Our projects are dedicated to solving the problems of reducing armpit odor and inhibiting the bacteria that producing unpleasant odor. Our project is mainly based on bacteriostatic and aromatic substance PLA and 2-PE production. PLA is the major bacteriostatic material[1], and we have built two models about the yield and inhibiting mechanism of PLA by an innovative method and developed a special product for its application.

Kinetic Model for Simulating and Predicting the Killing Mechanism of PLA to Staphylococcus epidermidis

The aim of developing the model in this part is to analyze the effect of different concentrations of PLA on the number of Staphylococcus epidermidis which is the main source of bacteria producing armpit odor. On this basis, we determined the optimal PLA concentration so that the population of Staphylococcus epidermidis can be significantly inhibited within the desired time.

Introduction

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?

Our model show

Fig.1 Simulation of the number of bacteria colonies under different concentration of PLA.

a=1.0885, b=2.4360, c=1.7748,

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
△Cpn	The overflow concentration of PLA

The above is the final model. When concentration of PLA in our system is less than the critical value Cp*, it won’t have any inhibition on the satin, so we built a kinetic equation to describe the growth of Staphylococcus epidermidis in natural condition. On the other hand, when the concentration of PLA reaches the critical value, we use β△Cpn to simulate Mortality of Staphylococcus epidermidis, and the survival of Staphylococcus epidermidis is expressed by combining the first formula. For complete analysis, please reading Method as follows.

Method

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 0 mmol/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 Fig.2.

Fig.2 Natural growth of Staphylococcus epidermidis with an initial OD=1 Abs

It can be seen from Fig.2 that the growth trend of Staphylococcus epidermidis 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 a1,b1,c1 are 2.854x108 , 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 Fig.3.

Fig.3 Fitting the natural growth of Staphylococcus epidermidis by Guassian function

At the first glance of Fig.3 ，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 x(1)=x1 , where x1 denotes the number of bacteria measured at t=1. The result is plotted as shown in Fig.4. The reason why Fig.3 and Fig.4 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 x1. Fig.3 is the curve of Guassian function starting from x(1)=a1 , which doesn’t not equal to the measured number of bacteria at time t=1, whereas Fig.4 is the curve of Guassian function starting from x1 , the actual number of bacteria measured at time t=1 . Although the initial points of the curves in Fig.3 and Fig.4 have only slightly difference, the gap between the two curve becomes larger and lagers as the time goes on.

Fig.4 The computational curve of the system (2) under the identified parameters from Eq.(1)

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 Fig.5, which also shows that the overall fitting performance is not satisfying. Note that Fig.3 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.

Fig.5 The computational curve of the system (2) under the re-identified parameters

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 a,b,c are 1.0885, 2.4360, 1.7748, respectively. The computational value of the system (4) under the optimal parameters is plotted in Fig.6 in comparison with the experimental data.

Fig.6 The computational curve of the system (4) under its optimal parameters

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 in system. The initial value of OD of Staphylococcus epidermidis is set to 1 Abs, and the PLA concentrations are set to be 5 mmol/L, 7 mmol/L, 10.5 mmol/L respectively. Similar to the previous experiment, the number of bacteria was measured every hour for a total time of 8 hours.

Fig.7 Growth of Staphylococcus epidermidis with an initial OD=1 Abs under different PLA concentrations

In Fig.7, we can see that the number of Staphylococcus epidermidis in 5 mmol/L PLA is consistent with that in 0 mmol/L PLA, which means that the 5 mmol/L of PLA has no inhibiting effect on the growth of Staphylococcus epidermidis. However, 7 mmol/L and 10.5 mmol/L of PLA had obvious inhibiting effects on the growth of Staphylococcus epidermidis, and the inhibition of 10.5 mmol/L of PLA on the bacterial growth was much greater than 7 mmol/L. We get the following conclusions:
① The strain was partly inhibited by PLA only when the PLA concentration is higher a certain value Cp* , which is referred to as a critical concentration.
② When the concentration of PLA, denoted by Cp, is higher than the critical concentration Cp*, 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 strength as the increase of overflow PLA concentration, i.e., Cp - Cp*, 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 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 system
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

Fig.8 Validate the system (7) under an initial OD value of 1Abs but different PLA concentrations

To verify the accuracy of our model, we had used 8.5 mmol/L and 11.25 mmol/L PLA to react with Staphylococcus epidermidis with an initial OD value of 1 Abs. 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 Fig.8. We can conclude from the figure that our model has good performance in predicting the growth of Staphylococcus epidermidis under the effect of PLA.

（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 0 mmol/L, 7 mmol/L, 8.5 mmol/L, 10 mmol/L, respectively. The results are depicted in Fig.9 and Fig.10.

Fig.9 Growth of Staphylococcus epidermidis with an initial OD value of 0.5 Abs under different concentrations of PLA

Fig.10 Growth of Staphylococcus epidermidis with an initial OD value of 2 Abs under different concentrations of PLA

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.

（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 Staphylococcus 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.

Fig.11 Growth of Staphylococcus epidermidis with an initial OD value of 1 Abs under 11.25 mmol/L of PLA

Achievements

Since the information this model provides about the mechanism of the PLA is important for many biological experiments, our model is valuable and innovative, and it can be served as a reference for other teams who are interested in the related study of this area. It is worthy mention that the inhibiting mechanisms of PLA on strain haven’t been clearly known so far and there is no existing model to represent the dynamics of PLA effect on strain growth. It is for the first time a model is proposed to describe this effect in our work.

And we got also some conclusions as follows.
Through the analysis of our model, we concluded that：
① Greater than 6.3987 mmol/L of PLA will have a killing effect on Staphylococcus epidermidis.
② 11.25 mmol/L of PLA will force the population of Staphylococcus epidermidis to persistently decrease, and inhibit all the strains within an acceptable time.
③ Under the same concentration of PLA, the number of strains inhibited by PLA is in proportion to the total number of current strains, and therefore, under the same concentration of PLA, the growth trend of the strain is the same for different initial OD values. It means that the time for completely inhibiting the strains is independent to the initial OD value of the strains, which was also proved by experiments.
④ Any kind of bacteria can grow naturally in a certain range of concentration of PLA under the critical value, which can be used in screening and isolation of strains.

Kinetic Model for the Production of PLA by the Engineered Strain

Introduction

In this part, we will develop a model to describe the kinetics of the production of PLA by the engineered bacteria. This model will allow us to understand the effect of the biomass on the final concentration of PLA synthesized by the engineered bacteria，and therefore effectively solve the following problem:
Whether our system can produce the desired concentration of PLA?

Method

To develop the model for describing the formation of PLA, we conducted three groups of experiments under substrate-sufficient condition. Our experiment used three groups of engineered bacteria whose initial OD values is 15 Abs, 22.5 Abs, and 30 Abs to produce PLA, and then measured production every 20 minutes for a total of 120 minutes. The concentrations of PLA as a function of time were plotted in the concentrations of PLA as a function of time were plotted in Fig.12.

Fig.12 The production of PLA by engineering bacteria under different initial OD values

Under substrate-sufficient condition, the biomass becomes the main factor affecting the formation rate and the final concentration of PLA. In this context, we use Monod[2] equation as the basic model for describing the formation kinetics of PLA by the strain.
Where P denotes the concentration of the production PLA, X denotes the biomass (the number of engineered bacteria), mp is the maximum specific formation rate and Km is the half-velocity constant.
Note that the biomass X can be seems as a constant for each experiment since the total fermentation time is only 2 hours. When X is set to be constant, the system (8) implies that the formation rate is also a constant, i.e., the concentration of PLA increase linearly as time goes on, which however doesn’t match the trend as shown in Figure. Therefore, the system (8) needs to be modified.
By further analyzing the experimental data, we can found that the final concentrations of PLA are almost the same for the three experiments, although the concentrations of PLA are different at the early stages of the three experiments. We guess that PLA might be an inhibitory product in the synthesis of PLA. When the concentration of PLA reaches some critical value, the synthesis of PLA ceases. According to the above analysis, we modified the system (8) as follows.

Parameter	Meanings
mp	The maximum specific formation rate of PLA
Km	Half-velocity constant
P(t)	The concentration of PLA at time t
Pm	The maximum concentration of PLA that can be produced

where P* denotes the critical concentration of PLA beyond which the synthesis of PLA ceases.
The parameters in (9) are identified based on the experimental data, and the comparison between the computational values from the system (9) and the experimental data are shown in Fig.13.

Fig.13 The comparison of the computational curve and the experimental data about the production of PLA by engineering.

According to the above model, we can obtain that, under substrate-sufficient condition, the synthesis rate of PLA gradually decreases as the accumulation of PLA. When the concentration of PLA reaches Pm, PLA is no longer produced and maintains a stable yield. Besides, the value of Pm doesn’t depends on the biomass, i.e., it is a constant, and its values is Pm=16.9201. That is, the maximum concentration of PLA produced by the engineered bacteria is 16.9201 mmol/L, which satisfies the aforementioned optimal PLA concentration for killing Staphylococcus epidermidis.

Achievement

(1) Simulating the PLA production process，which let us know the effect of the biomass on the production of PLA and when the required amount of PLA can be obtained, saving time and materials to facilitate commercialization.
(2) It illustrates the mechanism why the amount of PLA could be the same under substrate-sufficient conditions even through the formation rates of PLA might be significantly different at the early stages of the production process under different biomass.

Integrating the two models for simulating the mixed growth process

Introduction

The final purpose of our modelling is to simulate the growth trend of Staphylococcus epidermidis in the environment where PLA is gradually produced by the engineered strains, and analyze the effect of the biomass of the engineered strains on the growth trend of Staphylococcus epidermidis. To this end, we integrate the proposed models (the model (7) and the model (9)) as a whole system, which is described by the following system of differential equations.

By increasing the biomass X of the engineered strain from 2 Abs to 10 Abs with a step size of 2 Abs, we carried out 5 groups of numerical simulations for the system (9). The results were plotted in Fig.14, where the dash lines are the number of Staphylococcus epidermidis, and the solid lines are the concentrations of PLA produced by the engineered strain.

Fig.14 The biomass trajectories of Staphylococcus epidermidis and the concentration of PLA produced by the engineered strain

It can be seen from Fig.14 that the accumulation of PLA becomes faster as the biomass X of the engineered strain increases. Thus, higher value of X leads to higher concentration of PLA. As a result, the inhibitory strength of PLA on the growth of Staphylococcus epidermidis becomes more and more significant as the biomass increases. When the biomass of the engineered strain reaches 6 Abs, the natural growth of Staphylococcus epidermidis is completely overwhelmed by the killing effect of PLA. In this context, the number of Staphylococcus epidermidis declines rapidly as time goes on, and the population of Staphylococcus epidermidis drops to zero within 2 hours. In other words, 6 Abs biomass of the engineered strain is enough to inhibit Staphylococcus epidermidis within an acceptable time.

Application of models

Our first model applies to all kinds of bacteria within a certain range, as long as we change the individual parameter values. And through the information provided to us by the above models and the discussion within the group, we use the two models together. We are able to know how many engineered bacteria we need to put in the product that can inhibiting the desired amounts of bacteria effectively. Besides, those models are connected with time, which also allows us regulate the system working time.
The value of our model lies in its innovativeness and practicality, and no one has built it in this field till now. In addition to guiding our projects, there are many other utilizing values of our models to be developed, such as strain screening and experimental guidance for other teams working on similar projects.

Reference

[1] Ohhira I, Kuwaki S, Morita H, et al. Identification of 3-Phenyllactic Acid As a Possible Antibacterial Substance Produced by Enterococcus faecalis TH10[J]. Biocontrol Science. 2004, 9(3):77-81.
[2] Monod, J. (1949). The growth of bacterial cultures. Annual Reviews in M, 3(Xl), 371–394.

Team:FJNU-China/Model

Models

Overview

Kinetic Model for Simulating and Predicting the Killing Mechanism of PLA to Staphylococcus epidermidis

Introduction

Our model show

Method

Achievements

Kinetic Model for the Production of PLA by the Engineered Strain

Introduction

Method

Achievement

Integrating the two models for simulating the mixed growth process

Introduction

Application of models

Reference