Molecular motion in cell membranes: Analytic study of fence-hindered random walks

V. M. Kenkre, L. Giuggioli, and Z. Kalay

Phys. Rev. E 77, 051907 – Published 13 May 2008

Abstract

A theoretical calculation is presented to describe the confined motion of transmembrane molecules in cell membranes. The study is analytic, based on Master equations for the probability of the molecules moving as random walkers, and leads to explicit usable solutions including expressions for the molecular mean square displacement and effective diffusion constants. One outcome is a detailed understanding of the dependence of the time variation of the mean square displacement on the initial placement of the molecule within the confined region. How to use the calculations is illustrated by extracting (confinement) compartment sizes from experimentally reported published observations from single particle tracking experiments on the diffusion of gold-tagged $G$ -protein coupled $μ$ -opioid receptors in the normal rat kidney cell membrane, and by further comparing the analytical results to observations on the diffusion of phospholipids, also in normal rat kidney cells.

Received 5 February 2008

DOI:https://doi.org/10.1103/PhysRevE.77.051907

Authors & Affiliations

V. M. Kenkre, L. Giuggioli, and Z. Kalay

Consortium of the Americas for Interdisciplinary Science and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA

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Vol. 77, Iss. 5 — May 2008

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Images

Figure 1
Illustrative comparison of our analytic result for the mean square displacement with Monte Carlo simulations, showing excellent agreement. Parameters are $f = 0.01$ , $F = 2$ $H = 4$ . Solid curves show the result of the simulations, averaged over $20 000$ runs. Dashed lines are the analytical results. They are almost indistinguishable from one another. The inset shows the behavior at short times.Reuse & Permissions
Figure 2
Time dependence of the mean square displacement and the instantaneous diffusion coefficient for various magnitudes of the confinement effect. Plotted as a function of time is the normalized $y (t)$ (left) and $D (t) ∕ D$ (right), averaged over all initial locations in the compartment, for different values of $D_{e f f} ∕ D$ : 0.001 (solid line), 0.01 (dashed line), 0.1 (dashed-dotted line), 0.5 (dotted line). All quantities considered are made dimensionless appropriately.Reuse & Permissions
Figure 3
Comparison of diffusion observations of phospholipids reported in Ref. 28 to our theoretical predictions based on three disparate values of the compartment size $L$ used in our Eq. (21). One of the values used is that deduced in Ref. 28, $230 nm$ . There is very good agreement of theory (solid line) and observations as shown. Each of the other two values differs by an order of magnitude: $2300 nm$ (dashed line) and $23 nm$ (dotted line). In both these cases theory shows poor agreement with experiment. We have used $D = 4.6 μ m^{2} s^{- 1}$ and $D_{e f f} = 1.2 μ m^{2} s^{- 1}$ .Reuse & Permissions
Figure 4
Theoretical predictions for various initial placements of the molecule compared to observations reported in Ref. 28. Circles and triangles denote observations. Lines represent our theory. Dashed line shows the average of the mean square displacement over all initial positions. For the solid lines, the initial position is $x_{0} = 0$ , $L ∕ 8$ , $L ∕ 4$ , $3 L ∕ 8$ , and $7 L ∕ 16$ , going upwards from the lowest curve in the plot. Values of $D$ and $D_{e f f}$ that correspond to the observations are as in Fig. 3 and the deduced compartment size is $L = 230 nm$ .Reuse & Permissions
Figure 5
Effect of initial placement of the random walker at site $p$ within a compartment on the time dependence of the effective transfer rate which, in the continuum limit, would be proportional to the time-dependent effective diffusion coefficient. Each compartment has 11 sites, i.e., $H = 10$ , and $f ∕ F = 0.01$ . Averaging equally over all initially localized placements results, as shown by the dashed-dotted line, in a monotonically decreasing transfer rate. However, interesting structures appear for initial placements at the center of , end of, and elsewhere in the compartment as shown. See text for explanation.Reuse & Permissions

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