Computing the Diamagnetic Susceptibility and Diamagnetic

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Computing the Diamagnetic Susceptibility and Diamagnetic Anisotropy of Membrane Proteins from Structural Subunits Mahnoush Babaei, Isaac C Jones, Kaushik Dayal, and Meagan S Mauter J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b01251 • Publication Date (Web): 18 Apr 2017 Downloaded from http://pubs.acs.org on April 23, 2017

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Journal of Chemical Theory and Computation

Computing the Diamagnetic Susceptibility and Diamagnetic Anisotropy of Membrane Proteins from Structural Subunits

Mahnoush Babaei,1 Isaac C. Jones, 1 Kaushik Dayal, 1 and Meagan S. Mauter,1,2*

Author Affiliations: 1

Department of Civil and Environmental Engineering, Carnegie Mellon University, 5000

Forbes Ave., Pittsburgh PA 15213. 2

Department of Engineering and Public Policy, Carnegie Mellon University, 5000

Forbes Ave., Pittsburgh PA 15213. *Author to Whom Correspondence Should Be Addressed e-mail: [email protected] phone: (412) 268-5688 Keywords: membrane protein, magnetic susceptibility, diamagnetic anisotropy, magnetic energy

1

ACS Paragon Plus Environment


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Abstract The behavior of large, complex molecules in the presence of magnetic fields is experimentally challenging to measure and computationally intensive to predict. This work proposes a novel, mixed-methods approach for efficiently computing the principal magnetic susceptibility and diamagnetic anisotropy of membrane proteins.

The

hierarchical primary (amino acid), secondary (alpha helical and beta sheet), and tertiary (alpha helix and beta barrel) structure of transmembrane proteins enables analysis of a complex molecule using discrete subunits of varying size and resolution. The proposed method converts the magnetic susceptibility tensor for all protein subunits to a unit coordinate system and sums them to build the magnetic susceptibility tensor for the membrane protein. Using this approach, we calculate the diamagnetic anisotropy for all transmembrane proteins of known structure and investigate effect of different subunit resolution on the resulting predictions of diamagnetic anisotropy. We demonstrate that amino acid residues with aromatic side groups exhibit higher diamagnetic anisotropies. On average, high percentages of aromatic amino acid subunits, a beta barrel tertiary structure, and a small volume are correlated with high volumetric diamagnetic anisotropy.

Finally, we demonstrate that accounting for the spatial position of the

residues with respect to one-another is critical to accurately computing the magnetic properties of the complex protein molecule.

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Introduction Membrane proteins (MPs) provide highly specialized sensing, transport, and catalytic processes that facilitate cellular communication with the external environment. Recent progress in 2D MP crystallization has facilitated their structural characterization and incorporation into engineered devices.1–4 Expanding the quality, efficiency, and scalability of techniques for 2D MP crystallization will further facilitate their application as responsive elements in functional devices.5–12 A promising route toward manipulating the crystallization process is to apply an external magnetic field during MP self-assembly.13 This technique was used in crystallization of Aquaporin-0 (Aqp0), a water pore expressed in the fiber cells of the mammalian lens, and Outer membrane protein F (OmpF), a porin expressed in Escherichia coli that allows for the passive diffusion of hydrophilic small molecules.

We observed significant

improvements in both short- and long- range order for both proteins, though the effect was stronger for the OmpF than for Aqp0. The difference is attributed to the higher diamagnetic anisotropy (DA) of OmpF protein. Unfortunately, the absence of a rigorous method for determining the DA of complex molecules has made it difficult to assess the generalizability of this technique. A robust method for rapidly assessing the DA of structurally diverse MPs would provide mechanistic insight into the relationship with protein structure and experimental insight into the magnetic field strength necessary to facilitate crystallization across a diverse set of MPs. It would also aid determination of orientation restraints in protein nuclear magnetic resonance experiments.14 3



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Existing methods for determining the magnetic properties of complex molecules, like MPs, include both experimental and computational approaches. Force based methods were the first experimental approaches for measuring the magnetic properties of materials.

The Gouy method,15 and its correlaries,16,17 estimate the volumetric

diamagnetic susceptibility (DS) of materials by relating DS to the change in the mass of the sample upon application of a magnetic field.

The Faraday method18 has an

arrangement similar to Gouy balance, but allows for smaller samples sizes. Here, the sample is placed in an inhomogeneous magnetic field and the mass susceptibility of the sample is determined by monitoring the change in force exerted on the sample upon application of the magnetic field.

Finally, for large isotropic single crystals, the

Krishnan’s critical torque method19 relates crystal’s response to torsion in the presence of a magnetic field to the molar anisotropy in the horizontal plane of the suspended crystal. A clear limitation of this method is its application to materials that do not readily form single crystals of considerable size. Additional experimental approaches for measuring the DS and DA of samples include induction methods using supraconducting quantum interference devices,20–23 SQUID, to measure magnetic flux quanta as the sample passes through the detector coil or nuclear magnetic resonance (NMR)24,25 methods to measure the differences in chemical shift between a sample and a reference in sample tubes of different shape factors. In short, experimental techniques are capable of directly measuring the DA and DS of materials19,26,27, but these methods are labor intensive, can be error prone, and are often experimentally limited by the properties of the sample, the purity of the sample, or the bulk nature of the measurements.28

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Computational techniques, on the other hand, are capable of screening the magnetic properties of molecules with well characterized chemical sequences. Most computational techniques use ab initio methods to estimate DS of simple molecules by applying molecular orbital theory29,30 or valence electron wave functions31–34 at the atomic35–37 or bond38,39 levels. Each of these methods estimates the interactions between atoms and their electrons in the magnetic field by adopting different approaches such as HartreeFock approximation method40,41 for solving the Schrodinger equation.

The

computational intensity of these approaches have primarily limited their application to structural moieties or small compounds.42–44 For larger or more complex molecular structures, these techniques are complimented by a set of semi-empirical methods, such as Kader’s method45, that weight and sum the DS of atoms or structural moieties. These methods do not incorporate information about the relative position of atoms and bonds within the molecule, and are therefore expected to perform poorly for large, highly ordered molecules, including many MPs. Another approach is to use coupled quantum equations to provide computational estimates of NMR chemical shifts and DS for complex chemical structures.46–48 A range of computational methods are employed, including ab initio approaches such as HartreeFock and density functional theory; gauge-independent atomic orbital (GIAO)49–54 and individual gauge for localized orbital (IGLO) approaches48,55, and gradient or analytical derivative methods.51

Initially these computational methods were limited to small

molecules, but current approaches have been adapted for biomolecules56–58 and other large molecules.59–63

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A critical shortcoming of computational NMR methods is the challenge of estimating the molecular DA. Though recent work on molecular rings and small linear structures use magnetically induced current density functions to provide insight into electron mobility, chemical shifts, and DA,58,64–66 these functions are currently limited in application to a small set of molecular structures. In short, we lack a robust technique for rapidly estimating the DA of complex molecules with potentially high degrees of internal order, such as protein systems.56,59 Few of these experimental or computational techniques for estimating DS and DA have been applied to MP systems.56,59 Experimental measurements of protein DS and DA have been widely performed on proteins that crystallize in three dimensions, but the difficulty of crystallizing in 2D has stymied similar measurements on MPs.67–70

Ab initio

techniques have been applied to estimate the DS and DA of peptide bonds,71,72 but their computational intensity limits their study of entire proteins. Finally, the few papers that empirically estimate the DA of the proteins neglect information about the relative position of protein subunits, instead assuming that all amino acids in the protein sequence are parallel.71 As a result of these gaps in the research, there is little understanding of how primary, secondary, or tertiary structure influences MP DA. Clarifying this relationship in an empirical study would better inform the level of structural resolution necessary when computationally estimating the protein’s DA. This paper presents a mixed-methods approach for calculating the DS and DA of MPs. Using experimentally and computationally derived estimates of the DS of structural subunits from the published literature, we compute the magnetic properties of MPs at the

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primary, secondary, and tertiary structure levels.

We then apply these estimates to

investigate the value of information associated with estimating DA using amino acid subunits, and to calculate the magnetic energy, relative to thermal energy kT, required to align MPs in a magnetic field. While this paper develops this technique using MPs, it is broadly applicable for other large molecules, including globular, fibrous, and disordered proteins, where the exact positions of atoms and the magnetic properties of their subunits are known. Theory Magnetization occurs whenever a magnetic field is imposed on a substance other than a vacuum. Depending on the atomic composition, the substance will exhibit either a diamagnetic or a paramagnetic response. In presence of an applied magnetic field of magnitude H, the magnetic induction, B, is given by Equation 1.

(1)

B = H+4πM

The magnetic induction, also known as the magnetic flux density, is the cumulative effect of both applied magnetic field and the magnetic moment per unit volume, M, due to the induced magnetic field in the substance. For most materials at room temperature, there is an approximately linear relationship between the applied magnetic field intensity and the magnetic moment per unit volume of the substance. This relationship is described by the material’s volumetric magnetic susceptibility, , and is defined as the dimensionless proportion of the magnetic moment per unit volume to the applied magnetic field (Equation 2). The molar magnetic susceptibility (cm3/mol) is then obtained by

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multiplying the normalized volumetric susceptibility by the molar volume of the substance.

=

(2)

While the magnitude of magnetic susceptibility indicates the degree of magnetization of a substance, its sign indicates the type of magnetic behavior. A positive susceptibility indicates a paramagnetic response for the material, while a negative sign implies diamagnetic behavior.

In other words, diamagnetism describes orbital motion of

electrons in the substance that oppose the applied magnetic field. The applied magnetic field causes the free electrons in the bonds to circulate and induce a current. This current will subsequently produce an electromagnetic field, which works in the same direction as the applied magnetic field inside the ring and in the opposite direction outside of it. While magnetic susceptibility is reported as a scalar value for isotropic materials, asymmetrical structural features often introduce anisotropies in the material’s magnetic susceptibility. The magnetic susceptibility along the principle axes of the material are represented as tensors, and their corresponding principle directions are used to determine the local coordinates of the substance. Diamagnetic anisotropy is then defined as the difference between maximum and minimum principle magnetic susceptibilities of the material.

For instance, the diamagnetic anisotropy of an asymmetric structure with

< < is calculated using Equation 3. ∆ = −

(3)

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In a molecule with axisymmetric structure, and principle magnetic susceptibility of ≈ < , the in-plane and normal to the plane susceptibilities, respectively shown

by ∥ and are determined using Equations 4 and 5.

∥ = ( + )

(4)

=

(5)

For this structure the diamagnetic anisotropy (DA) is then defined using Equation 6.

∆ = − ∥

(6)

Methods Selection of Membrane Proteins: We calculate the DA for membrane proteins of known structure from the list of resolved 3D structures included in the mpstruc database as of September 1, 2016.73 Structures in this database are classified into three main groups of alpha helices, beta barrel and monotopic, which in this paper are considered as the different types of tertiary structures for the MPs. To establish local coordinates of a peptide bond, the proposed method requires the coordinates of at least three atoms from that peptide bond. Of the 642 available MP structures, 7 have coordinates only for the alpha-carbons in the structure, which makes them unfit for the proposed method. Another 7 MPs exceed the size limit for the .pdb file format (more than 62 chains and/or 99999 atom lines), and were excluded from the

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present analysis. In total, we calculate the DA of 132 beta-barrel, 451 alpha-helical, and 45 monotopic MPs. The average number of residues for beta-barrel, alpha-helical and monotopic structures is 850, 1096 and 1215, respectively. A complete list of MPs included in the calculations is provided as a .csv file embedded in SI Section 4, as is a list of excluded MPs. Specification of Protein Subunits: The DA of the MP is calculated by determining the principle DS of protein subunits, transforming the principle axes of each subunit from local to global coordinates (Figure 1A), and summing over all subunits. We calculate the DA of MPs using molecular moiety subunits of either peptides or peptides plus moieties with aromatic rings. We compare these results to the DA of each MP using the DA of amino acid subunits that we estimate as described below. Each subunit includes data on the spatial relationship between atoms, the principle DS of the subunit, and the principle direction of the magnetic susceptibility tensors, or the eigenvectors. The spatial relationship between subunits within the macromolecule was obtained from the protein data bank (PDB) file for each MP. The principle magnetic susceptibilities, or a combination of mean magnetic susceptibility and diamagnetic anisotropy, for each subunit were obtained from previously published literature.19,71,74–81 The primary factors in selecting between estimates of subunit DS are the structure of the macromolecule and the desired accuracy. While bond-level subunits would appear to be the minimum logical unit, selecting bonds omits the effect of electron resonance within the molecule.82 Instead, we select molecular moieties as the minimum subunit such that the effect of resonance electrons is already included in their magnetic properties. Although this approach will not give us the same accurate results as the first principal

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methods, it closely matches the available experimental values with much lower computational costs, which meets the purpose of this paper. For the “peptide bond” case, we approximate the MP as a series of peptide bonds and omit all other moieties. We perform these calculations with and without accounting for the position of the peptide bond in the MP.

This latter case reflects the state-of-art in current computational

approaches for estimating the DA of MPs, and the present approach represents an improvement on that state-of-art. For the “peptide bond plus aromatic ring” case, we also incorporate the DS and position of molecular moieties into the calculation of DS tensors. Because the DS of moieties spans several orders of magnitude, we consider only those moieties with DS values on or above the order of the peptide bond. For the MPs investigated in this study, all of the moieties considered are peptide bonds or molecular moieties containing aromatic ring structures. Amino acids, or the MP primary structure, was also evaluated as a minimum subunit. Here, data availability was limited to experimentally measured values for the mean molar magnetic susceptibility of a subset of 14 amino acids.

The mean molar magnetic

susceptibility for the remaining 6 amino acids was estimated using the RealTime Predictor™program.83 Like the experimentally reported values, this program does not report the tensor values for DS. Thus, we calculate the magnetic susceptibility tensors for the 20 major amino acids using their full molecular structure. Specification of Amino-Acid Subunits: To establish the magnetic susceptibility tensor for amino acid subunits, we divided the amino acid into smaller molecular subunits with available magnetic properties. We then apply a similar approach to that of calculating

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MP DA by determining the principle DS of the molecular subunits, transforming the principle axes of each subunit from local to global coordinates (Figure 1A), and summing over all subunits. In order to preserve the accuracy of magnetic properties stemming from electron resonance, amino acids were sub-divided into molecular groups that maximized the number of atoms per group. Further information on the selection of molecular groups is available in the SI Section 1.

Whenever available, data from

previous literature was used for molecular groups such as alkane bonds or peptide bonds. However, many side groups could not be accurately represented as a series of single molecular bonds due to electron resonance or to unique atoms such as nitrogen, oxygen, or sulfur for which data was unavailable. Instead, we approximate the principle magnetic susceptibility of these side groups by resolving each side chain into smaller groups of bonds, as previously described. After collecting all the required data for each molecule, the calculations were repeated 10 times for each type of amino acid.

The average values for principle magnetic

susceptibilities are then reported and used in the further calculations. To calculate the magnetic properties of the larger protein structures, using amino acid subunits, we determine the local coordinates and magnetic susceptibility tensor for each subunit within the structure. We define the local coordinates of each amino acid as the local coordinates for the peptide bond of that amino acid. The magnetic susceptibility tensor in local coordinates is equal to the magnetic susceptibility tensor for the subunit from the subunit calculation step. The ability to use secondary (alpha helix and beta strand) and tertiary (alpha helix and beta barrel in this paper) structures as subunits for the diamagnetic anisotropy

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calculations was then investigated using the calculated DA from the primary subunits. 229 beta strands and 215 alpha helices with different lengths and sequence of amino acids were used for investigating the secondary structure. We calculate the DA of MPs using different molecular moiety subunits and compare the results of calculating the DA using primary (amino acid) subunits, secondary (alpha helix, beta strand) subunits, and tertiary (alpha helix, beta barrel) structure. Calculation of Diamagnetic Anisotropy: Using the spatial coordination of atoms, a set of local axes for each subunit was established. As shown in Figure 1B, for an example of benzene ring, the local axes were defined to be the same as the principle magnetic susceptibility direction for each subunit. Complete information on the principle magnetic susceptibilities and their corresponding direction for all subunits is provided in SI Section 1. The data for principle magnetic susceptibility of each subunit, named as K1, K2 and K3, was then used to establish the principle magnetic susceptibility tensor (Equation 7).

=0 0

0 0

0 0

(7)

In the next step, the magnetic susceptibility tensor for each subunit, which was defined in the system of its local coordinates, was transformed to a single global coordinate. Considering and to be the global and local axes as shown in Figure 1a, the rotation matrix was defined as in Equation 8 and applied on the susceptibility tensor as Equation 9,

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x'2

B

A

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x3

cos-1Q31

x'3

x'3

x'1

x'1

cos-1Q33

C x1

x'3

x2 x'1

-1

cos Q12

x'2

B x'2

Figure 1. (a) Transformation between the local and global coordinate system and the physical interpretation of elements of the rotation matrix. (b) Local coordinates system defined for a benzene ring subunit based on the principle magnetic susceptibility directions. (c) Visualization of an applied magnetic field and the induced magnetization for a single benzene ring.

= cos ( , )

(8)

!" = !"!"! # "

(9)

Here, Q is the rotation matrix used for transforming the magnetic susceptibility tensor from local to global coordinate system (Figure 1A). Throughout this paper, K is defined as the magnetic susceptibility tensor in local coordinates while is the magnetic susceptibility tensor in the global coordinate, for the same subunit. Although the choice of global coordinate was arbitrary, it remained consistent for all the subunits during the calculation. Here, the global Cartesian coordinate system was chosen

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as the global coordinate system for the membrane protein. As shown in Figure 1C, the applied magnetic field on the membrane protein is considered along the axes . The transformed magnetic susceptibility tensors for all the subunits were then summed to compute the net magnetic susceptibility tensor for the entire membrane protein structure. The DA was then determined by finding the eigenvectors and eigenvalues of the total magnetic susceptibility tensor for the protein structure. The eigenvalues for this tensor are invariants values of a tensor and represent characteristic properties of the material, which in here are the magnetic susceptibilities. DA is thus defined as the difference between the lowest and highest magnetic susceptibility value. Membrane Protein Volume Calculation: The volume for a single MP is calculated using a rolling probe method84 and the spatial coordinates available in PDB file. In this method, a virtual probe or ball of a given radius is rolled around the van der Waals surface of the molecule. The probe size can be set manually and by decreasing the probe radius, we can get a more detailed volume of the molecule. The radius we use in our calculations is set to 5 Angstroms. The values for susceptibility and anisotropy of all molecular moieties from previous literature are reported in molar basis with the units [$% /%'(]. However, to compare the magnetic energy of MP structures, we need to use their volumetric diamagnetic anisotropy. For converting the molar diamagnetic anisotropy to volumetric diamagnetic anisotropy, we use the following approach. $% ×23×456 % %'()* ∆ +, $-. / 0 78888889 %'()* ∆ +, :; / 0 %'( %'(

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B

% / ? @A ×C %'()* ∆ +, :; / 0 7888888889 D'(E%F ∆ +, :; !G+%F,.+',(F.." %'(

Results and Discussion Existing computational methods for estimating the DS and DA of MPs fail either because MP size exceeds limits for ab initio calculations or because empirical formulations have not incorporated the spatial relationship between atoms. The method evaluated in this work leverages previously reported values for the principle magnetic susceptibility tensors of bonds and select molecules determined through ab initio calculations, and sums them in manner spatially consistent with their orientation in a large, complex molecule. We compare the effect of subunit choice on the calculated values for the DS and DA of 583 MPs. The results for 45 monotopic MPs are not presented here but are available in SI Table S3. To validate the proposed computational method for calculating the DA of different molecular structures, this computational method is first applied to estimate the DA of amino acids using molecular moiety subunits. Experimental data for the mean magnetic susceptibility of 14 of the 20 major amino acids was available in the literature, providing a valid point of comparison for our method. For the other six, we compare the computed mean magnetic susceptibility to that estimated by the RealTime Predictor™ software package.

The average absolute deviation between the experimental and RealTime

Predictor™ values for mean magnetic susceptibility were 5.4%, though these deviations range from 2.6% to 24% (SI Section 2).

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In contrast, the mean magnetic susceptibility of amino acids calculated using the proposed method slightly overestimate experimental values (average absolute deviation is 5.0%; deviations ranging between 0.74% to 12%) (Figure 2A; SI Section 2). Calculated and experimentally measured values diverge most significantly for amino acids with aromatic components (i.e. HIS, TYR, PHE, TRP), with the average absolute deviation 10%.

When experimental values were unavailable, we compared calculated mean

magnetic susceptibilities to those estimated by the RealTime Predictor™. In all cases, the calculated mean molar magnetic susceptibility is within 16% of both the experimentally measured and RealTime Predictor™ values for the 20 major amino acids.

Figure 2. Magnetic properties of amino acids. (A) Comparison between experimental, RealTime Predictor™, and calculated values of mean molar magnetic susceptibility for the 20 major amino acids. Additional detail on the comparison between experimental and RealTime Predictor™ values is reported in SI Section 2 (B) Comparison of the calculated molar diamagnetic anisotropy of amino acids using different structural subunits. Amino acid subunits are calculated using the values for experimental mean magnetic susceptibility whenever available, but in the absence of that data use the values estimated from RealTime Predictor™. Numerical comparison of the values for mean magnetic susceptibility is presented in SI Table S2.

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We also compare the choice of subunit size and complexity when calculating the DA of the 20 major amino acids (Figure 2B).

The first approach follows Worcester71 by

assuming a constant value for the peptide bond and ignoring the contributions of the side groups in the calculation of DA (Figure 2B, Peptide Subunit). This is an adequate approximation for amino acids where the DS of the side group is small compared to that of the peptide bond. This approximation fails, however, for side groups containing aromatic, amide, or acidic moieties with significant electron resonance. For instance, this baseline calculation underestimates the DA of amino acids with aromatic moieties a factor of 6 to 17. The second approach expands on the first by including the contributions to DS from both the peptide bond and any aromatic moieties, while also accounting for the bond angles of these aromatic groups relative to the peptide bond (Peptide + Aromatic Ring Subunits, Figure 2B). However, for non-aromatic VAL, ASN, ASP, GLU, and GLN amino acids, the electron resonance and the position of the moiety with respect to the peptide bond was significant enough to cause moderate deviation from calculations only considering the peptide bond. Finally, the third approach accounts for all side group contributions and their relative positions in the calculations (Amino Acid Subunits, Figure 2B). While this third approach is more computationally intensive, it is necessary to fully describe the magnetic properties of amino acids.

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Figure 3. Effect of subunit selection on calculated diamagnetic properties of membrane proteins. (A) Calculated molar diamagnetic anisotropy of alpha helical and beta barrel membrane proteins using different amino acid subunit approximations. (B) Total volumetric diamagnetic anisotropy of alpha helical and beta barrel membrane proteins using different amino acid subunit approximations. (C) Total molar diamagnetic anisotropy of individual alpha helices and beta strands using amino acid subunits.

Using the amino acid subunit approximations reported in Figure 2B, we then calculate the molar DAs of all MPs of known structure73 (132 beta barrel and 451 alpha helical membrane proteins) (Figure 3A). Using the previously described method for calculating volume, we also obtain the molar volume of each MP. We observe that molar volume scales linearly with the number of amino acid residues in the MP over the wide range of membrane protein sizes and structures (SI Figure 1). Thus, we do not normalize the molar diamagnetic anisotropy by number of residues, since it would be equivalent to the volumetric diamagnetic anisotropy. In each case, we differentiate results for alpha helix and beta barrel tertiary structures. For the amino acid subunit case, we also display results that omit the position-based calculation method developed in this work, which we title “sequence-based” since it only

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incorporates the amino acid sequence, not the relative position of those amino acids with respect to one another. For each case we highlight the calculated value for a single beta barrel protein, OmpF, and a single alpha helical protein, Aqp0, to draw parallels with our previous experimental work.13 The calculated DA values of the 628 MPs (including the monotopic structures) investigated in this study are individually reported in SI Table S3. As expected, calculations of position-based molar and volumetric DA using peptide subunit inputs are significantly lower than those using the peptide plus aromatic subunit approximation or using the full amino acid subunit (Figures 3A and 3B). While the mean values for the peptide plus aromatic subunits approximate those of the full aromatic subunits, the difference for individual proteins can be significant if those proteins contain significant quantities of ASP, GLU, or GLN amino acids. In both cases, the change in diamagnetic anisotropy is evident in both alpha helical and beta-barrel structures.

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Figure 4. Effect of primary and secondary structure on the calculated diamagnetic properties of membrane proteins and the distribution of different groups of primary structures in the membrane proteins. (A) Distribution of aromatic, amide and acidic, and other residues in membrane proteins with different tertiary structures. (B) Effect of aromatic residues on calculated volumetric diamagnetic anisotropy of membrane proteins with different tertiary structures. (C) Effect of amide and acidic residues on calculated volumetric diamagnetic anisotropy of membrane proteins with different tertiary structures. (D) Effect of secondary structures of a beta strand and alpha helix on the diamagnetic properties of membrane proteins.

Comparison of the position-based and sequence-based methods indicates a critical need for the proposed approach to calculate the magnetic properties of complex molecules. The DA values that rely only on the amino acid sequence and omit the spatial relationship between amino acids significantly overestimate both the molar and volumetric DA (Figures 3A and 3B). Thus, while the sequence-based methods 21



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significantly simplify the calculation of DA, they are grossly inaccurate and cannot be used in subsequent calculations. Finally, these results provide insight into the molecular composition and structure of MPs that enhance MP DA. Understanding the effect and relative importance of MP primary, secondary, and tertiary structure of MPs may be useful in estimating the magnetic properties of proteins with unresolved secondary and tertiary structures. At the primary structure level, we hypothesized that MPs with higher percentages of aromatic, amide, and acidic amino acid residues would correlate with higher DA of the membrane protein. Our calculations suggest that this hypothesis is only clearly supported for aromatic residues. There is little variation in the percentage of aromatic residues across MPs with different tertiary structures, and aromatic residues account for less than a third of all amino acid residues in MPs of known structure (Figure 4A). MPs with a large percentage of aromatic amino acids have a significantly higher volumetric DA than those MPs with a low number (Figure 4B). This effect is observed separately for both beta barrel and alpha helical MPs, and is also consistent across the entire population of transmembrane and monotopic MPs of known structure. Thus, the significantly higher anisotropic response of the aromatic ring moieties (Figure 2B) influences the magnetic properties of the entire membrane protein structure (Figure 4B). Contrary to our hypotheses, however, we observed no meaningful relationship between the percentage of amide and acidic amino acid residues and the volumetric DA (Figure 4C). Amide and acidic amino acids residues also account for less than a third of all amino acid residues in MPs of known structure (Figure 4A), though they are moderately overrepresented in beta barrel proteins compared to alpha helical tertiary structures. This

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observation explains the greater increase in calculated DA of beta barrel MPs when using the DA values for the full amino acid subunits in the calculations (Figures 3A and 3B). These results suggest that the primary structure may be useful for qualitative comparison of the magnetic properties of MPs with only partially resolved structures, but that the comparisons are likely to have greater validity if the tertiary structure of the proteins is known. To determine the influence of secondary structure on MP DA, we isolate each of the alpha helices and beta strands from the 18 of MPs, which were chosen randomly from the list of our 628 structures (Figure 3C). At the secondary structure level, the difference in the molar DA of beta strands and alpha helices is very small. Thus, the secondary structure of the protein appears to have little effect on the overall molar DA. At the tertiary level, we observe significant differences in the molar and volumetric DA of alpha helical and beta barrel MPs. The median molar DA of MPs with beta barrel tertiary structures is significantly higher than alpha helical tertiary structures, and this trend is preserved when normalized to the number of residues in the MP, as well as to the volume of the MP. The difference in diamagnetic anisotropy of beta barrel and alpha helical MPs is a direct result of tertiary structure, as evidenced by the statistically equivalent median values of molar and volumetric DA for the sequence-based (primary structure only) calculations (Figure 3A and 3B). In short, large proteins with a high density of aromatic amino acid subunits and beta barrel tertiary structure are likely to exhibit the highest DA. Proteins with high DA also exhibit higher magnetic energy, ∆E, for a protein crystal of N protein or oligomer units (Equation 10).

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∆E =

IJ∆KLM NO

Page 24 of 42

(10)

Here, V is the unit molecular volume, ∆χ is the volumetric diamagnetic anisotropy, B is

the magnetic field strength, and μR is the permeability of vacuum. At field strengths or

volumes sufficient to satisfy the condition ∆E >>k L T, the magnetic field energy exceeds the thermal energy of the system and the protein or protein crystal is magnetically stabilized.

Conclusions We propose a novel mixed-method technique for facile prediction of MP DA where the spatial coordinates of the atoms are well known.

This technique demonstrates that

accounting for the spatial coordinates and orientation of moieties is critical to accurately predicting the magnetic properties of proteins and other complex molecules. Although this technique will not give results with the same level of accuracy as the first principal methods, it closely matches the available experimental values with much lower computational costs.

As such, we believe that this method augments existing

computational methods as a screening tool for diamagnetic properties of membrane proteins. In addition, this paper develops generalized relationships between MP structure and DA. High percentages of aromatic amino acid subunits, a tertiary beta barrel structure, and a small volume are predictors of high volumetric DA. Indeed, these relationships may enable qualitative predictions of the DA of barrel shaped MPs in the absence of precise information about all aspects of the protein’s structure. However, as the tertiary structure

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becomes more complex, the ability to predict the DA using the information about primary and tertiary structure weakens. This work can be used to assess the feasibility of magnetically directed MP crystallization, provide design guidance in the fabrication of artificial macromolecules with high diamagnetic anisotropy, and inform the selection of supporting lipid or blockcopolymer bilayers. In this final case, it may be possible to increase the magnetic energy of crystals formed of proteins with low DA by selecting supporting block copolymers with high DA. Finally, this work may also be extended to soluble membrane proteins, where the computational approach remains valid.

Associated Content Supporting Information SI Section 1: Calculating the Magnetic Properties of Amino Acid Subunits; SI Section 2: Comparison Between RealTime Predictor™ and Experimental Values for Diamagnetic Anisotropy of Amino Acids; SI Section 3: Determining the Molar Volume of Membrane Proteins; SI Section 4:

Magnetic Properties of Membrane Proteins; SI Section 5:

Membrane Proteins Used for Extracting the Spatial Coordinates of the Secondary Structures.

Author Information Corresponding Author Tel: 412-480-9960 E-mail: [email protected] Notes

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The authors declare no competing financial interest.

Acknowledgments We thank Manish Kumar for his helpful insights.

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◊

●

† Error of 5~10% ◊ Error > 10%

Peptide Subunits Peptide + Aromatic Ring Subunits Amino Acid Subunits

−8

−6

−4

−2

ACS Paragon Plus Environment 0

CYS SER THR GLY ALA MET LEU PRO ARG ILE LYS VAL ASN ASP GLU GLN HIS TYR PHE TRP

−10

◊

†

Molar Diamagnetic Anisotropy [cm3/mol]

−12

Calculated Experiment RealTime Predictor™

PHE TYR ARG TRP

●

−14

−5

GLY ALA SER ASP THR ASN CYS PRO VAL GLU GLN LEU ILE MET HIS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

10 Journal of Chemical Theory and ×Computation B−10

−5

LYS

Mean Magnetic Susceptibility [cm3/mol]

10 Page 39 of× 42 A −16

−6

Beta Barrel Alpha Helix OmpF Aqp0

−10 −4 −5 −2

0

Peptide Bond Subunits

Peptide Bond + Aromatic Ring Subunits

Amino Acid Subunits

0

Amino AcidParagon ACS Subunits (Sequence− Based)

Page ×40 of 42 10−6 −5

−1.4

Plus

−1.2 −1

Beta Barrel Alpha Helix OmpF Aqp0

−4

−3

−0.8 −0.6

−2

−0.4 −1

−0.2 0

Peptide Bond Environment Subunits

Peptide Bond + Aromatic Ring Subunits

Amino Acid Subunits

Amino Acid Subunits (Sequence− Based)

0

Sequence−Based Volumetric Diamagnetic Anisotropy [dimensionless]

Position−Based Molar Diamagnetic Anisotropy [cm3/mol]

−15

−8

Sequence−Based Molar Diamagnetic Anisotropy [cm3/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

−6 Journal of Chemical Theory Computation × 10 × 10−2 B and

× 10−3

Position−Based Volumetric Diamagnetic Anisotropy [dimensionless]

A

× 10 B and Journal of Chemical Theory Computation

A Page 41 of 42

−7

D

× 10−7

Volumetric Diamagnetic Anisotropy [dimensionless]

Beta Barrel Alpha Helix All Membrane Proteins

5 4 3 2

>26

24~26

22~24

Percent of Amide and Acidic Residues in the Protein [%]

−6

−8

−6

−4

−2

Beta Strand

Alpha Helix

>20

18~20

Percent of Aromatic Residues in the Protein [%]

× 10

ACS Paragon Plus Environment 0 20~22

18~20

16~18

14~16

12~14

10~12

0

8~10

1

16~18

Other Residues

14~16

Amide and Acidic Residues

Computing the Diamagnetic Susceptibility and Diamagnetic

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