Analysis of symmetries in models of multi-strain infections

In mathematical studies of the dynamics of multi-strain diseases caused by antigenically diverse pathogens, there is a substantial interest in analytical insights. Using the example of a generic model of multi-strain diseases with cross-immunity between strains, we show that a significant understanding of the stability of steady states and possible dynamical behaviours can be achieved when the symmetry of interactions between strains is taken into account. Techniques of equivariant bifurcation theory allow one to identify the type of possible symmetry-breaking Hopf bifurcation, as well as to classify different periodic solutions in terms of their spatial and temporal symmetries. The approach is also illustrated on other models of multi-strain diseases, where the same methodology provides a systematic understanding of bifurcation scenarios and periodic behaviours. The results of the analysis are quite generic, and have wider implications for understanding the dynamics of a large class of models of multi-strain diseases.     

Bifurcation,stability diagrams,and varying diffusion coefficients in reaction-diffusion equations

The effect of keeping all the parameters constant, except the diffusion coefficients, in a pair of reaction-diffusion equations is studied. It is shown that the stability of the constant solution and the bifurcation points can be easily established by constructing a simple stability diagram. The possible qualitatively different diagrams are enumerated.     

Diffusivity, viscosity, and cluster formation in protein solutions

The diffusion coefficient and viscosity of lysozyme solutions were measured at 25°C in various buffers with and without sodium chloride. Measurements were made over the entire concentration range in each case and were extended into the supersaturated region. The results show that diffusion coefficient behavior depends strongly on the buffer used and the ionic strength of the solution, which means the amount of sodium chloride used in buffer solution. Viscosity measurements indicate a small degree of time dependence, with the viscosity increasing with solution age.     

Rapid identification of multi-strain HBV infection in patient by high-throughput DNA sequencing

Hepatitis B is a well-known risk factor for the development of liver cancer and is closely associated with patient morbidity and mortality. Viral mutants and variants have the potential to evade immune response and prolong infection, and thus it is crucial to develop a methodology for the rapid identification of multi-strain hepatitis infections in patients. Here we describe a method based on selective region amplification of viral genome and deep sequencing, which may be used for rapid identification of multi-strain hepatitis B virus (HBV) infection in patients. The method works even with significantly low amounts of patients’ serum samples, where the wet-lab procedures take about 1.5 days, followed by a quick bioinformatic analysis to reveal the final results. Our method can potentially be applied to the rapid and reliable identification of multi-strain HBV infection and help improve treatment regiments.     

Bifurcation analysis of piecewise smooth ecological models

The aim of this paper is the study of the long-term behavior of population communities described by piecewise smooth models (known as Filippov systems). Models of this kind are often used to describe populations with selective switching between alternative habitats or diets or to mimic the evolution of an exploited resource where harvesting is forbidden when the resource is below a prescribed threshold. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations.     

Bifurcation analysis of continuous biochemical reactor models

The validity of a biochemical reactor model often is evaluated by comparing transient responses to experimental data. Dynamic simulation can be a rather inefficient and ineffective tool for analyzing bioreactor models that exhibit complex nonlinear behavior. Bifurcation analysis is a powerful tool for obtaining a more efficient and complete characterization of the model behavior. To illustrate the power of bifurcation analysis, the steady-state and transient behavior of three continuous bioreactor models consisting of a small number of ordinary differential equations are investigated. Several important features, as well as potential limitations, that are difficult to ascertain via dynamic simulation are disclosed through the bifurcation analysis. The results motivate the use of dynamic simulation and bifurcation analysis as complementary tools for analyzing the nonlinear behavior of bioreactor models.     

Mice,microbes and models of infection

We urgently need animal models to study infectious disease. Mice are susceptible to a similar range of microbial infections as humans. Marked differences between inbred strains of mice in their response to pathogen infection can be exploited to analyse the genetic basis of infections. In addition, the genetic tools that are available in the laboratory mouse, and new techniques to monitor the expression of bacterial genes in vivo, make it the principal experimental animal model for studying mechanisms of infection and immunity.     

Bifurcation analysis of periodic SEIR and SIR epidemic models

The bifurcations of the periodic solutions of SEIR and SIR epidemic models with sinusoidally varying contact rate are investigated. The analysis is carried out with respect to two parameters: the mean value and the degree of seasonality of the contact rate. The corresponding portraits in the two-parameter space are obtained by means of a numerical continuation method. Codimension two bifurcations (degenerate flips and cusps) are detected, and multiple stable modes of behavior are identified in various regions of the parameter space. Finally, it is shown how the parametric portrait of the SEIR model tends to that of the SIR model when the latent period tends to zero.     

On state-space reduction in multi-strain pathogen models, with an application to antigenic drift in influenza A

Many pathogens exist in phenotypically distinct strains that interact with each other through competition for hosts. General models that describe such multi-strain systems are extremely difficult to analyze because their state spaces are enormously large. Reduced models have been proposed, but so far all of them necessarily allow for coinfections and require that immunity be mediated solely by reduced infectivity, a potentially problematic assumption. Here, we suggest a new state-space reduction approach that allows immunity to be mediated by either reduced infectivity or reduced susceptibility and that can naturally be used for models with or without coinfections. Our approach utilizes the general framework of status-based models. The cornerstone of our method is the introduction of immunity variables, which describe multi-strain systems more naturally than the traditional tracking of susceptible and infected hosts. Models expressed in this way can be approximated in a natural way by a truncation method that is akin to moment closure, allowing us to sharply reduce the size of the state space, and thus to consider models with many strains in a tractable manner. Applying our method to the phenomenon of antigenic drift in influenza A, we propose a potentially general mechanism that could constrain viral evolution to a one-dimensional manifold in a two-dimensional trait space. Our framework broadens the class of multi-strain systems that can be adequately described by reduced models. It permits computational, and even analytical, investigation and thus serves as a useful tool for understanding the evolution and ecology of multi-strain pathogens.     

Polymer models of protein stability, folding, and interactions

The unfolded state and flexible linkers in the folded structure play essential roles in protein stability and folding and protein-protein interactions. Intrinsic to these roles is the fact that unfolded proteins and flexible linkers sample many different conformations. Polymer models may capture this and complement experiments in elucidating the contributions of the unfolded state and flexible linkers. Here I review what can be predicted from these models and how well these predictions match experiments. For example, Gaussian chain models give quantitatively reasonable predictions of the effects of residual charge-charge interactions in the unfolded state and qualitatively reasonable results for the effects of spatial confinement and macromolecular crowding on protein stability. A wormlike chain model has met with success in quantifying the effects of flexible linkers in binding affinity enhancement and in regulatory switches. In future developments, more realistic models may emerge from molecular dynamics simulations, and these models will guide experiments to advance our understanding of the unfolded state and flexible linkers.     

Binding to F-actin guides cadherin cluster assembly,stability, and movement

The cadherin extracellular region produces intercellular adhesion clusters through trans- and cis-intercadherin bonds, and the intracellular region connects these clusters to the cytoskeleton. To elucidate the interdependence of these binding events, cadherin adhesion was reconstructed from the minimal number of structural elements. F-actin–uncoupled adhesive clusters displayed high instability and random motion. Their assembly required a cadherin cis-binding interface. Coupling these clusters with F-actin through an α-catenin actin-binding domain (αABD) dramatically extended cluster lifetime and conferred direction to cluster motility. In addition, αABD partially lifted the requirement for the cis-interface for cluster assembly. Even more dramatic enhancement of cadherin clustering was observed if αABD was joined with cadherin through a flexible linker or if it was replaced with an actin-binding domain of utrophin. These data present direct evidence that binding to F-actin stabilizes cadherin clusters and cooperates with the cis-interface in cadherin clustering. Such cooperation apparently synchronizes extracellular and intracellular binding events in the process of adherens junction assembly.     

Morphology,ultrastructure, and germ cell cluster formation in ovarioles of aphids

The structure of aphid ovaries, including ovipare and virginopare morphs of five species, was investigated by light and electron microscopy. Aphids contain telotrophic meroistic ovarioles. The amount and distribution of cytoplasmic components of nurse cells, nutritive cords, and young oocytes are nearly identical to those known from scale insects and heteropterans. Each ovariole has a constant number of nurse cells and oocytes. In ovaries of ovipare morphs, the nurse cell nuclei enlarge by endomitosis (n = 2⁸n?2¹⁰n), whereas in virginopare morphs the nurse cell nuclei remain small (n = 2²n?2⁴n). Furthermore, in virginoparae the previtellogenic growth of oocytes is highly reduced, and vitellogenesis and chorionogenesis are blocked totally. Embryogenesis starts immediately after the shortened previtellogenic growth. In each ovariole, all germ cell descendants belong to one germ cell cluster that follows the 2ⁿ rule. The cluster normally contains 2⁵ = (32) cells, but other mostly smaller numbers also occur. In contrast to polytrophic meroistic ovarioles, more than one cell of each cluster will develop into an oocyte. In Drepanosiphum platanoides, 16 (2^n?1) nurse cells and 16 (2^n?1) oocytes exist in each cluster, whereas, in Metopolophium dirhodum, 8 (2^n?2) oocytes and 24 (2^n?1 + 2^n?2) nurse cells are normally found. In many ovarioles of Macrosiphum rosae, 21 nurse cells nourish 11 oocytes. Models of germ cell cluster formation in aphid ovaries are discussed.     

Aggregation,stability, and oscillations in different models for host-macroparasite interactions

Aggregation is generally recognized as an important factor in the dynamics of host-macroparasite interactions and it has been found relevant in stabilizing the dynamics toward an equilibrium coexistence. In this paper we review the models of Anderson and May (1978, J. Anim. Ecol. 47, 219-247, 249-267) and compare them with some more recently developed models, which incorporate explicit mechanisms (multiple infections or host heterogeneity) for generating aggregation and different degrees of mathematical accuracy. We found that the stabilization yielded by aggregation depends strongly on the mechanism producing the aggregation: multiple infections are much less stabilizing than when aggregation is assumed to be fixed from the outside, while the opposite holds for host heterogeneity. We also give analytical estimates of the period of oscillations occurring when the equilibrium is unstable. Finally, we explore in these models the role of aggregation in host regulation and in determining a threshold value for parasite establishment.     

Global stability of population models

Local stability seems to imply global stability for population models. To investigate this claim, we formally define apopulation model. This definition seems to include the one-dimensional discrete models now in use. We derive a necessary and sufficient condition for the global stability of our defined class of models. We derive an easily testable sufficient condition for local stability to imply global stability. We also show that if a discrete model is majorized by one of these stable population models, then the discrete model is globally stable. We demonstrate the utility of these theorems by using them to prove that the regions of local and global stability coincide for six models from the literature. We close by arguing that these theorems give a method for demonstrating global stability that is simpler and easier to apply than the usual method of Liapunov functions.     

Cooperative cluster formation, DNA bending and base-flipping by O6-alkylguanine-DNA alkyltransferase

O(6)-Alkylguanine-DNA alkyltransferase (AGT) repairs mutagenic O(6)-alkylguanine and O(4)-alkylthymine adducts in DNA, protecting the genome and also contributing to the resistance of tumors to chemotherapeutic alkylating agents. AGT binds DNA cooperatively, and cooperative interactions are likely to be important in lesion search and repair. We examined morphologies of complexes on long, unmodified DNAs, using analytical ultracentrifugation and atomic force microscopy. AGT formed clusters of ≤11 proteins. Longer clusters, predicted by the McGhee-von Hippel model, were not seen even at high [protein]. Interestingly, torsional stress due to DNA unwinding has the potential to limit cluster size to the observed range. DNA at cluster sites showed bend angles (～0, ～30 and ～60°) that are consistent with models in which each protein induces a bend of ～30°. Distributions of complexes along the DNA are incompatible with sequence specificity but suggest modest preference for DNA ends. These properties tell us about environments in which AGT may function. Small cooperative clusters and the ability to accommodate a range of DNA bends allow function where DNA topology is constrained, such as near DNA-replication complexes. The low sequence specificity allows efficient and unbiased lesion search across the entire genome.     

Gene cloning, purification, and characterization of two cyanobacterial NifS homologs driving iron-sulfur cluster formation

Iron-sulfur proteins are essential in the photosynthetic system and many other biological processes. We have isolated and characterized enzymes driving the formation of iron-sulfur clusters from Synechocystis sp. PCC6803. Two genes (slr0387 and sll0704), showing similarity to nifS of Azotobacter vinelandii, were cloned, and their gene products (SsCsdl and SsCsd2) were purified. They catalyzed the desulfuration of L-cysteine. Reconstitution of a [2Fe-2S] cluster of cyanobacterial ferredoxin proceeded much faster in the presence of L-cysteine and either of these enzymes than when using sodium sulfide. These results suggest that SsCsdl and SsCsd2 facilitate the iron-sulfur cluster assembly by producing inorganic sulfur from L-cysteine. Synechocystis sp. PCC6803 has no gene coding for a protein with similarity to the N-terminal domain of NifU of A. vinelandii, which is believed to cooperate with NifS to assemble iron-sulfur clusters. Thus, the cluster formation in the cyanobacterium probably proceeds through a mechanism that is different from that in A. vinelandii.     

Helix formation and stability in membranes

In this article we review current understanding of basic principles for the folding of membrane proteins, focusing on the more abundant alpha-helical class. Membrane proteins, vital to many biological functions and implicated in numerous diseases, fold into their active conformations in the complex environment of the cell bilayer membrane. While many membrane proteins rely on the translocon and chaperone proteins to fold correctly, others can achieve their functional form in the absence of any translation apparatus or other aides. Nevertheless, the spontaneous folding process is not well understood at the molecular level. Recent findings suggest that helix fraying and loop formation may be important for overall structure, dynamics and regulation of function. Several types of membrane helices with ionizable amino acids change their topology with pH. Additionally we note that some peptides, including many that are rich in arginine, and a particular analogue of gramicidin, are able passively to translocate across cell membranes. The findings indicate that a final protein structure in a lipid-bilayer membrane is sequence-based, with lipids contributing to stability and regulation. While much progress has been made toward understanding the folding process for alpha-helical membrane proteins, it remains a work in progress. This article is part of a Special Issue entitled: Emergence of Complex Behavior in Biomembranes edited by Marjorie Longo.