Using Sedimentation Equilibrium for the Characterization of Self-Associating System in Analytical Ultracentrifuge

The dimerization of α-chymotrypsin was studied as an example of a typical self-associating system to demonstrate how such a reaction can be characterized. It has also been proposed that α-chymotrypsin serves as a model system to ensure proper operation of the analytical ultracentrifuge.

An ideal model system would be a well-characterized association where a single equilibrium reaction occurs, but with little impact as a result of solution conditions including temperature, ionic strength, and pH.

However, a well characterized α-chymotrypsin has been applied to test the consistency of instrument measurements over time. α-chymotrypsin shows a monomer-dimer equilibrium at approximately pH 4; however, the association constant can change with different lots and different buffer conditions. Therefore, a standard lot number and buffer is a good idea for repeated analysis.

In 0.2 M KCl, 0.01 M acetate, pH 4.4 at 20°C, it has been reported that the dimerization constant varies approximately two-fold between lots of α-chymotrypsin supplied by Worthington Biochemicals with a maximum of 44×103L/mol.

This variation seems to be a function of lot number rather than experimental error, as there was less than 10% variation over a four-year period.

Materials and Methods

Using the Beckman Coulter Proteomelab XL-A analytical ultracentrifuge, a commercial α-chymotrypsin sample was examined for self-associative behavior by sedimentation equilibrium.

The protein was run at three different concentrations, which were 0.2, 0.4 and 0.6mg/mL in 10mM NaOAc, 0.2M NaF, pH 4.0 at 283nm and 20°C without further purification.

The literature value of 0.736mL/g was used as the partial specific volume. In the calculation, a buffer density of 1.001g/mL was also used. A nonlinear least-squares curve-fitting algorithm integrated in the Proteomelab XL-A Data Analysis Software, was used to assess the sedimentation equilibrium data.

The self-association model (equation 1) enables analysis of either a single ideal species or up to four associating species, depending on which parameters in the equation are allowed to vary during convergence. Data were studied both as single and multiple data files.

clip_image002_0001

where Ar = absorbance at radius r

Amonomer ,r0 = absorbance of the monomer at the reference radius r0

M = monomer molecular weight

n2 = stoichiometry for species 2

Ka,2 = association constant for the monomer-n-mer equilibrium of species 2

n3 = stoichiometry for species 3

Ka,3 = association constant for the monomer-n-mer equilibrium of species 3

n4 = stoichiometry for species 4

Ka,4 = association constant for the monomer-n-mer equilibrium of species 4

E = baseline offset

B = second virial coefficient for nonideality

The association constant was converted to units of M-1 using an extinction coefficient E1%280 of 20.4 and assuming was used to a value of twice that for the dimer, that is:

clip_image004_0001

Results and Discussion

This article presents a stepwise approach for establishing the self-associative behavior.

Step 1 – An equilibrium gradient (absorbance versus radius) is transformed into a plot of Mw,app vs. absorbance provides data about the system’s associative order.

During the transformation, a segment of data points, usually 10 to 40, are moved across the radial path one data point at a time and calculating Mw,app from the slope of a ln(A) vs. r2 plot of this subset.

A plot of Mw,app vs. absorbance (considered the midpoint of each segment) produces a sequence of connecting lines whose slope corresponds to the molecular weight. The plot shape can give an approximation of the self-associative behavior with regard to concentration.

This assumes that the sample obeys the Beer-Lambert Law where concentration and absorbance are proportional.

Mw,app does not change with absorbance for material behaving as one ideal species. On the other hand, the plot for an associating system curves upwards with increasing absorbance.

The first estimation of the system’s associative order is provided by dividing the molecular weight at the highest absorbance by the molecular weight at the lowest absorbance; that is, the monomer molecular weight M1.

With regard to α-chymotrypsin (Figure 1a), the material seems to be associating as a monomer-dimer, but assembly to a higher order aggregate can be realized. At this stage, it is impossible to differentiate between the two.

Generally, a system can be made to assemble more completely by running it at higher concentrations, but that system will start to show nonideality when pushed to increased concentrations, which can obscure the maximum associative state.

A second technique that is more qualitative and provides data regarding the system’s homogeneity is basically a plot of ln (A) vs. r2. It gives a straight line with a slope proportional to the molecular weight.

For a single ideal species, the line stays straight across the whole radial path. However, for an associating system, the line will deflect in an upwards direction, as a result of the presence of higher molecular weight aggregates redistributing to the bottom of the cell.

The linear nature of the ln (A) vs. r2 plot for chymotrypsin shows how this kind of method can be misleading (Figure 1b). Deviations of below 10% can be difficult to see with the naked eye.

Mw,app vs. concentration plot of α-chymotrypsin at pH 4.0. The appearance of the gradient increasing to the next multiple of monomer molecular weight (21,600g/mol, Ref. 5) suggests the dimer as a likely associative state.The equilibrium gradient depicted as a plot of ln(A) vs. r2. The apparent linear nature of this plot suggests an ideal species, which runs contrary to the earlier evidence. Due to the inherent insensitivity of this approach, this type of diagnostic is more common in detecting the absence of homogeneity; i.e., deviations from a straight line indicate that a sample is definitely not behaving ideally.

Figure 1a. Mw,app vs. concentration plot of α-chymotrypsin at pH 4.0. The appearance of the gradient increasing to the next multiple of monomer molecular weight (21,600g/mol, Ref. 5) suggests the dimer as a likely associative state. Figure 1b. The equilibrium gradient depicted as a plot of ln(A) vs. r2. The apparent linear nature of this plot suggests an ideal species, which runs contrary to the earlier evidence. Due to the inherent insensitivity of this approach, this type of diagnostic is more common in detecting the absence of homogeneity; i.e., deviations from a straight line indicate that a sample is definitely not behaving ideally. Image credt: Beckman Coulter

Step 2 – The data are fit to a single ideal species model. Besides providing an estimate of the apparent weight-average molecular weight (Mw,app ), the pattern from the residuals (the points off the best-fit curve) can give an understanding of the system’s behavior.

A residual pattern for α-chymotrypsin consistent with an associating system and a Mw,app (30,012) > M1 (21,600) is shown in Figure 2. This data helps validate the plot of Mw,app vs. absorbance.

The equilibrium fit results of α-chymotrypsin modeled as a single ideal species. The fitted parameter for the weight-average molecular weight (Mw,app), estimated at 30,012g/mol, was found to be higher than the monomer molecular weight, suggesting aggregation. The residuals from the best-fit curve reveal a systematic pattern indicative of an aggregating system.

Figure 2a. The equilibrium fit results of α-chymotrypsin modeled as a single ideal species. The fitted parameter for the weight-average molecular weight (Mw,app), estimated at 30,012g/mol, was found to be higher than the monomer molecular weight, suggesting aggregation. The residuals from the best-fit curve reveal a systematic pattern indicative of an aggregating system. Image credt: Beckman Coulter

The residuals plot scaled to make the pattern indicating association more recognizable.

Figure 2b. The residuals plot scaled to make the pattern indicating association more recognizable. Image credt: Beckman Coulter

Step 3 – The data are fit to a more complex self-associating model. As well as identifying the stoichiometry, the system’s association constant may also be estimated. This step is also applied for testing the system’s reversibility.

Irrespective of initial concentration or rotor speed, a reversible self-associating system should give the same association constant. Multiple data files are used to evaluate more complicated models, assuming that individual files of data have previously tested for any abnormal behavior.

As well as enabling convergence on a global least-squares minimum, multiple data files can collectively span the system’s associative range.

However, with regard to complex models, one limitation is that the greater number of parameters used to describe the model may pose issues during a fit. For instance, if there are too many parameters being allowed to vary at one time, the statistical significance of the fitted values can be compromised for the sake of a fit.

Due to this, it is important to restrict the number of parameters that are allowed to vary during a fit. As a rule of thumb, the stoichiometry and the monomer molecular weight are usually constrained to their known or suspected values during a fit.

Under denaturing conditions, M1 estimates can be made on the analytical ultracentrifuge. Then, the other parameters are established by enabling each parameter to vary over a series of iterative fits.

Figure 3a shows α-Chymotrypsin modeled as a monomer-dimer system. In this example, the baseline offset term (E) was constrained at zero. This term corrects for any residual components that contribute to the system’s absorbance.

The baseline offset term is generally determined by overspeeding the equilibrium run and then reading the absorbance of the depleted region gradient, which is called the meniscus-depletion technique.

Two of the files show a pronounced slope, as shown in Figure 3a. Figure 3b shows the baseline correction term and the fit is seen to dramatically improve.

The fit is assessed based on the randomness of the residuals, the goodness of fit statistic, the relative tightness of the confidence limits, the magnitude of the residuals, and by checking certain fit parameters for physical significance.

In this instance, the fitted values for the baseline offset were validated by meniscus depletion at 45,000rpm for approximately six hours.

Multiple data files (three concentrations) showing the deviation from a best-fit curve of a monomer-dimer associative model. In this example, the baseline offset term (E) was constrained at zero and the residual pattern is skewed for two of the data files

Figure 3a. Multiple data files (three concentrations) showing the deviation from a best-fit curve of a monomer-dimer associative model. In this example, the baseline offset term (E) was constrained at zero and the residual pattern is skewed for two of the data files. Image credt: Beckman Coulter

The same example with the baseline offset term allowed to vary. In this case, the residuals for all three files are shown to afford a random scatter, indicating a good fit to the model. Figure 3c. The best-fit curve and the residual plot for each of the three files from the same monomer-dimer fit of Figure 3b.

Figure 3b. The same example with the baseline offset term allowed to vary. In this case, the residuals for all three files are shown to afford a random scatter, indicating a good fit to the model. Figure 3c. The best-fit curve and the residual plot for each of the three files from the same monomer-dimer fit of Figure 3b. Image credt: Beckman Coulter

The best-fit curve and the residual plot for each of the three files from the same monomer-dimer fit of Figure 3b.

Figure 3c. The best-fit curve and the residual plot for each of the three files from the same monomer-dimer fit of Figure 3b. Image credt: Beckman Coulter

Table 1 shows the values estimated for the association constant. The literature values for the association constant are found to vary from one lot to another, which has been attributed to the incomplete participation of the monomer in the equilibrium.

With additional purification, the association constant is increased and readings are more consistent. At this stage, the right model seems to be a monomer-dimer system, albeit higher order models should also be assessed.

Concentrations/ Buffer System

Kabs2

95% Confidence Limits

Kconc3

( X 10-3 L/mol)

Klit4

Kabs

0.2, 0.4, and 0.6 mg/mL;

10 mM NaOAc,

0.2 M NaF, pH 4.0

2.56

2.17–3.03

67.3

44.4

35.7

14.9

27.4

Step 4 – In this step, α-Chymotrypsin was modeled as a monomer-trimer system. The fit is noticeably worse than for the simple dimerization, as depicted in Figure 4.

The residuals are nonrandom and the magnitude of the residuals is high. The manner in which the data dips away from the best-fit curve suggests that fits to higher order systems will lead to yet worse fits.

The conclusion based on this brief exercise is that α-chymotrypsin at pH 4.0 acts as a reversible monomer-dimer self-associating system.

  • Multiple data files and the Proteomelab XL-A Data Analysis Software were used to curve-fit the sedimentation equilibrium gradients.
  • The association constant as estimated from a best-fit curve in terms of absorbance units utilizing clip_image020_0000= 0.736mL/g and ρ = 1.001g/mL.
  • The association constant converted into units of M-1 utilizing e = 44,064L/mol-cm.
  • Different literature values for the association constant (determined at pH4.4),indicating lot-to-lot variation.

Shows a relatively poor fit when the same three data files are fit to a monomer-trimer self-associating system under similar fit conditions.

Figure 4. Shows a relatively poor fit when the same three data files are fit to a monomer-trimer self-associating system under similar fit conditions. Image credt: Beckman Coulter

Acknowledgements

Produced from content authored by Paul Voelker and Don McRorie, Beckman Coulter.

References

  1. Teller, D. C. Characterization of proteins by sedimentation equilibrium in the analytical ultracentrifuge. Methods in Enzymology, Vol. 27, pp. 346-441. Editors-in-chief: S. P. Colowick and N. O. Kaplan. New York, Academic Press, 1973
  2. Aune, K. C., Timasheff, S. N. Dimerization of a-chymotrypsin. I. pH dependence in the acid region. Biochemistry 10, 1609-1616 (1971)
  3. Aune, K. C., Goldsmith, L. C., Timasheff, S. N. Dimerization of a-chymotrypsin. II. Ionic strength and temperature dependence. Biochemistry 10, 1617-1622 (1971)
  4. Miller, D. D., Horbett, T. A., Teller, D. C. Reevaluation of the activation of bovine chymotrypsinogen A. Biochemistry 10, 4641-4648 (1971)
  5. Handbook of Biochemistry Selected Data for Molecular Biology, pp. C-10 and C-74. 2nd Ed. Cleveland, OH, Chemical Rubber Co., 1970.
  6. Johnson, M. L., Correia, J. J., Yphantis, D. A., Halvorson, H. R. Analysis of data from the analytical ultracentrifuge by nonlinear least-squares techniques. Biophys. J. 36, 575-588 (1981)
  7. McRorie, D. K., Voelker, P. J. Self-Associating Systems in the Analytical Ultracentrifuge. Fullerton, CA, Beckman Instruments, Inc., 1993.

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Last updated: May 16, 2020 at 12:36 PM

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