Overview
The Logistic Model suitable for the action of antimicrobial peptides is selected from the traditional growth model, which correlates \(\mathrm{t_0}\) with the antimicrobial peptide concentration C at the maximum change rate of bacterial population to make corrections. The problem of how to determine the range of parameter values is solved in the case of insufficient experimental data. Finally, according to the experimental data of similar antimicrobial peptides, we provide a referential concentration for the wet experiment.
Introdution
We used BLP-7 to inhibit the growth of Propionibacterium acnes. To find the optimum concentration, an antibacterial model was established.
Bacteria growth generally has four stages:
- Lag phase
- Logarithmic phase
- Stable phase
- decline phase
If the antimicrobial peptide works, deviation of the growth curve of the first three stages will be observed.
Model Construction
Select the original model
The commonly used models are: logistic model, gompertz model and richards model. According to the literature [1] the logistic model is more suitable
Logistic:
$$ N_t=A_2+(A_1-A_2)/(1+(t/t_0)^p) $$
\(\mathrm{N_t}\): logarithmic value of the colony
\(\mathrm{A_1}\): initial value of colony
\(\mathrm{A_2}\): final value of colony
\(\mathrm{t_0}\): the time corresponding to the maximum bacterial growth rate
\(\mathrm{p}\): power index
\(\mathrm{t}\): time
The growth curve of bacteria at different concentrations of antibacterial peptides. (Lg is the number of colonies that per milliliter of bacterial suspension can form)
Figure 1: Lg-Time
Figure 2: Lg-Time
From the figures above, we have made following analyses.
At low concentration, antibacterial peptides mainly inhibit growth, making the lag phase longer. What's more, the higher the concentration is, the longer the lag period would be. But after a long period of time, the bacterium increased quantitively on the macro level.
At high concentration, they have lethal effect: the number of bacteria will be greatly reduced in a short time and will remain a low concentration.
Thus, we assume that as the concentration increases, the number of bacteria decreases faster.
Model adjustment
Introduction of the antimicrobial peptide concentration \(\mathrm{C}\)
According to the literature[1] data, fit the image of \(\mathrm{t_0}\) and \(\mathrm{C}\) from 0 to 2 MIC
Figure 3: Time-Concentration
The concentration corresponding to the peak value is \(\mathrm{c_0}\), which we called the critical concentration.
When \(\mathrm{C}\)< \(\mathrm{c_0}\), \(\mathrm{t_0}\) is positively correlated with \(\mathrm{C}\) conforming the suppression conjecture .
When \(\mathrm{C}\)> \(\mathrm{c_0}\), \(\mathrm{t_0}\) is negatively correlated with \(\mathrm{C}\), conforming to the lethal effect conjecture.
Thus, we propose the following function to fit:
$$
T_0=a\times e^{-(\frac{C-c_0}{c_1}) ^2 } $$
Explore the rationality
When the value of \(\mathrm{c_1}\) is fixed, the value of \(\mathrm{a}\) changes
Figure 4: Time-Concentration
Figure4 indicates:
\(\mathrm{t_0}\) increases with \(\mathrm{a}\)
Before \(\mathrm{c_0}\), \(\mathrm{t_0}\) will increase with \(\mathrm{a}\), indicating that BLP-7 has better sterilization effect.
After \(\mathrm{c_0}\), \(\mathrm{t_0}\) will increase with \(\mathrm{a}\), indicating that BLP-7 has worse sterilization effect.
Thus, we assume \(\mathrm{a}\) is an indirect parameter to describe the characteristics of BLP-7.
When the value of \(\mathrm{a}\) is fixed at 25.01, the value of \(\mathrm{c_1}\) changes
Figure 5: Time-Concentration
\(\mathrm{t_0}\) increases with \(\mathrm{c}\)
Before \(\mathrm{c_0}\), \(\mathrm{t_0}\) will increase with \(\mathrm{c}\), indicating that BLP-7 has better sterilization effect.
After \(\mathrm{c_0}\), \(\mathrm{t_0}\) will increase with \(\mathrm{c}\), indicating that BLP-7 has worse sterilization effect.
From above, we can see:
when \(\mathrm{a}\) increases, \(\mathrm{c_1}\) has to decrease to fit the data
In another word, the \(\mathrm{a}\) and \(\mathrm{c_1}\) can inversely present the characteristics of BLP-7.
Thus, the function we assumed
$$ T_0=a\times e^{-(\frac{c-C_0}{c_1}) ^2 } $$
Figure 6: t0-c
Conclusion:
This model is suitable for different combinations of antimicrobial peptides and bacteria.
The parameters \(\mathrm{a}\) and \(\mathrm{c_1}\) may change, but there will always be a group of coincidences for experimental data analysis
Obtain the value range of \(\mathrm{a}\) and \(\mathrm{c_1}\)
1.According to the document (2) Antimicrobial peptide, Chinodracine, is applied to P. acnes
| Values / \(\mu\)M | 200 | 100 | 50 | 25 | 12.5 | 6.25 | 3.13 |
|---|---|---|---|---|---|---|---|
| Growth | - | - | - | - | - | + | + |
'+' : bacterial grow;\(\quad\) '-' : do not grow \(\qquad\qquad\)The observation time is 16 hours
Make assumptions :
Before the time it reaches \(\mathrm{t_0}\), it is impossible to judge whether the number of bacteria has increased or decreased.
When the peptide concentration \(\mathrm{C}\)>\(\mathrm{c_0}\), the bacteria reach the minimum value. It requires \(\mathrm{2t_0}\).
the range of \(\mathrm{c_0}\) is 6.25~12.5.
For \(\mathrm{C}=6.25\)
$$ T_0=a\times e^{-(\frac{6.25-C_0}{c_1}) ^2 }<16 $$
For \(\mathrm{C}=12.5\)
$$ T_0=a\times e^{-(\frac{12.5-C_0}{c_1})^2 }<16 $$
because the literature considers about that when \(\mathrm{c_0}=12.5\), the number of bacteria would decrease after observation.
2\(\mathrm{t_0}>16\) $$ T_0=a\times e^{-(\frac{12.5-C_0}{c_1})^2 }<\frac{16}{2} $$
$$ \frac{6.25}{\sqrt{\ln(\frac{a}{16})}}<c_1<\frac{6.25}{\sqrt{\ln(\frac{a}{8})}} $$ If \(a = 24\), the range of \(c_1\) is from 5.9629 to 9.46628
| Values / \(\mu\)M | 0.22 | 0.11 | 0.044 | 0.0195 | 0.0044 |
|---|---|---|---|---|---|
| Growth | + | + | + | + | + |
'+' : bacterial grow;\(\quad\) '-' : do not grow \(\qquad\qquad\)The observation time is 24 hours
The reasons may be the following two points:
- The culture time \(t< t_0\), then the antibacterial effect can not be observed.
- The concentration \(C\) is too low to show lethal effect.
Because the two antimicrobial peptides act on the same bacteria, the value of Chinodracine is meaningful referance.
If \(a = 24\), \(c_1\) ranges from 5.9629 to 9.46628.
After doing in the math, we konw that to achieve the inhibitory effect, \(C\) should be at least 3.31\(\mu\)M.
Conclusion
The model can be applied to most calculation of bacterial growth under the antimicrobial peptides $$ T_0=a*e^{-((c-C_0)/c_1 )^2 } $$
Lacking fundamental experimental data, as long as we get one \(c_3\) of inhibiting effect, one \(c_4\) of killing effect and the observation time T, the possible range of the corresponding function can be given based on the equation below and the possible growth-curve can be predicted.
$$ \sqrt{(\frac{(c_3-c_4)^2}{\ln(\frac{a}{T})})}<C_1<\sqrt{(\frac{(c_3-c_4)^2}{\ln(\frac{2\cdot a}{T})})} $$
- According to the concentration of antimicrobial peptide, the time required to reach the maximum change in the number of bacteria can be predicted, and the required concentration can also be deduced by time.
Future Direction
- Get more experiment data to revise the model.
- The correctness of the model is verified with the experimental data in the literature[1].
References
[1] Pu YH, Sun LJ, Wang YL, et al. Inhibitory Activity of a Novel Antimicrobial Peptide AMPNT-6 from Bacillus subtilis against Listeria monocytogenes in Shrimp[J]. Science and Technology of Food Industry, 2013, 34(013):94-98.
[2] Wu Y, Zhang GX. Inhibitory Effect of the Antimicrobial Peptide Chionodracine Against Propionibacterium Acnes and Its Anti-inflammatory Effect on Acne Vulgaris[J]. Guangdong Chemical, 2021,48(04):26-28.