Homeworks are due at the beginning of class one week after they were posted. Solutions will be posted two days after the homeworks are submitted, and homeworks will be returned a week after they are submitted.
HW 1 (due 1/27), solutionsHW 3 (due 2/12), solutions
- Clarification for problem 5: "In problem 5 of the HW (i.e. problem 8.4 of PBoC) what I am asking you to do is to work out a partial differential equation for the probability that the polymer ends at position x after N steps. This derivation was essentially sketched in class. The clarification that I want to propose is this: p(x,N) is the generic solution and a special name (G(x,N), known as the Green function is given when the "initial condition" is a spike (or delta function) like we have chosen. Remember, we said that G(x,0)=delta(x-x0). Further, I described G(x,N) as a "counter" that tells us the number of configurations that are of length N and end at position x. More precisely, G(x,N) should be thought of as the statistical weight of the polymer configurations with N links that end at x. Because of the absorbing boundary conditions, once we find the result for G(x,N) (or if you prefer to call it p(x,N)), we need to normalize the solution as I discussed in class."
HW 4 (due 2/17), solutions
- Reading:
- For problem 1: Fiebig A., Keren K., and Theriot J. (2006) Fine-scale time-lapse analysis of the biphasic, dynamic behaviour of the two Vibrio cholerae chromosomes. Molec. Microbiol. 60(5): 1164-1178.
- For problem 3: Rosenfeld N., Young J.W., Alon U., Swain P.S., and Elowtiz M.B. (2005) Gene regulation at the single-cell level. Science, 307: 1962-1965.
- The data for problem 1. A clarification about this spreadsheet: "On problem 1 of HW4 you will need to use data from the Fiebig et al. experiments. We have provided you both the "raw" data which comes from taking the data points from their graphs as well as a second version of the data that is properly normalized such that the integral of p(x,N) over the length (or radius) is one. You will probably want to use the second version of the data when you make your fits."
- There is a mistake in fig. 19.31 in PBoC--this is the corrected version of the figure that you'll need to reproduce as part of the homework.
- Please note the corrected version of eqn. 19.52 given in the homework.
- Part A of the homework requires you to solve differential equations numerically using Matlab or some other program of your choice. The TAs will be giving tutorials on solving equations numerically in Matlab on Sunday, 2/22, from 5-7 pm in the third-floor multimedia room in SFL, and on Monday 2/23 during Steph's and Maja's usual office hours (7-9pm, SFL 229). The code we will be going over is: HO_ODE.m and HO_Tutorial_main.m
- Clarification of 1(a) of part A: In (a) of HW 5 part A we ask you to derive the dynamical equations for the repressilator and then to put these equations in dimensionless form--the dynamical equations we're referring to are found in equation 19.45 in section 19.3.3 of PBoC, and their dimensionless form is given in equation 19.46.
- Reading for part B:
- Golding I., Paulsson J., Zawilski S.M., and Cox E.C. (2005) Real-time kinetics of gene activity in individual bacteria. Cell 123 (16): 1025-36.
- Gregor T., Bialek W., de Ruyter van Stevenick R.R., Tank D.W., and Wieschaus E. (2005) Diffusion and scaling during early embryonic pattern formation. PNAS 102 (51) 18403-18407.
- Bergmann S., Sandler O., Sberro H., Shnider S., Schejter E., Shilo B.-Z., and Barkai N. (2007) Pre-steady state decoding of the bicoid morphogen gradient. PLoS Biology 5(2): e46.
- PBoC 11.5