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This repository contains the source code for the polymer simulations, the source code for exactnumerical calculations that determine the steady state conformation of a linear phantom polymer, and severalMathematica notebook that illustrates analytic calculations, for the article "Polymer folding through active processes recreates features of genome organization" published in Proceedings of the National Academy of Sciences of the United States of America 120, e2221726120. File Description Active-Polymer-Conformation.zip Mathematica notebooks that illustrate the analytical calculations, Julia code for calculating the steady-state conformation of an active polymer, and Julia code for inferring a candidate distribution of independent active processes that best reproduces a given target steady state. active-polymers-main.zip Python code for running Brownian dynamics simulations of active Rouse chains without self-avoidance. HiC_analysis.zip Python code for analyzing Hi-C data. polychrom_analysis.zip Python scripts interfacing with polychrom (https://github.com/open2c/polychrom) to run and analyze active polymer simulations with self-avoidance. &nbsp

In growing populations, the fate of mutations depends on their competitive ability against the ancestor and their ability to colonize new territory. Here we present a theory that integrates both aspects of mutant fitness by coupling the classic description of one-dimensional competition (Fisher equation) to the minimal model of front shape (KPZ equation). We solved these equations and found three regimes, which are controlled solely by the expansion rates, solely by the competitive abilities, or by both. Collectively, our results provide a simple framework to study spatial competition., Comment: 5 pages, 4 figures

Turing patterns emerge from a spatially uniform state following a linear instability driven by diffusion. Features of the eventual pattern (stabilized by non-linearities) are already present in the initial unstable modes. On a uniform flat surface or perfect sphere, the unstable modes and final patterns are degenerate, reflecting translational/rotational symmetry. This symmetry can be broken, e.g. by a bump on a flat substrate or by deforming a sphere. As the diffusion operator on a two dimensional manifold depends on the underlying curvature, the degeneracy of the initial unstable mode is similarly reduced. Different shapes can pin different modes. We adapt methods of conformal mapping and perturbation theory to analytically examine how bumps and ripples entrain modes of the diffusion operator on cylinders and spheres. We confirm these results numerically, and provide closed form expressions that describe how non-uniformities in curvature pin diffusion-driven instabilities and the resulting patterns., Comment: arXiv:1901.09900 is a substantial re-write such that replacement isn't correct

Thermal fluctuations in non-equilibrium steady states generically lead to power law decay of correlations for conserved quantities. Embedded bodies which constrain fluctuations in turn experience fluctuation induced forces. We compute these forces for the simple case of parallel slabs in a driven diffusive system. The force falls off with slab separation $d$ as $k_B T/d$ (at temperature $T$, and in all spatial dimensions), but can be attractive or repulsive. Unlike the equilibrium Casimir force, the force amplitude is non-universal and explicitly depends on dynamics. The techniques introduced can be generalized to study pressure and fluctuation induced forces in a broad class of non-equilibrium systems., Comment: 5 pages, 2 figures

We show that roughness or surface modulations change the distance dependence of (power-law) interactions between curved objects at proximity. The modified scaling law is then simply related to the order of the first non-vanishing coefficient of the Taylor expansion of the distribution of separations between the surfaces. The latter can in principle be estimated by scanning measurements, or computed for well characterized modulations, and then used to predict short-distance scaling behavior in disparate experiments. For example, we predict that the radiative heat transfer between a rough sphere and a plate approaches a constant with decreasing separation. Similar saturation is expected for the Casimir force between dielectric or metallic surfaces with appropriate modulations over distinct length scales., Comment: 5 pages, 2 figures

As borders between different regions, lines are an important element of natural images. Already at the level of the mammalian primary visual cortex (V1), neurons respond best to lines of a given orientation. We reduce a set of images to linear segments and analyze their statistical properties. In particular, appropriately defined Fourier spectra show more power in their transverse component than in the longitudinal one. We then characterize filters that are best suited for extracting information from such images, and find some qualitative consistency with neural connections in V1. We also demonstrate that such filters are efficient in reconstructing missing lines in an image.

We investigate the probability for appearance of knots in self-avoiding loops (SALs) on a cubic lattice. A set of N-step loops is generated by attempting to combine pairs of (N/2)-step self-avoiding walks constructed by a dimerization method. We demonstrate that our method produces unbiased samples of SALs, and study the knot formation probability as a function of loop size. Our results corroborate the conclusions of Yao et. al. with loops generated by a Monte Carlo method., Comment: RevTeX4, 4 pages, 4 eps figures

Inspired by patterns observed in mixtures of microtubules and molecular motors, we propose continuum equations for the evolution of motor density, and microtubule orientation. The chief ingredients are the transport of motors along tubules, and the alignment of tubules in the process. The macroscopic equations lead to aster and vortex patterns in qualitative agreement with experiments. While the early stages of evolution of tubules are similar to coarsening of spins following a quench, the rearrangement of motors leads to arrested coarsening at low densities. Even in one dimension, the equations exhibit a variety of interesting behaviors, such as symmetry breaking, moving fronts, and motor localization.

The elastic moduli of vortex crystals in anisotropic superconductors are frequently involved in the investigation of their phase diagram and transport properties. We provide a detailed analysis of the harmonic eigenvalues (normal modes) of the vortex lattice for general values of the magnetic field strength, going beyond the elastic continuum regime. The detailed behavior of these wavevector-dependent eigenvalues within the Brillouin zone (BZ), is compared with several frequently used approximations that we also recalculate. Throughout the BZ, transverse modes are less costly than their longitudinal counterparts, and there is an angular dependence which becomes more marked close to the zone boundary. Based on these results, we propose an analytic correction to the nonlocal continuum formulas which fits quite well the numerical behavior of the eigenvalues in the London regime. We use this approximate expression to calculate thermal fluctuations and the full melting line (according to Lindeman's criterion) for various values of the anisotropy parameter., Comment: 17 pages, 13 figures

Earlier work on dynamical critical phenomena in the context of magnetic hysteresis for uniaxial (scalar) spins, is extended to the case of a multicomponent (vector) field. From symmetry arguments and a perturbative renormalization group approach (in the path integral formalism), it is found that the generic behavior at long time and length scales is described by the scalar fixed point (reached for a given value of the magnetic field and of the quenched disorder), with the corresponding Ising-like exponents. By tuning an additional parameter, however, a fully rotationally invariant fixed point can be reached, at which all components become critical simultaneously, with O(n)-like exponents. Furthermore, the possibility of a spontaneous non-equilibrium transverse ordering, controlled by a distinct fixed point, is unveiled and the associated exponents calculated. In addition to these central results, a didactic ``derivation'' of the equations of motion for the spin field are given, the scalar model is revisited and treated in a more direct fashion, and some issues pertaining to time dependences and the problem of multiple solutions within the path integral formalism are clarified., Comment: Some brief clarifications added in the text; to appear in Phys. Rev. E

We study analytically and numerically the winding of directed polymers of length $t$ around each other or around a rod. Unconfined polymers in pure media have exponentially decaying winding angle distributions, the decay constant depending on whether the interaction is repulsive or neutral, but not on microscopic details. In the presence of a chiral asymmetry, the exponential tails become non universal. In all these cases the mean winding angle is proportional to $\ln t$. When the polymer is confined to a finite region around the winding center, e.g. due to an attractive interaction, the winding angle distribution is Gaussian, with a variance proportional to $t$. We also examine the windings of polymers in random systems. Our results suggest that randomness reduces entanglements, leading to a narrow (Gaussian) distribution with a mean winding angle of the order of $\sqrt{\ln t}$., Comment: 31 pages LaTeX and 14 postscript figures

These notes are for the proceedings of the NATO Advanced Study Insitutue on "Scale Invariance, Interfaces, and Non--Equilibrium Dynamics", held at the Isaac Newton Institute, 20--30 June 1994. The four lectures address a number of issues related to dynamic fluctuations of lines in non-equilibrium circumstances. The first two are devoted to the critical behavior of contact lines and flux lines depinning from impurities. It is emphasized that anisotropies in the medium lead to different universality classes. The importance of nonlinearities for moving lines are discussed in the context of flux lines and polymers in the last two lectures. A dynamic form birefringence is predicted for drifting polymers., Comment: 24 pages, Plain TeX, figures included via epsf.sty and packed with uufiles, MIT-CMT-EK-94-3

Contents: A. Introduction B. High Temperature Expansions for the Ising Model C. Characteristic Functions and Cumulants D. The One Dimensional Chain E. Directed Paths and the Transfer Matrix F. Moments of the Correlation Function G. The Probability Distribution in Two Dimensions H. Higher Dimensions I. Random Signs J. Other Realizations of DPRM K. Quantum Interference of Strongly Localized Electrons L. The Locator Expansion and Forward Scattering Paths M. Magnetic Field Response N. Unitary Propagation O. Unitary Averages P. Summing all Paths in High Dimensions Q. The Ising Model on a Square Lattice R. Singular Behavior S. The Two Dimensional Spin Glass T. Results for the Two Dimensional Spin Glass, Comment: 75 pages. Lectures presented at 1994 Les Houches Summer School ``Fluctuating Geometries in Statistical Mechanics and Field Theory'' (also available at http://xxx.lanl.gov/lh94/ )

We consider a model of a reconstructed crystal surface, first considered by Villain and Vilfan (Europhys. Lett. 12, p. 523 (1990) and Surf. Sci. 257, p. 368 (1991)) for the gold (110) surface, in which roughening occurs via the formation of anisotropic steps traversing the entire length of the crystal. The model is studied by a mapping to a spin--1/2 Fermion system in 1+1 dimensions, which, in the absence of islands, is precisely the Hubbard model. We consider a general $\pbyo$ reconstruction, in the presence of inter--step interactions and closed islands. Our analysis predicts the existence of a new type of rough phase, with incommensurate correlations in the reconstruction order parameter and unusual momentum space singularities at a characteristic ``Fermi momentum'' and its harmonics, analagous to the Luttinger liquid of one--dimensional Fermions. The general phase structure for $p>1$ is as follows: for $p>2$, there is a flat ordered (FO), a rough incommensurate (RI), and a flat incommensurate phase (FI). The FO--RI and FO--FI transitions are of the commensurate to incommensurate type, and the FI--RI transition is in the Kosterlitz--Thouless (KT) universality class. For $p=2$, the FI phase is replaced by a flat disordered phase (FD), and there may be a new rough disordered phase (RD). The FO--FD transition is now of Ising type, and the FD--RD and RI--RD transitions are in the KT universality class., 33 pages including 12 figures

A model is proposed for the evolution of the profile of a growing interface. The deterministic growth is solved exactly, and exhibits nontrivial relaxation patterns. The stochastic version is studied by dynamic renormalization-group techniques and by mappings to Burgers's equation and to a random directed-polymer problem. The exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations. Predictions are made for more dimensions.

Quantum and thermal fluctuations of electromagnetic waves are the cornerstone of quantum and statistical physics, and inherent to such phenomena as thermal radiation and van der Waals forces. While the basic principles are the material of elementary texts, recent experimental and technological advances have made it necessary to come to terms with counterintuitive consequences of electromagnetic fluctuations at short scales -- in the so called {\it near-field} regime. We focus on three manifestations of such behavior: {\bf (i)} The Stefan--Boltzmann law describes thermal radiation from macroscopic bodies, but fails to account for magnitude, polarization and coherence of radiation from small objects (say compared to the skin depth). {\bf (ii)} The heat transfer between two bodies at similar close proximity is dominated by evanescent waves, and can be several orders of magnitude larger than the classical contribution due to propagating waves. {\bf (iii)} Casimir/van der Waals interactions are a dominant force between objects at sub-micron separation; the non-equilibrium analogs of this force (for objects at different temperatures) have not been sufficiently explored (at least experimentally). To explore these phenomena we introduce the tool of fluctuational quantum electrodynamics (QED) originally introduced by Rytov in the 1950s. Combined with a scattering formalism, this enables studies of heat radiation and transfer, equilibrium and non-equilibrium forces for objects of different material properties, shapes, separations and arrangements., Comment: accepted for publication in Annual Reviews of Condensed Matter Physics

In addition to an innate immune system that battles pathogens in a nonspecific fashion, higher organisms, such as humans, possess an adaptive immune system to combat diverse (and evolving) microbial pathogens. Remarkably, the adaptive immune system mounts pathogen-specific responses, which can be recalled upon reinfection with the same pathogen. It is difficult to see how the adaptive immune system can be preprogrammed to respond specifically to a vast and unknown set of pathogens. Although major advances have been made in understanding pertinent molecular and cellular phenomena, the precise principles that govern many aspects of an immune response are largely unknown. We discuss complementary approaches from statistical mechanics and cell biology that can shed light on how key components of the adaptive immune system, T cells, develop to enable pathogen-specific responses against many diverse pathogens. The mechanistic understanding that emerges has implications for how host genetics may influence the development of T cells with differing responses to the human immunodeficiency virus (HIV) infection., Ragon Institute of MGH, MIT and Harvard, National Institutes of Health (U.S.) (Grant 1PO1-AI071195-01), National Science Foundation (U.S.) (Grant DMR-12-06323), National Institutes of Health (U.S.) (Director's Pioneer Award)

We study the shift of rotational levels of a diatomic polar molecule due to its van der Waals (vdW) interaction with a gently curved dielectric surface at temperature $T$, and submicron separations. The molecule is assumed to be in its electronic and vibrational ground state, and the rotational degrees are described by a rigid rotor model. We show that under these conditions retardation effects and surface dispersion can be neglected. The level shifts are found to be independent of $T$, and given by the quantum state averaged classical electrostatic interaction of the dipole with its image on the surface. We use a derivative expansion for the static Green's function to express the shifts in terms of surface curvature. We argue that the curvature induced line splitting is experimentally observable, and not obscured by natural line widths and thermal broadening., Comment: 8 pages, 2 figures, version accepted for publication in Phys. Rev. A

Generation of potent antibodies by a mutation-selection process called affinity maturation is a key component of effective immune responses. Antibodies that protect against highly mutable pathogens must neutralize diverse strains. Developing effective immunization strategies to drive their evolution requires understanding how affinity maturation happens in an environment where variants of the same antigen are present. We present an in silico model of affinity maturation driven by antigen variants which reveals that induction of cross-reactive antibodies often occurs with low probability because conflicting selection forces, imposed by different antigen variants, can frustrate affinity maturation. We describe how variables such as temporal pattern of antigen administration influence the outcome of this frustrated evolutionary process. Our calculations predict, and experiments in mice with variant gp120 constructs of the HIV envelope protein confirm, that sequential immunization with antigen variants is preferred over a cocktail for induction of cross-reactive antibodies focused on the shared CD4 binding site epitope., Ragon Institute of MGH, MIT and Harvard, Center for HIV/AIDS Vaccine Immunology and Immunogen Discovery (Grant UM1AI100663)

Broadly neutralizing HIV antibodies (bnAbs) are typically highly somatically mutated, raising doubts as to whether they can be elicited by vaccination. We used 454 sequencing and designed a novel phylogenetic method to model lineage evolution of the bnAbs PGT121–134 and found a positive correlation between the level of somatic hypermutation (SHM) and the development of neutralization breadth and potency. Strikingly, putative intermediates were characterized that show approximately half the mutation level of PGT121–134 but were still capable of neutralizing roughly 40–80% of PGT121–134 sensitive viruses in a 74-virus panel at median titers between 15- and 3-fold higher than PGT121–134. Such antibodies with lower levels of SHM may be more amenable to elicitation through vaccination while still providing noteworthy coverage. Binding characterization indicated a preference of inferred intermediates for native Env binding over monomeric gp120, suggesting that the PGT121–134 lineage may have been selected for binding to native Env at some point during maturation. Analysis of glycan-dependent neutralization for inferred intermediates identified additional adjacent glycans that comprise the epitope and suggests changes in glycan dependency or recognition over the course of affinity maturation for this lineage. Finally, patterns of neutralization of inferred bnAb intermediates suggest hypotheses as to how SHM may lead to potent and broad HIV neutralization and provide important clues for immunogen design., National Institutes of Health (U.S.) (Bioinformatics and Integrative Genomics Fellowship), National Institutes of Health (U.S.) (Center for Excellence in Genomic Science)

We develop a small distance expansion for the radiative heat transfer between gently curved objects, in terms of the ratio of distance to radius of curvature. A gradient expansion allows us to go beyond the lowest-order proximity transfer approximation. The range of validity of such expansion depends on temperature as well as material properties. Generally, the expansion converges faster for the derivative of the transfer than for the transfer itself, which we use by introducing a near-field adjusted plot. For the case of a sphere and a plate, the logarithmic correction to the leading term has a very small prefactor for all materials investigated., National Science Foundation (U.S.) (NSF Grant No. DMR-12-06323), United States. Dept. of Energy (DOE grant No. DE-FG02-02ER45977), Deutsche Forschungsgemeinschaft (DFG grant No.KR3844/1-1)

We study polymers attached to spherical (circular) or paraboloidal (parabolic) probes in three (two) dimensions. Both self-avoiding and random walks are examined numerically. The behavior of a polymer of size R[subscript 0] attached to the tip of a probe with radius of curvature R, differs qualitatively for large and small values of the ratio s=R[subscript 0]/R. We demonstrate that the scaled compliance (inverse force constant) S/R[2 over 0], and scaled mean position of the polymer end-point (x[subscript ⊥])/R can be expressed as a function of s. Scaled compliance is anisotropic, and quite large in the direction parallel to the surface when R[subscript 0]~R. The exponent γ, characterizing the number of polymer configurations, crosses over from a value of γ[subscript 1] – characteristic of a planar boundary – at small s to one reflecting the overall shape of the probe at large s. For a spherical probe the crossover is to an unencumbered polymer, while for a parabolic probe we cannot rule out a new exponent., National Science Foundation (U.S.) (Grant DMR-08-03315)

The binding of a transcription factor (TF) to a DNA operator site can initiate or repress the expression of a gene. Computational prediction of sites recognized by a TF has traditionally relied upon knowledge of several cognate sites, rather than an ab initio approach. Here, we examine the possibility of using structure-based energy calculations that require no knowledge of bound sites but rather start with the structure of a protein–DNA complex. We study the PurR Escherichia coli TF, and explore to which extent atomistic models of protein–DNA complexes can be used to distinguish between cognate and noncognate DNA sites. Particular emphasis is placed on systematic evaluation of this approach by comparing its performance with bioinformatic methods, by testing it against random decoys and sites of homologous TFs. We also examine a set of experimental mutations in both DNA and the protein. Using our explicit estimates of energy, we show that the specificity for PurR is dominated by direct protein–DNA interactions, and weakly influenced by bending of DNA., National Science Foundation (U.S.) (Grant DMR-08- 03315), Deutsche Forschungsgemeinschaft (DFG) (Grant VI237/1), NEC Research Support Fund, National Institutes of Health. National Centers for Biomedical Computing (Informatics for Integrating Biology and the Bedside), National Institutes of Health (U.S.) (3U54LM008748-04S1)

A widely used method for estimating Casimir interactions [H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948)] between gently curved material surfaces at short distances is the proximity force approximation (PFA). While this approximation is asymptotically exact at vanishing separations, quantifying corrections to PFA has been notoriously difficult. Here we use a derivative expansion to compute the leading curvature correction to PFA for metals (gold) and insulators (SiO$_2$) at room temperature. We derive an explicit expression for the amplitude $\hat\theta_1$ of the PFA correction to the force gradient for axially symmetric surfaces. In the non-retarded limit, the corrections to the Casimir free energy are found to scale logarithmically with distance. For gold, $\hat\theta_1$ has an unusually large temperature dependence., Comment: 4 pages, 2 figures