## Model for

Unregulated Gene Expression

## A model for this simple system shown above can then be written, keeping the assumptions in mind. To start with, we can consider the two variables that are of importance to us in determining the level of gene expression. These are the mRNA and protein levels (since the DNA levels in a cell are assumed to be constant, they are not of interest).

Therefore, let us write the differential equation for mRNA first -

mRNA is produced from DNA, and degraded spontaneously. Therefore, at any instant of time, the rate of change of mRNA can be written as -

Note, that here, we have not written the reaction where mRNA is being converted to protein, since mRNA is not actually being consumed there or being produced. 1 molecule of mRNA simply produces 1 molecule of protein (assumption).

Further, it has to be noted that the [DNA] and [mRNA] terms appear in the equation since in writing the model, we assume that mass action kinetics are valid, ie, the rate of the reaction is equal to the rate constant times the concentration of the reactant, raised to a power equal to the number of molecules of the reactant.

Now, we know that the DNA concentration remains constant and does not change over time. Therefore, the [DNA] term can be included in the constant itself, to give

Now, the dynamics of the protein can be similarly written as

And that is it! We’ve just written down our first model, for a gene being expressed from a constitutive promoter. Now that we have our model, we can simulate these and find out the dynamics.

Simulation basically means solving the differential equations to get the variation of the component (mRNA, protein) with time. This can be done by hand for the equations above. However, as models get more complex, implicit equations appear, which are much more difficult to solve by hand. Thus, it is essential to get the hang of modelling software such as MATLAB or R, which solve differential equations and simulate the model for a specified period of time.

Thus, we write down the model on MATLAB here, and simulate it for a time period of 200 time units. The values of the constants used for alpha, gamma etc and the MATLAB code for the same can be found on the github library link given below. The plot obtained is as follows -

Changing the parameters for production and degradation rates can give different kinds of graphs, and can be explored by simply changing the values of alpha, gamma, K etc in the model and simulating the same. However, as we can see here, the mRNA and protein levels both rise to a certain fixed value. This is known as the steady state value.

However, we can make a further simplification in this model. Generally, the mRNA dynamics are faster than the protein dynamics. This means that mRNA levels approach their steady state value faster than proteins do. Therefore, we can say make the assumption and further simplification that before the protein dynamics start to come into play, the date of change of mRNA is zero. This is known as the “quasi steady state assumption”.

Therefore at steady state,

Thus,

Now, we can replace the value of [mRNA] in equation (2) with the value given above, to get -

We can now try to simulate and plot the graph for the protein levels, and compare the time series of the two models -

Therefore, we can see that by making the assumption that mRNA is already at steady state at the start of time, the protein levels begin to rise faster than the earlier model. However, the steady state value for protein remains the same. This is because we have only simplified the model by changing the time scale and assuming that at the given time scale, mRNA dynamics are at steady state. We have not changed the steady state per se.

## Model for

Regulated Gene Expression