IGEM:IMPERIAL/2009/M3/Modelling/analysis/1

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


 * Lambda cI promoter
 * It is a strong inducible and non-leaky promoter.
 * Assumption on cI binding. Are u including a study of leakiness


 * Restriction enzymes DpnII and TaqI
 * The production of restriction enzymes DpnII and TaqI is directly dependent on temperature, which acts as a switch when threshold is exceeded.
 * DpnII and TaqI are produced in parallel when there is an increase in temperature. Therefore, temperature acts as a switch for their production.
 * In addition, they are produced when functional cI protein binds to the lambda cI promoter; therefore their production is also dependent on the concentration of functional lambda cI.
 * 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.

=Predictions=
 * Cell death
 * We assume that cell death is a function of restriction enzyme concentration, which is related to temperature and protein cI concentration.
 * Each cell has a constant probability of dying due to the action of the restriction enzymes in any unit of time.
 * This probability of dying is a function of the internal concentration of restriction enzymes only.
 * A hill function is chosen because initially, there is very little killing with the increase of temperatures. When the threshold temperature is reached, we start killing cells.


 * If the temperatures are below threshold, little restriction enzymes are being produced; hence population of live cells will increase exponentially as we have assumed that 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.

=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.

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