Measuring the Elasticity of Poly‐l‐Proline Helices with Terahertz Spectroscopy

Abstract The rigidity of poly‐l‐proline is an important contributor to the stability of many protein secondary structures, where it has been shown to strongly influence bulk flexibility. The experimental Young's moduli of two known poly‐l‐proline helical forms, right‐handed all‐cis (Form I) and left‐handed all‐trans (Form II), were determined in the crystalline state by using an approach that combines terahertz time‐domain spectroscopy, X‐ray diffraction, and solid‐state density functional theory. Contrary to expectations, the helices were found to be considerably less rigid than many other natural and synthetic polymers, as well as differing greatly from each other, with Young's moduli of 4.9 and 9.6 GPa for Forms I and II, respectively.

The ability of ap rotein to maintain proper secondary structure [1] is reflected in its elasticity, [2] which represents the tendency of the system to structurally deform under external forces.S everal studies have invoked elasticity to explain the origins of observed protein properties,i ncluding structural stability, [3] mechanical strength, [4] and catalytic activity. [5] Despite the great relevance of elasticity,i ts quantification in large biomolecules has proven to be an elusive goal owing to the difficulties associated with measuring stress-strain curves for these materials. [6] Common methods for studying protein flexibility include X-ray crystallography and NMR spectroscopy, [7] but neither is able to provide specific values for the elastic parameters,and only general inferences can be drawn. To overcome the experimental limitations,s everal indirect approaches have been developed that attempt to relate amino acid sequences in various domains to bulk elasticity,but these often rely on simple empirical data (i.e., packing density and intermolecular contacts), rather than actual stress-strain probe measurements. [5a, 8] Apromising alternative experimental technique is vibrational spectroscopy because it offers the advantage of being able to probe the stress (energy) and resulting strain (motion) of particular vibrational modes. [9] A drawback is that traditional vibrational methods (e.g., midinfrared spectroscopy [10] )probe only the motions localized to individual bonds.This may yield elastic information about the specific bond, but does not provide knowledge of the sample in its entirety.Therefore,different types of vibrations must be considered.
Te rahertz time-domain spectroscopy (THz-TDS) is apowerful tool for accessing sub-150 cm À1 vibrational modes that involve large-amplitude global molecular motions,and it has proven useful for characterizing biomolecules such as cellulose and DNA. [11] These low-frequencymotions include both external (rotation and translation) and internal (torsion) vibrations of condensed-phase sample components,m eaning that both the bulk and localized stress-strain relationships can be simultaneously explored. Thev ibrational force constants determined through THz-TDS are ad irect measure of the elastic properties of the studied material, yielding immediately useful elastic constants such as the Youngs modulus through classical relationships to Hookes law.This approach was used in this work to characterize the two helical conformations of the poly-l-proline polypeptide and evaluate its rigidity as compared to other polymers.
Poly-l-proline,acomponent of collagen, [12] is considered to be arigid peptide sequence and it is often found in proteins where it is believed to add mechanical stability to secondary structure. [13] Proline is unique amongst naturally occurring amino acids as the only residue able to readily form both cis and trans configurations about its peptide bond linkages, [14] thereby permitting two different helical structures to exist for poly-l-proline. [15] Theall-cis right-handed helix (Form I, PP-I) is tightly wound, [16] while the all-trans left-handed helix (Form II, PP-II) adopts al ess dense geometry ( Figure 1). [17] Theavailability of these similar, yet fundamentally different, poly-l-proline helices makes them excellent choices for exploring the connection between molecular structure,l owfrequency vibrational motions,and bulk elastic constants.
While the rigidity of poly-l-proline chains has been explored within protein structures, [13c] no studies of the actual elasticity have been performed. This dearth of information is in part due to alack of atomic-level structural data for either helix. Herein, the terahertz vibrations of PP-I and PP-II were assigned and analyzed by using structures determined from ac ombination of experimental powder Xray diffraction (PXRD) and solid-state density functional theory (ss-DFT) calculations.C ollectively,t hese techniques enable quantification of the elastic properties of this large biopolymer.
Thel ow-temperature (78 K) THz-TDS vibrational spectra ( Figure 2) of solid PP-I and PP-II (1-10 kDa;f or experimental details,s ee the Supporting Information) were acquired over a2 0-150 cm À1 (0.6-4.5 THz, 7.6 GHz resolution) spectral window with aC herenkov-radiation based source, [18] thereby permitting features to be identified beyond the reach of more commonly available instruments. [19] The THz-TDS spectra of both samples contain distinct features that are specific to the conformation of each poly-l-proline helix and also to the three-dimensional arrangement of the helices in the solid state.T he analysis of the vibrational data began with full redetermination of the complete crystal structures of both PP-I and PP-II.
Powder X-ray diffraction measurements (90 K) were performed on both samples (Figure 3), and the results revealed numerous Bragg reflections unique to each solid, surpassing the quality of those previously reported. [16,17] The samples were free from any noticeable cross-contamination, as evidenced by the lack of reflections from the complementary form in both patterns.D espite the high-quality PXRD patterns,s uch data alone were not sufficient for complete structural determinations with atomic precision, and utilization of computational methods was necessary to arrive at detailed solutions.
In the case of PP-I, initial crystal structures were constructed using the previously published interatomic distances and angles, [16] but with the solid-state packing arrangements and strand orientations varied (for details,s ee the Supporting Information). After full ss-DFT optimization, the PP-I crystal was found to have monoclinic P2 1 symmetry in agreement with estimates made by Shmueli and Tr aub ( Figure 4). [20] Theu nit cell contains as ingle all-cis poly-lproline helix that makes three complete turns over the course of 10 residues,w ith the helical axis corresponding to the crystallographic b-axis.This arrangement results in an infinite matrix of neighboring helices oriented parallel to each other and extending throughout the entire crystalline solid.
Thes tructure of PP-II is similar to that reported previously, [17] with the most obvious advancement being inclusion of hydrogen atom positions.P P-II crystallizes in the hexagonal P3 2 space group,a nd similar to the PP-I structure,t he unit cell contains as ingle all-trans helix with three proline residues corresponding to as ingle helical turn (Figure 4). TheP P-II helices are arranged parallel to each other in order to minimize void space,but the more extended PP-II helix enables more efficient packing than PP-I, thereby  maximizing London dispersion interactions.T his is particularly important because both poly-l-proline structures lack any hydrogen bond donors,m eaning that interhelix interactions are due entirely to London dispersion and dipolar forces.
With the two poly-l-proline structures solved, calculation of the vibrational eigenvectors and eigenvalues could be performed to enable assignment of specific modes for determination of the elastic properties of the helices.C onsidering PP-II first, where the higher crystalline symmetry results in al ower number of IR-active vibrational modes, acorrelation between experiment and theory can be observed (Figure 2, bottom). Thel ower symmetry of PP-I results in af ar greater number of IR-active vibrational modes and ahigher spectral density in the low-frequencyregion, but the major contributing modes can still be assigned. Thes ub-150 cm À1 vibrational motions of both forms,d etermined by visualization of the eigenvector displacements,a re primarily rotations and torsions of the pyrrolidine rings that result in complex spring-like elongation and contraction of the helix ( Figure 5). Specifically,the 68.15 cm À1 mode (exp.66.6 cm À1 ) in PP-I and the 100.10 cm À1 mode in PP-II (exp.98.1 cm À1 )are most representative of the prototypical helical compressionextension motion, thus making them prime candidates for Youngs modulus determination through the use of vibrational force constants.
Youngs modulus (Y)i su sed to describe the rigidity of solids,w ith ah igher value being indicative of am ore rigid structure (Y rubber % 0.01 GPa, [21] Y iron bar % 200 GPa [22] ). The equation for Youngs modulus relates the stress s ðÞto the strain e ðÞand is commonly calculated from the slope of an experimental stress-strain curve, where F is the force exerted on the material, L 0 and A 0 are the equilibrium length and area of the material, respectively,and DL is the change in the equilibrium length of the sample.I t

Angewandte Chemie
Communications becomes clear that rearranging the equation for Youngs modulus results in af orm of Hookes law, which is av alid assumption when considering small stresses and strains. [23] Thec onnection between Hookes law and Youngs modulus can be leveraged through vibrational spectroscopy,s ince it relates the vibrational frequency( n)o f aharmonic oscillator with areduced mass (m), to an analogue of the classical force constant (k), Thel ow-frequencym otions accessible by THz-TDS are large-amplitude vibrations of the entire bulk structure,a nd therefore can be used to determine Youngs modulus for the solid.
Using the observed terahertz frequencies and calculated m values,the experimental force constants could be determined for the two assigned spring modes,thereby ultimately yielding the Youngs moduli of the different polyproline helices (Table 1). Thee lastic properties of the two polyproline helices were verified computationally by using ab initio methods (not through vibrational force constants). [24] The results (Table 1) show that the values for Youngs moduli calculated entirely from first principles are in very good agreement with those determined by using the experimental terahertz vibrational frequencies.A dditionally,a san independent check of the applied theory,the Youngs modulus of crystalline polyethylene was calculated by using the same methods,a nd the results (Y = 14.63 GPa) matched well with previously published data [25] (Y = 15.8 GPa).
Contrary to suggestions in the literature, [26] the elasticity results indicate that poly-l-proline is actually considerably less rigid than many other common polymeric materials, [25,27] although it is more rigid than poly-l-alanine (Table 2). [28] The significantly different rigidities of the two forms of poly-lproline,w ith PP-II showing an approximately 96 %l arger Youngs modulus than PP-I, is due to differences in the peptide bond geometries between the two structures.T he near orthogonal orientation of the cis peptide bond with respect to the helical axis in PP-I means that any change in it leads to al arge alteration in the overall helical length. This was confirmed computationally by comparing the geometries of the two helices after manually increasing the helical axes from their equilibrium lengths by 10 %a nd subsequently allowing the structures to relax within the constraint of fixed helical length. Theresults showed that both the covalent bond lengths and dihedral angles were distorted in PP-I (average absolute change per o fd istortion of 0.002 a nd 2.6788 8, respectively) by am uch smaller degree than PP-II (average absolute change per o fd istortion of 0.024 a nd 13.088 8, respectively), despite the same relative change in helical length. Additionally,t he calculations provided some insight into the energy required for conversion of the more stable PP-II structure into the PP-I form (DG 298K = 4.39 kJ mol À1 per residue). The1 0% elongation of the PP-II helix and concomitant dihedral angle changes resulted in an energy increase within the polypeptide of 10.02 kJ mol À1 per o f distortion, which serves as ap reliminary indicator of the barrier opposing the formation of PP-I. These results are consistent with previous studies of the poly-l-proline transformation, which found that large activation energy barriers exist along the conversion coordinate. [15,29] Themeasurement of biopolymer elasticity through acombined approach of THz-TDS experiments and ss-DFT simulations enables quantification of molecular rigidities to be achieved in arelatively straightforward way.This methodology has yielded the previously unmeasured Youngs moduli of the widespread poly-l-proline polypeptide in both its helical forms,a nd revealed them to be considerably more elastic than expected. This prompts contemplation of their role as an analytical standard for rigidity. [30] This method could, in principle,b ea pplied to other biological systems of any phase;h owever an ordered crystalline environment greatly facilitates interpretation of the spectral data. Ultimately,r eliable quantification of biomolecular elasticity promotes ac omplete understanding of the factors affecting protein stability and the mechanisms associated with structural change. [33]   Poly-l-alanine [28] 2.4-2.9 Poly-l-proline Form I [a] 4.9 AE 0.2 Poly-l-proline Form II [a] 9.6 AE 0.1 Poly(methylm ethacrylate) Single Helix [27a] 11 Polyethylene [25] 15.