A pulse of chemoattractant molecules is released a distance from a bacterium

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Physical foundations of the living cell · Physical foundations of the living cell

Forces, Machines and Signals, Ch Diffusion: friction, capture and the crowded cytoplasm

Problem

A pulse of chemoattractant molecules is released a distance L=100 μmL = 100\,\mathrm{\mu m} from a bacterium, in water where the molecules have D=1×10−9 m2 s−1D = 1\times10^{-9}\,\mathrm{m^2\,s^{-1}}. Once the pulse arrives, the local concentration at the bacterium settles to c∞=1 nMc_\infty = 1\,\mathrm{nM}. The bacterium is a sphere of radius a=1 μma = 1\,\mathrm{\mu m} carrying N=1500N = 1500 receptor patches, each of radius s=1 nms = 1\,\mathrm{nm}, distributed over its surface. ((a)) Using the spreading of a point release, estimate the time tt for the pulse to travel the distance LL by diffusion, taking t∼L2/2Dt \sim L^2/2D. ((b)) Once the local concentration reaches c∞c_\infty, find the steady capture rate II (molecules per second) using the receptor patch capture formula I=4πDac∞ Ns/(Ns+πa)I = 4\pi D a c_\infty\,Ns/(Ns+\pi a). ((c)) The counting-limit estimate met in Chapter 1 gives the fractional precision of a concentration measurement from counting NcountN_{\rm count} captured molecules as δc/c∼1/Ncount\delta c/c \sim 1/\sqrt{N_{\rm count}}. Find the averaging time τneeded\tau_{\rm needed} over which the bacterium must integrate captures to measure c∞c_\infty to 10%10\% precision, and compare it with the diffusion delivery time tt found in part (a).

Browse the problem library