Analysis of reaction-diffusion systems with anomalous subdiffusion

Application of the galactomannan gel from Cassia grandis seeds for biomedical purposes: Study of the incorporation of collagenases and their release profile

The galactomannan-based gel from Cassia grandis seeds was used to incorporate Penicillium sp. UCP 1286 and commercial collagenases. Experiments were carried out according to a 23-full factorial design to identify the most significant parameters for the incorporation process. The pH of the incorporation solution (pHi), stirring time (t), and initial protein concentration in the crude extract (PCi) were selected as the three independent variables, and the efficiency of collagenase incorporation (E) and collagenolytic activity (CA) after 360 min as the responses. pHi and PCi showed positive statistically significant effects on E, while CA was positively influenced by pHi and t, but negatively by PCi. The fungi collagenase was released from the gel following a pseudo-Fickian behavior. Additionally, no <76 % of collagenase was efficiently incorporated into the gel retaining a high CA (32.5-69.8 U/mL). The obtained results for the commercial collagenase (E = 93.88 %, CA = 65.8 U/mL, and n = 0.10) demonstrated a pseudo-Fickian behavior similar to the fungi-collagenase. The results confirm the biotechnological potential of the gel as an efficient matrix for the incorporation of catalytic compounds; additionally, the incorporation of collagenases was achieved by retaining the proteases CA and releasing them in a controlled manner.

What determines sub-diffusive behavior in crowded protein solutions?

The aqueous environment inside cells is densely packed. A typical cell has a macromolecular concentration in the range 90-450 g/L, with 5%-40% of its volume being occupied by macromolecules, resulting in what is known as macromolecular crowding. The space available for the free diffusion of metabolites and other macromolecules is thus greatly reduced, leading to so-called excluded volume effects. The slow diffusion of macromolecules under crowded conditions has been explained using transient complex formation. However, sub-diffusion noted in earlier works is not well characterized, particularly the role played by transient complex formation and excluded volume effects. We have used Brownian dynamics simulations to characterize the diffusion of chymotrypsin inhibitor 2 in protein solutions of bovine serum albumin and lysozyme at concentrations ranging from 50 to 300 g/L. The predicted changes in diffusion coefficient as a function of crowder concentration are consistent with NMR experiments. The sub-diffusive behavior observed in the sub-microsecond timescale can be explained in terms of a so-called cage effect, arising from rattling motion in a local molecular cage as a consequence of excluded volume effects. By selectively manipulating the nature of interactions between protein molecules, we determined that excluded volume effects induce sub-diffusive dynamics at sub-microsecond timescales. These findings may help to explain the diffusion-mediated effects of protein crowding on cellular processes.

Mathematical Modelling of the Spatial Distribution of a COVID-19 Outbreak with Vaccination Using Diffusion Equation

The formulation of mathematical models using differential equations has become crucial in predicting the evolution of viral diseases in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China, which causes a severe and potentially fatal respiratory syndrome. Since then, it has been declared a pandemic by the World Health Organization and has spread around the globe. A reaction−diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process, in which different substances are transformed, and a diffusion process, which causes their distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic using the bias of reaction−diffusion equations. Both local and global asymptotic stability conditions for the equilibria were determined using a Lyapunov function, and the nature of the stability was determined using the Routh−Hurwitz criterion. Furthermore, we consider the conditions for the existence and uniqueness of the model solution and show the spatial distribution of the model compartments when the basic reproduction rate R0<1 and R0>1. Thereafter, we conducted a sensitivity analysis to determine the most sensitive parameters in the proposed model. We demonstrate the model’s effectiveness by performing numerical simulations and investigating the impact of vaccination, together with the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. Therefore, we offer to the public health policymakers a better understanding of COVID-19 management.

The Impact of Membrane Protein Diffusion on GPCR Signaling

Spatiotemporal signal shaping in G protein-coupled receptor (GPCR) signaling is now a well-established and accepted notion to explain how signaling specificity can be achieved by a superfamily sharing only a handful of downstream second messengers. Dozens of Gs-coupled GPCR signals ultimately converge on the production of cAMP, a ubiquitous second messenger. This idea is almost always framed in terms of local concentrations, the differences in which are maintained by means of spatial separation. However, given the dynamic nature of the reaction-diffusion processes at hand, the dynamics, in particular the local diffusional properties of the receptors and their cognate G proteins, are also important. By combining some first principle considerations, simulated data, and experimental data of the receptors diffusing on the membranes of living cells, we offer a short perspective on the modulatory role of local membrane diffusion in regulating GPCR-mediated cell signaling. Our analysis points to a diffusion-limited regime where the effective production rate of activated G protein scales linearly with the receptor–G protein complex’s relative diffusion rate and to an interesting role played by the membrane geometry in modulating the efficiency of coupling.

Line-FRAP, A Versatile Method to Measure Diffusion Rates In Vitro and In Vivo

The crowded cellular milieu affect molecular diffusion through hard (occluded space) and soft (weak, non-specific) interactions. Multiple methods have been developed to measure diffusion coefficients at physiological protein concentrations within cells, each with its limitations. Here, we show that Line-FRAP, combined with rigours data analysis, is able to determine diffusion coefficients in a variety of environments, from in vitro to in vivo. The use of Line mode greatly improves time resolution of FRAP data acquisition, from 20-100 Hz in the classical mode to 800 Hz in the line mode. This improves data analysis, as intensity and radius of the bleach at the first post-bleach frame is critical. We evaluated the method on different proteins labelled chemically or fused to YFP in a wide range of environments. The diffusion coefficients measured in HeLa and in E. coli were ~2.5-fold and 15-fold slower than in buffer, and were comparable to previously published data. Increasing the osmotic pressure on E. coli further decreases diffusion, to the point at which proteins virtually stop moving. The method presented here, which requires a confocal microscope equipped with dual scanners, can be applied to study a large range of molecules with different sizes, and provides robust results in a wide range of environments and protein concentrations for fast diffusing molecules.

G protein-coupled receptor-G protein interactions: a single-molecule perspective

G protein-coupled receptors (GPCRs) regulate many cellular and physiological processes, responding to a diverse range of extracellular stimuli including hormones, neurotransmitters, odorants, and light. Decades of biochemical and pharmacological studies have provided fundamental insights into the mechanisms of GPCR signaling. Thanks to recent advances in structural biology, we now possess an atomistic understanding of receptor activation and G protein coupling. However, how GPCRs and G proteins interact in living cells to confer signaling efficiency and specificity remains insufficiently understood. The development of advanced optical methods, including single-molecule microscopy, has provided the means to study receptors and G proteins in living cells with unprecedented spatio-temporal resolution. The results of these studies reveal an unexpected level of complexity, whereby GPCRs undergo transient interactions among themselves as well as with G proteins and structural elements of the plasma membrane to form short-lived signaling nanodomains that likely confer both rapidity and specificity to GPCR signaling. These findings may provide new strategies to pharmaceutically modulate GPCR function, which might eventually pave the way to innovative drugs for common diseases such as diabetes or heart failure.

Fractional cable equation for general geometry: A model of axons with swellings and anomalous diffusion

Different experimental studies have reported anomalous diffusion in brain tissues and notably this anomalous diffusion is expressed through fractional derivatives. Axons are important to understand neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease, and Parkinson's disease. Indeed, abnormal accumulation of proteins and organelles in axons is a hallmark of these diseases. The diffusion in the axons can become anomalous as a result of this abnormality. In this case the voltage propagation in axons is affected. Another hallmark of different neurodegenerative diseases is given by discrete swellings along the axon. In order to model the voltage propagation in axons with anomalous diffusion and swellings, in this paper we propose a fractional cable equation for a general geometry. This generalized equation depends on fractional parameters and geometric quantities such as the curvature and torsion of the cable. For a cable with a constant radius we show that the voltage decreases when the fractional effect increases. In cables with swellings we find that when the fractional effect or the swelling radius increases, the voltage decreases. Similar behavior is obtained when the number of swellings and the fractional effect increase. Moreover, we find that when the radius swelling (or the number of swellings) and the fractional effect increase at the same time, the voltage dramatically decreases.

Non-Isothermal Effectiveness Factor for Catalytic Particles with Non-Fickian Diffusion

In this note, the effects of the non-Fickian diffusion on the prediction of effectiveness factor in non-isothermal porous catalytic slab considering external transport resistances are studied. A Green’s function formulation is used to solve the fractional diffusion-reaction model assuming non-Fickian diffusion to describe the internal mass transport in the porous catalytic particle. Evaluation of the effectiveness factor considering linear and nonlinear reaction rates was developed under isothermal and non-isothermal conditions. In both cases, numerical simulations show the relation existent between the anomalous diffusion with the performance of the catalytic slabs.

Diffusion-limited encounter rate in a three-dimensional lattice of connected compartments studied by Brownian-dynamics simulations

We considered the rate at which a diffusing particle encounters a target in a three-dimensional lattice of compartments with semipermeable walls. This work expands a previous theory [Li et al., Phys. Rev. Lett. 113, 028303 (2014)] for the encounter rate in the dilute limit of targets to the general case of any density of targets. We also used Brownian dynamics simulations to evaluate the approximations in the analytical theory. We find that the largest errors in the analytical theory are on the order of 10%. This work therefore demonstrates an analytical theory capable of describing the encounter rates in compartmentalized environments for any level of confinement and any target density.

Calculated rates of diffusion-limited reactions in a three-dimensional network of connected compartments: application to porous catalysts and biological systems

We describe the diffusion limit for reaction rates in a three-dimensional system of connected compartments. This model exhibits the length-scale dependent diffusion that can be observed in many heterogeneous environments, such as porous catalysts and biological environments. We obtain a simple analytical expression for the diffusion limit applicable to any scale of the compartment confinement. This diffusion limit exceeds the classic Smoluchowski diffusion limit that was derived for homogeneous environments but is often applied to biological reactions in heterogeneous environments. We expect our new diffusion limit to provide a more appropriate upper bound on reaction rates in biological systems, porous structures, and other heterogeneous environments where obstacles create local confinement.

Computational modeling reveals optimal strategy for kinase transport by microtubules to nerve terminals

Intracellular transport of proteins by motors along cytoskeletal filaments is crucial to the proper functioning of many eukaryotic cells. Since most proteins are synthesized at the cell body, mechanisms are required to deliver them to the growing periphery. In this article, we use computational modeling to study the strategies of protein transport in the context of JNK (c-JUN NH2-terminal kinase) transport along microtubules to the terminals of neuronal cells. One such strategy for protein transport is for the proteins of the JNK signaling cascade to bind to scaffolds, and to have the whole protein-scaffold cargo transported by kinesin motors along microtubules. We show how this strategy outperforms protein transport by diffusion alone, using metrics such as signaling rate and signal amplification. We find that there exists a range of scaffold concentrations for which JNK transport is optimal. Increase in scaffold concentration increases signaling rate and signal amplification but an excess of scaffolds results in the dilution of reactants. Similarly, there exists a range of kinesin motor speeds for which JNK transport is optimal. Signaling rate and signal amplification increases with kinesin motor speed until the speed of motor translocation becomes faster than kinase/scaffold-motor binding. Finally, we suggest experiments that can be performed to validate whether, in physiological conditions, neuronal cells do indeed adopt such an optimal strategy. Understanding cytoskeletal-assisted protein transport is crucial since axonal and cell body accumulation of organelles and proteins is a histological feature in many human neurodegenerative diseases. In this paper, we have shown that axonal transport performance changes with altered transport component concentrations and transport speeds wherein these aspects can be modulated to improve axonal efficiency and prevent or slowdown axonal deterioration.

Single-molecule diffusion in a periodic potential at a solid-liquid interface

We used single-molecule tracking experiments to observe the motion of small hydrophobic fluorescent molecules at the interface between water and a solid surface that exhibited periodic chemical patterns. The dynamics were characterized by non-ergodic, continuous time random walk statistics. The step-size distributions displayed enhanced probability of steps to periodic distances, consistent with theoretical predictions for diffusion in an atomic/molecular scale periodic potential. Surprisingly, this general behavior was observed here for surfaces exhibiting characteristic length scales three orders of magnitude larger than atomic/molecular dimensions, and may provide a new way to understand and control solid-liquid interfacial diffusion for molecular targeting applications.

FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW

A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

Revisiting point FRAP to quantitatively characterize anomalous diffusion in live cells

Fluorescence recovery after photobleaching (FRAP) is widely used to interrogate diffusion and binding of proteins in live cells. Herein, we apply two-photon excited FRAP with a diffraction limited bleaching and observation volume to study anomalous diffusion of unconjugated green fluorescence protein (GFP) in vitro and in cells. Experiments performed on dilute solutions of GFP reveal that reversible fluorophore bleaching can be mistakenly interpreted as anomalous diffusion. We derive a reaction-diffusion FRAP model that includes reversible photobleaching, and demonstrate that it properly accounts for these photophysics. We then apply this model to investigate the diffusion of GFP in HeLa cells and polytene cells of Drosophila larval salivary glands. GFP exhibits anomalous diffusion in the cytoplasm of both cell types and in HeLa nuclei. Polytene nuclei contain optically resolvable chromosomes, permitting FRAP experiments that focus separately on chromosomal or interchrosomal regions. We find that GFP exhibits anomalous diffusion in chromosomal regions but diffuses normally in regions devoid of chromatin. This observation indicates that obstructed transport through chromatin and not crowding by macromolecules is a source of anomalous diffusion in polytene nuclei. This behavior is likely true in other cells, so it will be important to account for this type of transport physics and for reversible photobleaching to properly interpret future FRAP experiments on DNA-binding proteins.

Motion analysis of light-powered autonomous silver chloride nanomotors

Powered by UV light, nano/micrometer-sized silver chloride particles exhibit autonomous movement and form "schools" in aqueous solution, i.e. regions in which the number density of particles is significantly higher than the global average. In this paper, the silver chloride particles in such a system are classified by their proximity to other AgCl particles--be they isolated, coupled or schooled--and their motion paths are tracked and analyzed. By plotting time-averaged mean squared displacements of each particle over various time intervals from 0.1 s to 15.0 s, we discover different diffusive behaviors for the three classes of silver chloride particles.

Lateral dynamics of proteins with polybasic domain on anionic membranes: a dynamic Monte-Carlo study

Positively charged polybasic domains are essential for recruiting multiple signaling proteins, such as Ras GTPases and Src kinase, to the negatively charged cellular membranes. Much less, however, is known about the influence of electrostatic interactions on the lateral dynamics of these proteins. We developed a dynamic Monte-Carlo automaton that faithfully simulates lateral diffusion of the adsorbed positively charged oligopeptides as well as the dynamics of mono- (phosphatidylserine) and polyvalent (PIP(2)) anionic lipids within the bilayer. In agreement with earlier results, our simulations reveal lipid demixing that leads to the formation of a lipid shell associated with the peptide. The computed association times and average numbers of bound lipids demonstrate that tetravalent PIP(2) interacts with the peptide much more strongly than monovalent lipid. On the spatially homogeneous membrane, the lipid shell affects the behavior of the peptide only by weakly reducing its lateral mobility. However, spatially heterogeneous distributions of monovalent lipids are found to produce peptide drift, the velocity of which is determined by the total charge of the peptide-lipid complex. We hypothesize that this predicted phenomenon may affect the spatial distribution of proteins with polybasic domains in the context of cell-signaling events that alter the local density of monovalent anionic lipids.

Agent-based simulation of reactions in the crowded and structured intracellular environment: Influence of mobility and location of the reactants

Background

In this paper we apply a novel agent-based simulation method in order to model intracellular reactions in detail. The simulations are performed within a virtual cytoskeleton enriched with further crowding elements, which allows the analysis of molecular crowding effects on intracellular diffusion and reaction rates. The cytoskeleton network leads to a reduction in the mobility of molecules. Molecules can also unspecifically bind to membranes or the cytoskeleton affecting (i) the fraction of unbound molecules in the cytosol and (ii) furthermore reducing the mobility. Binding of molecules to intracellular structures or scaffolds can in turn lead to a microcompartmentalization of the cell. Especially the formation of enzyme complexes promoting metabolic channeling, e.g. in glycolysis, depends on the co-localization of the proteins.

Results

While the co-localization of enzymes leads to faster reaction rates, the reduced mobility decreases the collision rate of reactants, hence reducing the reaction rate, as expected. This effect is most prominent in diffusion limited reactions. Furthermore, anomalous diffusion can occur due to molecular crowding in the cell. In the context of diffusion controlled reactions, anomalous diffusion leads to fractal reaction kinetics. The simulation framework is used to quantify and separate the effects originating from molecular crowding or the reduced mobility of the reactants. We were able to define three factors which describe the effective reaction rate, namely f ^diff for the diffusion effect, f ^volume for the crowding, and f ^access for the reduced accessibility of the molecules.

Conclusions

Molecule distributions, reaction rate constants and structural parameters can be adjusted separately in the simulation allowing a comprehensive study of individual effects in the context of a realistic cell environment. As such, the present simulation can help to bridge the gap between in vivo and in vitro kinetics.

Transient anomalous subdiffusion: effects of specific and nonspecific probe binding with actin gels

When signaling molecules diffuse through the cytosol, they encounter a wide variety of obstacles that hinder their mobility in space and time. Some of those factors include, but are not limited to, interactions with mobile and immobile targets or obstacles. Besides finding a crowded environment inside the cell, macromolecules assemble into molecular complexes that drive specific biological functions adding additional complexity to their diffusion. Thus, simple models of diffusion often fail to explain mobility through the cell interior, and new approaches are needed. Here we used fluorescent correlation spectroscopy to measure diffusion of three molecules of similar size with different surface properties diffusing in actin gels. The fluorescent probes were (a) quantum dots, (b) yellow-green fluorescent spheres, and (c) the beta isoform of Ca(2+) calmodulin-dependent protein kinase II tagged with green fluorescent protein. We compared various models for fitting the autocorrelation function (ACF) including single component, two-component, and anomalous diffusion. The two-component and anomalous diffusion models were superior and were largely indistinguishable based on a goodness of fit criteria. To better resolve differences between these two models, we modified the ACF to observe temporal variations in diffusion. We found in both simulated and experimental data a transient anomalous subdiffusion between two freely diffusing regimes produced by binding interactions of the diffusive tracers with actin gels.