Sedimentation Equilibrium

Circular Dichroism DSC Laser Light Scattering Sedimentation Equilibrium Sedimentation Velocity

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Sedimentation equilibrium (SE-AUC) is an analytical ultracentrifugation method for measuring protein molecular masses in solution and for studying protein-protein interactions. It is particularly valuable for:

  • establishing whether the native state of a protein is a monomer, dimer, trimer, etc.
  • measuring the equilibrium constant (Kd) for association of proteins which reversibly self-associate to form oligomers
  • determining whether proteins weakly self-associate at high concentrations (up to ~200 mg/mL) and characterizing their repulsive or attractive interactions (e.g. by the second virial coefficient B22)
  • measuring the stoichiometry of complexes between two or more different proteins (e.g. a soluble receptor and its ligand or an antigen-antibody pair), or between a protein and a non-protein ligand
  • measuring the equilibrium constants for reversible protein-protein and protein-ligand interactions (approximate Kd range 1 nanomolar to 1 millimolar)

In sedimentation equilibrium the sample is spun in an analytical ultracentrifuge at a speed high enough to force the protein toward the outside of the rotor, but not high enough to cause the sample to form a pellet. As the centrifugal force produces a gradient in protein concentration across the centrifuge cell, diffusion acts to oppose this concentration gradient. Eventually an exact balance is reached between sedimentation and diffusion, and the concentration distribution reaches an equilibrium. This equilibrium concentration distribution across the cell is then measured while the sample is spinning, using either absorbance or refractive index detection in our Beckman XL-I (picture).

The key point about sedimentation equilibrium is that the concentration distribution at equilibrium depends only on molecular mass, and is entirely independent of the shape of the molecule. The precision of the molecular masses determined by this technique is usually 1-2%. 

Furthermore, for proteins which self-associate to oligomers, or for mixtures of molecules that bind to one another, the overall distribution will also be in chemical equilibrium for the association process, and therefore will reflect the higher molecular weight of the associated states and their proportion in the sample. 

Example 1: Characterizing solution molar mass of a protein at concentrations up to ~120 mg/mL

It is well known that self-association or non-specific molecular interactions in protein products at high concentrations can significantly impact solution viscosity, which in turn affects drug delivery and product manufacturability. It is also essential to minimize aggregation and particle formation (to have colloidal stability).

Aggregation, particle formation, solubility, and/or viscosity are often highly correlated with the solution second virial coefficient, B22, which quantifies the non-specific attractive or repulsive interactions between protein molecules (the solution “non-ideality”). However B22 is typically measured at concentrations below 10-20 mg/mL (“semi-dilute” solutions), and it is unclear whether this single parameter is truly sufficient to describe the behavior at formulated concentrations of ≥100 mg/mL, which are now common for therapeutic proteins.

The data in the graph below illustrate a new SE-AUC approach using samples of very low volume (~20 μL) at concentrations up to ~120 mg/mL. The small sample volume allows equilibration to occur in only a few hours, increasing throughput, and together with the use of refractive index (RI) detection and low rotor speeds it keeps the concentration gradients across the SE cell within a workable range despite the high protein concentrations.

The graph shows the ratio of the apparent weight-average molar mass at each concentration for a moderately-sized (20-70 kDa) therapeutic protein relative to that observed at infinite dilution. We see that the molar mass decreases strongly with concentration rather than increasing due to reversible self-association. Indeed near 120 mg/mL Mapp is nearly 4-fold lower than at low concentration.

Qualitatively the graph above tells us these protein molecules exhibit strong repulsive interactions, but how do we quantitate that? The simplest model is the first-order virial expansion:

1/Mapp = 1/M0 + 2B22c                      [1]

where the second virial coefficient B22 measures the strength of the net repulsion (positive values) or attraction (negative values). The graph below re-plots the data in a form where equation 1 predicts a straight line with a slope proportional to B22. While the data below ~20 mg/mL can be reasonably approximately by a straight line (the dashed green line), clearly at higher concentrations the data curve upward strongly (the repulsion between molecules grows stronger). This means that extrapolating the low-concentration data to high concentrations would grossly under-estimate the strength of the repulsive interactions at higher concentrations.

The full range of data were fitted to a more complex, third-order virial expansion:

1/Mapp = 1/M0 + 2B22c + 3B33c2 + 4B44c3                   [2]

That fit is shown as the solid black curve in the graph above. For this sample the third virial coefficient B33 turns out to be zero (within measurement error), but a large positive fourth virial coefficient B44 is necessary to fit the full range of data.

This approach and these data are discussed in more detail in this poster presented at WCBP 2015.

Example 2: Is a Sequence Homolog a True Structure Homolog?

Tumor necrosis factor alpha (TNF) was the first known member of a family of signaling molecules involved in inflammation, apoptosis, and many other important functions. A hallmark of this family is that these proteins normally occur as trimers in solution.

A potential new member of this family was identified on the basis of sequence homology. However, when it was expressed in E. coli and refolded from inclusion bodies, it appeared to be a monomer based on its elution relative to standards on size-exclusion chromatography (SEC). Did this mean it was not truly a member of this family, or simply that it was not correctly refolded, or was the mass estimate from SEC wrong?

The graph below shows some sedimentation equilibrium data for this molecule, showing the concentration as a function of position within the cell as monitored by absorbance at 230 nm. Note that the total amount of protein for this experiment was <10 micrograms.

raw sedimentation equilibrium (SE-AUC) data for a TNF homolog

The next graph (below) shows that data re-plotted as the natural log of absorbance vs. radius2/2. In this type of plot a single species gives a straight line whose slope is proportional to mass.  The light blue line indicates the theoretical slope calculated for the monomer mass (~17 kDa). The dark blue line (mostly hidden behind the data points) has the theoretical slope for the trimer mass. This plot therefore makes it obvious that this protein is indeed a trimer, and therefore it is indeed a homolog of TNF (and presumably is correctly folded).

linearized plot of the TNF homolog SE-AUC data

Although the results for only a single sample and rotor speed are shown here, in general to quantitatively characterize a protein and whether it self-associates we run 3-9 samples over a broad range of loading concentrations and at two or more rotor speeds, and these data are then simultaneously ("globally") analyzed.

Example 3: Functional Characterization of a Monoclonal Antibody

The function of many proteins is to bind to other proteins, and sedimentation equilibrium is a very powerful tool for studying such binding interactions.

The graph below summarizes the data (points) and fitted curves for 8 experiments on mixtures of a monoclonal antibody and its ~25 kDa protein antigen. The data sets cover experiments at different mixing ratios of antibody to antigen, and by using scans at either 280 or 230 nm they also cover a wide range of concentrations. (Note that this entire set of experiments used only ~80 micrograms of antibody.)

overlay of sedimentation equilibrium data for antibody-antigen mixtures and fitted curves

To analyze these data  an appropriate binding model is needed. The model shown below is the simplest one possible for an antibody with two binding sites, and simply assumes that both sites have the same binding affinity and bind independently of one another (no cooperativity and no steric blocking of one site by antigen bound to the other).

antibody-antigen binding model

In fitting these data one is essentially asking: Is there a single value of the dissociation constant, K1, that can explain all 8 experiments? The solid lines in the graph above represent the best fit of this model, with K1 = 48 nanomolar, and the fact that the lines follow the data points quite well shows that this is a good fit. Importantly, this good fit also implies that both binding sites on the antibody are active, and active simultaneously. This data analysis was done using custom software available only at Alliance Protein Laboratories.

The value of K1 is actually quite well determined, with statistical analysis indicating we can be 95% confident the true value is between 43 and 52 nM (a 5% standard error, or only 60 cal/mol in terms of binding energy!) While this statistical analysis probably overestimates the true precision at least several-fold, nonetheless it is clear this approach can give very precise binding affinities.

Importantly, this approach could be used to quantitatively compare different antibodies, different lots of the same antibody, loss of activity of aged samples, etc.

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