Mathematical Biochemistry in a Nutshell

David Fell

Innsbruck 2 March 2014

This precourse is designed for those less familiar with the mathematical aspects of modelling and systems biology. The aim is not to improve anyone's competence in mathematics, but to provide a basic introduction to some of the main mathematical concepts and computational techniques used to represent and model biological systems. The selection of topics is designed to underpin the main course and consists of:

Functions, parameters and variables
Calculating time courses from ordinary differential equations
Matrices and vectors
Mathematical representation of cellular systems
Sensitivity analysis - illustrated with metabolic control analysis

Useful tools

For the exercises, it will be useful to have the following available:

A plotting program, such as
- Gnuplot, described at http://www.gnuplot.info/. Versions are available for for Linux (from your repository), Windows and Mac.
- Matlab (or Octave, but Octave uses Gnuplot).
A spreadsheet program such as
- Excel, with the solver add-in activated.
- LibreOffice Calc. (There is an nlp-solver add-in, but I can't get it to activate.)

1. Functions, parameters and variables

The slides are available to download. Only the problem answers are missing!
Download enzymes.zip and unzip it in a folder/directory. For the gnuplot examples to work, this directory will have to be gnuplot's working directory. Contents are:
1. MMplot.xls/.ods are examples of function plotting with a spreadsheet, as needed for the exercise on slide 6. Much easier and far superior is gnuplot. Open gnuplot and type:
  - ```
  gnuplot> mm(s) = s*V/(Km + s)
  gnuplot> V = 100
  gnuplot> Km = 2
  gnuplot> set xlabel '[S]'
  gnuplot> set ylabel 'rate'
  gnuplot> plot [s=0:5*Km] mm(s)
  gnuplot> Km = 4
  gnuplot> replot
  gnuplot> 
```
2. twoenz.gnu generates the two enzyme steady state graph shown in the slides. To plot the graph, open gnuplot and type
  - ```
  gnuplot> load 'twoenz.gnu' 
```
3. viewlogcont.gnu and viewcontmap.gnu plot the sum of squared residuals against K_m and V as in the lecture slide. The mouse will rotate the 3-D plot.
4. enzfitter.gnu determines K_m and V for the data in enz1.dat. Open enzfitter.gnuwith a text editor to see the commands you will need to modify to fit the data in enz2.dat.
5. EnzFitter.xls is an Excel spreadsheet that fits K_m and V. It contains the same data as enz1.dat; the initial estimates are in cells B3 and D3. Go to cell G14 and then select the 'solver' menu item to minimise this cell by altering cells B3 and D3. (This sheet does not work in other spreadsheet software as they don't have the solver add-in.)
If you've got time on your hands, follow the gnuplot exercise 2 at http://mudshark.brookes.ac.uk/Meetings/Delhi2013/Practicals/Practical_1.

2. Calculating time courses

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.
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.
Can you add a column that computes the product concentration at each time point?