Chemically limited reactions on a percolation cluster

The subcellular dynamics of GPCR signaling

G protein-coupled receptors (GPCRs) are the largest family of membrane receptors and mediate the effects of a multitude of extracellular cues, such as hormones, neurotransmitters, odorants and light. Because of their involvement in numerous physiological and pathological processes and their accessibility, they are extensively exploited as pharmacological targets. Biochemical and structural biology investigations have clarified the molecular basis of GPCR signaling to a high level of detail. In spite of this, how GPCRs can efficiently and precisely translate extracellular signals into specific and well-orchestrated biological responses in the complexity of a living cell or organism remains insufficiently understood. To explain the high efficiency and specificity observed in GPCR signaling, it has been suggested that GPCR might signal in discrete nanodomains on the plasma membrane or even form stable complexes with G proteins and effectors. However, directly testing these hypotheses has proven a major challenge. Recent studies taking advantage of innovative optical methods such as fluorescence resonance energy transfer (FRET) and single-molecule microscopy have begun to dig into the organization of GPCR signaling in living cells on the spatial (nm) and temporal (ms) scales on which cell signaling events are taking place. The results of these studies are revealing a complex and highly dynamic picture, whereby GPCRs undergo transient interaction with their signaling partners, membrane lipids and the cytoskeleton to form short-lived signaling nanodomains both on the plasma membrane and at intracellular sites. Continuous exchanges among such nanodomains via later diffusion as well as via membrane trafficking might provide a highly sophisticated way of controlling the timing and location of GPCR signaling. Here, we will review the most recent advances in our understanding of the organization of GPCR signaling in living cells, with a particular focus on its dynamics.

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

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

Anomalous versus slowed-down Brownian diffusion in the ligand-binding equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

Dynamic organizing principles of the plasma membrane that regulate signal transduction: commemorating the fortieth anniversary of Singer and Nicolson's fluid-mosaic model

The recent rapid accumulation of knowledge on the dynamics and structure of the plasma membrane has prompted major modifications of the textbook fluid-mosaic model. However, because the new data have been obtained in a variety of research contexts using various biological paradigms, the impact of the critical conceptual modifications on biomedical research and development has been limited. In this review, we try to synthesize our current biological, chemical, and physical knowledge about the plasma membrane to provide new fundamental organizing principles of this structure that underlie every molecular mechanism that realizes its functions. Special attention is paid to signal transduction function and the dynamic aspect of the organizing principles. We propose that the cooperative action of the hierarchical three-tiered mesoscale (2-300 nm) domains--actin-membrane-skeleton induced compartments (40-300 nm), raft domains (2-20 nm), and dynamic protein complex domains (3-10 nm)--is critical for membrane function and distinguishes the plasma membrane from a classical Singer-Nicolson-type model.

Confining domains lead to reaction bursts: reaction kinetics in the plasma membrane

Confinement of molecules in specific small volumes and areas within a cell is likely to be a general strategy that is developed during evolution for regulating the interactions and functions of biomolecules. The cellular plasma membrane, which is the outermost membrane that surrounds the entire cell, was considered to be a continuous two-dimensional liquid, but it is becoming clear that it consists of numerous nano-meso-scale domains with various lifetimes, such as raft domains and cytoskeleton-induced compartments, and membrane molecules are dynamically trapped in these domains. In this article, we give a theoretical account on the effects of molecular confinement on reversible bimolecular reactions in a partitioned surface such as the plasma membrane. By performing simulations based on a lattice-based model of diffusion and reaction, we found that in the presence of membrane partitioning, bimolecular reactions that occur in each compartment proceed in bursts during which the reaction rate is sharply and briefly increased even though the asymptotic reaction rate remains the same. We characterized the time between reaction bursts and the burst amplitude as a function of the model parameters, and discussed the biological significance of the reaction bursts in the presence of strong inhibitor activity.

Membrane mechanisms for signal transduction: the coupling of the meso-scale raft domains to membrane-skeleton-induced compartments and dynamic protein complexes

Virtually all biological membranes on earth share the basic structure of a two-dimensional liquid. Such universality and peculiarity are comparable to those of the double helical structure of DNA, strongly suggesting the possibility that the fundamental mechanisms for the various functions of the plasma membrane could essentially be understood by a set of simple organizing principles, developed during the course of evolution. As an initial effort toward the development of such understanding, in this review, we present the concept of the cooperative action of the hierarchical three-tiered meso-scale (2-300 nm) domains in the plasma membrane: (1) actin membrane-skeleton-induced compartments (40-300 nm), (2) raft domains (2-20 nm), and (3) dynamic protein complex domains (3-10nm). Special attention is paid to the concept of meso-scale domains, where both thermal fluctuations and weak cooperativity play critical roles, and the coupling of the raft domains to the membrane-skeleton-induced compartments as well as dynamic protein complexes. The three-tiered meso-domain architecture of the plasma membrane provides an excellent perspective for understanding the membrane mechanisms of signal transduction.

Quantifying the effects of elastic collisions and non-covalent binding on glutamate receptor trafficking in the post-synaptic density

One mechanism of information storage in neurons is believed to be determined by the strength of synaptic contacts. The strength of an excitatory synapse is partially due to the concentration of a particular type of ionotropic glutamate receptor (AMPAR) in the post-synaptic density (PSD). AMPAR concentration in the PSD has to be plastic, to allow the storage of new memories; but it also has to be stable to preserve important information. Although much is known about the molecular identity of synapses, the biophysical mechanisms by which AMPAR can enter, leave and remain in the synapse are unclear. We used Monte Carlo simulations to determine the influence of PSD structure and activity in maintaining homeostatic concentrations of AMPARs in the synapse. We found that, the high concentration and excluded volume caused by PSD molecules result in molecular crowding. Diffusion of AMPAR in the PSD under such conditions is anomalous. Anomalous diffusion of AMPAR results in retention of these receptors inside the PSD for periods ranging from minutes to several hours in the absence of strong binding of receptors to PSD molecules. Trapping of receptors in the PSD by crowding effects was very sensitive to the concentration of PSD molecules, showing a switch-like behavior for retention of receptors. Non-covalent binding of AMPAR to anchored PSD molecules allowed the synapse to become well-mixed, resulting in normal diffusion of AMPAR. Binding also allowed the exchange of receptors in and out of the PSD. We propose that molecular crowding is an important biophysical mechanism to maintain homeostatic synaptic concentrations of AMPARs in the PSD without the need of energetically expensive biochemical reactions. In this context, binding of AMPAR with PSD molecules could collaborate with crowding to maintain synaptic homeostasis but could also allow synaptic plasticity by increasing the exchange of these receptors with the surrounding extra-synaptic membrane.

One of the most accepted theories of information storage in neurons is that it is partially localized in the strength of synaptic contacts. Evidence suggests that at the cellular level, in combination with other cellular mechanisms, this is implemented by increasing or decreasing the concentration of a particular type of membrane molecules. Two opposing mechanisms have to coexist in synapses to allow them to store information. On one hand, synapses have to be flexible, to allow the storage of new memories. On the other hand, synapses have to be stable to preserve previously learned information. Although much is known about the molecular identity of synapses, the biophysical mechanisms by which molecules can enter, leave and remain in the synapse are unclear. Our modeling work uses fundamental biophysical principles to quantify the effects of molecular collisions and biochemical reactions. Our results show that molecular collisions alone, between the diffusing proteins with anchored molecules in the synapse, can replicate known experimental results. Molecular collision in combination with biochemical binding can be fundamental biophysical principles used by synapses for the formation and preservation of memories.

Analysis of reaction-diffusion systems with anomalous subdiffusion

Reaction-diffusion equations are the cornerstone of modeling biochemical systems with spatial gradients, which are relevant to biological processes such as signal transduction. Implicit in the formulation of these equations is the assumption of Fick's law, which states that the local diffusive flux of species i is proportional to its concentration gradient; however, in the context of complex fluids such as cytoplasm and cell membranes, the use of Fick's law is based on empiricism, whereas evidence has been mounting that such media foster anomalous subdiffusion (with mean-squared displacement increasing less than linearly with time) over certain length scales. Particularly when modeling diffusion-controlled reactions and other systems where the spatial domain is considered semi-infinite, assuming Fickian diffusion might not be appropriate. In this article, two simple, conceptually extreme models of anomalous subdiffusion are used in the framework of Green's functions to demonstrate the solution of four reaction-diffusion problems that are well known in the biophysical context of signal transduction: fluorescence recovery after photobleaching, the Smolochowski limit for diffusion-controlled reactions in solution, the spatial range of a diffusing molecule with finite lifetime, and the collision coupling mechanism of diffusion-controlled reactions in two dimensions. In each case, there are only subtle differences between the two subdiffusion models, suggesting how measurements of mean-squared displacement versus time might generally inform models of reactive systems with partial diffusion control.

Elucidating anomalous protein diffusion in living cells with fluorescence correlation spectroscopy-facts and pitfalls

Anomalous protein diffusion has been frequently observed in intracellular fluids and on membranes of living cells. Indeed, a large variety of specimen, from bacteriae to mammalian cells, and several non-invasive measurement techniques, e.g. fluorescence correlation spectroscopy, have revealed that the mean square displacement (MSD) of proteins in vivo is often characterized by an anomalous power-law increase mean value of tau(t)(2) mean value of ~ t(alpha) with 0.5 < alpha </= 0.8. Here, we review these results with a particular focus on fluorescence correlation spectroscopy, and we report on possible causes of variations of the anomaly degree alpha. Moreover, we highlight generic consequences of anomalous diffusion that are likely to play an important role in the cellular context.

Effect of macromolecular crowding on reaction rates: a computational and theoretical study

The effect of macromolecular crowding on the rates of association reactions are investigated using theory and computer simulations. Reactants and crowding agents are both hard spheres, and when two reactants collide they form product with a reaction probability, p(rxn). A value of p(rxn) < 1 crudely mimics the fact that proteins must be oriented properly for an association reaction to occur. The simulations show that the dependence of the reaction rate on the volume fraction of crowding agents varies with the reaction probability. For reaction probabilities close to unity where most of encounters between reactants lead to a reaction, the reaction rate always decreases as the volume fraction of crowding agents is increased due to the reduced diffusion coefficient of reactants. On the other hand, for very small reaction probabilities where, in most of encounters, the reaction does not occur, the reaction rate increases with the volume fraction of crowding agents--in this case, due to the increase probability of a recollision. The Smoluchowski theory refined with the radiation boundary condition and the radial distribution function at contact is in quantitative agreement with simulations for the reaction rate constant and allows the quantitative analysis of both effects separately.

Signal transduction at point-blank range: analysis of a spatial coupling mechanism for pathway crosstalk

The plasma membrane provides a physical platform for the orchestration of molecular interactions and biochemical conversions involved in the early stages of receptor-mediated signal transduction in living cells. In that context, we introduce here the concept of spatial coupling, wherein simultaneous recruitment of different enzymes to the same receptor scaffold facilitates crosstalk between different signaling pathways through the local release and capture of activated signaling molecules. To study the spatiotemporal dynamics of this mechanism, we have developed a Brownian dynamics modeling approach and applied it to the receptor-mediated activation of Ras and the cooperative recruitment of phosphoinositide 3-kinase (PI3K) by activated receptors and Ras. Various analyses of the model simulations show that cooperative assembly of multimolecular complexes nucleated by activated receptors is facilitated by the local release and capture of membrane-anchored signaling molecules (such as active Ras) from/by receptor-bound signaling proteins. In the case of Ras/PI3K crosstalk, the model predicts that PI3K is more likely to be recruited by activated receptors bound or recently visited by the enzyme that activates Ras. By this mechanism, receptor-bound PI3K is stabilized through short-range, diffusion-controlled capture of active Ras and Ras/PI3K complexes released from the receptor complex. We contend that this mechanism is a means by which signaling pathways are propagated and spatially coordinated for efficient crosstalk between them.

Sampling the cell with anomalous diffusion - the discovery of slowness

Diffusion-mediated searching for interaction partners is an ubiquitous process in cell biology. Transcription factors, for example, search specific DNA sequences, signaling proteins aim at interacting with specific cofactors, and peripheral membrane proteins try to dock to membrane domains. Brownian motion, however, is affected by molecular crowding that induces anomalous diffusion (so-called subdiffusion) of proteins and larger structures, thereby compromising diffusive transport and the associated sampling processes. Contrary to the naive expectation that subdiffusion obstructs cellular processes, we show here by computer simulations that subdiffusion rather increases the probability of finding a nearby target. Consequently, important events like protein complex formation and signal propagation are enhanced as compared to normal diffusion. Hence, cells indeed benefit from their crowded internal state and the associated anomalous diffusion.

Macromolecular crowding and its influence on possible reaction mechanisms in photosynthetic electron flow

The diffusion of plastoquinol and its binding to the Qo site of the cyt bf complex in the course of photosynthetic electron transport was studied by following the sigmoidal flash-induced re-reduction kinetics of P700 after previous oxidation of the intersystem electron carriers. The data resulting from these experiments were matched with a simulation of electron transport using Monte Carlo techniques. The simulation was able to account for the experimental observations. Two different extreme cases of reaction mechanism at the Qo site were compared: a diffusion limited collisional mechanism and a non-diffusion limited tight binding mechanism. Assuming a tight binding mechanism led to best matches due to the high protein density in thylakoids. The varied parameters resulted in values well within the range of published data. The results emphasise the importance of structural characteristics of thylakoids in models of electron transport.

Probing the nanoscale viscoelasticity of intracellular fluids in living cells

We have used fluorescence correlation spectroscopy to determine the anomalous diffusion properties of fluorescently tagged gold beads in the cytoplasm and the nucleus of living cells. From the extracted mean-square displacement v(tau) approximately tau(alpha), we have determined the complex shear modulus G(omega) approximately omega(alpha) for both compartments. Without treatment, all tested cell lines showed a strong viscoelastic behavior of the cytoplasm and the nucleoplasm, highlighting the crowdedness of these intracellular fluids. We also found a similar viscoelastic response in frog egg extract, which tended toward a solely viscous behavior upon dilution. When cells were osmotically stressed, the diffusion became less anomalous and the viscoelastic response changed. In particular, the anomality changed from alpha approximately 0.55 to alpha approximately 0.66, which indicates that the Zimm model for polymer solutions under varying solvent conditions is a good empirical description of the material properties of the cytoplasm and the nucleoplasm. Since osmotic stress may eventually trigger cell death, we propose, on the basis of our observations, that intracellular fluids are maintained in a state similar to crowded polymer solutions under good solvent conditions to keep the cell viable.

Kinetics of bimolecular reactions in model bilayers and biological membranes. A critical review

The quantitative study of the probability of molecular encounters giving rise to a reaction in membranes is a challenging discipline. Model systems, model in the sense that they use model bilayers and model reactants, have been widely used for this purpose, but the methodologies employed for the analysis of the results obtained in experiments, and for experimental design, are so disparate that a concerned experimentalist has difficulty in deciding about the value of each approach. This review intends to examine the several approaches that can be found in the literature showing, when feasible, the weakness, strengths and limits of application of each of them. There is not, so far, a full experimental validation of the most promising theories for the analysis of reactions in two dimensions, what leaves open a large field for new research. The major challenge resides in the time range in which the processes take place, but the possibilities of the existing techniques for these studies are far from exhausted. We review also the attempts of several authors to quantitatively analyze the kinetics of reactions in biological membranes. Especially in this field, the recently developed microspectroscopies enclose a still unexplored potential.

Molecular-dynamics simulations for nonclassical kinetics of diffusion-controlled bimolecular reactions

Molecular-dynamics simulations are presented for the diffusion-controlled bimolecular reaction A+B<==>C in two and three dimensions. The reactants and solvent molecules are modeled as spheres interacting via continuous potential-energy functions. The interaction potential between two reactants contains a deep well that results in a reaction. When the solvent concentration is low and the reactant dynamics is essentially ballistic, the system reaches equilibrium rapidly, and the reaction follows classical kinetics with exponential decay to the equilibrium. When the solvent concentration is high the particles enter the normal diffusion regime quickly and nonclassical behavior is observed, i.e., the reactant concentrations approach equilibrium as t(-d/2) where d is the dimensionality of space. When the reaction well depth is large, however, the reaction becomes irreversible within the simulation time. In this case the reactant concentrations decay as t(-d/4). Interestingly this behavior is also observed at intermediate times for reversible reactions.

First-passage method for the study of the efficiency of a two-channel reaction on a lattice

We study the efficiency of a two-channel reaction between two walkers on a finite one-dimensional periodic lattice. The walkers perform a combination of synchronous and asynchronous jumps on the lattice and react instantaneously when they meet at the same site (first channel) or upon position exchange (second channel). We develop a method based on a conditional first-passage problem to obtain exact results for the mean number of time steps needed for the reaction to take place as well as for higher order moments. Previous results obtained in the framework of a difference equation approach are fully confirmed, including the existence of a parity effect. For even lattices the maximum efficiency corresponds to a mixture of synchronous events and a small amount of asynchronous events, while for odd lattices the reaction time is minimized by a purely synchronous process. We provide an intuitive explanation for this behavior. In addition, we give explicit expressions for the variance of the reaction time. The latter displays a similar even-odd behavior, suggesting that the parity effect extends to higher order moments.

In a mirror dimly: tracing the movements of molecules in living cells

The random movement of molecules (diffusion) is fundamental to most cellular processes, including enzymatic reactions, signalling, protein-protein interaction, as well as domain and pattern formation. Despite playing a central role, diffusion is, to a large extent, under-appreciated in the cell biology community. One reason for this is that diffusion is rather challenging to study in living cells. This article is intended to explain, at least in part, how we can go about studying diffusion of molecules in living cells, why it is important and how it provides us with important clues about biological systems. As the title 'In a mirror dimly' suggests, we do this by monitoring faint light emitted by fluorescent probes or proteins using advanced optics (e.g. mirrors) and electronics. The data are then fitted and interpreted with mathematical and physical models, providing a glimpse into the world of molecules.

Monte Carlo simulations of receptor dynamics: insights into cell signaling

Many receptor-level processes involve the diffusion and reaction of receptors with other membrane-localized molecules. Monte Carlo simulation is a powerful technique that allows us to track the motions and discrete reactions of individual receptors, thus simulating receptor dynamics and the early events of signal transduction. In this paper, we discuss simulations of two receptor processes, receptor dimerization and G-protein activation. Our first set of simulations demonstrates how receptor dimerization can create clusters of receptors via partner switching and the relevance of this clustering for receptor cross-talk and integrin signaling. Our second set of simulations investigates the activation and desensitization of G-protein coupled receptors when either a single agonist or both an agonist and an antagonist are present. For G-protein coupled receptor systems in the presence of an agonist alone, the dissociation rate constant of agonist is predicted to affect the ratio of G-protein activation to receptor phosphorylation. Similarly, this ratio is affected by the antagonist dissociation rate constant when both agonist and antagonist are present. The relationship of simulation predictions to experimental findings and potential applications of our findings are also discussed.

Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells

Macromolecular crowding dramatically affects cellular processes such as protein folding and assembly, regulation of metabolic pathways, and condensation of DNA. Despite increased attention, we still lack a definition for how crowded a heterogeneous environment is at the molecular scale and how this manifests in basic physical phenomena like diffusion. Here, we show by means of fluorescence correlation spectroscopy and computer simulations that crowding manifests itself through the emergence of anomalous subdiffusion of cytoplasmic macromolecules. In other words, the mean square displacement of a protein will grow less than linear in time and the degree of this anomality depends on the size and conformation of the traced particle and on the total protein concentration of the solution. We therefore propose that the anomality of the diffusion can be used as a quantifiable measure for the crowdedness of the cytoplasm at the molecular scale.

Enhancement of the recollision rate in diffusion-influenced reactions in an inhomogeneous medium

Brownian dynamics simulations were performed to determine the first collision and recollision rates of spherical reagent particles in a reaction medium made heterogeneous by the presence of randomly located inert spherical obstacles in a continuum solvent. The recollision rate vp (and hence the overall reactive collision rate when activation energy is high) was always enhanced by the presence of obstacles, the degree of enhancement increasing with the volume fraction occupied by obstacles (phi) and with decreasing reagent concentration phiR. Enhancement increased with obstacle size at high phiR, and fell with increasing obstacle size at low phiR. The vp-phiR data follow a power law, where the scaling factor betap decreased with decreasing obstacle size and increasing phi, and the prefactor kp initially increased with phi and then fell (except for large obstacles). The behavior of betap appears to be largely due to the obstacles reducing the probability that reagent particles escape from each other after collision, while the dominant factors responsible for the behavior of kp appear to be initially the effect of obstacles in enhancing effective local reagent concentration, and then (for small obstacles), their reduction of the reagent-particle coordination number. As the energy of activation falls, the reactive collision rate becomes less influenced by the reagent recollision rate and more influenced by the rate of first collision. For low-activation-energy reactions, the presence of obstacles depresses the reactive collision rate if reagent concentration is low or if the obstacles are small and their concentration high. The fall in the reactive collision rate with decreasing activation energy is steeper, the lower the reagent concentration and the smaller the obstacles.

Stabilizing Turing patterns with subdiffusion in systems with low particle numbers

The role of subdiffusion in the formation of spatial Turing patterns with particle number fluctuations is studied. It is demonstrated for a generic activator-inhibitor system that for normal diffusion the particle number fluctuations stabilize the homogenous steady state in a regime where the mean-field analysis already predicts stable spatial patterns. In contrast, pattern formation is stabilized considerably even for very low particle numbers when the activator moves subdiffusively while the inhibitor diffuses normally. In particular, this also holds true when the subdiffusive activator spreads faster than the inhibitor on small time scales. Possible applications to pattern formation in cell biology are discussed.