Tau Inclusions in Alzheimer's, Chronic Traumatic Encephalopathy and Pick's Disease. A Speculation on How Differences in Backbone Polarization Underlie Divergent Pathways of Tau Aggregation

Tau-related dementias appear to involve specific to each disease aggregation pathways and morphologies of filamentous tau assemblies. To understand etiology of these differences, here we elucidate molecular mechanism of formation of tau PHFs based on the PMO theory of misfolding and aggregation of pleiomorphic proteins associated with neurodegenerative diseases. In this model, fibrillization of tau is initiated by the coupled binding and folding of the MTB domains that yields antiparallel homodimers, in analogy to folding of split inteins. The free energy of binding is minimized when the antiparallel alignment brings about backbone-backbone H-bonding between the MTBD segments of similar “strand” propensities. To assess these propensities, a function of the NMR shielding tensors of the Cα atoms is introduced as the folding potential function FPi; the Cα tensors are obtained by the quantum mechanical modeling of protein secondary structure (GIAO//B3LYP/D95**). The calculated FPi plots show that the “strand” propensities of the MBTD segments, and hence the homodimer's register, can be affected by the relatively small changes in the environment's pH, as a result of protonation of MBTD's conserved histidines. The assembly of the antiparallel tau dimers into granular aggregates and their subsequent conversion into the parallel cross-β structure of paired helical filaments is expected to follow the same path as the previously described fibrillization of Aβ. Consequently, the core structure of the nascent tau fibril is determined by the register of the tau homodimer. This model accounts for the reported differences in (i) fibril-core structure of in vivo and in vitro filaments, (ii) cross-seeding of isoforms, (iii) effects of reducing/non-reducing conditions, (iv) effects of PHF6 mutations, and (v) homologs' aggregation properties. The proposed model also suggests that in contrast to Alzheimer's and chronic traumatic encephalopathy disease, the assembly of tau prions in Pick's disease would be facilitated by a moderate drop in pH that accompanies e.g., transit in the endosomal system, inflammation response or an ischemic injury

Protein folding, misfolding and aggregation: The importance of two-electron stabilizing interactions

<div><p>Proteins associated with neurodegenerative diseases are highly pleiomorphic and may adopt an all-α-helical fold in one environment, assemble into all-β-sheet or collapse into a coil in another, and rapidly polymerize in yet another one via divergent aggregation pathways that yield broad diversity of aggregates’ morphology. A thorough understanding of this behaviour may be necessary to develop a treatment for Alzheimer’s and related disorders. Unfortunately, our present comprehension of folding and misfolding is limited for want of a physicochemical theory of protein secondary and tertiary structure. Here we demonstrate that electronic configuration and hyperconjugation of the peptide amide bonds ought to be taken into account to advance such a theory. To capture the effect of polarization of peptide linkages on conformational and H-bonding propensity of the polypeptide backbone, we introduce a function of shielding tensors of the C^α atoms. Carrying no information about side chain-side chain interactions, this function nonetheless identifies basic features of the secondary and tertiary structure, establishes sequence correlates of the metamorphic and pH-driven equilibria, relates binding affinities and folding rate constants to secondary structure preferences, and manifests common patterns of backbone density distribution in amyloidogenic regions of Alzheimer’s amyloid β and tau, Parkinson’s α-synuclein and prions. Based on those findings, a split-intein like mechanism of molecular recognition is proposed to underlie dimerization of Aβ, tau, αS and PrP^C, and divergent pathways for subsequent association of dimers are outlined; a related mechanism is proposed to underlie formation of PrP^Sc fibrils. The model does account for: (i) structural features of paranuclei, off-pathway oligomers, non-fibrillar aggregates and fibrils; (ii) effects of incubation conditions, point mutations, isoform lengths, small-molecule assembly modulators and chirality of solid-liquid interface on the rate and morphology of aggregation; (iii) fibril-surface catalysis of secondary nucleation; and (iv) self-propagation of infectious strains of mammalian prions.</p></div

Morphology of Aβ aggregation on the NIBC monolayers on gold: The combined effects of cysteine chirality and wedge-like shape of Aβ paranuclei.

<p><b>(a)</b> The putative complex of the Aβ paranucleus on the monolayer surface involves backbone H-bonding of the extended H14-K16 segment to the isobutyryl carbonyls of N-isobutyrylcysteines [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref041" target="_blank">41</a>]. The positively charged side chains of His-14 and Lys-16 extend in the same direction. <b>(b)</b> The topology of the self-assembled cysteine monolayer on gold [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref302" target="_blank">302</a>]: The monolayer comprises pairs of long files of N-isobutyrylcysteines. <b>(c)</b> A model of the surface of <i>L</i>-NIBC monolayer [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref302" target="_blank">302</a>], H14 and K16 side chain interactions with the <i>L</i>-NIBC isobutyryl carbonyls, and the putative tight packing of two Aβ paranuclei. The binding of the H14 side chain to the isobutyryl C = O of the neighbouring row of cysteines is unimpeded; the K16 side chain encounters an impediment but its length and flexibility make the binding possible. Thus, the tight packing of paranuclei is achieved via parallel alignment of the H14-K16 segments which directs all the complexed paranuclei to ‘wedge out’ in one direction i.e. promotes annular stacking and formation of large ring structures. <b>(d)</b> A model of the surface of <i>D</i>-NIBC monolayer [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref302" target="_blank">302</a>], H14 and K16 side chain interactions with the <i>D</i>-NIBC isobutyryl carbonyls, and the putative tight packing of two Aβ paranuclei. Here the H14 side chain cannot bind to the isobutyryl C = O of the neighbouring row of cysteines; it encounters an impediment and lacks flexibility and length needed to overcome the hindrance. Thus, the tight packing on this surface is achieved via antiparallel alignment of the H14-K16 segments which directs the neighbouring paranuclei to ‘wedge out’ in opposite directions and thereby promotes tubular stacking and formation of elongated bar structures.</p

Effect of pH on conformational and H-bonding propensity of the polypeptide backbone.

<p><b>(A) Conformational equilibria of islet amyloid polypeptide (amylin)</b> [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref157" target="_blank">157</a>]: (a) The <i>FP</i>_<i>i</i> plot of amylin at high pH and the putative β-sheet conformation of the peptide. Assuming σ^His≡σ^H° in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.t001" target="_blank">Table 1</a>, the segment L16-T30 has β-hairpin potential in the aqueous buffer: L16-N21 = ‒0.0690 (C₅ strand), N22-I26 = 0.6667 (turn), and L27-T30 = ‒0.0964 (C₅ strand); (b) The <i>FP</i>_<i>i</i> plot for amylin at low pH and the putative helical conformation of the peptide. Assuming σ^His≡σ^H+ in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.t001" target="_blank">Table 1</a>, the segment T9-N22 has unambiguous α-helical potential in the aqueous buffer, = 0.1746. <b>(B) Acid-induced loop-to-helix transition in influenza hemagglutinin</b> [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref158" target="_blank">158</a>]. The trimeric glycoprotein hemagglutinin from influenza virus acts as a fusogen at low pH of endocytic vesicles. The activity is contingent on a large-scale structural rearrangement crucial for delivering the viral contents into host cells. The rearrangement involves inter alia conversion of B-loop of hemagglutinin (residues 55–76) into a long α-helix: (a) The <i>FP</i>_<i>i</i> vs. Δ<i>FP</i>_{<i>i-1→i+1</i>} plot for B-loop at high pH. The segment has a helical <i>FP</i>_<i>i</i> profile but very low and it is accordingly unstructured; (b) The <i>FP</i>_<i>i</i> vs. Δ<i>FP</i>_{<i>i-1→i+1</i>} plot for B-loop at low pH. The helical data cluster is now shifted into the stable ‘helix’ region, > 0. <b>(C) Partial unfolding of the translocation-domain helices of the α-pore forming diphteria toxin at low pH</b> [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref159" target="_blank">159</a>]. The toxin invades a cell by crossing the endosome membrane via the process mediated by the C-terminal segment of the all-α domain T (translocation domain) and triggered by the reduced pH in the endosomal lumen: (a) At the standard physiological pH, the toxin is well-structured and the C-terminal segment of the domain T is buried under the helices TH1-TH4, PDB ID 1f0l; (b) At low pH, the TH2, TH3 and TH4 helices become completely disordered, PDB ID 4ow6 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref159" target="_blank">159</a>]. As a result, the C-terminal helices are exposed and can interact with the membrane; (c) The <i>FP</i>_<i>i</i> vs. Δ<i>FP</i>_{<i>i-1→i+1</i>} plot for the domain T at the neutral pH. The plot shows the characteristic pattern of a soluble multi-helix bundle, cf. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.g015" target="_blank">Fig 15A(e)</a>, where α-helices incorporate ‘C₅ strand’ segments necessary to stabilize the compact tertiary structure; (d) The <i>FP</i>_<i>i</i> vs. Δ<i>FP</i>_{<i>i-1→i+1</i>} plot for the domain T at low pH: the folding potential <i>FP</i>_<i>i</i> becomes more positive for the entire domain (the ‘C₅ strand’ segments are ‘titrated out’ which destabilizes the tertiary structure); (e)-(h) The <i>FP</i>_<i>i</i> vs. Δ<i>FP</i>_{<i>i-1→i+1</i>} plots for the α-helices TH3 and TH4 at the neutral pH, panels (e) and (g), and at the low pH, panels (f) and (h). In both cases the folding potential shifts from <i>FP</i>_<i>i</i>≤0 in panels (e) and (g) (the α-helices which are stable in the interior of a compact structure) to <i>FP</i>_<i>i</i> ≥0.3 in panels (f) and (h) (the α-helices which are unstable except in a highly polarizing environment cf. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.g007" target="_blank">Fig 7</a>). <b>(D) Acid-induced unfolding and aggregation of transthyretin</b> [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref038" target="_blank">38</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref162" target="_blank">162</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref172" target="_blank">172</a>]: (a) The superposed <i>FP</i>_<i>i</i> plots for transthyretin at pH >7 and pH 4, and the DSSP assignments for the native monomer (homotetramer subunit, β sandwich fold). The red and purple outlines mark the strand and turn segments most affected by the reduction of pH: <i>FP</i>_<i>i</i> of the V30-F33 segment (βB strand) shifts from ‘C₅ strand’ to ‘helix’ propensity, and the two ‘turn’ segments (D-E≡D* and E-F≡E*) are destabilized. In addition, <i>FP</i>_<i>i</i> of βC and βE segments shifts from ‘C₅ strand’ to ‘C_7eq strand’ propensity. These changes destabilize the outer β sheet of transthyretin βC↓βB↑βE↓βF↑; (b) Putative amyloidogenic monomers of transthyretin generated by step-by-step dismantling and rearrangement of transthyretin β sandwich consistent with the destabilizing <i>FP</i>_<i>i</i> shifts and the solid state NMR [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref162" target="_blank">162</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref164" target="_blank">164</a>], spin-labelling [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref165" target="_blank">165</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref166" target="_blank">166</a>], immunoreactivity [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref167" target="_blank">167</a>] and H/D exchange (HXMS) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref168" target="_blank">168</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref169" target="_blank">169</a>] studies; (c) The pathway of acid-induced unfolding and aggregation of transthyretin [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref038" target="_blank">38</a>], and the topology of annular octamers [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref170" target="_blank">170</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref172" target="_blank">172</a>] and protofibrils [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref169" target="_blank">169</a>].</p

Conformational diversity in the binary complexes of extended oligopeptide strands.

<p>The geometries and energies obtained by quantum-mechanical modeling of the two-stranded β-sheets <b>6</b>–<b>8</b> (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#sec002" target="_blank">Computational Methods</a>). The individual strands in these complexes optimize either to the C₅ or the C_7eq (2₇-ribbon) geometries, and their conformations are same in the antiparallel complexes (C₅↑C₅↓ or C_7eq↑C_7eq↓) and mixed in the parallel complexes (C_7eq↑C₅↑); <i>the antiparallel complexes with mixed strand conformations (C</i>_<i>7eq</i><i>↑C</i>_<i>5</i><i>↓) are unstable in unconstrained optimizations</i>. <b>(A)</b> The antiparallel complexes of the tetrapeptides (AcNH-Ala₃-NH₂)₂ displaying the edge-to-edge topoisomerism: the assembly creates either two or one large H-bonded (HB) ring. <b>1a</b>: the C₅↑C₅↓ complex with two large HB rings; <b>1b</b>: the C_7eq↑C_7eq↓ complex with one large HB ring; <b>1c</b>: the C₅↓C₅↑ complex with one large HB ring; <b>1d</b>: the C_7eq↓C_7eq↑ complex with two large HB rings. <b>(B)</b> The parallel complexes of the hexapeptides (AcNH-Ala₅-NH₂)₂ displaying the edge-to-edge topoisomerism: here all the H-bonded rings are equivalent but complex formation involves the edges with either two or three intrachain H-bonds. <b>2a</b>: the C_7eq↓C₅↓ complex involving the edges with two intrachain H-bonds; <b>2b</b>: the C₅↓C_7eq↓ complex involving the edges with three intrachain H-bonds. The large difference in the energy of the edge-to-edge topoisomers is not observed in the case of the binary complexes of the oligopeptides with the odd number of the peptide bonds. <b>(C)</b> The relative energies of the <b>3a</b>: C₅↓C₅↑, <b>3b</b>(≡<b>2a</b>): C_7eq↓C₅↓, and <b>3c</b>: C_7eq↑C_7eq↓ complexes of the hexapeptides (AcNH-Ala₅-NH₂)₂. <b>(D)</b> The segments comprising two consecutive strands form stable β-hairpins (antiparallel assembly) when the two strands are either (a) both highly polarized (C₅↑C₅↓) or (b) both moderately polarized (C_7eq↑C_7eq↓) (color-coding as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.g001" target="_blank">Fig 1</a>). In contrast, when one strand is highly polarized and the other is moderately polarized, these segments are expected to form (c) β-solenoid coils (parallel assembly, C_7eq↑C₅↑) or (d) unstable β-hairpins (antiparallel assembly C_7eq↑C₅↓) which are prone to convert into β-arches; similarly (e) when one strand is highly polarized and the other is least-polarized (the configuration described by a large contribution of the structure I, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.g001" target="_blank">Fig 1C</a>), the segment may form a hairpin (C₅↑C₅*↓) which is also prone to convert into β-arch.</p

as a measure of conformational and H-bonding propensity of the polypeptide backbone.

<p><b>(A</b>) <b>as a probe of backbone mobility</b>: (a) The mean temperature factors <i>B</i>_<i>i</i> of the backbone N atoms in α-helices vs. mean <i>FP</i>_<i>i</i> of those helices, (α), in the xylanase from <i>Thermoascus auranticus</i>, PDB ID 1i1wA [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref115" target="_blank">115</a>]. Helical residues are assigned according to the Swiss-PDBViewer: helix <b>A</b> 6–12, <b>B</b> 24–27, <b>C</b> 32–38, <b>D</b> 51-54, <b>E</b> 64–76, <b>F</b> 93–96, <b>G</b> 101–117, <b>H</b> 143–147, <b>I</b> 151–163, <b>J</b> 182–197, <b>K</b> 215–227, <b>L</b> 245–257, <b>M</b> 292-301; (b) The mean temperature factors <i>B</i>_<i>i</i> of the backbone N atoms in β strands vs. mean <i>FP</i>_<i>i</i>of those strands, (β). The strand residues are assigned according to the Swiss-PDBViewer and DSSP protocol implemented in the RCSB PDB database: <b>N</b> 17–22, <b>O</b> 41–46, <b>P</b> 79–81, <b>Q</b> 124–127, <b>R</b> 132–134, <b>S</b> 138–140, <b>T</b> 168–173, <b>U</b> 202–206, <b>V</b> 208–210, <b>W</b> 232–236, <b>X</b> 239–242, <b>Y</b> 264–266, <b>Z</b> 279–281. The trendlines are obtained by fitting 2^nd and 4^th order polynomial functions. <b>(B) and the energy of backbone-backbone H-bonding</b>. Δ(Δ<i>G</i>_f) vs. <i>FP</i>_<i>i</i> for the single-site amide-to-ester X(i)ξ substitutions (Δ(Δ<i>G</i>_f) = Δ<i>G</i>_f,WT‒Δ<i>G</i>_f,X(i)ξ) in Pin1 WW domain. The data shown for the mutants in which the perturbed amide donates, but does not accept, a hydrogen bond (thermal, GdnHCl, pH 7.0; PDB ID 2kcf) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref116" target="_blank">116</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref117" target="_blank">117</a>]. The trendline is obtained by fitting 4^th order polynomial function. <b>(C) and the amyloid fibril-forming capacity of oligopeptides</b>. The data for the total of 942 unique hexapeptide structures are taken from the WALTZ-DB database of amyloid forming peptides [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref118" target="_blank">118</a>]; excluding the Pro-containing peptides, the sample comprises 240 amyloidogenic and 702 non-amyloidogenic hexapeptides. The mean <i>FP</i>_<i>i</i> of each hexapeptide, (peptide), is defined as the mean of <i>FP</i>_<i>i</i> of the two central residues. The amyloid fibril-forming capacity of the hexapeptides with within a specific 0.05 range (‒0.750±0.025 etc.) is defined as the frequency of the amyloidogenic peptides within this range in the entire amyloidogenic sample (240 entries), normalized by the frequency of both amyloidogenic and non-amyloidogenic peptides within the same range in the total hexapeptide sample (942 entries).</p

Electronic configuration of the polypeptide backbone and stability of large-to-small hydrophobic variants.

<p>The figure presents the Δ(Δ<i>G</i>_f) vs. plots for the single-site Xaa(i)Ala mutations in helices and sheets (Δ(Δ<i>G</i>_f) = ΔG_f,WT‒Δ<i>G</i>_f,Xaa(i)Ala, = ⅓ (<i>FP</i>_<i>i-1</i>+ <i>FP</i>_<i>i</i> + <i>FP</i>_<i>i+1</i>); Xaa = F, I, L, M, T, V, W, Y). The ‘helix’ and ‘strand’ residues are assigned based on the DSSP protocol implemented in the RCSB PDB database; the trendlines are obtained by fitting polynomial functions [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref125" target="_blank">125</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref131" target="_blank">131</a>]: <b>(A)</b> The Δ(Δ<i>G</i>_f) data for the α-helices in: (a) staphylococcal nuclease (GdnHCl, pH 7.0 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref125" target="_blank">125</a>], PDB ID 1nuc), villin headpiece subdomain (thermal, pH 7.0 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref126" target="_blank">126</a>], PDB ID 1yri, the data set of the highest (shown) is not included in the calculation of the trendline), and acyl coenzyme A binding protein (GdnHCl, pH 5.3 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref087" target="_blank">87</a>], PDB ID 2abd); (b) bacteriophage T4 lysozyme (thermal, pH 3.0 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref127" target="_blank">127</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref128" target="_blank">128</a>], PDB ID 2lzm, the complete scattergram, including the four data sets of the highest , is shown in the insert). <b>(B)</b> The Δ(Δ<i>G</i>_f) data for the β-sheet strands in: (a) fibronectin type III domains of human tenascin TNfn3 (3^rd module, PDB ID 1ten) and fibronectin FNfn10 (10^th module, PDB ID 1fnf) (thermal, urea, GdnHSCN, pH 5.0) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref129" target="_blank">129</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref130" target="_blank">130</a>]; (b) immunophilin FKBP12 (urea, pH 7.5 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref131" target="_blank">131</a>], PDB ID 2ppn, the data set of the highest (shown) is not included in the calculation of the trendline); (c) staphylococcal nuclease (GdnHCl, pH 7.0 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref125" target="_blank">125</a>], PDB ID 1nuc, the data set of the highest (not shown) is not included in the calculation of the trendline).</p

Electronic configuration of the polypeptide backbone versus complexity of tertiary structure and order of oligomerization in soluble and integral membrane proteins.

<p>The <i>FP</i>_<i>i</i> vs. Δ<i>FP</i>_{<i>i-1→i+1</i>} plots, cf. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.g005" target="_blank">Fig 5</a>, may characterize the relationship between distribution of backbone density and organization of globular structure. For α-helices, narrow distribution of the folding potential values and the slope values, Δ<i>FP</i>_{<i>i-1→i+1</i>} ~0, generates compact clusters of the data points. As the complexity of α structure increases and the exposure to the medium decreases, this clusters shift from <i>FP</i>_<i>i</i> >0 to <i>FP</i>_<i>i</i> <0 region in the case of soluble proteins, see the panels in <b>(A)</b>, and in the opposite direction, from <i>FP</i>_<i>i</i> <0 to <i>FP</i>_<i>i</i> >0 region in the case of integral membrane proteins, see the panels in <b>(B)</b>. The same trends are discernible in the <i>FP</i>_<i>i</i> vs. Δ<i>FP</i>_{<i>i-1→i+1</i>} plots for the antiparallel β sheets that assemble ‘C₅ strands’ or ‘C_7eq strands’ and ‘<i>FP</i>_<i>i</i>>>0 turns’, panels <b>(C)</b> and <b>(D)</b>, even though large differences in the folding potential values and wide distribution of the slope values, from Δ<i>FP</i>_{<i>i-1→i+1</i>} ~ ‒1 to Δ<i>FP</i>_{<i>i-1→i+1</i>} ~ 1, generate in this case circular distribution of data points. <b>(A)</b> (a) The peripheral stalk helix of F₀F₁ ATP synthase from <i>E</i>. <i>coli</i> (b₂ subunit in the top diagram in the right-hand panel), UniProt # P0ABA0: residues A32-K122 (the hinge region, the dimerization region, and the C-terminal δ-domain-binding region); (b) One of the two 160Å-long helices of colicin Ia (residues R351-K470) that span the periplasmic space, linking the receptor-binding domain to the other domains, PDB ID 1cii; (c) The parallel α-helical coiled coil cortexillin I, PDB ID 1d7m; (d) The Leu-zipper segment of the GCN4 bZIP protein PDB ID 2dgc: the ‘helix’ cluster is shifted to the <i>FP</i>_<i>i</i> ~0 region; (e) The pattern of the multi-helix bundle which functions as an enzyme within a heterooligomeric complex: the ‘helix’ cluster is partly shifted into the <i>FP</i>_<i>i</i><0 region. The structure shown is the catalytic domain of the guanine nucleotide exchange factor (Ddl homology (DH) domain) of human PAK-interacting exchange protein, PDB ID 1by1. <b>(B)</b> (a) Transmembrane segment of the peripheral stalk of F₀F₁ ATP synthase from <i>E</i>. <i>coli</i> (b₂ subunit in the diagram in the centre, color-coded orange), UniProt # P0ABA0: residues A11-A31; (b) Transmembrane α-helices of human glycophorins A, B, C and E: UniProt #’s P02724, P06028, P15421, and P04921. The glycophorin helices apparently are brought together in the membrane by the dimerization of the extra-membrane domains; (c) The transmembrane α subunit of the membrane associated acetylcholine receptor from <i>Torpedo marmorata</i>, PDB ID 2bg9; the five transmembrane subunits of this receptor do not form a tight oligomer structure and are held in place by the extra-membrane subunits; (d) Bacteriorhodopsin, a membrane protein (light-driven proton pump) from <i>Halobacterium salinarum</i>; the biological assembly is the homotrimer, PDB ID 1fbb; (e) Glycerol facilitator (GlpF), a membrane channel protein of the aquaporin family; the biological assembly is a homotetramer, PDB ID 1fx8; (f) Subunit <i>c</i> of F₀F₁ ATP synthase from <i>E</i>. <i>coli</i> (color-coded blue in the diagram in the centre): the biological assembly is a homodecamer, PDB ID 1ijp. <b>(C)</b> (a) The single-sheet protein, the central β-sheet (residues A81-D205) of the <i>Borrelia burgdorferi</i> spirochete antigen, outer surface protein A (OspA), PDB ID 2g8c; (b) The β sandwich C-terminal domain of the α-amylase from <i>Geobacillus stearothermophilus</i>, PDB ID 1qho; (c) The strands βF(H88-A97) and βH(S115-T123) of the homotetramer of transthyretin, buried in the tetramer interior, PDB ID 5l4j. <b>(D)</b> (a) 16-stranded β barrel, the monomeric integral outer-membrane porin OmpF from <i>E</i>. <i>coli</i>, PDB ID 1opr; (b) 16-stranded β barrel, the trimeric integral outer-membrane porin OmpG from <i>E</i>. <i>coli</i>, PDB ID 2f1c.</p

<i>FP</i>_<i>i</i> as a probe of the three-dimensional structure of proteins.

<p><b>(A)</b> The patterns in the plots of Δ<i>FP</i>_{<i>i-1→i+1</i>} (Eq. 2) vs. the residue number, characteristic of the archetypal ‘helix’, ‘strand’ and ‘turn’. <b>(B)</b> Characteristic clusters of the data sets in the plots of <i>FP</i>_<i>i</i> vs. the ‘slope’ of <i>FP</i>_<i>i</i>, Δ<i>FP</i>_{<i>i-1→i+1</i>}, which correspond to the three archetypal elements of the secondary structure: e.g. the presence of the archetypal ‘helix’ will be marked by a compact cluster of data sets in the center of the plot. The ordinate of this cluster will vary since the optimal <i>FP</i>_<i>i</i> value for ‘helix’ depends on the medium’s capacity to polarize the protein, <i>vide infra</i>. Note that ‘strand’ and ‘turn’ have each two avatars: (i) ‘C₅ strand’ and ‘C_7eq strand’, and (ii) ‘ <i>FP</i>_<i>i</i>>>0 turn’ (defined here as the three- or five-residue segment that incorporates Gly in the centre) and ‘ <i>FP</i>_<i>i</i><<0 turn’. <b>(C)</b> The presence of the archetypal antiparallel ‘sheet’ would be marked by a circular distribution of data sets that combines the ‘C₅ strand’/‘turn’ or ‘C_7eq strand’/‘turn’ clusters while the presence of the parallel ‘sheet’ would be marked by a combination of the ‘C₅ strand’ and ‘C_7eq strand’ clusters, cf. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.g002" target="_blank">Fig 2</a>. This is illustrated by examples of <i>de novo</i> designed three-stranded antiparallel β-sheets (three-stranded β meanders), two- and three-stranded parallel β-sheets, and two-stranded parallel β-sheets embedded in left-handed coils from the C-terminal domains of the penicillin binding protein PBP2x from <i>Streptococcus pneumoniae</i>, PDB ID 1k25: (a) KGEWTFVNGKYTVSINGKKITVSI, ~50% in β structure, H₂O, pH 3, 25°C (C₅↑C₅↓C₅↑-meander) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref071" target="_blank">71</a>]; (b) TWIQNGSTKWYQNGSTKIYT, 20–30% in β structure, H₂O, pH 3.25, 10°C (C₅↑C₅↓C₅↑-meander) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref072" target="_blank">72</a>]; (c) RGWSLQNGKYTLNGKTMEGR, ~35% in β structure, 10%D₂O/H₂O or D₂O, pH 5, 0–10°C (C_7eq↑C_7eq↓C_7eq↑-meander) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref073" target="_blank">73</a>]; (d) C₅↑C_7eq↑-parallel sheet, cf. the <i>FP</i>_<i>i</i> plot. The C-termini of two strands are connected by the D-prolyl-1,1-dimethyl-1,2-diaminoethane unit (diamine linker D-Pro-DADME), ~64% ‘folding-core’ residues (F5-V8 and R11-L14) in β structure at 10°C, 10%D₂O/H₂O, 100 mM sodium acetate buffer, pH 3.8 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref074" target="_blank">74</a>]; (e) C_7eq↑C₅↑C_7eq↑-parallel sheet, cf. the <i>FP</i>_<i>i</i> plot. The C-termini of strands 1 and 2 are connected by the diamine D-Pro-DADME while the N-termini of strands 2 and 3 are connected by the diacid formed from (1<i>R</i>,2<i>S</i>)-cyclohexanedicarboxylic acid (CHDA) and Gly, 4°C, 10%D₂O/H₂O, 2.5 mM sodium [D₃]acetate buffer, pH 3.8 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref075" target="_blank">75</a>]; (f) the C_7eq strands from two C₅↑C_7eq↑-parallel sheets in the left-handed coils of PBP2x from <i>Streptococcus pneumoniae</i>, PDB ID 1k25; (g) the C₅ strands from two C₅↑C_7eq↑-parallel sheets in the left-handed coils of PBP2x, PDB ID 1k25.</p

Electronic configuration of the polypeptide backbone and secondary structure propensity.

<p><b>(A) Experimental α-helix propensities</b>: (a) The averaged relative α-helix propensity data obtained in the site-directed mutagenesis studies of both peptides and proteins, adjusted so that Δ(Δ<i>G</i>_f) = 0 for Ala and Δ(Δ<i>G</i>_f) = 1 for Gly [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref101" target="_blank">101</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref108" target="_blank">108</a>], vs. the NMR shielding tensors <b>σ</b>(C^α)^Xaa (3₁₀-helix AcG(Xaa)GGGNH₂; GIAO//B3LYP/D95**, cf. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#sec002" target="_blank">Computational Methods</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.s006" target="_blank">S1 Table</a>): ♦ glycine and amino acids whose C^β and C^γ are the methyl, methylene or methine groups, r² = 0.83; ▲proline; ◊ any other amino acids including three highly fluorinated amino acids, r² = 0.52 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref107" target="_blank">107</a>]; trendlines obtained by fitting 2^nd order polynomial functions; (b) The Lifson-Roig propagation free energies for the amino acids whose C^β and C^γ are the methyl, methylene or methine groups, in 88% methanol-water [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref109" target="_blank">109</a>]; (c) The Lifson-Roig propagation free energies for the same set of amino acids in 40% (cyan) and 90% (navy) trifluoroethanol-water [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref109" target="_blank">109</a>]. The propensities are determined at the sites in the helices interior. <b>(B) Experimental β-sheet propensities</b> from site-directed mutagenesis (kcal mol^-1, Δ(Δ<i>G</i>_f) = 0 for Gly in (D) and Δ(Δ<i>G</i>_f) = 0 for Ala in (E), (F) and (G)) vs. calculated NMR shielding tensors <b>σ</b>(C^α)^Xaa (AcGGGGGXaaNHMe in β-hairpin (Ib turn); GIAO//B3LYP/D95**, cf. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#sec002" target="_blank">Computational Methods</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.s006" target="_blank">S1 Table</a>): (a) zinc-finger β-hairpin, site 3, r² = 0.89 (edge strand, the guest site is not H-bonded) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref110" target="_blank">110</a>]; (b) Ig binding B1 domain of streptococcal protein G, r² = 0.83 (variant E42A/D46A/T53A, site 44, edge strand, the guest site is H-bonded) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref111" target="_blank">111</a>]; (c) Ig binding B1 domain of streptococcal protein G, r² = 0.84 (variant I6A/T44A/T51S/T55/S, site 53, central strand) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref112" target="_blank">112</a>]; (d) Ig binding B1 domain of streptococcal protein G, r² = 0.76 (I6A/T44A, site 53, central strand) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref113" target="_blank">113</a>,<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref114" target="_blank">114</a>]. Δ(Δ<i>G</i>_f) for Pro in (b), (c) and (d) set at the minimum value of 3 kcal mol^-1 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0180905#pone.0180905.ref112" target="_blank">112</a>]; trendlines obtained by fitting 4^th order polynomial functions.</p