High-resolution models of motion of macromolecules in cell membranes (original) (raw)
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Time Series Analysis of Particle Tracking Data for Molecular Motion on the Cell Membrane
Bulletin of Mathematical Biology, 2009
Biophysicists use single particle tracking (SPT) methods to probe the dynamic behavior of individual proteins and lipids in cell membranes. The mean squared displacement (MSD) has proven to be a powerful tool for analyzing the data and drawing conclusions about membrane organization, including features like lipid rafts, protein islands, and confinement zones defined by cytoskeletal barriers. Here, we implement time series analysis as a new analytic tool to analyze further the motion of membrane proteins. The experimental data track the motion of 40 nm gold particles bound to Class I major histocompatibility complex (MHCI) molecules on the membranes of mouse hepatoma cells. Our first novel result is that the tracks are significantly autocorrelated. Because of this, we developed linear autoregressive models to elucidate the autocorrelations. Estimates of the signal to noise ratio for the models show that the autocorrelated part of the motion is significant. Next, we fit the probability distributions of jump sizes with four different models. The first model is a general Weibull distribution that shows that the motion is characterized by an excess of short jumps as compared to a normal random walk. We also fit the data with a chi distribution which provides a natural estimate of the dimension d of the space in which a random walk is occurring. For the biological data, the estimates satisfy 1 < d < 2, implying that particle motion is not confined to a line, but also does not occur freely in the plane. The dimension gives a quantitative estimate of the amount of nanometer scale obstruction met by a diffusing molecule. We introduce a new distribution and use the generalized extreme value distribution to show that the biological data also have an excess of long jumps as compared to normal diffusion. These fits provide novel estimates of the microscopic diffusion constant. Previous MSD analyses of SPT data have provided evidence for nanometer-scale confinement zones that restrict lateral diffusion, supporting the notion that plasma membrane organization is highly structured. Our demonstration that membrane protein motion is autocorrelated and is characterized by an excess of both short and long jumps reinforces * Corresponding author.
Biophysical Journal, 2008
We propose here an approach for the analysis of single-molecule trajectories which is based on a comprehensive comparison of an experimental data set with multiple Monte Carlo simulations of the diffusion process. It allows quantitative data analysis, particularly whenever analytical treatment of a model is infeasible. Simulations are performed on a discrete parameter space and compared with the experimental results by a nonparametric statistical test. The method provides a matrix of p-values that assess the probability for having observed the experimental data at each setting of the model parameters. We show the testing approach for three typical situations observed in the cellular plasma membrane: i), free Brownian motion of the tracer, ii), hop diffusion of the tracer in a periodic meshwork of squares, and iii), transient binding of the tracer to slowly diffusing structures. By plotting the p-value as a function of the model parameters, one can easily identify the most consistent parameter settings but also recover mutual dependencies and ambiguities which are difficult to determine by standard fitting routines. Finally, we used the test to reanalyze previous data obtained on the diffusion of the glycosylphosphatidylinositol-protein CD59 in the plasma membrane of the human T24 cell line.
Non-Brownian diffusion in lipid membranes: Experiments and simulations
Biochimica et Biophysica Acta (BBA) - Biomembranes, 2016
The dynamics of constituents and the surface response of cellular membranes-also in connection to the binding of various particles and macromolecules to the membrane-are still a matter of controversy in the membrane biophysics community, particularly with respect to crowded membranes of living biological cells. We here put into perspective recent single particle tracking experiments in the plasma membranes of living cells and supercomputing studies of lipid bilayer model membranes with and without protein crowding. Special emphasis is put on the observation of anomalous, non-Brownian diffusion of both lipid molecules and proteins embedded in the lipid bilayer. While single component, pure lipid bilayers in simulations exhibit only transient anomalous diffusion of lipid molecules on nanosecond time scales, the persistence of anomalous diffusion becomes significantly longer ranged on the addition of disorder-through the addition of cholesterol or proteins-and on passing of the membrane lipids to the gel phase. Concurrently, experiments demonstrate the anomalous diffusion of membrane embedded proteins up to macroscopic time scales in the minute time range. Particular emphasis will be put on the physical character of the anomalous diffusion, in particular, the occurrence of ageing observed in the experiments-the effective diffusivity of the measured particles is a decreasing function of time. Moreover, we present results for the time dependent local scaling exponent of the mean squared displacement of the monitored particles. Recent results finding deviations from the commonly assumed Gaussian diffusion patterns in protein crowded membranes are reported. The properties of the displacement autocorrelation function of the lipid molecules are discussed in the light of their appropriate physical anomalous diffusion models, both for non-crowded and crowded membranes. In the last part of this review we address the upcoming field of membrane distortion by elongated membrane-binding particles. We discuss how membrane compartmentalisation and the particle-membrane binding energy may impact the dynamics and response of lipid membranes. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.
Inferring diffusion in single live cells at the single-molecule level
2013
Abstract The movement of molecules inside living cells is a fundamental feature of biological processes. The ability to both observe and analyse the details of molecular diffusion in vivo at the single-molecule and single-cell level can add significant insight into understanding molecular architectures of diffusing molecules and the nanoscale environment in which the molecules diffuse.
Diffusion within the Cytoplasm: A Mesoscale Model of Interacting Macromolecules
Biophysical Journal, 2014
Recent experiments carried out in the dense cytoplasm of living cells have highlighted the importance of proteome composition and nonspecific intermolecular interactions in regulating macromolecule diffusion and organization. Despite this, the dependence of diffusion-interaction on physicochemical properties such as the degree of poly-dispersity and the balance between steric repulsion and nonspecific attraction among macromolecules was not systematically addressed. In this work, we study the problem of diffusion-interaction in the bacterial cytoplasm, combining theory and experimental data to build a minimal coarse-grained representation of the cytoplasm, which also includes, for the first time to our knowledge, the nucleoid. With stochastic molecular-dynamics simulations of a virtual cytoplasm we are able to track the single biomolecule motion, sizing from 3 to 80 nm, on submillisecond-long trajectories. We demonstrate that the size dependence of diffusion coefficients, anomalous exponents, and the effective viscosity experienced by biomolecules in the cytoplasm is fine-tuned by the intermolecular interactions. Accounting only for excluded volume in these potentials gives a weaker size-dependence than that expected from experimental data. On the contrary, adding nonspecific attraction in the range of 1-10 thermal energy units produces a stronger variation of the transport properties at growing biopolymer sizes. Normal and anomalous diffusive regimes emerge straightforwardly from the combination of high macromolecular concentration, poly-dispersity, stochasticity, and weak nonspecific interactions. As a result, small biopolymers experience a viscous cytoplasm, while the motion of big ones is jammed because the entanglements produced by the network of interactions and the entropic effects caused by poly-dispersity are stronger.
Quantifying the Influence of the Crowded Cytoplasm on Small Molecule Diffusion
Cytosolic crowding can influence the thermodynamics and kinetics of in vivo chemical reactions. Most significantly, proteins and nucleic acid crowders reduce the accessible volume fraction, ϕ, available to a diffusing substrate, thereby reducing its effective diffusion rate, D eff , relative to its rate in bulk solution. However, D eff can be further hindered or even enhanced, when long-range crowder/diffuser interactions are significant. To probe these effects, we numerically estimated D eff values for small, charged molecules in representative, cytosolic protein lattices up to 0.1 × 0.1 × 0.1 μm 3 in volume via the homogenized Smoluchowski electro-diffusion equation. We further validated our predictions against D eff estimates from ϕ-dependent analytical relationships, such as the Maxwell−Garnett (MG) bound, as well as explicit solutions of the time-dependent electro-diffusion equation. We find that in typical, moderately crowded cell cytoplasm (ϕ ≈ 0.8), D eff is primarily determined by ϕ; in other words, diverse protein shapes and heterogeneous distributions only modestly impact D eff. However, electrostatic interactions between diffusers and crowders, particularly at low electrolyte ionic strengths, can substantially modulate D eff. These findings help delineate the extent that cytoplasmic crowders influence small molecule diffusion, which ultimately may shape the efficiency and timing of intracellular signaling pathways. More generally, the quantitative agreement between computationally expensive solutions of the time-dependent electro-diffusion equation and its comparatively cheaper homogenized form suggest that the latter is a broadly effective model for diffusion in wide-ranging, crowded biological media. ■ INTRODUCTION Intracellular biochemical reactions commonly rely on the diffusion of charged, small molecules 1 between regions where substrates are stored or synthesized to where they are ultimately utilized. These intracellular reactions can be strongly influenced by effective diffusion rates of their substrates, 2 which may be depressed by up to an order of magnitude relative to unrestricted diffusion in bulk solution. 3,4 In part, protein-, carbohydrate-, and nucleic acid−based " crowders " restrict the intracellular volume accessible to diffusing substrates and thereby reduce their rates of transport (reviewed here 5). However, additional factors beyond volume exclusion can influence the mobility of molecular diffusers. These factors include the distribution and activity of proteins or charged surfaces of organelles that selectively bind substrates, as well as the substrate's shape and charge. 3,6−8 Improved quantitative predictions that delineate the relative contributions of these factors on shaping biomolecule diffusion is a challenging, but necessary, endeavor for understanding biochemical reactions and signaling in vitro and in vivo. 9−11 A wide range of experimental and simulation approaches have provided considerable insight into the effective diffusion rates of small molecules involved in intracellular signaling (see reviews 3,12). Methods such as fluorescence or raster image correlation spectroscopy, 13,14 and nuclear magnetic resonance 15−17 have yielded refined estimates for diffusion rates of small molecules such as calcium, magnesium, and AMP in crowded intracellular media. A common theme emerging from these studies is that small molecule diffusion rates are substantially smaller than measurements in crowder-free solutions and are frequently anisotropic, owing to the organization of intracellular structures, such as actin filament lattices. In part, these reductions in diffusion rates can be attributed to a reduced free volume fraction, 3 although the nature of substrate/crowder interactions can substantially strongly modulate the effective rate. 6 However, given the considerable variations in the sizes, molecular composition, and electrostatic properties of cytoplasmic crowders, isolating the contributions of each factor via experimental means is challenging. 9 In this regard, computational modeling is a strong complement to experimental techniques for describing the molecular underpinnings of experimentally observed diffusion rates and their influence on intracellular signaling. 9,18 A variety of computational approaches for modeling diffusion in crowded domains has been reported in the literature. 18 The most common of which are based on explicit, particle-based representations of the diffuser or both diffuser and crowders, lattice-based models that assume discrete, finite-length hops for
Diffusion in Condensed Matter, 2005
Nature is always in motion. As simple as it is, this statement is true in the sense that numerous phenomena in living systems are characterized by nonequilibrium. Our muscles are in constant need of energy provided by the metabolic pathway, the blood flows in our veins as long as the heart keeps going, and yet old cells are constantly being replaced by fresh ones as a typical life time of a cell is on the order of one day. The state of living beings can therefore only rarely be described by equilibrium. However, even if a true equilibrium were possible in living systems, we would find spontaneous thermal fluctuations to occur around the equilibrium state, again implying that the matter were moving in time.
Computer Simulations of Protein Diffusion in Compartmentalized Cell Membranes
Biophysical Journal, 2009
The diffusion of proteins in the cell membrane is investigated using computer simulations of a two-dimensional model. The membrane is assumed to be divided into compartments, with adjacent compartments separated by a barrier of stationary obstacles. Each compartment contains traps represented by stationary attractive disks. Depending on their size, these traps are intended to model either smaller compartments or binding sites. The simulations are intended to model the doublecompartment model, which has been used to interpret single molecule experiments in normal rat kidney cells, where five regimes of transport are observed. The simulations show, however, that five regimes are observed only when there is a large separation between the sizes of the traps and large compartments, casting doubt on the double compartment model for the membrane. The diffusive behavior is sensitive to the concentration and size of traps and the strength of the barrier between compartments suggesting that the diffusion of proteins can be effectively used to characterize the structure of the membrane.
Diffusion through thin membranes: Modeling across scales
Physical Review E, 2016
From macroscopic to microscopic scales it is demonstrated that diffusion through membranes can be modeled using specific boundary conditions across them. The membranes are here considered thin in comparison to the overall size of the system. In a macroscopic scale the membrane is introduced as a transmission boundary condition, which enables an effective modeling of systems that involve multiple scales. In a mesoscopic scale, a numerical lattice-Boltzmann scheme with a partial-bounceback condition at the membrane is proposed and analyzed. It is shown that this mesoscopic approach provides a consistent approximation of the transmission boundary condition. Furthermore, analysis of the mesoscopic scheme gives rise to an expression for the permeability of a thin membrane as a function of a mesoscopic transmission parameter. In a microscopic model, the mean waiting time for a passage of a particle through the membrane is in accordance with this permeability. Numerical results computed with the mesoscopic scheme are then compared successfully with analytical solutions derived in a macroscopic scale, and the membrane model introduced here is used to simulate diffusive transport between the cell nucleus and cytoplasm through the nuclear envelope in a realistic cell model based on fluorescence microscopy data. By comparing the simulated fluorophore transport to the experimental one, we determine the permeability of the nuclear envelope of HeLa cells to enhanced yellow fluorescent protein.