Insights About CHAPS Aggregation Obtained by Spin Label EPR Spectroscopy
Supplementary Material
1.- Calculation of apparent order parameters from EPR spectra
The apparent order parameter is related to both the dynamics and the angular amplitude of motion of the z-axis of the nitroxide molecular frame, pointing along the unpaired electron p-orbital1. Its values lie in the range 0< Sapp< 1.
It can be calculated as: 1
where
; , with Amax and Amin determined from the spectra as shown in Figure S1 (a); and, , , and .
The parameters Aczz = 3.29 mT, Acxx= 0.59 mT, and Acyy= 0.54 mT are the single crystal values of the 14N hyperfine coupling tensor.1
2.- Analysis of the EPR spectra of 12-SASL obtained at different CHAPS concentrations by Principal Factor Analysis.
The main assumption is that the EPR spectrum at each concentration is a unique lineal combination of Nfundamental spectra (the basis of the vectorial space), each one arising from a different physical environmentofthe spin label, which does not change with concentration.2For the kth concentration in the series, the measured spectrum yk(Bi) (derivative of the EPR absorption at the magnetic field Bi) can be written as a linear combination:
(1)
where f j(Bi) is the value of the unknown j fundamental spectrum at field Bi, ck jis its relative weight at the kth CHAPS concentration (), and Ndenotes the number of of independent components needed to describe the whole series of spectra as lineal combinations of the fundamental ones. Defining ‘‘data’’ and ‘‘fundamental’’
matrices S and F by Ski=yk(Bi) and Fji= f j(Bi) respectively, we note that Eq. (1) can be viewed as one representation of the matrix S in a vector space. To be more specific, for lBi’s and t measured spectra at different concentrations, there are an infinite number of t×n matrices K and n×l matrices Fsuch that
S= KF(2)
But, by hypothesis, the only physically meaningful such decomposition of S is that in Eq. (1), which remains to be determined.
The notion of an abstract vector space implies the existence of a set of basis vectors FIforming the rows of matrix F in Eq. (2). One such set is furnished by the eigenvectors Qiof the Hermitian t×t covariance matrix Z = STS (ST is the traspose of matrix S), such that
ZQi = iQi(3)
Note that, from Eq. (1), there would be exactly n nonzero eigenvalues if there were no experimental errors. The presence of errors introduces unphysical additional eigenvalues. Further details of the calculations can be found in the references2-4.
There are several criteria allowingone to determine thenumber N of independent spectra, i.e. the dimension of the basis set. Statistical criteria,as experimental real error and factor indicator, were not used, because in our case(due to the normalization of the spectra) the error is not uniformely distributed, leading to overvaluationof the basis dimension.2 Instead, the regular behavior of the relative weights ckj for each abstract component j as a function of concentration k was used as a criterium to determine the basis dimension.Figure S2 shows the outputrelative weights ckjfor the first 4 abstract components as a function of CHAPS concentration.
It can be seen that the relative weights for components 1, 2 and 3 show a regular behavior as CHAPS concentration is increased. Instead, the 4th component shows an erratic variation of its weight, with strong oscillations. The 5th and subsequent components display a similar erratic behavior (not shown). These facts were taken into account to determine that the basis of independent spectra needed to describe the whole set of experimental data has only N =3 components.
In order to obtain the set of three fundamental spectra with physical significance, the process of “target transformation”3,4was performed. To this aim, it is necessary to know the values of the relative weightsckj for at least N - 1 components for some of the experimental spectra. As the first spectrumof the series is identical to that obtained for 12-SASL in a CHAPS free solution, it is clear that this is one of the pure components. Based upon physical criteria, and considering that the relative weights should be always positive and lower than 1, we could determine the shape of the spectra of the second and third components (Fig. 5 in the paper) and the evolution of the relative weight of each of the threecomponents as a function of surfactant concentration (Fig.6 in the paper). Changes up to ±3% in the proposed weights yield undetectable modifications in the results, but higher changes cause negative weights, which are not physically acceptable.
Then, by least squares fits of the 20 experimental spectra (k = 1 to 20) we determined thecoefficientsckj (j=1 to 3) representing the relative weights of each component at the surfactant concentration k that lead to reproduce the experimental spectrum as a linear combination of the basis spectra (Figure 6 in the paper). Figures 3 (a) and (b) show all the spectra with the corresponding fits. It can be seen that the three fundamental spectra are able to give a very good reproduction of the spectra at all CHAPS concentrations.
3.- Reorientational dynamics of secondary CHAPS micelles does not affect the EPR spectrum of 12-SASL
As discussed by Marsh and Horvath5, the characteristic rotational frequency for motional averaging in a EPR spectrum of a spin label is related to the anisotropy of the hyperfine tensor5 as , where Ris the rotational correlation time, and (Azz - Axx) is the maximum anisotropy of the 14N-hyperfine splitting, expressed in angular frequency units.5 Thus, rotational frequencies much lower than 108s-1, i.e. rotational correlation times well above 10-8s, would cause no averaging effects on the EPR spectrum. The Stokes- Einstein-Debye equation , where allowed us to calculate R0 =2.15 nm as the radius of a spherical aggregate having a rotational correlation time of 10-8s in water at room temperature. A rough estimation assuming constant density and spherical micelle shape yields that aggregation of 4 primary micelles is needed to obtain a radius of 2.3 nm, larger than R0. In this way, secondary micelles formed by aggregation of 4 or more primary ones will rotate slowly enough so that its collective motion would cause negligible effects in the EPR spectrum of a spin label incorporated to its structure. Thus, spectral averaging effects would be caused only by internal degrees of freedom of the spin label within the secondary micelle structure.
References
(1) Griffith O.H., Jost P. Lipid Spin Labels in Biological Membranes. In Spin Labeling: Theory and applications, Berliner L.J., Ed.; Academic Press: New York,1976; pp 454-523.
(2) Malinowski, E.R.; Howery, D.G. Factor Analysis in Chemistry; Wiley: New York, 1980.
(3) Koropecki, R. R.; Alvarez, F.; Arce, R. Infrared study of the Si-H stretching band in a-SiC:H, J. Appl. Phys.1991, 69, 7805-7811.
(4) Harmann, H. H., Modern Factor Analysis, 3rd ed. The University of Chicago Press, Chicago, 1976.
(5) Marsh, D.; Horvath, L.I. Structure, dynamics and composition of the lipid-protein interface. Perspectives from spin-labelling. Biochim. Biophys. Acta1998, 1376, 267-296.
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