Multi-Systems Model
Fulfilled the first of our objectives, we could aim the next target: Predict the behaviour of more complex constructs, using more than one Quorum Sensing System!
Using the parameters results, more specifically α, we can try to predict the interaction between two or more systems when used together, being able to forecast their behaviour in different applications. For this, at first, we need to increase the complexity of our model, making a combination of two systems of equations. So, as usual in math, we start with the simplest case scenario: Two systems producing two different proteins.
In this new proposal, some new parameters and interactions emerge, due to new properties of the interaction: Now we have not only the promoter crosstalk, but also the chemical crosstalk, the interaction between the Homoserine-lactone and the receptor of different systems.
Therefore, using the first model as base, we can go directly to the visual representation of the system, so we can visualize it. In the likes of the first model, we have the following notation:
- Sphere – Variable of the system
- Line – Relation between the systems
- Arrow - indicates the directionality of the system (back of the arrow decreases and front of the arrow grows
- Bar (In the line) – Indicates no decreasing of the variable with the line
Also, the term above the line/in the right of the line, is the parameter that conducts the relation.
In this representation, we have two generic Systems, X and Y, each one producing a protein, Px and Py, respectively. Each parameter/variable has an index indicating which system they belong to. With that in mind, we can construct the model itself:
$$ \frac{dS_{X}}{dt} = \alpha_{Sx} - \mu_{Sx} S \\ \frac{dH_{X}}{dt} = \alpha_{Hx} S + \theta C_{XX} + \theta C_{YX} - H( \mu_{Hx} H + \beta_{XX} R_{X} \beta_{YX} R_{Y})\\ \frac{dR_{X}}{dt} = \alpha_{Rx} + \theta C_{XX} + \theta C_{XY} - R( \mu_{Rx} R + \beta_{XX} H_{X} + \beta_{XY} H_{Y}) \\ \frac{dS_{Y}}{dt} = \alpha_{Sy} - \mu_{Sy} S \\ \frac{dH_{Y}}{dt} = \alpha_{Hy} S + \theta C_{YY} + \theta C_{XY} - H( \mu_{Hy} H + \beta_{XY} R_{X} + \beta_{YY} R_{Y})\\ \frac{dR_{Y}}{dt} = \alpha_{Ry} + \theta C_{YY} + \theta C_{YX} - R( \mu_{Ry} R + \beta_{YX} H_{X} + \beta_{YY} H_{Y}) \\ \frac{dC_{XX}}{dt} = \beta_{XX} R_{X} H_{X} - C (\mu_{C} + \theta_{XX})\\ \frac{dC_{XY}}{dt} = \beta_{XY} R_{X} H_{Y} - C (\mu_{C} + \theta_{XY})\\ \frac{dC_{YY}}{dt} = \beta_{YY} R_{Y} H_{Y} - C (\mu_{C} + \theta_{XY})\\ \frac{dC_{YX}}{dt} = \beta_{YX} R_{Y} H_{X} - C (\mu_{C} + \theta`_{YY})\\ \frac{dP_{X}}{dt} = \alpha_{PX} C_{XX} + \alpha_{PX_{XY}}C_{XY} + \alpha_{PX_{YY}} C_{YY} + \alpha_{PX_{YX}}C_{YX} - P_{X} (\mu_{Px})\\ \frac{dP_{Y}}{dt} =\alpha_{PY} C_{XX} + \alpha_{PY_{XY}}C_{XY} + \alpha_{PY_{YY}} C_{YY} + \alpha_{PY_{YX}}C_{YX} - P_{Y} (\mu_{Py})\\ $$Now, before running the simulations with that model, we need to do some assumptions about the parameters. At first, let’s consider that αSx and αSy are the same, as well as αRx and αRy. Also, that consideration is valid for all the μ’s, θ¬Cxx and θ¬Cyy, θ¬Cxy and θ¬Cyx, βCxx and βCyy, βCxy and βCyx.
Considering this system, we now can escalate to a more generic system of equations, with N systems integrated! For that, we use this first combination, with two systems, and then generalizes to N. Where we find the following ODE’s: