392 Questions

takes a time t, and explain why this results in a peaked distribution to the experi­

mentally measured wait time distribution for the ~30°–​40° step.

8.18 The probability P that a virus at a distance d away from the center of a spherical cell of

radius r will touch the cell membrane can be approximated by a simple diffusion-​to-​

capture model that predicts P ≈ r/​d.

a

By assuming a Poisson distribution to the occurrence of this touching event,

show that the mean number of touches of a virus undergoes oscillating between

the membrane and its release point before diffusing away completely is (r/​2d 2)

(2d–​r).

b

If a cell extracted from a tissue sample infected with the virus resembles an

oblate ellipsoid of major axis of 10 μm and minor axis of 5 μm with a measured

mean number of membrane touches of 4.1 ± 3.0 (±standard deviation) of a given

virus measured using single-​particle tracking from single-​molecule fluorescence

imaging, estimate how many cells there are in a tissue sample of volume 1 mL.

c

If the effective Brownian diffusion coefficient of a virus is 7.5 μm² s⁻¹, and the

incubation period (the time between initial viral infection and subsequent release

of fresh viruses from an infected cell) is 1–​3 days, estimate the time taken to

infect all cells in the tissue sample, stating any assumptions that you make.

8.19 For 1D diffusion of a single-​membrane protein complex, the drag force F is related to

the speed v by F =​ γv where γ is frictional drag coefficient, and after a time t, the mean

square displacement is given by 2Dt where D is the diffusion coefficient.

a

Show that if all the kinetic energy of the diffusing complex is dissipated in moving

around the surrounding lipid fluid, then D if given by the Stokes–​Einstein rela­

tion D =​ kBT/​γ.

b

For diffusion of a particular protein complex, two models were considered for the

conformation of protein subunits, either a tightly packed cylinder in the mem­

brane in which all subunits pack together to generate a roughly uniform circular

cross-​section perpendicular to the lipid membrane itself or a cylindrical shell

model having a greater radius for the same number of subunits, in which the

subunits form a ring cross-​section leaving a central pore. Using stepwise photo­

bleaching of YFP-​tagged protein complexes in combination with single-​particle

tracking, the mean values of D were observed to vary as ~1/​N where N was

the estimated number of protein subunits per complex. Show with reasoning

whether this supports a tightly packed or a cylindrical shell model for the com­

plex. (Hint: use the equipartition theorem and assume that the sole source of the

complex’s kinetic energy is thermal.)

8.20 In a photobleaching experiment to measure the protein stoichiometry of a molecular

complex, in which all proteins in the complex were labeled by a fluorescent dye, by

counting the number of photobleach steps present, a preliminary estimate using a

Chung–​Kennedy filter with window width of 15 data points suggested a stoichiom­

etry of four protein molecules per complex, while subsequent analysis of the same

data using a window width of eight data points suggest a stoichiometry of six protein

molecules, while further analyses with window widths in the range three to seven

datapoints suggested a stoichiometry of ~12 molecules while using a window width

of two data points or using no Chung–​Kennedy filter but just the pairwise difference

of single consecutive data points suggested stoichiometries of ~15 and ~19, respect­

ively. What’s going on?

REFERENCES

KEY REFERENCE

Berg, H.C. (1983). Random Walks in Biology. Princeton University Press, Princeton, NJ.