Team:BNU-China/Model/flagellin
Flagellar Filament Model
Introduction
Our plan to display flagellar filaments on the yeast surface consists of mainly three steps:
- Display flagellin monomers on the yeast surface.
- Put the yeasts obtained in step one in a solution with sufficiently high concentrations of flagellin monomers and certain anions (known as good “salting-outers”) to promote spontaneous nucleation on the yeast surface.
- Transfer the yeasts obtained in step two, which have flagellin seeds on their surface, to a new buffer with physiological ionic strength and pH that also has a sufficiently high concentration of flagellin monomers. Under this condition, the flagellin seeds on the yeasts’ surface will elongate into flagellar filaments.
If we want to use this system to display enzymes, we only need to fuse the desired enzymes (often along the same chain of reactions) with the flagellin monomers, and then the obtained yeasts from the above steps will have a much larger number of enzymes displayed on it than ordinary yeasts.
We shall give a brief explanation of the above design. First note that although there is a lot of literature about Salmonella flagellins but very little about E. coli, for safety reasons we can only use E. coli flagellins in our design. However, since the flagellar polymerization of Salmonella and that of E. coli are very similar (Kondoh and Hotani, 1974), we can transfer some of the previous results concerning Salmonella flagellar filaments to E. coli.
Lino (1974) gave a review of the assembly of Salmonella flagellins in vitro. There are several characteristics worth noting. First, in vitro assembly is possible even when the flagellin seed is attached to a cell body, which confirms the fundamental validity of our design. Second, the assembly of flagellins in vitro has two steps, nucleation and elongation. Nucleation forms the seeds upon which elongation takes place. Under physiological ionic strength and pH, flagellin monomers can grow on seeds to form flagellar filaments, but they cannot nucleate into seeds. Spontaneous nucleation can only happen in the presence of a high concentration of certain anions (often referred to as good "salting-outers", e.g. $SO_4^{2-}$, $F^-$, citrate ions) and neutral pH. The second and third step of our design correspond to the nucleation process and the elongation process respectively.
A key issue here is the competition between the process occurring on the yeast surface and that in the solution. Once nucleated, fragments of filaments will not be able to bind with one another to form new filaments (Lino, 1974). Therefore, only free flagellin monomers in the solution are useful for the polymerization on the yeast surface. Seeds formed in the solution are not only useless, but also reduce the concentration of flagellin monomers, which decelerates assembly on the yeast surface. Hence, our goal is to promote polymerization on the yeast surface and avoid it in the solution.
Our design prohibits assembly in the solution in the elongation step, namely step three. The condition in the buffer, i.e. physiological ionic strength and pH, only allows elongation, and nucleation cannot occur. Then, the flagellin monomers can only assemble on the existing seeds on the yeast surface, which is exactly what we want.
However, in step two, the nucleation step, the assembly in the solution cannot be completely avoided. We are therefore interested in how we can adjust the conditions to best promote polymerization on the yeast surface, and how many seeds will form in the end. Our model tries to answer these questions.
Model Formulation
Tables are provided for all the variables and parameters in the model at the end of this section for your reference.
For reasons stated in the section above, our model focuses on the nucleation process under conditions specified in step two. Elongation takes place alongside nucleation, and our model has to describe these two processes both on the yeast surface and in the solution. It is worth noting that depolymerization plays a relatively minor role in the assembly of flagellar filaments compared to microtubules, so our model will not put too much stress on this process like we did in the microtubule model.
There are some very sophisticated models describing protein polymerization. For instance, Kashchiev (2015) gave a comprehensive analysis of the polymerization process using nucleation theory, which, with all the correct parameters given, can predict the concentration of flagellar filaments of every length. However, in the simulation of step two of our design, we are only interested in the number of seeds formed on the yeast surface. The lengths of the filaments are not of our concern, so we do not distinguish between filaments of different lengths.
We treat the nucleation process as a one-step reaction between $s$ monomers:
\[ \ce{s \ monomer -> seed} \]
Then, according to the law of mass action, we have
\[ \frac{\mathrm{d}C_n}{\mathrm{d}t}=k_nC_m^s \]
where $C_n$ is the concentration of nuclei (a.k.a. seeds) in the solution, $k_n$ is the nucleation rate constant and $C_m$ is the concentration of monomers in the solution. Here, filaments of different lengths are all treated as seeds and are considered together in the variable $C_n$. As will be seen the third differential equation, their main effect on the system is reducing $C_m$ by elongation. Note that the elongation of filaments in vitro is not infinite, because there is a probability that an error may occur at the distal end of the filament which can terminate further growth. However, it usually takes about an hour for nucleation to finish (Wakabayashi et al., 1969), and the typical length of the filaments formed in that time is less than $1 \ \mu\text{m}$, which is much shorter than $50 \ \mu\text{m}$, the observed maximal length of flagellar filaments assembled in vitro (Lino, 1974). This indicates that within the simulation period, the termination of elongation can be neglected. Therefore, there is no term in the above equation for the removal of nuclei from the system, which means that once a seed is formed, it stays in the system and consumes monomers by elongation until the end of our simulation.
Similarly, we have
\[ \frac{\mathrm{d}C_{ns}}{\mathrm{d}t}=k_{ns}C_m^{s-1}(C_s-C_{ns}) \]
where $C_{ns}$ is the concentration of nucleated display sites on yeasts, $k_{ns}$ is the nucleation rate constant on display sites, and $C_s$ is the total concentration of display sites. Due to different conditions on the yeast surface and the lower mobility of monomers displayed there, there should be a minor difference between $k_n$ and $k_{ns}$. This difference can only be determined in experiments, but due to time constraints, we have not been able to conduct this experiment, so in our simulation we will take $k_n = k_{ns}$.
The above formulation has been used in many previous models (e.g. Wakabayashi et al. (1969), Michaels et al. (2014)). Kashchiev (2015) gave a more detailed theory of the nucleation process, but confirmed that the approximation in the above form is valid.
We shall next examine the elongation process. Asakura (1968) established that during elongation, the concentration of monomers follows Michaelis–Menten kinetics, where seeds (regardless of length) are seen as the enzyme and monomers are seen as the substrate. This process can be illustrated as follows
\[ \ce{monomer + seed <=>[K_m] monomer-seed ->[V] seed} \]
where $K_m$ is the dissociation constant of the $\text{monomer-seed}$ complex, and $V$ is the rate at which $\text{monomer-seed}$ transforms into a new $\text{seed}$ ready for further elongation. By rapid equilibrium approximation, $K_m$ can be taken as the Michaelis constant. This gives us
\[ \frac{\mathrm{d}C_m}{\mathrm{d}t}=-\frac{k_eC_nC_m}{K_m+C_m}-\frac{k_{es}C_{ns}C_m}{K_m+C_m}-s\frac{\mathrm{d}C_n}{\mathrm{d}t}-(s-1)\frac{\mathrm{d}C_{ns}}{\mathrm{d}t} \]
where $k_e$ and $k_{es}$ are the turnover numbers of the elongation process in the solution and on the yeast surface respectively, and $K_m$ is the Michaelis constant. By the same argument as above, we set $k_{es} = k_e$. The first two terms correspond to the decrease resulting from the elongation process, while the last two terms correspond to the nucleation process.
Putting the obtained equations together, we get
\[ \left\{ \begin{aligned} \frac{\mathrm{d}C_n}{\mathrm{d}t}&=k_nC_m^s\\ \frac{\mathrm{d}C_{ns}}{\mathrm{d}t}&=k_{ns}C_m^{s-1}(C_s-C_{ns})\\ \frac{\mathrm{d}C_m}{\mathrm{d}t}&=-\frac{k_eC_nC_m}{K_m+C_m}-\frac{k_{es}C_{ns}C_m}{K_m+C_m}-s\frac{\mathrm{d}C_n}{\mathrm{d}t}-(s-1)\frac{\mathrm{d}C_{ns}}{\mathrm{d}t}\end{aligned}\qquad\qquad\begin{aligned}c_n(0)&=0\\c_{ns}(0)&=0\\c_m(0)&=c_{m0} \end{aligned} \right. \]
Many of the parameters in these equations are not directly accessible in the literature, and since we do not have enough time to measure them in experiment, we shall estimate them based on the existing data in the literature, which is explained in detail in the following section. If you are not interested in the derivation of these parameters, you may skip the next section.
| Variable | Description | Units |
|---|---|---|
| $C_n$ | Concentration of seeds in the solution | $\text{mol / L}$ |
| $C_{ns}$ | Concentration of seeds on the yeasts’ surface | $\text{mol / L}$ |
| $C_m$ | Concentration of flagelin monomers in the solution | $\text{mol / L}$ |
| Parameter | Description | Estimation | Source |
|---|---|---|---|
| $k_n$ | The nucleation rate constant in the solution | 0.5301 | Kashchiev, 2015; Kondoh and Hotani, 1974; Wakabayashi et al., 1969; O’Brien et al., 1972; Champness, 1971 (see Parameter Estimation section for derivation) |
| $k_{ns}$ | The nucleation rate constant on the yeast surface | 0.5301 | Set to be equal to $k_n$ |
| $s$ | The number of monomers needed to form a seed | 3.3 | Wakabayashi et al., 1969 (see Parameter Estimation section for explanation) |
| $k_e$ | The turnover number of elongation in the solution | 0.03669 | Kashchiev, 2015; Kondoh and Hotani, 1974; Wakabayashi et al., 1969; O’Brien et al., 1972; Champness, 1971 (see Parameter Estimation section for derivation) |
| $k_{es}$ | The turnover number of elongation on the yeast surface | 0.03669 | Set to be equal to $k_e$ |
| $K_m$ | The Michaelis constant of elongation | $2.8 \times 10^{-4} \ \text{mol / L}$ | Kondoh and Hotani, 1974 |
| $C_{m0}$ | The initial concentration of flagellin monomers | Varying from 1×9.667×10−5 mol/L to 10×9.667×10−5 mol/L, to be analyzed later in Numerical Simulation section | Wakabayashi et al., 1969 (see Parameter Estimation section for explanation) |
| $C_s$ | The total concentration of display sites | Varying from 1×4.430×10−10 mol/L to 100×4.430×10−10 mol/L, to be analyzed later in Numerical Simulation section | Estimated from experience (see Parameter Estimation section for derivation) |
Parameter Estimation
The Estimation of $s$
Recall that $s$ is the stoichiometric number in the reaction
\[ \ce{s \ monomer -> seed} \]
Wakabayashi et al. (1969) gave an estimation of $s=3.3$ from their experimental data. Our using this estimation directly instead of rounding it to an integer is justified by Kashchiev (2015), which illustrated that $k_nC_m^s$ is in essence not the nucleation rate, but an approximation of the forward flux in Kashchiev’s more sophisticated model. Therefore, $s$ need not be an integer, and using the more exact result $s=3.3$ is actually more accurate. Also note that although this estimation is for Salmonella flagellins, it is also basically valid for the E. coli case due to their great similarity in polymerization (Kondoh and Hotani, 1974).
The Estimation of $C_{m0}$
$C_{m0}$ is the initial concentration of flagellin monomers. In our simulation, we refered to the concentration given by Wakabayashi et al. (1969), which is around $5 \ \text{mg / ml}$. Transforming mass concentration to molar concentration gives us $C_{m0} = 9.667 \times 10^{-5} \ \text{mol / L}$. Therefore we decided to run our simulation with $C_{m0}$ varying from $1 \times 9.667 \times 10^{-5} \ \text{mol / L}$ to $10 \times 9.667 \times 10^{-5} \ \text{mol / L}$. Note that $K_m = 2.8 \times 10^{-4} \ \text{mol / L}$, and that the rate of elongation reaches half its maximum speed at this concentration. The range we chose was roughly from one-third to 3 times of $K_m$.
The Estimation of $C_s$
$C_s$ is the total concentration of display sites. From our fellow teammates in wet lab, we learned that the typical concentration of yeasts obtained after cultivation is $1.107 \times 10^{-14} \ \text{mol / L}$. From literature (Shibasaki et al., 2009) we found that although the average number of protein molecules displayed on each yeast depends on the type of the displayed protein, it typically falls between around $10^4$ to $10^5$. In our simulation later, we shall assume that $4 \times 10^4$ flagellin monomers are displayed on each yeast on average. This gives us $C_s = 4.430 \times 10^{-10} \ \text{mol / L}$. Notice that this concentration is far smaller than the value of $C_{m0}$, which implies that the monomers in the solution can probably accommodate a lot more yeasts. In practice, we can concentrate the yeast suspension to obtain higher concentrations. Therefore, we decided to run our simulation with $C_s$ varying from $1 \times 4.430 \times 10^{-10} \ \text{mol / L}$ to $100 \times 4.430 \times 10^{-10} \ \text{mol / L}$.
The Estimation of $k_n$ and $k_e$
We estimated $k_n$ and $k_e$ based on the data in Wakabayashi et al. (1969) and Kondoh and Hotani (1974), assisted by the theory provided by Kashchiev (2015).
Wakabayashi et al. (1969) conducted several spontaneous nucleation experiments with Salmonella flagellin and obtained some useful data. Using these data, we derived the corresponding parameters $k_n$ and $k_e$ for Salmonella. (In this process, we used the fact that a flagellar filament consists of 11 longitudinal rows of flagellins (O’Brien et al., 1972) and that the axial spacing between flagellins is $5 \ \text{nm}$ (Champness, 1971), hence the citation in Table 2.) To transfer these data to the case of E. coli, we utilized the comparison between Salmonella and E. coli flagellin polymerization given by Kondoh and Hotani (1974). They gave the $K_m$ and $k_e$ for Salmonella and E. coli respectively. Note that the data here was obtained under conditions similar to our elongation step (around physiological ionic strength and pH), while the data given by Wakabayashi et al. was obtained under conditions allowing spontaneous nucleation (with high concentrations of salting-outers). Since our model mainly concerns the nucleation step, we cannot directly use this data. However, the information provided by Kondoh and Hotani (1974) tells us that the $k_e$ of E. coli is about one-tenth of that of Salmonella. If we assume that this relationship holds in the change of conditions between the elongation step and the nucleation step, we can transfer the $k_e$ we obtained for Salmonella under nucleation conditions from Wakabayashi et al. (1969) to $k_e$ of E. coli under the same conditions.
What about $k_n$? Intuitively, we know that since E. coli is slower in elongation than Salmonella (Kondoh and Hotani, 1974), it must be slower in nucleation as well. But by how much? To calculate the $k_n$ of E. coli from that of Salmonella, we must find the underlying mechanistic consistency between elongation and nucleation. The theory of Kashchiev (2015) revealed the mechanism underlying protein polymerization and guided us in this transferring process. It viewed nucleation as a “harder” kind of elongation. Before the seed is formed, the structure of the subnuclei is highly unstable, and once the nucleation is complete, the elongation based on the formed seed is much easier. Based on the equations it proposed, we derived the corresponding $k_n$ of E. coli from the data we already had from Wakabayashi et al. (1969) and Kondoh and Hotani (1974).
Although we derived the above parameters strictly from literature, they are still only rough estimates. Due to time constraints, we were not able to conduct experiments to measure them, but in the future such measurements will provide more accurate estimations of these parameters. Also, note that the solution of our equations varies continuously with the parameters, so small deviations from the actual values will not affect the general characteristics of our results.
Numerical Simulation
We are interested in how we can adjust the parameters $C_s$ (which is determined by the concentration of yeasts) and $C_{m0}$ (which is the initial concentration of monomers) to achieve the best possible results. Here, we say that the result is good if a high percentage of display sites on the yeasts have nucleated into seeds. We define an index $\theta = \frac{C_{ns}}{C_s}$ to measure the performance of the system.
To find the parameters that maximize $\theta$, we did a parametric sweep in the range specified in Table 2. (See Parameter Estimation section for derivation of the search range.) For each combination of parameters, we ran a numerical simulation of the ODEs in a time range of 10 hours, which is about the maximum acceptable reaction time given by our teammates in wet lab. We plotted the resulting $\theta$ in the following graph.
Figure 1 Parametric sweep to find the best $C_{m0}$ and $C_s$ to maximize $\theta$.
See Parameter Estimation section for derivation of the search range
We can see that $\theta$ barely changes when $C_s$ changes, which is reasonable since $C_s << C_{m0}$. $\theta$ increases almost linearly with $C_{m0}$ and does not seem to hit its limit in our parameter range. However, notice that the highest $C_{m0}$ here is about $50 \ \text{mg / ml}$, which is very high indeed. A higher concentration is not only extremely hard to obtain, but might also make the buffer so dense that the normal properties of the reaction is compromised. Therefore, we suggest that in practice, it is sufficient for us to get the concentrations of monomers and yeasts as high as experimentally convenient.
To see how different variables in our equations change over time, we plotted one simulation with $C_s = 4.430 \times 10^{-8} \ \text{mol / L}$ and $C_{m0} = 9.667 \times 10^{-5}$ in the graphs below. To see the full process, we set the simulation time to 20 hours.
Figure 2 The change of the concentration of seeds in the solution over time. $C_s = 4.430 \times 10^{-8} \ \text{mol / L}$, $C_{m0} = 9.667 \times 10^{-5}$
Figure 3 The change of the concentration of seeds on the yeasts’ surface over time.
$C_s = 4.430 \times 10^{-8} \ \text{mol / L}$, $C_{m0} = 9.667 \times 10^{-5}$
Figure 4 The change of the concentration of flagelin monomers in the solution over time.
$C_s = 4.430 \times 10^{-8} \ \text{mol / L}$, $C_{m0} = 9.667 \times 10^{-5}$
Figure 5 The change of the percentage of nucleated sites on the yeast surface over time.
$C_s = 4.430 \times 10^{-8} \ \text{mol / L}$, $C_{m0} = 9.667 \times 10^{-5}$
We can see that the concentration of monomers $C_m$ gradually decays to zero as the concentration of seeds in the solution ($C_n$) and on the yeast surface ($C_{ns}$) gradually increase until around 10 hours. Notice that at hour 10 there are still some monomers left in the solution, but the number of seeds has stopped increasing, meaning that after this point, the main process becomes elongation.
$\theta$ also virtually ceases to increase after 10 hours, which implies that this is an appropriate time length for the nucleation step.
Finally, let us put the value of $\theta$ into perspectives. At 10 hours, $\theta = 1.2 \times 10^{-3}$. Each yeast has around $4 \times 10^4$ display sites (Shibasaki et al., 2009), giving about 48 flagellin seeds formed on each yeast on average. Note that polymerization of flagellar filaments in vitro can reach up to $50 \ \mu\text{m}$ (Lino, 1974), that a flagellar filament consists of 11 longitudinal rows of flagellins (O’Brien et al., 1972), and that the axial spacing between flagellins is $5 \ \text{nm}$ (Champness, 1971). If we attach an enzyme on each monomer, the number of enzymes displayed on each yeast will increase by $48 \times 11 \times \frac{50 \times 10^{-6} \ \text{m}}{5 \times 10^{-9} \ \text{m}} = 5.28 \times 10^6$, which is 100 times that of an ordinary yeast display system.
Conclusion and Discussion
We have two main results from the model we developed.
The first is that by assembling flagellar filaments on the yeast surface, we can increase the display capacity by as much as 100 times that of an ordinary yeast display system. Note that in our simulation, there were no external interventions. In practice, when most of the monomers in the solution have polymerized and are no longer useful, we may change the buffer into a new one with abundant monomers. In this way, we can increase the display capacity even more. Also note that the flagellar filaments formed in the solution can be recycled and turned into monomers again by imposing conditions such as strong acidity.
The second is that within an experimentally reasonable range, the percentage of successful nucleation among display sites increases almost linearly with respect to the concentration of flagellin monomers, and that the process completes at around 10 hours. This conclusion provides directions for wet lab experiments, which is to make the concentrations of yeasts and flagellin monomers as high as experimentally convenient, and to culture the system for about 10 hours.
As mentioned above, some of the parameters in the simulation were derived from literature. To better reflect reality, experiments can be conducted to measure these parameters. However, the qualitative results of our model will most likely hold.
If in future efforts, an estimation of the length distribution of the filaments formed on the yeast surface is needed, one can refer to the model proposed by Kashchiev (2015), who gave equations that could predict the concentrations of filaments of every length.
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