Stochastic model of virus and defective interfering particle spread across mammalian cells with immune response
Neil R. Clark, Karla A. Tapia, Aditi Dandapani, Benjamin D. MacArthur, Carolina Lopez, Avi Ma`ayan
SStochastic model of virus and defective interfering particle spreadacross mammalian cells with immune response
Neil R. Clark, Karla A. Tapia, Aditi Dandapani, BenjaminD. MacArthur, Carolina Lopez, and Avi Ma‘ayan ∗ Department of Pharmacology and Systems Therapeutics,Mount Sinai School of Medicine, New York, NY 10029, USA Department of Pathobiology, University of Pennsylvania, Philadelphia, PA 19104, USA Department of Applied Mathematics,Columbia University, New York, NY 10027, USA School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK (Dated: September 17, 2018) a r X i v : . [ q - b i o . P E ] A ug bstract Much of the work on modeling the spread of viral infections utilized partial differential equa-tions. Traveling-wave solutions to these PDEs are typically concentrated on velocities and theirdependence on the various parameters. Most of the investigations into the dynamical interactionof virus and defective interfering particles (DIP), which are incomplete forms of the virus thatreplicate through co-infection, have followed the same lines. In this work we present an agentbased model of viral infection with consideration of DIP and the negative feedback loop introducedby interferon production as part of the host innate immune response. The model is based on highresolution microscopic images of plaques of dead cells we took from mammalian cells infected withSendai virus with low and high DIP content. In order to investigate the effects discrete stochasticmicroscopic mechanisms have on the macroscopic growth of viral plaques, we generate an agent-based model of viral infection. The two main aims of this work are to: (i) investigate the effects ofdiscrete microscopic randomness on the macroscopic growth of viral plaques; and (ii) examine thedynamic interactions between the full length virus, DIP and interferon, and interpret what maybe the evolutionary role of DIP. We find that we can explain the qualitative differences betweenour stochastic model and deterministic models in terms of the fractal geometry of the resultingplaques, and that DIP have a delaying effect while the interaction between interferon and DIP hasa slowing effect on the growth of viral plaques, potentially contributing to viral latency. ∗ [email protected] . INTRODUCTION The modeling of the dynamics of viral infection across host cells is a classical problem inthe field of population dynamics and dispersal. Partial differential equations (PDEs) modelsof such systems have a long history; notably Skellam [1] was the first to apply PDEs to therandom dispersal of biological populations. Such models apply the continuum assumptionwhereby populations of individuals are represented as scalar concentration fields which obeyPDEs. Many of these models are not analytically solvable, and generally, simple solutionssuch as traveling wave, are commonly used to describe the dispersal of virus across hostcells. Here we study the spatial and stochastic effects of the dispersal of virus amongst animmobile space of host cells with the use of an agent-based model. In this context we examinethe dynamic interactions of the virus with defective interfering particles (DIP), which areincomplete forms of the virus, described in detail below, and the interferon production byhost cells’ immune response.The system analyze here is made of a continuous monolayer of host cells and a distributionof full-length and defective viral particles as well as interferon molecules. The virus spreadsby infecting cells, replicating, then releasing it’s yield upon killing the host cell; this yieldof virus particles are then free to diffuse and infect neighboring cells, generating a growingplaque of dead cells. The host cells’ immune response can detect defective viral particles andthis results in the release interferon molecules that locally reduce the probability of furtherviral infection. The addition of the negative feedback loop through the interferon responseby host cells due to DIP detection was not previously modeled by others.A typical approach to model the dispersal of a virus across host cells is to use the con-tinuum assumption whereby the distribution of particles and cells are represented as scalarconcentration fields which are solutions to differential equations. A system of differentialequations which embodies the hypothesized significant mechanisms is derived and studiedfor insight into the population dynamics. This approach has been popular in the field ofimmunology modeling to enhance understanding of HIV-1 infection and other pathogens[2], and to explore the idea that DIP could be used for HIV therapy [3], while in [4] theauthors investigated the population dynamics of virus and DIP in serial passage cultureswith recurrence relations.An extension of this approach is to study the spatio-temporal dynamics of a population3pread. The continuum assumption being made even on the scale of whole organisms [1]being among the first to do this. Typically partial differential equations such as, ∂P∂t = D∇ P + αP (1)where D is the dispersal rate and α is the intrinsic growth rate, form the basis of thesemodels. One of the first applications of this class of models to study virus-DIP infectionwas by Frank [5], however this model did not give a full treatment of the spatio-temporaldevelopment and did not include the immune response. A PDE model was developed by Yinand McCaskill [6], wherein the spread of the virus was represented as a reaction diffusionsystem, V . H k − (cid:42)(cid:41) − k I k −→ Y . V (2)where V , I and H represent the concentration of virus, infected host cells and uninfectedhost cells, and k , k − , and k represent the rates of viral infection, desorption and cell death.The authors devised the corresponding PDE model, looked for traveling wave solutions, andconsidered the dependence of the velocity upon the parameters of the model. A similarapproach was taken in [7]. Haseltine [8] took this approach one step further by fitting theirmodel to images of growing viral plaques, while also concentrating on the velocity. Morerecently Amore [9] expanded upon reaction-diffusion models by including the delay timebetween infection of a cell and the release of viral progeny. The preoccupation with velocitypermeates most of the literature on this subject, however here we are mainly concerned withthe qualitative details of the spatio-temporal dynamics of viral infections, it’s implications forthe dynamical significance of the stochastic nature of the physical and biological mechanisms,focusing on the possible role for DIP in the context of their detection by the host and theresultant interferon immune response.Here, rather than velocity, we investigate the qualitative dynamic effects of the mecha-nisms of virus dispersal, adding the important variables of interferon and DIP. DIP werediscovered in the 1950’s as incomplete forms of the influenza virus that interfere with viralreplication [10, 11]. They were subsequently observed for almost all RNA viruses such asrabies [12], sendai (SeV) [13], polio [14], sindbis [15], vesicular stomatitis virus (VSV) [16],and measles [17, 18]. It was discovered that DIP interfere with viral replication by overloading the viral replication machinery because shorter DIP replicate faster compared withthe production of full-length virus [19]. It was also discovered that DIP can be detected4y the host, promoting interferon production leading to a robust immune response [19–21].Many DIP can only replicate through co-infection with the full virus leading to a parasiticor predator-pray type relationship.DIP are a conserved biological phenomenon with no known function. Given that manyspecies of virus are often found to co-exist with their corresponding DIP, it is reasonableto suppose that they could be performing a biological function that confer an evolutionaryadvantage, or otherwise exist in some kind of evolutionary equilibrium with the virus. How-ever, this is not currently known or proposed, and the above properties, being ostensiblydetrimental to the virus, do not signal an obvious function or mutual evolutionary advan-tage. Here we shall consider, through a generic model, the dynamical interaction betweenvirus, DIP, host cells and their innate immune response. We demonstrate that DIP canhave a delaying effect on the spread of virus, and interferon can have a slowing effect. Weprovide some insight into these specific relationships. Our model leads us to a more gen-eral consideration of the continuum assumption behind PDE approaches to the modeling ofvirus spread in an immobile space of host cells. The most significant result of this part ofour investigation is that, in this system, a discrete stochastic model may have qualitativelydifferent solutions than the deterministic, traveling-wave solutions of reaction-diffusion PDEmodels. In which case it is important to determine which type of model is most appropriatefor this biological system, and this determination may extend understanding of the mostimportant mechanisms in the spread of viral infection in a host.Our model is primarily based on high-resolution microscopy images we took of stationaryprimate kidney cell line in culture infected with Sendai virus (SeV) with or without DIP. Weconstruct a stochastic agent based model which does not rely on the continuum assumption,and retains the discrete and random nature of viral infection and decay. We explore thequalitative properties of the solutions for various values of the parameters. By comparingthe output of our agent based model to PDE models, we observe that the stochastic modelof plaques are growing with an accelerating speed. The mechanism by which this occurs isexplored quantitatively in terms of the fractal geometry of the model plaques. The dynamiceffect of DIP and the interferon response is gauged qualitatively, and it is found that DIPcan delay the growth of viral plaques while interferon can slow their growth. Hence, theknown biological properties of DIP could potentially explain their moderating effect on viralplaque growth. 5 . Biological and experimental background We aim to model the spread of an RNA virus, its DIP and the interferon response ofthe immune system, through a monolayer of living mammalian cells in a dish. As the virusspreads, a region of dead cells is formed called a plaque . Most of our results will concernthe properties of these plaques. The construction of the model is based on an abstrac-tion of the mechanisms of virus spread which we based on microscopic images we collectedexperimentally from stained plaques. In this experiment LLCMK2 cells were grown in Dul-becco’s modified Eagle’s medium (Gibco-BRL) supplemented with 10% fetal calf serum(FCS; Gibco-BRL), Sodium pyruvate, L glutamine and gentamicin. The cells were plated in24 well plates. Confluent monolayers were infected with 200 infectious particles of SeV (lowDIP) alone or SeV together with 2000 DIP, or mock infected. After 1h incubation at 37C thecells were overlaid with 500ul of agar melted in infection media containing 0.025 mg of trypsin(Worthington). The infected LLCMK2 cells were then fixed with 4% paraformaldehyde at48 or 96 hrs post infection and blocked overnight at 4C with PBS/BSA 1%. Cells were thenstained with a monoclonal anti-SeV NP antibody (clone 3F11) for 45 min at room tem-perature, washed twice with PBS/BSA 1%, and incubated with a peroxidase-conjugatedsecondary antibody (Jackson ImmunoResearch) for 45 min at room temperature. Afterwashing, the staining was developed using the AEC Substrate kit according to manufac-turer’s recommendations (BD Pharmingen). Picture scanning of the wells (10x) was takenusing a Zeiss Axioplan2IE microscope and montage stitching done with the Metamorphsoftware (MDS Analytical Technologies) at the MSSM-Microscope Share Resource Facility(Fig.1). This figure shows the roughly circular outlines of the monolayer of cells, where thosestained red have been killed by the virus. We can immediately see from these images thatDIP appear to have arrested the growth of viral plaques.The abstraction of the mechanisms of spread of infection that we envisage is that initialinfection is nucleated, being seeded by the infection of an individual cell with an individualvirus particle. The virus replicates internally for some time before the cell is killed andthe virus yield is released. The released viral particles then diffuse freely until they eitherdecay or they infect a healthy neighboring cell and thereby spread the infection. Thisabstraction is consistent with the observation of distinct plaques of dead cells which growby an expansion of their boundaries, which are quite sharp and irregular. A low virus yield6 O (10) particles) and significant decay rate ( O (10 − s − ) ), and the discrete random natureof infection, replication and decay, suggests that stochastic effects may be significant at theintercellular scale. In this case we would not make the continuum assumption. However,the local growth of even macroscopic viral plaques is expected to be generated by the samemicroscopic mechanisms as the intercellular spread. We investigate the stochastic effectsat the intercellular scale on the macroscopic growth of viral plaques by developing a modelwhich incorporates them explicitly. Furthermore we examine the qualitative effects of DIPand the interferon immune response on the dynamics of viral infection in our model. B. Model construction
The model is agent-based, such that the hypothesized significant biological entities arerepresented as agents, and the significant biological and physical mechanisms are representedas the rules of behavior of the agents. The model results in stochastic simulations of indi-vidual viral plaques in which each individual viral particle, interferon molecule and cells isretained explicitly. We begin by describing the agents, their states and their correspondingbiological entities before describing the rules and the parameters.The monolayer of cells is represented by a square lattice. Each element of the latticecan be in two states corresponding to a living or dead cell. This lattice also serves as thediscretization of space in which the viral particles, DIP and interferon molecules are located.The total number of each type of particle at each lattice point at each time is stored. Theviral particles can be in two states, internal and external. In the external state the particlesare free to diffuse on the lattice, however, internal particles correspond to those which areresiding inside living cells. Agents representing interferon molecules are secreted by livingcells and are never internal in the model. Figure 2 shows a schematic description of theseagents.The rules of behavior of the agents encode the following biological and physical mech-anisms: diffusion, infection, decay, replication, interferon secretion and cell killing. First,diffusion is incorporated by making free agents perform a discrete random walk on the lat-tice, the timestep is chosen to be consistent with the prescribed diffusion coefficient. Ateach discrete timestep viral particles undergo an independent Bernouli trial which deter-mines weather they decay or, if the cell at it’s current lattice position is alive, the viral7article may infect the cell and thereby become internal. The probabilities of the Bernoulitrials are set to be consistent with the prescribed viral infection and decay rates. Internalviral particles replicate via a Poisson process with rates consistent with the prescribed yieldand lifetime of an infected cell. The replication obeys the known logic of the interaction ofvirus and DIP such that (i) virus alone - replicates at rate r (ii) DIP alone - no replication(ii) virus and DIP confection - virus replicates at rate r/ ρ , DIP replicate at rate r, where ρ ≈ O (20). This parameter has experimental backing in from Yount et al. [19]. The effectof this parameter is not investigated here since it has experimental backing and exploringits effect on the model is beyond the scope of this study. Cells infected with DIP can de-tect these particles, and this leads to secretion of interferon molecules at a prescribed rate,treated as a free parameter. The effect of the interferon concentration is to locally reducethe probability of viral infection. We model this with a hill-function: p i = p i, βI (3)Where p i, and β are constant parameters and I is the concentration of interferon. Finally,when the virus/DIP has replicated up to it’s yield, the cell dies and the internal particlesbecome external.The model parameters dictate the spatial dimensions of the lattice, the diffusion coeffi-cients, and the rates of the various processes. We set the grid spacing to be equal to theapproximate cell spacing in our experimental monolayer, 20 µm , and the grid size to 400cells , so that we can investigate macroscopic plaques. A base set of parameters, shown intable I, is chosen consistent with You and Yin [7], then perturbations around this set aremade in runs of the model. The initial condition is for a single infected cell in the center ofthe lattice which seeds the growth of an individual plaque. The lattice has periodic bound-ary conditions however the model is not run for enough time for the effects of the boundaryto have an effect on the results. II. RESULTS
Figure 11 shows the development of an individual model plaque: the spatial distributionof the dead cells, the free virus, DIP and interferon at several times. The model plaqueshave a compact morphology with irregular boundaries. The free virus and DIP reside pre-8ominantly on the periphery of the plaque, and so the plaque grows by an expansion of it’sboundaries. Most previous approaches to this type of problem have been concerned with thevelocity of traveling wave solutions. However, here we are concerned with the qualitativesolutions to our model and their difference from traveling waves. Later we consider thequalitative effects of DIP and interferon.First, we examine the growth of the number of dead cells in a plaque for various parametersets. Figure 4 shows the number of dead cells in model plaques against time, where the grayand black curves correspond to runs in which interferon was present and left out of themodel respectively. We observe an initial phase of fast growth, where the plaque is O (10)cells in number. This fast growth is dominated by nucleation events. There are few killedcells in the vicinity, and hence the growth is not significantly limited by the presence of deadcells.At later times, when the plaque is O (100) cells in number, the growth curves change toa power-law, when no-interferon is introduced in the model. Growth with a monotonicallyreducing exponent in the presence of interferon is observed. The line in the figure indicatesthe slope of a power law quadratic in time; we can see that the model plaques are growingfaster than this rate. In the following section we concentrate on the power-law growth.A power-law growth is perhaps not surprising for a diffusion-limited growth such as themodel we present here. However, the exponent of the power-law is of interest to us becauseit involves a qualitative difference to the traveling wave solutions. Subsequently we shalladdress the effect of the DIP and interferon on the overall behavior of the model. A. Accelerating plaque growth
In the absence of interferon, the growth curves of model plaques shown in figure 4 obey apower-law with an exponent greater than 2 which means that their mean radius acceleratestheir expansion. A plaque with a simple geometry growing by linear expansion, as is the casefor a traveling wave, would grow as t , where t is time. We aim to interpret this differencein terms of the geometry of our model plaques with the aim of illuminating the importanceof the microscopic stochasticity for the qualitative nature of the growth of macroscopic viralplaques. We aim to do this with a phenomenological argument.Part of the abstraction for the mechanism of plaque growth is that the virus resides on9he periphery of the plaque and the plaques grow when the infected cells on the peripherydie and release viral particles to infect nearby cells. One explanation for the observedqualitative difference in the observed exponent of the plaque power-law growth is that theirregular geometry of the plaques gives them a larger perimeter for a given area and thereformore infected cells for a given number of dead cells, resulting in a greater exponent in thepower-law growth.This would rely on a fractal plaque geometry, where the fluctuations in the plaque bound-ary are scale-free, and increase in their range of scales as the plaque develops; in which casethe plaque would have the following fractal area-perimeter (A-p) relationship: A ∝ p D (4)where D is the fractal dimension of the plaque boundary.We begin by demonstrating the fractal nature of the model plaques. Figure 5 showsthe area-perimeter relationship for model plaques in the absence of interferon. We see thatin each case the exponent is greater than 2, indicating a fractal dimension of the plaqueboundaries which is greater than unity. In order to further demonstrate the scaling natureof the plaque boundary fluctuations, we calculate the radial coordinates, origin at the centerof the lattice, of infected cells on the boundary, and plot the radial against the angularcomponent in a Cartesian plot 6 at three different times in the development of an individualplaque. In each case there are fluctuations at the spatial scale of the lattice spacing. However,we can see that the range of scales increases with time as the fluctuation curve extends overan ever larger range of scales. It may appear from the images of the plaques that theybecome more circular with time, this is because the range of fluctuations scaled with themean radius of the plaque decays. However, the absolute scale of the fluctuations increases.This is illustrated in figure 7 which shows the root-mean-squared fluctuation intensity as afunction of time for an individual representative model plaque.Next we consider the relationship between the number of infected cells, n i and the plaqueperimeter p . Direct proportionality is unlikely because there is a band of infected cells offinite width which follows the perimeter, so we would expect fluctuations in the perimeterof the order of this width to be smoothed out, such that n i ∝ p γ . (5)10ut because we observe this band to be thin, we may expect an exponent, γ , close to butsmaller than unity. In figure 8 we plot the number of infected cells against the plaqueperimeter for all model plaques which developed in the absence of interferon. We observean exponent which is marginally less than unity in each case.To test the hypothesis that the fractal geometry of our plaques contributes to their accel-erating growth, we analyze the expected exponent of the growth given the fractal dimensionof the plaque and the relationship between the number of infected cells and the perimeter.If we suppose that the rate of change of the number of dead cells in a plaque, Q , is inproportion to the number of infected cells, n i , ignoring the time delay between infection andcell death, dQ ( t ) dt ∝ n i (6)and we take the relationships that follow from the fractal geometry of the plaques, n i ∝ p γ (7)along with equation 4, then if we take Q to simply be proportional to the plaque area wecan write, dQ ( t ) dt ∝ Q ( t ) γD , (8)which we can integrate to obtain, Q ( t ) ∝ t − γD . (9)For each model plaque, in the absence of interferon, we plot the number of infected cellsagainst the perimeter, the number of dead cells against the perimeter, and the number ofdead cells against time (see supplementary figure 9) and least-squares fit to estimate theexponents D and β , and compare the predicted growth exponent from equation 9. Thiscomparison is shown in figure 10, where we can see that the degree of correlation is partiallylimited by the errors in the estimation of the exponents D and γ . However, it is enough tosuggest that the fractal nature of the plaque geometry can at least partially account for theexponent of the plaque growth curve.Another potential explanation for the accelerating velocity of plaque growth is meancurvature effects. As the plaque grows the curvature of the mean boundary falls - it theplaque growth velocity depended on this curvature then this could potentially account forthe acceleration. To test this we ran the model with a different initial condition: a line of11ells from the top of the lattice to the bottom were set to be infected intimal; the resultingplaque was a plane front which spread to the outer edges of the lattice (see figure IV). Themean plane front has constant curvature, i.e., zero. Hence, any acceleration cannot be dueto curvature of the mean front. Figure IV shows the accelerating growth of this plaque,which appears to be due to the development of fluctuations in its boundary geometry ratherthan the mean curvature. III. THE EFFECT OF DIP AND INTERFERON
In our wet-lab experiment DIP outnumbered virus particles by about an order of mag-nitude in order to generate a significant effect of the infection. In the light of this, and thefact that DIP cannot replicate alone, in order to simulate the effects of DIP on an individualplaque we set the initial conditions such that the plaque develops in a randomly uniform dis-tribution of internal DIP. In order to isolate the effect of the DIP, we examined the growth ofplaques with various concentrations of DIP in the absence of interferon. Figure 13 shows therate of growth of plaques where the same parameters are used (see supplementary figures)except 5%, 20%, and 30% DIP. We observe that the strongest effect of DIP appears to be adelaying effect on the growth of the plaques such that the 30% growth curve remains abouta factor of two smaller than the 5%.In order to further investigate the effect of DIP, we plot the effect of the concentration ofDIP upon the exponent of the plaque growth curve, the fractal dimension and the exponentof the power-law relation between the number of infected cells and the perimeter (see figure14). We see that the fractal dimension increases with DIP, and that the value of β falls.These two effects approximately cancel out such that the exponent of the growth curve is notsignificantly affected. It appears that the presence of DIP is felt in the early developmentof the plaque when it is made of O (100) cells in number, at that time the delay is induced.Figure 15 shows model plaques with parameters set with various interferon secretionrates, decay rates, and strengths of effect on live cell (see equation 3), while keeping allother parameters the same. The aim in varying the interferon relevant parameters wasto observe the various qualitative changes to the model plaque growth curves due to thedynamic effects of interferon. Broadly speaking the interferon slows the growth of plaquesfrom a power-law to a curve with a monotonically decreasing exponent - possibly a modified12ogarithmic growth. However, depending on the balance of the parameters governing thestrength of the effect of interferon on neighboring cells, and the interferon decay rate, theplaques can display a bi-phasic growth curve in which the curve is initially concave but asthe concentration of interferon reaches saturation, the curve can return to a power-law form. IV. CONCLUSIONS
In the construction and execution of our model we have addressed two main aims: (i)to investigate the qualitative differences in the simulated plaque growth resulting from de-terministic PDE models and our discrete stochastic agent-based model; (ii) explore thequalitative dynamic effects of DIP and interferon on the growth of viral plaques.We revealed that the agent-based model produces plaques which grow faster than quadrat-ically in time, this in contrast to most previous work on such systems which look for traveling-wave solutions and then focus on velocities. We found that the fractal geometry of theplaques in the agent-based model can at least partly explain the difference between theexponents.The model indicated that DIP have a delaying effect on the growth of model plaques,and that large amount of DIP relative to virus particles are required to have an appreciableeffect. DIP appear hamper the growth of viral infections, as we observed that DIP arrestthe growth of our experimental SeV plaques. The model results show that the impededgrowth of viral plaques due to DIP can at least partly be a dynamical effect which onlydepends on the known biological properties of DIP. It is tempting to tentatively propose thehypothesis that DIP dynamically impede the growth of viral infection, and that this couldbe performing a useful function for the virus in moderating its spread so it does not kill thehost before it is provided with the opportunity to jump host.The final aim was to investigate the effects of the interferon immune response. We founda range of qualitatively different growth curves, the form of which depended critically on theparameters governing the secretion of interferon and the protective effect of interferon onuninfected cells. Broadly speaking, considering the interferon response in the model sloweddown the power-law growth to one with a monotonically decreasing exponent, similar to alogarithmic growth. As such, interferon has a much more dramatic slowing effect on thegrowth rate of viral plaques. 13 cknowledgments
We thank Drs. Charles Peskin from NYU, Jacob Yount from the Rockefeller University,and James G. Wetmur from MSSM for useful discussions. This work was supported by NIHgrants 5P50GM071558-03, 1R01DK088541-01A1, and KL2RR029885-0109. [1] J. Skellam, Biometrika , 196 (1951).[2] A. Perelson, Nature Reviews Immunology , 28 (2002).[3] G. Nelson and A. Perelson, Math Biosci. , 127 (1995).[4] T. Stauffer and J. Yin, Virology journal , 257 (2010).[5] S. Frank, Journal of Theoretical Biology , 279 (2000).[6] J. Yin and J. McCaskill, Biophysical journal , 1540 (1992).[7] L. YOU and J. YIN, Journal of theoretical biology , 365 (1999).[8] E. Haseltine, V. Lam, J. Yin, and J. Rawlings, Bulletin of Mathematical Biology , 1730(2008).[9] D. Amor and J. Fort, Physical Review E , 061905 (2010).[10] P. VON MAGNUS, Adv Virus Res. , 59 (1954).[11] K. Paucker and W. Henle, Virology , 198 (1958).[12] H. Clark, N. Parks, and W. Wunner, J Gen Virol , 245 (1981).[13] D. Kolakofsky, Cell , 547 (1976).[14] C. Cole, D. Smoler, E. Wimmer, and D. Baltimore, J. Virol , 478 (1971).[15] T. Shenk and V. Stollar, Virology , 162 (1973).[16] M. Schubert, J. Keene, R. Lazzarini, and S. Emerson, Cell , 103 (1978).[17] W. Hall, S. Martin, and E. Gould, Med Microbiol Immunol. , 155 (1974).[18] B. Rima, W. Davidson, and S. Martin, J Gen Virol , 89 (1977).[19] J. Yount, T. Kraus, C. Horvath, T. Moran, and C. Lopez, J Immunol , 4503 (2006).[20] P. Marcus and M. Sekellick, Nature , 815 (1977).[21] C. Lopez, J. Yount, T. Hermesh, and T. Moran, J. Virol. , 4538 (2006). ABLE I: Base set of model parameters, which is based on [7], and the decay rates are taken from[5]. Parameter Symbol ValueVirus infection rate k ,V . × − ml / hourDIP infection rate k ,D . × − ml / hourInfected cell death rate k , . × − hour − Virus decay rate k ,V . × − s − DIP decay rate k ,D k ,V Interferon decay rate k ,D k ,V Diffusion coefficient D . × − cm / hourlattice spacing dx µ mtimestep dt dx D = 1 . × sLifetime of infected cell L = k − Y
50 virus copiesrate of virus replication R V Y ield/Lif etime rate of virus replication R D Y ield/Lif etime rate of interferon sectretion S rate of virus replicationStrength of interferon α IG. 1: Viral plaques grown in vitro . High resolution images including mock treated cells areavailable from http://amp.pharm.mssm.edu/dip-high-res-images.zip.FIG. 2: A schematic illustration of the agents which represent the biological entities in the agent-based model. (cid:61)
200 t (cid:61)
400 t (cid:61)
800 t (cid:61) (cid:61)
FIG. 3: A depiction of the distribution of killed cells, free virus, free DIP, and free interferon for arepresentative model plaque at four times.
50 100 200 500 1000 t (cid:144) timesteps110100100010 Dead cells t FIG. 4: The number of dead cells in individual plaques plotted against time. The black and graycurves correspond to plaques growing in the absence and presence of interferon respectively. Theparameter sets for these growth curves can be found in the supplementary materials. (cid:180) perimeter (cid:144) cells10100100010 Dead cells t FIG. 5: The number of dead cells plotted against the plaque perimeter. The number of deadcells is directly proportional the plaque area, so the scaling of this curve is the same as for thearea-perimeter relationship and can be used to estimate the fractal dimension of the plaques. (cid:45) (cid:45) (cid:45) Θ t (cid:61) (cid:61) (cid:61) FIG. 6: The polar coordinated of infected cells residing on the plaque boundary for an individualplaque at three different times. The plot shows the increasing range of scales of the fluctuations inthe shape of the model plaque boundary. (cid:144) timesteps0.1110Intensity absoluterelative FIG. 7: The root-mean-squared intensity of the fluctuations in a representative, individual plaqueboundary, both absolute, and scaled with respect to the mean radius of the plaque. The fluctuationsdecay with respect to the mean radius of the plaque, such that the plaques appear more circularand they grow larger, however the absolute intensity of the fluctuations increases as the modelplaque grows, in a scale-free manner.
10 50 100 500 1000 50001 (cid:180) perimeter (cid:144) cells5010050010005000 n i t FIG. 8: The number of infected cells plotted against plaque the perimeter of individual plaquesgrowing in the absence of interferon. (cid:144) timesteps1000200050001 (cid:180) (cid:180) Dead cells 1000 1000050002000 30001500 150007000 perimeter (cid:144) cells100050020020003001501500700 n i (cid:144) cells1000200050001 (cid:180) (cid:180) Dead cells
FIG. 9: The curves which are least-squares fitted and upon which the estimations plotted in figure10 are based. .20 0.25 0.30 0.35 0.40 Exponent0.10.20.30.4Estimate FIG. 10: The power-law exponent predicted from equation 9, which is based on the fractal geometryof the plaques, plotted against the actual power-law exponent estimated by least-squares fitting. t (cid:61)
200 timesteps t (cid:61)
400 timesteps t (cid:61)
800 timesteps t (cid:61) FIG. 11: A depiction of the distribution of dead cells in the spreading of a plaque which was startedwith the initial condition where a line of cells from the top to the bottom of the lattice are infectedwith virus. (cid:144) timesteps50 000dead cells t FIG. 12: The number of dead cells in the growth of a plaque.
000 1500 t (cid:144) timesteps200050001 (cid:180) (cid:180) Dead cells
FIG. 13: The number of dead cells in plaques with identical parameter sets (see supplementarymaterials), but growing in the presence of different concentrations of DIP. The green, red and blackcurves correspond to 5%, 20%, and 30% of cells infected with DIP respectively. (cid:37)
DIP2.202.222.242.262.282.30Growth exponent 5 10 15 20 25 30 (cid:37)
DIP1.31.41.51.61.71.8D 5 10 15 20 25 30 (cid:37)
DIP0.550.600.650.700.750.800.85 Γ FIG. 14: The effect of the DIP concentration on the exponent in the plaque growth curve power-law,the fractal dimension of the plaque boundary, and the relation bet6ween the number of infectedcells and the plaque perimeter.
20 50 100 200 500 1000 t (cid:144) timesteps10100100010 Dead cells