APh161: Physical Biology of the CellHomework 4Due Date: Tuesday, February 17, 2009“We must travel in the direction of our fear.” – John BerrymanReading:Read the first part of chap. 19 of PBoC, pgs. 721-746.1. Chromosome Geography in Prokaryotes.(a) Read the paper by Fiebig et al. in order to understand the chromo-some geography experiments they have performed and write a one paragraph“Scientific American” style description of the experiment.(b) Repeat in detail with careful explanations all of the steps leading upto eqn. 8.50. This means you need to show your work for determining theFourier coefficients, explain the normalization in the denominator, etc.(c) In this part of the problem I would like you to fit the model from part(b) to experimental data for Vibrio cells. First, observe the geometry of theVibrio cells (though treat them as straight cylinders rather than curved cells)and determine their length and their width. Now, make two plots using theVibrio data provided with this homework as an Excel spreadsheet. One plotshould be of the distribution of the fluorescence marker along the long axisand the other of the distribution along the short axis. Each of the figuresshould include both the provided data points with error bars and a curveshowing the best fit of the model from (b) to the data. To do this fit youwill need to use Matlab or Mathematica or some other computer program ofyour choice.Your fit should depend upon the parameters x0and N (assume the nakedDNA value of the Kuhn length). Comment on the best-fit values of thoseparameters (are they reasonable?), and give a description of how the plotschange as the position of the marker gets farther from the tether point. Does1confinement matter along either of these directions? Note that the way I amhaving you model the data is an approximation. In particular, you are usingthe one-dimensional solution for the distribution twice, once for each of thetwo perpendicular directions, but both representing the same polymer. Inreality, you would want to solve the full 3D problem (which is much tougher).2. Statistical Mechanics of Gene Regulation.- from chap. 19 of PBoC, work out probs. 19.1 and 19.2 (a).3. Protein Concentration by Dilution.In this problem we consider the concentration of mRNA or proteins as afunction of time in dividing cells. This exercise provides some of the concep-tual tools we will need to write down rate equations describing gene expres-sion. In particular, the point of this problem is to work out the concentrationof a protein given that we start with a single parental cell that has N copiesof this protein (in the experiments of Rosenfeld et al. this is a fluorescently-labeled transcription factor). At some point while the culture is growing, theproduction of the protein is stopped by providing a chemical in the mediumand then the number of copies per cell is reduced as a result of dilution asthe cells divide.(a) Work out a differential equation for the change in protein concen-tration as a function of the time that has elapsed since production of theprotein was stopped. Solve the equation and relate the decay constant tothe cell cycle time. Note that here we are only interested in the dilutionthat results from the original N copies of the protein being partitioned intoan ever-larger number of daughter cells, not in the dilution that occurs aseach individual cell lengthens in preparation for the next round of division.Note also that in this part we’re interested in a continuous model—you’lllook at the discrete version in part (b). HINT: there are two ways to ap-proach this problem. You can consider the change in the concentration asa function of the change in the number of cells into which the original Nproteins are partitioned. Or you can note that for a bacterium like E. coli,it is a reasonable assumption to imagine that the cell diameter is unchangedand that the size is controlled by the cell length, such that the change in2volume with time is simply the change in length with time times a constantprefactor; then consider the change in protein concentration as a function ofthe change in the total volume into which the original N proteins are diluted.(b) We can repeat a calculation like that given above using a discretelanguage in which the number of proteins per cell is a discrete integer. Imag-ine that before cell division, the number of copies of a given transcriptionfactor in the cell is N. In particular, for every cell doubling, the number ofproteins is reduced by a factor of 2. Using such a picture, write a formula forthe average number of proteins per cell as a function of the number of celldivisions and relate this result to that obtained in part (a). Furthermore,by using the fact that 2 = exp (ln 2), reconcile the discrete and continuouspictures precisely.(c) Interestingly, the model used in part (b) opens the door to one of themost important themes in physics, namely, that of fluctuations. In particular,as the cells divide from one generation to the next, each daughter does notreally get N/2 copies of the protein since the dilution effect is a stochasticprocess. Rather the partitioning of the N proteins into daughter cells duringdivision follows the binomial distribution. Analyzing these fluctuations canactually lead to a quantification of the number of copies of a protein in a cell.In this part of the problem, work out the expected fluctuations after eachdivision by noting that the fluctuations can be written asq< (N1− N2)2>,where N1and N2are the number of proteins that end up in daughter cells 1and 2 respectively. Show thatq< (N1− N2)2> =√N (hint: you’ll need touse the binomial theorem.)Next, look at the Rosenfeld paper and explain how measuring fluores-cence variations can be used to calibrate the exact number of copies of thefluorescent protein in a cell. Assume that the fluorescence intensity in eachcell can be written as I = αN, where α is some calibration factor and N thenumber of proteins. Make a plot ofq< (I1− I2)2> versus Itotand explainhow to get the calibration factor α from this
View Full Document
Unlocking...