Minimal mesoscale model for protein-mediated vesiculation in clathrin-dependent endocytosis

Boundary Flow-Induced Membrane Tubulation Under Turgor Pressures

During clathrin-mediated endocytosis in yeast cells, a small patch of flat membrane is deformed into a tubular shape. It is generally believed that the tubulation is powered by actin polymerization. However, studies based on quantitative measurement of the actin molecules suggest that they are not sufficient to produce the forces to overcome the high turgor pressure inside of the cell. In this paper, we model the membrane as a viscous 2D fluid with elasticity and study the dynamic membrane deformation powered by a boundary lipid flow under osmotic pressure. We find that in the absence pressure, the lipid flow drives the membrane into a spherical shape or a parachute shape. The shapes over time exhibit self-similarity. The presence of pressure transforms the membrane into a tubular shape that elongates almost linearly with time and the self-similarity between shapes at different times is lost. Furthermore, the width of the tube is found to scale inversely to the cubic root of the pressure, and the tension across the membrane is negative and scales to the cubic root squared of the pressure. Our results demonstrate that boundary flow powered by myosin motors, as a new way to deform the membrane, could be a supplementary mechanism to actin polymerization to drive endocytosis in yeast cells.

Enzyme-Regulated Non-Thermal Fluctuations Enhance Ligand Diffusion and Receptor-Mediated Endocytosis

Active enzymes during catalyzing chemical reactions, have been found to generate significant mechanical fluctuations, which can influence the dynamics of their surroundings. These phenomena open new avenues for controlling mass transport in complex and dynamically inhomogeneous environments through localized chemical reactions. To explore this potential, we studied the uptake of transferrin molecules in retinal pigment epithelium (RPE) cells via clathrin-mediated endocytosis. In the presence of enzyme catalysis in the extracellular matrix, we observed a significant enhancement in the transport of fluorophore-tagged transferrin inside the cells. Fluorescence correlation spectroscopy measurements showed substantial increase in transferrin diffusion in the presence of active fluctuations. This study sheds light on the possibility that enzyme-substrate reactions within the extracellular matrix may induce long-range mechanical influences, facilitating targeted material delivery within intracellular milieu more efficiently than passive diffusion. These insights are expected to contribute to the development of better therapeutic strategies by overcoming limitations imposed by slow molecular diffusion under complex environments.

Asymmetric crowders and membrane morphology at the nexus of intracellular trafficking and oncology

A definitive understanding of the interplay between protein binding/migration and membrane curvature evolution is emerging but needs further study. The mechanisms defining such phenomena are critical to intracellular transport and trafficking of proteins. Among trafficking modalities, exosomes have drawn attention in cancer research as these nano-sized naturally occurring vehicles are implicated in intercellular communication in the tumor microenvironment, suppressing anti-tumor immunity and preparing the metastatic niche for progression. A significant question in the field is how the release and composition of tumor exosomes are regulated. In this perspective article, we explore how physical factors such as geometry and tissue mechanics regulate cell cortical tension to influence exosome production by co-opting the biophysics as well as the signaling dynamics of intracellular trafficking pathways and how these exosomes contribute to the suppression of anti-tumor immunity and promote metastasis. We describe a multiscale modeling approach whose impact goes beyond the fundamental investigation of specific cellular processes toward actual clinical translation. Exosomal mechanisms are critical to developing and approving liquid biopsy technologies, poised to transform future non-invasive, longitudinal profiling of evolving tumors and resistance to cancer therapies to bring us one step closer to the promise of personalized medicine.

Free energy calculations for membrane morphological transformations and insights to physical biology and oncology

In this chapter, we aim to bridge basic molecular and cellular principles surrounding membrane curvature generation with rewiring of cellular signals in cancer through multiscale models. We describe a general framework that integrates signaling with other cellular functions like trafficking, cell-cell and cell-matrix adhesion, and motility. The guiding question in our approach is: how does a physical change in cell membrane configuration caused by external stimuli (including those by the extracellular microenvironment) alter trafficking, signaling and subsequent cell fate? We answer this question by constructing a modeling framework based on stochastic spatial continuum models of cell membrane deformations. We apply this framework to explore the link between trafficking, signaling in the tumor microenvironment, and cell fate. At each stage, we aim to connect the results of our predictions with cellular experiments.

How clathrin-coated pits control nanoparticle avidity for cells

The paramount relevance of clathrin-coated pits (CCPs) to receptor-mediated endocytosis of nanoparticles, extracellular vesicles, and viruses has made them the focus of many studies; however, the role of CCP geometry in the ligand-receptor interactions between multivalent nanoparticles and cells has not been investigated. We hypothesized the general dependence of nanoparticle binding energy on local membrane curvature to be expandable to the specific case of ligand-functionalized nanoparticles binding cell membranes, in the sense that membrane structures whose curvature matches that of the particle (e.g., CCPs) signficantly contribute to binding avidity. We investigated this hypothesis with nanoparticles that bind multivalently to angiotensin II receptor type 1, which is subject to clathrin-mediated endocytosis. When we used cholesterol extraction to prevent the action of CCPs, we found a 67 to 100-fold loss in avidity. We created a theoretical model that predicts this decrease based on the loss of ligand-receptor interactions when CCPs, which perfectly match nanoparticle geometry, are absent. Our findings shed new light on how cells "see" nanoparticles. The presence or absence of CPPs is so influential on how cells interact with nanoparticles that the number of particles required to be visible to cells changes by two orders of magnitude depending on CCP presence.

Mem3DG: Modeling membrane mechanochemical dynamics in 3D using discrete differential geometry

Biomembranes adopt varying morphologies that are vital to cellular functions. Many studies use computational modeling to understand how various mechanochemical factors contribute to membrane shape transformations. Compared with approximation-based methods (e.g., finite element method [FEM]), the class of discrete mesh models offers greater flexibility to simulate complex physics and shapes in three dimensions; its formulation produces an efficient algorithm while maintaining coordinate-free geometric descriptions. However, ambiguities in geometric definitions in the discrete context have led to a lack of consensus on which discrete mesh model is theoretically and numerically optimal; a bijective relationship between the terms contributing to both the energy and forces from the discrete and smooth geometric theories remains to be established. We address this and present an extensible framework, Mem3DG, for modeling 3D mechanochemical dynamics of membranes based on discrete differential geometry (DDG) on triangulated meshes. The formalism of DDG resolves the inconsistency and provides a unifying perspective on how to relate the smooth and discrete energy and forces. To demonstrate, Mem3DG is used to model a sequence of examples with increasing mechanochemical complexity: recovering classical shape transformations such as 1) biconcave disk, dumbbell, and unduloid; and 2) spherical bud on spherical, flat-patch membrane; investigating how the coupling of membrane mechanics with protein mobility jointly affects phase and shape transformation. As high-resolution 3D imaging of membrane ultrastructure becomes more readily available, we envision Mem3DG to be applied as an end-to-end tool to simulate realistic cell geometry under user-specified mechanochemical conditions.

Data driven and biophysical insights into the regulation of trafficking vesicles by extracellular matrix stiffness

Biomechanical signals from remodeled extracellular matrix (ECM) promote tumor progression. Here, we show that cell-matrix and cell-cell communication may be inherently linked and tuned through mechanisms of mechanosensitive biogenesis of trafficking vesicles. Pan-cancer analysis of cancer cells' mechanical properties (focusing primarily on cell stiffness) on substrates of varied stiffness and composition elucidated a heterogeneous cellular response to mechanical stimuli. Through machine learning, we identified a fingerprint of cytoskeleton-related proteins that accurately characterize cell stiffness in different ECM conditions. Expression of their respective genes correlates with patient prognosis across different tumor types. The levels of selected cytoskeleton proteins indicated that cortical tension mirrors the increase (or decrease) in cell stiffness with a change in ECM stiffness. A mechanistic biophysical model shows that the tendency for curvature generation by curvature-inducing proteins has an ultrasensitive dependence on cortical tension. This study thus highlights the effect of ECM stiffness, mediated by cortical tension, in modulating vesicle biogenesis.

Quantification of Curvature Sensing Behavior of Curvature-Inducing Proteins on Model Wavy Substrates

Curvature-inducing proteins are involved in a variety of membrane remodeling processes in the cell. Several in vitro experiments have quantified the curvature sensing behavior of these proteins in model lipid systems. One such system consists of a membrane bilayer laid atop a wavy substrate (Hsieh in Langmuir 28:12838-12843, 2012). In these experiments, the bilayer conforms to the wavy substrate, and curvature-inducing proteins show preferential segregation on the wavy membrane. Using a mesoscale computational membrane model based on the Helfrich Hamiltonian, here we present a study which analyzes the curvature sensing characteristics of this membrane-protein system, and elucidates key physical principles governing protein segregation on the wavy substrate and other in vitro systems. In this article we compute the local protein densities from the free energy landscape associated with membrane remodeling by curvature-inducing proteins. In specific, we use the Widom insertion technique to compute the free energy landscape for an inhomogeneous system with spatially varying density and the results obtained with this minimal model show excellent agreement with experimental studies that demonstrate the association between membrane curvature and local protein density. The free energy-based framework employed in this study can be used for different membrane morphologies and varied protein characteristics to gain mechanistic insights into protein sorting on membranes.

Dynamic interplay between cell membrane tension and clathrin-mediated endocytosis

Deformability of the plasma membrane, the outermost surface of metazoan cells, allows cells to be dynamic, mobile and flexible. Factors that affect this deformability, such as tension on the membrane, can regulate a myriad of cellular functions, including membrane resealing, cell motility, polarisation, shape maintenance, membrane area control and endocytic vesicle trafficking. This review focuses on mechanoregulation of clathrin-mediated endocytosis (CME). We first delineate the origins of cell membrane tension and the factors that yield to its spatial and temporal fluctuations within cells. We then review the recent literature demonstrating that tension on the membrane is a fast-acting and reversible regulator of CME. Finally, we discuss tension-based regulation of endocytic clathrin coat formation during physiological processes.

Nonaxisymmetric Shapes of Biological Membranes from Locally Induced Curvature

In various biological processes such as endocytosis and caveolae formation, the cell membrane is locally deformed into curved morphologies. Previous models to study membrane morphologies resulting from locally induced curvature often only consider the possibility of axisymmetric shapes-an indeed unphysical constraint. Past studies predict that the cell membrane buds at low resting tensions and stalls at a flat pit at high resting tensions. In this work, we lift the restriction to axisymmetry to study all possible membrane morphologies. Only if the resting tension of the membrane is low, we reproduce axisymmetric membrane morphologies. When the resting tension is moderate to high, we show that 1) axisymmetric membrane pits are unstable and 2) nonaxisymmetric ridge-shaped structures are energetically favorable. Furthermore, we find the interplay between intramembrane viscous flow and the rate of induced curvature affects the membrane's ability to transition into nonaxisymmetric ridges and axisymmetric buds. In particular, we show that axisymmetric buds are favored when the induced curvature is rapidly increased, whereas nonaxisymmetric ridges are favored when the curvature is slowly increased. Our results hold relevant implications for biological processes such as endocytosis and physical phenomena like phase separation in lipid bilayers.

Cell-type-specific differences in KDEL receptor clustering in mammalian cells

In eukaryotic cells, KDEL receptors (KDELRs) facilitate the retrieval of endoplasmic reticulum (ER) luminal proteins from the Golgi compartment back to the ER. Apart from the well-documented retention function, recent findings reveal that the cellular KDELRs have more complex roles, e.g. in cell signalling, protein secretion, cell adhesion and tumorigenesis. Furthermore, several studies suggest that a sub-population of KDELRs is located at the cell surface, where they could form and internalize KDELR/cargo clusters after K/HDEL-ligand binding. However, so far it has been unclear whether there are species- or cell-type-specific differences in KDELR clustering. By comparing ligand-induced KDELR clustering in different mouse and human cell lines via live cell imaging, we show that macrophage cell lines from both species do not develop any clusters. Using RT-qPCR experiments and numerical analysis, we address the role of KDELR expression as well as endocytosis and exocytosis rates on the receptor clustering at the plasma membrane and discuss how the efficiency of directed transport to preferred docking sites on the membrane influences the exponent of the power-law distribution of the cluster size.

Emergent membrane morphologies in relaxed and tense membranes in presence of reversible adhesive pinning interactions

The morphologies of cell membranes, and specifically the local curvature distributions are determined either by its intrinsic components such as lipids and membrane-associated proteins or by the adhesion forces due to membrane interactions with the cytoskeleton, extracellular matrix (ECM) and other cells in the tissue, as well as physical variables such as membrane and frame tensions. We present a computational analysis for a model of pinned membranes based on the dynamically triangulated Monte Carlo (MC) model for membranes. We show that membrane adhesion to ECM or a substrate promotes curvature generation on cell membranes, and this process depends on the excess area, or equivalently membrane tension, and the density of adhesion sites. This biophysics based model predicts adhesion induced biogenesis of microvesicles in cell membranes. For a moderate density of adhesion sites and high excess membrane area, an increase in membrane tension can result in the formation of microvesicles and tubules on the membrane. We also demonstrate the significance of intrinsically curved proteins in promoting vesiculation on pinned membranes. The results presented here are relevant to the understanding of microvesicle biogenesis and curved membrane topographies due to physical factors such as substrate stiffness and ECM interactions.

Biophysics of membrane curvature remodeling at molecular and mesoscopic lengthscales

At the micron scale, where cell organelles display an amazing complexity in their shape and organization, the physical properties of a biological membrane can be better-understood using continuum models subject to thermal (stochastic) undulations. Yet, the chief orchestrators of these complex and intriguing shapes are a specialized class of membrane associating often peripheral proteins called curvature remodeling proteins (CRPs) that operate at the molecular level through specific protein-lipid interactions. We review multiscale methodologies to model these systems at the molecular as well as at the mesoscopic and cellular scales, and also present a free energy perspective of membrane remodeling through the organization and assembly of CRPs. We discuss the morphological space of nearly planar to highly curved membranes, methods to include thermal fluctuations, and review studies that model such proteins as curvature fields to describe the emergent curved morphologies. We also discuss several mesoscale models applied to a variety of cellular processes, where the phenomenological parameters (such as curvature field strength) are often mapped to models of real systems based on molecular simulations. Much insight can be gained from the calculation of free energies of membranes states with protein fields, which enable accurate mapping of the state and parameter values at which the membrane undergoes morphological transformations such as vesiculation or tubulation. By tuning the strength, anisotropy, and spatial organization of the curvature-field, one can generate a rich array of membrane morphologies that are highly relevant to shapes of several cellular organelles. We review applications of these models to budding of vesicles commonly seen in cellular signaling and trafficking processes such as clathrin mediated endocytosis, sorting by the ESCRT protein complexes, and cellular exocytosis regulated by the exocyst complex. We discuss future prospects where such models can be combined with other models for cytoskeletal assembly, and discuss their role in understanding the effects of cell membrane tension and the mechanics of the extracellular microenvironment on cellular processes.

Clathrin polymerization exhibits high mechano-geometric sensitivity

How tension modulates cellular transport has become a topic of interest in the recent past. However, the effect of tension on clathrin assembly and vesicle growth remains less understood. Here, we use the classical Helfrich theory to predict the energetic cost that clathrin is required to pay to remodel the membrane at different stages of vesicle formation. Our study reveals that this energetic cost is highly sensitive to not only the tension in the membrane but also to the instantaneous geometry of the membrane during shape evolution. Our study predicts a sharp reduction in clathrin coat size in the intermediate tension regime (0.01-0.1 mN m-1). Remarkably, the natural propensity of the membrane to undergo bending beyond the Ω shape causes a significant decrease in the energy needed from clathrin to drive vesicle growth. Our studies in mammalian cells confirm a reduction in clathrin coat size in an increased tension environment. In addition, our findings suggest that the two apparently distinct clathrin assembly modes, namely coated pits and coated plaques, observed in experimental investigations might be a consequence of varied tensions in the plasma membrane. Overall, the mechano-geometric sensitivity revealed in this study might also be at play during the polymerization of other membrane remodeling proteins.

Shape transitions during clathrin-induced endocytosis

Endocytosis is among the most common transport mechanisms which cells employ to receive macromolecules, the so-called cargo, from its extra cellular environment. Clathrin-mediated endocytosis (CME), in particular, involves the cytoplasmic protein clathrin which induces formation and internalization of clathrin-coated membrane buds that contain extra-cellular cargo. Decades of experimental work have established that the morphology of the clathrin coat evolves with time and induces its curvature on the membrane bud; but energetics of the process remain unclear. Recent experiments by Avinoam et al. [Science 348, 1369 (2015)SCIEAS0036-807510.1126/science.aaa9555] reported that the area of the clathrin coat remains fixed while its curvature increases with time and also the clathrin molecules in the coat turn over rapidly. We show that these observations challenge existing models of coated membrane bud formation. We analyze their data to bring out certain features consistent with the underlying lattice structure of the coat. We hypothesize that membrane curvature inhibits clathrin deposition and propose a kinetic model that explains the area distribution of clathrin coats. We also show that their data on shape evolution of the coated membrane bud can be approximately understood from simple geometric considerations. However, the energetics of the coat formation which controls the kinetics of the process remains a puzzle.

Curvature-undulation coupling as a basis for curvature sensing and generation in bilayer membranes

We present coarse-grained molecular dynamics simulations of the epsin N-terminal homology domain interacting with a lipid bilayer and demonstrate a rigorous theoretical formalism and analysis method for computing the induced curvature field in varying concentrations of the protein in the dilute limit. Our theory is based on the description of the height-height undulation spectrum in the presence of a curvature field. We formulated an objective function to compare the acquired undulation spectrum from the simulations to that of the theory. We recover the curvature field parameters by minimizing the objective function even in the limit where the protein-induced membrane curvature is of the same order as the amplitude due to thermal undulations. The coupling between curvature and undulations leads to significant predictions: (i) Under dilute conditions, the proteins can sense a site of spontaneous curvature at distances much larger than their size; (ii) as the density of proteins increases the coupling focuses and stabilizes the curvature field to the site of the proteins; and (iii) the mapping of the protein localization and the induction of a stable curvature is a cooperative process that can be described through a Hill function.

Cargo binding promotes KDEL receptor clustering at the mammalian cell surface

Transmembrane receptor clustering is a ubiquitous phenomenon in pro- and eukaryotic cells to physically sense receptor/ligand interactions and subsequently translate an exogenous signal into a cellular response. Despite that receptor cluster formation has been described for a wide variety of receptors, ranging from chemotactic receptors in bacteria to growth factor and neurotransmitter receptors in mammalian cells, a mechanistic understanding of the underlying molecular processes is still puzzling. In an attempt to fill this gap we followed a combined experimental and theoretical approach by dissecting and modulating cargo binding, internalization and cellular response mediated by KDEL receptors (KDELRs) at the mammalian cell surface after interaction with a model cargo/ligand. Using a fluorescent variant of ricin toxin A chain as KDELR-ligand (eGFP-RTA^H/KDEL), we demonstrate that cargo binding induces dose-dependent receptor cluster formation at and subsequent internalization from the membrane which is associated and counteracted by anterograde and microtubule-assisted receptor transport to preferred docking sites at the plasma membrane. By means of analytical arguments and extensive numerical simulations we show that cargo-synchronized receptor transport from and to the membrane is causative for KDELR/cargo cluster formation at the mammalian cell surface.

Thermodynamic free energy methods to investigate shape transitions in bilayer membranes

The conformational free energy landscape of a system is a fundamental thermodynamic quantity of importance particularly in the study of soft matter and biological systems, in which the entropic contributions play a dominant role. While computational methods to delineate the free energy landscape are routinely used to analyze the relative stability of conformational states, to determine phase boundaries, and to compute ligand-receptor binding energies its use in problems involving the cell membrane is limited. Here, we present an overview of four different free energy methods to study morphological transitions in bilayer membranes, induced either by the action of curvature remodeling proteins or due to the application of external forces. Using a triangulated surface as a model for the cell membrane and using the framework of dynamical triangulation Monte Carlo, we have focused on the methods of Widom insertion, thermodynamic integration, Bennett acceptance scheme, and umbrella sampling and weighted histogram analysis. We have demonstrated how these methods can be employed in a variety of problems involving the cell membrane. Specifically, we have shown that the chemical potential, computed using Widom insertion, and the relative free energies, computed using thermodynamic integration and Bennett acceptance method, are excellent measures to study the transition from curvature sensing to curvature inducing behavior of membrane associated proteins. The umbrella sampling and WHAM analysis has been used to study the thermodynamics of tether formation in cell membranes and the quantitative predictions of the computational model are in excellent agreement with experimental measurements. Furthermore, we also present a method based on WHAM and thermodynamic integration to handle problems related to end-point-catastrophe that are common in most free energy methods.

Membrane Mechanics of Endocytosis in Cells with Turgor

Endocytosis is an essential process by which cells internalize a piece of plasma membrane and material from the outside. In cells with turgor, pressure opposes membrane deformations, and increases the amount of force that has to be generated by the endocytic machinery. To determine this force, and calculate the shape of the membrane, we used physical theory to model an elastic surface under pressure. Accurate fits of experimental profiles are obtained assuming that the coated membrane is highly rigid and preferentially curved at the endocytic site. The forces required from the actin machinery peaks at the onset of deformation, indicating that once invagination has been initiated, endocytosis is unlikely to stall before completion. Coat proteins do not lower the initiation force but may affect the process by the curvature they induce. In the presence of isotropic curvature inducers, pulling the tip of the invagination can trigger the formation of a neck at the base of the invagination. Hence direct neck constriction by actin may not be required, while its pulling role is essential. Finally, the theory shows that anisotropic curvature effectors stabilize membrane invaginations, and the loss of crescent-shaped BAR domain proteins such as Rvs167 could therefore trigger membrane scission.

Cells use endocytosis to intake molecules and to recycle components of their membrane. Even in its simplest form, endocytosis involves a large number of proteins with often redundant functions that are organized into a microscopic force-producing “machine”. Knowing how much force is needed to induce a membrane invagination is essential to understand how this endocytic machine may operate. We show that experimental membrane shapes are well described theoretically by a thin sheet elastic model including a difference of pressure across the membrane due to turgor. This allows us to integrate the different contributions that shape the membrane, and to compute the forces opposing membrane deformation. This calculation provides an estimate of the pulling force that must be generated by the actin machinery in yeast. We also identify a membrane instability that could lead to vesicle budding.

On the modeling of endocytosis in yeast

The cell membrane deforms during endocytosis to surround extracellular material and draw it into the cell. Results of experiments on endocytosis in yeast show general agreement that 1) actin polymerizes into a network of filaments exerting active forces on the membrane to deform it, and 2) the large-scale membrane deformation is tubular in shape. In contrast, there are three competing proposals for precisely how the actin filament network organizes itself to drive the deformation. We use variational approaches and numerical simulations to address this competition by analyzing a meso-scale model of actin-mediated endocytosis in yeast. The meso-scale model breaks up the invagination process into three stages: 1) initiation, where clathrin interacts with the membrane via adaptor proteins; 2) elongation, where the membrane is then further deformed by polymerizing actin filaments; and 3) pinch-off. Our results suggest that the pinch-off mechanism may be assisted by a pearling-like instability. We rule out two of the three competing proposals for the organization of the actin filament network during the elongation stage. These two proposals could be important in the pinch-off stage, however, where additional actin polymerization helps break off the vesicle. Implications and comparisons with earlier modeling of endocytosis in yeast are discussed.

Endocytic proteins drive vesicle growth via instability in high membrane tension environment

Clathrin-mediated endocytosis (CME) is a key pathway for transporting cargo into cells via membrane vesicles; it plays an integral role in nutrient import, signal transduction, neurotransmission, and cellular entry of pathogens and drug-carrying nanoparticles. Because CME entails substantial local remodeling of the plasma membrane, the presence of membrane tension offers resistance to bending and hence, vesicle formation. Experiments show that in such high-tension conditions, actin dynamics is required to carry out CME successfully. In this study, we build on these pioneering experimental studies to provide fundamental mechanistic insights into the roles of two key endocytic proteins-namely, actin and BAR proteins-in driving vesicle formation in high membrane tension environment. Our study reveals an actin force-induced "snap-through instability" that triggers a rapid shape transition from a shallow invagination to a highly invaginated tubular structure. We show that the association of BAR proteins stabilizes vesicles and induces a milder instability. In addition, we present a rather counterintuitive role of BAR depolymerization in regulating the shape evolution of vesicles. We show that the dissociation of BAR proteins, supported by actin-BAR synergy, leads to considerable elongation and squeezing of vesicles. Going beyond the membrane geometry, we put forth a stress-based perspective for the onset of vesicle scission and predict the shapes and composition of detached vesicles. We present the snap-through transition and the high in-plane stress as possible explanations for the intriguing direct transformation of broad and shallow invaginations into detached vesicles in BAR mutant yeast cells.

Phenomenology based multiscale models as tools to understand cell membrane and organelle morphologies

An intriguing question in cell biology is "how do cells regulate their shape?" It is commonly believed that the observed cellular morphologies are a result of the complex interaction among the lipid molecules (constituting the cell membrane), and with a number of other macromolecules, such as proteins. It is also believed that the common biophysical processes essential for the functioning of a cell also play an important role in cellular morphogenesis. At the cellular scale-where typical dimensions are in the order of micrometers-the effects arising from the molecular scale can either be modeled as equilibrium or non-equilibrium processes. In this chapter, we discuss the dynamically triangulated Monte Carlo technique to model and simulate membrane morphologies at the cellular scale, which in turn can be used to investigate several questions related to shape regulation in cells. In particular, we focus on two specific problems within the framework of isotropic and anisotropic elasticity theories: namely, (i) the origin of complex, physiologically relevant, membrane shapes due to the interaction of the membrane with curvature remodeling proteins, and (ii) the genesis of steady state cellular shapes due to the action of non-equilibrium forces that are generated by the fission and fusion of transport vesicles and by the binding and unbinding of proteins from the parent membrane.

Structural basis for membrane binding and remodeling by the exomer secretory vesicle cargo adaptor

Cargo adaptor subunits of vesicle coat protein complexes sort transmembrane proteins to distinct membrane compartments in eukaryotic cells. The exomer complex is the only cargo adaptor known to sort proteins at the trans-Golgi network into secretory vesicles. Exomer function is regulated by the Arf1 GTPase, a master regulator of trafficking at the Golgi. We report the structure of exomer bound to two copies of Arf1. Exomer interacts with each Arf1 molecule via two surfaces, one of which is a noncanonical interface that regulates GTP hydrolysis. The structure uncovers an unexpected membrane-proximal hydrophobic element that exomer uses in cooperation with Arf1 to remodel membranes. Given the constrained motion of the exomer hinge region, we envision that exomer dynamically positions multiple membrane insertion elements to drive membrane fission. In contrast to other known cargo adaptors, exomer therefore couples two functions, cargo sorting and membrane fission, into a single complex.

Multiscale computational models in physical systems biology of intracellular trafficking

In intracellular trafficking, a definitive understanding of the interplay between protein binding and membrane morphology remains incomplete. The authors describe a computational approach by integrating coarse-grained molecular dynamics (CGMD) simulations with continuum Monte Carlo (CM) simulations of the membrane to study protein-membrane interactions and the ensuing membrane curvature. They relate the curvature field strength discerned from the molecular level to its effect at the cellular length-scale. They perform thermodynamic integration on the CM model to describe the free energy landscape of vesiculation in clathrin-mediated endocytosis. The method presented here delineates membrane morphologies and maps out the free energy changes associated with membrane remodeling due to varying coat sizes, coat curvature strengths, membrane bending rigidities, and tensions; furthermore several constraints on mechanisms underlying clathrin-mediated endocytosis have also been identified, Their CGMD simulations have revealed the importance of PIP2 for stable binding of proteins essential for curvature induction in the bilayer and have provided a molecular basis for the positive curvature induction by the epsin N-terminal homology (EIMTH) domain. Calculation of the free energy landscape for vesicle budding has identified the critical size and curvature strength of a clathrin coat required for nucleation and stabilisation of a mature vesicle.

Mesoscale computational studies of membrane bilayer remodeling by curvature-inducing proteins

Biological membranes constitute boundaries of cells and cell organelles. These membranes are soft fluid interfaces whose thermodynamic states are dictated by bending moduli, induced curvature fields, and thermal fluctuations. Recently, there has been a flood of experimental evidence highlighting active roles for these structures in many cellular processes ranging from trafficking of cargo to cell motility. It is believed that the local membrane curvature, which is continuously altered due to its interactions with myriad proteins and other macromolecules attached to its surface, holds the key to the emergent functionality in these cellular processes. Mechanisms at the atomic scale are dictated by protein-lipid interaction strength, lipid composition, lipid distribution in the vicinity of the protein, shape and amino acid composition of the protein, and its amino acid contents. The specificity of molecular interactions together with the cooperativity of multiple proteins induce and stabilize complex membrane shapes at the mesoscale. These shapes span a wide spectrum ranging from the spherical plasma membrane to the complex cisternae of the Golgi apparatus. Mapping the relation between the protein-induced deformations at the molecular scale and the resulting mesoscale morphologies is key to bridging cellular experiments across the various length scales. In this review, we focus on the theoretical and computational methods used to understand the phenomenology underlying protein-driven membrane remodeling. Interactions at the molecular scale can be computationally probed by all atom and coarse grained molecular dynamics (MD, CGMD), as well as dissipative particle dynamics (DPD) simulations, which we only describe in passing. We choose to focus on several continuum approaches extending the Canham - Helfrich elastic energy model for membranes to include the effect of curvature-inducing proteins and explore the conformational phase space of such systems. In this description, the protein is expressed in the form of a spontaneous curvature field. The approaches include field theoretical methods limited to the small deformation regime, triangulated surfaces and particle-based computational models to investigate the large-deformation regimes observed in the natural state of many biological membranes. Applications of these methods to understand the properties of biological membranes in homogeneous and inhomogeneous environments of proteins, whose underlying curvature fields are either isotropic or anisotropic, are discussed. The diversity in the curvature fields elicits a rich variety of morphological states, including tubes, discs, branched tubes, and caveola. Mapping the thermodynamic stability of these states as a function of tuning parameters such as concentration and strength of curvature induction of the proteins is discussed. The relative stabilities of these self-organized shapes are examined through free-energy calculations. The suite of methods discussed here can be tailored to applications in specific cellular settings such as endocytosis during cargo trafficking and tubulation of filopodial structures in migrating cells, which makes these methods a powerful complement to experimental studies.

Defining the free-energy landscape of curvature-inducing proteins on membrane bilayers

Curvature-sensing and curvature-remodeling proteins, such as Amphiphysin, Epsin, and Exo70, are known to reshape cell membranes, and this remodeling event is essential for key biophysical processes such as tubulation, exocytosis, and endocytosis. Curvature-inducing proteins can act as curvature sensors; they aggregate to membrane regions matching their intrinsic curvature; as well as induce curvature in cell membranes to stabilize emergent high curvature, nonspherical, structures such as tubules, discs, and caveolae. A definitive understanding of the interplay between protein recruitment and migration, the evolution of membrane curvature, and membrane morphological transitions is emerging but remains incomplete. Here, within a continuum framework and using the machinery of Monte Carlo simulations, we introduce and compare three free-energy methods to delineate the free-energy landscape of curvature-inducing proteins on bilayer membranes. We demonstrate the utility of the Widom test particle (or field) insertion methodology in computing the excess chemical potentials associated with curvature-inducing proteins on the membrane-in particular, we use this method to track the onset of morphological transitions in the membrane at elevated protein densities. We validate this approach by comparing the results from the Widom method with those of thermodynamic integration and Bennett acceptance ratio methods. Furthermore, the predictions from the Widom method have been tested against analytical calculations of the excess chemical potential at infinite dilution. Our results are useful in precisely quantifying the free-energy landscape, and also in determining the phase boundaries associated with curvature-induction, curvature-sensing, and morphological transitions. This approach can be extended to studies exploring the role of thermal fluctuations and other external (control) variables, such as membrane excess area, in shaping curvature-mediated interactions on bilayer membranes.

Counterion-mediated pattern formation in membranes containing anionic lipids

Most lipid components of cell membranes are either neutral, like cholesterol, or zwitterionic, like phosphatidylcholine and sphingomyelin. Very few lipids, such as sphingosine, are cationic at physiological pH. These generally interact only transiently with the lipid bilayer, and their synthetic analogs are often designed to destabilize the membrane for drug or DNA delivery. However, anionic lipids are common in both eukaryotic and prokaryotic cell membranes. The net charge per anionic phospholipid ranges from -1 for the most abundant anionic lipids such as phosphatidylserine, to near -7 for phosphatidylinositol 3,4,5 trisphosphate, although the effective charge depends on many environmental factors. Anionic phospholipids and other negatively charged lipids such as lipopolysaccharides are not randomly distributed in the lipid bilayer, but are highly restricted to specific leaflets of the bilayer and to regions near transmembrane proteins or other organized structures within the plane of the membrane. This review highlights some recent evidence that counterions, in the form of monovalent or divalent metal ions, polyamines, or cationic protein domains, have a large influence on the lateral distribution of anionic lipids within the membrane, and that lateral demixing of anionic lipids has effects on membrane curvature and protein function that are important for biological control.

Exo70 generates membrane curvature for morphogenesis and cell migration

Dynamic shape changes of the plasma membrane are fundamental to many processes, ranging from morphogenesis and cell migration to phagocytosis and viral propagation. Here, we demonstrate that Exo70, a component of the exocyst complex, induces tubular membrane invaginations toward the lumen of synthetic vesicles in vitro and generates protrusions on the surface of cells. Biochemical analyses using Exo70 mutants and independent molecular dynamics simulations based on Exo70 structure demonstrate that Exo70 generates negative membrane curvature through an oligomerization-based mechanism. In cells, the membrane-deformation function of Exo70 is required for protrusion formation and directional cell migration. Exo70 thus represents a membrane-bending protein that may couple actin dynamics and plasma membrane remodeling for morphogenesis.

Coarse-Grained Models for Protein-Cell Membrane Interactions

The physiological properties of biological soft matter are the product of collective interactions, which span many time and length scales. Recent computational modeling efforts have helped illuminate experiments that characterize the ways in which proteins modulate membrane physics. Linking these models across time and length scales in a multiscale model explains how atomistic information propagates to larger scales. This paper reviews continuum modeling and coarse-grained molecular dynamics methods, which connect atomistic simulations and single-molecule experiments with the observed microscopic or mesoscale properties of soft-matter systems essential to our understanding of cells, particularly those involved in sculpting and remodeling cell membranes.

Robust polarity establishment occurs via an endocytosis-based cortical corralling mechanism

Cell polarity can be established via the spatial coordination of the opposing membrane trafficking activities of endocytosis and exocytosis.

Formation of a stable polarity axis underlies numerous biological processes. Here, using high-resolution imaging and complementary mathematical modeling we find that cell polarity can be established via the spatial coordination of opposing membrane trafficking activities: endocytosis and exocytosis. During polarity establishment in budding yeast, these antagonistic processes become apposed. Endocytic vesicles corral a central exocytic zone, tightening it to a vertex that establishes the polarity axis for the ensuing cell cycle. Concomitantly, the endocytic system reaches an equilibrium where internalization events occur at a constant frequency. Endocytic mutants that failed to initiate periodic internalization events within the corral displayed wide, unstable polarity axes. These results, predicted by in silico modeling and verified by high resolution in vivo studies, identify a requirement for endocytic corralling during robust polarity establishment.

Mesoscale simulations of curvature-inducing protein partitioning on lipid bilayer membranes in the presence of mean curvature fields

The membrane-surface migration of curvature-inducing proteins in response to membrane curvature gradients has been investigated using Monte Carlo simulations of a curvilinear membrane model based on the Helfrich Hamiltonian. Consistent with theoretical and experimental data, we find the proteins that generate curvature can also sense the background membrane curvature, wherein they preferentially partition to the high curvature regions. The partitioning strength depends linearly on local membrane curvature and the slope (or the coupling constant) of the partitioning probability versus mean curvature depends on the membrane bending rigidity and instantaneous curvature field caused by different proteins. Our simulation study allows us to quantitatively characterize and identify the important factors affecting the coupling constant (slope), which may be difficult to determine in experiments. Furthermore, the membrane model is used to study budding of vesicles where it is found that in order to stabilize a mature vesicle with a stable 'neck-region' (or stable membrane overhangs), the area (extent) of the intrinsic curvature region needs to exceed a threshold-critical value. The migration and partitioning of curvature-inducing proteins in a budding vesicle with a stable neck (with a characteristic negative value of the Gaussian curvature) is investigated.

Structural systems biology and multiscale signaling models

We review current advances in experimental as well as computational modeling and simulation approaches to structural systems biology, whose overall aim is to build quantitative models of signaling networks while retaining the crucial elements of molecular specificity. We briefly discuss the current and emerging experimental and computational methods, particularly focusing on hybrid and multiscale methods, and highlight several applications in cell signaling with quantitative and predictive capabilities. The scope of such models range from delineating protein-protein interactions to describing clinical implications.

Multiscale Modeling of Functionalized Nanocarriers in Targeted Drug Delivery

Targeted drug delivery using functionalized nanocarriers (NCs) is a strategy in therapeutic and diagnostic applications. In this paper we review the recent development of models at multiple length and time scales and their applications to targeting of antibody functionalized nanocarriers to antigens (receptors) on the endothelial cell (EC) surface. Our mesoscale (100 nm-1 μm) model is based on phenomenological interaction potentials for receptor-ligand interactions, receptor-flexure and resistance offered by glycocalyx. All free parameters are either directly determined from independent biophysical and cell biology experiments or estimated using molecular dynamics simulations. We employ a Metropolis Monte Carlo (MC) strategy in conjunction with the weighted histogram analysis method (WHAM) to compute the free energy landscape (potential of mean force or PMF) associated with the multivalent antigen-antibody interactions mediating the NC binding to EC. The binding affinities (association constants) are then derived from the PMF by computing absolute binding free energy of binding of NC to EC, taking into account the relevant translational and rotational entropy losses of NC and the receptors. We validate our model predictions by comparing the computed binding affinities and PMF to a wide range of experimental measurements, including in vitro cell culture, in vivo endothelial targeting, atomic force microscopy (AFM), and flow chamber experiments. The model predictions agree closely and quantitatively with all types experimental measurements. On this basis, we conclude that our computational protocol represents a quantitative and predictive approach for model driven design and optimization of functionalized NCs in targeted vascular drug delivery.

Systems biology and physical biology of clathrin-mediated endocytosis

In this review, we describe the application of experimental data and modeling of intracellular endocytic trafficking mechanisms with a focus on the process of clathrin-mediated endocytosis. A detailed parts-list for the protein-protein interactions in clathrin-mediated endocytosis has been available for some time. However, recent experimental, theoretical, and computational tools have proved to be critical in establishing a sequence of events, cooperative dynamics, and energetics of the intracellular process. On the experimental front, total internal reflection fluorescence microscopy, photo-activated localization microscopy, and spinning-disk confocal microscopy have focused on assembly and patterning of endocytic proteins at the membrane, while on the theory front, minimal theoretical models for clathrin nucleation, biophysical models for membrane curvature and bending elasticity, as well as methods from computational structural and systems biology, have proved insightful in describing membrane topologies, curvature mechanisms, and energetics.

Connecting the virtual world of computers to the real world of medicinal chemistry

Drug discovery involves the simultaneous optimization of chemical and biological properties, usually in a single small molecule, which modulates one of nature's most complex systems: the balance between human health and disease. The increased use of computer-aided methods is having a significant impact on all aspects of the drug-discovery and development process and with improved methods and ever faster computers, computer-aided molecular design will be ever more central to the discovery process.