Baculovirus transit through insect cell membranes: A mechanistic approach

Highlights

• •A novel mechanistic model of the early stages of viral infection.
• •Excellent fit to experimental evidence.
• •The maximum number of virions that Sf9 cells can carry: 55 viruses/cell, is reported.
• •Cells that carry virions on their surface, in their interior, or both are distinguished.
• •Analytical mathematical solution renders satisfactory results.

Abstract

Baculovirus systems are used for various purposes, but the kinetics of the infection process is not fully understood yet. We investigated the dynamics of virion movement from a medium toward the interior of insect cells and established a mechanistic model that shows an excellent fit to experimental results. It also makes possible a description of the viral dynamics on the cell surface. A novel measurement method was used to distinguish between infected cells that carry virions on their surfaces, cells that carry virions in their interior, and those carrying virions both inside and on their surface. The maximum number of virions carried by a cell: 55 viruses/cell, and the time required for viral internalization, 0.8$h$, are reported. This information is particularly useful for assessing the infection efficacy and the required number of virions needed to infect a given cell population. Although our model specifically concerns the infection process of Sf9 insect cells by baculovirus, it describes general features of viral infection. Some of the model features may eventually be applicable in the studies towards palliation of the COVID-19 outbreak.

Nomenclature

Abbreviations

wt AcV1

wild type Anti-Baculovirus Envelope gp64 Monoclonal Antibody

wt AcV1-PE

wild type specific label for GP64

BV

baculovirus

BEVS

baculovirus expression vector system

MOI

multiplicity of infection

$\mathit{NC}$

nucleocapsid

TOI

time of infection

FCM

flow computed cytometry

GP64

major envelope protein of the BV

$\mathit{hpi}$

hours post infection

PFU

plaque forming unit

Sf9

(CCTCC-GDC0008) insect cell line used

SYBR

nucleic acid dye

FSC-H

forward scattering

SSC-H

side scattering

TCID50

50% tissue culture Infective Dose

PBS

phosphate-buffered saline

SYBR Green I

specific nucleic acid dye

SHIV

simian/human immunodeficiency virus

wt AcMNPV, CCTCC GDV114

virus used (wild type Autographa californica multicapsid nucleopolyhedrovirus)

Symbols

$A\left(t\right)$

virions attached to cell surface [NC/mL]

$C$

total cell concentration, [cell/mL]

$C_i$

concentration of infected cells, [cell/mL]

$C_{\mathit{id}}$

concentration of dark cells (Eq. (4)), [cell/mL]

$C_{\mathit{iw}}$

concentration of infected cells measured without permeabilization [cell/mL]

$C_{\mathit{ip}}$

concentration of infected cells measured with permeabilization [cell/mL]

$C_{\mathit{ipf}}$

final measured concentration of infected cells [cell/mL]

$C_0$

initial cell concentration, [cell/mL]

$C_v$

concentration of viable, or not infected cells, [cell/mL]

$I\left(t\right)$

number of internalized viruses at time $t$ , [cell/mL]

$k$

kinetic constant for virion removal from the medium, Eq. (6), [mL/$\left(NC\times \mathrm{h}\right)$]

$k_i^{\text{'}}$

kinetic constant for cell infection, [mL/$\left(N^{2}C\times h\right)$]

$k_i$

modified constant for cell infection, ($k_i^{\text{'}}\times$N), [mL/$\left(NC\times h\right)$]

$k_d$

proportionality constant for dark cell appearance, defined in Eq. (19), [1/$h$]

$k_g$

kinetic constant of biomass growth [1/$h$]

$k_w$

Proportionality constant in Eq. (19), [cell/$\mathit{NC}$]

N

maximum number of virions that can attach to a cell

$r_a\left(t\right)$

rate of change of the attached virions function $A\left(t\right)$ , [$\mathit{NC}$/$\left(\mathrm{m}\mathrm{L}\times h\right)$]

$r_i\left(t\right)$

rate of viral internalization, [$\mathit{NC}$/$\left(\mathrm{m}\mathrm{L}\times h\right)$]

$r_x\left(t\right)$

rate of virion removal from the medium [$\mathit{NC}$/$\left(\mathrm{m}\mathrm{L}\times h\right)$]

$t$

time, [$h$]

$t_e$

time of internalization, [$h$]

$V$

concentration of virions in the medium [$\mathit{NC}$/mL]

$V_f$

final measured concentration of virions, [$\mathit{NC}$/mL]

$V_0$

initial concentration of virions, [$\mathit{NC}$/mL]

$x\left(t\right)$

concentration of viruses removed from the medium, [$\mathit{NC}$/mL]

$\mathit{Wc}$

scaling factor, [mL/cell]

$\mathit{Wx}$

scaling factor [mL/$\mathit{NC}$]

$Z$

cost function

$\beta$

($N{C}_0$), Eq. (21)

$\delta$

$k\left(N{·C}_0-V_0\right)$, Eq. (21)

1. Introduction

The baculovirus (BV) expression vector system (BEVS) is of great interest for the production of recombinant proteins in a wide range of fields—from basic science research to the development of biomedical applications and the production of bio-insecticides (Felberbaum, 2015, Kost et al., 2005, Mehalko and Esposito, 2016, Shang et al., 2017). The production of recombinant proteins in BEVS is greatly affected by the characteristics and kinetics of the viral infection process—including, for instance, the multiplicity of infection (MOI), time of infection (TOI), cell cycle, cell line selection, and culture state (Ge et al., 2014, Kioukia et al., 1995, Lecina et al., 2006, Licari and Bailey, 1992, Monteiro et al., 2017, Monteiro et al., 2014, Monterio et al., 2016, Nwe et al., 2006, Saito et al., 2002, Zhang et al., 2005, Enden et al., 2005), but the early events of viral infection, such as the attachment of virions to their receptors and the kinetics of their entry into the cell, are still not entirely understood. This gap hampers the development of efficient and specific genetically engineered proteins and gene therapy vectors based on BV – a goal that has lately gained much interest (Licari and Bailey, 1992, Felberbaum, 2015, Kost et al., 2005, Mehalko and Esposito, 2016, Shang et al., 2017). Since baculovirus was used for efficient gene transfer to mammalian cells in the 1990s (Boyce and Bucher, 1996, Hofmann et al., 1995), several effective baculovirus-based gene therapy vectors had been created and tested, demonstrating favorable therapeutic efficacy in laboratory and preclinical phase studies (Ono et al., 2018, Mansouri and Berger, 2018, Makkonen et al., 2015, Van Oers et al., 2015). Although the entry processes and the mechanisms involving BVs infection of insect cells or transduction of mammalian cells remain poorly understood, it has been shown that a similar mechanism may mediate GP64 binding to permissive host cells (Blissard and Theilmann, 2018, Oflynn et al., 2012, Dong and Blissard, 2012) and that BVs enter insect and mammalian cells through a clathrin-mediated endocytosis (Kataoka et al., 2012, Long et al., 2006). Thus, comprehensively characterizing the kinetics underlying the early steps of BVs infection of insect cells is crucial both for improving the production of recombinant proteins in insect cells and for designing therapy vectors and enhancing gene transduction efficacy in therapeutic studies.

Mathematical modeling and simulations may serve as effective predictive tools in pursuing this goal (Licari and Bailey, 1992, Dee and Shuler, 1997, Zhang et al., 2005, Enden et al., 2005), as they allow dynamical simulation of the process and estimating variables that are hard to measure directly, such as the rate of virion internalization after its attachment to the cell. Quantitative models were also employed to simulate the infection dynamics of infectious diseases. Padmanabhan and Dixit (2011) constructed a kinetic model of hepatitis C virus (HCV) which accounted explicitly for the dependence of HCV entry target cells on CD81 expression. Iwami et al. (2012) optimized a mathematical model to describe the kinetics of the simian/human immunodeficiency virus (SHIV) infection which improved the understanding of SHIV and human immunodeficiency virus type-1 pathogenesis. Such models, and the virtual experiments that they facilitate, are useful for interpreting the mechanistic aspects of the viral infection process and may save time, manpower, and physical resources in predicting their outcomes. The main purpose of the current study was to construct such a model, focusing on the early stages of a high-MOI BV infection of insect cells (Zhang et al., 2005, Wong et al., 1996), based on and tested against experimental results.

Our model embodies an interdisciplinary integration of microbiology, biology, engineering and mathematics and sets a basic mechanistic approach to the early stages of the infection process. It disregards the effect of non-infective viral mutants which might enter the cell without infecting it. It also does not address the effectiveness of infection, which may vary depending on the elapsed storage time, mainly due to viral aggregation within the viral stocks (Jorio et al., 2006, Shen et al., 2002). As we used the same storage time in all experiments, particle aggregation did not affect the measured results. Consequently, we disregarded storage time in our model. Based on detailed and accurate measurements, our model enhances our understanding of the early BV infection process and can be potentially extended to describe the early stages of other infection processes, such as gene delivery by viruses in mammalian cells. It also provides a basis for modelling the production of foreign proteins. The present model is a first step towards the long-term goal of establishing a more elaborated model of the production process, aimed at saving time, manpower, and physical resources.

2. Materials and methods

2.1. Cells and virions

The insect cell line used in this study was Sf9 (CCTCC-GDC0008). The cells were routinely maintained in 25-cm² culture T-flasks (Corning, NY) at 27 °C in SF-SFM medium (Wmbio, Shuzhou, China). For suspension cultures, the Sf9 cells were incubated at 100 rpm in triplicate in 250 mL Erlenmeyer flasks containing 60 mL of cell suspension and supplemented with 0.05% (w/v) Pluronic F-68 (Sigma, St. Louis, MO). Wild-type Autographa californica nucleopolyhedrovirus (wt AcMNPV, CCTCC GDV114) was kindly provided by Professor Qi Yipeng (Wuhan University, China).

2.2. Experimental procedure

Cells from the T-flasks were transferred to suspension cultures in Erlenmeyer flasks to decrease the lag phase in subsequent stages. When the cells in the pre-culture of the first Erlenmeyer flask reached the late exponential phase (after approximately 96$h$, depending on the initial cell density), the cultures were transferred to 12 Erlenmeyer flasks (control, i.e., mock-infected, and three different MOIs, all in triplicates) and diluted with fresh medium to a cell concentration of approximately 1.2 × 10⁶ cells/mL. After 12$h$, these new cell cultures were considered to be in their early exponential phases, reaching a density of approximately 1.7 × 10⁶ cells/mL. At this point, the cultures were infected with MOIs of 114, 219, or 421 BVs per cell. Note that the units of MOI used here are the ratio of viral particles to cells, as counted by a flow cytometer (Becton Dickinson, San Jose, TX), while, traditionally, it is measured as plaque-forming units (PFUs) per cell or as TCID₅₀/cell. The ratio of FCM counts to end-point dilution assay titers ranges from 1.0 to 9.4, with an average of 3.7 and a standard deviation of 2.4 (Jorio et al., 2006). Thus, the MOIs of 100, 200 and 400 BVs/cell used in present experiments were equivalent to 25, 50 and 100 TCID₅₀/cell.

2.3. Analytical methods

2.3.1. Flow cytometric detection of baculovirus titer

An improved method of FCM was used to quantify baculovirus titer, as described previously (Jorio et al., 2006, Shen et al., 2002, Brussaard, 2004). Prior to the FCM analysis, in the final 1-mL phosphate-buffered saline (PBS, pH 8.0) solution, virion samples were fixed with 0.1% (w/v) paraformaldehyde for 30 min at 4 °C and then stained with a specific nucleic acid dye, SYBR Green I, at a commercially available concentration (1 × ${10}^{-4}$) for 10 min at 80 °C. Yellow-green fluorescence microspheres (1 μm in diameter, Molecular Probes, Eugene, OR) were added (as an internal reference to the BV) to the control sample (Brussaard et al., 2000). All samples were analyzed using a FACS Calibur flow cytometer (Becton Dickinson, San Jose, TX) and the threshold was set on green fluorescence (FL1-H) to eliminate the background interference.

Comparing the number of virion particles with yellow-green fluorescent microspheres, the BV titer was determined by FCM using CellQuest software (Becton Dickinson, San Jose, TX). To avoid the coincidence of viral particles (i.e., two or more particles simultaneously being within the sensing zone), the samples were diluted such that the total event rate was kept below 500 events/s.

2.3.2. Determination of infected and uninfected cells

Cells were counted with a hemocytometer using trypan blue (0.5% w/v) exclusion to distinguish dead from live cells. To experimentally monitor the early stages of the infection process and detect infection in cells to which virions attach, we directly tracked GP64—the major envelope protein of the BV, which plays a crucial role in the early stages of infection. GP64 is essential for the attachment of virions to cells in a culture and for their transport between cells in infected tissues (Monsma et al., 1996), as it is involved in the binding of the virions to host cell receptors (Hefferon et al., 1999) and acts to locally reduce the membrane pH, which triggers membrane fusion. Once the virus has entered the cell by endocytosis, GP64 is necessary for stripping the nucleocapsid from its capsid proteins and for releasing the virion into the cytosol (Kingsley et al., 1999, Markovic et al., 1998, Plonsky et al., 1999). To label GP64 specifically and thereby detect only infected cells, we used the monoclonal antibody AcV1, which binds to GP64 but does not reduce its ability to attach to cells (Kataoka et al., 2012, Hohmann and Faulkner, 1983, Volkman and Goldsmith, 1985, Dhungel et al., 2013, Granio et al., 2009). To differentiate between virions that are attached to the cell surface from those that have already entered it, we either permeabilized the cells or left them intact (see methods) assuming that AcV1 labeled BVs attaches to the cell surface in both permeabilized and non-permeabilized cells, but it penetrates only into permeabilized cells. Thus, cells can be categorized according to the following criteria: cells carrying viruses on their surface only, cells carrying viruses both on their surface and in their interior, and cells carrying virions only in their interior (where in the latter all the attachment sites have been used and all the virions have made their ingress into the cells). Two types of measurements were done on each sample, rendering different readings. Fig. 1 shows diagrammatically which cells are detected in each reading. AcV1, conjugated with a fluorescent PE.

Fig. 1.

Three cells in the different states considered are shown: A cell carrying viruses on its surface only, a cell carrying viruses both on its surface and in its interior, and a cell carrying virions only in its interior. Two types of measurements were done on the same sample producing different readings: (A) With cell permeabilization (Surface as well as internalized viruses are detected) and (B) Without cell permeabilization (Only virions on the surface are tagged).

(AcV1-PE), was used specifically against the GP64 envelope glycoprotein of the BVs, thereby labeling attached and, in the case of permeabilized cells, internalized viruses (Jacobberger et al., 1986). The infected cells were then distinguished from uninfected cells using FCM analysis. A cell was considered “infected” once the first virion attached to its surface. Thus, we identified infected cells by their high antibody fluorescence, while all other cells were considered to be uninfected (and are not distinguishable from the “dark cells” described in the text). The results of the FCM measurements of the concentrations of infected cells were analyzed by CellQuest software (Becton Dickinson, San Jose, CABD) and interpreted using the subtraction method (Sladek and Jacobberger, 1993).

The samples were taken along time from cultures infected with MOIs of 114, 219, or 421 BVs/cell (see “Infection” above), and these samples were placed into 5-mL tubes by centrifugation (5 min at 1200 rpm). Then, the supernatants were aspirated, and the remaining cells were fully re-suspended by adding 3 mL of PBS (pH: 7.2) with vortex. After washing the cell samples twice, the cell samples were fixed by adding 100 μL of a paraformaldehyde solution (1% v/v final concentration) and vortexing, then stored at 4 °C in the dark for 30 min. To count the number of permeabilized infected cells, the samples were washed once with 3 mL PBS (pH: 7.2) and then added with 100 μL of a permeabilization reagent (reagent B, Fix & Perm kit, Invitrogen, Grand Island, NY) and 5 μL of AcV1-PE (eBioscience, Santiago, CA). To count the number of non-permeabilized infected cells, the samples were stained with 5 μL AcV1-PE without adding the permeabilization reagents. After a 30 min incubation at 4 °C in the dark, the cell samples were washed again. Finally, to obtain the optimal fluorescence signal, the cell samples were examined by FCM within 30 min. The infected and uninfected cells were detected using the scatter plot with forward scattering (FSC-H) versus side scattering (SSC-H), and then the ratio of infected to total cells was analyzed by the BD CellQuest software. Control cell samples were treated using the same steps, but without adding AcV1-PE (data not shown).

3. Results, modeling and discussion

3.1. Attachment of virions to cells

We began by establishing the capacity of Sf9 cells to carry Autographa californica multicapsid nucleopolyhedrovirus (AcMNPV) – an important reference for MOI optimization. We detected the virion titer by flow cytometry (FCM), using the fluorescent dye SYBR Green I to stain the nucleic acids of the virions (Jorio et al., 2006, Shen et al., 2002, Brussaard, 2004, Brussaard et al., 2000). The Sf9 cells were infected with AcMNPV at MOIs of 114, 219, or 421 BVs/cell, and the obtained virion and cell profiles are shown in Fig. 2 . It should be kept in mind that generally baculovirus infections are reported in units of TCID₅₀/cell. In such case the usual range of MOI in a Baculovirus infection is 10–50. In the present case we are using FCM measurements. The ratio of FCM counts to end-point dilution assay titers ranges from 1.0 to 9.4, with an average of 3.7 and a standard deviation of 2.4 (Jorio et al., 2006). Thus, the MOIs of 100, 200 and 400 BVs/cell used in present experiments were almost equivalent to 25, 50 and 100 TCID₅₀/cell.

Fig. 2.

Main experimental data. Changes in the cell density (empty data points, left ordinate) and virion titer (filled data points, right ordinate), from 0 to 9 $\mathit{hpi}$, at MOIs of 114 (a), 219 (b), and 421 (c) BVs/cell. Cell growth is observed only during the first 4 h post infection. Each data point represents the mean ± SD of three independent experiments.

The maximum number of virions that can attach to a cell, N, is defined here as:

$$N=⌊V\left(0\right)-V_f⌋/C_{\mathit{ipf}}$$ (1)

where it is assumed that $\mathrm{M}\mathrm{O}\mathrm{I}\ge N$.

Here $V_f$ is the final measured concentration of virions in the medium, $C_{\mathit{ipf}}$ is the final measured concentration of infected cells 9 h post-infection ($\mathit{hpi}$), as shown in Fig. 2 and Table 1 . The sub-index p indicates cell samples which were treated with permeabilizing reagents., $C_{\mathit{ipf}}$ is also an upper bound of the concentration of the infected cells at any time point. For MOIs = 114, 219, and 421, we found N = 52.8, 55.3, and 58 attachment sites per cell, respectively. In principle it is accepted here that N is a property of the cell, which means that a cell has finite attachment sites that should be identical for all cells and under all conditions. In our experimental results we found a small difference between the three MOIs, with a standard deviation lower than 5% for a mean N of 55, which may be considered as experimentally acceptable. Thus, we suggest using $N$ = 55 for Autographa californica at any MOI. When this model is applied to anti-tumor therapy with baculovirus vectors, N would be the basis for calculating the upper limit of viral concentration required to deliver a foreign gene to target cells. The optimal dosage per cell should be between zero and N. The desired dose per cell (MOI) can be defined as that which maximizes the expectancy of treated cells with minimal costs.

Table 1.

Experimental data on free viruses in the medium and the infected cell concentrations for each MOI.

	MOI: 114 (C0 = 1.69 × 106 cells mL−1)		MOI: 219 (C0 = 1.65 × 106 cells mL−1)		MOI: 421 (C0 = 1.65 × 106 cells mL−1)
$\mathit{hpi}$	Infected cell titer	Virus titer	Infected cell titer	Virus titer	Infected cell titer	Virus titer
h	(×106 cells mL−1)	(×108 BVs/mL)	(×106 cells mL−1)	(×108 BVs/mL)	(×106 cells mL−1)	(×108 BVs/mL)
0	0.000	1.912	0.000	3.611	0.000	6.949
1	0.992	1.377	1.060	3.064	1.080	6.277
2	1.255	1.197	1.323	2.862	1.330	6.142
3	1.463	1.057	1.504	2.777	1.563	6.008
4	1.544	1.003	1.607	2.69	1.645	5.931
5	1.586	0.986	1.653	2.667	1.719	5.938
6	1.559	1.014	1.634	2.663	1.694	5.965
7	1.613	0.969	1.696	2.698	1.707	5.934
8	1.620	1.012	1.674	2.671	1.705	5.947
9	1.704	1.011	1.700	2.67	1.734	5.944

3.2. Kinetic modelling

3.2.1. Growth kinetics of non-infected cells

In the initial stage of the study, we characterized the kinetics of non-infected Sf9 cells, which were grown under the same conditions as those used in the infection experiments (see Methods in the Attachments), but without BV infection. It was found that the simple first-order growth rate fits satisfactorily the collected data. The use of more sophisticated kinetic forms is therefore unnecessary. The kinetic expression for the growth of non-infected cells is:

$$\mathit{dC}/dt=k_{g}C$$ (2)

$$C\left(0\right)=C_0$$

The solution of this equation is:

$$C\left(t\right)=C_{0}exp\left(k_g·t\right)$$ (3)

where $C$ is the cell concentration, $t$ is time, and $k_g$ is the growth rate factor. Regression of expression (3), based on all experimental data points in a 60$h$ experiment, yielded $k_g=0.022\left[h^{-1}\right]$. This kinetic rate constant was used in the mathematical model described below. The value is lower than data reported previously for these cells, but it should be kept in mind that anyway the growth rate of the cells is sensibly lower that the infection rate. This consideration is reinforced further by the success of the assumption of constant cell concentration rate presented in Section 3.5.

3.2.2. Dynamics of the early stages of infection

The dynamics of the early stages of the infection process in Sf9 cells are shown in Fig. 3 for MOIs = 114, 219, and 421. The percentage of infected cells, as evaluated without permeabilization, $C_{\mathit{iw}}$, increased until 4 $\mathit{hpi}$, reached a maximum of 79%, and then slowly decreased to 65% at 9 $\mathit{hpi}$. This maximum will be explained below in terms of the balance between virus attachment and virus internalization. In contrast, the percentage of the infected cells, as evaluated when permeabilizing reagents were used in the analysis, $C_{\mathit{ip}}$, increased monotonically, reached a maximum of 97%, and then, either remained unchanged, or moderately increased to a plateau. This effect was observed with only minor differences between the three examined MOIs (although higher MOIs resulted in a slightly higher $C_{\mathit{ip}}$), suggesting that almost all cells were infected by or before approximately 6 $\mathit{hpi}$. $C_{\mathit{ip}}$ denotes the number of infected cells carrying virions on their surface and/or their interior. $C_{\mathit{ip}}$ is an essential indicator in pathopoiesis of viruses. The higher $C_{\mathit{ip}}$, the severe the incidence of diseases. In the early stage of virus infection, the rate of change of $C_{\mathit{ip}}$ is similar to the transduction dynamics of target cells with viral vectors in an ex vivo therapy. In the same way the transduction efficiency of viral vectors plays a fundamental role in improving the therapeutic efficacy of a gene therapy program (Asad et al., 2017, Kalesnykas et al., 2017, Sinn et al., 2017, Espirituramirez et al., 2018); $C_{\mathit{ip}}$ could be considered a key parameter in gene therapy treatment based on viral vectors.

Fig. 3.

The infection process. Evolution of the concentrations of infected cells as detected in non-permeabilized cells (empty data points) and permeabilized cells (filled data points), from 0 to 9 $\mathit{hpi}$, at MOIs of 114 (red), 219 (black) and 421 (blue) BVs/cell. The number of infected cells increases monotonically to approximately 100% when measured on permeabilized cells but only reaches approximately 70% when measured on cells without permeabilization. Each data point represents the mean ± SD of three independent experiments. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The difference between the values of $C_{\mathit{ip}}$ and $C_{\mathit{iw}}$ corresponds to the apparition of a new type of cells: those that have already internalized the maximum possible number of virions, N. In an FCM analysis, these cells show a very weak fluorescence, similar to that of uninfected cells, because they do not internally carry an AcV1 conjugated with a fluorescent phycoerythrin (PE). Therefore, it is difficult to distinguish between them and uninfected cells, unless they are treated with permeabilizing agent – which would make them strongly fluorescent. If we denote their concentration as $C_{\mathit{id}}$, then

$${C_{\mathit{id}}=C}_{\mathit{ip}}-C_{\mathit{iw}}$$ (4)

The profiles shown in Fig. 3 indicate that $C_{\mathit{id}}$ remains very low during the first hour post infection, but then begins to increase; this slow initial response (as manifested by the coincidence of $C_{\mathit{ip}}$ and $C_{\mathit{iw}}$) is presumably because either none or very few of the infected cells have internalized virions to their full capacity (N) at this stage. $C_{\mathit{id}}$ is expected to continue to increase and eventually reach values close to $C_{\mathit{ip}}$ as $C_{\mathit{iw}}$ reaches very low values due to full internalization of all the virions that had attached to the cell surface. $C_{\mathit{id}}$ indicates the amount of cells with fully internalized virions, and the change in $C_{\mathit{id}}$ reflects the infection stage. The higher $C_{\mathit{id}}$, the more advanced is the infection stage.

3.2.3. Modeling the dynamics of the early stages of infection

To model the dynamics of the early infection process, we used the following variables: x [virions/mL], the concentration of virions removed from the medium; $C_v$ [cells/mL], the concentration of uninfected cells; and $C_i$ [cells/mL], the concentration of infected cells. We considered the total concentration of cells in the system to be equal to the sum of concentrations of the uninfected and infected cells:

$$C=C_v+C_i$$ (5)

The infection process is viewed here as two serial steps: the first is the virion–cell encounter and its attachment to the cell surface, as defined in Eqs. (6), (9); the second step consists of the internalization of the virion into the cell, assumed here to last a certain time, $t_e$, as reflected in Eqs. (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (19), (21). In addition, we assume that (1) all attachment sites are identical and the behavior of all virions is essentially identical; (2) the internalization time $t_e$ is identical for all cell-virion couplings; (3) after infection, virions continue to attach to cells as long as the saturation point (N) is not reached; (4) each attachment site can be used only once; (5) the time between consecutive virion attachments to the same cell is shorter than $t_e$. The latter bears the simplifying assumption that after the first virion attachment to each untreated cell, it will appear as “infected” until all its N attachment sites are used and all its surface virions internalized.

3.2.4. Kinetics of virion attachment

One of the simplest ways to depict macroscopically the mechanism of virion attachment to cells is to assume that the rate of attachment is proportional to both the number of free virions and the number of unused attachment sites on the cell.

Denote the initial virion concentration in the medium as $V_0$ and the number of virions removed from a unit volume of the medium (via attachment to cells) as $x\left(t\right)$. The concentration of free virions in the medium is therefore, $V=V_0-x$. Assuming that the rate of virus attachments is proportional both to free virus concentration and to the concentration of attachment sites, the differential equation describing the adsorption rate of virions can be written as:

$$\mathit{dx}/dt=k·\left[V_0-x\left(t\right)\right]·\left[N·C\left(t\right)-x\left(t\right)\right]=r_{x}0<t$$ (6)

where $C\left(t\right)=C_v\left(t\right)+C_i\left(t\right)$ is the total cell concentration $t$ hours after TOI, and

$$\mathrm{a}\mathrm{t}t=0\left(=TOI\right),x\left(0\right)=0.$$ (7)

Eq. (6) therefore states that the rate of virus attachment is proportional both to the number of free virus and to the number of free attachments sites. In the equation, $k$ [mL/(attachment sites × $h$)] is the attachment kinetic constant, and the number of free attachment sites per unit volume in the cell culture $t$ hours after TOI equals $\left[N·C\left(t\right)-x\left(t\right)\right]$.

3.2.5. Cell infection as detected in permeabilized cells

The material balance on the infected cells in the system yields:

$$d{C}_{\mathit{ip}}/dt=k_i^{\text{'}}·\left(V_0-x\right)·N·C_{v}0<t$$ (8)

which assumes that rate of increase of infected cells concentration is proportional to both the free virus and viable cells concentrations.

In Eq. (8), $C_{\mathit{ip}}$ denotes the concentration of infected cells, treated with permeabilizing reagents, while $k_i^{\text{'}}$ is the kinetic constant of infection. Defining $k_i=k_i^{\text{'}}·N$, we obtain

$$d{C}_{\mathit{ip}}/dt=k_i·\left(V_0-x\right)·C_{v}0<t$$ (9)

where $k_i=k_i$ [mL/(number of virions × $h$)].

$$\mathrm{A}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{I},C_{\mathit{ip}}\left(0\right)=0$$ (10)

A balance on the pool of viable cells will include a cell growth term, already shown in Eq. (1) to be a first-order process with a kinetic constant $k_g$, and a second term for viable cell depletion by infection:

$$d{C}_v/dt=k_g·C_v-k_i·\left(V_0-x\right)·C_v$$ (11)

$$\mathrm{A}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{I},C_v\left(0\right)=C\left(0\right)=C_0$$ (12)

where $C_0$ for each MOI is shown in Table 1.

Eq. (11) indicates that the accumulation of viable cells is balanced by the difference between the generation of new cells and the depletion of the pool of viable cells due to infection. Eqs. (5), (6), (7), (8), (9), (10), (11), (12) were solved to obtain concentration profiles of the virions removed from the medium, $x$, infected cells, $C_i$, and viable cells, $C_v$. The results of the complete model are detailed below.

3.2.6. Cell infection as detected in non-permeabilized cells

In this case, cells containing only internalized virions cannot be distinguished from uninfected cells, since AcV1 conjugated with a fluorescent PE cannot enter into non-permeabilized cells. Assuming that the internalization time, $t_e$, is identical for all virions, we can calculate $A\left(t\right)$, i.e., the number of virions (per unit volume of the medium) attached to cell surfaces (and, therefore, detectable) at any given time. This value is equal to $x\left(t\right)$ for $t<t_e$, and is equal to the difference between the number of virions that were attached to the cells between $\left(t-t_e\right)$ and $t$ for $t_e\le t$, since virions that arrived earlier than $\left(t-t_e\right)$ have already been internalized and cannot be detected. Therefore,

$$A\left(t\right)=\left\{\begin{array}{cc}x\left(t\right)& 0\le t<t_e\hfill \\ x\left(t\right)-x\left(t-t_e\right)& t_e\le t\hfill \end{array}\right)$$ (13)

Denoting by $I\left(t\right)$ the number of internalized viruses at time $t$ per unit volume of the medium, we obtain:

$$I\left(t\right)=\left\{\begin{array}{cc}0& 0\le t<t_e\hfill \\ x\left(t-t_e\right)& t_e\le t\hfill \end{array}\right)$$ (14)

The rate of change of $A\left(t\right)$, $r_a\left(t\right)$ , at time $t$ , is:

$$r_a\left(t\right)=d/dt\left\{\begin{array}{cc}x\left(t\right)& 0\le t<t_e\hfill \\ x\left(t\right)-x\left(t-t_e\right)& t_e\le t\hfill \end{array}\right)$$ (15)

And, following, Eq. (6):

$$r_a\left(t\right)=\left\{\begin{array}{cc}{r}_x\left(t\right)& 0\le t<t_e\hfill \\ r_x\left(t\right)-r_x\left(t-t_e\right)& t_e\le t\hfill \end{array}\right)$$ (16)

The rate of viral internalization, $r_i\left(t\right)$ , at a time $t$ , is:

$$\begin{array}{c}{r}_i\left(t\right)=dI\left(t\right)/dt=d/dt\left\{\begin{array}{cc}0& 0\le t<t_e\hfill \\ x\left(t-t_e\right)& t_e\le t\hfill \end{array}\right)\\ =\left\{\begin{array}{cc}0& 0\le t<t_e\hfill \\ k·\left[V_0-x\left(t-t_e\right)\right]·\left[N·C\left(t-t_e\right)-x\left(t-t_e\right)\right]& t_e\le t\hfill \end{array}\right)\end{array}$$ (17)

The next step is to associate the virion attachment and internalization processes to the measured infected cell concentrations, $C_{\mathit{ip}}$ and $C_{\mathit{iw}}$. This association, expressed in Eq. (4), can be obtained by using the difference between the two above mentioned variables, $C_{\mathit{id}}$, which represents the concentration of the infected non-permeabilized cells that have already internalized all the attached virions. The relationship between virion internalization and $C_{\mathit{id}}$ is by no means simple. The internalization rate, $r_i$, is the result of a very large number of attachment events of viruses to cells. At a given time, the states of these cells vary considerably, ranging from non-infected (a very small fraction) to infected cells that already have attached various numbers of virions, up to N. The darkening of any detected cell occurs only after the cell has accumulated N attachment events and a time t_e has passed since the last such event. Considering the multiple possibilities in the process of occupying a cell to its maximum capacity N, two extreme patterns can be imagined: in one extreme, the distribution of virions among the cells is completely homogeneous: at any point in time, the number of virions in each cell is identical and, consequently, no cell darkening is observed until the very final stages of the process. In such a case, the profile of $C_{\mathit{id}}\left(t\right)$ would be zero until the very end of the process. On the other extreme, the virions are distributed among the cells in an orderly fashion, such that attachment to a new cell does not occur before the previously infected cell is fully saturated, i.e., at any time $t⩾t_e,C_{\mathit{id}}\left(t\right)=⌊\frac{x\left(t-t_e\right)}{N}⌋$.

Where $⌊x⌋$ denotes the “whole value function” (also called the “floor function”), which returns the largest integer that is smaller than or equals to x (rounded down). It follows that, for any $t\ge t_e$,

$$C_{\mathit{id}}\left(t\right)=C_{\mathit{ip}}\left(t\right)-C_{\mathit{iw}}\left(t\right)\le \lfloor \frac{x\left(t-t_e\right)}{N}\rfloor \le \frac{x\left(t-t_e\right)}{N}$$ (18)

A crude approximation to $C_{\mathit{id}}\left(t\right)$ is made here by assuming that the concentration of darkened cells is proportional to $x\left(t-t_e\right)$ for $t\ge t_e$. Noting that, for $t<t_e,C_{\mathit{iw}}\left(t\right)=C_{\mathit{ip}}\left(t\right)$, we obtain:

$$C_{\mathit{id}}\left(t\right)=C_{\mathit{ip}}\left(t\right)-C_{\mathit{iw}}\left(t\right)=\left\{\begin{array}{c}0\\ k_{w}x\left(t-t_e\right)\end{array}\right)\begin{array}{c}0\le t<t_e\\ t_e\le t\end{array}$$ (19)

3.3. Calculations

The solution of this model, represented in equations (6–19), was fitted to the experimental data and the kinetic parameters k and $k_i$, $k_w$, and $t_e$ were determined. The optimization was accomplished using the MATLAB and Polymath codes. The optimized parameters are shown in Table 2 .

Table 2.

Optimal values of the model parameters for each MOI.

MOI $[NC/cell]$	$t_e$$[h]$	$k_w$$[cell/NC]$	$k_i$$[cc/\left(NC\times h\right)]$	$K$$[cc/\left(NC\times h\right)]$
114	0.8	0.0055	5.1 × 10−9	5.3 × 10−9
219	0.79	0.0048	2.9 × 10−9	2.7 × 10−9
421	0.67	0.0045	1.6 × 10−9	1.6 × 10−9

As the table indicates, both $k$ and $k_i$ change with MOI. The similarity of the values of $k$ and $k_i$ at each MOI indicates that the sensitivity of the system to the instantaneous number of available attachment sites is weak, as we further discuss below (in the “constant $C$” model). In contrast, both $k_w$ and $t_e$ show a very small dependence on MOI. This indicates that once initiated, the infection process is not influenced by the viral concentration in the medium.

Eqs. (6), (7), (8), (9), (10), (11), (12), (13) were solved numerically using MATLAB and the parameters were evaluated with respect to the measured data in the time interval $0\le t\le 3h$. Fig. 4 shows the measured and calculated values of $x$, i.e., the number of virions bound to the cells along time, and Fig. 5 shows the predicted and measured profiles of $C_{\mathit{ip}}$, $C_{\mathit{iw}}$, and $C_v$ for each of the three MOIs. An excellent fit was found between the model and the experimental data in all cases, supporting our basic assumptions and the mechanistic approach. In Figs. 6 a and 6b, we used the calculated parameters to extrapolate the numerical solution for MOI = 219 to $t=9h$. The solution compares well with the experimental results even at this extended range. Similar results were obtained for MOIs = 114 and 421 (data not shown).

Fig. 4.

Modeling results. Comparison between the experimentally measured data points of $x$, the virions bound to the cells (either on their surface or internalized), and the corresponding predicted values (continuous lines) at MOIs of 114 (a), 219 (b), and 421 (c). The model offers a satisfactory fit to the experimental findings. Each data point represents the mean ± SD of three independent experiments. Panel (d) demonstrates an extrapolation of the numerical solution (for MOI = 219) to $t=9h$. Solid lines represent the complete model (Eqs. (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20)) and dashed lines represent the model predictions, assuming a constant cell concentration.

Fig. 5.

Modeling results. Comparison of the experimentally measured (points) and calculated (continuous lines) values of $C_{\mathit{ip}}$ (permeabilized infected cells), $C_v$ (viable cells) and $C_{\mathit{iw}}$ (non-permeabilized infected cells) at MOIs of 114 (a), 219 (b), and 421 (c). The model fits satisfactorily to the experimental findings. Each data point represents the mean ± SD of three independent experiments. Panel d demonstrates an extrapolation of the numerical solution (for MOI = 219) to $t=9h$. Solid lines represent the complete model (Eqs. (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20)) and dashed lines represent the model predictions assuming a constant cell concentration.

Fig. 6.

Virus dynamics. (a) Temporal profiles of virions bound to the cells, $x\left(t\right)$; virions on the surface of the cells, $A\left(t\right)$, and internalized viruses, $I\left(t\right)$. (b) Comparison of the rate of virion removal from the medium, $r_x$; the rate of virus internalization, $r_i$; and the rate of change of virions found on the surface of the cells, $r_a$, based on the approximation in Eq. (20).

3.3.1. Parameter evaluation and sensitivity analysis

Eq. (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19) were solved numerically and the obtained solution was used to evaluate the model parameters ${k,k}_i,k_w,{\mathrm{a}\mathrm{n}\mathrm{d}t}_e$ by minimizing the sum of the squares of the differences between the model predictions and the respective experimental measurements of $C_{\mathit{ip}}$, $C_{\mathit{iw}}$, and $x$ in the interval $0<t\le 3h$. Since the virion and cell concentrations differ by three orders of magnitude, the convergence criterion was applied with respect to the scaled variables. The variable $x$ was scaled with respect to $(N\times C_0)$, which is approximately the maximal possible number of attachment sites on all the cells (see Eq. (22) below). Each of the cell concentrations $C_{\mathit{ip}}$ and $C_{\mathit{iw}}$ was scaled with respect to $C_0$.

Scarcity of experimental data hindered the implementation of statistical analysis to determine confidence intervals of the evaluated parameters. Instead, the sensitivity of the model to parameter perturbations was analysed by evaluating the relative change in the cost function in response to parameter perturbations. The values of ${k,k}_i,k_w,{\mathrm{a}\mathrm{n}\mathrm{d}t}_e$ were perturbed sequentially downwards and upwards from their base-case values. The relative effects of these perturbations on a cost function,

$$Z=\sqrt{\sum _{i}Wc{\left(C_{\mathit{ip}}-\overline{C_{\mathit{ip}}}\right)}^2+\sum _{i}Wc{\left(C_{\mathit{iw}}-\overline{C_{\mathit{iw}}}\right)}^2+\sum _{i}Wx{\left(x_i-\overline{x_i}\right)}^2}$$

are demonstrated in Table 3 for MOI = 114.

Table 3.

Sensitivity analysis.

$k\left[mL/\left(NC\times h\right)\right]$	$k_i\left[mL/\left(NC\times h\right)\right]$	$k_w\left[cell/NC\right]$	$t_e$ [h]	Relative change in cost function (%)
5.3 × 10−9	5.1 × 10−9	0.0055	0.8	0 (base values)
5.3 × 10−9 + 0.3 × 10−9	5.1 × 10−9	0.0055	0.8	2.2
5.3 × 10−9 − 0.3 × 10−9	5.1 × 10−9	0.0055	0.8	2.15
5.3 × 10−9	5.1 × 10−9 + 0.3 × 10−9	0.0055	0.8	2.2669
5.3 × 10−9	5.1 × 10−9 − 0.3 × 10−9	0.0055	0.8	1.6424
5.3 × 10−9	5.1 × 10−9	0.0055 + 0.0005	0.8	9.4741
5.3 × 10−9	5.1 × 10−9	0.0055 − 0.0005	0.8	14.0371
5.3 × 10−9	5.1 × 10−9	0.0055	0.8 + 0.1	1.3534
5.3 × 10−9	5.1 × 10−9	0.0055	0.8 − 0.1	1.7015

Z was minimized with respect to the parameters $k,k_i,k_w$, and $t_e.$ Here the barred variables are the experimentally measured variables, $i$ is the time index; $\mathit{Wc}={C_0}^{-1}$ and $\mathit{Wx}=1/\left(C_0\times N\right)$ are the scaling factors. Clearly, the model is sensitive to changes in $k_w$. On the other hand, the general agreement between the fitted values and the respective experimental values, suggests that the assumptions on which the mechanistic model is based on are satisfactory. In particular the homogeneity of $t_e$ also supports this assertion. However, both $t_e$ and $k_w$ are practically independent of the concentration of virions in the medium. The results for $t_e$, which are within the range of time usually observed in cell infection processes, confirm our assumption that $t_e$ can be related to as a constant. The fact that all examined MOIs render the same $t_e$ indicates that the internalization process is a property dependent on biological processes occurring on the cell surface and in its interior regardless of the virus concentration in the medium. Hefferon et al. (1999) found that Orgyia pseudotsugata multicapsid nucleopolyhedrovirus (OpMNPV) virions entered Spodoptera frugiperda Sf9 cells with a half-time of approximately 12.5 min. Their results differed from the results in the current study with the AcMNPV-Sf9 system shown in Table 2. This indicates that different Baculovirus may have different internalization half-time value. Since the internalization time for each specific virus-host system is expected to be a characteristic constant, it seems that viral vectors with shorter internalization times would have some advantages in gene therapy, since they enter host cells more rapidly while the cells are robust. Therefore, the estimation of the internalization time for any potential virus-host combinations may be very important. The very low value of the constant $k_w$ indicates that the actual accumulation rate of dark cells $d{C}_{\mathit{id}}/dt$ is substantially lower than the floor function shown in Eq. (18), and yet follows a similar pattern, given the quality of the fit.

3.4. Simulation of the dynamics on the cell surface

Based on the results shown above, a crude approximation to $d{C}_{\mathit{id}}/dt$ can be made assuming that the generation rate of darkened cells is proportional to $r_i$:

$$d{C}_{\mathit{id}}/dt=k_w·r_i\left(t\right)=k_w\left\{\begin{array}{cc}0& 0\le t<t_e\hfill \\ k·\left[V_0-x\left(t-t_e\right)\right]·\left[N·C\left(t-t_e\right)-x\left(t-t_e\right)\right]& t_e\le t\hfill \end{array}\right)$$ (20)

As an example, Fig. 6 a demonstrates the typical dynamics of the system variables $x\left(t\right)$ , $A\left(t\right)$ , and $I\left(t\right)$ in the case of MOI = 114 (see below), using the obtained values of the parameters $t_e$ and $k_w$ (shown in Table 3).

During $0\le t<t_e$ , $A\left(t\right)$ coincides (and increases) with $x\left(t\right)$. At $t=t_e$ (the onset of internalization), $I\left(t\right)$ begins to increase and the number of viruses on the surface, $A\left(t\right)$, begins to decrease sharply. Eventually, $A\left(t\right)$ will approach zero as $I\left(t\right)$ converges to $x$. Despite the abrupt transition at $t_e$, due to the discontinuity of our model at that time, the general form of the profiles portrays a clear picture of the dynamics of the virions at the surface of the cells.

Fig. 6(b) shows the rates of virion attachment $r_x$, virion internalization $r_i$, and the rate of change in the number of virions attached to the cell surface, $r_a$.

A noticeable discontinuity of $r_a$ is observed at $t_e=$ 0.8, due to the first occurrence of internalization. The shift in the internalization rate at $t=t_e$ is in agreement with our experimental observations, which were based on the measured profiles of $C_{\mathit{ip}}$ and $C_{\mathit{iw}}$. The model also indicates that, near 4 $\mathit{hpi}$, the difference between $r_i$ and $r_a$ becomes very small, as saturation is approached and the process nears convergence.

Despite the simplifications made, the model provides a reasonable prediction and can be considered a first approximation to the rate of viral internalization, on which experimental data are scant. Some of the features in the present model may eventually be applicable in the studies towards palliation of the COVID-19 outbreak (Zhu et al., 2020, Binnicker, 2020)

3.5. Approximation based on constant cell concentrations

The low growth factor, $k_g=0.022$, suggests that the cell concentration, $C$ may be treated as nearly constant at the range $0\le t\le 3h$. This observation is further supported by the experimental behavior of the cell concentration shown in Fig. 6. In this section, we neglect the changes in $C$ and consider it to be a constant equal to $C_0$. In this case, $k_g$ is set to zero and Eqs. (6), (7), (8), (9), (10), (11), (12) can be integrated analytically to yield a closed-form solution for $x$ , $C_{\mathit{ip}}$, $C_{\mathit{id}}$, $C_{\mathit{iw}}$, and $C_v$.

$$x\left(t\right)=\beta V_0\left[exp\left(\delta t\right)-1\right]/\left[\beta ·exp\left(\delta t\right)-V_0\right]$$ (21)

where $\beta =N{C}_0$; $\delta =k_c\left(N{C}_0-V_0\right)$, and $k_c$ is equivalent to $k$ in Eq. (5).

The closed-form solutions of $C_{\mathit{ip}}$, $C_v$, $C_{\mathit{id}}$ and $C_{\mathit{iw}}$ are shown in the Appendix A.

Next, we used the closed-form solution with the parameters shown in Table 3. The simplifying assumption introduces very little change in the solution and can apparently be used safely. Moreover, the closeness of the dashed line (representing the constant-C model) to the solid line in Fig. 6 proves that the error in using the constant-$C$ model introduces very small errors (possibly within experimental uncertainty).

Note that the asymptotic value of $x\left(t\right)$,

$$\underset{t\to \infty }{\lim }x\left(t\right)=\beta =N{C}_0,$$ (22)

is independent of the kinetic constant $k_c$. In fact, this is the total number of attachment sites on the cells, as might be expected. As defined above, $x\left(t\right)$ is the number of virions per unit volume that have been captured by cells between $0$ and $t$. It reaches maximum when all cells are saturated by N virions. N is a constant, specific to each virus-host system

The approximated solution, based on a constant value of the total cell concentration, appears to be successful for our set of initial conditions. The solution of Eq. (6) can be replaced by the closed-form expression given above (Eq. (21)) without noticeable changes in the predictability capacity of the model.

4. Conclusions

In the present multi-disciplinary research, we propose a mathematical model of the early-stage kinetics of Sf9 cell infection by BV, based on and tested against experimental results. To the best of our knowledge, this is the first time that the monoclonal antibody AcV1 is used to label GP64 in order to distinguish between infected and uninfected cells at different stages of virion attachment/internalization using FCM. We used MOIs of 114, 219, and 421. The maximum number of virions that can attach to a cell, N, which is crucial for planning a high MOI process, was evaluated for Autographa californica and found to be 55. In general, N is a significant reference for investigating the required dosage of the viruses in gene therapy schemes. The virus-host dependent constant $t_e$ is an important characteristic of the internalization process. Some variables, such as $C_{\mathit{ip}}$, $C_{\mathit{id}}$ and $x\left(t\right)$, in our model have great significance in describing the kinetics of processes like protein production and gene therapy by baculovirus vectors.

The mathematical solution of our model shows a satisfactory fit to the experimental measurements of virion removal from the medium and the concentration profiles of the infected cells and explains the first stage dynamics in cell infection. An FCM-mediated comparison of the infection dynamics with and without cell permeabilization enabled us to distinguish between the rate of virion attachment and the rate of internalization. We mathematically obtained an adequate approximation of the dynamic behavior of the virions on the surface of the cells and presented their transit rate profiles. A simplifying approximation of a constant cell concentration allowed us to obtain a closed-form solution for the profile of attached virions, which may be used to render a considerable simplification in the calculations. The proposed mechanistic approach may inspire modeling of other virus–cell systems independent of infection efficiency. It may contribute to modeling the development of novel recombinant proteins, drugs, and gene therapy vectors in mammalian cells.

Declaration of Competing Interest

The authors declare no competing interests, financial or other.

Appendix A.

Constant cell-based calculations

$${C_{\mathit{ip}}\left(t\right)=C}_0·\left(1-exp\left\{k_i\left(\beta -V_0\right)\left[t-ln\left|\left[\beta exp\left(\delta t\right)-V_0\right]/\left(\beta -V_0\right)\right|/\delta \right]\right\}\right)$$ (A1)

$$C_v\left(t\right)=C_0·exp\left\{k_i\left(\beta -V_0\right)\left[t-ln\left|\left[\beta exp\left(\delta t\right)-V_0\right]/\left(\beta -V_0\right)\right|/\delta \right]\right\}$$ (A2)

Under our assumptions we have:

$$C_{\mathit{id}}\left(t\right)=\left\{\begin{array}{cc}0& 0\le t\le t_e\hfill \\ k_{w}x\left(t-t_e\right)& t_e<t\hfill \end{array}\right)$$ (19)

Hence, applying [A1], [A2] we obtain closed-form solution for $C_{\mathit{id}}\left(t\right)$ and $C_{\mathit{iw}}$, respectively:

$$C_{\mathit{id}}\left(t\right)=\left\{\begin{array}{cc}0& 0\le t\le t_e\hfill \\ k_w\beta V_0\left[exp\left(\delta \left(t-t_e\right)\right)-1\right]/\left[\beta ·exp\left(\delta \left(t-t_e\right)\right)-V_0\right]& t_e\le t\hfill \end{array}\right)$$ (A3)

$$\begin{array}{c}{C}_{\mathit{iw}}\left(t\right)=C_{\mathit{ip}}\left(t\right)-C_{\mathit{id}}\left(t\right)\hfill \\ =C_0·\left(1-exp\left\{k_i\left(\beta -V_0\right)\left[t-ln\left|\left[\beta exp\left(\delta t\right)-V_0\right]/\left(\beta -V_0\right)\right|/\delta \right]\right\}\right)\hfill \\ -\left\{\begin{array}{c}0\\ k_w\beta V_0\left[exp\left(\delta \left(t-t_e\right)\right)-1\right]/\left[\beta ·exp\left(\delta \left(t-t_e\right)\right)-V_0\right]\end{array}\right)\begin{array}{c}0\le t\le t_e\\ t_e<t\end{array}\hfill \end{array}$$ (A4)