SAWdoubler: a program for counting self-avoiding walks

This article presents SAWdoubler, a package for counting the total number Z(N) of self-avoiding walks (SAWs) on a regular lattice by the length-doubling method, of which the basic concept has been published previously by us. We discuss an algorithm for the creation of all SAWs of length N, efficient storage of these SAWs in a tree data structure, and an algorithm for the computation of correction terms to the count Z(2N) for SAWs of double length, removing all combinations of two intersecting single-length SAWs. We present an efficient numbering of the lattice sites that enables exploitation of symmetry and leads to a smaller tree data structure; this numbering is by increasing Euclidean distance from the origin of the lattice. Furthermore, we show how the computation can be parallelised by distributing the iterations of the main loop of the algorithm over the cores of a multicore architecture. Experimental results on the 3D cubic lattice demonstrate that Z(28) can be computed on a dual-core PC in only 1 hour and 40 minutes, with a speedup of 1.56 compared to the single-core computation and with a gain by using symmetry of a factor of 26. We present results for memory use and show how the computation is made to fit in 4 Gbyte RAM. It is easy to extend the SAWdoubler software to other lattices; it is publicly available under the GNU LGPL license.Comment: 29 pages, 3 figure

Hidden dependence of spreading vulnerability on topological complexity

Many dynamical phenomena in complex systems concern spreading that plays out on top of networks with changing architecture over time -- commonly known as temporal networks. A complex system's proneness to facilitate spreading phenomena, which we abbreviate as its `spreading vulnerability', is often surmised to be related to the topology of the temporal network featured by the system. Yet, cleanly extracting spreading vulnerability of a complex system directly from the topological information of the temporal network remains a challenge. Here, using data from a diverse set of real-world complex systems, we develop the `entropy of temporal entanglement' as a novel and insightful quantity to measure topological complexities of temporal networks. We show that this parameter-free quantity naturally allows for topological comparisons across vastly different complex systems. Importantly, by simulating three different types of stochastic dynamical processes playing out on top of temporal networks, we demonstrate that the entropy of temporal entanglement serves as a quantitative embodiment of the systems' spreading vulnerability, irrespective of the details of the processes. In being able to do so, i.e., in being able to quantitatively extract a complex system's proneness to facilitate spreading phenomena from topology, this entropic measure opens itself for applications in a wide variety of natural, social, biological and engineered systems.Comment: 15 pages, 9 figures, to appear in Phys. Rev.

Exact enumeration of Hamiltonian walks on the 4 × 4 × 4 cube and applications to protein folding

Abstract Hamiltonian walks on lattices are model systems for compact polymers such as proteins. Here we enumerate exactly the number of Hamiltonian walks on the 4 × 4 × 4 cube and give estimates up to the 7 × 7 × 7 cube through Monte Carlo methods. We find that the number of configurations grows faster with chain length than previously anticipated. Finally, we discuss uniqueness of ground states in the HP model for protein folding

Computational study of remodeling in a nucleosomal array

Chromatin remodeling complexes utilize the energy of ATP hydrolysis to change the packing state of chromatin, e.g. by catalysing the sliding of nucleosomes along DNA. Here we present simple models to describe experimental data of changes in DNA accessibility along a synthetic, repetitive array of nucleosomes during remodeling by the ACF enzyme or its isolated ATPase subunit, ISWI. We find substantial qualitative differences between the remodeling activities of ISWI and ACF. To understand better the observed behavior for the ACF remodeler, we study more microscopic models of nucleosomal arrays

Exact enumeration of self-avoiding walks on BCC and FCC lattices

Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method. Analysis of the enumeration results yields values for the exponents γ and ν which are in agreement with, but less accurate than, those obtained earlier from enumeration results on the simple cubic lattice. The non-universal growth constant and amplitudes are accurately determined, yielding for the BCC lattice μ = 6.530 520(20), A = 1.1785(40), and D = 1.0864(50), and for the FCC lattice μ = 10.037 075(20), A = 1.1736(24), and D = 1.0460(50)

Multiplexing Genetic and Nucleosome Positioning Codes: A Computational Approach

<div><p>Eukaryotic DNA is strongly bent inside fundamental packaging units: the nucleosomes. It is known that their positions are strongly influenced by the mechanical properties of the underlying DNA sequence. Here we discuss the possibility that these mechanical properties and the concomitant nucleosome positions are not just a side product of the given DNA sequence, e.g. that of the genes, but that a mechanical evolution of DNA molecules might have taken place. We first demonstrate the possibility of multiplexing classical and mechanical genetic information using a computational nucleosome model. In a second step we give evidence for genome-wide multiplexing in <i>Saccharomyces cerevisiae</i> and <i>Schizosacharomyces pombe</i>. This suggests that the exact positions of nucleosomes play crucial roles in chromatin function.</p></div

Mechanisms underlying multiplexing.

<p>(A) Energy landscape (black dashed curve) of a sequence with highly optimized nucleosome affinity at −5 bp, produced by MMC at very low temperature (15 K). The colored curves are landscapes for three synonymous mutants (three different codon frames) that are optimized via sMMC for high affinity at position 0. The maximum cannot be turned into a minimum in this case, signaling that multiplexing would not be possible on genomes if they were selected for highest nucleosome affinity. (B) Distribution of AA, TT and TA dinucleotides around minor groove bending site −25 bp for the shifted nucleosome from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0156905#pone.0156905.g002" target="_blank">Fig 2C</a> bottom. Dashed blue curves: natural preferences (attained through MMC), red curves: distribution obtained from sMMC on that particular stretch of the YAL002W gene; both simulations are performed at 100 K. Though not optimal, sMMC brings in AA at a position close to its preferred position (amplitude almost 1) indicating the plasticity of the mechanical code.</p

Mechanical energy landscape along a 500 bp stretch of the YAL002W gene of <i>S. cerevisiae</i>.

<p>(A) Elastic energy of the nucleosome model as a function of position obtained from a Monte Carlo simulation at 50 K. (B) Effective energy including excluded volume between nucleosomes. In both, (A) and (B), the vertical lines indicate experimentally determined nucleosome positions from the unique nucleosome map [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0156905#pone.0156905.ref025" target="_blank">25</a>]. (C) The top graph shows a fraction of the original landscape from (A), the five landscapes below are produced via sMMC with the nucleosome positioned at the corresponding dashed vertical line. The minima can be shifted freely on top of genes, proving that multiplexing is possible.</p

Nucleosomal DNA model with bp step dependent mechanical properties.

<p>(A) The rigid base-pair model is forced, using 28 constraints (indicated by red spheres), into a lefthanded superhelical path that mimics the DNA conformation in the nucleosome crystal structure [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0156905#pone.0156905.ref004" target="_blank">4</a>]. (B) Fraction of dinucleotides GC and AA/TT/TA at each position along the nucleosome model found in 10 million high affinity sequences produced by MMC at 100 K. The solid and dashed lines indicate minor and major groove bending sites; the nucleosome dyad is at 0 bp. The model recovers the basic nucleosome positioning rules [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0156905#pone.0156905.ref001" target="_blank">1</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0156905#pone.0156905.ref003" target="_blank">3</a>]. (C) Same as (B), but on top of 1200 coding sequences (produced by sMMC). The same periodic signals are found albeit with a smaller amplitude.</p