Identifying systematic errors in a power spectral analysis of simulated lipid membranes

Mesoscale simulation of biomembranes with FreeDTS

We present FreeDTS software for performing computational research on biomembranes at the mesoscale. In this software, a membrane is represented by a dynamically triangulated surface equipped with vertex-based inclusions to integrate the effects of integral and peripheral membrane proteins. Several algorithms are included in the software to simulate complex membranes at different conditions such as framed membranes with constant tension, vesicles and high-genus membranes with various fixed volumes or constant pressure differences and applying external forces to membrane regions. Furthermore, the software allows the user to turn off the shape evolution of the membrane and focus solely on the organization of proteins. As a result, we can take realistic membrane shapes obtained from, for example, cryo-electron tomography and backmap them into a finer simulation model. In addition to many biomembrane applications, this software brings us a step closer to simulating realistic biomembranes with molecular resolution. Here we provide several interesting showcases of the power of the software but leave a wide range of potential applications for interested users.

Nonuniversal impact of cholesterol on membranes mobility, curvature sensing and elasticity

Biological membranes, composed mainly of phospholipids and cholesterol, play a vital role as cellular barriers. They undergo localized reshaping in response to environmental cues and protein interactions, with the energetics of deformations crucial for exerting biological functions. This study investigates the non-universal role of cholesterol on the structure and elasticity of saturated and unsaturated lipid membranes. Our study uncovers a highly cooperative relationship between thermal membrane bending and local cholesterol redistribution, with cholesterol showing a strong preference for the compressed membrane leaflet. Remarkably, in unsaturated membranes, increased cholesterol mobility enhances cooperativity, resulting in membrane softening despite membrane thickening and lipid compression caused by cholesterol. These findings elucidate the intricate interplay between thermodynamic forces and local molecular interactions that govern collective properties of membranes.

Molecular insights into the effects of focused ultrasound mechanotherapy on lipid bilayers: Unlocking the keys to design effective treatments

Administration of focused ultrasounds (US) represents an attractive complement to classical therapies for a wide range of maladies, from cancer to neurological pathologies, as they are non-invasive, easily targeted, their dosage is easy to control, and they involve low risks. Different mechanisms have been proposed for their activity but the direct effect of their interaction with cell membranes is not well understood at the molecular level. This is in part due to the difficulty of designing experiments able to probe the required spatio-temporal resolutions. Here we use Molecular Dynamics (MD) simulations at two resolution levels and machine learning (ML) classification tools to shed light on the effects that focused US mechanotherapy methods have over a range of lipid bilayers. Our results indicate that the dynamic-structural response of the membrane models to the mechanical perturbations caused by the sound waves strongly depends on the lipid composition. The analyses performed on the MD trajectories contribute to a better understanding of the behavior of lipid membranes, and to open up a path for the rational design of new therapies for the long list of diseases characterized by specific lipid profiles of pathological membrane cells.

Cristae formation is a mechanical buckling event controlled by the inner membrane lipidome

Cristae are high curvature structures in the inner mitochondrial membrane (IMM) that are crucial for ATP production. While cristae-shaping proteins have been defined, analogous mechanisms for lipids have yet to be elucidated. Here we combine experimental lipidome dissection with multi-scale modeling to investigate how lipid interactions dictate IMM morphology and ATP generation. When modulating phospholipid (PL) saturation in engineered yeast strains, we observed a surprisingly abrupt breakpoint in IMM topology driven by a continuous loss of ATP synthase organization at cristae ridges. We found that cardiolipin (CL) specifically buffers the IMM against curvature loss, an effect that is independent of ATP synthase dimerization. To explain this interaction, we developed a continuum model for cristae tubule formation that integrates both lipid and protein-mediated curvatures. The model highlighted a snapthrough instability, which drives IMM collapse upon small changes in membrane properties. We also showed that CL is essential in low oxygen conditions that promote PL saturation. These results demonstrate that the mechanical function of CL is dependent on the surrounding lipid and protein components of the IMM.

Membrane free-energy landscapes derived from atomistic dynamics explain nonuniversal cholesterol-induced stiffening

All lipid membranes have inherent morphological preferences and resist deformation. Yet adaptations in membrane shape can and do occur at multiple length scales. While this plasticity is crucial for cellular physiology, the factors controlling the morphological energetics of lipid bilayers and the dominant mechanisms of membrane remodeling remain to be fully understood. An ongoing debate regarding the universality of the stiffening effect of cholesterol underscores the challenges facing this field, both experimentally and theoretically, even for simple lipid mixtures. On the computational side, we have argued that enhanced-sampling all-atom molecular dynamics simulations are uniquely suited for the quantification of membrane conformational energetics, as they minimize a priori assumptions and permit analysis of bilayers in deformed states. To showcase this approach, we examine reported inconsistencies between alternative experimental measurements of bending moduli for cholesterol-enriched membranes. Specifically, we analyze lipid bilayers with different chain saturation and compute free-energy landscapes for curvature deformations distributed over areas from ∼5 to ∼60 nm2 . These enhanced simulations, totaling over 100 μs of sampling time, enable us to directly quantify both bending and tilt moduli and to dissect the contributing factors and molecular mechanisms of curvature generation at each length scale. Our results show that the effects of cholesterol on bending rigidity are lipid-specific and suggest that this specificity arises from differences in the torsional dynamics of the acyl chains. In summary, we demonstrate that quantitative relationships can now be established between lipid structure and bending energetics, paving the way for addressing open fundamental questions in cell membrane mechanics.

Membrane free-energy landscapes derived from atomistic dynamics explain nonuniversal cholesterol-induced stiffening

All lipid membranes have inherent morphological preferences and resist deformation. Yet adaptations in membrane shape can and do occur at multiple length scales. While this plasticity is crucial for cellular physiology, the factors controlling the morphological energetics of lipid bilayers and the dominant mechanisms of membrane remodeling remain unclear. An ongoing debate regarding the universality of the stiffening effect of cholesterol underscores the challenges facing this field, both experimentally and theoretically, even for simple lipid mixtures. On the computational side, we have argued that enhanced- sampling all-atom molecular dynamics simulations are uniquely suited for quantification of membrane conformational energetics, not only because they minimize a-priori assumptions, but also because they permit analysis of bilayers in deformed states. To showcase this approach, we examine reported inconsistencies between alternative experimental measurements of bending moduli for cholesterol-enriched membranes. Specifically, we analyze lipid bilayers with different chain saturation, and compute free-energy landscapes for curvature deformations distributed over areas from ∼5 to ∼60 nm 2 . These enhanced simulations, totaling over 100 microseconds of sampling time, enable us to directly quantify both bending and tilt moduli, and to dissect the contributing factors and molecular mechanisms of curvature generation at each length scale. Our results show that cholesterol effects are lipid-specific, in agreement with giantvesicle measurements, and explain why experiments probing nanometer scale lipid dynamics diverge. In summary, we demonstrate that quantitative structure-mechanics relationships can now be established for heterogenous membranes, paving the way for addressing open fundamental questions in cell membrane mechanics.

Significance

Elucidating the energetics and mechanisms of membrane remodeling is an essential step towards understanding cell physiology. This problem is challenging, however, because membrane bending involves both large-scale and atomic-level dynamics, which are difficult to measure simultaneously. A recent controversy regarding the stiffening effect of cholesterol, which is ubiquitous in animal cells, illustrates this challenge. We show how enhanced molecular-dynamics simulations can bridge this length-scale gap and reconcile seemingly incongruent observations. This approach facilitates a conceptual connection between lipid chemistry and membrane mechanics, thereby providing a solid basis for future research on remodeling phenomena, such as in membrane trafficking or viral infection.

Determination of elastic parameters of lipid membranes from simulation under varied external pressure

Many cellular processes such as endocytosis, exocytosis, and vesicle trafficking involve membrane deformations, which can be analyzed in the framework of the elastic theories of lipid membranes. These models operate with phenomenological elastic parameters. A connection between these parameters and the internal structure of lipid membranes can be provided by three-dimensional (3D) elastic theories. Considering a membrane as a 3D layer, Campelo et al. [F. Campelo et al., Adv. Colloid Interface Sci. 208, 25 (2014)10.1016/j.cis.2014.01.018] developed a theoretical basis for the calculation of elastic parameters. In this work we generalize and improve this approach by considering a more general condition of global incompressibility instead of local incompressibility. Crucially, we find an important correction to the theory of Campelo et al., which if not taken into account leads to a significant miscalculation of elastic parameters. With the total volume conservation taken into account, we derive an expression for the local Poisson's ratio, which determines how the local volume changes upon stretching and permits a more precise determination of elastic parameters. Also, we substantially simplify the procedure by calculating the derivatives of the moments of the local tension with respect to stretching instead of calculating the local stretching modulus. We obtain a relation between the Gaussian curvature modulus as a function of stretching and the bending modulus, showing that these two elastic parameters are not independent, as was previously assumed. The proposed algorithm is applied to membranes composed of pure dipalmitoylphosphatidylcholine (DPPC), dioleoylphosphatidylcholine (DOPC), and their mixture. The following elastic parameters of these systems are obtained: the monolayer bending and stretching moduli, spontaneous curvature, neutral surface position, and local Poisson's ratio. It is shown that the bending modulus of the DPPC/DOPC mixture follows a more complex trend than predicted by the classical Reuss averaging, which is often employed in theoretical frameworks.

Generation of nanoscopic membrane curvature for membrane trafficking

Curved membranes are key features of intracellular organelles, and their generation involves dynamic protein complexes. Here we describe the fundamental mechanisms such as the hydrophobic insertion, scaffolding and crowding mechanisms these proteins use to produce membrane curvatures and complex shapes required to form intracellular organelles and vesicular structures involved in endocytosis and secretion. For each mechanism, we discuss its cellular functions as well as the underlying physical principles and the specific membrane properties required for the mechanism to be feasible. We propose that the integration of individual mechanisms into a highly controlled, robust process of curvature generation often relies on the assembly of proteins into coats. How cells unify and organize the curvature-generating factors at the nanoscale is presented for three ubiquitous coats central for membrane trafficking in eukaryotes: clathrin-coated pits, caveolae, and COPI and COPII coats. The emerging theme is that these coats arrange and coordinate curvature-generating factors in time and space to dynamically shape membranes to accomplish membrane trafficking within cells.

Determination of Elastic Parameters of Lipid Membranes with Molecular Dynamics: A Review of Approaches and Theoretical Aspects

Lipid membranes are abundant in living organisms, where they constitute a surrounding shell for cells and their organelles. There are many circumstances in which the deformations of lipid membranes are involved in living cells: fusion and fission, membrane-mediated interaction between membrane inclusions, lipid-protein interaction, formation of pores, etc. In all of these cases, elastic parameters of lipid membranes are important for the description of membrane deformations, as these parameters determine energy barriers and characteristic times of membrane-involved phenomena. Since the development of molecular dynamics (MD), a variety of in silico methods have been proposed for the determination of elastic parameters of simulated lipid membranes. These MD methods allow for the consideration of details unattainable in experimental techniques and represent a distinct scientific field, which is rapidly developing. This work provides a review of these MD approaches with a focus on theoretical aspects. Two main challenges are identified: (i) the ambiguity in the transition from the continuum description of elastic theories to the discrete representation of MD simulations, and (ii) the determination of intrinsic elastic parameters of lipid mixtures, which is complicated due to the composition-curvature coupling effect.

Tilt Modulus of Bilayer Membranes Self-Assembled from Rod-Coil Diblock Copolymers

Quantitatively understanding membrane fission and fusion requires a mathematical model taking their underlying elastic degrees of freedom, such as the molecule's tilt, into account. Hamm-Kozlov's model is such a framework that includes a tilt modulus along with the bending modulus and Gaussian modulus. This paper investigates the tilt modulus of liquid-crystalline bilayer membranes by applying self-consistent field theory. Unlike the widely used method in molecular dynamics simulation which extracts the tilt modulus by simulating bilayer buckles with various single modes, we introduce a tilt constrain term in the free energy to stabilize bilayers with various tilt angles. Fitting the energy curve as a function of the tilt angle to Hamm-Kozlov's elastic energy allows us to extract the tilt modulus directly. Based on this novel scheme and focused on the bilayers self-assembled from rod-coil diblock copolymers, we carry out a systematic study of the dependence of the tensionless A-phase bilayer's tilt modulus on the microscopic parameters.

Bending Modulus of Lipid Membranes from Density Correlation Functions

The bending modulus κ quantifies the elasticity of biological membranes in terms of the free energy cost of increasing the membrane corrugation. Molecular dynamics (MD) simulations provide a powerful approach to quantify κ by analyzing the thermal fluctuations of the lipid bilayer. However, existing methods require the identification and filtering of non-mesoscopic fluctuation modes. State of the art methods rely on identifying a smooth surface to describe the membrane shape. These methods introduce uncertainties in calculating κ since they rely on different criteria to select the relevant fluctuation modes. Here, we present a method to compute κ using molecular simulations. Our approach circumvents the need to define a mesoscopic surface or an orientation field for the lipid tails explicitly. The bending and tilt moduli can be extracted from the analysis of the density correlation function (DCF). The method introduced here builds on the Bedeaux and Weeks (BW) theory for the DCF of fluctuating interfaces and on the coupled undulatory (CU) mode introduced by us in previous work. We test the BW-DCF method by computing the elastic properties of lipid membranes with different system sizes (from 500 to 6000 lipid molecules) and using coarse-grained (for POPC and DPPC lipids) and fully atomistic models (for DPPC). Further, we quantify the impact of cholesterol on the bending modulus of DPPC bilayers. We compare our results with bending moduli obtained with X-ray diffraction data and different computer simulation methods.

Unravelling hierarchical levels of structure in lipid membranes

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A set of hierarchical levels of structure can be proposed for lipid bilayers.

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The composition of a lipid bilayer is identified as a fluid version of primary structure.

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The interaction between leaflets is taken as the secondary structure of lipid bilayers.

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A method to identify membrane domains and their interaction is proposed as a third level of structure.

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The highly specific lipid perturbation around embedded macromolecules is taken as the quaternary level of structure.

In analogy with the hierarchical levels typically used to describe the structure of nucleic acids or proteins and keeping in mind that lipid bilayers are not just mere envelopers for biological material but directly responsible for many important functions of life, it is discussed here how membrane models can also be interpreted in terms of different hierarchies in their structure. Namely, lipid composition, interaction between leaflets, existence and interaction of domains arising from the coordinate behavior of lipids and their properties, plus the manifest and specific perturbation of the lipid organization around macromolecules embedded in a membrane are hereby used to define the primary, secondary, tertiary and quaternary structures, respectively. Molecular Dynamics simulations are used to illustrate this proposal. Alternative levels of organization and methods to define domains can be proposed but the final aim is to highlight the paradigm arising from this description which is expected to have significant consequences on deciphering the underlying factors governing membranes and their interactions with other molecules.