# Team:Tec-Monterrey/Modeling

## Abstract

We built a mathematical model for the dynamics in a lytic infection process for the production of a specific protein on the basis of chemical reaction networks. We aim to optimize the protein expression as a function of the multiplicity of infection (MOI).

## Defining parameters

It was considered a simple chemical reaction network with a quasi-steady state approximation for the interaction between the baculovirus and the sf9. This led to a first approximation in the rate constant in the infection process. More will be discussed below. The parameters that will be used in the development of the model are the following:

C_{U} |
Non-infected Cell concentration. |

C_{1} |
Infected Cell concentration. |

V |
Baculovirus concentration. |

C_{P} |
Complex concentration (Non-infected Cell+Virus). |

N |
Effective baculovirus production rate. |

d |
Dead rate due to infection. |

P |
Natural saturation population. |

r |
Natural Malthusian parameter for the Sf9. |

k_{1} |
Rate constant for the complex formation. |

k_{-1} |
Rate constant for the complex degradation. |

k_{2} |
Rate constant for the infection process. |

K |
Quasi-steady-state rate constant in the infection. |

Φ_{V} |
Average virus production per infected cell. |

Φ_{P} |
Average protein production per infected cell. |

Τ_{IV} |
Time lapse between the infection of the cell and the beginning of the virus production. |

Τ_{IP} |
Time lapse between the infection of the cell and the beginning of the protein production. |

Τ_{D} |
Time lapse between the infection of the cell and cell's death. |

## Chemical Reaction Network Modeling

We consider as a first approach that only one virion is required for the infection process of a **Sf9**cell. An uninfected cell.**C _{U}** interacts with virions

**V**to create a complex

**C**. Some of them will be infected due to this interaction, i.e. the virion will cross the membrane and begin the infection process, the rest will stay in this complex form or will dissolve:

_{P}(1)

The **Sf9** cells will continue to replicate naturally as long as it doesn't get infected. After the infection, the corresponding malthusian parameter will decrease dramatically and, in fact, can be neglected after the infection: Combining both processes, we obtain the following rates for the **C _{U}** and the complex

**C**:

_{P}(2)

(3)

After infection, the cells will die because of it with an specific death rate **d**. After it's death, virions will be released, but only a certain fraction of them will continue to cause infection, because of this we consider an effective rate of virions production **N**. This is represented in the following reaction:

(4)

From the reactions in (1) and the rates described in the previous paragraph, we get the following equations:

(5)

(6)

## Quasi-steady state assumption

Since the complex is a temporary interaction, in the order of seconds, due to regular movement, the changes in **C _{P}** are in a much faster time-scale than the rest of the system, which develops during a couple of days, therefore it may be treated with a quasi-steady state approximation:

(7)

If we substitute (7) back into (2), (5) and (6), we get:

(8)

(9)

(10)

where we have defined **K≡(k _{1}k_{2})/(k_{-1}+k_{2})**.It should be noticed that this is equivalent to the following system:

(11)

Where **k** serves as a parameter that describes the affinity between the baculovirus and the **Sf9**, mainly because of the interactions in the membrane of the cell. On the other side, **N** tells how effective is the production of new virions due to the lytic infection. It contains the information of the internal process in the cell, as well as the externalization.

## A second approach: Delay differential equations

We also consider a second approach in the term of the virions production. The virions are not produced instantaneously after the infection of a cell. We consider a period **Τ _{IV}** after infection before starting the production. Also, the cell only produces virions while they are alive, from this we have to consider an average period

**Τ**from the moment the cell is infected until the moment that it finally dies.

_{D}Consider an instant **t _{I}**, at this moment we consider that a certain amount of

**dC**new healthy cells are infected. By definition this differential is given by

_{I}**dC**. We will work under the assumption that every cell has the same behavior after infection. They should produce only while they are alive, from this the production rate should have a positive value only in [

_{I}=C'_{I}(t_{I})dt_{I}**t**+

_{I}**Τ**,

_{IV}**t**+

_{I}**Τ**] and be zero everywhere else, from this we know we are looking for a production rate of the form:

_{D}(12)

where **ψ(t)>=0**. If $\Phi$ is the average virus production per infected cell we expect that:

(13)

The shape of the curve **ψ _{V}** is given by the mechanism of production. To obtain this shape by experimental ways may be too difficult to accomplish in this project but instead we proposed different shapes, based on the knowledge of the promoters used.

This function is expected to increase as the virus production begins until it reaches a maximum of production and later, it would decrease as the metabolic functions in the cell decline. The skewness of the function will depend whether if an early promoter is used or a late promoter.

From the previous assumptions the consider a displaced normalized Bessel Function of the first kind from **0** until its first root. The Bessel function can be defined as:

(14)

Here, **Γ(t)** represents the gamma function, defined by:

(15)

For the normalization we consider the first value **j _{v}>0** where

**j**is zero. We scale the interval [

_{v}(x)**0,j**] unto [

_{v}**τ**]

_{IV},τ_{D}We define the normalization factor **α _{Vv}** as the value of the following integral:

(16)

With this definition it can be observed that the function **ψ(t)=(Φ _{V}/α_{Vv})J_{V}(tj_{v}/(τ_{V}-τ_{IV}))** satisfies the desired condition. Moreover, the parameter

**v**can be used to modify the skewness of the function, having a positive skewness when

**0< v< 1**and a negative skewness if

**1 < n**.

From this argument it follows that in a certain instant $t_I$, the virus production is increased by a certain amount **f(t-t _{I})C'_{I}(t_{I})dt_{I}**, where:

(17)

But we have to consider the contributions made in all the previous moments until a given point in time to know the total virus production. Considering the infection instants as a continuum, we can rewrite the equation (9) as:

(18)

## Protein Expresssion

For the protein expression we can obtain a similar equation through an analogous argument. Clearly the average protein production per infected cell will differ from the one of the virus; also the protein may take a different time to begin its production, therefore we consider a different time lapse $\tau_{IP}$ between the infection and the production. Beside this, we consider a degradation rate because of the interaction of the protein with the remainings of the cell:

(19)

where **g(t-t _{I})** is the function defined by

(20)

Here we defined **α _{Pσ}** in a similar manner as before:

(21)

Notice that the parameter for the Bessel functions in both expressions don't need to be the same, moreover, they can be expected to be different since the mechanisms for both are, in general, different.

The final system of equations is the following:

(22)

(23)

(24)

(25)

## Numerical Analysis

For the numerical analysis, the computational package of MATLAB is used. For the Bessel function we use a routine already built in MATLAB. The evaluation of **j _{v}** is performed by a numerical method based on Newton-Raphson and for the calculation of the normalization factor

**α**an algorithm of Gauss-Legendre quadrature is used.

For the Malthusian parameter, we performed the logistical regression as specified in {Modern Regression Methods} (Thomas 1997), the data of a healthy culture of **Sf9** in the laboratory is used. A logistic behavior is assumed as the following:

(26)

where **C(t)** is the number of cell in a given moment and **P** is an expected saturation population. Because of experimental complications, **P** was estimated as 1.1 times the greatest value obtained of **C** experimentally.

The resulting model gives the following curve, compared to the original:

For the normalization, we consider different intervals and give several values for the roots of the Bessel function and the corresponding normalization factors for the definition of **ψ(t)/Φ = (1/α _{v})J_{v}(tj_{v}/(τ_{D}-τ_{I}))**, measuring the time in hours:

## Final comments

The average virus production **Φ _{V}** and average protein production

**Φ**are still to be calculated experimentally to be included in the model. After estimating this quantities, a numerical derivative can be used to evaluate the derivatives in the system of differential equation with experimental values. A linear regression is to be used to obtain an estimation of

_{P}**K**.

After calculating every parameter, the equations from (22) to (25) can be used to predict numerically the development of a given system, from different initial conditions and, from this, find the optimal setting for experimental trials and later escalation.

## References

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- Ryan P. Thomas: Modern Regression Methods, (Wiley series in probability and statistics. Applied probability and statistics). "A Wiley-Interscience publication" (1997).
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- Rohrmann, G.F: Baculovirus Molecular Biology, third edition, Department of Mi- crobiology Oregon State University (2013).
- Hu, Yu-Chen and Bentley, William E : A kinetic and statistical-thermodynamic model for baculovirus infection and virus-like particle assembly in suspended insect cells (1999).
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- Invitrogen life technologies: Bac-to-Bac Baculovirus Expression System (2013).