T-1782A
T e c h n i c a l ␣ I n f o r m a t i o n
Analytical Ultracentrifugation
...............................................
α-Chymotrypsin: Characterization of a Self-Associating System in
the Analytical Ultracentrifuge
Paul Voelker and Don McRorie
Beckman Instruments, Inc.
Introduction
We have studied the dimerization of α-chymotrypsin as an example of a typical self-associating
system in order to show how such a reaction may be
characterized. α-Chymotrypsin has also been suggested as a model system to ensure the proper operation of the analytical ultracentrifuge.
An ideal model system should be a well-characterized association where a single equilibrium reaction is being observed with little effect from solution
conditions such as pH, ionic strength and temperature. Such a system probably does not exist. However, α-chymotrypsin has been well-characterized
and has been used to test the consistancy of instrument measurements over time.(1) At about pH 4,
α-chymotrypsin exhibits a monomer-dimer equilibrium, but the association constant can vary with different lots and different buffer conditions.(2-4)
Therefore, it is a good idea to have a standard lot
number and buffer for repeated examination. In 0.01
M acetate, 0.2 M KCl, pH 4.4 at 20°C, the dimerization constant has been reported to vary about twofold between lots of a-chymotrypsin purchased from
Worthington Biochemicals with a maximum of
44 × 103 L/mol.(4) The variation appears to be a
function of lot number and not of experimental error
since the same lot showed less than 10% variation
over a period of four years.
Materials and Methods
A commercial sample of α-chymotrypsin
(Worthington Biochemical Corp., lot # 37K093)
was evaluated for self-associative behavior by
sedimentation equilibrium using an Optima™ XL-A
analytical ultracentrifuge from Beckman. The
protein was run at three concentrations (0.2, 0.4
and 0.6 mg/mL) in 10 mM NaOAc, 0.2 M NaF,
pH 4.0 at 283 nm and 20°C without further
purification. The literature value of 0.736 mL/g
was used as the partial specific volume.(5) A buffer
density of 1.001 g/mL was also used in the
calculation. Sedimentation equilibrium data were
evaluated using a nonlinear least-squares curvefitting algorithm(6) contained in the XL-A Data
Analysis Software. The self-association model
shown in equation 1 permits 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
analyzed both as single and multiple data files.
BECKMAN
[ln( Amonomer,r
Ar,total = e
0
)+
(1 − vρ)ω 2
M(r 2 −r 02 )− BM( Atotal,r − Atotal,r )
0
2RT
[ln( Amonomer,r
+e
[ln( Amonomer,r
+e
[ln( Amonomer,r
+e
0
0
0
)+ln( K a ,2 )+
]
(1 − vρ)ω 2
n2 M(r 2 −r 02 ) − BM( Atotal,r − Atotal,r 0 )
2RT
(1 − vρ)ω 2
)+ln( K a ,3 )+
n3 M(r 2 −r 02 ) − BM( Atotal,r − Atotal,r 0 )
2RT
)+ln( K a ,4 )+
]
(1)
]
(1 − vρ)ω 2
n4 M(r 2 −r 02 ) − BM( Atotal,r − Atotal,r )
0
2RT
]
+E
where Ar
= absorbance at radius r
Amonomer,r0 = absorbance of the monomer at
the reference radius r0
M
= monomer molecular weight
= stoichiometry for species 2
n2
Ka,2
= association constant for the
monomer-n-mer equilibrium
of species 2
n3
Ka,3
The association constant was converted to units of
plot of this subset. A plot of Mw,app vs. absorbance
(taken as the midpoint of each segment) creates a
series of connecting lines whose slope is proportional to the molecular weight. The shape of the plot
can provide an estimate of self-associative behavior
with respect to concentration. This assumes that the
sample obeys the Beer-Lambert Law where absorbance and concentration are proportional.
For material behaving as a single ideal species,
Mw,app does not vary with absorbance. The plot for
an associating system, on the other hand, curves upward with increasing absorbance. Dividing the molecular weight at the maximum absorbance by the
molecular weight at the minimum absorbance
(i.e., the monomer molecular weight, M1) provides
a first approximation of the associative order of the
system. In the case of α-chymotrypsin (Figure 1a),
the material appears to be associating as a monomer-dimer, although assembly to a higher order aggregate is possible. At this point it is not possible to
discriminate between the two.
In general, a system can be made to assemble
more fully by running it at higher concentrations. It
should be noted, however, that systems will begin to
exhibit nonideality (from excessive crowding or
charge effects) when pushed to higher concentrations and that this can obscure the highest associative state.
A second, more qualitative method that can provide information about the homogeneity of the system is simply a plot of ln (A) vs. r2 as mentioned
above. This plot yields a straight line with a slope
proportional to the molecular weight. For a single
ideal species, the line remains straight over the en-
n4
Ka,4
E
B
(5)
M-1 using an extinction coefficient E1%
280 of 20.4,
and assuming a value of twice that for the dimer; i.e.,
Kconc = Kabs ×
εl
2
(2)
where Kconc = the association constant in M-1
Kabs = the association constant in terms of
absorbance (estimated directly from
a best-fit curve of a monomer-dimer
self-associating system)
ε
= the extinction coefficient in
L/mol-cm [44,064 L/mol-cm for
α-chymotrypsin monomer, given an
absorptivity (a) of 2.04 L/g-cm and a
monomer molecular weight of
21,600 g/mol(5)]
l
= the pathlength in cm (1.2 for a
12-mm centerpiece)
Results and Discussion
A stepwise approach used to determine the self-associative behavior of α-chymotrypsin is presented
here. A general approach to more complicated systems is beyond the scope of this paper and is dealt
with in a separate publication.(7)
Step 1. Transforming an equilibrium gradient
(absorbance vs. radius) into a plot of Mw,app vs. absorbance provides information about the associative
order of the system. The transformation involves
moving a segment of data points (typically 10–40)
across the radial path one data point at a time and
calculating Mw,app from the slope of a ln(A) vs. r2
2
= stoichiometry for species 3
= association constant for the monomer-nmer equilibrium of species 3
= stoichiometry for species 4
= association constant for the monomern-mer equilibrium of species 4
= baseline offset
= second virial coefficient for nonideality
0.04
Mw,app
Residuals
35000
0.02
0.00
-0.02
0.8
30000
Absorbance
0.6
25000
0.1
0.2
0.3
0.4
0.5
0.4
Concentration
0.2
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 (21,600 g/mol, Ref. 5) suggests the
dimer as a likely associative state.
Data: CHYM1C.RA1_abs.
Model: ideal1
Chi^2= 7.1964E-5
Ao
0.076746 0.001528
H
2.376925E-5
****
M
3.0012 331.4
Xo
6.8920 ****
E
0
****
XLA information:
temp: 20 Speed: 20000
RHO : 1 V-bar: 0.736
Initial guesses:
Ao : 0.0952
H : 0.00002
M : 27658
Xo : 6.892
E:0
0.0
6.9
7.0
7.1
Radius (cm)
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 30,012 g/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.
0.0
-0.5
ln(A)
-1.0
-1.5
-2.0
0.04
-2.5
-3.0
50
45
r2
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.
Residuals
0.02
tire radial path. For an associating system, however,
the line will deflect upward, due to the presence of
higher molecular weight aggregates redistributing to
the cell bottom. The apparent linear nature of the ln
(A) vs. r2 plot for chymotrypsin illustrates how this
type of technique can be misleading (see Figure 1b).
Deviations of less than 10% can be difficult to perceive with the naked eye.
Step 2. The data are fit to a single ideal species
model. In addition to getting an estimate of the apparent weight-average molecular weight (Mw,app),
the pattern from the residuals (the points off the
best-fit curve) can provide insight into the behavior
0.00
-0.02
6.8
7.0
7.2
Radius (cm)
Figure 2b. The residuals plot scaled to make the
pattern indicating association more recognizable.
3
of the system. Figure 2 shows a residual pattern for
α-chymotrypsin consistent with an associating system and a Mw,app (30,012) > M1 (21,600). This information also helps to confirm the plot of Mw,app
vs. absorbance.
Step 3. The data are fit to a more complex selfassociating model. In addition to identifying the stoichiometry, the association constant of the system
can also be estimated. This step is also used to test
the reversibility of the system. A reversible self-associating system should yield the same association
constant independent of rotor speed or initial concentration. The more complex models are usually
evaluated with multiple data files (assuming individual data files have been pretested for any aberrant behavior). In addition to facilitating convergence on a global least-squares minimum, multiple
data files carry the advantage of collectively spanning the associative range of the system.
One of the caveats of dealing with more complex models, however, is that the larger number of
parameters that are used to describe the model may
cause problems during a fit. For example, if too
many parameters are allowed to vary at one time,
the statistical significance of the fitted values can be
compromised for the sake of a fit. For this reason,
the number of parameters allowed to vary during a
fit should be kept to a minimum. As a general rule,
the monomer molecular weight and the stoichiometry are typically constrained to their known (or suspected) values during a fit. Estimates of M1 can be
made on the analytical ultracentrifuge under denaturing conditions. The other parameters are then determined by successively allowing each one to vary
over a series of iterative fits.
α-Chymotrypsin was modeled as a monomerdimer system, as shown in Figure 3a. In this first example, the term for the baseline offset (E) was constrained at zero. The baseline offset term corrects
for any residual components that contribute to the
absorbance of the system. This term is usually measured by overspeeding the equilibrium run and reading the absorbance of the depleted region of the gradient (the meniscus-depletion method). As shown in
Figure 3a, two of the files have a pronounced slope
to them. The baseline correction term is included in
Figure 3b, and the fit is seen to improve dramatically. The fit is evaluated on the basis of the randomness of the residuals, the magnitude of the residuals (expressed in terms of standard deviations,
i.e., the average absorbance collected at each radial
position), by the relative tightness of the confidence
limits, by the goodness of fit statistic and by check-
15
Residuals
10
5
0
-5
-10
5.90
5.95
6.00
6.05
6.10
6.15
Radius (cm)
8
6
Residuals
4
2
0
-2
-4
6.40
6.45
6.50
6.55
6.60
6.65
Radius (cm)
10
Residuals
5
0
-5
-10
6.85
6.90
6.95
7.00
7.05
7.10
7.15
Radius (cm)
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.
4
Residuals
8
6
4
8
6
4
2
0
-2
-4
-6
-8
2.4
2.2
2.0
1.8
0
Absorbance
Residuals
2
-2
-4
1.6
1.4
1.2
1.0
0.8
-6
0.6
0.4
-8
5.90
5.95
6.00
6.05
6.10
0.2
6.15
5.9
6.1
6.0
Radius (cm)
Radius (cm)
Residuals
7
6
5
4
6
4
2
0
-2
-4
-6
1.6
2
1.4
1
1.2
0
Absorbance
Residuals
3
-1
-2
1.0
0.8
-3
0.6
-4
0.4
-5
0.2
6.40
6.45
6.50
6.55
6.60
6.65
6.4
Radius (cm)
6.5
6.6
Residuals
Radius (cm)
4
3
4
2
0
-2
-4
-6
2
0.8
0
-1
0.6
Absorbance
Residuals
1
-2
-3
0.4
-4
0.2
-5
6.85
6.90
6.95
7.00
7.05
7.10
7.15
0.0
Radius (cm)
6.9
7.0
7.1
Radius (cm)
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 monomerdimer fit of Figure 3b.
5
ing some of the fit parameters for physical significance. The fitted values for the baseline offset in
this example were confirmed by meniscus depletion
at 45,000 rpm for about 6 hours.
The values estimated for the association constant are presented in Table 1. The literature values
for the association constant are shown to vary from
lot to lot. This has been attributed to incomplete
participation of the monomer in the equilibrium
(incompetent monomer); with further purification,
there is an increase in the association constant and
more consistent readings.(4) At this point the correct
model appears to be a monomer-dimer system,
although higher order models should also be
evaluated.
Step 4. α-Chymotrypsin was modeled as a
monomer-trimer system, and as shown in Figure 4,
the fit is markedly worse than for the simple dimerization. The residuals are nonrandom and the magnitude of the residuals is high. The way in which the
data dips away from the best-fit curve indicates
that fits to higher order systems will result in even
worse fits.
The conclusion based on this brief exercise is
that α-chymotrypsin at pH 4.0 behaves as a reversible monomer-dimer self-associating system.
Table 1. The Monomer-Dimer Self-Association Constants for α-Chymotrypsin1
Concentrations/
Buffer System
Kabs2
95% Confidence
Limits
Kconc3
(× 10-3 L/mol)
Klit4
67.3
44.4
35.7
14.9
27.4
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
1
The sedimentation equilibrium gradients were curve-fit using multiple data files and the XL-A Data
Analysis Software.
2 The association constant as estimated from a best-fit curve in terms of absorbance units using
= 0.736 mL/g and ρ = 1.001 g/mL.
v
3 The association constant converted into units of M-1 using ε = 44,064 L/mol-cm.
4 Different literature values for the association constant (determined at pH 4.4), reflecting lot-to-lot variation.
6
Residuals
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 αchymotrypsin. I. pH dependence in the acid region. Biochemistry 10, 1609-1616 (1971)
3. Aune, K. C., Goldsmith, L. C., Timasheff, S. N.
Dimerization of α-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.
20
10
0
-10
-20
-30
-40
2.6
2.4
2.2
2.0
Absorbance
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
5.9
6.1
6.0
Residuals
Radius (cm)
20
10
0
-10
-20
-30
1.6
1.4
Absorbance
1.2
1.0
0.8
0.6
0.4
0.2
6.4
6.5
6.6
Radius (cm)
Residuals
20
10
0
-10
-20
0.8
Absorbance
0.6
0.4
0.2
0.0
6.9
7.0
7.1
Radius (cm)
Figure 4. Shows a relatively poor fit when the same
three data files are fit to a monomer-trimer selfassociating system under similar fit conditions.
7
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