A fractional calculus approach to self-similar protein dynamics

Existence and uniqueness of well-posed fractional boundary value problem

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer's fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.

Analysis of Velocity Autocorrelation Functions from Molecular Dynamics Simulations of a Small Peptide by the Generalized Langevin Equation with a Power-Law Kernel

Internal motions play an essential role in the biological functions of proteins and have been the subject of numerous theoretical and spectroscopic studies. Such complex environments are associated with anomalous diffusion where, in contrast to the classical Brownian motion, the relevant correlation functions have power law decays with time. In this work, we investigate the presence of long memory stochastic processes through the analysis of atomic velocity autocorrelation functions. Analytical expressions of the velocity autocorrelation function spectrum obtained through a Mori-Zwanzig projection approach were shown to be compatible with molecular dynamics simulations of a small helical peptide (8-polyalanine).

Signature of functional enzyme dynamics in quasielastic neutron scattering spectra: The case of phosphoglycerate kinase

We present an analysis of high-resolution quasi-elastic neutron scattering spectra of phosphoglycerate kinase which elucidates the influence of the enzymatic activity on the dynamics of the protein. We show that in the active state the inter-domain motions are amplified and the intra-domain asymptotic power-law relaxation ∝t-α is accelerated, with a reduced coefficient α. Employing an energy landscape picture of protein dynamics, this observation can be translated into a widening of the distribution of energy barriers separating conformational substates of the protein.

Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations

Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equations from the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equations based on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equations for the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equations are unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem

In this paper, we investigate a class of boundary value problems involving Caputo fractional derivative $ {{}^C\mathcal{D}^{\alpha}_{a}} $ of order $ \alpha \in (2, 3) $, and the usual derivative, of the form $ \begin{equation*} ({{}^C\mathcal{D}^{\alpha}_{a}}x)(t)+p(t)x'(t)+q(t)x(t) = g(t), \quad a\leq t\leq b, \end{equation*} $ for an unknown $ x $ with $ x(a) = x'(a) = x(b) = 0 $, and $ p, \; q, \; g\in C^2([a, b]) $. The proposed method uses certain integral inequalities, Banach's Contraction Principle and Krasnoselskii's Fixed Point Theorem to identify conditions that guarantee the existence and uniqueness of the solution (for the problem under study) and that allow the deduction of Ulam-Hyers and Ulam-Hyers-Rassias stabilities.

Multiscale relaxation dynamics and diffusion of myelin basic protein in solution studied by quasielastic neutron scattering

We report an analysis of high-resolution quasielastic neutron scattering spectra from Myelin Basic Protein (MBP) in solution, comparing the spectra at three different temperatures (283, 303, and 323 K) for a pure D₂O buffer and a mixture of D₂O buffer with 30% of deuterated trifluoroethanol (TFE). Accompanying experiments with dynamic light scattering and Circular Dichroism (CD) spectroscopy have been performed to obtain, respectively, the global diffusion constant and the secondary structure content of the molecule for both buffers as a function of temperature. Modeling the decay of the neutron intermediate scattering function by the Mittag-Leffler relaxation function, ϕ(t) = E_α(-(t/τ)^α) (0 < α < 1), we find that trifluoroethanol slows down the relaxation dynamics of the protein at 283 K and leads to a broader relaxation rate spectrum. This effect vanishes with increasing temperature, and at 323 K, its relaxation dynamics is identical in both solvents. These results are coherent with the data from dynamic light scattering, which show that the hydrodynamic radius of MBP in TFE-enriched solutions does not depend on temperature and is only slightly smaller compared to the pure D₂O buffer, except for 283 K, where it is much reduced. In accordance with these observations, the CD spectra reveal that TFE induces essentially a partial transition from β-strands to α-helices, but only a weak increase in the total secondary structure content, leaving about 50% of the protein unfolded. The results show that MBP is for all temperatures and in both buffers an intrinsically disordered protein and that TFE essentially induces a reduction in its hydrodynamic radius and its relaxation dynamics at low temperatures.

Subdiffusive-Brownian crossover in membrane proteins: a generalized Langevin equation-based approach

In this work, we propose a generalized Langevin equation-based model to describe the lateral diffusion of a protein in a lipid bilayer. The memory kernel is represented in terms of a viscous (instantaneous) and an elastic (noninstantaneous) component modeled through a Dirac δ function and a three-parameter Mittag-Leffler type function, respectively. By imposing a specific relationship between the parameters of the three-parameter Mittag-Leffler function, the different dynamical regimes-namely ballistic, subdiffusive, and Brownian, as well as the crossover from one regime to another-are retrieved. Within this approach, the transition time from the ballistic to the subdiffusive regime and the spectrum of relaxation times underlying the transition from the subdiffusive to the Brownian regime are given. The reliability of the model is tested by comparing the mean-square displacement derived in the framework of this model and the mean-square displacement of a protein diffusing in a membrane calculated through molecular dynamics simulations.

Coupled fractional differential systems with random effects in Banach spaces

The purpose of this work is to investigate the existence of solutions for a system of random differential equations involving the Riemann–Liouville fractional derivative. The existence result is established by means of a random abstract formulation to Sadovskii’s fixed point theorem principle [A. Baliki, J. J. Nieto, A. Ouahab and M. L. Sinacer, Random semilinear system of differential equations with impulses, Fixed Point Theory Appl. 2017 2017, Paper No. 27] combined with a technique based on vector-valued metrics and convergent to zero matrices. An example is also provided to illustrate our result.

Optimal solution of the fractional order breast cancer competition model

In this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body's natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique.

Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation

The article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are studied. It is shown that the nonlocal finite-difference scheme is conditionally stable and converges to the first order. Using the fractional Riccati equation as an example, the computational accuracy of the numerical method is analyzed. It is shown that with an increase in the nodes of the computational grid, the order of computational accuracy tends to unity, i.e., to the theoretical value of the order of accuracy.

Existence and data dependence results for fractional differential equations involving atangana-baleanu derivative

In the current paper, we consider multi-derivative nonlinear fractional differential equations involving Atangana-Baleanu fractional derivative. We investigate the fundamental results about the existence, uniqueness, boundedness and dependence of the solution on various data. The analysis is based on a fractional integral operator due to T. R. Prabhakar involving generalized Mittag-Leffler function, the Krasnoselskii's fixed point theorem and Gronwall-Bellman inequality with continuous functions.

Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

New Study of the Existence and Dimension of the Set of Solutions for Nonlocal Impulsive Differential Inclusions with a Sectorial Operator

In this article, we are interested in a new generic class of nonlocal fractional impulsive differential inclusions with linear sectorial operator and Lipschitz multivalued function in the setting of finite dimensional Banach spaces. By modifying the definition of PC-mild solutions initiated by Shu, we succeeded to determine new conditions that sufficiently guarantee the existence of the solutions. The results are obtained by combining techniques of fractional calculus and the fixed point theorem for contraction maps. We also characterize the topological structure of the set of solutions. Finally, we provide a demonstration to address the applicability of our theoretical results.

GENAVOS: A New Tool for Modelling and Analyzing Cancer Gene Regulatory Networks Using Delayed Nonlinear Variable Order Fractional System

Gene regulatory networks (GRN) are one of the etiologies associated with cancer. Their dysregulation can be associated with cancer formation and asymmetric cellular functions in cancer stem cells, leading to disease persistence and resistance to treatment. Systems that model the complex dynamics of these networks along with adapting to partially known real omics data are closer to reality and may be useful to understand the mechanisms underlying neoplastic phenomena. In this paper, for the first time, modelling of GRNs is performed using delayed nonlinear variable order fractional (VOF) systems in the state space by a new tool called GENAVOS. Although the tool uses gene expression time series data to identify and optimize system parameters, it also models possible epigenetic signals, and the results show that the nonlinear VOF systems have very good flexibility in adapting to real data. We found that GRNs in cancer cells actually have a larger delay parameter than in normal cells. It is also possible to create weak chaotic, periodic, and quasi-periodic oscillations by changing the parameters. Chaos can be associated with the onset of cancer. Our findings indicate a profound effect of time-varying orders on these networks, which may be related to a type of cellular epigenetic memory. By changing the delay parameter and the variable order functions (possible epigenetics signals) for a normal cell system, its behaviour becomes quite similar to the behaviour of a cancer cell. This work confirms the effective role of the miR-17-92 cluster as an epigenetic factor in the cancer cell cycle.

Variable-order particle dynamics: formulation and application to the simulation of edge dislocations

This study presents the application of variable-order (VO) fractional operators to modelling the dynamics of edge dislocations under the effect of a static state of shear stress. More specifically, a particle dynamic approach is used to simulate the microscopic structure of a material where the constitutive atoms or molecules are modelled via discrete masses and their interaction via inter-particle forces. VO operators are introduced in the formulation in order to capture the complex linear-to-nonlinear dynamic transitions following the translation of dislocations as well as the creation and annihilation of bonds between particles. Remarkably, the motion of the dislocation does not require any a priori assumption in terms of either possible trajectory or sections of the model that could undergo the nonlinear transition associated with the creation and annihilation of bonds. The model only requires the definition of the initial location of the dislocations. Results will show that the VO formulation is fully evolutionary and capable of capturing both the sliding and the coalescence of edge dislocations by simply exploiting the instantaneous response of the system and the state of stress. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.

Existence and finite-time stability of a unique almost periodic positive solution for fractional-order Lasota–Wazewska red blood cell models

In this paper, we are concerned with a class of fractional-order Lasota–Wazewska red blood cell models. By applying a fixed point theorem on a normal cone, we first obtain the sufficient conditions for the existence of a unique almost periodic positive solution of the considered models. Then, considering that all of the red blood cells in animals survive in a finite-time interval, we study the finite-time stability of the almost periodic positive solution by using some inequality techniques. Our results and methods of this paper are new. Finally, we give numerical examples to show the feasibility of the obtained results.

Applications of variable-order fractional operators: a review

Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

Truncated Lévy motion through path integrals and applications to nondiffusive suprathermal ion transport

Fractional Levy motion has been derived from its generalized Langevin equation via path integrals in earlier works and has since proven to be a useful model for nonlocal and non-Markovian processes, especially in the context of nondiffusive transport. Here, we generalize the approach to treat tempered Lévy distributions and derive the propagator and diffusion equation of truncated asymmetrical fractional Levy motion via path integrals. The model now recovers exponentially tempered tails above a chosen scale in the propagator, and therefore finite moments at all orders. Concise analytical expressions for its variance, skewness, and kurtosis are derived as a function of time. We then illustrate the versatility of this model by applying it to simulations of the turbulent transport of fast ions in the TORPEX basic plasma device.

Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion

In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator–prey systems with anomalous diffusion.

Existence and Iterative Method for Some Riemann Fractional Nonlinear Boundary Value Problems

In this paper, we prove the existence and uniqueness of solution for some Riemann–Liouville fractional nonlinear boundary value problems. The positivity of the solution and the monotony of iterations are also considered. Some examples are presented to illustrate the main results. Our results generalize those obtained by Wei et al (Existence and iterative method for some fourth order nonlinear boundary value problems. Appl. Math. Lett. 2019, 87, 101–107.) to the fractional setting.

An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation

The q -homotopy analysis transform method ( q -HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with two of them having Jacobi elliptic functions. The numerical simulations have been conducted to verify that the proposed scheme is reliable and accurate. Moreover, the uniqueness and convergence analysis for the projected problem is also presented. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyze the complex problems arising in science and technology.

Modeling and analysis of fractional order DC-DC converter

Due to the non-idealities of commercial inductors, the demand for a better model that accurately describe their dynamic response is elevated. So, the fractional order models of Buck, Boost and Buck-Boost DC-DC converters are presented in this paper. The detailed analysis is made for the two most common modes of converter operation: Continuous Conduction Mode (CCM) and Discontinuous Conduction Mode (DCM). Closed form time domain expressions are derived for inductor currents, voltage gain, average current, conduction time and power efficiency where the effect of the fractional order inductor is found to be strongly present. For example, the peak inductor current at steady state increases with decreasing the inductor order. Advanced Design Systems (ADS) circuit simulations are used to verify the derived formulas, where the fractional order inductor is simulated using Valsa Constant Phase Element (CPE) approximation and Generalized Impedance Converter (GIC). Different simulation results are introduced with good matching to the theoretical formulas for the three DC-DC converter topologies under different fractional orders. A comprehensive comparison with the recently published literature is presented to show the advantages and disadvantages of each approach.

Fractional integral-like processing in retinal cones reduces noise and improves adaptation

In the human retina, rod and cone cells detect incoming light with a molecule called rhodopsin. After rhodopsin molecules are activated (by photon impact), these molecules activate the rest of the signalling process for a brief period of time until they are deactivated by a multistage process. First, active rhodopsin is phosphorylated multiple times. Following this, they are further inhibited by the binding of molecules called arrestins. Finally, they decay into opsins. The time required for each of these stages becomes progressively longer, and each stage further reduces the activity of rhodopsin. However, while this deactivation process itself is well researched, the roles of the above stages in signal (and image) processing are poorly understood. In this paper, we will show that the activity of rhodopsin molecules during the deactivation process can be described as the fractional integration of an incoming signal. Furthermore, we show how this affects an image; specifically, the effect of fractional integration in video and signal processing and how it reduces noise and the improves adaptability under different lighting conditions. Our experimental results provide a better understanding of vertebrate and human vision, and why the rods and cones of the retina differ from the light detectors in cameras.

Entropy Production Associated with Aggregation into Granules in a Subdiffusive Environment

We study the entropy production that is associated with the growing or shrinking of a small granule in, for instance, a colloidal suspension or in an aggregating polymer chain. A granule will fluctuate in size when the energy of binding is comparable to k B T , which is the "quantum" of Brownian energy. Especially for polymers, the conformational energy landscape is often rough and has been commonly modeled as being self-similar in its structure. The subdiffusion that emerges in such a high-dimensional, fractal environment leads to a Fokker-Planck Equation with a fractional time derivative. We set up such a so-called fractional Fokker-Planck Equation for the aggregation into granules. From that Fokker-Planck Equation, we derive an expression for the entropy production of a growing granule.

Dynamic Neutron Scattering by Biological Systems

Dynamic neutron scattering directly probes motions in biological systems on femtosecond to microsecond timescales. When combined with molecular dynamics simulation and normal mode analysis, detailed descriptions of the forms and frequencies of motions can be derived. We examine vibrations in proteins, the temperature dependence of protein motions, and concepts describing the rich variety of motions detectable using neutrons in biological systems at physiological temperatures. New techniques for deriving information on collective motions using coherent scattering are also reviewed.

Anomalous Behavior of Hyaluronan Crosslinking Due to the Presence of Excess Phospholipids in the Articular Cartilage System of Osteoarthritis

Lubrication of articular cartilage is a complex multiscale phenomenon in synovial joint organ systems. In these systems, synovial fluid properties result from synergistic interactions between a variety of molecular constituent. Two molecular classes in particular are of importance in understanding lubrication mechanisms: hyaluronic acid and phospholipids. The purpose of this study is to evaluate interactions between hyaluronic acid and phospholipids at various functionality levels during normal and pathological synovial fluid conditions. Molecular dynamic simulations of hyaluronic acid and phospholipids complexes were performed with the concentration of hyaluronic acid set at a constant value for two organizational forms, extended (normal) and coiled (pathologic). The results demonstrated that phospholipids affect the crosslinking mechanisms of hyaluronic acid significantly and the influence is higher during pathological conditions. During normal conditions, hyaluronic acid and phospholipid interactions seem to have no competing mechanism to that of the interaction between hyaluronic acid to hyaluronic acid. On the other hand, the structures formed under pathologic conditions were highly affected by phospholipid concentration.

Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results

With fractional differential equations (FDEs) rising in popularity and methods for solving them still being developed, approximations to solutions of fractional initial value problems (IVPs) have great applications in related fields. This paper proves an extension of Picard’s Iterative Existence and Uniqueness Theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous term satisfies the usual Lipschitz’s condition. As an application of our method, we have provided several numerical examples.

Modulation of hemoglobin dynamics by an allosteric effector

Hemoglobin (Hb) is an extensively studied paradigm of proteins that alter their function in response to allosteric effectors. Models of its action have been used as prototypes for structure‐function relationships in many proteins, and models for the molecular basis of its function have been deeply studied and extensively argued. Recent reports suggest that dynamics may play an important role in its function. Relatively little is known about the slow, correlated motions of hemoglobin subunits in various structural states because experimental and computational strategies for their characterization are challenging. Allosteric effectors such as inositol hexaphosphate (IHP) bind to both deoxy‐Hb and HbCO, albeit at different sites, leading to a lowered oxygen affinity. The manner in which these effectors impact oxygen binding is unclear and may involve changes in structure, dynamics or both. Here we use neutron spin echo measurements accompanied by wide‐angle X‐ray scattering to show that binding of IHP to HbCO results in an increase in the rate of coordinated motions of Hb subunits relative to one another with little if any change in large scale structure. This increase of large‐scale dynamics seems to be coupled with a decrease in the average magnitude of higher frequency modes of individual residues. These observations indicate that enhanced dynamic motions contribute to the functional changes induced by IHP and suggest that they may be responsible for the lowered oxygen affinity triggered by these effectors.

MESOSCOPIC MODELING OF STOCHASTIC REACTION-DIFFUSION KINETICS IN THE SUBDIFFUSIVE REGIME

Subdiffusion has been proposed as an explanation of various kinetic phenomena inside living cells. In order to fascilitate large-scale computational studies of subdiffusive chemical processes, we extend a recently suggested mesoscopic model of subdiffusion into an accurate and consistent reaction-subdiffusion computational framework. Two different possible models of chemical reaction are revealed and some basic dynamic properties are derived. In certain cases those mesoscopic models have a direct interpretation at the macroscopic level as fractional partial differential equations in a bounded time interval. Through analysis and numerical experiments we estimate the macroscopic effects of reactions under subdiffusive mixing. The models display properties observed also in experiments: for a short time interval the behavior of the diffusion and the reaction is ordinary, in an intermediate interval the behavior is anomalous, and at long times the behavior is ordinary again.

Spectral decomposition of nonlinear systems with memory

We present an alternative approach to the analysis of nonlinear systems with long-term memory that is based on the Koopman operator and a Lévy transformation in time. Memory effects are considered to be the result of interactions between a system and its surrounding environment. The analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale. On average, the time evolution of a mode follows a Mittag-Leffler function, and the system can be described using the fractional calculus. The general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation. When analyzing data from an ill-defined (black-box) system, the spectral decomposition in terms of Mittag-Leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods. Consequently, the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators, or operators with long-term memory in time, when model details are unknown.

A modified Sips distribution for use in adsorption isotherms and in fractal kinetic studies

A modified Sips' energy distribution is proposed to account for Langmuir–Freundlich adsorption isotherms and Mittag–Leffler fractal kinetics.

Effect of viscogens on the kinetic response of a photoperturbed allosteric protein

By covalently binding a photoswitchable linker across the binding groove of the PDZ2 domain, a small conformational change can be photo-initiated that mimics the allosteric transition of the protein. The response of its binding groove is investigated with the help of ultrafast pump-probe IR spectroscopy from picoseconds to tens of microseconds. The temperature dependence of that response is compatible with diffusive dynamics on a rugged energy landscape without any prominent energy barrier. Furthermore, the dependence of the kinetics on the concentration of certain viscogens, sucrose, and glycerol, has been investigated. A pronounced viscosity dependence is observed that can be best fit by a power law, i.e., a fractional viscosity dependence. The change of kinetics when comparing sucrose with glycerol as viscogen, however, provides strong evidence that direct interactions of the viscogen molecule with the protein do play a role as well. This conclusion is supported by accompanying molecular dynamics simulations.

Adaptation of Extremophilic Proteins with Temperature and Pressure: Evidence from Initiation Factor 6

In this work, we study dynamical properties of an extremophilic protein, Initiation Factor 6 (IF6), produced by the archeabacterium Methanocaldococcus jannascii, which thrives close to deep-sea hydrothermal vents where temperatures reach 80 °C and the pressure is up to 750 bar. Molecular dynamics simulations (MD) and quasi-elastic neutron scattering (QENS) measurements give new insights into the dynamical properties of this protein with respect to its eukaryotic and mesophilic homologue. Results obtained by MD are supported by QENS data and are interpreted within the framework of a fractional Brownian dynamics model for the characterization of protein relaxation dynamics. IF6 from M. jannaschii at high temperature and pressure shares similar flexibility with its eukaryotic homologue from S. cerevisieae under ambient conditions. This work shows for the first time, to our knowledge, that the very common pattern of corresponding states for thermophilic protein adaptation can be extended to thermo-barophilic proteins. A detailed analysis of dynamic properties and of local structural fluctuations reveals a complex pattern for "corresponding" structural flexibilities. In particular, in the case of IF6, the latter seems to be strongly related to the entropic contribution given by an additional, C-terminal, 20 amino-acid tail which is evolutionary conserved in all mesophilic IF6s.

On the reliability of NMR relaxation data analyses: a Markov Chain Monte Carlo approach

The analysis of NMR relaxation data is revisited along the lines of a Bayesian approach. Using a Markov Chain Monte Carlo strategy of data fitting, we investigate conditions under which relaxation data can be effectively interpreted in terms of internal dynamics. The limitations to the extraction of kinetic parameters that characterize internal dynamics are analyzed, and we show that extracting characteristic time scales shorter than a few tens of ps is very unlikely. However, using MCMC methods, reliable estimates of the marginal probability distributions and estimators (average, standard deviations, etc.) can still be obtained for subsets of the model parameters. Thus, unlike more conventional strategies of data analysis, the method avoids a model selection process. In addition, it indicates what information may be extracted from the data, but also what cannot.

Multiple positive solutions to nonlinear boundary value problems of a system for fractional differential equations

By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by −D0+ν1y1(t) = λ1a1(t)f(y1(t), y2(t)), − D0+ν2y2(t) = λ2a2(t)g(y1(t), y2(t)), where D0+ν is the standard Riemann-Liouville fractional derivative, ν1, ν2 ∈ (n − 1, n] for n > 3 and n ∈ N, subject to the boundary conditions y1(i)(0) = 0 = y2(i)(0), for 0 ≤ i ≤ n − 2, and [D0+αy1(t)]t=1 = 0 = [D0+αy2(t)]t=1, for 1 ≤ α ≤ n − 2, or y1(i)(0) = 0 = y2(i)(0), for 0 ≤ i ≤ n − 2, and [D0+αy1(t)]t=1 = ϕ1(y1), [D0+αy2(t)]t=1 = ϕ2(y2), for 1 ≤ α ≤ n − 2, ϕ1, ϕ2 ∈ C([0,1], R). Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.

Fractional dynamical model for the generation of ECG like signals from filtered coupled Van-der Pol oscillators

In this paper, an incommensurate fractional order (FO) model has been proposed to generate ECG like waveforms. Earlier investigation of ECG like waveform generation is based on two identical Van-der Pol (VdP) family of oscillators, which are coupled by time delays and gains. In this paper, we suitably modify the three state equations corresponding to the nonlinear cross-product of states, time delay coupling of the two oscillators and low-pass filtering, using the concept of fractional derivatives. Our results show that a wide variety of ECG like waveforms can be simulated from the proposed generalized models, characterizing heart conditions under different physiological conditions. Such generalization of the modelling of ECG waveforms may be useful to understand the physiological process behind ECG signal generation in normal and abnormal heart conditions. Along with the proposed FO models, an optimization based approach is also presented to estimate the VdP oscillator parameters for representing a realistic ECG like signal.

Group superballistic diffusion: bimodal velocity inducing coexistence of two states in a corrugated plane

We consider anomalous diffusion of a particle moving in a tilted periodic potential in the presence of Lévy noise and nonlinear friction. Using Monte Carlo simulations, we have found some interesting characteristics of diffusion in such a nonlinear system: when the noise intensity is weak and the external force is close to the critical value at which local minima of the potential just vanish, the nonmonotonic behavior of the effective diffusion index and the superballistic diffusion are observed. This is due to the bimodal nature of the velocity distribution, and thus the test particles exist in either a running state or a long-tailed behind state in the spatial coordinate; the latter is disintegrated into small pieces of the probability peaks. We provide a relation between the group diffusion coefficient and the phase diffusion coefficient. It is shown that the distance between the above two-state centers increasing with time plays the definitive role in the superballistic group diffusion.

Anomalous statistics of random relaxations in random environments

We comprehensively analyze the emergence of anomalous statistics in the context of the random relaxation (RARE) model [Eliazar and Metzler, J. Chem. Phys. 137, 234106 (2012)], a recently introduced versatile model of random relaxations in random environments. The RARE model considers excitations scattered randomly across a metric space around a reaction center. The excitations react randomly with the center, the reaction rates depending on the excitations' distances from this center. Relaxation occurs upon the first reaction between an excitation and the center. Addressing both the relaxation time and the relaxation range, we explore when these random variables display anomalous statistics, namely, heavy tails at zero and at infinity that manifest, respectively, exceptionally high occurrence probabilities of very small and very large outliers. A cohesive set of closed-form analytic results is established, determining precisely when such anomalous statistics emerge.

Calculating Position-Dependent Diffusivity in Biased Molecular Dynamics Simulations

Calculating transition rates and other kinetic quantities from molecular simulations requires knowledge not only of the free energy along the relevant coordinate but also the diffusivity as a function of that coordinate. A variety of methods are currently used to map the free-energy landscape in molecular simulations; however, simultaneous calculation of position-dependent diffusivity is complicated by biasing forces applied with many of these methods. Here, we describe a method to calculate position-dependent diffusivities in simulations including known time-dependent biasing forces, which relies on a previously proposed Bayesian inference scheme. We first apply the method to an explicitly diffusive model, and then to an equilibrium molecular dynamics simulation of liquid water including a position-dependent thermostat, comparing the results to those of an established method. Finally, we test the method on a system of liquid water, where oscillations of the free energy along the coordinate of interest preclude sufficient sampling in an equilibrium simulation. The adaptive biasing force method permits roughly uniform sampling along this coordinate, while the method presented here gives a consistent result for the position-dependent diffusivity, even in a short simulation where the adaptive biasing force is only partially converged.