Deterministic Models

From the previous exercises on choosing suitable parameters and ODEs for our models, we have produced some simple models for our systems. The models are all generated using [http://www.mathworks.com/products/simulink/ SIMULINK]. Simulink has proven to be very useful in controlling variable inputs and it allows dynamic modeling. The user interface is friendly and easy to use.

E7 production system

The goal of this model is to predict the dynamic response of the E7 production system and to see how a change in the Lactose Input would affect the way the cell responses and the speed and shape of response.

Assumptions:
1) The cell behaves like a well-mixed vessel or reactor.
2) Lactose input into the cell is constant and well-distributed.
3) The temperature and other conditions such as fluid medium concentrations remain constant throughout the time period.

Results:

E7 Production SIMULINK Model

E7 Production Modeling Results

Lactose Step Function

Analysis:
Modeling of the System shows that Lactose induction is essential to produce E7 and a variation of Lactose input can result in different yields of E7. As seen in the figures above, the concentration of E7 produced increases until it reaches a final value without any oscillations. the profile of the response has a sigmoidal or "S" shape witha derivative that goes through a maximum at an inflection point and reduces to zero at the steady state value. It is also observed that the production of E7 begins almost instantaenously upon the addition of Lactose and the system possess no momentum.

The modeling exercise has confirmed that the lactose input would have an impact on the system's production of E7 and that the impact is certainly significant.

Lysis Production system

The goal of this model is to predict the dynamic response of the Lysis protein production system. There are two kinds of inputs in this system, Ai-2 and Iron ions. The response to the system under 4 different conditions would be examined and they are:
1)Both Inputs are Present
2)Both Inputs are Not Present
3)Only the Ai-2 input is present
4)Only the Fe ion input is present

Assumptions:
1) The cell behaves like a well-mixed vessel or reactor.
2) Lactose input into the cell is constant and well-distributed.
3) The temperature and other conditions such as fluid medium concentrations remain constant throughout the time period.

Results:

Lysis Production SIMULINK Model

Lysis Prodcution Modeling Results

Iron Step Function

Ai-2 Step Function

Analysis:
Here we define 1 as a certain threshold e.g. (<250 µM) that when the lysis protein reaches it, lysis in the cell definitely occurs. Although we wish for an absolute AND gate, where 0 will have no lysis production at all, simulation on biological systems shows that such results are impossible. We cna observe that for the response, it is a sigmoidal response that does not taper off to a steady state value even after the time period has been extended for 1440 min. It is also observed that the response of the system to any input seems to take a while before the actual production begins. We can infer that the phosphorylations processes and AND gate within the system all takes time to process before the Lysis protein is produced.

Both addition Fe ions and Ai-2 alone would induce a certain level of lysis production. However when both are available, the lysis protein production would be higher. In the presence of Ai-2 alone, Lysis will most likely still occur albeit at a slower rate compared to situation when both Fe and Ai-2 are present. With the addition of iron and Ai-2 together the rate of lysis production is still significantly much higher compared to Ai-2 alone.

Therefore, by giving the logical output ‘1’ as a suitable threshold value of Lysis protein production higher than that of Ai-2 induction alone, we would still be able to obtain an AND gate based on our definition.

The model also gives us the warning that the Lysis production system may not be as fast to respond to the inputs as we desire. There in lies the compromise that if we wish for a very fast Lysis protein production response, we might have to sacrifice the specificity that an AND gate system may provide.

New Lysis Production System

New Lysis Production SIMULINK model

lysis output

The above graph shows the simulation done for 300min. The Ai-2 input was done at time = 250 min. As seen, the production lysis protein jumps upon AI-2 addition.

lysis output

The next graph shows that the system reaches steady state around 100 min after AI-2 induction.

After our wetlab experience, we were unable to synthesize our lysis production system with an AND gate detection. A simpler system comprising of only Ai-2 detection to trigger lysis production was created. The new detection system is easier to synthesize but the draw back is that there is a loss of specificity. The SIMULINK model for this particular system was also created and tested. As expected, the lysis production is much faster than the previous AND gate system. This was because the regulation to produce lysis proteins was less.

Updates

A new lysis production system model has been added to complement our wetlab results.