# Rule of thumb for time it takes a protein to diffuse across the cell

Value 0.01 sec Bacteria Escherichia coli "Physical Biology of the Cell", Rob Phillips, Jane Kondev and Julie Theriot (2009). Page 110 Taking X, distance to be traveled, as 1µm (BNID 100 002) and D, diffusion coefficient in cytoplasm, as 10µm^2/sec (D in cytoplasm is in the range of 5-15µm^2/sec, GFP-BNID 100 193, 40kda dextran, similar weight to protein, BNID 100198). According to equation of diffusion (in 3 dimensions): t diffusion=X^2/(6×D). 1µm^2/60µm^2/sec=0.017sec≈0.01sec. For time it takes a protein to diffuse across a HeLa cell see BNID 105 339. In cytoplasm there are solutes and D is smaller than in water (100µm^2/sec). D in water can be calculated from the Einstein-Stokes eq. D=KBT/(6×π×Ƞ×R) where R=2.5nm, typical protein radius. KB=Boltzmann's constant, Ƞ=viscosity, 0.001 Pa×sec for water. (1.38×10^-23Kg×m^2×sec^-2×K^-1×300K)/ (6×3.14×0.001Kg×m^-1×sec^-1×2.5×10^-9m)=8.8×10^-11m^2/sec= 88µm^2/sec≈100µm^2/sec Please see "Physical biology of the cell" 2nd edition 2013 p.128 3rd paragraph: "Estimate: moving proteins from here to there-For molecules and assemblies that move passively within the cells, the time scale can be estimated using equation 3.18 (t(diffusion)≈X^2/D, where D is the diffusion constant). For a protein with a 5nm diameter, the diffusion constant in water is roughly 100µm^2/s - this estimate can be obtained from the Stokes-Einstein equation (to be discussed in more detail in Chapter 13 in equation 13.62 on p.531), which gives the diffusion constant of a sphere of radius R moving through a fluid of viscosity Ƞ at temperature T as D=KB/(6πȠR), The timescale for such a typical protein to diffuse a distance of [investigators’] standard ruler (that is, across an E. coli) is: t(E. coli)≈[L(E. coli)]^2/D≈1µm^2/100µm^2/sec≈0.01sec. (eq.3.19)." Uri M 103801