BIOE97033 : Modelling in Biology - Chemical Breaking of Glucose - Glycolitic Oscillations - Biology Assignment Help

Assignment Task :

Question 1: Population dynamics of one species 

Consider the following non-dimensionalised version of an ecological model of an insect population in a forest: 

(a) Find the fixed points of this system, draw the flow of trajectories on the line and deduce the stability of the fixed points. All initial conditions (except one) converge to a certain value of the population of insects. If r = 0.6 and k = 4, which value is this? What are the attractors for this system?  

(b) What do the parameters r and k represent?

(d) Use fsolve to calculate numerically the fixed points for k = 13 and r = {0.25, 0.8}. How would the behaviour of the system change if r is increased from 0.25 to 0.8? How many bifurcation(s) occur? What type of bifurcation(s) do we have here?

Question 2: Positive feedback in glycolysis  

The energy that cells need to function is obtained mainly from glycolysis, that is, the chemical breaking of glucose (elementary sugar). This cellular process involves several biochemical reactions, and one of its crucial features is the positive feedback exerted by adenosine diphosphate (ADP) on the enzyme phosphofructokinase (PFK). This effect is typically analyzed by looking at the glycolitic model shown below. Remark: although this model is an oversimplification of the glycolisis system, which contains tens of reactions and species, it is the minimal model that can describe its main dynamic features. 

In the simplified model, vin is the rate of glucose supply (considered constant in this exercise), vP FK is the enzymatic rate of PFK, and vout is the rate of product consumption. The corresponding non-dimentionalised model is given as:

The parameters k1 and k2 are positive, and the term k1xy2 describes the allosteric effect of ADP on the enzyme PFK. 

1. Compute the fixed point(s) of the model in (2).  

2. Write MATLAB code to numerically integrate (2) using the solver ode45. To do this, create a script Glyco.m and a function file GlycoFun.m. Glyco.m should apply the solver ode45 to the function GlycoFun.m and produce a plot of the time evolution of x(t) and y(t) on the same graph. 

Using your code, solve the system for vin = 0.5, k1 = 1, and k2 = {0.05, 0.15, 2}. Draw a separate plot of the time evolution of x(t) and y(t) for each value of k2. You are free to choose the initial condition (but remember that species concentrations must be always positive). Explain the observed behaviours for the different values of k2. Integrate the script and function code as an appendix to your coursework PDF document.  

3. (a) For vin = 0.5, k1 = 1 and k2 = {0.05, 0.15, 2}, plot in Matlab the nullclines of (2) by creating a x vs y plot with suitable ranges (again, create one plot for each value of k2).

(b) On each plot, indicate the direction of the flow on the nullclines and briefly explain it.

(c) On each plot, overlay on the plot of the nullclines the phase plane trajectories obtained from the simulations in Part 2 above.  

4. For vin = 0.5, k1 = 1 compute the critical value of k2 that leads to glycolitic oscillations both (a) analytically (by finding analytically the value of k2 at which the steady state of the system in (2) becomes locally unstable) and (b) numerically (by sweeping values of k2 and checking the behaviour corresponding to the time simulations of the system in (2)).

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