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:

Chemical Reaction Network Modeling

We consider as a first approach that only one virion is required for the infection process of a Sf9cell. An uninfected cell.C_U interacts with virions V to create a complex C_P. 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:

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:

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:

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

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:

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

where we have defined K≡(k₁k₂)/(k_-1+k₂).It should be noticed that this is equivalent to the following system:

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

Consider an instant t_I, at this moment we consider that a certain amount of dC_I new healthy cells are infected. By definition this differential is given by dC_I=C'_I(t_I)dt_I. 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 [t_I+Τ_IV,t_I+Τ_D] and be zero everywhere else, from this we know we are looking for a production rate of the form:

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

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:

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

For the normalization we consider the first value j_v>0 where j_v(x) is zero. We scale the interval [0,j_v] unto [τ_IV,τ_D]

We define the normalization factor α_Vv as the value of the following integral:

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:

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:

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:

where g(t-t_I) is the function defined by

Here we defined α_Pσ in a similar manner as before:

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:

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:

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_P 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 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

1. Ingalls, B: Mathematical Modelling in Systems Biology: An Introduction Applied Mathematics, University of Waterloo.
2. Ryan P. Thomas: Modern Regression Methods, (Wiley series in probability and statistics. Applied probability and statistics). "A Wiley-Interscience publication" (1997).
3. Invitrogen life technologies: Guide to Baculovirus Expression Vector Systems (BEVS) and Insect Cell Culture Techniques (2002).
4. Rohrmann, G.F: Baculovirus Molecular Biology, third edition, Department of Mi- crobiology Oregon State University (2013).
5. 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).
6. Zhang, You H. and Enden, Giora and Merchuk, José C: Insect cellsBaculovirus system:Factors affecting growth and low MOI infection (2005).
7. Zhang, Hong and Georgescu, Paul: The influence of the multiplicity of infection upon the dynamics of a crop-pestpathogen model with defence mechanisms (2014).
8. Zhang, You H. and Enden, Giora and Merchuk, José C: A model of the dynamics of insect cell infection at low multiplicity of infection (2005).
9. Invitrogen life technologies: Bac-to-Bac Baculovirus Expression System (2013).