Coarse-Grained Models for Protein-Cell Membrane Interactions

Supercomputing in the biological sciences: Toward Zettascale and Yottascale simulations

Molecular simulations of biological systems tend to be significantly more compute-intensive than those in materials science and astrophysics, due to important contributions of long-range electrostatic forces and large numbers of time steps (>1E9) required. Simulations of biomolecular complexes of microseconds to milliseconds are considered state-of-the-art today. However, these time scales are miniscule in comparison to physiological time scales relevant to molecular machine activity, drug action, and elongation cycles for protein synthesis, RNA synthesis, and DNA synthesis (seconds to days). While an exascale supercomputer has simulated an entire virus for nanoseconds, this supercomputer would need to be 10 billion times faster to simulate that virus for 3 hours of physiological time, demonstrating the insatiable need for computing power. With growing interest in computational drug design from the pharmaceutical sector, the biological sciences are positioned to be an industry driver in computing.

Direct and cell-mediated EV-ECM interplay

Extracellular vesicles (EV) are a heterogeneous group of lipid particles excreted by cells. They play an important role in regeneration, development, inflammation, and cancer progression, together with the extracellular matrix (ECM), which they constantly interact with. In this review, we discuss direct and indirect interactions of EVs and the ECM and their impact on different physiological processes. The ECM affects the secretion of EVs, and the properties of the ECM and EVs modulate EVs' diffusion and adhesion. On the other hand, EVs can affect the ECM both directly through enzymes and indirectly through the modulation of the ECM synthesis and remodeling by cells. This review emphasizes recently discovered types of EVs bound to the ECM and isolated by enzymatic digestion, including matrix-bound nanovesicles (MBV) and tissue-derived EV (TiEV). In addition to the experimental studies, computer models of the EV-ECM-cell interactions, from all-atom models to quantitative pharmacology models aiming to improve our understanding of the interaction mechanisms, are also considered. STATEMENT OF SIGNIFICANCE: Application of extracellular vesicles in tissue engineering is an actively developing area. Vesicles not only affect cells themselves but also interact with the matrix and change it. The matrix also influences both cells and vesicles. In this review, different possible types of interactions between vesicles, matrix, and cells are discussed. Furthermore, the united EV-ECM system and its regulation through the cellular activity are presented.

Molecular Dynamics Simulations Help Determine the Molecular Mechanisms of Lasioglossin-III and Its Variant Peptides' Membrane Interfacial Interactions

Lasioglossin-III (LL-III) is a potent broad-spectrum antimicrobial peptide used in diverse antimicrobial applications. In this work, coarse-grained and all-atom molecular dynamics simulation strategies were used in tandem to interpret the molecular mechanisms involved in the interfacial dynamics of LL-III and its recombinant variants during interactions with diverse cell membrane systems. Our results indicate that the membrane charges act as the driving force for initiating the membrane-peptide interactions, while the hydrophobic or van der Waals forces help to reinforce the membrane-peptide bindings. The optimized charge-hydrophobicity ratio of the LL-III peptides helps ensure their high specificity toward bacterial membranes compared to mammalian membrane systems, which also helps explain our experimental observations. Overall, we hope that our work gives new insight into the antimicrobial action of LL-III peptides and that the adopted simulation strategy will help other scientists and engineers extract maximal information from complex molecular simulations using minimal computational power.

Optimizing Coarse-Grained Models for Large-Scale Membrane Protein Simulation

Coarse-grained (CG) models have been developed for studying membrane proteins at physiologically relevant scales. Such methods, including popular CG lipid models, exhibit stability and efficiency at moderate scales, but they can become impractical or even unusable beyond a critical size due to various technical issues. Here, we report that these scale-dependent issues can arise from progressively slower relaxation dynamics and become confounded by unforeseen instabilities observed only at larger scales. To address these issues, we systemically optimized a 4-site solvent-free CG lipid model that is suitable for conducting micron-scale molecular dynamics simulations of membrane proteins under various membrane properties. We applied this lipid model to explore the long-range membrane deformation induced by a large mechanosensitive ion channel, PIEZO. We show that the optimized CG models are powerful in elucidating the structural and dynamic interplay between PIEZO and the membrane. Furthermore, we anticipate that our methodological insights can prove useful for resolving issues stemming from scale-dependent limitations of similar CG methodologies.

Lipids and proteins: Insights into the dynamics of assembly, recognition, condensate formation. What is still missing?

Lipid membranes and proteins, which are part of us throughout our lives, have been studied for decades. However, every year, new discoveries show how little we know about them. In a reader-friendly manner for people not involved in the field, this paper tries to serve as a bridge between physicists and biologists and new young researchers diving into the field to show its relevance, pointing out just some of the plethora of lines of research yet to be unraveled. It illustrates how new ways, from experimental to theoretical approaches, are needed in order to understand the structures and interactions that take place in a single lipid, protein, or multicomponent system, as we are still only scratching the surface.

Effect of Force Field Resolution on Membrane Mechanical Response and Mechanoporation Damage under Deformation Simulations

Damage induced by transient disruption and mechanoporation in an intact cell membrane is a vital nanoscale biomechanical mechanism that critically affects cell viability. To complement experimental studies of mechanical membrane damage and disruption, molecular dynamics (MD) simulations have been performed at different force field resolutions, each of which follows different parameterization strategies and thus may influence the properties and dynamics of membrane systems. Therefore, the current study performed tensile deformation MD simulations of bilayer membranes using all-atom (AA), united-atom (UA), and coarse-grained Martini (CG-M) models to investigate how the damage biomechanics differs across atomistic and coarse-grained (CG) simulations. The mechanical response and mechanoporation damage were qualitatively similar but quantitatively different in the three models, including some progressive changes based on the coarse-graining level. The membranes yielded and reached ultimate strength at similar strains; however, the coarser systems exhibited lower average yield stresses and failure strains. The average failure strain in the UA model was approximately 7% lower than the AA, and the CG-M was 20% lower than UA and 27% lower than AA. The CG systems also nucleated a higher number of pores and larger pores, which resulted in higher damage during the deformation process. Overall, the study provides insight on the impact of force field-a critical factor in modeling biomolecular systems and their interactions-in inspecting membrane mechanosensitive responses and serves as a reference for justifying the appropriate force field for future studies of more complex membranes and more diverse biomolecular assemblies.

Fractional Extended Diffusion Theory to capture anomalous relaxation from biased/accelerated molecular simulations

Biased and accelerated molecular simulations (BAMS) are widely used tools to observe relevant molecular phenomena occurring on time scales inaccessible to standard molecular dynamics, but evaluation of the physical time scales involved in the processes is not directly possible from them. For this reason, the problem of recovering dynamics from such kinds of simulations is the object of very active research due to the relevant theoretical and practical implications of dynamics on the properties of both natural and synthetic molecular systems. In a recent paper [A. Rapallo et al., J. Comput. Chem. 42, 586-599 (2021)], it has been shown how the coupling of BAMS (which destroys the dynamics but allows to calculate average properties) with Extended Diffusion Theory (EDT) (which requires input appropriate equilibrium averages calculated over the BAMS trajectories) allows to effectively use the Smoluchowski equation to calculate the orientational time correlation function of the head-tail unit vector defined over a peptide in water solution. Orientational relaxation of this vector is the result of the coupling of internal molecular motions with overall molecular rotation, and it was very well described by correlation functions expressed in terms of weighted sums of suitable time-exponentially decaying functions, in agreement with a Brownian diffusive regime. However, situations occur where exponentially decaying functions are no longer appropriate to capture the actual dynamical behavior, which exhibits persistent long time correlations, compatible with the so called subdiffusive regimes. In this paper, a generalization of EDT will be given, exploiting a fractional Smoluchowski equation (FEDT) to capture the non-exponential character observed in the relaxation of intramolecular distances and molecular radius of gyration, whose dynamics depend on internal molecular motions only. The calculation methods, proper to EDT, are adapted to implement the generalization of the theory, and the resulting algorithm confirms FEDT as a tool of practical value in recovering dynamics from BAMS, to be used in general situations, involving both regular and anomalous diffusion regimes.

Computational Design of Peptides for Biomaterials Applications

Computer-aided molecular design and protein engineering emerge as promising and active subjects in bioengineering and biotechnological applications. On one hand, due to the advancing computing power in the past decade, modeling toolkits and force fields have been put to use for accurate multiscale modeling of biomolecules including lipid, protein, carbohydrate, and nucleic acids. On the other hand, machine learning emerges as a revolutionary data analysis tool that promises to leverage physicochemical properties and structural information obtained from modeling in order to build quantitative protein structure-function relationships. We review recent computational works that utilize state-of-the-art computational methods to engineer peptides and proteins for various emerging biomedical, antimicrobial, and antifreeze applications. We also discuss challenges and possible future directions toward developing a roadmap for efficient biomolecular design and engineering.

Development, Validation, and Applications of Nonbonded Interaction Parameters between Coarse-Grained Amino Acid and Water Models

Interactions between amino acids and water play an important role in determining the stability and folding/unfolding, in aqueous solution, of many biological macromolecules, which affects their function. Thus, understanding the molecular-level interactions between water and amino acids is crucial to tune their function in aqueous solutions. Herein, we have developed nonbonded interaction parameters between the coarse-grained (CG) models of 20 amino acids and the one-site CG water model. The nonbonded parameters, represented using the 12-6 Lennard Jones (LJ) potential form, have been optimized using an artificial neural network (ANN)-assisted particle swarm optimization (PSO) (ANN-assisted PSO) method. All-atom (AA) molecular dynamics (MD) simulations of dipeptides in TIP3P water molecules were performed to calculate the Gibbs hydration free energies. The nonbonded force-field (FF) parameters between CG amino acids and the one-site CG water model were developed to accurately reproduce these energies. Furthermore, to test the transferability of these newly developed parameters, we calculated the hydration free energies of the analogues of the amino acid side chains, which showed good agreement with reported experimental data. Additionally, we show the applicability of these models by performing self-assembly simulations of peptide amphiphiles. Overall, these models are transferable and can be used to study the self-assembly of various biomaterials and biomolecules to develop a mechanistic understanding of these processes.

Simulation study of membrane bending by protein crowding: a case study with the epsin N-terminal homology domain

The mechanisms by which peripheral membrane proteins generate curvature is currently an active area of research. One of the proposed mechanisms is amphipathic insertion or the 'wedge' mechanism in which the protein shallowly inserts an amphipathic helix inside the membrane to drive the curvature. However, recent experimental studies have challenged the efficiency of the 'wedge' mechanism as it requires unusual protein densities. These studies proposed an alternative mechanism, namely 'protein-crowding', in which the lateral pressure generated by the random collisions among the membrane bound proteins drives the bending. In this study, we employ atomistic and coarse-grained molecular dynamics simulations to investigate the effects of amphipathic insertion and protein crowding on the membrane surface. Considering epsin N-terminal homology (ENTH) domain as a model protein, we show that amphipathic insertion is not essential for membrane bending. Our results suggest that ENTH domains can aggregate on the membrane surface by employing another structured region (H₃ helix). And this protein crowding decreases the cohesive energy of the lipid tails which causes a significant decrease in the membrane bending rigidity. The ENTH domain can generate a similar degree of membrane curvature irrespective of the activity of its H₀ helix. Our results are consistent with the recent experimental results.

Globular Proteins and Where to Find Them within a Polymer Brush-A Case Study

Protein adsorption by polymerized surfaces is an interdisciplinary topic that has been approached in many ways, leading to a plethora of theoretical, numerical and experimental insight. There is a wide variety of models trying to accurately capture the essence of adsorption and its effect on the conformations of proteins and polymers. However, atomistic simulations are case-specific and computationally demanding. Here, we explore universal aspects of the dynamics of protein adsorption through a coarse-grained (CG) model, that allows us to explore the effects of various design parameters. To this end, we adopt the hydrophobic-polar (HP) model for proteins, place them uniformly at the upper bound of a CG polymer brush whose multibead-spring chains are tethered to a solid implicit wall. We find that the most crucial factor affecting the adsorption efficiency appears to be the polymer grafting density, while the size of the protein and its hydrophobicity ratio come also into play. We discuss the roles of ligands and attractive tethering surfaces to the primary adsorption as well as secondary and ternary adsorption in the presence of attractive (towards the hydrophilic part of the protein) beads along varying spots of the backbone of the polymer chains. The percentage and rate of adsorption, density profiles and the shapes of the proteins, alongside with the respective potential of mean force are recorded to compare the various scenarios during protein adsorption.

Temporally Coherent Backmapping of Molecular Trajectories From Coarse-Grained to Atomistic Resolution

Coarse-graining offers a means to extend the achievable time and length scales of molecular dynamics simulations beyond what is practically possible in the atomistic regime. Sampling molecular configurations of interest can be done efficiently using coarse-grained simulations, from which meaningful physicochemical information can be inferred if the corresponding all-atom configurations are reconstructed. However, this procedure of backmapping to reintroduce the lost atomistic detail into coarse-grain structures has proven a challenging task due to the many feasible atomistic configurations that can be associated with one coarse-grain structure. Existing backmapping methods are strictly frame-based, relying on either heuristics to replace coarse-grain particles with atomic fragments and subsequent relaxation or parametrized models to propose atomic coordinates separately and independently for each coarse-grain structure. These approaches neglect information from previous trajectory frames that is critical to ensuring temporal coherence of the backmapped trajectory, while also offering information potentially helpful to producing higher-fidelity atomic reconstructions. In this work, we present a deep learning-enabled data-driven approach for temporally coherent backmapping that explicitly incorporates information from preceding trajectory structures. Our method trains a conditional variational autoencoder to nondeterministically reconstruct atomistic detail conditioned on both the target coarse-grain configuration and the previously reconstructed atomistic configuration. We demonstrate our backmapping approach on two exemplar biomolecular systems: alanine dipeptide and the miniprotein chignolin. We show that our backmapped trajectories accurately recover the structural, thermodynamic, and kinetic properties of the atomistic trajectory data.

Study of trioleoylglycerol two-layer and adiposome cross-section mimicking four-layer systems through atomic-level simulations

Adiposomes are artificially prepared lipid droplet (LD)-mimetic structures, which, unlike LDs, do not harbor proteins. The dynamics of interaction between triacylglycerols (TAGs), drug molecule, and phospholipids in adiposomes is currently not well-established. Trioleoylglycerol (TOG) molecule was divided into three parts: two oleoyl tails and one 2-monooleoylglycerol (MOG). Forcefield parameters for two oleoyl tails were adopted from the AMBER18 repository while that of the MOG forcefield was taken from the literature. Charge correction was performed on the MOG forcefield before its utilization. After charge correction, the resulting TOG molecule had zero charge. TOG bilayer (2L) and tetralayer (4L) systems were prepared and simulated. TOG bilayer (2L) systems-modeled from two different initial conformations, the TOG3 conformation and the TOG2:1 conformation-showed that TOG2:1 conformation was more prevailing irrespective of the starting conformation and was subsequently used in further simulations. The hydrated TOG 2L system showed TOG-water solution solubility of 0.051 mol L-1 which is near experimental values. This validated the correct parameterization of the TOG molecule. The simulations of 4L systems showed stable membrane behaviors toward the end of simulations. It was also observed that in the 4L system, the TOG molecules showed the formation of micelles with the drug molecule. Almost six TOGs remained continuously in contact with the drug molecule throughout the simulation. The availability of charge-corrected TOG parameterization is expected to equip future studies with a framework for molecular dynamics simulations of adiposomes and/or LDs at the atomic level.

COGRIMEN: Coarse-Grained Method for Modeling of Membrane Proteins in Implicit Environments

The presented methodology is based on coarse-grained representation of biomolecules in implicit environments and is designed for the molecular dynamics simulations of membrane proteins and their complexes. The membrane proteins are not only found in the cell membrane but also in all membranous compartments of the cell: Golgi apparatus, mitochondria, endosomes and lysosomes, and they usually form large complexes. To investigate such systems the methodology is proposed based on two independent approaches combining the coarse-grained MARTINI model for proteins and the effective energy function to mimic the water/membrane environments. The latter is based on the implicit environment developed for all-atom simulations in the IMM1 method. The force field solvation parameters for COGRIMEN were initially calculated from IMM1 all-atom parameters and then optimized using Genetic Algorithms. The new methodology was tested on membrane proteins, their complexes and oligomers. COGRIMEN method is implemented as a patch for NAMD program and can be useful for fast and brief studies of large membrane protein complexes.

Simulation study of domain formation in a model bacterial membrane

Recent experimental studies revealed that functional membrane microdomains (FMMs) are formed in prokaryotic cells which are structurally and functionally similar to the lipid rafts formed in eukaryotic cells. In this study, we employ coarse-grained molecular dynamics simulations to investigate the mechanism of domain formation and its physiochemical properties in a model methicillin-resistant staphylococcus aureus (MRSA) cell membrane. We find that domains are formed through lateral segregation of staphyloxanthin (STX), a carotenoid which shields the bacteria from the host's immune because of its antioxidant nature. Simulation results suggest that membrane integrity increases with the size of the domain, which is assessed by computing bond order parameter of the lipid tails, membrane expansion modulus and water permeability across the membrane. Various membrane domain proteins such as flotillin-like protein floA and penicillin binding protein (PBP2a) preferentially bind with the STX and accumulate in the membrane domain which is consistent with the recent experimental results.

Kinetics of self-assembly of inclusions due to lipid membrane thickness interactions

Self-assembly of proteins on lipid membranes underlies many important processes in cell biology, such as, exo- and endo-cytosis, assembly of viruses, etc. An attractive force that can cause self-assembly is mediated by membrane thickness interactions between proteins. The free energy profile associated with this attractive force is a result of the overlap of thickness deformation fields around the proteins which can be calculated from the solution of a boundary value problem. Yet, the time scales over which two inclusions coalesce has not been explored, even though the evolution of particle concentrations on membranes has been modeled using phase-field approaches. In this paper we compute this time scale as a function of the initial distance between two inclusions by viewing their coalescence as a first passage time problem. The mean first passage time is computed using Langevin dynamics and a partial differential equation, and both methods are found to be in excellent agreement. Inclusions of three different shapes are studied and it is found that for two inclusions separated by about hundred nanometers the time to coalescence is hundreds of milliseconds irrespective of shape. An efficient computation of the interaction energy of inclusions is central to our work. We compute it using a finite difference technique and show that our results are in excellent agreement with those from a previously proposed semi-analytical method based on Fourier-Bessel series. The computational strategies described in this paper could potentially lead to efficient methods to explore the kinetics of self-assembly of proteins on lipid membranes.

A survey of multiscale modeling: Foundations, historical milestones, current status, and future prospects

Research problems in the domains of physical, engineering, biological sciences often span multiple time and length scales, owing to the complexity of information transfer underlying mechanisms. Multiscale modeling (MSM) and high-performance computing (HPC) have emerged as indispensable tools for tackling such complex problems. We review the foundations, historical developments, and current paradigms in MSM. A paradigm shift in MSM implementations is being fueled by the rapid advances and emerging paradigms in HPC at the dawn of exascale computing. Moreover, amidst the explosion of data science, engineering, and medicine, machine learning (ML) integrated with MSM is poised to enhance the capabilities of standard MSM approaches significantly, particularly in the face of increasing problem complexity. The potential to blend MSM, HPC, and ML presents opportunities for unbound innovation and promises to represent the future of MSM and explainable ML that will likely define the fields in the 21st century.

Molecular simulations of lipid membrane partitioning and translocation by bacterial quorum sensing modulators

Quorum sensing (QS) is a bacterial communication process mediated by both native and non-native small-molecule quorum sensing modulators (QSMs), many of which have been synthesized to disrupt QS pathways. While structure-activity relationships have been developed to relate QSM structure to the activation or inhibition of QS receptors, less is known about the transport mechanisms that enable QSMs to cross the lipid membrane and access intracellular receptors. In this study, we used atomistic MD simulations and an implicit solvent model, called COSMOmic, to analyze the partitioning and translocation of QSMs across lipid bilayers. We performed umbrella sampling at atomistic resolution to calculate partitioning and translocation free energies for a set of naturally occurring QSMs, then used COSMOmic to screen the water-membrane partition and translocation free energies for 50 native and non-native QSMs that target LasR, one of the LuxR family of quorum-sensing receptors. This screening procedure revealed the influence of systematic changes to head and tail group structures on membrane partitioning and translocation free energies at a significantly reduced computational cost compared to atomistic MD simulations. Comparisons with previously determined QSM activities suggest that QSMs that are least likely to partition into the bilayer are also less active. This work thus demonstrates the ability of the computational protocol to interrogate QSM-bilayer interactions which may help guide the design of new QSMs with engineered membrane interactions.

Coarse Grained Molecular Dynamic Simulations for the Study of TNF Receptor Family Members' Transmembrane Organization

The Tumor Necrosis Factor (TNF) and the TNF receptor (TNFR) superfamilies are composed of 19 ligands and 30 receptors, respectively. The oligomeric properties of ligands, both membrane bound and soluble, has been studied most. However, less is known about the oligomeric properties of TNFRs. Earlier reports identified the extracellular, membrane-distal, cysteine-rich domain as a pre-ligand assembly domain which stabilizes receptor dimers and/or trimers in the absence of ligand. Nevertheless, recent reports based on structural nuclear magnetic resonance (NMR) highlight the intrinsic role of the transmembrane domains to form dimers (p75NTR), trimers (Fas), or dimers of trimers (DR5). Thus, understanding the structural basis of transmembrane oligomerization may shed light on the mechanism for signal transduction and the impact of disease-associated mutations in this region. To this end, here we used an in silico coarse grained molecular dynamics approach with Martini force field to study TNFR transmembrane homotypic interactions. We have first validated this approach studying the three TNFR described by NMR (p75NTR, Fas, and DR5). We have simulated membrane patches composed of 36 helices of the same receptor equidistantly distributed in order to get unbiassed information on spontaneous proteins assemblies. Good agreement was found in the specific residues involved in homotypic interactions and we were able to observe dimers, trimers, and higher-order oligomers corresponding to those reported in NMR experiments. We have, applied this approach to study the assembly of disease-related mutations being able to assess their impact on oligomerization stability. In conclusion, our results showed the usefulness of coarse grained simulations with Martini force field to study in an unbiased manner higher order transmembrane oligomerization.

Multiscale modeling of protein membrane interactions for nanoparticle targeting in drug delivery

Nanoparticle (NP)-based imaging and drug delivery systems for systemic (e.g. intravenous) therapeutic and diagnostic applications are inherently a complex integration of biology and engineering. A broad range of length and time scales are essential to hydrodynamic and microscopic molecular interactions mediating NP (drug nanocarriers, imaging agents) motion in blood flow, cell binding/uptake, and tissue accumulation. A computational model of time-dependent tissue delivery, providing in silico prediction of organ-specific accumulation of NPs, can be leveraged in NP design and clinical applications. In this article, we provide the current state-of-the-art and future outlook for the development of predictive models for NP transport, targeting, and distribution through the integration of new computational schemes rooted in statistical mechanics and transport. The resulting multiscale model will comprehensively incorporate: (i) hydrodynamic interactions in the vascular scales relevant to NP margination; (ii) physical and mechanical forces defining cellular and tissue architecture and epitope accessibility mediating NP adhesion; and (iii) subcellular and paracellular interactions including molecular-level targeting impacting NP uptake.

Interplay between ligand mobility and nanoparticle geometry during cellular uptake of PEGylated liposomes and bicelles

We explore the cellular uptake process of PEGylated liposomes and bicelles by investigating their membrane wrapping process using large-scale molecular dynamics simulations. We find that due to the mobility of ligands on the liposome/bicelle, the membrane wrapping process of a PEGylated liposome/bicelle can be divided into two stages, whose transition is determined by a critical wrapping fraction fc; it is reached when all the ligands are exhausted and bound to receptors within the cell membrane. Before this critical scenario is approached, the grafted polyethylene glycol (PEG) polymers aggregate together within the membrane-wrapped region of the liposome/bicelle, driven by ligand-receptor binding. For wrapping fractions f > fc, membrane wrapping cannot proceed unless a compressive membrane tension is provided. By systematically varying the membrane tension and PEG molar ratio, we establish phase diagrams about wrapping states for both PEGylated liposomes and bicelles. According to these diagrams, we find that the absolute value of the compressive membrane tension required by a fully wrapped PEGylated bicelle is smaller than that of the PEGylated liposome, indicating that the PEGylated bicelle is easily internalized by cells. Further theoretical analysis reveals that compared to a liposome, the flatter surface at the top of a bicelle makes it energetically more favored beyond the critical wrapping fraction fc. Our simulations confirm that the interplay between ligand mobility and NP geometry can significantly change the understanding about the influence of NP geometry on the membrane wrapping process. It can help us to better understand the cellular uptake process of the PEGylated liposome/bicelle and to improve the design of lipid-like NPs for drug delivery.

The plasma membrane as a mechanochemical transducer

Cells are constantly submitted to external mechanical stresses, which they must withstand and respond to. By forming a physical boundary between cells and their environment that is also a biochemical platform, the plasma membrane (PM) is a key interface mediating both cellular response to mechanical stimuli, and subsequent biochemical responses. Here, we review the role of the PM as a mechanosensing structure. We first analyse how the PM responds to mechanical stresses, and then discuss how this mechanical response triggers downstream biochemical responses. The molecular players involved in PM mechanochemical transduction include sensors of membrane unfolding, membrane tension, membrane curvature or membrane domain rearrangement. These sensors trigger signalling cascades fundamental both in healthy scenarios and in diseases such as cancer, which cells harness to maintain integrity, keep or restore homeostasis and adapt to their external environment. This article is part of a discussion meeting issue 'Forces in cancer: interdisciplinary approaches in tumour mechanobiology'.

Single-Nanometer Changes in Nanopore Geometry Influence Curvature, Local Properties, and Protein Localization in Membrane Simulations

Nanoporous surfaces are used in many applications in intracellular sensing and drug delivery. However, the effects of such nanostructures on cell membrane properties are still far from understood. Here, we use coarse-grained molecular dynamics simulations to show that nanoporous substrates can stimulate membrane-curvature effects and that this curvature-sensing effect is much more sensitive than previously thought. We define a series of design parameters for inducing a nanoscale membrane curvature and show that nanopore taper plays a key role in membrane deformation, elucidating a previously unexplored fabrication parameter applicable to many nanostructured biomaterials. We report significant changes in the membrane area per lipid and thickness at regions of curvature. Finally, we demonstrate that regions of the nanopore-induced membrane curvature act as local hotspots for an increased bioactivity. We show spontaneous binding and localization of the epsin N-terminal homology (ENTH) domain to the regions of curvature. Understanding this interplay between the membrane curvature and nanoporosity at the biointerface helps both explain recent biological results and suggests a pathway for developing the next generation of cell-active substrates.

Computational Modeling of Realistic Cell Membranes

Cell membranes contain a large variety of lipid types and are crowded with proteins, endowing them with the plasticity needed to fulfill their key roles in cell functioning. The compositional complexity of cellular membranes gives rise to a heterogeneous lateral organization, which is still poorly understood. Computational models, in particular molecular dynamics simulations and related techniques, have provided important insight into the organizational principles of cell membranes over the past decades. Now, we are witnessing a transition from simulations of simpler membrane models to multicomponent systems, culminating in realistic models of an increasing variety of cell types and organelles. Here, we review the state of the art in the field of realistic membrane simulations and discuss the current limitations and challenges ahead.

Membrane Wrapping Efficiency of Elastic Nanoparticles during Endocytosis: Size and Shape Matter

Using coarse-grained molecular dynamics simulations, we systematically investigate the receptor-mediated endocytosis of elastic nanoparticles (NPs) with different sizes, ranging from 25 to 100 nm, and shapes, including sphere-like, oblate-like, and prolate-like. Simulation results provide clear evidence that the membrane wrapping efficiency of NPs during endocytosis is a result of competition between receptor diffusion kinetics and thermodynamic driving force. The receptor diffusion kinetics refer to the kinetics of receptor recruitment that are affected by the contact edge length between the NP and membrane. The thermodynamic driving force represents the amount of required free energy to drive NPs into a cell. Under the volume constraint of elastic NPs, the soft spherical NPs are found to have similar contact edge lengths to rigid ones and to less efficiently be fully wrapped due to their elastic deformation. Moreover, the difference in wrapping efficiency between soft and rigid spherical NPs increases with their sizes, due to the increment of their elastic energy change. Furthermore, because of its prominent large contact edge length, the oblate ellipsoid is found to be the least sensitive geometry to the variation in NP's elasticity among the spherical, prolate, and oblate shapes during the membrane wrapping. In addition, simulation results indicate that conflicting experimental observations on the efficiency of cellular uptake of elastic NPs could be caused by their different mechanical properties. Our simulations provide a detailed mechanistic understanding about the influence of NPs' size, shape, and elasticity on their membrane wrapping efficiency, which serves as a rational guidance for the design of NP-based drug carriers.

Lipid composition and salt concentration as regulatory factors of the anion selectivity of VDAC studied by coarse-grained molecular dynamics simulations

The voltage-dependent anion channel (VDAC) is a mitochondrial outer membrane protein whose fundamental function is to facilitate and regulate the flow of metabolites between the cytosol and the mitochondrial intermembrane space. Using coarse-grained molecular dynamics simulations, we investigated the dependence of VDAC selectivity towards small inorganic anions on two factors: the ionic strength and the lipid composition. In agreement with experimental data we found that VDAC becomes less anion selective with increasing salt concentration due to the screening of a few basic residues that point into the pore lumen. The molecular dynamics simulations provide insight into the regulation mechanism of VDAC selectivity by the composition in the lipid membrane and suggest that the ion distribution is differently modulated by POPE compared to the POPC bilayer. This occurs through the more persistent interactions of acidic residues located at both rims of the β-barrel with head groups of POPE which in turn impact the electrostatic potential and thereby the selectivity of the pore. This mechanism occurs not only in POPE single component membranes but also in a mixed POPE/POPC bilayer by an enrichment of POPE over POPC lipids on the surface of VDAC. Thus we show here that computationally-inexpensive coarse-grained simulations are able to capture, in a semi-quantitative way, essential features of VDAC anion selectivity and could pave the way toward a molecular level understanding of metabolite transport in natural membranes.

Dynamic stability of nano-fibers self-assembled from short amphiphilic A6D peptides

Self-assembly of A₆D amphiphilic peptides in explicit water is studied by using coarse-grained molecular dynamics simulations. It is observed that the self-assembly of randomly distributed A₆D peptides leads to the formation of a network of nano-fibers. Two other simulations with cylindrical nano-fibers as the initial configuration show the dynamic stability of the self-assembled nano-fibers. As a striking feature, notable fluctuations occur along the axes of the nano-fibers. Depending on the number of peptides per unit length of the nano-fiber, flat-shaped bulges or spiral shapes along the nano-fiber axis are observed at the fluctuations. Analysis of the particle distribution around the nano-fiber indicates that the hydrophobic core and the hydrophilic shell of the nano-structure are preserved in both simulations. The size of the deformations and their correlation times are different in the two simulations. This study gives new insights into the dynamics of the self-assembled nano-structures of short amphiphilic peptides.

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.

Understanding receptor-mediated endocytosis of elastic nanoparticles through coarse grained molecular dynamic simulation

For nanoparticle (NP)-based drug delivery platforms, the elasticity of the NPs has a significant influence on their blood circulation time and cellular uptake efficiency. However, due to the complexity of the endocytosis process and the inconsistency in the definition of elasticity for NPs in experiments, the understanding about the receptor-mediated endocytosis process of elastic NPs is still limited. In this work, we developed a coarse-grained molecular dynamics (CGMD) model for elastic NPs. The energy change of the elastic NPs can be precisely controlled by the bond, area, volume and bending potentials of this CGMD model. To represent liposomes with different elasticities, we systematically varied the bending rigidity of elastic NPs in CGMD simulations. Additionally, we changed the radius of the elastic NPs to explore the potential size effect. Through virtual nano-indentation tests, we found that the effective stiffness of elastic NPs was determined by their bending rigidity and size. Afterwards, we investigated the receptor-mediated endocytosis process of elastic NPs with different sizes and bending rigidities. We found that the membrane wrapping of soft NPs was faster than that of the stiff ones at the early stage, due to the NP deformation induced large contact area between the NPs and the membrane. However, because of the large energy penalties induced by the NP deformation, the membrane wrapping speed of soft NPs slows down during the late stage. Eventually, the soft NPs are wrapped less efficiently than the stiff ones during the membrane wrapping process. Through systematic CGMD simulations, we found a scaling law between the cellular uptake efficiency and the phenomenal bending rigidity of elastic NPs, which agrees reasonably well with experimental observations. Furthermore, we observed that the membrane wrapping efficiencies of soft and stiff NPs with large sizes were close to each other, due to the stronger ligand-receptor binding force and smaller difference in the stiffness of elastic NPs. Our computational model provides an effective tool to investigate the receptor-mediated endocytosis of elastic NPs with well controlled mechanical properties. This study can also be applied to guide the design of NP-based drug carriers with high efficacy, by utilizing their elastic properties.

Self-Assembly Behavior of Amphiphilic Janus Dendrimers in Water: A Combined Experimental and Coarse-Grained Molecular Dynamics Simulation Approach

Amphiphilic Janus dendrimers (JDs) are repetitively branched molecules with hydrophilic and hydrophobic components that self-assemble in water to form a variety of morphologies, including vesicles analogous to liposomes with potential pharmaceutical and medical application. To date, the self-assembly of JDs has not been fully investigated thus it is important to gain insight into its mechanism and dependence on JDs’ molecular structure. In this study, the aggregation behavior in water of a second-generation bis-MPA JD was evaluated using experimental and computational methods. Dispersions of JDs in water were carried out using the thin-film hydration and ethanol injection methods. Resulting assemblies were characterized by dynamic light scattering, confocal microscopy, and atomic force microscopy. Furthermore, a coarse-grained molecular dynamics (CG-MD) simulation was performed to study the mechanism of JDs aggregation. The obtaining of assemblies in water with no interdigitated bilayers was confirmed by the experimental characterization and CG-MD simulation. Assemblies with dendrimersome characteristics were obtained using the ethanol injection method. The results of this study establish a relationship between the molecular structure of the JD and the properties of its aggregates in water. Thus, our findings could be relevant for the design of novel JDs with tailored assemblies suitable for drug delivery systems.

Aggregation of polyethylene glycol polymers suppresses receptor-mediated endocytosis of PEGylated liposomes

The PEGylated liposome, composed of an aqueous core and a fluid state lipid bilayer shell, is one of the few Food and Drug Administration (FDA) approved drug delivery platforms. To prevent the absorption of serum proteins, the surface of a liposome is decorated by hydrophilic and bio-compatible polyethylene glycol (PEG) polymers, which can significantly extend the blood circulation time of liposomes. In this work, with the help of dissipative particle dynamics (DPD) simulations, we explore how the tethered PEG polymers will affect the membrane wrapping process of PEGylated liposomes during endocytosis. Specifically, we compare the membrane wrapping process of a PEGylated rigid nanoparticle (NP) with a PEGylated liposome under identical conditions. Due to the mobility of grafted PEG polymers on the liposome's surface, the complete wrapping of a PEGylated liposome can be dramatically delayed and blocked, in comparison with a PEGylated rigid NP. For the first time, we observe the aggregation of PEG polymers in the contact region between a PEGylated liposome and the membrane, which in turn leads to a ligand-free region on the surface of the liposome during endocytosis. Subsequently, the partially wrapped PEGylated liposome can be bounced back to a less wrapped state. Through free energy analysis, we find that the aggregation of PEG polymers during the membrane wrapping process of a PEGylated liposome introduces a dramatic free energy penalty of about ∼800k_BT, which is almost twice that of a PEGylated rigid NP. Here k_B and T are the Boltzmann constant and temperature, respectively. Such a large energy barrier and the existence of a ligand-free region on the surface of PEGlylated liposomes prevent their membrane wrapping, thereby reducing the chance of internalization by tumor cells. Therefore, our DPD simulation results provide a possible explanation for the inefficient cellular uptake of PEGylated liposomes. In addition, we suggest that by increasing the repulsive interactions between grafted PEG polymers it might be possible to limit their aggregation, and in turn, facilitate the internalization of PEGylated liposomes. The current study provides fundamental insights into the endocytosis of PEGylated liposomes, which could help to design this platform with high efficacy for drug delivery.

Lipid molecules can induce an opening of membrane-facing tunnels in cytochrome P450 1A2

Cytochrome P450 1A2 (P450 1A2, CYP1A2) is a membrane-bound enzyme that oxidizes a broad range of hydrophobic substrates. The structure and dynamics of both the catalytic and trans-membrane (TM) domains of this enzyme in the membrane/water environment were investigated using a multiscale computational approach, including coarse-grained and all-atom molecular dynamics. Starting from the spontaneous self-assembly of the system containing the TM or soluble domain immersed in randomized dilauroyl phosphatidylcholine (DLPC)/water mixture into their respective membrane-bound forms, we reconstituted the membrane-bound structure of the full-length P450 1A2. This structure includes a TM helix that spans the membrane, while being connected to the catalytic domain by a short flexible loop. Furthermore, in this model, the upper part of the TM helix interacts directly with a conserved and highly hydrophobic N-terminal proline-rich segment of the catalytic domain; this segment and the FG loop are immersed in the membrane, whereas the remaining portion of the catalytic domain remains exposed to aqueous solution. The shallow membrane immersion of the catalytic domain induces a depression in the opposite intact layer of the phospholipids. This structural effect may help in stabilizing the position of the TM helix directly beneath the catalytic domain. The partial immersion of the catalytic domain also allows for the enzyme substrates to enter the active site from either aqueous solution or phospholipid environment via several solvent- and membrane-facing tunnels in the full-length P450 1A2. The calculated tunnel dynamics indicated that the opening probability of the membrane-facing tunnels is significantly enhanced when a DLPC molecule spontaneously penetrates into the membrane-facing tunnel 2d. The energetics of the lipid penetration process were assessed by the linear interaction energy (LIE) approximation, and found to be thermodynamically feasible.

Computational approaches to cell-nanomaterial interactions: keeping balance between therapeutic efficiency and cytotoxicity

Owing to their unique properties, nanomaterials have been widely used in biomedicine since they have obvious inherent advantages over traditional ones. However, nanomaterials may also cause dysfunction in proteins, genes and cells, resulting in cytotoxic and genotoxic responses. Recently, more and more attention has been paid to these potential toxicities of nanomaterials, especially to the risks of nanomaterials to human health and safety. Therefore, when using nanomaterials for biomedical applications, it is of great importance to keep the balance between therapeutic efficiency and cytotoxicity (i.e., increase the therapeutic efficiency as well as decrease the potential toxicity). This requires a deeper understanding of the interactions between various types of nanomaterials and biological systems at the nano/bio interface. In this review, from the point of view of theoretical researchers, we will present the current status regarding the physical mechanism of cytotoxicity caused by nanomaterials, mainly based on recent simulation results. In addition, the strategies for minimizing the nanotoxicity naturally and artificially will also be discussed in detail. Furthermore, we should notice that toxicity is not always bad for clinical use since causing the death of specific cells is the main way of treating disease. Enhancing the targeting ability of nanomaterials to diseased cells and minimizing their side effects on normal cells will always be hugely challenging issues in nanomedicine. By combining the latest computational studies with some experimental verifications, we will provide special insights into recent advances regarding these problems, especially for the design of novel environment-responsive nanomaterials.

Self-Assembly of Amphiphilic Dendrimers: The Role of Generation and Alkyl Chain Length in siRNA Interaction

An ideal nucleic-acid transfection system should combine the physical and chemical characteristics of cationic lipids and linear polymers to decrease cytotoxicity and uptake limitations. Previous research described new types of carriers termed amphiphilic dendrimers (ADs), which are based on polyamidoamine dendrimers (PAMAM). These ADs display the cell membrane affinity advantage of lipids and preserve the high affinity for DNA possessed by cationic dendrimers. These lipid/dendrimer hybrids consist of a low-generation, hydrophilic dendron (G2, G1, or G0) bonded to a hydrophobic tail. The G2-18C AD was reported to be an efficient siRNA vector with significant gene silencing. However, shorter tail ADs (G2-15C and G2-13C) and lower generation (G0 and G1) dendrimers failed as transfection carriers. To date, the self-assembly phenomenon of this class of amphiphilic dendrimers has not been molecularly explored using molecular simulation methods. To gain insight into these systems, the present study used coarse-grained molecular dynamics simulations to describe how ADs are able to self-assemble into an aggregate, and, specifically, how tail length and generation play a key role in this event. Finally, explanations are given for the better efficiency of G2/18-C as gene carrier in terms of binding of siRNA. This knowledge could be relevant for the design of novel, safer ADs with well-optimized affinity for siRNA.

Spatial Rearrangement and Mobility Heterogeneity of an Anionic Lipid Monolayer Induced by the Anchoring of Cationic Semiflexible Polymer Chains

We use Monte Carlo simulations to investigate the interactions between cationic semiflexible polymer chains and a model fluid lipid monolayer composed of charge-neutral phosphatidyl-choline (PC), tetravalent anionic phosphatidylinositol 4,5-bisphosphate (PIP₂), and univalent anionic phosphatidylserine (PS) lipids. In particular, we explore how chain rigidity and polymer concentration influence the spatial rearrangement and mobility heterogeneity of the monolayer under the conditions where the cationic polymers anchor on the monolayer. We find that the anchored cationic polymers only sequester the tetravalent PIP₂ lipids at low polymer concentrations, where the interaction strength between the polymers and the monolayer exhibits a non-monotonic dependence on the degree of chain rigidity. Specifically, maximal anchoring occurs at low polymer concentrations, when the polymer chains have an intermediate degree of rigidity, for which the PIP₂ clustering becomes most enhanced and the mobility of the polymer/PIP₂ complexes becomes most reduced. On the other hand, at sufficiently high polymer concentrations, the anchoring strength decreases monotonically as the chains stiffen-a result that arises from the pronounced competitions among polymer chains. In this case, the flexible polymers can confine all PIP₂ lipids and further sequester the univalent PS lipids, whereas the stiffer polymers tend to partially dissociate from the monolayer and only sequester smaller PIP₂ clusters with greater mobilities. We further illustrate that the mobility gradient of the single PIP₂ lipids in the sequestered clusters is sensitively modulated by the cooperative effects between anchored segments of the polymers with different rigidities. Our work thus demonstrates that the rigidity and concentration of anchored polymers are both important parameters for tuning the regulation of anionic lipids.

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.

Force Field Development for Lipid Membrane Simulations

With the rapid development of computer power and wide availability of modelling software computer simulations of realistic models of lipid membranes, including their interactions with various molecular species, polypeptides and membrane proteins have become feasible for many research groups. The crucial issue of the reliability of such simulations is the quality of the force field, and many efforts, especially in the latest several years, have been devoted to parametrization and optimization of the force fields for biomembrane modelling. In this review, we give account of the recent development in this area, covering different classes of force fields, principles of the force field parametrization, comparison of the force fields, and their experimental validation. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

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.

Protein modeling and molecular dynamics simulation of the two novel surfactant proteins SP-G and SP-H

Surfactant proteins are well known from the human lung where they are responsible for the stability and flexibility of the pulmonary surfactant system. They are able to influence the surface tension of the gas–liquid interface specifically by directly interacting with single lipids. This work describes the generation of reliable protein structure models to support the experimental characterization of two novel putative surfactant proteins called SP-G and SP-H. The obtained protein models were complemented by predicted posttranslational modifications and placed in a lipid model system mimicking the pulmonary surface. Molecular dynamics simulations of these protein-lipid systems showed the stability of the protein models and the formation of interactions between protein surface and lipid head groups on an atomic scale. Thereby, interaction interface and strength seem to be dependent on orientation and posttranslational modification of the protein. The here presented modeling was fundamental for experimental localization studies and the simulations showed that SP-G and SP-H are theoretically able to interact with lipid systems and thus are members of the surfactant protein family.

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.

Endocytosis of PEGylated nanoparticles accompanied by structural and free energy changes of the grafted polyethylene glycol

Nanoparticles (NPs) are in use to efficiently deliver drug molecules into diseased cells. The surfaces of NPs are usually grafted with polyethylene glycol (PEG) polymers, during so-called PEGylation, to improve water solubility, avoid aggregation, and prevent opsonization during blood circulation. The interplay between grafting density σp and grafted PEG polymerization degree N makes cellular uptake of PEGylated NPs distinct from that of bare NPs. To understand the role played by grafted PEG polymers, we study the endocytosis of 8 nm sized PEGylated NPs with different σp and N through large scale dissipative particle dynamics (DPD) simulations. The free energy change Fpolymer of grafted PEG polymers, before and after endocytosis, is identified to have an effect which is comparable to, or even larger than, the bending energy of the membrane during endocytosis. Based on self-consistent field theory Fpolymer is found to be dependent on both σp and N. By incorporating Fpolymer, the critical ligand-receptor binding strength for PEGylated NPs to be internalized can be correctly predicted by a simple analytical equation. Without considering Fpolymer, it turns out impossible to predict whether the PEGylated NPs will be delivered into the diseased cells. These simulation results and theoretical analysis not only provide new insights into the endocytosis process of PEGylated NPs, but also shed light on the underlying physical mechanisms, which can be utilized for designing efficient PEGylated NP-based therapeutic carriers with improved cellular targeting and uptake.

Anisotropic spontaneous curvatures in lipid membranes

Symmetry restrictions due to fluidity require the strain energy in the Helfrich theory of lipid membranes to be locally isotropic in nature. Although this framework is suitable for modeling the interaction of membranes with proteins that generate spherical curvature such as clathrin, there are other important membrane-bending proteins such as BIN-amphiphysin-Rvs proteins that form a cylindrical coat with different curvatures in the longitudinal and the circumferential directions. In this work, we present a detailed mathematical treatment of the theory of lipid membranes incorporating anisotropic spontaneous curvatures. We derive the associated Euler-Lagrange equations and the edge conditions in a generalized setting that allows spatial heterogeneities in the properties of the membrane-protein system. We employ this theory to model the constriction of a membrane tubule by a cylindrical scaffold. In particular, we highlight the role of the equilibrium equation in the tangential plane in regulating the spatial variation of the surface tension field.