Analytical ultracentrifugation

Historical Analytical Ultracentrifuge (Model E)

Typical temporal course of the sedimentation bands, which were measured by means of an absorption detector as part of a sedimentation speed experiment (SV) with bovine serum albumin at 40,000 rpm and 20 ° C.

The analytical ultracentrifugation (AUC) is an analysis method which detects the movement and position of suspended particles in a centrifugal field by means of optical measuring methods. Most often, sedimentation and diffusion coefficients are determined, whereby conclusions can be drawn about the shape, size, density or mass of the examined particles or biomolecules. The technology was largely developed by The Svedberg .

description

In a centrifugal field, centrifugal force , frictional force and buoyancy force act on particles. These forces and thus the sedimentation speed depend on the size, shape and density of the particles, as well as experimental parameters such as the speed of the centrifuge , distance to the axis of rotation and solvent properties. In addition, small particles are subject to Brownian motion , which is another influencing factor in a measurement. The associated diffusion coefficient is determined by the temperature, the selected solvent and the size and shape of the particle.

An integral part of the process is the use of an optical measurement process that enables the temporal and / or spatial distribution of the particles to be recorded during centrifugation. The aim of the measurement process is to utilize the thermodynamic and hydrodynamic phenomena in such a way that conclusions can be drawn about frequently distributed variables such as shape, density, size / mass or the association behavior of DNA , proteins or nanoparticles . In addition, the recorded optical properties can be linked to the determined geometric parameters.

construction

An analytical ultracentrifuge is an ultracentrifuge that has been converted so that the sample is still accessible to an optical detector during the centrifugation process in a vacuum. Special measuring cells in a special rotor are used for this. The measuring cells typically consist of sector-shaped sample containers that are closed by optically transparent glass windows. The optical detectors used are based on principles that allow the measurement signal to be associated with the concentration of the species being examined (e.g. Lambert-Beer's law ). Absorption and interference detectors are currently commercially available.

In-house developments by users in science and industry also include detectors based on Schlieren optics and fluorescence .

Theoretical description

sedimentation

The sedimentation coefficient describes the relationship between the sedimentation speed and the used centrifugal acceleration:

${\ displaystyle s = {\ frac {v} {\ omega ^ {2} r}} = {\ frac {m (1 - {\ frac {\ rho _ {s}} {\ rho _ {p}}} )} {f}}}$

The individual symbols stand for the following quantities :

${\ displaystyle v}$ - Sedimentation speed (positive values correspond to sedimentation, negative values correspond to flotation )
${\ displaystyle r}$ - Distance of the particle to the axis of rotation / rotor axis
${\ displaystyle \ omega}$ - angular velocity
${\ displaystyle m}$ - mass of the particle
${\ displaystyle \ rho _ {p}}$ - density of the particle
${\ displaystyle \ rho _ {s}}$ - Density of the solvent
${\ displaystyle f}$ - Friction factor according to Stoke's law .

This means that the sedimentation coefficient in the case of ideal sedimentation only depends on the material and solvent parameters and not on the radial position and the speed of the centrifuge. For particles showing negligible Brownian motion , the sedimentation coefficient can be used to calculate a trajectory for known initial conditions and angular velocity . In addition, it can be determined from the particle trajectory and the centrifugal field. ${\ displaystyle s}$

If particles with different sedimentation coefficients are measured in one experiment, the sedimentation bands are broadened.

diffusion

In addition to the deterministic forces that occur, the particles within an AUZ experiment are also subject to the irregular Brownian motion . These microscopic movements consequently lead to a macroscopic net particle flow contrary to a concentration gradient according to Fick's law with diffusion coefficient and particle concentration : ${\ displaystyle {\ vec {J}}}$ ${\ displaystyle D}$ ${\ displaystyle c}$

${\ displaystyle {\ vec {J}} = - D {\ vec {\ nabla}} c}$

According to the Einstein-Smoluchowski relationship , the diffusion coefficient can be understood as:

${\ displaystyle D = {\ frac {k_ {B} T} {f}}}$

It describes the absolute temperature and the Boltzmann constant . The diffusion coefficient therefore depends on size and shape. In an AUZ experiment, sedimenting particles lead to a particle flow opposite to the sedimentation direction, which in turn leads to a broadening of the sedimentation bands. ${\ displaystyle T}$ ${\ displaystyle k_ {B}}$

Lamb equation

In order to consider an entire system of particles in a centrifugal field, a change must be made from the microscopic observation of the particle path to a macroscopic observation analogous to the diffusion-driven particle flow. Using cylindrical coordinates and the continuity equation , the equation named after Ole Lamm, a PhD student at The Svedberg , can be set up for a single species:

${\ displaystyle {\ frac {\ partial c} {\ partial t}} = D \ left [\ left ({\ frac {\ partial ^ {2} c} {\ partial r ^ {2}}} \ right) + {\ frac {1} {r}} \ left ({\ frac {\ partial c} {\ partial r}} \ right) \ right] -s \ omega ^ {2} \ left [r \ left ({ \ frac {\ partial c} {\ partial r}} \ right) + 2c \ right]}$

The time denotes . ${\ displaystyle t}$

The numerical solutions of this differential equation thus make it possible to describe time- and location-dependent sedimentation and diffusion processes in a sector-shaped measuring cell and thus also to analyze measurement data with regard to and . This also includes the possibility of being able to distinguish the two broadening mechanisms of the sedimentation bands ( diffusion and polydispersity of the particle size distribution ) in the analysis. ${\ displaystyle s}$ ${\ displaystyle D}$

Svedberg equation

Since and both depend on the size and shape of the examined particle, an unknown parameter can be eliminated by combining both coefficients. The equation obtained by replacing is named after The Svedberg : ${\ displaystyle s}$ ${\ displaystyle {\ ce {D}}}$ ${\ displaystyle f}$

${\ displaystyle M = {\ frac {s \ cdot N_ {A} k_ {B} T} {D \ cdot (1 - {\ frac {\ rho _ {s}} {\ rho _ {p}}}) }}}$

It denotes the molar mass and the Avogadro constant . Knowing about and and the corresponding density (often the replacement with the partial specific volume is also made) of a z. B. protein it is thus possible to determine the molar mass independently of the shape. ${\ displaystyle M}$ ${\ displaystyle N_ {A}}$ ${\ displaystyle s}$ ${\ displaystyle {\ ce {D}}}$ ${\ displaystyle {\ frac {1} {\ rho _ {p}}} = {\ bar {\ nu}}}$ ${\ displaystyle {\ bar {\ nu}}}$

Measurement methods

In the course of the history of the AUZ, a large number of special measurement modes have been developed. The most important basic types include the sedimentation velocity experiment (SV) and the sedimentation equilibrium experiment (SE). Both experiments are typically started with a homogeneous concentration distribution of the particles in the measuring cell that are to be examined. While a static concentration distribution within the measuring cell is measured in the SE, in the SV the sedimentation and often also the diffusion dynamics are resolved.

Sedimentation Equilibrium Experiment (SE)

In the SE, it is neither determined nor directly. At comparatively low speeds, the equilibrium between diffusion-driven and sedimentation-driven particle flow is awaited. Since this equilibrium exists in every radius position, there is a static concentration distribution within the measuring cell. From this exponential concentration distribution, the molar mass can be determined using the Svedberg equation and with knowledge of the particle density. If there is interaction between the particles, the second virial coefficient can also be determined. However, this typically requires several measurements at different speeds and output concentrations. With sufficient measurement time, the sedimentation equilibrium represents the end point of every sedimentation velocity experiment. ${\ displaystyle s}$ ${\ displaystyle D}$

A special form of SE is the use of a solvent density gradient . The density of the particles can be determined.

Sedimentation Velocity Experiment (SV)

The SV ( sedimentation velocity ) tracks the radial and temporal change in the particle concentration from the beginning of the experiment. From this data, using the numerical solutions of the Lamm equation, distributions of and can be determined, which can be converted into particle size distributions . SV is usually preferred over SE, as this is typically superior to SE in terms of information content. This is because and can be analyzed separately. In addition, particles with different sedimentation properties can be better separated from one another in the analysis of an SV data set, so that polydisperse particle size distributions can also be determined. For spherical particles, the sedimentation or diffusion coefficient can be converted directly into a particle size using the definition of and . In the case of shape anisotropic particles, equivalent diameters can be used. ${\ displaystyle s}$ ${\ displaystyle D}$ ${\ displaystyle s}$ ${\ displaystyle D}$ ${\ displaystyle s}$ ${\ displaystyle D}$

In a special form of the SV, the change in the particle concentration over time is tracked at a fixed radius position; this enables very broad particle size distributions to be recorded, whereby the diffusion properties then have to be dispensed with.

Individual evidence

↑ ^a ^b The Nobel Prize in Chemistry 1926. Retrieved March 10, 2020 (American English).

↑ ^a ^b ^c ^d ^e ^f ^g Analytical Ultracentrifugation . 2016, doi : 10.1007 / 978-4-431-55985-6 .

↑ ^a ^b ^c ^d ^e Analytical Ultracentrifugation of Polymers and Nanoparticles (= Springer Laboratory ). Springer-Verlag, New York 2006, ISBN 978-3-540-23432-6 , doi : 10.1007 / b137083 .

↑ Optima AUC Analytical Ultracentrifuge - Beckman Coulter. Retrieved March 10, 2020 .

^ ^A ^b Jean-Michel Claverie, Henri Dreux, René Cohen: Sedimentation of generalized systems of interacting particles. I. Solution of systems of complete lamb equations . In: Biopolymers . tape 14 , no. 8 , 1975, ISSN 1097-0282 , pp. 1685-1700 , doi : 10.1002 / gdp . 1975.360140811 .

↑ Borries Demeler, Hashim Saber: Determination of Molecular Parameters by Fitting Sedimentation Data to Finite-Element Solutions of the Lamm Equation . In: Biophysical Journal . tape 74 , no. 1 , January 1998, pp. 444–454 , doi : 10.1016 / S0006-3495 (98) 77802-6 , PMID 9449345 , PMC 1299397 (free full text) - ( elsevier.com [accessed April 13, 2020]).

↑ Schuck, Peter: Sedimentation velocity analytical ultracentrifugation: discrete species and size distributions of macromolecules and particles . CRC Press, Boca Raton, FL, ISBN 978-1-4987-6895-5 ( [1] [accessed April 13, 2020]).

[:2-1] The Nobel Prize in Chemistry 1926. Retrieved March 10, 2020 (American English).

[:0-2] ↑ ^a ^b ^c ^d ^e ^f ^g Analytical Ultracentrifugation . 2016, doi : 10.1007 / 978-4-431-55985-6 .

[:3-3] Analytical Ultracentrifugation of Polymers and Nanoparticles (= Springer Laboratory ). Springer-Verlag, New York 2006, ISBN 978-3-540-23432-6 , doi : 10.1007 / b137083 .

[4] Optima AUC Analytical Ultracentrifuge - Beckman Coulter. Retrieved March 10, 2020 .