Correlated diffusion in lipid bilayers.

Lipid membranes are complex quasi-two-dimensional fluids, whose importance in biology and unique physical/materials properties have made them a major target for biophysical research. Recent single-molecule tracking experiments in membranes have caused some controversy, calling the venerable Saffman-Delbrück model into question and suggesting that, perhaps, current understanding of membrane hydrodynamics is imperfect. However, single-molecule tracking is not well suited to resolving the details of hydrodynamic flows; observations involving correlations between multiple molecules are superior for this purpose. Here dual-color molecular tracking with submillisecond time resolution and submicron spatial resolution is employed to reveal correlations in the Brownian motion of pairs of fluorescently labeled lipids in membranes. These correlations extend hundreds of nanometers in freely floating bilayers (black lipid membranes) but are severely suppressed in supported lipid bilayers. The measurements are consistent with hydrodynamic predictions based on an extended Saffman-Delbrück theory that explicitly accounts for the two-leaflet bilayer structure of lipid membranes.

Dynamic correlations in lipid bilayer membranes over finite time intervals.

Recent single-molecule measurements [Schoch et al., Proc. Natl. Acad. Sci. U. S. A. 118, e2113202118 (2021)] have observed dynamic lipid-lipid correlations in membranes with submicrometer spatial resolution and submillisecond temporal resolution. While short from an instrumentation standpoint, these length and time scales remain long compared to microscopic molecular motions. Theoretical expressions are derived to infer experimentally measurable correlations from the two-body diffusion matrix appropriate for membrane-bound bodies coupled by hydrodynamic interactions. The temporal (and associated spatial) averaging resulting from finite acquisition times has the effect of washing out correlations as compared to naive predictions (i.e., the bare elements of the diffusion matrix), which would be expected to hold for instantaneous measurements. The theoretical predictions are shown to be in excellent agreement with Brownian dynamics simulations of experimental measurements. Numerical results suggest that the experimental measurement of membrane protein diffusion, in complement to lipid diffusion measurements, might help to resolve the experimental ambiguities encountered for certain black lipid membranes.

Motion of objects embedded in lipid bilayer membranes: Advection and effective viscosity.

An interfacial regularized Stokeslet scheme is presented to predict the motion of solid bodies (e.g., proteins or gel-phase domains) embedded within flowing lipid bilayer membranes. The approach provides a numerical route to calculate velocities and angular velocities in complex flow fields that are not amenable to simple Faxén-like approximations. Additionally, when applied to shearing motions, the calculations yield predictions for the effective surface viscosity of dilute rigid-body-laden membranes. In the case of cylindrical proteins, effective viscosity calculations are compared to two prior analytical predictions from the literature. Effective viscosity predictions for a dilute suspension of rod-shaped objects in the membrane are also presented.

Integrated rate laws for processive and distributive enzymatic turnover.

Recently derived steady-state differential rate laws for the catalytic turnover of molecules containing two substrate sites are reformulated as integrated rate laws. The analysis applies to a broad class of Markovian dynamic models, motivated by the varied and often complex mechanisms associated with DNA modifying enzymes. Analysis of experimental data for the methylation kinetics of DNA by Dam (DNA adenine methyltransferase) is drastically improved through the use of integrated rate laws. Data that are too noisy for fitting to differential predictions are reliably interpreted through the integrated rate laws.

Lipid diffusion in the distal and proximal leaflets of supported lipid bilayer membranes studied by single particle tracking.

Supported lipid bilayers (SLBs) have been studied extensively as simple but powerful models for cellular membranes. Yet, potential differences in the dynamics of the two leaflets of a SLB remain poorly understood. Here, using single particle tracking, we obtain a detailed picture of bilayer dynamics. We observe two clearly separate diffusing populations, fast and slow, that we associate with motion in the distal and proximal leaflets of the SLB, respectively, based on fluorescence quenching experiments. We estimate diffusion coefficients using standard techniques as well as a new method based on the blur of images due to motion. Fitting the observed diffusion coefficients to a two-leaflet membrane hydrodynamic model allows for the simultaneous determination of the intermonolayer friction coefficient and the substrate-membrane friction coefficient, without any prior assumptions on the strengths of the relevant interactions. Remarkably, our calculations suggest that the viscosity of the interfacial water confined between the membrane and the substrate is elevated by ∼104 as compared to bulk water. Using hidden Markov model analysis, we then obtain insight into the transbilayer movement of lipids. We find that lipid flip-flop dynamics are very fast, with half times in the range of seconds. Importantly, we find little evidence for membrane defect mediated lipid flip-flop for SLBs at temperatures well above the solid-to-liquid transition, though defects seem to be involved when the SLBs are cooled down. Our work thus shows that the combination of single particle tracking and advanced hydrodynamic modeling provides a powerful means to obtain insight into membrane dynamics.

On the generality of Michaelian kinetics.

The reversible Michaelis-Menten equation is shown to follow from a very broad class of steady-state kinetic models involving enzymes that adopt a unique free (i.e., not complexed to substrate/product) state in solution. In the case of enzymes with multiple free states/conformations (e.g., fluctuating, hysteretic, or co-operative monomeric enzymes), Michaelian behavior is still assured if the relative steady-state populations of free enzyme states are independent of substrate and product concentration. Prior models for Michaelian behavior in multiple conformer enzymes are shown to be special cases of this single condition.

Theory of single molecule emission spectroscopy.

A general theory and calculation framework for the prediction of frequency-resolved single molecule photon counting statistics is presented. Expressions for the generating function of photon counts are derived, both for the case of naive "detection" based solely on photon emission from the molecule and also for experimentally realizable detection of emitted photons, and are used to explicitly calculate low-order photon-counting moments. The two cases of naive detection versus physical detection are compared to one another and it is demonstrated that the physical detection scheme resolves certain inconsistencies predicted via the naive detection approach. Applications to two different models for molecular dynamics are considered: a simple two-level system and a two-level absorber subject to spectral diffusion.

Extracting enzyme processivity from kinetic assays.

A steady-state analysis for the catalytic turnover of molecules containing two substrate sites is presented. A broad class of Markovian dynamic models, motivated by the action of DNA modifying enzymes and the rich variety of translocation mechanisms associated with these systems (e.g., sliding, hopping, intersegmental transfer, etc.), is considered. The modeling suggests an elementary and general method of data analysis, which enables the extraction of the enzyme's processivity directly and unambiguously from experimental data. This analysis is not limited to the initial velocity regime. The predictions are validated both against detailed numerical models and by revisiting published experimental data for EcoRI endonuclease acting on DNA.

Strong influence of periodic boundary conditions on lateral diffusion in lipid bilayer membranes.

The Saffman-Delbrück hydrodynamic model for lipid-bilayer membranes is modified to account for the periodic boundary conditions commonly imposed in molecular simulations. Predicted lateral diffusion coefficients for membrane-embedded solid bodies are sensitive to box shape and converge slowly to the limit of infinite box size, raising serious doubts for the prospects of using detailed simulations to accurately predict membrane-protein diffusivities and related transport properties. Estimates for the relative error associated with periodic boundary artifacts are 50% and higher for fully atomistic models in currently feasible simulation boxes. MARTINI simulations of LacY membrane protein diffusion and LacY dimer diffusion in DPPC membranes and lipid diffusion in pure DPPC bilayers support the underlying hydrodynamic model.

Determination of biomembrane bending moduli in fully atomistic simulations.

The bilayer bending modulus (Kc) is one of the most important physical constants characterizing lipid membranes, but precisely measuring it is a challenge, both experimentally and computationally. Experimental measurements on chemically identical bilayers often differ depending upon the techniques employed, and robust simulation results have previously been limited to coarse-grained models (at varying levels of resolution). This Communication demonstrates the extraction of Kc from fully atomistic molecular dynamics simulations for three different single-component lipid bilayers (DPPC, DOPC, and DOPE). The results agree quantitatively with experiments that measure thermal shape fluctuations in giant unilamellar vesicles. Lipid tilt, twist, and compression moduli are also reported.

Calculating hydrodynamic interactions for membrane-embedded objects.

A recently introduced numerical scheme for calculating self-diffusion coefficients of solid objects embedded in lipid bilayer membranes is extended to enable calculation of hydrodynamic interactions between multiple objects. The method is used to validate recent analytical predictions by Oppenheimer and Diamant [Biophys. J. 96, 3041 2009] related to the coupled diffusion of membrane embedded proteins and is shown to converge to known near-field lubrication results as objects closely approach one another; however, the present methodology also applies outside of the limiting regimes where analytical results are available. Multiple different examples involving pairs of disk-like objects with various constraints imposed on their relative motions demonstrate the importance of hydrodynamic interactions in the dynamics of proteins and lipid domains on membrane surfaces. It is demonstrated that the relative change in self-diffusion of a membrane embedded object upon perturbation by a similar proximal solid object displays a maximum for object sizes comparable to the Saffman-Delbrück length of the membrane.

Fluctuating hydrodynamics of multicomponent membranes with embedded proteins.

A simulation method for the dynamics of inhomogeneous lipid bilayer membranes is presented. The membrane is treated using stochastic Saffman-Delbrück hydrodynamics, coupled to a phase-field description of lipid composition and discrete membrane proteins. Multiple applications are considered to validate and parameterize the model. The dynamics of membrane composition fluctuations above the critical point and phase separation dynamics below the critical point are studied in some detail, including the effects of adding proteins to the mixture.

Continuum simulations of biomembrane dynamics and the importance of hydrodynamic effects.

Traditional particle-based simulation strategies are impractical for the study of lipid bilayers and biological membranes over the longest length and time scales (microns, seconds and longer) relevant to cellular biology. Continuum-based models developed within the frameworks of elasticity theory, fluid dynamics and statistical mechanics provide a framework for studying membrane biophysics over a range of mesoscopic to macroscopic length and time regimes, but the application of such ideas to simulation studies has occurred only relatively recently. We review some of our efforts in this direction with emphasis on the dynamics in model membrane systems. Several examples are presented that highlight the prominent role of hydrodynamics in membrane dynamics and we argue that careful consideration of fluid dynamics is key to understanding membrane biophysics at the cellular scale.

Single molecule counting statistics for systems with periodic driving.

We extend the generating function approach for calculation of event statistics observed in single molecule spectroscopy to cases where the single molecule evolves under explicitly time-dependent and periodic perturbation. Floquet theory is used to recast the generating function equations for the periodically driven system into effective equations devoid of explicit time-dependence. Two examples are considered, one employing simple stochastic dynamics and the other quantum dynamics, to demonstrate the versatility and numerical accuracy of the methodology.

Thermal fluctuations in shape, thickness, and molecular orientation in lipid bilayers. II. Finite surface tensions.

We investigate the role of lipid chemical potential on the shape, thickness, and molecular orientation (lipid tilting relative to the monolayer surface normal) of lipid bilayers via a continuum-level model. We predict that decreasing the chemical potential at constant temperature, which is associated with an increase in surface tension via the Gibbs-Duhem relation, leads both to the well known reduction in thermal membrane undulations and also to increasing fluctuation amplitudes for bilayer thickness and molecular orientation. These trends are shown to be in good agreement with molecular simulations, however it is impossible to achieve full quantitative agreement between theory and simulation within the confines of the present model. We suggest that the assumption of lipid volume incompressibility, common to our theoretical treatment and other continuum models in the literature, may be partially responsible for the quantitative discrepancies between theory and simulation.

Determining biomembrane bending rigidities from simulations of modest size.

Thermal fluctuations of lipid orientation are analyzed to infer the bending rigidity of lipid bilayers directly from molecular simulations. Compared to the traditional analysis of thermal membrane undulations, the proposed method is reliable down to shorter wavelengths and allows for determination of the bending rigidity using smaller simulation boxes. The requisite theoretical arguments behind this analysis are presented and verified by simulations spanning a diverse range of lipid models from the literature.

Effective temperature of red-blood-cell membrane fluctuations.

Biologically driven nonequilibrium fluctuations are often characterized by their non-Gaussianity or by an "effective temperature", which is frequency dependent and higher than the ambient temperature. We address these two measures theoretically by examining a randomly kicked particle, with a variable number of kicking motors, and show how these two indicators of nonequilibrium behavior can contradict. Our results are compared with new experiments on shape fluctuations of red-blood cell membranes, and demonstrate how the physical nature of the motors in this system can be revealed using these global measures of nonequilibrium.

Dynamic scaling in phase separation kinetics for quasi-two-dimensional membranes.

We consider the dynamics of phase separation in lipid bilayer membranes, modeled as flat two-dimensional liquid sheets within a bulk fluid, both in the creeping flow approximation. We present scaling arguments that suggest asymptotic coarsening in these systems is characterized by a length scale R(t) ~ t(1/2) for critical (bicontinuous) phase separation and R(t) ~t(1/3) for off-critical concentrations (droplet morphology). In this limit, the bulk fluid is the primary source of dissipation. We also address these questions with continuum stochastic hydrodynamic simulations. We see evidence of scaling violation in critical phase separation, where isolated circular domains coarsen slower than elongated ones. However, we also find a region of apparent scaling where R(t) ~ t(1/2) is observed. This appears to be due to the competition of thermal and hydrodynamic effects. We argue that the diversity of scaling exponents measured in experiment and prior simulations can in part be attributed to certain measurements lying outside the asymptotic long-length-scale regime, and provide a framework to help understand these results. We also discuss a few simple generalizations to confined membranes and membranes in which inertia is relevant.

The intermediate scattering function for lipid bilayer membranes: from nanometers to microns.

A numerical scheme based upon established hydrodynamic and elastic considerations is introduced and used to predict the intermediate scattering function for lipid bilayer membranes. The predictions span multiple wavelength regimes, including those studied by dynamic light scattering (DLS; microns) and neutron spin-echo (NSE) spectroscopy (10-100 nm). The results validate a recent theory specific to the NSE regime and expose slight inaccuracies associated with the theoretical results available in the DLS regime. The assumptions that underlie both our numerical methods and the related theoretical predictions are reviewed in detail to explain when certain results can be applied to experiment and where caution must be exercised.

Thermal fluctuations in shape, thickness, and molecular orientation in lipid bilayers.

We present a unified continuum-level model for bilayer energetics that includes the effects of bending, compression, lipid orientation (tilting relative to the monolayer surface normal), and microscopic noise (protrusions). Expressions for thermal fluctuation amplitudes of several physical quantities are derived. These predictions are shown to be in good agreement with molecular simulations.