Team:Peking/ProjectApplication

Introduction

In order to the estimate the effect of killing and improve the application methods, other than varied tests in laboratory, we also developed a macro-level model to examine whether our E. coli will work well considering spatial distribution and diffusion. We described interaction and diffusion of elements in the fresh water system by using Partial Differential Equations (PDEs). Even though we didn’t know the analytical expression of interaction of elements in water, we can propose a possible interaction function, and analyze its spatial properties, in order to find an optimized way to pour bacterium solution.

Methods

Finite-difference methods are applied in solving PDEs in this model.

Diffuision

Differentials to space coordinates are given by Fick Law. The length of a time step is approximated to Δt = 1min, and the discretization of space coordinates is approximated to Δs = 1dm (s = x, y, z), considering the scale of boundary is 10²m and the time scale is 10¹ days.

Interaction

A, E, L indicate the concentration of cyanobacteria, our programmed E. coli, lysozyme which kills cyanobacteria. We also introduced a variable which described the organic nutrition released by cracked cyanobacteria (indicated by N), which contributed to respiration of E. coli. Interacitons between them are given below.

The first term in Equ(1) describes the growth of cyanobacteria by logistic model, in which K1 indicate the growth rate of cyanobacteria and k2 indicate the steady value. The second term means the killing effect by lysozyme. Equ(2) describes the accumulation and consumption of organic nutrition. Equ(3) describes the reproduction and natural mortality of our E. coli and Equ(4) describes secretion and reduction of lysozyme.

There are several factors which are not considered in equations above such as nutrition accumulation by cyanobacteria which dies naturally, degradation of lysozyme, evaluation by binding part and so on, because they are either too small compared with other items, or they are irrelevant to the issue that we care. As this model is focused on the application in macro-scale water system, accurate parameters are not required meanwhile not available, considering the form of equations could not be proved. So some parameters were given by speculation to satisfy our expactation.

Spatial Analysis

We controlled the total volume of bacteriuma solution, and we tried several basic mode of pouring: at one point, at several points, along a line or evenly on the whole surface. Then we measured the time scale of cyanobacteria elimination. From this, we could find out if mode of pouring

Results

Parameters manipulation

Ignoring the diffusion, the PDEs were converted to Ordinary Differential Equations (ODE).

Speculatively the equations have two typical forms of solution, indicating the success and failure of the project. For there is a positive feedback mechanism by introducing organic nutrition, oscillatory solution could not exist.

Through the test, we noticed that the reasonable type of the solution is the only "success" (Fig. 1) regardless of reasonable variation of K₉ and K₅. The other parameters were not the crucial factor of the type of the solution as the magnitude of order are estimable, for example, K₆, the growth rate of E. coli.

Spatial analysis

We set a 11m×11m×10m test space with well-distributed cyanobacteria. Then we tested 4 basic mode of pouring the bacteriuma solution. At the center point of the surface (Fig. 2A), at four points near the border (Fig. 2B), along with a line (Fig. 2C) and on the whole surface (Fig. 2D).

We found that the time scales of cyanobacteria elimination are almost the same in the four conditions. We thought it is because of the positive feedback mechanic by introducing the organic nutrition and limited border. Although we didn't get instructional results from this test, more reasonable result we will get if changing the form of interaction and border condition more reasonable and this model will contribute more to our project in application.