M.A.A. da Silva

New approach to dynamical Monte Carlo methods: application to an epidemic model

In this work we introduce a new approach to dynamical Monte Carlo methods to simulate Markovian processes. We apply this approach to formulate and study an epidemic generalized SIRS model. The results are in excellent agreement with the forth order Runge–Kutta Method in a region of deterministic solution. We also show that purely local interactions reproduce a poissonian-like process at mesoscopic level. The simulations for this case are checked self-consistently using a stochastic version of the Euler Method.

Deterministic folding: The role of entropic forces and steric specificities

The inverse folding problem of proteinlike macromolecules is studied by using a lattice Monte Carlo (MC) model in which steric specificities (nearest-neighbors constraints) are included and the hydrophobic effect is treated explicitly by considering interactions between the chain and solvent molecules. Chemical attributes and steric peculiarities of the residues are encoded in a 10-letter alphabet and a correspondent “syntax” is provided in order to write suitable sequences for the specified target structures; twenty-four target configurations, chosen in order to cover all possible values of the average contact order χ (0.2381⩽χ⩽0.4947 for this system), were encoded and analyzed. The results, obtained by MC simulations, are strongly influenced by geometrical properties of the native configuration, namely χ and the relative number φ of crankshafts-type structures: For χ<0.35 the folding is deterministic, that is, the syntax is able to encode successful sequences: The system presents larger encodability, mi...

Temporal Duration and Event Size Distribution at the Epidemic Threshold

The epidemic event, seen as a nonequilibrium dynamic process, is studied through a simple stochastic system reminiscent of the classical SIR model. The system is described in terms of global and local variables and was mainly treated by means of Monte Carlo simulation; square lattices N×N, with N=23, 51, 100, 151, and 211 were used. Distinct extensive runs were performed and then classified as corresponding to epidemic or non-epidemic phase. They were examined with detail through the analysis of the event duration and event size; illustrations, such as density-like plots in the space of the models parameters, are provided. The epidemic/non-epidemic phase presents smaller/larger relative fluctuations, whereas closer to the threshold the uncertainty reaches its highest values. Far enough from the threshold, the distribution φ(t) of the events time duration t shows a step-like appearance. However at the threshold line it shows an exponential behavior of the form φ (t) ∼ exp (-ωt); the same behavior is observed for the event size distribution. These results help to explain why the approach to epidemic threshold would be hard to anticipate with standard census data.

Exact solution of an anisotropic 2D random walk model with strong memory correlations

Over the last decade, there has been progress in understanding one-dimensional non-Markovian processes via analytic, sometimes exact, solutions. The extension of these ideas and methods to two and higher dimensions is challenging. We report the first exactly solvable two-dimensional (2D) non-Markovian random walk model belonging to the family of the elephant random walk model. In contrast to Levy walks or fractional Brownian motion, such models incorporate memory effects by keeping an explicit history of the random walk trajectory. We study a memory driven 2D random walk with correlated memory and stops, i.e. pauses in motion. The model has an inherent anisotropy with consequences for its diffusive properties, thereby mixing the dominant regime along one dimension with a subdiffusive walk along a perpendicular dimension. The anomalous diffusion regimes are fully characterized by an exact determination of the Hurst exponent. We discuss the remarkably rich phase diagram, as well as several possible combinations of the independent walks in both directions. The relationship between the exponents of the first and second moments is also unveiled.

Entropic force and folding of macromolecules

Abstract Effective forces between monomers of a polymeric chain are a widely employed simplification, in order to describe the hydrophobic effect in the folding mechanism. Alternatively, in this work, we consider the hydrophobic interaction as an entropic force, driving the chain into a shape of minimum surface area. A Monte Carlo simulation is used in the calculations, revealing that when the interaction between the chain and the solvent is established directly, the chain conformation presents important properties: compactness and malleability are the two more important ones.

Superdiffusion driven by exponentially decaying memory

A superdiffusive random walk model with exponentially decaying memory is reported. This seems to be a self-contradictory statement, since it is well known that random walks with exponentially decaying temporal correlations can be approximated arbitrarily well by Markov processes and that central limit theorems prohibit superdiffusion for Markovian walks with finite variance of step sizes. The solution to the apparent paradox is that the model is genuinely non-Markovian, due to a time-dependent decay constant associated with the exponential behavior.

Geometrical effects on folding of macromolecules

Geometrical effects on folding of macromolecules are investigated using linear chains with tetrahedral structure and hard-core interactions among its monomers; extra self-avoidance, namely, nontopological neighbor, is also considered. Our results were obtained by exact calculations using chains with small number N of monomers (up to 16) and by Monte Carlo simulation, using the ensemble growth method (EGM), for larger N. For some cases we provide a comparative study using two types of lattice and three different models. The original number of angle choices, ζ=3 (coordination number), is shown to be effectively reduced to ζeff=2.760, and the radius of gyration and end-to-end distance, for finite chains (N⩽140), scales with the number of monomers as Nν, where ν≅2/3. This is significantly larger than the corresponding value for the self-avoiding walk model, ν≅0.6. The relative frequency of monomer pair contacts was obtained by the exact Gibbs ensemble, involving all possible configurations. The same calculati...

Basic ingredients for mathematical modeling of tumor growth in vitro: Cooperative effects and search for space

Based on the literature data from HT-29 cell monolayers, we develop a model for its growth, analogous to an epidemic model, mixing local and global interactions. First, we propose and solve a deterministic equation for the progress of these colonies. Thus, we add a stochastic (local) interaction and simulate the evolution of an Eden-like aggregate by using dynamical Monte Carlo methods. The growth curves of both deterministic and stochastic models are in excellent agreement with the experimental observations. The waiting times distributions, generated via our stochastic model, allowed us to analyze the role of mesoscopic events. We obtain log-normal distributions in the initial stages of the growth and Gaussians at long times. We interpret these outcomes in the light of cellular division events: in the early stages, the phenomena are dependent each other in a multiplicative geometric-based process, and they are independent at long times. We conclude that the main ingredients for a good minimalist model of tumor growth, at mesoscopic level, are intrinsic cooperative mechanisms and competitive search for space.

Narrow log-periodic modulations in non-Markovian random walks

What are the necessary ingredients for log-periodicity to appear in the dynamics of a random walk model? Can they be subtle enough to be overlooked? Previous studies suggest that long-range damaged memory and negative feedback together are necessary conditions for the emergence of log-periodic oscillations. The role of negative feedback would then be crucial, forcing the system to change direction. In this paper we show that small-amplitude log-periodic oscillations can emerge when the system is driven by positive feedback. Due to their very small amplitude, these oscillations can easily be mistaken for numerical finite-size effects. The models we use consist of discrete-time random walks with strong memory correlations where the decision process is taken from memory profiles based either on a binomial distribution or on a delta distribution. Anomalous superdiffusive behavior and log-periodic modulations are shown to arise in the large time limit for convenient choices of the models parameters.

Scaling analysis of random walks with persistence lengths: Application to self-avoiding walks.

We develop an approach for performing scaling analysis of N-step random walks (RWs). The mean square end-to-end distance, 〈R[over ⃗]_{N}^{2}〉, is written in terms of inner persistence lengths (IPLs), which we define by the ensemble averages of dot products between the walkers position and displacement vectors, at the jth step. For RW models statistically invariant under orthogonal transformations, we analytically introduce a relation between 〈R[over ⃗]_{N}^{2}〉 and the persistence length, λ_{N}, which is defined as the mean end-to-end vector projection in the first step direction. For self-avoiding walks (SAWs) on 2D and 3D lattices we introduce a series expansion for λ_{N}, and by Monte Carlo simulations we find that λ_{∞} is equal to a constant; the scaling corrections for λ_{N} can be second- and higher-order corrections to scaling for 〈R[over ⃗]_{N}^{2}〉. Building SAWs with typically 100 steps, we estimate the exponents ν_{0} and Δ_{1} from the IPL behavior as function of j. The obtained results are in excellent agreement with those in the literature. This shows that only an ensemble of paths with the same length is sufficient for determining the scaling behavior of 〈R[over ⃗]_{N}^{2}〉, being that the whole information needed is contained in the inner part of the paths.