# Distributions

Most chromatography analysis programs calculate the properties of a sample for a single peak (i.e., a range of elution volumes). For example, one might determine the average molar mass, intrinsic viscosity, or hydrodynamic radius for a peak. This is a very powerful way to characterize different fractions of an eluting sample.

Another powerful way to interpret data from a fractionated sample is to consider distributions of measured quantities. For example, one might want to know the fraction of a sample that has a molar mass within a certain range. The way to do this is to calculate a cumulative distribution of molar masses.

To calculate a cumulative distribution, the following steps are taken:

1. The total mass of the eluting sample is calculated. If the concentration is measured at each eluting slice, then the total mass is equal to the sum of the concentration measurements times the eluting slice volume.
2. The molar mass is calculated for each eluting slice using a DAWN or miniDAWN MALS detector.
3. The resulting molar masses and the measured concentration at that slice are then sorted by ascending mass.
4. The weight fraction of sample with a certain molar mass or lower is then calculated by summing all of the concentrations up to the specified molar mass, multiplying this sum by the volume of the eluting slice, then dividing by the total mass of the sample.

The result of this procedure for a fractionated BSA sample is shown in the figure below. As can be seen, there is a "stair-step" pattern, with the steps occurring at the mass for each oligomer. It is easy to determine the fraction of sample in each oligomer simply by looking at the step height.

The differential distribution is exactly that. One way to calculate a differential distribution is to take the differential of the cumulative distribution. This can be seen clearly in the overlay below of the cumulative and differential distributions of a fractionated BSA sample. The differential distribution provides a revealing visual representation of the weight fraction of a sample within a certain molar mass range, and provides information on the ultimate resolution of the entire fractionation, data collection, and analysis system.

The calculation of differential distributions has traditionally been a tricky affair. The simplest approach is to fit the resulting data to a monotonic function, and then use the analytical expression for the fit function to calculate the differential distribution directly. This technique has severe limitations (a.k.a. it simply does not work), particularly for narrow peaks, oligomer sequences, and nonstandard fractionation techniques. Wyatt Technology has implemented a proprietary adaptive binning algorithm to accurately calculate differential distributions without fitting the data. The results are truly amazing. It is a snap to calculate differential distributions for narrow standards and oligomer sequences.

ASTRA capitalizes on this important functionality by presenting users with the powerful distribution analysis procedure. As can be seen in the screen shot, the fraction of a sample in a certain mass range can be ascertained simply by dragging a region in the combined distribution graph. In this case, the weight fraction of BSA monomer, dimer, and trimer can be determined and reported. In addition, when combined with ASTRA's number density calculations, the number fraction of a sample in a certain size range can be determined with the distribution analysis procedure.

#### Why the band broadening correction is essential for distributions:

Calculated distributions present an accurate map of the computed value for each eluting slice. As shown in the discussion on band broadening, setting a peak region can compensate for the "grimace" in molar mass across the peak; the average molar mass is correct. There is no way to compensate for the "grimace" in the distribution plot, however. Without the band broadening correction, the values from a sharp, monodisperse peak are artificially spread out. There is an apparent loss in resolution. With the band broadening correction, the true distribution becomes apparent, and resolution is restored. The animation below of BSA oligomers measured with and without the band broadening correction demonstrates clearly that the band broadening correction is necessary to determine accurate distributions.

#### 13 Key Questions to Ask

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Wyatt Technology is the recognized leader in light scattering instrumentation and software for determining the absolute molar mass, size, charge and interactions of macromolecules and nanoparticles in solution.

Wyatt's line of multi-angle static light scattering products couple to size exclusion chromatography (SEC-MALS), field-flow fractionation (FFF-MALS), and stop-flow composition-gradient systems (CG-MALS). Our dynamic light scattering (DLS) products operate in traditional cuvette as well as on-line and automated, high-throughput modes. We also offer unique instruments for electrophoretic light scattering (MP-PALS), differential refractometry, and differential viscosity.