Mutant fixation in the presence of a natural enemy

The literature about mutant invasion and fixation typically assumes populations to exist in isolation from their ecosystem. Yet, populations are part of ecological communities, and enemy-victim (e.g. predator-prey or pathogen-host) interactions are particularly common. We use spatially explicit, computational pathogen-host models (with wild-type and mutant hosts) to re-visit the established theory about mutant fixation, where the pathogen equally attacks both wild-type and mutant individuals. Mutant fitness is assumed to be unrelated to infection. We find that pathogen presence substantially weakens selection, increasing the fixation probability of disadvantageous mutants and decreasing it for advantageous mutants. The magnitude of the effect rises with the infection rate. This occurs because infection induces spatial structures, where mutant and wild-type individuals are mostly spatially separated. Thus, instead of mutant and wild-type individuals competing with each other, it is mutant and wild-type "patches" that compete, resulting in smaller fitness differences and weakened selection. This implies that the deleterious mutant burden in natural populations might be higher than expected from traditional theory.

The role of migration in mutant dynamics in fragmented populations

Mutant dynamics in fragmented populations have been studied extensively in evolutionary biology. Yet, open questions remain, both experimentally and theoretically. Some of the fundamental properties predicted by models still need to be addressed experimentally. We contribute to this by using a combination of experiments and theory to investigate the role of migration in mutant distribution. In the case of neutral mutants, while the mean frequency of mutants is not influenced by migration, the probability distribution is. To address this empirically, we performed in vitro experiments, where mixtures of GFP-labelled ("mutant") and non-labelled ("wid-type") murine cells were grown in wells (demes), and migration was mimicked via cell transfer from well to well. In the presence of migration, we observed a change in the skewedness of the distribution of the mutant frequencies in the wells, consistent with previous and our own model predictions. In the presence of de novo mutant production, we used modelling to investigate the level at which disadvantageous mutants are predicted to exist, which has implications for the adaptive potential of the population in case of an environmental change. In panmictic populations, disadvantageous mutants can persist around a steady state, determined by the rate of mutant production and the selective disadvantage (selection-mutation balance). In a fragmented system that consists of demes connected by migration, a steady-state persistence of disadvantageous mutants is also observed, which, however, is fundamentally different from the mutation-selection balance and characterized by higher mutant levels. The increase in mutant frequencies above the selection-mutation balance can be maintained in small ( N < N c ) demes as long as the migration rate is sufficiently small. The migration rate above which the mutants approach the selection-mutation balance decays exponentially with N / N c . The observed increase in the mutant numbers is not explained by the change in the effective population size. Implications for evolutionary processes in diseases are discussed, where the pre-existence of disadvantageous drug-resistant mutant cells or pathogens drives the response of the disease to treatments.

Effect of Human Behavior on the Evolution of Viral Strains During an Epidemic

It is well known in the literature that human behavior can change as a reaction to disease observed in others, and that such behavioral changes can be an important factor in the spread of an epidemic. It has been noted that human behavioral traits in disease avoidance are under selection in the presence of infectious diseases. Here, we explore a complementary trend: the pathogen itself might experience a force of selection to become less “visible,” or less “symptomatic,” in the presence of such human behavioral trends. Using a stochastic SIR agent-based model, we investigated the co-evolution of two viral strains with cross-immunity, where the resident strain is symptomatic while the mutant strain is asymptomatic. We assumed that individuals exercised self-regulated social distancing (SD) behavior if one of their neighbors was infected with a symptomatic strain. We observed that the proportion of asymptomatic carriers increased over time with a stronger effect corresponding to higher levels of self-regulated SD. Adding mandated SD made the effect more significant, while the existence of a time-delay between the onset of infection and the change of behavior reduced the advantage of the asymptomatic strain. These results were consistent under random geometric networks, scale-free networks, and a synthetic network that represented the social behavior of the residents of New Orleans.

Aspirin's effect on kinetic parameters of cells contributes to its role in reducing incidence of advanced colorectal adenomas, shown by a multiscale computational study

Aspirin intake has been shown to lead to significant protection against colorectal cancer, for example with an up to twofold reduction in colorectal adenoma incidence rates at higher doses. The mechanisms contributing to protection are not yet fully understood. While aspirin is an anti-inflammatory drug and can thus influence the tumor microenvironment, in vitro and in vivo experiments have recently shown that aspirin can also have a direct effect on cellular kinetics and fitness. It reduces the rate of tumor cell division and increases the rate of cell death. The question arises whether such changes in cellular fitness are sufficient to significantly contribute to the epidemiologically observed protection. To investigate this, we constructed a class of mathematical models of in vivo evolution of advanced adenomas, parameterized it with available estimates, and calculated population level incidence. Fitting the predictions to age incidence data revealed that only a model that included colonic crypt competition can account for the observed age-incidence curve. This model was then used to predict modified incidence patterns if cellular kinetics were altered as a result of aspirin treatment. We found that changes in cellular fitness that were within the experimentally observed ranges could reduce advanced adenoma incidence by a sufficient amount to account for age incidence data in aspirin-treated patient cohorts. While the mechanisms that contribute to the protective effect of aspirin are likely complex and multi-factorial, our study demonstrates that direct aspirin-induced changes of tumor cell fitness can significantly contribute to epidemiologically observed reduced incidence patterns.

Latency reversal plus natural killer cells diminish HIV reservoir in vivo

HIV is difficult to eradicate due to the persistence of a long-lived reservoir of latently infected cells. Previous studies have shown that natural killer cells are important to inhibiting HIV infection, but it is unclear whether the administration of natural killer cells can reduce rebound viremia when anti-retroviral therapy is discontinued. Here we show the administration of allogeneic human peripheral blood natural killer cells delays viral rebound following interruption of anti-retroviral therapy in humanized mice infected with HIV-1. Utilizing genetically barcoded virus technology, we show these natural killer cells efficiently reduced viral clones rebounding from latency. Moreover, a kick and kill strategy comprised of the protein kinase C modulator and latency reversing agent SUW133 and allogeneic human peripheral blood natural killer cells during anti-retroviral therapy eliminated the viral reservoir in a subset of mice. Therefore, combinations utilizing latency reversal agents with targeted cellular killing agents may be an effective approach to eradicating the viral reservoir.

A hybrid stochastic-deterministic approach to explore multiple infection and evolution in HIV

To study viral evolutionary processes within patients, mathematical models have been instrumental. Yet, the need for stochastic simulations of minority mutant dynamics can pose computational challenges, especially in heterogeneous systems where very large and very small sub-populations coexist. Here, we describe a hybrid stochastic-deterministic algorithm to simulate mutant evolution in large viral populations, such as acute HIV-1 infection, and further include the multiple infection of cells. We demonstrate that the hybrid method can approximate the fully stochastic dynamics with sufficient accuracy at a fraction of the computational time, and quantify evolutionary end points that cannot be expressed by deterministic models, such as the mutant distribution or the probability of mutant existence at a given infected cell population size. We apply this method to study the role of multiple infection and intracellular interactions among different virus strains (such as complementation and interference) for mutant evolution. Multiple infection is predicted to increase the number of mutants at a given infected cell population size, due to a larger number of infection events. We further find that viral complementation can significantly enhance the spread of disadvantageous mutants, but only in select circumstances: it requires the occurrence of direct cell-to-cell transmission through virological synapses, as well as a substantial fitness disadvantage of the mutant, most likely corresponding to defective virus particles. This, however, likely has strong biological consequences because defective viruses can carry genetic diversity that can be incorporated into functional virus genomes via recombination. Through this mechanism, synaptic transmission in HIV might promote virus evolvability.

The evolution of human immunodeficiency virus within patients is an important part of the disease process. In particular, the presence of mutants that are resistant against anti-viral drugs can result in challenges to the long-term control of the infection. To study disease progression, computer simulations have been useful. However, in some cases these simulations can be difficult because of the complexity of the model. Here, we use a computational complexity reducing algorithm to simulate mutant dynamics in large populations, which can approximate the full model at a fraction of the time. The use of this algorithm allows us to study different transmission methods, viral processes that occur between virus strains within individual cells, and important quantities such as the mutant distribution or the probability of mutant existence at a given infected cell population size. We find that the direct synaptic cell-to-cell transmission of the virus through virological synapses can have strong biological consequences because it can promote potentially defective viruses that carry genetic diversity which can be incorporated into functional virus genomes during infection. Through this process, synaptic transmission in human immunodeficiency virus might promote virus evolvability.

Multi-scale network targeting: A holistic systems-biology approach to cancer treatment

The vulnerabilities of cancer at the cellular and, recently, with the introduction of immunotherapy, at the tissue level, have been exploited with variable success. Evaluating the cancer system vulnerabilities at the organismic level through analysis of network topology and network dynamics can potentially predict novel anti-cancer drug targets directed at the macroscopic cancer networks. Theoretical work analyzing the properties and the vulnerabilities of the multi-scale network of cancer needs to go hand-in-hand with experimental research that uncovers the biological nature of the relevant networks and reveals new targetable vulnerabilities. It is our hope that attacking cancer on different spatial scales, in a concerted integrated approach, may present opportunities for novel ways to prevent treatment resistance.

Network models and the interpretation of prolonged infection plateaus in the COVID19 pandemic

Non-pharmaceutical intervention measures, such as social distancing, have so far been the only means to slow the spread of SARS-CoV-2. In the United States, strict social distancing during the first wave of virus spread has resulted in different types of infection dynamics. In some states, such as New York, extensive infection spread was followed by a pronounced decline of infection levels. In other states, such as California, less infection spread occurred before strict social distancing, and a different pattern was observed. Instead of a pronounced infection decline, a long-lasting plateau is evident, characterized by similar daily new infection levels. Here we show that network models, in which individuals and their social contacts are explicitly tracked, can reproduce the plateau if network connections are cut due to social distancing measures. The reason is that in networks characterized by a 2D spatial structure, infection tends to spread quadratically with time, but as edges are randomly removed, the infection spreads along nearly one-dimensional infection “corridors”, resulting in plateau dynamics. Further, we show that plateau dynamics are observed only if interventions start sufficiently early; late intervention leads to a “peak and decay” pattern. Interestingly, the plateau dynamics are predicted to eventually transition into an infection decline phase without any further increase in social distancing measures. Additionally, the models suggest that a second wave becomes significantly less pronounced if social distancing is only relaxed once the dynamics have transitioned to the decline phase. The network models analyzed here allow us to interpret and reconcile different infection dynamics during social distancing observed in various US states.

Quantifying the dynamics of viral recombination during free virus and cell-to-cell transmission in HIV-1 infection

Recombination has been shown to contribute to human immunodeficiency virus-1 (HIV-1) evolution in vivo, but the underlying dynamics are extremely complex, depending on the nature of the fitness landscapes and of epistatic interactions. A less well-studied determinant of recombinant evolution is the mode of virus transmission in the cell population. HIV-1 can spread by free virus transmission, resulting largely in singly infected cells, and also by direct cell-to-cell transmission, resulting in the simultaneous infection of cells with multiple viruses. We investigate the contribution of these two transmission pathways to recombinant evolution, by applying mathematical models to in vitro experimental data on the growth of fluorescent reporter viruses under static conditions (where both transmission pathways operate), and under gentle shaking conditions, where cell-to-cell transmission is largely inhibited. The parameterized mathematical models are then used to extrapolate the viral evolutionary dynamics beyond the experimental settings. Assuming a fixed basic reproductive ratio of the virus (independent of transmission pathway), we find that recombinant evolution is fastest if virus spread is driven only by cell-to-cell transmission and slows down if both transmission pathways operate. Recombinant evolution is slowest if all virus spread occurs through free virus transmission. This is due to cell-to-cell transmission 1, increasing infection multiplicity; 2, promoting the co-transmission of different virus strains from cell to cell; and 3, increasing the rate at which point mutations are generated as a result of more reverse transcription events. This study further resulted in the estimation of various parameters that characterize these evolutionary processes. For example, we estimate that during cell-to-cell transmission, an average of three viruses successfully integrated into the target cell, which can significantly raise the infection multiplicity compared to free virus transmission. In general, our study points towards the importance of infection multiplicity and cell-to-cell transmission for HIV evolution.

Role of high-dose exposure in transmission hot zones as a driver of SARS-CoV-2 dynamics

Epidemiological data about SARS-CoV-2 spread indicate that the virus is not transmitted uniformly in the population. The transmission tends to be more effective in select settings that involve exposure to relatively high viral dose, such as in crowded indoor settings, assisted living facilities, prisons or food processing plants. To explore the effect on infection dynamics, we describe a new mathematical model where transmission can occur (i) in the community at large, characterized by low-dose exposure and mostly mild disease, and (ii) in so-called transmission hot zones, characterized by high-dose exposure that can be associated with more severe disease. The model yields different types of epidemiological dynamics, depending on the relative importance of hot zone and community transmission. Interesting dynamics occur if the rate of virus release/deposition from severely infected people is larger than that of mildly infected individuals. Under this assumption, we find that successful infection spread can hinge upon high-dose hot zone transmission, yet the majority of infections are predicted to occur in the community at large with mild disease. In this regime, residual hot zone transmission can account for continued virus spread during community lockdowns, and the suppression of hot zones after community interventions are relaxed can cause a prolonged lack of infection resurgence following the reopening of society. This gives rise to the notion that targeted interventions specifically reducing virus transmission in the hot zones have the potential to suppress overall infection spread, including in the community at large. Epidemiological trends in the USA and Europe are interpreted in light of this model.

Beyond the pair approximation: Modeling colonization population dynamics

The process of range expansion (colonization) is one of the basic types of biological dynamics, whereby a species grows and spreads outwards, occupying new territories. Spatial modeling of this process is naturally implemented as a stochastic cellular automaton, with individuals occupying nodes on a rectangular grid, births and deaths occurring probabilistically, and individuals only reproducing onto unoccupied neighboring spots. In this paper we derive several approximations that allow prediction of the expected range expansion dynamics, based on the reproduction and death rates. We derive several approximations, where the cellular automaton is described by a system of ordinary differential equations that preserves correlations among neighboring spots (up to a distance). This methodology allows us to develop accurate approximations of the population size and the expected spatial shape, at a fraction of the computational time required to simulate the original stochastic system. In addition, we provide simple formulas for the steady-state population densities for von Neumann and Moore neighborhoods. Finally, we derive concise approximations for the speed of range expansion in terms of the reproduction and death rates, for both types of neighborhoods. The methodology is generalizable to more complex scenarios, such as different interaction ranges and multiple-species systems.

Effect of feedback regulation on stem cell fractions in tissues and tumors: Understanding chemoresistance in cancer

While resistance mutations are often implicated in the failure of cancer therapy, lack of response also occurs without such mutants. In bladder cancer mouse xenografts, repeated chemotherapy cycles have resulted in cancer stem cell (CSC) enrichment, and consequent loss of therapy response due to the reduced susceptibility of CSCs to drugs. A particular feedback loop present in the xenografts has been shown to promote CSC enrichment in this system. Yet, many other regulatory loops might also be operational and might promote CSC enrichment. Their identification is central to improving therapy response. Here, we perform a comprehensive mathematical analysis to define what types of regulatory feedback loops can and cannot contribute to CSC enrichment, providing guidance to the experimental identification of feedback molecules. We derive a formula that reveals whether or not the cell population experiences CSC enrichment over time, based on the properties of the feedback. We find that negative feedback on the CSC division rate or positive feedback on differentiated cell death rate can lead to CSC enrichment. Further, the feedback mediators that achieve CSC enrichment can be secreted by either CSCs or by more differentiated cells. The extent of enrichment is determined by the CSC death rate, the CSC self-renewal probability, and by feedback strength. Defining these general characteristics of feedback loops can guide the experimental screening for and identification of feedback mediators that can promote CSC enrichment in bladder cancer and potentially other tumors. This can help understand and overcome the phenomenon of CSC-based therapy resistance.

A comprehensive in vivo and mathematic modeling-based kinetic characterization for aspirin-induced chemoprevention in colorectal cancer

Accumulating evidence suggests that aspirin has anti-tumorigenic properties in colorectal cancer (CRC). Herein, we undertook a comprehensive and systematic series of in vivo animal experiments followed by 3D-mathematical modeling to determine the kinetics of aspirin's anti-cancer effects on CRC growth. In this study, CRC xenografts were generated using four CRC cell lines with and without PIK3CA mutations and microsatellite instability, and the animals were administered with various aspirin doses (0, 15, 50, and 100 mg/kg) for 2 weeks. Cell proliferation, apoptosis and protein expression were evaluated, followed by 3D-mathematical modeling analysis to estimate cellular division and death rates and their impact on aspirin-mediated changes on tumor growth. We observed that aspirin resulted in a dose-dependent decrease in the cell division rate, and a concomitant increase in the cell death rates in xenografts from all cell lines. Aspirin significantly inhibited cell proliferation as measured by Ki67 staining (P < 0.05-0.01). The negative effect of aspirin on the rate of tumor cell proliferation was more significant in xenograft tumors derived from PIK3CA mutant versus wild-type cells. A computational model of 3D-tumor growth suggests that the growth inhibitory effect of aspirin on the tumor growth kinetics is due to a reduction of tumor colony formation, and that this effect is sufficiently strong to be an important contributor to the reduction of CRC incidence in aspirin-treated patients. In conclusion, we provide a detailed kinetics of aspirin-mediated inhibition of tumor cell proliferation, which support the epidemiological data for the observed protective effect of aspirin in CRC patients.

Role of high-dose exposure in transmission hot zones as a driver of SARS-CoV2 dynamics

Epidemiological data on the spread of SARS-CoV-2 in the absence and presence of various non-pharmaceutical interventions indicate that the virus is not transmitted uniformly in the population. Transmission tends to be more effective in select settings that involve exposure to relatively high viral dose, such as in crowded indoor settings, assisted living facilities, prisons, or food processing plants. To explore the effect on infection dynamics, we describe a new mathematical model where transmission can occur (i) in the community at large, characterized by low dose exposure and mostly mild disease, and (ii) in so called transmission hot zones, characterized by high dose exposure that can be associated with more severe disease. Interestingly, we find that successful infection spread can hinge upon high-dose hot zone transmission, yet the majority of infections are predicted to occur in the community at large with mild disease. This gives rise to the prediction that targeted interventions that specifically reduce virus transmission in the hot zones (but not in the community at large) have the potential to suppress overall infection spread, including in the community at large. The model can further reconcile seemingly contradicting epidemiological observations. While in some locations like California, strict stay-home orders failed to significantly reduce infection prevalence, in other locations, such as New York and several European countries, stay-home orders lead to a pronounced fall in infection levels, which remained suppressed for some months after re-opening of society. Differences in hot zone transmission levels during and after social distancing interventions can account for these diverging infection patterns. These modeling results warrant further epidemiological investigations into the role of high dose hot zone transmission for the maintenance of SARS-CoV-2 spread.

Patterns of the COVID-19 pandemic spread around the world: exponential versus power laws

We have analysed the COVID-19 epidemic data of more than 174 countries (excluding China) in the period between 22 January and 28 March 2020. We found that some countries (such as the USA, the UK and Canada) follow an exponential epidemic growth, while others (like Italy and several other European countries) show a power law like growth. Regardless of the best fitting law, many countries can be shown to follow a common trajectory that is similar to Italy (the epicentre at the time of analysis), but with varying degrees of delay. We found that countries with 'younger' epidemics, i.e. countries where the epidemic started more recently, tend to exhibit more exponential like behaviour, while countries that were closer behind Italy tend to follow a power law growth. We hypothesize that there is a universal growth pattern of this infection that starts off as exponential and subsequently becomes more power law like. Although it cannot be excluded that this growth pattern is a consequence of social distancing measures, an alternative explanation is that it is an intrinsic epidemic growth law, dictated by a spatially distributed community structure, where the growth in individual highly mixed communities is exponential but the longer term, local geographical spread (in the absence of global mixing) results in a power law. This is supported by computer simulations of a metapopulation model that gives rise to predictions about the growth dynamics that are consistent with correlations found in the epidemiological data. Therefore, seeing a deviation from straight exponential growth may be a natural progression of the epidemic in each country. On the practical side, this indicates that (i) even in the absence of strict social distancing interventions, exponential growth is not an accurate predictor of longer term infection spread, and (ii) a deviation from exponential spread and a reduction of estimated doubling times do not necessarily indicate successful interventions, which are instead indicated by a transition to a reduced power or by a deviation from power law behaviour.

Mutant Evolution in Spatially Structured and Fragmented Expanding Populations

Mutant evolution in spatially structured systems is important for a range of biological systems, but aspects of it still require further elucidation. Adding to previous work, we provide a simple derivation of growth laws that characterize the number of mutants of different relative fitness in expanding populations in spatial models of different dimensionalities. These laws are universal and independent of "microscopic" modeling details. We further study the accumulation of mutants and find that, with advantageous and neutral mutants, more of them are present in spatially structured, compared to well-mixed colonies of the same size. The behavior of disadvantageous mutants is subtle: if they are disadvantageous through a reduction in division rates, the result is the same, and it is the opposite if the disadvantage is due to a death rate increase. Finally, we show that in all cases, the same results are observed in fragmented, nonspatial patch models. This suggests that the patterns observed are the consequence of population fragmentation, and not spatial restrictions per se We provide an intuitive explanation for the complex dependence of disadvantageous mutant evolution on spatial restriction, which relies on desynchronized dynamics in different locations/patches, and plays out differently depending on whether the disadvantage is due to a lower division rate or a higher death rate. Implications for specific biological systems, such as the evolution of drug-resistant cell mutants in cancer or bacterial biofilms, are discussed.

Aspirin and the chemoprevention of cancers: A mathematical and evolutionary dynamics perspective

Epidemiological data indicate that long-term low dose aspirin administration has a protective effect against the occurrence of colorectal cancer, both in sporadic and in hereditary forms of the disease. The mechanisms underlying this protective effect, however, are incompletely understood. The molecular events that lead to protection have been partly defined, but remain to be fully characterized. So far, however, approaches based on evolutionary dynamics have not been discussed much, but can potentially offer important insights. The aim of this review is to highlight this line of investigation and the results that have been obtained. A core observation in this respect is that aspirin has a direct negative impact on the growth dynamics of the cells, by influencing the kinetics of tumor cell division and death. We discuss the application of mathematical models to experimental data to quantify these parameter changes. We then describe further mathematical models that have been used to explore how these aspirin-mediated changes in kinetic parameters influence the probability of successful colony growth versus extinction, and how they affect the evolution of the tumor during aspirin administration. Finally, we discuss mathematical models that have been used to investigate the selective forces that can lead to the rise of mismatch-repair deficient cells in an inflammatory environment, and how this selection can be potentially altered through aspirin-mediated interventions. This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Analytical and Computational Methods > Analytical Methods Analytical and Computational Methods > Computational Methods.

Evolutionary dynamics of culturally transmitted, fertility-reducing traits

Human populations in many countries have undergone a phase of demographic transition, characterized by a major reduction in fertility at a time of increased resource availability. A key stylized fact is that the reduction in fertility is preceded by a reduction in mortality and a consequent increase in population density. Various theories have been proposed to account for the demographic transition process, including maladaptation, increased parental investment in fewer offspring, and cultural evolution. None of these approaches, including formal cultural evolutionary models of the demographic transitions, have addressed a possible direct causal relationship between a reduction in mortality and the subsequent decline in fertility. We provide mathematical models in which low mortality favours the cultural selection of low-fertility traits. This occurs because reduced mortality slows turnover in the model, which allows the cultural transmission advantage of low-fertility traits to outrace their reproductive disadvantage. For mortality to be a crucial determinant of outcome, a cultural transmission bias is required where slow reproducers exert higher social influence. Computer simulations of our models that allow for exogenous variation in the death rate can reproduce the central features of the demographic transition process, including substantial reductions in fertility within only one to three generations. A model assuming continuous evolution of reproduction rates through imitation errors predicts fertility to fall below replacement levels if death rates are sufficiently low. This can potentially explain the very low preferred family sizes in Western Europe.

Effect of synaptic cell-to-cell transmission and recombination on the evolution of double mutants in HIV

Recombination in HIV infection can impact virus evolution in vivo in complex ways, as has been shown both experimentally and mathematically. The effect of free virus versus synaptic, cell-to-cell transmission on the evolution of double mutants, however, has not been investigated. Here, we do so by using a stochastic agent-based model. Consistent with data, we assume spatial constraints for synaptic but not for free-virus transmission. Two important effects of the viral spread mode are observed: (i) for disadvantageous mutants, synaptic transmission protects against detrimental effects of recombination on double mutant persistence. Under free virus transmission, recombination increases double mutant levels for negative epistasis, but reduces them for positive epistasis. This reduction for positive epistasis is much diminished under predominantly synaptic transmission, and recombination can, in fact, lead to increased mutant levels. (ii) The mode of virus spread also directly influences the evolutionary fate of double mutants. For disadvantageous mutants, double mutant production is the predominant driving force, and hence synaptic transmission leads to highest double mutant levels due to increased transmission efficiency. For advantageous mutants, double mutant spread is the most important force, and hence free virus transmission leads to fastest invasion due to better mixing. For neutral mutants, both production and spread of double mutants are important, and hence an optimal mixture of free virus and synaptic transmission maximizes double mutant fractions. Therefore, both free virus and synaptic transmission can enhance or delay double mutant evolution. Implications for drug resistance in HIV are discussed.

Passenger mutations can accelerate tumour suppressor gene inactivation in cancer evolution

Carcinogenesis is an evolutionary process whereby cells accumulate multiple mutations. Besides the 'driver mutations' that cause the disease, cells also accumulate a number of other mutations with seemingly no direct role in this evolutionary process. They are called passenger mutations. While it has been argued that passenger mutations render tumours more fragile due to reduced fitness, the role of passenger mutations remains understudied. Using evolutionary computational models, we demonstrate that in the context of tumour suppressor gene inactivation (and hence fitness valley crossing), the presence of passenger mutations can accelerate the rate of evolution by reducing overall population fitness and increasing the relative fitness of intermediate mutants in the fitness valley crossing pathway. Hence, the baseline rate of tumour suppressor gene inactivation might be faster than previously thought. Conceptually, parallels are found in the field of turbulence and pattern formation, where instabilities can be driven by perturbations that are damped (disadvantageous), but provide a richer set of pathways such that a system can achieve some desired goal more readily. This highlights, through a number of novel parallels, the relevance of physical sciences in oncology.

Environmental spatial and temporal variability and its role in non-favoured mutant dynamics

Understanding how environmental variability (or randomness) affects evolution is of fundamental importance for biology. The presence of temporal or spatial variability significantly affects the competition dynamics in populations, and gives rise to some counterintuitive observations. In this paper, we consider both birth-death (BD) or death-birth (DB) Moran processes, which are set up on a circular or a complete graph. We investigate spatial and temporal variability affecting division and/or death parameters. Assuming that mutant and wild-type fitness parameters are drawn from an identical distribution, we study mutant fixation probability and timing. We demonstrate that temporal and spatial types of variability possess fundamentally different properties. Under temporal randomness, in a completely mixed system, minority mutants experience (i) higher than neutral fixation probability and a higher mean conditional fixation time, if the division rates are affected by randomness and (ii) lower fixation probability and lower mean conditional fixation time if the death rates are affected. Once spatial restrictions are imposed, however, these effects completely disappear, and mutants in a circular graph experience neutral dynamics, but only for the DB update rule in case (i) and for the BD rule in case (ii) above. In contrast to this, in the case of spatially variable environment, both for BD/DB processes, both for complete/circular graph and both for division/death rates affected, minority mutants experience a higher than neutral probability of fixation. Fixation time, however, is increased by randomness on a circle, while it decreases for complete graphs under random division rates. A basic difference between temporal and spatial kinds of variability is the types of correlations that occur in the system. Under temporal randomness, mutants are spatially correlated with each other (they simply have equal fitness values at a given moment of time; the same holds for wild-types). Under spatial randomness, there are subtler, temporal correlations among mutant and wild-type cells, which manifest themselves by cells of each type 'claiming' better spots for themselves. Applications of this theory include cancer generation and biofilm dynamics.

Object-Label-Order Effect When Learning From an Inconsistent Source

Learning in natural environments is often characterized by a degree of inconsistency from an input. These inconsistencies occur, for example, when learning from more than one source, or when the presence of environmental noise distorts incoming information; as a result, the task faced by the learner becomes ambiguous. In this study, we investigate how learners handle such situations. We focus on the setting where a learner receives and processes a sequence of utterances to master associations between objects and their labels, where the source is inconsistent by design: It uses both "correct" and "incorrect" object-label pairings. We hypothesize that depending on the order of presentation, the result of the learning may be different. To this end, we consider two types of symbolic learning procedures: the Object-Label (OL) and the Label-Object (LO) process. In the OL process, the learner is first exposed to the object, and then the label. In the LO process, this order is reversed. We perform experiments with human subjects, and also construct a computational model that is based on a nonlinear stochastic reinforcement learning algorithm. It is observed experimentally that OL learners are generally better at processing inconsistent input compared to LO learners. We show that the patterns observed in the learning experiments can be reproduced in the simulations if the model includes (a) an ability to regularize the input (and also to do the opposite, i.e., undermatch) and (b) an ability to take account of implicit negative evidence (i.e., interactions among different objects/labels). The model suggests that while both types of learners utilize implicit negative evidence in a similar way, there is a difference in regularization patterns: OL learners regularize the input, whereas LO learners undermatch. As a result, OL learners are able to form a more consistent system of image-utterance associations, despite the ambiguous learning task.

Multiple infection of cells changes the dynamics of basic viral evolutionary processes

The infection of cells by multiple copies of a given virus can impact viral evolution in a variety of ways, yet some of the most basic evolutionary dynamics remain underexplored. Using computational models, we investigate how infection multiplicity affects the fixation probability of mutants, the rate of mutant generation, and the timing of mutant invasion. An important insight from these models is that for neutral and disadvantageous phenotypes, rare mutants initially enjoy a fitness advantage in the presence of multiple infection of cells. This arises because multiple infection allows the rare mutant to enter more target cells and to spread faster, while it does not accelerate the spread of the resident wild-type virus. The rare mutant population can increase by entry into both uninfected and wild-type-infected cells, while the established wild-type population can initially only grow through entry into uninfected cells. Following this initial advantageous phase, the dynamics are governed by drift or negative selection, respectively, and a higher multiplicity reduces the chances that mutants fix in the population. Hence, while increased infection multiplicity promotes the presence of neutral and disadvantageous mutants in the short-term, it makes it less likely in the longer term. We show how these theoretical insights can be useful for the interpretation of experimental data on virus evolution at low and high multiplicities. The dynamics explored here provide a basis for the investigation of more complex viral evolutionary processes, including recombination, reassortment, as well as complementary/inhibitory interactions.

Effect of aspirin on tumour cell colony formation and evolution

Aspirin is known to reduce the risk of colorectal cancer (CRC) incidence, but the underlying mechanisms are not fully understood. In a previous study, we quantified the in vitro growth kinetics of different CRC tumour cell lines treated with varying doses of aspirin, measuring the rate of cell division and cell death. Here, we use these measured parameters to calculate the chances of successful clonal expansion and to determine the evolutionary potential of the tumour cell lines in the presence and absence of aspirin. The calculations indicate that aspirin increases the probability that a single tumour cell fails to clonally expand. Further, calculations suggest that aspirin increases the evolutionary potential of an expanding tumour cell colony. An aspirin-treated tumour cell population is predicted to result in the accumulation of more mutations (and is thus more virulent and more difficult to treat) than a cell population of the same size that grew without aspirin. This indicates a potential trade-off between delaying the onset of cancer and increasing its evolutionary potential through chemoprevention. Further work needs to investigate to what extent these findings apply to in vivo settings, and to what degree they contribute to the epidemiologically documented aspirin-mediated protection.

Musical trends and predictability of success in contemporary songs in and out of the top charts

We analyse more than 500 000 songs released in the UK between 1985 and 2015 to understand the dynamics of success (defined as 'making it' into the top charts), correlate success with acoustic features and explore the predictability of success. Several multi-decadal trends have been uncovered. For example, there is a clear downward trend in 'happiness' and 'brightness', as well as a slight upward trend in 'sadness'. Furthermore, songs are becoming less 'male'. Interestingly, successful songs exhibit their own distinct dynamics. In particular, they tend to be 'happier', more 'party-like', less 'relaxed' and more 'female' than most. The difference between successful and average songs is not straightforward. In the context of some features, successful songs pre-empt the dynamics of all songs, and in others they tend to reflect the past. We used random forests to predict the success of songs, first based on their acoustic features, and then adding the 'superstar' variable (informing us whether the song's artist had appeared in the top charts in the near past). This allowed quantification of the contribution of purely musical characteristics in the songs' success, and suggested the time scale of fashion dynamics in popular music.

Quantitative study of color category boundaries

We use World Color Survey (WCS) data to design quantitative methods to study color categorization, with the focus on the "geometric" properties of categories, in particular, on studying their shape, and creating a consistent methodology to identify category boundaries. We introduce the notion of "No Man's Land" and "Some Man's Land" to distinguish color chips that belong to no color category and those that belong to some color category. We introduce a "color-stimulus-strength" function that characterizes color boundaries. While categories may come in a variety of shapes, and their boundaries are nonuniform and can vary in thickness, there are universal patterns that emerge. For example, the boundary-to-category-mass ratio is a decreasing function of category strength (i.e., stronger categories have relatively thinner boundaries), and boundary mass obeys a "square root"-like law as a function of category mass (i.e., roughly speaking, color categories behave like 2D circles). We further identify a relationship between color boundaries and Shannon's entropy, which can be calculated by using the field data of the WCS. We find that depending on the informational content of a given chip, it can belong to three distinct types: (I) strongly belonging to a color category; (II) belonging to a boundary between two or more categories; (III) not belonging to a category or a boundary. The last two cases can be interpreted in terms of evolution and temporal dynamics of color categories.

Spatial evolution of regularization in learned behavior of animals

Stochastic population dynamics of learned traits are studied, where individual learners behave according to a reinforcement learner model, which is a nonlinear version of the Bush-Mosteller model. Depending on a regularization parameter (parameter a), the learners may possess different degrees of overmatching (regularization behavior, 0 ≤ a < 1), frequency matching (corresponding to a=1), or undermatching behavior (a > 1). Both non-spatial and spatial models are considered, to study the interplay of individual heterogeneity of behavior, spatial and temporal effects of learning, and the possibility of emergence of regional culture. In non-spatial models, we observe that populations of individuals learning from each other converge to a universally shared, deterministic rule (either rule "1" or rule "0"), only if they to some extent possess the ability to generalize (a < 1). Otherwise, a low-coherence solution where both rules are used intermittently by everyone, is achieved. If the evolution of the regularization ability is included, then we find that a initially evolves toward lower values, and a shared solution is established when everyone reliably uses the same rule. The spatial (2D) model has two well known limiting cases: if a=0 (the strongest degree of regularization), the model converges to a threshold voter model, and if a=1 (frequency matching), it is equivalent to the discrete diffusion equation. If 0 < a < 1 (the case where individuals regularize), spatial patterns emerge, where patches of different usage of the rule are formed. Smaller values of a lead to sharper and longer lived patches. Values of a < 1 close to unity result in probabilistic outcomes where patches only survive if they are attached to the boundary. Analytical treatment of the 1D case reveals the existence of approximate equilibria that have front structure, where spatially intermittent deterministic usage of one and the other rule are separated by interfaces whose analytical form is derived.

Cooperation-based branching as a mechanism of evolutionary speciation

When performing complex tasks, coexistence of organisms in a shared environment can be achieved by means of different strategies. For example, individuals can evolve to complete all parts of the complex task, choosing self-sufficiency over cooperation. On the other hand, they may choose to split parts of the task and share the products for mutual benefit, such that distinct groups of the organisms specialize on a subset of elementary tasks. In contrast to the existing theory of specialization and task sharing for cells in multicellular organisms (or colonies of social insects), here we describe a mechanism of evolutionary branching which is based on cooperation and division of labor, and where selection happens at the individual level. Using a class of mathematical models and the methodology of adaptive dynamics, we investigate the conditions for such branching into distinct cooperating subgroups to occur. We show that, as long as performing multiple tasks is associated with additional cost, branching occurs for a wide parameter range, and this scenario is stable against the invasion of cheaters. We hypothesize that over time, this can lead to evolutionary speciation. Examples from bacterial evolution and the connection with the Black Queen Hypothesis are discussed. It is our hope that the theory of diversification rooted in cooperation may inspire further ecological research to identify more evolutionary examples consistent with this speciation mechanism.

Optimizing homeostatic cell renewal in hierarchical tissues

In order to maintain homeostasis, mature cells removed from the top compartment of hierarchical tissues have to be replenished by means of differentiation and self-renewal events happening in the more primitive compartments. As each cell division is associated with a risk of mutation, cell division patterns have to be optimized, in order to minimize or delay the risk of malignancy generation. Here we study this optimization problem, focusing on the role of division tree length, that is, the number of layers of cells activated in response to the loss of terminally differentiated cells, which is related to the balance between differentiation and self-renewal events in the compartments. Using both analytical methods and stochastic simulations in a metapopulation-style model, we find that shorter division trees are advantageous if the objective is to minimize the total number of one-hit mutants in the cell population. Longer division trees on the other hand minimize the accumulation of two-hit mutants, which is a more likely evolutionary goal given the key role played by tumor suppressor genes in cancer initiation. While division tree length is the most important property determining mutant accumulation, we also find that increasing the size of primitive compartments helps to delay two-hit mutant generation.

Cells in multicellular organisms are organized hierarchically. A stem cell gives rise to a chain of dividing and progressively differentiating offspring. At the end of this chain (called a lineage) are terminally differentiated cells that perform their function and undergo programmed cell death, to be replaced by new divisions of less differentiated cells. Here we are interested in the design of such lineages. At one extreme, one can imagine that a loss of terminally differentiated cells only results in divisions of cells in close hierarchical proximity to them, giving rise to very short division trees. On the other hand, it is possible that a long chain of increasingly primitive cells gets activated in response to the loss of differentiated cells. We expect that an important type of selection pressure acting upon tissue design is the minimization of mutations that happen in the course of everyday tissue maintenance (homeostasis). For example, tumor suppressor gene inactivation (two consecutive mutations) is an early rate-limiting step in many cancers. Using mathematical and computational methods, we find that the length of division trees is anti-correlated with the likelihood of double mutations, and lengthening the trees may provide an evolutionary advantage to the organism by delaying the onset of cancer.

The effect of spatial randomness on the average fixation time of mutants

The mean conditional fixation time of a mutant is an important measure of stochastic population dynamics, widely studied in ecology and evolution. Here, we investigate the effect of spatial randomness on the mean conditional fixation time of mutants in a constant population of cells, N. Specifically, we assume that fitness values of wild type cells and mutants at different locations come from given probability distributions and do not change in time. We study spatial arrangements of cells on regular graphs with different degrees, from the circle to the complete graph, and vary assumptions on the fitness probability distributions. Some examples include: identical probability distributions for wild types and mutants; cases when only one of the cell types has random fitness values while the other has deterministic fitness; and cases where the mutants are advantaged or disadvantaged. Using analytical calculations and stochastic numerical simulations, we find that randomness has a strong impact on fixation time. In the case of complete graphs, randomness accelerates mutant fixation for all population sizes, and in the case of circular graphs, randomness delays mutant fixation for N larger than a threshold value (for small values of N, different behaviors are observed depending on the fitness distribution functions). These results emphasize fundamental differences in population dynamics under different assumptions on cell connectedness. They are explained by the existence of randomly occurring “dead zones” that can significantly delay fixation on networks with low connectivity; and by the existence of randomly occurring “lucky zones” that can facilitate fixation on networks of high connectivity. Results for death-birth and birth-death formulations of the Moran process, as well as for the (haploid) Wright Fisher model are presented.

We study the influence of randomness on evolutionary dynamics, assuming that a newly arising mutant may experience a different set of environments compared to the wild type. We calculate the mean conditional fixation time of the mutant under different assumptions on spatial interactions, and show that randomness has a strong impact on the fixation time. In particular, it delays the fixation of mutants on 1D circles and accelerates it on complete graphs (the so called mass action, or complete mixing, model). This result holds for advantageous, disadvantageous, and neutral (on average) mutants. The reason for this pattern is quite intuitive: in a rigid, 1D structure, randomness can by chance put a “roadblock” and disrupt mutant spread, causing significant delay. In higher dimensions, there are many ways for a mutant to spread, and it is difficult to block all of them by chance; on the other hand, randomness can enhance fixation by providing an “easier” path. The effects of a random environment are important in biological models such as bacterial growth or cancer initiation/progression.

Effect of cell cycle duration on somatic evolutionary dynamics

Cellular checkpoints prevent damage and mutation accumulation in tissue cells. DNA repair is one mechanism that can be triggered by checkpoints and involves temporary cell cycle arrest and thus delayed reproduction. Repair‐deficient cells avoid this delay, which has been argued to lead to a selective advantage in the presence of frequent damage. We investigate this hypothesis with stochastic modeling, using mathematical analysis and agent‐based computations. We first model competition between two cell types: a cell population that enters temporary cell cycle arrest, corresponding to repair (referred to as arresting cells), and one that does not enter arrest (referred to as nonarresting cells). Although nonarresting cells are predicted to grow with a faster rate than arresting cells in isolation, this does not translate into a selective advantage in the model. Interestingly, the evolutionary properties of the nonarresting cells depend on the measure (or observable) of interest. When examining the average populations sizes in competition simulations, nonarresting and arresting cells display neutral dynamics. The fixation probability of nonarresting mutants, however, is lower than predicted for a neutral scenario, suggesting a selective disadvantage in this setting. For nonarresting cells to gain a selective advantage, additional mechanisms must be invoked in the model, such as small, repeated phases of tissue damage, each resulting in a brief period of regenerative growth. The same properties are observed in a more complex model where it is explicitly assumed that repair and temporary cell cycle arrest are dependent on the cell having sustained DNA damage, the rate of which can be varied. We conclude that repair‐deficient cells are not automatically advantageous in the presence of frequent DNA damage and that mechanisms beyond avoidance of cell cycle delay must be invoked to explain their emergence.

Determining the control networks regulating stem cell lineages in colonic crypts

The question of stem cell control is at the center of our understanding of tissue functioning, both in healthy and cancerous conditions. It is well accepted that cellular fate decisions (such as divisions, differentiation, apoptosis) are orchestrated by a network of regulatory signals emitted by different cell populations in the lineage and the surrounding tissue. The exact regulatory network that governs stem cell lineages in a given tissue is usually unknown. Here we propose an algorithm to identify a set of candidate control networks that are compatible with (a) measured means and variances of cell populations in different compartments, (b) qualitative information on cell population dynamics, such as the existence of local controls and oscillatory reaction of the system to population size perturbations, and (c) statistics of correlations between cell numbers in different compartments. Using the example of human colon crypts, where lineages are comprised of stem cells, transit amplifying cells, and differentiated cells, we start with a theoretically known set of 32 smallest control networks compatible with tissue stability. Utilizing near-equilibrium stochastic calculus of stem cells developed earlier, we apply a series of tests, where we compare the networks' expected behavior with the observations. This allows us to exclude most of the networks, until only three, very similar, candidate networks remain, which are most compatible with the measurements. This work demonstrates how theoretical analysis of control networks combined with only static biological data can shed light onto the inner workings of stem cell lineages, in the absence of direct experimental assessment of regulatory signaling mechanisms. The resulting candidate networks are dominated by negative control loops and possess the following properties: (1) stem cell division decisions are negatively controlled by the stem cell population, (2) stem cell differentiation decisions are negatively controlled by the transit amplifying cell population.

Quantitative approach for defining basic color terms and color category best exemplars

A new method is presented that identifies basic color terms (BCTs) from color-naming data. A function is defined that measures how well a term is understood by a communicating population. BCTs are then separated from other color terms by a threshold value applied to this function. A new mathematical algorithm is proposed and analyzed for determining the best exemplar associated with each BCT. Using data provided by the World Color Survey, comparisons are made between the paper's methods and those from other studies. These comparisons show that the paper's new definition of "basicness" mostly agrees with the typical definition found in the color categorization literature, which was originally due to Kay and colleagues. The new definition, unlike the typical one, has the advantage of not relying on syntactic or semantic features of languages or color lexicons. This permits the methodology developed to be generalizable and applied to other category domains for which a construct of "basicness" could have an important role.

Genotype by random environmental interactions gives an advantage to non-favored minor alleles

Fixation probability, the probability that the frequency of a newly arising mutation in a population will eventually reach unity, is a fundamental quantity in evolutionary genetics. Here we use a number of models (several versions of the Moran model and the haploid Wright-Fisher model) to examine fixation probabilities for a constant size population where the fitness is a random function of both allelic state and spatial position, despite neither allele being favored on average. The concept of fitness varying with respect to both genotype and environment is important in models of cancer initiation and progression, bacterial dynamics, and drug resistance. Under our model spatial heterogeneity redefines the notion of neutrality for a newly arising mutation, as such mutations fix at a higher rate than that predicted under neutrality. The increased fixation probability appears to be due to rare alleles having an advantage. The magnitude of this effect can be large, and is an increasing function of the spatial variance and skew in fitness. The effect is largest when the fitness values of the mutants and wild types are anti-correlated across environments. We discuss results for both a spatial ring geometry of cells (such as that of a colonic crypt), a 2D lattice and a mass-action (complete graph) arrangement.

Mathematical Modeling of Learning from an Inconsistent Source: A Nonlinear Approach

Continuing the discussion of how children can modify and regularize linguistic inputs from adults, we present a new interpretation of existing algorithms to model and investigate the process of a learner learning from an inconsistent source. On the basis of this approach is a (possibly nonlinear) function (the update function) that relates the current state of the learner with an increment that it receives upon processing the source's input, in a sequence of updates. The model can be considered a nonlinear generalization of the classic Bush-Mosteller algorithm. Our model allows us to analyze and present a theoretical explanation of a frequency boosting property, whereby the learner surpasses the fluency of the source by increasing the frequency of the most common input. We derive analytical expressions for the frequency of the learner, and also identify a class of update functions that exhibit frequency boosting. Applications to the Feature-Label-Order effect in learning are presented.

Aspirin-Induced Chemoprevention and Response Kinetics Are Enhanced by PIK3CA Mutations in Colorectal Cancer Cells

This study was designed to determine how aspirin influences the growth kinetics and characteristics of cultured colorectal cancer cells that harbor a variety of different mutational backgrounds, including PIK3CA- and KRAS-activating mutations, and the presence or absence of microsatellite instability. Colorectal cancer cell lines (HCT116, HCT116 + Chr3/5, RKO, SW480, HCT15, CACO2, HT29, and SW48) were treated with pharmacologically relevant doses of aspirin (0.5-10 mmol/L) and evaluated for proliferation and cell-cycle distribution. These parameters were fitted to a mathematical model to quantify the effects and understand the mechanism(s) by which aspirin modifies growth in colorectal cancer cells. We also evaluated the effects of aspirin on key G₀-G₁ cell-cycle genes that are regulated by the PI3K-Akt pathway. Aspirin decelerated growth rates and disrupted cell-cycle dynamics more profoundly in faster growing colorectal cancer cell lines, which tended to be PIK3CA mutants. Additionally, microarray analysis of 151 colorectal cancer cell lines identified important cell-cycle regulatory genes that are downstream targets of PIK3 and were also dysregulated by aspirin treatment (PCNA and RB1). Our study demonstrated what clinical trials have only speculated, that PIK3CA-mutant colorectal cancers are more sensitive to aspirin. Aspirin inhibited cell growth in all colorectal cancer cell lines regardless of mutational background, but the effects were exacerbated in cells with PIK3CA mutations. Mathematical modeling combined with bench science revealed that cells with PIK3CA-mutations experience significant G₀-G₁ arrest and explains why patients with PIK3CA mutant colorectal cancers may benefit from aspirin use after diagnosis. Cancer Prev Res; 10(3); 208-18. ©2017 AACR.

The role of cell location and spatial gradients in the evolutionary dynamics of colon and intestinal crypts

BACKGROUND

Colon and intestinal crypts serve as an important model system for adult stem cell proliferation and differentiation. We develop a spatial stochastic model to study the rate of somatic evolution in a normal crypt, focusing on the production of two-hit mutants that inactivate a tumor suppressor gene. We investigate the effect of cell division pattern along the crypt on mutant production, assuming that the division rate of each cell depends on its location.

RESULTS

We find that higher probability of division at the bottom of the crypt, where the stem cells are located, leads to a higher rate of double-hit mutant production. The optimal case for delaying mutations occurs when most of the cell divisions happen at the top of the crypt. We further consider an optimization problem where the "evolutionary" penalty for double-hit mutant generation is complemented with a "functional" penalty that assures that fully differentiated cells at the top of the crypt cannot divide.

CONCLUSION

The trade-off between the two types of objectives leads to the selection of an intermediate division pattern, where the cells in the middle of the crypt divide with the highest rate. This matches the pattern of cell divisions obtained experimentally in murine crypts.

REVIEWERS

This article was reviewed by David Axelrod (nominated by an Editorial Board member, Marek Kimmel), Yang Kuang and Anna Marciniak-Czochra. For the full reviews, please go to the Reviewers' comments section.

Near Equilibrium Calculus of Stem Cells in Application to the Airway Epithelium Lineage

Homeostatic maintenance of tissues is orchestrated by well tuned networks of cellular signaling. Such networks regulate, in a stochastic manner, fates of all cells within the respective lineages. Processes such as symmetric and asymmetric divisions, differentiation, de-differentiation, and death have to be controlled in a dynamic fashion, such that the cell population is maintained at a stable equilibrium, has a sufficiently low level of stochastic variation, and is capable of responding efficiently to external damage. Cellular lineages in real tissues may consist of a number of different cell types, connected by hierarchical relationships, albeit not necessarily linear, and engaged in a number of different processes. Here we develop a general mathematical methodology for near equilibrium studies of arbitrarily complex hierarchical cell populations, under regulation by a control network. This methodology allows us to (1) determine stability properties of the network, (2) calculate the stochastic variance, and (3) predict how different control mechanisms affect stability and robustness of the system. We demonstrate the versatility of this tool by using the example of the airway epithelium lineage. Recent research shows that airway epithelium stem cells divide mostly asymmetrically, while the so-called secretory cells divide predominantly symmetrically. It further provides quantitative data on the recovery dynamics of the airway epithelium, which can include secretory cell de-differentiation. Using our new methodology, we demonstrate that while a number of regulatory networks can be compatible with the observed recovery behavior, the observed division patterns of cells are the most optimal from the viewpoint of homeostatic lineage stability and minimizing the variation of the cell population size. This not only explains the observed yet poorly understood features of airway tissue architecture, but also helps to deduce the information on the still largely hypothetical regulatory mechanisms governing tissue turnover, and lends insight into how different control loops influence the stability and variance properties of cell populations.

Tissue stability is the basic property of healthy organs, and yet the mechanisms governing the stable, long-term maintenance of cell numbers in tissues are poorly understood. While more and more signaling pathways are being discovered, for the most part it remains unknown how they are being put together by different cell types into complex, nonlinear, hierarchical control networks that, on the one hand, reliably maintain constant cell numbers, and on the other hand, quickly adjust to oversee the robust response to tissue damage. Theoretical approaches can fill the gap by being able to reconstruct the underlying control network, based on the observations about the aspects of cellular dynamics. We argue that while many hypothetical networks may be capable of basic cell lineage maintenance, some are much more efficient from the viewpoint of variance minimization. Thus, we developed a new methodology that can test various control networks for stability, variance, and robustness. In the example of the airway epithelium that we highlight, it turns out that the evolutionary selected, actual architecture coincides with the mathematically optimal solution that minimizes the fluctuations of cell numbers at homeostasis.

In Vivo HIV-1 Cell-to-Cell Transmission Promotes Multicopy Micro-compartmentalized Infection

HIV-1 infection is enhanced by adhesive structures that form between infected and uninfected T cells called virological synapses (VSs). This mode of transmission results in the frequent co-transmission of multiple copies of HIV-1 across the VS, which can reduce sensitivity to antiretroviral drugs. Studying HIV-1 infection of humanized mice, we measured the frequency of co-transmission and the spatiotemporal organization of infected cells as indicators of cell-to-cell transmission in vivo. When inoculating mice with cells co-infected with two viral genotypes, we observed high levels of co-transmission to target cells. Additionally, micro-anatomical clustering of viral genotypes within lymphoid tissue indicates that viral spread is driven by local processes and not a diffuse viral cloud. Intravital splenic imaging reveals that anchored HIV-infected cells induce arrest of interacting, uninfected CD4(+) T cells to form Env-dependent cell-cell conjugates. These findings suggest that HIV-1 spread between immune cells can be anatomically localized into infectious clusters.

Evolution of genetic instability in heterogeneous tumors

Genetic instability is an important characteristic of cancer. While most cancers develop genetic instability at some stage of their progression, sometimes a temporary rise of instability is followed by the return to a relatively stable genome. Neither the reasons for these dynamics, nor, more generally, the role of instability in tumor progression, are well understood. In this paper we develop a class of mathematical models to study the evolutionary competition dynamics among different sub-populations in a heterogeneous tumor. We observe that despite the complexity of this multi-component and multi-process system, there is only a small number of scenarios expected in the context of the evolution of instability. If the penalty incurred by unstable cells (the decrease in the growth due to deleterious mutations) is high compared with the gain (the production rate of advantageous mutations), then instability does not evolve. In the opposite case, instability evolves and comes to dominate the system. In the intermediate parameter regime, instability is generated but later gives way to stable clones. Moreover, the model also informs us of the patterns of instability for cancer lineages corresponding to different stages of progression. It is predicted that mutations causing instability are merely "passengers" in tumors that have undergone only a small number of malignant mutations. Further down the path of carcinogenesis, however, unstable cells are more likely to give rise to the winning clonal wave that takes over the tumor and carries the evolution forward, thus conferring a causal role of the instability in such cases. Further, each individual clonal wave (i.e. cells harboring a fixed number of malignant driver mutations) experiences its own evolutionary history. It can fall under one of three types of temporal behavior: stable throughout, unstable to stable, or unstable throughout. Which scenario is realized depends on the subtle (but predictable) interplay among mutation rates and the death toll associated with the instability. The modeling approach provided here sheds light onto important aspects of the evolutionary dynamics of instability, which may be relevant to treatment scenarios that target instability or damage repair.

The Role of Symmetric Stem Cell Divisions in Tissue Homeostasis

Successful maintenance of cellular lineages critically depends on the fate decision dynamics of stem cells (SCs) upon division. There are three possible strategies with respect to SC fate decision symmetry: (a) asymmetric mode, when each and every SC division produces one SC and one non-SC progeny; (b) symmetric mode, when 50% of all divisions produce two SCs and another 50%-two non-SC progeny; (c) mixed mode, when both the asymmetric and two types of symmetric SC divisions co-exist and are partitioned so that long-term net balance of the lineage output stays constant. Theoretically, either of these strategies can achieve lineage homeostasis. However, it remains unclear which strategy(s) are more advantageous and under what specific circumstances, and what minimal control mechanisms are required to operate them. Here we used stochastic modeling to analyze and quantify the ability of different types of divisions to maintain long-term lineage homeostasis, in the context of different control networks. Using the example of a two-component lineage, consisting of SCs and one type of non-SC progeny, we show that its tight homeostatic control is not necessarily associated with purely asymmetric divisions. Through stochastic analysis and simulations we show that asymmetric divisions can either stabilize or destabilize the lineage system, depending on the underlying control network. We further apply our computational model to biological observations in the context of a two-component lineage of mouse epidermis, where autonomous lineage control has been proposed and notable regional differences, in terms of symmetric division ratio, have been noted-higher in thickened epidermis of the paw skin as compared to ear and tail skin. By using our model we propose a possible explanation for the regional differences in epidermal lineage control strategies. We demonstrate how symmetric divisions can work to stabilize paw epidermis lineage, which experiences high level of micro-injuries and a lack of hair follicles as a back-up source of SCs.

Characterizing inhibited tumor growth in stem-cell-driven non-spatial cancers

Healthy human tissue is highly regulated to maintain homeostasis. Secreted negative feedback factors that inhibit stem cell division and stem cell self-renewal play a fundamental role in establishing this control. The appearance of abnormal cancerous growth requires an escape from these regulatory mechanisms. In a previous study we found that for non-solid tumors if feedback inhibition on stem cell self-renewal is lost, but the feedback on the division rate is still intact, then the tumor dynamics are characterized by a relatively slow sub-exponential growth that we called inhibited growth. Here we characterize the cell dynamics of inhibited cancer growth by modeling feedback inhibition using Hill equations. We find asymptotic approximations for the growth rates of the stem cell and differentiated cell populations in terms of the strength of the inhibitory signal: stem cells grow as a power law t(1/k+1),and the differentiated cells grow as t(1/k), where k is the Hill coefficient in the feedback law regulating cell divisions. It follows that as the tumor grows, undifferentiated cells take up an increasingly large fraction of the population. Implications of these results for specific cancers including CML are discussed. Understanding how the regulatory mechanisms that continue to operate in cancer affect the rate of disease progression can provide important insights relevant to chronic or other slow progressing types of cancer.

The benefits of treating undetectable tumors

Cancer prevention is predicted to result in more positive therapeutic outcomes than post-diagnostic interventions, and so may be a viable option for future personalized medicine.

The role of the bi-compartmental stem cell niche in delaying cancer

In recent years, by using modern imaging techniques, scientists have found evidence of collaboration between different types of stem cells (SCs), and proposed a bi-compartmental organization of the SC niche. Here we create a class of stochastic models to simulate the dynamics of such a heterogeneous SC niche. We consider two SC groups: the border compartment, S1, is in direct contact with transit-amplifying (TA) cells, and the central compartment, S2, is hierarchically upstream from S1. The S1 SCs differentiate or divide asymmetrically when the tissue needs TA cells. Both groups proliferate when the tissue requires SCs (thus maintaining homeostasis). There is an influx of S2 cells into the border compartment, either by migration, or by proliferation. We examine this model in the context of double-hit mutant generation, which is a rate-limiting step in the development of many cancers. We discover that this type of a cooperative pattern in the stem niche with two compartments leads to a significantly smaller rate of double-hit mutant production compared with a homogeneous, one-compartmental SC niche. Furthermore, the minimum probability of double-hit mutant generation corresponds to purely symmetric division of SCs, consistent with the literature. Finally, the optimal architecture (which minimizes the rate of double-hit mutant production) requires a large proliferation rate of S1 cells along with a small, but non-zero, proliferation rate of S2 cells. This result is remarkably similar to the niche structure described recently by several authors, where one of the two SC compartments was found more actively engaged in tissue homeostasis and turnover, while the other was characterized by higher levels of quiescence (but contributed strongly to injury recovery). Both numerical and analytical results are presented.

Analysis of stochastic stem cell models with control

Understanding the dynamics of stem cell lineages is of central importance both for healthy and cancerous tissues. We study stochastic population dynamics of stem cells and differentiated cells, where cell decisions, such as proliferation vs. differentiation decisions, or division and death decisions, are under regulation from surrounding cells. The goal is to understand how different types of control mechanisms affect the means and variances of cell numbers. We use the assumption of weak dependencies of the regulatory functions (the controls) on the cell populations near the equilibrium to formulate moment equations. We then study three different methods of closure, showing that they all lead to the same results for the highest order terms in the expressions for the moments. We derive simple explicit expressions for the means and the variances of stem cell and differentiated cell numbers. It turns out that the variance is expressed as an algebraic function of partial derivatives of the controls with respect to the population sizes at the equilibrium. We demonstrate that these findings are consistent with the results previously obtained in the context of particular systems, and also present two novel examples with negative and positive control of division and differentiation decisions. This methodology is formulated without any specific assumptions on the functional form of the controls, and thus can be used for any biological system.

The duality of spatial death-birth and birth-death processes and limitations of the isothermal theorem

Evolutionary models on graphs, as an extension of the Moran process, have two major implementations: birth-death (BD) models (or the invasion process) and death-birth (DB) models (or voter models). The isothermal theorem states that the fixation probability of mutants in a large group of graph structures (known as isothermal graphs, which include regular graphs) coincides with that for the mixed population. This result has been proved by Lieberman et al. (2005 Nature 433, 312-316. (doi:10.1038/nature03204)) in the case of BD processes, where mutants differ from the wild-types by their birth rate (and not by their death rate). In this paper, we discuss to what extent the isothermal theorem can be formulated for DB processes, proving that it only holds for mutants that differ from the wild-type by their death rate (and not by their birth rate). For more general BD and DB processes with arbitrary birth and death rates of mutants, we show that the fixation probabilities of mutants are different from those obtained in the mass-action populations. We focus on spatial lattices and show that the difference between BD and DB processes on one- and two-dimensional lattices is non-small even for large population sizes. We support these results with a generating function approach that can be generalized to arbitrary graph structures. Finally, we discuss several biological applications of the results.

Curcumin mediates chemosensitization to 5-fluorouracil through miRNA-induced suppression of epithelial-to-mesenchymal transition in chemoresistant colorectal cancer

Resistance to cytotoxic chemotherapy is a major cause of mortality in colorectal cancer (CRC) patients. Chemoresistance has been linked primarily to a subset of cancer cells undergoing epithelial-mesenchymal transition (EMT). Curcumin, a botanical with antitumorigenic properties, has been shown to enhance sensitivity of cancer cells to chemotherapeutic drugs, but the molecular mechanisms underlying this phenomenon remain unclear. Effects of curcumin and 5-fluorouracil (5FU) individually, and in combination, were examined in parental and 5FU resistant (5FUR) cell lines. We performed a series of growth proliferation and apoptosis assays in 2D and 3D cell cultures. Furthermore, we identified and analyzed the expression pattern of a subset of putative EMT-suppressive microRNAs (miRNAs) and their downstream target genes regulated by curcumin. Chemosensitizing effects of curcumin were validated in a xenograft mouse model. Combined treatment with curcumin and 5FU enhanced cellular apoptosis and inhibited proliferation in both parental and 5FUR cells, whereas 5FU alone was ineffective in 5FUR cells. A group of EMT-suppressive miRNAs were upregulated by curcumin treatment in 5FUR cells. Curcumin suppressed EMT in 5FUR cells by downregulating BMI1, SUZ12 and EZH2 transcripts, key mediators of cancer stemness-related polycomb repressive complex subunits. Using a xenograft and mathematical models, we further demonstrated that curcumin sensitized 5FU to suppress tumor growth. We provide novel mechanistic evidence for curcumin-mediated sensitization to 5FU-related chemoresistance through suppression of EMT in 5FUR cells via upregulation of EMT-suppressive miRNAs. This study highlights the potential therapeutic usefulness of curcumin as an adjunct in patients with chemoresistant advanced CRC.

Complex role of space in the crossing of fitness valleys by asexual populations

The evolution of complex traits requires the accumulation of multiple mutations, which can be disadvantageous, neutral or advantageous relative to the wild-type. We study two spatial (two-dimensional) models of fitness valley crossing (the constant-population Moran process and the non-constant-population contact process), varying the number of loci involved and the degree of mixing. We find that spatial interactions accelerate the crossing of fitness valleys in the Moran process in the context of neutral and disadvantageous intermediate mutants because of the formation of mutant islands that increase the lifespan of mutant lineages. By contrast, in the contact process, spatial structure can accelerate or delay the emergence of the complex trait, and there can even be an optimal degree of mixing that maximizes the rate of evolution. For advantageous intermediate mutants, spatial interactions always delay the evolution of complex traits, in both the Moran and contact processes. The role of the mutant islands here is the opposite: instead of protecting, they constrict the growth of mutants. We conclude that the laws of population growth can be crucial for the effect of spatial interactions on the rate of evolution, and we relate the two processes explored here to different biological situations.