Reactions on cell membranes: comparison of continuum theory and Brownian dynamics simulations

Motion of a Brownian particle in the presence of reactive boundaries

We study the one-dimensional motion of a Brownian particle inside a confinement described by two reactive boundaries which can partially reflect or absorb the particle. Understanding the effects of such boundaries is important in physics, chemistry, and biology. We compute the probability density of the particle displacement exactly, from which we derive expressions for the survival probability and the mean absorption time as a function of the reactive coefficients. Furthermore, using the Feynman-Kac formalism, we investigate the local time profile, which is the fluctuating time spent by the particle at a given location, both till a fixed observation time and till the absorption time. Our analytical results are compared to numerical simulations, showing perfect agreement.

The Influence of Molecular Reach and Diffusivity on the Efficacy of Membrane-Confined Reactions

Signaling by surface receptors often relies on tethered reactions whereby an enzyme bound to the cytoplasmic tail of a receptor catalyzes reactions on substrates within reach. The overall length and stiffness of the receptor tail, the enzyme, and the substrate determine a biophysical parameter termed the molecular reach of the reaction. This parameter determines the probability that the receptor-tethered enzyme will contact the substrate in the volume proximal to the membrane when separated by different distances within the membrane plane. In this work, we develop particle-based stochastic reaction-diffusion models to study the interplay between molecular reach and diffusion. We find that increasing the molecular reach can increase reaction efficacy for slowly diffusing receptors, whereas for rapidly diffusing receptors, increasing molecular reach reduces reaction efficacy. In contrast, if reactions are forced to take place within the two-dimensional plasma membrane instead of the three-dimensional volume proximal to it or if molecules diffuse in three dimensions, increasing molecular reach increases reaction efficacy for all diffusivities. We show results in the context of immune checkpoint receptors (PD-1 dephosphorylating CD28), a standard opposing kinase-phosphatase reaction, and a minimal two-particle model. The work highlights the importance of the three-dimensional nature of many two-dimensional membrane-confined interactions, illustrating a role for molecular reach in controlling biochemical reactions.

A spatio-temporal model reveals self-limiting FcɛRI cross-linking by multivalent antigens

Aggregation of cell surface receptor proteins by multivalent antigens is an essential early step for immune cell signalling. A number of experimental and modelling studies in the past have investigated multivalent ligand-mediated aggregation of IgE receptors (FcɛRI) in the plasma membrane of mast cells. However, understanding of the mechanisms of FcɛRI aggregation remains incomplete. Experimental reports indicate that FcɛRI forms relatively small and finite-sized clusters when stimulated by a multivalent ligand. By contrast, modelling studies have shown that receptor cross-linking by a trivalent ligand may lead to the formation of large receptor superaggregates that may potentially give rise to hyperactive cellular responses. In this work, we have developed a Brownian dynamics-based spatio-temporal model to analyse FcɛRI aggregation by a trivalent antigen. Unlike the existing models, which implemented non-spatial simulation approaches, our model explicitly accounts for the coarse-grained site-specific features of the multivalent species (molecules and complexes). The model incorporates membrane diffusion, steric collisions and sub-nanometre-scale site-specific interaction of the time-evolving species of arbitrary structures. Using the model, we investigated temporal evolution of the species and their diffusivities. Consistent with a recent experimental report, our model predicted sharp decay in species mobility in the plasma membrane in response receptor cross-linking by a multivalent antigen. We show that, due to such decay in the species mobility, post-stimulation receptor aggregation may become self-limiting. Our analysis reveals a potential regulatory mechanism suppressing hyperactivation of immune cells in response to multivalent antigens.

A model-based analysis of tissue targeting efficacy of nanoparticles

Tissue targeting is a critical challenge for systemic delivery of drug nanocarriers. To overcome this challenge, major research efforts have been undertaken to design ligand-conjugated nanoparticles. However, limited work has been done to quantitatively assess the effectiveness of such approach. In this work, using a mechanistic spatio-temporal model, I investigate the effectiveness of ligand-directed tissue targeting. By applying an approach from the colloidal filtration theory, I develop a Brownian dynamics model of nanoparticle-cell interaction. The model incorporates a single cell and its surrounding flow field. It considers both specific (receptor-mediated) and non-specific (bare cell surface-mediated) recognition of nanoparticles subject to convective and diffusive motion. Using the model, I investigate how the specific and non-specific interactions compare in determining the overall targeting efficacy. My analysis provides some interesting findings that contradict the general notion that effective targeting is possible based upon the differential receptor expression in cancer and non-cancer cells. I show that such strategy may yield only a marginal gain in the targeting efficacy. Moreover, non-specific interaction may have an important influence on particle recognition by cells even at high receptor expression levels.

A multiscale modeling study of particle size effects on the tissue penetration efficacy of drug-delivery nanoparticles

BACKGROUND

Particle size is a key parameter for drug-delivery nanoparticle design. It is believed that the size of a nanoparticle may have important effects on its ability to overcome the transport barriers in biological tissues. Nonetheless, such effects remain poorly understood. Using a multiscale model, this work investigates particle size effects on the tissue distribution and penetration efficacy of drug-delivery nanoparticles.

RESULTS

We have developed a multiscale spatiotemporal model of nanoparticle transport in biological tissues. The model implements a time-adaptive Brownian Dynamics algorithm that links microscale particle-cell interactions and adhesion dynamics to tissue-scale particle dispersion and penetration. The model accounts for the advection, diffusion, and cellular uptakes of particles. Using the model, we have analyzed how particle size affects the intra-tissue dispersion and penetration of drug delivery nanoparticles. We focused on two published experimental works that investigated particle size effects in in vitro and in vivo tissue conditions. By analyzing experimental data reported in these two studies, we show that particle size effects may appear pronounced in an in vitro cell-free tissue system, such as collagen matrix. In an in vivo tissue system, the effects of particle size could be relatively modest. We provide a detailed analysis on how particle-cell interactions may determine distribution and penetration of nanoparticles in a biological tissue.

CONCLUSION

Our work suggests that the size of a nanoparticle may play a less significant role in its ability to overcome the intra-tissue transport barriers. We show that experiments involving cell-free tissue systems may yield misleading observations of particle size effects due to the absence of advective transport and particle-cell interactions.

Architecture and biogenesis of plus-strand RNA virus replication factories

Plus-strand RNA virus replication occurs in tight association with cytoplasmic host cell membranes. Both, viral and cellular factors cooperatively generate distinct organelle-like structures, designated viral replication factories. This compartmentalization allows coordination of the different steps of the viral replication cycle, highly efficient genome replication and protection of the viral RNA from cellular defense mechanisms. Electron tomography studies conducted during the last couple of years revealed the three dimensional structure of numerous plus-strand RNA virus replication compartments and highlight morphological analogies between different virus families. Based on the morphology of virus-induced membrane rearrangements, we propose two separate subclasses: the invaginated vesicle/spherule type and the double membrane vesicle type. This review discusses common themes and distinct differences in the architecture of plus-strand RNA virus-induced membrane alterations and summarizes recent progress that has been made in understanding the complex interplay between viral and co-opted cellular factors in biogenesis and maintenance of plus-strand RNA virus replication factories.

Dynamics of enzymatic digestion of elastic fibers and networks under tension

We study the enzymatic degradation of an elastic fiber under tension using an anisotropic random-walk model coupled with binding-unbinding reactions that weaken the fiber. The fiber is represented by a chain of elastic springs in series along which enzyme molecules can diffuse. Numerical simulations show that the fiber stiffness decreases exponentially with two distinct regimes. The time constant of the first regime decreases with increasing tension. Using a mean field calculation, we partition the time constant into geometrical, chemical and externally controllable factors, which is corroborated by the simulations. We incorporate the fiber model into a multiscale network model of the extracellular matrix and find that network effects do not mask the exponential decay of stiffness at the fiber level. To test these predictions, we measure the force relaxation of elastin sheets stretched to 20% uniaxial strain in the presence of elastase. The decay of force is exponential and the time constant is proportional to the inverse of enzyme concentration in agreement with model predictions. Furthermore, the fragment mass released into the bath during digestion is linearly related to enzyme concentration that is also borne out in the model. We conclude that in the complex extracellular matrix, feedback between the local rate of fiber digestion and the force the fiber carries acts to attenuate any spatial heterogeneity of digestion such that molecular processes manifest directly at the macroscale. Our findings can help better understand remodeling processes during development or in disease in which enzyme concentrations and/or mechanical forces become abnormal.

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.

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.

Spatial regulation and the rate of signal transduction activation

Of the many important signaling events that take place on the surface of a mammalian cell, activation of signal transduction pathways via interactions of cell surface receptors is one of the most important. Evidence suggests that cell surface proteins are not as freely diffusible as implied by the classic fluid mosaic model and that their confinement to membrane domains is regulated. It is unknown whether these dynamic localization mechanisms function to enhance signal transduction activation rate or to minimize cross talk among pathways that share common intermediates. To determine which of these two possibilities is more likely, we derive an explicit equation for the rate at which cell surface membrane proteins interact based on a Brownian motion model in the presence of endocytosis and exocytosis. We find that in the absence of any diffusion constraints, cell surface protein interaction rate is extremely high relative to cytoplasmic protein interaction rate even in a large mammalian cell with a receptor abundance of a mere two hundred molecules. Since a larger number of downstream signaling events needs to take place, each occurring at a much slower rate than the initial activation via association of cell surface proteins, we conclude that the role of co-localization is most likely that of cross-talk reduction rather than coupling efficiency enhancement.

Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da approximately >0.1 (tested in the receptor density range of 10(2)-10(5)/microm(2)) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.