CS904 Computational Biology – Second Assignment
Modelling Single Cell dynamics
DUE DATE: 19th February at noon
If tasks refer to problems from labs, please see for more details there.
TASK 1 [15 points]
Provide your code and results for Problem 1 (Channel gating) of Lab 4. Use
+ = 0.1/ms and k
− = 0.3/ms.
a) Use a reasonably long enough time interval T to capture enough events,
and state the value of T in your write-up. Find a suitable way of plotting the
opening/closing of the channel, for the purpose of visualisation perhaps for a
shorter time interval.
b) State the average open and closed time from your simulation experiment
and compare them to the expected values (hτOi =
k− ;hτC i =
c) Plot the distribution of open and closed times from your simulation experiment
and compare them to their theoretically expected distributions (see
section on Dwell times from script).
Hint: If creating the distribution of closed times, for example, note that a
channel that has been closed for a total duration of 1 sec, also contributes to
the counts of channels that have been closed for 0.1 , 0.2 , . . . , 0.9 seconds, if
you were to sample at an interval of 0.1 seconds (width of your histogram bins).
Choose a reasonable sampling interval and normalise the histogram so that you
can plot it together with the theoretical distribution. The latter can be plotted
as a line.
Task 2 [15 points]
Provide your code and results for Problem 2 from Lab 4 (Modelling Calcium
a) Produce simulation runs (until M reaches Mmax = 1000, or a sensible
value which makes plotting not too difficult!) and plots which demonstrate
when the channel is open or closed and how the number of intracellular ions,
Hint: You can either use different sets of propensity function depending on
whether your channel is open or closed, or you can use the following, where
propensities automatically evaluate to 0 if a transition cannot take place (the
lab script uses a mixture of both, which is lacking in rigour; thanks to Scott for
pointing this out).
α1 = k
+C; α2 = k
−O; α3 = k
Task 3 [40 points]
Provide your code and results for Problem 3 from Lab 4 (Simulating intracellular
diffusion of Calcium ions)
Note: Scaling space and time in b) to reasonable dimensions is not easy, and
it looks that the values given in the lab script do not allow to show the effect of
subdiffusion. The effective diffusion constant (D = 5 × 10−6 cm2
, units correct
as stated) was determined experimentally, so should not be changed. Instead,
change the size of the bounding box determined by yt, xl and xr (yb = 0).
Determine the maximum distance xmax travelled by the ions in a) and try for
example yt =
xmax , xl = −
xmax and xr = +1
xmax. If you have already
come up with a different approach to demonstrate subdiffusion due to spatial
hindrance, then keep that (even if it is based on scaling D), stating what exactly
Task 4 [20 points]
Provide your code and results for Problem 1a from Lab 5 (Simulation model of
the Kai cyanobacterial clock)
Plot timecourses of the variables T, S, D, U, and A. Confirm that the period
of your clock is approximately 21 hours.
Task 5 [10 points]
This task is related to the Hungarian algorithm from lecture 12.
a) Try to devise a mathematical formula for a suitable cost function to solve
the assignment problem of matching corresponding cells in subsequent time
frames of a 2D image time series. The parameters to consider are vector x, the
position of cell centres, and I the average intensity of each cell. The assumption
is that for a corresponding pair of cells x and I will not change drastically.
Hint: The distance function used to define similarity between data points in the
previous diffusion mapping might come in handy.
b) The Hungarian algorithm requires a square cost matrix. What would
you do if you deal with images where cells are entering or leaving the field? In
words, describe a strategy (no more than 4-5 sentences).