Numerical Validation of IFT in the Analysis of Protein–Surfactant

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Numerical Validation of IFT in the Analysis of Protein−Surfactant Complexes with SAXS and SANS J. Matthew Franklin,†,§,# Lalitanand N. Surampudi,‡ Henry S. Ashbaugh,‡ and Danilo C. Pozzo*,§ †

Department of Chemical Engineering, Michigan State University, East Lansing, Michigan 48825, United States Department of Chemical and Biomolecular Engineering, Tulane University, New Orleans, Louisiana 70118, United States § Department of Chemical Engineering, University of Washington, Seattle, Washington 98195, United States ‡

S Supporting Information *

ABSTRACT: The use of the indirect Fourier transform methods for evaluating structural parameters directly in real space with small-angle scattering measurements is validated for the analysis of protein−surfactant complexes. An efficient Monte Carlo approach rapidly generates in silico structures based on a realistic pearl-necklace model for denatured proteins decorated with surfactant micelles. Corresponding scattering profiles are calculated and averaged over a large number of possible configurations for each structure. IFT algorithms are then used to calculate the corresponding pair-distance distribution function, and structural information is extracted directly without model fitting. The extracted parameters are compared and correlated with the known structure of the simulated complexes to assess the quality of the information that can be reliably obtained from these systems. The average extension, nearest-neighbor micelle distance, and average number of associated micelles are all accurately extracted through IFT calculations. Improved and simple approaches to reliably extract the average extension of the complex and the total number of associated micelles are presented.

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INTRODUCTION

Many experimental techniques are available to probe a wide range of length and time scales and to fully understand selfassembly and interactions in protein−surfactant complexes. Circular dichroism (CD) uses polarized light to measure chiral structures in biological materials. Far-UV CD probes the secondary structure of proteins and peptides and, for example, can be used to determine the relative fraction of α-helices, βsheets, and coils.8 Far-UV is also used to investigate conformational changes occurring in proteins during denaturation. Unfortunately, this technique cannot be used to determine the spatial distribution of these structural features.7,8 Near-UV CD analyzes tertiary structure and measures the degree of immobilization of aromatic residues during denaturation.8 Various nuclear magnetic resonance (NMR) techniques, like Global Protein State NMR9 and Pulsed Field Gradient NMR diffusometry,10 also allow scientists to explore conformational states and interactions in protein−surfactant systems. Unfortunately, most NMR techniques (e.g., NOESY, HMQC, HMBC, and HSQC) can only provide complete structural information on small protein systems where peak overlap issues are not significant. Furthermore, denaturation with surfactants can further complicate the NMR characterization of protein

Many processes and applications exploit the properties and functions of structures that emerge when amphiphilic molecules and proteins interact in aqueous solutions. Laundry detergent formulations often incorporate enzymes that must remain active in the presence of high concentrations of surfactant.1 Food properties including rheology, taste, and mouth-feel are often modified by controlling protein−surfactant or protein− lipid interactions.2,3 Biological pharmaceuticals, often incorporating proteins or peptides, are frequently formulated for delivery in liposomes and other lipid or surfactant nanostructures.4 Protein−surfactant complexes are also integral parts of workhorse techniques in analytical macromolecule separations such as polyacrylamide gel electrophoresis and related techniques.5 Despite a long history of utilizing protein− surfactant complexes in numerous applications, important questions still remain concerning the structure, dynamics, and underlying chemistry. Therefore, the detailed analysis of structure in protein−surfactant complexes will contribute to increase our understanding of complex interactions that emerge between proteins and surfactants. Ideally, this analysis is approached through a combination of methods that provide complementary insight into the structure of protein−surfactant complexes such that this information is used to improve existing technologies or to generate new ones.1,6,7 © 2012 American Chemical Society

Received: July 13, 2012 Revised: August 5, 2012 Published: August 5, 2012 12593

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Figure 1. Schematic description of the methods used to validate structural parameters of protein−surfactant complexes obtained through IFT. Blue arrows correspond to the typical experimental approach used to calculate P(r) in real protein−surfactant systems. Red arrows correspond to the approach used in this work to critically assess the information content of the P(r) obtained using IFT with simulated protein−surfactant structures.

deuterium substitution is known to significantly enhance hydrogen bonding interactions, and these effects could result in stronger binding interactions that are known to affect the thermodynamic and mechanical properties of proteins.17,18 Small angle X-ray and neutron scattering experiments also allow simultaneous experiments (e.g., spectroscopy, rheology, etc.) and external sample manipulation (e.g., application of shear, electric and magnetic fields) to be performed in situ during structure analysis.15,19 These techniques are frequently used to study biological macromolecules because of the versatility in environmental conditions and the commensurate length scales that are probed (0.1−500 nm). Scattering measurements also average structures over long time periods and over large particle ensembles, thus producing statistically meaningful structural information. Finally, high resolution is readily achieved to subnanometer accuracy allowing measurement of average micelle sizes.14 The current consensus structure for protein−surfactant complexes is the “pearl-necklace” model, where surfactant micelles (pearls) decorate an unfolded protein chain (necklace). These structures have been identified in numerous protein−surfactant complexes, and it seems to be the most common structure that is identified in SANS, SAXS, and cryoTEM.12,20,21 Unfortunately, there are no analytical models that could be used for fitting the form factor of these structures that simultaneously describe relevant intrinsic features such as chain flexibility, micelle separation distance, and steric and electrostatic interactions between micelles in a complex and between different protein−surfactant complexes.22 Therefore, most SAXS and SANS analyses of protein−surfactant complexes use indirect Fourier transformation (IFT) methods that allow for model-independent analysis of any dispersed structure in real space.23 IFT algorithms are used to calculate a real space pair-distance distribution function, P(r), from the scattering function in reciprocal space.23 For heterogeneous particles, this function is interpreted as the probability of two ends of a vector of length r having the same scattering length density contrast with respect to the solvent (Δρ) multiplied by the square of r (eq 1).

structures. In contrast, cryogenic electron microscopy (cryoTEM) has been used to provide a direct view of the structure in several protein−surfactant systems.11,12 These images provide a powerful direct observation of key structural features in protein−surfactant complex systems and have helped immensely to elucidate their shape in solution. However, cryoTEM also requires freezing that can lead to artifacts and it is difficult to obtain quantitative structural analyses because images are a two-dimensional projection of a three-dimensional system. It is also frequently difficult, time-consuming, and expensive to collect, analyze, and average enough images to obtain statistically significant parameters for many systems. Small angle X-ray scattering (SAXS) and neutron scattering (SANS) allow for direct structural measurements with nanometer-scale resolution on protein−surfactant dispersions without requiring any freezing or sample manipulation. Nevertheless, the extraction of accurate parameters from scattering profiles requires careful interpretation of the data. This work focuses specifically on evaluating the accuracy of structural data obtained from indirect Fourier transformation methods, a technique frequently used to interpret small-angle X-ray and neutron scattering (SAXS and SANS) experiments for protein−surfactant complexes.12−16 SANS and SAXS provide complementary structural information highlighting different parts of the structure and allowing for self-consistent quantitative measurements. Since Xrays interact primarily with electron clouds, the scattering length density (ρ), a measure of the scattering power, is a function of the local electron density and proportional to the atomic number of the atoms. In contrast, neutrons interact with the nuclei and this interaction varies non-monotonically with atomic number. The neutron−atom interaction and corresponding ρ also depend on the specific isotope type, providing the particularly strong contrast with hydrogen (H1) and deuterium (H2) isotopes. Differences in the nature of the scattering interaction allow SAXS and SANS to offer two complementary views of the same structure for rigorous and self-consistent analysis. On the other hand, potential structural effects can sometimes arise from the replacement of hydrogen atoms with deuterium in the solvent or within proteins. Therefore, isotope-substitution effects must be carefully considered when directly comparing SAXS and SANS data, as they could potentially lead to structural differences. Specifically,

P(r ) = 12594

∫V Δρ(r1)·Δρ(r1 − r) dr1

r2

(1)

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generate physically accurate protein−surfactant structures and scattering profiles. However, protein−surfactant complexes are typically too large to be able to generate enough statistically distinct structures and conformations to provide meaningful results with reasonable computational resources. Currently, MD methods take multiple days to simulate tens of nanoseconds for a system with less than 106 atoms. Since the goal of this work is only to validate the physical interpretation of IFT profiles, the simulation of detailed structures that properly and accurately account for all interactions (i.e., covalent, van der Waals, electrostatic, and steric interactions) is not necessary to achieve this objective. Here, we focus on assessing the extraction of structural parameters from P(r) profiles obtained through IFT methods and to identify potential pitfalls in the use of this method. Instead of full atomistic simulations of protein−surfactant complex structures, we employ a mixed-scale approach in which atomistic MD simulations are used to generate individual micelle structures and a coarse-grained Monte Carlo simulation of a polymer random walk are used to build thousands of independent “pearl-necklace” structures decorated with these micelles. Therefore, the only interactions that are fully taken into account in our mixed-scale method are the excluded volume interactions between micelles and the intermolecular interactions between surfactants located within individual micelles. The excluded volume effect is accounted for by rejecting any protein−surfactant complex configurations with overlapping micelles. For each scattering calculation, 500 protein−surfactant complexes were independently generated using the random walk algorithm. These configurations represent the ensemble of structural fluctuations averaged in typical SAXS and SANS experiments by virtue of the macroscopic size of the beam and the duration of the measurements. The scattering profile sufficiently converges after 500 configurations (see Supporting Information). Scattering profiles are then calculated with Crysol and Cryson and averaged for all of the simulated structures. The corresponding P(r) is then inverted through IFT with the program GNOM.24 The structural parameters that are obtained from the interpretation of P(r) are then correlated to the known structural parameters of the simulated structures in order to validate the use of the IFT method and to identify potential pitfalls in the interpretation of experimental P(r) functions.

This function is effectively the real-space analogue of the scattering profile, containing equivalent structural information. Because it is in real space, it is easier to interpret the P(r) directly to obtain structural parameters without having to assume or guess a structural model or form-factor. The P(r) can also be calculated from any known distribution of scattering length density (ρ) or particle shape. The particle shape information can come, for example, from geometric models or from computer simulations. IFT methods calculate P(r) through Fourier inversion of the scattering profile using eq 2: P(r ) =

1 2π 2

∫0

∞

I(q)qr sin(qr ) dq

(2)

where q is the scattering vector and I(q) is the scattering intensity. Several algorithms for performing this inversion have been published in the literature.23,24 Nevertheless, when properly done the results should be independent of the specific algorithm that is used. The use of IFT to extract structural information from protein−surfactant complexes has been routinely applied throughout the literature.12−16,20,21 The method has never been validated, however, for accuracy and the interpretation of different features in P(r) for systems consisting of protein− surfactant complexes. This lack of validation can lead to inappropriate interpretation of P(r) functions and raises questions about the quality of the information that is being extracted. Moreover, it is also possible that all the usable information contained in the P(r) may not be extracted leading to incomplete interpretation of results. Here, we simulate a range of protein−surfactant structures and evaluate P(r) to systematically interrogate the accuracy of the IFT method and guide future research in the analysis of protein−surfactant complexes via SANS and SAXS. Figure 1 depicts the process that we use to validate the interpretation of P(r) for protein− surfactant complexes using the “pearl-necklace” model. Blue arrows indicate the typical process used by experimentalists to interpret experimental profiles with IFT. In contrast, we follow two separate paths shown by red arrows to simulate structures with known structural features and to also calculate scattering profiles from these simulations and invert them with IFT. This fully in silico procedure allows us to directly compare structural parameters obtained from IFT to the known parameters of the simulated structures. This assessment is very important because structural information is always lost when performing scattering experiments (i.e., the phase problem). Furthermore, the information quality can also degrade through the numerical Fourier transformation process, even though this is a welldefined mathematical transform, because extrapolation of real experimental data toward zero and infinity is always necessary.25 Scattering profiles can be numerically calculated given any molecular resolution structure by appropriately accounting for all interatomic correlations and all possible particle orientations. The ATSAS suite, developed by the BIO-SAXS group at EMBL-Hamburg, is an established set of programs for performing scattering analysis on biological systems. The programs Crysol26 and Cryson27 calculate scattering profiles corresponding to SAXS and SANS experiments, respectively, from any atomic-level structure in Protein Data Bank (PDB) file format. In this work, hundreds of protein−surfactant complexes are generated in silico and scattering profiles are subsequently calculated and averaged. Full atomistic molecular dynamics simulations could, in principle, also be used to

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MODEL DESCRIPTION We performed all-atom simulations of sodium alkyl sulfate micelles in water using the GROMACS 4 molecular dynamics package.28 Initially spherical micelles were constructed with aggregation numbers for sodium octyl sulfate (SC8S), sodium decyl sulfate (SC10S), and sodium dodecyl sulfate (SC12S) of 21, 40, and 64, matching experimental results for micelles in water at ambient conditions.29 No added salt was considered. Water was modeled using the TIP3P30 description, while the General Amber Force Field31 was used to model the surfactants with headgroup partial charges taken from the work of Bruce and co-workers.32,33 The number of waters included in each simulation was 2625, 4326, and 6542 for the octyl, decyl, and dodecyl surfactant simulations, respectively. Energy minimization was performed before and after hydration of the micelle in a cubic box with periodic boundary conditions. These minimized structures were simulated for 2 ns for equilibration, followed by a 10 ns production run to obtain micelle snapshots. 12595

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Figure 2. Structure of SC12S, SC10S, and SC8S micelles from MD simulations and the corresponding SAXS/SANS P(r) calculations from GNOM. The scattering calculations correspond to averages from 83, 92, and 100 micelle configurations for SC8S, SC10S, and SC12S sampled at random from 10 ns of simulation time.

extended contour length of a protein, because these typically maintain a significant fraction of secondary structure in protein−surfactant complexes.7,8 LK represents the length of freely jointed segments in the random walk. We define LK as a percentage of LC for each structure. Larger LK values represent more rigid protein backbones. S corresponds to the spacing of micelles along the path of the random walk and the value is defined as a multiple of the micelle diameter. For each protein− surfactant structure that is calculated, we generate 500 independent configurations to account for ensemble and time averages that occur in real scattering experiments. This number of configurations is determined to be sufficiently large for the averaged scattering profile to converge (Supporting Information). This efficient simulation method allows us to sample a large parameter space for different protein−surfactant nanostructures while only using a modest amount of computing power. In total, 23 500 independent protein−surfactant conformations are generated, corresponding to 47 different parameter sets (i.e., surfactant type, LC, LK, and S). Example conformations for protein−surfactant complexes are shown with the corresponding SANS P(r) in Figure 3. In the calculation of scattering profiles with Crysol (SAXS) and Cryson (SANS), the contribution of the diffuse Na+ counterion shell, the protein backbone, and possible water hydration layers near the micelles are all neglected. For SANS calculations with Cryson, the surfactants are assumed to be fully hydrogenated and the solvent to be pure D2O. To increase the density of points in the scattering profile, two scattering calculations are performed over different q-ranges and the results are concatenated. This results in a q-range of 0.01 to 0.5 nm−1 with 386 data points for SAXS data and a q-range of 0.04 to 0.5 nm−1 with 99 data points for SANS data. After averaging the scattering profiles of 500 independent configurations, the P(r) is calculated using indirect Fourier transformation with the program GNOM. The regularization parameter, α, represents the Lagrange multiplier in IFT calculations by GNOM. With a priori structural information, α can be simply chosen to yield a reasonable P(r).24 Choosing α requires one to balance curve smoothness over detailed features. In this work, the α value is

A time step of 2 fs was used to integrate the equations of motion. Simulations were performed in the isothermal−isobaric ensemble at 300 K and atmospheric pressure. The temperature and pressure were maintained using the Nosé-Hoover thermostat34 and Parrinello-Rahman barostat,35 respectively. Particle Mesh Ewald summation36,37 was used to evaluate electrostatic interactions and the LINCS algorithm38 was used to constrain the geometry of water and fix the carbon−hydrogen bond lengths on the surfactant alky chains. From the simulation trajectory, 100 evenly spaced snapshots of the micelle were sampled and the origin of the simulation box was repositioned at the micelle center-of-mass for scattering analysis. Frames that included free surfactant monomers were discarded, leaving 83 frames for SC8S, 92 for SC10S, and 100 for SC12S. The selected frames were used to calculate average SAXS and SANS profiles of a single micelle for the three surfactants using Crysol and Cryson, respectively. The structures and corresponding P(r) functions are shown in Figure 2. The SAXS profiles show two peaks while the SANS profiles show a single peak typical of spheroidal surfactant micelles of hydrogenated surfactants. These types of profiles are typical for particles with homogeneous (SANS) and heterogeneous (SAXS) radial scattering length density distributions.39 The single micelle frames are then randomly selected to build the structures of the protein−surfactant complexes. Previous work showed that, in most cases, the structure of the micelles attached to proteins is almost identical to that of free micelles.14,15 The aggregation number of SDS, however, can be different in the presence of proteins than in free micelles.13 Nevertheless, we expect our approach to generate sufficiently realistic protein−surfactant complexes that are necessary to validate the IFT methods. The protein is assumed, reasonably, to be completely denatured by the surfactant. As such, we simulate the protein as a random-walk coil that is decorated with micelles. The protein−surfactant complexes are generated with a fixed contour length (LC), Kuhn length (LK), and distance between micelles (S). LC is the total length of the random walk, and it simulates the extension of the protein backbone. Note that the LC of simulated structures would not typically match the fully 12596

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between neighboring micelles can also be estimated from the third peak of the SAXS P(r) and second peak from the SANS P(r) of the protein−surfactant structure (see Figure 4), as suggested in previous work.14 Despite the detailed information that is contained in the scattering profiles, a simple method for estimating the total number of micelles associated to a given protein complex has not been reported in the literature for these systems. We propose a simple and accurate method to extract this information from the partial integration of the P(r) function for protein−surfactant complexes. This method is based on the fact that the shape of the P(r) function for low distances is primarily determined by the shape of the associated micelles. For pearl-necklace structures, this region can usually be isolated from the rest of the P(r). The integral of P(r) is also directly related to the scattering intensity at zero angle, I(0).39 The ratio of areas of the intermicelle to intramicelle features then correlates to the total number of micelles on the structure. A detailed description and justification for using this method is given in the Supporting Information. Figure 5 shows the

Figure 3. P(r) plots of pearl-necklace structures for different surfactant micelles corresponding to simulated SANS data using Cryson and typical snapshot configurations (out of 500 total). LC = 40 nm, S = 2DMicelle, and LK = 0.1 LC.

set to 0.05 for all calculations which produced the P(r) plots with all of the structural features being investigated while retaining smooth curve characteristics. The largest particle dimension (LMax) can be measured directly from the P(r) function as the point where values reach zero or cross the x-axis. However, for many flexible (or polydisperse) systems, P(r) is found to decay very smoothly to zero at high r (Figures 2−4). This creates a significant tail

Figure 5. Schematic of integration region used to calculate the number of micelles associated to the proteins. The ratio of the full integration of P(r) to the partial integration (dark area) directly correlates to the average number of micelles (N) in the protein−surfactant complex.

aforementioned P(r) integration regions for SANS and SAXS data. This integration method can be performed on experimental P(r) functions without requiring absolute intensity data. However, the proposed method does require that only protein−surfactant complexes are present in the sample, because free micelles in solution will contribute to the single-micelle region in the PDDF causing an underestimation of the number of associated micelles to the protein. Therefore, careful dialysis must be performed to ensure that background surfactant concentrations are below the CMC. The micelle structures from hydrogenated surfactants in SAXS experiments can be modeled as spherical core−shell structures with a heterogeneous scattering length density distribution, ρcore < ρshell and ρcore ≈ ρsolvent. In contrast, the same structures are well represented as spheres with homogeneous scattering length density with ρcore < ρsolvent in SANS experiments with D2O buffers. Therefore, the two structures appear very different when the two P(r) values are calculated (Figures 4 and 5). This effect is critically important for the correct interpretation of SAXS or SANS data from IFT, because it could lead to significant confusion in the extraction of structural parameters.

Figure 4. Typical P(r) of pearl-necklace structures calculated from the simulated SAXS and SANS data for SC8S, SC10S, and SC12S with LC = 40 nm, S = 2DMicelle, and LK = 0.1LC. The structural parameters that can be directly extracted from the P(r) of protein−surfactant complexes are highlighted for SC12S.

region that often causes uncertainty in the extraction of accurate values for LMax for these complexes. In addition, basis functions used in IFT inversion are often oscillatory causing small oscillations at large distances. This causes the exact crossover point to be very sensitive to the maximum distance (rMax) used to define the upper bound in the IFT calculation. To overcome these limitations in the extraction of LMax, we propose a more accurate method based on fitting a linear function to the decay region and extrapolating this line to zero probability. Figure 4 shows an example P(r) with the extrapolation crossover point as opposed to the real crossover point. The extrapolated crossover point provides a good measure of the average LMax (see Results section). The average distance 12597

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Figure 6. Correlation between the average overall extension extracted from P(r) calculations of SAXS (A) and SANS (B) profiles and the real average extension obtained directly from the simulated structures. The P(r) distance is estimated by extrapolating the linear segment of the decaying region of the P(r) (see Figure 4). Dashed lines correspond to parity and the solid line corresponds to the best linear fit.

Figure 7. Correlation between average distance between two micelles. The geometric distance is calculated from the actual structure, and the P(r) distance is estimated from the position of the third peak for SAXS (A) and the second peak for SANS (B) P(r), respectively. Dashed lines correspond to parity and solid line corresponds to the best linear fit.

Figure 8. Correlation between the calculated number of associated micelles (N) from the P(r) and the real number of micelles from the simulation. Additional details of the method are provided in the Supporting Information. Dashed lines correspond to parity and solid line corresponds to the best linear fit.

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RESULTS

estimated more accurately than by determining the exact crossover point. The region used for this linear extrapolation is determined by identifying a linear range of at least 5 nm that is close to the crossover point. This method simulates the procedure that could be used to interpret a P(r) function obtained from experiments. Micelle Nearest-Neighbor Distance. The nearest-neighbor micelle distance, LSep, is calculated from the simulated structures by averaging the distance between nearest-neighbor micelles. For each micelle in a protein−surfactant structure, the

Following the process described in Figure 1, the structural parameters from the P(r) functions and the actual “pearlnecklace” models are extracted and compiled into Figures 6−8. Average Extension of Protein−Surfactant Complexes. The average particle dimension from the P(r) function and the average model structure correlate very well for both SAXS and SANS data (Figure 6). By extrapolating P(r) to zero in the linear region (Figure 4), the average particle dimension can be 12598

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center-to-center distance to the next nearest micellenot necessarily located along the chainis calculated and values are averaged for all micelles in the structure. This parameter is also quantified in the P(r) from the position of the third peak for the SAXS data and the second peak for the SANS data. The correlation between the extracted parameter and the real value is shown in Figure 7. A strong 1:1 correlation is obtained in both SAXS and SANS data, but these are weaker than those determined for the maximum extension (LMax). The correlation of the next-nearest neighbor distance in SANS is also slightly stronger than in SAXS. Average Number of Micelles in Pearl-Necklace Complexes. The average number of surfactant micelles associated to a pearl-necklace structure (N) is a valuable piece of structural information. The first peaks of SAXS and SANS P(r) are correlated to the intramicelle structure, while the other features correspond to intermicellar interactions. Upon integration of the P(r), the ratio of the intermicelle features to the intramicellar region correlates directly to the number of micelles in the structure for SANS data (Figure 8). However, the correlation of the ratio of integrals to the known value of N in the SAXS data is much weaker. The correlation is also particularly weak for the smaller surfactant micelles (SC8S), and when this data is removed, the correlation quality improves significantly. The justification for the use of this method for estimating N, as described in the Supporting Information, is strictly valid for micelles that have a homogeneous distribution of ρ. Therefore, it is not surprising that the correlation is stronger for hydrogenated surfactants in simulated SANS data (condition met) than it is with SAXS data (condition not met). In hydrogenated surfactants, the sulfate groups have a much larger electron density than the hydrocarbon tails. Therefore, this results in a heterogeneous distribution of electron density contrast for hydrogenated micelles characterized with SAXS. In SANS, the largest contrast is between the hydrogenated surfactant tails and the deuterated solvent. Therefore, these micelles and the corresponding protein−surfactant complexes can be considered to be homogeneous “particles”. This fundamental difference in contrast is also the explanation for why the two techniques result in the observation of a single correlation peak corresponding to the intramicelle scattering in SANS and two correlation peaks in SAXS profiles of the same structure.

chains. Hydrophobicity and charge distribution calculations from known protein sequences could be used to map potential micelle locations based on local physicochemical properties.40 Nonbonded electrostatic and van der Waals interaction potentials could also be incorporated using similar Monte Carlo methodologies or with computationally efficient coarsegrained MD simulation to accurately model the influence of surfactant interactions in more realistic denatured proteins while still being able to calculate statistically meaningful scattering profiles. This approach could even be used to simulate the early stages of protein denaturation, which is known to be a complex process.41 With these and other improvements, new approaches to fit experimental data from these systems could be developed.

CONCLUSIONS A multiscale simulation approach is used to validate and verify the accuracy of the structural information that is extracted from experimental P(r) functions in protein−surfactant complexes with pearl-necklace structures. The results demonstrate that accurate structural information can be extracted reliably from these scattering measurements. The overall particle dimension (LMax), average nearest-neighbor distance (LSep), and the average number of micelles (N) are all accurately calculated, validating measurements used to study protein−surfactants using SAXS and SANS for structures described by the pearlnecklace model. The calculation of the number of micelles (N) from the P(r) is a new result that could prove to be especially useful in future small-angle scattering studies. In addition, this initial approach suggests that more accurate “pearl-necklace” models could also be developed to take into account the nonuniform spatial distribution of micelles along the protein backbone, electrostatic interactions between micelles, and any potential contribution of the polypeptide

VARIABLE DEFINITIONS: ρ, scattering length density; P(r) = ⟨∫ V Δρ(r1) ·Δρ(r1−r) dr1⟩r̂2, pair-distance distribution function; DMicelle, average micelle diameter; LC, contour length, the total distance along the random walk path; LK, Kuhn length, the length of the random walk segments; S, distance between micelles along the random walk path; LMax, largest particle dimension, distance between the two farthest apart atoms; rMax, upper bound of the IFT calculation; N, number of micelles in structure; LSep, average separation distance between micelles in the simulated structures. This corresponds to center-to-center distance to the next nearest micelle and, unlike S, is not necessarily defined along the contour of the polypeptide chain. The values are averaged for all micelles in the structure and for all simulated configurations.

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ASSOCIATED CONTENT

S Supporting Information *

A mathematical justification for calculating the number of micelles from the pearl-necklace structure and the basis for selecting 500 structures. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address #

Whitaker Fellow at European Molecular Biology Laboratory, EMBL-Hamburg. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J. Matthew Franklin gratefully acknowledges funding provided through the University of Washington Amgen Scholars undergraduate research experience program. We are also grateful to Prof. James Pfaendtner for providing computing resources used to perform these calculations and for his thoughtful comments. We also acknowledge support from the National Science Foundation for funding through the grant No. EEC-0824347.

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REFERENCES

(1) Otzen, D. Protein−surfactant interactions: A tale of many states. Biochemi. Biophys. Acta 2011, 1814, 562−591.

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dx.doi.org/10.1021/la3028379 | Langmuir 2012, 28, 12593−12600