Enhanced Sampling of Phase Transitions in Coarse-Grained Lipid Bilayers

Machine learning in molecular biophysics: Protein allostery, multi-level free energy simulations, and lipid phase transitions

Machine learning (ML) techniques have been making major impacts on all areas of science and engineering, including biophysics. In this review, we discuss several applications of ML to biophysical problems based on our recent research. The topics include the use of ML techniques to identify hotspot residues in allosteric proteins using deep mutational scanning data and to analyze how mutations of these hotspots perturb co-operativity in the framework of a statistical thermodynamic model, to improve the accuracy of free energy simulations by integrating data from different levels of potential energy functions, and to determine the phase transition temperature of lipid membranes. Through these examples, we illustrate the unique value of ML in extracting patterns or parameters from complex data sets, as well as the remaining limitations. By implementing the ML approaches in the context of physically motivated models or computational frameworks, we are able to gain a deeper mechanistic understanding or better convergence in numerical simulations. We conclude by briefly discussing how the introduced models can be further expanded to tackle more complex problems.

Atomistic analysis of nematic phase transition in 4-cyano-4'-n-alkyl biphenyl liquid crystals: Sampling for the first-order phase transition and the free-energy decomposition

Molecular dynamics simulations were conducted using the generalized replica exchange method (gREM) on the 4-cyano-4'-n-alkyl biphenyl (nCB) system with n = 5, 6, 7, and 8, which exhibits a nematic-isotropic (NI) phase transition. Sampling near the phase transition temperature in systems undergoing first-order phase transitions, such as the NI phase transition, is demanding due to the substantial energy gap between the two phases. To address this, gREM, specifically designed for first-order phase transitions, was utilized to enhance sampling near the NI phase transition temperature. Free-energy calculations based on the energy representation (ER) theory were employed to characterize the NI phase transition. ER evaluates the insertion free energy of the nCB molecule for both nematic and isotropic phases, revealing a change in the temperature dependence across the NI phase transition. Further decomposition into energetic and entropic terms quantitatively shows the balance between these contributions at the NI phase transition temperature.

Machine Learning-Assisted Phase Transition Temperatures from Generalized Replica Exchange Simulations of Dry Martini Lipid Bilayers

Accurate estimation of phase transition temperatures has been a longstanding challenge for molecular simulations. Recently, the generalized Replica Exchange technique for estimating phase transition temperatures has allowed for improved sampling of the phase transition; however, it requires a significant number of simultaneous replicas both inside and outside of the transition region leading to costly computational expense. In this work, the recently developed machine learning-assisted lipid phase analysis technique for learning the phase of individual lipids has been combined with generalized Replica Exchange Molecular Dynamics to reduce the overall computational expense of evaluating transition temperatures. This technique is then applied to eight different Dry Martini lipids to demonstrate its ability to describe transition temperatures as a function of chain length and tail saturation.

Simulating the nematic-isotropic phase transition of liquid crystal model via generalized replica-exchange method

The nematic-isotropic (NI) phase transition of 4-cyano-4'-pentylbiphenyl was simulated using the generalized replica-exchange method (gREM) based on molecular dynamics simulations. The effective temperature is introduced in the gREM, allowing for the enhanced sampling of configurations in the unstable region, which is intrinsic to the first-order phase transition. The sampling performance was analyzed with different system sizes and compared with that of the temperature replica-exchange method (tREM). It was observed that gREM is capable of sampling configurations at sufficient replica-exchange acceptance ratios even around the NI transition temperature. A bimodal distribution of the order parameter at the transition region was found, which is in agreement with the mean-field theory. In contrast, tREM is ineffective around the transition temperature owing to the potential energy gap between the nematic and isotropic phases.

Interdomain exchange and the flip-flop of cholesterol in ternary component lipid membranes and their effects on heterogeneous cholesterol diffusion

Cell membranes are heterogeneous with a variety of lipids, cholesterol, and proteins and are composed of domains of different compositions. Such heterogeneous environments make the transport of cholesterol complicated: cholesterol not only diffuses within a particular domain but also travels between domains. Cholesterol also flip-flops between upper and lower leaflets such that cholesterol may reside both within leaflets and in the central region between two leaflets. How the presence of multiple domains and the interdomain exchange of cholesterol would affect the cholesterol transport, however, remains elusive. In this study, therefore, we perform molecular dynamics simulations up to 100μs for ternary component lipid membranes, which consist of saturated lipids (dipalmitoylphosphatidylcholine, DPPC), unsaturated lipids (dilinoleylphosphatidylcholine, DIPC), and cholesterol. The ternary component membranes in our simulations form two domains readily: DPPC and DIPC domains. We find that the diffusion of cholesterol molecules is much more heterogeneous and non-Gaussian than expected for binary component lipid membranes of lipids and cholesterol. The non-Gaussian parameter of the cholesterol molecules is about four times larger in the ternary component lipid membranes than in the binary component lipid membranes. Such non-Gaussian and heterogeneous transport of cholesterol arises from the interplay among the interdomain kinetics, the different diffusivity of cholesterol in different domains, and the flip-flop of cholesterol. This suggests that in cell membranes that consist of various domains and proteins, the cholesterol transport can be very heterogeneous. We also find that the mechanism of the interdomain exchange differs for different domains: cholesterol tends to exit the DIPC domain along the central region of the membrane for the DIPC-to-DPPC transition, while the cholesterol is likely to exit the DPPC domain within the membrane leaflet for the DPPC-to-DIPC transition. Also, the interdomain exchange kinetics of cholesterol for the DPPC-to-DIPC transition is up to 7.9 times slower than the DIPC-to-DPPC transition.

A Data-Driven Dimensionality Reduction Approach to Compare and Classify Lipid Force Fields

Molecular dynamics simulations of all-atom and coarse-grained lipid bilayer models are increasingly used to obtain useful insights for understanding the structural dynamics of these assemblies. In this context, one crucial point concerns the comparison of the performance and accuracy of classical force fields (FFs), which sometimes remains elusive. To date, the assessments performed on different classical potentials are mostly based on the comparison with experimental observables, which typically regard average properties. However, local differences of the structure and dynamics, which are poorly captured by average measurements, can make a difference, but these are nontrivial to catch. Here, we propose an agnostic way to compare different FFs at different resolutions (atomistic, united-atom, and coarse-grained), by means of a high-dimensional similarity metrics built on the framework of Smooth Overlap of Atomic Position (SOAP). We compare and classify a set of 13 FFs, modeling 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) bilayers. Our SOAP kernel-based metrics allows us to compare, discriminate, and correlate different FFs at different model resolutions in an unbiased, high-dimensional way. This also captures differences between FFs in modeling nonaverage events (originating from local transitions), for example, the liquid-to-gel phase transition in dipalmitoylphosphatidylcholine (DPPC) bilayers, for which our metrics allows us to identify nucleation centers for the phase transition, highlighting some intrinsic resolution limitations in implicit versus explicit solvent FFs.

Evaluating Coarse-Grained MARTINI Force-Fields for Capturing the Ripple Phase of Lipid Membranes

Phospholipids, which are an integral component of cell membranes, exhibit a rich variety of lamellar phases modulated by temperature and composition. Molecular dynamics (MD) simulations have greatly enhanced our understanding of phospholipid membranes by capturing experimentally observed phases and phase transitions at molecular resolution. However, the ripple (P_β') membrane phase, observed as an intermediate phase below the main gel-to-liquid crystalline transition with some lipids, has been challenging to capture with MD simulations, both at all-atom and coarse-grained (CG) resolutions. Here, with an aggregate ∼2.5 μs all-atom and ∼122 μs CGMD simulations, we systematically assess the ability of six CG MARTINI 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) lipid and water force-field (FF) variants, parametrized to capture the DPPC gel and fluid phases, for their ability to capture the P_β' phase, and compared observations with those from an all-atom FF. Upon cooling from the fluid phase to below the phase transition temperature with smaller (380-lipid) and larger (>2200-lipid) MARTINI and all-atom (CHARMM36 FF) DPPC lipid bilayers, we observed that smaller bilayers with both all-atom and MARTINI FFs sampled interdigitated P_β' and ripple-like states, respectively. However, while all-atom simulations of the larger DPPC membranes exhibited the formation of the P_β' phase, MARTINI membranes did not sample interdigitated ripple-like states at larger system sizes. We then demonstrated that the ripple-like states in smaller MARTINI membranes were kinetically trapped structures caused by finite size effects rather than being representative of true P_β' phases. We showed that a MARTINI FF variant that could capture the tilted L_β' gel phase, a prerequisite for stabilizing the P_β' phase, was unable to capture the rippled phase upon cooling. Our study reveals that the current MARTINI FFs (including MARTINI3) may require specific reparametrization of the interaction potentials to stabilize lipid interdigitation, a characteristic of the ripple phase.

Simulation of fluid/gel phase equilibrium in lipid vesicles

Simulation of single component dipalmitoylphosphatidylcholine (DPPC) coarse-grained DRY-MARTINI lipid vesicles of diameter 10 nm (1350 lipids), 20 nm (5100 lipids) and 40 nm (17 600 lipids) is performed using statistical temperature molecular dynamics (STMD), to study finite size effects upon the order-disorder gel/fluid transition. STMD obtains enhanced sampling using a generalized ensemble, obtaining a flat energy distribution between upper and lower cutoffs, with little computational cost over canonical molecular dynamics. A single STMD trajectory of moderate length is sufficient to sample 20+ transition events, without trapping in the gel phase, and obtain well averaged properties. Phase transitions are analyzed via the energy-dependence of the statistical temperature, T_S(U). The transition temperature decreases with decreasing diameter, in agreement with experiment, and the transition changes from first order to borderline first-second order. The size- and layer-dependence of the structure of both stable phases, and of the pathway of the phase transition, are determined. It is argued that the finite size effects are primarily caused by the disruption of the gel packing by curvature. Inhomogeneous states with faceted gel patches connected by unusual fluid seams are observed at high curvature, with visually different structure in the inner and outer layers due to the different curvatures. Thus a simple physical picture describes phase transitions in nanoscale finite systems far from the thermodynamic limit.

Intermolecular Forces Model for Lipid Microbubble Shells

Lipid-coated microbubbles are currently used clinically as ultrasound contrast agents for echocardiography and radiology and are being developed for many new diagnostic and therapeutic applications. Accordingly, there is a growing need to engineer specific formulations by employing rational design to guide lipid selection and processing. This approach requires a quantitative relationship between lipid chemistry and interfacial properties of the microbubble shell. Just such a model is proposed here on the basis of lateral Coulomb and van der Waals interactions between lipid head- and tailgroups, using previous coarse graining and force fields developed for molecular dynamics simulations. The model predicts with sufficient accuracy the monolayer permeability, the elasticity as a function of either lipid composition or temperature, and the equilibrium spreading surface tension of the lipid onto an air/water interface. In the future, the intermolecular forces model could be employed to elucidate more complex phenomena and to engineer novel microbubble formulations.

Role of Computational Methods in Going beyond X-ray Crystallography to Explore Protein Structure and Dynamics

Protein structural biology came a long way since the determination of the first three-dimensional structure of myoglobin about six decades ago. Across this period, X-ray crystallography was the most important experimental method for gaining atomic-resolution insight into protein structures. However, as the role of dynamics gained importance in the function of proteins, the limitations of X-ray crystallography in not being able to capture dynamics came to the forefront. Computational methods proved to be immensely successful in understanding protein dynamics in solution, and they continue to improve in terms of both the scale and the types of systems that can be studied. In this review, we briefly discuss the limitations of X-ray crystallography in studying protein dynamics, and then provide an overview of different computational methods that are instrumental in understanding the dynamics of proteins and biomacromolecular complexes.

Regimes of Complex Lipid Bilayer Phases Induced by Cholesterol Concentration in MD Simulation

Cholesterol is essential to the formation of phase-separated lipid domains in membranes. Lipid domains can exist in different thermodynamic phases depending on the molecular composition and play significant roles in determining structure and function of membrane proteins. We investigate the role of cholesterol in the structure and dynamics of ternary lipid mixtures displaying phase separation using molecular dynamics simulations, employing a physiologically relevant span of cholesterol concentration. We find that cholesterol can induce formation of three regimes of phase behavior: 1) miscible liquid-disordered bulk, 2) phase-separated, domain-registered coexistence of liquid-disordered and liquid-ordered domains, and 3) phase-separated, domain-antiregistered coexistence of liquid-disordered and newly identified nanoscopic gel domains composed of cholesterol threads we name "cholesterolic gel" domains. These findings are validated and discussed in the context of current experimental knowledge, models of cholesterol spatial distributions, and models of ternary lipid-mixture phase separation.