Chapter 3 – Making Light Work in Biology  105

The lateral width of the Slimfield excitation field at the sample is given by Equation

3.56 of wc =​ f λ/​πws. Here, ws =​ 0.1 mm, f =​ 2 mm, but now λ refers instead to the

excitation wavelength, so:

wc =​ (2 × 10^–3) × (488 × 10^–​9)/​(π x 0.1 × 10^–​3) =​ 3.1 × 10^–​6, or ~3 µm. Therefore, since

this width is < 10 µm, we can assume the cell membrane is flat over this Slimfield

excitation field and thus diffusion within its membrane in that region is in the

2D focal plane of the microscope. To calculate the diffusion coefficient D in the

membrane we use the Stokes–​Einstein relation of Equation 2.11 D = ​kBT/​γ where

kB is the Boltzmann constant (1.38 × 10^–​23 J/​k), T the absolute temperature (~300 K

for room temperature), and γ is the frictional drag coefficient (such that the drag

force =​ speed multiplied by this frictional drag coefficient), thus:

D =​ (1.38 × 10^–​23 × 300)/​(1.1 × 10^–​7) =​ 3.7 × 10^–​14 m²/​s

The maximum sampling time Δt to avoid blurring of the PSF image for GFP (see

Equation 2.12) occurs roughly when the mean squared displacement of the

diffusing integrated membrane protein is the same as the PSF width w for the

GFP molecule, so w² = ​2DnΔt where the spatial dimensionality n here is 2. So:

Δt =​ (212 × 10^–​9)²/​(4 × 3.7 × 10^–​14) =​ 0.3 s or 300 ms

The frictional drag coefficient scales linearly with viscosity, so the equivalent Δt

in the cytoplasm will be lower by a factor of ~100cp/​1cp, or 100, so the equiva­

lent value of D in the cytoplasm will be higher by a factor of 100 compared to

the membrane. However, the cytoplasmic diffusion is in 3D not 2D, so n =​ 3, thus

the equivalent Δt will be smaller by a factor of 100 × 3/​2 or 150, giving Δt =​ 2 ms.

b

To diffuse from the center of a lipid raft to the edge is a distance d of 100 nm. If

we equate this to the root mean squared distance for diffusion in 2D this implies

d² = ​2DnΔttot so the total time taken for this to occur is:

Δttot =​ (100 × 10^–​9)²/​(2 × 3.7 × 10^–​14 × 2) = ​6.8 × 10^–​5 s, or 680 ms. So, a single GFP

molecule is very likely to have bleached before it reaches the raft edge as this is

more than 20-​fold larger than the typical 30 ms photobleaching time indicated.

This illustrates one issue with using Slimfield, or indeed any high intensity fluores­

cence microscopy, in that the high laser excitation intensities required mean that

the total time before photobleaching occurs is sometimes too short to enable

monitoring of longer duration biological processes. One way to address this

issue here could be to use the longer sampling time of 300 ms we estimated for

the membrane diffusion, since to retain the same brightness of a GFP molecule

might then require 100-​fold less excitation intensity of the laser, assuming the

fluorescence emission output scales linearly with excitation intensity, and so the

GFP might photobleach after more like ~100 × 0.3 s =​ 30 s. But then we would

struggle to be able to observe the initial more rapid 3D diffusion unblurred in the

cytoplasm with this much longer sampling time. Another alternative approach

could be to use the rapid 2 ms sampling time throughout, but when the protein

is integrated into the membrane to stroboscopically illuminate it, so to space out

the 30 ms/​2 ms or ~15 image frames of fluorescence data that we have before

photobleaching occurs over the ~680 ms required for the diffusion process to the

edge of the raft. But one issue here would be synchronization of the software in

real time to the strobing control for the laser excitation so that the strobing starts

automatically only when the protein integrates into the membrane. Technically

non-​trivial to achieve, as they say, but it illustrates that measuring biological

processes across multiple time and length scales does still present demanding

instrumentation and analysis challenges for biophysics!