APh161: Physical Biology of the CellHomework 6Due Date: Tuesday, March 1, 2005“To do successful research, you don’t need to know everything. You justneed to know of one thing that isn’t known.” - Arthur Schawlow, quoted in”Lasers, Spectroscopy and New Ideas” edited by Yen and LevensonReading: Chaps. 10 and 12 of Howard.1. Diffusion to Capture: The Hard WayIn class I gave an intuitive derivation of the problem of diffusion to capturewithout ever solving the diffusion equation. In this problem, I want you towork out the features of diffusion to capture with a perfect absorber usingthe full machinery of the diffusion equation.(a) Recall that we wish to solve for the steady-state condition in which weprescribe a far-field concentration c0and assume that the absorber (a sphereof radius a) is a perfect absorber (c(a) = 0). Write the diffusion equation inspherical coordinates at steady-state (i.e. ∂c/∂t = 0).(b) Show that the resulting concentration profile is of the formc(r) = A +Br, (1)and use the conditions c(a) = 0 and c(∞) = c0to determine the constantsA and B.(c) Compute the flux at the surface of the sphere and then use this to eval-uate dn/dt and confirm the expression for the diffusive speed limit that Idiscussed in class.(d) Recall Prof. Bob Austin’s (Princeton Physics) quip that ”physics isn’tworth a damn unless you put in some numbers”. Let’s put in some numbers1and actually evaluate the diffusive speed limit for several cases of interest.In particular, let’s work out the rate for actin monomers to be incorporatedonto a growing actin filament and for oxygen arriving at hemoglobin. Thatis, make an estimate of the size and diffusion constant for G-actin and O2and compare the rates that you find with the konfor the actin polymeriza-tion reaction and for the uptake of oxygen by hemoglobin. Of course, youwill have to make some assumptions about c0- try the critical concentrationfor actin and for oxygen, maybe you can find some reasonable numbers onthe web. The discussion on pgs. 308 and 309 of Howard give an interestingdiscussion of the diffusion-limited speed limit. Note that we are making asimplifying assumption by treating the growing filament and the hemoglobinas stationary.2. Microtubule Dynamics.There have been a number of models that have been set forth to examinethe intriguing character of cytoskeletal dynamics. In this problem, I willwalk you through one phenomenological model for steady-state microtubuledynamics that was introduced by Dogterom and Leibler in Phys. Rev. Lett.70, 1347 (1993) - the paper is on the course website. Note that there is amore interesting class of models that include GTP hydrolysis explicitly (seePhys. Rev. E54, 5538 (1996).) As an example of the type of data we aretrying to come to terms with, see fig. 1 and fig. 6 of Fygenson et al., Phys.Rev. E50, 1579 (1994). Fig. 1 shows a record of the length of a singlemicrotubule as a function of time and reveals the series of ”catastrophes”and ”rescues” as the polymer changes its length.(a) Deduce eqns. 1 and 2 of the Dogterom paper - in particular, note thatthey are thinking of a probability distribution p+(n, t) and p−(n, t) whichgives the probability of finding a microtubule of length n that is growing (+)or shrinking (−). Write a master equation like we did in class for p+(n, t)and p−(n, t) by noting that there are 4 things that can happen to changethe probability at each instant. Consider the + case - (i) the n − 1 polymercan grow and become an n polymer - characterized by a rate v+, (ii) the npolymer can grow and become and n + 1 polymer - also characterized by arate v+, (iii) the n+ polymer can switch from growing to shrinking with arate f+−and (iv) the n− polymer can switch from shrinking to growing with2a rate f−+. What I am arguing is that if you account for all four of thesepossibilities you will have the correct master equation. Use a Taylor expan-sion on stuff like p+(n − 1, t) − p+(n, t) to obtain the equations as written inthe Dogterom paper.(b) Solve these equations in the steady state (i.e. ∂p±(n, t)/∂t = 0) and showthat the relevant parameter isσ =v+f−+− v−f+−v+v−. (2)Explain what all of this means. When I say ”solve”, what I mean is find allof the p±(n). What you have done is to find the distribution of lengths.(c) Use fig. 1 from the Fygenson paper above to estimate the relevant pa-rameters v+, v−, f+−, f−+, and then find the average length of the polymerswhich are predicted by this simple model. To find the average length you willneed to sum over all lengths with their appropriate probability. The Fygen-son data in their fig. 1 captures some key ideas - the slopes of the growth anddecay regions tell you about the on and off rates, and the durations of thegrowth and decay periods tell you something about the parameters f+−andf−+. Note that by fitting the dynamical data, you are deducing/predictingsomething about the distribution of lengths.3. ATP: A Feel for the Numbers.a) (From Lehninger Principles of Biochemistry). A 68kg (150lb) adultrequires a caloric intake of 2000kcal (8360 kJ) of food per day. The food ismetabolized and the free energy is used to synthesize ATP, which then pro-vides energy for the body’s daily chemical and mechanical work. Assumingthat the efficiency of conversion of food energy into ATP is 50%, calculatethe weight of ATP used by a human in 24 hours. How many ATP moleculesis this? Take this a bit further by making a crude estimate of the number ofmitochondria in a human being and by assuming some reasonable numberof ATP synthase molecules per mitochondrion. Use this to see whether thenumber of ATP synthesized each day is consistent with the rate at which theATP synthase molecules can be producing it. (RP to class: this last bit is3very crude and I am not entirely confident that it will all work out.)b) The Stokes drag formula (see Howard’s book if you don’t know about this)says that the drag force on a sphere under low Reynolds number conditions isgiven by Fdrag= −6πηav, where π is the ratio of the circumference of a circleto its diameter, η is the fluid’s viscosity, a is the radius of the sphere and v isthe particle’s velocity. Consider a spherical bacterium! Taking our cue fromE. coli which swims (by turning its flagellae) at a speed of about 20 µm/sec,figure out how many ATPs it must consume per second in order to travelat this speed, assuming that all
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