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1.  Tissue architecture, feedback regulation, and resilience to viral infection 
Journal of theoretical biology  2013;340:131-138.
Tissue homeostasis is one of the central requirements for the existence of multicellular organisms, and is maintained by complex feedback regulatory processes. Homeostasis can be disturbed by diseases such as viruses and tumors. Here, we use mathematical models to investigate how tissue architecture influences the ability to maintain tissue homeostasis during viral infections. In particular, two different tissue designs are considered. In the first scenario, stem cells secrete negative feedback factors that influence the balance between stem cell self-renewal and differentiation. In the second scenario, those feedback factors are not produced by stem cells but by differentiated cells. The model shows a tradeoff. If feedback factors are produced by stem cells, then a viral infection will lead to a significant reduction in the number of differentiated cells leading to tissue pathology, but the number of stem cells is not affected at equilibrium. In contrast, if the feedback factors are produced by differentiated cells, a viral infection never reduces the number of tissue cells at equilibrium because the feedback mechanism compensates for virus-induced cells death. The number of stem cells, however, becomes elevated, which could increase the chance of these stem cells to accumulate mutations that can drive cancer. Interestingly, if the virus interferes with feedback factor production by cells, uncontrolled growth can occur in the presence of the virus even in the absence of genetic lesions in cells. Hence, the optimal design would be to produce feedback factors by both stem and differentiated cells in quantities that strike a balance between protecting against tissue destruction and stem cell elevation during infection.
PMCID: PMC4102259  PMID: 24056215
virus dynamics; mathematical models; tissue architecture; tissue design; stem cells
2.  Minimizing the risk of cancer: tissue architecture and cellular replication limits 
Normal somatic cells are capable of only a limited number of divisions, which prevents unlimited cell proliferation and the onset of tumours. Cancer cells find ways to circumvent this obstacle, typically by expressing the enzyme telomerase and less often by alternative recombination strategies. Given this fundamental link between cellular replication limits and cancer, it is important to understand how a tissue's architecture affects the replicative capacity of a cell population. We define this as the average number of remaining divisions at equilibrium. The lower the replication capacity, the lower the chances to escape the replication limit during abnormal growth when a tumour develops. In this paper, we examine how the replication capacity is influenced by defining characteristics of cell lineages, such as the number of intermediate cell compartments, self-renewal capability of cells and division rates. We describe an optimal tissue architecture that minimizes the replication capacity of dividing cells and thus the risk of cancer. Interestingly, some of the features that define an optimal tissue architecture have been documented in a variety of tissues, suggesting that they may have evolved as a cancer-protecting strategy in multicellular organisms.
PMCID: PMC3730689  PMID: 23825115
cancer; telomeres; cell lineages
3.  Virus dynamics in the presence of synaptic transmission 
Mathematical biosciences  2013;242(2):161-171.
Traditionally, virus dynamics models consider populations of infected and target cells, and a population of free virus that can infect susceptible cells. In recent years, however, it has become clear that direct cell-to-cell transmission can also play an important role for the in vivo spread of viruses, especially retroviruses such as human T lymphotropic virus-1 (HTLV-1) and Human immundeficeincy virus (HIV). Such cell-to-cell transmission is thought to occur through the formation of virological synapses that are formed between an infected source cell and a susceptible target cell. Here we formulate and analyze a class of virus dynamics models that include such cell-cell synaptic transmission. We explore different ”strategies” of the virus defined by the number of viruses passed per synapse, and determine how the choice of strategy influences the basic reproductive ratio, R0, of the virus and thus its ability to establish a persistent infection. We show that depending on specific assumptions about the viral kinetics, strategies with low or intermediate numbers of viruses transferred may correspond to the highest values of R0. We also explore the evolutionary competition of viruses of different strains, which differ by their synaptic strategy, and show that viruses characterized by synaptic strategies with the highest R0 win the evolutionary competition and exclude other, inferior, strains.
PMCID: PMC4122664  PMID: 23357287
4.  Infection of HIV-specific CD4 T helper cells and the clonal composition of the response 
Journal of theoretical biology  2012;304:143-151.
A hallmark of human immunodeficiency virus is its ability to infect CD4+ T helper cells, thus impairing helper cell responses and consequently effector responses whose maintenance depends on help (such as killer T cells and B cells). In particular, the virus has been shown to infect HIV specific helper cells preferentially. Using mathematical models, we investigate the consequence of this assumption for the basic dynamics between HIV and its target cells, assuming the existence of two independently regulated helper cell clones, directed against different epitopes of the virus. In contrast to previous studies, we examine a relatively simple scenario, only concentrating on the interactions between the virus and its target cells, not taking into account any helper-dependent effector responses. Further, there is no direct competition for space or antigenic stimulation in the model. Yet, a set of interesting outcomes is observed that provide further insights into factors that shape helper cell responses. Despite the absence of competition, a stronger helper cell clone can still exclude a weaker one because the two clones are infected by the same pathogen, an ecological concept called “apparent competition”. Moreover, we also observe “facilitation”: if one of the helper cell clones is too weak to become established in isolation, the presence of a stronger clone can provide enhanced antigenic stimulation, thus allowing the weaker clone to persist. The dependencies of these outcomes on parameters is explored. Factors that reduce viral infectivity and increase the death rate of infected cells promote coexistence, which is in agreement with the observation that stronger immunity correlates with broader helper cell responses. The basic model is extended to explicitly take into account helper-dependent CTL responses and direct competition. This study sheds further light onto the factors that can influence the clonal composition of HIV-specific helper cell responses, which has implications for the overall pattern of disease progression.
PMCID: PMC4082790  PMID: 22480435
5.  On the relative fitness of early and late stage Simian Immunodeficiency Virus isolates 
Theoretical population biology  2007;72(3):426-435.
Simian immunodeficiency virus (SIV) has been shown to evolve from a relatively slowly replicating and mildly cytopathic virus early in the infection (SIVMneCL8) to a faster replicating and more cytopathic virus at later stages of the infection (SIVMne170). It has recently been demonstrated that the early and mildly cytopathic variant SIVMneCL8 out-competed the late and highly cytopathic strain SIVMne170 in cell culture experiments, because the fitness disadvantage derived from the higher cytopathicity was not matched by a sufficient increase in the viral replication rate. However, in another set of experiments where the life span of cells in culture was artificially limited, the late and more cytopathic virus won the competition, because under this condition cytopahticity was not an important determinant of viral fitness. It was hypothesized that the limited life span experiment reflected the immune-mediated high turnover environment in vivo more accurately, and that the presence of immune responses accounts for the selection of the cytopathic strain SIVmne170 during later stages of the infection. This paper investigates the effect of immune responses, in particular cytotoxic T lymphocyte (CTL) responses, on the competition dynamics between these two SIV strains with the help of mathematical models. Model analysis and parameter estimates derived from previously published data on SIV growth kinetics suggest that the SIV-specific CTL response might not be the driving force that leads to the selection of the cytopathic strain SIVMne170 during later stages of the infection. This implies that more complex evolutionary mechanisms might have to be invoked in order to explain the emergence of these strains in vivo. One possibility is that the ability of multiple virus particles to infect the same cell (coinfection) might be a prerequisite for the emergence of the cytopathic strain SIVMne170 as the disease progresses.
PMCID: PMC4058051  PMID: 17499829
6.  Infection dynamics in HIV-specific CD4 T cells: does a CD4 T cell boost benefit the host or the virus? 
Mathematical biosciences  2007;209(1):14-29.
Recent experimental data have shown that HIV specific CD4 T cells provide a very important target for HIV replication. We use mathematical models to explore the effect of specific CD4 T cell infection on the dynamics of virus spread and immune responses. Infected CD4 T cells can provide antigen for their own stimulation. We show that such auto-catalytic cell division can significantly enhance virus spread, and can also provide an additional reservoir for virus persistence during anti-viral drug therapy. In addition, the initial number of HIV-specific CD4 T cells is an important determinant of acute infection dynamics. A high initial number of HIV-specific CD4 T cells can lead to a sudden and fast drop of the population of HIV-specific CD4 T cells which results quickly in their extinction. On the other hand, a low initial number of HIV-specific CD4 T cells can lead to a prolonged persistence of HIV specific CD4 T cell help at higher levels. The model suggests that boosting the population of HIV-specific CD4 T cells can increase the amount of virus-induced immune impairment, lead to less efficient anti-viral effector responses, and thus speed up disease progression, especially if effector responses such as CTL have not been sufficiently boosted at the same time.
PMCID: PMC4058052  PMID: 17379260
7.  An HIV-1 Replication Pathway Utilizing Reverse Transcription Products That Fail To Integrate 
Journal of Virology  2013;87(23):12701-12720.
Integration is a central event in the replication of retroviruses, yet ≥90% of HIV-1 reverse transcripts fail to integrate, resulting in accumulation of unintegrated viral DNA in cells. However, understanding what role, if any, unintegrated viral DNA plays in the natural history of HIV-1 has remained elusive. Unintegrated HIV-1 DNA is reported to possess a limited capacity for gene expression restricted to early gene products and is considered a replicative dead end. Although the majority of peripheral blood CD4+ T cells are refractory to infection, nonactivated CD4 T cells present in lymphoid and mucosal tissues are major targets for infection. Treatment with cytokine interleukin-2 (IL-2), IL-4, IL-7, or IL-15 renders CD4+ T cells permissive to HIV-1 infection in the absence of cell activation and proliferation and provides a useful model for infection of resting CD4+ T cells. We found that infection of cytokine-treated resting CD4+ T cells in the presence of raltegravir or with integrase active-site mutant HIV-1 yielded de novo virus production following subsequent T cell activation. Infection with integration-competent HIV-1 naturally generated a population of cells generating virus from unintegrated DNA. Latent infection persisted for several weeks and could be activated to virus production by a combination of a histone deacetylase inhibitor and a protein kinase C activator or by T cell activation. HIV-1 Vpr was essential for unintegrated HIV-1 gene expression and de novo virus production in this system. Bypassing integration by this mechanism may allow the preservation of genetic information that otherwise would be lost.
PMCID: PMC3838139  PMID: 24049167
8.  Dynamics of Cellular Responses to Radiation 
PLoS Computational Biology  2014;10(4):e1003513.
Understanding the consequences of exposure to low dose ionizing radiation is an important public health concern. While the risk of low dose radiation has been estimated by extrapolation from data at higher doses according to the linear non-threshold model, it has become clear that cellular responses can be very different at low compared to high radiation doses. Important phenomena in this respect include radioadaptive responses as well as low-dose hyper-radiosensitivity (HRS) and increased radioresistance (IRR). With radioadaptive responses, low dose exposure can protect against subsequent challenges, and two mechanisms have been suggested: an intracellular mechanism, inducing cellular changes as a result of the priming radiation, and induction of a protected state by inter-cellular communication. We use mathematical models to examine the effect of these mechanisms on cellular responses to low dose radiation. We find that the intracellular mechanism can account for the occurrence of radioadaptive responses. Interestingly, the same mechanism can also explain the existence of the HRS and IRR phenomena, and successfully describe experimentally observed dose-response relationships for a variety of cell types. This indicates that different, seemingly unrelated, low dose phenomena might be connected and driven by common core processes. With respect to the inter-cellular communication mechanism, we find that it can also account for the occurrence of radioadaptive responses, indicating redundancy in this respect. The model, however, also suggests that the communication mechanism can be vital for the long term survival of cell populations that are continuously exposed to relatively low levels of radiation, which cannot be achieved with the intracellular mechanism in our model. Experimental tests to address our model predictions are proposed.
Author Summary
The effect of low-dose radiation on cells and tissues is a public health concern, because the human population is exposed to low-dose ionizing radiation coming from a variety of sources, such as cosmic rays, soil radioactivity, environmental contaminations, and various medical procedures. At low doses of radiation, phenomena are observed that do not occur at higher doses, such as radioadaptive responses as well as low-dose hyper-radiosensitivity (HRS) and increased radioresistance (IRR), which are so far not fully understood. Each of these phenomena have been investigated separately, and specific mechanisms have been suggested to explain them. Using mathematical models that are successfully fitted to experimental data under a variety of conditions, we show that a set of basic and documented assumptions about cellular responses to low-dose radiation can explain all three low-dose phenomena, indicating that they are inter-related. According to the model, these phenomena are brought about by the multi-factorial interactions that underlie the population dynamics of the cells involved, and this provides a new framework to understand these responses, and to evaluate the risk to human health posed by exposure to low-dose radiation.
PMCID: PMC3983039  PMID: 24722167
9.  Relative contribution of free-virus and synaptic transmission to the spread of HIV-1 through target cell populations 
Biology Letters  2013;9(1):20121049.
Human immunodeficiency virus can spread through target cells by transmission of cell-free virus or directly from cell-to-cell via formation of virological synapses. Although cell-to-cell transmission has been described as much more efficient than cell-free infection, the relative contribution of the two transmission pathways to virus growth during multiple rounds of replication remains poorly defined. Here, we fit a mathematical model to previously published and newly generated in vitro data, and determine that free-virus and synaptic transmission contribute approximately equally to the growth of the virus population.
PMCID: PMC3565528  PMID: 23269844
human immunodeficiency virus; mathematical model; virological synapse
10.  Dependence of the Firearm-Related Homicide Rate on Gun Availability: A Mathematical Analysis 
PLoS ONE  2013;8(7):e71606.
In the USA, the relationship between the legal availability of guns and the firearm-related homicide rate has been debated. It has been argued that unrestricted gun availability promotes the occurrence of firearm-induced homicides. It has also been pointed out that gun possession can protect potential victims when attacked. This paper provides a first mathematical analysis of this tradeoff, with the goal to steer the debate towards arguing about assumptions, statistics, and scientific methods. The model is based on a set of clearly defined assumptions, which are supported by available statistical data, and is formulated axiomatically such that results do not depend on arbitrary mathematical expressions. According to this framework, two alternative scenarios can minimize the gun-related homicide rate: a ban of private firearms possession, or a policy allowing the general population to carry guns. Importantly, the model identifies the crucial parameters that determine which policy minimizes the death rate, and thus serves as a guide for the design of future epidemiological studies. The parameters that need to be measured include the fraction of offenders that illegally possess a gun, the degree of protection provided by gun ownership, and the fraction of the population who take up their right to own a gun and carry it when attacked. Limited data available in the literature were used to demonstrate how the model can be parameterized, and this preliminary analysis suggests that a ban of private firearm possession, or possibly a partial reduction in gun availability, might lower the rate of firearm-induced homicides. This, however, should not be seen as a policy recommendation, due to the limited data available to inform and parameterize the model. However, the model clearly defines what needs to be measured, and provides a basis for a scientific discussion about assumptions and data.
PMCID: PMC3724776  PMID: 23923062
11.  Synaptic transmission and the susceptibility of HIV infection to anti-viral drugs 
Scientific Reports  2013;3:2103.
Cell-to-cell viral transmission via virological synapses has been argued to reduce susceptibility of the virus population to anti-viral drugs through multiple infection of cells, contributing to low-level viral persistence during therapy. Using a mathematical framework, we examine the role of synaptic transmission in treatment susceptibility. A key factor is the relative probability of individual virions to infect a cell during free-virus and synaptic transmission, a currently unknown quantity. If this infection probability is higher for free-virus transmission, then treatment susceptibility is lowest if one virus is transferred per synapse, and multiple infection of cells increases susceptibility. In the opposite case, treatment susceptibility is minimized for an intermediate number of virions transferred per synapse. Hence, multiple infection via synapses does not simply lower treatment susceptibility. Without further experimental investigations, one cannot conclude that synaptic transmission provides an additional mechanism for the virus to persist at low levels during anti-viral therapy.
PMCID: PMC3696900  PMID: 23811684
12.  Evolutionary dynamics of giant viruses and their virophages 
Ecology and Evolution  2013;3(7):2103-2115.
Giant viruses contain large genomes, encode many proteins atypical for viruses, replicate in large viral factories, and tend to infect protists. The giant virus replication factories can in turn be infected by so called virophages, which are smaller viruses that negatively impact giant virus replication. An example is Mimiviruses that infect the protist Acanthamoeba and that are themselves infected by the virophage Sputnik. This study examines the evolutionary dynamics of this system, using mathematical models. While the models suggest that the virophage population will evolve to increasing degrees of giant virus inhibition, it further suggests that this renders the virophage population prone to extinction due to dynamic instabilities over wide parameter ranges. Implications and conditions required to avoid extinction are discussed. Another interesting result is that virophage presence can fundamentally alter the evolutionary course of the giant virus. While the giant virus is predicted to evolve toward increasing its basic reproductive ratio in the absence of the virophage, the opposite is true in its presence. Therefore, virophages can not only benefit the host population directly by inhibiting the giant viruses but also indirectly by causing giant viruses to evolve toward weaker phenotypes. Experimental tests for this model are suggested.
PMCID: PMC3728950  PMID: 23919155
Evolutionary dynamics; giant viruses; mathematical models; virophages
13.  Methylation kinetics and CpG-island methylator phenotyope status in colorectal cancer cell lines 
Biology Direct  2013;8:14.
Hypermethylation of CpG islands is thought to contribute to carcinogenesis through the inactivation of tumor suppressor genes. Tumor cells with relatively high levels of CpG island methylation are considered CpG island methylator phenotypes (CIMP). The mechanisms that are responsible for regulating the activity of de novo methylation are not well understood.
We quantify and compare de novo methylation kinetics in CIMP and non-CIMP colon cancer cell lines in the context of different loci, following 5-aza-2’deoxycytidine (5-AZA)-mediated de-methylation of cells. In non-CIMP cells, a relatively fast rate of re-methylation is observed that starts with a certain time delay after cessation of 5-AZA treatment. CIMP cells, on the other hand, start re-methylation without a time delay but at a significantly slower rate. A mathematical model can account for these counter-intuitive results by assuming negative feedback regulation of de novo methylation activity and by further assuming that this regulation is corrupted in CIMP cells. This model further suggests that when methylation levels have grown back to physiological levels, de novo methylation activity ceases in non-CIMP cells, while it continues at a constant low level in CIMP cells.
We propose that the faster rate of re-methylation observed in non-CIMP compared to CIMP cells in our study could be a consequence of feedback-mediated regulation of DNA methyl transferase activity. Testing this hypothesis will involve the search for specific feedback regulatory mechanisms involved in the activation of de novo methylation.
Reviewers’ report
This article was reviewed by Georg Luebeck, Tomasz Lipniacki, and Anna Marciniak-Czochra
PMCID: PMC3691599  PMID: 23758948
Methylation kinetics; Methylator phenotype; Methylation rates; Mathematical modeling
14.  Stem Cell Control, Oscillations, and Tissue Regeneration in Spatial and Non-Spatial Models 
Normal human tissue is organized into cell lineages, in which the highly differentiated mature cells that perform tissue functions are the end product of an orderly tissue-specific sequence of divisions that start with stem cells or progenitor cells. Tissue homeostasis and effective regeneration after injuries requires tight regulation of these cell lineages and feedback loops play a fundamental role in this regard. In particular, signals secreted from differentiated cells that inhibit stem cell division and stem cell self-renewal are important in establishing control. In this article we study in detail the cell dynamics that arise from this control mechanism. These dynamics are fundamental to our understanding of cancer, given that tumor initiation requires an escape from tissue regulation. Knowledge on the processes of cellular control can provide insights into the pathways that lead to deregulation and consequently cancer development.
PMCID: PMC3625858  PMID: 23596567
tissue regeneration; cell linage control; tissue stability; mathematical models; cancer
15.  Accelerated crossing of fitness valleys through division of labor and cheating in asexual populations 
Scientific Reports  2012;2:917.
Complex traits can require the accumulation of multiple mutations that are individually deleterious. Their evolution requires a fitness valley to be crossed, which can take relatively long time spans. A new evolutionary mechanism is described that accelerates the emergence of complex phenotypes, based on a “division of labor” game and the occurrence of cheaters. If each intermediate mutation leads to a product that can be shared with others, the complex type can arise relatively quickly as an emergent property among cooperating individuals, without any given individual having to accumulate all mutations. Moreover, the emergence of cheaters that destroy cooperative interactions can lead to the emergence of individuals that have accumulated all necessary mutations on a time scale that is significantly faster than observed in the absence of cooperation and cheating. Application of this mechanism to somatic and microbial evolution is discussed, including evolutionary processes in tumors, biofilms, and viral infections.
PMCID: PMC3512085  PMID: 23209877
16.  Effect of Synaptic Transmission on Viral Fitness in HIV Infection 
PLoS ONE  2012;7(11):e48361.
HIV can spread through its target cell population either via cell-free transmission, or by cell-to-cell transmission, presumably through virological synapses. Synaptic transmission entails the transfer of tens to hundreds of viruses per synapse, a fraction of which successfully integrate into the target cell genome. It is currently not understood how synaptic transmission affects viral fitness. Using a mathematical model, we investigate how different synaptic transmission strategies, defined by the number of viruses passed per synapse, influence the basic reproductive ratio of the virus, R0, and virus load. In the most basic scenario, the model suggests that R0 is maximized if a single virus particle is transferred per synapse. R0 decreases and the infection eventually cannot be maintained for larger numbers of transferred viruses, because multiple infection of the same cell wastes viruses that could otherwise enter uninfected cells. To explain the relatively large number of HIV copies transferred per synapse, we consider additional biological assumptions under which an intermediate number of viruses transferred per synapse could maximize R0. These include an increased burst size in multiply infected cells, the saturation of anti-viral factors upon infection of cells, and rate limiting steps during the process of synapse formation.
PMCID: PMC3499495  PMID: 23166585
17.  Complex Spatial Dynamics of Oncolytic Viruses In Vitro: Mathematical and Experimental Approaches 
PLoS Computational Biology  2012;8(6):e1002547.
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.
Author Summary
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.
PMCID: PMC3375216  PMID: 22719239
18.  Increased burst size in multiply infected cells can alter basic virus dynamics 
Biology Direct  2012;7:16.
The dynamics of viral infections have been studied extensively in a variety of settings, both experimentally and with mathematical models. The majority of mathematical models assumes that only one virus can infect a given cell at a time. It is, however, clear that especially in the context of high viral load, cells can become infected with multiple copies of a virus, a process called coinfection. This has been best demonstrated experimentally for human immunodeficiency virus (HIV), although it is thought to be equally relevant for a number of other viral infections. In a previously explored mathematical model, the viral output from an infected cell does not depend on the number of viruses that reside in the cell, i.e. viral replication is limited by cellular rather than viral factors. In this case, basic virus dynamics properties are not altered by coinfection.
Here, we explore the alternative assumption that multiply infected cells are characterized by an increased burst size and find that this can fundamentally alter model predictions. Under this scenario, establishment of infection may not be solely determined by the basic reproductive ratio of the virus, but can depend on the initial virus load. Upon infection, the virus population need not follow straight exponential growth. Instead, the exponential rate of growth can increase over time as virus load becomes larger. Moreover, the model suggests that the ability of anti-viral drugs to suppress the virus population can depend on the virus load upon initiation of therapy. This is because more coinfected cells, which produce more virus, are present at higher virus loads. Hence, the degree of drug resistance is not only determined by the viral genotype, but also by the prevalence of coinfected cells.
Our work shows how an increased burst size in multiply infected cells can alter basic infection dynamics. This forms the basis for future experimental testing of model assumptions and predictions that can distinguish between the different scenarios.
This article was reviewed by RJdeB, RMR and MK.
PMCID: PMC3482397  PMID: 22569346
Multiple infection of cells; Increased burst size; HIV; Mathematical models; Virus dynamics
19.  Multiploid Inheritance of HIV-1 during Cell-to-Cell Infection ▿ †  
Journal of Virology  2011;85(14):7169-7176.
During cell-to-cell transmission of human immunodeficiency virus type 1 (HIV-1), many viral particles can be simultaneously transferred from infected to uninfected CD4 T cells through structures called virological synapses (VS). Here we directly examine how cell-free and cell-to-cell infections differ from infections initiated with cell-free virus in the number of genetic copies that are transmitted from one generation to the next, i.e., the genetic inheritance. Following exposure to HIV-1-expressing cells, we show that target cells with high viral uptake are much more likely to become infected. Using T cells that coexpress distinct fluorescent HIV-1 variants, we show that multiple copies of HIV-1 can be cotransmitted across a single VS. In contrast to cell-free HIV-1 infection, which titrates with Poisson statistics, the titration of cell-associated HIV-1 to low rates of overall infection generates a constant fraction of the newly infected cells that are cofluorescent. Triple infection was also readily detected when cells expressing three fluorescent viruses were used as donor cells. A computational model and a statistical model are presented to estimate the degree to which cofluorescence underestimates coinfection frequency. Lastly, direct detection of HIV-1 proviruses using fluorescence in situ hybridization confirmed that significantly more HIV-1 DNA copies are found in primary T cells infected with cell-associated virus than in those infected with cell-free virus. Together, the data suggest that multiploid inheritance is common during cell-to-cell HIV-1 infection. From this study, we suggest that cell-to-cell infection may explain the high copy numbers of proviruses found in infected cells in vivo and may provide a mechanism through which HIV preserves sequence heterogeneity in viral quasispecies through genetic complementation.
PMCID: PMC3126592  PMID: 21543479
20.  Heterogeneity in chronic myeloid leukaemia dynamics during imatinib treatment: role of immune responses 
Previous studies have shown that during imatinib therapy, the decline of chronic myeloid leukaemia BCR-ABL transcript numbers involves a fast phase followed by a slow phase in averaged datasets. Drug resistance leads to regrowth. In this paper, variation of treatment responses between patients is examined. A significant positive correlation is found between slopes of the fast and the slow phase of decline. A significant negative correlation is found between slopes of the slow phase of decline and the regrowth phase. No correlation is found between slopes of the fast phase of decline and the regrowth phase. A mathematical model that is successfully fitted to diverse clinical profiles explains these correlations by invoking the immune response as a key determinant of tumour decline during treatment. Boosting immunity during drug therapy could enhance the response to treatment in patients.
PMCID: PMC2871874  PMID: 20147328
mathematical models; population dynamics; immune system; dynamics; cancer treatment dynamics
21.  ODE models for oncolytic virus dynamics 
Journal of theoretical biology  2010;263(4):530-543.
Replicating oncolytic viruses are able to infect and lyse cancer cells and spread through the tumor, while leaving normal cells largely unharmed. This makes them potentially useful in cancer therapy, and a variety of viruses have shown promising results in clinical trials. Nevertheless, consistent success remains elusive and the correlates of success have been the subject of investigation, both from an experimental and a mathematical point of view. Mathematical modeling of oncolytic virus therapy is often limited by the fact that the predicted dynamics depend strongly on particular mathematical terms in the model, the nature of which remain uncertain. We aim to address this issue in the context of ODE modeling, by formulating a general computational framework that is independent of particular mathematical expressions. By analyzing this framework, we find some new insights into the conditions for successful virus therapy. We find that depending on our assumptions about the virus spread, there can be two distinct types of dynamics. In models of the first type (the “fast spread” models), we predict that the viruses can eliminate the tumor if the viral replication rate is sufficiently high. The second type of models is characterized by a suboptimal spread (the “slow spread” models). For such models, the simulated treatment may fail, even for very high viral replication rates. Our methodology can be used to study the dynamics of many biological systems, and thus has implications beyond the study of virus therapy of cancers.
PMCID: PMC2839021  PMID: 20085772
22.  Effect of different modes of viral spread on the dynamics of multiply infected cells in human immunodeficiency virus infection 
Infection of individual cells with more than one HIV particle is an important feature of HIV replication, which may contribute to HIV pathogenesis via the occurrence of recombination, viral complementation and other outcomes that influence HIV replication and evolutionary dynamics. A previous mathematical model of co-infection has shown that the number of cells infected with i viruses correlates with the ith power of the singly infected cell population, and this has partly been observed in experiments. This model, however, assumed that virus spread from cell to cell occurs only via free virus particles, and that viruses and cells mix perfectly. Here, we introduce a cellular automaton model that takes into account different modes of virus spread among cells, including cell to cell transmission via the virological synapse, and spatially constrained virus spread. In these scenarios, it is found that the number of multiply infected cells correlates linearly with the number of singly infected cells, meaning that co-infection plays a greater role at lower virus loads. The model further indicates that current experimental systems that are used to study co-infection dynamics fail to reflect the true dynamics of multiply infected cells under these specific assumptions, and that new experimental techniques need to be designed to distinguish between the different assumptions.
PMCID: PMC3033025  PMID: 20659927
HIV; multiple infection; mathematical model; spatial; virus spread
23.  Use of oncolytic viruses for the eradication of drug-resistant cancer cells 
Targeted therapy using small-molecule inhibitors is a promising new therapy approach against cancer, but drug-resistant mutants present an obstacle to success. Oncolytic virus therapy, where viruses replicate specifically in cancer cells and kill them, is another promising therapy approach against cancer. While encouraging results have been observed in clinical trials, consistent success has not been possible so far. Based on a computational framework, I report that even if oncolytic virus therapy fails to eradicate a cancer, it can have the potential to eradicate the sub-population of drug-resistant cancer cells. Once this has occurred, targeted drug therapy can be used to induce cancer remission. For this to work, a drug resistance mutation must confer a certain fitness cost to the cell, as has been documented in the literature. The reason for this finding is that in the presence of a shared virus, the faster growing (drug-sensitive) cell population produces an amount of virus that is too much for the slower growing (drug-resistant) cell population to survive. This is derived from a population dynamic principle known as apparent competition. Therefore, a sequential combination of oncolytic virus and targeted therapies can overcome major weaknesses of either approach alone.
PMCID: PMC2658788  PMID: 18664430
mathematical models; oncolytic virus therapy; apparent competition
24.  Immunity and protection by live attenuated HIV/SIV vaccines 
Virology  2008;378(2):299-305.
Live attenuated virus vaccines have shown the greatest potential to protect against simian immunodeficiency virus (SIV) infection, a model for human immunodeficiency virus (HIV). Immunity against the vaccine virus is thought to mediate protection. However, it is shown computationally that the opposite might be true. According to the model, the initial growth of the challenge strain, its peak load, and its potential to be pathogenic is higher if immunity against the vaccine virus is stronger. This is because the initial growth of the challenge strain is mainly determined by virus competition rather than immune suppression. The stronger the immunity against the vaccine strain, the weaker its competitive ability relative to the challenge strain, and the lower the level of protection. If the vaccine virus does protect the host against a challenge, it is because the competitive interactions between the viruses inhibit the initial growth of the challenge strain. According to these arguments, an inverse correlation between the level of attenuation and the level of protection is expected, and this has indeed been observed in experimental data.
PMCID: PMC2613491  PMID: 18586297
HIV; vaccine; live attenuated; protection; theoretical biology; dynamics; mathematical model
25.  Model-driven Approaches for in Vitro Combination Therapy using ONYX-015 Replicating Oncolytic Adenovirus 1 
Replicating genetically modified adenoviruses have shown promise as a new treatment approach against cancer. Recombinant adenoviruses replicate only in cancer cells which contain certain mutations, such as the loss of functional p53, as is the case in the virus ONYX-015. The successful entry of the viral particle into target cells is strongly dependent on the presence of the main receptor for adenovirus, the coxsackie- and adeno-virus receptor (CAR). This receptor is frequently down-regulated in highly malignant cells, rendering this population less vulnerable to viral attack. It has been shown that use of MEK inhibitors can up-regulate CAR expression, resulting in enhanced adenovirus entry into the cells. However, inhibition of MEK results in G1 cell cycle arrest, rendering infected cells temporarily unable to produce virus. This forces a tradeo1. While drug mediated up-regulation of CAR enhances virus entry into cancer cells, the consequent cell cycle arrest inhibits production of new virus particles and the replication of the virus. Optimal control-based schedules of MEK inhibitor application should increase the efficacy of this treatment, maximizing the overall tumor toxicity by exploiting the dynamics of CAR expression and viral production. We introduce a mathematical model of these dynamics and show simple optimal control based strategies which motivate this approach.
PMCID: PMC2712658  PMID: 17095020
Oncolytic Viruses; ONYX-015; Optimal Control; Mathematical Modeling

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