Team:Imperial College London/M3/Modelling/Analysis

=Assumptions=
 * The protein cI is constitutively expressed by the strong promoter J23114.


 * Restriction enzymes DpnII and TaqI
 * At higher temperatures, DpnII and TaqI will not undergo conformation changes or undergo denaturation. Therefore, they will work at optimal levels from 28°C to 42°C.
 * No saturation of enzyme restriction sites will occur. Restriction enzyme concentration will always be in access of the substrate.


 * Cell growth
 * Assume that cell growth prior killing by thermoinduction is not constrained by availability of resources. (This correlates to the classic assumptions for exponential phase)


 * Cell death
 * We assume that the rate of cell death is a function of restriction enzyme concentration at the time. Ie, the system has no memory.
 * More precisely, we assume that each cell has a constant probability of dying due to the action of the restriction enzymes in any unit of time.
 * It is this probability of dying that is modelled as is a function of the internal concentration of restriction enzymes only.
 * A hill function is often chosen because initially, there is very little killing with the increase of temperatures. Then, when threshold temperature is reached, killing is expected to be the dominating feature.

=Predictions=


 * If the temperatures are below threshold, little restriction enzymes are being produced; hence population of live cells will increase exponentially. (Cell growth is not constrained by resources such as food and space.)


 * If the temperatures are above threshold, the population growth of live cells will be constrained by the concentration of restriction enzyme.


 * Initially, cells will grow exponentially as the restriction enzymes take time to be produced. When enough restriction enzymes are being produced, cell death will begin, and an exponential decrease in cell population will be observed.


 * When we increase the temperature, we produce more restriction enzymes at a faster rate. Hence, at higher temperatures, the exponential decrease in live cell numbers will occur earlier. Furthermore, the maximum live cell population at a given temperature will also be lower with increasing temperatures, because a greater concentration of restriction enzyme means more cell death in the same amount of time.


 * If the lambda cI promoter is strong, there will be a higher rate of cell death than cell growth; we will see a decrease in live cell population. However, if lambda cI promoter is weak, there might not be enough killing, so we will only see a reduction in growth of cell population, ie: a slower increase in population.

=The Model=

We model the behaviour of our system with 6 differential equations. The first two are concerned with the production of protein cI, which represses killing. The next two tells us the production of restriction enzymes. Equation (5) describes the cell population, while equation (6) tells us the total number of dead cells. This will allow us to relate our model to the live and dead cells assay performed.

In this model, the effects of Dam methylase are not included.

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