= Lateral mobility of membrane proteins
{title2=$\langle r^2\rangle=4Dt$}
<Lateral mobility of membrane proteins> is movement along a membrane surface. It differs from <membrane transport> through the boundary and from reversing <membrane topology>. Attachment, crowding and barriers can restrict <diffusion>; the <fluid mosaic model> does not require every <membrane protein> to wander freely.<The Cell: A Molecular Approach, 2nd edition by Geoffrey M. Cooper/Structure of the Plasma Membrane>[Cooper, The Cell: A Molecular Approach, 2nd edition, Structure of the Plasma Membrane, Mobility of Membrane Proteins: lateral movement and restrictions]{ref}
An original local model treats a small, approximately flat patch as an unbounded plane with uniform <diffusion coefficient> $D$, no drift and no trapping. Represent the two displacement coordinates by independent <Brownian motions>, each with zero <expected value> and <variance> $2Dt$. Since $r^2=x^2+y^2$, addition of the two coordinate expectations gives
$$
\langle r^2\rangle=4Dt,\qquad r_{\mathrm{rms}}=\sqrt{4Dt}.
$$
This is a root-mean-square displacement, not distance traveled along a jagged trajectory or the displacement of every <protein>. For a stipulated $D=0.08\,\mu\mathrm{m}^2\mathrm{s}^{-1}$ and $t=5\,\mathrm{s}$, $\langle r^2\rangle=1.6\,\mu\mathrm{m}^2$ and $r_{\mathrm{rms}}\approx1.265\,\mu\mathrm{m}$. Reaching a stipulated root-mean-square scale $L=2\,\mu\mathrm{m}$ gives $t=L^2/(4D)=12.5\,\mathrm{s}$. These model values are not measured <diffusion coefficients> or mixing times from the historical experiment.
Multiplying the target displacement scale by two multiplies the model time by four. This quadratic scaling is a useful way to see why local encounters and redistribution over an entire <biological cell> need not happen on the same timescale. A curved, bounded <cell membrane>, restricted paths and different mobile populations require more than this flat-patch model; disappearance of an initial pattern alone does not determine $D$.
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