Simulations of (an)isotropic diffusion on curved biological surfaces

Lipid membranes supported by polydimethylsiloxane substrates with designed geometry

The membrane curvature of cells and intracellular compartments continuously adapts to enable cells to perform vital functions, from cell division to signal trafficking. Understanding how membrane geometry affects these processes in vivo is challenging because of the biochemical and geometrical complexity as well as the short time and small length scales involved in cellular processes. By contrast, in vitro model membranes with engineered curvature would provide a versatile platform for this investigation and applications to biosensing and biocomputing. Here, we present a strategy that allows fabrication of lipid membranes with designed shape by combining 3D micro-printing and replica-molding lithography with polydimethylsiloxane to create curved micrometer-sized scaffolds with virtually any geometry. The resulting supported lipid membranes are homogeneous and fluid. We demonstrate the versatility of the system by fabricating structures of interesting combinations of mean and Gaussian curvature. We study the lateral phase separation and how local curvature influences the effective diffusion coefficient. Overall, we offer a bio-compatible platform for understanding curvature-dependent cellular processes and developing programmable bio-interfaces for living cells and nanostructures.

Diffusion Analyses along Mean and Gaussian-Curved Membranes with CurD

Curved cellular membranes are both abundant and functionally relevant. While novel tomography approaches reveal the structural details of curved membranes, their dynamics pose an experimental challenge. Curvature especially affects the diffusion of lipids and macromolecules, yet neither experiments nor continuum models distinguish geometric effects from those caused by curvature-induced changes in membrane properties. Molecular simulations could excel here, yet despite community interest toward curved membranes, tools for their analysis are still lacking. Here, we satisfy this demand by introducing CurD, our novel and openly available implementation of the Vertex-oriented Triangle Propagation algorithm to the study of lipid diffusion along membranes with mean and/or Gaussian curvature. This approach, aided by our highly optimized implementation, computes geodetic distances significantly faster than conventional implementations of path-finding algorithms. Our tool, applied to coarse-grained simulations, allows for the first time the analysis of curvature effects on diffusion at size scales relevant to physiological processes such as endocytosis. Our analyses with different membrane geometries reveal that Gaussian curvature plays a surprisingly small role on lipid motion, whereas mean curvature; i.e., the packing of lipid headgroups largely dictates their mobility.

Analyzing Photoactivation with Diffusion Models to Study Transport in the Endoplasmic Reticulum Network

Photoactivation is a paradigm consisting in local molecular fluorescent activation by laser illumination in a chosen region (source) while measuring the concentration at a target region. Data-driven modeling is concerned with the following questions: how from the measurement in these two regions is it possible to infer the properties of molecular propagation? How is it possible to use such responses to infer motions occurring in networks such as the endoplasmic reticulum? In this book chapter, we shall review the data-driven analysis based on diffusion-transport models and numerical simulations to interpret the photoactivation dynamics and extract biophysical parameters. We will discuss modeling approaches to reconstruct local network properties from photoactivation transients.

Dancing with the Stars: Using Image Analysis to Study the Choreography of the Endoplasmic Reticulum and Its Partners and of Movement Within Its Tubules

In this chapter, approaches to the image analysis of the choreography of the plant endoplasmic reticulum (ER) labeled with fluorescent fusion proteins ("stars," if you wish) are presented. The approaches include the analyses of those parts of the ER that are attached through membrane contact sites to moving or non-moving partners (other "stars"). Image analysis is also used to understand the nature of the tubular polygonal network, the hallmark of this organelle, and how the polygons change over time due to tubule sliding or motion. Furthermore, the remodeling polygons of the ER interact with regions of fundamentally different topologies, the ER cisternae, and image analysis can be used to separate the tubules from the cisternae. ER cisternae, like polygons and tubules, can be motile or stationary. To study which parts are attached to non-moving partners, such as domains of the ER that form membrane contact sites with the plasma membrane/cell wall, an image analysis approach called persistency mapping has been used. To study the domains of the ER that move rapidly and stream through the cell, image analysis of optic flow has been used. However, optic flow approaches confuse the movement of the ER itself with the movement of proteins within the ER. As an overall measure of ER dynamics, optic flow approaches are of value, but their limitation as to what exactly is "flowing" needs to be specified. Finally, there are important imaging approaches that directly address the movement of fluorescent proteins within the ER lumen or in the membrane of the ER. Of these, fluorescence recovery after photobleaching (FRAP), inverse FRAP (iFRAP), and single particle tracking approaches are described.

Midpoint projection algorithm for stochastic differential equations on manifolds

Stochastic differential equations projected onto manifolds occur in physics, chemistry, biology, engineering, nanotechnology, and optimization, with interdisciplinary applications. Intrinsic coordinate stochastic equations on the manifold are sometimes computationally impractical, and numerical projections are therefore useful in many cases. In this paper a combined midpoint projection algorithm is proposed that uses a midpoint projection onto a tangent space, combined with a subsequent normal projection to satisfy the constraints. We also show that the Stratonovich form of stochastic calculus is generally obtained with finite bandwidth noise in the presence of a strong enough external potential that constrains the resulting physical motion to a manifold. Numerical examples are given for a wide range of manifolds, including circular, spheroidal, hyperboloidal, and catenoidal cases, higher-order polynomial constraints that give a quasicubical surface, and a ten-dimensional hypersphere. In all cases the combined midpoint method has greatly reduced errors compared to other methods used for comparison, namely, a combined Euler projection approach and a tangential projection algorithm. We derive intrinsic stochastic equations for spheroidal and hyperboloidal surfaces for comparison purposes to verify the results. Our technique can handle multiple constraints, which allows manifolds that embody several conserved quantities. The algorithm is accurate, simple, and efficient. A reduction of an order of magnitude in the diffusion distance error is found compared to the other methods and an up to several orders of magnitude reduction in constraint function errors.

Quantifying Molecular Dynamics within Complex Cellular Morphologies using LLSM-FRAP

Quantifying molecular dynamics within the context of complex cellular morphologies is essential toward understanding the inner workings and function of cells. Fluorescence recovery after photobleaching (FRAP) is one of the most broadly applied techniques to measure the reaction diffusion dynamics of molecules in living cells. FRAP measurements typically restrict themselves to single-plane image acquisition within a subcellular-sized region of interest due to the limited temporal resolution and undesirable photobleaching induced by 3D fluorescence confocal or widefield microscopy. Here, an experimental and computational pipeline combining lattice light sheet microscopy, FRAP, and numerical simulations, offering rapid and minimally invasive quantification of molecular dynamics with respect to 3D cell morphology is presented. Having the opportunity to accurately measure and interpret the dynamics of molecules in 3D with respect to cell morphology has the potential to reveal unprecedented insights into the function of living cells.

Getting around the cell: physical transport in the intracellular world

Eukaryotic cells face the challenging task of transporting a variety of particles through the complex intracellular milieu in order to deliver, distribute, and mix the many components that support cell function. In this review, we explore the biological objectives and physical mechanisms of intracellular transport. Our focus is on cytoplasmic and intra-organelle transport at the whole-cell scale. We outline several key biological functions that depend on physically transporting components across the cell, including the delivery of secreted proteins, support of cell growth and repair, propagation of intracellular signals, establishment of organelle contacts, and spatial organization of metabolic gradients. We then review the three primary physical modes of transport in eukaryotic cells: diffusive motion, motor-driven transport, and advection by cytoplasmic flow. For each mechanism, we identify the main factors that determine speed and directionality. We also highlight the efficiency of each transport mode in fulfilling various key objectives of transport, such as particle mixing, directed delivery, and rapid target search. Taken together, the interplay of diffusion, molecular motors, and flows supports the intracellular transport needs that underlie a broad variety of biological phenomena.

The Unfolded Protein Response: Detecting and Responding to Fluctuations in the Protein-Folding Capacity of the Endoplasmic Reticulum

Most of the secreted and plasma membrane proteins are synthesized on membrane-bound ribosomes on the endoplasmic reticulum (ER). They require engagement of ER-resident chaperones and foldases that assist in their folding and maturation. Since protein homeostasis in the ER is crucial for cellular function, the protein-folding status in the organelle's lumen is continually surveyed by a network of signaling pathways, collectively called the unfolded protein response (UPR). Protein-folding imbalances, or "ER stress," are detected by highly conserved sensors that adjust the ER's protein-folding capacity according to the physiological needs of the cell. We review recent developments in the field that have provided new insights into the ER stress-sensing mechanisms used by UPR sensors and the mechanisms by which they integrate various cellular inputs to adjust the folding capacity of the organelle to accommodate to fluctuations in ER protein-folding demands.

Conventional analysis of movement on non-flat surfaces like the plasma membrane makes Brownian motion appear anomalous

Cells are neither flat nor smooth, which has serious implications for prevailing plasma membrane models and cellular processes like cell signalling, adhesion and molecular clustering. Using probability distributions from diffusion simulations, we demonstrate that 2D and 3D Euclidean distance measurements substantially underestimate diffusion on non-flat surfaces. Intuitively, the shortest within surface distance (SWSD), the geodesic distance, should reduce this problem. The SWSD is accurate for foldable surfaces but, although it outperforms 2D and 3D Euclidean measurements, it still underestimates movement on deformed surfaces. We demonstrate that the reason behind the underestimation is that topographical features themselves can produce both super- and subdiffusion, i.e. the appearance of anomalous diffusion. Differentiating between topography-induced and genuine anomalous diffusion requires characterising the surface by simulating Brownian motion on high-resolution cell surface images and a comparison with the experimental data.

Quantitative Analysis of Hepatitis C NS5A Viral Protein Dynamics on the ER Surface

Exploring biophysical properties of virus-encoded components and their requirement for virus replication is an exciting new area of interdisciplinary virological research. To date, spatial resolution has only rarely been analyzed in computational/biophysical descriptions of virus replication dynamics. However, it is widely acknowledged that intracellular spatial dependence is a crucial component of virus life cycles. The hepatitis C virus-encoded NS5A protein is an endoplasmatic reticulum (ER)-anchored viral protein and an essential component of the virus replication machinery. Therefore, we simulate NS5A dynamics on realistic reconstructed, curved ER surfaces by means of surface partial differential equations (sPDE) upon unstructured grids. We match the in silico NS5A diffusion constant such that the NS5A sPDE simulation data reproduce experimental NS5A fluorescence recovery after photobleaching (FRAP) time series data. This parameter estimation yields the NS5A diffusion constant. Such parameters are needed for spatial models of HCV dynamics, which we are developing in parallel but remain qualitative at this stage. Thus, our present study likely provides the first quantitative biophysical description of the movement of a viral component. Our spatio-temporal resolved ansatz paves new ways for understanding intricate spatial-defined processes central to specfic aspects of virus life cycles.

Dancing with the Stars: Using Image Analysis to Study the Choreography of the Endoplasmic Reticulum and Its Partners and of Movement Within Its Tubules

In this chapter, approaches to the image analysis of the choreography of the plant endoplasmic reticulum (ER) labeled with fluorescent fusion proteins ("stars," if you wish) are presented. The approaches include the analyses of those parts of the ER that are attached through membrane contact sites to moving or nonmoving partners (other "stars"). Image analysis is also used to understand the nature of the tubular polygonal network, the hallmark of this organelle, and how the polygons change over time due to tubule sliding or motion. Furthermore, the remodeling polygons of the ER interact with regions of fundamentally different topology, the ER cisternae, and image analysis can be used to separate the tubules from the cisternae. ER cisternae, like polygons and tubules, can be motile or stationary. To study which parts are attached to nonmoving partners, such as domains of the ER that form membrane contact sites with the plasma membrane/cell wall, an image analysis approach called persistency mapping has been used. To study the domains of the ER that are moving rapidly and streaming through the cell, the image analysis of optic flow has been used. However, optic flow approaches confuse the movement of the ER itself with the movement of proteins within the ER. As an overall measure of ER dynamics, optic flow approaches are of value, but their limitation as to what exactly is "flowing" needs to be specified. Finally, there are important imaging approaches that directly address the movement of fluorescent proteins within the ER lumen or in the membrane of the ER. Of these, fluorescence recovery after photobleaching (FRAP), inverse FRAP (iFRAP), and single particle tracking approaches are described.

An extended model of vesicle fusion at the plasma membrane to estimate protein lateral diffusion from TIRF microscopy images

BACKGROUND

Characterizing membrane dynamics is a key issue to understand cell exchanges with the extra-cellular medium. Total internal reflection fluorescence microscopy (TIRFM) is well suited to focus on the late steps of exocytosis at the plasma membrane. However, it is still a challenging task to quantify (lateral) diffusion and estimate local dynamics of proteins.

RESULTS

A new model was introduced to represent the behavior of cargo transmembrane proteins during the vesicle fusion to the plasma membrane at the end of the exocytosis process. Two biophysical parameters, the diffusion coefficient and the release rate parameter, are automatically estimated from TIRFM image sequences, to account for both the lateral diffusion of molecules at the membrane and the continuous release of the proteins from the vesicle to the plasma membrane. Quantitative evaluation on 300 realistic computer-generated image sequences demonstrated the efficiency and accuracy of the method. The application of our method on 16 real TIRFM image sequences additionally revealed differences in the dynamic behavior of Transferrin Receptor (TfR) and Langerin proteins.

CONCLUSION

An automated method has been designed to simultaneously estimate the diffusion coefficient and the release rate for each individual vesicle fusion event at the plasma membrane in TIRFM image sequences. It can be exploited for further deciphering cell membrane dynamics.

A biochemo-mechano coupled, computational model combining membrane transport and pericellular proteolysis in tissue mechanics

We present a computational model for the interaction of surface- and volume-bound scalar transport and reaction processes with a deformable porous medium. The application in mind is pericellular proteolysis, i.e. the dissolution of the solid phase of the extracellular matrix (ECM) as a response to the activation of certain chemical species at the cell membrane and in the vicinity of the cell. A poroelastic medium model represents the extra cellular scaffold and the interstitial fluid flow, while a surface-bound transport model accounts for the diffusion and reaction of membrane-bound chemical species. By further modelling the volume-bound transport, we consider the advection, diffusion and reaction of sequestered chemical species within the extracellular scaffold. The chemo-mechanical coupling is established by introducing a continuum formulation for the interplay of reaction rates and the mechanical state of the ECM. It is based on known experimental insights and theoretical work on the thermodynamics of porous media and degradation kinetics of collagen fibres on the one hand and a damage-like effect of the fibre dissolution on the mechanical integrity of the ECM on the other hand. The resulting system of partial differential equations is solved via the finite-element method. To the best of our knowledge, it is the first computational model including contemporaneously the coupling between (i) advection-diffusion-reaction processes, (ii) interstitial flow and deformation of a porous medium, and (iii) the chemo-mechanical interaction impelled by the dissolution of the ECM. Our numerical examples show good agreement with experimental data. Furthermore, we outline the capability of the methodology to extend existing numerical approaches towards a more comprehensive model for cellular biochemo-mechanics.

Dynamics and Structure-Function Relationships of the Lamin B Receptor (LBR)

The lamin B receptor (LBR) is a multi-spanning membrane protein of the inner nuclear membrane that is often employed as a "reporter" of nuclear envelope dynamics. We show here that the diffusional mobility of full-length LBR exhibits significant regional variation along the nuclear envelope, consistent with the existence of discrete LBR microdomains and the occurrence of multiple, asymmetrically-spaced anastomoses along the nuclear envelope-endoplasmic reticulum interface. Interestingly, a commonly used fusion protein that contains the amino-terminal region and the first transmembrane domain of LBR exhibits reduced mobility at the nuclear envelope, but behaves similarly to full-length LBR in the endoplasmic reticulum. On the other hand, carboxy-terminally truncated mutants that retain the first four transmembrane domains and a part or the whole of the amino-terminal region of LBR are generally hyper-mobile. These results suggest that LBR dynamics is structure and compartment specific. They also indicate that native LBR is probably "configured" by long-range interactions that involve the loops between adjacent transmembrane domains and parts of the amino-terminal region.

Deciphering chemokine properties by a hybrid agent-based model of Aspergillus fumigatus infection in human alveoli

The ubiquitous airborne fungal pathogen Aspergillus fumigatus is inhaled by humans every day. In the lung, it is able to quickly adapt to the humid environment and, if not removed within a time frame of 4–8 h, the pathogen may cause damage by germination and invasive growth. Applying a to-scale agent-based model of human alveoli to simulate early A. fumigatus infection under physiological conditions, we recently demonstrated that alveolar macrophages require chemotactic cues to accomplish the task of pathogen detection within the aforementioned time frame. The objective of this study is to specify our general prediction on the as yet unidentified chemokine by a quantitative analysis of its expected properties, such as the diffusion coefficient and the rates of secretion and degradation. To this end, the rule-based implementation of chemokine diffusion in the initial agent-based model is revised by numerically solving the spatio-temporal reaction-diffusion equation in the complex structure of the alveolus. In this hybrid agent-based model, alveolar macrophages are represented as migrating agents that are coupled to the interactive layer of diffusing molecule concentrations by the kinetics of chemokine receptor binding, internalization and re-expression. Performing simulations for more than a million virtual infection scenarios, we find that the ratio of secretion rate to the diffusion coefficient is the main indicator for the success of pathogen detection. Moreover, a subdivision of the parameter space into regimes of successful and unsuccessful parameter combination by this ratio is specific for values of the migration speed and the directional persistence time of alveolar macrophages, but depends only weakly on chemokine degradation rates.

Inference of protein kinetics by stochastic modeling and simulation of fluorescence recovery after photobleaching experiments

MOTIVATION

Fluorescence recovery after photobleaching (FRAP) is a functional live cell imaging technique that permits the exploration of protein dynamics in living cells. To extract kinetic parameters from FRAP data, a number of analytical models have been developed. Simplifications are inherent in these models, which may lead to inexhaustive or inaccurate exploitation of the experimental data. An appealing alternative is offered by the simulation of biological processes in realistic environments at a particle level. However, inference of kinetic parameters using simulation-based models is still limited.

RESULTS

We introduce and demonstrate a new method for the inference of kinetic parameter values from FRAP data. A small number of in silico FRAP experiments is used to construct a mapping from FRAP recovery curves to the parameters of the underlying protein kinetics. Parameter estimates from experimental data can then be computed by applying the mapping to the observed recovery curves. A bootstrap process is used to investigate identifiability of the physical parameters and determine confidence regions for their estimates. Our method circumvents the computational burden of seeking the best-fitting parameters via iterative simulation. After validation on synthetic data, the method is applied to the analysis of the nuclear proteins Cdt1, PCNA and GFPnls. Parameter estimation results from several experimental samples are in accordance with previous findings, but also allow us to discuss identifiability issues as well as cell-to-cell variability of the protein kinetics.

IMPLEMENTATION

All methods were implemented in MATLAB R2011b. Monte Carlo simulations were run on the HPC cluster Brutus of ETH Zurich.

CONTACT

lygeros@control.ee.ethz.ch or lygerou@med.upatras.gr

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

High-order spectral/hp element discretisation for reaction-diffusion problems on surfaces: Application to cardiac electrophysiology

We present a numerical discretisation of an embedded two-dimensional manifold using high-order continuous Galerkin spectral/hp elements, which provide exponential convergence of the solution with increasing polynomial order, while retaining geometric flexibility in the representation of the domain. Our work is motivated by applications in cardiac electrophysiology where sharp gradients in the solution benefit from the high-order discretisation, while the computational cost of anatomically-realistic models can be significantly reduced through the surface representation and use of high-order methods. We describe and validate our discretisation and provide a demonstration of its application to modelling electrochemical propagation across a human left atrium.

Single-cell imaging of mechanotransduction in endothelial cells

Endothelial cells (ECs) are constantly exposed to chemical and mechanical microenvironment in vivo. In mechanotransduction, cells can sense and translate the extracellular mechanical cues into intracellular biochemical signals, to regulate cellular processes. This regulation is crucial for many physiological functions, such as cell adhesion, migration, proliferation, and survival, as well as the progression of disease such as atherosclerosis. Here, we overview the current molecular understanding of mechanotransduction in ECs associated with atherosclerosis, especially those in response to physiological shear stress. The enabling technology of live-cell imaging has allowed the study of spatiotemporal molecular events and unprecedented understanding of intracellular signaling responses in mechanotransduction. Hence, we also introduce recent studies on mechanotransduction using single-cell imaging technologies.

Parameter importance in FRAP acquisition and analysis: a simulation approach

Fluorescence recovery after photobleaching (FRAP) is a widespread technique used to determine intracellular reaction and diffusion parameters. In recent years, due to technical advances and an increasing number of mathematical models for analysis, there was a resurging interest in FRAP applications. However, care has to be taken when inverting parameters from such data. We study potential influences on FRAP acquisition and analysis like initial fluorescence distribution, membrane passage, and geometrical aspects. Monte Carlo simulations are employed for the investigation of reaction-diffusion processes to additionally include cases in which no analytical description is available. To assess the importance of influencing factors we apply a sensitivity method based on elementary effects providing an estimate for the global parameter space. The combination of simulations and sensitivity measure helps us to predict ranges of parameters used in acquisition and analysis for which a reliably inversion of reaction-diffusion parameters is possible. Using this approach, we show that FRAP data are highly susceptible to misinterpretation. However, by identifying the parameters of susceptibility, our analysis provides the means for taking measures to significantly improve FRAP data interpretation and analysis.

Modeling and simulation of biological systems from image data

This essay provides an introduction to the terminology, concepts, methods, and challenges of image-based modeling in biology. Image-based modeling and simulation aims at using systematic, quantitative image data to build predictive models of biological systems that can be simulated with a computer. This allows one to disentangle molecular mechanisms from effects of shape and geometry. Questions like "what is the functional role of shape" or "how are biological shapes generated and regulated" can be addressed in the framework of image-based systems biology. The combination of image quantification, model building, and computer simulation is illustrated here using the example of diffusion in the endoplasmic reticulum.

Coupling Image Restoration and Segmentation: A Generalized Linear Model/Bregman Perspective

We introduce a new class of data-fitting energies that couple image segmentation with image restoration. These functionals model the image intensity using the statistical framework of generalized linear models. By duality, we establish an information-theoretic interpretation using Bregman divergences. We demonstrate how this formulation couples in a principled way image restoration tasks such as denoising, deblurring (deconvolution), and inpainting with segmentation. We present an alternating minimization algorithm to solve the resulting composite photometric/geometric inverse problem. We use Fisher scoring to solve the photometric problem and to provide asymptotic uncertainty estimates. We derive the shape gradient of our data-fitting energy and investigate convex relaxation for the geometric problem. We introduce a new alternating split-Bregman strategy to solve the resulting convex problem and present experiments and comparisons on both synthetic and real-world images.

Cell-free transmission of human adenovirus by passive mass transfer in cell culture simulated in a computer model

Viruses spread between cells, tissues, and organisms by cell-free and cell-cell transmissions. Both mechanisms enhance disease development, but it is difficult to distinguish between them. Here, we analyzed the transmission mode of human adenovirus (HAdV) in monolayers of epithelial cells by wet laboratory experimentation and a computer simulation. Using live-cell fluorescence microscopy and replication-competent HAdV2 expressing green fluorescent protein, we found that the spread of infection invariably occurred after cell lysis. It was affected by convection and blocked by neutralizing antibodies but was independent of second-round infections. If cells were overlaid with agarose, convection was blocked and round plaques developed around lytic infected cells. Infected cells that did not lyse did not give rise to plaques, highlighting the importance of cell-free transmission. Key parameters for cell-free virus transmission were the time from infection to lysis, the dose of free viruses determining infection probability, and the diffusion of single HAdV particles in aqueous medium. With these parameters, we developed an in silico model using multiscale hybrid dynamics, cellular automata, and particle strength exchange. This so-called white box model is based on experimentally determined parameters and reproduces viral infection spreading as a function of the local concentration of free viruses. These analyses imply that the extent of lytic infections can be determined by either direct plaque assays or can be predicted by calculations of virus diffusion constants and modeling.

Exact on-lattice stochastic reaction-diffusion simulations using partial-propensity methods

Stochastic reaction-diffusion systems frequently exhibit behavior that is not predicted by deterministic simulation models. Stochastic simulation methods, however, are computationally expensive. We present a more efficient stochastic reaction-diffusion simulation algorithm that samples realizations from the exact solution of the reaction-diffusion master equation. The present algorithm, called partial-propensity stochastic reaction-diffusion (PSRD) method, uses an on-lattice discretization of the reaction-diffusion system and relies on partial-propensity methods for computational efficiency. We describe the algorithm in detail, provide a theoretical analysis of its computational cost, and demonstrate its computational performance in benchmarks. We then illustrate the application of PSRD to two- and three-dimensional pattern-forming Gray-Scott systems, highlighting the role of intrinsic noise in these systems.

Virtual Cell: computational tools for modeling in cell biology

The Virtual Cell (VCell) is a general computational framework for modeling physicochemical and electrophysiological processes in living cells. Developed by the National Resource for Cell Analysis and Modeling at the University of Connecticut Health Center, it provides automated tools for simulating a wide range of cellular phenomena in space and time, both deterministically and stochastically. These computational tools allow one to couple electrophysiology and reaction kinetics with transport mechanisms, such as diffusion and directed transport, and map them onto spatial domains of various shapes, including irregular three-dimensional geometries derived from experimental images. In this article, we review new robust computational tools recently deployed in VCell for treating spatially resolved models.

Lateral diffusion on tubular membranes: quantification of measurements bias

Single Particle Tracking (SPT) is a powerful technique for the analysis of the lateral diffusion of the lipid and protein components of biological membranes. In neurons, SPT allows the study of the real-time dynamics of receptors for neurotransmitters that diffuse continuously in and out synapses. In the simplest case where the membrane is flat and is parallel to the focal plane of the microscope the analysis of diffusion from SPT data is relatively straightforward. However, in most biological samples the membranes are curved, which complicates analysis and may lead to erroneous conclusions as for the mode of lateral diffusion. Here we considered the case of lateral diffusion in tubular membranes, such as axons, dendrites or the neck of dendritic spines. Monte Carlo simulations allowed us to evaluate the error in diffusion coefficient (D) calculation if the curvature is not taken into account. The underestimation is determined by the diameter of the tubular surface, the frequency of image acquisition and the degree of mobility itself. We found that projected trajectories give estimates that are 25 to 50% lower than the real D in case of 2D-SPT over the tubular surface. The use of 3D-SPT improved the measurements if the frequency of image acquisition was fast enough in relation to the mobility of the molecules and the diameter of the tube. Nevertheless, the calculation of D from the components of displacements in the axis of the tubular structure gave accurate estimate of D, free of geometrical artefacts. We show the application of this approach to analyze the diffusion of a lipid on model tubular membranes and of a membrane-bound GFP on neurites from cultured rat hippocampal neurons.

Shaping the synaptic signal: molecular mobility inside and outside the cleft

Rapid communication in the brain relies on the release and diffusion of small transmitter molecules across the synaptic cleft. How these diffuse signals are transformed into cellular responses is determined by the scatter of target postsynaptic receptors, which in turn depends on receptor movement in cell membranes. Thus, by shaping information transfer in neural circuits, mechanisms that regulate molecular mobility affect nearly every aspect of brain function and dysfunction. Here we review two facets of molecular mobility that have traditionally been considered separately, namely extracellular and intra-membrane diffusion. By focusing on the interplay between these processes we illustrate the remarkable versatility of signal formation in synapses and highlight areas of emerging understanding in the molecular physiology and biophysics of synaptic transmission.

System analysis shows distinct mechanisms and common principles of nuclear envelope protein dynamics

The ER–inner nuclear membrane trafficking of 15 integral membrane proteins followed by FRAP shows distinct ATP- and Ran-dependent translocation mechanisms.

The nuclear envelope contains >100 transmembrane proteins that continuously exchange with the endoplasmic reticulum and move within the nuclear membranes. To better understand the organization and dynamics of this system, we compared the trafficking of 15 integral nuclear envelope proteins using FRAP. A surprising 30-fold range of mobilities was observed. The dynamic behavior of several of these proteins was also analyzed after depletion of ATP and/or Ran, two functions implicated in endoplasmic reticulum–inner nuclear membrane translocation. This revealed that ATP- and Ran-dependent translocation mechanisms are distinct and not used by all inner nuclear membrane proteins. The Ran-dependent mechanism requires the phenylalanine-glycine (FG)-nucleoporin Nup35, which is consistent with use of the nuclear pore complex peripheral channels. Intriguingly, the addition of FGs to membrane proteins reduces FRAP recovery times, and this also depends on Nup35. Modeling of three proteins that were unaffected by either ATP or Ran depletion indicates that the wide range in mobilities could be explained by differences in binding affinities in the inner nuclear membrane.

Abstractions and Middleware for Petascale Computing and Beyond

As high-performance computing moves to the petascale and beyond, a number of algorithmic and software challenges need to be addressed. This paper reviews the main performance-limiting factors in today’s high-performance computing software and outlines a possible new programming paradigm to address them. The proposed paradigm is based on abstract parallel data structures and operations that encapsulate much of the complexity of an application, but still make communication overhead explicit. The authors argue that all numerical simulations can be formulated in terms of the presented abstractions, which thus define an abstract semantic specification language for parallel numerical simulations. Simulations defined in this language can automatically be translated to source code containing the appropriate calls to a middleware that implements the underlying abstractions. Finally, the structure and functionality of such a middleware are outlined while demonstrating its feasibility on the example of the parallel particle-mesh library (PPM).

A Lagrangian particle method for reaction-diffusion systems on deforming surfaces

Reaction-diffusion processes on complex deforming surfaces are fundamental to a number of biological processes ranging from embryonic development to cancer tumor growth and angiogenesis. The simulation of these processes using continuum reaction-diffusion models requires computational methods capable of accurately tracking the geometric deformations and discretizing on them the governing equations. We employ a Lagrangian level-set formulation to capture the deformation of the geometry and use an embedding formulation and an adaptive particle method to discretize both the level-set equations and the corresponding reaction-diffusion. We validate the proposed method and discuss its advantages and drawbacks through simulations of reaction-diffusion equations on complex and deforming geometries.

Experimental-computational evaluation of human bone marrow stromal cell spreading on trabecular bone structures

The clinical application of macro-porous scaffolds for bone regeneration is significantly affected by the problem of insufficient cell colonization. Given the wide variety of different scaffold structures used for tissue engineering it is essential to derive relationships for cell colonization independent of scaffold architecture. To study cell population spreading on 3D structures decoupled from nutrient limitations, an in vitro culture system was developed consisting of thin slices of human trabecular bone seeded with Human Bone Marrow Stromal Cells, combined with dedicated microCT imaging and computational modeling of cell population spreading. Only the first phase of in vitro scaffold colonization was addressed, in which cells migrate and proliferate up to the stage when the surface of the bone is covered as a monolayer, a critical prerequisite for further tissue formation. The results confirm the model's ability to represent experimentally observed cell population spreading. The key advantage of the computational model was that by incorporating complex 3D structure, cell behavior can be characterized quantitatively in terms of intrinsic migration parameters, which could potentially be used for predictions on different macro-porous scaffolds subject to additional experimental validation. This type of modeling will prove useful in predicting cell colonization and improving strategies for skeletal tissue engineering.

Dynamin-dependent membrane drift recruits AMPA receptors to dendritic spines

The surface expression and localization of AMPA receptors (AMPARs) at dendritic spines are tightly controlled to regulate synaptic transmission. Here we show that de novo exocytosis of the GluR2 AMPAR subunit occurs at the dendritic shaft and that new AMPARs diffuse into spines by lateral diffusion in the membrane. However, membrane topology restricts this lateral diffusion. We therefore investigated which mechanisms recruit AMPARs to spines from the shaft and demonstrated that inhibition of dynamin GTPase activity reduced lateral diffusion of membrane-anchored green fluorescent protein and super-ecliptic pHluorin (SEP)-GluR2 into spines. In addition, the activation of synaptic N-methyl-d-aspartate (NMDA) receptors enhanced lateral diffusion of SEP-GluR2 and increased the number of endogenous AMPARs in spines. The NMDA-invoked effects were prevented by dynamin inhibition, suggesting that activity-dependent dynamin-mediated endocytosis within spines generates a net inward membrane drift that overrides lateral diffusion barriers to enhance membrane protein delivery into spines. These results provide a novel mechanistic explanation of how AMPARs and other membrane proteins are recruited to spines by synaptic activity.

Anomalous diffusion induced by cristae geometry in the inner mitochondrial membrane

Diffusion of inner membrane proteins is a prerequisite for correct functionality of mitochondria. The complicated structure of tubular, vesicular or flat cristae and their small connections to the inner boundary membrane impose constraints on the mobility of proteins making their diffusion a very complicated process. Therefore we investigate the molecular transport along the main mitochondrial axis using highly accurate computational methods. Diffusion is modeled on a curvilinear surface reproducing the shape of mitochondrial inner membrane (IM). Monte Carlo simulations are carried out for topologies resembling both tubular and lamellar cristae, for a range of physiologically viable crista sizes and densities. Geometrical confinement induces up to several-fold reduction in apparent mobility. IM surface curvature per se generates transient anomalous diffusion (TAD), while finite and stable values of projected diffusion coefficients are recovered in a quasi-normal regime for short- and long-time limits. In both these cases, a simple area-scaling law is found sufficient to explain limiting diffusion coefficients for permeable cristae junctions, while asymmetric reduction of the junction permeability leads to strong but predictable variations in molecular motion rate. A geometry-based model is given as an illustration for the time-dependence of diffusivity when IM has tubular topology. Implications for experimental observations of diffusion along mitochondria using methods of optical microscopy are drawn out: a non-homogenous power law is proposed as a suitable approach to TAD. The data demonstrate that if not taken into account appropriately, geometrical effects lead to significant misinterpretation of molecular mobility measurements in cellular curvilinear membranes.

Convolution-based one and two component FRAP analysis: theory and application

The method of fluorescence redistribution after photobleaching (FRAP) is increasingly receiving interest in biological applications as it is nowadays used not only to determine mobility parameters per se, but to investigate dynamic changes in the concentration or distribution of diffusing molecules. Here, we develop a new simple convolution-based approach to analyze FRAP data using the whole image information. This method does not require information about the timing and localization of the bleaching event but uses the first image acquired directly after photobleaching to calculate the intensity distributions, instead. Changes in pools of molecules with different velocities, which are monitored by applying repetitive FRAP experiments within a single cell, can be analyzed by means of a global model by assuming two global diffusion coefficients with changing portions. We validate the approach by simulation and show that translocation of the YFP-fused PH-domain of phospholipase Cdelta1 can be quantitatively monitored by FRAP analysis in a time-resolved manner. The new FRAP data analysis procedure may be applied to investigate signal transduction pathways using biosensors that change their mobility. An altered mobility in response to the activation of signaling cascades may result either from an altered size of the biosensor, e.g. due to multimerization processes or from translocation of the sensor to an environment with different viscosity.

FAST MONTE CARLO SIMULATION METHODS FOR BIOLOGICAL REACTION-DIFFUSION SYSTEMS IN SOLUTION AND ON SURFACES

Many important physiological processes operate at time and space scales far beyond those accessible to atom-realistic simulations, and yet discrete stochastic rather than continuum methods may best represent finite numbers of molecules interacting in complex cellular spaces. We describe and validate new tools and algorithms developed for a new version of the MCell simulation program (MCell3), which supports generalized Monte Carlo modeling of diffusion and chemical reaction in solution, on surfaces representing membranes, and combinations thereof. A new syntax for describing the spatial directionality of surface reactions is introduced, along with optimizations and algorithms that can substantially reduce computational costs (e.g., event scheduling, variable time and space steps). Examples for simple reactions in simple spaces are validated by comparison to analytic solutions. Thus we show how spatially realistic Monte Carlo simulations of biological systems can be far more cost-effective than often is assumed, and provide a level of accuracy and insight beyond that of continuum methods.

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.

Analysis and interpretation of two-dimensional single-particle tracking microscopy measurements: effect of local surface roughness

Methodological advances in light microscopy have made it possible to record the motions of individual lipid and protein molecules resident in the membrane of living cells down to the nanometer level of precision in the x, y plane. Such measurement of a single molecule's trajectory for a sufficiently long period of time or the measurement of multiple molecules' trajectories for a shorter period of time can in principle provide the necessary information to derive the particle's macroscopic two-dimensional-diffusion coefficient-a quantity of vital biological interest. However, one drawback of the light microscopy procedures used in such experiments is their relatively poor discriminatory capability for determining spatial differences along the z axis in comparison to those in the x, y plane. In this study we used computer simulation to examine the likely effect of local surface roughness over the nanometer to micrometer scale on the determination of diffusion constants in the membrane bilayer by the use of such optical-microscope-based single-particle tracking (SPT) procedures. We specifically examined motion of a single molecule along (i) a locally planar and (ii) a locally rough surface. Our results indicate a need for caution in applying overly simplistic analytical strategies to the analysis of data from SPT measurements and provide upper and lower bounds for the likely degree of error introduced on the basis of surface roughness effects alone. Additionally we present an empirical method based on an autocorrelation function approach that may prove useful in identifying the existence of surface roughness and give some idea of its extent.

Analysis of membrane-localized binding kinetics with FRAP

Interactions between plasma membrane-associated proteins on interacting cells are critical for many important biological processes. Few experimental techniques, however, can accurately determine the association and the dissociation rates between such interacting pairs when the two molecules diffuse on apposing membranes or lipid bilayers. In this study, we give a theoretical description of how and when fluorescence recovery after photobleaching (FRAP) experiments can be used to quantify these reaction rates. We analyze the effect of binding on FRAP recovery curves with a reaction-diffusion model and systematically identify different regimes in the parameter space of the association and the dissociation constants for which the full model simplifies into equivalent one-parameter models. Based on this analysis, we propose an experimental protocol that may be used to identify the kinetic parameters of binding in the appropriate parameter regime. We present simulated experiments illustrating our protocol and lay down guidelines for parameter estimation.

Curvature coupling dependence of membrane protein diffusion coefficients

We consider the lateral diffusion of a protein interacting with the curvature of the membrane. The interaction energy is minimized if the particle is at a membrane position with a certain curvature that agrees with the spontaneous curvature of the particle. We employ stochastic simulations that take into account both the thermal fluctuations of the membrane and the diffusive behavior of the particle. In this study, we neglect the influence of the particle on the membrane dynamics, thus the membrane dynamics agrees with that of a freely fluctuating membrane. Overall, we find that this curvature coupling substantially enhances the diffusion coefficient. We compare the ratio of the projected or measured diffusion coefficient and the free intramembrane diffusion coefficient, which is a parameter of the simulations, with analytical results that rely on several approximations. We find that the simulations always lead to a somewhat smaller diffusion coefficient than that from our analytical approach. A detailed study of the correlations of the forces acting on the particle indicates that the diffusing inclusion tries to follow favorable positions on the membrane such that forces along the trajectory are on average smaller than they would be for random particle positions.

Role of three-dimensional bleach distribution in confocal and two-photon fluorescence recovery after photobleaching experiments

The quantitative analysis of fluorescence perturbation experiments such as fluorescence recovery after photobleaching (FRAP) requires suitable analytical models to be developed. When diffusion in 3D environments is considered, the description of the initial condition produced by the perturbation (i.e., the photobleaching of a selected region in FRAP) represents a crucial aspect. Though it is widely known that bleaching profiles approximations can lead to errors in quantitative FRAP measurements, a detailed analysis of the sources and the effects of these approximations has never been conducted until now. In this study, we measured the experimental 3D bleaching distributions obtained in conventional and two-photon excitation schemes and analyzed the deviations from the ideal cases usually adopted in FRAP experiments. In addition, we considered the non-first-order effects generated by the high energy pulses usually delivered in FRAP experiments. These data have been used for finite-element simulations mimicking FRAP experiments on free diffusing molecules and compared with FRAP model curves based on the ideal bleach distributions. The results show that two-photon excitation more closely fits ideal bleaching patterns even in the event of fluorescence saturation, achieving estimations of diffusion coefficients within 20% accuracy of the correct value.

Diffusion on a Curved Surface Coupled to Diffusion in the Volume: Application to Cell Biology

An algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume, a problem often arising in cell biology. It applies to pixilated surfaces obtained from experimental images and performs at low computational cost. In the method, the Laplace-Beltrami operator is approximated locally by the Laplacian on the tangential plane and then a finite volume discretization scheme based on a Voronoi decomposition is applied. Convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2. The method is applied to a cell-biological problem where a signaling molecule, G-protein Rac, cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior. Simulations on realistic cell geometry are performed to validate, and determine the accuracy of, a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching. The method is implemented within the Virtual Cell computational framework freely accessible at www.vcell.org.

Inferring the lifetime of endosomal protein complexes by fluorescence recovery after photobleaching

Cellular signal transduction is dynamic, with signaling proteins continually associating and dissociating into and from protein complexes. Here we present a fluorescence recovery after photobleaching technique to determine the lifetime of protein complexes on intracellular vesicles. We use Bayesian inference based on a model that includes the diffusion of cytosolic proteins and their interaction with membrane-bound receptors. Our analysis is general: we incorporate prior information on protein diffusion, measurement error in determining fluorescence intensities, corrections for photobleaching, and variation in the concentration of receptors between vesicles. We apply our method to the complexes formed on endosomes by G-protein-coupled receptors and the protein beta-arrestin. The lifetime of these complexes determines the recycling rate of the receptors. We find in mammalian cells that the bradykinin type 2 receptor and beta-arrestin2 complex has a lifetime of approximately 2 min, while the angiotensin II type 1A receptor and beta-arrestin2 complex has a lifetime of approximately 6 min. As well as allowing quantitative comparisons between experiments, our method provides in vivo parameters for systems biology simulations of signaling networks.

Diffusion on ruffled membrane surfaces

We present a position Langevin equation for overdamped particle motion on rough two-dimensional surfaces. A Brownian dynamics algorithm is suggested to evolve this equation numerically, allowing for the prediction of effective (projected) diffusion coefficients over corrugated surfaces. In the case of static surface roughness, we find that a simple area-scaling prediction for the projected diffusion coefficient leads to seemingly quantitative agreement with numerical results. To study the effect of dynamic surface evolution on the diffusive process, we consider particle diffusion over a thermally fluctuating elastic membrane. Surface fluctuation has the effect of increasing the effective diffusivity toward a limiting annealed-surface value discussed previously. We argue that protein motion over cell surfaces spans a variety of physical regimes, making it impossible to identify a single approximation scheme appropriate to all measurements of interest.

Continuous fluorescence microphotolysis and correlation spectroscopy using 4Pi microscopy

Continuous fluorescence microphotolysis (CFM) and fluorescence correlation spectroscopy (FCS) permit measurement of molecular mobility and association reactions in single living cells. CFM and FCS complement each other ideally and can be realized using identical equipment. So far, the spatial resolution of CFM and FCS was restricted by the resolution of the light microscope to the micrometer scale. However, cellular functions generally occur on the nanometer scale. Here, we develop the theoretical and computational framework for CFM and FCS experiments using 4Pi microscopy, which features an axial resolution of approximately 100 nm. The framework, taking the actual 4Pi point spread function of the instrument into account, was validated by measurements on model systems, employing 4Pi conditions or normal confocal conditions together with either single- or two-photon excitation. In all cases experimental data could be well fitted by computed curves for expected diffusion coefficients, even when the signal/noise ratio was small due to the small number of fluorophores involved.

Modulation of lateral diffusion in the plasma membrane by protein density

The rate of lateral diffusion of proteins over micron-scale distances in the plasma membrane (PM) of mammalian cells is much slower than in artificial membranes [1, 2]. Different models have been advanced to account for this discrepancy. They invoke either effects on the apparent viscosity of cell membranes through, for example, protein crowding [3, 4], or a role for cortical factors such as actin or spectrin filaments [1]. Here, we use photobleaching to test specific predictions of these models [5]. Neither loss of detectable cortical actin nor knockdown of spectrin expression has any effect on diffusion. Disruption of the PM by formation of ventral membrane sheets or permeabilization induces aggregation of membrane proteins, with a concomitant increase in rates of diffusion for the nonaggregated fraction. In addition, procedures that directly increase or decrease the total protein content of the PM in live cells cause reciprocal changes in lateral diffusion rates. Our data imply that slow diffusion over micron-scale distances is an intrinsic property of the membrane itself and that the density of proteins within the membrane is a significant parameter in determining rates of lateral diffusion.

Theoretically predicted effects of Gaussian curvature on lateral diffusion of membrane molecules

Lateral diffusion on curved biological membranes has been studied theoretically and experimentally. However, how membrane geometries influence the diffusion process remains unclear. Here we show the significance of Gaussian curvature by numerically solving the diffusion equation in a geodesic polar coordinate system with regard to several types of surfaces including elliptic and hyperbolic paraboloids. On surfaces where Gaussian curvature has positive and negative values, diffusion is slower and faster than on the plane, respectively. The deviation from the normal diffusion on the plane tends to get larger as the absolute value of Gaussian curvature increases. Diffusion is anisotropic at a surface region where the normal curvature is anisotropic and Gaussian curvature has nonzero values. The anisotropy can be classified into several types according to whether diffusion is the fastest or the slowest in the principal directions. In the case of diffusion on spheroids, the limited area of a closed surface reduces the diffusion rate so greatly that the slowdown effects of positive values of Gaussian curvature are concealed. Analysis of the diffusion equation suggests that Gaussian curvature causes slowed or accelerated diffusion and anisotropic diffusion in any type of surface. Furthermore, it is discussed the degree to which Gaussian curvature influences diffusive phenomena taking place in real membranes through such effects. These results provide a different image of biological membranes that lateral diffusion of membrane molecules is usually anisotropic and the diffusion rate kaleidoscopically changes according to place.

Hybrid simulations of lateral diffusion in fluctuating membranes

In this paper we introduce a method to simulate lateral diffusion of inclusions in a fluctuating membrane. The regarded systems are governed by two dynamic processes: the height fluctuations of the membrane and the diffusion of the inclusion along the membrane. While membrane fluctuations can be expressed in terms of a dynamic equation which follows from the Helfrich Hamiltonian, the dynamics of the diffusing particle is described by a Langevin or Smoluchowski equation. In the latter equations, the curvature of the surface needs to be accounted for, which makes particle diffusion a function of membrane fluctuations. In our scheme these coupled dynamic equations, the membrane equation and the Langevin equation for the particle, are numerically integrated to simulate diffusion in a membrane. The simulations are used to study the ratio of the diffusion coefficient projected on a flat plane and the intramembrane diffusion coefficient for the case of free diffusion. We compare our results with recent analytical results that employ a preaveraging approximation and analyze the validity of this approximation. A detailed simulation study of the relevant correlation functions reveals a surprisingly large range where the approximation is applicable.

Modeling synaptic dynamics driven by receptor lateral diffusion

The synaptic weight between a pre- and a postsynaptic neuron depends in part on the number of postsynaptic receptors. On the surface of neurons, receptors traffic by random motion in and out from a microstructure called the postsynaptic density (PSD). In the PSD, receptors can be stabilized at the membrane when they bind to scaffolding proteins. We propose a mathematical model to compute the postsynaptic counterpart of the synaptic weight based on receptor trafficking. We take into account the receptor fluxes at the PSD, which can be regulated by neuronal activity, and the interactions of receptors with the scaffolding molecules. Using a Markovian approach, we estimate the mean and the fluctuations of the number of bound receptors. When the number of receptors is large, a deterministic system is also derived. Moreover, these equations can be used, for example, to fit fluorescence-recovery-after-photobleaching experiments to determine, in living neurons, the chemical binding constants for the receptors/scaffolding molecules interaction at synapses.