The first aim of our modeling is to estimate AHL production rate κm in marine cells. The synthesis of AHLs by marine bacteria involves the transcription of certain genes when the secondary structure of RNA thermometers was melted and then output Lux I promotes AHL production by catalyzing the reaction of the two AHL precursor molecules, S-adenosylmethionine (SAM) and hexanoyl-ACP (acyl carrier protein).

The reaction can be summarized as,

Where S, C6, A and M are SAM, hexanoyl-ACP, 3-oxohexanoyl-homoserine lactone (AHL) and the waste product, respectively.

In marine construct, output of Lux I occurs at a constant rate L0 as indicated in Figure 1 and decays at a rate of λL. Noted that λL was not the natural decay rate, the degradation of LuxI was accelerated by introducing LVA tail in marine cells.

We further assumed that the concentrations of SAM and, C6-ACP are at a constant levels S0 and C0, respectively. Hence, we may model the production rate of AHL by,

The background noise production rate is donated as kd and ka is the AHL production rates due to the initiation process.

Compared to the timescales of interest, cell division time, these internal biochemical reactions occur very quickly, and it is reasonable to assume that equation (1) is in equilibrium (mathematically, it means dL/dt =0). Combining the result with (2) gives

When polar and coast cells were initiated by diffusing AHLs, they would start to produce chromoproteins. However, it is very hard for us to determine how many chromoproteins that those cells produced unless we disassembled the cells and purified proteins multiple times. Thus, for this reason, the second attempt of this modeling practice was to investigate the initiation status in polar and coast cells.

To achieve this thought, we need to find the diffusion pattern of AHL toward/from the medium. We introduced the factor η=σ⋅A/Vc based on the work of Garcia-Ojalvo (2003) and ETH-Zurich 2013 igem project (2013), with σ representing the membrane permeability, A is surface area and Vc is the cell volume, and ηext=δ/Vext with Vext being the total extracellular volume. Hence,

in which Ae was external AHL concentration and Am was internal concentration of AHL in marine cells.

The marine cell concentration nm change rate donated as,

Where K is the carrying capacity and α is the intrinsic growth rate.

We assumed that, for the location of coast and polar cells, external AHL concentration was consistent with Ae. Hence,

The production of chromoproteins depend on both the concentration of AHL (A) and growing stage. We use a Hill function (Eq. (6) and Eq. (7)) to model the AHL-dependence of chromoprotein synthesis. The Hill function provides cooperative switching from the synthesis-off to the synthesis-on states, via two parameters a and m,

Those two cooperative switch function gave the insight of the producing rate of different chromoproteins in polar and coast constructs.