Collaboration

In our collaboration with Fujian Agriculture and Forestry University (FAFU), we helped them develop a mathematical model of the PSB-assisted phytoextraction process to verify the validity of the new technology in real life application. Their corresponding page can be accessed here. We also produced the structural images of some key proteins for them. The corresponding page of FAFU can be accessed here.

Because of the polarity of microtubule, the two kinds of monomer we display on the yeast may lead to different experiment results, so we decided to display the both monomers onto the yeast, one of which was helped by our collaborators. Besides, we need to see the effect of mGFP on our tubulins, FAFU also take part in the work.

BNU:Modeling

Overview

Heavy metal pollution of the soil is plaguing vast areas of lands across the world. In a process called phytoextraction, hyperaccumulator plants can extract the heavy metals in the soil and gradually make the soil return to its natural conditions. An existing method to accelerate this process is to apply phosphate-solubilizing bacteria (PSB) to the soil around the plant. These kinds of bacteria can produce organic acid and release heavy metals from their insoluble states in order for hyperaccumulator plants to better absorb them. However, the growth process of these plants are typically inhibited by high concentrations of heavy metals, which results in low biomass, decelerating the phytoextraction process.

The FAFU team aims to engineer bacillus megaterium, a type of PSB, to produce heavy-metal-binding proteins to counter the downside of its phosphate-solubilizing effect by lowering the concentration of free heavy metal ions in the soil. This modification can promote the accumulation of hyperaccumulator plants’ biomass, which, as mentioned above, will accelerate the phytoextration process.

Different from lab conditions, this process is far more complicated in real-life application in the soil. To help our FAFU collaborators examine the feasibility of their design, we developed a mathematical model to describe and simulate this process.

Model Formulation

Our main focus is to examine whether the designed system can effectively lower the concentration of heavy metal ions in the soil. There are 3 main processes at play here: the phytoextraction process, the organic acid production process, and the binding of heavy metal ions with proteins produced by the engineered PSB. To analyze this system, we built the following system of ordinary differential equations.

\[ \left\{ \begin{aligned} \frac{\mathrm{d}c_m}{\mathrm{d}t}&=-r_ac_{m,eq}\\ \frac{\mathrm{d}c_p}{\mathrm{d}t}&=-r_dc_{p,eq}+a_1\frac{\mathrm{d}m}{\mathrm{d}t}\\ \frac{\mathrm{d}c_h}{\mathrm{d}t}&=mr_h\\ \frac{\mathrm{d}m}{\mathrm{d}t}&=m(1-\frac{m}{K(c_{h,eq})}) \end{aligned} \qquad\qquad \begin{aligned} c_m(0)&=c_{m0}\\ c_p(0)&=c_{p0}\\ c_h(0)&=c_{h0}\\ m(0)&=m_0 \end{aligned} \right. \]

The above equations are for the case where proteins are displayed on the surface of PSB. For the case where the proteins are secreted into the environment, we only need to change the second equation to

\[ \frac{\mathrm{d}c_p}{\mathrm{d}t}=-r_dc_{p,eq}+a_1'm \]

For clarity, a table of variables and a table of parameters are provided at the end of this section for your reference.

We shall now explain in detail the formulation of these equations. First, notice that for the concentrations $c_m$ (heavy metal ions), $c_p$ (protein) and $c_h$ (hydrogen ions), there are the corresponding $c_{m,eq}$, $c_{p,eq}$ and $c_{h,eq}$ in the right side of the equation. The former ones are the imaginary “initial” concentrations, while the latter ones are the concentrations at the equilibrium state. To further expound this idea, we present the three major chemical reactions in the rhizospheric system.

\[ \begin{align}& \ce{M^2+ + pro <=> pro(M^2+)} \\ & \ce{H+ + pro <=> pro(H+)} \\ & \ce{MN + 2H+ -> H2N + M^2+} \end{align} \]

The reaction equations written above are highly schematic. $M^{2+}$ represents a typical type of heavy metal ion like $Pb^{2+}$. $pro$ stands for a heavy-metal-binding protein like MBP. $N$ is a general anion which combines with $M^{2+}$ to form insoluble salt. The first reaction is the binding process of heavy metal ions and proteins. From the literature provided by FAFU, we learned that when pH is low, hydrogen ions compete with heavy metal ions for protein binding sites, hence the second reaction. We assume one binding site of ions for each protein molecule. The third reaction captures the PSB’s ability of secreting organic acids into the soil to release heavy metals from their insoluble forms. We denote the equilibrium constants of these equations by $K_1$, $K_2$ and $K_3$. We assume that temperature remains mostly stable in our simulation, which makes $K_1$, $K_2$ and $K_3$ constant over time.

From our analysis in the overview, we can see that the external influences that can change the balance of this chemical system are the absorption of $M^{2+}$ of the plants, the display and degradation of $pro$, and the increase of $H^+$ brought about by PSB. The change of the concentrations of other reactants is an indirect result from the chemical balance shift. The mixture of external influences and internal balance shifts brings much difficulty to our analysis.

To address this problem, we noticed that at our scale of study which is typically in hours, the chemical reactions are comparatively microscopic, meaning that the system can be viewed as in equilibrium at all times. We can therefore imagine an “initial” state for every equilibrium state and separate the analysis of the chemical process and that of the external changes. With the help of $K_1$, $K_2$ and $K_3$, we can also get a one-to-one map between the “initial” states and the equilibrium states.

\[ (c_{m,eq}, c_{p,eq}, c_{h,eq}) = f(c_m,c_p,c_h;K_1,K_2,K_3) \]

Now we are ready to expound the differential equations given in the beginning.

The first equation describes the phytoextraction process. The change in the imaginary “initial” concentration of heavy metals ($c_m$) is only affected by the absorption of the plant, which is assumed to be proportional to its actual concentration in the soil ($c_{m,eq}$). $r_a$ is the absorption rate of the plant.

Similarly, the change in $c_p$ is only related to the degradation of the actual available proteins in the environment ($c_{p,eq}$) and the new proteins produced by the engineered PSB. $r_d$ is the degradation rate. In the protein production part, since FAFU designed two parts, one for surface display and one for secretion, we also have two possible terms to describe them. For display, one PSB is only able to display a fixed amount of proteins in its lifetime, therefore the main source of protein production is the reproduction of PSB, hence the $a_1\frac{\mathrm{d}m}{\mathrm{d}t}$ term. For secretion, PSB is constantly secreting proteins, hence the $a_1’\frac{\mathrm{d}m}{\mathrm{d}t}$ term. However, we only have done the simulation for the display version.

The third equation is about the change in pH. In our simulation time period, we only consider the effect of PSBs’ secreting organic acids. $r_h$ is the acid production rate, which we have taken as a constant.

The fourth equation is just the ordinary logistic growth model with a varied carrying capacity $K(c_{h,eq})$ for the PSB (whose population density is $m$) in the soil. Assuming temperature and other relevant conditions are constant, we see it as a function of acidity. It is a quadratic function with respect to $\text{pH}=-\mathrm{lg}c_{h,eq}$ obtained from fitting experimental data in the literature. It reaches its highest point $K_{max}$ at $\text{pH}=6$ i.e. $c_{h,eq}=1\times 10^{-6}$ and decreases to $0$ at $\text{pH}=2.69$ and $\text{pH}=9.76$.

Table 1 Table of variables in the model for FAFU.

Variable	Description	Unit
$c_m$	The concentration of heavy metal ions (before chemical equilibrium).	$\text{mol / L}$
$c_{m,eq}$	The concentration of heavy metal ions (at chemical equilibrium).	$\text{mol / L}$
$c_p$	The concentration of proteins (before chemical equilibrium).	$\text{mol / L}$
$c_{p,eq}$	The concentration of proteins (at chemical equilibrium).	$\text{mol / L}$
$c_h$	The concentration of hydrogen ions (before chemical equilibrium).	$\text{mol / L}$
$c_{h,eq}$	The concentration of hydrogen ions (at chemical equilibrium).	$\text{mol / L}$
$m$	The population density of PSB.	$\text{mg / kg}$

Table 2 Table of parameters in the model for FAFU.

Parameter	Description	Estimation
$c_{m0}$	The initial concentration of heavy metal ions.	$5 \ \times 10^{-6}\text{mol / L}$
$c_{p0}$	The initial concentration of proteins.	$0 \ \text{mol / L}$
$c_{h0}$	The initial concentration of hydrogen ions.	$1 \ \times 10^{-7}\text{mol / L}$
$m_0$	The initial population density of PSB.	$100 \ \text{mg / kg}$
$r_a$	The absorption rate of the plant.	$0.01$
$r_d$	The rate of degradation of the proteins.	$0.0058$
$a_1$	The amount of proteins displayed for a unit of PSB.	$3 \ \times 10^{-9}\text{(mol / L) / (mg / kg)}$
$r_h$	The rate of organic acid production of PSB.	$1 \ \times 10^{-10} \text{mol / (L} \cdot \text{h)}$
$K_1$	The equilibrium constant of reaction (1).	$300 \ \text{(mol / L)}^{-1}$
$K_2$	The equilibrium constant of reaction (2).	$900 \ \text{(mol / L)}^{-1}$
$K_3$	The equilibrium constant of reaction (3).	$3 \ \times 10^8\text{(mol / L)}^{-1}$
$K_{max}$	The maximum carrying capacity of PSB.	$1000 \ \text{mg / kg}$

Numerical Simulation

Due to time constraints, we were not able to measure some parameters in experiments. However, since we are primarily interested in the qualitative dynamics of the system, the estimations in the above table can still provide reasonable conditions for our simulation.

We ran a numerical simulation of the differential equations using MATLAB for a time period of 100 hours. The simulation results are as follows.

Figure 1 The simulation result of $c_{m,eq}$, the concentration of heavy metal ions

Figure 2 The simulation result of $c_{p,eq}$, the concentration of proteins

Figure 3 The simulation result of $c_{h,eq}$, the concentration of hydrogen ions

Figure 4 The simulation result of $m$, the population density of PSB

We can see a steady drop of heavy metal ion concentration which drops to more than half of its initial value over less than 5 days, which confirms the plausibility of FAFU’s design.

There are also some other interesting points in the graphs. We can see an almost constant decrease of pH from about 7 to about 5 in 100 hours, which shows the PSB’s ability to improve the soil conditions. Changing the pH also affects the population density of PSB. As pH goes below the optimum value 6, $m$ begins to decrease, making $c_{p,eq}$ decrease along with it.

Conclusion

Our model gave a schematic description of the dynamics of FAFU’s design in real-life application. The results confirmed the feasibility of their work.

Furthermore, the framework provided above can be used in further analysis. With some modifications, the model can also be used to simulate the case where secreted proteins are used instead of displayed ones, and where the original PSB (without the specifically designed proteins) instead of engineered PSB is used. Comparing these results can help guide future efforts in experiment and application.

BNU:Protein Structure

Our fellow collaborators at FAFU asked us to analyze the tertiary structure of some of the proteins they are using with their sequences. We used Swiss model to predict their structures and obtained the following pictures.

Figure 5 The protein structures of DsbA (fusion protein-periplasm protein), DsbA + MBP (Hg), DsbA + MBP (Pb), GST (glutathione-S-transferase) - MT (metallothionein) (Cd), Lpp – OmpA + MBP (Hg), Lpp – OmpA + MBP (Pb), MBP (Hg metal binding peptide engineered from MerR), MBP (Pb metal binding peptide engineered from PbrR)

FAFU:Experiment

FAFU helped us transfer PYD1-α and PYCα-β-mGFP into EBY100 and INVSc1 respectively. They obtained the monoclonal cultures as the following picture showed and sequenced some of the yeasts. As a result, they sent us the two kinds of vectors we need.

Figure 6 the monoclonal cultures of PYD1-α and PYCα-β-mGFP FAFU obtained
(PYD1-α in the first row and PYCα-β-mGFP in the second row)

Figure 7 the monoclonal cultures of PYD1-α and PYCα-β-mGFP FAFU sent us
(PYD1-α in the right and PYCα-β-mGFP in the left)

We inducible expressed the two kinds of vectors and found out they expressed the correct proteins. Especially for the PYCα-β-mGFP, we detected the fluorescence signal of mGFP, which meant mGFP worked well in our system.

Figure 8 the fluorescence signal of mGFP we detected on PYCα-β-mGFP