Growth Model

The Growth Model

Background and Assumptions

As the literature indicated, bacteria and cyanobacteria are living together in the soil, both known as crucial microorganisms during the development of biological soil crust. Bacteria secrete organic acids and powerful exoenzymes that dissolve inorganic/organic nitrogen and phosphorus for cyanobacteria to use; they also make many other essential compounds like vitamins accessible. On the other side, Nostoc sp. and even more cyanobacteria can fix CO₂ and N₂ from the air, thus offering organic matter and extra nitrogen as fundamental needs for bacteria^[1]. Such a mutualism is the central relationship between them that determines their survival in the soil. In some sense, they are interdependent in the nutrition-lacking soil. Although some cyanobacteria and Bacillus in lakes may create toxins to repress the growth of each other, many studies have proved that Nostoc sp. and Bacillus subtilis do coexist in the soil and cooperate to make sandy soil a suitable environment for lives^[2].

Therefore, we put the application scenario of the growth model in a desert soil environment, and the goal of the model is to simulate the growth of two kinds of organisms under the action of resource competition in a certain artificially managed soil.Through simulation, we explored the formation and existence of the symbiotic relationship between Bacillus Subtilis and Nostoc sp. Our growth model focuses on the above relationship, emphasizing the nutritional constraint. For simplicity, we also make the following principal assumptions:

We ignore other organisms and most environmental changes (such as rain, wind, CO₂ and or so), aiming to simulate merely the interations between Bacillus Subtilis and Nostoc sp. Since the soil environment is really complicated, it is impossible to take everything into account. However, in regard of the nitrogen and phosphrus fixing, we may use the average rate considering all microorganisms in the soil.
We assume that the two microorganisms and all related nutrients are distributed evenly in the soil. In other words, they are living in a solution-like system and spatial heterogeneity is not considered. Thus, we can use biomass concentration and nutrient concentrition to describe this system quantitively.
“Nutrient” means matter that bacteria and cyanobacteria can directly and quickly, or “utilizable nutrients”. C, N, P exist in plenty of forms in the soil. Literature call them “dissolved nitrogen”, “soluable phosphorus” and so on. Only low molecular weight (LMW) or ions can be quickly utilized. For example, amino acids and soluable phosphate are “utilizable nutrients”, while humus and Ca₃(PO₄)₂ are not, which need enzymatical decomposition.
We agree that after Nostoc sp. die, the cell matter is quickly decomposed by bacteria and return to “utilizable nutrients”, while Bacillus Subtilis don’t do this^[3].
Another assumptions of established models we used.

We are not considering spores, because they are not actually metabolizing. In other words, they neither produce or consume nutrients and EPS.

We are not considering spore germination and metabolism of spores, because they are not actually metabolizing and the environment can not meet the suitable conditions for germination. In other words, they neither produce or consume nutrients and EPS.

Mathematical Model

Equations

Main variables of interest are:

$N_i$ ($\mathrm{g/L}$): biomass concentration (cell dry weight per unit volume);
$R_j$ ($\mathrm{g/L}$): nutrient concentrition (mass of the element per unit volume).

In the following equations built on classic resource competition model (Theoretical Ecology: Principles and Applications) , subscript $i$ equals to 1 for Bacillus subtilis and 2 for Nostoc sp. ; subscript $j$ equals to $c$ for carbon, $n$ for nitrogen and $p$ for phosphorus, respectively.

Tab. 1 equations
No.	Equation	Descriptions
1	$\dfrac{\mathrm{d}N_i}{\mathrm{d}t}=Birth_i - Death_i$	For both microorganisms, the net growth rate $\frac{\mathrm{d}N_i}{\mathrm{d}t}$ is defined by birth rate minus death rate.
2	$Birth_i=f_i r_i N_i$	Birth rate $Birth_i$ is modified by a nutrient-constraint factor $f_i(R)$ which is not more than 1.
3	$Death_i=m_i N_i$	Growth and death rates are proportional to current population $N_i$ quantity with a ratio.
4	$f_i(R)=\min\limits_j\{\dfrac{R_j}{R_j+K_{ij}N_i}\}$	The constraint factor is built according to Monod growth kinetics with Liebig's law of the minimum.
5	$\dfrac{\mathrm{d}R_j}{\mathrm{d}t}=Income_j-\sum\limits_i Q_{ij} Grow_i$ or $\dfrac{\mathrm{d}N_i}{\mathrm{d}t}$	The concentration of nutrients are dynamically regulated by production and consumption by microorganisms.

Here we will detailedly explain our innovation in the last two equations.

Nutrient-constraint factor

$f_i(R)$ looks quite similar to Michaelis-Menten equation, which is used by many former iGem teams. When $R_j$ goes to infinity, $f_i(R)$ approaches to 1, thus have no effect on the birth rate; otherwise, the birth rate seriously decrease, even leading that net growth rate is less than 0.
$f_i(R)$ reflects how much each nutrient limits growth. Inspired by Liebig's law of the minimum, the overall effect is determined by the most lacking nutrient. Even if carbon and nitrogen is sufficient, when phosphrous is deficient, microorganisms cannot grow.
$K_{ij}$ is a half-saturation constant like Michaelis-Menten constant. However, most literatures ignored that the amount of nutrient needed obviously depends on biomass concentration. Therefore, we adopt half-saturation constant per unit biomass concentration, forming a variable and more realistic half-saturation constant. This is actually based on [4]'s research.

Dynamic change of nutrients

In order to monitor the nutrition limitation ($f_i$) encountered by microorganisms, we established the dynamic change equation of resources according to the physiological metabolism of microorganisms in the environment. The detailed income and output is shown in equation 7:

$$ \begin{align} \dfrac{\mathrm{d}R_c}{\mathrm{d}t}&=-Re_1\cdot N_1+NPh\cdot N_2-Q_{1c}Grow_1-Q_{2c}\dfrac{\mathrm{d}N_2}{\mathrm{d}t}\\ \dfrac{\mathrm{d}R_n}{\mathrm{d}t}&=N_{sol}\cdot N_1 + N_{fix}\cdot N_2-Q_{1n}Grow_1-Q_{2n}\dfrac{\mathrm{d}N_2}{\mathrm{d}t}\\ \dfrac{\mathrm{d}R_p}{\mathrm{d}t}&=P_{sol}\cdot N_1-Q_{1p}Grow_1-Q_{2p}\dfrac{\mathrm{d}N_2}{\mathrm{d}t} \end{align} $$

Based on assumption 4, the loss of nutrients is proportional to net growth rate ($\frac{\mathrm{d}N_2}{\mathrm{d}t}$) for Nostoc sp. and birth rate for *B. Subtilis* ($Grow_1$). So dead *B. Subtilis* (or spores) don't give its cell matter back to $R_c$, $R_n$ and $R_p$.
Obviously, every resource will be consumed as a part of an organism when it grows. $Q_{ij}$ means consumption rate of nutrient $j$ per unit biomass concentration.
- Aerobic respiration is alway ongoing, so we have to use the ratio between C consumption and biomass increase as $Q$.
The consumption of carbon resources includes the growth and reproduction of microorganisms and the dissipation of respiration; At the same time, photosynthesis from Nostoc sp. will supplement carbon source.
- $Re_1$ represents the consumption rate of carbon source per unit biomass B.S;
- $NPh$ means the net fixation rate of organic carbon obtained by photosynthesis rate per unit biomass vegetable Nostoc sp. ,which offsets its respiration.
The formation of microorganisms is the main way to consume N resources, and the supplement comes from nitrogen mineralization mediated by Bacillus Subtilis (decomposition of organic matter by extracellular enzymes) and nitrogen fixation by Nostoc sp. using N~2~ in the atmosphere. We have not considered the nitrogen cycle process caused by other environmental factors, although they often play a potentially important role.
- $N_{sol}$ represents the nitrogen mineralization rate per unit biomass Bacillus Subtilis;
- $N_{fix}$ is the nitrogen fixation rate per unit biomass Nostoc sp.
The consumption of P element is similar to that of N element, and Bacillus Subtilis takes the secretion of organic acids and extracellular enzymes to mediate the solubilization of phosphate minerals and organophosphorus in the environment as the supplement of resources.
- $P_{sol}$ represents the rate of producing available P per unit biomass Bacillus Subtilis.

Parameters' Values

All $K$ and $Q$ are dimensionless. Units of rates are all $\mathrm{g/(L\cdot min)}$ .

See Parameters Here

Tab. 2 parameters

Name	Value	Descriptions	References
$K_{1c}, K_{1n}, K_{1p},K_{2n}$	1.1460,0.638,0.057,0.051	The paper is using Contois model for Bacillus Subtilis and pseudomonas aeruginosa.	^[5]
$K_{2c},K_{2p}$	0,0.0187	$K_{2c}$ given by the paper equals to 0; $K2p$ for Cyanothece sp. is given.	^[6]
$Q_{1c}$	0.6528	Calculated from the slope of the “nutrient cosumption”-$\mu$ curve. It means the reciprocal of “growth yield”.	^[7]
$Q_{1n},Q_{1p}$	0.0734,0.00607	Calculated from mass ratio between elements. $Q_{1n}$: P limited scenario; vice versa.	^[8]
$Q_{2c}$	0.2523	We do’t have access to autotropical carbon consumption/biomass concentration. We just used elemental mass fraction.	^[9]
$Q_{2n}$	0.0846	As for N and P, we use their proportion in cell dry weight because there is no extra loss.	^[9]
$Q_{2p}$	0.00248	Ditto. Species: “blue-green algae”	^[10]
$Re_1$	1.34e-4	We calculated it from “maintenance coefficient”, intercept of the “nutrient cosumption”-$\mu$ curve.	^[7]
$NPh$	9.27e-3	Using the luminosity on a sunny day, we determined it based on a luminosity-net photosynthesis rate model $NPh=NPh_{max}\dfrac{I}{K_s+I+I^2/K_i}$	^[11]
$N_{fix}$	7.38e-6	Calculated from control group N-fixing rate and convert it to rate per unit biomass concentration.	^[12]
$N_{sol}$	2.60e-4	Calculated from average N-fixing rate of mixed soil microorganisms under different biomass concentration.	^[13]
$P_{sol}$	2.15e-5	Calculated from Bacillius megaterium P solubilizing rate of poultry bones.	^[14]

Result Analysis

Growth curve

We use the data obtained from the literature and the experimental analysis results to provide a set of parameters with high reliability for the growth model. After trial, we used $0.1 \mathrm{g/L}$ and $0.2 \mathrm{g/L}$ as the initial biomass of Bacillus subtilis and Nostoc sp., respectively. The following analysis uses this initial value. Using the parameters and initial values mentioned above, the following is the result of growth simulation by the model:
图。

In the beginning, both B. subtilis and Nostoc sp. will be limited by P resources, and the two organisms will grow slowly temporarily. However, P concentration will continue to increase with the phosphorus dissolution of B. subtilis. Therefore, the restriction factor on B. subtilis quickly changed from P to N and reduced the restriction on Nostoc sp. of P to a certain extent. Nostoc sp. keeps growing faster, leading to large consumption of N, and the resource of N tends to decline on the 10th day. B. subtilis was restrained by competition of N and $f_1$ decreases slowly. Therefore, its biomass gradually decreased after reaching its peak.

With the consumption of N, N replaces P to limit the growth of Nostoc sp. on the 16th day, and the growth rate of Nostoc sp. decreases (an obvious turning point on the 16th day). The biomass of B. subtilis reached the lowest, and the N limitation became smaller, $f_1$ increases slowly. This allows B. subtilis to grow again and makes the amount of N resources rise. Eventually, both B. subtilis and Nostoc sp. can achieve a steady growth trend, and all resources can also rise steadily.

It can be seen that both organisms have passed through P and N restriction periods from the results, although this may be related to the initial value of resources and $K$. However, in the absence of special instructions, in the subsequent parameter analysis, we default that both organisms will pass through P first and then N limited period.

Parameters sensitivity

To gain insight into the growth mechanisms, we explored the influence of the parameters on the ultimate biomass of B. subtilis and N sp., and then analyzed the stability of the whole system. This part also guides a more detailed sensitivity analysis. When comparing groups of data, the coefficient of variation (CV) can reflect the absolute value of data dispersion while eliminating the influence of measurement scale and dimension. CV analysis is carried out to observe the influence of parameters. It is expressed as:

cv=\dfrac{\sigma}{\mu} \tag{8}

where $\sigma$ represents standard variance and $\mu$ represents the average value. For example, dataset {9,10,11} has the same CV as dataset {90,100,110}, while the latter one's variance is 10 times larger. We adopt a Monte Carlo simulation to calculate CV. For different parameters, we randomly sample from a normal distribution whose mean value $\mu$ is the literature value of each parameter, and variance σ is small. After trial, we set $\sigma$ as 5%. Then they are inputted into the model and it outputs biomass of the two species on some certain days. These outputs are fed into Eq. 8 to obtain CV of this parameter. The results are shown in Table 3 and Fig. 9,10:

Tab. 3 CV of m1
Microorganism	5 days	10 days	15 days	20 days	25 days
B. subtilis	0.485	1.373	4.374	7.259	11.324
Nostoc. sp	0.140	0.500	1.203	2.236	5.268

Tab. 3 Coefficient of Variance of $m_1$. Influence of $m_1$ is too large, we list it separately to better show the influence of other factors.

Expansion of parameters range

By analyzing the sensitivity of parameters, we find that the model has an obvious response to the change of some parameters. To extend the available range of the model, we will deeply understand the influence of these sensitive parameters on the model to study the performance of the model under different environmental conditions and biological species.

$m_1$

m1 describes the natural mortality rate of B. subtilis (including sporulation), and it is worth noting that after B. subtilis dies/sporulation, this part of resources cannot be utilized temporarily. The biomass of B. subtilis and Nostoc sp. decreased with the increase of $m_1$. This is because the increase of $m_1$ limits the growth of B. subtilis and promotes the loss of resources.

$P_{sol}$ and $N_{sol}$

Overview：$N_{sol}$ and $P_{sol}$ affect the input rate of resources into the system, thus affecting the interaction process of total resources, growth rate and biomass. Experiments (cv, simulation) show that changing $N_{sol}$ and $P_{sol}$ can significantly change the relationship between B. subtilis and Nostoc sp.

$P_{sol}$

We analyzed $P_{sol}$ by study the derivative of biomass to it. With the increase of phosphorus dissolution rate, the derivative of B. subtilis biomass was first positive, then negative, and finally approached 0, while the derivative of Nostoc sp. biomass first increased and then decreased, which was always positive.
Explanation: A faster phosphorus dissolution rate could alleviate the P-limitation experienced by B. subtilis and Nostoc sp. at an early stage. After exceeding a certain threshold, due to the early arrival of the N-limiting stage, the fast-growing Nostoc sp. inhibits the growth of B. subtilis by competing for N, and this effect will gradually flatten out.

$N_{sol}$

When the nitrogen mineralization rate is low, the derivative of B. subtilis biomass to nitrogen mineralization rate is positive, and the derivative of Nostoc sp. is negative. With the increase of nitrogen mineralization rate, the derivative gradually approaches and reaches 0. The length of this process is related to the time of the target point.
Explanation: A higher nitrogen mineralization rate allowed B. subtilis and Nostoc sp. to enter the N-limiting stage later. B. subtilis without timely N-limiting would have a higher growth rate and inhibited the growth of Nostoc sp. by competing for P. When the nitrogen mineralization rate increases to a certain value, the N-limiting stage will be completely lifted and the growth condition will not be changed.

$K_{1N}$

Overview:$K$ affects the growth rate under resource constraints, which leads to the cross-influence of growth rate and biomass/resources.

$K_{1n}$

$K_{1n}$ affects the growth of B. subtilis under N restriction. It can be seen from fig. 1 that the smaller $K_{1n}$ is, the higher the initial coincidence degree of curves is(). this is because the smaller $K_{1n}$ is, the later B. subtilis enters the n-restriction stage, and then $K_{1n}$ affects the growth rate. However, the early change of B. subtilis biomass did not cause an obvious change of P, so it had no obvious effect on the growth of Nostoc sp.. The rapid growth of B. subtilis in the early stage will consume more N, although after the subsequent growth slows down, more N will be dissolved by the larger B. subtilis biomass.

It is obvious here that B. subtilis biomass and N resources fluctuate in the medium term, which is caused by the short-term rapid growth of cyanobacteria after entering the N limitation period. This phenomenon is similar to that in standard parameter output results.

$K_{2n}$

$K_{2n}$ affects the growth of Nostoc sp. under N restriction. Before 15 days, the change of $K_{2n}$ had little effect on growth and resources, because the growth of Nostoc sp. was limited by P resources before 15 days. After 15 days, the smaller $K_{2n}$ is, the faster the growth of Nostoc sp. will be, the faster the consumption of N and P will be, and the growth of B. subtilis will be squeezed. However, due to the rapid consumption of resources, the biomass will be smaller later, and the amount of P and N resources will be lower in the end.

$K_{1p}$

$K_{1p}$ affects the growth of B. subtilis under P restriction. With the decrease of $K_{1p}$, B. subtilis will have a faster growth rate because of the decrease of P restriction. But at the same time, P resources will be greatly reduced in a short time, so the growth of Nostoc sp. will slow down. Then, depending on the larger biomass of B. subtilis to dissolve P, P resources will increase with the decrease of KP1, and the rapid growth of Nostoc sp. will also increase, and the growth of B. subtilis will be inhibited by the competition for N.

$K_{2p}$

The larger the $K_{2p}$ is, the greater the P restriction on Nostoc sp., so it will have a slower growth rate in the early stage. At the same time, due to the competitive relationship of P, B. subtilis will have a higher growth rate. This also makes P resources more, so in the later period, the biomass of Nostoc sp. will be more than when $K_{2p}$ is small.

$Q_{1n}$

Obviously, the lower $Q_{1n}$ will cause the decrease of n resource consumption, and Nostoc sp. will grow faster after entering the n limit, and the resources will accumulate continuously. The rapid consumption of p will slightly increase the inhibition of Nostoc sp. However, after the biomass of B. subtilis accumulated to a certain extent, both N and P increased rapidly, so that B. subtilis and Nostoc sp. could reach higher biomass.

$Q_{1p}$

The smaller the $Q_{1p}$, the more P will be accumulated in the early stage. Bacillus Subtilis will grow faster temporarily, and then the fast-growing Nostoc sp consumes a lot of N, which inhibits the growth of Bacillus Subtilis. The smaller $Q_{1p}$ is, the more obvious this phenomenon is, while the larger will not appear. Therefore, with the increase of $Q_{1p}$, the growth of both will first increase and then decrease. Therefore, we find that: $Q_{1p}$ is too small to cause Nostoc sp to grow too fast, which is not conducive to resource accumulation; Too much $Q_{1p}$ limits the growth of Bacillus Subtilis and is not conducive to resource accumulation.

polysaccharide production

Experiments proved that the polysaccharide yield of Bacillus subtilis can be up to 2.06 times as high as before by transferring different combinations of pgmA and galU. We predicted the real-time yield of polysaccharide by the growth model and the Luedeking-Piret equation, so as to predict whether the yield of exopolysaccharide can be significantly improved in the early stage. The yield of polysaccharide is very important for the stability and functionality of the system.

Under normal circumstances, there was no obvious difference in polysaccharide yield between them in the early stage. Comparing Fig. 45 and 46, we can find that the polysaccharide yield of B. subtilis is obviously improved in the early stage after the synthetic biological transformation, and the polysaccharide yield of B. subtilis will be more than that of Nostoc sp. in a long period of time. This indicates that the function of B. subtilis will be greatly improved in the early stage. This is in line with our project design. From the early stage when B. subtilisplayed a leading role, it will gradually transition to the period when Nostoc sp. played its role.

Conclusion and Prospects

We found that Bacillus Subtilis and Nostoc sp. can successfully so-survive in the soil maily due to their high Complementarity in function. They can effectively increase the content of bioavailable nutrients in the soil while growing. This proves that our symbiotic system can improve the soil effectively. EPS can play an effective role when living things can survive stably.Through parameter sensitivity analysis, we found that the system was highly sensitive to the change of $m_1$, which indicated that the change of mortality rate of *Bacillus Subtilis* caused by external disturbance and environmental change had a large impact on the system.
Though being a moderately good simulation of the growth of the two microorganisms, our growth model is not flawless due to neglecting various environmental factors and needs perfection. By extending the range of parameters, we think that the model can still be simulated for different environments and biological systems. Therefore, our model has great potential and prospects.

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Team:XJTU-China/Growth-Model

XJTU-China

The Growth Model

Table of Contents

Background and Assumptions

Mathematical Model

Equations

Nutrient-constraint factor

Dynamic change of nutrients

Parameters' Values

Result Analysis

Growth curve

Parameters sensitivity

Expansion of parameters range

$m_1$

$P_{sol}$ and $N_{sol}$

$P_{sol}$

$N_{sol}$

$K_{1N}$

$K_{1n}$

$K_{2n}$

$K_{1p}$

$K_{2p}$

$Q_{1n}$

$Q_{1p}$

polysaccharide production

Conclusion and Prospects

Project

Parts

Lab

HP and Team

Sponsors

Team:XJTU-China/Growth-Model

The Growth Model

Table of Contents

Background and Assumptions

Mathematical Model

Equations

Nutrient-constraint factor

Dynamic change of nutrients

Parameters' Values

Result Analysis

Growth curve

Parameters sensitivity

Expansion of parameters range

$m_1$

$P_{sol}$ and $N_{sol}$

$P_{sol}$

$N_{sol}$

$K_{1N}$

$K_{1n}$

$K_{2n}$

$K_{1p}$

$K_{2p}$

$Q_{1n}$​

$Q_{1p}$​​

polysaccharide production

Conclusion and Prospects

Project

Parts

Lab

HP and Team

Sponsors

$Q_{1n}$

$Q_{1p}$