HST.410J/6.021J Lecture 6February 27, 2007• number of solute particles << number of solvent particles• motion of solute determined by collisions with solvent (ignore solute-solute interactions)• focus on 1 solute particle, assume motions of others are statistically identicalEvery τ seconds, solute particle gets hit by solvent particle.In response, solute particle is equally likely to move +l or −l.τ = mean free time; l = mean free pathx0 l 2l 3lRandom Walk Model00.20.40.60.8100.20.40.60.81-7-5-3-1135701234567W(m,n)mnFigure from Weiss, T. F. Cellular Biophysics, Vol. I. Cambridge, MA: MIT Press, 1996.Courtesy of MIT Press. Used with permission.Harvard-MIT Division of Health Sciences and TechnologyHST.410J: Projects in Microscale Engineering for the Life Sciences, Spring 2007Course Directors: Prof. Dennis Freeman, Prof. Martha Gray, and Prof. Alexander AranyosiFigure from Weiss, T. F. Cellular Biophysics, Vol. I. Cambridge, MA: MIT Press, 1996.Courtesy of MIT Press. Used with permission.–3 –2 –1 0 321 x 50% 68% 95% 99%–3 –2 –1 0 321 x 50% 68% 95% 99%Apply Fick's law to dye demonstration initially dye no dye c(x,t=0) Find c(x,t) for t>0 φ(x,t=0) x Fick's law:� � ∂ c(x,t)φ(x,t) = − D ∂ x • provides information about time "t" only� • need new information to get from time "t" to time "t+Δt" c(x,t) If there is net flux out of a region,� then the concentration in that region must fall.� If there is net flux into a region,� φ(x,t) then the concentration in that region must rise. → conservation of solute x_ _ _ _ Continuity Equation c(x+Δx,t) 2 x x+Δx volume AΔx area A area A φ(x,t) φ(x+Δx,t) x x+Δx Amount of solute entering� Change in amount of solute� through edges during (t,t+Δt) in volume from t to t+Δt left right time 2 time 1 φ(x,t+Δ2t) A Δt − φ(x+Δx,t+Δ2t) A Δt = c(x+Δ2x,t+Δt) A Δx − c(x+Δ2x,t) A Δx equal if solute is neither created nor destroyed φ(x,t+Δ_ 2t) − φ(x+Δx,t+Δ_ 2t) c(x+Δ_ 2x,t+Δt) − c(x+Δ_ 2x,t) = Δx Δt Take limit as Δx→0 and Δt→0 ∂ φ(x,t) ∂ c(x,t)− = ∂ x ∂ tImportance of Scale 2 = x1/2 ; D = 10−5 cm2 for small solutes (e.g., Na+)t1/2 D s t1/2 x1/2 1� membrane sized� 10 nm� µsec� 10 cell sized� 10 µm� 1� dime sized 10 mm 105 sec ≈ 1 day sec�