New insights into diffusion in 3D crowded media by Monte Carlo simulations: effect of size, mobility and spatial distribution of obstacles

Intracellular Macromolecular Crowding within Individual Stress Fibers Analyzed by Fluorescence Correlation Spectroscopy

Introduction

The diffusion of cell components such as proteins is crucial to the function of all living cells. The abundance of macromolecules in cells is likely to cause a state of macromolecular crowding, but its effects on the extent of diffusion remain poorly understood.

Methods

Here we investigate the diffusion rate in three distinct locations in mesenchymal cell types, namely the open cytoplasm, the stress fibers in the open cytoplasm, and those below the nucleus using three kinds of biologically inert green fluorescent proteins (GFPs), namely a monomer, dimer, and trimer GFP. Fluorescence correlation spectroscopy (FCS) was used to determine the diffusion coefficients.

Results

We show that diffusion tends to be lowered on average in stress fibers and is significantly lower in those located below the nucleus. Our data suggest that the diffusive properties of GFPs, and potentially other molecules as well, are hindered by macromolecular crowding. However, although the size dependence on protein diffusion was also studied for monomer, dimer, and trimer GFPs, there was no significant difference in the diffusion rates among the GFPs of these sizes. These results could be attributed to the lack of significant change in protein size among the selected GFP multimers.

Conclusion

The data presented here would provide a basis for better understanding of the complex protein diffusion in the nonuniform cytoplasm, shedding light on cellular responses to mechanical stress, their local mechanical properties, and reduced turnover in senescent cells.

Activity-induced diffusion recovery in crowded colloidal suspensions

We show that the forces generated by active enzyme molecules are strong enough to influence the dynamics of their surroundings under artificial crowded environments. We measured the behavior of polymer microparticles in a quasi-two-dimensional system under aqueous environment, at various area fraction values of particles. In the presence of enzymatic activity, not only was the diffusion of the suspended particles enhanced at shorter time-scales, but the system also showed a transition from subdiffusive to diffusive dynamics at longer time-scale limits. Similar observations were also recorded with enzyme-functionalized microparticles. Brownian dynamics simulations have been performed to support the experimental observations.

Unusual Aspects of Charge Regulation in Flexible Weak Polyelectrolytes

This article reviews the state of the art of the studies on charge regulation (CR) effects in flexible weak polyelectrolytes (FWPE). The characteristic of FWPE is the strong coupling of ionization and conformational degrees of freedom. After introducing the necessary fundamental concepts, some unconventional aspects of the the physical chemistry of FWPE are discussed. These aspects are: (i) the extension of statistical mechanics techniques to include ionization equilibria and, in particular, the use of the recently proposed Site Binding-Rotational Isomeric State (SBRIS) model, which allows the calculation of ionization and conformational properties on the same foot; (ii) the recent progresses in the inclusion of proton equilibria in computer simulations; (iii) the possibility of mechanically induced CR in the stretching of FWPE; (iv) the non-trivial adsorption of FWPE on ionized surfaces with the same charge sign as the PE (the so-called "wrong side" of the isoelectric point); (v) the influence of macromolecular crowding on CR.

Cellular aggregation dictates universal spreading behaviour of a whole-blood drop on a paper strip

HYPOTHESIS

The complex spreading dynamics of blood on paper matrix is likely to be quantitatively altered with variations in the fractional occupancy of red blood cells in the whole blood (haematocrit). Here, we presented an apparently surprising observation that a finite volume blood drop undergoes a universal time-dependent spreading on a filter paper strip that is virtually invariant with its hematocrit level within physiologically healthy regime, though distinctively distinguishable from the spreading laws of blood plasma and water.

EXPERIMENTS

Our hypothesis was ascertained by performing controlled wicking experiments on filter papers of different grades. Spreading of human blood samples of different haematocrit levels ranging between 15% and 51% and the plasma separated from therein were traced by combined high-speed imaging and microscopy. These experiments were complemented with a semi-analytical theory to decipher the key physics of interest.

RESULTS

Our results unveiled the exclusive influence of the obstructing cellular aggregates in the randomly distributed hierarchically structured porous pathways and deciphered the role of the networked structures of the various plasma proteins that induced hindered diffusion. The resulting universal signatures of spontaneous dynamic spreading, delving centrally on the fractional reduction in the interlaced porous passages, provide novel design basis for paper-microfluidic kits in medical diagnostics and beyond.

Molecular Dynamics Simulations of Rhodamine B Zwitterion Diffusion in Polyelectrolyte Solutions

Polyelectrolytes continue to find wide interest and application in science and engineering, including areas such as water purification, drug delivery, and multilayer thin films. We have been interested in the dynamics of small molecules in a variety of polyelectrolyte (PE) environments; in this paper, we report simulations and analysis of the small dye molecule rhodamine B (RB) in several very simple polyelectrolyte solutions. Translational diffusion of the RB zwitterion has been measured in fully atomistic, 2 μs long molecular dynamics simulations in four different polyelectrolyte solutions. Two solutions contain the common polyanion sodium poly(styrene sulfonate) (PSS), one with a 30-mer chain and the other with 10 trimers. The other two solutions contain the common polycation poly(allyldimethylammonium) chloride (PDDA), one with two 15-mers and the other with 10 trimers. RB diffusion was also simulated in several polymer-free solutions to verify its known experimental value for the translational diffusion coefficient, D_RB, of 4.7 × 10^-6 cm²/s at 300 K. RB diffusion was slowed in all four simulated PE solutions, but to varying degrees. D_RB values of 3.07 × 10^-6 and 3.22 × 10^-6 cm²/s were found in PSS 30-mer and PSS trimer solutions, respectively, whereas PDDA 15-mer and trimer solutions yielded values of 2.19 × 10^-6 and 3.34 × 10^-6 cm²/s. Significant associations between RB and the PEs were analyzed and interpreted via a two-state diffusion model (bound and free diffusion) that describes the data well. Crowder size effects and anomalous diffusion were also analyzed. Finally, RB translation along the polyelectrolytes during association was characterized.

Dynamics of Opposing Polymer Brushes: A Computer Simulation Study

Opposing polymer brush systems were synthesized and investigated by molecular modeling. Chains were restricted to a face-centered cubic lattice with the excluded volume interactions only. The system was confined between two parallel impenetrable walls, with the same number of chains grafted to each surface. The dynamic properties of such systems were studied by Monte Carlo simulations based on the dynamic lattice liquid model and using a highly efficient parallel machine ARUZ, which enabled the study of large systems and long timescales. The influence of the surface density and mean polymer length on the system dynamic was discussed. The self-diffusion coefficient of the solvent depended strongly on the degree of polymerization and on the polymer concentration. It was also shown that it is possible to capture changes in solvent mobility that can be attributed to the regions of different polymer densities.

MonteCarbo: A software to generate and dock multifunctionalized ring molecules

MonteCarbo is an open-source software to construct simple 5-, 6-, and 7-membered ring multifunctionalized monosaccharides and nucleobases and dock them into the active site of carbohydrate-active enzymes. The core bash script executes simple orders to generate the Z-matrix of the neutral molecule of interest. After that, a Fortran90 code based on a pseudo-random number generator (Monte Carlo method) is executed to assign dihedral angles to the different rotamers present in the structure (ring and rotating functional groups). The program also has a generalized internal coordinates (GIC) implementation of the Cremer and Pople puckering coordinates ring. Once the structures are generated and optimized, a second code is ready to execute in serial the docking of multiple conformers in the active site of a wide family of enzymes.

Spatio-temporal modeling of the crowding conditions and metabolic variability in microbial communities

The metabolic capabilities of the species and the local environment shape the microbial interactions in a community either through the exchange of metabolic products or the competition for the resources. Cells are often arranged in close proximity to each other, creating a crowded environment that unevenly reduce the diffusion of nutrients. Herein, we investigated how the crowding conditions and metabolic variability among cells shape the dynamics of microbial communities. For this, we developed CROMICS, a spatio-temporal framework that combines techniques such as individual-based modeling, scaled particle theory, and thermodynamic flux analysis to explicitly incorporate the cell metabolism and the impact of the presence of macromolecular components on the nutrients diffusion. This framework was used to study two archetypical microbial communities (i) Escherichia coli and Salmonella enterica that cooperate with each other by exchanging metabolites, and (ii) two E. coli with different production level of extracellular polymeric substances (EPS) that compete for the same nutrients. In the mutualistic community, our results demonstrate that crowding enhanced the fitness of cooperative mutants by reducing the leakage of metabolites from the region where they are produced, avoiding the resource competition with non-cooperative cells. Moreover, we also show that E. coli EPS-secreting mutants won the competition against the non-secreting cells by creating less dense structures (i.e. increasing the spacing among the cells) that allow mutants to expand and reach regions closer to the nutrient supply point. A modest enhancement of the relative fitness of EPS-secreting cells over the non-secreting ones were found when the crowding effect was taken into account in the simulations. The emergence of cell-cell interactions and the intracellular conflicts arising from the trade-off between growth and the secretion of metabolites or EPS could provide a local competitive advantage to one species, either by supplying more cross-feeding metabolites or by creating a less dense neighborhood.

Microbial communities play a key role in biogeochemical cycles, bioremediation, and human health. In crowded microbial systems such as biofilms and cellular aggregates, the close proximity between individual cells reduces the free space for the nutrients diffusion. To model the heterogeneous nature of these microbial systems, we developed CROMICS, a framework that integrates the information about the metabolic capabilities of each individual cell as well as the size and location of cells and macromolecules in the medium. The interactions among the individuals arise naturally through competition for or the exchange of metabolites. We show how the presence of mutants and a reduced diffusion in crowded environments can perturb the local availability of nutrients and therefore modify the dynamics of a microbial community. The discovered mechanisms underlying the microbial interactions in crowded systems together with the developed framework represent a valuable starting point for future studies of the interplay of human microbiome and host metabolism, the pathogen invasion, and the evaluation of antibiotic effectiveness.

Tracer diffusion in crowded solutions of sticky polymers

Macromolecular diffusion in strongly confined geometries and crowded environments is still to a large extent an open subject in soft matter physics and biology. In this paper, we employ large-scale Langevin dynamics simulations to investigate how the diffusion of a tracer is influenced by the combined action of excluded-volume and weak attractive crowder-tracer interactions. We consider two species of tracers, standard hard-core particles described by the Weeks-Chandler-Andersen (WCA) repulsive potential and core-softened (CS) particles, which model, e.g., globular proteins, charged colloids, and nanoparticles covered by polymeric brushes. These systems are characterized by the presence of two length scales in the interaction and can show waterlike anomalies in their diffusion, stemming from the inherent competition between different length scales. Here we report a comprehensive study of both diffusion and structure of these two tracer species in an environment crowded by quenched configurations of polymers at increasing density. We analyze in detail how the tracer-polymer affinity and the system density affect transport as compared to the emergence of specific static spatial correlations. In particular, we find that, while hardly any differences emerge in the diffusion properties of WCA and CS particles, the propensity to develop structural order for large crowding is strongly frustrated for CS particles. Surprisingly, for large enough affinity for the crowding matrix, the diffusion coefficient of WCA tracers display a nonmonotonic trend as their density is increased when compared to the zero affinity scenario. This waterlike anomaly turns out to be even larger than what observed for CS particle and appears to be rooted in a similar competition between excluded-volume and affinity effects.

Molecular transport in systems containing binding obstacles

We studied the movement of particles in crowded environments by means of extensive Monte Carlo simulations. The dynamic lattice liquid model was employed for this purpose. It is based on the cooperative movement concept and allows the study of systems at high densities. The cooperative model of molecular transport is assumed: the motion of all moving particles is highly correlated. The model is supposed to mimic lateral motion in a membrane and therefore the system is two-dimensional with moving objects and traps placed on a triangular lattice. In our study the interaction (binding) of traps with moving particles was assumed. The conditions in which subdiffusive motion appeared in the system were analysed. The influence of the strength of binding on the dynamic percolation threshold was also shown. The differences in the dynamics compared to systems with impenetrable obstacles and with systems without correlation in motion were presented and discussed. It was shown that in the case of correlated motion the influence of deep traps is similar to that of impenetrable obstacles. If the traps are shallow a recovery to normal diffusion was observed for longer time periods.

Consequences of Heterogeneous Crowding on an Enzymatic Reaction: A Residence Time Monte Carlo Approach

Translational diffusion of a free substrate in crowded metabolically active spaces such as cell cytoplasm or mitochondrial matrix is punctuated by collisions and nonspecific interactions with soluble/immobile macromolecules/macrostructures in a variety of shapes/sizes. It is not understood how such disruptions alter enzyme reaction kinetics in such spaces. A novel Monte Carlo (MC) technique, "residence time MC", has been developed to study the kinetics of a simple enzyme-substrate reaction in a crowded milieu using a single immobile enzyme in the midst of diffusing substrates and products. The reaction time lost while the substrate nonspecifically interacts or is transiently trapped with ambient macromolecules is quantified by introducing the residence time "tau". Tau scales with the size of crowding macromolecules but makes the knowledge of their shape redundant. The residence time thus presents a convenient parameter to realistically mimic the sticky surroundings encountered by a diffusing substrate in heterogeneously crowded physiological spaces. Results reveal that for identical substrate concentration and excluded volume, increase in tau significantly diminished enzymatic product yield and reaction rate, slowed down substrate/product diffusion, and prolonged their relaxation times. A smooth transition from the anomalous subdiffusive motion to normal diffusion at long time limits was observed irrespective of the value of tau. The predictions from the model are shown to be in qualitative agreement with in vitro experimental data revealing the rate of alkaline phosphatase-catalyzed hydrolysis of p-nitrophenyl phosphate in the midst of 40/500/2000 kDa dextrans. Our findings from the residence time MC model also attempt to rationalize previously unexplained experimental observations in crowded enzyme kinetics literature. Furthermore, major insights to emerge from this study are the reasons why free diffusion of the substrate in crowded physiological spaces is detrimental to enzyme function. It is argued that organized enzyme clusters such as "metabolon" may perhaps exist to regulate the substrate translocation in such sticky physiological spaces to maintain optimal enzyme function. In summary, this work provides key insights explaining why absence of substrate channeling can dramatically slow down enzyme reaction rate in crowded metabolically active spaces.

Macromolecular diffusion in crowded media beyond the hard-sphere model

The effect of macromolecular crowding on diffusion beyond the hard-core sphere model is studied. A new coarse-grained model is presented, the Chain Entanglement Softened Potential (CESP) model, which takes into account the macromolecular flexibility and chain entanglement. The CESP model uses a shoulder-shaped interaction potential that is implemented in the Brownian Dynamics (BD) computations. The interaction potential contains only one parameter associated with the chain entanglement energetic cost (Ur). The hydrodynamic interactions are included in the BD computations via Tokuyama mean-field equations. The model is used to analyze the diffusion of a streptavidin protein among different sized dextran obstacles. For this system, Ur is obtained by fitting the streptavidin experimental long-time diffusion coefficient Dlongversus the macromolecular concentration for D50 (indicating their molecular weight in kg mol-1) dextran obstacles. The obtained Dlong values show better quantitative agreement with experiments than those obtained with hard-core spheres. Moreover, once parametrized, the CESP model is also able to quantitatively predict Dlong and the anomalous exponent (α) for streptavidin diffusion among D10, D400 and D700 dextran obstacles. Dlong, the short-time diffusion coefficient (Dshort) and α are obtained from the BD simulations by using a new empirical expression, able to describe the full temporal evolution of the diffusion coefficient.

Macromolecular crowding directs the motion of small molecules inside cells

It is now well established that cell interiors are significantly crowded by macromolecules, which impede diffusion and enhance binding rates. However, it is not fully appreciated that levels of crowding are heterogeneous, and can vary substantially between subcellular regions. In this article, starting from a microscopic model, we derive coupled nonlinear partial differential equations for the concentrations of two populations of large and small spherical particles with steric volume exclusion. By performing an expansion in the ratio of the particle sizes, we find that the diffusion of a small particle in the presence of large particles obeys an advection–diffusion equation, with a reduced diffusion coefficient and a velocity directed towards less crowded regions. The interplay between advection and diffusion leads to behaviour that differs significantly from Brownian diffusion. We show that biologically plausible distributions of macromolecules can lead to highly non-Gaussian probability densities for the small particle position, including asymmetrical and multimodal densities. We confirm all our results using hard-sphere Brownian dynamics simulations.

Unravelling the complexity of signalling networks in cancer: A review of the increasing role for computational modelling

Cancer induction is a highly complex process involving hundreds of different inducers but whose eventual outcome is the same. Clearly, it is essential to understand how signalling pathways and networks generated by these inducers interact to regulate cell behaviour and create the cancer phenotype. While enormous strides have been made in identifying key networking profiles, the amount of data generated far exceeds our ability to understand how it all "fits together". The number of potential interactions is astronomically large and requires novel approaches and extreme computation methods to dissect them out. However, such methodologies have high intrinsic mathematical and conceptual content which is difficult to follow. This review explains how computation modelling is progressively finding solutions and also revealing unexpected and unpredictable nano-scale molecular behaviours extremely relevant to how signalling and networking are coherently integrated. It is divided into linked sections illustrated by numerous figures from the literature describing different approaches and offering visual portrayals of networking and major conceptual advances in the field. First, the problem of signalling complexity and data collection is illustrated for only a small selection of known oncogenes. Next, new concepts from biophysics, molecular behaviours, kinetics, organisation at the nano level and predictive models are presented. These areas include: visual representations of networking, Energy Landscapes and energy transfer/dissemination (entropy); diffusion, percolation; molecular crowding; protein allostery; quinary structure and fractal distributions; energy management, metabolism and re-examination of the Warburg effect. The importance of unravelling complex network interactions is then illustrated for some widely-used drugs in cancer therapy whose interactions are very extensive. Finally, use of computational modelling to develop micro- and nano- functional models ("bottom-up" research) is highlighted. The review concludes that computational modelling is an essential part of cancer research and is vital to understanding network formation and molecular behaviours that are associated with it. Its role is increasingly essential because it is unravelling the huge complexity of cancer induction otherwise unattainable by any other approach.

Gradient-driven diffusion and pattern formation in crowded mixtures

Gradient-driven diffusion in crowded, multicomponent mixtures is a topic of high interest because of its role in biological processes such as transport in cell membranes. In partially phase-separated solutions, gradient-driven diffusion affects microstructure, which in turn affects diffusivity; a key question is how this complex coupling controls both transport and pattern formation. To examine these mechanisms, we study a two-dimensional multicomponent lattice gas model, where "tracer" molecules diffuse between a source and a sink separated by a solution of sticky "crowder" molecules that cluster to form dynamically evolving obstacles. In the high-temperature limit, crowders and tracers are miscible, and transport may be predicted analytically. At intermediate temperatures, crowders phase separate into clusters that drift toward the tracer sink. As a result, steady-state tracer diffusivity depends nonmonotonically on both temperature and crowder density, and we observe a variety of complex microstructures. In the low-temperature limit, crowders rapidly aggregate to form obstacles that are kinetically arrested; if crowder density is near the percolation threshold, resulting tracer diffusivity shows scaling behavior with the same scaling exponent as the random resistor network model. Though highly idealized, this simple model reveals fundamental mechanisms governing coupled gradient-driven diffusion, phase separation, and microstructural evolution in crowded mixtures.

Comparison of different models of motion in a crowded environment: a Monte Carlo study

In this paper we investigate the motion of molecules in crowded environments for two dramatically different types of molecular transport. The first type is realized by the dynamic lattice liquid model, which is based on a cooperative movement concept and thus, the motion of molecules is highly correlated. The second one corresponds to a so-called motion of a single agent where the motion of molecules is considered as a random walk without any correlation with other moving elements. The crowded environments are modeled as a two-dimensional triangular lattice with fixed impenetrable obstacles. Our simulation results indicate that the type of transport has an impact on the dynamics of the system, the percolation threshold, critical exponents, and on molecules' trajectories.

An analytical method for disentangling the roles of adhesion and crowding for random walk models on a crowded lattice

Migration of cells and molecules in vivo is affected by interactions with obstacles. These interactions can include crowding effects, as well as adhesion/repulsion between the motile cell/molecule and the obstacles. Here we present an analytical framework that can be used to separately quantify the roles of crowding and adhesion/repulsion using a lattice-based random walk model. Our method leads to an exact calculation of the long time Fickian diffusivity, and avoids the need for computationally expensive stochastic simulations.

Simulation of Molecular Transport in Systems Containing Mobile Obstacles

In this paper, we investigate the movement of molecules in crowded environments with obstacles undergoing Brownian motion by means of extensive Monte Carlo simulations. Our investigations were performed using the dynamic lattice liquid model, which was based on the cooperative movement concept and allowed to mimic systems at high densities where the motion of all elements (obstacles as well as moving particles) were highly correlated. The crowded environments are modeled on a two-dimensional triangular lattice containing obstacles (particles whose mobility was significantly reduced) moving by a Brownian motion. The subdiffusive motion of both elements in the system was analyzed. It was shown that the percolation transition does not exist in such systems in spite of the cooperative character of the particles' motion. The reduction of the obstacle mobility leads to the longer caging of liquid particles by mobile obstacles.

Connecting the dots: the effects of macromolecular crowding on cell physiology

The physicochemical properties of cellular environments with a high macromolecular content have been systematically characterized to explain differences observed in the diffusion coefficients, kinetics parameters, and thermodynamic properties of proteins inside and outside of cells. However, much less attention has been given to the effects of macromolecular crowding on cell physiology. Here, we review recent findings that shed some light on the role of crowding in various cellular processes, such as reduction of biochemical activities, structural reorganization of the cytoplasm, cytoplasm fluidity, and cellular dormancy. We conclude by presenting some unresolved problems that require the attention of biophysicists, biochemists, and cell physiologists. Although it is still underappreciated, macromolecular crowding plays a critical role in life as we know it.

Homogenization Theory for the Prediction of Obstructed Solute Diffusivity in Macromolecular Solutions

The study of diffusion in macromolecular solutions is important in many biomedical applications such as separations, drug delivery, and cell encapsulation, and key for many biological processes such as protein assembly and interstitial transport. Not surprisingly, multiple models for the a-priori prediction of diffusion in macromolecular environments have been proposed. However, most models include parameters that are not readily measurable, are specific to the polymer-solute-solvent system, or are fitted and do not have a physical meaning. Here, for the first time, we develop a homogenization theory framework for the prediction of effective solute diffusivity in macromolecular environments based on physical parameters that are easily measurable and not specific to the macromolecule-solute-solvent system. Homogenization theory is useful for situations where knowledge of fine-scale parameters is used to predict bulk system behavior. As a first approximation, we focus on a model where the solute is subjected to obstructed diffusion via stationary spherical obstacles. We find that the homogenization theory results agree well with computationally more expensive Monte Carlo simulations. Moreover, the homogenization theory agrees with effective diffusivities of a solute in dilute and semi-dilute polymer solutions measured using fluorescence correlation spectroscopy. Lastly, we provide a mathematical formula for the effective diffusivity in terms of a non-dimensional and easily measurable geometric system parameter.

Calculating the Fickian diffusivity for a lattice-based random walk with agents and obstacles of different shapes and sizes

Random walk models are often used to interpret experimental observations of the motion of biological cells and molecules. A key aim in applying a random walk model to mimic an in vitro experiment is to estimate the Fickian diffusivity (or Fickian diffusion coefficient), D. However, many in vivo experiments are complicated by the fact that the motion of cells and molecules is hindered by the presence of obstacles. Crowded transport processes have been modeled using repeated stochastic simulations in which a motile agent undergoes a random walk on a lattice that is populated by immobile obstacles. Early studies considered the most straightforward case in which the motile agent and the obstacles are the same size. More recent studies considered stochastic random walk simulations describing the motion of an agent through an environment populated by obstacles of different shapes and sizes. Here, we build on previous simulation studies by analyzing a general class of lattice-based random walk models with agents and obstacles of various shapes and sizes. Our analysis provides exact calculations of the Fickian diffusivity, allowing us to draw conclusions about the role of the size, shape and density of the obstacles, as well as examining the role of the size and shape of the motile agent. Since our analysis is exact, we calculate D directly without the need for random walk simulations. In summary, we find that the shape, size and density of obstacles has a major influence on the exact Fickian diffusivity. Furthermore, our results indicate that the difference in diffusivity for symmetric and asymmetric obstacles is significant.

A Lattice-Boltzmann scheme for the simulation of diffusion in intracellular crowded systems

Background

The intracellular environment is a complex and crowded medium where the diffusion of proteins, metabolites and other molecules can be decreased. One of the most popular methodologies for the simulation of diffusion in crowding systems is the Monte Carlo algorithm (MC) which tracks the movement of each particle. This can, however, be computationally expensive for a system comprising a large number of molecules. On the other hand, the Lattice Boltzmann Method (LBM) tracks the movement of collections of molecules, which represents significant savings in computational time. Nevertheless in the classical manifestation of such scheme the crowding conditions are neglected.

Methods

In this paper we use Scaled Particle Theory (SPT) to approximate the probability to find free space for the displacement of hard-disk molecules and in this way to incorporate the crowding effect to the LBM. This new methodology which couples SPT and LBM is validated using a kinetic Monte Carlo (kMC) algorithm, which is used here as our "computational experiment".

Results

The results indicate that LBM over-predicts the diffusion in 2D crowded systems, while the proposed coupled SPT-LBM predicts the same behaviour as the kinetic Monte Carlo (kMC) algorithm but with a significantly reduced computational effort. Despite the fact that small deviations between the two methods were observed, in part due to the mesoscopic and microscopic nature of each method, respectively, the agreement was satisfactory both from a qualitative and a quantitative point of view.

Conclusions

A crowding-adaptation to LBM has been developed using SPT, allowing fast simulations of diffusion-systems of different size hard-disk molecules in two-dimensional space. This methodology takes into account crowding conditions; not only the space fraction occupied by the crowder molecules but also the influence of the size of the crowder which can affect the displacement of molecules across the lattice system.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-015-0769-8) contains supplementary material, which is available to authorized users.

An analytical correlated random walk model and its application to understand subdiffusion in crowded environment

Subdiffusion in crowded environment such as movement of macromolecule in a living cell has often been observed experimentally. The primary reason for subdiffusion is volume exclusion by the crowder molecules. However, other effects such as hydrodynamic interaction may also play an important role. Although there are a large number of computer simulation studies on understanding molecular crowding, there is a lack of theoretical models that can be connected to both experiment and simulation. In the current work, we have formulated a one-dimensional correlated random walk model by connecting this to the motion in a crowded environment. We have found the exact solution of the probability distribution function of the model by solving it analytically. The parameters of our model can be obtained either from simulation or experiment. It has been shown that this analytical model captures some of the general features of diffusion in crowded environment as given in the previous literature and its prediction for transient subdiffusion closely matches the observations of a previous study of computer simulation of Escherichia coli cytoplasm. It is likely that this model will open up more development of theoretical models in this area.

Nonmonotonic diffusion of particles among larger attractive crowding spheres

We study the diffusive motion of particles among fixed spherical crowders. The diffusers interact with the crowders through a combination of a hard-core repulsion and a short-range attraction. The long-time effective diffusion coefficient of the diffusers is found to depend nonmonotonically on the strength of their attraction to the crowders. That is, for a given concentration of crowders, a weak attraction to the crowders enhances diffusion. We show that this counterintuitive fact can be understood in terms of the mesoscopic excess chemical potential landscape experienced by the diffuser. The roughness of this excess chemical potential landscape quantitatively captures the nonmonotonic dependence of the diffusion rate on the strength of crowder-diffuser attraction; thus, it is a purely static predictor of dynamic behavior. The mesoscopic view given here provides a unified explanation for enhanced diffusion effects that have been found in various systems of technological and biological interest.

Macromolecular crowding effect upon in vitro enzyme kinetics: mixed activation-diffusion control of the oxidation of NADH by pyruvate catalyzed by lactate dehydrogenase

Enzyme kinetics studies have been usually designed as dilute solution experiments, which differ substantially from in vivo conditions. However, cell cytosol is crowded with a high concentration of molecules having different shapes and sizes. The consequences of such crowding in enzymatic reactions remain unclear. The aim of the present study is to understand the effect of macromolecular crowding produced by dextran of different sizes and at diverse concentrations in the well-known reaction of oxidation of NADH by pyruvate catalyzed by L-lactate dehydrogenase (LDH). Our results indicate that the reaction rate is determined by both the occupied volume and the relative size of dextran obstacles with respect to the enzyme present in the reaction. Moreover, we analyzed the influence of macromolecular crowding on the Michaelis-Menten constants, vmax and Km. The obtained results show that only high concentrations and large sizes of dextran reduce both constants suggesting a mixed activation-diffusion control of this enzymatic reaction due to the dextran crowding action. From our knowledge, this is the first experimental study that depicts mixed activation-diffusion control in an enzymatic reaction due to the effect of crowding.

Particle laden fluid interfaces: dynamics and interfacial rheology

We review the dynamics of particle laden interfaces, both particle monolayers and particle+surfactant monolayers. We also discuss the use of the Brownian motion of microparticles trapped at fluid interfaces for measuring the shear rheology of surfactant and polymer monolayers. We describe the basic concepts of interfacial rheology and the different experimental methods for measuring both dilational and shear surface complex moduli over a broad range of frequencies, with emphasis in the micro-rheology methods. In the case of particles trapped at interfaces the calculation of the diffusion coefficient from the Brownian trajectories of the particles is calculated as a function of particle surface concentration. We describe in detail the calculation in the case of subdiffusive particle dynamics. A comprehensive review of dilational and shear rheology of particle monolayers and particle+surfactant monolayers is presented. Finally the advantages and current open problems of the use of the Brownian motion of microparticles for calculating the shear complex modulus of monolayers are described in detail.

The shape of protein crowders is a major determinant of protein diffusion

As a model for understanding how molecular crowding influences diffusion and transport of proteins in cellular environments, we combined experimental and theoretical approaches to study the diffusion of proteins in highly concentrated protein solutions. Bovine serum albumin and γ-Globulin were chosen as molecular crowders and as tracers. These two proteins are representatives of the main types of plasma protein and have different shapes and sizes. Solutions consisting of one or both proteins were studied. The self-diffusion coefficients of the fluorescently labeled tracer proteins were measured by means of fluorescence correlation spectroscopy at a total protein concentration of up to 400 g/L. γ-Globulin is found to have a stronger influence as a crowder on the tracer self-diffusion coefficient than Bovine serum albumin. Brownian dynamics simulations show that the excluded volume and the shape of the crowding protein have a significantly stronger influence on translational and rotational diffusion coefficients, as well as transient oligomerization, than hydrodynamic or direct interactions. Anomalous subdiffusion, which is not observed at the experimental fluorescence correlation spectroscopy timescales (>100 μs), appears only at very short timescales (<1 μs) in the simulations due to steric effects of the proteins. We envision that the combined experimental and computational approach employed here can be developed to unravel the different biophysical contributions to protein motion and interaction in cellular environments by systematically varying protein properties such as molecular weight, size, shape, and electrostatic interactions.

Spatio-temporal anomalous diffusion imaging: results in controlled phantoms and in excised human meningiomas

Recently, we measured two anomalous diffusion (AD) parameters: the spatial and the temporal AD indices, called γ and α, respectively, by using spectroscopic pulse gradient field methods. We showed that γ quantifies pseudo-superdiffusion processes, while α quantifies subdiffusion processes. Here, we propose γ and α maps obtained in a controlled heterogeneous phantom, comprised of packed micro-beads in water and in excised human meningiomas. In few words, α maps represent the multi-scale spatial distribution of the disorder degree in the system, while γ maps are influenced by local internal gradients, thus highlighting the interface between compartments characterized by different magnetic susceptibility. γ maps were already obtained by means of AD stretched exponential imaging and α-type maps have been recently achieved for fixed rat brain with the aim of highlighting the fractal dimension of specific brain regions. However, to our knowledge, the maps representative of the spatial distribution of α and γ obtained on the same controlled sample and in the same excised tissue have never been compared. Moreover, we show here, for the first time, that α maps are representative of the spatial distribution of the disorder degree of the system. In a first phase, γ and α maps of controlled phantom characterized by an ordered and a disordered rearrangement of packed micro-beads of different sizes in water and by different magnetic susceptibility (Δχ) between beads and water were obtained. In a second phase, we investigated excised human meningiomas of different consistency. Results reported here, obtained at 9.4T, show that α and γ maps are characterized by a different image contrast. Indeed, unlike γ maps, α maps are insensible to (Δχ) and they are sensible to the disorder degree of the microstructural rearrangement. These observations strongly suggest that AD indices α and γ reflect some additional microstructural information which cannot be obtained using conventional diffusion methods based on Gaussian diffusion. Moreover, α and γ maps obtained in excised meningiomas seem to provide more microstructural details above those obtained with conventional DTI analysis, which could be used to improve the classification of meningiomas based on their consistency.