ARTICLE pubs.acs.org/jchemeduc
A Simple Spreadsheet Program To Simulate and Analyze the Far-UV Circular Dichroism Spectra of Proteins Luciano A. Abriata* Instituto de Biologia Molecular y Celular de Rosario (IBR-CONICET), Rosario, Argentina
bS Supporting Information ABSTRACT: A simple algorithm was implemented in a spreadsheet program to simulate the circular dichroism spectra of proteins from their secondary structure content and to fit R-helix, β-sheet, and random coil contents from experimental far-UV circular dichroism spectra. The physical basis of the method is briefly reviewed within the context of protein structure, and the mathematical aspects of the approach are given together with ready-to-use code and examples. The work illustrates the use of simple yet powerful programming tools for simulations and data analysis in class and provides a means to explore the circular dichroism spectra of proteins and the applications of this spectroscopic technique to the study of proteins. KEYWORDS: Graduate Education/Research, Upper-Division Undergraduate, Biochemistry, Polymer Chemistry, ComputerBased Learning, Distance Learning/Self Instruction, Biophysical Chemistry, Computational Chemistry, Proteins/Peptides, Spectroscopy
F
ar-UV circular dichroism (CD) is a simple yet powerful tool for determining the secondary structure composition of proteins. This is based in the sensitivity of CD in the 190 250 nm region to the chirality of each peptide bond, ultimately dictated by the local secondary structure at each residue along the chain. Structures with R-helix, β-sheet, and random coil conformations produce distinct spectral features in CD, and the spectrum of a protein is, to a good approximation, the average of the basis spectra for pure structures weighed by the content of each structure type. For the interested instructors and students, excellent books and reviews covering the physical basis of CD and its applications to the study of proteins abound.16 As pointed out by Urbach, despite its exceptional ease of use and connection to central topics in protein chemistry, CD remains rare in the undergraduate curriculum.7 In his recent article, Urbach introduces the theory and practice of CD spectroscopy discussing its importance in the curriculum and the instrumentation required for a set of experiments.7 Several experiments to incorporate CD into lab sessions have been also described by Bondesen and Schuh,8 including the qualitative comparison of CD spectra for globular proteins with different structures, the characterization of a molten globule, and the study of the kinetics and thermodynamics of processes involving changes in secondary structure, such as unfolding and helix sheet transitions. Recently, Zhang and co-workers have also introduced an interesting laboratory activity where a thermal denaturation study of myoglobin using CD is described to assess protein stability and two methods of data analysis are applied.9 Complementing experiments, simulations are interesting sources of knowledge that help build experience on different systems and allow for possible scenarios to be evaluated and dissected.10,11 Simulations also provide direct contact with figures and units of measurement, in much the same way as when processing Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
experimental data. Setting up simulations involves symbolic operations with equations, aiding the development of mathematical skills. Finally, simulations might be the only possibility to examine a concept when there is no access to the proper instrumentation or when little time is available in the curriculum. There are a vast number of different programs available to teach and to learn about nature from simulations, including the SIR (simulations and interactive resources) series published in this Journal by Martin.12 A set of spreadsheet-based tools is presented to simulate the CD spectra in the far-UV region and to fit experimental spectra to obtain secondary structure contents. Similar to other articles published in this Journal, Microsoft Excel was employed where the code is written as VBA scripts. The spreadsheet is available as Supporting Information.
’ THEORETICAL BACKGROUND Protein Secondary Structure and CD Spectra
The relationships between protein CD spectra and the corresponding secondary structure contents have been extensively discussed in books and reviews.26,13,14 Estimating structure contents from CD spectra is mostly an empirical task, which has been tackled with several approaches involving methods based on least-squares fittings, neural networks, singular-value decomposition, and so forth.2,1520 All of these methods share the need for a set of basis spectra, either of peptides with known pure structures or proteins for which high-resolution structures are available. Also, all of these methods assume that (i) only peptide bonds give CD signals in the far-UV region (with Published: June 22, 2011 1268
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negligible contributions from side chains), (ii) the effects (CD signals) of each secondary structure element are linearly additive and tertiary structure exerts no effect, (iii) each secondary structure type can be described with only one CD spectrum, and (iv) the three-dimensional structures of reference proteins or peptides in the basis set are retained in solution.2 One of the first methods for the analysis of protein CD spectra, still widely used today, was introduced in the late 1960s by Greenfield and Fasman.21,22 The rationale behind their simple approach is that, to a first approximation, the far-UV CD spectrum of a protein can be regarded as a linear combination of the spectra corresponding to the three most common secondary structure types, that is, R, β, and random coil elements. The CD spectra of pure components (the “basis spectra”) can be obtained by a variety of methods, ultimately involving structural information derived from high-resolution X-ray structures of peptides and proteins. In their first work, the authors employed as basis set the CD spectra of a poly-L-lysine peptide acquired under different conditions, in which it adopted either pure R, β, or coil structures.21 The work by Greenfield and Fasman soon became a citation classic, where an interesting anecdote recalls the fact that their fitting program was written in the unfriendly Fortran language and ran on what was a supercomputer at that time, which took up a large room.23 Even though the original method by Greenfield and Fasman has been superseded by novel algorithms, it remains clear, educational, and tractable at the final stages of undergraduate and beginning of doctoral studies. The newest methods for CD data analysis involve large data sets and complex mathematical operations with unclear connections to the underlying physics and chemistry. Thus, the same theoretical foundations and basis spectra from Greenfield and Fasman’s work have been adopted here to simulate and analyze the CD spectra of proteins, but delivered as a simple and ready-to-use spreadsheet program with open source code written in the friendly BASIC language. Simulating Protein CD Spectra from Secondary Structure Content
The assumption that a CD spectrum is the linear combination of the spectra of pure secondary structures, can be expressed mathematically as ½θðλÞ ¼ χR ½θRðλÞ + χβ ½θβðλÞ + χrc ½θrcðλÞ
for λE½190:::250 nm
ð1Þ where [θ(λ)] is the normalized CD intensity at wavelength λ; χR, χβ, and χrc are the fractions of R-helix, β-sheet, and random coil, respectively; and [θR(λ)], [θβ(λ)], and [θrc(λ)] are the normalized residue ellipticities of pure-R, pure-β, and pure-coil structures at wavelength λ, respectively. In this expression and throughout the text, the brackets indicate normalized CD intensities, that is, intensity per molar concentration, per residue, and per cm cell path length. Once [θ(λ)] is obtained from eq 1, the experimental value accounting for protein size (in number of residues n), protein concentration (C, in units of mol/L), and cell path (b, in cm) can be calculated as θðcalc, λÞ ¼ bnC½θðλÞ 106
polypeptide adopts an R helical conformation, at pH 7 it is a random coil, and heating followed by recooling at pH 11.1 results in β-sheet structures.21 The (λ,θ) data points for poly-L-lysine structures available in literature were fitted to 10th-order polynomials, so that the θ values could be retrieved at any continuous value of λ in the spectral region from 190 to 250 nm. The use of 10th-order polynomials ensures that all the important curvatures are reproduced. Table S1 (in the Supporting Information) shows the coefficients for the polynomials that interpolate the pure R, β, and coil basis spectra. Fitting Protein CD Spectra To Obtain Secondary Structure Contents
In practice, one will most commonly deal with the reverse problem, that is, determining the coefficients χR, χβ, and χrc, which are the fractions of R-helix, β-sheet, and random coil structure in the protein. Such procedure involves fitting the experimental spectrum to eq 1, after proper normalization per protein concentration and size and cell path. Normalization of the experimental data is achieved by ½θðexp, λÞ ¼ θðexp, λÞ =ðbnC106 Þ
Fitting [θ(exp,λ)] in a least-squares sense is finding the set of χR, χβ, and χrc values that minimizes the following error function,24 E, E¼
∑λ f½θðexp, λÞ ðχR ½θRðλÞ + χβ ½θβðλÞ + χrc ½θrcðλÞÞg2
ð4Þ
where the summation runs through all the values of λ, and reflects the sum of the squared differences between experimental and back-predicted CD intensities. Given the form of eq 4, it follows that the only global minimum for E can be found by setting its first derivative to zero for each of the parameters χR, χβ, and χrc.24 For the R-helix content ∂E ¼ 2 ð½θðexp, λÞ χR ½θRðλÞ χβ ½θβðλÞ χrc ½θrcðλÞ Þð ½θRðλÞ Þ ∂χR
∑
¼0
ð5Þ
and hence
∑ð½θðexp, λÞ χR ½θRðλÞ χβ½θβðλÞ χrc½θrcðλÞ Þð ½θRðλÞÞ ¼ 0
ð6aÞ
Distributing and reordering terms, gives
∑χR ½θRðλÞ2 + ∑χβ ½θβðλÞ½θRðλÞ + ∑χrc ½θrcðλÞ ½θRðλÞ ¼ ∑½θðexp, λÞ ½θRðλÞ
ð6bÞ
In the summations, the secondary structure contents are constants that can be taken out, giving
∑
∑
∑
χR ½θRðλÞ 2 + χβ ½θβðλÞ ½θRðλÞ + χrc ½θrcðλÞ ½θRðλÞ
ð2Þ
Different sets of the basis spectra required in eq 1 are available in the literature.2,3 In this work, the basis set derived by Greenfield and Fasman from the CD spectra of poly-L-lysine under different pH and temperature conditions is used. At pH 10.8, this
ð3Þ
¼
∑½θðexp, λÞ½θRðλÞ
ð6cÞ
Working similarly for the β-sheet and random coil contents, the following set of three equations with three unknowns is 1269
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Figure 1. The spreadsheet program for simulating and fitting protein circular dichroism spectra, available in the Supporting Information.
derived:
∑
∑
∑
χR ½θRðλÞ 2 + χβ ½θβðλÞ ½θRðλÞ + χrc ½θrcðλÞ ½θRðλÞ ¼
∑½θðexp, λÞ½θRðλÞ
∑
ð7Þ
∑
∑
χR ½θRðλÞ ½θβðλÞ + χβ ½θβðλÞ 2 + χrc ½θrcðλÞ ½θβðλÞ ¼
∑½θðexp, λÞ ½θβðλÞ
∑
ð8Þ
∑
∑
χR ½θRðλÞ ½θrcðλÞ + χβ ½θβðλÞ ½θrcðλÞ + χrc ½θrcðλÞ 2 ¼
∑½θðexp, λÞ½θrcðλÞ
ð9Þ
Any suitable method can be employed to solve the equations system, here Cramer’s rule in matricial form is used owing to its simplicity for a 3 3 system24
∑½θRðλÞ 2 ∑½θβðλÞ ½θRðλÞ ∑½θrcðλÞ ½θRðλÞ χR ∑½θRðλÞ ½θβðλÞ ∑½θβðλÞ 2 ∑½θrcðλÞ ½θβðλÞ χβ ∑½θRðλÞ ½θrcðλÞ ∑½θβðλÞ ½θrcðλÞ ∑½θrcðλÞ 2 χrc ∑½θðexp, λÞ ½θRðλÞ ¼ ∑½θðexp, λÞ ½θβðλÞ ð10Þ ∑½θðexp, λÞ ½θrcðλÞ
or simply M 3 X = V. Solving for vector X yields its components: XR ¼ DR =D Xβ ¼ Dβ =D Xrc ¼ Drc =D
ð11Þ
in which D is the determinant of the 3 3 matrix on the left of eq 10 (M), and the Di are the determinants of the same matrix with each column replaced by the vector of constant terms V. The quality of the fit can be assessed in a number of ways, including the simplest root-mean-square deviation (rmsd) and Pearson’s correlation coefficient24 between experimental and back-simulated data. These values should be near 0 and near 1, respectively, in a perfect fit. Another important checkpoint is the visual inspection of the residuals24 (experimental back-predicted), which should distribute randomly around 0 in a good fit.
’ PROGRAM IMPLEMENTATION The algorithms presented above for spectral simulation and fitting were implemented in Microsoft Excel and Visual Basic for Applications, 2003 edition and are ready for use by instructors and students. The spreadsheet, available as Supporting Information, contains six dedicated cells: three cells where secondary structure contents are defined for simulations and three cells for sample-specific parameters used for simulation or fitting (number of residues and molar concentration of protein, and cell path). Four columns are dedicated to spectra: A for the wavelengths, C for experimental data, B for simulated spectra (either from scratch or after a fit is performed), and D for residuals calculated after the fitting (Figure 1). Two command buttons contain the code. One command button simulates a spectrum in the second column in the range of wavelengths defined in the first column using the 6 parameters provided in the upper cells and plots the simulated spectrum in red on the upper graph. The other command button fits the data in column C, displays the resulting structure contents, r2, and root-mean-square deviation (rmsd) and outputs the backcalculated spectrum in column B and the residuals in column D, updating the plots for back-prediction (red on the upper graph) and residuals (lower graph). ’ PROGRAM APPLICATIONS Sample Simulations of CD Spectra
The basis spectra of polylysine in R helical, β-sheet, and random coil conformations are shown in Figure 2A, simulated with the spreadsheet program and scaled as if corresponding to a 20 μM solution of a 180 residue-long protein in a cuvette with a path of 0.1 cm. The main features of each basis spectrum are evident, such as a maximum under 195 nm and two minima at 210 and 222 nm in the spectra of R-helices, a maximum at 195 nm and a minimum at 217 nm in the spectra of β-sheets, and a negative peak under 200 nm followed by no features above 210 nm in the case of random coils. Simulated spectra for three hypothetical proteins of the same size and concentration, containing only R-helices and random coils in ratios of 60:40, 50:50, and 40:60 are shown in Figure 2B. The resulting spectra reveal that the most marked differences occur at around 190 and 222 nm, wavelengths that could be used in principle to qualitatively assess the quantity of R helical content. Indeed, the 222 nm feature has been utilized.25 The 1270
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Figure 2. (A) Spectra of polylysine in pure R-helix, β-sheet, and random coil conformations. (B) Simulated spectra for three proteins with variable R-helix and random coil contents. (C) Simulated spectra for proteins with 10% random coil and variable quantities of R-helices and β-sheets.
Table 2. Parameters Simulated for the Unfolding Process of a Hypothetical Proteina [U]/(mol L1)
f[U]
Xhelix
Xcoil
0
0.000
0.800
0.200
1
0.002
0.798
0.202
2
0.018
0.786
0.214
3 4
0.119 0.500
0.705 0.400
0.295 0.600
5
0.881
0.095
0.905
6
0.982
0.014
0.986
7
0.998
0.002
0.998
8
1.000
0.000
1.000
The protein has 80% R-helix and 20% random coil structures in the native, folded state.
a
Figure 3. (A) Experimental and fitted CD spectrum of hen lysozyme (5 μM, 129 residues, 0.1 cm path length) and (B) the corresponding X-ray structure (2vb1).
Table 1. Results of Fitting the Experimental CD Spectrum of Lysozyme versus the Structure Contents Derived from the X-ray Structure CD fit
X-ray (2vb1)
Alpha helix (%) Beta Sheet (%)
49.7 12.9
50 7
Random coil (%)
37.4
43
Type
simulations performed here explain the origins of the method based on the CD at 222 nm. It should be noted that the presence of β-sheet structures would result in bias toward higher R-helix content than the real content because the β-sheets absorb at this wavelength (see Figure 2A). Simulated spectra for other hypothetical proteins of the same size and concentration, containing a small constant quantity of random coil and varying quantities of R-helices and β-sheets are shown in Figure 2C. This set of spectra illustrates how hard it is to resolve by naked eye the mixed contributions from R-helices and β-sheets, which can be much better accomplished by numerical algorithms. Fitting Real-World CD Data
The experimental CD spectrum of hen lysozyme was analyzed using the fitting routine of the spreadsheet program. This routine
works exactly as described. The experimental spectrum and the back-simulations from the best fit are shown in Figure 3A, and the fitted structure contents are summarized in Table 1. The fitted structural content nicely match the structural content derived from a high-resolution X-ray structure of this protein (Figure 3B). Both the X-ray structure and the CD fits reveal a high quantity (50%) of R helical content, a significant quantity of coils (including loops and turns), and some residual β-sheet structure. For the latter two elements, the difference of 6% observed between the CD fit and the X-ray structure can be attributed to slight differences in the solution and crystal structures of the proteins or to the method of data analysis itself. However, the differences observed here are on the order of the average difference reported by most methods for sheet and coil structures, that is, around 5%.2,3 Simulating Protein Unfolding by a Denaturant
Protein folding is a central topic in modern biophysics. Most enzymes are only active in their folded forms; thus, unfolding is tightly controlled inside living cells and is undesirable in the case of industrial enzymes. Moreover, in the last decades, unfolded and misfolded states of proteins have been linked to a number of diseases such as Alzheimer and Parkinson. Reviews on protein folding, unfolding, and misfolding are available for the interested reader.2629 Circular dichroism is a simple and sensitive tool to investigate folding and unfolding processes,2,3 as well as fluorescence spectroscopy. In particular, far-UV CD spectroscopy can be used 1271
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Figure 4. (A) Simulated CD spectra for a protein being unfolded by urea (details in the text). (B) Difference between CD spectra at 0 and 8 M urea, showing sensitivity maxima at nearly 190 and 220 nm. (C) CD intensity profile at 220 nm, point of maximum CD difference in a clean region.
to probe changes in secondary structure along the unfolding process and has been extensively used to monitor the folded state of proteins. In this simple example, a two-state unfolding process driven by urea is simulated for a hypothetical protein, whose native state has 80% R-helix and 20% random coil, to predict the expected far-UV CD spectral features of the transition. Two-state unfolding experiments near equilibrium conditions show that the fraction of unfolded protein f[U] follows a sigmoidal dependence against the concentration of denaturant, urea in our example [U], with characteristic [U]1/2 (concentration of urea at which f[U] = 0.5) and sharpness s for the transition around [U]1/2: f½U ¼
1 1 + esð½U1=2 ½UÞ
ð12aÞ
Equation 12a is a phenomenological description for the unfolding transition. The value of [U]1/2 increases with the stability of the protein against urea and is typically in the range from 1 to 4 M. Running the unfolding simulation with a value of [U]1/2 = 4 M sets the transition approximately at the middle of the maximum possible concentration of urea in solution and a sharpness s = 2 L/mol produces smooth changes of f[U] at low and high urea concentrations and the steepest changes around [U]1/2: f½U ¼
1 1 + eð2 L=molÞð4 mol=L ½UÞ
ð12bÞ
This set of parameters produces a curve with three defined stages amenable for teaching (Table 2, columns 1 and 2). At any point along the titration, the resulting CD spectrum, CD[U], is assumed to be the sum of the spectra of the folded and unfolded protein, weighed by the corresponding populations in terms of the fraction of folded and unfolded protein: CD½U ¼ f½U CDunfold + ð1 f½U ÞCDfold
ð13Þ
The CD spectrum of the unfolded form, CDunfold, is that of pure random coil, whereas the spectrum of the folded protein, CDfold, is that corresponding to 80% R-helix and 20% coil. Plugging in the corresponding equations gives CD½U ¼ f½U CDcoil + ð1 f½U Þð0:8CDhelix + 0:2CDcoil Þ ð14Þ
Distributing and rearranging gives CD½U ¼ f½U CDcoil + 0:8CDhelix + 0:2CDcoil 0:8f½U CDhelix 0:2f½U CDcoil CD½U ¼ ð0:2 + 0:8f½U ÞCDcoil + 0:8ð1 f½U ÞCDhelix
ð15Þ ð16Þ
Or simply CD½U ¼ Xcoil CDcoil + Xhelix CDhelix
ð17Þ
with Xcoil = 0.2 + 0.8f[U] and Xhelix = 0.8(1 f[U]). The fractions of unfolded protein in a hypothetical unfolding curve were simulated at nine urea concentrations from 0 to 8 M (Table 2). The corresponding Xcoil and Xhelix parameters were calculated and then the nine spectra were simulated. The simulated spectra are shown in Figure 4A, and the difference between the spectra without and with 8 M urea is shown in Figure 4B. A maximum in the signal change is observed at 220 nm, the wavelength used in the profile traced in Figure 4C. This profile naturally follows the shape of eq 12b and would be the starting point for a series of thermodynamic analyses if it came from experimental data.
’ CONCLUSION A spreadsheet program for the simulation and analysis of protein circular dichroism data based on the original work by Greenfield and Fasman is presented. Even though this method has been superseded by several algorithms, most of which are available in commercial programs, this implementation is highly pedagogical for two main reasons. First, a thorough theoretical description of the methods used is given, closely linked to the underlying physics and chemistry. Second, students interested in programming can easily view and edit the source code, which is written in the amenable BASIC language. A set of activities is provided in the Supporting Information that can complement or replace experimental sessions and help instructors develop material for lessons. The program could also be used to process data obtained from experimental activities such as those proposed by Bondesen and Schuh,8 or in the simulation mode before and after lab sessions. A number of other practical activities can be derived, for example, the kinetics of protein folding could be simulated or the effect of using different spectral regions on the fits could be assessed. 1272
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Overall, this is a useful and highly pedagogical instrument for teaching and learning about simulations, protein structure, spectroscopy, and data analysis.
’ ASSOCIATED CONTENT
bS
Supporting Information A ready-to-use spreadsheet program (with all the needed code to simulate and fit CD data), a spreadsheet with sample experimental spectra, and Table S1 (containing the parameters required to simulate the CD spectra of pure structures). This material is available via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ REFERENCES (1) Rodger, A.; Norden, B. Circular Dichroism and Linear Dichroism. In Oxford Chemistry Masters; Oxford University Press: Oxford, 1997; ISBN 019855897X. (2) Circular Dichroism and the Conformational Analysis of Biomolecules, 1st ed.; Fasman, G. D., Ed.; Plenum Press: New York, 1996; ISBN 0306451425. (3) Rodger, A. Methods Enzymol. 1993, 226, 232. (4) Adler, A. J.; Greenfield, N. J.; Fasman, G. D. Methods Enzymol. 1973, 27, 675–735. (5) Johnson, W. C., Jr. Annu. Rev. Biophys. Chem. 1988, 17, 145–166. (6) Johnson, W. C., Jr. Methods Biochem. Anal. 1985, 31, 61–163. (7) Urbach, A. R. J. Chem. Educ. 2010, 87 (9), 891–893. (8) Bondesen, B. A.; Schuh, M. D. J. Chem. Educ. 2001, 78 (9), 1244. (9) Mehl, A. F.; Crawford, M. A.; Zhang, L. J. Chem. Educ. 2009, 86 (5), 600. (10) Blackburn, M. J. Chem. Educ. 1995, 72 (6), 553. (11) Robinson, W. R. J. Chem. Educ. 2000, 77 (1), 17. (12) Martin, J. S. J. Chem. Educ. 2002, 79 (5), 639. (13) Woody, R. W. Circular Dichroism of Peptides. In The Peptides; Hruby, V. J., Ed.; Academic Press: New York, 1985; Vol. 7, pp 15114. (14) Yang, J. T. Methods Enzymol. 1986, 30, 208–269. (15) Provencher, S. W.; Gl€ockner, J. Biochemistry 1981, 20, 33–37. (16) Hennessey, J. P., Jr.; Johnson, W. C., Jr. Biochemistry 1981, 20, 1085–1094. (17) Manavalan, P.; Johnson, W. C., Jr. Anal. Biochem. 1987, 167, 76–85. (18) B€ohm, G.; Muhr, R.; Jaenicke, R. Protein Eng. 1992, 5, 191–195. (19) Dalmas, B.; Hunter, G. J.; Bannister, W. H. Biochem. Mol. Biol. Int. 1994, 34, 17–26. (20) Sreerama, N.; Woody, R. W. Anal. Biochem. 1993, 209, 32–44. (21) Davidson, B.; Fasman, G. D. Biochemistry 1967, 6 (6), 1616–1629. (22) Greenfield, N. J.; Fasman, G. D. Biochemistry 1969, 8, 4108–4116. (23) Greenfield, N. CC/Life Sci. 1982, No. 26, 28–28. (24) Bevington, P. Robinson, D. K. Data Reduction and Error Analysis for the Physical Sciences, 3rd ed.; McGraw-Hill: New York, 2003; ISBN 9780072472271. (25) The 190 nm feature alone is not used much, since the region under 200 nm is typically hard to record accurately. (26) Berg, J. M.; Tymoczko, J. L.; Stryer, L. Protein Structure and Function. In Biochemistry, 5th ed.; W. H. Freeman and Co.: New York, 2002. (27) Selkoe, D. J. Nature 2003, 6968 (426), 900–904. (28) Chiti, F.; Dobson, C. Annu. Rev. Biochem. 2006, 75, 333–366. (29) Robson, B.; Vaithilingham, A. Prog. Mol. Biol. Transl. Sci. 2008, 84, 161–202. 1273
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