Background

For multiple considerations such as biosafety, the concentration of Escherichia coli that secretes the requisite substances must be controlled within an appropriate range. At this point, a oscillation system can be devised, which involves two types of organisms, namely prey and predator, designated as c1 and c2 respectively.

c1, as the prey, is the bacterium used to express the target gene (glucose control gene), and the signal it secretes can guarantee the survival of the predator (inhibiting the expression of the suicide gene). When the density of the prey bacteria is not high, the suicide gene of the predator is activated, maintaining the predator bacteria at a relatively low density. Once the density of the prey bacteria reaches a certain level, the suicide gene of the predator is suppressed, and thus its density also attains a certain magnitude. The signal secreted by the predator reaches a certain concentration and enters the prey bacteria, initiating the suicide module of the prey bacteria. A model is designed for this multi -component situation, listing the effects of each component of the model, and using differential equations to conduct quantitative calculations of the eventual states of the two microorganisms. This is a relatively complex process, considering both the secretion process within the body and its influence on the mortality rate. Both the predator and the prey contain suicide genes. The prey inhibits the expression of the predator's death gene, while the predator promotes the expression of the prey's death gene.

For the sake of simplification, we no longer analyze the specific formation mechanisms of c1 and c2, but rather conduct the analysis based on experimental data.

Model construction

The whole relation are considered first ^[1]:

Here, we know that c1 and c2 represent predator and hunter, respectively, with indicates a particular substance secreted. Among them, A₂ is a special substance secreted by c2, LuxI, which inhibits the expression of suicide genes to improve survival rate, while A₁ is the c1-secreted factor LasI that promotes suicidal effects.

For a detailed analysis of the mechanism, please refer to the figure below

First of all, considering that the two colonies actually share certain resources in the medium, a certain competitive relationship can be satisfied.

The first part shows the normal situation of competitive reproduction, and the growth of both is roughly consistent with the Logistic model, but the death is subject to its own growth.

The death factor can be paused by A_e1, A_e2 is what these secreted substances look like in vitro. For the sake of simplification, we do not analyze the generation rate in detail, and consider uppercase A_ei. Synthesis is only proportional to the number of colony populations, while consumption roughly follows the law of diffusion and is only related to concentration differences:

For the impact of mortality, we believe that the Hill reaction process with P_i is the core step. Meanwhile, the mortality can be summarized as:

At the same time, considering the mass action law and the convenience of measuring the concentration in vivo and in vitro, P1 and P2 can be A_ei. instead of representation, after integration, a system of differential equations is obtained^[2]:

Model solving

For the solution of ordinary differential equations, numerical methods are often used to solve them. Especially for these nonlinear systems of differential equations. The Rugga-Yutta method can be used to solve the first order ordinary differential equations. Optimization is performed by partitioning regions into initial values.

Here's how to do it: consider four functions.

Among them, the four functions are below.

At the same time, this system of differential equations involves several undetermined coefficients, which needs to be selected according to the actual situation.

Model result

As shown in the figure, after initial fluctuations, the numbers of predators and predators will show a periodic stabilization. This phenomenon is not only consistent with the actual observation results, but also shows that the QS system has a certain self-regulation ability, and can pull the initial deviation value back to the normal range. This robustness of the system provides a guarantee for the smooth conduct of subsequent operations.

Background

In material engineering and medical research, the use of engineered bacteria such as E. coli to colonize the body and secrete therapeutic substances is a key strategy. In this paper, we will introduce a two-part model to simulate the adhesion, reproduction, dipeptide secretion, circulation and decomposition of E. coli in vivo. Finally, we will obtain the concentration curve of the compound in the body. Considering that E. coli will proliferate and secrete products after adhesion in the body, we will comprehensively consider the second module, that is, the process in which E. coli adheres to the mucosa of the small intestine after entering the organism, synthesizes and secretes dipeptides.

Model construction

The model can be divided into two parts: 1. Reproduction process of E. coli in vivo: The Logistic growth model is used to describe the reproduction dynamics of E. coli. 2. The process of dipeptide secretion into the body: The one-compartment model of pharmacokinetics was used to characterize the process of dipeptide secretion and absorption.

Part 1: The reproduction of E. coli
the Logistic model was used to accurately describe the reproductive behavior of E. coli on the mucosa of the small intestine. The differential equation of the Logistic model is expressed as follows:

N is the number of E. coli.
k is the growth rate constant.
K is the environmental carrying capacity, that is, the maximum number of E. coli that can be held on the mucosa of the small intestine.

Part 2: Secretion and Absorption of Dipeptides
After E. coli have colonized and multiplied, they begin to secrete dipeptides. It is assumed that the rate of secretion of dipeptides is proportional to the number of E. coli, and the absorption process of dipeptides in the body is described using a one-chamber model (as shown in the figure below).

The basic equation of the one-chamber model is:

- X the dipeptide concentration in the body.
-Q₁ The rate of input of the dipeptide is assumed to be proportional to the number of E. coli, N, Q₁= 1* N.
-Q₂ is the output rate of the dipeptide, proportional to the concentration of the dipeptide X, Q₂= k_x* X^[3].

Therefore, the above equation can be expressed as:

m, k_xis the constant.

Model solving

By combining the above equations, the dynamic behavior of the whole system can be solved, and the concentration change curve of dipeptide in vivo can be obtained. The specific steps are as follows:

1. Initialization parameters and initial conditions, including.
2. Solve simultaneous differential equations using numerical methods such as Runge-Kutta method.
3. The curve of dipeptide concentration with time was analyzed and plotted.

Model result

Through the above model, we can describe in detail the reproduction of Escherichia coli in vivo, the secretion of dipeptides and their distribution and metabolism in vivo. This provides an important theoretical basis and reference for the study and optimization of treatment methods based on engineered bacteria. Through the above steps and codes, we can simulate and analyze the reproduction of E. coli in the body, the secretion of dipeptides and their dynamic changes in the body. This model has important application value in bioengineering and medical research.

Background

In order to enable the engineered bacteria to play a better role, it can be attached to a specific plane. (Refer to the description of the wet experimental group) On this basis, the analysis of the specific adhesion of cells is worth exploring. In actual production, if we want to get some specific cells to touch and separate each other, we tend to understand the distribution of cells given the molecules on the cell surface. There are two receptors on the cell surface, Ag and Nb, and these two substances will interact to achieve the effect of adhesion ^[3].

By constructing the distribution of antibody antigens on the surface, it can be inferred that it will exhibit some specific structures in the body. At this time, you can consider using a Monte Carlo algorithm similar to cellular automata, so that the cell iterates according to the surrounding situation, and finally get the possible situation^[5].

Model construction

When considering the quantitative measurement of adhesion modules, it is necessary to consider the construction of fitness functions to measure the optimal state of each cell. First, the initial values are set according to the different types of cells. Consider N pairs of Ag, Nb antigens, and antibodies. Each type of cell has one or more antigens of a specific kind. Set the adhesion function, and for a certain location, count the eight cells that are nearby. If there is a corresponding antigen-antibody directly adjacent to the cell, this cell is scored. Otherwise, no points will be scored. If there are many pairs of conjugated antigen antibodies, then the score is accumulated. If it is on the diagonal, then mark 0.5 points. This integral system can be regarded as a fitness function, according to the distribution of surface antigens and antibodies, determine the degree of adhesion, and use a number to represent, this value as the degree of evolutionary advantage.

Next, evolution takes place, that is, making the cell "move" in the direction of greater adhesion, and thus move in the direction actually expected. This is where iteration comes in. The iterative process can be broken down into steps like this
·Step1 For each iteration, each cell randomly generates a different state i than the original.
·Step2 Calculate the fitness function s(i) for the new state
·Step3 Calculate the change in score and write it as ΔS
·Step4 Generate a random number between 0 and 1 r
·Step5 if r＜e^ΔS/kT ,and accept the new state; Otherwise retain the current state. In other words, the result optimization must be greedy to accept, and the result deterioration probability to accept.

However, as the number of iterations increases, T slowly decreases, and eventually, a definitive image will be obtained

The Monte Carlo algorithm can be used for specific solutions. The Monte Carlo algorithm first gives the initial position of a few cells, and then begins to evolve according to the scoring rules.

Model solving

Monte Carlo algorithm solving can be divided into the following steps: First initialize the grid situation, each grid is randomly distributed with different kinds of cells

The Monte Carlo solution can be divided into the following steps:
·First, the grid is initialized, and each grid is randomly distributed with different kinds of cells
·Second,to define the evolution rules: here, the adherence function is used to measure the results. Each cell receives a corresponding evolution score based on its surroundings,Represents the KTH evolution result of the ij position
·Third, cell changes change the results: in order, different cells iterate according to their surroundings. Always move in the direction of increasing the score. Based on this, we set a stop criterion, where we believe that the final result is when the total score stops increasing

Model result

The actual cells may not only be attracted to each other, but also be disturbed by different factors such as resource competition and resource perception.


1. Effects of cell survival pheromones and resource looting on distribution

Sensing radius is large ( 240 * cell radius):

Most of the cells are squeezed in the container wall due to resource looting, and a small amount of pos and neg exist in the middle of the container. The distance between the two pairs is large, and the distance within the same pair is compact.

Medium Sensing radius = 30 * cell radius:

Pos-neg pairs and neu cells were distributed in the container in separate families, and there was an obvious band gap between the pos-neg cell family and the neu cell family.

Short The sensing radius (7.5 * cell radius):

The survival pressure was low, all cells were distributed disordered, and a small number of isolated pos or neg cells existed.

2: Effect of cell ratio on distribution
When in one step (sensing radius = 30 * cell radius), adjust the ratio of the two cells to 1:1, 1:2, 1:3, and the results are as follows:

It was found that basically one neg cell was surrounded by 1 /2 /3 pos cells, which was consistent with the initial ratio.