Sources of anomalous diffusion on cell membranes: a Monte Carlo study

Diffusion anisotropy in collagen gels and tumors: the effect of fiber network orientation

The interstitial matrix is comprised of cross-linked collagen fibers, generally arranged in nonisotropic orientations. Spatial alignment of matrix components within the tissue can affect diffusion patterns of drugs. In this study, we developed a methodology for the calculation of diffusion coefficients of macromolecules and nanoparticles in collagenous tissues. The tissues are modeled as three-dimensional, stochastic, fiber networks with varying degrees of alignment. We employed a random walk approach to simulate diffusion and a Stokesian dynamics method to account for hydrodynamic hindrance. We performed our analysis for four different structures ranging from nearly isotropic to perfectly aligned. We showed that the overall diffusion coefficient is not affected by the orientation of the network. However, structural anisotropy results in diffusion anisotropy, which becomes more significant with increase in the degree of alignment, the size of the diffusing particle, and the fiber volume fraction. To test our model predictions we performed diffusion measurements in reconstituted collagen gels and tumor xenografts. We measured fiber alignment and diffusion with second harmonic generation and multiphoton fluorescent recovery after photobleaching techniques, respectively. The results showed for the first time in tumors that the structure and orientation of collagen fibers in the extracellular space leads to diffusion anisotropy.

Monte Carlo investigation of diffusion of receptors and ligands that bind across opposing surfaces

Studies of receptor diffusion on a cell surface show a variety of behaviors, such as diffusive, sub-diffusive, or super-diffusive motion. However, most studies to date focus on receptor molecules diffusing on a single cell surface. We have previously studied receptor diffusion to probe the molecular mechanism of receptor clustering at the cell-cell junction between two opposing cell surfaces. Here, we characterize the diffusion of receptors and ligands that bind to each other across two opposing cell surfaces, as in cell-cell and cell-bilayer interactions. We use a Monte Carlo method, where receptors and ligands are simulated as independent agents that bind and diffuse probabilistically. We vary receptor-ligand binding affinity and plot the molecule-averaged mean square displacement (MSD) of ligand molecules as a function of time. Our results show that MSD plots are qualitatively different for flat and curved interfaces, as well as between the cases of presence and absence of directed transport of receptor-ligand complexes toward a specific location on the interface. Receptor-ligand binding across two opposing surfaces leads to transient sub-diffusive motion at early times provided the interface is flat. This effect is entirely absent if the interface is curved, however, in this instance we observe sub-diffusive motion. In addition, a decrease in the equilibrium value of the MSD occurs as affinity increases, something which is absent for a flat interface. In the presence of directed transport of receptor-ligand complexes, we observe super-diffusive motion at early times for a flat interface. Super-diffusive motion is absent for a curved interface, however, in this case we observe a transient decrease in MSD with time prior to equilibration for high-affinity values.

Mathematical modeling of K-Ras nanocluster formation on the plasma membrane

K-Ras functions as a critical node in the mitogen-activated protein kinase (MAPK) pathway that regulates key cellular functions including proliferation, differentiation, and apoptosis. Following growth factor receptor activation K-Ras.GTP forms nanoclusters on the plasma membrane through interaction with the scaffold protein galectin-3. The generation of nanoclusters is essential for high fidelity signal transduction via the MAPK pathway. To explore the mechanisms underlying K-Ras.GTP nanocluster formation, we developed a mathematical model of K-Ras-galectin-3 interactions. We designed a computational method to calculate protein collision rates based on experimentally determined protein diffusion rates and diffusion mechanisms and used a genetic algorithm to search the values of key model parameters. The optimal estimated model parameters were validated using experimental data. The resulting model accurately replicates critical features of K-Ras nanoclustering, including a fixed ratio of clustered K-Ras.GTP to monomeric K-Ras.GTP that is independent of the concentration of K-Ras.GTP. The model reproduces experimental results showing that the cytosolic level of galectin-3 determines the magnitude of the K-Ras.GTP clustered fraction and illustrates that nanoclustering is regulated by key nonequilibrium processes. Our kinetic model identifies a potential biophysical mechanism for K-Ras nanoclustering and suggests general principles that may be relevant for other plasma-membrane-localized proteins.

Anomalous transport and nonlinear reactions in spiny dendrites

We present a mesoscopic description of the anomalous transport and reactions of particles in spiny dendrites. As a starting point we use two-state Markovian model with the transition probabilities depending on residence time variable. The main assumption is that the longer a particle survives inside spine, the smaller becomes the transition probability from spine to dendrite. We extend a linear model presented in Fedotov [Phys. Rev. Lett. 101, 218102 (2008)] and derive the nonlinear Master equations for the average densities of particles inside spines and parent dendrite by eliminating residence time variable. We show that the flux of particles between spines and parent dendrite is not local in time and space. In particular the average flux of particles from a population of spines through spines necks into parent dendrite depends on chemical reactions in spines. This memory effect means that one cannot separate the exchange flux of particles and the chemical reactions inside spines. This phenomenon does not exist in the Markovian case. The flux of particles from dendrite to spines is found to depend on the transport process inside dendrite. We show that if the particles inside a dendrite have constant velocity, the mean particle's position <x(t)> increases as t(μ) with μ<1 (anomalous advection). We derive a fractional advection-diffusion equation for the total density of particles.

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.

Hierarchical organization of the plasma membrane: investigations by single-molecule tracking vs. fluorescence correlation spectroscopy

Single-molecule tracking and fluorescence correlation spectroscopy (FCS) applied to the plasma membrane in living cells have allowed a number of unprecedented observations, thus fostering a new basic understanding of molecular diffusion, interaction, and signal transduction in the plasma membrane. It is becoming clear that the plasma membrane is a heterogeneous entity, containing diverse structures on nano-meso-scales (2-200 nm) with a variety of lifetimes, where certain membrane molecules stay together for limited durations. Molecular interactions occur in the time-dependent inhomogeneous two-dimensional liquid of the plasma membrane, which might be a key for plasma membrane functions.

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.

Lipid raft-mediated regulation of G-protein coupled receptor signaling by ligands which influence receptor dimerization: a computational study

G-protein coupled receptors (GPCRs) are the largest family of cell surface receptors; they activate heterotrimeric G-proteins in response to ligand stimulation. Although many GPCRs have been shown to form homo- and/or heterodimers on the cell membrane, the purpose of this dimerization is not known. Recent research has shown that receptor dimerization may have a role in organization of receptors on the cell surface. In addition, microdomains on the cell membrane termed lipid rafts have been shown to play a role in GPCR localization. Using a combination of stochastic (Monte Carlo) and deterministic modeling, we propose a novel mechanism for lipid raft partitioning of GPCRs based on reversible dimerization of receptors and then demonstrate that such localization can affect GPCR signaling. Modeling results are consistent with a variety of experimental data indicating that lipid rafts have a role in amplification or attenuation of G-protein signaling. Thus our work suggests a new mechanism by which dimerization-inducing or inhibiting characteristics of ligands can influence GPCR signaling by controlling receptor organization on the cell membrane.

Subdiffusion in time-averaged, confined random walks

Certain techniques characterizing diffusive processes, such as single-particle tracking or molecular dynamics simulation, provide time averages rather than ensemble averages. Whereas the ensemble-averaged mean-squared displacement (MSD) of an unbounded continuous time random walk (CTRW) with a broad distribution of waiting times exhibits subdiffusion, the time-averaged MSD, delta2, does not. We demonstrate that, in contrast to the unbounded CTRW, in which delta2 is linear in the lag time Delta, the time-averaged MSD of the CTRW of a walker confined to a finite volume is sublinear in Delta, i.e., for long lag times delta2 approximately Delta1-alpha. The present results permit the application of CTRW to interpret time-averaged experimental quantities.

Directional persistence and the optimality of run-and-tumble chemotaxis

E. coli does chemotaxis by performing a biased random walk composed of alternating periods of swimming (runs) and reorientations (tumbles). Tumbles are typically modelled as complete directional randomisations but it is known that in wild type E. coli, successive run directions are actually weakly correlated, with a mean directional difference of approximately 63 degrees. We recently presented a model of the evolution of chemotactic swimming strategies in bacteria which is able to quantitatively reproduce the emergence of this correlation. The agreement between model and experiments suggests that directional persistence may serve some function, a hypothesis supported by the results of an earlier model. Here we investigate the effect of persistence on chemotactic efficiency, using a spatial Monte Carlo model of bacterial swimming in a gradient, combined with simulations of natural selection based on chemotactic efficiency. A direct search of the parameter space reveals two attractant gradient regimes, (a) a low-gradient regime, in which efficiency is unaffected by directional persistence and (b) a high-gradient regime, in which persistence can improve chemotactic efficiency. The value of the persistence parameter that maximises this effect corresponds very closely with the value observed experimentally. This result is matched by independent simulations of the evolution of directional memory in a population of model bacteria, which also predict the emergence of persistence in high-gradient conditions. The relationship between optimality and persistence in different environments may reflect a universal property of random-walk foraging algorithms, which must strike a compromise between two competing aims: exploration and exploitation. We also present a new graphical way to generally illustrate the evolution of a particular trait in a population, in terms of variations in an evolvable parameter.

FFAT rescues VAPA-mediated inhibition of ER-to-Golgi transport and VAPB-mediated ER aggregation

The VAMP-associated proteins termed VAP are a small gene family of proteins characterised by the presence of an N-terminal major sperm protein (MSP) domain. The P56S mutation of the B isoform (VAPB) has been linked to late-onset amyotrophic lateral sclerosis (ALS8) and its expression causes formation of large ER aggregates. Overexpression of the wild-type A isoform (VAPA) but not the B isoform (VAPB), inhibited ER-to-Golgi transport of membrane proteins. This transport block by VAPA was primarily due to decreased segregation of membrane cargo into ER vesicles. We also found that VAPA inhibited lateral diffusion of membrane proteins, most likely through its stable association with microtubules. The MSP domain of VAP is known to interact with the FFAT motif (two phenylalanines in an acidic tract) of proteins involved in sterol regulation. Overexpression of FFAT restored ER-to-Golgi transport and lateral diffusion of membrane proteins, and resolved the large ER aggregates in VAPB-P56S. Application of a FFAT peptide restored in vitro ER vesicle budding and disrupted VAP-microtubule association. Thus, overexpression of the two VAP isoforms causes retention of ER membrane proteins by impeding lateral diffusion and their incorporation into transport vesicles. This inhibitory effect can be relieved by expression of the FFAT motif.

The spatiotemporal pattern of Src activation at lipid rafts revealed by diffusion-corrected FRET imaging

Genetically encoded biosensors based on fluorescence resonance energy transfer (FRET) have been widely applied to visualize the molecular activity in live cells with high spatiotemporal resolution. However, the rapid diffusion of biosensor proteins hinders a precise reconstruction of the actual molecular activation map. Based on fluorescence recovery after photobleaching (FRAP) experiments, we have developed a finite element (FE) method to analyze, simulate, and subtract the diffusion effect of mobile biosensors. This method has been applied to analyze the mobility of Src FRET biosensors engineered to reside at different subcompartments in live cells. The results indicate that the Src biosensor located in the cytoplasm moves 4–8 folds faster (0.93±0.06 µm²/sec) than those anchored on different compartments in plasma membrane (at lipid raft: 0.11±0.01 µm²/sec and outside: 0.18±0.02 µm²/sec). The mobility of biosensor at lipid rafts is slower than that outside of lipid rafts and is dominated by two-dimensional diffusion. When this diffusion effect was subtracted from the FRET ratio images, high Src activity at lipid rafts was observed at clustered regions proximal to the cell periphery, which remained relatively stationary upon epidermal growth factor (EGF) stimulation. This result suggests that EGF induced a Src activation at lipid rafts with well-coordinated spatiotemporal patterns. Our FE-based method also provides an integrated platform of image analysis for studying molecular mobility and reconstructing the spatiotemporal activation maps of signaling molecules in live cells.

Fluorescence biosensors have been widely used to report the spatial and temporal activity of target molecules in live cells. However, biosensors can move independently of the target molecule and carry its signal to other subcellular locations. Therefore, the observed images appear to be the combination of the target molecular activity and the artifacts introduced by the movement of the biosensors (mainly due to diffusion). The intriguing question is how to estimate and exclude the movement effect of biosensors from the observed fluorescent images and to reconstruct the real activity map of the target molecules. The Src molecule plays important roles in cell adhesion, migration, and cancer invasion. In this paper, we developed a novel computational method to analyze and simulate the movement of the Src biosensor, which was then subtracted from the original fluorescent images. With this computational method, we observed discrete clusters of high Src activity at relatively stationary locations on the plasma membrane. Therefore, our results highlight the coordination of molecular activities in space and time. In addition to Src, our computational method can be used to reconstruct the activity map of other signaling molecules.

Using plasma membrane nanoclusters to build better signaling circuits

Cellular signaling pathways do not simply transmit data; they integrate and process signals to operate as switches, oscillators, logic gates, memory modules and many other types of control system. These complex processing capabilities enable cells to respond appropriately to the myriad of external cues that direct growth and development. The idea that crosstalk and feedback loops are used as control systems in biological signaling networks is well established. Signaling networks are also subject to exquisite spatial regulation, yet how spatial control modulates signal outputs is less well understood. Here, we explore the spatial organization of two different signal transduction circuits: receptor tyrosine kinase activation of the mitogen-activated protein kinase module; and glycosylphosphatidylinositol-anchored receptor activation of phospholipase C. With regards to these pathways, recent results have refocused attention on the crucial role of lipid rafts and plasma membrane nanodomains in signal transmission. We identify common design principals that highlight how the spatial organization of signal transduction circuits can be used as a fundamental control mechanism to modulate system outputs in vivo.

Nonergodicity mimics inhomogeneity in single particle tracking

Most statistical theories of anomalous diffusion rely on ensemble-averaged quantities such as the mean squared displacement. Single molecule tracking measurements require, however, temporal averaging. We contrast the two approaches in the case of continuous-time random walks with a power-law distribution of waiting times psi(t) proportional to t{-1-alpha}, with 0<alpha<1, lacking the mean. We show that, contrary to what is expected, the temporal averaged mean squared displacement leads to a simple diffusive behavior with diffusion coefficients that strongly differ from one trajectory to another. This distribution of diffusion coefficients renders a system inhomogeneous: an ensemble of simple diffusers with different diffusion coefficients. Taking an ensemble average over these diffusion coefficients results in an effective diffusion coefficient K{eff} approximately T{alpha-1} which depends on the length of the trajectory T.

Diffusion of nanoparticles in monolayers is modulated by domain size

Langmuir monolayers are often used as simple models for biological membranes. The possibility to change their composition and phase state in a very controlled manner as well as access to a large observation area makes them a versatile tool for the investigation of membrane-related interactions. Inspired by experiments in our group, we investigate the interaction of single, partially charged nanoparticles with lipid microdomains by Monte Carlo simulations. Condensed domains in inhomogeneous Langmuir monolayers exhibit an electric dipole field interacting attractively with the nanoparticle's dipole moment. With increasing domain size, the resulting electric field changes from single dipole to semi-infinite domain characteristics, significantly influencing the motion of the particle. Small immobile domains (R = 1 microm) confine the movement of the tracer to the boundary of the domain whereas for large domains (R > or = 10 microm) its motion is only temporarily hindered. This suggests a powerful mechanism for controlling diffusive transport in lipid membranes.

Ras nanoclusters: combining digital and analog signaling

Cellular signaling pathways respond to external inputs to drive pivotal cellular decisions. Far from being mere data relay systems, signaling cascades form complex interacting networks with multiple layers of feedback and feed-forward control loops regulated in both space and time. While it may be intuitively obvious that this complexity allows cells to assess and respond appropriately to a myriad of external cues, untangling the wires to understand precisely how complex networks function as control and computational systems presents a daunting challenge to theoretical and experimental biologists alike. In this review we have focused on activation of the canonical MAP kinase cascade by receptor tyrosine kinases (RTKs) in order to examine some of the fundamental design principles used to build biological circuits and control systems. In particular, we explore how cells can reconfigure signaling cascades to generate distinct biological outputs by utilizing the unique spatial constraints available in biological membranes.

Ras nanoclusters: molecular structure and assembly

H-, N- and K-ras4B are lipid-anchored, peripheral membrane guanine nucleotide binding proteins. Recent work has shown that Ras proteins are laterally segregated into non-overlapping, dynamic domains of the plasma membrane called nanoclusters. This lateral segregation is important to specify Ras interactions with membrane-associated proteins, effectors and scaffolding proteins and is critical for Ras signal transduction. Here we review biological, in vitro and structural data that provide insight into the molecular basis of how palmitoylated Ras proteins are anchored to the plasma membrane. We explore possible mechanisms for how the interactions of H-ras with a lipid bilayer may drive nanocluster formation.

Tracking single lipase molecules on a trimyristin substrate surface using quantum dots

The mobility of single lipase molecules has been analyzed using single molecule tracking on a trimyristin substrate surface. This was achieved by conjugating lipases to quantum dots and imaging on spin-coated trimyristin surfaces by means of confocal laser scanning microscopy. Image series of single lipase molecules were collected, and the diffusion coefficient was quantified by analyzing the mean square displacement of the calculated trajectories. During no-flow conditions, the lipase diffusion coefficient was (8.0+/-5.0)x10(-10) cm2/s. The trajectories had a "bead on a string" appearance, with the lipase molecule restricted in certain regions of the surface and then migrating to another region where the restricted diffusion continued. This gave rise to clusters in the trajectories. When a flow was applied to the system, the total distance and average step length between the clusters increased, but the restricted diffusion in the cluster regions was unaffected. This can be explained by the lipase operating in two different modes on the surface. In the cluster regions, the lipase is likely oriented with the active site toward the surface and hydrolyzes the substrate. Between these regions, a diffusion process is proposed where the lipase is in contact with the surface but affected by the external flow.

Single-particle tracking of membrane protein diffusion in a potential: simulation, detection, and application to confined diffusion of CFTR Cl- channels

Confined diffusion of membrane receptors and lipids can result from intramembrane barriers, skeletal interactions, rafts, and other phenomena. We simulated single-particle diffusion in two dimensions in an arbitrary potential, V(r), based on summation of random and potential gradient-driven motions. Algorithms were applied and verified for detection of potential-driven diffusion, and for determination of V(r) from radial particle density distributions, taking into account experimental uncertainties in particle position and finite trajectory recording. Single-particle tracking (SPT) analysis of the diffusion of cystic fibrosis transmembrane conductance regulator (CFTR) Cl(-) channels in mammalian cells revealed confined diffusion with diffusion coefficient approximately 0.004 microm(2)/s. SPT data fitted closely to a springlike attractive potential, V(r) = kr(2), but not to other V(r) forms such as hard-wall or viscoelastic-like potentials. The "spring constant", k, determined from SPT data was 2.6 +/- 0.8 pN/microm, and not altered significantly by modulation of skeletal protein architecture by jasplakinolide. However, k was reduced by a low concentration of latrunculin, supporting the involvement of actin in the springlike tethering of CFTR. Confined diffusion of membrane proteins is likely a general phenomenon suitable for noninvasive V(r) analysis of force-producing mechanisms. Our data provide the first measurement of actin elasticity, to the best of our knowledge, that does not involve application of an external force.