Team:NCTU Formosa/Modeling/Model 1

Our system uses soil factors as input and combines the conditions of soil with our models to predict changes in nitrogen, phosphorus and potassium levels that can increase turmeric productivity. To accurately determine a way to improve productivity and estimate the results, our models need to account for several variables that could affect microbial distribution, and thus nutrient levels and plant growth. We first created growth curves of our target phosphate solubilizing bacteria, Bacillus subtilis, under different temperature, pH and salinity values in our growth model. We then explored the effects that bacteria have on not only themselves but also the amount of soluble nutrients, and vice versa, in our bacterial relationships model. Finally, our productivity model connects the effects of nitrogen, phosphorus and potassium as well as temperature, pH and salinity to resulting growth of turmeric. Click below to learn more about each of our models!

Growth Model

Figure1: Growth model progress

Excess soluble phosphates are the result of excess Bacillus subtilis (and other PSBs) in soil. Bacteriocins present a natural and effective way of inhibiting growth of subtilis, but we must first know how fast the subtilis in soil is growing as well as how much to kill. To start, we researched the growth of 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 bacillus by considering the most important soil factors that would affect its growth. After consulting with soil microbiomes expert Professor Young of National Chung Hsin University, we determined the factors that influence subtilis growth the greatest are temperature, pH and salinity. To model the effects that these different conditions have on subtilis growth, we observed the relationship between each individual factor and subtilis.

Temperature to Bacteria Growth Curve

Figure2: Temperature growth curve model progress

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

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

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

$$T_{min}=284; T_{max}=326; a=1.915e^{-5}; b=0.105$$

$$f(t)=t$$

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

Table 1: Variable and Parameter used in temperature model.
Symbol	Unit	Explanation
$T$	[K]	The input temperature value
$T_{max}$	[K]	Maximum heat resistance
$T_{min}$	[K]	Minimum heat resistance
$t$	[min]	Time

Figure3: Temperature growth curve

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

pH to Bacteria Growth Curve

Figure4: pH growth curve model progress

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

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

Again, we cultured subtilis in different pH solutions to find the parameter values specific to our target. The resulting equation is

$$pH_{min}=3.3; pH_{max}=9.72; pH_{opt}=5.82; c=1.97e^{-4}; d=0.4035; e=8.054e^{-4}$$

$$f(t)=t$$

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

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

Figure5: pH growth curve

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

Salinity to Bacteria Growth Curve

Figure6: Salinity growth curve model progress

For salinity, we discovered that a simple logistic growth curve was enough to serve as our base equation. Logistic growth is defined by the equation

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

Culturing subtilis in different salinities yielded the following specific parameters in the equation:

$$f=-7.617e^{-3}; g=1.853e^{-3}; h=1.24e^{-2}$$

$$f(t)=t$$

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

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

Figure7: Salinity growth curve

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

Figure8: Merge growth curve model progress

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

Merge growth curve

With our three factors all accounted for, we needed to identify the weighting of each factor before merging the equations we found into one model. To do this, we randomly selected 3 sets of different temperature, pH and salinity and input the growth results into the Simulink program found in Matlab. Through this we could determine how much impact each factor has on subtilis growth relative to the other two factors and adjust the parameters accordingly. Our resulting combined model has the ability to predict subtilis growth rate under a wide range of temperature, pH and salinity.