Finite-size scaling investigation of the liquid-liquid critical point in ST2 water and its stability with respect to crystallization.

The liquid-liquid critical point scenario of water hypothesizes the existence of two metastable liquid phases--low-density liquid (LDL) and high-density liquid (HDL)--deep within the supercooled region. The hypothesis originates from computer simulations of the ST2 water model, but the stability of the LDL phase with respect to the crystal is still being debated. We simulate supercooled ST2 water at constant pressure, constant temperature, and constant number of molecules N for N ≤ 729 and times up to 1 µs. We observe clear differences between the two liquids, both structural and dynamical. Using several methods, including finite-size scaling, we confirm the presence of a liquid-liquid phase transition ending in a critical point. We find that the LDL is stable with respect to the crystal in 98% of our runs (we perform 372 runs for LDL or LDL-like states), and in 100% of our runs for the two largest system sizes (N = 512 and 729, for which we perform 136 runs for LDL or LDL-like states). In all these runs, tiny crystallites grow and then melt within 1 µs. Only for N ≤ 343 we observe six events (over 236 runs for LDL or LDL-like states) of spontaneous crystallization after crystallites reach an estimated critical size of about 70 ± 10 molecules.

Parallel folding pathways in the SH3 domain protein.

The transition-state ensemble (TSE) is the set of protein conformations with an equal probability to fold or unfold. Its characterization is crucial for an understanding of the folding process. We determined the TSE of the src-SH3 domain protein by using extensive molecular dynamics simulations of the Go model and computing the folding probability of a generated set of TSE candidate conformations. We found that the TSE possesses a well-defined hydrophobic core with variable enveloping structures resulting from the superposition of three parallel folding pathways. The most preferred pathway agrees with the experimentally determined TSE, while the two least preferred pathways differ significantly. The knowledge of the different pathways allows us to design the interactions between amino acids that guide the protein to fold through the least preferred pathway. This particular design is akin to a circular permutation of the protein. The finding motivates the hypothesis that the different experimentally observed TSEs in homologous proteins and circular permutants may represent potentially available pathways to the wild-type protein.

Cascading Failures in Interdependent Networks with Multiple Supply-Demand Links and Functionality Thresholds.

Various social, financial, biological and technological systems can be modeled by interdependent networks. It has been assumed that in order to remain functional, nodes in one network must receive the support from nodes belonging to different networks. So far these models have been limited to the case in which the failure propagates across networks only if the nodes lose all their supply nodes. In this paper we develop a more realistic model for two interdependent networks in which each node has its own supply threshold, i.e., they need the support of a minimum number of supply nodes to remain functional. In addition, we analyze different conditions of internal node failure due to disconnection from nodes within its own network. We show that several local internal failure conditions lead to similar nontrivial results. When there are no internal failures the model is equivalent to a bipartite system, which can be useful to model a financial market. We explore the rich behaviors of these models that include discontinuous and continuous phase transitions. Using the generating functions formalism, we analytically solve all the models in the limit of infinitely large networks and find an excellent agreement with the stochastic simulations.

Cascading failures in interdependent networks with finite functional components.

We present a cascading failure model of two interdependent networks in which functional nodes belong to components of size greater than or equal to s. We find theoretically and via simulation that in complex networks with random dependency links the transition is first order for s≥3 and continuous for s=2. We also study interdependent lattices with a distance constraint r in the dependency links and find that increasing r moves the system from a regime without a phase transition to one with a second-order transition. As r continues to increase, the system collapses in a first-order transition. Each regime is associated with a different structure of domain formation of functional nodes.

Publisher's Note: Cascading failures in interdependent networks with finite functional components [Phys. Rev. E 94, 042304 (2016)].

This corrects the article DOI: 10.1103/PhysRevE.94.042304.

Identifying the protein folding nucleus using molecular dynamics.

Molecular dynamics simulations of folding in an off-lattice protein model reveal a nucleation scenario, in which a few well-defined contacts are formed with high probability in the transition state ensemble of conformations. Their appearance determines folding cooperativity and drives the model protein into its folded conformation. Amino acid residues participating in those contacts may serve as "accelerator pedals" used by molecular evolution to control protein folding rate.

Clustering of identical oligomers in coding and noncoding DNA sequences.

We develop a quantitative method for analyzing repetitions of identical short oligomers in coding and noncoding DNA sequences. We analyze sequences presently available in the GenBank separately for primate, mammal, vertebrate, rodent, invertebrate and plant taxonomic partitions. We find that some oligomers "cluster" more than they would if randomly distributed, while other oligomers "repel" each other. To quantify this degree of clustering, we define clustering measures. We find that (i) clustering significantly differs in coding and noncoding DNA; (ii) in most cases, monomers, dimers and tetramers cluster in noncoding DNA but appear to repel each other in coding DNA. (iii) The degree of clustering for different sources (primates, invertebrates, and plants) is more conserved among these sources in the case of coding DNA than in the case of noncoding DNA. (iv) In contrast to other oligomers, we find that trimers always prefer to cluster. (v) Clustering of each particular oligomer is conserved within the same organism.

Thermodynamically important contacts in folding of model proteins.

We introduce a quantity, the entropic susceptibility, that measures the thermodynamic importance-for the folding transition-of the contacts between amino acids in model proteins. Using this quantity, we find that only one equilibrium run of a computer simulation of a model protein is sufficient to select a subset of contacts that give rise to the peak in the specific heat observed at the folding transition. To illustrate the method, we identify thermodynamically important contacts in a model 46-mer. We show that only about 50% of all contacts present in the protein native state are responsible for the sharp peak in the specific heat at the folding transition temperature, while the remaining 50% of contacts do not affect the specific heat.

Effect of bond lifetime on the dynamics of a short-range attractive colloidal system.

We perform molecular dynamics simulations of short-range attractive colloid particles modeled by a narrow (3% of the hard sphere diameter) square well potential of unit depth. We compare the dynamics of systems with the same thermodynamics but different bond lifetimes, by adding to the square well potential a thin barrier at the edge of the attractive well. For permanent bonds, the relaxation time tau diverges as the packing fraction phi approaches a threshold related to percolation, while for short-lived bonds, the phi dependence of tau is more typical of a glassy system. At intermediate bond lifetimes, the phi dependence of tau is driven by percolation at low phi , but then crosses over to glassy behavior at higher phi . We also study the wave vector dependence of the percolation dynamics.

Liquid-liquid phase transitions for soft-core attractive potentials.

Using event-driven molecular dynamics simulations, we study a three-dimensional one-component system of spherical particles interacting via a discontinuous potential combining a repulsive square soft core and an attractive square well. In the case of a narrow attractive well, it has been shown that this potential has two metastable gas-liquid critical points. Here we systematically investigate how the changes of the parameters of this potential affect the phase diagram of the system. We find a broad range of potential parameters for which the system has both a gas-liquid critical point C1 and a liquid-liquid critical point C2. For the liquid-gas critical point we find that the derivatives of the critical temperature and pressure, with respect to the parameters of the potential, have the same signs: they are positive for increasing width of the attractive well and negative for increasing width and repulsive energy of the soft core. This result resembles the behavior of the liquid-gas critical point for standard liquids. In contrast, for the liquid-liquid critical point the critical pressure decreases as the critical temperature increases. As a consequence, the liquid-liquid critical point exists at positive pressures only in a finite range of parameters. We present a modified van der Waals equation which qualitatively reproduces the behavior of both critical points within some range of parameters, and gives us insight on the mechanisms ruling the dependence of the two critical points on the potential's parameters. The soft-core potential studied here resembles model potentials used for colloids, proteins, and potentials that have been related to liquid metals, raising an interesting possibility that a liquid-liquid phase transition may be present in some systems where it has not yet been observed.

Waterlike anomalies for core-softened models of fluids: two-dimensional systems.

We use molecular-dynamics simulations in two dimensions to investigate the possibility that a core-softened potential can reproduce static and dynamic anomalies found experimentally in liquid water: (i) the increase in specific volume upon cooling, (ii) the increase in isothermal compressibility upon cooling, and (iii) the increase in the diffusion coefficient with pressure. We relate these anomalies to the shape of the potential. We obtain the phase diagram of the system and identify two solid phases: a square crystal (high-density phase) and a triangular crystal (low-density phase). We also discuss the relation between the anomalies observed and the polymorphism of the solid. Finally, we compare the phase diagram of our model system with experimental data, noting especially the line of temperatures of maximum density, the line of pressures of maximum diffusion constant, and the line of temperatures of minimum isothermal compressibility.

Evidence for an unusual dynamical-arrest scenario in short-ranged colloidal systems.

Extensive molecular dynamics simulation studies of particles interacting via a short-ranged attractive square-well potential are reported. The calculated loci of constant diffusion coefficient D in the temperature-packing fraction plane show a reentrant behavior, i.e., an increase of diffusivity on cooling, confirming an important part of the high volume-fraction dynamical-arrest scenario earlier predicted by theory for particles with short-ranged potentials. The more efficient localization mechanism induced by the short-range bonding provides, on average, additional free volume as compared to the hard-sphere case and results in faster dynamics.

Metastable liquid-liquid phase transition in a single-component system with only one crystal phase and no density anomaly.

We investigate the phase behavior of a single-component system in three dimensions with spherically-symmetric, pairwise-additive, soft-core interactions with an attractive well at a long distance, a repulsive soft-core shoulder at an intermediate distance, and a hard-core repulsion at a short distance, similar to potentials used to describe liquid systems such as colloids, protein solutions, or liquid metals. We showed [Nature (London) 409, 692 (2001)] that, even with no evidence of the density anomaly, the phase diagram has two first-order fluid-fluid phase transitions, one ending in a gas-low-density-liquid (LDL) critical point, and the other in a gas-high-density-liquid (HDL) critical point, with a LDL-HDL phase transition at low temperatures. Here we use integral equation calculations to explore the three-parameter space of the soft-core potential and perform molecular dynamics simulations in the interesting region of parameters. For the equilibrium phase diagram, we analyze the structure of the crystal phase and find that, within the considered range of densities, the structure is independent of the density. Then, we analyze in detail the fluid metastable phases and, by explicit thermodynamic calculation in the supercooled phase, we show the absence of the density anomaly. We suggest that this absence is related to the presence of only one stable crystal structure.

Confirmation of anomalous dynamical arrest in attractive colloids: a molecular dynamics study.

Previous theoretical simulation and experimental studies have indicated that particles with a short-ranged attraction exhibit a range of dynamical arrest phenomena. These include very pronounced reentrance in the dynamical arrest curve, a logarithmic singularity in the density correlation functions, and the existence of "attractive" and "repulsive" glasses. Here we carry out extensive molecular dynamics calculations on dense systems interacting via a square-well potential. This is one of the simplest systems with the required properties, and may be regarded as canonical for interpreting the phase diagram, and now also the dynamical arrest. We confirm the theoretical predictions for reentrance, logarithmic singularity, and give a direct evidence of the existence, independent of theory, of two distinct glasses. We now regard the previous predictions of these phenomena as having been established.

Dynamics of viscous penetration in percolation porous media.

We investigate the dynamics of viscous penetration in two-dimensional percolation networks at criticality for the case in which the ratio between the viscosities of displaced and injected fluids is very large. We report extensive numerical simulations that indicate that the scaling exponents for the breakthrough time distribution are the same as the previously reported values computed for the case of unit viscosity ratio. Our results are consistent with the possibility that viscous displacement through critical percolation networks constitutes a single universality class, independent of the viscosity ratio. We also find that the distributions of mass and breakthrough time of the invaded clusters have the same scaling form, but with different critical exponents.

Discrete molecular dynamics simulations of peptide aggregation.

We study the aggregation of peptides using the discrete molecular dynamics simulations. Specifically, at temperatures above the alpha-helix melting temperature of a single peptide, the model peptides aggregate into a multilayer parallel beta-sheet structure. This structure has an interstrand distance of 4.8 A and an intersheet distance of 10 A, which agree with experimental observations. Our model explains these results as follows: hydrogen-bond interactions give rise to the interstrand spacing in beta sheets, while Go interactions between side chains make beta strands parallel to each other and allow beta sheets to pack into layers. An important feature of our results is that the aggregates contain free edges, which may allow for further aggregation of model peptides to form elongated fibrils.

Average time spent by Lévy flights and walks on an interval with absorbing boundaries.

We consider a Lévy flyer of order alpha that starts from a point x(0) on an interval [O,L] with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Lévy flights and Lévy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Lévy processes converge to the continuous approximations in all cases except the case of alpha-->2, and the cases of x(0)-->0 and x(0)-->L. For alpha>2, when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if x(0) is placed in the vicinity of absorbing boundaries, the average total length has a minimum at alpha=1, corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.

Fractals in biology and medicine.

Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material. Furthermore, we discuss the finding that the exponent alpha quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.

Scaling features of noncoding DNA.

We review evidence supporting the idea that the DNA sequence in genes containing noncoding regions is correlated, and that the correlation is remarkably long range--indeed, base pairs thousands of base pairs distant are correlated. We do not find such a long-range correlation in the coding regions of the gene, and utilize this fact to build a Coding Sequence Finder Algorithm, which uses statistical ideas to locate the coding regions of an unknown DNA sequence. Finally, we describe briefly some recent work adapting to DNA the Zipf approach to analyzing linguistic texts, and the Shannon approach to quantifying the "redundancy" of a linguistic text in terms of a measurable entropy function, and reporting that noncoding regions in eukaryotes display a larger redundancy than coding regions. Specifically, we consider the possibility that this result is solely a consequence of nucleotide concentration differences as first noted by Bonhoeffer and his collaborators. We find that cytosine-guanine (CG) concentration does have a strong "background" effect on redundancy. However, we find that for the purine-pyrimidine binary mapping rule, which is not affected by the difference in CG concentration, the Shannon redundancy for the set of analyzed sequences is larger for noncoding regions compared to coding regions.