The modeling consists of three parts:

1. modeling of the plane distribution of protons
2. the aggregation circuit of quorum sensing.
3. the generator model.

Part 1 The Plane 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 utilize 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 the distribution of protons and get the solutions of diffusion according to Fick's Laws of Diffusion: $$j=-D\frac{\partial C}{\partial x}$$ where $J$ is the diffusion flux. $J$ measures the amount of substance that will flow through a small area during time interval. $D$ is the diffusion coefficient or diffusivity in dimensions of $length^2 time^{−1}$ $C$ is the concentration in dimensions $X$ is the position. Method Formation of hydrogen ions’ concentration gradient In our assumption, we first design a concentration gradient of hydrogen ions’ concentration as in any other traditional practice in the water channel. We filled two pools with waste water of different pH values. Then, just link the two by a water channel. As the channel may be rather small comparing to the pools, we may simulate that the pH value of two pools keep the stable pH value of its own in a long time. In the end, the hydrogen ions’ concentration gradient can be constructed in the water channel. The formulation can be listed below. Supposing that the length of water channel may be $d$ The y-axis is along the gradient, while the x-axis is perpendicular to y-axis. When the concentration is steady, we shall assume that the concentration close to acidic water is a, and the concentration close to alkaline water is $b$. Hence, concentration $C$ $$C(x,0,t)=a$$ $$C(x,d,t)=b$$ The diffusion is defined by $$\frac{\partial C}{\partial t}=-D\nabla ^2C$$ $$\frac{\partial C}{\partial t}=-D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\partial^2 C}{\partial y^2}\right)$$ When the concentration is steady, differential of C by t is zero. The equation is as followed： $$\frac{\partial C}{\partial t}=0$$ $$\frac{\partial^2 C}{\partial x^2}+\frac{\partial^2 C}{\partial y^2}=0$$ The marginal conditions can be deduced as followed: $$C(x,0)=a$$ $$C(x,d)=b$$ The equation can be solved as followed: $$C(x,y)=a+\frac{(b-a)}{d}\times y$$

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: Fig.1 The plane distribution of protons in the culture medium
We divide the wasted water into two parts, the acid one and the alkaline one. Since the diffusion channel is quite small compared to the source of wasted water, we will not take the channel into our consideration. Thus, under the assumption of pH=6 in the acid part and pH=8 in the other part, and taking the source into consideration, we can get a new equation when t tends to infinitely great: $$C(r,\infty)=\lim_{t \to \infty}C_0\left(1-\frac{r}{\sqrt{DT\pi}}\right)=C_0$$ Therefore, when equilibrium is reached, pH in the acid part is always 6 and pH in the other part is 8. When these two parts neturalize after diffusion, the concentration of H+ decreases rapidly.
Fig. 3 distribution of H+ in the medium

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. The regulatory network: Fig.3 Regulatory network of aggregation circuit.

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 0.00288

Parameters	Brief Introduction	Value	Unit	Reference
$r_1$	Concentration of transcribed mRNAs of PureI	-	M	-
$r_2$	Concentration of transcribed mRNAs of PrLux	-	M	-
$r_3$	Concentration of transcribed mRNAs of P lamda CI	-	M	-
$r_4$	Concentration of transcribed mRNAs of PLac	-	M	-
$r_5$	Concentration of transcribed mRNAs of vgb	-	M	-
$r_6$	Concentration of transcribed mRNAs of fdfH	-	M	-
$\gamma$	Degradation rate of mRNA	0.00288	$s^{-1}$	Alon, Uri. "An Introduction to Systems Biology Design Principles of Biological Circiuts." London: Chapman & Hall/CRC, 2007.
$\delta_1$	Degradation rate of LuxI	0.00288	$s^{-1}$	Ref:iGEM Aberdeen Scotland Team
$\delta_2$	Degradation rate of AHL	0.00288	$h^{-1}$	Ref:iGEM ZJU Team
$\delta_3$	Degradation rate of LuxR	0.0231	$min^{-1}$	Ref: iGEM2008 Grenoble Team
$\delta_4$	Degradation rates protein A	0.00289	$s^{-1}$	Ref:iGEM ZJU Team
$\delta_5$	Degradation rate of CI	1/30+1/44	$min^{-1}$	Ref: iGEM2007 PKU Team
$\delta_6$	Degradation rate of LacI	0.046	$min^{-1}$	Ref: iGEM2011 Grenoble Team
$\delta_7$	Degradation rate of CsgD	-	-	-
$\delta_8$	Degregation rate of Prlux	0.0058	$s^{-1}$	Ref: iGEM2008 Groningen Team
$k$	Translation rate	0.1	$s^{-1}$	Assumption
$k_1$	dissociation contant	$9.63\times 10^{-5}$	$s^{-1}$	Assumption
$k_2$	dissociation contant	$9.63\times 10^{-5}$	$s^{-1}$	Assumption
$k_3$	dissociation contant	$9.63\times 10^{-5}$	$s^{-1}$	Assumption
$k_4$	dissociation contant	$9.63\times 10^{-5}$	$s^{-1}$	Assumption
$k_5$	dissociation contant	$9.63\times 10^{-5}$	$s^{-1}$	Assumption
$n$	Hill coefficient	2	-	Assumption
$m_1$	Max transcription rates of PureI	$6.55\times 10^{-5}$	$nM\cdot s^{-1}$	Assumption
$m_2$	Max transcription rates of PrLux	$7.89\times 10^{-5}$	$nM\cdot s^{-1}$	Assumption
$m_3$	Max transcription rates of P lamda CI	$6.55\times 10^{-5}$	$nM\cdot s^{-1}$	Assumption
$m_4$	Max transcription rates of PLac	$6.55\times 10^{-5}$	$nM\cdot s^{-1}$	Assumption
$m_5$	Max transcription rates of Plux	$2.36\times 10^{-5}$	$nM\cdot s^{-1}$	Assumption
$w_1$	Celluar death rate of CcdB	$4\times10^{-3}$		$nM^{-1}\cdot s h^{-1}$	You, L., et al. Nature, 2004. 428(6985): p. 868-71
$w_2$	Degregation rate of CcdB	2	$h^{-1}$	You, L., et al. Nature, 2004. 428(6985): p. 868-71
$w_3$	Transcription rate of CcdB	5	$h^{-1}$	You, L., et al. Nature, 2004. 428(6985): p. 868-71
$w_4$	Transcription rate of AHL	$4\times 10^{-7}$	$nM\cdot L \cdot s ^h{-1}$	You, L., et al. Nature, 2004. 428(6985): p. 868-71
$N$	Cell Density	-	-	-
$Nm$	Cell Density	0.9	$h^{-1}$	-
$x_2$	AHL intercellular diffusion	0.001	$mm^{2}\cdot s^{-1}$	-
$\alpha$	Transcription rate of Pvgd	-	$M\cdot s^{-1}$	Subhayu Basu,et al. Nature, 2005
$\beta$	Translation rate of fdhF	-	$M\cdot s^{-1}$	Subhayu Basu,et al. Nature, 2005

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 For the different oxygen distributed in the biofilm and different hydrogen ions’ distribution in the panel , we can solve these equations at different oxygen concentration and hydrogen ions’ gradient. The nonlinearity of the Hill function makes it a big obstacle for us to use ODEs for solving equations of the bacteria batteries’ pathway.

Simulation

In the above circuit, E.coli forms biofilm but what we concern is the activity of inner bacteria. Thus, this part of simulations focuses on the equilibrium of all substances。 Notice that we care about the final state of the bacteria in different concentration of hydrogen ions and oxygen condition in biofilms. The situation can be reached to the final steady without any change. We can assume that all of the variables differential by time can be equaled to zero. That is, let $$\frac{d[X]}{dt}=0$$ [X] stands for every substances of equation (1) –(14) Since all equations above are all constructed, the connection between CsgD and concentration of H+ can be derived from other substances when the state can be reached to equilibrium.The formulation can be calculated followed:
[IPTG] is external concentration of IPTG and its value sets to be zero during simulation. As we applied Pvgb to construct the biofilms, we should analyze the different concentration of oxygen which may inhibit or improve the activity of CsgD. The data below is derived from ZJU-China igem 2011.
From the figure, we can get the relationship between the distribution of oxygen in the biofilm and expression rate of GFP, RFP and YFP. Clearly, the behavior of this system can be determined by the two promoters, i.e., PureI and Pvgb. However, as the hill equations cannot be solved by OEDs. Then, we hope to illustrate the behaviors of this system precisely. We will discuss this in two cases. a) Under the assumption that the concentration of H+ where the distance from bacteria’s aggregating panel is not high enough to active the circuit to form biofilm, and the expression of Pvgb on the high concentration of oxygen is very low. So [H]=0,and a=0.Now the change of concentration of CsgD is shown in the following figure. b) Under the assumption that the concentration of H+ where the distance from bacteria’s aggregating panel is high enough to active the circuit to form biofilm, and the expression of Pvgb on the high concentration of oxygen is very enough. Now the change of concentration of CsgD is shown in following figure.