1 Basics of Modelling Enzyme Reactions

1 Understanding Euler Integration

1. Open the spreadsheet file euler.ods (on Linux/the ScrumPy Virtual Machine) or euler.xls (with Excel). Check that you understand how the column of substrate concentration values is calculated.

2. Using the tabs to select the sheets for t=0.01 and t=1.0, look at how the solutions change with increase of the Km, or decrease of the Vm.
3. Can you add a column that computes the product concentration at each time point?

2 Plotting enzyme kinetic functions

1. Open gnuplot. (In Linux/the virtual machine, open a terminal window and type `gnuplot'.

2. Write a function for the reversible Michaelis-Menten equation for the reaction A -> B and assign the parameters as follows:

   gnuplot> rmm1(A,B)=Vm1*(A-B/Keq1)/Km1a/(1+A/Km1a+B/Km1b)
   gnuplot> Km1a = 1
   gnuplot> Km1b = 1
   gnuplot> Keq1 = 2
   gnuplot> Vm1=10

3. Now do a 3-D plot for a range of values of substrate and product with the command:

   gnuplot> splot [A=0:10] [B=0:10] rmm1(A,B)

4. A plot window will open. You can view the plot from different angles by holding the left mouse button down and steering the pointer round the plot window. If the vertical rate axis is rather short, holding both mouse buttons down (?central button on a 3-button mouse?) and moving the cursor up will change the z axis magnification, and side to side will change the x-y magnification. Have you lost track of which axis is which? Label your plot as follows:

   gnuplot> set xlabel "[A]"
   gnuplot> set ylabel "[B]"
   gnuplot> set zlabel "v"
   gnuplot> replot

5. Now do a 2-D plot of the effect of varying B at constant A:

   gnuplot> set xlabel "B"
   gnuplot> set ylabel "v"
   gnuplot> A = 1
   gnuplot> plot [B=0:2.5] rmm1(A,B)

6. Let's add a second enzyme that uses B and converts it to C:

   gnuplot> rmm2(B,C)=Vm2*(B-C/Keq2)/Km2b/(1+B/Km2b+C/Km2c)
   gnuplot> Vm2 = 10
   gnuplot> Km2b = 1
   gnuplot> Km2c = 1
   gnuplot> Keq2 = 2
   gnuplot> C = 1
   gnuplot> plot [B=0:2.5] rmm2(B,C)

7. Plot the responses of the two enzymes simultaneously:

   gnuplot> plot [B=0:2.5] rmm1(A,B), rmm2(B,C)

8. From the graph, answer the following questions:
   1. (After changing parameters you can use the "replot" command to update the graph)
   2. What will happen to the concentration if B in the system if it were initially 0.5 when the enzymes were added?
   3. What will happen to the concentration if B in the system if it were initially 1.0 when the enzymes were added?
   4. When the concentration of B reaches the value at which the two curves cross, what happens then? (We are assuming that the concentrations of A and C are constant.)
   5. What is the value of B at the crossing point? Record this and keep the value for tomorrow.
9. Set Vm1 to different values in the range 0.1 to 20. What happens? Try other parameter sets, e.g. Vm1 = 10, Km1b=10, Keq1 = 100, Vm2 = 3. Also, weaken the product inhibition on reaction 1 by making Km1b even larger.
10. Now repeat the above using the irreversible forms of both equations:

    gnuplot> mm1(A) = Vmax1*A/(A+Km1)
    gnuplot> mm2(B) = Vmax2*B/(B+Km2)

11. What is the effect of varying these parameters on the steady-state concentration of B in the reversible and irreversible cases ?