Overview
With the ambitious of exploiting modeling to guide, validate, clarify and expand our project, the MODEL group of Nanjing-China builds a Four-in Model: PolyP in Design, PolyP in Bacteria, PolyP in Gut and PolyP in Application, which covers molecular simulation, mathematic models and stochastic models. To make our novel models understood by everyone, we document our work in great detail, in which some stories and experience are shared. To make it convenient for readers, we provide an overview table to shed light on our Four-in Model. You can choose the part you are most interested in.
| Four-in Models | |||
| Name | Kind | Highlight | Relation with the project |
| PolyP in Design | molecular simulation | simulation of the binding between enzymes and polyPs in biosynthetic process | 1)PolyPs with different lengths were docked to two phosphotransferases, PPK1 and PPK2B, to compare the binding behaviors and predicted binding energy of them. 2)Results guide how the project should be implemented with the best choice of the enzyme for polyP biosynthesis. |
|---|---|---|---|
| PolyP in Bacteria | mathematical model | a novel model for bacterial growth in synthetic wastewater (PA) medium | 1)Experimental data was used for curve fitting to test the model. 2)Fitting results validate that the plasmid design worked and the model will help lab members to predict bacterial growth. |
| PolyP in Gut | stochastic model | a great way to describe the relationships between polyP and gut bacteria based on computer programming | 1)Coefficients can be given by stochastic simulation based on programming. 2)Results can help predict the possible effect of polyP on gut bacteria community composition. |
| PolyP in Application | mathematical model | a polyP-associated coherent feedforward loop for medicine | 1)Simulative changes in concentrations of ions, active enzymes and stable polyP provide insight into transforming polyP into both IBD drugs and health products. |
PolyP in Design
Introduction
Because the lab group wanted to biosynthesize polyP, we helped with the choice of the enzyme by molecular simulation. As the major enzymes participating in the metabolism of polyP, PPK1 and PPK2, which respectively play a role in the synthesis and degradation of polyP, both belong to polyphosphate kinases (PPKs).But there’s an exception that PPK2B of C. glutamicum shows preference for polyP synthesis . With the goal of guiding experimental choice of the enzyme, we used Autodock4 to dock polyPs with different lengths to PPK1 and PPK2B. Comparing the binding behaviors and predicted binding energy of these two enzymes, we concluded that it is PPK1 that has an advantage in polyP synthesis.
Hypothesis
Interactions between PPK1 and polyPs should be stronger than interactions between PPK2B and polyPs, considering that PPK1 is more common among bacteria.
Methods
In the preparation stage, the accurate 3D structure of PPK1 was acquired from Protein Data Bank and the structure of PPK2B was predicted by SWISS-MODEL, using homology modeling. As for the structure of polyPs with different lengths, we obtained them from PubChem.
In the docking procedure, firstly we prepared the PDBQT files of proteins and ligands by adding hydrogens, computing charges and assigning AD4 type for each atom. For ligands, there was an additional step to choose roots and torsion trees for them.Secondly, we set map types for ligands, created a searching space called gridbox which contained all the atoms of proteins, exported it as a GPF file and ran autogrid.Thirdly, we chose the semi-flexible docking, set the number of genetic algorithm runs as 50, created a DPF file and ran Autodock.Finally, the received DLG file was analyzed to choose the conformation of ligands docking into enzymes,according to binding energy and docking sites.
Figure1 The schematic workflow of actual methods.
Assumption
We assumed that the conformation of various ligands could be changed, while the conformation of enzymes remained rigid. So we chose the semi-fexible docking to simulate binding behaviors.
Model
The final simulated models were modified and visualized through Pymol as shown in Figure 1 and 2.
(A) (B)
(C) (D)
(E) (F)
Figure 1 Docking results of PPK1: Pictures on the left show the most probable binding sites for polyP3(A), polyP5(C)and polyP20(E), while pictures on the right provide a magnified look at the binding sites and label the amino acids that are responsible for interacting with polyP3(B), polyP5(D) and polyP20(F). The yellow dotted lines represent hydrogen bonds between proteins and polyPs. And interacting amino acids are shown in orange while the bonds that directly interact with polyP are emphasized as yellow sticks.
(A) (B)
(C) (D)
(E) (F)
Figure 2 Docking results of PPK2B: Pictures on the left show the most probable binding sites for polyP3(A), polyP5(C)and polyP20(E), while pictures on the right provide a magnified look at the binding sites and exhibit the amino acids that are responsible for interacting with polyP3(B), polyP5(D) and polyP20(F). The yellow dotted lines represent hydrogen bonds between proteins and polyPs. And interacting amino acids are shown in orange while the bonds that directly interact with polyP are emphasized as yellow sticks.
Results and Discussion
| Binding energy | PolyP3 | PolyP5 | PolyP20 |
| PPK1 | -3.84 | -2.50 | -2.19 |
| PPK2B | -3.54 | -1.95 | -1.49 |
Table 1 Predicted binding energy for every docking simulation.
The predicted binding energy from DLG files is listed in Table 1. Three conclusions can be drawn with these results.
1. There are basic amino acids, such as arginine and lysine, in the docking sites, which indicates electrostatic interactions between the enzyme and phosphate groups during the elongation.
2. Because Autodock4 predicts the binding energy according to semi empirical free energy force field, the binding energy partly represents the binding affinities between enzymes and ligands. Therefore, predicted binding energy between enzymes and polyPs increases as the polyP gets longer, indicating that the interaction and thereby catalytic ability weaken as the polyP chain grows.
3. With a lower binding energy, the binding between PPK1 and polyP is tighter than that between PPK2 and polyP with the same length, supporting our hypothesis. Therefore, we suggested the lab group to choose PPK1 for polyP synthesis.
PolyP in Bacteria
Introduction
Having synthesized polyP with bacteria (see details in Experiments part), our lab members will promote their research if bacterial growth can be predicted. Hence, it is high time for MODEL to focus on polyP in bacteria.
Overall, based on Gompertz model, we established mathematical models for both EP/EPVM-transformed(vectors containing ppk1 sequences) and V-transformed (vector not containing ppk1 sequences) bacterial growth under two conditions, respectively:
1.LB medium, a common medium for bacterial culture
2.PA medium, an enrichment medium facilitating polyP synthesis
With analysis and comparison, not only have we depicted the bacterial growth curve, but we also helped our team validate the design of the plasmid and gain insight into how PA medium might influence bacteria.
Data
Thanks to our perseverant lab partners, we got rich data from three experiments though the data varied a lot due to different conditions and different measurement strategies (see details in Experiments part). As shown in Figure 1, OD600 representing bacteria density seems reasonable while OD700 associated with polyP synthesis lives a tough life.
（a） (b) (c)
Figure 1 Pchip interpolation results with raw experimental data. (a), (b) and (c) are based on the first, the second and the last, respectively. The legend applies to all subfigures.
Therefore, we laid emphasis on bacterial growth, i.e., OD600.
Assumptions
1.The value of OD600 is proportional to bacteria density within the range of 0-3. Thus, those higher than 3 are discarded.
2.In our equations, the variable, t, is obtained by dividing the real time by one hundred, i.e.tModel=tMeasurement/100. Such a trick does facilitate curve fitting.
Model
y=A0*exp(-exp(-k0*(t-t0))) （Ⅰ）
y=A1*exp(-exp(-k1*(t-t1)))-(A2*exp(-exp(-k2*(t-t2)))-A2*exp(-exp(k2*t2))) （Ⅱ）
| Sign | Meaning |
| exp(x) | ex |
| Variable | Meaning |
| t | Time |
| y | The value of OD600 in either LB or PA, which can be viewed as bacteria density |
| Parameter | Meaning |
| A0 | The value of the upper asymptote, i.e. the maximum of bacteria density in normal conditions (LB) |
| k0 | growth-rate coefficient affecting the slope |
| t0 | time at inflection, i.e. the moment with the maximal instantaneous growth rate |
| A1,A2 | determining the final steady bacterial density in PA medium |
| k1,k2 | quantitative comparison between them is related to global growth rate in PA medium |
| t1,t2 | additional parameters in the model for PA culture |
Table 1. The meanings of the signs, variables and parameters.
Equation (Ⅰ) and (Ⅱ) are models for bacterial growth in LB and PA, respectively. The model for PA is named as Serendipity Model (abbreviated S Model) while the other is still called Gompertz Model (abbreviated G Model). To make it understood by everyone, stories behind models will be told later.
Parameters and Methods
The meanings of the parameters have been listed in Table 1. The values were obtained with MATLAB Curve Fitting Toolbox and they will be shown in RESULTS. As for Fitting Options, A0, k0, t0, A1, A2, k1 and k2 were set as positive numbers due to their biological meanings.
Story
G Model: Spices Lead To a Feast
The story of G Model resembles preparations for a feast with spices. It is not hard to image how awful it will be if no spices are added to the ingredients. Here, model selection and reparametrization act as spices in terms of the feast of growth curve.
Why G Model?
When it comes to the bacterial growth curve, it is universally recognized that Logistic-Verhulst Model (abbreviated L-V Model) dominates the field.
(a) (b) (c)
Figure 2 The graphs of y (a), y’ (b) and y’’ (c) against time. Dots are inflection points.
Also belonging to a sigmoid model, G Model caught our eyes with its increase rate: G Model has an asymmetrical first derivative image whereas L-V Model exhibits a symmetrical one (Figure 2(b)). Thus, G Model may have an advantage in depicting bacterial growth with different increase rate before and after certain points. This coincides with the information from data interpolation (the upper panels in Figure 1（a）and (c)).
How does the reparametrization work?
After deciding on the model type, we were confronted with many model forms[1]. Taking simplicity, interpretability and practicability into consideration, we adopted a three-parameter form as equation (Ⅱ).
Table 2 Properties of G Model and L-V Model. Different colors, i.e. blue and red, are used to highlight the similarities between two models, respectively.
♦Simplicity
Less parameters demonstrate less (least) experimental data in comparison with those four-parameter forms.
♦Interpretability
Not only can all the parameters be endowed with biological sense (Table 1), but some properties can also be interpreted well by analogy to L-V Model (Table 2). Interestingly, it seems that logarithm plays a role in describing the constraints of surroundings in G Model while subtraction and division perform the duty in L-V Model. Such similarities also help to make the model better understood.
♦Practicability
Lab members can acquire some properties directly (Table 2).
S Model: All Roads Lead To Rome
Serendipity is a hopeful word with the meaning of luck, surprise and happiness. To be honest, the establishment of the model for PA was once in a dilemma until we tried a new way to describe the influences of the medium. All roads lead to Rome though we once made a detour.
Before Serendipity
Inspired by the optimization of L-V Model in others’ attempt[2], we begun with adding a “PA” function (i.e. the influence of PA medium on the bacterial increase) to the first-order differential equation.
However, our attempts seemed in vain as many types of “p(t)” either worsened the fitting results or challenged the computation ability of MATLAB.
With Serendipity
Puzzled but persistent, we happened to obtain new data, ΔOD600, by the theoretical values in LB with G Model minus the experimental data in PA (equations Ⅲ).
It was clear in Figure 3(a) that discrete ΔOD600 derived from the first data indicated a sigmoid curve. Thus, we examined if G Model could be applied to ΔOD600. In particular, we made the curves pass through the origin as equation (Ⅳ) and it was named as G’ Model.
Serendipity occurred.
The fitting succeeded (Figure 3(b)), verified by R-squares, SSE and RMSE (Table 3). Besides , the last data also coincided with G’ Model (Figure 3(c) , 3(d) and Table 3).
（a） (b)
（c） (d)
Figure 3 ΔOD600-associated diagrams. The upper and the lower panels are derived from the first and the third data, respectively.
Table 3 Fitting evaluation of G’ Model for △OD600. Better fitting results correspond to R-squares closer to 1 and others closer to 0.
After Serendipity
Not indulging ourselves in surprise, we took action to exploit △OD600to build a direct model for OD600. Naturally, equation (Ⅴ) was first adopted through addition and subtraction based on ΔOD600. And we named it S’ Model.
| Parameter | Meaning |
| A0,k0,t0,Δt | Parameters associated with G Model in LB, as mentioned above |
| A, k, t* | Representing influences of PA on the bacterial growth (Since it is not the final outcome, details are not analyzed here.) |
However, S’ Model was not the Mr. Right. The fitting results (see details in Results) were not satisfactory. Besides, itwas largelydependent on experiments because it would be on strike unless the data in LB was available (just like our second data).
Finally, we turned to S Model (equation (Ⅱ)), which retains the form of the subtraction of two G Models.
y=A1*exp(-exp(-k1*(t-t1)))-(A2*exp(-exp(-k2*(t-t2))))-A2*exp(-exp(k2*t2))) (Ⅱ)
Results
G Model
（a） (b)
Figure 4 Fitting results of G Model for LB. (a) and (b) are outcomes derived from 1st and 3rd data, respectively. Inflection points are exhibited with pentagram.
Table 4 Parameters and fitting evaluation of G Model for LB. Better fitting results correspond to R-squares closer to 1 and others closer to 0.
S Model
(a)
(b)
(c)
Figure 5 Fitting results of S Model for PA. (a), (b) and (c) are the outcomes derived from 1st, 2nd and 3rd data, respectively. Curves for V are on the left and those for EP/EPVM are on the right. In each subplot, black dots are experimental data. Red lines on the left and blue lines on the right are the results of S Model. Other lines are simulative results with alternative methods or models.
Table 5. Parameters and fitting evaluation of S Model for PA. Better fitting results correspond to R-squares closer to 1 and others closer to 0.
Discussion
As shown in Figure 4 and Figure 5, both models simulate the bacterial growth well. R-squares are larger than 0.99 in all the fitting of G Model. Although S Model does not stand out in fitting evaluation, its results are better than G Model and S’ Model with successful simulation of fluctuations.
When cultured in LB medium, V-transformed bacteria show a larger k0 and a smaller x0 (Table 4), i.e. a higher growth rate and an earlier inflection point. This reflects that the additional genetic parts may burden engineered bacteria when there are not enough resources for their “talents”.
When it comes to PA medium, parameters of S Model provide information on the ultimately steady density and the competence in utilizing phosphorus. Thanks to the positive values of A1, A2, k1 and k2, the ultimate bacterial density can be calculated as below. (The curves do not exhibit a decrease stage probably because we built models based on experimental data measured in increase stage.)
As shown in Figure 6, bacteria with ppk1 (EP or EPVM) own higher ultimate densities than their respective blank bacteria, which demonstrates that former ones have an advantage in surviving on phosphorus. More importantly, what validates our redesign from EP to EPVM is that the ultimate density increases from 0.4795 to 0.5173.
Figure 6. Predictive curves of 2nd and 3rd experiments with a long time scale. The ultimately steady bacterial densities are noted. The first experiment is not included because there were improper measurement settings (see details in lab part).
It is common knowledge that enrichment medium promotes metabolically fastidious organisms to grow with specific growth factors. PA medium is not an exception. Hence, we deduce that those fastidious bacteria are more likely to exhibit a LB-like growth curve because of better adaptation to additional factors. Since k1 and k2 have to do with curve slopes, ratios of them are focused on (Figure 7a and b). Clearly, there are abrupt fluctuations in early periods with ratios lower than 1while others with ratios higher than 1look sigmoid during increasing stage. Because ratios derived from our fitting results are all smaller than 1, we further analyze such situations as illustrated in Figure 7b. As k1 is closer to k2, the fluctuation is less evident and the whole curve is more similar to a sigmoid curve (in increasing stage). Turning to our project, we find that not only do ratios of blank bacteria decrease a lot but ratios of functional ones also increase (Figure 7c). It further proves that the well-designed EPVM-transformed bacteria are more competitive in PA medium and more skilled in polyP synthesis.
(a) (b) (c)
Figure 7 Figures focusing on ratios of k1 to k2. (a) shows curves with ratios from 1:16 to 16:1 and (b) targets at ratios smaller than 1. (c) presents ratios from our fitting results. In (a) and (b), the values of the parameters are set as below: A1=A2=1, t1=t2=1, k1=0.5, and k2 can be calculated with k1 and ratios.
Experience
To our knowledge, MODEL group of Nanjing-China is the first to build a novel mathematical model for polyP-synthetic bacteria in PA medium. It not only depicts the bacterial growth with acceptable accuracy but also provides some parameters to evaluate the influence of PA medium on bacterial growth. Most importantly, it assures our team that the design did work.
Although our methods are not so striking, it is worth sharing our experience behind the innovative result.
1.Lab is the most reliable friend for Model. If we did not keep in touch with lab partners, we would never know that data larger than 3 was not credible. And it was these data that caused abnormal fluctuations which disturbed us a lot.
2.Details determine success or failure. Without attention on the differences in growth rate, G Model might not be selected. Without comparison between G and L-V, such a useful three-parameter form might not be adopted. Without tricks in G’ Model, the secret of ΔOD600 might not be unveiled. Without the modification of S’ Model, S Model might not be built.
3.Perseverance contributes to successful models, but not stubbornness. It was our flexible attempt on ΔOD600 that played a vital role in the whole work.
References
[1] Tjørve, Kathleen M. C., and Even Tjørve. “The Use of Gompertz Models in Growth Analyses, and New Gompertz-Model Approach: An Addition to the Unified-Richards Family.” PLOS ONE, vol. 12, no. 6, 2017.
[2] Stein, Richard R., et al. “Ecological Modeling from Time-Series Inference: Insight into Dynamics and Stability of Intestinal Microbiota.” PLOS Computational Biology, vol. 9, no. 12, 2013.
PolyP in Gut
Introduction
Recent research showed that polyP may enhance the epithelial barrier function[1][2], which makes polyP a potential drug to treat IBD, as IBD patients usually suffer from severe intestinal injury. Meanwhile, after having a discussion with NJMU-China, we received the suggestion that the relation between polyP and intestinal bacteria may be an interesting point. With literature research conducted, we found that there is a correlation between polyP and intestinal microbiome[3][4]. Inspired by the information above, we MODEL group came up with the idea of using stochastic models to simulate the impact of polyP on the intestinal bacteria, which may help explore its effects on IBD.
Model
1.General Design
As shown in Figure 1(a), we created a 50*50 grid, comprised of 2500 squaresin total. Colors were distributed to these squares, representing the phyla of the bacteria which takes up a particular spot. We MODEL named it as Grid Model of Intestinal Bacteria Competition. The aim of the grid model is to simulate the changes of the bacteria in gut. For each generation, the colors of these squares changed according to an algorithm which we will discuss later. We considered each generation of the model as a result of bacteria competition, in which some bacteria may beat others because they are more adaptable to the current environment. Here we focused on the ratio of each phylum(shown by the colors) after certain generations, as the ratio can be a direct way to show the changes of the intestinal bacteria distribution.
Now, we will discuss the logic of the algorithm that decided the shift of colors:
(1) Randomly generate the bacteria with no preference
(2) 2500 squares go through “Battles” for each generation(All “Battles” start at the same time, so there won’t be mutual interference)
(3) In each battle, a weighted randomization, which bases on respective competitiveness, decides who will be the winner
(4) Renew the grid, and return to the second step.
For example, in Figure 1(b), we zoom into a particular square in the bottom right area of the grid. The squares within the white frame include the owner of the central spot and its 8 neighbors, between which the battle will happen. It’s worth mentioning that those squares on the edge or in the corner may have less than 8 neighbors, but they will still go through battles just like others.
（a） (b)
Figure 1 The general design of the grid model
(a)The model is comprised of 2500 squares, arranged in 50 rows and 50 columns. Colors of the squares show the phyla of the bacteria.(b)The bacteria that will take up the central square(here is the red one) in next generation is among the competitors within the white frame.
However, we found the draft version of the model may lead to infinite expansion of one superior type. To prevent this, we then introduced the restrain factor. In “Battle”, restrain factor can lower the competitiveness of each square according to the number of squares which have the same color. The formulas are shown as below:
Cx'=Cx-n*R
R=min{CA, CB, CC}/8-0.01
CX’: Modified competitiveness
CX: Initial competitiveness of the type x
n:Number of the squares in the same color
R: Restrain factor
In order to prevent CX’ from being negative, we gave the equation of R as it was shown above. Since n may reach eight, we selected the minimum value among CA, CB, CC,divided it by eight, then subtracted 0.01 from it, so that every CX’ will not be negative.
After the modification, the result is shown in Figure 2. The number of squares of each color reaches a relative balance at about 200th generations. So we choose 200th generation as the breakpoint of the model in the following tests.
Figure 2 The number of squares changes with generation
2.DSS and PolyP
We aimed to simulate how DSS (Dextran Sulfate Sodium Salt, able to induce colitis) and polyP would influence the gut microbiome. So after finishing the draft version of our grid model, parameters about DSS and polyP were introduced. Considering that DSS and polyP may have different influences on different phyla of bacteria, we gave the equations as below:
CX’=KDP-X*(CX-n*R)
KDP-X=1+m*(KD-X-KP-X)
KDP-X: Influence of DSS and polyP on type x
KD-X: Influence of DSS on type x
KP-X: Influence of polyP on type x
m: Coefficient of the combined effect of DSS and polyP
Assumptions
1. The influence of DSS and polyP is proportional to the concentration.
2. The bacteria which belongs to the same phylum have similar sensitivity against DSS and polyP.
Methods
As the grid model is a stochastic model, a lot of mathematic calculation is needed to get enough results. Then we took the average of the results because stochastic models may vary a certain range due to its randomness. So we used Python as a tool to complete our task by programming. By adding self-correction module to the program, the model can adjust its parameters according to the fitting result, and eventually output the required data.
Results
As two phyla of bacteria, Firmicutes and Bacteroidetes, are dominant in mice gut, we mainly focused on their proportions, with other low-proportion bacteria included in one general group named “Others”. With few data possessed, we finished the design and major structure of the Grid Model of Intestinal Bacteria Competition. We aimed to provide our lab group guidance that whether the certain concentration of DSS-polyP will cause alteration of intestinal flora, and we also aimed to raise a novel idea for future teams and modelers in terms of simulating the intestinal bacteria succession.
1.Blank Control Group
Figure 3 Results of the simulation of blank control
In this part, we successfully optimized our model to make it evolve into the similar status as the data provided by fecal bacteria sequencing. As shown in the figure above, two phyla of intestinal bacteria in the majority respectively represent Firmicutes (the red bar) and Bacteroidota (the blue bar). We know that under normal circumstances, Bacteroidota, or Bacteroidetes, are highly adjusted to the gastrointestinal tract, also, they can perform metabolic conversions that are essential for the host. Firmicutes, represented by the blue bar, are also important intestinal bacteria.
2.DSS-polyP Group
Figure 4 Results of the simulation of DSS-polyP Group
In this part, we use the model to simulate the combined effect of DSS and polyP which acts on community composition of intestinal bacteria. Here we can see that comparing to the Blank Control Group, the percentage of Firmicutes is dropping, while the percentage of Bacteroidota is increasing. Research showed that Firmicutes may be associated with reduced low-grade inflammation in obesity[5]. This finding led us to the explanation that when ratio of Firmicutes arises in gut, there could be potential risk of uncontrolled inflammation, which may stimulate IBD. Meanwhile, with the treatment of polyP, the proportion of Firmicutes decreased, illustrating that polyP can have possible impact on treating IBD via influencing intestinal microbiome.
Data and Parameters
| Parameters | Values |
| CA, CB, CC | [11.7, 10.54, 9.32] |
| KD-A, KD-B, KD-C | [0.036,-0.036,-0.006] |
| KP-A, KP-B, KP-C | [-0.204,0.224,-0.04] |
| KP-A, KP-B, KP-C | [-0.204,0.224,-0.04] |
| m | -0.5488 |
•The data of fecal bacteria sequencing was provided by Mr. Guo’s laboratory.
•The program package is provided in the link below:
“link”(click here to download the code)
[1] Segawa, S., et al., Probiotic-Derived Polyphosphate Enhances the Epithelial Barrier Function and Maintains Intestinal Homeostasis through Integrin-p38 MAPK Pathway. Plos One, 2011. 6(8).
[2] Kashima, S., et al., Polyphosphate, an active molecule derived from probiotic Lactobacillus brevis, improves the fibrosis in murine colitis. Translational Research, 2015. 166(2): p. 163-175.
[3] McMeechan, A., et al., Inactivation of ppk differentially affects virulence and disrupts ATP homeostasis in Salmonella enterica serovars Typhimurium and Gallinarum. Research in Microbiology, 2007. 158(1): p. 79-85.
[4] Pina-Mimbela, R., et al., Polyphosphate kinases modulate Campylobacter jejuni outer membrane constituents and alter its capacity to invade and survive in intestinal epithelial cells in vitro. Emerging Microbes & Infections, 2015. 4.
[5] Chakraborti CK. New-found link between microbiota and obesity. World J Gastrointest Pathophysiol. 2015;6(4):110-119. doi:10.4291/wjgp.v6.i4.110
PolyP in Application
Introduction
Polyphosphate, especially those with enhanced stability via ionic bonds, can play a role in reversing IBD in vivo. Thus, it is stable polyP that has the potential for medical application.
Thanks to chemical properties, it seems reasonable for Mg2+ to stabilize the long chains. Considering the Mg2+-dependent efficiency of PPK1, we MODEL group abstracted a coherent feed-forward loop (CFL). in bacteria. As illustrated in Figure 1, polyP-Mg production is regulated by an “AND” gate consisting of Mg2+ and PPK1. Their relation can be summarized as below:
a) Mg2+ enhances the enzymatic activity of PPK1.
b) PPK1 catalyzes the polyP synthesis.
c) Mg2+is also required for the functional polyP, i.e. polyP-Mg.
It seems that Mg2+ acts as an input signal to affect the output of polyP-Mg.
Figure 1 Schematic design of polyP in application model. The basic CFL and the polyP-associated one is shown on the left and right, respectively. X, Y and Z represent different substances and correspond to Mg2+, PPK1 and polyP-Mg, respectively. The arrows stand for the enhancing effect. The “AND” gate means that both effects are required for the final production.
With the ultimate goal of transforming polyP into either medicine or health products, polyP Neo is on the way to explore smart strategies for polyP utilization. Thanks to NJMU members who mentioned others’ success in orally administration of engineered bacteria[2], we envisioned installing the bacterial factory in the intestine to exploit the CFL. Based on ODEs, we simulated how Mg2+ concentrations might control the levels of active PPKs and polyP-Mg.
Model
The Basic CFL
Our work was based on mathematical equations describing the basic CFL as below. The competition between substance concentrations and it active threshold makes it more dynamic.
The meanings of the variables, parameters and functions are described in Table 1.
Table 1 The meanings of the variables, parameters and functions in the basic CFL
PolyP-Mg Associated CFL
(The meaning of X, Y and Z is equal to Mg2+, active PPK1 and polyP-Mg, respectively.)
In terms of polyP-Mg associated CFL, biological properties were taken into consideration besides mathematical forms.
Overall, we focused on three circumstances:
(i) the healthy user with Mg2+ fluctuations (H),
(ii)the IBD user (D)and (iii) the user suffering a relapse but better than D (R).
The following described the biochemical processes:
(PPK1* is the active form of PPK1 and parameters are different rate constants.)
And functions in the basic CFL were reified as below:
Table 2 The meanings of the additional parameters in the polyP-Mg associated CFL.
Assumptions
1. About the three circumstances
Different concentrations of Mg2+ reflect three circumstances.
Since IBD often impedes intestinal absorption, [Mg2+] can be higher in IBD patients’ guts than healthy ones. And the disease courses can be indicated by specific concentrations.
2. About the biochemical processes
1) Equation (Ⅰ) focuses on PPK1* metabolism.
i. The quantity of PPK1 is assumed constant thanks to the frequent supplement of bacteria as health products or treatment agents.
ii. The binding between the ion and the enzyme is so tight that it is hard to inactivate PPK1*. So the former reaction is irreversible.
iii. The degradation of PPK1 is neglected for the same reason in i.
2) Equation (Ⅱ) focuses on the catalysis of PPK1*.
i. ATP is sufficient
ii. The enzyme is much less than ATP so that PPK1* binds with ATP once PPK1* is produced. Thus, the two-substrate reaction can be viewed as a one-substrate reaction with an adjusted enzyme, PPK1*-ATP.
Michaelis-Menten equation is applied to this process.
iii. Michaelis-Menten equation is applied to this process.
3) Equation (Ⅲ) focuses on the fate of polyP-Mg.
i. Similar to the binding between Mg2+ and PPK1, polyP-Mg production is irreversible.
ii. The degradation occurs only when Mg2+ ii.is insufficient (such as H) due to
the stability of polyP-Mg.
iii. Compared with other substrates, polyP and its derivates are so rare that polyP-Mg is produced very quickly as long as the thresholds are exceeded. In this case, the following equation can be utilized:
[polyPn-1]=[polyPn]=[polyP-Mg] (Ⅳ)
3. About the reified functions
1) About θ(I >= kIJ)
i. Without Mg2+ over the threshold, PPK1 cannot work at all so that polyP is not synthesized.
ii. Also, no stable polyP-Mg can be produced.
Therefore, the influence is all-or-nothing (1 or 0), depending on the comparison with the active threshold.
2) About [X](t) (in E.coli)
It is a stair-like function.
i. The consumption of Mg2+ is ignored in comparison with its total quantity.
ii. In terms of the ion transportation from the intestinal tract to the bacterial cytoplasm, it is assumed that the ion concentrations are proportional in two environments. Thus, Mg2+ differences mentioned in i. can be applied to E.coli.
iii. The time scale of changes in [Mg2+] is much faster compared with the whole treatment/health-care course so that the ion concentration changes steeply as stairs.
3) About [Y](t)
Law of mass action is applied to reified f2(t) and g2(t). Because of the fixed [PPK1], pY is actually the product of kY and [PPK1].
4) About [Z](t)
f3(t) and g3(t) are based on M-M equation and equation (Ⅳ).
To be specific, the rate of reaction (Ⅱ) is described with M-M equation:
Data and parameters
As a bold attempt, no data was available and the values of the parameters were set up according to computational feasibility and biological sense.
Table 3 The values of the parameters.
Results
Under H circumstance: A Smart Filter And A Latent Detector
（a） (b) (c)
Figure 2 The results of the simulation of the H circumstance are shown as plots depicting the substance concentrations against time. X, Y and Z are described in three individual plots or in one plot. (a), (b) and (c) are different in the duration of the X activation (i.e. the time when [X] reaches 1) which is manipulated to be 0.3, 0.5 and 0.7,respectively.
The simulation of H circumstance is shown in Figure 2. The duration of the reach to X’s threshold is focused on. Two main conclusions can be drawn and both indicate medical significances:
i. There is no Z production because of inadequate Y as long as X’s reach to its threshold does not last too long. In our work, [Y] fails to rise to (Figure 2 (a) and (b)) or maintain at (Figure 2 (c)) its active threshold. With a larger duration value, [Y] will exceed kYZ and Z will appear subsequently.
Since longer durations can be a symbol of higher risks of IBD, such a pattern shapes the device as a smart filter ignoring transient fluctuations but handling dubious abnormality of X . Therefore, it not only makes better use of phosphorus resources but also comforts the user that the exotic residents will not disturb the harmonious microbiome.
ii. Although it decreases as a whole, Y remains within a period after the regression of X. The closer it is to the turning point, the more Y there is.
It seems that a latent detector is installed in the gut. If the fluctuations capable to induce Y production are warnings of IBD, Y will respond to the battle with a background level. Moreover, it caters for sustainability that the level is inversely proportional to the interval between the fluctuation and the disease attack and the interval indicates the state of the illness to some extent.
Under D and R circumstance: A Powerful Producer And A Shrewd Monitor
Figure 3 The results of the simulation of D and R circumstance are shown as plots depicting the substance concentrations against time. X, Y and Z are described in three individual plots or in one plot. (a), (b) and (c) are different in the interval between the recovery and the relapse which is manipulate to be 0.5, 1.0 and 1.5, respectively.
Since the disease development is a sine qua non of the relapse, we linked D and R temporally, which provided a new target, the interval between D and R. Results are shown in Figure 3. Also, there are two conclusions and relative significances:
i. In contrast to the silence under H circumstance, Z does appear with the higher [X] as a symbol of serious conditions.
Thus, this system has the potential to become a powerful producer.
ii.
Figure 4 The changes of [Y] (left) and [Z] (right) against time with different intervals between D and R
The Z level in the relapse is related to the interval between D and R as a result of the correlation of Y with the interval. As illustrated in Figure 4, with the same setting of the D circumstance, changes are identical before the relapse. Nevertheless, as the relapse occurs earlier and earlier, the decline stage is shorter and shorter. And it impacts Y more directly because the decrease is connected tightly with kYZ, which then determines the fate of Z.
It is no exaggeration to say that the Mg-associated CFL acts as a shrewd monitor protecting patients from the relapse. The more rapid the relapse, the higher background levels of Y and Z there will be. Therefore, both efficacy and sustainability are taken into consideration.
Conclusion
Satisfying safety and efficacy, our polyP-Mg associated CFL acts as a filter, a detector, a producer and a monitor. Although this model is just theoretical and more efforts, such as manipulating Mg2+ concentration with adjuvant drugs, should be made to find proper parameters, our work is a bold attempt to bring the team insight into new medical application of polyP.
References
[1] Fujiya, Mikihiro, et al. “Long-Chain Polyphosphate Is a Potential Agent for Inducing Mucosal Healing of the Colon in Ulcerative Colitis.” Clinical Pharmacology & Therapeutics, vol. 107, no. 2, 2020, pp. 452–461.
[2] Vandenbroucke, K., et al. “Orally Administered L. Lactis Secreting an Anti-TNF Nanobody Demonstrate Efficacy in Chronic Colitis.” Mucosal Immunology, vol. 3, no. 1, 2010, pp. 49–56.