Summer course Systems Biology (August 2009)

Matlab Introduction
Modeling mRNA transcription

An easy way of writing differential equations ODE given a set of reactions is to multiply the terms on the back arrow by the rate of the reaction (constant on top of the arrow) with the appropriate sign (postive if the entity is being produced or negative if it is being degraded). If a mathematical software is not available it is always possible to obtain data points to describe the behavior of an ODE. However this technique is limited by the step size:


 * if the step size is too small the computer might round it up to zero.
 * if the step size is too big we lose resolution.

Thus for the reaction

$$ \xrightarrow{\alpha} mRNA \xrightarrow{\beta}$$

we get the following ODE:

$$ \frac{dmRNA}{dt}=+\alpha - \beta \cdot mRNA $$

Note that the rate of production alpha is in fact being multiplied by 1. This is how we represent that the rate of production of mRNA is independent of other elements in the reaction.

If we wanted to model the dynamics of more than one mRNA more equations can be added to the system. However, when using Matlab the new equations can only be added in column arrays (see matlab code for an example).

Modeling transcription and translation

We have the following reactions

$$ \xrightarrow{\alpha_m} mRNA \xrightarrow{\beta_m}$$

$$ mRNA \xrightarrow{\alpha_p} Protein \xrightarrow{\beta_p}$$

And thus the following ODEs:

$$ \frac{dmRNA}{dt}=+\alpha_m - \beta_m \cdot mRNA $$

$$ \frac{dProtein}{dt}=+\alpha_p \cdot mRNA - \beta_p \cdot Protein

$$

Parameter fitting

The parameters in these equations, in this case, our rates of production or degradation of different entities cannot easily be calculated. However it is possible to approach their values by different methods.

Least squares

Residuals

In Matlab there are tools to fit parameters like fmin search or cftool (curved fit tool)