Evolution of pathogens towards low R0 in heterogeneous populations

Maximization of the basic reproduction ratio or R(0) is widely believed to drive the emergence of novel pathogens. The presence of exploitable heterogeneities in a population, such as high variance in the number of potentially infectious contacts, increases R(0) and thus pathogens that can exploit heterogeneities in the contact structure have an advantage over those that do not. However, exploitation of heterogeneities results in a more rapid depletion of the potentially susceptible neighbourhood for an infected host. Here a simple model of pathogen evolution in a heterogeneous environment is developed and placed in the context of HIV transmission. In this model, it is shown that pathogens may evolve towards lower R(0), even if this results in pathogen extinction. For sufficiently high transmissibility, two locally stable strategies exist for an evolving pathogen, one that exploits heterogeneities and results in higher R(0), and one that does not, and results in lower R(0). While the low R(0) strategy is never evolutionarily stable, invading strains with higher R(0) will also converge to the low R(0) strategy if not sufficiently different from the resident strain. Heterogenous transmission is increasingly recognized as fundamental to epidemiological dynamics and the evolution of pathogens; here, it is shown that the ability to exploit heterogeneity is a strategy that can itself evolve.     

Bursts of Vertex Activation and Epidemics in Evolving Networks

The dynamic nature of contact patterns creates diverse temporal structures. In particular, empirical studies have shown that contact patterns follow heterogeneous inter-event time intervals, meaning that periods of high activity are followed by long periods of inactivity. To investigate the impact of these heterogeneities in the spread of infection from a theoretical perspective, we propose a stochastic model to generate temporal networks where vertices make instantaneous contacts following heterogeneous inter-event intervals, and may leave and enter the system. We study how these properties affect the prevalence of an infection and estimate , the number of secondary infections of an infectious individual in a completely susceptible population, by modeling simulated infections (SI and SIR) that co-evolve with the network structure. We find that heterogeneous contact patterns cause earlier and larger epidemics in the SIR model in comparison to homogeneous scenarios for a vast range of parameter values, while smaller epidemics may happen in some combinations of parameters. In the case of SI and heterogeneous patterns, the epidemics develop faster in the earlier stages followed by a slowdown in the asymptotic limit. For increasing vertex turnover rates, heterogeneous patterns generally cause higher prevalence in comparison to homogeneous scenarios with the same average inter-event interval. We find that is generally higher for heterogeneous patterns, except for sufficiently large infection duration and transmission probability.     

Network theory and SARS: predicting outbreak diversity

Many infectious diseases spread through populations via the networks formed by physical contacts among individuals. The patterns of these contacts tend to be highly heterogeneous. Traditional "compartmental" modeling in epidemiology, however, assumes that population groups are fully mixed, that is, every individual has an equal chance of spreading the disease to every other. Applications of compartmental models to Severe Acute Respiratory Syndrome (SARS) resulted in estimates of the fundamental quantity called the basic reproductive number R0--the number of new cases of SARS resulting from a single initial case--above one, implying that, without public health intervention, most outbreaks should spark large-scale epidemics. Here we compare these predictions to the early epidemiology of SARS. We apply the methods of contact network epidemiology to illustrate that for a single value of R0, any two outbreaks, even in the same setting, may have very different epidemiological outcomes. We offer quantitative insight into the heterogeneity of SARS outbreaks worldwide, and illustrate the utility of this approach for assessing public health strategies.     

Sheep Movement Networks and the Transmission of Infectious Diseases

Background and Methodology

Various approaches have been used to investigate how properties of farm contact networks impact on the transmission of infectious diseases. The potential for transmission of an infection through a contact network can be evaluated in terms of the basic reproduction number, R ₀. The magnitude of R ₀ is related to the mean contact rate of a host, in this case a farm, and is further influenced by heterogeneities in contact rates of individual hosts. The latter can be evaluated as the second order moments of the contact matrix (variances in contact rates, and co-variance between contacts to and from individual hosts). Here we calculate these quantities for the farms in a country-wide livestock network: >15,000 Scottish sheep farms in each of 4 years from July 2003 to June 2007. The analysis is relevant to endemic and chronic infections with prolonged periods of infectivity of affected animals, and uses different weightings of contacts to address disease scenarios of low, intermediate and high animal-level prevalence.

Principal Findings and Conclusions

Analysis of networks of Scottish farms via sheep movements from July 2003 to June 2007 suggests that heterogeneities in movement patterns (variances and covariances of rates of movement on and off the farms) make a substantial contribution to the potential for the transmission of infectious diseases, quantified as R ₀, within the farm population. A small percentage of farms (<20%) contribute the bulk of the transmission potential (>80%) and these farms could be efficiently targeted by interventions aimed at reducing spread of diseases via animal movement.     

Spatial and Temporal Analysis of Contact Rates in Female White-Tailed Deer

Abstract: White-tailed deer (Odocoileus virginianus) are important game mammals and potential reservoirs of diseases of domestic livestock; thus, diseases of deer are of great concern to wildlife managers. Contact, either direct or indirect, is necessary for disease transmission, but we know little about the ecological contexts that promote intrasexual contact among deer. Using pair-wise direct contacts estimated from Global Positioning System collar locations and joint utilization distributions (JUDs), we assessed habitats in which contacts occur to test whether direct contact rates among female white-tailed deer in different social groups differs among land-cover types. We also tested whether contact rates differed among seasons, lunar phases, and times of day. We obtained locations from 27 female deer for periods of 0.5–17 months during 2002–2006. We designated any simultaneous pair of locations for 2 deer <25 m apart as a direct contact. For each season, we used compositional analysis to compare land-cover types where 2 deer had contact to available land-cover weighted by their JUD. We used mixed-model logistic regression to test for effects of season, lunar phase, and time of day on contact rates. Contact rates during the gestation season were greater than expected from random use in forest and grassland cover, whereas contact rates during the fawning period were greater in agricultural fields than in other land-cover types. Contact rates were greatest during the rut and lowest in summer. Diel patterns of contact rates varied with season, and contact rates were elevated during full moon compared to other lunar periods. Both spatial and temporal analyses suggest that contact between female deer in different social groups occurs mainly during feeding, which highlights the potential impact of food distribution and habitat on contact rates among deer. By using methods to associate contacts and land-cover, we have created beneficial tools for more elaborate and detailed studies of disease transmission. Our methods can offer information necessary to develop spatially realistic models of disease transmission in deer.     

Adaptive evolution of the insulin gene in caviomorph rodents

Insulin is a conservative molecule among mammals, maintaining both its structure and function. Rodents that belong to the Suborder Hystricognathi represent an exception, having a very divergent molecule with unusual physiological properties. In this work, we analyzed the evolutionary pattern of the insulin gene in caviomorph rodents (South American hystricomorph rodents). We found that these rodents have higher rates of nonsynonymous:synonymous substitutions (d(N)/d(S)) than nonhystricomorph rodents and that values are heterogeneous inside the group. We estimated codons under positive selection, specifically the second binding site (A13 and B17) and others related with hexamerization (B18, B20, and B22). In the monomer structure, all selected sites formed a single patch around the second binding site. In the hexamer structure, these amino acids were grouped into three major patches. In this structure, contacts between B chains involved all selected sites (except B18), and between faces in the center of the molecule, all contacts were among selected sites. While there is no clear hypothesis regarding the cause of this drastic change, experimental evidence does show that this group of rodents has some peculiarities in growth function, and, whether coincidental or not, these changes appeared together with important changes in life-history traits.     

Molecular mechanisms of protein-protein recognition: whether the surface placed charged residues determine the recognition process?

We studied the structure and composition of contact areas in 812 different kind dimeric protein-protein complexes from Brookhaven data base (PDB ) in order to reveal their pecularities with regard to protein-protein recognition. We have found, that the large portion of complexes (approximately 70%) have oppositely charged residues in the contact areas (interfaces) on the subunits surfaces, which form electrostatic contacts - R:E, R:D, K:E, K:D, H:E, H:D. These results are consistent with the current view that high rate complex formation may be driven by the long-range electrostatic interaction between charged AA residues of subunits surfaces. However, there are many complexes among the studied ones (approximately 30%), which have no electrostatic contacts at all in their contact area. Thus a question arises: what forces account for high complex formation rates (i.e. for the distant orienting of subunits before encounter) by forming complexes where the surface contact areas lack electrostatic contacts? We believe that the long-range orienting electrostatic interaction of subunits may account for all cases of efficient complex formation if one drops the traditional view that protein subunits interact mainly through their surfaces. We suggest that the distant orienting being due to the electrostatic interaction between the whole aggregates of partial electric charges of atoms of each complex subunits. Our preliminary model calculations (unpublished) made for ribonuclease dimer (does not have electrostatic contacts) conform this suggestion.     

Effects of Joint Space Use and Group Membership on Contact Rates Among White-Tailed Deer

Abstract: Establishment and spread of infectious diseases are controlled by the frequency of contacts among hosts. Although managers can estimate transmission coefficients from the relationship between disease prevalence and age or time, they may wish to quantify or compare contact rates before a disease is established or while it is at very low prevalence. Our objectives were to quantify direct and indirect contacts rates among white-tailed deer (Odocoileus virginianus) and to compare these measures of contact rate with simpler measures of joint space use. We deployed Global Positioning System (GPS) collars on 23 deer near Carbondale, Illinois, USA, from 2002 to 2005. We used location data from the GPS collars to measure pairwise rates of direct and indirect contact, based on a range of proximity criteria and time lags, as well as volume of intersection (VI) of kernel utilization distributions. We analyzed contact rates at a given distance criterion and time lag using mixed-model logistic regression. Direct contact rates increased with increasing VI and were higher in autumn—spring than in summer. After accounting for VI, the estimated odds of direct contact during autumn—spring periods were 5.0–22.1-fold greater (depending on the proximity criterion) for pairs of deer in the same social group than for between-group pairs, but for direct contacts during summer the within:between-group odds ratio did not differ significantly from 1. Indirect contact rates also increased with VI, but the effects of both season and pair-type were much smaller than for direct contacts and differed little as the time lag increased from 1–30 days. These results indicate that simple measures of joint space use are insufficient indices of direct contact because group membership can substantially increase contacts at a given level of joint space use. With indirect transmission, however, group membership had a much smaller influence after accounting for VI. Relationships between contact rates and season, VI, and pair-type were generally robust to changes in the proximity criterion defining a contact, and patterns of indirect contacts were affected little by the choice of time lag from 1–30 days. The use of GPS collars provides a framework for testing hypotheses about the form of contact networks among large mammals and comparing potential direct and indirect contact rates across gradients of ecological factors, such as population density or landscape configuration.     

Comparisons of the molecular evolutionary process at rbcL and ndhF in the grass family (Poaceae)

We examine rate heterogeneity among evolutionary lineages of the grass family at two plasmid loci, ndhF and rbcL, and we introduce a method to determine whether patterns of rate heterogeneity are correlated between loci. We show both that rates of synonymous evolution are heterogeneous among grass lineages and that are heterogeneity is correlated between loci at synonymous sites. At nonsynonymous sites, the pattern of rate heterogeneity is not correlated between loci, primarily due to an aberrant pattern of rate heterogeneity at nonsynonymous sites of rbcL. We compare patterns of synonymous rate heterogeneity to predictors based on the generation time effect and the speciation rate hypotheses. Although there is some evidence for generation time effects, neither generation time effects nor speciation rates appear to be sufficient to explain patterns of rate heterogeneity in the grass plastid sequences.      

Mixing in age-structured population models of infectious diseases

Infectious diseases are controlled by reducing pathogen replication within or transmission between hosts. Models can reliably evaluate alternative strategies for curtailing transmission, but only if interpersonal mixing is represented realistically. Compartmental modelers commonly use convex combinations of contacts within and among groups of similarly aged individuals, respectively termed preferential and proportionate mixing. Recently published face-to-face conversation and time-use studies suggest that parents and children and co-workers also mix preferentially. As indirect effects arise from the off-diagonal elements of mixing matrices, these observations are exceedingly important. Accordingly, we refined the formula published by Jacquez et al. [19] to account for these newly-observed patterns and estimated age-specific fractions of contacts with each preferred group. As the ages of contemporaries need not be identical nor those of parents and children to differ by exactly the generation time, we also estimated the variances of the Gaussian distributions with which we replaced the Kronecker delta commonly used in theoretical studies. Our formulae reproduce observed patterns and can be used, given contacts, to estimate probabilities of infection on contact, infection rates, and reproduction numbers. As examples, we illustrate these calculations for influenza based on "attack rates" from a prospective household study during the 1957 pandemic and for varicella based on cumulative incidence estimated from a cross-sectional serological survey conducted from 1988-94, together with contact rates from the several face-to-face conversation and time-use studies. Susceptibility to infection on contact generally declines with age, but may be elevated among adolescents and adults with young children.     

The size of epidemics in populations with heterogeneous susceptibility

We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.     

Epidemiological models for sexually transmitted diseases

The classical models for sexually transmitted infections assume homogeneous mixing either between all males and females or between certain subgroups of males and females with heterogeneous contact rates. This implies that everybody is all the time at risk of acquiring an infection. These models ignore the fact that the formation of a pair of two susceptibles renders them in a sense temporarily immune to infection as long as the partners do not separate and have no contacts with other partners. The present paper takes into account the phenomenon of pair formation by introducing explicitly a pairing rate and a separation rate. The infection transmission dynamics depends on the contact rate within a pair and the duration of a partnership. It turns out that endemic equilibria can only exist if the separation rate is sufficiently large in order to ensure the necessary number of sexual partners. The classical models are recovered if one lets the separation rate tend to infinity.This work has been supported by Deutsche Forschungsgemeinschaft     

Plasma membrane—endoplasmic reticulum contact sites regulate phosphatidylcholine synthesis

Synthesis of phospholipids, sterols and sphingolipids is thought to occur at contact sites between the endoplasmic reticulum (ER) and other organelles because many lipid‐synthesizing enzymes are enriched in these contacts. In only a few cases have the enzymes been localized to contacts in vivo and in no instances have the contacts been demonstrated to be required for enzyme function. Here, we show that plasma membrane (PM)—ER contact sites in yeast are required for phosphatidylcholine synthesis and regulate the activity of the phosphatidylethanolamine N‐methyltransferase enzyme, Opi3. Opi3 activity requires Osh3, which localizes to PM–ER contacts where it might facilitate in trans catalysis by Opi3. Thus, membrane contact sites provide a structural mechanism to regulate lipid synthesis.     

MHC allele-specific molecular features determine peptide/HLA-A2 conformations that are recognized by HLA-A2-restricted T cell receptors

The structures of alphabeta TCRs bound to complexes of class I MHC molecules and peptide show that the TCRs make multiple contacts with the alpha1 and alpha2 helixes of the MHC. Previously we have shown that the A6 TCR in complex with the HLA-A2/Tax peptide has 15 contact sites on HLA-A2. Single amino acid mutagenesis of these contact sites demonstrated that mutation of only three amino acids clustered on the alpha1 helix (R65, K66, A69) disrupted recognition by the A6 TCR. In the present study we have asked whether TCRs that recognize four other peptides presented by HLA-A2 interact with the MHC in identical, similar, or different patterns as the A6 TCR. Mutants K66A and Q155A had the highest frequency of negative effects on lysis. A subset of peptide-specific CTL also selectively recognized mutants K66A or Q155A in the absence of exogenous cognate peptides, indicating that these mutations affected the presentation of endogenous peptide/HLA-A2 complexes. These findings suggest that most HLA-A2-restricted TCRs recognize surfaces on the HLA-A2/peptide complex that are dependent upon the side chains of K66 and Q155 in the central portion of the peptide binding groove. Crystallographic structures of several peptide/HLA-A2 structures have shown that the side chains of these critical amino acids that make contact with the A6 TCR also contact the bound peptide. Collectively, our results indicate that the generalized effects of changes at these critical amino acids are probably due to the fact that they can be directly contacted by TCRs as well as influence the binding and presentation of the bound peptides.     

Heterogeneous Nucleation and Crowding in Sickle Hemoglobin: An Analytic Approach

Sickle hemoglobin nucleation occurs in solution as a homogeneous process or on existing polymers in a heterogeneous process. We have developed an analytic formulation to describe the solution crowding and large nonideality that affects the heterogeneous nucleation of sickle hemoglobin by using convex particle theory. The formulation successfully fits the concentration and temperature dependence of the heterogeneous nucleation process over 14 orders of magnitude. Unlike previous approaches, however, the new formulation can also accurately describe the effects of adding nonpolymerizing agents to the solution. Without additional adjustable parameters, the model now describes the data of M. Ivanova, R. Jasuja, S. Kwong, R. W. Briehl, and F. A. Ferrone, (Biophys. J. 2000, 79:1016-1022), in which up to 50% of the sickle hemoglobin is substituted by cross-linked hemoglobin A, which does not polymerize, and which substitution causes the rates to decrease by 10⁵. The success of this approach provides insight into the polymerization process: from the size-dependence of the contact energy deduced here, it also appears that various contacts of unknown origin are energetically significant in the heterogeneous nucleation process.     

Come a little bit closer! Lipid droplet-ER contact sites are getting crowded

Not so long ago, contact sites between the endoplasmic reticulum (ER) and lipid droplets (LDs) were largely unexplored on a molecular level. In recent years however, numerous proteins have been identified that are enriched or exclusively located at the interfaces between LDs and the ER. These comprise members of protein classes typically found in diverse types of contacts, such as organelle tethers and lipid transfer proteins, but also proteins that have no similarities to known contact site machineries. This structurally heterogeneous group of contact site residents might be required to fulfill unique aspects of LD-ER contact biology, such as de novo LD biogenesis, and maintenance of lipidic connections between LDs and ER. Here, we summarize the current knowledge on the molecular components of this special organelle contact site, and discuss their features and functions.     

Effects of heterogeneous and clustered contact patterns on infectious disease dynamics

The spread of infectious diseases fundamentally depends on the pattern of contacts between individuals. Although studies of contact networks have shown that heterogeneity in the number of contacts and the duration of contacts can have far-reaching epidemiological consequences, models often assume that contacts are chosen at random and thereby ignore the sociological, temporal and/or spatial clustering of contacts. Here we investigate the simultaneous effects of heterogeneous and clustered contact patterns on epidemic dynamics. To model population structure, we generalize the configuration model which has a tunable degree distribution (number of contacts per node) and level of clustering (number of three cliques). To model epidemic dynamics for this class of random graph, we derive a tractable, low-dimensional system of ordinary differential equations that accounts for the effects of network structure on the course of the epidemic. We find that the interaction between clustering and the degree distribution is complex. Clustering always slows an epidemic, but simultaneously increasing clustering and the variance of the degree distribution can increase final epidemic size. We also show that bond percolation-based approximations can be highly biased if one incorrectly assumes that infectious periods are homogeneous, and the magnitude of this bias increases with the amount of clustering in the network. We apply this approach to model the high clustering of contacts within households, using contact parameters estimated from survey data of social interactions, and we identify conditions under which network models that do not account for household structure will be biased.     

P1 and P7 plasmid partition: ParB protein bound to its partition site makes a separate discriminator contact with the DNA that determines species specificity.

The cis-acting P1 and P7 parS sites are responsible for active partition of P1 and P7 plasmids to daughter cells. The two sites are similar but function only with ParB proteins from the correct species. Using hybrid ParB proteins and hybrid parS sites, we show that specificity is determined by contacts between bases that lie within two parS hexamer boxes and a region in the ParB C-terminus. We refer to these contacts as discriminator contacts. The P7 discriminator contacts were mapped to 3 and 2 bp respectively within the two parS hexamer boxes, and a 10 amino acid region of P7 ParB. Similarly placed residues of different sequence are responsible for the P1 discriminator contact. The discriminator contacts are distinct from previously identified DNA binding contacts which involve different ParB and parS regions. Deletion of the ParB C-terminus that makes the discriminator contact does not diminish in vitro binding but renders it species independent. The discriminator contact is therefore a negative function, interfering with binding of the wrong ParB, but not providing energy for the binding of the correct one. Similar discriminator contacts might be responsible for specificities seen among families of eukaryotic DNA binding proteins that share common binding motifs.     

Contact Density Affects Protein Evolutionary Rate from Bacteria to Animals

The density of contacts or the fraction of buried sites in a protein structure is thought to be related to a protein’s designability, and genes encoding more designable proteins should evolve faster than other genes. Several recent studies have tested this hypothesis but have found conflicting results. Here, we investigate how a gene’s evolutionary rate is affected by its protein’s contact density, considering the four species Escherichia coli, Saccharomyces cerevisiae, Drosophila melanogaster, and Homo sapiens. We find for all four species that contact density correlates positively with evolutionary rate, and that these correlations do not seem to be confounded by gene expression level. The strength of this signal, however, varies widely among species. We also study the effect of contact density on domain evolution in multidomain proteins and find that a domain’s contact density influences the domain’s evolutionary rate. Within the same protein, a domain with higher contact density tends to evolve faster than a domain with lower contact density. Our study provides evidence that contact density can increase evolutionary rates, and that it acts similarly on the level of entire proteins and of individual protein domains. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Desolvation Barrier Effects Are a Likely Contributor to the Remarkable Diversity in the Folding Rates of Small Proteins

The variation in folding rate among single-domain natural proteins is tremendous, but common models with explicit representations of the protein chain are either demonstrably insufficient or unclear as to their capability for rationalizing the experimental diversity in folding rates. In view of the critical role of water exclusion in cooperative folding, we apply native-centric, coarse-grained chain modeling with elementary desolvation barriers to investigate solvation effects on folding rates. For a set of 13 proteins, folding rates simulated with desolvation barriers cover ∼ 4.6 orders of magnitude, spanning a range essentially identical to that observed experimentally. In contrast, folding rates simulated without desolvation barriers cover only ∼ 2.2 orders of magnitude. Following a Hammond-like trend, the folding transition-state ensemble (TSE) of a protein model with desolvation barriers generally has a higher average number of native contacts and is structurally more specific, that is, less diffused, than the TSE of the corresponding model without desolvation barriers. Folding is generally significantly slower in models with desolvation barriers because of their higher overall macroscopic folding barriers as well as slower conformational diffusion speeds in the TSE that are ≈ 1/50 times those in models without desolvation barriers. Nonetheless, the average root-mean-square deviation between the TSE and the native conformation is often similar in the two modeling approaches, a finding suggestive of a more robust structural requirement for the folding rate-limiting step. The increased folding rate diversity in models with desolvation barriers originates from the tendency of these microscopic barriers to cause more heightening of the overall macroscopic folding free-energy barriers for proteins with more nonlocal native contacts than those with fewer such contacts. Thus, the enhancement of folding cooperativity by solvation effects is seen as positively correlated with a protein's native topological complexity.