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Growth Curve Model

Excess soluble phosphates are the result of excess Bacillus subtilis (and other phosphate solubilizing bacterias) in soil. Bacteriocins present a natural and effective way of inhibiting growth of B. subtilis, but we must first know how fast the B. subtilis in soil is growing as well as how much to kill. To start, we researched the growth of B. subtilis, as well as bacteria in general, to get an idea of how our target grows in soil. As we were unable to find any literature that specifically models the growth of bacteria in soil, we decided to create a modified basic growth curve of B. subtilis by considering the most important soil factors that would affect its growth. After consulting with soil microbiomes expert Professor Young of National Chung Hsing University, we determined the factors that influence B. subtilis growth the greatest are temperature, pH and salinity. To model the effects that these different conditions have on B. subtilis growth, we observed the relationship between each individual factor (Temperature, pH, and Salinity) and B. subtilis.

Figure 1: Growth model progress

Temperature to Bacteria Growth Curve

Figure 2: Temperature growth curve model progress

For temperature, we began with the Ratkowsky equation, which describes effect of temperature on general bacterial growth rate, modeled as

$$R_{temp}(T)=a\cdot[(T-T_{min})\cdot(1-e^{(b\cdot(T-T_{max}))})]^2$$

$$T_{min}=284K;\: T_{max}=326K$$

To simulate the growth of B. subtilis under certain conditions, we use Logistic regression function here

$$Logistic\: Function:\: f(t)=\frac{C}{1+A\cdot e^{-B\cdot t}}$$

Then we combine the growth rate under certain condition with Logistic regression function

$$f(R_{temp},t)=\frac{C}{1+A\cdot e^{-R_{temp}\cdot t}}$$

Table 1: Variable and Parameter used in temperature model.
Symbol	Unit	Explanation
$R_{temp}(T)$	[O.D._600nm/sec]	Bacterial growth rate under different temperature
$T$	[K]	The input temperature value
$T_{max}$	[K]	Maximum heat resistance
$T_{min}$	[K]	Minimum heat resistance
$t$	[min]	Time
$A$	[O.D._600nm/sec]	Initial bacterial growth rate
$B$	[O.D._600nm/sec]	Maximum bacterial growth rate
$C$	[O.D._600nm]	Maximum amount of bacteria under certain conditions

To modify this curve into a B. subtilis-specific model, we cultured B. subtilis under different temperatures to identify the unknown parameters a and b.

Figure 3: The growth curve of Bacillus subtilis under different temperature control

From this, we got the parameters a and b

$$a=1.915\cdot 10^{-5};\: b=0.105$$

Figure 4: Simulate growth curve of B. subtilis under 37°C.

Our final equation models how B. subtilis grows under different temperature conditions.

pH to Bacteria Growth Curve

Figure 5: pH growth curve model progress

As with temperature, the pH model began as a general equation describing bacterial growth rate under different pH values. We started with the cardinal pH equation

$$R_{pH}(pH)=\frac{c\cdot(pH-pH_{min})\cdot(pH-pH_{max})}{d\cdot((pH-pH_{min})\cdot(pH-pH_{max})-e\cdot(pH-pH_{opt})^2)}$$

$$pH_{min}=3.3;\: pH_{max}=9.72;\: pH_{opt}=5.82$$

To simulate the growth of B. subtilis under certain conditions, we use Logistic regression function here

$$Logistic\: Function:\: f(t)=\frac{C}{1+A\cdot e^{-B\cdot t}}$$

Then we combine the growth rate under certain condition with Logistic regression function

$$f(R_{pH},t)=\frac{C}{1+A\cdot e^{-R_{pH}\cdot t}}$$

Table 2: Variable and Parameter used in pH model.
Symbol	Unit	Explanation
$pH$	None	The input pH value
$pH_{max}$	None	Maximum pH value resistance
$pH_{min}$	None	Minimum pH value resistance
$pH_{opt}$	None	Optimized pH value
$t$	[min]	Time

Again, we cultured B. subtilis in different pH solutions to find the parameter values specific to our target.

Figure 6: The growth curve of Bacillus subtilis under different pH control

From this, we got the parameters c, d, and e

$$c=1.97\cdot 10^{-4};\: d=0.4035;\: e=8.054\cdot 10^{-4}$$

Figure 7: Simulate growth curve of B. subtilis under pH=7.0.

This model accounts for any effects that changes in pH may have on B. subtilis growth in soil.

Salinity to Bacteria Growth Curve

Figure 8: Salinity growth curve model progress

For salinity, we discovered that a simple logistic growth curve was enough to serve as our base equation. According to our references, we derived the equation describing bacterial growth rate under different salinity, model as

$$R_{sal}(sal)=(f\cdot sal^2)+(g\cdot sal)+h$$

To simulate the growth of B. subtilis under certain conditions, we use Logistic regression function here

$$Logistic\: Function:\: f(t)=\frac{C}{1+A\cdot e^{-B\cdot t}}$$

Then we combine the growth rate under certain condition with Logistic regression function

$$f(R_{sal},t)=\frac{C}{1+A\cdot e^{-R_{sal}\cdot t}}$$

Table 3: Variable and Parameter used in salinity model.
Symbol	Unit	Explanation
$sal$	[mM]	The input salinity
$t$	[min]	Time

Culturing B. subtilis in different salinities to yield the specific parameters in salinity equation.

Figure 9: The growth curve of Bacillus subtilis under different salinity control

Thus, we got the parameters as following

$$f=-7.617\cdot 10^{-3};\: g=1.853\cdot 10^{-3};\: h=1.24\cdot 10^{-2}$$

Figure 10: Simulate growth curve of B. subtilis under 0.17mM salinity.

This yields the final piece of our bacterial soil growth model, and explains any consequences of shifts in salinity of our soil.

Complete Growth Curve Model

Figure 11: Merge growth curve model progress

With our three factors all accounted for, we needed to identify the weights of each factor ($\alpha,\beta,\gamma$) before merging the equations we found into one model. To do this, we randomly selected 3 sets of different temperature, pH and salinity.

Figure 12: The growth curve of Bacillus subtilis under the multiple control

Finally, we input the growth results into Simulink program, model as

$$f^*=\alpha\cdot f(R_{temp},t)+\beta\cdot f(R_{pH},t)+\gamma\cdot f(R_{sal},t)$$

$$\alpha=-7.617\cdot 10^{-3};\: \beta=1.853\cdot 10^{-3};\: \gamma=1.24\cdot 10^{-2}$$

Figure 13: Simulate growth curve of B. subtilis under multi-factor of 37°C, pH=7.0, and 0.17mM salinity.

Through this we could determine how much impact each factor has on B. subtilis growth relative to the other two factors and adjust the parameters accordingly. Our resulting combined model has the ability to predict B. subtilis growth rate under a wide range of temperature, pH and salinity.

Figure 14: Calibration of MIC prediction model.

References

1. Ishimine, Y., et al. (2004). "Effects of planting date on emergence, growth and yield of turmeric (Curcuma longa L.) in Okinawa Prefecture, Southern Japan." 48(1): 10-16.

2. Krulwich, T. A., et al. (1985). "Buffering capacity of bacilli that grow at different pH ranges." Journal of Bacteriology 162(2): 768-772.

3. Lambert, R. J. (2011). "A new model for the effect of pH on microbial growth: an extension of the Gamma hypothesis." J Appl Microbiol 110(1): 61-68.

4. Ratkowsky, D. A., et al. (1983). "Model for bacterial culture growth rate throughout the entire biokinetic temperature range." J Bacteriol 154(3): 1222-1226.

5. Rousk, J., et al. (2011). Bacterial Salt Tolerance is Unrelated to Soil Salinity Across an Arid Agroecosystem Salinity Gradient.

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