research papers
The zipper groups of the amyloid state of proteins
^{a}Department of Chemistry and Biochemistry, Howard Hughes Medical Institute, UCLADOE Institute for Genomics and Proteomics, University of California at Los Angeles, Box 951570, Los Angeles, CA 900951570, USA
^{*}Correspondence email: jstroud@mbi.ucla.edu
Fibrous proteins in the amyloid state are found both associated with numerous diseases and in the normal functions of cells. Amyloid fibers contain a repetitive spine, commonly built from a pair of βsheets whose βstrands run perpendicular to the fiber direction and whose side chains interdigitate, much like the teeth of a zipper. In fiber spines known as homosteric zippers, identical protein segments sharing identical packing environments make the two βsheets. In previous work based on atomic resolution crystal structures of homosteric zippers derived from a dozen proteins, the symmetries of homosteric zippers were categorized into eight classes. Here, it is shown through a formal derivation that each homosteric zipper class corresponds to a unique set of symmetry groups termed `zipper groups'. Furthermore, the eight previously identified classes do not account for all of the 15 possible zipper groups, which may be categorized into the complete set of ten classes. Because of their foundations in group theory, the 15 zipper groups provide a mathematically rigorous classification for homosteric zippers.
Keywords: amyloid spine; steric zippers; zipper groups; symmetry; group theory; amyloid fibers.
1. Introduction
Amyloid fibers were first found associated with denatured proteins and diseases (Eisenberg & Jucker, 2012), but have more recently been discovered in a variety of normal cellular processes (Chapman et al., 2002; Si et al., 2003; Fowler et al., 2006; Maji et al., 2009; Kato et al., 2012). Apparently, evolution has harnessed this tight but reversible mode of protein association for numerous biological functions.
Although models of amyloid fibers are varied (Nelson & Eisenberg, 2006), several models contain a structural feature known as the amyloid spine (Fig. 1a). The spine consists of a pair of βsheets that run the length of the fiber (Nelson et al., 2005). The βstrands of the two spinal βsheets are small adhesive segments of potentially larger polypeptide chains that constitute the fiber (Sambashivan et al., 2005). βHydrogen bonding within the spinal βsheets mediates βstrand adhesion along the spine axis and provides cooperative forces that bestow fibers with high thermodynamic stability (Nelson et al., 2005; Tsemekhman et al., 2007). Tightly interdigitated side chains of the βstrands (Fig. 1b) bind the two βsheets together in a spine geometry termed a `steric zipper' (Nelson et al., 2005; Sawaya et al., 2007). Along with numerous examples from singlecrystal diffraction (Eisenberg & Jucker, 2012), the steric zipper model is consistent with models derived from other types of data. For instance, a fiberdiffraction model of polyglutamine satisfies the requirements for a steric zipper (Sikorski & Atkins, 2005), as do models of amyloidβ fibrillar oligomers derived from powder diffraction (Stroud et al., 2012).
A variety of steric zipper symmetries have emerged from Xray crystallographic studies of the adhesive segments of amyloid fibers (Nelson et al., 2005; Sawaya et al., 2007; Wiltzius et al., 2009; Colletier et al., 2011). In these studies the spinal βsheets have identical βstrands running nearly perpendicular to the spine axis (Fig. 1c). Additionally, all of the βstrands in a given spine have identical packing environments. Given the growing catalog of amyloid interactions, it is useful to enumerate the symmetries of these interactions, just as it was useful to enumerate the space groups: all possible ways to pack identical objects. Here, I derive and illustrate all possible steric zipper interactions between identical protein segments wherein all segments share identical packing environments: the socalled homosteric zippers.
In all steric zipper Xray structures determined to date, two βsheets form the spine (Sawaya et al., 2007). The spinal βsheets of a zipper are identical in that (i) both βsheets are parallel (i.e. all βstrands run in the same direction) or antiparallel (alternate βstrands run in opposite directions) and (ii) both βsheets have the same βhydrogenbonding pattern such that identical residues make the same βhydrogenbonding contacts in both βsheets. Additionally, these atomic resolution structures reveal steric zippers that consist exclusively of βstrands of a single sequence, called `homosteric zippers' (Eisenberg & Jucker, 2012). Homosteric zippers differ from `heterosteric zippers', in which the βstrands of the spine have more than one sequence.
Sawaya and coworkers classified the known homosteric zipper structures using three class constraints to specify zipper features: (i) how each βstrand interacts with its nearestneighboring βstrands in the same βsheet (`parallel' or `antiparallel'), (ii) whether the same sides of both βsheets are up (`up–up') or whether one is up and other is down (`up–down') or whether both can be rotated by 180° around an axis perpendicular to the βsheets to yield identical βsheets (`up=down') and (iii) whether identical (`facetoface') or opposite (`facetoback') faces of the βsheets create the zipper interface or whether the βsheets can both be flipped around an axis parallel to the spine axis to yield the same βsheets (`face=back'). Eight classes of homosteric zippers arise from the possible combinations of these class constraints (Sawaya et al., 2007).
The grouptheoretic treatment herein establishes a mathematically rigorous classification of 15 zipper groups and shows that the full set of homosteric zipper classes should expand to ten. To enumerate all homosteric zipper classes, I describe their symmetries in terms of a coordinate system in which the homosteric zipper spine is oriented with its βhydrogen bonds (backbone C=O and N—H groups) running nearly parallel to the y axis and with its βstrands running nearly parallel to the z axis (Fig. 1c). The spine axis is parallel to the y axis, lying in the interface between the βsheets at a defined location in the coordinate system (Fig. 1f).
By enumerating all possible combinations of symmetry operations (Table 1), I demonstrate the existence of 15 distinct symmetry groups, termed `zipper groups' (Table 2). Each of the eight homosteric zipper classes identified by Sawaya and coworkers corresponds to a subset of the zipper groups, accounting for 12 of the 15 zipper groups. Two novel classes of homosteric zippers correspond to the three remaining zipper groups (Fig. 2), extending the number of homosteric zipper classes to ten.

2. Methods: derivation of the 15 homosteric zipper groups
2.1. The zippergroup positions
I represent the homosteric zipper lattice by a set of translations, {e_{3}, x_{3}, y_{3}, xy_{3}} (1), such that one βsheet is centered on e_{3} and another on x_{3}, with the spine axis halfway between e_{3} and x_{3} and running parallel to the y axis (Fig. 1f),
βStrands that occupy a zipper lattice can be in one of four orientations (2) that correspond to a reference (E_{3}) or π rotations (I_{3}, J_{3} and K_{3}) around each of the three principal axes (x, y and z, respectively),
A multicolored box (shown as if unfolded in Fig. 1g) illustrates these orientations. Different sides of the box are visible depending on the orientation (Fig. 1h).
Combining the translations {e_{3}, x_{3}, y_{3}, xy_{3}} with rotations {E_{3}, I_{3}, J_{3} and K_{3}} produces positions. For example, J_{3} and x_{3} combine to produce the position Jx. Positions are represented as 4 × 4 projective transformation matrices. An example is shown in (3):
The resulting positions (E, I, J, K, x, Ix, Jx, …, Kxy) form a finite group under matrix multiplication wherein the translation component (elements [1 4], [2 4] and [3 4]) of the product matrix is under modulo 1. This modulo operation simply implies that a translation as large or larger than a moves to a position in a different unit cell.
2.2. Ten generators produce the 15 zipper groups
Every symmetry group available to zipper lattices may be produced from ten generators (Table 1) combined such that no more than one generator with a given translation (e_{3}, x_{3}, y_{3} or xy_{3}) is used in each combination. This requirement simply implies that no two segments may occupy the same lattice position. The resulting 15 distinct zipper groups are layer groups of nonenantiomorphic objects expanded to include multiple settings for several of the layer groups (Table 2).
To illustrate a zipper group, I place multicolored boxes that represent specific orientations (Fig. 1h) into a lattice. An example is Fig. 1(i), which illustrates two repeats of a GNNQQNY homosteric zipper (PDB entry 1yjp ; Nelson et al., 2005) which belongs to zipper group 1b. Fig. 1(f) depicts one of the zipper group 1b lattice.
2.3. Zipper groups have layergroup symmetry
Although the zippergroup settings are different from the standard layergroup settings, the zipper groups have layergroup symmetry. For example, the generators for zipper group 1b (E and Jx) combine to generate layer group 8 with C_{2}^{y} This symmetry is demonstrated by changing the basis such that the translation components of the two positions shift by (−¼ 0 0)^{T} (4),
Different zipper groups may have the same layergroup symmetry (Table 2). For example, zipper groups 1b and 7a both have layer group 9 symmetry. These two groups are distinguished by their settings, which places symmetry axes at different locations in the two zipper groups. Zipper group 1b has 2_{1} symmetry along the spine axis at x = ¼. In contrast, zipper group 7a has 2_{1} symmetries coincident with the centers of the βsheets at x = 0 and x = ½. Because of the stereochemistry of βhydrogen bonding, some symmetries along the z axis in zipper groups 9, 10a and 10b are imperfect.
3. Discussion
3.1. Nomenclature of the zipper groups reflects their relation to the homosteric zipper classes
Because zippergroup symmetries satisfy homosteric zipper classes, the names of the zipper groups have been assigned to reflect the names of homosteric zipper classes (Table 2). For example, zipper groups 1a and 1b satisfy the homosteric zipper class 1 constraints (parallel, up–up, face=back). Fig. 2 illustrates the relationships between zipper groups, their generators and the homosteric zipper classes.
3.2. Zipper groups predict novel homosteric zipper symmetries
A complete enumeration of zipper groups (Table 2) reveals that three zipper groups (9, 10a and 10b) remain after accounting for all eight previously identified homosteric zipper classes (Sawaya et al., 2007). Zipper groups 9, 10a and 10b satisfy the novel combination of parallel, face=back. Zipper group 9 differs from 10a and 10b in that both βsheets of zipper group 9 run in the same direction along z (a class constraint termed `headtohead'; Fig. 2). Zipper groups 10a and 10b have βsheets that run in opposite directions along z (`headtotail'). Although not yet observed in atomic structures of homosteric zippers, a parallel, face=back βsheet has been observed in the atomic resolution crystal structure of the mcLVFFA macrocyclic βsheet mimic (Liu et al., 2011). In this structure (Fig. 1j), the sequence LVFFA makes a parallel, face=back βsheet, suggesting that LVFFA has the potential to form zipper group 9, 10a or 10b.
3.3. Rationale for the preponderance of certain zipper groups
Some zipper groups are observed in crystal structures more frequently than expected, perhaps because symmetry influences the stability of the amyloid spine. For example, the symmetries of several zipper groups require that interacting βstrands from different βsheets be in the same plane, where the plane is perpendicular to the spine axis. Such zipper groups are described as `eclipsed' (Table 2). Of the 15 zipper groups, seven are eclipsed. Yet among all 44 published crystallographic homosteric zippers (Table 3), eclipsed zipper groups are rare, comprising only seven of the 44 zippers. This bias towards `staggered' zipper groups (groups that are not eclipsed) may arise from the profile method used to identify amyloidogenic segments, which employs the zipper group 1b NNQQNY zipper as a template (Nelson et al., 2005; Thompson et al., 2006). However, a strong bias towards staggered zippers exists even when group 1b zippers are excluded, with only seven of the 24 remaining zippers being eclipsed. The bias towards staggered zippers may reflect the fact that staggered zippers more readily interdigitate than eclipsed zippers, increasing the surface complementarity and hence the energetic favorability of sheet adhesion.

3.4. Zippergroup pseudosymmetry is observed in some crystal structures
In experimental structures, the strict layergroup symmetries of several zipper groups are broken by a shift along the z axis (termed `zshift') of one βsheet relative to the other (Table 2, `Alternate Symmetry'). This zshift produces a 2_{1} screw from C_{2}^{z} of zipper groups 3, 8 and 9. Similarly, the zshift creates 2_{1} symmetry from C_{2}^{x} of zipper groups 6a and 6b. The potential for some of these 2_{1} screw axes in x and z has been recognized previously (Sawaya et al., 2007). In zipper groups 5a and 5b, the zshift completely removes any symmetries in x and z. Because zipper groups are defined by generators with only x and y translations, the zshift does not influence the ability of the affected zipper groups to satisfy homosteric zipper classes.
4. Conclusion
Here, I develop a mathematically rigorous classification of homosteric zippers using group theory to derive the 15 zipper groups that specify all possible symmetries available to homosteric zippers. Zipper groups extend previous work in which eight symmetry classes of homosteric zipper spines were identified from crystal structures and from an intuitive analysis of the ways that pairs of βsheets can interact (Sawaya et al., 2007). Zipper groups may be categorized such that the complete homosteric zipper classification developed by Sawaya and coworkers expands to ten classes. Subsequent to the work of Sawaya and coworkers, a sheet satisfying this expanded set of symmetries was observed in a crystal structure (Liu et al., 2011). I anticipate that structures of amyloid spines belonging to the new homosteric zipper symmetries will be discovered in the future.
Acknowledgements
The author thanks Dr David Eisenberg for extensive suggestions to improve the manuscript and Drs Lukasz Salwinski and Michael Sawaya for discussions. This work was supported by National Institutes of Health Grants AG 029430 and FGM077789A, and the Howard Hughes Medical Institute.
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