Exome-based case-control association study using extreme phenotype design reveals novel candidates with protective effect in diabetic retinopathy.

UNLABELLED: Diabetic retinopathy (DR) is a common clinical expression of diabetes mellitus-induced vasculopathy and is a major cause of vision loss. Significant gaps remain in our understanding of the molecular pathoetiology of DR, and it is hoped that human genetic approaches can reveal novel targets especially since DR is a heritable trait. Previous studies have focused on genetic risk factors of DR but their results have been mixed. In this study, we hypothesized that the use of the extreme phenotype design will increase the power of a genomewide search for "protective" genetic variants. We enrolled a small yet atypical cohort of 43 diabetics who did not develop DR a decade or more after diagnosis (cases), and 64 diabetics with DR (controls), all of similar ethnic background (Saudi). Whole-exome sequencing of the entire cohort was followed by statistical analysis employing combined multivariate and collapsing methods at the gene level, to identify genes that are enriched for rare variants in cases vs. CONTROLS: Three genes (NME3, LOC728699, and FASTK) reached gene-based genome-wide significance at the 10(-08) threshold (p value = 1.55 × 10(-10), 6.23 × 10(-10), 3.21 × 10(-08), respectively). Our results reveal novel candidate genes whose increased rare variant burden appears to protect against DR, thus highlighting them as attractive candidate targets, if replicated by future studies, for the treatment and prevention of DR. Extreme phenotype design when implemented in sequencing-based genome-wide case-control studies has the potential to reveal novel candidates with a smaller cohort size compared to standard study designs.

Slow kinetics of Brownian maxima.

We study extreme-value statistics of Brownian trajectories in one dimension. We define the maximum as the largest position to date and compare maxima of two particles undergoing independent Brownian motion. We focus on the probability P(t) that the two maxima remain ordered up to time t and find the algebraic decay P ∼ t(-ß) with exponent ß = 1/4. When the two particles have diffusion constants D(1) and D(2), the exponent depends on the mobilities, ß = (1/π) arctan sqrt[D(2)/D(1)]. We also use numerical simulations to investigate maxima of multiple particles in one dimension and the largest extension of particles in higher dimensions.

Jamming and tiling in fragmentation of rectangles.

We investigate a stochastic process where a rectangle breaks into smaller rectangles through a series of horizontal and vertical fragmentation events. We focus on the case where both the vertical size and the horizontal size of a rectangle are discrete variables. Because of this constraint, the system reaches a jammed state where all rectangles are sticks, that is, rectangles with minimal width. Sticks are frozen as they cannot break any further. The average number of sticks in the jammed state, S, grows as S≃A/sqrt[2πlnA] with rectangle area A in the large-area limit, and remarkably, this behavior is independent of the aspect ratio. The distribution of stick length has a power-law tail, and further, its moments are characterized by a nonlinear spectrum of scaling exponents. We also study an asymmetric breakage process where vertical and horizontal fragmentation events are realized with different probabilities. In this case, there is a phase transition between a weakly asymmetric phase where the length distribution is independent of system size and a strongly asymmetric phase where this distribution depends on system size.

Phase transition with nonthermodynamic states in reversible polymerization.

We investigate a reversible polymerization process in which individual polymers aggregate and fragment at a rate proportional to their molecular weight. We find a nonequilibrium phase transition despite the fact that the dynamics are perfectly reversible. When the strength of the fragmentation process exceeds a critical threshold, the system reaches a thermodynamic steady state where the total number of polymers is proportional to the system size. The polymer length distribution has a sharp exponential tail in this case. When the strength of the fragmentation process falls below the critical threshold, the steady state becomes nonthermodynamic as the total number of polymers grows sublinearly with the system size. Moreover, the length distribution has an algebraic tail and the characteristic exponent varies continuously with the fragmentation rate.

Extinction and survival in two-species annihilation.

We study diffusion-controlled two-species annihilation with a finite number of particles. In this stochastic process, particles move diffusively, and when two particles of opposite type come into contact, the two annihilate. We focus on the behavior in three spatial dimensions and for initial conditions where particles are confined to a compact domain. Generally, one species outnumbers the other, and we find that the difference between the number of majority and minority species, which is a conserved quantity, controls the behavior. When the number difference exceeds a critical value, the minority becomes extinct and a finite number of majority particles survive, while below this critical difference, a finite number of particles of both species survive. The critical difference Δ_{c} grows algebraically with the total initial number of particles N, and when N≫1, the critical difference scales as Δ_{c}∼N^{1/3}. Furthermore, when the initial concentrations of the two species are equal, the average number of surviving majority and minority particles, M_{+} and M_{-}, exhibit two distinct scaling behaviors, M_{+}∼N^{1/2} and M_{-}∼N^{1/6}. In contrast, when the initial populations are equal, these two quantities are comparable M_{+}∼M_{-}∼N^{1/3}.

Efficiency of competitions.

League competition is investigated using random processes and scaling techniques. In our model, a weak team can upset a strong team with a fixed probability. Teams play an equal number of head-to-head matches and the team with the largest number of wins is declared to be the champion. The total number of games needed for the best team to win the championship with high certainty T grows as the cube of the number of teams N , i.e., T approximately N(3). This number can be substantially reduced using preliminary rounds where teams play a small number of games and subsequently, only the top teams advance to the next round. When there are k rounds, the total number of games needed for the best team to emerge as champion, T(k), scales as follows, T(k) approximately N(gamma(k)) with gamma(k) = [1-(2/3)(k+1)](-1). For example, gamma(k)=95,2719,8165 for k=1,2,3 . These results suggest an algorithm for how to infer the best team using a schedule that is linear in N. We conclude that league format is an ineffective method of determining the best team, and that sequential elimination from the bottom up is fair and efficient.

Condensates in driven aggregation processes.

We investigate aggregation driven by mass injection. In this stochastic process, mass is added with constant rate r and clusters merge at a constant total rate 1 , so that both the total number of clusters and the total mass steadily grow with time. Analytic results are presented for the three classic aggregation rates K{i,j} between clusters of size i and j . When K{i,j}=const , the cluster size distribution decays exponentially. When K{i,j} proportional, i+j or K{i,j} proportional, ixj , there are two phases: (i) a condensate phase with a condensate containing a finite fraction of the mass in the system as well as finite clusters and (ii) a cluster phase with finite clusters only. For K{i,j} proportional, i+j , the cluster size distribution, c{k} , has a power-law tail, c{k} approximately k;{-gamma} in either phase. The exponent is a nonmonotonic function of the injection rate gamma=r(r-1) in the condensate phase r<2 and gamma=r in the cluster phase r>2 .

Alignment of rods and partition of integers.

We study dynamical ordering of rods. In this process, rod alignment via pairwise interactions competes with diffusive wiggling. Under strong diffusion, the system is disordered, but at weak diffusion, the system is ordered. We present an exact steady-state solution for the nonlinear and nonlocal kinetic theory of this process. We find the Fourier transform as a function of the order parameter, and show that Fourier modes decay exponentially with the wave number. We also obtain the order parameter in terms of the diffusion constant. This solution is obtained using iterated partitions of the integer numbers.

Kinetics of aggregation with choice.

We generalize the ordinary aggregation process to allow for choice. In ordinary aggregation, two random clusters merge and form a larger aggregate. In our implementation of choice, a target cluster and two candidate clusters are randomly selected and the target cluster merges with the larger of the two candidate clusters. We study the long-time asymptotic behavior and find that as in ordinary aggregation, the size density adheres to the standard scaling form. However, aggregation with choice exhibits a number of different features. First, the density of the smallest clusters exhibits anomalous scaling. Second, both the small-size and the large-size tails of the density are overpopulated, at the expense of the density of moderate-size clusters. We also study the complementary case where the smaller candidate cluster participates in the aggregation process and find an abundance of moderate clusters at the expense of small and large clusters. Additionally, we investigate aggregation processes with choice among multiple candidate clusters and a symmetric implementation where the choice is between two pairs of clusters.

Kinetic theory of random graphs: from paths to cycles.

The structural properties of evolving random graphs are investigated. Treating linking as a dynamic aggregation process, rate equations for the distribution of node to node distances (paths) and of cycles are formulated and solved analytically. At the gelation point, the typical length of paths and cycles, l , scales with the component size k as l approximately k(1/2) . Dynamic and finite-size scaling laws for the behavior at and near the gelation point are obtained. Finite-size scaling laws are verified using numerical simulations.

Power-law velocity distributions in granular gases.

The kinetic theory of granular gases is studied for spatially homogeneous systems. At large velocities, the equation governing the velocity distribution becomes linear, and it admits stationary solutions with a power-law tail, f (v) approximately v(-sigma) . This behavior holds in arbitrary dimension for arbitrary collision rates including both hard spheres and Maxwell molecules. Numerical simulations show that driven steady states with the same power-law tail can be realized by injecting energy into the system at very high energies. In one dimension, we also obtain self-similar time-dependent solutions where the velocities collapse to zero. At small velocities there is a steady state and a power-law tail but at large velocities, the behavior is time dependent with a stretched exponential decay.

Scaling exponents for ordered maxima.

We study extreme value statistics of multiple sequences of random variables. For each sequence with N variables, independently drawn from the same distribution, the running maximum is defined as the largest variable to date. We compare the running maxima of m independent sequences and investigate the probability S(N) that the maxima are perfectly ordered, that is, the running maximum of the first sequence is always larger than that of the second sequence, which is always larger than the running maximum of the third sequence, and so on. The probability S(N) is universal: it does not depend on the distribution from which the random variables are drawn. For two sequences, S(N)∼N(-1/2), and in general, the decay is algebraic, S(N)∼N(-σ(m)), for large N. We analytically obtain the exponent σ(3)≅1.302931 as root of a transcendental equation. Furthermore, the exponents σ(m) grow with m, and we show that σ(m)∼m for large m.

Exchange-driven growth.

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: (I) Growth-clusters grow indefinitely, (II) gelation-all mass is transformed into an infinite gel in a finite time, and (III) instant gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is Phi(x) approximately exp(-x(2-nu)), where nu is a homogeneity degree of the rate of exchange. At the borderline case nu=2, the distribution exhibits a generic algebraic tail, Phi(x) approximately x(-5). In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, T approximately [lnN](-(nu-2)), in the large system size limit N--> infinity. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.

Shattering transitions in collision-induced fragmentation.

We investigate the kinetics of nonlinear collision-induced fragmentation. We obtain the fragment mass distribution analytically by utilizing its traveling wave behavior. The system undergoes a shattering transition in which a finite fraction of the mass is lost to infinitesimal fragments (dust). The nature of the shattering transition depends on the fragmentation process. When the larger of the two colliding fragments splits, the transition is discontinuous and the entire mass is transformed into dust at the transition point. When the smaller fragment splits, the transition is continuous with the dust gaining mass steadily on the account of the fragments. At the transition point, the fragment mass distribution diverges algebraically for small masses, c(m) approximately m(-alpha), with alpha=1.201 91 em leader.

Finite-size fluctuations in interacting particle systems.

Fluctuations may govern the fate of an interacting particle system even on the mean-field level. This is demonstrated via a three species cyclic trapping reaction with a large, yet finite number of particles, where the final number of particles N(f) scales logarithmically with the system size N, N(f) approximately ln N. Statistical fluctuations, that become significant as the number of particles diminishes, are responsible for this behavior. This phenomenon underlies a broad range of interacting particle systems including in particular multispecies annihilation processes.

Size of outbreaks near the epidemic threshold.

The spread of infectious diseases near the epidemic threshold is investigated. Scaling laws for the size and the duration of outbreaks originating from a single infected individual in a large susceptible population are obtained. The maximal size of an outbreak n(*) scales as N(2/3) with N the population size. This scaling law implies that the average outbreak size [n]scales as N(1/3). Moreover, the maximal and the average duration of an outbreak grow as t(*) approximately N(1/3) and [t] approximately ln N, respectively.

Scaling, multiscaling, and nontrivial exponents in inelastic collision processes.

We investigate velocity statistics of homogeneous inelastic gases using the Boltzmann equation. Employing an approximate uniform collision rate, we obtain analytic results valid in arbitrary dimension. In the freely evolving case, the velocity distribution is characterized by an algebraic large-velocity tail, P(v,t) approximately v(-sigma). The exponent sigma(d,epsilon), a nontrivial root of an integral equation, varies continuously with the spatial dimension d and the dissipation coefficient epsilon. Although the velocity distribution follows a scaling form, its moments exhibit multiscaling asymptotic behavior. Furthermore, the velocity autocorrelation function decays algebraically with time, A(t)= approximately t(-alpha), with a nonuniversal dissipation-dependent exponent alpha=1/epsilon. In the forced case, the steady state Fourier transform is obtained via a cumulant expansion. Even in this case, velocity correlations develop and the velocity distribution is non-Maxwellian.

Extremal properties of random trees.

We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights approaches a traveling wave form. The wave front in the minimal case is governed by the small-extremal-height tail of the distribution, and conversely, the front in the maximal case is governed by the large-extremal-height tail of the distribution. We determine several statistical characteristics of the extremal height distribution analytically. In particular, the expected minimal and maximal heights grow logarithmically with the tree size, N, h(min) approximately v(min) ln N, and h(max) approximately v(max) ln N, with v(min)=0.373365ellipsis and v(max)=4.31107ellipsis, respectively. Corrections to this asymptotic behavior are of order O(ln ln N).

Entropic tightening of vibrated chains.

We investigate experimentally the distribution of configurations of a ring with an elementary topological constraint, a "figure-8" twist. Using a system far from thermal equilibrium, a vibrated granular chain, we show that configurations where one loop is small and the second is large are strongly preferred. Despite the highly non-equilibrium nature of the system, our results are consistent with recent predictions for equilibrium properties of topologically-constrained polymers. The dynamics of the tightening process weakly violates a (coarse-grained) detailed balance, indicating that the unexpected correspondence with an equilibrium entropic approach is not exact.

Statistics of superior records.

We study statistics of records in a sequence of random variables. These identical and independently distributed variables are drawn from the parent distribution ρ. The running record equals the maximum of all elements in the sequence up to a given point. We define a superior sequence as one where all running records are above the average record expected for the parent distribution ρ. We find that the fraction of superior sequences S(N) decays algebraically with sequence length N, S(N)~N(-ß) in the limit N→∞. Interestingly, the decay exponent ß is nontrivial, being the root of an integral equation. For example, when ρ is a uniform distribution with compact support, we find ß=0.450265. In general, the tail of the parent distribution governs the exponent ß. We also consider the dual problem of inferior sequences, where all records are below average, and find that the fraction of inferior sequences I(N) decays algebraically, albeit with a different decay exponent, I(N)~N(-α). We use the above statistical measures to analyze earthquake data.