Team:ShanghaiTech China/Model
Part 1
Population growth model of E. coli in the human body.
1、Basic and Developed kinetic Model
This part of the model reflects the competitive relationship between E. coli delivered into the human body through capsules and the original E. coli in the body. The Malthusian model is a mathematical model of population growth, in which the population growth rate is fixed. Our modeling utilizes the Logistic model to simulate population growth. The Logistic model is a comparatively competing mutual analysis model, which belongs to a block growth model and is closer to the actual situation than the Malthusian model. The Logistic model refers to a linear relationship between population growth rate and population density when a mathematical model is established based on the assumption of population growth rate.
E.g. The Logistic Model.
$\frac{dN}{dt}=r\times N\times(1-\frac{N}{{N}_{m}})\ \ \ \ \ \ (1)$
N: The number of the total population;
r: Intrinsic growth rate of the population;
Nm: Environmental capacity.
The expression of the Logistic model is easy to understand. The left side of the equation represents the rate of population growth. The $\frac{N}{{N}_{m}}$ value in the parentheses on the right side of the equation is equivalent to the correction of the growth rate in combination with the actual environment. That is, the larger the population and the greater the population and space occupation, the slower the population growth rate in case of limited environmental resources. The $\frac{N}{{N}_{m}}$ value can be understood as the space and resources that have been occupied by the existing population, and the $1-\frac{N}{{N}_{m}}$ value is the remaining resource that can sustain the growth of the population.
2、Proliferation of E. coli under the simulated model with time steps
Assume that the initial condition is as follows; the initial time is 0, and the time step is 1：
Use the software Origin to obtain a standard s-curve. In wet lab，the Nm value of a Petri dish to E. coli can be measured every dt, and the value of r of E. coli can be measured. Then, the two values can be used to simulate the estimation of the proliferation of E. coli. The difference between the estimated and actual values can then be compared by counting the E. coli in the actual dish. If the experiment is not much different, the model can be used to estimate the number of E. coli when the number of E. coli is difficult to count directly.
3、Proliferation of E. coli under the simulated model with function
The original function can be obtained by integrating the formula (1), and the original function can also describe the proliferation of E. coli well. The original function can also estimate the proliferation of E. coli in the future based on initial conditions.
integrating the formula (1)：
$N=\frac{{N}_{m}}{1+(\frac{{N}_{m}}{{N}_{0}}-1)\times e^{-r\times (t-t0)}}$
In the laboratory, we also measured the total bacterial count of pD+me E. coli by absorbance at several time points. There is a formula between the absorbance and the number of E. coli, Lambert-beer's law:
$A=ε\times b\times c$
ε，Molar absorption coefficient or molar absorptivity (L·mol-1·cm-1)；
b，Sample optical path (cm)；
c，Sample concentration (mol/L).
It can be seen that the number and concentration of E. coli, the concentration and the absorbance are a proportional function. Therefore, it is possible to set the absorbance as k times the number of E. coli and fit the curve with absorbance.
Fitting with software Origin:
The abscissa time is a linear coordinate, so the s-curve model corresponding to the linear coordinates is selected in the fitting. Corresponding to the Boltzmann model in the software Origin. Get the image and the function as follows：
The function obtained by fitting is：
$\mathrm{y}=\mathrm{{A}_{2}}+\frac{\mathrm{{A}_{1}}-\mathrm{{A}_{2}}}{1+e^{\frac{x-x 0}{d x}}}$
Compare with another equation:
$\mathrm{N}=\frac{\mathrm{{N}_{m}}}{1+\left(\frac{{N}_{m}}{{N}_{0}}-1\right) \times e^{-r(t-t 0)}}$
Remove the intercept $A_{2}$，And assume that the value of $t_{0}$ is 0. $N_{0}$ is the initial E. coli value. Obviously，The dependent variable y corresponds to $N_{m}$, and the independent variable x corresponds to $t$, so:
$$\mathrm{{N}_{m}}=\mathrm{k}(\mathrm{{A}_{1}}-\mathrm{{A}_{2}})$$
$$-\mathbf{r} \times \ln \left(\frac{{N}_{m}}{{N}_{0}}-1\right) \times \mathbf{t}=\mathrm{k}\left(\frac{x-x 0}{d t}\right)$$
k is the proportional coefficient brought by unit conversion and symbol. The individual values can be matched. And correlation coefficient R2 is 0.99587, approaching to 1.
On this basis, we re-fitting with a custom function fitting method, so that each parameter value is exempt from conversion and the physical meaning is clearer. The following results were obtained:
After fitting according to the custom function pattern, the respective coefficient sizes are as shown in the above table. The correlation coefficient R2=0.99587 is very close to 1, indicating that the custom function fits accurately. The E. coli proliferation model was completed.
Part 2
The relationship between the number of E. coli and the Fluorescence Intensity.
1、Principle analysis
Protein expression is replicated, transduced, translated, etc., and each process has multiple chemical reactions. In general, all reaction rates are unlikely to be the same, and a relatively low rate reaction must occur. The lowest rate response becomes the rate-determining step of the entire expression process. The single chemical reaction that becomes the rate-determining step allows the overall reaction model to be estimated simply by the reaction model of the rate-determining step because the entire expression process is constrained.
The rate-limiting step is also a single chemical reaction that is primarily related to the concentration of the substance and the concentration of the product involved in the reaction. Since the entire expression process is dynamically balanced, its rate should be stable, which means a stable expression rate can be maintained in a single bacterium.
Therefore, when there are multiple bacteria, the overall expression rate will be linearly related to the number of bacteria.
2、Combined with data (e.g.-pDawn-mEGFP)
As can be seen from the above figure, the linear fit of the number of E. coli As can be seen from the above figure, the linear fits the number of E. coli pDawn-mEGFP and expression intensity well. And correlation coefficient R2 is 0.9824, approaching to 1. This model is basically in line with the actual situation and can be used for calculation and prediction. When estimating the expression level by the number of E. coli, it is only necessary to use several experiments to determine the magnification coefficient $k_{mag}$ of the quantity and expression level of Escherichia coli.
Part 3
Other relevant parts of the model.
E. coli interspecies competition
Considering the presence of E. coli in the large intestine, there will be two E. coli populations after the introduction of new E. coli. This is to consider establishing a two-species competition model. The Lotka-Volterra model is a growth model that deals with the inter-species competition of multiple groups, which is the extension of the Logistic model.
E.g. The Lotka-Volterra Model.
$$\frac{\mathrm{d} \mathrm{{N}_{A}}}{d t}=\mathrm{r}_{1} \times \mathrm{{N}_{A}} \times \left(1-\frac{\mathrm{{N}_{A}}}{{N}_{m}}-k_{1} \times \frac{{N}_{B}}{{N}_{m}}\right) （2）$$
$N_{A}[t]$: the number of E. coli (class A) in the small intestine environment when it is stable; $N_{B}[t]$: The number of artificially modified E. coli (category B) in the capsule to reach the designated position of the small intestine; The released quantity is $N_{B}[t]$; $N_{A}[t]$ and $N_{B}[t]$ are functions as a function of time, and the initial values are set to $N_{A}[0]$, $N_{B}[0]$ respectively. Equation (2) has one more term “$k_{1}\times N_{B}/N_{m}$” in parentheses than equation (1), which reflects the effect of $N_{B}$ population on $N_{A}$. $k_{1}$ refers to the competition coefficient, that is each B class E. coli occupies a space and a resource of 1 time of each A class E. coli. Similarly, we can list the equation of $N_{B}$ based on Logistic-based model, in which k2 is also the competition coefficient. It refers that each A class E. coli possesses twice the space and resources of each B class E. coli.
$$\frac{\mathrm{d} \mathrm{{N}_{B}}}{d t}=\mathrm{r}_{2} \times \mathrm{{N}_{B}} \times \left(1-\frac{\mathrm{N}_{B}}{{N}_{m}}-k_{2} \times \frac{{N}_{A}}{{N}_{m}}\right) （3）$$
After obtaining the two equations (2) and (3), the continuous independent variable t can be divided into steps in dt when the code is implemented. Initial values such as $N_{A}$[0] and $N_{B}$[0] are input, and $r_{1}$, $r_{2}$, and $N_{m}$ are measured by experiments, and $k_{1}$ and $k_{2}$ are estimated to gradually simulate the population status of two E. coli.