Measurement of protein size in concentrated solutions by small angle X-ray scattering

Abstract

By simulations on the distance distribution function (DDF) derived from small angle X-ray scattering (SAXS) theoretical data of a dense monodisperse system, we found a quantitative mathematical correlation between the apparent size of a spherically symmetric (or nearly spherically symmetric) homogenous particle and the concentration of the solution. SAXS experiments on protein solutions of human hemoglobin and horse myoglobin validated the correlation. This gives a new method to determine, from the SAXS DDF, the size of spherically symmetric (or nearly spherically symmetric) particles of a dense monodisperse system, specifically for protein solutions with interference effects.

Keywords: distance distribution function; inter-particle interference; protein size; small angle X-ray scattering.

Figure 1

Theoretical scattering curves for monodisperse systems with different volume ratios based on a sphere interference model derived from Eqs. (1) and (2) with K = 1 and R = 1 nm. where K is a constant, R is the radius of the spherically symmetric (or nearly spherically symmetric) particle, V ₁ is the average volume offered to each particle, V ₀ is the total volume offered to the particles, V ₀/V ₁ is the well‐defined number of particles contained in V ₀, I is scattering intensity, q is the scattering vector, a.u. means arbitrary units.

Figure 2

Theoretical distance distribution function P(r) for monodisperse systems with different volume ratios based on a sphere interface model derived from theoretical scattering intensities in Figure 1 using Eq. (3). P(r) is related to the frequency of certain distances r within a particle. r ₂ is labeled at which P(r)=0.

Figure 3

Variation of the apparent r ₂ versus V ₀/V ₁ derived from distance distribution function data. r ₂ is labeled at which P(r)=0, V ₁ is the average volume offered to each particle, V ₀ is the total volume offered to the particles.

Figure 4

Variation of the apparent lnr ₂ versus (V ₀/V ₁)^0.52 based on the data derived from Figure 3. r ₂ is labeled at which P(r)=0, V ₁ is the average volume offered to each particle, V ₀ is the total volume offered to the particles.

Figure 5

Experimental scattering curves of human hemoglobin solutions at different concentrations measured at the X33 SAXS station at the storage ring DORIS III of the Deutsches Elektronen Synchrotron (DESY, Hamburg, Germany) with the incident X‐ray wavelength of 0.15 nm and a 1M PILATUS (Dectris, Baden, Switzerland) as detector. I is scattering intensity, q is the scattering vector, c is the concentration of samples.

Figure 6

Distance distribution function curves of human hemoglobin solutions at different concentrations c derived from the corresponding scattering curves in Figure 5. P(r) is related to the frequency of certain distances r within a particle.

Figure 7

Variation of the apparent lnr ₂ values (at which P(r)=0) obtained from Figure 6 versus the concentration c ^0.52 of human hemoglobin solutions. r ₂ is labeled at which P(r)=0, c is the concentration of samples.

Figure 8

Variation of the apparent lnr ₂ values versus the concentration c^0.52 of horse myoglobin solutions. SAXS experiments of horse myoglobin solutions were performed at the 1W2A SAXS station at Beijing Synchrotron Radiation Facility (BSRF, Beijing, China) with the incident X‐ray wavelength of 0.154 nm and a 1MF PILATUS (Dectris, Baden, Switzerland) as detector. r ₂ is labeled at which P(r)=0, c is the concentration of samples.