# Modeling our Design

### Overview

The main prediction of our model was based in the interview that we did with Prof. Margareth Capurro, where she denoted that there is a problem with the paratransgenesis strategy: "the delay between the detection part of the system and the elimination of the disease’s agent", as shown in the interview on our integrated HP page.

Then, based on Prof. Capurro suggestion we decided to investigate the protein production over time, so the best way to do this is utilizing differential equations. Also correlating the pathogen population with the protein concentration, mediated by some toxicity term. Now, utilizing the two detection systems, we developed two systems of equations, with the same basic elements: Protein, RNA and Pathogen population over time.

### Assumptions

For building the model we assumed some premises for both systems, thus we could have a simplified model that describe well what we wanted:

The regular mosquito gut microbiota is stable, so we can assume that the population of P. agglomerans is constant in the community. In the circuits, non-explicit expression control, like repression and induction, are not represented in the model, i.e., by default all the expressions, either protein or rna, are constitutive and not influenced by its own concentration. For the protein expression, translation and secretion are considered with no alterations over time. Also for both systems the pathogen behaviour is assumed as a regular populational growth. Thus, assuming those premises we reached in the systems of differential equations that describe our circuits.

### Building the Equations

Either systems have basically the same “types” of equations that will describe the system:

**Base system:**

- One equation for each gene in the system. This is another term that has a relevant variation in time. Each circuit in the systems will be represented by a differential equation, describing the expression of the mRNA (or the repressor) in time. So in these equations, we basically have a term, γ, representing a basal expression, which can be modulated by some repressor or inducer, and degradation rate, μ.
- An equation that describes the secreted protein, called P. The growth of secreted protein is dependant on the population of the chassis, a rate of secretion (it’s important to note that the protein secretion is an important part of the problem, since the protein needs to be in the medium to stop the pathogen life cycle), and a rate of translation and transcription. For the reduction there is the natural degradation rate, μ
_{p}, and the encounter of protein and the pathogen, as, once in the pathogen, the protein can’t "be used" again. Note that the encounter term is not required for EPIP4, for example, since the peptide does not interact directly with the pathogen. So we there is a "boolean" term, ξ, that just represents if the action is direct (1) or indirect (0). - The populational growth of the pathogen. This is a general equation that the describes the populational dynamics, with the population being represented by N in the equation. We used the logistic model of population growth, being the intrinsic growth represented by "r" and the carrying capacity "K", that can describe in the most generic form this dynamic. In some cases, like the malaria pathogen, Plasmodium sp., we don’t have an actual logistic growth, but a population without growth, due to the formation of ookinetes. For that cases, in the simulations, we can simply define the initial conditions with the population equal the carrying capacity or with a really small "r".

There is also a term of unnatural death, caused by the protein effect, represented with the encounter of the pathogen and the protein, with an term of toxicity, "λ". An assumption was made here, to simplify the model, that every time the encounter occurs the pathogen takes the protein from the medium. The toxicity term can, partially take care of this. However the toxicity ideally doesn’t consider this, since is not expressed in the decay of protein, in its equation.

So we have the following equation, that will be used in both systems:

Given these postulates, we have the following systems:

## Toehold Switch:

Use this reference table if you get confused when reading the equations and just want to remember the meaning of a term:

Parameter | Symbol |
---|---|

Protein of interest | P |

mRNA (Switch) of the protein | mR |

Trigger RNA | mT |

Pathogen population | N |

P. agglomerans population | Na |

Translation/Secretion rate | θ |

Complex formation/release constant | α |

Degradation rate | μ |

Protein effect direct(1) or indirect(0) | ξ |

Basal production | γ |

Toxicity | λ |

Intrinsic growth | r |

Pathogen carrying capacity in gut | K |

The first of the two systems was based in the Toehold Switch, which circuits you can see in this page. Like described before, we have an equation describing the protein, **P**, one describing the population of the pathogen, **N**, and other two equations that describe the circuits of the system, that is, expression of the protein mRNA, **mR** and the trigger mRNA, **mT**. Since to express the protein we need to have the formation of the Switch-Trigger complex, we then have an equation that describes this complex, **mRT**.
For an easier view of the system, we did a graphical representation of the equations:

The expression of the protein mRNA is simply the expression rate plus the dissociation of mRT complex (The product of the complex and the dissociation rate, α_{mRmT}), minus the degradation rate, μ_{mR}, and the complex mRT formation (The product of mT and the complex association rate, α_{mRT}. The expression of the Trigger RNA is very similar in almost all aspects, except by the inducer in the basal production.

The mRT complex is the complex with the linearized protein mRNA, so it can be expressed. Which is described by this simple kinetics:

So it can be described in the equation by the encounter of mR and MT times the formation constant minus the disassociation rate (mRT itself times the disassociation constant). There is also a degradation rate to consider:

The expression of the protein - expressed in the first equation - is determined by product of the population of P. agglomerans with all the other terms. Then we have the quantity of protein, which is the product of secretion/translation (We represented this rate with a single term, because considering those are, by our assumptions, fixedly the same in all the cases, so we don’t have to represent they separated (even if they are important parts), the encounter of the protein mRNA and the trigger RNA, by a term, ɑ, that represents a disassociation rate, which varies from 1 (Always keep the association) to 0 (always disassociate). Like described in base system, we also have a degradation term and the take out of the medium, i.e., encounter with the pathogen.

Like described before, the pathogen equation.

Looking to the modelling and our results, we managed to elaborate some questions the model might answer, beyond the initial question: Is the system sensitive enough so that it could work in real world? Those mRNA variations will activate the system or nothing will happen? With some simulations we could investigate those questions.

### Analysis and Simulation

For gaining some enlightenment about our system, we did some analysis in our model.

Initially we decided to do the calculus of the system stability. We found the following stable points:

So we can see that, for N*=0, K∗r < λ∗P*. So it's just needed to have a bigger toxicity or a bigger amount of protein, i.e., increase it expression. Also looking at the equilibrium point of P, we can see that the only terms we can change are mR and mT, therefore, their basal productions, γ_{mR} and γ_{mT}.

We then made some simulations of our model, so that we could have a better viewing of the problem and see if it works as expected

Simulation Parameters | |||
---|---|---|---|

Parameter Symbol | Values | Parameter Symbol | Values |

Na* | 500 P. agglomerans | μ_{P} |
2.13 10^{-2} |

θ | 2.85 min^{-1} |
ξ | 1 |

[I] | 2*0.2mM of IPTG | γ_{mR}=γ_{mT} |
3.17*150 copies∗cell^{-1}∗min^{-1} |

α_{mRmT} |
0.05 | λ | 0.01 |

α_{mRT} |
0.01 | r | 1 |

μ_{mR}=μ_{mT}=μ_{mRT} |
0.19 min^{-1} |
K* | 50 Plasmodium sp. |

Note: Our parameters were based in the works of Terenius, O., et al. (2012), Wang. S (2012), Miller, M. et al. (2012). We also used our work from our lab to define and refine some of then, like the amount of inducer.

*These terms are usually in a bigger scale, in thousands or millions, but for a better visualization we used in that scale.

As expected, when you have a bigger expression of protein, your population of Plasmodium go towards 0. But what if we had a protein that was too toxic, or where we could vary the terms. Is interesting to note that in the real biological system, when a mosquito gets in touch with infected blood, millions of Plasmodium specimens enter the mosquito. However, in the gut, they quickly go down to a “stable” population, so we used a bigger initial population then the carrying capacity.

So we decided to build a heatmap to check the variation of the Plasmodium population when we vary the toxicity and the basal expression of mR.

Note: The closer to red the bigger is the population, probably close to the carrying capacity. Then, closer to blue means the population tends to 0.

And then, a clear tendency can be seen , where increasing those terms will, again, make the population go to 0. But that relation coming in both sides actually shows that, for instance, a stronger promoter could be used to increase the protein production.

**Iron-Lactate Detector:**

Use this reference table if you get confused when reading the equations and just want to remember the meaning of a term:

Parameter | Symbol |
---|---|

Protein of interest | P |

mRNA of the protein | mR |

Repressor | R_{1,2,3} |

Pathogen population | N |

P. agglomerans population | Na |

Translation/Secretion rate | θ |

Protein effect direct(1) or indirect(0) | ξ |

Repressor-DNA association constant | ω |

Basal production | γ |

Complex formation/release constant | α |

Toxicity | λ |

Intrinsic growth | r |

Pathogen carrying capacity in gut | K |

The second system is the Iron-Lactate detector, you can learn more about it here. In this system we have the same equations for protein, **P**, pathogen population, **N**, and protein mRNA, **mR**, only with some changes in the terms, that will be discussed in each equation. But in the Iron-Lactate Biosensor we have 3 other circuits modulating the expression of **mR**, tetR, represented by **R _{1}**, and lldR, represented by

**R**, as repressor of

_{2}**mR**, and dtxR as a repressor of tetR.

There is also a scheme to make the visualization easier:

The expression here is initially very similar, with a regular expression, γ_{mR}, and the degradation rate, μ_{mR}. But here we have a different system, since there is the action of the repressors tetR, R_{1}, and lldR, R_{2}. For each repressor we have a term, ω_{1R} and ω_{2R} respectively, of association of the repressor and the DNA. In the case of lldR there’s also the association with lactate, which deactivates the repressor, times a constant of association, ɑ_{2l}.

The repressors have a very similar equation, since we considered all of their expression in one single term, assuming a low variation on taxes of RNA transcription and protein translation. So they are basically described a basal expression, γ, minus a degradation rate.

The only difference is in the repressor tetR, R_{1}, as it expression is controlled dtxR, R_{3}, in presence of iron ions, following the same logic of the repression expressed before.

The protein, **P**, and pathogen population, **N**, have basically the same equations. The only change in the protein equation is that know the traduction occurs directly from protein mRNA e not from an intermediary complex.