Analytical ultracentrifugation

The analytical ultracentrifugation (AUC) method was firstly introduced by Svedberg as a method for the analysis of Au-NPs in 1920. His pioneering work revealed the colloidal nature of Au-NPs as well as the existence of proteins and polymers as macromolecules containing a great number of covalently linked atoms. Since then the tremendous progresses in chemistry and biology have created a library of macromolecules of different compositions. But a question, which does not change since 1920, is how to characterize these molecules regarding their purities, sizes and molecular weights. Among the nowadays more preferred methods in polymer chemistry and biology such as different chromatography techniques or electrophoresis, the AUC still remains a very sensitive method for the characterization of the molecular weight distribution of polymers and protein structures. The AUC is used for the characterization of proteins regarding their agglomeration, interaction strength and stoichiometry98,99,100,101

, and nanoparticle size distributions with Ångstrom resolution¹⁰². However, it is still not widely used for the investigation of protein-nanoparticle hybrid structures. Due to the high sensitivity to nanoparticle size, shape and density AUC was applied to detect the adsorption of biomolecules on nanoparticles and the characterization of the resulting biohybrid. Calabretta et al. reported about a systematical change of the sedimentation coefficient (s)-value of Au-NPs in solution with increasing LacI protein concentration¹⁰³. The shift of the s-value to a smaller value could also be detected in Au-NP solution with DNA¹⁰⁴. The s-values decrease with increasing DNA length due to the density change of the Au-NP after DNA attachment. The above mentioned examples prove the power and versatility of AUC for the analysis of biological macromolecules, colloidal particles and their hybrid structures.

For an AUC measurement, the sample is dispersed in a solvent. The sample is placed in a sector-shaped ultracentrifugation cell and subjected to a gravitational field induced by a spinning centrifuge rotor leading to the sedimentation of the solute particles. The resulting gravitational field can reach up to 250000 x g, which means that a mass of 1 g will experience an apparent weight of 250 kg. The sedimentation of the particle is measured by the concentration change (c) with radius (r) and time (t). The c(r,t) profile is detected by an absorption and interference optics. The particle in a gravitational field is exposed to three forces, the gravitational force or sedimentation force (FS), buoyant force (FB) and frictional force (FF). As these three forces are in equilibrium, the particle will move with constant sedimentation velocity u. This movement leads to Equ. 16 for the calculation of the sedimentation coefficient

𝒔

Chapter 2 – Fundamentals velocity of the particle moving a radial pathway towards the bottom cell, 𝜔=angular velocity of the spinning rotor, r = distance of the particle from the axis of rotation, s = sedimentation coefficient.

The frictional coefficient f is well known and depends on the size and shape of the particle. In a simple approach of a spherical particle, the Stokes-Einstein (Equ. 17) and Stokes equations

In Equ. 16 the s value is a material specific value depending on the molecular masses, shapes and sizes of the sedimenting particles. The combination of equations 16 and 17 gives the molecular mass M of the particle in the famous Svedberg equation at given s, D and 𝜐̅ values:

𝑀 = 𝑠𝑅𝑇

𝐷(1 − 𝜐̅𝜌_𝑠) (19)

Furthermore, by the combination of equations 16, 17 and 18 the particle diameter dp can be obtained from the sedimentation coefficient assuming a hard sphere:

𝑑_𝑝= √18𝜂_𝑠𝑠

𝜌_𝑝− 𝜌_𝑠 (20)

s = viscosity of the solvent, ρp = density of the particle.

Based on the s value from an AUC experiment, material specific properties such as size dp and molecular mass M can be obtained.

A second approach to describe the principals of the AUC is the Lamm equation. This approach contains a more accurate characterization of the thermodynamic process in an AUC experiment.

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As mentioned before, the change of sample concentration as a function of radius and time c(r,t) is the basic information of an AUC experiment. The Lamm equation also describes this change and consists of a diffusion term with the diffusion coefficient D and sedimentation term with the sedimentation coefficient s, as shown in Equ. 21.

𝜕𝑐

In an experiment only two mass transport mechanisms of the particles will occur in the AUC cell, in fact, the mass transports via sedimentation and diffusion. Depending on the dominance of the first or second term in the Lamm equation, four basic types of experiments can be conducted in the AUC, namely the sedimentation velocity, sedimentation equilibrium, synthetic boundary and the density gradient experiments. In this thesis, only the sedimentation velocity experiment was used. In a sedimentation velocity experiment the solute particles sediment according to their mass/size, density and shape to the cell bottom at a sufficiently high angular velocity. This produces a depletion of particles near the sample meniscus (sharp vertical spike at 6.02 cm) and the formation of a sharp boundary between the depleted region and the uniform concentration of sedimenting solute in the plateau region, as shown in Fig. 16. The rbnd is the inflection point of the curve in the middle of the boundary region. As the s value is size dependent, a fractionation of particles of similar shape and density according to their sizes can be conducted.

The gravitational force is not constant within an AUC cell resulting in an increase of the force and the velocity u of the boundary with time. To take this effect into account, a differential

By integration of this expression with the radial position of the meniscus rM the Equ. 23 can be formed. s. Since the real samples contain multiple components or have a continuous distribution of sizes and molecular masses, only the rbnd values are used and the estimated s-values are weight

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average sedimentation coefficients. Thus, the determination of the complete differential sedimentation coefficient distribution g(s) is the more precise solution for these samples. The consequent definition of the sedimentation coefficient distribution g(s) or the integral form therefrom G(s)=c(s)/c0 is expressed in Equ. 24. distributions of different species. In this work, a time derivative method in the SEDFIT software yields the g*(s) distribution (the star indicates that the distribution is not diffusion corrected).

This method determines the time derivative of neighbouring radial scans c(r,t) recorded at different times leading to the following expression, which is a combination of equations 23 and 24: broadening effect becomes more pronounced for NPs of size below 30 nm or macromolecules of molar mass below roughly 100 kDa. Therefore, it is desirable to remove this diffusion broadening effect in the s-distribution for the calculation of a more precise particle size and molecular mass distribution. In this thesis, the method developed by Schuck is applied to the originally obtained g*(s) distribution for the determination of the diffusion corrected c(s) distribution. This method bases on the modeling of the Lamm equation using fits of a series of concentration profiles c(r,t) to the finite element solutions of the Lamm differential equation.

This modeling process involves an educated initial guess of the sedimentation coefficients, the frictional coefficients and the partial specific volume. The calculated data are adjusted to the experimental data by means of the maximum of entropy regularization. A disadvantage of Schuck’s method are the non-existing ’ghost peaks’ in the c(s) distribution for polydisperse samples.

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Fig. 16: AUC raw data from a sedimentation velocity experiment of an antigen solution recorded with an absorption optics at 280 nm wavelength¹⁰⁵. The boundary shifts to the cell bottom with increasing time.