A kinetic model has been developed for describing the effect of PLA on staphylococcus epidermidis that is the main bacteria that produces the armpit order.
    The most important contribution of our model is that it helps us to solve the following problems:
(1) What is the effect of the PLA on the number of Staphylococcus epidermidis over time?
(2) How will the bacteria grow at any given concentration of PLA?
(3) How much PLA should we put in our products to inhibit Staphylococcus epidermidis in a desired rate?

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    a=1.0885, b=2.4360, c=1.7748,

    Our model establishing was divided into two steps:
(1) Develop a basic kinetic model to simulate the growth of Staphylococcus epidermidis without PLA effect.
(2) Add a term onto the basic model to represent the effect of PLA on staphylococcus epidermidis growth.
To establish our model, we added different concentration of PLA to Staphylococcus epidermidis, then we measured the number of bacteria every hour for a total of 8 hours. Based on the experimental data, the model was developed as follows.
(1)Develop a basic model for describing the Staphylococcus epidermidis growth without PLA.
First, natural growth of Staphylococcus epidermidis without adding PLA (i.e., PLA concentration is 0mmol/L), was observed in experiment. The number of Staphylococcus epidermidis was measured every hour for a total time of 8 hours, which is depicted in Figure 1.

It can be seen from Figure 1 that the growth trend of Staphylococcus epidermis was basically similar to a Gaussian function, which is defined as:

    Here X(t) denotes the number of Staphylococcus epidermidis at time t. We firstly used the function defined in Eq.(1) as a function basis for representing the number of the bacteria as a function of the time t, and identified values of the involved parameters, which is implemented on Matlab. The values of the identified parameters are , 3.63 and 0.8511, respectively. The curve of the chosen Guassian function under the identified parameters is plotted in comparison with the experimental data as shown in Figure 2.

    At the first glance of Figure 2，Guassian function has already fitted the experimental data ‘very well’. We further took derivative of the Guassian function so that we can obtain the differential equation for describing the temporal changes of bacteria. We have:

    Using the kinetic model (2) under the identified parameters, we can simulate the growth of the bacteria starting from the initial point , where denotes the number of bacteria measured at . The result is plotted as shown in Figure 3. The reason why Figure 2 and Figure 3 have tremendous difference in fitting performance is that the system (2), which is derived from Gaussian function in (1), is very sensitive to the initial point . Figure 2 is the curve of Guassian function starting from , which doesn’t not equal to the measured number of bacteria at time , whereas Figure 3 is the curve of Guassian function starting from , the actual number of bacteria measured at time . Although the initial points of the curves in Figure 2 and Figure 3 have only slightly difference, the gap between the two curve becomes larger and lagers as the time goes on.

    We also re-identified the parameters by minimizing the distance between the computational value from the system (2) and the experimental data, the results is plotted in Figure 4, which also shows that the overall fitting performance is not satisfying. Note that Figure 2 has a good overall fitting performance by letting the initial point of the curve being not the experimental measured data, which is not allowed while using the model for prediction. The above discussion tells us that the kinetic model (2) derived from Guassian function in (1) is not suitable for predicting the number of bacteria, which also means that the standard Guassian function is not suitable. Therefore, we need to modify the model.

    In the system (1), the specific of growth rate is expressed as:

    Which is a linear function of the time t. We replace this term by a quadratic function as follows:

    And the system (2) with this new expression of specific growth rate becomes:

Using the experimental data, we identify the values of the parameters involved in the system (4). We obtained that the identified values of are 1.0885, 2.4360, 1.7748, respectively. The computational value of the system (4) under the optimal parameters is plotted in Figure 5 in comparison with the experimental data.

    So far, we have developed a basic kinetic model for describing the growth of the strain without PLA, which has a perfect prediction of the change of bacteria population over time.

(2) Extending the basic model to address the effect of PLA on Staphylococcus epidermidis growth.
    To extend the kinetic model (4) to describe the inhibiting effect of PLA on the bacteria, we further conducted a series of experiments to observe the change of bacteria population under different PLA concentration sin medium. The initial value of OD of Staphylococcus epidermidis is set to 1Abs, and the PLA concentrations are set to be 5mmol/L、7mmol/L、10.5mmol/L respectively. Similar to the previous experiment, the number of bacteria was measured every hour for a total time of 8 hours.

    In Figure 6, we can see that the number of Staphylococcus epidermidis in 5mmol/L PLA is consistent with that in 0mmol/L PLA, which means that the 5mmol/L of PLA has no inhibiting effect on the growth of Staphylococcus epidermidis. However, 7mmol/L and 10.5mmol/L of PLA had obvious inhibiting effects on the growth of Staphylococcus epidermidis, and the inhibition of 10.5mmol/L of PLA on the bacterial growth was much greater than 7mmol/L. We get the following conclusions:
① The strain was partly inhibited by PLA only when the PLA concentration is higher a certain value , which is referred to as a critical concentration.
② When the concentration of PLA, denoted by , is higher than the critical concentration , the strength of the inhibiting killing effect increases as PLA concentration increase. And it can be roughly inferred from the experimental data that the growth of inhibiting killing strength as the increase of overflow PLA concentration, i.e., , is faster than a linear growth, it is therefore preferred to adopt superlinear functions to define the relationship between inhibiting strength and overflow PLA concentration.
We choose two classes of superlinear functions: power functions and exponential functions, as the candidate functions for represent the specific inhibiting killing rate of PLA on the bacteria. By respectively introducing the two candidate functions in the basic model (4), we can give the kinetic model for describing the growth of the bacteria under the inhibiting effect of PLA as follows:

    The parameters in the systems (5) and (6) are respectively identified based on the conducted experiments and the fitting performances are compared between the two models. Since the fitting performance of Model (5) is superior in comparison with Model (6), we conclude that the power function is more suitable for describing the specific inhibitory strength of PLA on the bacteria. Finally, we can describe the growth of bacteria with or without PLA by the following unified kinetic model:

Symbol	Meanings
Cp	Concentration of PLA in the medium
Cp*	Critical concentration of PLA to inhibit bacteria
x(t)	The number of Staphylococcus epidermidis at time t
△Cpn	The overflow concentration of PLA

（3）Model validation

    To verify the accuracy of our model, we had used 8.5mmol/l and 11.25mmol/ l PLA to react with Staphylococcus epidermidis with an initial OD value of 1Abs. We then measured the residual amounts of bacteria every hour and compared the actual experimental data with the computational values from the system (7) as shown in Figure 7. We can conclude from the figure that our model has good performance in predicting the growth of Staphylococcus epidermidis under the effect of PLA.

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    （4）Analyzing bacteria growth under different operation conditions （vary PLA concentration or Initial OD value of Staphylococcus epidermidis）
    PLA concentration is a parameter in the system (7). By changing its value, we can analyze the growth trend under different PLA concentrations. Besides, the initial OD value of Staphylococcus epidermidis is an initial condition in our model, and we also can change its value to simulate the growth trend of Staphylococcus epidermidis under different initial values of OD.
    In our numerical experiments, we set the initial OD value of Staphylococcus epidermidis to be 0.5 Abs and 2 Abs, respectively. For each initial value of OD, we conduct a group of experiments, the concentration of PLA is set to be 0mmol/L, 7mmol/L, 8.5mmol/L, 10mmol/L, respectively. The results are depicted in Figure 8 and Figure 9.

    Through these two figures, we found that the effect of PLA in inhibiting the bacteria is independent to the initial OD value, which however is very sensitive to the change in PLA concentration.
    The above analysis told us that under the same concentration of PLA, the number of strains inhibited killed by PLA is in proportion to the total number of current strains. Which means our models fit to bacteria under any numbers.

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    （5）Calculating the minimal concentration of PLA required for a desired effect
    Through the analysis of our model, we concluded that：
① When the concentration of PLA is greater than 6.3987 mmol/L, it will have an inhibiting effect on the epidermidis.
② When the concentration of PLA reaches 11.25 mmol/L，the death rate of staphylococcus epidermidis outpaces the growth rate of the strain all the time, resulting a sustained decrease in the amount of the strain, which means 11.25 mmol/L is the minimal concentration of PLA required for a desired inhibiting effect.

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Achievements

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