Stochastic population balance modeling of influenza virus replication in vaccine production processes
Introduction
This work is concerned with modeling of influenza A virus replication in mammalian cell cultures. Focus is on vaccine production processes using adherent Madin-Darby canine kidney (MDCK) cells growing on the surface of microcarriers (Genzel et al., 2004).
Replication of influenza A viruses was thoroughly investigated on a mainly qualitative level and a number of excellent textbooks and reviews describe the details of intracellular virus dynamics (Flint et al., 2000, Ludwig et al., 1999, Nicholson et al., 1998, Whittaker et al., 1996). In vitro influenza A virus can infect many cell types that possess surface glycoproteins with sialic acid moieties (influenza A virus binding sites, or receptors; Julkunen et al., 2001). Following virus attachment to the surface receptors of cells, virions are incorporated by receptor-mediated endocytosis. In a sequence of steps viral genome is transferred to the nucleus, virus protein synthesis and virus genome replication starts and the virus life-cycle ends with the budding and release of newly generated virus particles. For a quantitative understanding of virus replication dynamics and the interaction of virus particles with their host cells mathematical modeling plays a crucial role (Nowak and May, 2000, Power and Nielsen, 1996, Reichl and Sidorenko, 2007). Moehler et al. (2005) presented a simple unstructured model for the replication of influenza A virus in MDCK cell cultures. A structured approach considering intracellular events of the influenza virus infection cycle was introduced by Sidorenko and Reichl (2004). Other approaches presented detailed models concerning the initial steps of infection, such as virus binding and endocytosis (Mittal and Bentz, 2001, Nunes-Correia and Ramalho-Santos, 1999). In all of these papers, virus spreading in populations of cells as well as differentiation of infected cells were not considered. This, however, may play a crucial role for the design and the optimization of vaccine production processes. In influenza vaccine production, for example, final yields of production strains often depend on the careful selection of infection conditions and multiplicity of infection (MOI).
Differentiation of MDCK cells infected with influenza A virus on a population level can be monitored experimentally using laser scanning microscopy as illustrated in Fig. 1 or flow cytometry as illustrated in Fig. 2. In both cases, the infection status of cells is correlated with the intracellular concentration of viral components (proteins M1 and NP) and, therefore, differences in fluorescence intensity (Fig. 1). In contrast to this uninfected cells show a very low degree of nonspecific fluorescence intensity.
To quantitatively reproduce flow cytometry data, a distributed population balance model is presented in this paper. In this model the differentiation of infected cells is explicitly taken into account giving rise to a distributed population of infected cells evolving in time. Similar approaches were reported for baculovirus infections of insect cells by Hu and Bentley (2000), Licari and Bailey (1992), and Power and Nielsen (1996). Considering segregated populations of infected cells allows the investigation of optimal conditions of virus production, in particular, to determine the initial number of infectious virus particles to be added after cell growth phase in bioreactors for obtaining maximum yields. Furthermore, Haseltine et al. (2005) developed a segregated population balance model of generic viral infections, incorporating both intracellular and extracellular events. In contrast to these studies, in the present paper a stochastic rather than a deterministic approach is applied. It takes into account the random nature of the process and, thus, allows a more realistic representation of process dynamics compared to deterministic models. A typical example is the initial phase of virus uptake when the number of virions is usually much lower than the number of cells in a bioreactor. In addition, with the stochastic approach, multidimensional distributions of the population of infected cells obtained by flow cytometry are easily taken into account. Although this is beyond the scope of the present paper it is an interesting perspective for more detailed modeling of the vaccine production processes in the future.
Lumped stochastic models without differentiation between infected cells using Monte Carlo (MC) techniques have been widely applied to study virus replication dynamics. Zhdanov (2004) presented a structured model for the replication of a generic virus in a single cell considering viral genome replication, synthesis of viral mRNA molecules and proteins, capsid assembly and virus release. A later study, additionally, considered the competition of host and viral mRNA molecules for the host translation machinery (Zhdanov, 2005). Ferreira et al. (2001) considered a two-dimensional cellular automaton model for the spread of herpes simplex virus infection in corneal tissue. Several stochastic models have been developed to simulate the dynamics of HIV (Tan and Wu, 1998, Wang et al., 2006). To our knowledge, no stochastic distributed population balance model of virus replication has been developed so far.
The purpose of the model is (a) to reproduce and interpret flow cytometry data for the intracellular dynamics of the population of infected cells, (b) to investigate the influence on MOI on virus yield, and (c) to develop strategies for the optimization of vaccine production processes.
Section snippets
Cell line and influenza virus
Adherent MDCK cells (ECACC, no. 84121903) were cultivated and infected as described previously (Genzel et al., 2004). Briefly, precultures for bioreactor cultivations were expanded in an incubator (5% , Heraeus, Germany) in T-flask or roller bottles in cell growth medium (GMEM, Gibco, #22100-093; 10% fetal calf serum, FCS, Gibco, #10270-106; 2 g/L peptone, International Diagnostics Group, #MC33). After 7 days, cells were washed several times in phosphate buffered saline
Model formulation
The model considers the interaction of virus and host-cells. Unlike the cellular automaton models of Beauchemin et al. (2005) and Ferreira et al. (2001), which imply that the infection spreads directly from one cell to another, the population of virions is considered explicitly. Let [cells/mL] denote the initial total concentration of cells and [virions/cell], [–] and [–] the concentration of free virions, infected and uninfected cells divided by . At the beginning of infection
Numerical procedure
Computations were carried out on a computer running SuSE Linux 8.2, by using a program written in the Fortran 77 programming language (GNU Fortran 3.3.1). The plots of simulation results were obtained with MATLAB (version 6.1).
Results
The set of model parameters used for simulations is presented in Table 1. The model correctly reproduces the experimental data for the virus dynamics at (Fig. 4a). It also allows the satisfactory reproduction of data sets for (Fig. 4b) and 3.0 virions/cell (Fig. 4c) with only slight differences between simulated and experimental results. Furthermore, the model predicts the dynamic behavior of the number of uninfected and infected cells (Fig. 5) and
Discussion
In this article a stochastic distributed model of influenza virus replication is presented, which considers not only the general behavior of the number of uninfected and infected cells, but also the intracellular dynamics of the differentiated population of infected cells. In contrast to the models of Hu and Bentley (2000) and Licari and Bailey (1992), which used the number of virions infecting a cell as an internal coordinate, the number of intracellular VEs is considered here. Therefore, the
Conclusions
In this study a simple stochastic distributed population balance model of influenza A virus replication was presented. The number of intracellular viral components expressed in virus equivalents was used as an internal coordinate. The model allows simulating the dynamics of virus and cell populations, estimating the amount of cellular resources required for virus replication, like binding sites, endosomes, free amino acids, comparing these requirements with the available resources in a cell,
Abbreviations
HA hemagglutinin, hemagglutination MC monte Carlo MOI multiplicity of infection ODE ordinary differential equation p.i. post infection RBC red blood cells VE virus equivalent
Notation
parameter of virus adsorption, dimensionless parameter of virus degradation, dimensionless fluorescence intensity per one VE, FU concentration of RBC, cells/mL F fluorescence intensity, FU fluorescence intensity of uninfected cells, FU j number of VEs per cell, VEs/cell J class number, dimensionless MOI MOI, virions/cell number of endosomes per cell, endosomes/cell number of virions released by a producing cell per hour, virions/(cell h) number of MC runs, dimensionless number of
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