the Gaussian, a change in membrane fluidity can be repre-
sented by a change in variance of the distribution. By the reproductive
property of Gaussian distributions, the vacancy size would also be altered.
The effect of this change on the probability of solute passage is dependent
on the ratio between solute size (V,) and mean vacancy size r. An increase
in fluidity may hamper diffusion of a solute if V, < r, whereas a fluidity
decrease would augment diffusion. For V, > 7, a fluidity change would
have the opposite effect. If V, <, =, or > r, no effect would obtain. There
are, then, critical values for Vi/r The influence of temperature on con-
duction velocity in squid axons illustrates how this model could affect
physiological systems.
1. The Model
Although the mobility of membrane components, and the function of mem-
brane bound transport proteins are becoming increasingly understood
(Kornberg & McConnel, 1971; Feinstein, Fernandez & Sha’afi, 1975), the
influence of membrane fluidity on diffusion remains largely unexplained.
Here, I suggest a theoretical effect of membrane fluidity on the diffusion of
various sized solutes.
For the purposes of this discussion, the only limitations to the movement
of solutes through a membrane will be the presence of barriers, presumably
molecules or portions of molecules within the membrane. An exact definition
of these barriers will not be attempted, other than to assume they exhibit
brownian motion. Of importance, however, is the size of the space or
t This is contribution No. 68 from the Tallahassee, Sopchoppy and Gulf Coast Marine
Biological Association.
$ Present address: Department of Physiology,
School of Medicine, Case Western
Reserve University, Cleveland, Ohio 44106, U.S.A.
293
294
J. R. CLARKE
vacancy between barriers, which may be transiently occupied by a solute.
The mean size of the vacancies, relative to the solute size, would then reflect
the probability of a solute passing through a membrane (Stein, 1967).
Vacancy size is modulated by the statistical movements of the membrane
barriers whose random displacements may be described by the Gaussian
distribution: dP 1
_ __ ,-3C(x-X)lal*
~X-J211a (1)
P is the probability of finding the barrier boundary x units from its mean
position X. In this case, a2 would define the freedom of movement of the
barrier. If ~7’ is very small, then the barrier is closely restricted to the vicinity
of its mean. If o2 were large, then the barrier would move more freely, e.g.
the membrane would exhibit some degree of fluidity.
If the distance between two such barriers defines a vacancy, the vacancy
also will exhibit Gaussian characteristics. The reproductive property of
normal distributions states that the mean vacancy size, Iz, will simply be the
mean distance between any two barriers, with the variance from V, cv2,
equal to the sum of the variances of the individual barriers.
To determine the probability, P, that a vacancy will accommodate a
solute of size Vi, the equation for the Gaussian must be integrated with
respect to V through the interval Vi to infinity;
-+[(~-~)/Q’l* ,jV.
P=iJ&-,e (2)
Solutions for P may be found in tables of areas under the standard normal
curve.
Through the following example one sees that a change in membrane
fluidity can alter the distribution of vacancy sizes within that membrane.
Part I, in Fig. 1, demonstrates three spherical membrane barriers which are,
on the average, two units apart, vibrating with a variance of 0.25 units from
their mean positions. Statistics for the vacancy size would thus be:
v = 2, +1 = 0.71. A and B represent solutes whose diameters are larger and
smaller, respectively, than v. Obviously, membrane I will more easily accom-
modate solute B than A. Membrane II is similar to I except the vibrating
molecules in this membrane have a variance of one rather than one quarter,
(V= 2, cTYII = 1*41), e.g. they are more fluid. Again the smaller solute is
more easily contained.
Figure 2 is a plot of the Gaussians for the above illustration. The figure
further demonstrates how these distributions affect the ability of a solute to
pass through the membranes. Let us suppose that solute A is 2f units in
diameter, and B is only 14 units across. For A to pass between two vibrating
FLUIDITY INDUCED CHANGES IN DIFFUSION
295
-_------ -bin-- - - - - - - - - - -
FIG. 1. Model for diffusion through membranes of differing fluidities. “A” and “B”
represent solutes with the option of diffusing through either of two membranes, I and II.
The density of the shaded areas within the membranes indicate the frequency of displace
ment of the six membrane constitutents from their mean positions, indicating that the
mean vacancy sizes between the membrane constituents are identical, p, = 6. In spite of
this, membrane II is more fluid than I. Solute “a”, which has a diameter greater than the
mean vacancy size, would therefore diffuse through membrane II faster than through I.
Solute “b”, being smaller than the mean, would diffuse preferentially through membrane I.
Vacancy size, V
FIG. 2. The frequency distribution of vacancy sizes in membranes of differing fluidities.
p is the mean vacancy size between two membrane constituents. For solute “a” to cross a
membrane, a vacancy 2 V,, must exist.
The
dotted area to the right = PuI,,j-PuA)
For solute “b” to pass, a vacancy must not be less than V,, (V 2 V,). Dotted area to
left = (1 --P~II,J-U -PC,,,) = PcI,,--PtrIB,I.
7.8.
10
296
J. R. CLARKE
molecules, the instantaneous vacancy size, V, must be greater than 2+ units.
Since the area under this frequency distribution is equal to the probability of
finding a vacancy greater than some given size, (2), we find a higher prob-
ability for A passing through the more fluid membrane-the membrane
described by II in Fig. 1. But, just as it is more probable for the vacancy size
to be greater than 23, it is also more likely for it to be less than I$. That
means the more fluid state (II) will, in comparison to membrane I, favor the
passage of A, and discourage the transmission of B.
By converting to the standard normal form, it is found from the tables
that Per,), the probability that a vacancy 2 23 units will exist at any one
moment in membrane I, is 0.24. For solute B to pass through the membrane,
the vacancy size need only be 1) units across. Accordingly, Po,) = 0.76.
Membrane II, on the other hand, yields a probability for movement of
solute A of O-36. Likewise, PC,,,) is 0.64. It is evident that the more fluid
TABLE 1
The e@ect of solute size (Vi) and
a2
on Pi
Membrane I (leas fluid)
Membrane II (more fluid)
AP
0.25 0.50 0.24 0.76
l*oo 2m 0.36 O-64
- - +0*12 -0.12
ama, Variance (“fluidity”) of the membrane; rr r2, variance of vacancy size as deter-
mined by ama; PA, Probability that a solute A (P = 2, V, = 2*) will encounter an acceptable
vacancy at any one instant; PB, As above for solute B (P = 2, V, = 14); AP, the change
in probability of acceptance of solute A or B in making a fluidity transition from that
found in membrane I to that in II.
membrane II allows greater freedom of motion to the larger solute A than
does membrane I (Pot,) = 0.36 as opposed to PC,*) = 0.24). However
membrane II is more restrictive to the smaller solute (PC,,,, = 0.64 vs.
P
oeJ = O-76). For this particular example, the given increase in fluidity
from membrane I to membrane II results in a 50% increase in freedom for
solute A, and 16% decrease for solute B.
The differential effect of the fluidity increment on solutes of various sizes
depends directly upon the percentage change of fluidity, and the difference
between the mean vacancy size and the vacancy size required for solute
passage, (Vi- V). The changes in P for the model membranes is attributable
to
a
in equation (2). As (Vi- 7) goes to zero, the integral in equation (5)
loses its sensitivity to
a.
In other words, if the required vacancy size is equal
to the mean vacancy size, there will always be a 50% chance of having a
FLUIDITY INDUCED CHANGES IN DIFFUSION
297
vacancy large enough for solute passage, regardless of the fluidity. Likewise,
if V, approaches zero, (the solute becomes very small), then the solute is able
to penetrate the membrane with ease, once again regardless of the fluidity.
Conversely, if Vi becomes very large, then the membrane will be at all times
impermeable.
There are, then, optimum values of Vi for the expression of this fluidity
effect; one for solutes smaller than the mean vacancy size, and another for
solutes greater in diameter than 7. The value of V that will yield a maximum
sensitivity to the fluidity effect, for a given change in fluidity, can be calculated
as follows.
The difference in P of membranes of differing fluidities is:
AP = Ii Mu, +rr) -.W, +,)I dv,
(3)
wheref(V, by,,) is the Gaussian function for vacancy sizes within membrane
II, distinguished by a variance of by,,. By maximizing the derivative of this
equation, we find :
For membrane I and II in Fig. 1, with variances of O-25 and 1.0, respectively,
the standard deviation (by) of the V distributions are 0.707 and 1.414. For
an increment in fluidity from that found in I to that in II, V, critical
= 2kO.97. For a change in variance from 0.5 to 2, the value for V most
influenced by the fluidity change is Vi critical = Vf 1.36.
For clarity, the above treatment has neglected the rate of vacancy forma-
tion, which would ordinarily increase with augmented membrane fluidity.
As long as solutes are free to fill forming vacancies, an increase in formation
rate would accelerate net flux. This acceleration would apply to all solutes,
however, and must be superimposed upon the selective nature of vacancy
size changes. This fundamental distinction between the non-specific effect of
vacancy formation rate, and the relative effect of vacancy size must be kept in
mind during both the theoretical and experimental development of this
problem.
A search for this phenomenon in artificial lipid membranes is suggested.
However, the work of some authors already lend themselves to interpretation
by this model. Easton & Swenberg (1975) have recently shown that the
propagation velocity in squid axon, dependent upon permeability of the
axon membrane, rises with a moderate increase in temperature. However, the
rise in animals adapted to cold (about 5 “C) is slower than that in animals
adapted to 25 “C. Since the membrane fluidity of cold acclimated squid
should be about that of warm acclimated animals at their respective environ-
298
J. R. CLARKE
mental temperature (Kerkut & Taylor, 1958), it is to be expected that at
room temperature the former membranes would be more fluid than the
latter. Therefore, according to the hypothesis presented in this paper, the
more fluid membrane would oppose the velocity increase instigated by a
temperature increase upon an ion transport system, if one assumes the ions
to be small compared to the separation of membrane components (Vi< 7).
The fluidizing effect of a temperature increase would actually hinder the
repolarizing flow of ions to the nerve exterior. For larger diffusants one
would expect an acceleration of diffusion with a temperature rise. This is a
well documented occurrence often attributed to decreases in lipid viscosity,
but which might be further affected by the fluidity change which is wholly
unrelated to viscosity.
2. Conclusions
Without careful controls, this phenomenon may be masked in real mem-
branes. Its actions should first be tested with solute permeation through
mono or bilayers, with the solute and membrane constituents of carefully
chosen size. A maximum effect is found when the solute size, and thus
required vacancy size, V,, is approximately &50x of P. Eyring & Jhon
(1969) suggest that vacancies in a liquid are of approximately molecular size.
In a liquid-crystalline membrane, however, it is more difficult to ascertain a
reasonable I? It could easily be on the order of a few angstroms, much
smaller than membrane phospholipids, which would mean that only ions,
e.g. Na+ and K+, may fall within the critical size range for Vi.
The oriented, linear arrangement of phospholipids, and the variable shape
of integral membrane proteins comprise a much more complicated “barrier”
to solute permeation than is alluded to here. However, any movement
through a membrane may be separated into a number of steps, each with a
probability of completion influenced by the statistical nature of membrane
vacancies. Thus, the effect of fluidity changes may be amplified. The dif-
ferential control of solute permeation by slight fluidity alterations may,
therefore, have a profound regulatory effect upon phyiological processes.
This research was supported in part by USPHS HL 09283.
REFERENCES
EASTON, D. M. & SWFNBERG, C. E. (1975). Am. J. Physiol. 229,1249.
EYRINO, H. & JHON, M. S. (1969). Significant Liquid Structures. New York: John Wiley &
sons.
FENSTEIN, M. D., FERNANDEZ, S. M. & SHA’AFI, R. I. (1975). Biochim. biophys. Acta
413,354.
KWCUT, A. G. &TAYLOR, K. J. B. (1958). Behavior
13,259.
KORNBERG, R. K. & MCCONNEL, H. M. (1971). Biochemistry
10, 1111.
STEIN, W. D. (1967). The Movement of Molecules Across Cell Membranes. New York:
Academic Press.
