Introduction

The high-throughput neutralization assay for dengue fever, CFNT, using the SRIP developed by the Wet-lab, is an essential component to achieve the goals of our project. However, at present, producing SRIP consistently is challenging and there is not sufficient understanding about it. Consequently, we cannot precisely determine the amount of resources needed to produce the volume of SRIP necessary for the societal implementation of CFNT.

In this context, the Dry-lab evaluated the production volume of SRIP, which is key to CFNT, using a model. From the model's results, we gained insights into SRIP and determined the parameters that can accurately predict the production volume of SRIP in a steady state. We also discussed candidates for parameters that could result in more accurate fitting.

Methods

Model structure

The ordinary differential equation (ODE) model was created in Python and solved using the scipy.solve_ivp[link] solver. The model developed by Dry-lab is based on the VLP production system and has been tuned for SRIP using actual experimental data. This model takes as input the SRIP producing cell, the serine this cell requires, and the DNA needed to produce SRIP. In turn, it outputs the quantity of SRIP produced.

Description of the simulated system

The following five substances appear in our model.

• SRIP Producing Cell
  • SRIP Producing Cell is a cell that produces SRIP and is created by transforming HEK293T with three plasmids that we have created.
• Serine
  • It is contained in the culture medium, is necessary for cell growth, and has a significant impact on the growth rate of cells.
• DNA
  • The three plasmids we have produced are introduced into HEK293T and then transcribed RNA in the HEK293T nucleus. Plasmids introduced into mammals are not replicated in the nucleus and do not increase[3].
• RNA
  • RNA transcribed from our transgenic plasmid DNA produces the components of SRIP. RNA
• SRIP
  • One of our intended productions. Used to detect dengue virus antibodies. (For a detailed explanation, go to wet-lab) It has an outer membrane similar to that of dengue virus, but is safer because it does not have the ability to self-replicate. It is produced by SRIP Producing Cell. It is completed when SRIP components produced by RNA assemble and assemble in the smooth endoplasmic reticulum. The SRIP produced is released into the culture medium and adheres when left where cells are present.

Model for the growth of SRIP Producing Cell

　SRIP-producing cells are made by transforming HEK293T with three types of plasmid DNA that we have produced.HEK293T was cultured by attaching to petri dishes under a Serine-containing culture medium, and Medium Change was performed every day to provide a constant supply of Serine. Serine is essential for cell growth and concentration affects cell proliferation rate. The transformation was performed on HEK293T at 10% confluency using plasmids which we know the concentration and the cells were cultured intact in the same petri dish after transformation. After SRIP is collected every day along with the culture medium, Medium Change is performed. SRIP-producing cells produce new transformed cells by cell division. The cells are damaged by the SRIP production process and become incapable of SRIP production after a certain time.

Models for SRIP Synthesis

The introduced plasmid DNA will be introduced approximately uniformly into each HEK293T cell. When the cells divide, the transfected plasmid is distributed in a manner that divides roughly equally between the two dividing cells. Excessive cell division results in a shortage of plasmid DNA, the three plasmid DNAs needed to become an SRIP-producing cell, but this almost never happens in a few cell divisions. The introduced plasmid DNA is transcribed inside the nucleus of HEK293T to produce RNA. RNA is partially degraded over time to produce the components of SRIP, which are eventually aggregated in the rough endoplasmic reticulum and released out of the cell. The higher the RNA concentration, the greater the amount of RNA degradation per hour.

Model construction

The model consists of four ordinary differential equations. These ordinary differential equations are derived from the equations describing the path.

The four ODEs were implemented in Python and solved using scipy.solve_ivp. The fitting to the experimental data was done using scipy.curve_fit.

About the data used

The data presented here are derived from measurements of the SRIP concentration in Vero cells, after the addition of SRIP solution produced by HEK293T cells. SRIP solution was collected every 24 hours over a total period of five days. The detailed methodology is outlined below and further information can be found on the Experiments page.

The overall methodology is divided into five steps:

1. Transfection into HEK293T cells
2. Collection of SRIP solution
3. Inoculation into Vero cells
4. Immunostaining
5. Measurement of Infectious Unit (Concentration of SRIP)

1. Transfection to HEK293T Cells

HEK293T cells were cultured in 10 cm dishes using DMEM medium supplemented with 10% FBS and PS. The cell confluency at the time of transfection was approximately 50%. Transfection was performed using Polyethylenimine and involved an average of 2.0 x 10^11 copies of three different plasmids.

2. Collection of SRIP Solution

SRIP was secreted into the supernatant of the cells transfected in step 1. We collected 9 mL of this supernatant every 24 hours and stored it at -80°C.

3. Inoculation into Vero Cells

Vero cells were seeded in a 96-well plate and inoculated with 100 µL of serially diluted SRIP solution collected daily. The cells were then incubated at 37°C for two days.

4. Immunostaining

The cells from step 3 were immunostained using Anti YFV-NS1 Protein Rabbit Poly and Alexa-Flour 488 Donkey anti-Rabbit IgG(H+L).

5. Measurement of Infectious Unit (Concentration of SRIP)

Fluorescent images of the stained cells were captured at a 5x magnification. The number of fluorescent spots was analyzed using ImageJ to calculate the daily concentration of the SRIP solution.

We gathered SRIP solution concentrations every 24 hours, allowing us to compute the cumulative SRIP production while disregarding adsorption and degradation of SRIP. This makes the data suitable for modeling the overall production of SRIP.

However, it should be noted that the data from sample-2 may be less reliable due to variations in cell growth or well-specific differences during the culture period. Consequently, excluding the data from sample-2 when fitting the model will likely yield a more accurate representation.

Assumptions

• Assumptions of plasmid. Plasmids are not replicated after transfection[3]. Based on the number of HEK293T cells to be transformed and the number of plasmids to be introduced, even if 80% of the plasmids are introduced into HEK293T cells, the number of plasmids per cell at the time of transformation is large enough to ensure that the divided cells will be able to produce SRIP. Therefore, both SRIP-producing cells have the ability to produce SRIP until about 10 cell divisions. In other words, the SRIP-producing cells simply double in doubling time. Since the doubling time of HEK293T is 24 hours, the half-life of the plasmid is 24 hours[4].From this, P = ln(2)/24 and k_DNA = ln(2)/24 is obtained .
• - Serine concentration is constant. In wet-lab, the culture medium was changed once a day to keep it approximately constant. In actual operation, the concentration of Serine can be more accurately maintained at a constant level by using flow culture or other methods [2].
• - Once produced, SRIP does not decrease by adsorption. In wet-lab, the SRIP concentration was reset by collecting the culture medium once a day, so SRIP reduction due to SRIP adsorption or degradation is unlikely to occur. In actual operation, SRIP can be kept more accurate and constant by using fluidized culture, etc. [2].

Conclusion

The system behavior

We determined the six constants shown in Table 1 using experimental data. The results are presented in the table below.

Using these parameters, we plotted the temporal evolution of the concentration of the SRIP Producing Cell, as shown in the figure below.

From these results, it was found that X becomes $X=0$ around $t=90.3$. As all variables and constants in this system are guaranteed to be non-negative, we stopped updating the ODEs for both X itself and SRIP that include the term X at $t=90$.

Considering the behavior of SRIP after X becomes 0 at time $t=90$, the ODE for SRIP at $t=90$ can be transformed as follows.

Furthermore, after X becomes $X \leq 0$, the behavior of X can be described by the following equation.

Therefore, for $t\geq 90$, both $X$ and $SRIP$ are in a steady state, with $X=0$ and $SRIP=SRIP|_{t=90}$. Using this result, the temporal evolution of $SRIP$ is plotted in the figure below.

Based on the above, the six parameters obtained using experimental data are realistic values and adequately represented the results of the model when it reached a steady state.

Discussion

The model, when observed for its temporal evolution using parameters fitted with experimental data, was able to accurately represent the experimental results for $t \geq 48$, but did not adequately follow the experimental values for $t \leq 48$. The Dry-lab focused on this discrepancy and proposed and examined the following three hypotheses.

1. Considering Time-Dependent k_sRNA

Firstly, we considered it beneficial for k_sRNA to be time-dependent. If k_sRNA is time-dependent, the model will inherently account for the viral life cycle, aligning well with intuition. We explain below the reasons for considering time-dependent k_sRNA to be advantageous.

To begin, we focus on the equation:

Here, RNA reaches its maximum value at initial stages, and given$K_{RNA} = 9.7 \times 10^{2}$, it implies $RNA > K_{RNA}$. This leads to a rapid increase in $\frac{dSRIP}{dt}$.

If the rapid increase in RNA at $t=1$ is problematic, the cause should be in the equation:

Given that RNA is close to zero at initial time, the impacting term would be $k_{sRNA} \times DNA$ . However, $DNA_{0}$ is a fixed value in experiments, making k_sRNA the term to be scrutinized.

Considering that k_sRNA is time-dependent according to certain references, it should be zero at the initial stages of SRIP production, i.e.,$k_{sRNA} = 0$ . Implementing this condition would make $k_{sRNA} \times DNA = 0$ and therefore $\frac{dRNA}{dt} = 0$ , keeping RNA constant. This should lead to a subdued increase in $\frac{dSRIP}{dt}$ and make the model more strongly concave down for $0 < t < 40$ .

It is natural to assume no RNA production for SRIP genome at initial stages when considering the viral life cycle. Hence, making k_sRNA time-dependent is justified.

2. Approximating $\frac{dSRIP}{dt}$ Using a Hill Function

Next, we believe that $\frac{dSRIP}{dt}$ should be approximated using a Hill function. This is reasonable if we consider that RNA shows positive cooperativity with structural proteins like capsomere. Details for this rationale are explained below.

Firstly, a more strongly concave down shape for the model implies it is more sigmoidal. In the biological domain, a famous sigmoidal curve is the Hill function. This function resembles the Michaelis-Menten equation when the Hill coefficient is one. Therefore, the term $\frac{RNA}{RNA + K_{RNA}}$ in the equation $\frac{dSRIP}{dt} = k_{vp} \times \left( \frac{RNA}{RNA + K_{RNA}} \right) \times X$ could be thought of as similar to a Hill function term.

If we assume the Hill function for $\frac{dSRIP}{dt}$ with $n = 1$ , the equation should be expressed as:$\frac{dSRIP}{dt} = k_{vp} \times \left( \frac{RNA^{n}}{RNA^{n} + K_{RNA}^{n}} \right) \times X$Given the interaction of RNA with structural proteins like capsomere, it is reasonable to think that the Hill coefficient $n$ would be greater than one.

3. Reducing the Time $n$ Where $\frac{dX}{dt} = 0$

Lastly, we argue that the time $t = T$ where $\frac{dX}{dt} = 0$ should be reduced. This is because a lower inflection point for SRIP's model would make the model more strongly concave down for $0 < t < 40$ . Further details are as follows:

In this model, $T$ fulfilling $\frac{dX}{dt} = 0$ corresponds to the time when SRIP-producing cell density is maximized. Examining the equation $\frac{dX}{dt} = P \times \left( \frac{X_{max} - X}{X_{max}} \right) \times \left( \frac{Ser}{Ser + K_{Ser}} \right) \times X - k_{t} \times SRIP$ , we note that $P$ and $X_{max}$ are constants. Initially, SRIP is zero, so the term impacting $X$ is $\frac{Ser}{Ser + K_{Ser}}$ . The parameter obtained from fitting is $K_{Ser}$ , and for $T$ to be smaller, $\frac{dX}{dt}$ needs to increase more rapidly, requiring$K_{Ser}$ to be as small as possible.

If fitting is performed without a lower bound on $K_{Ser}$ , it tends to zero, making $T$ smaller and improving the fit of the model to become more strongly concave down between $0 < t < 40$ .

In summary, the inadequate fit of the SRIP model during the initial phase $0 < t < 40$ can be attributed to the following key factors:

1. k_sRNA should be time-dependent.
2. 2. $\frac{dSRIP}{dt}$ should be approximated using a Hill function.
3. 3. The time $T$ where $\frac{dX}{dt} = 0$ should be reduced.

Outlook

As we venture further into the realms of molecular biology and virology, the importance of integrating mathematical models and experimental approaches becomes increasingly evident. The high-throughput neutralization assay for dengue fever, CFNT, stands as a testament to this integrated approach. With the challenges presented by the consistent production of SRIP, our Dry-lab's endeavors in mathematically modeling its production have proven instrumental.

In this section, we demonstrated the potential of mathematical techniques, especially the use of ODEs, to precisely predict the steady-state production volume of SRIP. But beyond mere prediction, these tools also offer a lens to scrutinize, refine, and enhance our understanding of the system. The suggested improvements for a more natural model further reinforce the idea that experimental biology and computational analysis are not just complementary; they are interdependent.

Moving forward, as we aim for the societal implementation of CFNT, the insights gained from these models will play a pivotal role. They will guide resource allocation, experiment design, and ultimately, the scalable production of SRIP. We stand at an exciting crossroad where traditional wet-lab techniques meet the precision and foresight of computational modeling, promising a brighter future in our fight against dengue fever.

Reference