Helical Parameters and Correlations of Tandem Leucine Rich Repeats in Proteins

Leucine rich repeats (LRRs) are present in over 20,000 proteins from viruses to eukaryotes. Most LRR units are 20-30 residues long and can be divided into a highly conserved segment and a variable segment. Eight classes have been recognized. Two to sixty-two units occur in tandem to form an LRR structure. The tertiary structures of these LRRs are helical, in which the β-strands of the highly conserved segments stack in parallel. This helix consists of a super helical arrangement of repeating structural units. We call it a coil of solenoids. We have used our program HELFIT to assign helical parameters to 642 LRRs of known structures of 114 proteins. We report these parameters and their correlations with eight classes of LRR, with the number of repeat units in the LRR, with oligomerization, and with ligand state of the LRR. The helical parameters of the eight LRR classes frequently overlap one another. However, the constant distance between parallel β-strands is the primary determinant of the helical parameters of the LRRs. When the repeat number, n, in LRRs is small, the LRR structures are more variable and, by inference, more flexible. In the LRRs with n ≥ 10, ∆z (the rise per repeat unit) of the “RI-like” and “Cysteine-containing” classes is smaller than those of “SDS22-like”, and “Plant-specific” classes. This difference is ascribed mainly to the difference in the structural units. The helical parameters of the LRRs unambiguously describe both right handed and left handed helices, helical dimers, and subdomains if they exist. Moreover, the helical parameters sensitively detect structural changes induced by protein, protein interactions, glycosylation, and/or mutation. Helical Parameters and Correlations of Tandem Leucine Rich Repeats in Proteins


Introduction
LRRs are present in over 20,000 proteins from viruses to eukaryotes, as listed in the data bases of PFAM, SMART, PROSITE, and InterPro [1][2][3][4]. All organisms for which sequence data are available have at least one LRR protein. Most LRR proteins are involved in protein, protein interactions, as observed in the plant immune response and the mammalian innate immune response [5][6][7][8][9]. Beyond immunity, extensive functional diversity occurs among LRR proteins, including their involvement in apoptosis, autophagy, ubiquitin related processes, nuclear mRNA transport, neuronal development, and the type III secretion system in pathogenic bacteria [10][11][12][13][14][15][16]. Furthermore, plant LRR proteins including LRR containing receptor-like kinases and LRR containing receptor like proteins act as signal amplifiers in the cases of tissue damage, of establishing symbiotic relationships, and of affecting developmental processes [17][18][19].
Two to sixty-two LRR units occur in tandem. Each unit is typically 20-30 residues long and can be divided into a highly conserved segment (HCS) and a variable segment (VS) [6,20,21]. The HCS consists of an eleven residue stretch, LxxLxLxxNxL, or a twelve residue stretch, LxxLxLxxCxxL, in which "L" is Leu, Ile, Val, or Phe; "N" is Asn, Thr, Ser, or Cys; and "C" is Cys, Ser or Asn. Eight classes of LRRs-"RI-like", "Cysteine-containing", "SDS22-like", "IRREKO", "Bacterial", "Plantspecific", "Typical", and "TpLRR"-have been recognized [6,[20][21][22]. They are characterized by different lengths and consensus sequences of the VS of their repeat units. Most of the known LRRs consist of only one class of units; however a few have a mixture of two classes. A super-motif of LRRs occurs in a group of LRR proteins including the subfamily of small LRR proteoglycan including biglycan, and decorin, and TLRs 7, 8, and 9 [23][24][25]. Moreover, the first, N-terminal LRR unit in the "Typical" or "IRREKO" LRR domains is frequently occupied by a "Bacterial" motif [21]. Crystal and/or solution structures of representatives of all the eight classes of LRR are available.
Three residues at positions 3 to 5, xLx, in the HCS form a short β-strand [5][6][7][8][9]. These β-strands from tandem LRR units stack parallel and then the LRRs form an arc or a helix. This helix is not simple but consists of a super helical arrangement of repeating structural units. We call it a coil of solenoids.
The concave face, consisting of the HCSs of each LRR unit, consists of a parallel β-sheet, one strand from each HCS unit. The convex face, consisting of the VS of each LRR unit, is made of a variety of secondary structures including the α-helix, 3 10 -helix, polyproline II helix, and an extended conformation or a tandem arrangement of β-turns ( Figure 1). The various secondary structures on the convex side are connected to the strands forming the β-sheet in its concave side by two loops ( Figure  1). The "ascending loop" links the C-terminal end of the β-strand in the HCS to the N-terminus of the characteristic secondary structure in the VS [9] (Figure 1). The "descending loop" links the C-terminal end of the characteristic secondary structure in the VS to the N-terminus of the β-strand in the HCS of the following unit. Most of the known LRR structures have an N-and/or C-cap that shields the hydrophobic core of the first LRR unit at the N-terminus and/or the last unit at the C-terminus [5][6][7][8][9]. In extracellular proteins or extracellular regions, these caps frequently consist of Cys clusters consisting of two or four Cys residues on the N-and C-terminal sides of the LRRs.
Non-LRR, Island Regions, IRs, interrupting LRRs are widely distributed; they are referred to as "islands" or "loop outs" [26][27][28]. A large number of plant LRR proteins including LRR containing receptor like kinases and LRR containing receptor like proteins have IRs [24,28]. In our analyses we treat the LRR regions on either side of its IR as distinct helices, as well as analyzing the entire LRR as a single helix.
Enkhbayar et al. [29][30][31] developed a program, HELFIT, that determines the helical parameters-helix axis, pitch (with handedness), radius, and number of points or units per turn-as well as RMSD. Only four points are required to define a helix. The trace of a helix with pitch=0.0Å is a circle; with N=exactly 1.0 the trace is a straight line.
Here we have determined the helical parameters for each LRR and evaluated correlation with class of LRR, with the number of repeat units in the LRR, with oligomerization, and with ligand state of the LRR to reveal some structural features of LRR structures. We demonstrate that the constant distance between parallel β-strands is the primary determinant of the helical parameters of the LRRs. When the repeat number in LRRs is small, the LRR structures are more variable and, by inference, more flexible.

Methods
A helix consisting of n repeat units may be characterized by helix axis, pitch (P), helix radius (R), and number of repeat units per turn (N). HELFIT [29][30][31] computes these parameters in which the helix axis is represented by the unit vector ( Figure 2A). These parameters also yield the rise per repeat unit (∆z=P/N) and the rotation per repeat unit (∆Φ=360 o /N). Moreover, HELFIT gives rmsd: where d i is the closest distances from data point to the trace of the helix. Here gives the regularity of helix independent of its length. If p is relatively large, the LRR is not well described by a helix and might be more easily visualized as another shape, e.g. an ellipse.
In most LRRs the three residues at positions 3-5 in the HCS form a short β-strand, which is almost completely conserved in all LRR structures [5][6][7][8][9]. These β-strands from adjacent repeat units form a parallel, pleated sheet on the inside, or concave, surface of the helix. As reference points for HELFIT we use the coordinates of the α-carbon (Cα) of the consensus leucine residue at position 4 (corresponding to
the middle of each β-strand) in individual LRR repeat units.
Three structures of LRRs with n=3 have been determined. In these cases we performed 3D circle fitting that directly fits a circle to a set of 3D data points, because only three points define a circle [32]. The 3D circle fitting, meaning P=0.0 and ∆z=0.0 as an a priori assumption determines the radius of the circle, the number per circle, the center of the circle, and the root mean square deviation from the best fit circle. The 3D circle fitting was also utilized for the estimation of the average distance between adjacent repeats by the helical parameters. Furthermore, the center of the circle was used for the calculation of the distance between the two monomers in TLR dimers.
For grouping of eight LRR classes of known LRR domains we performed similarity searches by FASTA and by BLAST using representative LRR domains clearly belonging to individual classes as query sequences. The representative LRR domains were selected from those for which the 3D structures have been solved. "RI-like" LRRs are from RI, "Cysteine-containing" from Skp2, "SDS22-like" from InlA, "IRREKO" from InlJ, "Bacterial" from YopM, "Plant-specific" from FLS2, "Typical" from LINGO-1, and "TpLRR" from BACCAC_03700.

Helical parameters
The correlation of ∆z, ∆φ, and R D is the average Cα(i)-Cα(i+1) distance between adjacent repeats, in which the Cα atoms are at position 4 in the β-strand [32]. D is a function of ∆z, ∆Φ, and R.
The data points of 2⋅Rsin (∆Φ/2) versus ∆z fall on a circle with radius D; although, in the "TpLRR" class the data deviate from this circle ( Figure 3). The circle fittings using in total 627 data points (except for those of the "TpLRR" proteins) gives D=5.02 ± 0.00 Å [32]. This D corresponds to the inter-strand distance that is in the range of 4.5 to 5.5 Å [146]; this distance allows the formation of hydrogen bonds between parallel strands. In a canonical, parallel β-sheet the strands are aligned at the same level parallel to the axes of the individual strands. This occurs in LRR structures; the observed D is consistent with the distance between strands in the canonical parallel β-sheet.
The D value in the "TpLRR" class is 5.52 ± 0.14 Å and thus is larger and more variable than those in other classes. The calculation of the Cα(i)-Cα(i+1) distance revealed D>6.9 Å between neighboring β-strands, as observed in FAEPRAA2165_0102, EUBVEN_01088, BACCAC_03700, and BACOVA_01565. This greater distance arises from water bridges in the hydrogen bonds between the strands.
Most of the LRRs have R=15 → 30 Å (Supplementary Table 1). However, five LRRs in human Trk-A shows very small R (=0.89 Å) [143], while LRRs 7-13 in BT_1240 has very large R (=277 Å). Even in the extreme cases the 2⋅R⋅sin (∆Φ/2) and ∆z values fall on the circle in figure 3. In the former the helix axis is nearly perpendicular to the β-strand axis. In contrast, in the latter the helix axis is nearly parallel to the β-strand axis.
The helical parameters of the eight LRR classes frequently overlap one another ( Figure 2B). The ∆z and 2⋅R⋅sin (∆Φ/2) values in the "Typical" class are distributed broadly (Figure 3).  but also the entire dimer forms a single helix. We found four examples. Mouse CD14, in which the LRR does not conform to any of the eight classes, forms a dimer in the crystallographic asymmetric unit as well as in solution [139]. Dimerization in the crystal is mediated by LRR residues in the loop between β-strands in the HCS of repeats 12 and 13. The C-terminal β-strands of the two β-sheets from the two monomers interact in an antiparallel fashion and form a large and continuous β-sheet encompassing the entire CD14 dimer. The two monomers are related by a two fold axis of rotation, perpendicular to their common helix axis. The CD14 monomer forms a left handed helix, as does the CD14 dimer ( Figure 5 and supplementary Table 1).

Structures of LRRs in
Human RI complexed with human angiogenin or ribonuclease I forms a dimer with human RI-angiogenin/ribonuclease I [36,37]. The whole molecule of human RI consists of only LRRs; all belong to "RIlike" class. The two complexes are held together by many hydrogen bonds formed between the N-terminal β-strands of the two human RI molecules, leading to the formation of an antiparallel β-sheet. The entire RI dimer adopts a right handed helix whose helical parameters are similar to those of its individual monomers. These two hRI molecules are related by an approximate two fold axis of rotation, nearly perpendicular to the common helical axis.
Mouse RI bound to mouse ribonuclease I forms a tetramer in the crystallographic asymmetric unit. The tetramer consists of two dimers that are related by a glide plane. The monomers of both dimers adopt a right handed helix whose parameters are quite similar to those in the dimer of human RI-angiogenin/ribonuclease I (Supplementary Table  1).
Frog LGR4, whose LRRs belong to the "RI-like" class, also forms a homo-dimer in the crystal and in solution [104]. The two monomers in the LGR4 dimer are in close proximity at their C-terminal sides and are related by a two fold axis of rotation-TLRs 1-6, TLR8, RP105, LGR4, LGR5, and Drosophila Toll, "TpLRR" class-FAEPRAA2165_01021 and [144]. The circle fitting analysis yields D =4.90 Å ± 0.01 Å that seems to be smaller than in LRRs with n ≥ 4. This smaller D might reflect interactions of side chains bridging N-and C-caps [144].

Helical dimers in CD14, RI, and LGR4
In the homo-dimers of some LRRs not only the individual monomer  LRRs with the repeat number n ≥ 10 in the four classes of "RI-like", "Cysteine-containing", SDS22-like, and "Plant-specific". Some LRRs in the "TpLRR" class deviate from the common circle; this can be attributed to a larger twist angle of β-sheets, a breaking of β-strands in the HCS; or a prism like shape for the LRRs. "R" is the helix radius of a helix consisting of n repeat units, "∆z" the rise per repeat unit, and "∆Φ" the rotation per repeat unit.    Table  1). The angle between the two helix axes (Ω) best characterizes the discontinuity (Supplementary Table 2). The LRRs with two subdomains may be grouped into four categories.
The overall shape of the entire LRR in TLRs 1-6, TLR8, RP105, and BAGEGG_03329, is well approximated as half or three quarters of an ellipse. For example, the LRRs in human TLR3 with n=25 consist of two subdomains, LRRs 1-12 and LRRs 13-25 ( Figure 6). The 25 units of TLR3 forms a nearly flat ellipse with p=0.23 Å. This p value is larger than p=0.09 Å of units 1-12 in subdomain 1 and of units 13-25 in subdomain 2. Similar characteristics are observed in other LRRs that consist of two subdomains. The angle, Ω, between the two helix axes of the subdomains ranges from 13.4 → 19.2 o (Supplementary Table 2). This tilt generates a larger p for the entire LRRs.  Table 2). The non-LRR IR has a cluster of four Cys residues with the pattern of Cx 6 Cx 29 Cx 7 C. The formation of a disulfide bridge in Cx 7 C suggests that the IR acts as an N-cap, further supporting the formation of the second subdomain. The first LRR domain in Drosophila Toll with n=19 forms a half ellipse [95]. Therefore, the LRR domain is divided into two subdomains.
EUBVEN_01088, FAEPRAA2165_01021, and BT_1240 contain LRRs in which the continuity of the parallel β-sheet is disrupted. The "TpLRR" domains in EUBVEN_01088 and FAEPRAA2165_01021 with n=14 are kinked at the central seventh and eighth repeats with 35 and 37 residues, respectively; Ω=71 → 88º ( Figure 6 and supplementary Table 2). BT_1240 contains 13 LRRs of which four repeats are similar to the "TpLRR" consensus. BT_1240 forms a homo-dimer in the crystal, in which the two molecules are in close proximity at the C-terminal sides. The LRR unit 7 is 30 residues long and disrupts the continuity of the parallel β-sheet; Ω=64 → 70º (Supplementary Table 2).
The LRR domains in human LGR4 and LGR5 with n=19, and frog LGR4 with n=17 are kinked between the central eleventh and twelfth repeats [104][105][106][107]; Ω=19.1 → 40.1 o . The units of LRR11 and of LRR12 do not strictly obey the LxxLxLxxNxL rule. Instead, the Asn residues are replaced by A309 and T332, respectively. As a result, the asparagine ladder (in which the Asn side chains from different turns are stacked and form hydrogen bond connecting turns) breaks in this region, thereby generating two longer β-strands [104][105][106][107].

Eight classes of LRR
When the repeat number, n, is ≤ 8, the helical parameters for individual monomers of the homo-oligomers are sometimes variable. The helix parameters were not determined in these cases.
The LRR domains of RanGAP from human and fungi, n=11, and tropomodulin from chicken and Caenorhabditis elegans, n=5, show little similarity to the RI LRRs [41][42][43][44][45]. However, their repeat units are in β-α conformations that are the same as those of the "RI-like" class. Correspondingly, their helical parameters are similar to those of the four "RI-like" LRR proteins.
The LRR domain in LEGL7 is similar to that in CARMIL (E-value=6.2 · 10 -4 in FASTA). We tentatively assign the LEGL7 LRR to the "RI-like" class. The LRR domain consists of eleven units. The consensus is VxxLxLxxNxLxxxSxxELxxxLAxIPxx with 29 residues. DSSP analysis indicates that the "ascending loops" in the eight repeats adopt a 3 10 -helix of one turn at the underlined residues and the VS adopts an α-helix with 6 → 10 residues. The LRR domain adopts a lefthanded helix; P=-26.9 Å, ∆z=-1.53 Å, ∆Φ=20.5º, N=17.6 units/repeat, and R=13.7 Å (Supplementary Table 1).
Most of their VSs adopt a tandem arrangement of three β-turns (which we call consecutive -turns) on the convex faces (Figure 1). The consecutive -turns form a flattish amphipathic structure with main chain hydrogen bonds in a linear arrangement. Adjacent β-turn repeat motifs form a regular parallel packing arrangement with the core hydrophobic residues pointing alternately in and out of the plane of the structure interlocking with the corresponding hydrophobic residues from the adjacent repeat. The consecutive -turns have already been reported in the underlined eight residues of the VS consensus of xxLPxGLLxGLxx seen in three LRR units in GPIbα with n=9 [125]. Thus, the structural units may be represented by consecutive -turns.
Phenylalanine at position 19 is internally buried and is surrounded by four Leu's, which form the hydrophobic core of the LRR structure. These Phe's of consecutive units form the spine of the LRR [89]. This Phe spine involves aromatic, aromatic interactions; the benzyl groups form stacks of Phe side chains. The first, N-terminal LRR unit in most of the "Typical" LRR domains is "Bacterial" LRR. The VS adopts the polyproline II helix instead of consecutive β-turns.  Table 1). The LRR domains may be grouped into a right handed helix, a left-handed helix, and a near flat.
The LRR domains in biglycan, decorin, and TLR8 consist of three or four tandem repeats of a super motif of STT in which "S" is a "Bacterial" LRR unit and "T" is a "Typical" LRR unit [23,24]. The LRR domains of biglycan and decorin with n=12 consist of four repeats of STT. Also the N-terminal LRR domain in TLR8, n=27, contains three repeats of STT. The helix parameters range over: P=36. 7  The LRR domain in Xcv3220, n=9, has high sequence similarity to those in both human LGR4 with E-value=7.7 · 10 -6 in FASTA and human LGR5 with E-value=7.0 · 10 -6 in BLAST. We assume that this Xcv3220 LRR domain belongs to the "Typical" class. Seven of the nine repeats are represented by the consensus of LxxLxLxxCxx(x/-) LxxLPxxLxxLxx with 23-24 residues. The structural unit is β-3 10 . The LRR domain adopts a left-handed helix; P=22.3 Å, z=0.59 Å, ∆Φ=-9.6º, N=37.6 units/repeat, and R=29.7 Å.
The consensus sequence of the "TpLRR" class is LxxLxLxxxLxxIgxxAFxx(C/N)xx. The repeat length is 23-25. Bacterial "TpLRR" proteins include BACOVA_04585, bacterial group 3 Ig-like proteins (BACCAC_03700 and BACOVA_01565), FAEPRAA2165_01021, EUBVEN_01088, and BT_1240. The repeat numbers range from 12 to 14. The tertiary structures of the six "TpLRR" proteins have been determined. The ascending loops consist mainly of one β-turn and short β-strand of two residues, while the most of the descending loops consist of a single β-turn (Figure 1). Most of their VSs adopt a tandem arrangement of two or three β-turns.
The helix parameters are variable among the six proteins and deviate significantly from those of the other classes (Supplementary  Table 1). This can be attributed to a larger twist angle of β-sheets, to a breaking of β-strands in the HCS, or to a prism like shape for the LRRs. The mean twist angle of β-sheets in LRR proteins belonging to the other classes was 3 → 8° [32]. LRRs in the "TpLRR" class sometimes do not form β-strands in the HCS.

Structural change induced by protein, protein interactions, glycosylation and/or mutation
In the LRRs the concave surface, the convex surface, N-terminal domain including N-cap, C-terminal domain including C-cap, or their combinations are involved in protein, protein interactions (Supplementary Table 3).
The LRRs of VLR and Slit2 belong to the "Typical" class, as noted. VLRB.2D, n=5, bound to a protein antigen, HEL, as well as nonbound VLR has been determined [83]. VLR uses nearly its entire concave surface to bind HEL. This binding causes large changes of the helical parameters; a decrease of P, ∆z, and ∆Φ, and an increase in R (Supplementary Table 3). The structures of D3 in human Slit2 have been determined in the free state and in the complex with the Ig domain from Robo1 [91]. As seen in the VLRB.2D-HEL complex, the Slit2 D3 domain uses its concave surface to bind Ig. The formation of the complex causes large changes of the helical parameters. Conversely, an increase of P, ∆z, and ∆Φ, and a decrease in R is observed (Supplementary Table  3). The same structural change occurs in the InlA-cadherin complex; although, it is minor ( Supplementary Tables 1 and 3) [56].
The crystal structures of free LRIM1 and of free APL1 and of their hetero-dimer complex have been determined [134]. The LRIM1/ APL1complex has a single intermolecular disulfide bond. LRIM1 and APL1 contain 11 and 15 repeat units, respectively. The HELFIT analysis detects structural change in both LRR domains due to the complex formation. The LRIM1 LRR domain in the complex shows a significant increase in P and ∆z and decrease in R (Supplementary Table 3). In contrast, the helical parameters in the APL1 LRR domain are reversed and have small changes in ∆z and R.
The structures of frog LGR4 have been determined in the free state and in complex with R-spondin [104]. The formation of the complex induces small structural changes and increases in P, ∆z, and R (Supplementary Table 3).
The structures of free RabGGTα and its complex bound to escort protein 1 have been determined [53,54]. The helical handedness of the LRR domain with n=6 reverses upon binding to REP (Supplementary Table 1).
The crystal structures of the complexes of Skp1-Skp2 and Skp1-Skp2-Cks1 are available [46,47]. The binding of Cks1 causes slight structural changes; an increase of P, ∆z, and ∆Φ, and a decrease in R. Moreover, the crystal structures of the complexes with Ran-GPPNHP-RanBP1-RanGAP and Ran-GDP-AlFx-RanBP1-RanGAP have been determined [43]. The helical parameters of these two complexes differ slightly from one another.
Many structures of GPIbα have been reported in the free state and in complexes with one or two α-thrombins, von Willebrand Factor A1, and PEP inhibitor [124][125][126][127][128][129][130][131]. We do not find a clear pattern of structural changes. However, it appears that the binding of two α-thrombins causes a structural change.
The crystal structures of NgR-4, n=10, in both the non-glycosylated state and N-glycosylated at Asn-82 and Asn-179, are available [88,89]. N-glycosylation induces a decrease of P, ∆z, and ∆Φ and an increase of N and of R (Supplementary Table 3). The G28E mutant of bovine coupling factor B, n=5, has a smaller R than does the wild type (Supplementary Table 1) [152].

Spatial arrangement of TLR dimers
The structures of the unliganded and ligand activated human TLR8 dimer were determined [119]. Helix parameters of the two monomers in the unliganded dimer are generally similar to those in the liganded dimer. However, upon ligand binding the spatial arrangement of the two monomers in the dimer changes. The distance between the two monomers at their C-terminal regions changes from 53 Å to 30 Å [119]. HELFIT calculates the distance and the angle between the two helical axes of the two monomers in the dimers, L and ψ. Correspondingly, the ligand binding drastically changes these two parameters; ∆L=-2.6 Å and ∆ψ=17º (Figure 7).
The structures of TLR3 complexed with dsRNA and with six different Fabs (Fab15 light and heavy chains, Fab12 light and heavy chains, and Fab1068 light and heavy chains) have been determined, as well as the structure of free TLR3 [112,113]. The formations of the complexes induce decreases in the helix radii. The dsRNA TLR3 dimer has L=74.5 Å and ψ=148º.
Furthermore, the hetero-dimers of human TLR1 and TLR2, and of mouse TLR2 and TLR6 have been determined [108,109]. The values of L and ψ are 44.0 Å and 168º in the TLR1-TLR2 dimer and 53.0 Å and 169º in the TLR2-TLR6 dimer, respectively.

The constant inter-strand distance is the primary determinant of the helical parameters of the LRRs
The HELFIT analysis demonstrates that the inter-strand distance (D) is constant in the seven classes except for the "TpLRR" class; D=5.02 ± 0.00 Å ( Figure 2B). In contrast, D in the "TpLRR" class is significantly larger; D=5.52 ± 0.14 Å. This can be attributed to non-uniformity of the inter-strand distance. The helix radius R in some LRRs belonging to the "TpLRR" class is extremely small or very large (Supplementary  Table 1). Thus, the overall shape of these LRRs of the "TpLRR" class is best described as a prism; while those of the other classes are helices or semi-ellipses.

The "ascending loops" influence the helical parameters of the LRRs
The LRRs in RI and LEGL7 belong to the "RI-like" class. The RI LRR adopts a right handed helix or a horse shoe shape. In contrast, the LEGL7 LRR adopts a left handed helix (Figure 4). The LRRs form β-strands on the concave face (in the HCS) and α-helix of 9 → 15 residues on the convex face (in the VS). Both of the "descending loops" also consist of one β-turn. On the other hand, the ascending loops of the LEGL7 LRR include a 310-helix of one turn; while the RI LRR does not adopt this helix in the loops. Some LRRs in NLRC4, InlK, Lmof2365_1307, AtTIR1, AtCOI1, TLRs 1-6, and TLR8 do not obey the LxxLxLxxNxL consensus, especially at the underlined residues. In addition, some LRRs in these proteins frequently form longer β-strands consisting of four to seven residues in the HCS. This affects not only the ascending loops but also the descending loops. Indeed, the right handed helices of NLRC4, AtTIR1, AtCOI1, TLRs 1-6 and TLR8 are distorted. Also the LRRs in InlK and Lmof2365_1307, which belong to the "SDS22-like" class, adopt left handed helices; while those in the normal "SDS22-like" class form right handed helices.
The HCS of the "TpLRR" class with ten residues is shorter than eleven or twelve residues in the "Typical" class. On the other hand, the VS of the "TpLRR" class is quite similar to that of the "Typical" class. Both VSs consist mainly of consecutive -turns ( Figure 1). However, the helical parameters indicate that the "Typical" class adopts a helix, while the "TpLRR" class resembles a prism. These observations can be attributed to a structural difference of the ascending loops and the inter-strand distance.
The helix parameters vary among the "Typical" LRRs. Seven of the ten repeats in NgR-4 and eleven of the fourteen repeats LINGO-1 conform to the "Typical" consensus. However, their structures differ from one another. The NgR-4 LRR domain adopts a right handed helix with ∆z=2.09 Å; while the LINGO-1 LRR arc is nearly flat; |∆z|=0.06 Å. The LINGO-1 LRRs show a structural periodicity that consists of four tandem repeats of three consecutive LRR units. The first and second VSs in the three LRR units adopt a β-strand plus consecutive β-turns, while in the last VS the corresponding β-strand breaks. In contrast, this periodicity is not seen in the corresponding residues in NgR-4. The different helical parameters may be due to a difference in structure of the ascending loops and the conformations on the convex face.
In conclusion, the above observations suggest that the helical parameters more strongly depend on the structures of the "ascending loops" than of the "descending loops". Moreover, the helical parameters of the LRRs are influenced by helical elements on the convex face and the uniformity of parallel strand stacking on the concave face.

LRR domains having small repeat number are highly flexible
The values R and ∆z versus n for all 642 LRRs show a lot of variation, if n is small (Figure 3). Here we describe some examples. The first is the family of VLR. The lysozyme bound VLRB.2D forms a tetramer in the crystal [83]. The two monomers adopt a right handed helix, while the other two monomers adopt a left handed helix. Sea lamprey VLRB, n=5, bound to the BclA protein forms a trimer [84]. One monomer forms a right handed and the other monomer forms a left handed helix. The third monomer is not visible in the electron density map of the crystal structure. Tetrameric VLR 2913 has a range of helical parameters in its four monomers (Supplementary Table 1).
The crystal structures of InlB, n=8, in wild type, mutant, and complexes have been determined [60][61][62][63][64]85]. The InlB molecules form a monomer, dimer, and hexamer in the crystals. The helical parameters show a range of variation. InlB with an additional LRR unit inserted, n=9, forms a trimer. All the monomers have similar helical parameters.
Right handed and left handed helices exist in the dimer of SERK1, n=5, bound to BRI1 [71] and in the tetramer of bovine Coupling Factor B, n=5 [152]. InlK, n=7, forms a homo-octamer in the crystal [68]. The helix parameters of these eight monomers are highly variable (Supplementary Table 1). Even with n=17 in RI, the helix parameters are significantly different among four vertebrate species: both P and ∆z decrease in order of pig, bovine, human, and mouse [34][35][36][37].
These observations strongly suggest that LRR domains with n ≤ 7 are more variable and are more flexible in solution. NMR data of LC1 with n=6 support this inference; within the β-sheets the consensus leucine residues at position 4 have relatively low order parameters [52].

Structural features of individual LRR classes
As noted, the helical parameters of the eight LRR classes frequently overlap one another. Moreover, when the repeat number in LRRs is small, the LRR structures are more variable. Taken together, the helical parameters of the LRRs with n ≥ 10 were compared to find the structural features ( Figure 3B and supplementary Figure 1). The "TpLRR" class deviates significantly from other classes. The "Typical" class has highly variable values. The rise per subunit, ∆z, of the "RI-like" and "Cysteinecontaining" classes is clearly smaller than those of the "SDS22-like", and "Plant-specific" classes. This is mainly ascribed to the difference in the structural units.

Structures of non-LRR, IRs
The structures of non-LRR, IRs may be considered in two groups. In the first group, IRs allow the continuity of the parallel β-sheet and the LRR units to form a single domain of a regular LRR structure. Examples include the IRs of BRI1, AtRPK2, BT_0210, and TLR8.
The BRI1 LRR domain has 25 units interrupted by a 70 residue IR between unit 21 and unit 22 [28]. B. thetaiotaomicron BT_0210 contains 18 units interrupted by a 76-residue IR between units 2 and 3. In the two LRR domains the helical handedness reverses (Supplementary Table 1). The IR forms a small domain those folds back into the interior of the helix, where it has extensive polar and hydrophobic interactions. The IR between repeat units 15 and 16 of TLR8 is 39 residues long [24]. The additional residues loop out from the expected helical path of a regular LRR before rejoining it some residues later [119]. Consequently, this IR is in the exterior of the helix, in contrast to the IRs in BR11 and BT_0210.
In the second case IRs disrupt the continuity of the parallel β-sheet and form two distinct subdomains. This case is observed in AtTMK1 [75] and Drosophila Toll [95]. The AtTMK1 LRR domain contains 15 repeats interrupted by a 41 residue IR between LRR11 and LRR12 [28], while the Toll LRR domain with 24 repeats has a 68 residue IR between LRR10 and LRR11 [95] .

Conclusions
We have assigned helical parameters to the 642 LRRs of known structure using the program HELFIT. These parameters and their correlations with class of LRR, with the number of repeat units in the LRR, and with oligomerization and ligand state of the LRR are described. The helical parameters of the eight LRR classes frequently overlap one another. However, the constant distance between parallel β-strands is the primary determinant of the helical parameters of the LRRs. When the repeat number, n, is small, the LRR structures are more variable and, by inference, more flexible. In the LRRs with n ≥ 10, rise per repeat unit, ∆z, of the "RI-like" and "Cysteine-containing" classes are smaller than those of "SDS22-like", and "Plant-specific" classes. This is ascribed to the difference in the structural units. The helical parameters of the LRRs unambiguously define both right handed and left handed helices, and helical dimers and subdomains if they exist. Moreover, the helical parameters sensitively detect structural changes induced by protein, protein interactions, glycosylation, and/or mutation.