Lamm Equation Research Articles

From the Svedberg ultracentrifuge to the physics of the rotating carbon nanotube

The classical and the quantum motion of a massive particle in the rotating carbon nanotube is considered. Photon is included. The spin motion described by the Bargmann-Michel-Telegdi equation is considered in the rotation tube and rotating system. The future crucial problem is the Lamm equation in the rotating Carbon nanotube.

The photons and gluons sedimentation in the Svedberg ultracentrifuge

We consider the case of rotating black body (including the rotation of the black bath with gluons) where photons and gluons can perform the settling and sedimentation. The Planck formula for photons must be, in this situation, replaced by the Exner solution of the Lamm equation for photons.

Approaching Effective Differential Centrifugal Fractionation by Combining Image Analysis with Analytical Ultracentrifugation.

Centrifugation is one of the most commonly used methods for separation in biology and chemistry. However, effective fractionation is not always easy to obtain, as preparative centrifuge experiments are mostly conducted in an empirical way, even when it is guided by the quantitative results from analytical ultracentrifuge (AUC). Very few works have been performed to enhance the fractionation resolution of the differential centrifugation method in a swing-out rotor. This is primarily due to the absence of a characterization tool for sedimentation in the preparative centrifuge. In this study, we utilized image analysis to map the particle concentration distribution throughout the preparative centrifuge tube, revealing an unexpected and abnormal sedimentation process. By characterizing the sedimentation coefficient distributions of the fractionated product via AUC, we demonstrated that the overall sedimentation efficiency in a swing-out preparative centrifuge was significantly reduced. Furthermore, effective fractionation was confined to the intermediate phase of the entire sedimentation process. We propose that the mechanism here is a combination of the inverse Boycott effect and droplet sedimentation. The actual sedimentation process within a preparative centrifuge can be described by modifying the Lamm equation phenomenologically, which simply results in an effective sedimentation coefficient. Our work builds a foundation for determining the optimal preparative centrifugation conditions for various systems.

Polysaccharide valproates: Structure - property relationships in solution

Polysaccharides are promising macromolecular platforms for use in the life sciences. Here, bioactive cellulose, pullulan, and dextran valproates are characterized hydrodynamically by sedimentation velocity and thermodynamically by sedimentation equilibrium analytical ultracentrifugation. Using sedimentation-diffusion analysis of sedimentation velocity experiments by numerical solution of the Lamm equation enabled the calculation of sedimentation and diffusion coefficients, and consequently molar masses. Sedimentation equilibrium experiments were then also used to determine the average molar masses. The corresponding set of data, with independently performed self-diffusion measurements by nuclear magnetic resonance spectroscopy, and together with size exclusion chromatography molar masses by coupling to refractive index-, viscometric-, and multi-angle laser light scattering detection, were subsequently correlated to each other by the hydrodynamic invariant and sedimentation parameter. We assess statistically most relevant average values of the molar masses of these polysaccharide valproates with relevant macromolecular conformational characteristics.

Biochemical and structural analyses reveal that the tumor suppressor neurofibromin (NF1) forms a high-affinity dimer

Biochemical and structural analyses reveal that the tumor suppressor neurofibromin (NF1) forms a high-affinity dimer

Prediction and analysis of analytical ultracentrifugation experiments for heterogeneous macromolecules and nanoparticles based on Brownian dynamics simulation

In the prediction of sedimentation profiles in analytical ultracentrifugation, the counterflow due to diffusion must be taken into account for a proper analysis of experimental data in the determination of molecular properties. This is usually achieved by numerical solution of the Lamm equation. This paper presents an alternative approach, in which the displacement of the solute in the cell, resulting from the opposite effects of ultracentrifugal force and diffusional drift, is described by Brownian dynamics simulation of the solute particles. The formalism is developed for heterogeneous solutes, composed of several species, and implemented in computational schemes and tools. The accuracy of the procedure is verified by comparison with other methods based on the Lamm equation, and its efficiency is illustrated. The possibilities offered by the Brownian dynamics methods in the determination of solute properties and sample composition are demonstrated.

Two-dimensional grid optimization for sedimentation velocity analysis in the analytical ultracentrifuge.

Sedimentation velocity experiments performed in the analytical ultracentrifuge are modeled using finite-element solutions of the Lamm equation. During modeling, three fundamental parameters are optimized: the sedimentation coefficients, the diffusion coefficients, and the partial concentrations of all solutes present in a mixture. A general modeling approach consists of fitting the partial concentrations of solutes defined in a two-dimensional grid of sedimentation and diffusion coefficient combinations that cover the range of possible solutes for a given mixture. An increasing number of grid points increase the resolution of the model produced by the subsequent analysis, with denser grids giving rise to a very large system of equations. Here, we evaluate the efficiency and resolution of several regular grids and show that traditionally defined grids tend to provide inadequate coverage in one region of the grid, while at the same time being computationally wasteful in other sections of the grid. We describe a rapid and systematic approach for generating efficient two-dimensional analysis grids that balance optimal information content and model resolution for a given signal-to-noise ratio with improved calculation efficiency. These findings are general and apply to one- and two-dimensional grids, although they no longer represent regular grids. We provide a recipe for an improved grid-point spacing in both directions which eliminates unnecessary points, while at the same time providing a more uniform resolution that can be scaled based on the stochastic noise in the experimental data.

Multi-speed sedimentation velocity simulations with UltraScan-III.

Recent developments in the UltraScan-III software make it possible to model multi-speed analytical ultracentrifugation sedimentation velocity experiments using finite-element solutions of the Lamm equation. Using simulated data, we demonstrate here how these innovations can be used to enhance the resolution of sedimentation velocity experiments when compared to single-speed experiments. Using heterogeneous systems covering as much as five orders of magnitude in molar mass and fivefold in anisotropy, we compare results from runs performed at multiple speeds to those obtained from single-speed experiments, fitted individually and analyzed globally over multiple speeds, and quantify resolution for sample heterogeneous in size and anisotropy. We also provide guidance on the design of multi-speed experiments and offer a program that can be used to deduce optimal spacing of rotor speeds and speed step durations when a few parameters from the experiment can be estimated. These include the meniscus position, the sedimentation coefficient of the largest species in a mixture, and a solute distribution. Our results show that errors observed in the determination of hydrodynamic parameters for system with great heterogeneity are markedly reduced when multi-speed analysis is employed.

Allowance for boundary sharpening in the determination of diffusion coefficients by sedimentation velocity: a historical perspective.

This review summarizes endeavors undertaken in the middle of last century to employ the Lamm equation for quantitative analysis of boundary spreading in sedimentation velocity experiments on globular proteins, thereby illustrating the ingenuity required to achieve that goal in an era when an approximate analytical solution of that nonlinear differential equation of second order provided the only means for its application. Application of procedures based on that approximate solution to simulated sedimentation velocity distributions has revealed a slight disparity (about 3%) between returned and input values of the diffusion coefficient-a discrepancy comparable with that of estimates obtained by current simulative analyses based on numerical solution of the Lamm equation. Although the massive technological developments in the gathering and treatment of sedimentation velocity data over the past three to four decades have changed dramatically the manner in which boundary spreading is analyzed, they have not led to any significant improvement in the accuracy of the diffusion coefficient thereby deduced.

Brownian dynamics simulations of analytical ultracentrifugation experiments exhibiting hydrodynamic and thermodynamic non-ideality.

Hydrodynamic and thermodynamic non-ideality are important phenomena when studying concentrated and interacting systems in analytical ultracentrifugation (AUC). Here we present an extended Brownian Dynamics (BD) based algorithm which incorporates hydrodynamic and thermodynamic non-ideality. It can serve as an independent and versatile approach for the theoretical description of interparticulate interactions in AUC, as it allows tracking the trajectory of individual particles. Concentration dependencies of the sedimentation and diffusion coefficient have been implemented and validated for the extended BD model. For monodisperse systems, it is shown that profiles obtained by BD are in excellent agreement with well-established Lamm equation solvers. Moreover, important limits and restrictions of current Lamm equation based analysis methods are discussed. In particular, BD allows modeling and evaluation of AUC data of non-ideal polydisperse systems. This is relevant as most nanoparticulate systems are polydisperse in size. Here, a simulation for a polydisperse system including concentration effects is presented for the first time.

A Comprehensive Brownian Dynamics-Based Forward Model for Analytical (Ultra)Centrifugation

In this study, it is demonstrated how Brownian dynamics (BD) simulations can be used as a comprehensive forward model for analytical ultracentrifugation (AUC) and analytical centrifugation experiments. BD simulations are a versatile alternative to the numerical solution of Lamm's equation. Time dependent concentration profiles can be accurately predicted by means of the BD simulations even for colloidal systems with multiple species. It is shown that in combination with Mie's theory for light scattering, BD simulations can be extended to generate data mimicking a multiwavelength detector for AUC. The forward model can be utilized as a comprehensive tool for developing and testing sophisticated data analyzing tools that can enhance the capabilities of using centrifugation-based techniques for characterizing nanoparticles. The model is validated using common analysis tools developed for the evaluation of sedimentation velocity data generated by a single wavelength extinction detector. The accuracy of the code is also proved using the recently developed and highly versatile High Dynamic Range-MULTIwavelength FITting tool for accurately predicting particle size distributions via gravitational sweep experiments performed in AUCs equipped with multiwavelength detectors.

Gravitational Sweep Sedimentation Velocity

Sedimentation velocity (SV) analytical ultracentrifugation is a classical biophysical technique for the determination of the size-distribution of macromolecules, macromolecular complexes, and nanoparticles. SV has traditionally been carried out at a constant rotor speed, which limits the range of sedimentation coefficients that can be detected in a single experiment. Recently we have introduced methods to implement experiments with variable rotor speeds, in combination with variable field solutions to the Lamm equation, with the application to expedite the approach to sedimentation equilibrium. Here, we describe the use of variable-field sedimentation analysis to increase the size-range covered in SV experiments by approximately 100-fold through the use of a quasi-continuous increase of rotor speed during the experiment. Such ‘gravitational sweep’ sedimentation approaches has previously been shown to be very effective in the study of nanoparticles with large size ranges. However, previously diffusion processes were not accounted for, therefore posing a lower limit of particle sizes and limiting the accuracy of the size distribution. In the present work, we combine variable field solutions to the Lamm equation with diffusion-deconvoluted sedimentation coefficient distributions c(s), which further extends the macromolecular size-range that can be observed in a single SV experiment while maintaining accuracy and resolution. In this way, approximately five orders of magnitudes of sedimentation coefficients, or eight orders of magnitude of particle mass, can be probed in a single experiment. This can be useful, for example, in the study of proteins forming large assemblies, for example, as in fibrillation process or capsid self-assembly, in studies of the interaction between very dissimilar sized macromolecular species, in the study of broadly distributed nanoparticles.

Variable Field Analytical Ultracentrifugation: II. Gravitational Sweep Sedimentation Velocity

Sedimentation velocity (SV) analytical ultracentrifugation is a classical biophysical technique for the determination of the size-distribution of macromolecules, macromolecular complexes, and nanoparticles. SV has traditionally been carried out at a constant rotor speed, which limits the range of sedimentation coefficients that can be detected in a single experiment. Recently we have introduced methods to implement experiments with variable rotor speeds, in combination with variable field solutions to the Lamm equation, with the application to expedite the approach to sedimentation equilibrium. Here, we describe the use of variable-field sedimentation analysis to increase the size-range covered in SV experiments by ∼100-fold with a quasi-continuous increase of rotor speed during the experiment. Such a gravitational-sweep sedimentation approach has previously been shown to be very effective in the study of nanoparticles with large size ranges. In the past, diffusion processes were not accounted for, thereby posing a lower limit of particle sizes and limiting the accuracy of the size distribution. In this work, we combine variable field solutions to the Lamm equation with diffusion-deconvoluted sedimentation coefficient distributions c(s), which further extend the macromolecular size range that can be observed in a single SV experiment while maintaining accuracy and resolution. In this way, approximately five orders of magnitude of sedimentation coefficients, or eight orders of magnitude of particle mass, can be probed in a single experiment. This can be useful, for example, in the study of proteins forming large assemblies, as in fibrillation process or capsid self-assembly, in studies of the interaction between very dissimilar-sized macromolecular species, or in the study of broadly distributed nanoparticles.

Variable-Field Analytical Ultracentrifugation: I. Time-Optimized Sedimentation Equilibrium

Sedimentation equilibrium (SE) analytical ultracentrifugation (AUC) is a gold standard for the rigorous determination of macromolecular buoyant molar masses and the thermodynamic study of reversible interactions in solution. A significant experimental drawback is the long time required to attain SE, which is usually on the order of days. We have developed a method for time-optimized SE (toSE) with defined time-varying centrifugal fields that allow SE to be attained in a significantly (up to 10-fold) shorter time than is usually required. To achieve this, numerical Lamm equation solutions for sedimentation in time-varying fields are computed based on initial estimates of macromolecular transport properties. A parameterized rotor-speed schedule is optimized with the goal of achieving a minimal time to equilibrium while limiting transient sample preconcentration at the base of the solution column. The resulting rotor-speed schedule may include multiple over- and underspeeding phases, balancing the formation of gradients from strong sedimentation fluxes with periods of high diffusional transport. The computation is carried out in a new software program called TOSE, which also facilitates convenient experimental implementation. Further, we extend AUC data analysis to sedimentation processes in such time-varying centrifugal fields. Due to the initially high centrifugal fields in toSE and the resulting strong migration, it is possible to extract sedimentation coefficient distributions from the early data. This can provide better estimates of the size of macromolecular complexes and report on sample homogeneity early on,which may be used to further refine the prediction of the rotor-speed schedule. In this manner, the toSE experiment can beadapted in real time to the system under study, maximizing both the information content and the time efficiency of SE experiments.

Sedimentation Velocity: A Classical Perspective.

Here we give an overview of the history of sedimentation velocity analysis focusing on seminal and fundamental contributions that derived from early ultracentrifugation studies. We introduce the concepts of nonequilibrium thermodynamics and outline the derivation of the Svedberg and the Lamm equations and the requirements for including both hydrodynamic and thermodynamic nonideality. We introduce the phenomenological equations for coupled flows as developed from the principles of nonequilibrium or irreversible thermodynamics and derive a form of the Lamm equation that incorporates cross-diffusion coefficients and coupled gradient terms. We give an historical overview of solutions to the Lamm equation including Fujita-MacCosham solutions and Claverie finite-element numerical solutions and discuss the software that have implemented these solutions. We discuss the three major optical systems (absorbance, interference, and fluorescence) and recently developed multiwavelength systems. We also suggest a number of experimental practices and guidelines for optimizing the determination of s and D and discuss the appropriate centerpiece components and their utility. This chapter complements other recent reviews submitted by the authors (Correia, Lyons, Sherwood, & Stafford, 2015; Stafford, 2015) and should be considered an effort to revive the importance of irreversible thermodynamics in the understanding and analysis of sedimentation velocity ultracentrifugation data.

Sedimentation in a Time-Varying Centrifugal Field for Rapid Attainment of Sedimentation Equilibrium

Sedimentation equilibrium (SE) analytical ultracentrifugation is a gold standard for the rigorous thermodynamic study of buoyant molecular weight and reversible interactions of macromolecules in solution. A significant drawback is the long experiment time, as it takes days to attain SE with standard solution columns. We have developed a new method for using a time-varying centrifugal field optimized such as to attain SE in significantly shorter time than usually required. Experimental data show that this permits long-column SE experiments to be carried out in times comparable to sedimentation velocity experiments, approximately fivefold shorter than standard SE. In contrast to the classical initial overspeeding method, which uses a single initial speed, we employ a freely varying rotor speed profile during an initial phase, for example, parameterized as a step-wise modulated exponential decay to the desired SE rotor speed. The rotor speed schedule is computationally optimized on the basis of numerical Lamm equation solutions for given macromolecular sedimentation parameter estimates, with the goal to provide a rapid attainment of equilibrium without the drawback of strong transient sample pre-concentration at the base of the solution column. The resulting rotor speed schedule frequently includes both over- and under-speeding sequences, and can be conveniently implemented on the Optima XLA/I analytical ultracentrifuge. We extended AUC data analysis models in SEDFIT to permit the analysis of concentration profiles in arbitrarily time-varying fields, to make it possible to exploit the migration in the initially high centrifugal field for estimates on macromolecular sedimentation parameters, which may be used in real-time to refine the prediction of the rotor speed schedule, so that the SE experiment can be optimized in both information content and time efficiency.

A Parametrically Constrained Optimization Method for Fitting Sedimentation Velocity Experiments

A method for fitting sedimentation velocity experiments using whole boundary Lamm equation solutions is presented. The method, termed parametrically constrained spectrum analysis (PCSA), provides an optimized approach for simultaneously modeling heterogeneity in size and anisotropy of macromolecular mixtures. The solutions produced by PCSA are particularly useful for modeling polymerizing systems, where a single-valued relationship exists between the molar mass of the growing polymer chain and its corresponding anisotropy. The PCSA uses functional constraints to identify this relationship, and unlike other multidimensional grid methods, assures that only a single molar mass can be associated with a given anisotropy measurement. A description of the PCSA algorithm is presented, as well as several experimental and simulated examples that illustrate its utility and capabilities. The performance advantages of the PCSA method in comparison to other methods are documented. The method has been added to the UltraScan-III software suite, which is available for free download from http://www.ultrascan.uthscsa.edu.

The Off-rate of Monomers Dissociating from Amyloid-β Protofibrils

The interconversion of monomers, oligomers, and amyloid fibrils of the amyloid-β peptide (Aβ) has been implicated in the pathogenesis of Alzheimer disease. The determination of the kinetics of the individual association and dissociation reactions is hampered by the fact that forward and reverse reactions to/from different aggregation states occur simultaneously. Here, we report the kinetics of dissociation of Aβ monomers from protofibrils, prefibrillar high molecular weight oligomers previously shown to possess pronounced neurotoxicity. An engineered binding protein sequestering specifically monomeric Aβ was employed to follow protofibril dissociation by tryptophan fluorescence, precluding confounding effects of reverse or competing reactions. Aβ protofibril dissociation into monomers follows exponential decay kinetics with a time constant of ∼2 h at 25 °C and an activation energy of 80 kJ/mol, values typical for high affinity biomolecular interactions. This study demonstrates the high kinetic stability of Aβ protofibrils toward dissociation into monomers and supports the delineation of the Aβ folding and assembly energy landscape.

Small-angle X-ray Solution Scattering Study of the Multi-aminoacyl-tRNA Synthetase Complex Reveals an Elongated and Multi-armed particle

In animal cells, nine aminoacyl-tRNA synthetases are associated with the three auxiliary proteins p18, p38, and p43 to form a stable and conserved large multi-aminoacyl-tRNA synthetase complex (MARS), whose molecular mass has been proposed to be between 1.0 and 1.5 MDa. The complex acts as a molecular hub for coordinating protein synthesis and diverse regulatory signal pathways. Electron microscopy studies defined its low resolution molecular envelope as an overall rather compact, asymmetric triangular shape. Here, we have analyzed the composition and homogeneity of the native mammalian MARS isolated from rabbit liver and characterized its overall internal structure, size, and shape at low resolution by hydrodynamic methods and small-angle x-ray scattering in solution. Our data reveal that the MARS exhibits a much more elongated and multi-armed shape than expected from previous reports. The hydrodynamic and structural features of the MARS are large compared with other supramolecular assemblies involved in translation, including ribosome. The large dimensions and non-compact structural organization of MARS favor a large protein surface accessibility for all its components. This may be essential to allow structural rearrangements between the catalytic and cis-acting tRNA binding domains of the synthetases required for binding the bulky tRNA substrates. This non-compact architecture may also contribute to the spatiotemporal controlled release of some of its components, which participate in non-canonical functions after dissociation from the complex.

Structural and Biophysical Characterization of the Interactions between the Death Domain of Fas Receptor and Calmodulin

The extrinsic apoptotic pathway is initiated by cell surface death receptors such as Fas. Engagement of Fas by Fas ligand triggers a conformational change that allows Fas to interact with adaptor protein Fas-associated death domain (FADD) via the death domain, which recruits downstream signaling proteins to form the death-inducing signaling complex (DISC). Previous studies have shown that calmodulin (CaM) is recruited into the DISC in cholangiocarcinoma cells, suggesting a novel role of CaM in Fas-mediated signaling. CaM antagonists induce apoptosis through a Fas-related mechanism in cholangiocarcinoma and other cancer cell lines possibly by inhibiting Fas-CaM interactions. The structural determinants of Fas-CaM interaction and the underlying molecular mechanisms of inhibition, however, are unknown. Here we employed NMR and biophysical techniques to elucidate these mechanisms. Our data show that CaM binds to the death domain of Fas (FasDD) with an apparent dissociation constant (Kd) of ~2 μM and 2:1 CaM:FasDD stoichiometry. The interactions between FasDD and CaM are endothermic and entropically driven, suggesting that hydrophobic contacts are critical for binding. We also show that both the N- and C-terminal lobes of CaM are important for binding. NMR and surface plasmon resonance data show that three CaM antagonists (N-(6-aminohexyl)-5-chloro-1-naphthalene sulfonamide, tamoxifen, and trifluoperazine) greatly inhibit Fas-CaM interactions by blocking the Fas-binding site on CaM. Our findings provide the first structural evidence for Fas-CaM interactions and mechanism of inhibition and provide new insight into the molecular basis for a novel role of CaM in regulating Fas-mediated apoptosis.