The modeling consists of three parts:
a) a modeling of the plane distribution of protons
b) the aggregation circuit of quorum sensing.

Part 1 The Spatial Distribution of Proton

Theory and Method

we use the promoter PureI which is sensitive to the concentration of H+ to start the process of quorum sensing of E.coli. As H+ is the foundation of following steps, we need to build an environment with a proper gradient of protons' concentration. Therefore, we use a pole which we can put acid into, and take it as the maximum concentration of acid (Fig.1). We utilized two poles, one of which is instilled by acid that can supply H+. With the diffusion of H+, a gradient of protons' concentration is able to form. A schematic view of the specific equipment is revealed as following. Correspondingly, We build (stimulated) the spatial distribution of protons and get the solutions of diffusion according to Fick's Laws of Diffusion: $$C\left(r,t\right)=C_0\mathrm{erfc}\left(\frac{r}{2\sqrt{Dt}}\right)$$ r is the radical distance from the pole; t is the time;C0 is the concentration of H+ at the pole(that is the concentration of acid(H+) that we add);the length is the length of diffusion used to measure the length that H+ diffuses in direction of x axis at time t. The length is called the diffusion length and provides a measure of how far the concentration has propagated in the x-direction by diffusion in time t .XX D is the diffusion coefficient; the diffusion coefficient of H+ is 9.1*10_8 m2/s, [1] http://www.ncbi.nlm.nih.gov/pubmed?term=15100022 erfc is complementary error function: $$\mathrm{erfc}=1-\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}\int_{x}^{\infty}e^{-t^2}}dt$$

Simulation

Take the first two terms of the complementary error function after Taylor series expansion as the approximate number:
As a quick approximation of the error function, the first 2 terms of the Taylor series can be used: $$C\left(r,t\right)=C_0\left(1-\frac{x}{\sqrt{Dt\pi}}\right)$$ We can get the image as followed:

Part 2 The Aggregation Circuit of Quorum-sensing

Basic Concepts and the Aggregation Circuit

The ODE formalism models the concentrations of RNAs, proteins, and other molecules by time-dependent variables with values contained in the set of nonnegative real numbers. Regulatory interactions take the form of functional and differential relations between the concentration variables. For a typical transcription-translation process, the ODEs modeling approach associates two ODEs with any given gene $i$; one modeling the rate of change of the concentration of the transcribed mRNA $r_i$, and the other describing the rate of change of the concentration of its corresponding translated protein $p_i$. Thus for our network with 3 genes we have: $$\frac{dr_i}{dt}=F\left(R_i(p_1), R_i(p_2), R_i(p_3)\right)-\gamma_ir_i$$ $$\frac{dp_i}{dt}=P_i(r_i)-\delta_ip_i$$ Where (1) describes transcription, (2) describes translation, and $i = 1,…,N$. The functions $R_i(p_j)$ describe the dependence of mRNA concentration on protein concentration $p_j$ (If protein $p_j$ has no effect on mRNA $r_i$, then correspond function is set to zero.) The functional $F(•$) in (1) is defined in terms of sums and products of functions $R_i$. Function $P_i$ in (2) describes the translation of the mRNA $r_i$ into a protein $p_i$. Parameters $\gamma_i$, $\delta_i (i = 1,…,N)$, represent the degradation parameters of the mRNAs and proteins produced by gene i. As is common, we shall assume that the degradation of proteins or mRNAs is not regulated, namely that it does not depend on the concentrations of other molecules in the cell. Function $R_i$ is assumed to be in the form of Hill function as usual (since our cases are all inhibitors, we shall denote the Hill function $h^-(p,K,n))$, and the function $P_i$ is taken to be a linear term proportional to the concentration of mRNA $r_i$. $$h^-\left(p_i;K_i,n_i\right)=\frac{K_i^{n_i}}{p_i^{n_i}+K_i^{n_i}}$$ where $K_i$ is the microscopic dissociation constant, and $n_i$ is Hill coefficient, describing cooperativity. The regulatory network:

Equations and Parameters

Based on the fundamental assumptions above, we use ODEs to stimulate the circuit above. Taking both time and space into account, we can get all the equations as followed: (r represents the radical distance of one point in this environment from the pole.) \begin{equation} \frac{d[H^+]}{dt}=\frac{\delta C(r,t)}{\delta t} \end{equation} \begin{equation} \frac{d[r_1]}{dt}=\frac{m_1}{1+\left(\frac{[H^+]}{K_1}\right)^n}-\gamma[r_1] \end{equation} \begin{equation} \frac{d[LuxI]}{dt}=k[r_1]\delta_1[LuxI] \end{equation} \begin{equation} \frac{d[AHL]}{dt}=\frac{V_1[LuxI]}{K_2+[LuxI]}-k_1[LuxR][AHL]-\delta _2[AHL] \end{equation} \begin{equation} \frac{d[LuxR]}{dt}=k[r_1]-k_1[LuxR][AHL]-\delta_3[LuxR] \end{equation} \begin{equation} \frac{d[A]}{dt}=k_1[LuxR][AHL]-\frac{m_2}{1+\left(\frac{[A]}{K_3}\right)^n}-\delta_4[A] \end{equation} \begin{equation} \frac{d[cI]}{dt}=k[r_2]+\alpha-\delta_5[cI] \end{equation} \begin{equation} \frac{d[r_3]}{dt}=\frac{m_3}{1+\left(\frac{[cI]}{K_4}\right)}-\gamma[r_3] \end{equation} \begin{equation} \frac{d[LacI]}{dt}=k[r_3]=k[IPTG]-\delta_6[LacI] \end{equation} \begin{equation} \frac{d[r_4]}{dt}=\frac{m_4}{1+\left(\frac{K_5}{[LacI]}\right)^n}-\gamma[r_4] \end{equation} \begin{equation} \frac{d[CsgD]}{dt}=k[r_4]-\delta_7[CsgD] \end{equation} \begin{equation} \frac{d[CsgD]}{dt}=k[r_4]-\delta_7[CsgD] \end{equation} The parameter in these equations are listed in the table 1: Table 1 Meaning Name Unit Value Reference Concentration of transcribed mRNAs of promoter ureI r1 M - - Concentration of transcribed mRNAs of promoter rLux r2 M - - Concentration of transcribed mRNAs of promoter r012 r3 M - - Concentration of transcribed mRNAs of promoter Lac r4 M - - mRNA degradation rates γ 0.00288 Degradation rates protein LuxI δ1 0.00289 Degradation rates protein AHL δ2 0.00289 Degradation rates protein LuxR δ3 0.00289 Degradation rates protein A δ4 0.00289 Degradation rates protein cI δ5 0.00289 Degradation rates protein LacI δ6 0.00289 Degradation rates protein CsgD δ7 0.00289 Translation rates k 0.100 K1 K2 K3 K4 K5 m1 M/s m2 M/s m3 M/s m4 M/s K’ Hill coefficient n - 2.00 assumped Transcription rates of promoter vgb(function of oxygen concentration) M/s - - For a certain depth of the biofilm, the concentration of oxygen is a constant in our model . Therefore we could solve these equations at different oxygen concentration and combine all the results to show how this system work. Because of the nonlinearity of the Hill functions, the solutions of a system of ordinary differential equations of a network of many genes cannot generally be determined by analytical means. http://www.ncbi.nlm.nih.gov/pubmed?term=15100022