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We investigate the H-theorem for a class of generalized kinetic equations with fractional time-derivative, hyperbolic term, and nonlinear diffusion. When the H-theorem is satisfied, we demonstrate that different entropic forms may emerge due to the equation's nonlinearity. We obtain the entropy production related to these entropies and show that its form remains invariant. Furthermore, we investigate some behaviors for these equations from both numerical and analytical perspectives, showing a large class of behaviors connected with anomalous diffusion and their effects on entropy.

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Key neuronal functions have been successfully replicated in various hardware systems. Noticeable examples are neuronal networks constructed from memristors, which emulate complex electrochemical biological dynamics such as the efficacy and plasticity of a neuron. Neurons are highly active cells, communicating with chemical and electrical stimuli, but also emit light. These so-called biophotons are suspected to be a complementary vehicle to transport information across the brain. Here, we show that a memristor also releases photons during its operation akin to the production of neuronal light. Critical attributes of biophotons, such as self-generation, stochasticity, spectral coverage, sparsity, and correlation with the neuron's electrical activity, are replicated by our solid-state approach. Importantly, our time-resolved analysis of the correlated current transport and photon activity shows that emission takes place within a nanometer-sized active area and relies on electrically induced single-to-few active electroluminescent centers excited with moderate voltage (<3 V). Our findings further extend the emulating capability of a memristor to encompass neuronal optical activity and allow to construct memristive atomic-scale devices capable of handling simultaneously electrons and photons as information carriers.

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The process of end-joining during nonhomologous repair of DNA double-strand breaks (DSBs) after radiation damage is considered. Experimental evidence has revealed that the dynamics of DSB ends exhibit subdiffusive motion rather than simple diffusion with rare directional movement. Traditional models often overlook the rare long-range directed motion. To address this limitation, we present a heterogeneous anomalous diffusion model consisting of subdiffusive fractional Brownian motion interchanged with short periods of long-range movement. Our model sheds light on the underlying mechanisms of heterogeneous diffusion in DSB repair and could be used to quantify the DSB dynamics on a time scale inaccessible to single particle tracking analysis. The model predicts that the long-range movement of DSB ends is responsible for the misrepair of DSBs in the form of dicentric chromosome lesions.

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The interplay of diffusion with phenomena like stochastic adsorption-desorption, absorption, and reaction-diffusion is essential for life and manifests in diverse natural contexts. Many factors must be considered, including geometry, dimensionality, and the interplay of diffusion across bulk and surfaces. To address this complexity, we investigate the diffusion process in heterogeneous media, focusing on non-Markovian diffusion. This process is limited by a surface interaction with the bulk, described by a specific boundary condition relevant to systems such as living cells and biomaterials. The surface can adsorb and desorb particles, and the adsorbed particles may undergo lateral diffusion before returning to the bulk. Different behaviors of the system are identified through analytical and numerical approaches.

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The crowded bacterial cytoplasm is comprised of biomolecules that span several orders of magnitude in size and electrical charge. This complexity has been proposed as the source of the rich spatial organization and apparent anomalous diffusion of intracellular components, although this has not been tested directly. Here, we use biplane microscopy to track the 3D motion of self-assembled bacterial Genetically Encoded Multimeric nanoparticles (bGEMs) with tunable size (20 to 50 nm) and charge (-2160 to +1800 e) in live Escherichia coli cells. To probe intermolecular details at spatial and temporal resolutions beyond experimental limits, we also developed a colloidal whole-cell model that explicitly represents the size and charge of cytoplasmic macromolecules and the porous structure of the bacterial nucleoid. Combining these techniques, we show that bGEMs spatially segregate by size, with small 20-nm particles enriched inside the nucleoid, and larger and/or positively charged particles excluded from this region. Localization is driven by entropic and electrostatic forces arising from cytoplasmic polydispersity, nucleoid structure, geometrical confinement, and interactions with other biomolecules including ribosomes and DNA. We observe that at the timescales of traditional single molecule tracking experiments, motion appears sub-diffusive for all particle sizes and charges. However, using computer simulations with higher temporal resolution, we find that the apparent anomalous exponents are governed by the region of the cell in which bGEMs are located. Molecular motion does not display anomalous diffusion on short time scales and the apparent sub-diffusion arises from geometrical confinement within the nucleoid and by the cell boundary.

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The development, storage, transport, and subcutaneous delivery of highly concentrated monoclonal antibody formulations pose significant challenges due to the high solution viscosity and low diffusion of the antibody molecules in crowded environments. These issues often stem from the self-associating behavior of the antibody molecules, potentially leading to aggregation. In this work, we used a dissipative particle dynamics-based coarse-grained model to investigate the diffusion behavior of IgG1 antibody molecules in aqueous solutions with 15 and 32 mM NaCl and antibody concentrations ranging from 10 to 400 mg/mL. We determined the coarse-grained interaction parameters by matching the calculated structure factor with the computational and experimental data from the literature. Our results indicate Fickian diffusion for antibody concentrations of 10 and 25 mg/mL and anomalous diffusion for concentrations exceeding 50 mg/mL. The anomalous diffusion was observed for ∼0.33 to 0.4 µs, followed by Fickian diffusion for all antibody concentrations. We observed a strong linear correlation between the diffusion behavior of the antibody molecules (diffusion coefficient D and anomalous diffusion exponent α) and the amount of aggregates present in the solution and between the amount of aggregates and the Coulomb interaction energy. The investigation of underlying mechanisms for anomalous diffusion revealed that in crowded environments at high antibody concentrations, the attractive interaction between electrostatically complementary regions of the antibody molecules could further bring the neighboring molecules closer to one another, ultimately resulting in aggregate formation. Further, the Coulomb attraction can continue to draw more molecules together, forming larger aggregates.

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Most nutrient uptake problems are modeled by the convection-diffusion equation (CDE) abiding by Fick's law. Because nutrients needed by plants exist in the soil solution as a form of ions and the soil is a typical fractal structure of heterogeneity, it makes the solute transport appear anomalous diffusion in soil. Taking anomalous diffusion as a transport process, we propose time and space fractional nutrient uptake models based on the classic Nye-Tinker-Barber model. There does not appear apparent sub-diffusion of nitrate in the time fractional model until four months and the time fractional models are appropriate for describing long-term dynamics and slow sorption reaction; the space fractional model can capture super-diffusion in short term and it is suitable for describing nonlocal phenomena and daily variations driven by transpiration and metabolism; the anomalous diffusion more apparently appears near the root surface in the modeling simulation.

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Hyper-ballistic diffusion is shown to arise from a simple model of microswimmers moving through a porous media while competing for resources. By using a mean-field model where swimmers interact through the local concentration, we show that a non-linear Fokker-Planck equation arises. The solution exhibits hyper-ballistic superdiffusive motion, with a diffusion exponent of four. A microscopic simulation strategy is proposed, which shows excellent agreement with theoretical analysis.

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The spatiotemporal evolution of earthquakes induced by fluid injections into the subsurface can be erratic owing to the complexity of the physical process. To effectively mitigate the associated hazard and to draft appropriate regulatory strategies, a detailed understanding of how induced seismicity may evolve is needed. In this work, we build on the well-established continuous-time random walk (CTRW) theory to develop a purely stochastic framework that can delineate the essential characteristics of this process. We use data from the 2003 and 2012 hydraulic stimulations in the Cooper Basin geothermal field that induced thousands of microearthquakes to test and demonstrate the applicability of the model. Induced seismicity in the Cooper Basin shows all the characteristics of subdiffusion, as indicated by the fractional order power-law growth of the mean square displacement with time and broad waiting-time distributions with algebraic tails. We further use an appropriate master equation and the time-fractional diffusion equation to map the spatiotemporal evolution of seismicity. The results show good agreement between the model and the data regarding the peak earthquake concentration close to the two injection wells and the stretched exponential relaxation of seismicity with distance, suggesting that the CTRW model can be efficiently incorporated into induced seismicity forecasting.

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Objective. Microstructure imaging based on diffusion magnetic resonance signal is an advanced imaging technique that enablesin vivomapping of the brain's microstructure. Superficial white matter (SWM) plays an important role in brain development, maturation, and aging, while fewer microstructure imaging methods address the SWM due to its complexity. Therefore, this study aims to develop a diffusion propagation model to investigate the microstructural characteristics of the SWM region.Approach. In this paper, we hypothesize that the effect of cell membrane permeability and the water exchange between soma and dendrites cannot be neglected for typical clinical diffusion times (20 ms

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Aquitards significantly affect groundwater flow in multi-aquifer systems through adjacent aquifer leakage. Despite this, studies focusing on their heterogeneity and the non-conventional diffusion patterns of their flow are still limited. In this study, a factional derivative approach was first extended to explore the time-dependent behavior of flow transport in the aquitard. Two analytical solutions were derived for specific discharges in independent aquitards under different boundary conditions. The findings revealed that aquitard flow exhibits obvious anomalous diffusion behaviors, characterized by slower decay and heavy-tailed specific discharge data. The fractional derivative model provided a more accurate representation of this behavior than traditional models, as evidenced by its superior agreement with experimental data. Moreover, a transient model for pumping tests in a leaky aquifer system was developed, incorporating the memory effect of anomalous flow and vertical heterogeneity in aquitards. Relevant semi-analytical solutions were derived to explore the impacts of memory factor ß and decay exponent of aquitard hydraulic conductivity (K) on the leakage aquifer system. Theoretical results demonstrated that stronger memory effect reduces drawdowns in the aquitard and confined aquifer during mid-to-late times. A larger dimensionless decay exponent (Ad) decreases aquitard drawdown and increases aquifer drawdown at late times. Sensitivity analysis showed aquitard drawdown experiences two peaks in sensitivity to ß and Ad at early- or mid-times, affected by memory effect and decay exponent of aquitard K, signifying maximal impact at these specific intervals. This study provides a practical model to effectively manage groundwater resources by accurately reflecting aquitard memory and heterogeneity effects.

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In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate.

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We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case.

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In this paper a relaxation model of signal attenuation in diffusion-weighted magnetic resonance imaging (dMRI) with varying diffusion coefficient in terms of fractal derivative is proposed, in which the diffusion coefficient is a power law of the effective diffusion time. The relaxation model provides measures of diffusion constant, fractal dimension of diffusive trajectory of water molecule and the time power-law behavior of the diffusion coefficient. The proposed model was used to describe the magnetic resonance attenuation signal of the bullfrog sciatic nerve, and the corresponding spectral entropy was calculated to detect the environmental complexity in bullfrog sciatic nerve for water molecular diffusion. The results showed that the fractal derivative relaxation model (the VDC model) can accurately depict the diffusion pattern of water molecules in complex heterogeneous biological media at large b values. The VDC model provides an alternative theoretical reference for biological tissue detection based on time-dependent diffusion of water molecules.

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The spatiotemporal configuration of genes with distal regulatory elements, and the impact of chromatin mobility on transcription, remain unclear. Loop extrusion is an attractive model for bringing genetic elements together, but how this functionally interacts with transcription is also largely unknown. We combine live tracking of genomic loci and nascent transcripts with molecular dynamics simulations to assess the spatiotemporal arrangement of the Sox2 gene and its enhancer, in response to a battery of perturbations. We find a close link between chromatin mobility and transcriptional status: active elements display more constrained mobility, consistent with confinement within specialized nuclear sites, and alterations in enhancer mobility distinguish poised from transcribing alleles. Strikingly, we find that whereas loop extrusion and transcription factor-mediated clustering contribute to promoter-enhancer proximity, they have antagonistic effects on chromatin dynamics. This provides an experimental framework for the underappreciated role of chromatin dynamics in genome regulation.

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Cell membranes phase separate into ordered Lo and disordered Ld domains depending on their compositions. This membrane compartmentalization is heterogeneous and regulates the localization of specific proteins related to cell signaling and trafficking. However, it is unclear how the heterogeneity of the membranes affects the diffusion and localization of proteins in Lo and Ld domains. Here, using Langevin dynamics simulations coupled with the phase-field (LDPF) method, we investigate several tens of milliseconds-scale diffusion and localization of proteins in heterogeneous biological membrane models showing phase separation into Lo and Ld domains. The diffusivity of proteins exhibits temporal fluctuations depending on the field composition. Increases in molecular concentrations and domain preference of the molecule induce subdiffusive behavior due to molecular collisions by crowding and confinement effects, respectively. Moreover, we quantitatively demonstrate that the protein partitioning into the Lo domain is determined by the difference in molecular diffusivity between domains, molecular preference of domain, and molecular concentration. These results pave the way for understanding how biological reactions caused by molecular partitioning may be controlled in heterogeneous media. Moreover, the methodology proposed here is applicable not only to biological membrane systems but also to the study of diffusion and localization phenomena of molecules in various heterogeneous systems.

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The evolution of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in humans has been monitored at an unprecedented level due to the public health crisis, yet the stochastic dynamics underlying such a process is dubious. Here, considering the number of acquired mutations as the displacement of the viral particle from the origin, we performed biostatistical analyses from numerous whole genome sequences on the basis of a time-dependent probabilistic mathematical model. We showed that a model with a constant variant-dependent evolution rate and nonlinear mutational variance with time (i.e., anomalous diffusion) explained the SARS-CoV-2 evolutionary motion in humans during the first 120 wk of the pandemic in the United Kingdom. In particular, we found subdiffusion patterns for the Primal, Alpha, and Omicron variants but a weak superdiffusion pattern for the Delta variant. Our findings indicate that non-Brownian evolutionary motions occur in nature, thereby providing insight for viral phylodynamics.

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Diffusion-weighted magnetic resonance imaging (dMRI) is a method of capturing the signal of water molecules diffusing in heterogeneous materials. Gaussian diffusion is interrupted when water mobility is hampered by obstructions in complex structures, and the dMRI signal decay does not match the single exponential decay in Brownian motion. In this study, a concise continuous time random-walk diffusion model is derived with less parameters than the continuous time random walk (CTRW) model and used to characterize the attenuation signal of brain tissue. The fitting results are compared with the CTRW model and the mono-exponential model reflecting the sub-diffusion and the long tail phenomenon of signal decay. Three sample experiments on rat brain and human brain are chosen to evaluate the validity in explaining the anomalous diffusion of water molecules in biological tissues, particularly in brain tissues in diverse directions, which also extends the applications of the concise continuous time random-walk diffusion model. Furthermore, we note that the concise continuous time random-walk diffusion model has practical advantages over the classical exponential model from the perspective of computational accuracy especially in the case of large b values, and has less parameters and is comparable to the CTRW model.

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In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior.

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The self-organization of the brain matrix of serotonergic axons (fibers) remains an unsolved problem in neuroscience. The regional densities of this matrix have major implications for neuroplasticity, tissue regeneration, and the understanding of mental disorders, but the trajectories of its fibers are strongly stochastic and require novel conceptual and analytical approaches. In a major extension to our previous studies, we used a supercomputing simulation to model around one thousand serotonergic fibers as paths of superdiffusive fractional Brownian motion (FBM), a continuous-time stochastic process. The fibers produced long walks in a complex, three-dimensional shape based on the mouse brain and reflected at the outer (pial) and inner (ventricular) boundaries. The resultant regional densities were compared to the actual fiber densities in the corresponding neuroanatomically-defined regions. The relative densities showed strong qualitative similarities in the forebrain and midbrain, demonstrating the predictive potential of stochastic modeling in this system. The current simulation does not respect tissue heterogeneities but can be further improved with novel models of multifractional FBM. The study demonstrates that serotonergic fiber densities can be strongly influenced by the geometry of the brain, with implications for brain development, plasticity, and evolution.