Abstract

COCCO is characterized by its ability to rapidly induce apoptosis immediately after viral entry. This self-elimination of infected cells prevents viral replication and the spread of infection to surrounding cells. To reproduce this characteristic of COCCO – immediate response after infection – we believed it was necessary to construct a mathematical model that could express the time-dependent dynamics of infected cells. In constructing the model, based on feedback from Human Practices, we aimed for a simple model design focused on minimal elements. This approach was taken to complement the knowledge of viral transmission that cannot be obtained from wet lab experiments, while also allowing for the quantitative verification of the COCCO concept.

Through this viral transmission modeling, we achieved the following:

• By mathematically analyzing this model, we understood that a characteristic parameter with a threshold determines whether an infection is established.
• By running simulations using data collected from wet lab experiments or scientific papers, we showed that the spread of viral infection can be attenuated by COCCO.
• We confirmed that controlling the intensity of apoptosis via COCCO is the most effective intracellular control method we can implement to overcome the spread of infection.

Introduction

COCCO is a system where cells infected by a virus immediately die via apoptosis, but it does not directly inhibit the production of virus particles within the cell. Therefore, it is suggested that it is crucial for apoptosis to occur before the infected cell releases virus particles into its surroundings. To address this property, we utilized a model [3] that incorporates the "infection age"—the time elapsed since a virus entered a target cell—into the TIV (Target cell-Infected cell-Virus) model [1][2], which is the simplest description of intercellular infection dynamics.

Furthermore, because COCCO affects viral particle dynamics within an infected cell, there are many phenomena we should not consider for our COCCO simulations, such as the spatial heterogeneity of cells, variations in infection modes, and existing immune mechanisms. A point of caution common in mathematical modeling is that a model including excessive conditions to perfectly replicate a real situation will have its reliability compromised. Taking this into account, we constructed a model that incorporates the minimum necessary conditions for COCCO's environment.

When estimating all parameters in a mathematical model simultaneously, there is a concern that the uncertainty of the estimated parameters increases due to the sheer number of them. To limit this high degree of freedom in parameters, we devised a method to ultimately estimate all parameters by first estimating only a subset of them using reduced equations corresponding to the available experimental data, and repeating this process for multiple reference datasets. This technique is a very powerful estimation process in biological mathematical modeling, which tends to have many equations and associated parameters.

In this part, we will discuss whether the intercellular spread of infection can be terminated in the body of a chicken with COCCO, and the efficacy of accelerating the death rate of infected cells by inducing apoptosis. Supplementary information that is not essential for following the discussion is summarized in the Appendices.

Model

Model Structure

We assume that the process of a virus propagating between cells is a transition of epidemiological states as shown below, and we use the following age-structured TIV model [3] as the mathematical model to describe this.

\begin{align} \frac{dT(t)}{dt} = & ~\lambda - dT(t) - \beta T(t)V(t) \tag{1} \\ \frac{\partial i(t, a)}{\partial t} = & - \frac{\partial i(t, a)}{\partial a} - \delta (a)i(t, a) \tag{2} \\ \frac{dV(t)}{dt} = & \int_0^\infty p(a)i(t, a) da - cV(t) \tag{3} \end{align}

Boundary Conditions:

\begin{equation} i(t,0) = \beta T(t)V(t) \tag{4} \end{equation}

Initial conditions:

\begin{equation} T(0) = T_{0}, ~i(0, a) = i_{0}(a), ~V(0) = V_{0} \tag{5} \end{equation}

The variables for the number of cells and virus particles, and the other parameters included in the partial differential equation above, are defined as follows.

Table 1. The variables in the differential equation system.

Name	Definition	Unit	Discription
$t$	Time	h	Time for the entire system.
$a$	Infection age	h	Time that has elapsed since a virus particle entered a target cell
$T(t)$	Number of target cells	cells / mL	A target cell is a cell that is completely uninfected by a virus.
$i(t,a)$	Number of infected cells per unit age of infection	cells / (mL・h)	An infected cell is a cell that has been invaded by a virus.
$V(t)$	Number of virus particles	HA / mL	Infectious virus particles outside of the cells are counted.

Table 2. The other parameters in the differential equation system.

Name	Definition	Unit	Discription
λ	Production rate of target cells	cells / (mL・h)	Target cells are produced at a constant rate λ.
d	Death rate of target cells	1 / h	Target cells die at a constant rate d.
β	Infection rate	mL / (HA ・h)	Target cells are infected by the virus at a constant rate β per virus particle.
δ(a)	Death rate of infected cells	1 / h	Infected cells die at a rate δ(a) that depends on the infection age a.
p(a)	Virus production rate	HA / cells	Infected cells produce virus particles at a rate p(a) that depends on the infection age a
$c$	Virus clearance rate	1 / h	Virus particles are cleared at a constant rate c
$T_{0}$	Initial number of target cells	cells / mL	Number of target cells at t=0
$i_{0}(a)$	Initial number of infected cells per unit infection age	cells / (mL・h)	Distribution of infected cells with respect to infection age a at t=0
$V_{0}$	Initial number of virus particles	HA / mL	Number of infectious extracellular virus particles at t=0

$V(t)$ is defined as the number of virus particles , but measured as HA titer.

Detailed Assumptions ▼

Analysis

Among iGEM projects, mathematical modeling is often used to fix parameters and run specific numerical simulations to see if a system shows the desired behavior. However, this approach alone doesn't reveal sudden changes in the system's behavior from parameter variations, nor does it clarify the parameter conditions needed for a proof-of-concept. Therefore, it's essential to discuss the system's structure and fundamental properties by mathematically analyzing the model.

In this section, we discuss a method to analytically determine whether an epidemic will be eradicated, using the age-structured TIV model mentioned above.

Hereafter, we set $\displaystyle T_{0}=\frac{\lambda}{d}$. This corresponds to the value at the equilibrium state that $T(t)$ reaches in a situation where no infected cells or virus particles exist, i.e., when $i(t, a),~V(t)=0$. Here, as the value that determines whether an infection can be established, $R_{0}$​ is defined in the following form (see Appendix A) [4].

\begin{equation} \label{eq:R0} R_{0}=N\cdot \frac{\beta T_{0}}{c} \tag{6} \end{equation}

Here, N is defined as follows (see Appendix B) [3].

\begin{equation} \label{eq:N} N=\int_0^\infty p(a)e^{-\int_0^a \delta (s)ds}da \tag{7} \end{equation}

$N$ signifies the burst size; that is, the total number of virus particles produced by a single infected cell from the time it's created until it dies [1][2][3]. Also, $\displaystyle \frac{1}{c}$ represents the average lifespan of a virus particle (see Appendix A) [2]. Therefore, the $R_{0}$ above is the basic reproduction number: the number of new infected cells produced by a single infected cell over its lifetime during the initial stage of an infection [4]. The basic reproduction number has a threshold of 1, which governs whether an epidemic is established or fails. If \(R_{0}\lt1\) , the infectious disease dies out, and if $R_{0}\gt1$, the infection spreads (see Appendix B) [4].

Results

Parameter Estimation

To discuss the effectiveness of COCCO, the model must be constructed with parameters that match the cellular and viral conditions to which COCCO is applied. In this section, we describe the method used to estimate the parameters in our model and calculate ​$R_{0}$. Since COCCO is a system that accelerates the death rate of cells that have taken in virus particles via apoptosis, we assumed in our model that the effect of COCCO appears only in $\delta (a)$. When estimating parameters, we used experimental data from either the Wet Lab or scientific papers. To ensure the target parameters could be accurately estimated, we selected experimental conditions that allowed for assumptions that would reduce the degrees of freedom of the parameters. For the estimation, we used the Python least_squares function, which performs a least-squares method.

To investigate the effect of COCCO on $\delta (a)$, we estimated it using a single-cycle infection experiment conducted in the Wet Lab. The experimental data was obtained by simultaneously infecting HEK293 cells with influenza virus A/Victoria/361/2011(H3N2) and determining the percentage of viable cells at each time point post-infection. Since the only available time points were 6 hpi and 12 hpi, we concluded that we could not estimate the form of $\delta (a)$ as a function of $a$. Therefore $\delta (a)$ was set as a constant. We estimated and compared the parameters for wild-type cells and for a stable cell line expressing PKR-APAF1, respectively. We named these respective parameters $\delta _{wild}$ and $\delta _{COCCO}$.

The value of $\delta$ was estimated to be 0.240 and 0.295 for wild-type cells and PKR-APAF1 expressing cells, respectively. We named these $\delta _{wild}$ and ​$\delta _{COCCO}$. Since an experiment involving multiple cycles of virus infection could not be performed in the Wet Lab, it was necessary to use data from the literature to estimate the other parameters.

We performed parameter estimation using data from cell infection experiments with influenza virus A/PR/8/34 (H1N1) and HEK293 cells [5][6], which is conducted under conditions similar to our Wet Lab infection experiment. The parameter estimation was performed stepwise: first, $\lambda$ and $d$; second, $\delta(a)$, $p(a)$, and $c$; and third, $\beta$.

δ(a) was set as a constant $\delta$, and p(a) was specifically defined as the following function [3]. The terms $p_{max}$, $a_{1}$, $b_{1}$​ are positive constants and are the parameters to be estimated.

\begin{equation} p(a)=p_{max}\left( 1-e^{-b_{1}(a-a_{1})} \right) \tag{8} \end{equation}

For the detailed method of parameter estimation, see Appendix C. The results were as follows.

Table 3. Estimated parameters

Name	Definition	Unit	Value
λ	Production rate of target cells	cells / (mL・h)	$5.64\times 10^{4}$
d	Death rate of target cells	1 / h	$5.97\times 10^{-3}$
β	Infection rate	mL / (HA ・h)	$6.61\times 10^{-3}$
δ(a)	Death rate of infected cells	1 / h	$6.18\times 10^{-2}$
p(a)	Virus production rate	HA / cells	\begin{align*} p_{\max}&=2.93\times 10^{2} \\ a_{1}&=5.00 \\ b_{1}&=1.10\times 10^{-3} \end{align*}
$c$	Virus Clearance rate	1 / h	$5.73\times 10^{3}$

From these parameters, when T₀ = 1.00×10⁶, we obtain the following:

\begin{equation} \begin{array}{cc} N=\int_0^\infty p(a)e^{-\int_0^a \delta (s)ds}da = 60.5, & \quad R_{0}=N\cdot \frac{\beta T_{0}}{c} = 69.9 \end{array} \end{equation}

Because the virus strain, an important experimental condition, is different, the $\delta _{wild}$ and $\delta _{COCCO}$ estimated from the Wet Lab data cannot be directly compared with the $\delta$ estimated from literature data. Therefore, we consider $\delta'$, which is the death rate when cells express PKR-APAF1 under the experimental conditions in the literature. Assuming that the effect of expressing PKR-APAF1 is the ratio $\frac{\delta_{\text{cocco}}}{\delta_{\text{wild}}}$, then:

\begin{equation} \delta' = \delta \cdot \frac{\delta_{\text{cocco}}}{\delta_{\text{wild}}} \tag{9} \end{equation}

Calculating with the parameter values above gives $\delta '=7.60\times 10^{-2}$, $N = 37.4$, and $R_{0}=43.2$

This result does not satisfy the condition for ending the infection, which is $R_{0} \lt 1$. However, because $R_{0}$ was reduced compared to HEK293 wild type cells, it is considered that COCCO certainly has the effect of inhibiting the spread of viral infection. Although the ratio $\frac{\delta_{\text{cocco}}}{\delta_{\text{wild}}}$ is about 1.2, $R_{0}$ is reduced to 61.8% of the original. Thus, $R_{0}$ is greatly affected by only a slight change in $\delta$, which suggests that the spread of infection can be efficiently prevented by inducing the death of infected cells. We will discuss this in detail in the next section.

Sensitivity Analysis

$R_{0}$​ is the value that determines whether an infection can be established, and as stated in equations (6) and (7), it is determined by the parameters $\beta$, $c$, $T_{0}$, $\delta (a)$, and $p(a)$. Since COCCO is a system that accelerates the death rate of cells that have taken in virus particles via apoptosis, the effect of COCCO corresponds to an increase in $\delta(a)$. However, methods of suppressing the infection by changing other parameters can also be considered. For example, decreasing $\beta$ corresponds to preventing new cell infections, and increasing c corresponds to promoting the clearance of infectious virus particles. To quantitatively discuss how much increasing $\delta(a)$, as COCCO does, contributes to suppressing $R_{0​}$ below 1, it is necessary to perform a parameter sensitivity analysis on the controllable parameters that constitute $R_{0}$​. We considered the sensitivity of $R_{0}$ to each parameter—that is, its local sensitivity—and compared their absolute values.

We calculated the local sensitivity of $R_{0}$​ with respect to $\beta$, $c$, and $T_{0}$​ using the following formula.

\begin{equation} E_{x}:=\frac{x}{R_{0}} \cdot \frac{\partial R_{0}}{\partial x} ~~(x\in \{\beta, c, T_{0}\}) \tag{10} \end{equation}

Next, we calculated the functional local sensitivity of $R_{0}$​ with respect to $p(a)$ using the following formula (see Appendix C).

\begin{equation} e_{p}(a):=\frac{p(a)}{N} e^{-\int_0^a \delta (s)ds} \tag{11} \end{equation}

Furthermore, we calculated the functional local sensitivity of $R_{0}$​ with respect to $\delta(a)$ using the following formula (see Appendix C).

\begin{equation} e_{\delta}(a):=\frac{\delta (s)}{N} \int_a^\infty p(u)e^{\int_0^u \delta (s)ds} du \tag{12} \end{equation}

$e_{p}(a)$ gives the distribution of the contribution for each infection age $a$, and by integrating this over all a, the local sensitivity of $R_{0}$​ to $p(a)$, $E_{p}$​, can be obtained.

\begin{equation} E_{p}:=\int_0^{\infty} e_{p}(a)da \tag{13} \end{equation}

Similarly, the local sensitivity of $R_{0}$​ with respect to $\delta(a)$ over all $a$, $E_{\delta}$​, is defined as:

\begin{equation} E_{\delta}:=\int_0^{\infty} e_{\delta}(a)da \tag{14} \end{equation}

Here, it is found by analysis that regardless of the values of the parameters, $E_{\beta}=E_{T_{0}}=E_{p}=1, ~E_{c}=-1$. $E_{\delta}$​ takes a negative value, but it is not generally -1. If we define $p(a)$ and $\delta(a)$ as in (8) and (9),

\begin{equation} E_{\delta}=-\left( 1+a_{1}\delta +\frac{\delta}{\delta +b_{1}} \right) \tag{15} \end{equation}

Since $\delta, ~a_{1}, ~b_{1}\lt0$, we have $E_{\delta} \lt -1$, and regardless of what values the parameters take,

\begin{equation} |E_{x}| \lt |E_{\delta}| \quad (x\in \{\beta, c, T_{0}, p\}) \tag{16} \end{equation}

This indicates that by increasing $\delta$ —that is, by inducing the death of infected cells— $R_{0}$​ can be reduced more efficiently than by other methods. Substituting the values from Table 3 for $\delta, ~a_{1}, ~b_{1}$​, we get $E_{\delta}=-2.29$, which shows that a change in $\delta$ has more than double the effect on $R_{0}$​ compared to the other parameters.

Discussion & Conclusion

To understand the effect of COCCO on intercellular virus infection, we analyzed a TIV model that considers the infection age. As a result, the following points were clarified:

• The basic reproduction number $R_{0}$​ determines whether an infection is established.
• The introduction of COCCO mitigates the spread of viral infection.
• In combating the spread of infection, promoting the death of virus-infected cells is the most effective method compared to others.

The reason for introducing the infection age in the first place was the importance of inducing apoptosis before an infected cell releases virus particles into its surroundings. In our mathematical model, this means that for the infection to end, the effect of $\delta(a)$ must exceed that of $p(a)$. However, in the Wet Lab data available this time, there were few time points, making it impossible to determine the functional form of $\delta(a)$, so we had no choice but to set $\delta$ as a constant. For this reason, we could not fully obtain the benefit of introducing the infection age. For a more accurate analysis of COCCO's effect, it is necessary to conduct single-cycle and multi-cycle infection experiments and to track virus titers and viable cell counts at frequent time points. In our current calculations, $R_{0}$ did not fall below the threshold of 1 even with the introduction of COCCO, but the results could potentially change with such an accurate analysis.

This method of constructing and analyzing a mathematical model has potential applications for various projects, as follows:

• By building a minimal model to determine the potential success of a project, the objective can be achieved without losing reliability or interpretability.
• By constructing a model that is not overly complex and analyzing it mathematically, the essential properties of the system can be revealed.
• Analyzing the threshold-like behavior of a system can provide a basis for judging whether the system will function effectively.
• By limiting the parameters to be estimated at one time and estimating them in stages using multiple experimental datasets, a highly reliable parameter estimation can be achieved.

Appendix

References

Abstract

The fusion protein used in COCCO must be able to recognize viral dsRNA and induce apoptosis. To achieve this, we designed a fusion protein in which some domains of RIG-I were replaced (see Design). Based on the structure and function of RIG-I, it is highly likely that a fusion protein created by simply substituting domains would not have the necessary function. Therefore, it was necessary to improve this fusion protein by introducing amino acid mutations in silico.

We achieved the following:

• We integrated multiple software programs to build EVOLVE(Enhanced Variant Optimization via in-silico Ligand and interface EValuation), a workflow for designing optimal mutant proteins.
• We evaluated the performance of the designed mutant protein through modeling and wet lab experiments.
• As protein modeling provided an opportunity to improve the fusion protein, we were able to adopt the RIG-I and APAF1 fusion protein as a component to realize COCCO.
• From a modeling perspective, improvement in the function of the fusion protein could be expected.

The EVOLVE we constructed is in a format that is easy for users to handle during environment setup and execution. Therefore, our EVOLVE will likely be essential for future iGEM teams to meaningfully design proteins with enhanced interaction capabilities in silico.

Introduction

RIG-I is an intracellular innate immune factor that recognizes dsRNA and is composed of CARD1, CARD2, Helicase1 (Hel1), Helicase2 (Hel2), Helicase2i (Hel2i), Pincer, and CTD domains [1][2][3]. The two CARD domains transmit a signal to the downstream protein MAVS [2]. In the absence of dsRNA, CARD2 is bound to Hel2i, and RIG-I does not transmit a signal downstream [1][2]. When dsRNA binds to the CTD and Helicase domains, the structure changes, CARDs are released, and the signal is transmitted downstream.

In COCCO, instead of the two CARDs of RIG-I, a fusion protein is used that links the CARD of APAF1, which transmits a signal to Caspase9 (see Design). If both the CARD of APAF1 and the Hel2i of RIG-I remain wild-type, it's believed that the CARD will be constantly exposed due to the low binding affinity between them. In this state, accidental protein aggregation could potentially induce apoptosis even without dsRNA, so safety must be ensured when introducing it into chickens. To address this issue, it's necessary to introduce amino acid mutations into either the CARD of APAF1 or Hel2i to strengthen the Hel2i-CARD interaction. However, when mutations are introduced into the CARD of APAF1, the CARD-Caspase9 interaction must not be lost. Similarly, when mutations are introduced into Hel2i, the Hel2i-dsRNA interaction must not be lost.

We have constructed a workflow called EVOLVE to design mutant proteins with interaction capabilities enhanced beyond the wild-type. EVOLVE consists of five steps:

1. Step1. Mutable Site Selection
2. Step2. Mutation Introduction
3. Step3: Structure Prediction
4. Step4. Visual Inspection
5. Step5. Docking & Stability Analysis.

There are two versions of EVOLVE, which differ in the method used in Step 2: Mutation Introduction. A description of EVOLVE will be provided in the following chapters.

In this part, we will discuss the design of the mutant protein via EVOLVE and the evaluation of the resulting fusion protein. Additionally, similarly to the previous part, information that is not necessarily required to be read is compiled in the Appendices.

Workflow

EVOLVE consists of the steps described above. Version 1 was used for designing mutants of Hel2i and CARD of APAF1 , while Version 2 was used for the Hel2i. The main difference between Versions 1 and 2 is the screening method for the Hel2i-CARD of APAF1 binding affinity—the property we wanted the fusion protein to acquire. In Version 1, this binding affinity is evaluated in Step 5: Docking & Stability Analysis. In contrast, Version 2 is designed to increase the binding affinity at the same time the mutation is introduced in Step 2: Mutation Introduction. Version 1 is appropriate for introducing mutations into a wild-type sequence to search for sequences with high binding affinity. On the other hand, version 2 is suitable for improving binding affinity while maintaining a known complex structure. In our modeling, we initially designed a mutant protein using Version 1, but later devised Version 2 because we believed it was better to maintain the existing 3D structure to achieve COCCO's function of dynamically and precisely switching signals.

When building EVOLVE, we were careful to create a workflow that is easy for users to handle. First, the software used in EVOLVE is written in Python, so it can be easily run on Google Colaboratory without a complex environment setup. Google Colaboratory is an environment where Python can be run for free in a Google browser. However, since the GUI for PyMOL [4] is not available on Google Colaboratory, we recommend installing the application.

In addition, the input and output for the software used in each step are either an amino acid sequence or a molecular 3D structure, and only the following two file formats are used:

• FASTA: A text-based file format commonly used for recording sequence information of proteins and nucleic acids.
• PDB: A text-based file format commonly used for recording 3D structural information of proteins and nucleic acids.

We created a user manual so that anyone can use EVOLVE.