Team:Tec-Monterrey/Outreach

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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:

CU	Non-infected Cell concentration.
C1	Infected Cell concentration.
V	Baculovirus concentration.
CP	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.
k1	Rate constant for the complex formation.
k-1	Rate constant for the complex degradation.
k2	Rate constant for the infection process.
K	Quasi-steady-state rate constant in the infection.
ΦV	Average virus production per infected cell.
ΦP	Average protein production per infected cell.
ΤIV	Time lapse between the infection of the cell and the beginning of the virus production.
ΤIP	Time lapse between the infection of the cell and the beginning of the protein production.
ΤD	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 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:

(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≡(k₁k₂)/(k_-1+k₂).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_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: