Modeling
Content
Part 1: AbstractPart 2: Experiments, Limitations, and Modeling Purposes
Part 3: The Basic Model
Part 4: Basics of the Stereochemistry Model
Part 5: References
Part 1: Abstract
In our project, we combined the TDPs (a type of protein that comes from tardigrade) and lyophilization to produce the dry powder of bacteria, so that the bacteria can be stored at room temperature for a long time. Outside the laboratory, the team can't have the professional equipment such as -80°C lab freezers or cold chain transportation. Our project is devoted to providing a new method to attain the survival rate and original state of bacteria, which means it can preserve the bacteria properly and help the bacteria transport conveniently, cheaply, and safely before being used. When the bacteria needs to be used, simply add the solution that we already provided, and the bacteria will be resuscitated. Therefore, the practical application of engineered bacteria will continue to develop based on this storage method.
Part 2: Experiments, Limitations, and Modeling Purposes
Our experiments involve drying preservations which mainly include two phases—the TDPs expression phase and the TDPs degradation phase. In the TDPs expression phase, expressions of TDPs are induced, and enough TDPs are accumulated to reach a level at which TDPs can protect the bacteria. After the bacteria are resuscitated, they enter the TDPs degradation phase in which TDPs will be degraded to resume bacteria’s full functions.
Although we have proved that TDPs can prolong the storage of bacteria and the degradation of TDPs can be achieved, there are obvious limitations to our experiments. The degradation time and the expression intensity of TDPs are not measured nor controlled, and we have only used J23100 for TDP promote, and B0034 for TDP RBS. More gene combinations should be tried. We hope to change TDP RBSs, TDP promoters, and even TDPs of different intensities.
Therefore, we designed our model to overcome the limitations of our experiments and to generate results that would guide us to better suit customers’ different demands on the preservation time, survival rate, and degradation time of TDPs.
To proceed with more detailed discussions, we constructed two model vectors. For both vectors, CAHS 106094 was chosen as the TDP, with a degradation signal peptide pdt added on its C-terminal. The pdt can be specifically degraded by Mf-Lon protease from Mycoplasma florum. The expression and degradation of TDPs are controlled by different kinds of inducer. The basic model focuses on the model vector with iPTG and Ara as inducers, while the stereochemistry model focuses on the other model vector with iPTG and Tet as inducers.
(Figure 1. The diagram of the degradation parts of CAHS 106094 with iPTG and Ara as inducers)
Table 1. Parts
Name | Short Description |
---|---|
P_{C} | The promoter before TDPs sequence |
RBS_{1} | The RBS before TDPs sequence |
RBS_{2} | The RBS before Mf-Lon sequence |
pdt | Signal peptide on the C-terminal of TDPs |
P_{Lac} | LacI promoter |
P_{bad} | AraC promoter |
TT | T7 terminator |
Note: Commercial Pbad indeed is a compose of two promoters. One is for AraC and the other is for Mf-Lon. So the arrow cannot 100% accurately represent the direction of P_{bad}.
Since promoters and RBSs could be easily customized, our models will focus on calculating:
1. The suitable expression intensity of TDPs determined by the intensity of P_{C} and RBS_{1}.
2. The suitable degradation time of TDPs is controlled by the intensities of RBS_{2} and pdt.
Our future envision of the working process is shown in figure 2:
(Figure 2. The future mock float chart)
Part 3: The Basic Model
The Basic Model offers a clear view of the practical application of our project. In this model, we determine the suitable RBS_{2} and pdt intensity based on the demanded degradation time of TDPs in a classical way—using the Hill equation and Michaelis–Menten equation with assumptions made to simply the actual biochemical processes.
3.1 The Expression Level Of TDPs at TDPs Expression Phase
For our model vector in figure 1, iPTG inducers are added to induce the expressions of CAHS. We need to calculate the expressions level of CAHS 106094(CAHS for short)—that is to say, the concentration of CAHS at final equilibrium. We will manipulate the intensity parameters of P_{C} and RBS_{1} to predict the final expression level of CAHS. This is consistent with our first modeling purpose.
3.1.1 Assumptions
1.If the degradation module for figure 1 isn’t induced by arabinose, the basal level expression of Mf-Lon protease is reduced to an extent of having negligible effects on existing CAHS.
2.The terminator “TT” can totally terminate the transcription.
3.The concentration of each component in the cell are at an equilibrium.
4.The LacI protein will form a dimer, then combine with O_{Lac} sequence to repress the expression of downstream gene.
5.Cell growth and division have no influence on the model.
6.LacI gene has negligible expression in bacteria genome.
7.The iPTG is stable and won’t be degraded.
8.After adding the iPTG into bacteria solution, the concentration of iPTG inside and outside the cells will get to the same level rapidly.
3.1.2 Justifications for Assumptions
During the preparation process of bacteria, each component has enough time to get to a stable state. The strength of genes reveals expression and incomplete transcription are inefficient, and these strengths could be continued reducing by changing the gene parts which are more efficient in further so that it could achieve the purpose that the expression of genes won’t affect the bacteria system. Since the components are already stable, the growth and division of cells won't affect the modeling too much. Also, the LacI within the genome is at a low activation state and the LacI gene in some of the commercialized vector have been removed, so that we could use the genome-editing technology to get rid of this gene to make the expression of the gene more controllable in future.
According to the literature, there are two adjacent O_{Lac} sequences in the wild type Lac operator sequence, the LacI will form a stable dimer then combine with one of the O_{Lac} sequences. What’s more, while the LacI dimer connecting the two O_{Lac} sequences forms a relatively unstable tetramer, then it will form a ring structure that can effectively inhibit the expression of the downstream gene. However, in the commercialized vector, we find that there only has one O_{Lac} sequence, so the tetramer isn’t possessed practical significance. Considering the instability of LacI tetramer, we’ll ignore it and just focus on the effect between LacI dimer and O_{Lac}. Also, the form of polymerizing can influence the order of the Hill reaction.
(Figure 3. LacI repressing downstream gene expressions)
3.1.3 Equations and Calculations
Equation 1: At first, LacI is only controlled by the constitutive promoter (P_{Lac}), so the concentration of LacI is:
([LacI] is the concentration of LacI, t as for time, α_{1} is the maximum start-up rate of P_{Lac}, δ_{1} is the constant of LacI’s degradation rate)
Equation 2: According to the hypothesis, the concentration of LacI is stable, so the rate-of-change for LacI protein is zero. Set the Equation 1 is equal to zero, the concentration of LacI is:
([LacI]_{0} is the concentration of LacI protein before adding the iPTG)
Equation 3: While inducing the expression of CAHS, the iPTG is required. iPTG can combine with LacI so that the LacI will lose the repression function. Here we abstract a chemical reaction from these kind of combination and its forward reaction’s constant is k1 :
Equation 4: According to the definition of chemical equilibrium reaction, we can calculate that :
([LacI·iPTG] is the concentration of LacI which combined with iPTG. [iPTG] is the concentration of iPTG)
Equation 5: According to the equations above, each LacI·iPTG is transformed by one LacI, so the concentration of LacI which isn’t conbined with iPTG is:
Equation 6: Simulate the equation 2, 4 and 5, it can calculated that after adding the iPTG, the concentration of LacI which still have the ability to combine with O_{Lac} is:
Equation 7: What’s more, the function of LacI is repress the expression of downstream gene, so its effect is similar to the negative regulation of Hill reaction:
([CAHS] is the concentration of CAHS, α_{2} is the maximum start-up rate of P_{C}, R_{1} is the intensity of RBS_{1}, ε_{2} is the basal level expression of CAHS in this gene circuit, δ_{2} is the constant of LacI’s degradation rate, X_{M1} is a constant of this Hill reaction and m_{1} is the order of Hill reaction)
Equation 8: According to the hypothesis, the concentration of CAHS is stable, so that the rate-of-change for CAHS is zero. Set the Equation 6 is equal to zero, the concentration of CAHS is:
For summary, we can calculated that the [CAHS] is the concentration of CAHS when we begin the lyophilization process. (It’ll also called as [CAHS]_{0} in the later content)
3.2 TDPs Degradation Time at TDPs Degradation Phase
In real life, bacteria with our CAHS 106084 vector in figure 1 could be stored and transported in dry powder. Before use, a cultured solution with arabinose as inducers will be added to resuscitate the bacteria. Mf-Lon expressions will spike, causing the degradation of CAHS which will resume bacteria’s full functions. The degradation time of CAHS is predicted in this section.
3.2.1 Assumptions
1. The metabolism of bacteria is paused during drying, and the synthesis and decompose of protein are stopped.
2. After adding the solution to dry powder, the bacteria will resuscitate immediately, and its volume is exactly as same as before drying and its state is fully restored.
3. Ignore the effects of trace amounts of iPTG that remain in cells, which means iPTG do not exist in the post-resuscitation system.
4. On the left-hand side of Figure 1, if the degradation module isn’t induced by iPTG, the basal level expression of CAHS is reduced, so it could be ignored.
5. After adding the arabinose, almost all the dimer of AraC has already combined with arabinose, so we ignored the effect to the system that made by the AraC which isn’t combined with arabinose.
6 .Cell growth and cell division have negligible effects on biochemical processes.
7. Ignored the expression of LacI and AraC gene in bacteria genome.
8. The arabinose won’t be degraded, after adding the solution which has arabinose inducers to resuscitate the bacteria, the concentration of arabinose inside and outside the cells will get to the same level rapidly.
9. Ignore the effect to P_{bad} which made by CAP.
3.2.2 Justifications of Assumptions
Under the dry state, free water content and metabolic rate are reduced. The post-rehydration form is similar to the state before drying due to the determination of cell morphology by the cell walls. Because the volume of bacteria is small and more solution is added, the small amount of iPTG is extremely diluted, which indicates iPTG having negligible influences on biochemical processes. The reveal expression of CAHS is similar to Mf-Lon in part 3.1 and the circumstances of AraC in the genome are similar to LacI.
Since the degradation time is very short, the growth and division of cells and the reduction of arabinose’s concentration caused by degrading can be ignored. This hypothesis is more realistic, especially when the customer wants the CAHS protein to degrade over one cell cycle. Subsequently, AraC dimers can repress the expression of the downstream gene, but after combining it with arabinose, it’ll exert a positive effect.
Therefore, after the adding of arabinose, the AraC will cause two different effects on P_{bad}. Because the majority of AraC and arabinose are combined after adding enough arabinose, to simplify the model, we do not consider the repress expression effect of a small amount of AraC not combined with Arabic sugar to the downstream genes. Moreover, the CAP which is controlled by glucose can regulate the P_{bad} promoter, but considering that the concentration of glucose and CAP can be kept constant during the whole process, this part will be ignored.
3.2.3 Equations and Calculations
Equation 9: At first, the expression module of AraC is similar to the LacI in part 3.1, so the concentration of AraC is:
([AraC] is the concentration of AraC protein, t as for time, α_{3} is the maximum start-up rate of the promoter before AraC, δ_{3} is the constant of LacI’s degradation rate)
Equation 10: The expression of AraC begin before drying, so that the concentration of AraC is in homeostasis. Set the Equation 6 is equal to zero, the concentration of AraC is:
([AraC] is the concentration of AraC protein before adding the arabinose)
Equation 11: At this time we add the solution with arabinose to resuscitate the bacteria, the combination between AraC and arabinose is similar to the combination between LacI and iPTG, the forward reaction’s constant is k2 :
Equation 12: so that we can calculated that:
([AraC·Ara] is the concentration of the AraC protein which combined with arabinose, [Ara] is the concentration of arabinose that we added)
Equation 13: According to the equations above, each AraC·Ara is transformed by one AraC, so that we can calculated that:
Equation 14: Simulate the equation 10, 12 and 13, we can calculated that after adding the arabinose, the concentration of AraC·Ara which can stimulate the Pbad is that:
Equation 15: What’s more, the function of AraC·Ara is to promote the expression of downstream gene, so that its effect is similar to the positive regulation of Hill reaction:
([Mf-Lon] is the concentration of protease, α4 is the maximum start-up rate of P_{bad}, R_{2} is the intensity of RBS_{2}, ε_{4} is the basal level expression of protease in this gene circuit, δ_{4} is the constant of Mf-Lon’s degradation rate, X_{M2} is a constant of this Hill reaction and m_{2} is the order of the Hill reaction)
Equation 16: It is worth mentioning that because of the short time of protein degradation, Mf-Lon can’t be directly considered to be in equilibrium, so the dynamic equation 15 should be retained. The Mf-Lon degrade the CAHS which contain the pdt tag specifically, this degradation process is close to a bio-chemical reaction and here we can refer to the Michaelis-Menten equation. However, in the Michaelis-Menten equation, the concentration of enzyme is assumed to be constant, but actually the concentration of Mf-Lon is fluctuating during the process. Hence, the CAHS is treated as a substrate on the basis of the Michaelis-Menten equation and is appropriately adjusted to be available:
(based on the assumption that the [CAHS]_{0} is the CAHS protein concentration found in equation 8, the maximum rate of enzymatic reaction that the V_{max} concentration Mf-Lon protease can achieve, and K_{M} is the meter constant of the protease in this reaction)
Synthesize all the equations above and we can see that, as long as we determined the value of each individual constant, brought the numerical results of equation 8 into equation 16 which means assign to [CAHS]_{0}, using matlab to calculate the split equation 15 and 16, we could know how the concentration of CAHS changed after resuscitated. And as long as the residue limit of CAHS is given, we could know how much time is required if the concentration of CAHS will below this limit. By changing the RBS_{2}, the value of R_{2} in equation 15, which affects the time that CAHS could be fully degraded. Similarly, changing the pdt will cause the change of V_{max} in equation 16 and also the final time.
Therefore, according to the model, it can be calculated that when the customer is given a time limitation for degradation of CAHS protein, the combination of RBS_{2} and pdt can be selected.
3.3 Numerical Simulation
3.3.1 Parameters
By research data and inference, we assigned quantitative values to the parameters.
All the parameters are measurable. However, limited by a variety of factors, we did not measure them. To prove that our model is workable, we assigned the following numbers to the parameters.
α_{1}: it describes the strength of P_{LacI} , and influences the expression level of LacI. We considered that as a regulator, the system does not need too much LacI, so P_{LacI} should be a weak promoter. Therefore, we assigned 10 to it.
α_{2}: it describes the strength of P_{C} , and influences the expression level of CAHS 106094 (a TDP). In our experiment, we used the part J23100 as P_{C}. Via the part library, we found that the relative strength of J23100 is 2547 (J23112 is 1)^{[1]}. Therefore, as assigned 2547 to it.
α_{3}: it describes the strength of the promoter of AraC, which indeed is a part of P_{bad}. The situation is similar to α_{1}. Therefore, we assigned 10 to it.
α_{4}: it describes the strength of the promoter of Mf-Lon, which indeed is another part of P_{bad}. Mf-Lon is a protease. As an enzyme, it obviously holds high efficiency. As a result, it is not needed to be overexpressed. Therefore, we assigned 100 to it.
δ_{1}: it describes the degradation of LacI. Usually, exogenous proteins are hard to be degraded. The concentration of them are halved mainly because of cell division, so the half-life period is about 20 minutes, which is a cell cycle of E. coli. Team William_and_Mary 2017 has derived that the half-life period of a protein equals to ln2 / δ ^{[2]}. Let it equal to 20, we can calculate that δ is 0.0347. However, LacI is an endogenous protein, so its degradation should be a little faster. Therefore, we assigned 0.0462 to it, which means the half-life period is 15 min.
δ_{2}: it describes the degradation of CAHS 106094, an exogenous protein. We assigned 0.0347 to it. The reason had been described above (δ_{1}).
δ_{3}: it describes the degradation of AraC, an endogenous protein. We assigned 0.0462 to it, the same as δ_{1}.
δ_{4}: it describes the degradation of CAHS 106094, an exogenous protein. We assigned 0.0347 to it, the same as δ_{2}.
ε_{2} and ε_{4}: they describe the constitutive expression of CAHS 106094 and Mf-Lon. As two famous regulation system, the background should be not too high. Moreover, in the future, we can choose some regulators that have lower constitutive expression. As a result, just as the assumptions of our model, we assigned 0 to them.
R_{1}: it describes the strength of the RBS before CAHS 106094. In our experiment, we used the part B0034. Via the part library, we found that the relative strength of B0034 is 11 (B0032 is 1)^{[3]}. Therefore, as assigned 11 to it. Notably, RBS_{1} is a variable in our design, so we will study the influence of R_{1} when R_{1} changes.
R_{2}: it describes the strength of the RBS before Mf-Lon. As we described in α_{4}, the system does not need too much Mf-Lon. Therefore, as assigned 10 to it, which a value a little smaller than R_{1}. Notably, RBS_{2} is a variable in our design, so we will study the influence of R_{2} when R_{2} changes.
k_{1}: the equilibrium constant of the combination of LacI and iPTG. The two substances have a good affinity, so k_{1} should be high. We assigned 10000 to it.
k_{2}: the equilibrium constant of the combination of AraC and arainose. The two substances have a good affinity, so k_{2} should be high. We assigned 10000 to it.
[iPTG]: the concentration of iPTG. In our experiment, we add 2 mM iPTG. We assigned 2 to it.
[Ara]: the concentration of arabinose. In our experiment, we add 2‰ L-arabinose. We assigned 2 to it.
m_{1}: The Hill series of the combination LacI and LacO. As there is only one LacO in pet28a vector, we hypothesized that the LacI dimer acted, but nor tetramer. Therefore, m1 should be 2.
m_{2}: The Hill series of the combination araC and Pbad. araC is dimer. m2 should be 2.
X_{M1}: A constant in the Hill function of the combination LacI and LacO. It is a measurable value, but we do not have information about it. Therefore, we assigned 1 to it.
X_{M2}: A constant in the Hill function of the combination AraC and P_{bad}. It is a measurable value, but we do not have information about it. Therefore, we assigned 1 to it.
V_{max}: A constant in the michaelis-menten equation of the enzymatic reaction of Mf-Lon. The constant represents the maximum reaction rate of the reaction at certain enzyme concentration. It is influenced by the character of Mf-Lon and the pdt tag. We reference the data of iGEM Team William_and_Mary 2017, the relative degradation rate of pdt A, B, C, E, F are 16, 11, 7, 2, 1, respectively ^{[4]}. In our experiment, we used pdtA to construct the gene circuit. Therefore, we assigned 1 to it. we assigned 16 to it.
K_{m}: A constant in the michaelis-menten equation of the enzymatic reaction of Mf-Lon. It is related the character of Mf-Lon. It is a measurable value, but we do not have information about it. The value should be analogous to the concentration of TDP. Therefore, we assigned 10000 to it.
3.3.2 Calculations
By Equation 7, we simulated the gene circuit when the left part is activated by iPTG. CAHS 106094 (TDP) expresses to protect the bacteria.
By Equation 8, we can calculate that when the system reachs equilibrium, the concentration of CAHS 106094 (TDP) is 807,310. Then we substituted the value into equation 15 and 16, so that we simulated the gene circuit when the right part is activated by arabinose. Mf-Lon promotes the degradation of CAHS 106094 (TDP). TDPs may influence the normal functions of engineered bacteria. To degrade TDPs will avoid the potential interference. By simulation, we can see the system will quickly degrade CAHS 106094 (TDP). One more thing that we need to say is the m files to solve the differential equations were written with the help of Team Jilin_China, which we described the attribution detailedly at partenership page.
Of course, the expression level of CAHS 106094 (TDP) need to be adjusted. If a customer wants to prolong the storage life, more TDP is needed. If another customer do not need so long a storage life or survival rate, less TDP is neeed. to achieve, we can change the promoter or RBS before CAHS 106094 (TDP) gene. Here by the model, we showed that if RBS_{1} (R_{1} ) is changed, the expression of CAHS 106094 (TDP) changes quickly.
Similarly, some customers need to quickly degrade TDPs, but others do not. By changing RBS_{2} (R_{2} ) or pdt tag, we can control the degradation speed.
Similarly, all simulations above prove that our gene circuit will work well and can be easily adjusted. In the future, the model can also be used to guide to design the circuit for different customers.
The m files are put here.
Part 4: Basics of the Stereochemistry Model
The model above shows how our modeling is used in the interaction of the project design with the customer clearly. And in order to better fit the actual situation and reflect more of the essence of this biochemical process, we have constructed another more complex model known as the stereochemistry model, which can depict the process more accurately. It not only makes the results more realistic but also provides a valuable reference for future generations to use lactose manipulators, arabinose manipulators, and other components.
In addition, we replaced arabinose-related AraC and P_{bad} with tetracycline-related TetR and P_{tet}, depicting another similar gene circuit that uses different gene regulatory components. Such depicts also show how our system works when different control elements are used. In the future, for customers who need to use different regulatory components in engineering bacteria, we can choose different regulatory units for controlling the expression and degradation of CAHS proteins, making it easier for customers to use their preferred components to build specific functional engineering bacteria by using the strains we offered.
Part 5: References
[1] Registey of Standard Biological Parts, Part:BBa_J23100 , http://parts.igem.org/Part:BBa_J23100
[2] iGEM Team William_and_Mary 2017, http://2017.igem.org/Team:William_and_Mary/Description
[3]Registey of Standard Biological Parts, Part:BBa_B0034, http://parts.igem.org/Part:BBa_B0034:Experience
[4] iGEM Team William_and_Mary 2017, http://2017.igem.org/Team:William_and_Mary/Degradation_Rates