Mathematical model

Overview

Here, we will discuss the dynamics and interactions of chemical species introduced in the research section. We have developed a simplified mathematical model describing our system and based on that, an interactive simulation that can be controlled in real time. Based on this, we created a simplified visualization available on our web page. With this, the intended working of our system can be easily demonstrated to any audience.

Simplifications

The first model that was constructed before our lab work even begun involves many harsh simplifications. Our aim was to get a general picture of how the system could work in ideal conditions and how stable it was.

We assumed that the species identified from our gene circuit are the only ones that affect the overall concentrations in our bacterial culture. The bonding of CI to OR sites is assumed to be insignificant compared to overall concentration. The model is also strictly deterministic and doesn’t take into account any noise. The phosphorylation, decay, bonding and production of proteins are assumed to be linear functions of concentration.

Equations [to be inserted]

Based on the assumptions made before, we arrived at following differential equations to describe the idealized dynamics of our system:

These equations describe the essential proteins our system (YF1, FixJ, Phosphorylated YF1, Phosphorylated FixJ, CI, TetR) Proteins are produced depending on the strength of promoter and ribosome binding site, and also when phosphorylated protein (denoted with phos) is dephosphorylated back to its original form. The concentration of all proteins is reduced by decaying, which depends on the concentration of protein in question.

Coefficients

P1, P2, PA and PB denote the relative strengths of promoters and Rbs1&Rbs2 the relative strengths of ribosome binding sites, which both affect the protein synthesis linearly. Each protein has its own degradation coefficient (denoted Deg). I(B) is the combined effect of blue light that affects the phosphorylation of YF1. The phosphorylation of FixJ is assumed to depend on phosphorylation coefficient C(phos) and the concentration of phosphorylated YF1. The dephosphorylation here depends on a respective dephosphorylation constant DP(1&2 for YF1 and FixJ).

Promoters

Lights

Here, the C's denote the respective promoter's maximum activity. The N in front of CI and TetR concentrations is a normalization coefficient, which is needed to adjust the concentration so that it decreases the promotor activity from 1 to 0 times the maximum. The fuctions definitions must also change so that they never take negative values, which would make no sense when talking about promoter activities.

Simulation

Lights

[Blue light affects phosphorylation]

[Red light affects the rate of production as an idealized, completely arbitrary linear coefficient]

Runge-Kutta method

The dynamics of our system were approximated using 4th order Runge-Kutta method for the differential equations in our mathematical model [insert Runge-Kutta description]

Parameters

[we used completely arbitrary estimates of actual parameters / we consulted the followin publications to obtain desider values: [insert list]]

Software implementation

A computational model was created based on our mathematical model and the Runge-Kutta approximation. We made a real-time plotting function to illustrate the dynamics with each timestep. We added two light switches so that the user can have an impact on our simulation in real time. This all was then further developed into a presentable, user-friendly form that is accessible from our website. The simulation itself was developed using Python and translated into Javascript for web implementation [moar coming]

Accuracy

No noise, arbitrary parameters, simplified pathways and reaction equations -> over idealized, requires a lot of measurement of appropriate parameters and tuning to be realistic and accurate. Ideal for demonstration of our idea [insert in-depth error analysis]