# 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 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 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 with a 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 instantaneously 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.

## Detection & Lysis Production system

The goal of this model is to predict the dynamic response of the Detection & Lysis 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.

### Detection & Lysis System Version 1

Detection & Lysis Production Version 1 SIMULINK model

Detection & Lysis Production Version 1: Lysis output after 300 min

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

Detection & Lysis Production Version 1: Lysis output after 1000 min

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

### Detection & Lysis System Version 2 (with AND biological gate)

Results:

Detection & Lysis Production Version 2 SIMULINK Model
Detection & Lysis Production Version 2 Modeling Results
Iron Step Function
Ai-2 Step Function

Detail 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 can 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 inputs are present, the lysis protein production would be higher. In the presence of Ai-2 alone, Lysis protein will still be synthesized 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 Detection & Lysis 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.

Conclusion:

To confirm that our Detection & Lysis concept works, we created a Detection & Lysis System Version 1 that requires only Ai-2 molecules to trigger downstream Lysis. In future, we will create Detection & Lysis System Version 2 with a biological AND gate detection that senses both Ai-2 and Fe molecules . In comparison, the Detection & Lysis System Version 1 is easier to synthesize but at a cost of lower specificity against EHEC detection. The SIMULINK model for this particular system was created and tested. In accordance with our hypothesis, simulation result shows that lysis production is much faster than the one with AND gate system. This was because the regulation to produce lysis proteins was less.