Team:BUCT-China/Model

Summary
Description
Introduction
Assumptions
Definitions
Establishment
Conclusions
References
Appendix

Model

Summary sheet

It is well known that many biological experiments are aimed at isolated cell communities. From the perspective of theoretical verification, the characteristic information of cells, including the total number of cells growing in real time, distribution location and the relative proportion of cells in different cycles, is very key and important. These information are related to cell type, culture medium type and concentration. Therefore, the basic idea of this study is to use mathematical and physical tools to build a prediction model of cell growth distribution, simulate the growth of real cells, and verify the results in combination with biological experiments, so as to not only predict the main laws of real cell growth, but also lay a good foundation for real-time global biophysical simulation in the next step.

Our team selected chicken muscle satellite cells as seed cells. Two-dimensional cellular automata and three-dimensional cellular automata are used to calculate the cell growth. In this paper, a computer model based on two-dimensional and three-dimensional simulation of cell growth dynamics is proposed. The cell evolution rules are designed in the Von Neumann extended neighborhood of cells, and a large number of experiments and analysis are carried out. Experiments show that the model has the ability to simulate cell growth dynamics on the premise of following the basic principles of model algorithm, and the simulation results are in good agreement with clinical data. Based on the concise and intuitive CA algorithm, by changing a few micro parameters, we can control the evolution and describe the development process of macro things. For data visualization, we chose to build the netlogo platform, and the modeling idea is consistent with the content of the platform. The final modeling results can be displayed intuitively and clearly through this platform.

Keywords: Two dimensional cellular automata Three dimensional cellular automata Random growth model MATLAB Netlogo

I. Problem description

1.1 Background

With the development of biotechnology, great changes are taking place in our lives, such as the food we eat. Artificial meat has been mentioned by more and more people as meat products in the future. It has richer nutrition, better taste and more affordable price. It is an important direction of food development in the future. Therefore, our team is studying a method to take chicken muscle satellite cells as seed cells to differentiate into "artificial meat" with high nutrients. Chicken derived muscle satellite cell is a kind of stem cell, which has the potential to differentiate into muscle, and muscle contains high content of protein. When such cells adhere to the phfa and collagen composite scaffold, the existence of collagen scaffold also provides a richer taste for "artificial meat". Because chicken derived muscle satellite cells are extracted from chicken embryos rather than live chickens, they have better animal welfare. More importantly, chicken has no religious restrictions, and has a higher protein content. The meat is more delicate, easy to digest and has a wider audience. This paper proves the feasibility and rationality of this artificial meat from the perspective of mathematical model.

1.2 Restatement of the Problems

There are many standards to measure the growth and fruit of a piece of artificial meat, such as the quality, taste and other aspects of artificial meat. However, from the perspective of mathematical modeling, the growth result of artificial meat is directly determined by its final weight. Because it is not disturbed by subjective factors, it can be quantified. Explain the results of the experiment with scientific and rigorous conclusions. However, there are many factors that determine the quality of artificial meat, such as the number and type of cells. The quality of different types of cells is different, such as nerve cells, muscle cells, white blood cells and so on. In this paper, we use cellular automata model to establish a set of general rule framework of signal exchange and cell synergy in the process of biological pattern formation, and then study the self-organization process from cells and their behavior to cell-cell interaction and then to the occurrence of biological pattern. The growth of chicken muscle satellite cells on scaffolds was simulated by this general model. In this paper, we try to apply cellular automata model to simulate the growth pattern caused by cell proliferation.

II. Introduction of algorithm

Cellular automata, a branch derived from artificial life, is a well deserved complex system and a dual system of natural life system. It provides an ideal calculation model for complex system modeling, simulation and control, and its ability to represent complex systems, It provides a new idea for the modeling and Simulation of cell growth process. Cell growth is a complex biochemical process. For a single cell, its biological activity is relatively simple, but the whole biochemical process of its formation shows complex behavior. Cell communities interact with the environment to develop and enrich their own structure or morphology, that is, self-organization process. The mechanism of this process is complex and the dynamic behavior is changeable. The diversity, randomness, uncertainty and strong nonlinearity of its evolution process show that this process conforms to the characteristics of complex systems. At present, the models established to study cell growth are not established from the perspective of complex system, so the models established fail to show the characteristics of complex system, which makes a large gap between these models and the actual cell growth. Based on the mechanism of biological cell growth and the derived dynamic differential equation, a cellular automata model for simulating biological cell growth is established, programmed and simulated according to the designed matlab rules. The simulation results show that cellular automata can better reproduce the biological cell growth process described by the dynamic differential equation, and can also visualize the evolution process. Cellular automata is a micro simulation model with completely discrete space, time and state. It has a completely different modeling idea from the traditional mathematical model. Cellular automata is from the perspective of complex systems, using artificial neural networks.

In this paper, cellular automata model of biology cell growth( CABCGM) is constructed to simulate the growth dynamics of chicken muscle satellite cells. Based on extended neighborhood, cellular automata unifies the definition and evolution rules of proliferating cells, non proliferating cells and necrotic cells. It is easy to conduct virtual test and Analysis on its ability or performance of simulating growth dynamics under the unified theoretical framework of cellular automata.

Cellular automata is the basic starting point of mathematical model to simulate the growth process of artificial meat. Studying the evolution rules of cellular automata as a tumor growth model is helpful to understand the growth law of artificial meat. Cellular automata simulating the growth dynamics of artificial meat will become a virtual experimental model or virtual research object, and provide a virtual experimental research platform for system science. The virtual simulation experiment of artificial meat on this platform is like the digital simulation experiment of nuclear explosion on computer. The research results have positive significance for the experiment and research of artificial meat and the understanding and understanding of growth mechanism.

For data visualization, we chose to build the Netlogo platform, and the modeling idea is consistent with the content of the platform. The final modeling results can be displayed intuitively and clearly through this platform. Netlogo is a multi-agent visual simulation software, in which each participant is defined as a turtle. The corresponding rules can be set according to the background programming to determine the change of turtle state to simulate the actual situation. In addition, in addition to simulating the actual process, Netlogo can also directly obtain the state change diagram to more intuitively explain the changes of each state.

III.Assumptions

There are many factors that affect security check. Our model is setting up and implemented in a relatively ideal environment.to simplify the problems and make it convenient for us to simulate, we make the following basic assumptions, each of which is properly justified.
    • Assume that all cells grow in the same external environment, and there is no inhibition of cell growth due to uneven distribution of external nutrients
    • Different types of cells are in different growth stages, and there are subtle differences in cell volume and mass. Therefore, we assume that all cells have the same mass and simplify the model. The final quality of artificial meat is determined only by the number of cells
    • According to the principle of cellular automata, the number of cells at the next time is only determined by the number of cells at the previous time

IV. Symbols and Definitions

V. Establishment of Model

5.1 Two dimensional cellular automata model

5.1.1 Selection of CA algorithm

At present, there are two main types of models for cell behavior simulation: (1) differential equations, that is, a series of dynamic differential equations based on the observation of cell behavior; (2) CA model, such as simulating tumor growth. Compared with differential equation, CA has outstanding advantages. But pure mathematical models fail to recognize and understand the cell growth process from the perspective of complex system, so the established models fail to show the evolution characteristics of the complex system of cell growth. As a branch of artificial life system, cellular automata is a complex system and a dual system of natural life system, which provides a basis for modeling the cell growth and evolution process Simulation provides an ideal life computing model, so we choose cellular automata to simulate the growth process of real isolated cells in this experiment.

5.1.2 Structural design

The evolution rules of CA have local characteristics. Each cell evolution of CA depends on the adjacent cell state. In other words, the state of the cell at the next moment is only related to the cell state of its neighborhood. CA evolves in parallel according to the evolution rules, that is, at any discrete time, all cells in the cell space evolve according to the rules at the same time. Each cell in cellular automata represents a biological cell, and each cell has different states. In cellular automata, the growth and evolution process of biological cells is the continuous transformation process of each cell of cellular automata between different state values. The CA based biological cell growth model is defined as a quintuple.
CABCGM = {T, cells, cellspace, neighbors, rules}
T is discrete time;
Cells are the basic elements of CABCGM, namely cells;
Cellspace is cell space;
Neighbors are the neighborhood of CABCGM, that is, the definition domain of evolution rules;
Rules is the evolution rules of CABCGM stipulate the evolution of cellular state, which is the core of CABCGM.

5.1.3 discrete time

The whole growth process of biological cells is discretized. For each discrete time, t = kT. Where k is the discrete time series and T is the discrete time interval.

5.1.4 Cell space

The culture area for cell growth is L = (Lx, Ly,), the colony cells are represented by a square cell in the grid, the cells have no internal structure, and their positions are represented by coordinates(x, y) (x ∈ (1, Lx), y ∈ (1, ly)). Cells proliferate, metabolize and die on the grid according to certain rules. Nutrients and metabolites are represented by particles on a square grid, and the HPP model is superimposed with a random motion to describe the diffusion of particles.

5.1.5 Cell neighborhood

Cell neighbor is the evolution scope of its evolution rules, that is, the definition domain of cell evolution rules. Cellular automata model has two common neighbor shapes: von Neumann neighbor and Moore neighbor. Von Neumann neighbors are composed of four cells in contact with the four edges of the central cell; Von Neumann neighbors add the cells in contact with the four corners of the central cell to the range of neighbors, which is composed of 8 cells.As shown in the figure below：

Von Neumann neighbor model on the left and Moore neighbor model on the right. Black grid points represent the central cell, and gray grid points represent the neighbors of the cell.

5.1.6 Cell The animation of cells

Define the state space of the system as S＝(0,1)=(null, cell). For each cell, a clock’ t’ is defined to determine cell metabolism and reproductive behavior.The transformation rules of the growth process can be expressed as:

Sf(t+1)＝1 (Sf(t)＝0) ＆(reproduction)
Sf(t+1) = 0 (no cell) or (S≠0＆the cell left）

Each cell in CA represents a cell, and each cell has a different state, so that the state of CA cell at time t is S. S takes two state values (0 and 1). Where S = 0 indicates that it is empty or cell death at time t; S = 1 indicates that the new cells occupy the space; Thus, in the CA model, the cell growth process is the process of continuous transformation of the state values of each cell unit of CA between 0 and 1.

The evolution rules of CA have local characteristics. The evolution of each cell of CA depends on the cell state adjacent to it. In other words, the state of the cell at the next moment is only related to the cell state of its neighborhood. CA evolves in parallel according to the evolution rules, that is, at any discrete time, all cells in the cell space evolve according to the rules at the same time. According to the growth mechanism of biological cells and the derived dynamic differential equation, the evolution rules of CABCGM are designed as follows:

By setting the above model parameters and model rules, we can simulate the cell growth under the two-dimensional cellular automata algorithm. The simulation platform and process are introduced in the three-dimensional cellular automata algorithm later.

5.2 Three dimensional cellular automata model

Cellular automata models have been widely used in 2D space, and the research of 3D cellular automata models has attracted more and more attention. In 3D cellular automata model, it is generally listed a L × L × L cube grid array. There are 6 neighbors of the grid point. The schematic diagram of the central grid point and its neighbor space is shown in Fig:Because cells grow in a three-dimensional environment, we need to improve the traditional cellular automata algorithm. This means that by changing the neighbors and evolution rules of cells, the cell algorithm is also applicable in three-dimensional conditions.

Because cells grow in a three-dimensional environment, we need to improve the traditional cellular automata algorithm. This means that by changing the neighbors and evolution rules of cells, the cell algorithm is also applicable in three-dimensional conditions.

The three-dimensional cell model adopts the same design idea as the two-dimensional cell, and adopts the same evolution rules and the probability of division and apoptosis. The only thing that needs to be changed is the cell space. From two-dimensional to three-dimensional, the cell neighbors are also changing. In order to simplify the problem, we still use Von Neumann neighborhood, but the location coordinates of cells change from two-dimensional coordinates to spatial coordinates. The colony cells are represented by a square cell in the grid, the cells have no internal structure, and their positions are represented by coordinates(x, y, z) (x ∈ {(1, Lx), y ∈ (1, ly), z ∈ (1, lz)})

The functions of the Netlogo simulation software we use are divided into the learning part on the left half and the prediction part on the right half, as shown in the figure:

In the learning part, the model has two parameters to learn PA and Pb. Manually input a list (the total number of cells every fixed time). Through the circular grid search method, do ten simulations under the corresponding conditions of each PA and Pb, take the mean value, calculate the MAPE of the simulation value and the real value, and draw the picture.In the learning part, the model has two parameters to learn PA and Pb. Manually input a list (the total number of cells every fixed time). Through the circular grid search method, do ten simulations under the corresponding conditions of each PA and Pb, take the mean value, calculate the MAPE of the simulation value and the real value, and draw the picture. Because we know that the interval of input data is 1 hour, but we are not sure how long the interval of simulation iteration is (although the shorter the interval is, the more accurate it is), the program assumes that each interval represents 1, 0.5 and 0.25 hours, and draws pictures respectively.

Take PA and Pb corresponding to the minimum MAPE and put them into the prediction area on the right (PA = 0.6, Pb = 0.5). According to the experimental data, it is approximately considered that the fiber diameter is 250um and the layer height is 150um. In this way, the side length of a cell can be regarded as 50um, which is convenient for calculation. Also enter the initial number of cells and the number of iterations. The program defaults to a cell representing 1000 cells. If the initial number is 14400, the program will round up and turn 15 cells red.

The final results are shown in the figure below:

VI. Conclusions

6.1 Advantages

• At present, three-dimensional cellular automata is rarely used in the field of biology. This paper opens up a new way and method in describing the growth of cells and simulating their specific growth conditions
• Three dimensional cell modeling is still a new research direction. Internationally, scholars from multiple disciplines often collaborate to promote this research. Taking cell growth as a complex dynamic system and applying system science and computer science to simulate or simulate and study its dynamic evolution process will help us find the mechanism of cell growth and development and understand the characteristics of cells as self-organizing systems and dynamic evolution systems.

6.2 Disadvantages

• The simple 0-1 cellular automata and CABCGM we designed are all three-dimensional cellular automata of Von Neumann neighborhood. We can further study Moore neighborhood three-dimensional cellular automata. In this way, we can consider 9 neighborhood model or 27 neighborhood model
• The simulation model and related experiments based on theory in this paper are a very ideal research, ignoring many details of cells as complex life phenomena, such as growth and differentiation, and focusing on the exploration of the application ability of this pure mathematical tool in the field of biological information processing. If the model is further studied and mature, it can accommodate more factors affecting the cell growth process, such as the concentration of nutrients in culture medium and the concentration of cell metabolic waste, so as to better reflect the essential characteristics of cell growth

6.3 Extension of the Model

This paper introduces the specific steps of using the idea of cellular automata to design the model. According to this step, this experiment establishes a simple cellular automata model based on the derived dynamic differential equation. The purpose is to prove that the derived nonlinear dynamic differential equation can be used as the benchmark for us to design the evolution rules of cellular automata,
On the basis of two-dimensional cellular automata, the parameters of the derived three-dimensional cellular automata are set and adjusted. Based on the principle of simplifying the cellular automata algorithm but still in line with the objective facts, it is transformed with the help of matla platform. The evolution results are very consistent with the actual cell growth. The fact that cellular automata can be used to test cell growth is verified.
The simulation model and related experiments based on theory in this paper are a very ideal research, ignoring many details of cells as complex life phenomena.For example, when considering the factors affecting cell growth and the role of nutrients and metabolites in cell growth, different growth patterns can be obtained by controlling nutrient concentration, time scale of reproduction, diffusion and metabolism, inhibition coefficient of metabolites, etc. Generally speaking, the higher the concentration of nutrients, the faster the diffusion of nutrients, the faster the cell reproduction, and the faster the growth of artificial meat,; The faster the cell grows, the faster its metabolites increase, the more obvious the inhibitory effect on cell growth will be; The inhibitory effect of metabolites on cell growth is mainly regulated by the concentration of metabolites, the diffusion rate of metabolites and the inhibitory coefficient of metabolites on cell growth.
Although the object of our research is artificial meat, we can regard artificial meat as a multicellular organism, so we can regard the simulation of artificial meat growth as the simulation of the growth and development process of a multicellular organism, so as to provide some similar comparison and Enlightenment for the research of pattern formation process in developmental biology.

VII.References

    • [1]Zhao F，Tao ZL．Cellular automata approach to biological pattern formation (I)：the aggregation pattern in Dictyostelium discoideum．J Biomed Eng，2006 ；23 (2 )∶ 304
    • [2] Chopard B, DrozM Cellular automata modeling of physical systems．Cambridge University Press，1998 ，Ch1 ，2，
    • [3] Zhao F，Tao ZL．Mechanical models of pattern formations in developmental biology．Advances in Mechanics，2003 ；33∶95