Antigenic cooperation in viral populations: Transformation of functions of intra-host viral variants.

In this paper, we study intra-host viral adaptation by antigenic cooperation - a mechanism of immune escape that serves as an alternative to the standard mechanism of escape by continuous genomic diversification and allows to explain a number of experimental observations associated with the establishment of chronic infections by highly mutable viruses. Within this mechanism, the topology of a cross-immunoreactivity network forces intra-host viral variants to specialize for complementary roles and adapt to the host's immune response as a quasi-social ecosystem. Here we study dynamical changes in immune adaptation caused by evolutionary and epidemiological events. First, we show that the emergence of a viral variant with altered antigenic features may result in a rapid re-arrangement of the viral ecosystem and a change in the roles played by existing viral variants. In particular, it may push the population under immune escape by genomic diversification towards the stable state of adaptation by antigenic cooperation. Next, we study the effect of a viral transmission between two chronically infected hosts, which results in the merging of two intra-host viral populations in the state of stable immune-adapted equilibrium. In this case, we also describe how the newly formed viral population adapts to the host's environment by changing the functions of its members. The results are obtained analytically for minimal cross-immunoreactivity networks and numerically for larger populations.

Detecting intrinsic global geometry of an obstacle via layered scattering.

Given a closed k-dimensional submanifold K, encapsulated in a compact domain M ⊂ E, k ≤ n - 2, we consider the problem of determining the intrinsic geometry of the obstacle K (such as volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular ϵ-neighborhood T ( K , ϵ ) of K in M. The geodesics that participate in this scattering emanate from the boundary ∂ M and terminate there after a few reflections from the boundary ∂ T ( K , ϵ ). However, the major problem in this setting is that a ray (a billiard trajectory) may get stuck in the vicinity of K by entering some trap there so that this ray will have infinitely many reflections from ∂ T ( K , ϵ ). To rule out such a possibility, we modify the geometry of a tube T ( K , ϵ ) by building it from spherical bubbles. We need to use ⌈ dim ⁡ ( K ) / 2 ⌉ many bubbling tubes { T ( K , ϵ ) } for detecting certain global invariants of K, invariants that reflect its intrinsic geometry. Thus, the words "layered scattering" are in the title. These invariants were studied by Hermann Weyl in his classical theory of tubes T ( K , ϵ ) and their volumes.

Elliptic Flowers: New Types of Dynamics to Study Classical and Quantum Chaos.

We construct examples of billiards where two chaotic flows are moving in opposite directions around a non-chaotic core or vice versa; the dynamics in the core are chaotic but flows that are moving in opposite directions around it are non-chaotic. These examples belong to a new class of dynamical systems called elliptic flowers billiards. Such systems demonstrate a variety of new behaviors which have never been observed or predicted to exist. Elliptic flowers billiards, where a chaotic (non-chaotic) core coexists with the same (chaotic/non-chaotic) type of dynamics in flows were recently constructed. Therefore, all four possible types of coexisting dynamics in the core and tracks are detected. However, it is just the beginning of studies of elliptic flowers billiards, which have already extended the imagination of what may happen in phase spaces of nonlinear systems. We outline some further directions of investigation of elliptic flowers billiards, which may bring new insights into our understanding of classical and quantum dynamics in nonlinear systems.

Using earth mover's distance for viral outbreak investigations.

BACKGROUND: RNA viruses mutate at extremely high rates, forming an intra-host viral population of closely related variants, which allows them to evade the host's immune system and makes them particularly dangerous. Viral outbreaks pose a significant threat for public health, and, in order to deal with it, it is critical to infer transmission clusters, i.e., decide whether two viral samples belong to the same outbreak. Next-generation sequencing (NGS) can significantly help in tackling outbreak-related problems. While NGS data is first obtained as short reads, existing methods rely on assembled sequences. This requires reconstruction of the entire viral population, which is complicated, error-prone and time-consuming. RESULTS: The experimental validation using sequencing data from HCV outbreaks shows that the proposed algorithm can successfully identify genetic relatedness between viral populations, infer transmission direction, transmission clusters and outbreak sources, as well as decide whether the source is present in the sequenced outbreak sample and identify it. CONCLUSIONS: Introduced algorithm allows to cluster genetically related samples, infer transmission directions and predict sources of outbreaks. Validation on experimental data demonstrated that algorithm is able to reconstruct various transmission characteristics. Advantage of the method is the ability to bypass cumbersome read assembly, thus eliminating the chance to introduce new errors, and saving processing time by allowing to use raw NGS reads.

Early-Time Exponential Instabilities in Nonchaotic Quantum Systems.

The majority of classical dynamical systems are chaotic and exhibit the butterfly effect: a minute change in initial conditions has exponentially large effects later on. But this phenomenon is difficult to reconcile with quantum mechanics. One of the main goals in the field of quantum chaos is to establish a correspondence between the dynamics of classical chaotic systems and their quantum counterparts. In isolated systems in the absence of decoherence, there is such a correspondence in dynamics, but it usually persists only over a short time window, after which quantum interference washes out classical chaos. We demonstrate that quantum mechanics can also play the opposite role and generate exponential instabilities in classically nonchaotic systems within this early-time window. Our calculations employ the out-of-time-ordered correlator (OTOC)-a diagnostic that reduces to the Lyapunov exponent in the classical limit but is well defined for general quantum systems. We show that certain classically nonchaotic models, such as polygonal billiards, demonstrate a Lyapunov-like exponential growth of the OTOC at early times with Planck's-constant-dependent rates. This behavior is sharply contrasted with the slow early-time growth of the analog of the OTOC in the systems' classical counterparts. These results suggest that classical-to-quantum correspondence in dynamics is violated in the OTOC even before quantum interference develops.

Local Immunodeficiency: Role of Neutral Viruses.

This paper analyzes the role of neutral viruses in the phenomenon of local immunodeficiency. We show that, even in the absence of altruistic viruses, neutral viruses can support the existence of persistent viruses and thus local immunodeficiency. However, in all such cases neutral viruses can maintain only bounded (relatively small) concentration of persistent viruses. Moreover, in all such cases the state of local immunodeficiency could only be marginally stable, while it is known that altruistic viruses can maintain stable local immunodeficiency. We also present an absolutely minimal cross-immunoreactivity network where a stable and robust state of local immunodeficiency can be maintained. It is now a challenge to synthetic biology to build such small networks with stable local immunodeficiency. Another important challenge for biology is to understand which types of viruses can play a role of persistent, altruistic and neutral ones and whether a role which a given virus plays depends on the structure (topology) of a given cross-immunoreactivity network.

QUENTIN: reconstruction of disease transmissions from viral quasispecies genomic data.

Motivation: Genomic analysis has become one of the major tools for disease outbreak investigations. However, existing computational frameworks for inference of transmission history from viral genomic data often do not consider intra-host diversity of pathogens and heavily rely on additional epidemiological data, such as sampling times and exposure intervals. This impedes genomic analysis of outbreaks of highly mutable viruses associated with chronic infections, such as human immunodeficiency virus and hepatitis C virus, whose transmissions are often carried out through minor intra-host variants, while the additional epidemiological information often is either unavailable or has a limited use. Results: The proposed framework QUasispecies Evolution, Network-based Transmission INference (QUENTIN) addresses the above challenges by evolutionary analysis of intra-host viral populations sampled by deep sequencing and Bayesian inference using general properties of social networks relevant to infection dissemination. This method allows inference of transmission direction even without the supporting case-specific epidemiological information, identify transmission clusters and reconstruct transmission history. QUENTIN was validated on experimental and simulated data, and applied to investigate HCV transmission within a community of hosts with high-risk behavior. It is available at https://github.com/skumsp/QUENTIN. Contact: pskums@gsu.edu or alexz@cs.gsu.edu or rahul@sfsu.edu or yek0@cdc.gov. Supplementary information: Supplementary data are available at Bioinformatics online.

Antigenic cooperation among intrahost HCV variants organized into a complex network of cross-immunoreactivity.

Hepatitis C virus (HCV) has the propensity to cause chronic infection. Continuous immune escape has been proposed as a mechanism of intrahost viral evolution contributing to HCV persistence. Although the pronounced genetic diversity of intrahost HCV populations supports this hypothesis, recent observations of long-term persistence of individual HCV variants, negative selection increase, and complex dynamics of viral subpopulations during infection as well as broad cross-immunoreactivity (CR) among variants are inconsistent with the immune-escape hypothesis. Here, we present a mathematical model of intrahost viral population dynamics under the condition of a complex CR network (CRN) of viral variants and examine the contribution of CR to establishing persistent HCV infection. The model suggests a mechanism of viral adaptation by antigenic cooperation (AC), with immune responses against one variant protecting other variants. AC reduces the capacity of the host's immune system to neutralize certain viral variants. CRN structure determines specific roles for each viral variant in host adaptation, with variants eliciting broad-CR antibodies facilitating persistence of other variants immunoreacting with these antibodies. The proposed mechanism is supported by empirical observations of intrahost HCV evolution. Interference with AC is a potential strategy for interruption and prevention of chronic HCV infection.

Some new surprises in chaos.

"Chaos is found in greatest abundance wherever order is being sought.It always defeats order, because it is better organized"Terry PratchettA brief review is presented of some recent findings in the theory of chaotic dynamics. We also prove a statement that could be naturally considered as a dual one to the Poincaré theorem on recurrences. Numerical results demonstrate that some parts of the phase space of chaotic systems are more likely to be visited earlier than other parts. A new class of chaotic focusing billiards is discussed that clearly violates the main condition considered to be necessary for chaos in focusing billiards.

Local Immunodeficiency: Combining Cross-Immunoreactivity Networks.

This article continues the analysis of the recently observed phenomenon of local immunodeficiency (LI), which arises as a result of antigenic cooperation among intrahost viruses organized into a network of cross-immunoreactivity (CR). We study here what happens as the result of combining (connecting) the simplest CR networks, which have a stable state of LI. It turned out that many possibilities occur, particularly resulting in a change of roles of some viruses in the CR network. Our results also give some indications about a boundary of the set of CR networks with stable state of LI in the entire collection of all possible CR networks. Such borderline CR networks are characterized by only a marginally stable (neutral rather than stable) state of the LI, or by the existence of such subnetworks in a CR network that evolve independently of each other (although being connected).

Introduction to Focus Issue: statistical mechanics and billiard-type dynamical systems.

Dynamical systems of the billiard type are of fundamental importance for the description of numerous phenomena observed in many different fields of research, including statistical mechanics, Hamiltonian dynamics, nonlinear physics, and many others. This Focus Issue presents the recent progress in this area with contributions from the mathematical as well as physical stand point.

Many faces of stickiness in Hamiltonian systems.

We discuss the phenomenon of stickiness in Hamiltonian systems. By visual examples of billiards, it is demonstrated that one must make a difference between internal (within chaotic sea(s)) and external (in vicinity of KAM tori) stickiness. Besides, there exist two types of KAM-islands, elliptic and parabolic ones, which demonstrate different abilities of stickiness.

Path lengths in protein-protein interaction networks and biological complexity.

We investigated the biological significance of path lengths in 12 protein-protein interaction (PPI) networks. We put forward three predictions, based on the idea that biological complexity influences path lengths. First, at the network level, path lengths are generally longer in PPIs than in random networks. Second, this pattern is more pronounced in more complex organisms. Third, within a PPI network, path lengths of individual proteins are biologically significant. We found that in 11 of the 12 species, average path lengths in PPI networks are significantly longer than those in randomly rewired networks. The PPI network of the malaria parasite Plasmodium falciparum, however, does not exhibit deviation from rewired networks. Furthermore, eukaryotic PPIs exhibit significantly greater deviation from randomly rewired networks than prokaryotic PPIs. Thus our study highlights the potentially meaningful variation in path lengths of PPI networks. Moreover, node eccentricity, defined as the longest path from a protein to others, is significantly correlated with the levels of gene expression and dispensability in the yeast PPI network. We conclude that biological complexity influences both global and local properties of path lengths in PPI networks. Investigating variation of path lengths may provide new tools to analyze the evolution of functional modules in biological systems.

Suppressing Fermi acceleration in a driven elliptical billiard.

We study dynamical properties of an ensemble of noninteracting particles in a time-dependent elliptical-like billiard. It was recently shown [Phys. Rev. Lett. 100, 014103 (2008)] that for the nondissipative dynamics, the particle experiences unlimited energy growth. Here we show that inelastic collisions suppress Fermi acceleration in a driven elliptical-like billiard. This suppression is yet another indication that Fermi acceleration is not a structurally stable phenomenon.

Graph fractal dimension and the structure of fractal networks.

Fractals are geometric objects that are self-similar at different scales and whose geometric dimensions differ from so-called fractal dimensions. Fractals describe complex continuous structures in nature. Although indications of self-similarity and fractality of complex networks has been previously observed, it is challenging to adapt the machinery from the theory of fractality of continuous objects to discrete objects such as networks. In this article, we identify and study fractal networks using the innate methods of graph theory and combinatorics. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colourings and graph descriptive complexity, and analyse the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework in evolutionary biology and virology by analysing networks of viral strains sampled at different stages of evolution inside their hosts. Our methodology revealed gradual self-organization of intra-host viral populations over the course of infection and their adaptation to the host environment. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

Local immunodeficiency: Minimal networks and stability.

Some basic aspects of the recently discovered phenomenon of local immunodeficiency (Skums et al. [1]) generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated. We prove that local immunodeficiency (LI) that is stable under perturbations already occurs in very small networks and under general conditions on their parameters. Therefore our results are applicable not only to Hepatitis C where CR networks are known to be large (Skums et al. [1]), but also to other diseases with CR. A major necessary feature of such networks is the non-homogeneity of their topology. It is also shown that one can construct larger CR networks with stable LI by using small networks with stable LI as their building blocks. Our results imply that stable LI occurs in networks with quite general topology. In particular, the scale-free property of a CR network, assumed in Skums et al. [1], is not required.

Dynamics of spatial averages.

We study the dynamics of spatial averages of spatially extended dynamical systems. We present various examples of lattice dynamical systems to show the possibility of different behaviors, including asymptotically constant, periodic, and non-periodic, of spatial averages. We explain that the fluctuation in spatial averages is caused by the transitivity and the lack of symmetry of the dynamics of local subsystems. (c) 1997 American Institute of Physics.

Variational principle for periodic trajectories of hyperbolic billiards.

We prove for some classes of hyperbolic billiards that the action functional has only one local minimum or only one local maximum for any finite admissible sequence of regular components of the boundary. This result suggests an effective algorithm for the search of all periodic trajectories of these billiards. (c) 1995 American Institute of Physics.

Mushrooms and other billiards with divided phase space.

We present the first natural and visible examples of Hamiltonian systems with divided phase space allowing a rigorous mathematical analysis. The simplest such family (mushrooms) demonstrates a continuous transition from a completely chaotic system (stadium) to a completely integrable one (circle). In the course of this transition, an integrable island appears, grows and finally occupies the entire phase space. We also give the first examples of billiards with a "chaotic sea" (one ergodic component) and an arbitrary (finite or infinite) number of KAM islands and the examples with arbitrary (finite or infinite) number of chaotic (ergodic) components with positive measure coexisting with an arbitrary number of islands. Among other results is the first example of completely understood (rigorously studied) billiards in domains with a fractal boundary. (c) 2001 American Institute of Physics.