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.

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.

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.

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.

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.

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.

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.

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.

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.