Team:Tsinghua/Model

Modelling
The Domino processing starts with one or more external signal, which is transmitted and amplified by E.coli population till finally a computation output achieved. This is accomplished within an intact biofilm that forms on agar medium plate, a system that comprises multidimensional features. To illustrate the whole process clear cut through quantitative analysis, a set of time (t) and space(r) related differential equations is established to satisfy the mathematical model of Domino E.coli Community.

Time/space related equations of concentration
In this case, three particular molecules made by Domino E.coli Community are responsible to fulfill this work: C6HSL (synthesized by luxI), cI repressor (expression product of cI gene) and C12HSL (synthesized by lasR), each amount of those is a time and space related function, say, A(r, t), B(r, t), C(r, t). Due to the diffusible property of small molecules A (C6HSL) and B (C12HSL), the flux J of each is determined not only by time, but also by gradient of its position (refer to Fick’s first law). Listed below are the simultaneous differential equations of the three concentrations.

Equations of biosynthesis and the matrix expression
Here we use A, I, C to refer to the amount of AHL, lacI, cI, while m_A, m_I, m_C stand for the mRNA level, respectively. Like all the rest of living organisms, biosynthesis within Domino E.coli individual is ruled by the central dogma, the process of transcription and translation. Since biological macromolecules are often controlled by synthesis and degradation, a set of parameters are introduced, k_i for synthesis rate constant, l_i for degradation rate constant. Following are the simultaneous differential equations of key biosynthetic process in Domino E.coli.

*mRNA transcription of cI is inhibited by the binding of lacI inhibitor to lacO operator, thus k₅/I^n_I' is preferred rather than k₅I^n_I'.

For conciseness and beauteousness, it can be rewritten into the matrix form:

The vector function X:

The domino circuit matrix L:

In this case, m_A and m_I are controlled by the same promoter pLux, thus k₁=k₃.
The Hill coefficient vector:

Solution to the time related equations

However, things are more complicated under physical circumstances, AHL would not increase limitlessly within individual bacteria, mRNA level would finally reach equilibrium. That’s what Hill coefficient vector deals with. For convenience of derivation, we might let H=[1,1,1,1,1,1,1,1,1,-1,1,1], but in practical situation, we have to take into account that copy number of promoter is limited, and transcriptional factor binding to promoter could be saturated as well. AHL bound luxR acts as a transcriptional factor that recovers plux promoter activity from HNS. Let pmax=k₁A, i.e. plux promoter is fully activated. Although some of the equations are analytic unsolvable, a software Matlab (ver. 2010b) is capable of circumvent this through arithmetic solution.

Figure m.1 is arithmetical solved using Matlab, which suggests a clear positive correlation of lacI and A, controlled by the same promoter plux. Notice the y axis stands for the relative amount, i.e. the total amount divided by the saturated amount. Figure m.2, on the other hand, represents a negative correlation amongst lacI, mRNA of cI and cI. lacI plays a role of bridge combining A and cI, which exhibits similar synthesis curve with A, and inhibitory effect on cI.

Spatial distribution of concentration

Equations of flux describes the transmission rate of signal molecules (A and B) in spatial dimensions. In the very occasion we consider situation only in one dimension ,x axis. According to Fick’s second law, the change of concentration with time is described below.

Circuits of A and B resemble thus it needs only to examine either of them, and A is preferred for the reason of inhibitory effect on cI synthesis. The origin of x axis refers to where the initial signal is received and domino effect is triggered. Along with A, cI protein level shifts in the opposite manner: when A is synthesized and growing up, the already saturated cI starts to degrade with its magnitude drops down (figure m.3 and figure m.4, plotted using Matlab 2010b).

Hybrid promoter and threshold

plasR/cI Hybrid promoter containing lasR and cI binding site will be activated when the concentration of signal B(C12HSL) exceeds the threshold and the expression of cI repressor is below the threshold. As mentioned above, signal A (C6HSL) and signal B (C12HSL) will rise with time until reach the peak, while cI repressor decreases all the way, antagonizing but the tendency of A. This antagonism effect is transmitted away from, considering the simplest situation, the origin of x axis upon the culture medium, forming a one dimensional wave. Meanwhile, a second signal B triggers another wave, which leads to no variation of cI repressor though, moving against forward direction from the terminus of x axis. When the frontal edges of two waves intersect, a region where B is above the threshold and cI below its own, could be visualized through the expression of reporter RFP.