Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations with Applications to Cell Biology

Mathematics Subject Classification: 65C05, 60G50, 39A10, 92C37

Biased diffusion in confined media: Test of the Fick-Jacobs approximation and validity criteria

We study biased, diffusive transport of Brownian particles through narrow, spatially periodic structures in which the motion is constrained in lateral directions. The problem is analyzed under the perspective of the Fick-Jacobs equation, which accounts for the effect of the lateral confinement by introducing an entropic barrier in a one-dimensional diffusion. The validity of this approximation, based on the assumption of an instantaneous equilibration of the particle distribution in the cross section of the structure, is analyzed by comparing the different time scales that characterize the problem. A validity criterion is established in terms of the shape of the structure and of the applied force. It is analytically corroborated and verified by numerical simulations that the critical value of the force up to which this description holds true scales as the square of the periodicity of the structure. The criterion can be visualized by means of a diagram representing the regions where the Fick-Jacobs description becomes inaccurate in terms of the scaled force versus the periodicity of the structure.

Wall mediated transport in confined spaces: Exact theory for low density

We present a theory for the transport of molecules adsorbed in slit and cylindrical nanopores at low density, considering the axial momentum gain of molecules oscillating between diffuse wall reflections. Good agreement with molecular dynamics simulations is obtained over a wide range of pore sizes, including the regime of single-file diffusion where fluid-fluid interactions are shown to have a negligible effect on the collective transport coefficient. We show that dispersive fluid-wall interactions considerably attenuate transport compared to classical hard sphere theory.

Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion

We present a new a-priori estimate for discrete coagulation fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a-priori estimate provides a global L2 bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

Confined brownian ratchets

We analyze the dynamics of Brownian ratchets in a confined environment. The motion of the particles is described by a Fick-Jakobs kinetic equation in which the presence of boundaries is modeled by means of an entropic potential. The cases of a flashing ratchet, a two-state model, and a ratchet under the influence of a temperature gradient are analyzed in detail. We show the emergence of a strong cooperativity between the inherent rectification of the ratchet mechanism and the entropic bias of the fluctuations caused by spatial confinement. Net particle transport may take place in situations where none of those mechanisms leads to rectification when acting individually. The combined rectification mechanisms may lead to bidirectional transport and to new routes to segregation phenomena. Confined Brownian ratchets could be used to control transport in mesostructures and to engineer new and more efficient devices for transport at the nanoscale.

Entropic transport in confined media: a challenge for computational studies in biological and soft-matter systems

Transport in small-scale biological and soft-matter systems typically occurs under confinement conditions in which particles proceed through obstacles and irregularities of the boundaries that may significantly alter their trajectories. A transport model that assimilates the confinement to the presence of entropic barriers provides an efficient approach to quantify its effect on the particle current and the diffusion coefficient. We review the main peculiarities of entropic transport and treat two cases in which confinement effects play a crucial role, with the appearance of emergent properties. The presence of entropic barriers modifies the mean first-passage time distribution and therefore plays a very important role in ion transport through micro- and nano-channels. The functionality of molecular motors, modeled as Brownian ratchets, is strongly affected when the motor proceeds in a confined medium that may constitute another source of rectification. The interplay between ratchet and entropic rectification gives rise to a wide variety of dynamical behaviors, not observed when the Brownian motor proceeds in an unbounded medium. Entropic transport offers new venues of transport control and particle manipulation and new ways to engineer more efficient devices for transport at the nanoscale.

Input diffusion and the evolution of production networks

What determines which inputs are initially considered and eventually adopted in the productionof new or improved goods? Why are some inputs much more prominent than others? We modelthe evolution of input linkages as a process where new producers first search for potentially usefulinputs and then decide which ones to adopt. A new product initially draws a set of 'essentialsuppliers'. The search stage is then confined to the network neighborhood of the latter, i.e., to theinputs used by the essential suppliers. The adoption decision is driven by a tradeoff between thebenefits accruing from input variety and the costs of input adoption. This has important implicationsfor the number of forward linkages that a product (input variety) develops over time. Inputdiffusion is fostered by network centrality ? an input that is initially represented in many networkneighborhoods is subsequently more likely to be adopted. This mechanism also delivers a powerlaw distribution of forward linkages. Our predictions continue to hold when varieties are aggregatedinto sectors. We can thus test them, using detailed sectoral US input-output tables. We showthat initial network proximity of a sector in 1967 significantly increases the likelihood of adoptionthroughout the subsequent four decades. The same is true for rapid productivity growth in aninput-producing sector. Our empirical results highlight two conditions for new products to becomecentral nodes: initial network proximity to prospective adopters, and technological progress thatreduces their relative price. Semiconductors met both conditions.

Atomistic Study of Water Confined in Silica.

In this work, we have used a combined of atomistic simulation methods to explore the effects of confinement of water molecules between silica surfaces. Firstly, the mechanical properties of water severe confined (~3A) between two silica alpha-quartz was determined based on first principles calculations within the density functional theory (DFT). Simulated annealing methods were employed due to the complex potential energry surface, and the difficulties to avoid local minima. Our results suggest that much of the stiffness of the material (46%) remains, even after the insertion of a water monolayer in the silica. Secondly, in order to access typical time scales for confined systems, classical molecular dynamics was used to determine the dynamical properties of water confined in silica cylindrical pores, with diameters varying from 10 to 40A. in this case we have varied the passivation of the silica surface, from 13% to 100% of SiOH, and the other terminations being SiOH2 and SiOH3, the distribution of the different terminations was obtained with a Monte Carlo simulation. The simulations indicates a lowering of the diffusion coefficientes as the diameter decreases, due to the structuration of hydrogen bonds of water molecules; we have also obtained the density profiles of the confined water and the interfacial tension.

Effect Of Confined Impinging Jet Mixing Processing Parameters On Poly (Lactic Acid) Nanoparticle Morphology

Biodegradable nanoparticles are at the forefront of drug delivery research as they provide numerous advantages over traditional drug delivery methods. An important factor affecting the ability of nanoparticles to circulate within the blood stream and interact with cells is their morphology. In this study a novel processing method, confined impinging jet mixing, was used to form poly (lactic acid) nanoparticles through a solvent-diffusion process with Pluronic F-127 being used as a stabilizing agent. This study focused on the effects of Reynolds number (flow rate), surfactant presence in mixing, and polymer concentration on the morphology of poly (lactic acid) nanoparticles. In addition to looking at the parameters affecting poly (lactic acid) morphology, this study attempted to improve nanoparticle isolation and purification methods to increase nanoparticle yield and ensure specific morphologies were not being excluded during isolation and purification. The isolation and purification methods used in this study were centrifugation and a stir cell. This study successfully produced particles having pyramidal and cubic morphologies. Despite successful production of these morphologies the yield of non-spherical particles was very low, additionally great variability existed between redundant trails. Surfactant was determined to be very important for the stabilization of nanoparticles in solution but appears to be unnecessary for the formation of nanoparticles. Isolation and purification methods that produce a high yield of surfactant free particles have still not been perfected and additional testing will be necessary for improvement.¿

Bifurcation diagrams for polymer blends with diffuse interfaces in confined and adaptive geometries

Dynamics of binary mixtures such as polymer blends, and fluids near the critical point, is described by the model-H, which couples momentum transport and diffusion of the components [1]. We present an extended version of the model-H that allows to study the combined effect of phase separation in a polymer blend and surface structuring of the film itself [2]. We apply it to analyze the stability of vertically stratified base states on extended films of polymer blends and show that convective transport leads to new mechanisms of instability as compared to the simpler diffusive case described by the Cahn- Hilliard model [3, 4]. We carry out this analysis for realistic parameters of polymer blends used in experimental setups such as PS/PVME. However, geometrically more complicated states involving lateral structuring, strong deflections of the free surface, oblique diffuse interfaces, checkerboard modes, or droplets of a component above of the other are possible at critical composition solving the Cahn Hilliard equation in the static limit for rectangular domains [5, 6] or with deformable free surfaces [6]. We extend these results for off-critical compositions, since balanced overall composition in experiments are unusual. In particular, we study steady nonlinear solutions of the Cahn-Hilliard equation for bidimensional layers with fixed geometry and deformable free surface. Furthermore we distinguished the cases with and without energetic bias at the free surface. We present bifurcation diagrams for off-critical films of polymer blends with free surfaces, showing their free energy, and the L2-norms of surface deflection and the concentration field, as a function of lateral domain size and mean composition. Simultaneously, we look at spatial dependent profiles of the height and concentration. To treat the problem of films with arbitrary surface deflections our calculations are based on minimizing the free energy functional at given composition and geometric constraints using a variational approach based on the Cahn-Hilliard equation. The problem is solved numerically using the finite element method (FEM).

Influence of adsorbate interaction on transport in confined spaces

This article provides a review of the recent theory of transport in nanopores developed in the author's laboratory. In particular the influence of fluid-solid interactions on the transport coefficient is examined, showing that such interactions reduce the value of the coefficient by almost an order of magnitude in comparison to the Knudsen theory for non-interacting systems. The activation energy and potential energy barriers for diffusion in smooth pores with a one-dimensional potential energy profile are also discussed, indicating the inadequacy of the commonly used assumption of proportionality between the activation energy and heat of adsorption or the minimum pore potential energy. A further feature affected by fluid-solid interactions is the nature of the reflection of fluid molecules colliding with a pore wall surface, varying from being nearly specular - such as in carbon nanotubes - to nearly diffuse for amorphous solids. Diffuse reflection leads to momentum loss and reduced transport coefficients. However, fluid-solid interactions do not affect the transport coefficient in the single-file diffusion regime when the surface reflection is diffuse, and the transport coefficient in this case is largely independent of the adsorbed density.

Feasibility And Reproducibility Of Diffusion-weighted Magnetic Resonance Imaging Of The Fetal Brain In Twin-twin Transfusion Syndrome.

The aim of this study is to test the feasibility and reproducibility of diffusion-weighted magnetic resonance imaging (DW-MRI) evaluations of the fetal brains in cases of twin-twin transfusion syndrome (TTTS). From May 2011 to June 2012, 24 patients with severe TTTS underwent MRI scans for evaluation of the fetal brains. Datasets were analyzed offline on axial DW images and apparent diffusion coefficient (ADC) maps by two radiologists. The subjective evaluation was described as the absence or presence of water diffusion restriction. The objective evaluation was performed by the placement of 20-mm(2) circular regions of interest on the DW image and ADC maps. Subjective interobserver agreement was assessed by the kappa correlation coefficient. Objective intraobserver and interobserver agreements were assessed by proportionate Bland-Altman tests. Seventy-four DW-MRI scans were performed. Sixty of them (81.1%) were considered to be of good quality. Agreement between the radiologists was 100% for the absence or presence of diffusion restriction of water. For both intraobserver and interobserver agreement of ADC measurements, proportionate Bland-Altman tests showed average percentage differences of less than 1.5% and 95% CI of less than 18% for all sites evaluated. Our data demonstrate that DW-MRI evaluation of the fetal brain in TTTS is feasible and reproducible.

Dependence of the crossover exponent with the diffusion rate in the generalized contact process model

We study how the crossover exponent, phi, between the directed percolation (DP) and compact directed percolation (CDP) behaves as a function of the diffusion rate in a model that generalizes the contact process. Our conclusions are based in results pointed by perturbative series expansions and numerical simulations, and are consistent with a value phi = 2 for finite diffusion rates and phi = 1 in the limit of infinite diffusion rate.

Reaction-diffusion stochastic lattice model for a predator-prey system

We have the purpose of analyzing the effect of explicit diffusion processes in a predator-prey stochastic lattice model. More precisely we wish to investigate the possible effects due to diffusion upon the thresholds of coexistence of species, i. e., the possible changes in the transition between the active state and the absorbing state devoid of predators. To accomplish this task we have performed time dependent simulations and dynamic mean-field approximations. Our results indicate that the diffusive process can enhance the species coexistence.

Evaluation of the diffusion capacity of calcium hydroxide pastes through the dentinal tubules

This study aimed to evaluate the diffusion capacity of calcium hydroxide pastes with different vehicles through dentinal tubules. The study was conducted on 60 extracted single-rooted human teeth whose crowns had been removed. The root canals were instrumented and divided into 4 groups according to the vehicle of the calcium hydroxide paste: Group I - distilled water; Group II - propylene glycol; Group III - 0.2% chlorhexidine; Group IV - 2% chlorhexidine. After placement of the root canal dressings, the teeth were sealed and placed in flasks containing deionized water. After 1, 2, 7, 15, 30, 45 and 60 days, the pH of the water was measured to determine the diffusion of calcium hydroxide through the dentinal tubules. The data were recorded and statistically compared by the Tukey test. The results showed that all pastes presented a similar diffusion capacity through dentin. Group IV did not present difference compared to group I. Group II presented difference compared to the other groups, as did Group III. In conclusion, groups I and IV presented a better diffusion capacity through dentin than groups II and III; 2% chlorhexidine can be used as a vehicle in calcium hydroxide pastes.