Quantifying the invasion risk of West Nile virus: Insights from a multi-vector and multi-host SEIR model

The invasion of vector-borne diseases depends on the type of specific features of the vector and hosts at play. Within the Culex pipiens complex, differences in ecology, biology, and vector competence can influence the risk of West Nile virus (WNV) outbreaks. To determine which life-history traits affect WNV invasion into susceptible communities the most, we constructed an epidemiological Susceptible-Exposed-Infectious-Recovered model with three vector (eco)types, Culex pipiens pipiens, Cx. pip. molestus, and their hybrids, and two vertebrate hosts, birds (as amplifying hosts) and humans (as dead-end hosts). We investigated how differences in feeding preferences and transmission rates influenced WNV transmission across different habitats and two seasons (Spring versus Summer), to investigate the impact of increasing mosquitoes on the WNV transmission risk. Our results showed that vector feeding preferences and the transmission rate between mosquitoes and birds were the parameters that most influenced WNV invasion risk. Even though our model did not predict WNV invasion across any of the studied environments, we found that natural habitats displayed the highest susceptibility to WNV invasion. Pipiens (eco)type acted as the primary vector in all habitats. Hybrids, contrary to common opinion, showed minimal involvement in WNV transmission. However, it is important to interpret our study results with caution due to the possibility of idealized spring and summer seasons being reflected in the field-collected data. Our study could be a tool to enhance current vector surveillance and control programs by targeting specific vector types in specific environments, especially in natural habitat, which are most responsive to environmental shifts. The joint approach based on epidemiological modelling based on field collected data can help to reduce wasted time and economic costs while maximizing the efficiency of local public health authorities.

Resolving the enigma of Iquitos and Manaus: A modeling analysis of multiple COVID-19 epidemic waves in two Amazonian cities

The two nearby Amazonian cities of Iquitos and Manaus endured explosive COVID-19 epidemics and may well have suffered the world's highest infection and death rates over 2020, the first year of the pandemic. State-of-the-art epidemiological and modeling studies estimated that the populations of both cities came close to attaining herd immunity (>70% infected) at the termination of the first wave and were thus protected. This makes it difficult to explain the more deadly second wave of COVID-19 that struck again in Manaus just months later, simultaneous with the appearance of a new P.1 variant of concern, creating a catastrophe for the unprepared population. It was suggested that the second wave was driven by reinfections, but the episode has become controversial and an enigma in the history of the pandemic. We present a data-driven model of epidemic dynamics in Iquitos, which we also use to explain and model events in Manaus. By reverse engineering the multiple epidemic waves over 2 y in these two cities, the partially observed Markov process model inferred that the first wave left Manaus with a highly susceptible and vulnerable population (≈40% infected) open to invasion by P.1, in contrast to Iquitos (≈72% infected). The model reconstructed the full epidemic outbreak dynamics from mortality data by fitting a flexible time-varying reproductive number [Formula: see text] while estimating reinfection and impulsive immune evasion. The approach is currently highly relevant given the lack of tools available to assess these factors as new SARS-CoV-2 virus variants appear with different degrees of immune evasion.

Evolutionary Analysis of TCGA Data Using Over- and Under- Mutated Genes Identify Key Molecular Pathways and Cellular Functions in Lung Cancer Subtypes

We identify critical conserved and mutated genes through a theoretical model linking a gene’s fitness contribution to its observed mutational frequency in a clinical cohort. “Passenger” gene mutations do not alter fitness and have mutational frequencies determined by gene size and the mutation rate. Driver mutations, which increase fitness (and proliferation), are observed more frequently than expected. Non-synonymous mutations in essential genes reduce fitness and are eliminated by natural selection resulting in lower prevalence than expected. We apply this “evolutionary triage” principle to TCGA data from EGFR-mutant, KRAS-mutant, and NEK (non-EGFR/KRAS) lung adenocarcinomas. We find frequent overlap of evolutionarily selected non-synonymous gene mutations among the subtypes suggesting enrichment for adaptations to common local tissue selection forces. Overlap of conserved genes in the LUAD subtypes is rare suggesting negative evolutionary selection is strongly dependent on initiating mutational events during carcinogenesis. Highly expressed genes are more likely to be conserved and significant changes in expression (>20% increased/decreased) are common in genes with evolutionarily selected mutations but not in conserved genes. EGFR-mut cancers have fewer average mutations (89) than KRAS-mut (228) and NEK (313). Subtype-specific variation in conserved and mutated genes identify critical molecular components in cell signaling, extracellular matrix remodeling, and membrane transporters. These findings demonstrate subtype-specific patterns of co-adaptations between the defining driver mutation and somatically conserved genes as well as novel insights into epigenetic versus genetic contributions to cancer evolution.

Novel evolutionary dynamics of small populations in breast cancer adjuvant and neoadjuvant therapy

Disseminated cancer cells (DCCs) are detected in the circulation and bone marrow of up to 40% of breast cancer (BC) patients with clinically localized disease. The formation of metastases is governed by eco-evolutionary interactions of DCCs with the tissue during the transition from microscopic populations to macroscopic disease. Here, we view BC adjuvant and neoadjuvant treatments in the context of small population extinction dynamics observed in the Anthropocene era. Specifically, the unique eco-evolutionary dynamics of small asexually reproducing cancer populations render them highly vulnerable to: (1) environmental and demographic fluctuations, (2) Allee effects, (3) genetic drift and (4) population fragmentation. Furthermore, these typically interact, producing self-reinforcing, destructive dynamics—termed the Extinction Vortex—eradicating the population even when none of the perturbations is individually capable of causing extinction. We propose that developing BC adjuvant and neoadjuvant protocols may exploit these dynamics to prevent recovery and proliferation of small cancer populations during and after treatment—termed “Eco-evolutionary rescue” in natural extinctions. We hypothesize more strategic application of currently available agents based on the extinction vulnerabilities of small populations could improve clinical outcomes.

Extraordinary curtailment of massive typhus epidemic in the Warsaw Ghetto

The highly dependent interplay of disease, famine, war, and society is examined based on an extreme period during World War II. Using mathematical modeling, we reassess events during the Holocaust that led to the liquidation of the Warsaw Ghetto (1941-1942), with the eventual goal of deliberately killing ~450,000, mostly Jewish residents, many through widespread starvation and a large-scale typhus epidemic. The Nazis justified genocide supposedly to control the spread of disease. This exemplifies humanity's ability to turn upon itself, based on racially guided epidemiological principles, merely because of the appearance of a bacterium. Deadly disease and starvation dynamics are explored using modeling and the maths of food ration cards. Strangely, the epidemic was curtailed and was brought to a sudden halt before winter, when typhus normally accelerates. A far more massive epidemic outbreak was prevented through the antiepidemic efforts by the often considered incompetent and corrupt ghetto leadership and the Herculean efforts of ghetto doctors.

Eradicating Metastatic Cancer and the Eco-Evolutionary Dynamics of Anthropocene Extinctions

Curative therapy for metastatic cancers is equivalent to causing extinction of a large, heterogeneous, and geographically dispersed population. Although eradication of dinosaurs is a dramatic example of extinction dynamics, similar application of massive eco-evolutionary force in cancer treatment is typically limited by host toxicity. Here, we investigate the evolutionary dynamics of Anthropocene species extinctions as an alternative model for curative cancer therapy. Human activities can produce extinctions of large, diverse, and geographically distributed populations. The extinction of a species typically follows a pattern in which initial demographic and ecological insults reduce the size and heterogeneity of the population. The surviving individuals, with decreased genetic diversity and often fragmented ecology, are then vulnerable to small stochastic perturbations that further reduce the population until extinction is inevitable. We hypothesize large, diverse, and disseminated cancer populations can be eradicated using similar evolutionary dynamics. Initial therapy is applied to reduce population size and diversity and followed by new treatments to exploit the eco-evolutionary vulnerability of small and/or declining populations. Mathematical models and computer simulations demonstrate initial reductive treatment followed immediately by demographic and ecological perturbations, similar to the empirically derived treatment of pediatric acute lymphocytic leukemia, can consistently achieve curative outcomes in nonpediatric cancers. SIGNIFICANCE: Anthropocene extinctions suggest a strategy for eradicating metastatic cancers in which initial therapy, by reducing the size and diversity of the population, renders it vulnerable to extinction by rapidly applied additional perturbations.

Network motifs and their origins

Modern network science is a new and exciting research field that has transformed the study of complex systems over the last 2 decades. Of particular interest is the identification of small "network motifs" that might be embedded in a larger network and that indicate the presence of evolutionary design principles or have an overly influential role on system-wide dynamics. Motifs are patterns of interconnections, or subgraphs, that appear in an observed network significantly more often than in compatible randomized networks. The concept of network motifs was introduced into Systems Biology by Milo, Alon and colleagues in 2002, quickly revolutionized the field, and it has had a huge impact in wider scientific domains ever since. Here, we argue that the same concept and tools for the detection of motifs were well known in the ecological literature decades into the last century, a fact that is generally not recognized. We review the early history of network motifs, their evolution in the mathematics literature, and their recent rediscoveries.

Signatures of competition and strain structure within the major blood-stage antigen of Plasmodium falciparum in a local community in Ghana

The concept of niche partitioning has received considerable theoretical attention at the interface of ecology and evolution of infectious diseases. Strain theory postulates that pathogen populations can be structured into distinct nonoverlapping strains by frequency-dependent selection in response to intraspecific competition for host immune space. The malaria parasite Plasmodium falciparum presents an opportunity to investigate this phenomenon in nature, under conditions of high recombination rate and extensive antigenic diversity. The parasite's major blood-stage antigen, Pf EMP1, is encoded by the hyperdiverse var genes. With a dataset that includes thousands of var DBLα sequence types sampled from asymptomatic cases within an area of high endemicity in Ghana, we address how var diversity is distributed within isolates and compare this to the distribution of microsatellite allelic diversity within isolates to test whether antigenic and neutral regions of the genome are structured differently. With respect to var DBLα sequence types, we find that on average isolates exhibit significantly lower overlap than expected randomly, but that there also exists frequent pairs of isolates that are highly related. Furthermore, the linkage network of var DBLα sequence types reveals a pattern of nonrandom modularity unique to these antigenic genes, and we find that modules of highly linked DBLα types are not explainable by neutral forces related to var recombination constraints, microsatellite diversity, sampling location, host age, or multiplicity of infection. These findings of reduced overlap and modularity among the var antigenic genes are consistent with a role for immune selection as proposed by strain theory. Identifying the evolutionary and ecological dynamics that are responsible for the nonrandom structure in P. falciparum antigenic diversity is important for designing effective intervention in endemic areas.

Evidence of strain structure in Plasmodium falciparum var gene repertoires in children from Gabon, West Africa

Existing theory on competition for hosts between pathogen strains has proposed that immune selection can lead to the maintenance of strain structure consisting of discrete, weakly overlapping antigenic repertoires. This prediction of strain theory has conceptual overlap with fundamental ideas in ecology on niche partitioning and limiting similarity between coexisting species in an ecosystem, which oppose the hypothesis of neutral coexistence. For Plasmodium falciparum, strain theory has been specifically proposed in relation to the major surface antigen of the blood stage, known as PfEMP1 and encoded by the multicopy multigene family known as the var genes. Deep sampling of the DBLα domain of var genes in the local population of Bakoumba, West Africa, was completed to define whether patterns of repertoire overlap support a role of immune selection under the opposing force of high outcrossing, a characteristic of areas of intense malaria transmission. Using a 454 high-throughput sequencing protocol, we report extremely high diversity of the DBLα domain and a large parasite population with DBLα repertoires structured into nonrandom patterns of overlap. Such population structure, significant for the high diversity of var genes that compose it at a local level, supports the existence of "strains" characterized by distinct var gene repertoires. Nonneutral, frequency-dependent competition would be at play and could underlie these patterns. With a computational experiment that simulates an intervention similar to mass drug administration, we argue that the observed repertoire structure matters for the antigenic var diversity of the parasite population remaining after intervention.

Modeling the Impact of White-Plague Coral Disease in Climate Change Scenarios

Coral reefs are in global decline, with coral diseases increasing both in prevalence and in space, a situation that is expected only to worsen as future thermal stressors increase. Through intense surveillance, we have collected a unique and highly resolved dataset from the coral reef of Eilat (Israel, Red Sea), that documents the spatiotemporal dynamics of a White Plague Disease (WPD) outbreak over the course of a full season. Based on modern statistical methodologies, we develop a novel spatial epidemiological model that uses a maximum-likelihood procedure to fit the data and assess the transmission pattern of WPD. We link the model to sea surface temperature (SST) and test the possible effect of increasing temperatures on disease dynamics. Our results reveal that the likelihood of a susceptible coral to become infected is governed both by SST and by its spatial location relative to nearby infected corals. The model shows that the magnitude of WPD epidemics strongly depends on demographic circumstances; under one extreme, when recruitment is free-space regulated and coral density remains relatively constant, even an increase of only 0.5°C in SST can cause epidemics to double in magnitude. In reality, however, the spatial nature of transmission can effectively protect the community, restricting the magnitude of annual epidemics. This is because the probability of susceptible corals to become infected is negatively associated with coral density. Based on our findings, we expect that infectious diseases having a significant spatial component, such as Red-Sea WPD, will never lead to a complete destruction of the coral community under increased thermal stress. However, this also implies that signs of recovery of local coral communities may be misleading; indicative more of spatial dynamics than true rehabilitation of these communities. In contrast to earlier generic models, our approach captures dynamics of WPD both in space and time, accounting for the highly seasonal nature of annual WPD outbreaks.

Coral reefs are deteriorating at alarming rates, with coral disease outbreaks increasing in prevalence and in space. Anomalously high ocean temperatures are thought to significantly contribute to this problem. We collected a unique and highly resolved dataset of a White Plague Disease (WPD) outbreak from the coral reef of Eilat (Israel, Red Sea). By fitting a novel epidemiological model to the data, we characterize the dynamics of WPD, and study the possible effects of future increasing sea-surface temperatures (SST) on disease dynamics. In contrast to earlier studies, our approach captures the dynamics of coral disease both in space and time, and accounts for the highly seasonal nature of the annual outbreaks. We also apply a novel combination of spatiotemporal statistics and null hypothesis approaches to study the disease progression. Model forecasts into the future show that for some scenarios even an increase of only 0.5°C in SST can cause epidemics to double in magnitude. Since the probability of infection is found to be negatively associated with coral density, this implies that the spatial nature of disease transmission can both enhance and restrict the magnitude of annual epidemics. The results have implications for designing management policies appropriate for coral reef conservation.

Population structuring of multi-copy, antigen-encoding genes in Plasmodium falciparum

The coexistence of multiple independently circulating strains in pathogen populations that undergo sexual recombination is a central question of epidemiology with profound implications for control. An agent-based model is developed that extends earlier ‘strain theory’ by addressing the var gene family of Plasmodium falciparum. The model explicitly considers the extensive diversity of multi-copy genes that undergo antigenic variation via sequential, mutually exclusive expression. It tracks the dynamics of all unique var repertoires in a population of hosts, and shows that even under high levels of sexual recombination, strain competition mediated through cross-immunity structures the parasite population into a subset of coexisting dominant repertoires of var genes whose degree of antigenic overlap depends on transmission intensity. Empirical comparison of patterns of genetic variation at antigenic and neutral sites supports this role for immune selection in structuring parasite diversity.

DOI:http://dx.doi.org/10.7554/eLife.00093.001

Malaria is an infectious disease that is estimated to kill more than half a million people every year, mostly young children in Africa. It is spread by mosquitos that are infected with Plasmodium parasites that attack red blood cells in the human body. Plasmodium falciparum, the species that is responsible for most of these deaths, causes malaria by entering red blood cells and releasing antigens that travel to the surface of the cells, where they change the adhesion properties. This causes the infected red blood cells to accumulate in small blood vessels, surface capillaries or the brain, which can have severe consequences for the person infected.

P. falciparum is particularly dangerous because of its ability to vary the antigens displayed on the cell surface: this process, known as antigenic variation, helps to maintain infections for extended periods of time by allowing the antigens to stay one step ahead of the immune system (a process known as immune escape). The origins of antigenic variation lie in the fact that each P. falciparum genome has a repertoire of between 50 and 60 var genes that code for the variability of a major antigen that is responsible for immune escape in malaria. Molecular sequencing has shown that local parasite populations contain thousands of different types of var genes: hence, meiotic recombination in the mosquito can create a vast number of combinations of var repertoires.

Artzy-Randrup et al. have developed a computational model of this highly diverse and complex system to address the following question: is a local pathogen population composed of largely random and ephemeral repertoires of these genes, or is it structured into independently circulating strains? Their model goes beyond previous models by including interactions within the local host population that arise as a result of indirect competition between different strains of the pathogen for available hosts: this competition is influenced by the history of infection and, therefore, by the patterns of immunity within the host population. Previous models included within-host processes but not these higher, local population-level interactions.

The model simulates the dynamics of all the unique combinations of var genes in a population of hosts, and shows that even with high rates of reproduction, the parasite population self-organizes into a limited number of coexisting strains: the distinct var repertoires of these strains only weakly overlap, suggesting that the immune response of the host population has been partitioned into distinct niches. By investigating genetic variation at both antigenic sites and regions of the genome that do not code for antigens, Artzy-Randrup et al. suggest that immune selection—the selection imposed on var repertoires by the build up of specific immunity at the population level—plays a central role in structuring parasite diversity.

The new model should lead to a better understanding of the epidemiology of Plasmodium and other pathogens that work in similar ways, including Trypanosoma brucei (sleeping sickness), Borellia burgdorferi (Lyme disease) and Giardia lamblia (gastroenteritis), and help with global efforts to eliminate malaria and other diseases.

DOI:http://dx.doi.org/10.7554/eLife.00093.002

Transmission intensity and drug resistance in malaria population dynamics: implications for climate change

Although the spread of drug resistance and the influence of climate change on malaria are most often considered separately, these factors have the potential to interact through altered levels of transmission intensity. The influence of transmission intensity on the evolution of drug resistance has been addressed in theoretical studies from a population genetics' perspective; less is known however on how epidemiological dynamics at the population level modulates this influence. We ask from a theoretical perspective, whether population dynamics can explain non-trivial, non-monotonic, patterns of treatment failure with transmission intensity, and, if so, under what conditions. We then address the implications of warmer temperatures in an East African highland, where, as in other similar regions at the altitudinal edge of malaria's distribution, there has been a pronounced increase of cases from the 1970s to the 1990s. Our theoretical analyses, with a transmission model that includes different levels of immunity, demonstrate that an increase in transmission beyond a threshold can lead to a decrease in drug resistance, as previously shown [1], but that a second threshold may occur and lead to the re-establishment of drug resistance. Estimates of the increase in transmission intensity from the 1970s to the 1990s for the Kenyan time series, obtained by fitting the two-stage version of the model with an explicit representation of vector dynamics, suggest that warmer temperatures are likely to have moved the system towards the first threshold, and in so doing, to have promoted the faster spread of drug resistance. Climate change and drug resistance can interact and need not be considered as alternative explanations for trends in disease incidence in this region. Non-monotonic patterns of treatment failure with transmission intensity similar to those described as the ‘valley phenomenon’ for Uganda can result from epidemiological dynamics but under poorly understood assumptions.

Connectivity, cycles, and persistence thresholds in metapopulation networks

Synthesising the relationships between complexity, connectivity, and the stability of large biological systems has been a longstanding fundamental quest in theoretical biology and ecology. With the many exciting developments in modern network theory, interest in these issues has recently come to the forefront in a range of multidisciplinary areas. Here we outline a new theoretical analysis specifically relevant for the study of ecological metapopulations focusing primarily on marine systems, where subpopulations are generally connected via larval dispersal. Our work determines the qualitative and quantitative conditions by which dispersal and network structure control the persistence of a set of age-structured patch populations. Mathematical modelling combined with a graph theoretic analysis demonstrates that persistence depends crucially on the topology of cycles in the dispersal network which tend to enhance the effect of larvae "returning home." Our method clarifies the impact directly due to network structure, but this almost by definition can only be achieved by examining the simplified case in which patches are identical; an assumption that we later relax. The methodology identifies critical migration routes, whose presence are vital to overall stability, and therefore should have high conservation priority. In contrast, "lonely links," or links in the network that do not participate in a cyclical component, have no impact on persistence and thus have low conservation priority. A number of other intriguing criteria for persistence are derived. Our modelling framework reveals new insights regarding the determinants of persistence, stability, and thresholds in complex metapopulations. In particular, while theoretical arguments have, in the past, suggested that increasing connectivity is a destabilizing feature in complex systems, this is not evident in metapopulation networks where connectivity, cycles, coherency, and heterogeneity all tend to enhance persistence. The results should be of interest for many other scientific contexts that make use of network theory.

Compensatory evolution in mitochondrial tRNAs navigates valleys of low fitness

A long-standing controversy in evolutionary biology is whether or not evolving lineages can cross valleys on the fitness landscape that correspond to low-fitness genotypes, which can eventually enable them to reach isolated fitness peaks. Here we study the fitness landscapes traversed by switches between different AU and GC Watson-Crick nucleotide pairs at complementary sites of mitochondrial transfer RNA stem regions in 83 mammalian species. We find that such Watson-Crick switches occur 30-40 times more slowly than pairs of neutral substitutions, and that alleles corresponding to GU and AC non-Watson-Crick intermediate states segregate within human populations at low frequencies, similar to those of non-synonymous alleles. Substitutions leading to a Watson-Crick switch are strongly correlated, especially in mitochondrial tRNAs encoded on the GT-nucleotide-rich strand of the mitochondrial genome. Using these data we estimate that a typical Watson-Crick switch involves crossing a fitness valley of a depth of about 10(-3) or even about 10(-2), with AC intermediates being slightly more deleterious than GU intermediates. This compensatory evolution must proceed through rare intermediate variants that never reach fixation. The ubiquitous nature of compensatory evolution in mammalian mitochondrial tRNAs and other molecules implies that simultaneous fixation of two alleles that are individually deleterious may be a common phenomenon at the molecular level.

Emergence and size of the giant component in clustered random graphs with a given degree distribution

Standard techniques for analyzing network models usually break down in the presence of clustering. Here we introduce a new analytic tool, the "free-excess degree" distribution, which extends the generating function framework, making it applicable for clustered networks (C>0). The methodology is general and provides a new expression for the threshold point at which the giant component emerges and shows that it scales as (1-C)(-1). In addition, the size of the giant component may be predicted even for more complicated scenarios such as the removal of a fixed fraction of nodes at random.

Size-structured demographic models of coral populations

The demographic processes of growth, mortality, and the recruitment of young individuals, are the major organizing forces regulating communities in open systems. Here we present a size-structured (rather than age-structured) population model to examine the role of these different processes in space-limited open systems, taking coral reefs as an example. In this flux-diffusion model the growth rate of corals depends both on the available free-space (i.e. density-dependence) and on the particular size of the coral. In our analysis we progressively study several different forms of growth rate functions to disentangle the effects of free space and size-dependence on the model's stability. Unlike Roughgarden et al. [1985. Demographic theory for an open marine population space-limited recruitment. Ecology 66(1), 54-67], whose principal result is that the growth of settled organisms is destabilizing, we find that size-dependent growth rate often has the potential to endow stability. This is particularly true, if the growth rate is dependent on available free space (i.e. density dependent), but examples are given for growth rates that even lack this property. Further insights into reef system fragility are found through studying the sensitivity of the model steady state to changes in recruitment.

Sympatric speciation under incompatibility selection

The existing theory of sympatric speciation assumes that a local population splits into two species under one-dimensional disruptive selection, which favors both of the opposite extreme values of a quantitative trait. Here we model sympatric speciation under selection that favors high values of either of the two independently inherited traits, each required to efficiently consume one of the two available resources, but acts, because of a tradeoff, against those possessing high values of both traits. Such two-dimensional incompatibility selection is similar to that involved in allopatric speciation. Using a hypergeometric phenotypic model, we show that incompatibility selection readily leads to sympatric speciation. In contrast to disruptive selection, two distinct modes of sympatric speciation exist under incompatibility selection: under strong tradeoffs both of the new species are specialists, each consuming its own resource, but under moderate tradeoffs speciation may be asymmetric and involve the origin of a specialist and a generalist species. Also, incompatibility selection may lead to irreversible specialization: under strong tradeoffs, the population speciates if it consists mostly of unspecialized individuals, but remains undivided if most of the individuals are specialized to consume one of the resources. Incompatibility selection appears to be more realistic than disruptive selection, implying that incompatibility between individually adaptive alleles or trait states drives both allopatric and sympatric speciation.

Generating uniformly distributed random networks

The analysis of real networks taken from the biological, social, and physical sciences often requires a carefully posed statistical null-hypothesis approach. One common method requires comparing real networks to an ensemble of random matrices that satisfy realistic constraints in which each different matrix member is equiprobable. We discuss existing methods for generating uniformly distributed (constrained) random matrices, describe their shortcomings, and present an efficient technique that should have many practical applications.