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  Introduction

  E.casso uses the communication between cells made available by quorum sensing. Quorum diffuses away from one cell to the other. Because we wanted to find out how the quorums produced by Brush E.coli diffuse in our model, we applied the reaction-diffusion system to our model and analyzed the distribution and propagation of quorums. This section investigates how the quorums diffuse from one cell to the other which will provide crucial data for our final simulation.

  Objectives

  Answer the following questions,
- How fast does quorum diffuse into adjacent E. coli as rapidly as we predicted?
- How does the produced quorum diffuse with respect to time and space?
- How much quorum does the source E. coli ultimately transfer to neighboring E. coli?

  Modeling Approach

  Fig 1 | (a) Approximation of the dimensions of E.coli as a square having equal area (b) Schematic representation of E.coli distribution model on a grid

  According to the statistics on E. coli¹, the shape of E.coli is oval with 200nm minor axis and 2um major axis. For simulating our random diffusion system in E.casso, we simply the E.coli shape as square having equal area with the area of oval. (Fig 1a)

  where d is the side length of the square

  Based on this calculation, we can assume a two dimensional distribution model in which E.colis are distributed on a grid that consists of squares of side 1.1 micrometer. (Fig 1b) In each square, there may exist a brush E.coli, a paint E.coli, or nothing.

  According to the statistics on E. coli¹, the speed of a small molecule in the cytoplasm is 50 nanometers per millisecond. Also, all molecules move by random walk during diffusion.²   Using Python, we ran a simulation on the random walk of a quorum molecule. It was assumed that each step takes 0.2 milliseconds. [python code]

  Fig 2 | A two-dimensional random walk of a quorum molecule simulated using python language. The simulation was repeated a hundred times.

  For each trial, we measured the time it took the molecule to go out of bounds. Averaging the time for a hundred trials yielded that a quorum molecule takes about 449 milliseconds to diffuse to an adjacent cell. (Fig 2) In other words, it takes about 0.45 seconds for a quorum molecule produced in one cell to diffuse into the center of a nearby cell. Since this is much smaller than the minute time-scale considered in our following analysis, we can ignore the diffusion speed.

  Our quorum diffusion system follows a mathematical description of the change in concentration of one or more substances in space as a result of diffusion or chemical reaction.

  where q(x, t) is the distribution of the number of a substance as a function of time and distance from a reference point, D is a diagonal matrix of Diffusion coefficients, and R is an equation of local reaction.

  In regard to quorum, the equation can be written as,

  where R(q) stands for the diffusion reaction. By Fick's second law⁴,

  The final equation becomes⁵,

  where diffusion coefficient of Autoinducer ⁶
  Initially, the concentration of quorum is zero. Hence, q(x,0) is zero. The boundaries conditions are that the rate of production of quorum is constant and that the concentration approaches zero as distance increases. From the previous results, we obtain that 40 quorums are produced per minute.

  Results & Conclusion

  Using the pdepe function which indicates partial differential equation in MATLAB and considering E. coli as a point in space, the following graph represents the concentration as a function of time and distance. [MATLAB code]

  Fig 3 | Quorum Diffusion from E.coli1 computed with 100 mesh points in (a) three and (b) two dimensions.

  As expected, q decreases as distance increases, and increases as time progresses. (Fig. 3)

  In the distribution model, the distinct between two horizontal and two vertical adjacent cells are 1.1um, and the distinct between four adjacent diagonal cells is1.56 um.(Fig 1) Therefore, the average distinct between any two adjacent cells is 1.33um. Essentially, q(1.3 um, t) tells us the concentration of quorum that diffuses from an E. coli to an adjacent cell as a function of time.

  If we plot this using MATLAB, we can represent as in the following graph the concentration of quorum that diffuses to an adjacent cell as a function of time. [MATLAB code]

  Fig 4 | The quorum concentration at a distance x away from E.coli as a function of time during the first (a) 10min and (b)100min