In static light scattering (SLS) experiments, the time-averaged intensity of scattered light is measured as a function of particle concentration and scattering angle. These measurements yield the weight-averaged molar mass (M_w) of the scattering particles. If the size of the scattering species is smaller than λ/20 it will behave as a point source. In this case, a simplified form of the Debye equation, based on Rayleigh-Gans-Debye theory, can be applied for vertically polarized light,

K c R θ = 1 M w + 2 B 22 c MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqWGlbWscqWGJbWyaeaacqWGsbGudaWgaaqaaiabeI7aXbqabaaaaOGaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaacqWGnbqtdaWgaaqaaiabdEha3bqabaaaaOGaey4kaSIaeGOmaiJaemOqai0aaSbaaSqaaiabikdaYiabikdaYaqabaGccqWGJbWyaaa@3DE0@

where

K = 4 π 2 [ n 0 ( d n / d c ) ] 2 N A λ 4 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4saSKaeyypa0tcfa4aaSaaaeaacqaI0aancqaHapaCdaahaaqabeaacqaIYaGmaaGaei4waSLaemOBa42aaSbaaeaacqaIWaamaeqaaiabcIcaOmaalyaabaGaemizaqMaemOBa4gabaGaemizaqMaem4yamgaaiabcMcaPiabc2faDnaaCaaabeqaaiabikdaYaaaaeaacqWGobGtdaWgaaqaaiabdgeabbqabaGaeq4UdW2aaWbaaeqabaGaeGinaqdaaaaacqGGSaalaaa@45D0@

c is the concentration of the scattering species in solution, R_θ is the Rayleigh ratio or the excess intensity of scattered light at θ scattering angle, N_A is Avogadro's number, λ is the wavelength of the laser light source, n₀ is the refractive index of the solvent, and B₂₂ is the second osmotic virial coefficient of the scattering species.

The M_w is determined by plotting values of Kc/R_θ versus particle concentration at a given scattering angle. This graphical representation is referred to as a Debye plot. The y-intercept determined through linear regression gives the reciprocal weight-averaged molar mass (1/M_w) of particles in solution.

Dynamic light scattering (DLS) measures fluctuations in the intensity of scattered light which arise from thermal motion of particles in the system. Analysis of these temporal fluctuations allows for estimation of the diffusion coefficient, D, which can be used to determine the apparent hydrodynamic radius (R_h) and particle size distribution (PSD) of the scattering species. The rate at which the scattered light fluctuates is directly related to the "speed" at which molecules move due to diffusion. The time autocorrelation function of the scattered light intensity G₂(t) provides a quantitative measure of the particle motion. Most theory is developed using the scattered electric field autocorrelation function g₁(t), which is easily calculated from the experimentally measured light intensity autocorrelation function by use the Siegert relationship,

g 1 ( t ) = [ G 2 ( t ) − G 2 ( ∞ ) G 2 ( ∞ ) ] 1 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aaSbaaSqaaiabigdaXaqabaGcdaqadaqaaiabdsha0bGaayjkaiaawMcaaiabg2da9maadmaajuaGbaWaaSaaaeaacqWGhbWrdaWgaaqaaiabikdaYaqabaWaaeWaaeaacqWG0baDaiaawIcacaGLPaaacqGHsislcqWGhbWrdaWgaaqaaiabikdaYaqabaWaaeWaaeaacqGHEisPaiaawIcacaGLPaaaaeaacqWGhbWrdaWgaaqaaiabikdaYaqabaWaaeWaaeaacqGHEisPaiaawIcacaGLPaaaaaaakiaawUfacaGLDbaadaahaaWcbeqcfayaamaalaaabaGaeGymaedabaGaeGOmaidaaaaaaaa@4840@

where G₂(∞) is the baseline of the experimentally measured intensity autocorrelation function.

For non-interacting, monodisperse particles that are small compared with the wavelength of light, the scattered electric field correlation function g₁(t) decays exponentially,

g₁(t) = exp(-t/τ)

where τ is the relaxation time of the scattered field correlation function, and can be related to the diffusion coefficient of the scattering species

τ = 1 D q 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiXdqNaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaacqWGebarcqWGXbqCdaahaaqabeaacqaIYaGmaaaaaaaa@340C@

with scattering vector, q, given by

q = (4πn₀/λ) sin (θ/2)

The apparent R_h can be determined using the Stokes-Einstein relationship, which relates the diffusion coefficient of a hard sphere at infinite dilution (D₀) to its hydrodynamic radius

D 0 = k T 6 π η R h MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraq0aaSbaaSqaaiabicdaWaqabaGccqGH9aqpjuaGdaWcaaqaaiabdUgaRjabdsfaubqaaiabiAda2iabec8aWjabeE7aOjabdkfasnaaBaaabaGaemiAaGgabeaaaaaaaa@3996@

where k is the Boltzmann constant and η is the solvent viscosity at absolute temperature, T. This simplified analysis ignores concentrative effects due to hydrodynamic and thermodynamic interactions which have been described elsewhere.

Information about relaxation time and apparent hydrodynamic radius distributions can be extracted from the data by regressing appropriate theoretical models to the autocorrelation function. For example, if a bimodal distribution is expected, a two-exponential fit is performed estimate the two relaxation times independently.

g₁(t) = a₁ exp(-t/τ₁) + a₂ exp(-t/τ₂)

Parameters a₁ and a₂ are expansion coefficients known as light intensity weighted amplitudes and τ₁ and τ₂ are relaxation times.

Generalizing from this approach to systems having multiple particle sizes or broad size distributions is straightforward in principle, but requires specialized computational approaches in practice. Each scattering species can be represented by a corresponding relaxation time, resulting in a sum of exponentials

g₁(t) = a₁ exp(-t/τ₁) + a₂ exp(-t/τ₂)+...

where a_n values represent the relative contributions (amplitudes) of each particle size. The regularized inverse Laplace transform program CONTIN is routinely utilized to perform such multi-exponential analyses of relaxation-time distributions 23. Under the assumption that the scattering particles behave as hard spheres in dilute solution, the relaxation time distribution obtained can be converted into an apparent hydrodynamic size distribution using the Stokes-Einstein relationship.

The theory and principles of asymmetric flow FFF have been extensively reviewed elsewhere 242526. In summary, this type of FFF is performed inside a thin, ribbon-like channel approximately 30 cm in length, 2 cm wide, and ranging in thickness up to 500 μm. Carrier fluid is pumped through the channel from the inlet end exhibiting a laminar flow profile. A cross-flow is induced perpendicular to the channel flow, which exits the channel through the bottom wall fitted with an ultrafiltration membrane. The cross-flow forces the sample components toward this "accumulation wall" of the channel, where a concentration gradient is established. Small particles with higher diffusion coefficients achieve equilibrium positions at higher levels in the channel than larger particles, and are thus transported through the channel more rapidly due to the parabolic profile of the channel flow. Consequently, small particles elute first, opposite the order of elution in SEC.

In the manner described above, FFF accomplishes the separation of different sized particles in polydisperse solutions for analysis by downstream detectors. Molar masses of the fractionated species are determined using a multi-angle light scattering (MALS) detector which collects on-line SLS measurements at multiple scattering angles. The linear Zimm method 27 is used to obtain molar mass by setting B₂₂ to zero and extrapolating to zero scattering angle.

The following protein standards were obtained in lyophilized powder form from Sigma Aldrich (St. Louis, MO): bovine serum albumin (BSA), alcohol dehydrogenase, β-amylase, apoferritin, and thyroglobulin. All protein standards were dissolved in Dulbecco's phosphate buffer solution (PBS), pH 6.5, also obtained in powder form from Sigma and dissolved in chromatography grade Optima water from Fisher Scientific (Fair Lawn, NJ). PBS was filtered with 0.1 μm filters before use.

Lyophilized glucagon powder was donated by Eli Lilly and Company (Indianapolis, IN). Solvents used were certified 0.01 N HCl or 0.01 N NaOH purchased from Fisher Scientific (Fair Lawn, NJ). Prior to the addition of glucagon, the small quantities of solvents used in these studies were filtered through 0.02 μm syringe filters. This step is necessary in order to produce dust-free diluents required for light scattering studies. The glucagon powder was then dissolved in the dust-free HCl or NaOH. Light scattering experiments were performed on the glucagon solutions, both before and after an additional filtration using either 0.1 or 0.2 μm solvent filters obtained from Millipore (Bedford, MA) and Whatman (Kent, UK). A process diagram detailing the solvent and glucagon solution preparations is shown in Figure 1. Glucagon concentrations were determined spectrophotometrically using E (1 mg/mL; 1 cm) = 2.38 at 278 nm.

SLS and DLS experiments were conducted with an ALV-GmbH (Langen, Germany) SP-125 Compact DLS/SLS Goniometer. The majority of experiments presented here utilized a vertically polarized 400 mW diode-pumped, solid-state Coherent DPSS532-400 laser (Coherent Inc., Santa Clara, CA) operating at 532 nm wavelength as the light source. Glucagon solutions were prepared as previously described, placed in the sample compartment, fixed to restrict movement, and equilibrated at 30°C by a thermostatted bath. In DLS experiments, the apparent hydrodynamic size and PSD of glucagon samples were determined by obtaining autocorrelation functions of the scattered light intensity, G₂(t), using an ALV-5000/E multiple tau digital correlator. Time correlation functions were analyzed using CONTIN software to determine R_h and PSD. The specific refractive index (dn/dc) of glucagon was measured using a Bellingham & Stanley (Kent, UK) 60/ED Abbe refractometer with the laser light source described. The values obtained for dn/dc at 532 nm were 0.175 in HCl and 0.189 in NaOH.

Asymmetric flow FFF experiments were carried out with an Eclipse F separation system from Wyatt Technology Corporation (WTC, Santa Barbara, CA). For protein standard analysis, 250 μm channel thickness and 10 kDa MWCO membranes were used, also obtained from WTC. Protein standards were prepared in PBS as previously described, and this buffer was used as the mobile phase. For glucagon experiments, 450 μm channel thickness and 1 kDa MWCO ultrafiltration membranes were used, obtained from WTC. Glucagon solutions were prepared in certified 0.01 N HCl as previously described, and this solvent was used as the mobile phase. In all experiments, samples were injected into the FFF channel using a manual Rheodyne (Rohnert Park, CA) injection port. Channel flow was set at 1 mL/min, and cross-flow rates were controlled using the Eclipse2 software, version 2.3 (WTC).

FFF experiments were coupled with MALS detection for molar mass determination. The MALS detector used was an 18-angle DAWN EOS (WTC), which employs a 685 nm wavelength 30 mW linearly polarized Ga-As laser light source. Molar mass calculations were made using ASTRA software (WTC). On-line concentration measurements were made with a Shimadzu (Columbia, MD) SPD-10AV UV spectrophotometer set at a wavelength of 278 nm to monitor the absorbance of light by glucagon.

Separation of BSA oligomers was demonstrated using the FFF/MALS method described, in order to validate this method as suitable for protein separation. BSA is a common protein with a monomer molecular weight of 67 kDa that is known to form well defined oligomers in solution. This experiment is routinely used to optimize FFF performance. Figure 2 shows an optimal BSA separation, with 70 μg of BSA injected using PBS as mobile phase. The relative light scattering intensity of BSA monomer, dimer, trimer, tetramer and higher order aggregates can be identified.

In order to demonstrate the ability of FFF/MALS to not only speciate aggregates, but also to provide absolute molecular weight determination, a mixture of five different protein standards was analyzed: BSA, alcohol dehydrogenase, β-amylase, apoferritin, and thyroglobulin. The light scattering signal for the five separated proteins is shown in Figure 3. The injected quantity of each protein is shown in Table 1, along with a comparison between expected molecular weights and the M_w inferred from the light scattering results. The results of these FFF/MALS experiments using protein standards demonstrate that FFF is an effective method for separating proteins and protein aggregates.

Filtered glucagon solutions were analyzed immediately after mixing by SLS to determine the weight-averaged molar mass (M_w) using the Debye equation (Eqn. 1). The excess Rayleigh ratio at 90° scattering angle, R₉₀, was measured over a range of known concentrations. By plotting values of Kc/R₉₀ over a range of concentrations, the M_w was determined from the inverse of the y-intercept value. Debye plots of Kc/R₉₀ as a function of concentration for acidic and alkaline glucagon solutions are shown in Figure 4 and Figure 5. The M_w of soluble glucagon in the acidic pH region was determined to be 3539 ± 372 g/mol. Glucagon in alkaline solutions exhibited a slightly higher M_w, 3831 ± 275 g/mol. This slightly higher alkaline M_w may indicate the presence of higher molecular weight aggregates in the alkaline solutions. In order to verify this, DLS was utilized to further probe these samples.

Relaxation time distributions before and after filtration for both acidic and alkaline glucagon were obtained from CONTIN analysis of light intensity autocorrelation functions. The particle R_h distribution functions were calculated from the relaxation time distributions using the Stokes-Einstein relationship. CONTIN analysis generated R_h distribution plots of unfiltered acidic and alkaline glucagon solutions are shown in Figure 6a and 6b. These unfiltered glucagon solutions in dilute acid and base showed bimodal R_h distributions with peak R_h values around 1 and 100 nm, with the 1 nm peak indicating monomeric glucagon. The monomeric radius of glucagon can be confirmed using an accepted equation for approximating the volume of a typical monomeric protein from the M_w28. Using a literature reported molecular weight of 3485 g/mol, which is based on the amino acid sequence of glucagon, the estimated monomeric radius is 1.02 nm.

DLS results indicate that acidic and alkaline glucagon solutions exhibit distinctly different behavior following filtration. Bimodal distributions were still present following the filtration of alkaline glucagon solutions (Fig. 6c), whereas the higher molecular weight aggregates were eliminated by filtration of acidic glucagon solutions (Fig. 6d). The bimodal distribution that remained after filtration of alkaline glucagon solutions showed the larger subpopulation peak hydrodynamic radius value shifting to a slightly smaller R_h of 61 nm from its original size of 100 nm.

These DLS results indicate the presence of higher molecular weight aggregates that can be removed by filtration from acidic but not alkaline solutions. The filter retention may be related to the relative charge of the filter membrane or to the relative rigidity of the aggregates. Since the filters used were designed to have low affinity for protein, the aggregate rigidity seems to be the more plausible explanation for the observed effects.

The effect of pre-existing aggregates on aggregation rate was further investigated by measuring the raw light scattering signal intensity of freshly prepared 4 mg/mL glucagon solutions in 0.01 N HCl over a period of three days (Fig. 7). The rapid increase in light scattering intensity for the unfiltered glucagon preparation suggests that the presence of initial aggregates serve to nucleate the aggregation process. Conversely, the filtered acid glucagon preparation with the absence of initial aggregates stayed stable in solution for 50 hours.

To further quantify the two subpopulations within the unfiltered samples, SEC experiments were attempted. SEC proved ineffective in monitoring the molecular weight distribution of aggregating glucagon systems (data not shown). Aggregates present in glucagon solutions injected into a column using 0.01 N HCl eluent caused fouling and backpressure problems in the SEC system, and tended to dissociate inside the column, emerging in the monomeric state. Thus, this method was abandoned as a means to monitor the aggregation process.

FFF/MALS experiments were performed to verify the presence of large initial aggregates upon initial mixing of glucagon in 0.01 N HCl, and to confirm that the large aggregates can be removed by filtration, as previously indicated by DLS results. The 100 nm aggregate population was not concentrated enough to be detectable at the low concentrations and small injection volumes typically used with chromatographic methods. Therefore 250 μL of a 14 mg/ml glucagon solution was injected into the FFF channel for these experiments both before and after filtration with a 0.1 μm filter. Elution of a large aggregate peak can be observed in the unfiltered sample, but not in the filtered sample (Fig. 8). The use of a UV detector downstream of the FFF unit allowed for the verification that the large particles are indeed protein, since this peak exhibits measurable UV absorbance at 278 nm.

A high concentration unfiltered glucagon solution in 0.01 N HCl was again analyzed by FFF, both before and after 22 hours of incubation at 22°C. Initially, weight averaged molar mass of the monomer was analyzed by MALS to be 3650 g/mol, with a large aggregate peak of 1.2 × 10⁵ g/mol. After the incubation period, the sample was observed to be partially gelled. When this sample was again analyzed by FFF/MALS, the molar mass of the large aggregate had increased significantly to 1.9 × 10⁶ g/mol. The histograms in Figure 9 show the relative molar mass distributions for the two cases. The large aggregates are on the order of 1% of total injected mass, but the molar mass distributions for these peaks cannot be further analyzed due to extremely low UV signal strength.