Team:HZAU-China/Model
Model
The distribution density of bacteria in a culture medium depends on two factors: motion and reproduction of bacteria. For convenience, we will discuss these two factors separately.
Dynamic model of bacteria motion
In this project, we are trying to control the motion of bacteria by light signal to generate a colony of a specific shape. The light signal can affect the bacteria motility by influencing the expression of a protein cheZ. There are two motion forms of bacteria: tumbling and swimming. By tumbling, bacteria can change swimming direction but not position; by swimming, bacteria will move forward and change the position. In fact, the ratio of tumbling to swimming is different from one bacterium to another. At the microscopic level, the angle of tumbling is random, so we have no idea about the movement direction of a bacterium in any time. But from a macroscopic viewpoint, the probability of all directions are equal. This property is similar to the diffusion of a chemical molecule. So, bacteria swimming can be treated as the diffusion of bacteria and the rate of diffusion is related to the expression of protein cheZ.
Figure 1. The 2D plane of bacteria diffusion.
In Figure 1, the simple square represents the area of bacteria in coordinate \((x,y)\) approximately. The number of bacteria in this area is \(S(x,y,t)\) at time t. So, we have
\(S(x,y,t)=\rho(x,y,t)*\Delta x\Delta y\). \((1)\)
In Equation \((1)\), \(\rho(x,y,t)\) represents the bacteria density in that area. \(\Delta x\) and \(\Delta y\) are the small increments along the \(x\) and \(y\) axes, respectively. Taking the derivative of Equation (1), we have
\(\frac {\partial S(x,y,t)}{\partial t}=\frac {\partial\rho(x,y,t)*\Delta x\Delta y}{\partial t}\). \((2)\)
The left side of Equation \((2)\) represents the change rate of \(S(x,y,t)\). Ignoring the reproduction of bacteria, the change rate depends only on the rate of bacteria diffusion. As shown in Figure 1, bacteria can move along the \(x\) axis and the \(y\) axis. Suppose the movements along the \(x\) and \(y\) axes are positive, we arrive at
\(\frac {\partial\rho (x,y,t)\Delta x\Delta y}{\partial t}=\phi_x(x,y,t)\Delta y- \phi_x(x+\Delta x,y,t)\Delta y+\phi_y(x,y,t)\Delta x-\phi_y(x,y+\Delta y,t)\Delta x\). \((3)\)
In Equation \((3)\), \(\phi_x\) and \(\phi_y\) represent the diffusion of bacteria in \(x\) and \(y\) axes, respectively. Dividing \(\Delta x*\Delta y\) on both sides of Equation \((3)\), and we can get
\(\frac {\partial\rho(x,y,t)}{\partial t}=\frac {\phi_x(x,y,t)-\phi_x(x+\Delta x,y,t)}{\Delta x}+\frac {\phi_y(x,y,t)-\phi_y(x,y+\Delta y,t)}{\Delta y}\). \((4)\)
Let \(\Delta x\) and \(\Delta y\) take a limit to 0, Equation \((4)\) will become
\(\frac {\partial\rho(x,y,t)}{\partial t}=\lim_{\Delta x\rightarrow 0}\frac {\phi_x(x,y,t)-\phi_x(x+\Delta x,y,t)}{\Delta x}+\lim_{\Delta x\rightarrow 0}\frac {\phi_y(x,y,t)-\phi_y(x,y+\Delta y,t)}{\Delta y}\). \((5)\)
This equation is equivalent to
\(\frac{\partial \rho}{\partial t}=-(\frac{\partial\phi_x}{\partial x}+\frac{\partial\phi_y}{\partial y})\). \((6)\)
In this equation, \(\phi_x\) and \(\phi_y\) are unknown. In fact, the diffusion rates along \(x\) and \(y\) axes are the same. Refer to the assumption of Fourier Heat Equation:
1. If the temperature is constant within an area, there is no flow of heat.
2. If temperature difference exists in adjacent areas, heat will flow from areas of high temperature to low temperature.
3. For the same kind of material, the bigger the temperature difference between two adjacent areas, the faster the heat flow between them.
We have three similar characters for the diffusion of bacteria colony:
1. If the density of bacteria is constant in an area, the movement of bacteria will not lead to the change of density.
2. If density difference exists in adjacent areas, bacteria will swim from high density to low density areas.
3. The bigger the density difference between two adjacent areas, the faster the diffusion rate between them.
So, the rate of diffusion is related with the difference of density:
\(\phi_x = -k\frac{\partial\rho}{\partial x}\), \((7)\)
\(\phi_y = -k\frac{\partial\rho}{\partial y}\). \((8)\)
Combining Equation\((6)\), Equation\((7)\) and Equation\((8)\), we can get
\(\frac{\partial\rho}{\partial t} = k(\frac{\partial^2\rho}{\partial x^2}+\frac{\partial^2\rho}{\partial y^2})\). \((9)\)
Equation (9) is the dynamic model of bacteria motion.
Dynamic model of bacteria reproduction
There is a famous model about the growth of bacteria under the conditions of limited space and nutrients, called logistic growth model:
\(\frac{d\rho}{dt} = \gamma\rho(1-\frac{\rho}{\rho_s})\). \((10)\)
In Equation \((10)\), \(\rho\) represents the density of bacteria, and \(\gamma\) is the growth rate constant, and \(\rho_s\) is the saturated density.
Dynamic model of both reproduction and motion (R-M model)
Combining equation \((9)\) and equation \((10)\), we have
\(\frac{\partial\rho}{\partial t}=k(\frac{\partial^2\rho}{\partial x^2}+\frac{\partial^2\rho}{\partial y^2})+\gamma\rho(1-\frac{\rho}{\rho_s})\). \((11)\)
Referring to the paper by Liu et al. [1], we have the parameter values:
$$k=200~1000\mu m^2/s,$$ $$\gamma = 3.89e-4s^{-1},$$ $$\rho_s = 1500cell/\mu m^2.$$
We will use the Finite Element Method to solve this PDE. Initial conditions are defined as
$$\rho=matrix.zeros(200,200),$$ $$\rho[100,100]=100,$$
where \(matrix.zeros(200, 200)\) means a zero matrix with \(200*200\) elements and the initial value of the element at the index \([100, 100]\) of this matrix is \(100\). And the boundary conditions are
$$\rho[0:,]=0,$$ $$\rho[100:,0]=0,$$ $$\rho[0,0:]=0,$$ $$\rho[0,100:]=0.$$
With the initial and boundary conditions, Equation \((11)\) was solved by writing python program with Numpy and the result was visualized by the Mayavi python package, as shown in the following video (Movie 1).
Movie 1. Visualization for the R-M model of bacteria diffusion.
R-M model in a restricted area
In Movie 1, we can see that bacteria colony will eventually become a round shape. If we add a restrictive condition to the area to limit the bacteria movement, what shape will the bacteria colony become? The way to add area limit in our project is using light signal to control the motility of bacteria. Namely, green light is given to a specific area and red light is given to other area; bacteria can only move in the green light area. In Equation (11), bacterial motility is reflected by parameter k. So the model will be the following equations:
\(\frac{\partial\rho}{\partial t}=k(\frac{\partial^2\rho}{\partial x^2}+\frac{\partial^2\rho}{\partial y^2})+\gamma\rho(1-\frac{\rho}{\rho_s})\), \((11)\)
\(k=f(x,y)\). \((12)\)
The parameter \(k\) in Equation \((11)\) indicates the diffusion rate of the bacteria colony. \(f(x,y)\) is the area limit function, and it is an image matrix. In this project, we choose Pikachu (a famous cartoon character in the hot AR game "Pokimon Go") as our target pattern. See Figure 2.
Figure 2. Pikachu,a famous cartoon character.
The black part in Figure 2 means that the parameter \(k\) is of normal value. That is, \(k=k_{normal}=200~1000\mu m^2/s\). The white part in Figure 2 means that the parameter \(k\) takes small value which will be set to \(0\). Because \(k\) is a matrix in Equation \((11)\), it can be solved out by using Figure 2:
$$k = img*k_{normal}/255.$$
Under the aforementioned initial and boundary conditions, using python program to solve these equations, we can get the following visualization video (Movie 2).
Movie 2. Visualization for the R-M model of bacteria diffusion in a restricted area using Pikachu as the target pattern.
From Movie 2 we can see that the target pattern is basically formed. However, such a pattern formation is just equivalent to making a cookie using a mould. In this project, the pattern formation process will be regulated by a computer by real-time communication.
R-M model with dynamic regulation
Area limit is a kind of static regulation, but dynamic regulation is further explored in our project. In dynamic regulation, we will compare the shape of the bacteria colony with the target picture by real-time communication between the computer simulated virtual space and the real space in the culture medium through light, and the result of comparison will be converted to light signals shedding on the bacteria colony to control bacteria diffusion. So, the dynamic model with regulation will like the following equations:
\(\frac{\partial\rho}{\partial t}=k(\frac{\partial^2\rho}{\partial x^2}+\frac{\partial^2\rho}{\partial y^2})+\gamma\rho(1-\frac{\rho}{\rho_s})\), \((11)\)
\(k=f(t,\rho,img)\). \((13)\)
Unlike Equation \((12)\), there are two additional parameters in Equation \((13)\). The two additional parameters are time \(t\) and density \(\rho\), respectively. That means parameter \(k\) will change in the pattern formation process by adjustment. Equation\((13)\) can be interpreted as:
\(\rho_1=\rho*\frac{255}{max(\rho)}\), \((14)\)
\(\rho_2=threshold(\rho_1,THRES\_BINARY\_INV)\), \((15)\)
\(edges=Canny(\rho_2)\), \((16)\)
\(edges_2=bitwise\_and(edges, mask=img)\), \((17)\)
\(k_1=dilate(edges_2.kernel)\), \((18)\)
\(k_2=bitwise\_and(k_1,mask=img)\), \((19)\)
\(k=k_2*\frac{k_{normal}}{255}\). \((20)\)
Equations \((14)\rightarrow(20)\) are written in OpenCV language. Equation \((14)\) is for transforming the matrix of bacteria colony to an image. Equation \((15)\) is for image binaryzation. Equation \((16)\) is for finding the edge of an image. Equation \((17)\) is for comparing the current image with the target image. Equation \((18)\) is for expanding the edge. Equation \((19)\) is to prevent overflow from the edge. Equation \((20)\) transforms the result of Equation \((19)\) to light signal. Under the aforementioned initial and boundary conditions, using python program to solve these equations, we can get the result.
Movie 3. Visualization for the R-M model of bacteria diffusion with dynamic regulation using Pikachu as the target pattern.
You can download all the modeling programs from here.
References
1. Chenli liu, et al. Sequential Establishment of Stripe Patterns in an Expanding Cell Population. Science 334, 238. (2011)
2. Subhayu Basu and Yoram Gerchman, A synthetic multicellular system for programmed pattern formation. Nature Vol 434. (2005)
