Team:Fudan/Model
<!DOCTYPE html>
Model
Summary
The core of our project is a quorum-sensing system. As a therapeutic method, although process of packaging, medicine taken, digestion, and excretion is important with no doubt, we focus on what engineering bacteria do in the host's destine, including at which speed bacteria population will grow, how much lactase they will produce and how long a life circle of the whole bacteria group. Most importantly, our therapy is aimed at young babies, so the clinical experiment is of extremely high danger. Therefore, to model how our system work is not only for demonstration and optimization but also for simulation in a practical situation in the future, if possible. What we model is the most important part of our project design, and so that it can demonstrate, verify and optimize our project as much as possible.
We divide our system into two stages, reproduction and digestion stages respectively. The same process will result in totally different effects for the function of intercellular communication in the quorum-sensing theory. In addition, our model is built with the thread of practical process instead of mathematical analytics in order to provide a more obvious and useful conclusion.
Beginning
As soon as the nissle bacteria come into the intestine, they firstly reproduce themselves based on the abundant nutrition there to ensure their subsistence. Unfortunately, however actually, only a little number of bacteria survive at that time. Therefore, how to earn nutrition from the environment as much as they want and compete against other kinds of local bacteria, even their 'partners' will become a question of questions. We simplify this problem based on a classical model.
The situation mentioned above leads to a stage that cells do hardly ever communicate with each other, including signal passing and direct contact, in order to achieve some group functions but pay their every effort to guarantee individuals' survival.
Quorum Sensing Signal
Quorum Sensing is a group system mainly based on intercellular communication, and as we chose, signal transmission is the most common method to implement communication. More specifically, our project chooses \(AHL\)(Acyl-homoserine lactone), a kind of macromolecule, to be the signal transmitted between bacteria.
\(AHL\) is considered to be the auto-inducer which is able to switch quorum sensing on/off. A significant feature of \(AHL\) is the free diffusion method by which it travels through cytomembrane, free diffusion. It can be produced inside the cells by every engineering bacterium we created, but only when the quantity increases to a fixed threshold value, could the concentration of \(AHL\) in cells rise to a high level. A series of processes will be triggered since plenty of quantitative change generates qualitative change.
Although the detailed process that \(AHL\) accumulates as the population of bacteria mushrooms from outside to cells inside is not designed artificially, we can utilize a simple model to simulate that.
Reproduction Stage
Now let's set about exploring details inside every individual. On the top of our system, \(LuxI\) and \(LuxR\) can be expressed at the same speed in order to produce a significant promoter primitive, a combination of \(AHL\) and \(LuxR\). For convenience, we use \(RA\) to represent the combination in the following context.
As we mentioned briefly above, in the reproduction stage, individuals of bacteria are devoted to competing for survival resources from the host intestine and seldom communicate with each other and participate in some group life circle. So in this physical stage, we do not take intercellular communication into account. Instead, we mainly consider biochemical reactions inside a cell, which are regarded to be totally the same in every cell of bacteria.
Useful results and demonstrations of this module are listed in MODEL3.
In order to provide a relatively safe and steady intracellular environment, a vital substance, the antimicrobial peptide, can not be ignored. The antimicrobial peptide is considered to be the key factor to control the population of engineering bacteria. By impacting the growth rate of bacteria, the antimicrobial peptide decides a dynamic change of bacteria's population. The antimicrobial peptide will be expressed at a solid speed in our model if some conditions are achieved. However, it's not a simple approximate alternative but can be demonstrated with the expression of (McbA ) and (McbBCD). A detailed procedure is displayed in MODEL3.
The toxin can not only be produced but also be transferred to the extracellular environment. According to results in past papers and our test of toxic potency, the antimicrobial peptide in the host's intestine will not influence other local bacteria notably. Therefore, it's reasonable to ignore this part's effect. On the other hand, owing to the fact that transport speed of the antimicrobial peptide is changeable, the concentration of intracellular antimicrobial peptide also does not maintain a stable state, which could control the growth rate of engineering bacteria. The factor that controls the toxic transferring speed is named \(McbEF\), whose expression is controlled by the key factor we mentioned above, \(RA\).
In Digestion Stage, transcription of \(McbEF\) is hardly ever inhibited, and the result is that \(McbEF\) can be expressed at a fairly high speed and furthermore, most of the antimicrobial peptide will be transferred outside cells, which contributes to creating a safe and comfortable, internal and external environment for bacteria to multiply rapidly. However, the increase does not last too long, since the quorum sensing system is switched on right now. We elaborate on this process in MODEL3, too.
Digestion Stage
After the bacteria have been multiplying for a period, a new stage appears to work. A large cluster of bacteria settles on the surface of the intestine, whose consequence is that concentration of \(AHL\) in the neighborhood is nearly the same as the intracellular environment. At that time, the transmission rate of \(AHL\) is relatively low because of a slight concentration difference. The reasons are explained in MODEL1.
2 \(RA\) molecule is combined to be a \(RA\) dimer in order to switch on the promoter \(luxpR\). It controls expression of 2 proteins in different plasmids, \(tetR\) and \(LacZ\). Owing to little leakage and low expression of \(luxpR\), \(tetR\) and \(LacZ\) are also yielded, however, at a rather low rate. And this part of the product can not take a significant effect on digestion. From the perspective of energy, it's thought that the energy of the cell is mainly used for multiplying in the Reproduction Stage while in the Digestion Stage for production. \(ptetR\) is not inhibited and \(McbEF\) can be expressed favorably because of the low expression of \(tetR\).
In the Digestion Stage, concentration of \(RA\) could reach a relatively high level as we said, and therefore promoter \(luxpR\) is switched on swiftly for the characteristic of \(luxpR\). Details and reasons about this will be discussed in MODEL2. Once \(luxpR\) is switched on, \(tetR\) protein, a transcription factor at the same time, will be expressed in quantity. \(luxpR\) is also designed to be the promoter of product \(LacZ\), and the yield rate reaches the most at the very stage, which is quantized in MODEL4. Thanks to the similar characteristic of promoter \(ptetR\) with \(luxpR\), a large quantity of \(tetR\) is able to inhibit the expression of \(McbEF\) as soon as possible. As a consequence, the growth rate of engineering bacteria will decrease rapidly, and also, the concentration of \(AHL\) outside will not be equal to zero right away. Furthermore, \(AHL\) inside will largely transfer to the outside and promoter \(luxpR\) will be switched off again. Hereto, quorum sensing completes one cycle, and the process repeats from the Reproduction Stage again. We elaborate on this part in MODEL3.
Production and Ending
All in all, the main purpose of our system is to produce lactase in a baby's intestine periodically and controllably. In this situation, the production module is the most important. Lactase is not expected to be expressed explosively, which might harm the local environment, and not too slowly, which can not take effect in a real situation. Actually, we can control \(LacZ\)'s expression speed via a key factor \(RA\), and furthermore, realizing a periodic control mechanism. Click MODEL4 to see production rate of \(LacZ\) in simulation.
Last but not least, while \(LacZ\) can be produced in an ideal way, how to finish a cycle is an unavoidable problem. From the perspective of a high level, it means how much time the medicine will spend in vivo, which is important for patients since a relatively long time staying may affect the host's intestinal environment, even though slightly, while a short period may not digest lactose totally. It's obvious that the population of engineering bacteria is a crucial factor to control the remaining time. If there is no nissle, there is no \(LacZ\) to be produced. As imagined, the population curve is damply periodic changing.
Model 1: Signal Transmission
When population of engineering bacteria is not large, signal molecule \(AHL\) outside cells is regarded as equal to zero, for the huge intestinal erosion force. So there is no signal transmission at that stage and \(AHL\) transmission rate is relatively high.
Transmission begins in the Digestion Stage. The population of engineering bacteria reaches a high level then and the difference between the concentration of internal \(AHL\) and external is not big. In addition, Bacteria is always densely clustered and when they multiply, the volume of bacteria cluster increases. Since \(AHL\)'s transport method is free diffusion, it seldom transfers outside the cell in this stage. To demonstrate that, we consider a sphere filled with numerous bacteria. We divide this sphere into two parts from the deep to the surface, a relatively small core sphere named the core and a thin sphere shell respectively. Given that \(AHL\) in the outside environment is less than that inside, \(AHL\) inside will travel outside in a quite short time, which is ignored. It's reasonable to think that the concentration of \(AHL\) in the core is uniform and steady while it's not in the shell. External \(AHL\) in the shell is easy to diffuse and to be diluted.
We assume that the ratio of the population of engineering bacteria at the shell to the core is the ratio of the transmission rate of in Reproduction Stage to the Digestion Stage. $$ N=kV\\ V=\frac{4}{3}\pi R^3 $$ Given that the ratio of the radium of the core to the shell \(k_r\), we can derive the basic correlations. $$ \frac{dN}{N}=3\frac{dr}{R}=3k_r\\ $$ Furthermore, correlation of transmission rate of \(AHL\) in two stage is found: $$ k_{diff-Re}=3*k_{diff-Pr} $$
Model 2: Preparation module
This module describes a series of processes from the very begging of the system, \(LuxI\) and \(LuxR\) are expressed at a fixed speed, to the production of a key protein, \(RA\), whose concentration decides what reactions will take place in the next stage. In different stages, reactions in this module are rather different, and therefore, we will discuss two stages respectively.
Especially, the 2 stages of this module provide meaningful results for our wetlab, since \(RA\) is a fairly basic substance that decide in which stage the quorum-sensing system will reach. Based on this, our experiment for promoter \(luxpR\) with its mutants and even \(ptetR\) can be carried out in a quantitative level. And furthermore, the results from wetlab can facilitate models in next module.
2.1 Preparation module in Reproduction Stage
Firstly, we talk about the concentration change rate of AHL. We have declared that LuxI can be expressed at a fixed speed \(\alpha_{con}\). And we assume that any substance inside the cell will be degraded and diluted at a common speed \(\mu\).
$$ \phi \stackrel{\alpha_{con}}{\longrightarrow} LuxI\\ \phi \stackrel{\alpha_{con}}{\longrightarrow} LuxR\\ Every Substance \stackrel{\mu}{\longrightarrow} \phi $$ Next, we assume that AHL precursor is abundant while the enzyme \(LuxI\) is scarce in the beginning. Classical Michaelis-Menten equation is described as the following equation.
$$ ProductionRate=\frac{vXS}{K_m+S} $$ X is for the concentration of enzyme, whatever binding or free state. S is for zymolyte, \(K_m\) is called Michaelis-Menten const, and v is for the speed of process from the simple combination of enzyme and substrate to a single product and free enzyme. In this stage, as we said, the AHL precursor is not limited, so S is considered to be infinity. Then the production rate of AHL and concentration of enzymes are positively linearly correlated.
$$ \frac{d[LacI]}{dt}=\alpha_{con}-\mu[LacI] $$ Make the two sides of equation above be zero, then we get concentration of \(LacI\) in the steady-state.
$$ [LacI]=\frac{\mu}{\alpha_{con}}\\ LacI~~+~~AHL precursor\longrightarrow AHL\\ $$ Therefore, we think \(AHL\) can be produced at a constant speed in single cell. We might as well make that speed const a new signal \(\alpha_{AHL}=\frac{v_{AHL}\mu}{\alpha_{con}}\).
Another vital point about \(AHL\) can not be ignored, the free diffusion method of transportation. Once bacteria reach the destination, concentration of \(AHL\) can be assumed to be zero, which means that a large quantity of \(AHL\) produced will be transferred outside. On the other hand, we have to consider that the population of engineering bacteria could be relatively low so that we can assume that the concentration of \(AHL\) outside cells is always zero at the very beginning of the whole cycle, an approximate assumption but not excessive. Based on this, the speed of \(AHL\) inside transferring to outside is linearly positively correlated with concentration of \(AHL\) inside the cell at the very time, which is for concentration difference actually.
$$ v_{AHL-out}=k_{diff}*[AHL] $$ Hereto, the concentration curve of \(AHL\) and \(LuxR\) on this stage can be expressed.
$$ \frac{d[AHL]}{dt}=\alpha_{AHL}-\mu*[AHL]-k_{RA-ON}*[AHL]*[LuxR]+k_{RA-OFF}*[RA]-v_{AHL-out}\\ \frac{d[LuxR]}{dt}=\alpha_{LuxR}-\mu*[LuxR]-k_{RA-ON}*[AHL]*[LuxR]+k_{RA-OFF}*[RA]\\ $$ Furthermore, according to mass action law, \(RA\), the final result can be described.
$$ AHL~~+~~LuxR~~\stackrel{k_{RA-ON}}{\longleftrightarrow} RA\\ \frac{d[RA]}{dt}=k_{RA-ON}*[AHL]*[LuxR]-k_{RA-OFF}*[RA]-\mu[RA] $$Conclusion
From Figure 2, we can obviously find that concentration of \(RA\) converges on around 0.5nM below 1nM in 5s, which means \(RA\) as a transcription factor precursor can not switch on transcription of promoter \(luxpR\), which we'll discuss at once. And it is also indicated that \(LuxR\) is relatively abundant while \(AHL\) is a limited factor. It can be proved for high transferring rate of \(AHL\).
2.2 Preparation module in Digestion Stage
As we discussed above, in Digestion Stage, the concentration of \(AHL\) inside and outside has nearly no difference, which leads to a rather low rate for transferring to the intestine environment. In our model, we audaciously set it to zero. With no changes of other parameters, the concentration of 3 substances can be described as below:
$$ \frac{d[AHL]}{dt}=\alpha_{AHL}-\mu*[AHL]-k_{RA-ON}*[AHL]*[LuxR]+k_{RA-OFF}*[RA]\\ \frac{d[LuxR]}{dt}=\alpha_{LuxR}-\mu*[LuxR]-k_{RA-ON}*[AHL]*[LuxR]+k_{RA-OFF}*[RA]\\ \frac{d[RA]}{dt}=k_{RA-ON}*[AHL]*[LuxR]-k_{RA-OFF}*[RA]-\mu[RA] $$Conclusion
We can observe that a big difference between Figure 3 and Figure 2. Most importantly, \(RA\) in this stage reached a relatively high level at about 0.83nM. That is to say, \(RA\) inside cells is enough to switch on promoter \(luxpR\) at the exact stage. At the same time, the concentration of \(AHL\) inside cells in the stable state increases while the concentration of \(LuxR\) decreases more quickly.
2.3 Vital Promoter-\(HS100\)
Now let's pay attention to the vital promoter \(luxpR-HS100\). When \(RA\) can be estimated, what interests us most is the promoter. Unfortunately, after our prediction-details and methods are in MODEL3, however, we found that wild and previous promoter was not useful in our system for their relatively high leakage, even previous promoter having been optimized in this respect. In this situation, based on the results form our models, the improvement of the promoter is necessary.
And our improvement finally achieves the objective and the achievement is following at Figure4. From Figure 4, we can find that the curve about the strength of \(luxpR\) correlated with density of transcription factor \(RA\) is very steep. Focus on that when density of \(RA\) is 0.5nM, strength of \(luxpR\) is only 0.0024nM/s, which is located in a gentle growth stage, but when density of \(RA\) is 5.6nM, strength of \(luxpR\) is 0.0633nM/s, which is not only located in a rapid growth stage, but also over 15 times than the Reproduction Stage.
Model 3: Toxin and Population Module
In this module, we will see how the antibacterial peptide takes effect on our quorum-sensing system. The internal principles are demonstrated by this model and finally, the population of nissle1917 can be predicted. Although mathematical simplification and assumptions might affect the model's prediction accuracy, the tendency of intermediate substances and their order of magnitudes are relatively precise. What we modeled and predicted influenced the selection and optimizations of the antibacterial peptide and relevant promoters.
3.1 Toxin and Population module in Reproduction Stage
The reaction of producing the antimicrobial peptide is not an elementary reaction. The antimicrobial peptide precursor named \(McbA\) and protein enzyme named \(McbBCD\) is a cluster of genes. The expression of this cluster of genes is complicated but we can simplify that. We assume that \(McbA\) is expressed at some fixed speed, while \(McbBCD\) genes are viewed as a single gene expressed at the other fixed speed.
$$ \phi\stackrel{\alpha_{McbA}}{\longrightarrow}McbA\\ \phi\stackrel{\alpha_{McbBCD}}{\longrightarrow}McbBCD $$ \(McbA\) and \(McbBCD\) can produce the antimicrobial peptide via an enzymatic reaction. So equation (2) is also used here. If there are no other methods to inhibit the antimicrobial peptide, it can be described as below.
$$ \frac{d[McbA]}{dt}=\alpha_{McbA}-\mu*[McbA]-\frac{v_{tox}*[McbBCD]*[McbA]}{k_{tox}+McbA}\\ \frac{d[McbBCD]}{dt}=\alpha_{McbBCD}-\mu*[McbBCD]\\ \frac{d[Tox]}{dt}=\frac{v_{tox}*[McbBCD]*[McbA]}{k_{tox}+McbA}-\mu*[Tox] $$ Now Let's talk about the population model of the engineering bacteria. Based on Logistic Population Model, the growth rate \(\alpha_N\) and the environment capability \(K_N\). In addition, the antibacterial peptide will lead to a significant decrease in population, so the poisonous coefficient is very high. In an experimental environment, the population can be described as below.
$$ \frac{dN}{dt}=r*N*(1-\frac{N}{K})-[Tox]*k_{poison} $$ Attention to the physical environment, local intestinal bacteria will compete against engineering bacteria for resources, even if it's proved that the latter will not do any harm to the former by our experiment. According to classical Lotka–Volterra Competition equation, the dynamic change of competitive population can be described as below:
$$ \frac{d[N]}{dt}=rN(1-\Sigma_i\frac{\alpha_{i}N_i}{K_i}) $$ Competition coefficient \(\alpha_i\) and environment capability \(K_i\) are considered at this equation. Because the total competition coefficient is a fixed value, we put this const into equation(11).
$$ \frac{dN}{dt}=r*N*(1-\frac{N}{K}-compete_N)-[Tox]*k_{poison} $$ The antibacterial peptide can be limited by a protein named \(McbEF\). Expression of \(McbEF\) is controlled by the promoter \(ptetR\), which can be introduced by \(tetR\). And \(tetR\) is a Inhibitory transcription factor, whose expression could be controlled by \(luxpR\). As we have discussed, in Reproduction Stage, strength of promoter \(luxpR\) at stable state is 0.000111 nM/s, which is the leakage strength of \(luxpR\). And furthermore, expression rate of \(tetR\) is fixed. According to \(Hill ~Equation\), expression of \(McbEF\) can be described. Besides, it's worth noting that the effect of \(tetR\) is very intense for promoter \(luxpR\), which decides the characteristic of quorum sensing system in some sense.
Conclusion
From Figure 5(1)-5(4), we can observe that bacteria multiply at an unbelievable speed to around environment capability, since \(McbEF\) can be expressed at a considerable speed and the antibacterial peptide can not accumulate. And finally, the population of engineering bacteria displays a steady and gentle descent for the influence from \(tetR\).
3.2 Toxin and Population module in Digestion Stage
In this stage, strength of \(HS100\) is relatively high, while other parameters do not change but initial values.
Conclusion
From Figure 6(1)-6(4), we can obtain that expression of \(McbEF\) is decreased, and therefore the antibacterial peptide can accumulate to a higher level. As a result, the population of bacteria decreases gently in the first period, while the antibacterial peptide accumulates to some level, it appears a rapid descent. But the population will not decrease to zero because the stage will change before that.
Model 4: Production Module
Finally, let's discuss about the product \(LacZ\) at this module. Expression of \(LacZ\) is controlled by a mutant \(luxpR\) named \(fus100\), whose strength is over 2 times than \(HS100\), while the leakage is higher than the latter. Expression rate of \(LacZ\) in a single cell is important with no doubt, but population of engineering bacteria is a factor that can not be ignored for what we focus on is the total yield of \(LacZ\).
$$ \frac{d[LacZ]}{dt}=(\alpha_{fus100}-\mu*[LacZ]+\frac{\beta_{fus100}}{1+(K_{luxpR}/[RA]^2)}\\ TotalYield=N*[LacZ] $$Conclusion
From Figure 7(1)-7(2), we can find that in the Reproduction Stage, total yield of \(LacZ\) is less than 0.5mM, since \(fus100\) is not switched on and population of bacteria is limited, while in Digestion Stage, in Figure 8(1)-8(2)total yield is over 7nM, which is able to take effect on digest lactose since there are a number of cycles and each cycle over 7nM lactose can be yielded.
Parameter Table
Reference
[1] Coyte, K.Z., J. Schluter and K.R. Foster, The ecology of the microbiome: Networks, competition, and stability. Science, 2015. 350(6261): p. 663-6.
[2] Ohkusa, T., et al., Long-Term Ingestion of Lactosucrose Increases Bifidobacterium sp. in Human Fecal Flora. Digestion, 1995. 56(5): p. 415-420
[3] S. B. Hsu, The Application of the Poincart-Transform to the Lotka-Volterra Model. Journal of Mathematical Biology, 1978. 6, 67-73
[4] Taylor, P. A., and P. J. Leb. Williams. 1975. Theoretical studies on the coexistence of competing species under continuous-flow conditions. Can. J. Microbiol. 21: 90-98.
[5] Din, M.O., et al., Synchronized cycles of bacterial lysis for in vivo delivery. Nature, 2016. 536(7614): p. 81-85.
[6] Barril, C., À. Calsina and J. Ripoll, On the Reproduction Number of a Gut Microbiota Model. Bulletin of Mathematical Biology, 2017. 79(11): p. 2727-2746.
[7] Melissa B. Miller and Bonnie L. Bassler, Quorum Sensing in Bacteria. Annual Reviews Microbiol, 2001. 55:165–99
[8] Melke, P., et al., A cell-based model for quorum sensing in heterogeneous bacterial colonies. PLoS Comput Biol, 2010. 6(6): p. e1000819.
[9] Chopp, D.L., et al., A mathematical model of quorum sensing in a growing bacterial biofilm. Journal of Industrial Microbiology & Biotechnology, 2002. 29(6): p. 339-346