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Experimental Protocol 1. The template can be prepared from a plasmid or amplified by PCR. 2. Preincubate the DNA (10 ng) for 5 min at 37 with DnaA protein (30–300 ng) in 20 l transcription buffer (25 mM HEPES, pH 7.6. 100 mM potassium acetate, 5 mM magnesium acetate, 4 mM DTT, 0.2% Triton X-100, 0.5 mg ml1 BSA). 3. Add RNA polymerase to a concentration of 66 nM, followed by another 5-min incubation. 4. Start transcription by adding nucleotides: [-32P]UTP, unlabeled UTP (60 M), ATP (200 M), and CTP and GTP (400 M each). If single-round transcripts are measured, heparin (200 g+ml1) is added before the addition of nucleotides. 5. Incubate mixtures for 5 to 15 min. The reactions are terminated by adding stop solution and processed as described earlier for DNase I footprints (steps 4–6). Repression of the dnaA promoter by ADP- and ATP-complexed DnaA has been measured using this technique.9
[31] Analysis of Transcription Factor Interactions at Sedimentation Equilibrium By Margaret A. Daugherty and Michael G. Fried Large numbers of proteins participate in the assemblies that regulate and catalyze transcription.1–5 Among methods available for characterizing their interactions, sedimentation equilibrium (SE) ultracentrifugation stands out as a direct and rigorous means of determining molecular masses, interaction stoichiometries, association constants (hence free energies of association), and the influences of low molecular weight effectors, ions, and crowding on the stabilities of protein complexes. The method is useful for characterizing molecules and complexes with masses between 100 and 50 106 Da in a wide variety of buffers and over a very large range of protein concentration. Relatively little material is required, and because 1
A. Hochschild and S. L. Dove, Cell 92, 597 (1998). T. I. Li and R. A. Young, Annu. Rev. Genet. 34, 77 (2000). 3 D. Beckett, J. Mol. Biol. 314, 335 (2001). 4 A. Dvir, J. W. Conaway, and R. C. Conaway, Curr. Opin. Genet. Dev. 11, 209 (2001). 5 G. Gill, Essays Biochem. 37, 33 (2001). 2
METHODS IN ENZYMOLOGY, VOL. 370
Copyright 2003, Elsevier Inc. All rights reserved. 0076-6879/03 $35.00
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the method is nondestructive, samples can often be recovered for further use. The availability of modern instrumentation and the development of improved analysis methods have resulted in an upsurge of interest in SE during the past decade. This article describes the application of SE techniques to the characterization of transcription factors and their interactions. Overview of the Sedimentation Equilibrium Method
The SE technique has been in use since the mid-1920s,6 and the theoretical basis of SE analysis has been under development continually since that time. As a result, the literature describing applications of the technique is vast. Many excellent reviews have been published during the past decade,7–13 and earlier work is also a valuable resource.14–16 This article focuses on three situations that are encountered most frequently in studies of transcription factors: self-association, heteroassociation, and the presence of inactive components. Analysis of such interactions provides crucial information about the role of protein–protein interactions in the assembly of transcription and transcription–regulatory complexes.17 At sedimentation equilibrium, the concentration of species i at a specified position (r) in the solution column is given7,9,12 by ci ðrÞ ¼ ci;0 exp i r2 r02 (1) Here ci,0 is the concentration of the ith species at the reference position r0 (typically close to the meniscus), the reduced molecular weight, i, is equal to Mi ð1 vi Þ!2 =2RT with Mi the molecular weight, vi the partial specific volume (in ml/g), the solvent density (g/ml), ! the rotor angular velocity
6
T. Svedburg and K. O. Pedersen, ‘‘The Ultracentrifuge.’’ Johnson Reprint Corp., New York, 1956. 7 D. K. McRorie and P. J. Voelker, ‘‘Self-associating Systems in the Analytical Ultracentrifuge’’ Beckman Instruments, Inc., Palo Alto, CA, 1993. 8 J. C. Hansen, J. Lebowitz, and B. Demeler, Biochemistry 33, 13155 (1994). 9 T. M. Laue, Methods Enzymol. 259, 427 (1995). 10 A. P. Minton, Progr. Colloid Polym. Sci. 107, 11 (1997). 11 G. Rivas, W. F. Stafford III, and A. P. Minton, Methods 19, 194 (1999). 12 T. M. Laue and W. F. Stafford III, Annu. Rev. Biophys. Biomol. Struct. 28, 75 (1999). 13 J. Lebowitz, M. Lewis, and P. Schuck, Protein Sci. 11, 2067 (2002). 14 H. K. Schachman, ‘‘Ultracentrifugation in Biochemistry’’ Academic Press, New York, 1959. 15 J. W. Williams, ‘‘Ultracentrifugation of Macromolecules’’ Academic Press, New York, 1972. 16 D. C. Teller, Methods Enzymol. 27, 346 (1973). 17 D. F. Senear, J. B. Ross, and T. M. Laue, Methods 16, 3 (1998).
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(¼ rpm/30), R the gas constant (8.314 107 erg mol1 K1), and T the absolute temperature.18 For a solution of several components, the total concentration at position r is given by X cðrÞ ¼ ci;0 exp i r2 r02 (2) i
Here the summation is over all species. This expression embodies both the power and the challenge of the SE method. For solutions with a small number19 of components, direct fitting of Eq. (2) to experimental data can yield relative concentrations and buoyant molecular weights of each component. Often the stoichiometry of an assembly can be deduced from its molecular weight whereas the reference concentrations of sedimenting species (ci,0) can provide data allowing calculation of the equilibrium constant(s) governing the behavior of the system (see later). The challenge comes from the fact that values of the ci,0 and i terms must be extracted by curve fitting. Variants of Eq. (2) must be chosen that are appropriate to the molecular system, but the quality and quantity of data can limit the ability to discriminate between competing models. This problem rapidly becomes severe as the number of species included in the model (and thus the number of parameters to be determined by fitting) is increased. Examples presented here demonstrate these features and some methods for coping with these challenges. Applications
Self-Association Increasing molecular weight with increasing [protein] is indicative of a mass action association.4 For a protein (A) undergoing reversible monomernmer self-association according to nA . An, with apparent association constant Kobs ¼ ½An =½A n ¼ cA n;0 =ðcA;0 Þn , Eq. (2) becomes20 cðrÞ ¼ cA;0 exp ½A ðr2 r02 Þ þ Kobs ðcA;0 Þn exp ½nA ðr2 r02 Þ
18
(3)
Some authors prefer to define the reduced molecular weight as i Mi (1 vi )!2/RT. This is larger by a factor of two than the i defined in the text and requires inclusion of an additional factor of 1/2 in Eq. (1). Often data will contain a concentration-independent component due to a constant absorbance or refractive index difference between reference and sample sectors. For this reason, expressions describing absorbance or refractive index gradients commonly contain an additional constant ‘‘baseline offset’’ term. 19 Typically 3, although under favorable conditions, systems containing more species are amenable to analysis. 20 Because the term cA,0 appears elsewhere in Eq. (3), the substitution of cAn,0 by Kobs(cA,0)n does not change the number of adjustable parameters.
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Fig. 1. Sedimentation equilibrium analyses of E. coli RNA polymerase holoenzyme. Data obtained at 4 in buffers consisting of 40 mM Tris, pH 7.9, 10 mM MgCl2, 0.1 mM dithiothreitol, 5% glycerol, and 300 mM KCl (curve A) or 100 mM KCl (curve B). Sample A was centrifuged at 9000 rpm; data are offset by 0.1 absorbance unit for clarity. The smooth curve is the global fit of the ideal single-species model [Eq. (2) with a single term] to six data sets, of which only one is shown here. The value of Mr returned by this analysis was 454,000 6000, in good agreement with that expected for the enzyme. Sample B was centrifuged at 7000 rpm. The smooth curve is the global fit of the monomer–dimer model [Eq. (3), with n ¼ 2] to six data sets, with monomer molecular weight set at 454,000. In both cases the curve-fitting residuals (upper panels) are small and lack obvious systematic dependence on radial position, demonstrating that the corresponding models are consistent with the mass distributions present in these samples. Data from Dyckman and Fried,30 with permission.
Often, data of reasonable quality will allow assignment of n and Kobs for a two-species model, even if the buoyant molecular weight of the monomer is unknown (Fig. 1). For more than two species, the situation can be more challenging. Shown in Fig. 2A are data for the self-association of the fulllength yeast TATA-binding protein.21,22 Here, the model that consistently fit the data, over a wide range of [salt], pH, temperature, and [protein], was one in which monomers, tetramers, and octamers were in equilibrium23: cðrÞ ¼ cTBP;0 exp ½TBP ðr2 r02 Þ þ K14 ðcTBP;0 Þ4 exp ½4TBP ðr2 r02 Þ
þ k18 ðcTBP;0 Þ8 exp ½8TBP ðr2 r02 Þ
21
(4)
M. A. Daugherty, M. Brenowitz, and M. G. Fried, J. Mol. Biol. 285, 1389 (1999). M. A. Daugherty, M. Brenowitz, and M. G. Fried, Biochemistry 39, 4869 (2000). 23 This model assumes that TBP (and hence vTBP) is unchanged by self-association. If is allowed to float, there are four adjustable parameters. This can be reduced to three if the value of is fixed at the known monomer molecular weight. 22
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Fig. 2. (A) Yeast TATA-binding protein at sedimentation equilibrium. The buffer contained 20 mM HEPES/KOH, pH 7.9, 1 mM EDTA, 1 mM dithiothreitol (DTT) plus KCl as indicated. Rotor speed: 16,000 rpm. Solid lines represent global fits of Eq. (4) to data sets obtained with three starting concentrations of TBP and two rotor speeds (16,000 and 24,000 rpm). In both cases the curve-fitting residuals (upper panels) are small and appear to be distributed randomly, demonstrating that the monomer–tetramer–octamer model is consistent with the mass distributions present in these samples. (B) Dependence of the mole fractions of monomeric, tetrameric, and octameric TBP species on TBP concentration and
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The tetrameric species was a minor component under all conditions examined; however, at low temperature, where [tetramer] was low (mole fraction 0.2; Fig. 2B), satisfactory fits were also obtained with a monomer–octamer association model [Eq. (4) without the middle term]. Models containing a term for dimeric TBP returned concentrations for this species that were indistinguishable from zero under all conditions tested.21,22 This outcome is intriguing because some TBP and related TFIID preparations have been shown to form dimers.24–26 These examples demonstrate the challenge of distinguishing between competing models and the importance of testing several sets of experimental conditions (T, [protein], [salt], etc.) before concluding that a particular association mechanism is correct. The equilibrium constants given in Eqs. (3) and (4) are in the concentration units (absorbance, refractive index difference) appropriate to the detection system of the centrifuge. Equation (5) converts absorbance-scaled values of Kobs to the more familiar molar concentration scale: ! en1 ln1 (5) Kmolar ¼ Kobs n Here n is the association stoichiometry and e and l are the molar extinction coefficient and the optical path length, respectively.27 For refractive index-scaled values, the corresponding equation is9 n1 n1 MA YT Kmolar ¼ Kobs (6) n CT Here MA is the monomer molecular weight, n is the assembly stoichiometry, and YT/CT is the conversion factor relating signal-to-weight concentration.28 24
R. A. Coleman, A. K. P. Taggart, L. R. Benjamin, and B. F. Pugh, J. Biol. Chem. 270, 13842 (1995). 25 A. K. P. Taggart and B. F. Pugh, Science 272, 1331 (1996). 26 K. M. Campbell, R. T. Ranallo, L. A. Stargell, and K. J. Lumb, Biochemistry 39, 2633 (2000). 27 Two optical path lengths are common: that of standard cells is 1.2 cm and that of short optical path cells is 0.3 cm. 28 For proteins, the refractive increment dn/dc is equivalent to 3.33 fringes ml/mg using light of 670 nm and a 1.2-cm optical path length4.
temperature. The buffer contained 20 mM HEPES/KOH, pH 7.9, 1 mM EDTA, 1 mM DTT, and 120 mM KCl. Mole fractions of the monomer (solid line), tetramer (dashed line), and octamer (dotted line) were calculated using association constants determined at sedimentation equilibrium. Data from Daugherty et al.,22 with permission.
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Heteroassociation For two different proteins A and B associating reversibly according to the mechanism nA + mB . AnBm, with apparent association constant Kobs ¼ ½An Bm =½A n ½B m , Eq. (2) becomes11 cðrÞ ¼ cA;0 exp ½A ðr2 r02 Þ þ cB;0 exp ½B ðr2 r02 Þ
þ cAB;0 exp ½AB ðr2 r02 Þ
(7)
Here there is one term for each molecular species, and the parameters characterizing the complex are cAB;0 ¼ Kobs ðcA;0 Þn ðcB;0 Þm and AB ¼ ðnMA þ mMB Þð1 vAB Þ!2 =2RT. The partial specific volume of the complex is usually approximated by the weight average: vAB ¼
nMA vA þ mMB vB nMA þ mMB
(8)
This is equivalent to the assumption that complex formation is not accompanied by a significant change in the partial specific volumes of the components. This is a robust assumption, as shown by the accuracy with which v may be estimated from amino acid composition for many proteins.29 However, if highly accurate measurement of MAB is required, experimental determination of v is worthwhile. A method for measuring v is discussed later. Under favorable conditions, the stoichiometry of a complex can be inferred from its molecular weight. For example, Fig. 3 shows data for the association of the Escherichia coli CAP (Mr 47,238/dimer) and lac repressor (Mr 154,520/tetramer) proteins. The fit to Eq. (7) returned Mr (complex) 240,800 14,100, a value indistinguishable from that predicted for a 2CAP dimer:1 repressor tetramer complex (Mr 248,996). The small residuals attest to the compatibility of this model to data, although their upward deviation at the bottom of the cell suggests that higher molecular weight species can form at very high [protein]. Inference of stoichiometry from molecular weight is more difficult when the interacting partners have similar Mr (in such a case, for example, the trimolecular complexes A2B and AB2 would have indistinguishable molecular weights). It is also difficult when one interacting partner is so much larger than the other that the difference in Mr(complex) expected for two possible stoichiometries is smaller than the uncertainty in the measured value of Mr. Thus, data30 for the binding of E. coli CAP (Mr 47,238/ 29
H. Durschlag, in ‘‘Thermodynamic Data for Biochemistry and Biotechnology’’ (H.-J. Hinz, ed.), p. 45. Springer-Verlag, Berlin, 1986. 30 D. Dyckman and M. G. Fried, J. Biol. Chem. 277, 19064 (2002).
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Fig. 3. Association of E. coli CAP and lac repressor proteins. A solution containing a 2.4 106 M CAP dimer and a 1.2 106 M lac repressor tetramer was centrifuged to equilibrium at 8000 rpm and 4 . The smooth curve is the least-squares fit of Eq. (7) to data, with Mr values of CAP and lac repressor fixed, and the vcomplex given by the weight average of the constituent proteins. This analysis gave Mr (complex) 240,820 14,140, consistent with a 2 CAP dimer:1 repressor tetramer complex. Data from Fried and Daugherty,36 with permission.
dimer) with RNA polymerase holoenzyme (Mr 455,000) gave Mr(complex) 1,020,000 57,400. The uncertainty in this value prevents direct inference of the stoichiometry. In such cases, a continuous variation experiment can often specify the stoichiometry or narrow the range of alternatives. Continuous variation (Job) analyses31 allow determination of the ratio of reactants that maximize product formation at constant total [reactant]. For complexes in which one molecule of type A is bound by several of B, this ‘‘optimal combining ratio’’ is equivalent to stoichiometry. For complexes containing several molecules of each type (AnBm), the stoichiometry is a multiple of the optimal combining ratio. This experiment requires a series of samples in which the sum of protein concentrations is constant, but the mole fraction of each protein varies between 0 and 1. These are obtained conveniently from stock solutions of each protein adjusted to the same molar concentration. Samples are prepared by combining different volumes of each stock, maintaining a constant final volume. Analysis of SE data using Eq. (7) provides a measure of the concentration of complex in each sample; this value is maximized at the optimal combining ratio. For the CAP–RNA polymerase system described earlier, the yield of complex 31
C. Y. Huang, Methods Enzymol. 87, 509 (1982).
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Fig. 4. Continuous variation (Job) plot for the CAP–RNA polymerase interaction. (Bottom) Dependence of normalized [complex] ( 95% confidence limits) on mole fraction of CAP. Normalized [complex] is the concentration of complex in a given sample divided by that in the sample with the greatest amount of association. The total protein concentration was fixed ([CAP] þ [RNA polymerase] ¼ 7.2 107M), but the mole fraction was allowed to vary as indicated. (Top) Molecular weights ( 95% confidence limits) returned by these analyses. The horizontal line indicates the molecular weight expected for a 2:2 CAP:polymerase complex (Mr 1.004 106). Data from Dyckman and Fried,30 with permission.
is maximized at a CAP mole fraction 0.5 [equivalent to a 1:1 combining ratio (Fig. 4, bottom) panel]. The constant molecular weight at all combining ratios is a characteristic of mechanisms in which only one kind of complex is formed (Fig. 4, top). Because the molecular weight of the complex is slightly more than twice that of the RNA polymerase monomer, the stoichiometry of the complex32 must be 2:2. For heteroassociation reactions, the conversion of apparent equilibrium constants from absorbance units to the molar concentration scale is shown as n m e e Kmolar ¼ Kobs A B lnþm1 (9) eA n B m Here Kobs is the equilibrium constant in absorbance units, n and m are the stoichiometries of species A and B, respectively, and eA, eB, and eAnBm 32
If the stoichiometry can be established from Mr and/or combining ratio, reanalysis of original data using a model in which monomer molecular weights and stoichiometries are fixed can allow the extraction of Kobs while minimizing the number of adjustable species in the fit.
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are the molecular extinction coefficients of species A, B, and the complex. If extinction coefficients do not change with association, eAn Bm ¼ neA þ meB . The conversion of apparent equilibrium constants from refractive index units (fringes) to the molar concentration scale is given by nþm1 ðMA Þn ðMB Þm YT Kmolar ¼ Kobs (10) MAn Bm CT As before, YT/CT is the conversion factor relating signal-to-weight concentration. Noninteracting components Equation (2) can describe independently sedimenting species as well as ones that associate. This is valuable, as it is unusual to find samples that are both highly pure and 100% active. An interacting system containing an additional independent species might be described as nA + mB + C . AnBm þ C; here component C can be an inactive fraction of one of the interacting species (A or B), or an unrelated contaminant. At sedimentation equilibrium the concentration distribution is given by cðrÞ ¼ cA;0 exp ½A ðr2 r02 Þ þ cB;0 exp ½B ðr2 r02 Þ þ cC;0 exp ½C ðr2 r02 Þ þ cAB;0 exp ½AB ðr2 r02 Þ
(11)
As described earlier, cAB;0 ¼ Kobs ðcA;0 Þn ðcB;0 Þm , but cC,0 does not depend33 on cA,0 or cB,0. If species C is an inactive fraction of component A, for example, analysis of SE data according to a model in which all components equilibrate [e.g., Eq. (7)] will result in values of Kobs that decrease with increasing [A] and with increasing rotor speed.9 Inclusion of a separate term for species C results in values of Kobs that are independent of reactant concentrations and rotor speed, as expected for equilibrium constants.34 As mentioned previously, distinguishing between alternative models becomes more difficult as the number of species increases. When this becomes a problem, data from other techniques (such as sedimentation velocity) can provide valuable guidance in selection of the simplest realistic model of the system.
33
Under dilute solution conditions. In concentrated solutions, steric interactions and other nonideal effects can prevent independent sedimentation. 34 The presence of inactive species increases the number of adjustable terms in the sedimentation equation that must be evaluated by fitting. Because acquiring a meaningful fit becomes more difficult as the number of species increases, systems containing several inactive species should be purified further before a detailed SE analysis is attempted.
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Thermodynamic Linkage Often, low molecular weight solutes affect the stabilities of transcription factor assemblies. Such solutes include physiological signals such as cAMP and cGMP but also buffer ions and osmolytes. When this is the case, solute binding and macromolecular assembly are linked thermodynamically.35 Sedimentation equilibrium analysis can provide quantitative information about the stoichiometries and consequences of solute participation in the protein assembly reaction. At the relatively slow rotor speeds appropriate for SE analyses of proteins, the concentrations of low molecular weight solutes are virtually unchanged from meniscus to the bottom of the solution column. This allows one to treat macromolecular equilibria as if they are taking place at a constant concentration of the low molecular weight solute. An example of this is analysis of the cAMP-dependent association of the E. coli CAP and lac repressor proteins, for which the mechanism 2 CAP þ nðcAMPÞ þ lac repressor.CAP2 cAMPn lac repressor (12) appears to operate.36 Here the equilibrium constant K ¼ ½ðCAP2 cAMPn Þ lac repressor =½cAMP n ½CAP 2 ½lac repressor], and the observable macromolecular equilibrium quotient Kobs ¼ ½ðCAP2 cAMPn Þ lac repressor =½CAP 2 ½lac repressor . cAMP stoichiometry37 can be evaluated using the relation @ log Kobs ¼n @ log ½cAMP
(13)
A graph of log Kobs as a function of log [cAMP] is shown in Fig. 5. The slope (1.8 0.3) indicates that 2 equivalents of cAMP are bound for each CAP2repressor complex formed. While these data do not specify the distribution of cAMP within the complex, in the absence of repressor, native CAP dimers bind one equivalent of cAMP over the concentration range 0 < [cAMP] 20 M spanned in this experiment.38 It is thus possible to speculate that each molecule of CAP in the (CAP2cAMP2)repressor complex binds one molecule of cAMP.
35
J. Wyman and S. J. Gill, ‘‘Binding and Linkage,’’ University Science Books, Mill Valley, CA, 1990. 36 M. G. Fried and M. A. Daugherty, J. Biol. Chem. 276, 11226 (2001). 37 For the reaction shown in Eq. (12), net cAMP uptake will yield n > 0, net release n < 0, and a cAMP-independent reaction, n ¼ 0. 38 T. Takahashi, B. Blazy, and A. Baudras, Biochemistry 19, 5124 (1980).
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Fig. 5. Dependence of log Kobs on log [cAMP] in formation of the CAP2cAMPnlac repressor complex. Values of log Kobs measured in the presence of cAMP are indicated by filled symbols. The horizontal line gives the value of log Kobs at [cAMP] ¼ 0. Error bars represent 95% confidence limits of plotted values. The slope (1.8 0.3) is a measure of the number (n) of cAMP molecules bound per molecule of complex formed. Data from Fried and Daugherty,36 with permission.
Temperature Dependence of Association Because the temperature of samples can be controlled with reasonable precision (typically 0.1 ), it is sometimes possible to extract thermodynamic information from the temperature dependence of equilibrium constants measured at sedimentation equilibrium.39 At any temperature, changes in standard free energy, enthalpy, and entropy that accompany a protein interaction are described by RT ln Kobs ¼ Go ¼ Ho TSo . Hydrophobic interactions, which remove large amounts of nonpolar surface from water, are characterized by positive values of Ho and So and negative values of Cop , the heat capacity difference.40 A nonzero standard heat capacity difference contributes to enthalpy and entropy differences as shown by Ho ¼ Ho Cop; ðT Þ T o o o S ¼ S Cp; ln
(14) (15)
Here, is a reference temperature and So and Ho are the enthalpy and entropy differences at that temperature. The temperature dependence of the association constant can be written41 39
For accurate measurements, the operating temperature of the centrifuge should be calibrated as described by S. Liu and W. F. Stafford III, Anal. Biochem. 224, 199 (1995). 40 R. S. Spolar, J.-H. Ha, and M. T. Record, Jr., Proc. Natl. Acad. Sci. USA 86, 8382 (1989). 41 E. C. W. Clarke and D. N. Glew, Trans. Faraday Soc. 62, 539 (1966).
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transcription factors at sedimentation equilibrium
R ln Kobs ¼
Go 1 1 T 1 þ ln þ Ho þ Cop; T T
361 (16)
An example of a graph of RlnKobs against temperature is shown in Fig. 6. Nonlinearity in such plots is evidence of nonzero values of Cop; . One caveat to the use of SE in investigating association thermodynamics is that analytical ultracentrifuges are typically limited to temperatures between 4 and 40 . This narrow range can result in inaccurate estimates of van’t Hoff enthalpy and heat capacity changes. An offsetting advantage is that data for individual association reactions can often be obtained in solutions in which more than one interaction is taking place (e.g., the monomer–tetramer–octamer association of TATA-binding protein; Fig. 2). Thus, while the SE approach may not be appropriate for all systems, the thermodynamic information that can be obtained under favorable conditions is often unique and is generally complementary to that provided by calorimetry. Sample Preparation
Volume The most widely used cells are the six-channel short-column cell, containing three samples and three reference solutions, and the two-channel long-column cell, holding one sample and one reference solution. For most purposes, centerpieces with a thickness of 12 mm are used, although a twochannel centerpiece with 3 mm thickness is also available. Six-channel,
Fig. 6. Temperature dependence of yeast TATA-binding protein self-association. Data for the overall monomer–octamer reaction is shown. Smooth curves are fits of Eq. (16) to data using ¼ 303 K (30 ) as reference temperature. Binding reactions were carried out in 20 mM HEPES/KOH, pH 7.9, 1 mM EDTA, 1 mM dithiotrietol supplemented with 120 (&), 300 (d), and 1 (m) M KCl. Data from Daugherty et al.,22 with permission.
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12-mm cells accommodate samples 120 l, whereas two-channel, 12-mm cells accept samples of 450 l. Much smaller sample volumes have been used successfully in both cell types, and techniques for doing so have been described.9 In addition, special very short-column centerpieces, requiring very small sample volumes, have been described.9,42,43 Short-column experiments take significantly less time to reach sedimentation equilibrium than long-column experiments.9,44 However, long-solution columns provide many more data points in a single scan and can allow a single scan to span a wider range of concentrations than the short-column format. These features are of value in data analysis, especially when samples are heterogeneous. Sample Concentration For experiments carried out with absorbance detection, the optimal signal/noise is obtained between 0 and 1 AU at the selected wavelength. To obtain a concentration gradient that spans that range, the starting absorbances are typically between 0.1 and 0.35. Studies that require higher concentrations can be performed without exceeding the optimal range of the detector by tuning the monochromator away from the max of the sample. Studies at very low concentrations may require greater sensitivity than that available with intrinsic chromophores. Such work is often possible if the molecule(s) of interest is labeled with an extrinsic chromophore that has a high extinction coefficient. In addition, labeling with an extrinsic chromophore can allow an individual protein to be detected in complex mixtures where the number of species detectable at 215 or 280 nm is large. An elegant example of this approach takes advantage of the changes in absorbance spectra that result from the biosynthetic substitution of tryptophan with analogues such as 5-hydroxytryptophan.17,45 Methods for the simultaneous analysis of multiple components with differing absorbance spectra have been described.46 For experiments carried out with interference detection, protein concentrations in the range of 0.1–5 mg/ml are appropriate. 42
J. J. Correia and D. A. Yphantis, in ‘‘Analytical Ultracentrifuation in Biochemistry and Polymer Science’’ (S. E. Harding, A. J. Rowe, and J. C. Horton, eds.), p. 237. Royal Society of Chemistry, Cambridge, 1992. 43 T. M. Laue, Technical Information, Vol. DS-835, Beckman Instruments, Palo Alto, CA, 1992. 44 G. Ralston, ‘‘Introduction to Analytical Ultracentrifugation.’’ Beckman Instruments, Inc, Fullerton, 1993. 45 T. M. Laue, D. F. Senear, S. Eaton, and J. B. Ross, Biochemistry 32, 2469 (1993). 46 M. S. Lewis, R. I. Shrager, and S. J. Kim, in ‘‘Modern Analytical Ultracentrifugation’’ (T. M. Schuster and T. M. Laue, eds.), p. 94, Birkhauser, Boston, 1994.
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Sample Integrity As described earlier, classical sedimentation equilibrium analysis is limited by the number of species that can be resolved. For this reason, the best samples are of high purity. Because centrifuge runs can be timeconsuming, we recommend prerun verification that sample proteins are intact, in the correct state of posttranslational modification, and exhibit appropriate activities (e.g., DNA binding, enzyme activity). Many proteins have the potential to form more than one noncovalent complex (e.g., a specific assembly and a nonspecific aggregate). As these are not usually detectable by SDS–PAGE, but are detectable readily by sedimentation velocity analysis,13,47–49 measurement of the samples’ s value distribution can provide guidance in the choice of models for analysis of SE data. Postrun analyses (e.g., enzyme assays, SDS–PAGE) are also valuable, as they allow detection of any changes in sample integrity during sedimentation. After a run, samples are recovered easily through the filling holes using a micropipettor equipped with a gel-loading tip or a 1-ml syringe fitted with a capillary tube. For cells that lack filling holes (e.g., early vintage six-channel cells), samples can be recovered by careful removal of one window. Absence of Nonprotein Contaminants Nonprotein sample components (e.g., nucleic acids, starches, dextrans, and polyacrylamide) can complicate the sedimentation equilibrium analysis of a protein by contributing to the absorbance (or refractive index) gradient, by binding the proteins(s) of interest, or affecting activity coefficients through macromolecular crowding. If these components are not part of the system under study, it is worthwhile taking steps to ensure that they are not present in significant concentrations. Buffer Because sedimentation equilibrium experiments can be performed under a very wide range of solution conditions, it is often possible to tailor buffer compositions to the needs of a given system. However, some buffer properties can affect the outcome of a SE experiment. Important among these are transparency, salt concentration, and ability to suppress proteolysis. Data acquisition requires the transmission of light so buffer components that absorb strongly at the experimental wavelength(s) should be 47
P. Schuck, M. A. Perugini, N. R. Gonzales, G. J. Howlett, and D. Schubert, Biophys. J. 82, 1096 (2002). 48 J. J. Correia, Methods Enzymol. 321, 81 (2000). 49 L. M. Carruthers, V. R. Schirf, B. Demeler, and J. C. Hansen, Methods Enzymol. 321, 66.
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avoided. Proteins far from their isoelectric pH values can have significant charge; electrostatic interactions can bias the equilibrium concentration gradient, resulting in incorrect apparent molecular weights. For moderately charged proteins, concentrations of 1:1 salts in the 100–300 mM range are often enough to suppress this effect. Attainment of sedimentation equilibrium can require many hours, giving proteolytic contaminants ample time to degrade samples. The treatment of samples with phehylmethylsulfonyl fluoride and/or inclusion of low molecular weight protease inhibitors (e.g., aprotinin, leupeptin) can minimize this problem. A critical requirement for analysis of multicomponent systems is that all macromolecules must be at exchange (dialysis) equilibrium with the buffer.50 A typical dialysis consists of two or more changes of solvent in a 1000:1 solvent- to-sample ratio. The dialysate is then used in the reference sectors of the centerpieces. Alternatively, samples can be equilibrated rapidly with buffer using centrifugal gel-filtration techniques.51,52 Rotor Speed and Duration of Run If a molecular weight range can be predicted for the system, a useful estimate of rotor speeds can be obtained from data of Chervenka,53 reprinted more recently in the manual by McRorie and Voelker.7 These speeds are especially useful for experiments with long solution columns, although we often find that they are slower than optimal for short-column experiments. However, they are a valuable point for departure, as data obtained at several rotor speeds can allow discrimination of polydispersity and reversible association.9 In addition, data sets obtained over a range of speeds can be usefully combined in a global analysis using programs such as NONLIN.54 Because attainment of equilibrium following a speed reduction can be slow, multispeed experiments in which rotor speeds increase can take less time to complete than ones in which rotor speeds decrease. The time needed to reach equilibrium is proportional to the square of the solution column height.9,12 Thus short-column experiments reach equilibrium much faster than long-column runs. Attainment of equilibrium can be verified by subtracting successive scans.55 In short-column experiments, 50
E. F. Casassa and H. Eisenberg, Adv. Prot. Chem. 19, 287 (1964). H. S. Penefsky, Methods Enzymol. 56, 527 (1979). 52 K. Struhl, in ‘‘Current Protocols in Molecular Biology’’ (F. M. Ausubel et al., eds.), p. 3.4.8, Wiley, New York, 1989. 53 C. H. Chervenka, ‘‘A Manual of Methods for the Analytical Ultracentrifuge,’’ Spinco Division, Beckman Instruments, Palo Alto, CA 1969. 54 M. L. Johnson, J. J. Correia, D. A. Yphantis, and H. R. Halvorson, Biophys. J. 36, 575 (1981). 55 The program MATCH by D. Yphantis and J. Lary (available on the web at ftp:// rasmb.bbri.org) facilitates this comparison. 51
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scans separated by 1 h are an adequate test; for intermediate- and longcolumn experiments scans should be separated by at least 6 h. Very large molecules and ones undergoing slow association reactions can require many hours to reach equilibrium; for these systems, longer intervals are sometimes needed. Under favorable conditions, the time required to reach equilibrium can be reduced by ‘‘overspeeding’’ (a brief period at a higher speed followed by gentle deceleration to the final equilibrium speed).56 Determination of Buffer Density The buffer density contributes to the buoyancy term, 1 v, for each sedimenting species. Because dilute aqueous buffers have densities close to 1 g/ml and typical, unmodified proteins have v 0.73 ml/g, a 1% error in solvent density results in 3% error in Mr. Thus, buffer density must be known with accuracy. The density of a solution can be estimated by summing density increments for each component.11 Extensive published data allow accurate calculation of densities for dilute aqueous buffers with common components.7,29,57 This procedure is automated by the program SEDNTERP (Table I). However, oscillating densimeters allow solvent densities to be measured at high precision over the range of temperatures accessible to the centrifuge.58 Because these values represent the actual densities of the buffer preparations, they should be used wherever possible. Determination of Partial Specific Volume An accurate value of the partial specific volume (v) is also needed for evaluation of the buoyancy term. For proteins without posttranslation modification and/or nonprotein cofactor, v can be estimated from the amino acid composition according to the relation59 X N i M i vi i vest ¼ X (17) Ni Mi i
where Ni is the number of residues of type i, Mi is the component molecular weight (amino acid molecular weight minus 18), and vi is the tabulated 56
D. E. Roark, Biophys. Chem. 5, 185 (1976). T. M. Laue, B. D. Shah, T. M. Ridgeway, and S. L. Pelletier, in ‘‘Analytical Ultracentrifugation in Biochemistry and Polymer Science’’ (S. E. Harding, A. J. Rowe, and J. C. Harding, eds.), p. 90. The Royal Society of Chemistry, Cambridge, England, 1992. 58 O. Kratky, H. Leopold, and H. Stabinger, Methods Enzymol. 27, 98 (1973). 59 E. J. Cohn and J. T. Edsall, ‘‘Proteins, Amino Acids and Peptides as Ions and Dipolar Ions,’’ Reinhold, New York, 1943. 57
366
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TABLE I Representative Software for Editing, Display, and Analysis of Sedimentation Equilibrium Dataa Application NONLINb Beckman software SEDEQ f TWOCOMP f Omega Analysisi Ultrascan II
Ultraspin
WinMatch WinREEDIT XLGRAPH SEDNTERPm a
Function
Platform (source)
Analysis of reversibly associating homogeneous systems Analysis of reversibly associating homogenous systems Analysis of up to three independently sedimenting species Analysis of two species that may reversibly self- or heteroassociate Analysis of SE data Nonlinear analysis of SE and sedimentation velocity data; estimation of solution density and protein v Analysis of SE data; 20 models including homogeneous and some heterogeneous systems Test for attainment of equilibrium Editing raw data files from XL-I or XL-A Data graphing and transformation of XL-A and XL-I data Estimation of solution density and protein v
PC,c,d Mac,d DECd PCe PC (DOS)g PC (DOS)h PCd PC, UNIX, LINUXj
PCk
PCc PCc PCd,l PCd,l
The Reversible Associations in Structural and Molecular Biology (RASMB) group provides a web-based resource for researchers interested in the use and advancement of analytical ultracentrifugation (http://www.bbri.org/rasmb). A software archive is maintained at that address. b M. L. Johnson, J. J. Correia, D. A. Y phantis, and H. R. Halvorson, Biophys. J. 36, 575 (1981). c http://vm.uconn.edu/wwwbiotc.uaf.html. d http://www.biochem.uthscsa.edu/auc/ e Available from Beckman Instruments, Inc. f G. Rivas, W. F. Stafford, III, and A. P. Minton, Methods 19, 194 (1999). g ftp://bbri.harvard.edu/rasmb/spin/ms_dos/sedeq-minton/sedeq.doc; ftp://bbri.harvard. edu/rasmb/spin/ms_dos/sedeq-minton/sedeq.exe h ftp://bbri.harvard.edu/rasmb/spin/ms_dos/twocomp-minton/sedeq.doc; ftp://bbri.harvard. edu/rasmb/spin/ms_dos/twocomp-minton/sedeq.exe i G. B. Ralston and M. B. Morris, in ‘‘Analytical Ultracentrifugation in Biochemistry and Polymer Science’’ (S. E. Harding, A. J. Rowe, and J. C. Horton, eds.), p. 243, Royal Society of Chemistry, Cambridge, 1992. j http://www.ultrascan.uthscsa.edu/. k http://www.mrc-cpe.cam.ac.uk/ultraspin/. l http://jphilo.mailway.com/default.htm. m T. M. Laue, B. D. Shah, T. M. Ridgeway, and S. L. Pelletier, in ‘‘Analytical Ultracentrifugation in Biochemistry and Polymer Science’’ (S. E. Harding, A. J. Rowe, and J. C. Harding, eds.), p. 90, The Royal Society of Chemistry, Cambridge, England, 1992.
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partial specific volume. Tabulated values of vi have been published in several places.7,29 Values of vi are tabulated at a reference temperature (fre quently 25 ). If the experimental temperature differs from the reference, the value of vest should be corrected as described.57 Values of vest obtained in this way are frequently accurate to 1–2%. Alternatively, if the sequence molecular weight NiMi is known, the remaining unknown, v, can be estimated from the experimental reduced molecular weight, . While both approaches assume that the component is not undergoing self-association (which can change v), the second method has the advantage that the v estimate is obtained for the exact solution conditions used in the experiment. For proteins of unknown amino acid composition, with substantial posttranslational modification, or nonprotein prosthetic groups, partial specific volume can be measured experimentally by performing parallel experiments in H2O- and D2O-containing buffers. The value of v can be obtained by simultaneous solution of the equations60 Mbouyant; H2 O ¼ Mð1 vH2 O Þ
(18)
Mbouyant; D2 O ¼ Mð1 vD2 O Þ
(19)
in which Mbouyant,H2O and Mbouyant,D2O are the observed buoyant molecular weights and H2O and D2O are the measured densities of the H2O- and D2O-containing buffers.61 Data Analysis Two general approaches have been used for the analysis of SE data. These are graphical methods62–66 and nonlinear least-squares fitting.67 Although each has advantages, graphical methods generally require transformation of data, which can complicate error analysis.68 Direct fitting
60
S. J. Edelstein and H. K. Schachman, Methods Enzymol. 27, 82 (1973). This calculation ignores changes in M that result from H-D exchange, which will depend on the identity of amino acids and nonprotein components of the molecule(s) in question. 62 D. E. Roark and D. A. Yphantis, Ann. N.Y. Acad. Sci. 164, 245 (1969). 63 R. H. Haschemeyer and W. F. Bowers, Biochemistry 9, 435 (1970). 64 G. J. Howlett, Chemtracts-Biochem. Mol. Biol. 11, 950 (1998). 65 D. J. Winzor and P. R. Wills, in ‘‘Modern Analytical Ultracentrifugation: Acquisition and Interpretation of Data for Biological and Synthetic Polymer Systems’’ (T. M. Schuster and T. M. Laue, eds.), p. 66. Birkhauser, Boston, 1994. 66 D. J. Winzor, M. P. Jacobsen, and P. R. Wills, Biochemistry 37, 2226 (1998). 67 M. L. Johnson and M. Straume, in ‘‘Modern Analytical Ultracentrifugation: Acquisition and Interpretation of Data for Biological and Synthetic Polymer Systems’’ (T. M. Schuster and T. M. Laue, eds.), p. 37, Birkhauser, Boston, 1994. 68 M. L. Johnson, Anal. Biochem. 206, 215 (1992). 61
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using functions that relate concentration to radial position avoids this drawback. Criteria for judging the adequacy and consistency of fitting results have been reviewed.67,68 The past decade has seen an upsurge in the development of software for analysis of SE data, and more programs are under development at the time of this writing. A representative list of currently available software is given in Table I. The most powerful programs allow simultaneous fitting of several data sets, with some parameters (e.g., c0,i) considered as local variables evaluated for each data set and some (e.g., Mi, Kobs) considered as global, evaluated for the entire data ensemble. Data collected at several rotor speeds, wavelengths, or starting concentrations can be combined, as can data obtained by absorbance and interference detectors. The analysis of large amounts of data obtained over a wide range of conditions makes it possible to discriminate between competing models describing the sedimentation behavior of moderately complex systems like those described earlier. Prospects The past decade has seen a renaissance in SE analysis, catalyzed by the availability of modern instrumentation and software for data analysis and the increasing availability of biologically important macromolecules. Several developments suggest that this trend will continue. Fluorescence detection69 has lowered the concentration requirements of the SE technique, allowing study of hard-to-acquire macromolecules and equilibria with large association constants. Multiwavelength analysis46 and methods for labeling proteins with chromophoric amino acid analogues (e.g., 5-hydroxytryptophan45,70) allow the simultaneous measurement of several proteins in complex mixtures, as well as the analysis of labeled proteins in the presence of several unlabeled species. Postcentrifugation fractionation methods10,71 allow the detection of components by tracer methods, including direct counting of radioisotopes, radioimmunoassay, and real-time polymerase chain reaction. With these techniques, SE analyses can be carried out on target species at very low concentrations, in highly complex mixtures, and in the presence of high background concentrations of other macromolecules. These developments will make possible the analysis of systems of life-like complexity, spanning the concentration ranges that occur in vivo. 69
T. M. Laue, A. L. Anderson, and B. J. Weber, in ‘‘Ultrasensitive Clinical Laboratory Diagnostics’’ (G. Cohn, ed.), Vol. 2985, p. 196. SPIE, Bellingham, WA, 1997. 70 J. B. Ross, D. F. Senear, E. Waxman, B. B. Kombo, E. Rusinova, Y. T. Huang, W. R. Laws, and C. A. Hasselbacher, Proc. Natl. Acad. Sci. USA 89, 12023 (1992). 71 S. Darawashe and A. P. Minton, Anal. Biochem. 220, 1 (1994).
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Acknowledgments We gratefully acknowledge valuable discussions with Drs. Jack Correia, Michael Johnson, Jacob Lebowitz, and Allen Minton and thank the many people who sent reprints and preprints.
[32] Single-Molecule Studies of DNA Architectural Changes Induced by Regulatory Proteins By Laura Finzi and David Dunlap Single-molecule techniques are now established approaches in the investigation of the molecular mechanisms involved in nucleic acids/protein interactions. The advantages of single-molecule approaches are related to the fact that the observation is performed on just one molecule at a time. This allows the study of the behavior of a molecule as it goes through different conformational states in time, thus revealing states that are averaged out in bulk experiments where unsynchronized molecules are observed simultaneously. The tethered particle motion (TPM) method is a general method used to measure the contour length of a suitably bead-labeled and surface-immobilized DNA fragment. As such, it can potentially be used in a variety of single-molecule kinetics experiments and can yield mechanistic information. It can be used to reveal and monitor, dynamically, events that induce large conformational changes in single DNA molecules, such as those induced by regulatory proteins, or to study the mechanisms by which enzymes move in a directed fashion along DNA (polymerases, helicases, etc). For example, it is now clear that TPM assays can provide much information about the mechanisms by which transcriptional regulatory proteins act.1–3 Furthermore, the effect of supercoiling on DNA/protein interactions or the change in DNA-linking number induced by regulatory proteins can be characterized in a TPM experiment using magnetic tweezers.4,5 This article concentrates on the protocol used to prepare most TPM experiments. This consists mainly of three stages: (i) preparation of DNA constructs suitably labeled with biotin at one end and digoxigenin at the 1
Finzi and Gelles, Science 267, 378, (1995). Lia et al., PNAS 100, 11373–11377 (2003). 3 J. Gelles, personal communication. 4 T. R. Strick, J.-F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette, Science 271, 1835 (1996). 5 T. R. Strick, V. Croquette, and D. Bensimon, Nature 404, 901 (2000). 2
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Experimental Protocol 1. The template can be prepared from a plasmid or amplified by PCR. 2. Preincubate the DNA (10 ng) for 5 min at 37 with DnaA protein (30–300 ng) in 20 l transcription buffer (25 mM HEPES, pH 7.6. 100 mM potassium acetate, 5 mM magnesium acetate, 4 mM DTT, 0.2% Triton X-100, 0.5 mg ml1 BSA). 3. Add RNA polymerase to a concentration of 66 nM, followed by another 5-min incubation. 4. Start transcription by adding nucleotides: [-32P]UTP, unlabeled UTP (60 M), ATP (200 M), and CTP and GTP (400 M each). If single-round transcripts are measured, heparin (200 g+ml1) is added before the addition of nucleotides. 5. Incubate mixtures for 5 to 15 min. The reactions are terminated by adding stop solution and processed as described earlier for DNase I footprints (steps 4–6). Repression of the dnaA promoter by ADP- and ATP-complexed DnaA has been measured using this technique.9
[31] Analysis of Transcription Factor Interactions at Sedimentation Equilibrium By Margaret A. Daugherty and Michael G. Fried Large numbers of proteins participate in the assemblies that regulate and catalyze transcription.1–5 Among methods available for characterizing their interactions, sedimentation equilibrium (SE) ultracentrifugation stands out as a direct and rigorous means of determining molecular masses, interaction stoichiometries, association constants (hence free energies of association), and the influences of low molecular weight effectors, ions, and crowding on the stabilities of protein complexes. The method is useful for characterizing molecules and complexes with masses between 100 and 50 106 Da in a wide variety of buffers and over a very large range of protein concentration. Relatively little material is required, and because 1
A. Hochschild and S. L. Dove, Cell 92, 597 (1998). T. I. Li and R. A. Young, Annu. Rev. Genet. 34, 77 (2000). 3 D. Beckett, J. Mol. Biol. 314, 335 (2001). 4 A. Dvir, J. W. Conaway, and R. C. Conaway, Curr. Opin. Genet. Dev. 11, 209 (2001). 5 G. Gill, Essays Biochem. 37, 33 (2001). 2
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the method is nondestructive, samples can often be recovered for further use. The availability of modern instrumentation and the development of improved analysis methods have resulted in an upsurge of interest in SE during the past decade. This article describes the application of SE techniques to the characterization of transcription factors and their interactions. Overview of the Sedimentation Equilibrium Method
The SE technique has been in use since the mid-1920s,6 and the theoretical basis of SE analysis has been under development continually since that time. As a result, the literature describing applications of the technique is vast. Many excellent reviews have been published during the past decade,7–13 and earlier work is also a valuable resource.14–16 This article focuses on three situations that are encountered most frequently in studies of transcription factors: self-association, heteroassociation, and the presence of inactive components. Analysis of such interactions provides crucial information about the role of protein–protein interactions in the assembly of transcription and transcription–regulatory complexes.17 At sedimentation equilibrium, the concentration of species i at a specified position (r) in the solution column is given7,9,12 by ci ðrÞ ¼ ci;0 exp i r2 r02 (1) Here ci,0 is the concentration of the ith species at the reference position r0 (typically close to the meniscus), the reduced molecular weight, i, is equal to Mi ð1 vi Þ!2 =2RT with Mi the molecular weight, vi the partial specific volume (in ml/g), the solvent density (g/ml), ! the rotor angular velocity
6
T. Svedburg and K. O. Pedersen, ‘‘The Ultracentrifuge.’’ Johnson Reprint Corp., New York, 1956. 7 D. K. McRorie and P. J. Voelker, ‘‘Self-associating Systems in the Analytical Ultracentrifuge’’ Beckman Instruments, Inc., Palo Alto, CA, 1993. 8 J. C. Hansen, J. Lebowitz, and B. Demeler, Biochemistry 33, 13155 (1994). 9 T. M. Laue, Methods Enzymol. 259, 427 (1995). 10 A. P. Minton, Progr. Colloid Polym. Sci. 107, 11 (1997). 11 G. Rivas, W. F. Stafford III, and A. P. Minton, Methods 19, 194 (1999). 12 T. M. Laue and W. F. Stafford III, Annu. Rev. Biophys. Biomol. Struct. 28, 75 (1999). 13 J. Lebowitz, M. Lewis, and P. Schuck, Protein Sci. 11, 2067 (2002). 14 H. K. Schachman, ‘‘Ultracentrifugation in Biochemistry’’ Academic Press, New York, 1959. 15 J. W. Williams, ‘‘Ultracentrifugation of Macromolecules’’ Academic Press, New York, 1972. 16 D. C. Teller, Methods Enzymol. 27, 346 (1973). 17 D. F. Senear, J. B. Ross, and T. M. Laue, Methods 16, 3 (1998).
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(¼ rpm/30), R the gas constant (8.314 107 erg mol1 K1), and T the absolute temperature.18 For a solution of several components, the total concentration at position r is given by X cðrÞ ¼ ci;0 exp i r2 r02 (2) i
Here the summation is over all species. This expression embodies both the power and the challenge of the SE method. For solutions with a small number19 of components, direct fitting of Eq. (2) to experimental data can yield relative concentrations and buoyant molecular weights of each component. Often the stoichiometry of an assembly can be deduced from its molecular weight whereas the reference concentrations of sedimenting species (ci,0) can provide data allowing calculation of the equilibrium constant(s) governing the behavior of the system (see later). The challenge comes from the fact that values of the ci,0 and i terms must be extracted by curve fitting. Variants of Eq. (2) must be chosen that are appropriate to the molecular system, but the quality and quantity of data can limit the ability to discriminate between competing models. This problem rapidly becomes severe as the number of species included in the model (and thus the number of parameters to be determined by fitting) is increased. Examples presented here demonstrate these features and some methods for coping with these challenges. Applications
Self-Association Increasing molecular weight with increasing [protein] is indicative of a mass action association.4 For a protein (A) undergoing reversible monomernmer self-association according to nA . An, with apparent association constant Kobs ¼ ½An =½A n ¼ cA n;0 =ðcA;0 Þn , Eq. (2) becomes20 cðrÞ ¼ cA;0 exp ½A ðr2 r02 Þ þ Kobs ðcA;0 Þn exp ½nA ðr2 r02 Þ
18
(3)
Some authors prefer to define the reduced molecular weight as i Mi (1 vi )!2/RT. This is larger by a factor of two than the i defined in the text and requires inclusion of an additional factor of 1/2 in Eq. (1). Often data will contain a concentration-independent component due to a constant absorbance or refractive index difference between reference and sample sectors. For this reason, expressions describing absorbance or refractive index gradients commonly contain an additional constant ‘‘baseline offset’’ term. 19 Typically 3, although under favorable conditions, systems containing more species are amenable to analysis. 20 Because the term cA,0 appears elsewhere in Eq. (3), the substitution of cAn,0 by Kobs(cA,0)n does not change the number of adjustable parameters.
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Fig. 1. Sedimentation equilibrium analyses of E. coli RNA polymerase holoenzyme. Data obtained at 4 in buffers consisting of 40 mM Tris, pH 7.9, 10 mM MgCl2, 0.1 mM dithiothreitol, 5% glycerol, and 300 mM KCl (curve A) or 100 mM KCl (curve B). Sample A was centrifuged at 9000 rpm; data are offset by 0.1 absorbance unit for clarity. The smooth curve is the global fit of the ideal single-species model [Eq. (2) with a single term] to six data sets, of which only one is shown here. The value of Mr returned by this analysis was 454,000 6000, in good agreement with that expected for the enzyme. Sample B was centrifuged at 7000 rpm. The smooth curve is the global fit of the monomer–dimer model [Eq. (3), with n ¼ 2] to six data sets, with monomer molecular weight set at 454,000. In both cases the curve-fitting residuals (upper panels) are small and lack obvious systematic dependence on radial position, demonstrating that the corresponding models are consistent with the mass distributions present in these samples. Data from Dyckman and Fried,30 with permission.
Often, data of reasonable quality will allow assignment of n and Kobs for a two-species model, even if the buoyant molecular weight of the monomer is unknown (Fig. 1). For more than two species, the situation can be more challenging. Shown in Fig. 2A are data for the self-association of the fulllength yeast TATA-binding protein.21,22 Here, the model that consistently fit the data, over a wide range of [salt], pH, temperature, and [protein], was one in which monomers, tetramers, and octamers were in equilibrium23: cðrÞ ¼ cTBP;0 exp ½TBP ðr2 r02 Þ þ K14 ðcTBP;0 Þ4 exp ½4TBP ðr2 r02 Þ
þ k18 ðcTBP;0 Þ8 exp ½8TBP ðr2 r02 Þ
21
(4)
M. A. Daugherty, M. Brenowitz, and M. G. Fried, J. Mol. Biol. 285, 1389 (1999). M. A. Daugherty, M. Brenowitz, and M. G. Fried, Biochemistry 39, 4869 (2000). 23 This model assumes that TBP (and hence vTBP) is unchanged by self-association. If is allowed to float, there are four adjustable parameters. This can be reduced to three if the value of is fixed at the known monomer molecular weight. 22
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Fig. 2. (A) Yeast TATA-binding protein at sedimentation equilibrium. The buffer contained 20 mM HEPES/KOH, pH 7.9, 1 mM EDTA, 1 mM dithiothreitol (DTT) plus KCl as indicated. Rotor speed: 16,000 rpm. Solid lines represent global fits of Eq. (4) to data sets obtained with three starting concentrations of TBP and two rotor speeds (16,000 and 24,000 rpm). In both cases the curve-fitting residuals (upper panels) are small and appear to be distributed randomly, demonstrating that the monomer–tetramer–octamer model is consistent with the mass distributions present in these samples. (B) Dependence of the mole fractions of monomeric, tetrameric, and octameric TBP species on TBP concentration and
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The tetrameric species was a minor component under all conditions examined; however, at low temperature, where [tetramer] was low (mole fraction 0.2; Fig. 2B), satisfactory fits were also obtained with a monomer–octamer association model [Eq. (4) without the middle term]. Models containing a term for dimeric TBP returned concentrations for this species that were indistinguishable from zero under all conditions tested.21,22 This outcome is intriguing because some TBP and related TFIID preparations have been shown to form dimers.24–26 These examples demonstrate the challenge of distinguishing between competing models and the importance of testing several sets of experimental conditions (T, [protein], [salt], etc.) before concluding that a particular association mechanism is correct. The equilibrium constants given in Eqs. (3) and (4) are in the concentration units (absorbance, refractive index difference) appropriate to the detection system of the centrifuge. Equation (5) converts absorbance-scaled values of Kobs to the more familiar molar concentration scale: ! en1 ln1 (5) Kmolar ¼ Kobs n Here n is the association stoichiometry and e and l are the molar extinction coefficient and the optical path length, respectively.27 For refractive index-scaled values, the corresponding equation is9 n1 n1 MA YT Kmolar ¼ Kobs (6) n CT Here MA is the monomer molecular weight, n is the assembly stoichiometry, and YT/CT is the conversion factor relating signal-to-weight concentration.28 24
R. A. Coleman, A. K. P. Taggart, L. R. Benjamin, and B. F. Pugh, J. Biol. Chem. 270, 13842 (1995). 25 A. K. P. Taggart and B. F. Pugh, Science 272, 1331 (1996). 26 K. M. Campbell, R. T. Ranallo, L. A. Stargell, and K. J. Lumb, Biochemistry 39, 2633 (2000). 27 Two optical path lengths are common: that of standard cells is 1.2 cm and that of short optical path cells is 0.3 cm. 28 For proteins, the refractive increment dn/dc is equivalent to 3.33 fringes ml/mg using light of 670 nm and a 1.2-cm optical path length4.
temperature. The buffer contained 20 mM HEPES/KOH, pH 7.9, 1 mM EDTA, 1 mM DTT, and 120 mM KCl. Mole fractions of the monomer (solid line), tetramer (dashed line), and octamer (dotted line) were calculated using association constants determined at sedimentation equilibrium. Data from Daugherty et al.,22 with permission.
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Heteroassociation For two different proteins A and B associating reversibly according to the mechanism nA + mB . AnBm, with apparent association constant Kobs ¼ ½An Bm =½A n ½B m , Eq. (2) becomes11 cðrÞ ¼ cA;0 exp ½A ðr2 r02 Þ þ cB;0 exp ½B ðr2 r02 Þ
þ cAB;0 exp ½AB ðr2 r02 Þ
(7)
Here there is one term for each molecular species, and the parameters characterizing the complex are cAB;0 ¼ Kobs ðcA;0 Þn ðcB;0 Þm and AB ¼ ðnMA þ mMB Þð1 vAB Þ!2 =2RT. The partial specific volume of the complex is usually approximated by the weight average: vAB ¼
nMA vA þ mMB vB nMA þ mMB
(8)
This is equivalent to the assumption that complex formation is not accompanied by a significant change in the partial specific volumes of the components. This is a robust assumption, as shown by the accuracy with which v may be estimated from amino acid composition for many proteins.29 However, if highly accurate measurement of MAB is required, experimental determination of v is worthwhile. A method for measuring v is discussed later. Under favorable conditions, the stoichiometry of a complex can be inferred from its molecular weight. For example, Fig. 3 shows data for the association of the Escherichia coli CAP (Mr 47,238/dimer) and lac repressor (Mr 154,520/tetramer) proteins. The fit to Eq. (7) returned Mr (complex) 240,800 14,100, a value indistinguishable from that predicted for a 2CAP dimer:1 repressor tetramer complex (Mr 248,996). The small residuals attest to the compatibility of this model to data, although their upward deviation at the bottom of the cell suggests that higher molecular weight species can form at very high [protein]. Inference of stoichiometry from molecular weight is more difficult when the interacting partners have similar Mr (in such a case, for example, the trimolecular complexes A2B and AB2 would have indistinguishable molecular weights). It is also difficult when one interacting partner is so much larger than the other that the difference in Mr(complex) expected for two possible stoichiometries is smaller than the uncertainty in the measured value of Mr. Thus, data30 for the binding of E. coli CAP (Mr 47,238/ 29
H. Durschlag, in ‘‘Thermodynamic Data for Biochemistry and Biotechnology’’ (H.-J. Hinz, ed.), p. 45. Springer-Verlag, Berlin, 1986. 30 D. Dyckman and M. G. Fried, J. Biol. Chem. 277, 19064 (2002).
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Fig. 3. Association of E. coli CAP and lac repressor proteins. A solution containing a 2.4 106 M CAP dimer and a 1.2 106 M lac repressor tetramer was centrifuged to equilibrium at 8000 rpm and 4 . The smooth curve is the least-squares fit of Eq. (7) to data, with Mr values of CAP and lac repressor fixed, and the vcomplex given by the weight average of the constituent proteins. This analysis gave Mr (complex) 240,820 14,140, consistent with a 2 CAP dimer:1 repressor tetramer complex. Data from Fried and Daugherty,36 with permission.
dimer) with RNA polymerase holoenzyme (Mr 455,000) gave Mr(complex) 1,020,000 57,400. The uncertainty in this value prevents direct inference of the stoichiometry. In such cases, a continuous variation experiment can often specify the stoichiometry or narrow the range of alternatives. Continuous variation (Job) analyses31 allow determination of the ratio of reactants that maximize product formation at constant total [reactant]. For complexes in which one molecule of type A is bound by several of B, this ‘‘optimal combining ratio’’ is equivalent to stoichiometry. For complexes containing several molecules of each type (AnBm), the stoichiometry is a multiple of the optimal combining ratio. This experiment requires a series of samples in which the sum of protein concentrations is constant, but the mole fraction of each protein varies between 0 and 1. These are obtained conveniently from stock solutions of each protein adjusted to the same molar concentration. Samples are prepared by combining different volumes of each stock, maintaining a constant final volume. Analysis of SE data using Eq. (7) provides a measure of the concentration of complex in each sample; this value is maximized at the optimal combining ratio. For the CAP–RNA polymerase system described earlier, the yield of complex 31
C. Y. Huang, Methods Enzymol. 87, 509 (1982).
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Fig. 4. Continuous variation (Job) plot for the CAP–RNA polymerase interaction. (Bottom) Dependence of normalized [complex] ( 95% confidence limits) on mole fraction of CAP. Normalized [complex] is the concentration of complex in a given sample divided by that in the sample with the greatest amount of association. The total protein concentration was fixed ([CAP] þ [RNA polymerase] ¼ 7.2 107M), but the mole fraction was allowed to vary as indicated. (Top) Molecular weights ( 95% confidence limits) returned by these analyses. The horizontal line indicates the molecular weight expected for a 2:2 CAP:polymerase complex (Mr 1.004 106). Data from Dyckman and Fried,30 with permission.
is maximized at a CAP mole fraction 0.5 [equivalent to a 1:1 combining ratio (Fig. 4, bottom) panel]. The constant molecular weight at all combining ratios is a characteristic of mechanisms in which only one kind of complex is formed (Fig. 4, top). Because the molecular weight of the complex is slightly more than twice that of the RNA polymerase monomer, the stoichiometry of the complex32 must be 2:2. For heteroassociation reactions, the conversion of apparent equilibrium constants from absorbance units to the molar concentration scale is shown as n m e e Kmolar ¼ Kobs A B lnþm1 (9) eA n B m Here Kobs is the equilibrium constant in absorbance units, n and m are the stoichiometries of species A and B, respectively, and eA, eB, and eAnBm 32
If the stoichiometry can be established from Mr and/or combining ratio, reanalysis of original data using a model in which monomer molecular weights and stoichiometries are fixed can allow the extraction of Kobs while minimizing the number of adjustable species in the fit.
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polymerase associated factors
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are the molecular extinction coefficients of species A, B, and the complex. If extinction coefficients do not change with association, eAn Bm ¼ neA þ meB . The conversion of apparent equilibrium constants from refractive index units (fringes) to the molar concentration scale is given by nþm1 ðMA Þn ðMB Þm YT Kmolar ¼ Kobs (10) MAn Bm CT As before, YT/CT is the conversion factor relating signal-to-weight concentration. Noninteracting components Equation (2) can describe independently sedimenting species as well as ones that associate. This is valuable, as it is unusual to find samples that are both highly pure and 100% active. An interacting system containing an additional independent species might be described as nA + mB + C . AnBm þ C; here component C can be an inactive fraction of one of the interacting species (A or B), or an unrelated contaminant. At sedimentation equilibrium the concentration distribution is given by cðrÞ ¼ cA;0 exp ½A ðr2 r02 Þ þ cB;0 exp ½B ðr2 r02 Þ þ cC;0 exp ½C ðr2 r02 Þ þ cAB;0 exp ½AB ðr2 r02 Þ
(11)
As described earlier, cAB;0 ¼ Kobs ðcA;0 Þn ðcB;0 Þm , but cC,0 does not depend33 on cA,0 or cB,0. If species C is an inactive fraction of component A, for example, analysis of SE data according to a model in which all components equilibrate [e.g., Eq. (7)] will result in values of Kobs that decrease with increasing [A] and with increasing rotor speed.9 Inclusion of a separate term for species C results in values of Kobs that are independent of reactant concentrations and rotor speed, as expected for equilibrium constants.34 As mentioned previously, distinguishing between alternative models becomes more difficult as the number of species increases. When this becomes a problem, data from other techniques (such as sedimentation velocity) can provide valuable guidance in selection of the simplest realistic model of the system.
33
Under dilute solution conditions. In concentrated solutions, steric interactions and other nonideal effects can prevent independent sedimentation. 34 The presence of inactive species increases the number of adjustable terms in the sedimentation equation that must be evaluated by fitting. Because acquiring a meaningful fit becomes more difficult as the number of species increases, systems containing several inactive species should be purified further before a detailed SE analysis is attempted.
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Thermodynamic Linkage Often, low molecular weight solutes affect the stabilities of transcription factor assemblies. Such solutes include physiological signals such as cAMP and cGMP but also buffer ions and osmolytes. When this is the case, solute binding and macromolecular assembly are linked thermodynamically.35 Sedimentation equilibrium analysis can provide quantitative information about the stoichiometries and consequences of solute participation in the protein assembly reaction. At the relatively slow rotor speeds appropriate for SE analyses of proteins, the concentrations of low molecular weight solutes are virtually unchanged from meniscus to the bottom of the solution column. This allows one to treat macromolecular equilibria as if they are taking place at a constant concentration of the low molecular weight solute. An example of this is analysis of the cAMP-dependent association of the E. coli CAP and lac repressor proteins, for which the mechanism 2 CAP þ nðcAMPÞ þ lac repressor.CAP2 cAMPn lac repressor (12) appears to operate.36 Here the equilibrium constant K ¼ ½ðCAP2 cAMPn Þ lac repressor =½cAMP n ½CAP 2 ½lac repressor], and the observable macromolecular equilibrium quotient Kobs ¼ ½ðCAP2 cAMPn Þ lac repressor =½CAP 2 ½lac repressor . cAMP stoichiometry37 can be evaluated using the relation @ log Kobs ¼n @ log ½cAMP
(13)
A graph of log Kobs as a function of log [cAMP] is shown in Fig. 5. The slope (1.8 0.3) indicates that 2 equivalents of cAMP are bound for each CAP2repressor complex formed. While these data do not specify the distribution of cAMP within the complex, in the absence of repressor, native CAP dimers bind one equivalent of cAMP over the concentration range 0 < [cAMP] 20 M spanned in this experiment.38 It is thus possible to speculate that each molecule of CAP in the (CAP2cAMP2)repressor complex binds one molecule of cAMP.
35
J. Wyman and S. J. Gill, ‘‘Binding and Linkage,’’ University Science Books, Mill Valley, CA, 1990. 36 M. G. Fried and M. A. Daugherty, J. Biol. Chem. 276, 11226 (2001). 37 For the reaction shown in Eq. (12), net cAMP uptake will yield n > 0, net release n < 0, and a cAMP-independent reaction, n ¼ 0. 38 T. Takahashi, B. Blazy, and A. Baudras, Biochemistry 19, 5124 (1980).
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Fig. 5. Dependence of log Kobs on log [cAMP] in formation of the CAP2cAMPnlac repressor complex. Values of log Kobs measured in the presence of cAMP are indicated by filled symbols. The horizontal line gives the value of log Kobs at [cAMP] ¼ 0. Error bars represent 95% confidence limits of plotted values. The slope (1.8 0.3) is a measure of the number (n) of cAMP molecules bound per molecule of complex formed. Data from Fried and Daugherty,36 with permission.
Temperature Dependence of Association Because the temperature of samples can be controlled with reasonable precision (typically 0.1 ), it is sometimes possible to extract thermodynamic information from the temperature dependence of equilibrium constants measured at sedimentation equilibrium.39 At any temperature, changes in standard free energy, enthalpy, and entropy that accompany a protein interaction are described by RT ln Kobs ¼ Go ¼ Ho TSo . Hydrophobic interactions, which remove large amounts of nonpolar surface from water, are characterized by positive values of Ho and So and negative values of Cop , the heat capacity difference.40 A nonzero standard heat capacity difference contributes to enthalpy and entropy differences as shown by Ho ¼ Ho Cop; ðT Þ T o o o S ¼ S Cp; ln
(14) (15)
Here, is a reference temperature and So and Ho are the enthalpy and entropy differences at that temperature. The temperature dependence of the association constant can be written41 39
For accurate measurements, the operating temperature of the centrifuge should be calibrated as described by S. Liu and W. F. Stafford III, Anal. Biochem. 224, 199 (1995). 40 R. S. Spolar, J.-H. Ha, and M. T. Record, Jr., Proc. Natl. Acad. Sci. USA 86, 8382 (1989). 41 E. C. W. Clarke and D. N. Glew, Trans. Faraday Soc. 62, 539 (1966).
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transcription factors at sedimentation equilibrium
R ln Kobs ¼
Go 1 1 T 1 þ ln þ Ho þ Cop; T T
361 (16)
An example of a graph of RlnKobs against temperature is shown in Fig. 6. Nonlinearity in such plots is evidence of nonzero values of Cop; . One caveat to the use of SE in investigating association thermodynamics is that analytical ultracentrifuges are typically limited to temperatures between 4 and 40 . This narrow range can result in inaccurate estimates of van’t Hoff enthalpy and heat capacity changes. An offsetting advantage is that data for individual association reactions can often be obtained in solutions in which more than one interaction is taking place (e.g., the monomer–tetramer–octamer association of TATA-binding protein; Fig. 2). Thus, while the SE approach may not be appropriate for all systems, the thermodynamic information that can be obtained under favorable conditions is often unique and is generally complementary to that provided by calorimetry. Sample Preparation
Volume The most widely used cells are the six-channel short-column cell, containing three samples and three reference solutions, and the two-channel long-column cell, holding one sample and one reference solution. For most purposes, centerpieces with a thickness of 12 mm are used, although a twochannel centerpiece with 3 mm thickness is also available. Six-channel,
Fig. 6. Temperature dependence of yeast TATA-binding protein self-association. Data for the overall monomer–octamer reaction is shown. Smooth curves are fits of Eq. (16) to data using ¼ 303 K (30 ) as reference temperature. Binding reactions were carried out in 20 mM HEPES/KOH, pH 7.9, 1 mM EDTA, 1 mM dithiotrietol supplemented with 120 (&), 300 (d), and 1 (m) M KCl. Data from Daugherty et al.,22 with permission.
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polymerase associated factors
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12-mm cells accommodate samples 120 l, whereas two-channel, 12-mm cells accept samples of 450 l. Much smaller sample volumes have been used successfully in both cell types, and techniques for doing so have been described.9 In addition, special very short-column centerpieces, requiring very small sample volumes, have been described.9,42,43 Short-column experiments take significantly less time to reach sedimentation equilibrium than long-column experiments.9,44 However, long-solution columns provide many more data points in a single scan and can allow a single scan to span a wider range of concentrations than the short-column format. These features are of value in data analysis, especially when samples are heterogeneous. Sample Concentration For experiments carried out with absorbance detection, the optimal signal/noise is obtained between 0 and 1 AU at the selected wavelength. To obtain a concentration gradient that spans that range, the starting absorbances are typically between 0.1 and 0.35. Studies that require higher concentrations can be performed without exceeding the optimal range of the detector by tuning the monochromator away from the max of the sample. Studies at very low concentrations may require greater sensitivity than that available with intrinsic chromophores. Such work is often possible if the molecule(s) of interest is labeled with an extrinsic chromophore that has a high extinction coefficient. In addition, labeling with an extrinsic chromophore can allow an individual protein to be detected in complex mixtures where the number of species detectable at 215 or 280 nm is large. An elegant example of this approach takes advantage of the changes in absorbance spectra that result from the biosynthetic substitution of tryptophan with analogues such as 5-hydroxytryptophan.17,45 Methods for the simultaneous analysis of multiple components with differing absorbance spectra have been described.46 For experiments carried out with interference detection, protein concentrations in the range of 0.1–5 mg/ml are appropriate. 42
J. J. Correia and D. A. Yphantis, in ‘‘Analytical Ultracentrifuation in Biochemistry and Polymer Science’’ (S. E. Harding, A. J. Rowe, and J. C. Horton, eds.), p. 237. Royal Society of Chemistry, Cambridge, 1992. 43 T. M. Laue, Technical Information, Vol. DS-835, Beckman Instruments, Palo Alto, CA, 1992. 44 G. Ralston, ‘‘Introduction to Analytical Ultracentrifugation.’’ Beckman Instruments, Inc, Fullerton, 1993. 45 T. M. Laue, D. F. Senear, S. Eaton, and J. B. Ross, Biochemistry 32, 2469 (1993). 46 M. S. Lewis, R. I. Shrager, and S. J. Kim, in ‘‘Modern Analytical Ultracentrifugation’’ (T. M. Schuster and T. M. Laue, eds.), p. 94, Birkhauser, Boston, 1994.
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363
Sample Integrity As described earlier, classical sedimentation equilibrium analysis is limited by the number of species that can be resolved. For this reason, the best samples are of high purity. Because centrifuge runs can be timeconsuming, we recommend prerun verification that sample proteins are intact, in the correct state of posttranslational modification, and exhibit appropriate activities (e.g., DNA binding, enzyme activity). Many proteins have the potential to form more than one noncovalent complex (e.g., a specific assembly and a nonspecific aggregate). As these are not usually detectable by SDS–PAGE, but are detectable readily by sedimentation velocity analysis,13,47–49 measurement of the samples’ s value distribution can provide guidance in the choice of models for analysis of SE data. Postrun analyses (e.g., enzyme assays, SDS–PAGE) are also valuable, as they allow detection of any changes in sample integrity during sedimentation. After a run, samples are recovered easily through the filling holes using a micropipettor equipped with a gel-loading tip or a 1-ml syringe fitted with a capillary tube. For cells that lack filling holes (e.g., early vintage six-channel cells), samples can be recovered by careful removal of one window. Absence of Nonprotein Contaminants Nonprotein sample components (e.g., nucleic acids, starches, dextrans, and polyacrylamide) can complicate the sedimentation equilibrium analysis of a protein by contributing to the absorbance (or refractive index) gradient, by binding the proteins(s) of interest, or affecting activity coefficients through macromolecular crowding. If these components are not part of the system under study, it is worthwhile taking steps to ensure that they are not present in significant concentrations. Buffer Because sedimentation equilibrium experiments can be performed under a very wide range of solution conditions, it is often possible to tailor buffer compositions to the needs of a given system. However, some buffer properties can affect the outcome of a SE experiment. Important among these are transparency, salt concentration, and ability to suppress proteolysis. Data acquisition requires the transmission of light so buffer components that absorb strongly at the experimental wavelength(s) should be 47
P. Schuck, M. A. Perugini, N. R. Gonzales, G. J. Howlett, and D. Schubert, Biophys. J. 82, 1096 (2002). 48 J. J. Correia, Methods Enzymol. 321, 81 (2000). 49 L. M. Carruthers, V. R. Schirf, B. Demeler, and J. C. Hansen, Methods Enzymol. 321, 66.
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avoided. Proteins far from their isoelectric pH values can have significant charge; electrostatic interactions can bias the equilibrium concentration gradient, resulting in incorrect apparent molecular weights. For moderately charged proteins, concentrations of 1:1 salts in the 100–300 mM range are often enough to suppress this effect. Attainment of sedimentation equilibrium can require many hours, giving proteolytic contaminants ample time to degrade samples. The treatment of samples with phehylmethylsulfonyl fluoride and/or inclusion of low molecular weight protease inhibitors (e.g., aprotinin, leupeptin) can minimize this problem. A critical requirement for analysis of multicomponent systems is that all macromolecules must be at exchange (dialysis) equilibrium with the buffer.50 A typical dialysis consists of two or more changes of solvent in a 1000:1 solvent- to-sample ratio. The dialysate is then used in the reference sectors of the centerpieces. Alternatively, samples can be equilibrated rapidly with buffer using centrifugal gel-filtration techniques.51,52 Rotor Speed and Duration of Run If a molecular weight range can be predicted for the system, a useful estimate of rotor speeds can be obtained from data of Chervenka,53 reprinted more recently in the manual by McRorie and Voelker.7 These speeds are especially useful for experiments with long solution columns, although we often find that they are slower than optimal for short-column experiments. However, they are a valuable point for departure, as data obtained at several rotor speeds can allow discrimination of polydispersity and reversible association.9 In addition, data sets obtained over a range of speeds can be usefully combined in a global analysis using programs such as NONLIN.54 Because attainment of equilibrium following a speed reduction can be slow, multispeed experiments in which rotor speeds increase can take less time to complete than ones in which rotor speeds decrease. The time needed to reach equilibrium is proportional to the square of the solution column height.9,12 Thus short-column experiments reach equilibrium much faster than long-column runs. Attainment of equilibrium can be verified by subtracting successive scans.55 In short-column experiments, 50
E. F. Casassa and H. Eisenberg, Adv. Prot. Chem. 19, 287 (1964). H. S. Penefsky, Methods Enzymol. 56, 527 (1979). 52 K. Struhl, in ‘‘Current Protocols in Molecular Biology’’ (F. M. Ausubel et al., eds.), p. 3.4.8, Wiley, New York, 1989. 53 C. H. Chervenka, ‘‘A Manual of Methods for the Analytical Ultracentrifuge,’’ Spinco Division, Beckman Instruments, Palo Alto, CA 1969. 54 M. L. Johnson, J. J. Correia, D. A. Yphantis, and H. R. Halvorson, Biophys. J. 36, 575 (1981). 55 The program MATCH by D. Yphantis and J. Lary (available on the web at ftp:// rasmb.bbri.org) facilitates this comparison. 51
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scans separated by 1 h are an adequate test; for intermediate- and longcolumn experiments scans should be separated by at least 6 h. Very large molecules and ones undergoing slow association reactions can require many hours to reach equilibrium; for these systems, longer intervals are sometimes needed. Under favorable conditions, the time required to reach equilibrium can be reduced by ‘‘overspeeding’’ (a brief period at a higher speed followed by gentle deceleration to the final equilibrium speed).56 Determination of Buffer Density The buffer density contributes to the buoyancy term, 1 v, for each sedimenting species. Because dilute aqueous buffers have densities close to 1 g/ml and typical, unmodified proteins have v 0.73 ml/g, a 1% error in solvent density results in 3% error in Mr. Thus, buffer density must be known with accuracy. The density of a solution can be estimated by summing density increments for each component.11 Extensive published data allow accurate calculation of densities for dilute aqueous buffers with common components.7,29,57 This procedure is automated by the program SEDNTERP (Table I). However, oscillating densimeters allow solvent densities to be measured at high precision over the range of temperatures accessible to the centrifuge.58 Because these values represent the actual densities of the buffer preparations, they should be used wherever possible. Determination of Partial Specific Volume An accurate value of the partial specific volume (v) is also needed for evaluation of the buoyancy term. For proteins without posttranslation modification and/or nonprotein cofactor, v can be estimated from the amino acid composition according to the relation59 X N i M i vi i vest ¼ X (17) Ni Mi i
where Ni is the number of residues of type i, Mi is the component molecular weight (amino acid molecular weight minus 18), and vi is the tabulated 56
D. E. Roark, Biophys. Chem. 5, 185 (1976). T. M. Laue, B. D. Shah, T. M. Ridgeway, and S. L. Pelletier, in ‘‘Analytical Ultracentrifugation in Biochemistry and Polymer Science’’ (S. E. Harding, A. J. Rowe, and J. C. Harding, eds.), p. 90. The Royal Society of Chemistry, Cambridge, England, 1992. 58 O. Kratky, H. Leopold, and H. Stabinger, Methods Enzymol. 27, 98 (1973). 59 E. J. Cohn and J. T. Edsall, ‘‘Proteins, Amino Acids and Peptides as Ions and Dipolar Ions,’’ Reinhold, New York, 1943. 57
366
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TABLE I Representative Software for Editing, Display, and Analysis of Sedimentation Equilibrium Dataa Application NONLINb Beckman software SEDEQ f TWOCOMP f Omega Analysisi Ultrascan II
Ultraspin
WinMatch WinREEDIT XLGRAPH SEDNTERPm a
Function
Platform (source)
Analysis of reversibly associating homogeneous systems Analysis of reversibly associating homogenous systems Analysis of up to three independently sedimenting species Analysis of two species that may reversibly self- or heteroassociate Analysis of SE data Nonlinear analysis of SE and sedimentation velocity data; estimation of solution density and protein v Analysis of SE data; 20 models including homogeneous and some heterogeneous systems Test for attainment of equilibrium Editing raw data files from XL-I or XL-A Data graphing and transformation of XL-A and XL-I data Estimation of solution density and protein v
PC,c,d Mac,d DECd PCe PC (DOS)g PC (DOS)h PCd PC, UNIX, LINUXj
PCk
PCc PCc PCd,l PCd,l
The Reversible Associations in Structural and Molecular Biology (RASMB) group provides a web-based resource for researchers interested in the use and advancement of analytical ultracentrifugation (http://www.bbri.org/rasmb). A software archive is maintained at that address. b M. L. Johnson, J. J. Correia, D. A. Y phantis, and H. R. Halvorson, Biophys. J. 36, 575 (1981). c http://vm.uconn.edu/wwwbiotc.uaf.html. d http://www.biochem.uthscsa.edu/auc/ e Available from Beckman Instruments, Inc. f G. Rivas, W. F. Stafford, III, and A. P. Minton, Methods 19, 194 (1999). g ftp://bbri.harvard.edu/rasmb/spin/ms_dos/sedeq-minton/sedeq.doc; ftp://bbri.harvard. edu/rasmb/spin/ms_dos/sedeq-minton/sedeq.exe h ftp://bbri.harvard.edu/rasmb/spin/ms_dos/twocomp-minton/sedeq.doc; ftp://bbri.harvard. edu/rasmb/spin/ms_dos/twocomp-minton/sedeq.exe i G. B. Ralston and M. B. Morris, in ‘‘Analytical Ultracentrifugation in Biochemistry and Polymer Science’’ (S. E. Harding, A. J. Rowe, and J. C. Horton, eds.), p. 243, Royal Society of Chemistry, Cambridge, 1992. j http://www.ultrascan.uthscsa.edu/. k http://www.mrc-cpe.cam.ac.uk/ultraspin/. l http://jphilo.mailway.com/default.htm. m T. M. Laue, B. D. Shah, T. M. Ridgeway, and S. L. Pelletier, in ‘‘Analytical Ultracentrifugation in Biochemistry and Polymer Science’’ (S. E. Harding, A. J. Rowe, and J. C. Harding, eds.), p. 90, The Royal Society of Chemistry, Cambridge, England, 1992.
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partial specific volume. Tabulated values of vi have been published in several places.7,29 Values of vi are tabulated at a reference temperature (fre quently 25 ). If the experimental temperature differs from the reference, the value of vest should be corrected as described.57 Values of vest obtained in this way are frequently accurate to 1–2%. Alternatively, if the sequence molecular weight NiMi is known, the remaining unknown, v, can be estimated from the experimental reduced molecular weight, . While both approaches assume that the component is not undergoing self-association (which can change v), the second method has the advantage that the v estimate is obtained for the exact solution conditions used in the experiment. For proteins of unknown amino acid composition, with substantial posttranslational modification, or nonprotein prosthetic groups, partial specific volume can be measured experimentally by performing parallel experiments in H2O- and D2O-containing buffers. The value of v can be obtained by simultaneous solution of the equations60 Mbouyant; H2 O ¼ Mð1 vH2 O Þ
(18)
Mbouyant; D2 O ¼ Mð1 vD2 O Þ
(19)
in which Mbouyant,H2O and Mbouyant,D2O are the observed buoyant molecular weights and H2O and D2O are the measured densities of the H2O- and D2O-containing buffers.61 Data Analysis Two general approaches have been used for the analysis of SE data. These are graphical methods62–66 and nonlinear least-squares fitting.67 Although each has advantages, graphical methods generally require transformation of data, which can complicate error analysis.68 Direct fitting
60
S. J. Edelstein and H. K. Schachman, Methods Enzymol. 27, 82 (1973). This calculation ignores changes in M that result from H-D exchange, which will depend on the identity of amino acids and nonprotein components of the molecule(s) in question. 62 D. E. Roark and D. A. Yphantis, Ann. N.Y. Acad. Sci. 164, 245 (1969). 63 R. H. Haschemeyer and W. F. Bowers, Biochemistry 9, 435 (1970). 64 G. J. Howlett, Chemtracts-Biochem. Mol. Biol. 11, 950 (1998). 65 D. J. Winzor and P. R. Wills, in ‘‘Modern Analytical Ultracentrifugation: Acquisition and Interpretation of Data for Biological and Synthetic Polymer Systems’’ (T. M. Schuster and T. M. Laue, eds.), p. 66. Birkhauser, Boston, 1994. 66 D. J. Winzor, M. P. Jacobsen, and P. R. Wills, Biochemistry 37, 2226 (1998). 67 M. L. Johnson and M. Straume, in ‘‘Modern Analytical Ultracentrifugation: Acquisition and Interpretation of Data for Biological and Synthetic Polymer Systems’’ (T. M. Schuster and T. M. Laue, eds.), p. 37, Birkhauser, Boston, 1994. 68 M. L. Johnson, Anal. Biochem. 206, 215 (1992). 61
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using functions that relate concentration to radial position avoids this drawback. Criteria for judging the adequacy and consistency of fitting results have been reviewed.67,68 The past decade has seen an upsurge in the development of software for analysis of SE data, and more programs are under development at the time of this writing. A representative list of currently available software is given in Table I. The most powerful programs allow simultaneous fitting of several data sets, with some parameters (e.g., c0,i) considered as local variables evaluated for each data set and some (e.g., Mi, Kobs) considered as global, evaluated for the entire data ensemble. Data collected at several rotor speeds, wavelengths, or starting concentrations can be combined, as can data obtained by absorbance and interference detectors. The analysis of large amounts of data obtained over a wide range of conditions makes it possible to discriminate between competing models describing the sedimentation behavior of moderately complex systems like those described earlier. Prospects The past decade has seen a renaissance in SE analysis, catalyzed by the availability of modern instrumentation and software for data analysis and the increasing availability of biologically important macromolecules. Several developments suggest that this trend will continue. Fluorescence detection69 has lowered the concentration requirements of the SE technique, allowing study of hard-to-acquire macromolecules and equilibria with large association constants. Multiwavelength analysis46 and methods for labeling proteins with chromophoric amino acid analogues (e.g., 5-hydroxytryptophan45,70) allow the simultaneous measurement of several proteins in complex mixtures, as well as the analysis of labeled proteins in the presence of several unlabeled species. Postcentrifugation fractionation methods10,71 allow the detection of components by tracer methods, including direct counting of radioisotopes, radioimmunoassay, and real-time polymerase chain reaction. With these techniques, SE analyses can be carried out on target species at very low concentrations, in highly complex mixtures, and in the presence of high background concentrations of other macromolecules. These developments will make possible the analysis of systems of life-like complexity, spanning the concentration ranges that occur in vivo. 69
T. M. Laue, A. L. Anderson, and B. J. Weber, in ‘‘Ultrasensitive Clinical Laboratory Diagnostics’’ (G. Cohn, ed.), Vol. 2985, p. 196. SPIE, Bellingham, WA, 1997. 70 J. B. Ross, D. F. Senear, E. Waxman, B. B. Kombo, E. Rusinova, Y. T. Huang, W. R. Laws, and C. A. Hasselbacher, Proc. Natl. Acad. Sci. USA 89, 12023 (1992). 71 S. Darawashe and A. P. Minton, Anal. Biochem. 220, 1 (1994).
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Acknowledgments We gratefully acknowledge valuable discussions with Drs. Jack Correia, Michael Johnson, Jacob Lebowitz, and Allen Minton and thank the many people who sent reprints and preprints.
[32] Single-Molecule Studies of DNA Architectural Changes Induced by Regulatory Proteins By Laura Finzi and David Dunlap Single-molecule techniques are now established approaches in the investigation of the molecular mechanisms involved in nucleic acids/protein interactions. The advantages of single-molecule approaches are related to the fact that the observation is performed on just one molecule at a time. This allows the study of the behavior of a molecule as it goes through different conformational states in time, thus revealing states that are averaged out in bulk experiments where unsynchronized molecules are observed simultaneously. The tethered particle motion (TPM) method is a general method used to measure the contour length of a suitably bead-labeled and surface-immobilized DNA fragment. As such, it can potentially be used in a variety of single-molecule kinetics experiments and can yield mechanistic information. It can be used to reveal and monitor, dynamically, events that induce large conformational changes in single DNA molecules, such as those induced by regulatory proteins, or to study the mechanisms by which enzymes move in a directed fashion along DNA (polymerases, helicases, etc). For example, it is now clear that TPM assays can provide much information about the mechanisms by which transcriptional regulatory proteins act.1–3 Furthermore, the effect of supercoiling on DNA/protein interactions or the change in DNA-linking number induced by regulatory proteins can be characterized in a TPM experiment using magnetic tweezers.4,5 This article concentrates on the protocol used to prepare most TPM experiments. This consists mainly of three stages: (i) preparation of DNA constructs suitably labeled with biotin at one end and digoxigenin at the 1
Finzi and Gelles, Science 267, 378, (1995). Lia et al., PNAS 100, 11373–11377 (2003). 3 J. Gelles, personal communication. 4 T. R. Strick, J.-F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette, Science 271, 1835 (1996). 5 T. R. Strick, V. Croquette, and D. Bensimon, Nature 404, 901 (2000). 2
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