Conceived and designed the experiments: DW HF NLK. Performed the experiments: DW AH JWL ZS NLK. Analyzed the data: DW AH HF NLK. Contributed reagents/materials/analysis tools: DW AH HF NLK. Wrote the paper: DW NLK.
The authors have declared that no competing interests exist.
Oncolytic viruses replicate selectively in tumor cells and can serve as targeted treatment agents. While promising results have been observed in clinical trials, consistent success of therapy remains elusive. The dynamics of virus spread through tumor cell populations has been studied both experimentally and computationally. However, a basic understanding of the principles underlying virus spread in spatially structured target cell populations has yet to be obtained. This paper studies such dynamics, using a newly constructed recombinant adenovirus type-5 (Ad5) that expresses enhanced jellyfish green fluorescent protein (EGFP), AdEGFPuci, and grows on human 293 embryonic kidney epithelial cells, allowing us to track cell numbers and spatial patterns over time. The cells are arranged in a two-dimensional setting and allow virus spread to occur only to target cells within the local neighborhood. Despite the simplicity of the setup, complex dynamics are observed. Experiments gave rise to three spatial patterns that we call “hollow ring structure”, “filled ring structure”, and “disperse pattern”. An agent-based, stochastic computational model is used to simulate and interpret the experiments. The model can reproduce the experimentally observed patterns, and identifies key parameters that determine which pattern of virus growth arises. The model is further used to study the long-term outcome of the dynamics for the different growth patterns, and to investigate conditions under which the virus population eliminates the target cells. We find that both the filled ring structure and disperse pattern of initial expansion are indicative of treatment failure, where target cells persist in the long run. The hollow ring structure is associated with either target cell extinction or low-level persistence, both of which can be viewed as treatment success. Interestingly, it is found that equilibrium properties of ordinary differential equations describing the dynamics in local neighborhoods in the agent-based model can predict the outcome of the spatial virus-cell dynamics, which has important practical implications. This analysis provides a first step towards understanding spatial oncolytic virus dynamics, upon which more detailed investigations and further complexity can be built.
Traditional chemotherapy of cancers is characterized by strong side effects, while showing a low success rate in the long term control of tumors. Besides small molecule inhibitors, which have shown great promise, oncolytic viruses present an emerging specific treatment approach. They are engineered viruses that spread from tumor cell to tumor cell, killing them in the process. Non-tumor cells are generally not infected. While clinical trials have given rise to promising results, reliable success remains elusive. Besides experiments, computational approaches provide a valuable tool to better understand the dynamics of virus spread through a growing population or tumor cells. Combining in vitro experimental approaches with computational models, we study the principles of virus spread through a spatially structured population of cells, which is of fundamental importance to understanding virus treatment of solid tumors. We describe different growth patterns that can occur, interpret them, and explore how they relate to the ability of the virus to induce tumor regression. We further define how these spatial dynamics relate to settings where cells and viruses mix more readily, such as in many cell culture experiments that are used to evaluate candidate viruses.
Oncolytic viruses replicate selectively in tumor cells and have been explored as a targeted treatment approach against cancers
Besides experimental research, mathematical and computational modeling has increasingly become a tool to study the dynamics of oncolytic viruses. Mathematical models can help us understand the emerging properties of cancer-virus interactions, to interpret experimental results, and to design new experiments. The first mathematical models of oncolytic virus therapy considered ordinary differential equations that described the basic interactions between a replicating virus and a growing population of tumor cells, and also immune responses
We still do not, however, have a good understanding of the basic principles that govern spatially restricted virus spread through a population of target cells, what outcomes can be expected, and what determines those outcomes. Obtaining such basic knowledge is a necessary foundation for building predictive models of virus therapy. This knowledge can be used as a basis for examining the effects of further biological complexities on the outcome of virus therapy, such as immune responses, tumor-microenvironment interactions, cellular heterogeneity, cell-cell interactions, among others. Therefore, the aim of this paper is to study the basic dynamics of virus spread through a spatially arranged population of growing cells in a simple setting. To achieve this, we have constructed an in vitro experimental system in which a fluorescent labeled virus spreads through a target cell population in a two-dimensional geometry, such that an infected source cell can transfer the virus only to target cells in the direct neighborhood. Besides quantifying the number of infected cells, this also allows us to track the emerging spatial patterns over time. We found three distinct patterns of virus spread and determined the frequency of their occurrence. An agent-based model was used to simulate these experiments and to interpret the data. The model can qualitatively reproduce the experimental observations and suggests key parameters that determine the different growth patterns. Using this model, we explore the implications of the observed growth patterns for the long-term outcome of the dynamics, and obtain insights about the conditions required for the virus to drive the target cell population extinct in this setting. This is a first step towards understanding the basic principles of virus spread and the correlates of successful virotherapy in spatially structured cell populations, and provides a basis for more detailed explorations and for the incorporation of other complexities that are relevant for virus-tumor dynamics in vivo.
In order to examine spatial virus spread in a relatively simple setting, we constructed a recombinant adenovirus type-5 (Ad5) that expresses enhanced jellyfish green fluorescent protein (EGFP), AdEGFPuci, and grows on human 293 embryonic kidney epithelial (293) cells
(A) HEK293-mCherry cells were infected at an MOI≪1 and tracked every 24 h beginning at 5 days post-infection when initial spread of infection had occurred. Three representative patterns of AdEGFPuci infection were observed after at least 13 days post infection (pattern (i)) or as long as 19 dpi (patterns (ii) and (iii)) as shown in the micrographs (100×). The left panels represent the detection of HEK293-mCherry cell nuclei in the culture (mCherry RFP), whereas the right panels depict the identical field of view of HEK293-mCherry cells viewed for green fluorescence to detect cells infected with AdEGFPuci (EGFP). (B) Limited pattern (iii) of HEK293-mCherry cells infected by AdEGFPuci. The top panels depict AdEGFPuci infected cells (EGFP-positive), the middle panels depict all the HEK293-mCherry cell nuclei in the culture (mCherry RFP), and the bottom panels is the merge of those panels, illustrating infected vs. uninfected cells. The panels on the left are micrographs taken at 100× magnification; the right panels encompass the boxed area of the left images at 200× magnification. The arrows in the right panels point to an AdEGFPuci infected cell (inf.) and an uninfected AdEGFPuci cell (un inf.) within the center of the virus infected region of HEK293-mCherry cells at 19 dpi. The scale bar in both
In the following sections, we simulate these experiments with a computational model. We examine conditions for the formation of different patterns and examine implications of the different growth patterns for the ability of the virus to eradicate the target cell population.
Our in silico studies of the interactions between oncolytic viruses and cancerous cells rely on the agent-based modeling technique, where each cell is represented as an “agent” occupying a certain position on a grid, and interacting with other cells according to some (probabilistic) rules. Our modeling approach is spatial, that is, it takes into account the spatial distribution of the uninfected and infected cells
The model, based on previous work
Parameter | Meaning |
R | Division probability of uninfected cells |
D | Death probability of uninfected cells |
B | Probability for an infected cell to transmit the virus |
A | Death probability of infected cells |
In this section, we explore the initial virus growth patterns. As starting conditions we assume that the grid is filled with uninfected cells and that a relatively small square of infected cells (30×30 spots) is placed in the middle of the grid (which overall contains 300×300 spots). The emerging growth pattern depends on parameters that influence the rate of virus spread, in particular the probability for an infected cell to die,
Each row represents one case, characterized by a certain parameter combination. The left graph shows the spatial pattern. Green indicates uninfected cells, red infected cells, and grey empty spots. The middle graph shows the number of infected cells over time. The different lines represents 100 different instances of the simulation with the same parameter combination. The right graph shows the square root of the number of infected cells over time, again showing lines for 100 different runs. If this graph is linear, we observe quadratic or “surface” growth. Two basic types of initial growth are observed. Cases (a) and (b) show the formation of a ring structure, characterized by an initial phase of quadratic growth, and a subsequent phase of linear growth. In (a) the death rate of infected cells is lower, thus requiring a longer period of time until a hollow ring is formed. Hence, the duration of the quadratic growth phase is relatively long, almost the entire duration of the simulation presented here. Eventually, however, it will transition to linear growth, as exhibited in case (b), characterized by a faster death rate of infected cells. Case (d) shows a disperse growth law. No ring structure is formed. The virus population leaves behind susceptible cells, thus leading to a mixed pattern and quadratic growth. The small graphs depict a variation of this outcome which occurs if the viral spread rate is at the low end of the spectrum. Due to a higher number of target cells at the edge of the infection area, significant number of infected cells are only found there, and we observe a ring-like structure with a mass of uninfected cells in the center (upper small graph). Over time, the dynamics develop into a mixed pattern with low levels of infected cells (lower small graph), or the infection goes extinct. Case (c) lies in between the hollow ring and the mixed pattern. Initially a ring structure is formed, resulting first in quadratic, then in linear growth. However, on rare occasions uninfected cells remain in the wake of the expanding virus population, thus leading to concentric rings. While this does not happen in all instances of the simulations, when it does occur, the growth law transitions back from linear to quadratic. Parameter values were chosen as follows. (a)
In
In
Finally, no expanding ring structure is formed in
In order to go beyond the qualitative comparison of model and data, we fit the model to two sets of experimental data, one showing an expanding hollow ring, and the other the disperse growth pattern. A least squares algorithm (see
The number of cells was determined by measuring the fluorescent area of the infected cell population, divided by the fluorescent area of individual infected cells, using Photoshop (see
The predicted spatial pattern is the result of an individual run of the agent-based model with the parameter combination obtained from the model fitting procedure. Snapshots in time are shown, representing days 7, 9, 11, and 13 post infection. (a) The area of green fluorescence is shown, expressed by the infected cells, thus documenting the spatial spread of the virus through the population target cells arranged in a two dimensional setting. (b) In the computer simulation, green indicates infected cells, red infected cells, and grey empty spots.
The predicted spatial pattern is the result of an individual run of the agent-based model with the parameter combination obtained from the model fitting procedure. Snapshots in time are shown, representing days 7, 10, 11, and 12 post infection. (a) The area of green fluorescence is shown, expressed by the infected cells, thus documenting the spatial spread of the virus through the population target cells arranged in a two dimensional setting. (b) In the computer simulation, green indicates infected cells, red infected cells, and grey empty spots.
Here, we explore the long term dynamics, investigating how the above described patterns play out and correlate with the overall outcome if both the uninfected and infected cell population can expand in space. We seek to define conditions under which the virus can eliminate the target cell population in this system. All simulations are started with a small number of infected cells placed in a compact vicinity into a larger space filled with uninfected cells, which is in turn embedded into an even larger “empty” space (for the exact initial conditions for particular cases, see appropriate figure legends). In contrast to the simulations reported above, here we go beyond the initial virus growth stage, and focus on time-scales where the population of target cells experiences significant changes (grows in size in the absence of infection). The outcomes of this system include extinction of the target cells and thus the virus; extinction of the virus and persistence of the target cells; coexistence of virus and target cells. The dependency of these outcomes on the parameters is shown in
The plot is the result of at least 104 instances of the simulations, where the log10 of the parameters was varied between −4 and 4. The simulations were started by placing a small number of infected cells (5×5 cells) into a larger space filled with infected cells (13×13 cells). Identical results were observed over a very large range of initial conditions (differences were only observed if the initial number of cells is such that immediate stochastic extinction is likely, which are not regions of interest with respect to our study). Blue indicates coexistence of virus and cells. Red and orange indicate extinction of the cells and thus the virus. Red is used if extinction occurs before the boundary of the system has been reached, while orange is used if extinction occurs after cells have reached the boundary of the system. Grey indicates extinction of the virus while cells persist. Above the white line and below the black line, the “local” equilibrium number of uninfected and infected cells, respectively, is greater than one. This derives from ordinary differential equations that describe the dynamics within local neighborhoods where all cells can interact with each other (see
For each pattern, four snapshots in time are shown. Green indicates uninfected cells, red infected cells, and grey empty patches. See corresponding capital letters in
See text for details. Parameter values and initial conditions are given in
In the 30×30 grid, the following outcomes are found (
Two types of target cell extinction can be observed, both associated with the initial “hollow ring” structure. According to pattern
Another type of outcome is the coexistence of infected and uninfected cells, which is shown in pattern
Finally, there are two types of virus extinction patterns. Pattern
Although the spatial stochastic predator-prey system studied here exhibits a variety of patterns, its dynamics can be understood by studying the local interactions of the agents. The idea of a “characteristic scale” has been proposed in the literature in the context of different predator-prey models
We start from the well-known system of ordinary differential equations that can be derived for our agent-based model if no spatial restrictions were in place, and reproduction and infection events were driven by laws of mass-action:
While these properties of the virus-cell system are well-known, it is usually thought that the ordinary differential equations can only be applied to a well-mixed system, and fail to describe a spatially-distributed system of cells. Contrary to this,
In equations (1–2), the number of uninfected cells at the equilibrium (the value
Thus, in the spatial system, target cell extinction is observed if the local equilibrium number of uninfected cells is less than one (regions A & B,
In a larger, 300×300, grid (
This analysis shows that increasing the grid size allows more complex outcomes to occur and increases the parameter region in which the cell populations persist. The additional patterns that emerge in larger grids are variations of those found in the smaller grid and involve non-equilibrium persistence, where extinction occurs locally, but movement through space allows cells to temporarily avoid extinction. These dynamics are well documented in the ecological literature
The long term outcomes shown in
To gain further understanding of the complex dynamics exhibited by the spatial stochastic agent-based simulations, we have implemented several analytical tools. While some of them such as the pair approximation method
A metapopulation approach is a modeling technique which allows us to study dynamics in a spatial setting. The model consists of a collection of
Restricting the model to a one-dimensional setting greatly speeds up computing times while allowing us to verify our main results in the context of this model. A full analysis of the stochastic metapopulation dynamics, including a two-dimensional setting, will be provided in a subsequent paper.
As initial conditions we assume that in a subset of adjacent local patches in the middle of the metapopulation, target cells are present at their infection-free equilibrium levels. The rest of the patches are initially empty. A small amount of infected cells are placed into the middle patch, and the infection is allowed to spread from there (See appropriate figures for exact initial conditions).
The plot is a result of at least 104 instances of the simulations, where log10 of the parameters was varied between
In full analogy with the agent-based simulations, the results of metapopulation simulations can be described by looking at the equilibrium values for infected and target cells in local patches. We solve equations (1–2) where the carrying capacity is given by that of a local patch, and plot the lines corresponding to
The advantage of a metapopulation model is that partial differential equation (PDE) techniques can be used to approximate some aspects of the dynamics, and to obtain further analytical insights into the behavior of the system. Although PDEs of reaction-diffusion type fail to describe stochastic phenomena such as extinction, they provide valid information in the regimes where one observes wave propagation. One can model the spatial dynamics by the following equations,
Using the above estimates, we can identify the conditions under which the wave of infected cells destroys the wave of target cells, leading to virus-mediated extinction (regime MA), as opposed to the scenario where the two waves travel together (boundary-mediated extinction, region MB). If the wave of infected cells travels faster than the wave of target cells (
Note that in this formula we set
The above considerations help explain the qualitative shape of the phase diagram for the agent-based model. While establishing the exact match between the agent-based model and the metapopulation model goes beyond the scope of this paper, the above arguments demonstrate that
the patterns observed hold both in agent-based and in metapopulation models,
the local equilibrium argument allows us to map out many important features of the system's phase space in both models (which is especially surprising in the case of the agent-based model),
the PDE approach allows us to explain (and predict quantitatively in the case of metapopulation modeling) the front-propagation aspects of the dynamics.
We used a newly designed virus, AdEGFPuci, replicating on 293 embrionic kidney epithelial cells in order to study the spatial dynamics of virus growth in a two-dimensional setting in which a source cell transmits the virus to target cells in the immediate vicinity. We found that when the virus is placed into the midst of a target cell population, the initial growth pattern can be divided into two categories. Either the infected cells expand as a wave and eventually form a hollow ring, or the hollow ring is not formed and we observe cells remaining in the core of the ring which can lead to a disperse growth pattern. An agent-based model was used to qualitatively simulate these experiments. The model produced the same types of patterns found in the experiments and model fitting to the experimental data has shown that our description is consistent with observation. The data, together with the model, gave rise to a number of insights. When a ring is formed, the initial virus growth is quadratic and then becomes linear. In the absence of a ring, the entire virus growth is quadratic. These growth patterns are obviously related to and have implications for the dynamics of viral plaque formation, which have been previously investigated mathematically from other angles
An important theoretical finding is that the outcome of the spatial system (extinction versus persistence) can be predicted well from the local dynamics, characterized by the interactions among neighboring cells. In our model, this is given by the neighborhood of 3×3 spots. This result suggests that ordinary differential equations, which describe mass action dynamics, can be a valid approach to study the correlates of successful virus therapy even for tumors that exhibit spatial structure. The results of such approaches have to be correctly interpreted to include the notion of local neighborhoods. If interpreted correctly, insights gained from such previous modeling studies are applicable to the treatment of spatial tumors. In addition, this result tells us that simple in vitro experiments (where viruses and cells mix well) can be used to compare the basic replication and spread efficiencies of different viruses, and that this directly correlates with their ability to fight a spatially structured tumor. Of course, the correlates of successful treatment derived from such simple modeling studies are qualitative in nature. That is, it is for example possible to predict the effect of increasing a certain parameter, such as the death rate of infected cells, on the outcome of treatment. However, predicting whether or not virus treatment will result in the eradication/control of tumor cells is a very difficult problem, and so far no modeling approach exists that is suited to perform this task. Not only do we lack sufficient biological information, the same biological processes can be described mathematically in different ways, adding uncertainty to these models. As a next step it will be important to examine whether the global spatial dynamics can still be predicted from local mass-action dynamics in models when further complexity is introduced, including three-dimensional spatial structures, immune responses, physical barriers to spread, or cell populations with differential susceptibility to infection. If our result holds, then relatively simple ordinary differential equation models can be used to guide the exploration of other, more complex modeling approaches, as well as the design of experiments aimed at evaluating candidate viruses.
To put this work in the context of the existing literature, we note that experimental research has made much progress in the construction of viruses, in elucidating the molecular biology of virus replication in tumor cells, and in investigating clinical trials and correlates of success for several different types of candidate viruses and different tumors. Yet, the complex and multi-factorial interactions between viruses, their target cells, and other relevant components make it difficult to predict the outcomes of such dynamics, an area where mathematical tools can be of great help to complement experimental analysis, interpret data, and make experimentally testable predictions. In the recent years, there have been several mathematical/computational studies which examined spatially explicit models in more or less complex settings, taking into account an array of relevant factors that might affect the
Due to the complexity of the experimentally observed dynamics, further questions remain that are subject to ongoing investigation. The most puzzling observation was that identical experimental conditions, using the same virus-target cell system, gave rise to different patterns of virus growth. This indicates the existence of so far unidentified factors that influence virus spread in this
The finding that different patterns can form in identical experimental conditions also blocks our ability to turn our model into a truly predictive one. This would require parameters to be measured independently in the experimental system, and to demonstrate that these parameters yield a satisfactory fit of the model to the data. However, the dynamics are most likely characterized by different parameter combinations in the context of the different observed patterns. Before we fully understand the reasons for the various patterns, parameterization of the model remains an impossible task. Nevertheless, the model analysis presented here does highlight the parameters that determine the different outcomes, which guides our search for the responsible mechanisms. Identifying these mechanisms still requires extensive work which goes beyond the scope of the current paper.
We conclude that the dynamics of virus spread in the simplest spatial setting can be very complicated. We interpreted these dynamics with a computational model, and shed light onto the meaning of the different spatial patterns observed. We further found theoretically that mass-action dynamics in local areas can be indicative of the outcome of virus spread in a spatially structured cell population. This suggests that previous insights, gained from the analysis of ordinary differential equations, remain relevant for the spread of viruses in a spatial setting.
Both an agent-based model and a metapopulation model were used to examine the spatial dynamics of virus spread through a population of growing target cells. Analytical and computational methods are described in detail in the
Human 293 embryonic kidney epithelial cells (HEK293 or Ad293 cells – Agilent Technologies) were grown in Dulbecco's modified Eagle's medium (DMEM) supplemented with 10% fetal bovine serum (FBS). All cells were cultured at 37°C, 95% humidity, and 5% CO2. HEK293-H2BmCherry cells stably express the core nuclear histone H2B fused to monomeric red fluorescent protein mCherry
AdEGFPuci infected cells were scanned under UV illumination daily (100× magnification) using an Axiovert 200 M phase-contrast fluorescent microscope (Zeiss, Germany). Each grid was scored for the presence of green (virus-infected) cells and developing plaques were monitored for at least 20 days. Fluorescent images of AdEGFPuci infection that gave rise to plaques were recorded at 24–48 h intervals and montages of the recorded plaques were compiled using Microsoft PowerPoint. The rate of plaque expansion from AdEGFPuci infections was quantified by measuring the area of GFP fluorescence observed per day using Adobe Photoshop version 7.0. Briefly, each AdEGFPuci plaque image recorded was analyzed in Photoshop by selecting a green color range of moderate to low intensity to ensure the majority of the GFP expressing cells in the plaque was encompassed. The number of pixels displaying the selected color range of green fluorescence was defined as the area of GFP fluorescence in each plaque assayed. We analyzed several of the plaques at least three times to determine the average area of GFP fluorescence in each plaque/day. The area of GFP fluorescence from a single infected AdEGFPuci cell was quantified similarly and used to estimate the number of green (virus-infected) cells in the developing plaques. To calculate the number of fluorescent cells in the plaques using Photoshop, first the area of an individual cell fluorescing above a threshold was obtained from analyzing four independent cells at day 7 (the first day plaque was recorded). The mean area of fluorescence from the entire plaque was then divided by the mean area of fluorescence from an individual cell to derive the number of cells in the plaque. Standard deviations were then calculated by standard methods.
Mathematical details underlying the results presented in this paper. This file contains details about mathematical and numerical methods that were used to analyze the computational models.
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