Within-host evolution of SARS-CoV-2: how often are de novo mutations transmitted from symptomatic infections?

Despite a relatively low mutation rate, the large number of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infections has allowed for substantial genetic change, leading to a multitude of emerging variants. Using a recently determined mutation rate (per site replication), as well as within-host parameter estimates for symptomatic SARS-CoV-2 infection, we apply a stochastic transmission-bottleneck model to describe the survival probability of de novo SARS-CoV-2 mutations as a function of bottleneck size and selection coefficient. For narrow bottlenecks, we find that mutations affecting per-target-cell attachment rate (with phenotypes associated with fusogenicity and ACE2 binding) have similar transmission probabilities to mutations affecting viral load clearance (with phenotypes associated with humoral evasion). We further find that mutations affecting the eclipse rate (with phenotypes associated with reorganization of cellular metabolic processes and synthesis of viral budding precursor material) are highly favoured relative to all other traits examined. We find that mutations leading to reduced removal rates of infected cells (with phenotypes associated with innate immune evasion) have limited transmission advantage relative to mutations leading to humoral evasion. Predicted transmission probabilities, however, for mutations affecting innate immune evasion are more consistent with the range of clinically estimated household transmission probabilities for de novo mutations. This result suggests that although mutations affecting humoral evasion are more easily transmitted when they occur, mutations affecting innate immune evasion may occur more readily. We examine our predictions in the context of a number of previously characterized mutations in circulating strains of SARS-CoV-2. Our work offers both a null model for SARS-CoV-2 mutation rates and predicts which aspects of viral life history are most likely to successfully evolve, despite low mutation rates and repeated transmission bottlenecks.

Shifts in Mutation Bias Promote Mutators by Altering the Distribution of Fitness Effects

AbstractRecent experimental evidence demonstrates that shifts in mutational biases-for example, increases in transversion frequency-can change the distribution of fitness effects of mutations (DFE). In particular, reducing or reversing a prevailing bias can increase the probability that a de novo mutation is beneficial. It has also been shown that mutator bacteria are more likely to emerge if the beneficial mutations they generate have a larger effect size than observed in the wild type. Here, we connect these two results, demonstrating that mutator strains that reduce or reverse a prevailing bias have a positively shifted DFE, which in turn can dramatically increase their emergence probability. Since changes in mutation rate and bias are often coupled through the gain and loss of DNA repair enzymes, our results predict that the invasion of mutator strains will be facilitated by shifts in mutation bias that offer improved access to previously undersampled beneficial mutations.

Shifts in mutation spectra enhance access to beneficial mutations

Biased mutation spectra are pervasive, with wide variation in the magnitude of mutational biases that influence genome evolution and adaptation. How do such diverse biases evolve? Our experiments show that changing the mutation spectrum allows populations to sample previously undersampled mutational space, including beneficial mutations. The resulting shift in the distribution of fitness effects is advantageous: Beneficial mutation supply and beneficial pleiotropy both increase, while deleterious load reduces. More broadly, simulations indicate that reducing or reversing the direction of a long-term bias is always selectively favored. Such changes in mutation bias can occur easily via altered function of DNA repair genes. A phylogenetic analysis shows that these genes are repeatedly gained and lost in bacterial lineages, leading to frequent bias shifts in opposite directions. Thus, shifts in mutation spectra may evolve under selection and can directly alter the outcome of adaptive evolution by facilitating access to beneficial mutations.

Multiple cohort study of hospitalized SARS-CoV-2 in-host infection dynamics: Parameter estimates, identifiability, sensitivity and the eclipse phase profile

Within-host SARS-CoV-2 modelling studies have been published throughout the COVID-19 pandemic. These studies contain highly variable numbers of individuals and capture varying timescales of pathogen dynamics; some studies capture the time of disease onset, the peak viral load and subsequent heterogeneity in clearance dynamics across individuals, while others capture late-time post-peak dynamics. In this study, we curate multiple previously published SARS-CoV-2 viral load data sets, fit these data with a consistent modelling approach, and estimate the variability of in-host parameters including the basic reproduction number, R₀, as well as the best-fit eclipse phase profile. We find that fitted dynamics can be highly variable across data sets, and highly variable within data sets, particularly when key components of the dynamic trajectories (e.g. peak viral load) are not represented in the data. Further, we investigated the role of the eclipse phase time distribution in fitting SARS-CoV-2 viral load data. By varying the shape parameter of an Erlang distribution, we demonstrate that models with either no eclipse phase, or with an exponentially-distributed eclipse phase, offer significantly worse fits to these data, whereas models with less dispersion around the mean eclipse time (shape parameter two or more) offered the best fits to the available data across all data sets used in this work. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Surviving environmental change: when increasing population size can increase extinction risk

Populations threatened by an abrupt environmental change-due to rapid climate change, pathogens or invasive competitors-may survive if they possess or generate genetic combinations adapted to the novel, challenging condition. If these genotypes are initially rare or non-existent, the emergence of lineages that allow a declining population to survive is known as 'evolutionary rescue'. By contrast, the genotypes required for survival could, by chance, be common before the environmental change. Here, considering both of these possibilities, we find that the risk of extinction can be lower in very small or very large populations, but peaks at intermediate population sizes. This pattern occurs when the survival genotype has a small deleterious effect before the environmental change. Since mildly deleterious mutations constitute a large fraction of empirically measured fitness effects, we suggest that this unexpected result-an intermediate size that puts a population at a greater risk of extinction-may not be unusual in the face of environmental change.

Host life-history traits influence the distribution of prophages and the genes they carry

Bacterial strains with a short minimal doubling time-'fast-growing' hosts-are more likely to contain prophages than their slow-growing counterparts. Pathogenic bacterial species are likewise more likely to carry prophages. We develop a bioinformatics pipeline to examine the distribution of prophages in fast- and slow-growing lysogens, and pathogenic and non-pathogenic lysogens, analysing both prophage length and gene content for each class. By fitting these results to a mathematical model of the evolutionary forces acting on prophages, we predict whether the observed differences can be attributed to different rates of lysogeny among the host classes, or other evolutionary pressures. We also test for significant differences in gene content among prophages, identifying genes that are preferentially lost or maintained in each class. We find that fast-growing hosts and pathogens have a greater fraction of full-length prophages, and our analysis predicts that induction rates are significantly reduced in slow-growing hosts and non-pathogenic hosts. Consistent with previous results, we find that several proteins involved in the packaging of new phage particles and lysis are preferentially lost in cryptic prophages. This article is part of the theme issue 'The secret lives of microbial mobile genetic elements'.

Selection bias in mutation accumulation

Mutation accumulation (MA) experiments, in which de novo mutations are sampled and subsequently characterized, are an essential tool in understanding the processes underlying evolution. In microbial populations, MA protocols typically involve a period of population growth between severe bottlenecks, such that a single individual can form a visible colony. While it has long been appreciated that the action of positive selection during this growth phase cannot be eliminated, it is typically assumed to be negligible. Here, we quantify the effect of both positive and negative selection in MA studies, demonstrating that selective effects can substantially bias the distribution of fitness effects (DFE) and mutation rates estimated from typical MA protocols in microbes. We then present a simple correction for this bias which applies to both beneficial and deleterious mutations, and can be used to correct the observed DFE in multiple environments. We use simulated MA experiments to illustrate the extent to which the MA-inferred DFE differs from the underlying true DFE, and demonstrate that the proposed correction accurately reconstructs the true DFE over a wide range of scenarios; we also provide an example of these corrections applied to experimental data. These results highlight that positive selection during microbial MA experiments is in fact not negligible, but can be corrected to gain a more accurate understanding of fundamental evolutionary parameters. This article is protected by copyright. All rights reserved.

Estimating the risk of pandemic avian influenza

Outbreaks of highly pathogenic strains of avian influenza (HPAI) cause high mortality in avian populations worldwide. When spread from avian reservoirs to humans, HPAI infections cause mortality in about 50% of human infections. Cases of human-to-human transmission of HPAI are relatively rare, and have, to date, only been reported in situations of close contact. These transmissions have resulted in isolated clusters of human HPAI infections, but have not yet caused a pandemic. Given the large number of human H5N1 HPAI infections to date, none of which has resulted in a pandemic, we estimate an upper bound on the probability of H5N1 pandemic emergence. We use this estimate to provide the likelihood of observing such a pandemic over the next decade. We then develop a more accurate parameter-based estimate of the emergence probability and predict the likelihood that, through rare mutations, an H5N1 influenza pandemic will emerge over the same time span.

Mosaic vaccination: How distributing different vaccines across a population could improve epidemic control

Although vaccination has been remarkably effective against some pathogens, for others, rapid antigenic evolution results in vaccination conferring only weak and/or short‐lived protection. Consequently, considerable effort has been invested in developing more evolutionarily robust vaccines, either by targeting highly conserved components of the pathogen (universal vaccines) or by including multiple immunological targets within a single vaccine (multi‐epitope vaccines). An unexplored third possibility is to vaccinate individuals with one of a number of qualitatively different vaccines, creating a “mosaic” of individual immunity in the population. Here we explore whether a mosaic vaccination strategy can deliver superior epidemiological outcomes to “conventional” vaccination, in which all individuals receive the same vaccine. We suppose vaccine doses can be distributed between distinct vaccine “targets” (e.g., different surface proteins against which an immune response can be generated) and/or immunologically distinct variants at these targets (e.g., strains); the pathogen can undergo antigenic evolution at both targets. Using simple mathematical models, here we provide a proof‐of‐concept that mosaic vaccination often outperforms conventional vaccination, leading to fewer infected individuals, improved vaccine efficacy, and lower individual risks over the course of the epidemic.

Robustness and predictability of evolution in bottlenecked populations

Deterministic and stochastic evolutionary processes drive adaptation in natural populations. The strength of each component process is determined by the population size: deterministic components prevail in very large populations, while stochastic components are the driving mechanisms in small ones. Many natural populations, however, experience intermittent periods of growth, moving through states in which either stochastic or deterministic processes prevail. This growth is often countered by population bottlenecks, which abound in both natural and laboratory populations. Here we investigate how population bottlenecks shape the process of adaptation. We demonstrate that adaptive trajectories in populations experiencing regular bottlenecks can be naturally scaled in time units of generations; with this scaling the time courses of adaptation, fitness variance, and genetic diversity all become relatively insensitive to the timing of population bottlenecks, provided the bottleneck size exceeds a few thousand individuals. We also include analyses at the genotype level to investigate the impact of population bottlenecks on the predictability and distribution of evolutionary pathways. Irrespective of the timing of population bottlenecks, we find that predictability increases with population size. We also find that predictability of the adaptive pathways increases in increasingly rugged fitness landscapes. Overall, our work reveals that both the adaptation rate and the predictability of evolutionary trajectories are relatively robust to population bottlenecks.

Quantifying the forces that maintain prophages in bacterial genomes

Genome sequencing has revealed that prophages, viral sequences integrated in a bacterial chromosome, are abundant, accounting for as much as 20% of the bacterial genome. These sequences can confer fitness benefits to the bacterial host, but may also instigate cell death through induction. Several recent investigations have revealed that the distribution of prophage lengths is bimodal, with a clear distinction between small and large prophages. Here we develop a mathematical model of the evolutionary forces affecting the prophage size distribution, and fit this model to three recent data sets. This approach offers quantitative estimates for the relative rates of lysogeny, induction, mutational degradation and selection acting on a wide class of prophage sequences. The model predicts that large prophages are predominantly maintained by the introduction of new prophage sequences through lysogeny, whereas shorter prophages can be enriched when they no longer encode the genes necessary for induction, but still offer selective benefits to their hosts.

Evolutionary stability of the lysis-lysogeny decision: Why be virulent?

Lytic viruses infect and kill host cells, producing a large number of viral copies. Temperate viruses, in contrast, are able to integrate viral genetic material into the host cell DNA, leaving a viable host cell. The evolutionary advantage of this strategy, lysogeny, has been demonstrated in complex environments that include spatial structure, oscillating population dynamics, or periodic environmental collapse. Here, we examine the evolutionary stability of the lysis-lysogeny decision, that is, we predict the long-term outcome of the evolution of lysogeny rates. We demonstrate that viruses with high rates of lysogeny are stable against invasion by more virulent viral strains even in simple environments, as long as the pool of susceptible hosts is not unlimited. This mirrors well-known results in both r-K selection theory and virulence evolution: although virulent viruses have a faster potential growth rate, temperate strains are able to maintain positive growth on a lower density of the limiting resource, susceptible hosts. We then outline scenarios in which the rate of lysogeny is predicted to evolve either toward full lysogeny or full lysis. Finally, we demonstrate conditions under which intermediate rates of lysogeny, as observed in temperate viruses in nature, can be sustained long-term. In general, intermediate lysogeny rates persist when the coupling between susceptible host density and virus density is relaxed.

Evolutionary emergence of infectious diseases in heterogeneous host populations

The emergence and re-emergence of pathogens remains a major public health concern. Unfortunately, when and where pathogens will (re-)emerge is notoriously difficult to predict, as the erratic nature of those events is reinforced by the stochastic nature of pathogen evolution during the early phase of an epidemic. For instance, mutations allowing pathogens to escape host resistance may boost pathogen spread and promote emergence. Yet, the ecological factors that govern such evolutionary emergence remain elusive because of the lack of ecological realism of current theoretical frameworks and the difficulty of experimentally testing their predictions. Here, we develop a theoretical model to explore the effects of the heterogeneity of the host population on the probability of pathogen emergence, with or without pathogen evolution. We show that evolutionary emergence and the spread of escape mutations in the pathogen population is more likely to occur when the host population contains an intermediate proportion of resistant hosts. We also show that the probability of pathogen emergence rapidly declines with the diversity of resistance in the host population. Experimental tests using lytic bacteriophages infecting their bacterial hosts containing Clustered Regularly Interspaced Short Palindromic Repeat and CRISPR-associated (CRISPR-Cas) immune defenses confirm these theoretical predictions. These results suggest effective strategies for cross-species spillover and for the management of emerging infectious diseases.

The probability that an epidemic will break out is highly dependent on the ability of the pathogen to acquire new adaptive mutations and to induce evolutionary emergence. Forecasting pathogen emergence thus requires a good understanding of the interplay between the epidemiology and evolution taking place at the onset of an outbreak. Here, we provide a comprehensive theoretical framework to analyze the impact of host population heterogeneity on the probability of pathogen evolutionary emergence. We use this model to predict the impact of the fraction of susceptible hosts, the inoculum size of the pathogen, and the diversity of host resistance on pathogen emergence. Our experiments using lytic bacteriophages and CRISPR-resistant bacteria support our theoretical predictions and demonstrate that manipulating the diversity of resistance alleles in a host population may be an effective way to limit the emergence of new pathogens.

Mathematical modelling of CRISPR-Cas system effects on biofilm formation

Clustered regularly interspaced short palindromic repeats (CRISPR), linked with CRISPR associated (Cas) genes, can confer adaptive immunity to bacteria, against bacteriophage infections. Thus from a therapeutic standpoint, CRISPR immunity increases biofilm resistance to phage therapy. Recently, however, CRISPR-Cas genes have been implicated in reducing biofilm formation in lysogenized cells. Thus CRISPR immunity can have complex effects on phage-host-lysogen interactions, particularly in a biofilm. In this contribution, we develop and analyse a series of dynamical systems to elucidate and disentangle these interactions. Two competition models are used to study the effects of lysogens (first model) and CRISPR-immune bacteria (second model) in the biofilm. In the third model, the effect of delivering lysogens to a CRISPR-immune biofilm is investigated. Using standard analyses of equilibria, stability and bifurcations, our models predict that lysogens may be able to displace CRISPR-immune bacteria in a biofilm, and thus suggest strategies to eliminate phage-resistant biofilms.

Prophage as a genetic reservoir: Promoting diversity and driving innovation in the host community

Sequencing of bacterial genomes has revealed an abundance of prophage sequences in many bacterial species. Since these sequences are accessible, through recombination, to infecting phages, bacteria carry an arsenal of genetic material that can be used by these viruses. We develop a mathematical model to isolate the effects of this phenomenon on the coevolution of temperate phage and bacteria. The model predicts that prophage sequences may play a key role in maintaining the phage population in situations that would otherwise favor host cell resistance. In addition, prophage recombination facilitates the existence of multiple phage types, thus promoting diverse co-existence in the phage-host ecosystem. Finally, because the host carries an archive of previous phage strategies, prophage recombination can drive waves of innovation in the host cell population.

The dynamics of mobile promoters: Enhanced stability in promoter regions

Mobile promoters are emerging as a new class of mobile genetic elements, first identified by examining prokaryote genome sequences, and more recently confirmed by experimental observations in bacteria. Recent datasets have identified over 40,000 putative mobile promoters in sequenced prokaryote genomes, however only one-third of these are in regions of the genome directly upstream from coding sequences, that is, in promoter regions. The presence of many promoter sequences in non-promoter regions is unexplained. Here we develop a general mathematical model for the dynamics of mobile promoters, extending previous work to capture the dynamics both within and outside promoter regions. From this general model, we apply rigorous model selection techniques to identify which parameters are statistically justified in describing the available mobile promoter data, and find best-fit values of these parameters. Our results suggest that high rates of horizontal gene transfer maintain the population of mobile promoters in promoter regions, and that once established at these sites, mobile promoters are rarely lost, but are commonly copied to other genomic regions. In contrast, mobile promoter copies in non-promoter regions are more numerous and more volatile, experiencing substantially higher rates of duplication, loss and diversification.

Backward bifurcations, turning points and rich dynamics in simple disease models

In this paper, dynamical systems theory and bifurcation theory are applied to investigate the rich dynamical behaviours observed in three simple disease models. The 2- and 3-dimensional models we investigate have arisen in previous investigations of epidemiology, in-host disease, and autoimmunity. These closely related models display interesting dynamical behaviors including bistability, recurrence, and regular oscillations, each of which has possible clinical or public health implications. In this contribution we elucidate the key role of backward bifurcations in the parameter regimes leading to the behaviors of interest. We demonstrate that backward bifurcations with varied positions of turning points facilitate the appearance of Hopf bifurcations, and the varied dynamical behaviors are then determined by the properties of the Hopf bifurcation(s), including their location and direction. A Maple program developed earlier is implemented to determine the stability of limit cycles bifurcating from the Hopf bifurcation. Numerical simulations are presented to illustrate phenomena of interest such as bistability, recurrence and oscillation. We also discuss the physical motivations for the models and the clinical implications of the resulting dynamics.

Effects of infection on honey bee population dynamics: a model

We propose a model that combines the dynamics of the spread of disease within a bee colony with the underlying demographic dynamics of the colony to determine the ultimate fate of the colony under different scenarios. The model suggests that key factors in the survival or collapse of a honey bee colony in the face of an infection are the rate of transmission of the infection and the disease-induced death rate. An increase in the disease-induced death rate, which can be thought of as an increase in the severity of the disease, may actually help the colony overcome the disease and survive through winter. By contrast, an increase in the transmission rate, which means that bees are being infected at an earlier age, has a drastic deleterious effect. Another important finding relates to the timing of infection in relation to the onset of winter, indicating that in a time interval of approximately 20 days before the onset of winter the colony is most affected by the onset of infection. The results suggest further that the age of recruitment of hive bees to foraging duties is a good early marker for the survival or collapse of a honey bee colony in the face of infection, which is consistent with experimental evidence but the model provides insight into the underlying mechanisms. The most important result of the study is a clear distinction between an exposure of the honey bee colony to an environmental hazard such as pesticides or insecticides, or an exposure to an infectious disease. The results indicate unequivocally that in the scenarios that we have examined, and perhaps more generally, an infectious disease is far more hazardous to the survival of a bee colony than an environmental hazard that causes an equal death rate in foraging bees.

Rates of transposition in Escherichia coli

The evolutionary role of transposable elements (TEs) is still highly controversial. Two key parameters, the transposition rate (u and w, for replicative and non-replicative transposition) and the excision rate (e) are fundamental to understanding their evolution and maintenance in populations. We have estimated u, w and e for six families of TEs (including eight members: IS1, IS2, IS3, IS4, IS5, IS30, IS150 and IS186) in Escherichia coli, using a mutation accumulation (MA) experiment. In this experiment, mutations accumulate essentially at the rate at which they appear, during a period of 80 500 (1610 generations × 50 lines) generations, and spontaneous transposition events can be detected. This differs from other experiments in which insertions accumulated under strong selective pressure or over a limited genomic target. We therefore provide new estimates for the spontaneous rates of transposition and excision in E. coli. We observed 25 transposition and three excision events in 50 MA lines, leading to overall rate estimates of u ∼ 1.15 × 10(-5), w ∼ 4 × 10(-8) and e ∼ 1.08 × 10(-6) (per element, per generation). Furthermore, extensive variation between elements was found, consistent with previous knowledge of the mechanisms and regulation of transposition for the different elements.

High cell density alters the ratio of type III to I collagen synthesis by fibroblasts

HUMAN fibroblasts in culture synthesise both type I and type III collagen(1), with type I accounting for 70-90% of the total(2). In culture, the rates at which these proteins are synthesised is constant and apparently rather rigidly controlled(3). However, the proportions of these collagens differs in cells cultured with increased amounts of serum (increased type III/I)(4) as well as in cells obtained from patients with certain diseases. Cells from patients with the Ehlers-Danlos type IV syndrome make little or no type III collagen(5,6), whereas cells from patients with osteogenesis imperfecta have an increased type III/I (refs 7, 8). We have found that cells from some patients with systemic sclerosis (scleroderma), have a reduced type III/I ratio. However, as previously reported, these cells grew to a lower density than control cells(9). We report here that normal fibroblasts from human and guinea pig skin produce proportionally more type III collagen at high cell density, probably because of a reduction in the synthesis of type I collagen.

Model and test in a fungus of the probability that beneficial mutations survive drift

Determining the probability of fixation of beneficial mutations is critically important for building predictive models of adaptive evolution. Despite considerable theoretical work, models of fixation probability have stood untested for nearly a century. However, recent advances in experimental and theoretical techniques permit the development of models with testable predictions. We developed a new model for the probability of surviving genetic drift, a major component of fixation probability, for novel beneficial mutations in the fungus Aspergillus nidulans, based on the life-history characteristics of its colony growth on a solid surface. We tested the model by measuring the probability of surviving drift in 11 adapted strains introduced into wild-type populations of different densities. We found that the probability of surviving drift increased with mutant invasion fitness, and decreased with wild-type density, as expected. The model accurately predicted the survival probability for the majority of mutants, yielding one of the first direct tests of the extinction probability of beneficial mutations.

Transmission dynamics of the recently-identified BYD virus causing duck egg-drop syndrome

Baiyangdian (BYD) virus is a recently-identified mosquito-borne flavivirus that causes severe disease in ducks, with extremely rapid transmission, up to 15% mortality within 10 days and 90% reduction in egg production on duck farms within 5 days of infection. Because of the zoonotic nature of flaviviruses, the characterization of BYD virus and its epidemiology are important public health concerns. Here, we develop a mathematical model for the transmission dynamics of this novel virus. We validate the model against BYD outbreak data collected from duck farms in Southeast China, as well as experimental data obtained from an animal facility. Based on our model, the basic reproductive number of BYD virus is high (R₀ = 21) indicating that this virus is highly transmissible, consistent with the dramatic epidemiology observed in BYDV-affected duck farms. Our results indicate that younger ducks are more vulnerable to BYD disease and that ducks infected with BYD virus reduce egg production (to about 33% on average) for about 3 days post-infection; after 3 days infected ducks are no longer able to produce eggs. Using our model, we predict that control measures which reduce contact between mosquitoes and ducks such as mosquito nets are more effective than insecticides.

Estimating the evidence of selection and the reliability of inference in unigenic evolution

Background

Unigenic evolution is a large-scale mutagenesis experiment used to identify residues that are potentially important for protein function. Both currently-used methods for the analysis of unigenic evolution data analyze 'windows' of contiguous sites, a strategy that increases statistical power but incorrectly assumes that functionally-critical sites are contiguous. In addition, both methods require the questionable assumption of asymptotically-large sample size due to the presumption of approximate normality.

Results

We develop a novel approach, termed the Evidence of Selection (EoS), removing the assumption that functionally important sites are adjacent in sequence and and explicitly modelling the effects of limited sample-size. Precise statistical derivations show that the EoS score can be easily interpreted as an expected log-odds-ratio between two competing hypotheses, namely, the hypothetical presence or absence of functional selection for a given site. Using the EoS score, we then develop selection criteria by which functionally-important yet non-adjacent sites can be identified. An approximate power analysis is also developed to estimate the reliability of inference given the data. We validate and demonstrate the the practical utility of our method by analysis of the homing endonuclease I-Bmol, comparing our predictions with the results of existing methods.

Conclusions

Our method is able to assess both the evidence of selection at individual amino acid sites and estimate the reliability of those inferences. Experimental validation with I-Bmol proves its utility to identify functionally-important residues of poorly characterized proteins, demonstrating increased sensitivity over previous methods without loss of specificity. With the ability to guide the selection of precise experimental mutagenesis conditions, our method helps make unigenic analysis a more broadly applicable technique with which to probe protein function.

Availability

Software to compute, plot, and summarize EoS data is available as an open-source package called 'unigenic' for the 'R' programming language at http://www.fernandes.org/txp/article/13/an-analytical-framework-for-unigenic-evolution.

Identifying and seeing beyond multiple sequence alignment errors using intra-molecular protein covariation

Background

There is currently no way to verify the quality of a multiple sequence alignment that is independent of the assumptions used to build it. Sequence alignments are typically evaluated by a number of established criteria: sequence conservation, the number of aligned residues, the frequency of gaps, and the probable correct gap placement. Covariation analysis is used to find putatively important residue pairs in a sequence alignment. Different alignments of the same protein family give different results demonstrating that covariation depends on the quality of the sequence alignment. We thus hypothesized that current criteria are insufficient to build alignments for use with covariation analyses.

Methodology/Principal Findings

We show that current criteria are insufficient to build alignments for use with covariation analyses as systematic sequence alignment errors are present even in hand-curated structure-based alignment datasets like those from the Conserved Domain Database. We show that current non-parametric covariation statistics are sensitive to sequence misalignments and that this sensitivity can be used to identify systematic alignment errors. We demonstrate that removing alignment errors due to 1) improper structure alignment, 2) the presence of paralogous sequences, and 3) partial or otherwise erroneous sequences, improves contact prediction by covariation analysis. Finally we describe two non-parametric covariation statistics that are less sensitive to sequence alignment errors than those described previously in the literature.

Conclusions/Significance

Protein alignments with errors lead to false positive and false negative conclusions (incorrect assignment of covariation and conservation, respectively). Covariation analysis can provide a verification step, independent of traditional criteria, to identify systematic misalignments in protein alignments. Two non-parametric statistics are shown to be somewhat insensitive to misalignment errors, providing increased confidence in contact prediction when analyzing alignments with erroneous regions because of an emphasis on they emphasize pairwise covariation over group covariation.

The adaptation rate of asexuals: deleterious mutations, clonal interference and population bottlenecks

The rate at which a population adapts to its environment is a cornerstone of evolutionary theory, and recent experimental advances in microbial populations have renewed interest in predicting and testing this rate. Efforts to understand the adaptation rate theoretically are complicated by high mutation rates, to both beneficial and deleterious mutations, and by the fact that beneficial mutations compete with each other in asexual populations (clonal interference). Testable predictions must also include the effects of population bottlenecks, repeated reductions in population size imposed by the experimental protocol. In this contribution, we integrate previous work that addresses each of these issues, developing an overall prediction for the adaptation rate that includes: beneficial mutations with probabilistically distributed effects, deleterious mutations of arbitrary effect, population bottlenecks, and clonal interference.

Reconciling conflicting clinical studies of antioxidant supplementation as HIV therapy: a mathematical approach

BACKGROUND

Small, highly reactive molecules called reactive oxygen species (ROS) play a crucial role in cell signalling and infection control. However, high levels of ROS can cause significant damage to cell structure and function. Studies have shown that infection with the human immunodeficiency virus (HIV) results in increased ROS concentrations, which can in turn lead to faster progression of HIV infection, and cause CD4+ T-cell apoptosis. To counteract these effects, clinical studies have explored the possibility of raising antioxidant levels, with mixed results.

METHODS

In this paper, a mathematical model is used to explore this potential therapy, both analytically and numerically. For the numerical work, we use clinical data from both HIV-negative and HIV-positive injection drug users (IDUs) to estimate model parameters; these groups have lower baseline concentrations of antioxidants than non-IDU controls.

RESULTS

Our model suggests that increases in CD4+ T cell concentrations can result from moderate levels of daily antioxidant supplementation, while excessive supplementation has the potential to cause periods of immunosuppression.

CONCLUSION

We discuss implications for HIV therapy in IDUs and other populations which may have low baseline concentrations of antioxidants.

Adaptation rates of lytic viruses depend critically on whether host cells survive the bottleneck

We use a branching process approach to estimate the substitution rate, the rate at which beneficial mutations occur and fix, in populations of lytic viruses whose growth is controlled by periodic population bottlenecks. Our model predicts that substitution rates, and by extension adaptation rates, are profoundly affected by the survival of infected host cells at the bottleneck. In particular, we find that direct transfer (or environmental) bottlenecks, in which some fraction of both free virus and host cells are preserved, are associated with relatively slow adaptation rates for the virus. In contrast, viruses can adapt much more quickly when only free virus is transferred to a new host population, as is typical in an epidemiological setting. Finally, when premature lysis of the host-cell population is induced at the bottleneck, we predict that adaptation rates for the virus will, in general, be faster still. These results hold irrespective of the life-history trait affected by the beneficial mutation. The substitution rates in the presence of environmental bottlenecks are predicted to be as much as an order of magnitude lower than in the other two cases.

The effects of population bottlenecks on clonal interference, and the adaptation effective population size

Clonal interference refers to the competition that arises in asexual populations when multiple beneficial mutations segregate simultaneously. A large body of theoretical and experimental work now addresses this issue. Although much of the experimental work is performed in populations that grow exponentially between periodic population bottlenecks, the theoretical work to date has addressed only populations of a constant size. We derive an analytical approximation for the rate of adaptation in the presence of both clonal interference and bottlenecks, and compare this prediction to the results of an individual-based simulation, showing excellent agreement in the parameter regime in which clonal interference prevails. We also derive an appropriate definition for the effective population size for adaptive evolution experiments in the presence of population bottlenecks. This "adaptation effective population size" allows for a good approximation of the expected rate of adaptation, either in the strong-selection weak-mutation regime, or when clonal interference comes into play. In the multiple mutation regime, when the product of the population size and mutation rate is extremely large, these results no longer hold.

The fixation probability of beneficial mutations

The fixation probability, the probability that the frequency of a particular allele in a population will ultimately reach unity, is one of the cornerstones of population genetics. In this review, we give a brief historical overview of mathematical approaches used to estimate the fixation probability of beneficial alleles. We then focus on more recent work that has relaxed some of the key assumptions in these early papers, providing estimates that have wider applicability to both natural and laboratory settings. In the final section, we address the possibility of future work that might bridge the gap between theoretical results to date and results that might realistically be applied to the experimental evolution of microbial populations. Our aim is to highlight the concrete, testable predictions that have arisen from the theoretical literature, with the intention of further motivating the invaluable interplay between theory and experiment.

Fixation probabilities depend on life history: fecundity, generation time and survival in a burst-death model

The burst-death model has been developed to describe the life history of organisms with variable generation times and a burst of a fixed number of offspring. The model also includes an optional constant clearance rate, such as washout from a chemostat, and the possibility of sustained periods of population growth followed by severe bottlenecks, as in serial passaging. In this model, a beneficial mutation can either increase the burst rate or the burst size, or reduce the clearance rate, thus increasing survival. In this article we examine the effects of these three possible mechanisms on both the Malthusian fitness and the fixation probability of the lineage. We find that equivalent relative increases in the burst rate or burst size confer equivalent increases in the Malthusian fitness of a lineage, whereas increasing survival typically has a more moderate effect on Malthusian fitness. In contrast, for beneficial mutations that confer the same increase in fitness, mutations that increase survival are the most likely to fix, followed by mutations that increase the burst rate. Mutations that increase the burst size are the least likely to fix. These results imply that mutant lineages with the highest Malthusian fitness are not, in many cases, the most likely to escape extinction.

The division of labor: genotypic versus phenotypic specialization

A model of the division of labor in simple evolving systems is explored to compare two strategies evident in natural populations: phenotypic specialization (such as differentiation by regulated gene expression) and genotypic specialization (such as co-infection by complementary covirus populations). While genotypic specialization is vulnerable to the chance extinction of an essential specialist type and to parasitism, phenotypic specialization is able to overcome these hurdles. When simple spatial effects are included, phenotypic specialization has further benefits, protecting against destructive dynamic patterns. Many of the advantages of phenotypic specialization, however, can only be realized when a high degree of relatedness within groups is ensured.

Drug-Sparing Regimens for HIV Combination Therapy: Benefits Predicted for “Drug Coasting”

Structured Treatment Interruptions (STI) during HIV drug therapy were thought to potentially reduce side effects and toxicity, boost immune involvement, and possibly lower the viral set-point. Clinical trials of STI regimens, however, have had mixed results. We use an established mathematical model of HAART to estimate possible therapeutic outcomes for STI and for other, similar patterns in HIV combination therapy. We perform an exhaustive search of patterns of up to 60 days, for triple-drug combinations involving accurate pharmacokinetics for 12 specific antiviral drugs. The results of this analysis are consistent with recent clinical trials which have demonstrated that STI-type patterns, involving therapy interruption of weeks or months, are rarely optimal. Our analysis predicts, however, that the benefit of treatment can often be improved by including very short drug-free periods, during which the patient effectively "coasts" for a day or two on adequate drug concentrations due to the long half-life of some pharmaceuticals. Our analysis predicts many cases in which this may be achieved without increasing the risk of drug-resistance. This suggests that "drug coasting" patterns, significantly shorter than STI patterns, may merit further clinical investigation in efforts to find drug-sparing regimens for HIV.

Optimal drug treatment regimens for HIV depend on adherence

Drug therapies aimed at suppressing the human immunodeficiency virus (HIV) are highly effective, often reducing the viral load to below the limits of detection for years. Adherence to such antiviral regimens, however, is typically far from ideal. We have previously developed a model that predicts optimal treatment regimens by weighing drug toxicity against CD4+ T-cell counts, including the probability that drug resistance will emerge. We use this model to investigate the influence of adherence on therapy benefit. For a drug with a given half-life, we compare the effects of varying the dose amount and dose interval for different rates of adherence, and compute the optimal dose regimen for adherence between 65% and 95%. Our results suggest that for optimal treatment benefit, drug regimens should be adjusted for poor adherence, usually by increasing the dose amount and leaving the dose interval fixed. We also find that the benefit of therapy can be surprisingly robust to poor adherence, as long as the dose interval and dose amount are chosen accordingly.

Fixation probabilities when generation times are variable: the burst death model

Estimating the fixation probability of a beneficial mutation has a rich history in theoretical population genetics. Typically, to attain mathematical tractability, we assume that generation times are fixed, while the number of offspring per individual is stochastic. However, fixation probabilities are extremely sensitive to these assumptions regarding life history. In this article, we compute the fixation probability for a "burst-death" life-history model. The model assumes that generation times are exponentially distributed, but the number of offspring per individual is constant. We estimate the fixation probability for populations of constant size and for populations that grow exponentially between periodic population bottlenecks. We find that the fixation probability is, in general, substantially lower in the burst-death model than in classical models. We also note striking qualitative differences between the fates of beneficial mutations that increase burst size and mutations that increase the burst rate. In particular, once the burst size is sufficiently large relative to the wild type, the burst-death model predicts that fixation probability depends only on burst rate.

Time Course of Early Response to Chemotherapy in Non-Small Cell Lung Cancer Patients with 18F-FDG PET/CT

PET and (18)F-FDG have the potential to follow the early metabolic response to chemotherapy in patients with non-small cell lung cancer and to predict success or failure of the therapy.

METHODS

We studied 16 patients with non-small cell lung cancer as they followed 2 courses of docetaxel and carboplatin. Each patient was studied weekly for 7 wk, and tissue activity was assessed by the amount of radioactivity retained 90 min after the intravenous injection of (18)F-FDG. In a prospective analysis, the linear least-squares method was used to evaluate the time course of metabolic activity in tumor and liver, bone marrow, and unaffected lung tissues; a metabolic response was defined as a response in which the slope of the regression was negative and significantly different from zero. Our hypothesis was that patients who exhibited a tumor metabolic response would survive longer than those who did not. In a retrospective examination of our data, we grouped our patients into those who survived <6 mo and those who survived longer and calculated the difference in the standardized uptake value (SUV) between day 7 and subsequent time points to determine the most appropriate timing of 2 PET studies in predicting response to therapy.

RESULTS

Fifteen of 16 patients completed the study. In the prospective study, 8 patients were classified as nonresponders as the slope of the regression of tumor SUV versus time was not different from zero; they all died within 35 wk of the end of their study. Seven patients were classified as responders; 5 survived and 2 died, one at 25 wk and the other at 76 wk. In the retrospective study, a decrease of 0.5 SUV between studies performed at 1 and 3 wk after the initiation of chemotherapy was predictive of those patients who survived >6 mo and in whom chemotherapy was presumably successful.

CONCLUSION

Patients with non-small cell lung cancer who had a positive outcome, as exhibited by prolonged survival, were those who showed a tumor metabolic response assessed using weekly (18)F-FDG PET studies. (18)F-FDG PET studies performed at 1 and 3 wk after the initiation of chemotherapy allowed prediction of the response to therapy.

Costs versus benefits: best possible and best practical treatment regimens for HIV

Current HIV therapy, although highly effective, may cause very serious side effects, making adherence to the prescribed regimen difficult. Mathematical modeling may be used to evaluate alternative treatment regimens by weighing the positive results of treatment, such as higher levels of helper T cells, against the negative consequences, such as side effects and the possibility of resistance mutations. Although estimating the weights assigned to these factors is difficult, current clinical practice offers insight by defining situations in which therapy is considered "worthwhile". We therefore use clinical practice, along with the probability that a drug-resistant mutation is present at the start of therapy, to suggest methods of rationally estimating these weights. In our underlying model, we use ordinary differential equations to describe the time course of in-host HIV infection, and include populations of both activated CD4(+) T cells and CD8(+) T cells. We then determine the best possible treatment regimen, assuming that the effectiveness of the drug can be continually adjusted, and the best practical treatment regimen, evaluating all patterns of a block of days "on" therapy followed by a block of days "off" therapy. We find that when the tolerance for drug-resistant mutations is low, high drug concentrations which maintain low infected cell populations are optimal. In contrast, if the tolerance for drug-resistant mutations is fairly high, the optimal treatment involves periods of reduced drug exposure which consequently boost the immune response through increased antigen exposure. We elucidate the dependence of the optimal treatment regimen on the pharmacokinetic parameters of specific antiviral agents.

Functionally Important Residues in the Peptidyl-prolyl Isomerase Pin1 Revealed by Unigenic Evolution

Pin1 is a phosphorylation-dependent member of the parvulin family of peptidyl-prolyl isomerases exhibiting functional conservation between yeast and man. To perform an unbiased analysis of the regions of Pin1 essential for its functions, we generated libraries of randomly mutated forms of the human Pin1 cDNA and identified functional Pin1 alleles by their ability to complement the Pin1 homolog Ess1 in Saccharomyces cerevisiae. We isolated an extensive collection of functional mutant Pin1 clones harboring a total of 356 amino acid substitutions. Surprisingly, many residues previously thought to be critical in Pin1 were found to be altered in this collection of functional mutants. In fact, only 17 residues were completely conserved in these mutants and in Pin1 sequences from other eukaryotic organisms, with only two of these conserved residues located within the WW domain of Pin1. Examination of invariant residues provided new insights regarding a phosphate-binding loop that distinguishes a phosphorylation-dependent peptidyl-prolyl isomerase such as Pin1 from other parvulins. In addition, these studies led to an investigation of residues involved in catalysis including C113 that was previously implicated as the catalytic nucleophile. We demonstrate that substitution of C113 with D does not compromise Pin1 function in vivo nor does this substitution abolish catalytic activity in purified recombinant Pin1. These findings are consistent with the prospect that the function of residue 113 may not be that of a nucleophile, thus raising questions about the model of nucleophilic catalysis. Accordingly, an alternative catalytic mechanism for Pin1 is postulated.

Natural variation in HIV infection: Monte Carlo estimates that include CD8 effector cells

Viral load and CD4 T-cell counts in patients infected with the human immunodeficiency virus (HIV) are commonly used to guide clinical decisions regarding drug therapy or to assess therapeutic outcomes in clinical trials. However, random fluctuations in these markers of infection can obscure clinically significant change. We employ a Monte Carlo simulation to investigate contributing factors in the expected variability in CD4 T-cell count and viral load due solely to the stochastic nature of HIV infection. The simulation includes processes that contribute to the variability in HIV infection including CD4 and CD8 T-cell population dynamics as well as T-cell activation and proliferation. The simulation results may reconcile the wide range of variabilities in viral load observed in clinical studies, by quantifying correlations between viral load measurements taken days or weeks apart. The sensitivity of variability in T-cell count and viral load to changes in the lifetimes of CD4 and CD8 T-cells is investigated, as well as the effects of drug therapy.

Improving estimates of the basic reproductive ratio: using both the mean and the dispersal of transition times

In both within-host and epidemiological models of pathogen dynamics, the basic reproductive ratio, R0, is a powerful tool for gauging the risk associated with an emerging pathogen, or for estimating the magnitude of required control measures. Techniques for estimating R0, either from incidence data or in-host clinical measures, often rely on estimates of mean transition times, that is, the mean time before recovery, death or quarantine occurs. In many cases, however, either data or intuition may provide additional information about the dispersal of these transition times about the mean, even if the precise form of the underlying probability distribution remains unknown. For example, we may know that recovery typically occurs within a few days of the mean recovery time. In this paper we elucidate common situations in which R0 is sensitive to the dispersal of transition times about their respective means. We then provide simple correction factors that may be applied to improve estimates of R0 when not only the mean but also the standard deviation of transition times out of the infectious state can be estimated.

Development of an unbiased statistical method for the analysis of unigenic evolution

Background

Unigenic evolution is a powerful genetic strategy involving random mutagenesis of a single gene product to delineate functionally important domains of a protein. This method involves selection of variants of the protein which retain function, followed by statistical analysis comparing expected and observed mutation frequencies of each residue. Resultant mutability indices for each residue are averaged across a specified window of codons to identify hypomutable regions of the protein. As originally described, the effect of changes to the length of this averaging window was not fully eludicated. In addition, it was unclear when sufficient functional variants had been examined to conclude that residues conserved in all variants have important functional roles.

Results

We demonstrate that the length of averaging window dramatically affects identification of individual hypomutable regions and delineation of region boundaries. Accordingly, we devised a region-independent chi-square analysis that eliminates loss of information incurred during window averaging and removes the arbitrary assignment of window length. We also present a method to estimate the probability that conserved residues have not been mutated simply by chance. In addition, we describe an improved estimation of the expected mutation frequency.

Conclusion

Overall, these methods significantly extend the analysis of unigenic evolution data over existing methods to allow comprehensive, unbiased identification of domains and possibly even individual residues that are essential for protein function.

Monte Carlo estimates of natural variation in HIV infection

We describe a Monte Carlo simulation of the within-host dynamics of human immunodeficiency virus 1 (HIV-1). The simulation proceeds at the level of individual T-cells and virions in a small volume of plasma, thus capturing the inherent stochasticity in viral replication, mutation and T-cell infection. When cell lifetimes are distributed exponentially in the Monte Carlo approach, our simulation results are in perfect agreement with the predictions of the corresponding systems of differential equations from the literature. The Monte Carlo model, however, uniquely allows us to estimate the natural variability in important parameters such as the T-cell count, viral load, and the basic reproductive ratio, in both the presence and absence of drug therapy. The simulation also yields the probability that an infection will not become established after exposure to a viral inoculum of a given size. Finally, we extend the Monte Carlo approach to include distributions of cell lifetimes that are less-dispersed than exponential.

Using information theory to search for co-evolving residues in proteins

MOTIVATION

Some functionally important protein residues are easily detected since they correspond to conserved columns in a multiple sequence alignment (MSA). However important residues may also mutate, with compensatory mutations occurring elsewhere in the protein, which serve to preserve or restore functionality. It is difficult to distinguish these co-evolving sites from other non-conserved sites.

RESULTS

We used Mutual Information (MI) to identify co-evolving positions. Using in silico evolved MSAs, we examined the effects of the number of sequences, the size of amino acid alphabet and the mutation rate on two sources of background MI: finite sample size effects and phylogenetic influence. We then assessed the performance of various normalizations of MI in enhancing detection of co-evolving positions and found that normalization by the pair entropy was optimal. Real protein alignments were analyzed and co-evolving isolated pairs were often found to be in contact with each other.

AVAILABILITY

All data and program files can be found at http://www.biochem.uwo.ca/cgi-bin/CDD/index.cgi

Mutual information in protein multiple sequence alignments reveals two classes of coevolving positions

Information theory was used to identify nonconserved coevolving positions in multiple sequence alignments from a variety of protein families. Coevolving positions in these alignments fall into two general categories. One set is composed of positions that coevolve with only one or two other positions. These positions often display direct amino acid side-chain interactions with their coevolving partner. The other set comprises positions that coevolve with many others and are frequently located in regions critical for protein function, such as active sites and surfaces involved in intermolecular interactions and recognition. We find that coevolving positions are more likely to change protein function when mutated than are positions showing little coevolution. These results imply that information theory may be applied generally to find coevolving, nonconserved positions that are part of functional sites in uncharacterized protein families. We propose that these coevolving positions compose an important subset of the positions in an alignment, and may be as important to the structure and function of the protein family as are highly conserved positions.

Drug resistance in an immunological model of HIV-1 infection with impulsive drug effects

We consider an SIR-type model of immunological behaviour for HIV dynamics, including the effects of reverse transcriptase inhibitors and other drugs which prevent cellular infection. We use impulsive differential equations to model drug behaviour. We classify different regimes according to whether the drug efficacy is negligible, intermediate or high. We consider two strains of the virus: a wild-type strain that can be controlled by both intermediate and high drug concentrations, and a mutant strain that can only be controlled by high drug concentrations. Drug regimes may take trajectories through one, two or all three regimes, depending on the dosage and the dosing schedule. We demonstrate that drug resistance arises at both intermediate and high drug levels. At low drug levels resistance does not emerge, but the total T cell count is proven to be significantly lower than in the disease-free state. At intermediate drug levels, drug resistance is guaranteed to emerge. At high drug levels, either the drug-resistant strain will dominate or, in the absence of longer-lived reservoirs of infected cells, both viral sub-populations will be cleared. In the latter case the immune system is maintained by a population of T cells which have absorbed sufficient quantities of the drug to prevent infection by even the drug-resistant strain. We provide estimates of a range of dosages and dosing schedules which would, if physiologically tolerable, theoretically eliminate free virus in this system. Our results predict that to control viral load, decreasing the interval between doses is more effective than increasing the dose.

Fixation probability favors increased fecundity over reduced generation time

The cornerstone of population genetics is a probabilistic understanding of the ultimate fate--survival or extinction--of rare mutations. If a mutation is beneficial, it enables its carrier to reproduce faster than native wild-type individuals. In classic derivations and in the considerable body of research that has followed, "faster" has been defined mathematically to mean "able to produce more surviving offspring per generation." Many organisms, however, may increase their reproductive rate by producing the same number of offspring in a shorter generation time: a mutant bacterium, for example, may complete the cell cycle and produce two offspring more quickly than the wild type. We find that the ultimate fixation probability of a mutation conferring a shorter generation time differs from that of a mutation conferring more offspring by a factor of 2 ln(2)-nearly 40%. This predicts a reduction in the overall substitution rate for any mutation that decreases the generation time: fixation probability is biased toward increased offspring number.

Distinct effects of protease and reverse transcriptase inhibition in an immunological model of HIV-1 infection with impulsive drug effects

We present an immunological model that considers the dynamics of CD4+ T cells interacting with free virions, reverse transcriptase inhibiting drugs and protease inhibiting drugs. We divide the T cells into multiple classes and use impulsive differential equations to describe the drug activity. As expected, we find that insufficient dosing of either drug corresponds to high viral load and a large population of infectious T cells. The model further predicts that, in the absence of physiological limits on tolerable drug concentrations, sufficiently frequent dosing with the reverse transcriptase inhibitor alone could theoretically maintain the CD4+ T cell count arbitrarily close to the T cell count in the uninfected immune system. However, for frequent dosing of the protease inhibitor alone, the limiting T cell populations may not be enough to maintain the immune system. Furthermore, frequent dosing of both drugs has the same net effect on the T cell population as frequent dosing of the reverse transcriptase inhibitor only. Thus, the two drug classes can have fundamentally different effects on the long-term dynamics and the reverse transcriptase inhibitor, in particular, plays a crucial role in maintaining the immune system. We also provide estimates for the dosing intervals of each drug that could theoretically sustain the T cell population at a desired level.

MECHANISMS OF SPERM COMPETITION: TESTING THE FAIR RAFFLE

Sperm competition is a major force of sexual selection, but its implications for mating system and life-history evolution are just beginning to be understood. Of particular importance is understanding the mechanisms of sperm competition. Models have been developed to determine if sperm competition operates in a fair raffle process, whereby each sperm from competing males has an equal chance of fertilizing a female's ova, or if it operates in a loaded raffle process, whereby one male's sperm has a fertilization advantage. These models require data on relative sperm and offspring (paternity) numbers of competing males. Here we develop a model based on maximum-likelihood methods for differentiating between the fair and loaded raffle processes. The model calculates the relative competitiveness of two males' sperm (loadings) as well as the economy of scale (nonlinear returns to sperm number). Previous models implicitly assumed that there is no economy of scale, which may not be the case when there is cooperation or interference among sperm from a given male. We demonstrate that our model has superior power-in some instances more than double-than previous models. We apply our model to an example of sperm competition in the guppy (Poecilia reticulata) and show that the system may be characterized by a loaded raffle attributable to effects of second male precedence.