Overview

Introduction

Mathematical Modeling is, in a nutshell, the process of representing a system using mathematical concepts and language. Even complex phenomena can be analyzed by expressing them in equations and reasoning about them. In natural sciences, Mathematical Modeling is often used not only to analyze the data from experiments and observations, but also to predict the outcomes beforehand by using simulations and other methods.

One of the most important concepts in synthetic biology is the DBTL cycle, an engineering cycle that repeats the four steps of Design, Build, Test, and Learn to improve a project.

In Mathematical Modeling, a model is built, simulated, and the results are examined to provide feedback to the design process.

Compared to wet experiments, Mathematical Modeling is easy to prepare, and results can be obtained repeatedly under different conditions.

These features allow Mathematical Modeling to accelerate the DBTL cycle and boost project improvement.

Furthermore, recent advances in computers and AI allow modeling to handle more complex systems than ever before, making mathematical modeling increasingly indispensable for synthetic biology.

Brief Summary of Our Project

In this project, we aimed to construct a rapid detection and secretion system in mammalian cells without transcription and translation. The SWIFT system consists of several subsystems, and we divided it into three main modules for simplification and compatibility.

SWIFT is composed of three systems: MESA, Secretion, and Amplification System. The project went through several engineering cycles and decided to use the three modules as components. This page mainly outlines the modeling of the three modules in the last cycle. The contributions of the dry lab in each cycle can be confirmed by referring to the engineering cycle.

Following is an overview of Dry Lab's main relationships with the project.

An overview of Dry Lab's main relationships with the project

(1) Wet / Measurements ⇒ MESA Module

The Wet Lab measurements revealed a leak of MESA in the absence of Ligand, so we developed a model from the previous model. The model was fitted to the measurements and confirmed to be consistent with the experimental data.

(2) MESA Module ⇒ Wet / Measurements

First, we evaluated the probability of our MESA design to penetrate the membrane and fedback to the design using TMHMM.First, TMHMM was used to design the MESA to increase its transmembrane potential. ¹Then, based on the ODE model for MESA, sensitivity analysis and other analyses were conducted to reduce the leaks of the MESA we used. As a result, a feedback was made to Wet Lab that it is important to reduce the receptor expression rate $μ$, decrease the binding rate constant $k_{1on}$, and increase the dissociation constant $K_{d}$. In addition, it was concluded that it is important to increase the activity $k_{cat}$ of the protease to improve the rapidity of MESA detection. See MESA Module for more information.

(3) MESA ⇒ Secretion (Need for Amplification Module)

MESA has been studied as a system that releases TF (Transcription Factor) into the cytoplasm for transcriptional control, but in SWIFT, the protein released by MESA is a protease instead of TF, and the protease functions as a signal (input) in a secretory control system to improve the rapidity (Description for details). However, Dry Lab analysis suggested that the amount of protease output from MESA was not sufficient. Therefore, we added a new Protease Amplification Module to the project design and verified the effectiveness of the Amplification Module in (3) in the Dry Lab. (For details of the analysis, please refer to the Secretion Module.)

(4) Wet / Measurements ⇒ Secretion Module

Using the measurements from a previous study, we tested the validity of two models we built, one for transcriptional control by TF and the other for secretory control by Protease. ²As a result, the correlation coefficients of the developed models were 0.95 in the case of TF and 0.90 in the case of Secretion, which verifies the validity of the models. (See Secrertion for more information.)

(5) Secretion Module ⇒ Wet / Measurements

From the results of the analysis of the model, it was found that by amplifying the amount of protease released from MESA by about 100 times and by increasing the amount of secretory protein stored and the transcription rate, it was possible to achieve a rapidity that could not be achieved by transcriptional control. Therefore, we proposed the addition of a Protease Amplification module and feedback to Project Design with suggestions for improvement, such as increasing the expression level of COP I and using cells with high secretory capacity.

We also investigated the relationship between the concentration of protease released from MESA and the amount of secreted protease, and found that SWIFT's secretory control system is characterized by a sharp increase in secretion at concentrations above a threshold value. Therefore, it is possible to make the system more sophisticated by changing $k_{cat}$ according to the amount of leakage.

(6) Amplification Module ⇒ WET / Measurements

In the Amplification Module, the ODE model was used to verify whether the designed protease was amplified in the Dry Lab since the Wet experiment could not be conducted in time. In addition, the disorder of the Linker structure in the Protease-Linker-AI Domain was evaluated using Alphafold2 to increase the amplification of the protease and reduce leakage. ³⁴For details, please refer to the Amplification Module.

Figure 4: Comparison graphs of plDDT at each base of TVMV^Thrr-AI and mutated TVMV^Thr-AI sequences.

Figure 5: Comparison of predicted three-dimensional structures using LocalColabFold.

(7) Software ⇒ Wet / Measurements

SWIFT uses a total three proteases in three modules, MESA, Secretion, and Amplification Module. Each of the three modules showed that the values of protease parameters such as $k_{cat}$ and $K_{m}$ can adjust SWIFT's performance. We were keenly aware of the lack of software that could quickly find these parameters during system design. Therefore, we created a new software program, "Proteameter", that allows users to quickly narrow down candidate proteases by specifying a range of parameters. This software can also be used by future iGEMers to use proteases. (See Software page for more information.)

MESA Module

MESA is a system that detects soluble Ligand and releases POI.

Figure 6: Role of MESA Module in SWIFT MESA detects Soluable Ligand (Input) and releases Protease (Output). The released Protease is used as Input for the Amplification Module.

To begin with, in order for our designed MESA to work, Dry lab used THMHH to increase the transmembrane potential (SeeEngineering MESA cycle 1 for more information)⁵.

Next, in order for SWIFT to detect ligand and secrete proteins quickly and with less leakage, MESA must have (1) a low amount of leakage protease and (2) a rapid process from ligand detection to protease output. MESA modeling was conducted to promote understanding of the system for these improvements.

We performed mapping of reaction pathways and constructed a new ODE model. The constructed model was validated to ensure that it shows biologically stable behavior and that it is consistent with the measurement results. Based on the validated models, analyses such as sensitivity analysis were conducted.

The results showed that (1) decrease in absolute expression level and increase of binding affinity (2) increase in $k_{cat}$ of Protease in MESA were effective in improving the system. Overall, the modeling of the MESA Module has aided the design of our project, clarified the behavior of the MESA system, and provided clear guidance for further improvements in engineering. (See Conclusion for more details)

Secretion Module

In the previous MESA, transcription factors such as tTA are released from MESA to promote transcription.⁶ However, the response based on transcriptional control of secretory proteins lacks speed because it requires transcription and translation, so we devised a more rapid system. By attaching an endoplasmic reticulum retention signal called KKYL to a secreted protein and cleaving KKYL with protease, secretion can be achieved without transcription or translation.² In our project, we have established a secretion system without transcription and translation by releasing Protease from MESA and cleaving KKYL with this protease.

We developed two ODE models, one with tTA as input and one with protease as input, for the cases in which tTA and protease are placed downstream of the MESA system, respectively. We compared the rate from the release of a substance downstream of MESA to the secretion of the target substance, and analyzed the factors involved in the rapid secretion of the system. Since the experiments indicated the possibility of leakage in the MESA system, we also examined the relationship between protease concentration and secretion to determine its resistance to leakage.

Simulations based on this model showed that our system can produce a rapidity that cannot be achieved by previous TF-based MESA systems by increasing the amount of secreted protein stored in the endoplasmic reticulum, the transcription rate, and the concentration of protease released from the MESA. Based on the relationship between concentration of protease and the amount of secreted protein, we proposed that it is possible to design, depending on the amount of MSA leakage, the secretion system to be resistant to leakage by changing the type of Protease ($k_{cat}$). If MESA leakage is high, using a Protease with a small $k_{cat}$ will reduce the leakage of secreted proteins.

Amplification Module

One of the things we aimed for in SWIFT is versatility and flexibility. We worked on modeling to enable the design of SWIFT that meets user needs, such as increasing the absolute expression level of target substances and improving the S/N ratio, by freely selecting Protease released by MESA and the reaction mechanism between Proteases.

The Amplification System used this time is one in which Protease (P1) released by MESA activates Protease (P2) prepared in advance inside the cell by cleaving the pair of P2 and its inhibitor domain. Once P1 cuts the linker of P2, it can cut other linkers without becoming inactive, enabling amplification as a whole⁷.

Figure 7: Protease 1 (P1) serves as the input, and it yields Protease 2 (P2) as the output.

First, we compared the expression levels of target substances with and without introducing the Amplification System using an ODE model to confirm the superiority of the Amplification System and to ensure that the absolute expression level is sufficient.

Furthermore, we utilized stereoscopic structure simulation to qualitatively evaluate the binding energy between P2 and Inhibitor Domain and the stability of linkers in the above system. This allows us to change the entire system in the direction desired by users. Specifically, by increasing the binding energy and improving linker stability, it is possible to suppress leakage at the expense of absolute expression level, and vice versa.

When creating a model, we felt a lack of Protease Database, so we newly constructed ProteaseDB. This allows us to narrow down Proteases based on conditions such as orthogonality when selecting Proteases and also makes important constants for Modeling referable. Therefore, it will be helpful for future SWIFT designers and iGEM Community. In addition, we incorporated Protease Amplification System into SWIFT to make SWIFT more useful for users in more combinations, and verified the results by Modeling.⁷

Conclusion

To accurately define the functions required for SWIFT, we initiated a process of modularization and refinement. Initially, the project comprised two modules: MESA and Secretion. The necessity to augment the project design became evident through the analysis of the Secretion Module in the Dry Lab, suggesting the addition of the Amplification Module. This enhancement fostered better collaboration among the individual modules and propelled SWIFT's practical functionality to new level.

We constructed ODE models for each of the modules - MESA, Amplification, and Secretion. The models' validity was verified against actual measurement results for MESA and Secretion. For the Amplification Module, the model evaluated and verified the feasibility of protease amplification. These model analyses provided us with a refined understanding of each module's behavior and offered suggestions for improvements regarding parameters that could be manipulated in the Wet Lab.

Additionally, during the design process, we recognized the need to quickly and efficiently narrow down parameters for the protease, specifically $k_{cat}$ and $K_{m}$. To address this need, we developed the software "Proteameter." With the input of a range of $k_{cat}$ and $K_{m}$ values, Proteameter generates a list of proteases candidate that fall within the specified parameter range. This software not only serves present iGEMers but can also accept the registration of new proteases, saving time in the quest to identify proteases with the desired parameters.

Through the construction of concise and well-represented models, close collaboration with the Wet Lab, and dialogues with experts, we have successfully established SWIFT as a rapid and scalable detection and secretion system.

Our work involves the recognition of each genetic part function, and integretion into modules for the whole system. The optimization of module-specific functionalities, and the enhancement of interactions between modules, have collectively improved the system's practicality, as elaborated and illustrated above. We hope that our persistent and steady approach throughout the project will provide valuable insights to iGEM Community teams through our wiki.

MESA

Purpose

MESA is a system that detects a specific soluble ligand and releases the protein of interest (POI).MESA is a detection module that can express synthetic receptors corresponding to the soluble Ligand to be detected, allowing detection of a wide variety of ligands. (See Description for details).

Figure 1: Role of MESA module in SWIFT　MESA detects the desired soluble ligand (Input) and releases Protease (Output). The released Protease is used as Input for the Secretion Module.

In order for the MESA Module to work better in SWIFT, it needs to be enhanced in terms of (1) fewer leaks and (2) quicker system response. Although MESA is a composite part using existing proteases and ectodomains, there is no modeling of MESA as a whole. Therefore, we first mapped the reaction pathway of MESA and constructed a new ODE Model.

Method and Modeling

Mapping Reaction

As mentioned earlier, there is no Modeling for MESA, so we first mapped the reaction pathway. A schematic of the reaction pathway is shown below:

Figure 2: MESA reaction pathway and wordlist The reaction in this figure can be divided into three parts.

Receptor dimerization by binding of ligand to the receptor

The dimerization brings the TC and PC chains into physical proximity, and PR cleaves CS.

The target substance is released into the cytoplasm by cleavage

Based on the above information, we formulated the reaction equation. MESA can be utilized when the Ectodomain forms either a hetero or homodimer. In the homodimeric case, the PC and TC chains have the same binding strength. However, in the heterodimeric case, the binding affinity between the PC and TC chains and ligands is different, resulting in a different binding order. In this modeling, we will deal with the heterodimeric case because we used rapamycin in the Wet team.

Simulation Model

Based on the above reaction pathway mapped, an ODE model was constructed in MATLAB. The simulations were performed and results were visualized using MATLAB R2023a and Simbiology Model Analyzer 6.4.1.

A simple model based on the Hetero dimerization reaction was designed using rapamycin as the experimental system, and parameters were set based on literature values.MESA is a reaction in which a receptor on the membrane of the TC and PC chains detects the ligand, and the cleavage sequence is cleaved by protease under the membrane.Cleavage of the cleavage site by Protease is usually regarded as a type of enzymatic reaction and is described in the figure below.

MESA can be viewed as the association and dissociation of protease (enzyme) and cleavage sequence (substrate) in this enzymatic reaction, replacing ligand. Based on this idea, we performed modeling. A schematic diagram of the MESA reaction as compared to a normal enzyme reaction is shown below.

The dissociation constant between rapamycin and FKBP is 0.2 nM, while the dissociation constant between rapamycin and FRB is very large, 26 μM¹. Therefore, we constructed the above model under the assumption that the binding between rapamycin and FRB is negligible.

Figure 4: MESA equation
　In the absence of ligand, no reaction occurs and the product of the receptor production rate $μ$, the degradation rate $δ$, and the receptor concentration $[R]$ is balanced; in the presence of ligand, the reaction described in Figure 3 occurs and the equation is shown above.

Assumption

The above model is based on mass action dynamics. To this end, several assumptions are made. It is assumed that the diffusion of protein is sufficiently rapid and concentrations are uniform for the various substances inside and outside the cell. The concentration of ligand is constant because ligand is universally present compared to intracellularly. Other than that, we do not consider the degradation of the complex of Ligand and Receptor; we assume that the non-reactive Target Chain R2 is rapidly degraded after the POI is cleaved, and that ligand and receptor are transformed into the target substance $H$ under a rate constant of $k_{cat}$ after becoming a trimer. Cleavage by protease released into the cytoplasm due to e.g., being separated from the protease chain is not considered since it is a very small amount. The association of target chains and Protease chains is not considered.

Intial Values

Results

Behavioral Stability

Figure 5: Time dependence of $R_{1}$, $R_{2}$, $H$ at $L$=5e-9 M
The $R_{1}$, $R_{2}$ react to produce the target substance $H$. The response of the target substance is rapid, reaching a half response in about 10 minutes. Each concentration converges and the model shows biologically valid behavior.

Model Validation and Development by Wet Measurement

The results from the WET showed that MESA leak is about 50% even in the absence of ligand. Therefore, we developed the above model by adding a term in which receptors $R_{1}$ and $R_{2}$ collide incidentally at a rate of $k_{12}$. We also added a term to convert the ligand concentration from the wet measurement to the ligand concentration, since the ligand concentration was relative in the above model. These were estimated from the measurement at wet and determined to be $k_{12}$=5.20e6 (M*min)^-1,$A$=2.00e-2.

Figure 6: Equations of the constructed model

The terms $k_{12}R_{1}R_{2}$ for conversion of ligand concentration and accidental collision of $R_{1}$ and $R_{2}$ have been added. $[L]_{ex}$ represents the sample concentration of Rapamcin in the wet lab experiment.

The model was developed and fitted as described above.

Figure 7:Relationship between ligand concentration and fluorescence intensity
(Theoretical curve (blue), experimental value (red))

From these results, the model was refined to be more consistent with the Wet Lab measurements. Based on the results of the above wet lab measurements, the MESA Module should be improved in the following three areas where it works better in SWIFT.

1. To suppress the leakage of MESA in the presence of Ligand microspheres
2. Enhanced rapidity
3. Reduction of leak in the absence of Ligand

For (3), the main cause of leakage is thought to be the accidental collision of receptors with each other due to the physical proximity of the protease chain and the target chain, which is cleaved by the protease. Therefore, it can be proposed that reducing the receptor density by decreasing the receptor expression level and strengthening the steric hindrance of the ligand are effective.

However, it is difficult to reduce leakage by changing the parameters that can be moved in MESA.

Therefore, in the Dry lab, we embarked on (1) and (2).

Analysis

In the following Analysis, $k=0$ and $A=0$ was used to distinguish between leaks in the presence of ligands.

Figure 8: Relationship between ligand concentration in MESA and tne concentration of produced target substance

The figure above shows that even in the presence of a small amount of ligand (L=1e-10 M), the response of MESA is about half of the peak response.The graph also idicates that at L=1e-8 M the MESA response is approaching Plateau.

Therefore, in future discussions, we will assume L=1e-10M to have a minute amount of Ligand and a sufficient amount of Ligand at L=1e-8 M.

In order to increase the sensitivity of MESA, it is necessary to improve the signal-to-noise ratio by reducing only the amount of MESA response in the presence of a small amount of ligand.

Therefore, we performed a sensitivity analysis with the goal of reducing the response only when Ligand is in micro-presence.

The results of the sensitivity analysis show that the parameters related to $k_{1on}$ and $k_{1off}$ change significantly when ligand saturation and Ligand trace are present. This result suggests that the S/N ratio of MESA can be improved by changing the binding affinity between rapamycin and FKBP in experiment.

(1) To Suppress the Leakage of MESA in the Presence of Ligand Microspheres

Figure 10: Concentration of products when $k_{1on}$(M×min)^-1is varied in the presence of ligand micro $L$=1e-10 M (left) and excess ligand $L$=1e-8 M (right).

Insight

Based on the sensitivity analysis and the results of the above analysis, it is effective to decrease $k_{1on}$ and increase the dissociation constant ($K_{1d}=k_{1off}/k_{1on}$) between Rapamycin and FKBP in order to improve the signal-to-noise ratio.

The expression rate $μ$ is also a sensitive parameter in sensitivity analysis. However, $μ$ is sensitive regardless of the ligand concentration, and increasing the expression level of MESA will increase the output of MESA, but will increase the leak in the absence of ligand (as shown by the wet measurement), so increasing $μ$ is not a recommended It is not a recommended option.

On the other hand, reducing $μ$ is effective because it decreases the receptor density and reduces leak due to accidental dimerization in the absence of ligand. However, when receptor expression is reduced, the absolute amount of Protease released by MESA is also reduced. In addition, it was suggested in the later part of the Secretion Modeling that the output of MESA alone is not sufficient to produce sufficient secretion in the Secretion Module described below.

Therefore, the Dry Lab suggested the need for the Protease Amplification Module (described below) to ensure sufficient amount of protease for secretion while reducing the expression of receptors and suppressing leakage, and improvements were made in the project.

(2) Enhanced Rapidity

To analyze the rapidity of MESA, we varied the protease cleavage efficiency $k_{cat}$, which was moderately sensitive in the sensitivity analysis, and the binding affinity $k_{2on}$,$k_{2off}$ between the FKBP/Rapamycin complex and FRB.

Figure 11: Relationship between the amount of target substance produced when $k_{cat}$(min^-1) of protease is varied

Figure 12: Relationship between the amount of target substance produced by varying $k_{2on}$(M×min)^-1,$k_{2off}$(M×min)^-1 of protease

Actually, $k_{2on}$ is a parameter related to binding affinity and depends on the sequence of the receptor, but $k_{cat}$ can be improved by changing the protease or mutating CS, which simplifies the work in the wet lab. Therefore, it is basically desirable to increase $k_{cat}$. You can use our DB "Proteameter" for selecting proteases to increase $k_{cat}$.

Insight

For rapidity it is desirable to increase the activity $k_{cat}$ of the protease (or enhance the binding affinity of the second bond).

Conclusion

All tests performed to determine if the model met biologically feasible behavior were successfully accomplished. Simulated receptor and product concentrations were stable over time. Simulation results were compared to measurements in wet and developed to include leak in the absence of ligand, and the Model was validated. Based on the validated model, sensitivity and other analyses were performed, and it was proposed that (1) a decrease in $μ$ (decrease in leak in the absence of ligand due to a decrease in $k_{12}$) and (2) a larger $K_{d1}=k_{1off}/k_{1on}$ (decrease in leak when ligand is small) are crucial for improving leak and signal-to-noise ratios of MESA. It was suggested that it is extremely important to increase $K_{d1}=k_{1off}/k_{1on}$ (leak decrease when Ligand is small). It was also found that (2) the amount of response of MESA and $k_{cat}$ and $k_{2on}$ must be somewhat large to speed up the response speed in order to enhance the promptness. Increasing $μ$ also increases the response amount, but is not recommended because it increases Leak. The inclusion of these improvements suggests that MESA can be used as a quick detection switch and that response performance can be improved by adjusting the parameters.

Secretion

Purpose

Secretion is one of the most important modules that support SWIFT's rapidity: in secretion using MESA, the material separated by the MESA reaction was initially TF(tTA), but we wondered whether this was not sufficient to ensure the rapidity required by SWIFT. Therefore, we proposed a system that secretes POI stored by KKYL without transcription and translation, built a model, and provided feedback from the dry lab to improve the design around the SWIFT secretion system. The purpose in the simulation in the Dry lab is the followings 2 points.

1. Building a model of the response around the endoplasmic reticulum (ER) and Golgi apparatus (GA), and comparing the rate and amount of secretion between the previous transcriptional control response pathway and SWIFT's secretory control response pathway. Based on the results, we will provide feedback to the project design.
2. Investigating the relationship between the concentration of protease released from MESA and the amount of secreted protein, and capturing the characteristics of the system. Based on the Wet results, MESA has the possibility of leaking, so figure out which parameters should be varied to set the system threshold at a higher concentration to make it more resistant to protease leaks.

Method and Modeling

Description of the Secretion

This section describes two models of SWIFT's secretion module: transcriptional control and secretory control. Since the retention of KKYL is made by vesicles containing proteins moving back and forth between the ER and cis-GA, we initially built a model that distinguishes between the ER and cis-GA. However, the parameters related to the interaction between ER and Cis-GA are difficult to measure in a wet experiments, and the variables are complex. Therefore, we created a simple model that reduces the number of variables by combining ER and Cis-Golgi as shown below.

Figure 13: Flowchart of secretion system

The model of transcriptional control

Figure 14: Mechanism of transcriptional-based system Transcription factors(i.e. tTA) released from MESA regulates transcription of DNA sequence of interest.

The reaction of MESA, a module of ligand detection, releases tTA to form a dimer. This reaction is described by the following ordinary differential equation: tTA monomer dimerizes with a rate constant of $k_1$, and tTA dimer dissociates with $k_2$. $k_2$ is much larger than $k_1$, and tTA monomer dimerizes quickly, so only tTA dimer is considered for degradation.

The tTA dimer binds to the enhancer (TRE) and promotes transcription of the downstream POI gene. The promoter upstream of the POI gene transcripts at a rate of $\mu_{bS1}$ even in the absence of tTA transcriptional activation.

The mRNA of the gene of POI is translated. While being translated, POI enters the ER and is transported to cis-GA. As described earlier, this model does not distinguish between ER and cis-GA, so we do not consider transport from ER to cis-GA; POI is transported from cis-GA under the rate constant $k_{S1}$.

POI transported from cis-GA goes through the Golgi layer to trans-GA, where it is secreted to the extracellular under the rate constant $k_{S2}$.

The model of secretory control

Figure 15: Mechanism of secretion system Protease released from MESA cut off certain Cleavage Site in constitutively expressed proteins which includes protein of interest (POI), ER retention signal and Furin Cleavage Site (FCS); POI-FCS released from ER proceeds to trans golgi network and Furin cleaves FCS. Finally, POI is secreted.

Protein to be secreted (POI) + FURIN Cleavage Site (FCS) + Transmembrane Domain (TM) + HRV3C Cleavage Site (HCS) + KKYL mRNA sequence is transcripted at a constant rate downstream of a constitutive promoter.

POI+FCS+TM+HCS+KKYL mRNA is translated entering the ER, where it is retained between ER and cis-GA by KKYL until the input of HRV3C protease from MESA. When the HRV3C protease released by MESA recognizes the HCS,and it cuts off the KKYL on the cytoplasmic side; the HRV3C protease is assumed to be in a steady state because the reaction rate of binding to the recognition sequence is sufficiently rapid compared to the cleavage reaction, thus this reaction is described by the Michaelis-Menten formula.

The POI + FCS + TM that has been cut off KKYL is no longer retained by KKYL in the ER and is transported through the Golgi layer to the trans-Golgi under the rate constant of $k_{S1c}$.

POI + FCS + TM transported to the trans-Golgi is cut the TM off by a protease called FURIN, which is originally present in the trans-Golgi. This reaction, like the HRV3C protease, is also assumed and is described by the Michaelis-Menten equation.

POI that has been cut off the TM is transported out of the cell under the rate constant of $k_{S2c}$.

Evaluate the consistency of a model

To verify that the model describes the reaction accurately, we compared the experimental results of previous KKYL studies with the simulation results of the models. In previous studies, SEAP was attached to KKYL, which was designed to be cut off by abscisic acid (ABA)-inducible TEV split protease. For comparison with transcriptional control, SEAP secretory experiments with ABA-induced transcriptional activity were also conducted, in which SEAP activity in the medium was measured 30 min, 45 min, 60 min, 90 min, 120 min, 180 min, 240 min, and 360 min after ABA addition, and data were collected four times for transcriptional control and KKYL respectively. The data were normalized to the maximum secretory control at 360 min for transcriptional control, and we used that data². We simulated with $[H]$ = $[Ta]$ = 3e-9 M and $S1$ initial concentration $[S1]$ = 5e-5 M and calculated the values of $[Si]$ after 30 min, 45 min, 60 min, 90 min, 120 min, 180 min, 240 min, and 360min for each model of transcriptional control and secretory control. The $[Si]$ values for transcriptional control after 360 min were set to 1 and the other data were normalized. We calculated the average of the four experimental data from each of the experiments in the previous studies, and calculated the correlation coefficient between our model values and the calculated average values for transcriptional control and secretory control, respectively. The results are shown below.

Figure 16: Comparison of experimental data and simulated data [transcription] Correlation coefficient between experimental data and simulated data is 0.95.

Figure 17: Comparison of experimental data and simulated data [secretion] Correlation coefficient between experimental data and simulated data is 0.90.

From these results, we conclude that the model we built is sufficient to describe the secretory module of SWIFT.

Results

Comparing Transcriptional Control and Secretory Control

Simulations were conducted based on the above models, and the initial concentrations of tTA and protease, the inputs from MESA, were set to 3e-9 M for both models. This was calculated from 4000 molecules/cell, the concentration of tTA in human HeLa cells.¹⁶ In addition, the rate of secretion depended on the initial $S1$ storage, and we varied it stepwise. The initial storage of $S1$ is set as the lower limit of the steady-state concentration of $S1$ when the concentration of protease released from MESA is set to 0, and the upper limit is set to 100 times that concentration. The steady state concentration of $S1$ is as follows.

Figure 18: Change in S1 concentration in the absence of protease input The initial storage of S1 was estimated as the concentration at which the production and degradation of S1 are balanced (steady state) in the absence of protease input. The initial storage was found to be of the order of $10 nM$

Based on these results, we set the lower limit of the initial storage of $S1$ to 1e-8 M and the upper limit to 1e-6 M.

Simulated results of changes of the amount of $S1$ secreted when the initial storage amount of $S1$ is varied stepwise are shown in the figure below.

Figure 19:Comparison of concentration of S1(POI in ER and cis-GA) between transcription control and secretion control For secretion, 3 concentrations of input are tested

As a result, the model of transcriptional control was more rapid. We hypothesized that the concentration of protease released from MESA may have been too low, so that secretory control was slower. We ran the same simulation again at a concentration of 300 nM of tTA and protease in the input. At the same time, we also performed a simulation in which the transcription rate of $S1$ was changed stepwise instead of the initial concentration of $S1$. The results are shown below.

(a)

(b) Figure 20: Comparison of concentration of S1(POI in ER and cis-GA) between transcription control and secretion control
(a) The initial concentration of $S1$ is varied from 1e-8M to 1e-6 M
(b) The transcription speed of $S1$ is varied from 1e-14 M/min to 1e-11 M/min.

Increasing the concentration of protease released from MESA ensured rapidity of secretory control. Also, when comparing the case of increasing the initial concentration of POI with the case of increasing the transcription rate, the case of increasing the transcription rate shows a more long-term rapidity. However, a researcher in the field of intracellular trafficking (see HP page for details) raised the concern that increasing the transcription rate too much increases the possibility of leakage. The risk of leakage is lower when increasing the initial concentration of $S1$, but the risk of leakage is also increased if the promoter strength is increased to increase the initial concentration. Therefore, we recommend increasing the expression of COP I, which is related to the retention of KKYL. Also, using cell types with high secretory capacity is a good strategy. These cell types are said to be more resistant to ER stress, which would allow for more rapid secretion of SWIFT.¹⁷

In addition, we found that the amount of protease released from MESA is an important factor, so we proposed and built a model to amplify protease. See the Amplification section of Model for more information.

Insight

By comparing transcriptional control and secretory control, the following two insight were obtained.

1. The concentration of TF and protease released from MESA is as low as 3 nM, which does not ensure rapid secretory control. However, if the concentration of protease can be amplified 100 times by combining with a module for amplification of protease, it is possible to achieve a rapidity that cannot be achieved by transcriptional control. Therefore, the protease amplification module is essential for SWIFT.
2. In secretory control, the rapidity can be increased by increasing the initial concentration of secretory proteins or the transcriptional speed. Increasing the transcription rate will ensure a longer period of rapidity, but will increase the risk of leakage; increasing the expression level of COP I or using cells with high secretory capacity will increase rapidity while suppressing leakage.

These results were feedbacked to project design.

Relationship between protease concentration and the amount of secreted protein

We investigated how the amount of secretion changes with the concentration of protease released from MESA. The model used was secretory control, and a graph was drawn with the protease concentration on the horizontal axis and the amount of secretion after 400 min of input on the vertical axis. The graph is shown below.

Figure 21:The amount of secreted protease with change in the concentration of protease released from MESA

The graph shows a characteristic of a sharp increase in secretion in a certain concentration range. This means that a small amount of leakage from MESA does not cause secretion, but when the input exceeds a certain protease concentration (threshold), secretion increases like a switch. However, since the amount of leakage from MESA varies depending on the receptor type and expression level, it is desirable to be able to adjust the threshold concentration. So, we tried to examine the changes in the above graph by changing $k_cat$ in steps. We used $k_{catH}$ as a parameter because we use HRV3C protease. $k_{cat}$ can be easily adjusted by changing the type of protease and its CS, etc., and is not a parameter related to ER stress like $\mu_bs$, so we chose this as the parameter to change. The results are as follows.

Figure 22: The amount of secreted protease with change in $k_{cat}$ in steps ($k_{catH}$ = 142.8, 71.4, 34.7, 17.85 min^-1)

$k_{cat}$ = 142.8 min^-1 (blue line) is the same as the first graph; as $k_{cat}$ is decreased, the system becomes more resistant to leaks at higher concentrations. However, since the horizontal axis of the graph is logarithmic, the graph rises more sharply when $k_{cat}$ is higher, meaning that the system is more sensitive to changes in protease concentration.

It is difficult to say which value of $k_{cat}$ to set, but if there is a lot of leakage of MESA or if you want to suppress the leakage of secretion as much as possible, it is recommended to set a lower value of $k_{cat}$. On the other hand, if the concentration of protease released from MESA is low or if you want to ensure a large amount of secretion, a high value of $k_{cat}$ is recommended.

Insight

The following two Insights were obtained from the relationship between Protease concentration and secretion.

1. The model of secretory control showed a behavior in which secretion increases sharply when Protease input is above a certain concentration range (threshold). Since this indicates that secretion is unlikely to occur when there is leakage from MESA below the threshold. We thought that if the threshold could be adjusted, the design could be modified according to the amount of leakage from MESA.
2. By decreasing the $k_{cat}$ of the input Protease, it was found that it is possible to suppress secretion against more leakage from the MESA. On the other hand, higher $k_{cat}$ was found to be more sensitive to changes in Protease concentration. Therefore, we conclude that it is better to set a lower $k_{cat}$ value if there is a lot of leakage from MESA or if you want to suppress the leakage of Secretion as much as possible, and to set a higher $k_{cat}$ value if the concentration of Protease released from MESA is low or if you want to ensure a large amount of secretion. However, it is not easy to find a protease with the $k_{cat}$ you want to set up, so we created a protease database that allows you to find a protease by $k_{cat}$ as well. Please see the software page for details.

Conclusion

Since we were able to confirm the consistency of our models for secretory control and transcriptional control with experimental data from previous studies, we simulated these two models and compared their rapidity. As a result, we found that by amplifying the amount of protease released from MESA about 100 times and by increasing the amount of secretory control protein or transcription rate, we could achieve a rapidity that could not be achieved by transcriptional control. Based on these results, we proposed the protease amplification module and feedback to project design with improvement ideas such as increasing the expression level of COP I and using cells with high secretory capacity. We also investigated the relationship between the concentration of protease released from MESA and the amount of secreted protease and found that SWIFT's secretory control system has the characteristic that the amount of secreted protease increases sharply at concentrations above a threshold value. We also found that we can adjust that threshold by varying Protease's $k_{cat}$, so the system can be made more sophisticated by changing $k_{cat}$ according to the amount of leakage.

Amplification

Purpose

The Amplification module plays a crucial role in enhancing system performance by amplifying the output from MESA and ensuring an ample input supply to Secretion. In this module, Protease 1 (P1) serves as the input, and it yields Protease 2 (P2) as the output. It's worth noting that P1 and P2 are distinct types of proteases, and this action operates independently.Please refer to the description page for further details.

The inhibited P2 is actually in a state where the active site is occupied by a soluble inhibitor. The linker keeps the inhibitor close to the active site, thus maintaining the inhibition state.

According to the analysis of the dry lab so far, it is suggested that by increasing the output of the MESA module, the Secretion module can be made into a faster and more practical system than the existing transcription activation system. Also, it is desirable to lower the absolute output of MESA to increase the S/N ratio. Therefore, the mechanism to amplify the protease signal is essential for SWIFT. For this reason, we proposed to add an amplification module that amplifies the protease concentration change between the MESA module and the Secretion module, and feed back the results of the dry lab to the design of SWIFT. The purposes of the simulation in the dry lab are as follows: ①We model the reactions inside the amplification system and examine the relationship between input and output to see if the amplification system is practical from the perspectives of magnification and S/N ratio. ②We investigate the optimal linker sequence and feed it back to the design. From Human Practice, we learned that accidental collisions between linker parts and proteases among molecules cause unintended on-output. Therefore, we investigate how to moderately increase degradation resistance and reduce background output.

Method and Modeling

Modeling Amplification System

The modeling of the amplification system was constructed with reference to previous research ⁷. In this system, we designate S as the substrate, and P as the product generated from S by the action of P2. To investigate the behavioral changes with the amplification system compared to its absence, we considered two scenarios: one where the input protease(P1) directly produces P from S (the primitive system) and another where P1 activates the amplification system, resulting in the production of P from S by P2 (the amp system).

The reaction network is depicted in Figure 23.

Figure 23: Primitive Module And Amp Module

Here is an overview of the reactions in the amp system:

1. P1 cleaves the linker of P2-AI. According to our assumption, this process follows Michaelis-Menten kinetics.
2. P2 and AI generated from the cleavage of P2-AI degrade following mass-balance kinetics.
3. P2 produces P from S, but its activity is competitively inhibited by the free AI.

These reactions lead to a following set of nonlinear differential equations:

On the other hand, the reactions in the primitive system are simpler. P1 produces P from S, as described by the following differential equations.

The simulations were performed and results were visualized using MATLAB R2023a and Simbiology Model Analyzer 6.4.1. The program was executed on a Windows 11-based PC with 32GB of RAM and an AMD Ryzen 5700X processor.

The parameter values used are listed in the table below.

Editing and Adjusting Parts

Our Human Practice to Dr. Hidaka presented the fear that the linker in the AmplificationSystem could lead to leakage, so we wondered if the DNA sequence of the linker could be adjusted to suppress leakage. In addition, in order to increase versatility, we considered making the entire AmplificationSystem customizable so that it can be adjusted to increase the absolute expression level, which is inextricably linked to the leakage.

Dr. Arai gave us the following two methods to mutate and evaluate the AmplificationSystem sequence.

1. evaluate the binding energy of P2 and its Inhibitor Domain
2. evaluate the disorder of Inhibitor Domain and ligand. We examined and studied each of these two methods.

1.Evaluation of Binding Energy between P2 and Inhibitor Domain

Here, the term "binding" refers to the direct interaction between P2 and the Inhibitor Domain, rather than binding mediated by the linker. A higher binding energy can prevent P2 from becoming active without linker cleavage, thereby reducing leaks. Conversely, lower binding energy can increase the absolute expression level.

First, the three-dimensional structures of the Protease and Inhibitor Domain to be used are determined either through wet experiments or prior research. Subsequently, Gromacs is used to calculate the binding free energy. By comparing the free energies of multiple sequences, we can qualitatively assess them without the need for exhaustive wet experimentation, allowing us to identify sequences with desired properties.

However, we have not had significant computational resources to execute this method.

2. Evaluation of Linker Disorder Including the Cleavage Site

In the Amplification System, the linker includes the P1 Cleavage Site, and it is cleaved only when it contacts P1. The ease of linker cleavage is determined by its disorder, as explained in Professor Arai's work. Additionally, Professor Arai mentioned that disorder correlates with the accuracy of structure prediction when predicting from sequences.

Therefore, we tested around 20 different linker patterns near the Cleavage Site based on the sequences used in the paper. To compare the prediction accuracy in the Cleavage Site region, we used a computer equipped with 64GB of RAM, an Intel i-12700K, and an NVIDIA RTX A4000 running Ubuntu 22.04 on WSL2. We used Local Colab Fold 1.5.2⁴ for structure prediction and assessed the accuracy using pLDDT values.

Furthermore, we compared the predicted structures of the mutated sequences to ensure that molecules generated from mutated sequences retain the same functionality as the original.

Results

Evaluation of Amplification Factor

The simulation was executed as originally designed, with P1 as hrv3c and P2 as TVMV. The input from MESA was set to [P1] = 3e-9 M, which remained constant.

Initially, a comparison was made between input and output. The results of P2 released by the amplification system are shown in Figure 24. Simultaneously, a comparison was made between the output when using the amplification module and when not using it for the same input. These results are presented in Figure 25.

Figure 24: Relationship between Input (P1) And Output (P2) in The Amp System.
[P1] was kept constant at 3e-9 M. P2 is the protease released by the Amplification module. The time course of P1 and P2 is plotted.

Figure 25: Comparison of P Concentration Changes in The Amp System And Primitive System.
The concentration change of the P2 cleavage product when using the amplification system (amp) and when not using it (primitive) for [P1] constant at 3e-9 M is plotted.

As shown in Figure 24, the concentration of the output protease exceeded that of the input protease early in the process. Additionally, as shown in Figure 25, the amplification system delayed the onset but produced more the protease as the final product (P) at relatively early times compared to not using the amplification system.

Next, we considered methods to increase the amplification factor. To achieve this, we varied kcat1 and Km1 to expedite the cleavage of the final product, recording the [P2] at 30 minutes for each case in the amp system. We chose 30 minutes as we wanted to know the [P2] at a relatively early time point. The range of parameter variations was chosen to align with the range of real proteases in a protease database.

Figure 26: [P2] after 30 Minutes for Each k_cat and Km
The range in which the parameters were varied was done with reference to the protease database to be included in the range of real proteases.

As shown in Figure 26, a higher kcat and a lower Km result in a higher [P2]. Km has a wider range of possible values and has a significant impact on the results when changed. In response to these findings, we examined the concentration change of the output protease when using a different protease (HIV protease) with a lower Km. This result is presented in Figure 27.

Figure 27: Concentration of P2 When P1 Is Altered
P2 concentration when using HRV3C as P1 and when using HIV protease as P1 in the amp system

As depicted in Figure 27, changing the input protease P1 led to an increase in the concentration of the output protease P2 from the amplification module. Additionally, we assessed the impact of protease changes on the production rate of the final product P. In the Secretion section, it was demonstrated that if the MESA output were approximately [P1] = 3e-7 M, the Secretion module would function as a sufficiently fast system. Therefore, when [P1] was fixed at 3e-7 M in the primitive system, it was used as the ideal condition and plotted for reference alongside the amplification system. Please note that this condition assumes a time of 0 until the MESA output reaches a steady state, which will not occur in reality.

Figure 28: Comparison of Concentration Changes in The Final Product
P concentration when [P1] is fixed at 3e-7 M in the primitive system (ideal input, blue line, primitive.P), when P1 is HRV3C and [P1] is 3e-9 M in the amp system (yellow, amp.P), and when P1 is HIV protease and [P1] is 3e-9 M in the amp system (red, amp_1.P)

As shown in Figure 28, changing the protease resulted in an increase in the production rate of the final product, approaching the ideal condition.

Insights on the Evaluation of Amplification Factor

The amplification module demonstrated that it can immediately produce an output protease concentration higher than the input protease concentration. Additionally, it was shown that the concentration of the final product can surpass that of the non-amplified system within a relatively short time. This indicates that the amp system is practical on a time scale.

Furthermore, the choice of a protease with a lower $K_{m}$ can increase the amplification factor within a certain time frame.

Input/Output Ratio

The response of the system to changes in input was also examined. The concentration of P2 at 120 minutes after P1 is released was investigated when the P1 concentration was logarithmically spaced between 5e-12 M and 5e-9 M. The results are shown in Figures 29 and 30.

Figure 29: Relationship between Input And Output (120 Minutes after The Release of P1) in The Amplification Module

Figure 30: Relationship between Input And Output (120 Minutes after The Release of P1) in The Amplification Module [Low Input]

It was found that increasing the P1 concentration also increases the P2 concentration. While the input and output have a roughly linear relationship at low P1 concentrations, the output plateaus at a constant value as P1 concentration becomes high.

Insight

The output of the amplification module has limitations, which need to be considered in use cases where high output is required. One potential reason for the upper limit on output could be the expression level of P2-AI.

Furthermore, it's essential to be aware that the signal may not be amplified across all input concentration ranges. In the low-input range, such as 1e-10 M to 1e-9 M, there is a monotonically increasing relationship between input and output, making the amplification system functional.

However, in the high-input range, such as 2.0e-9 M to 3.0e-9 M, the output becomes constant, and amplification cannot be achieved.

On the other hand, in the low-concentration input range, a suitable input range was identified where the relationship between input (x) and output (y) follows a linear relationship (y = ax) without background output. This behavior can make　analysis more manageable.

Adjusting of Linker Sequence

TVMV^Thr-AI has a linker structure consisting of TVMV Protease, Thr-CS, TVMV-AI, and HisTag in this order. To manipulate the disorder of Thr-CS, 24 different linker patterns were tested between TVMV protease and Thr-CS, as well as between Thr-CS and TVMV-AI.

The evaluation of disorder was based on the prediction accuracy in the THR-CS region.

Figure 31:Comparison graphs of plDDT at each base of TVMV^Thr-AI and mutated TVMV^Thr-AI sequences. From left to right, the one with the lowest plDDT value in the Thr-CS interval, before mutation, and the one with the highest plDDT value in the Thr-CS interval.

Figure 32: Comparison of predicted three-dimensional structures using LocalColabFold.

Insight

It was found that varying the linker sequences could induce significant changes in the prediction accuracy of the cleavage site (CS) and, consequently, the disorder. Additionally, comparing predicted three-dimensional structures allowed for estimating if there were any functional changes in the molecules. However, for the actual functionality, feedback from wet experiments is essential, and this remains a future prospect.

Conclusion

A crucial amplification module was designed and assessed for its feasibility in constructing a system connecting MESA and Secretion.

In the ODE model, it was demonstrated that the amplification module successfully amplified input as expected, and it was also shown that the amplification factor could be adjusted by selecting the appropriate protease. However, it was revealed that the amplification factor depends on the input concentration range, and while our design works well within the appropriate concentration range, care must be taken during the design phase for more general implementations.

Furthermore, by adjusting the linker sequences around the CS of P2, it was observed that the prediction accuracy of CS could vary significantly. Since the accuracy of predicting three-dimensional structures correlates with the stability of the site, this approach can evaluate whether the CS in the mutated sequence is stable or unstable compared to a specific base sequence. Since increasing the stability of CS can reduce the leakage of the Amplification system, and decreasing the stability can increase the absolute expression of Protease, it can be said that Dry can reduce the number of experimental iterations by giving the trend sought for CS derived from the actual data obtained in Wet.