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polypeptide helices are a key secondary structure motif in a wide range of proteins @xcite . it is well known that some amino acids ( e.g. , alanine ) exhibit a stronger helix propensity than others @xcite , but the fact that the helical structure is so abundant in proteins is still intriguing . from a thermodynamic point of view , there are at least two possible limits in which helices compete with other structure prototypes . towards high temperature , one expects the transition to a random coil @xcite , which should become entropically favored as the temperature increases . towards low temperature , however , helices may themselves compete with other enthalpically stable conformations . however , in the most interesting regime , namely intermediate , physiological temperatures , stability may be determined by a delicate balance between enthalpy and entropy @xcite . we here unravel this balance quantitatively for the emergence of helical structure in a particularly well studied series of unsolvated polyalanine based peptides ac - ala@xmath0-lysh@xmath1 , @xmath2=4 - 8 @xcite . we consider explicitly not just the helical part of conformational space , but actually the much larger , general low - energy conformational space of the peptides , of which helices are a part . in this paper we show : ( i ) a comprehensive search of the conformational space for ac - ala@xmath0-lysh@xmath1 , @xmath2=4 - 8 ; ( ii ) harmonic free energy calculations for several structural candidates ; ( iii ) outlook on the role of anharmonicities in the potential energy surface ; and ( iv ) a theoretical comparison for longer model peptides , considering only helical motifs , with a different termination , in order to clarify the impact of lys on the soft vibrational modes . our key finding is that there is a significant vibrational entropic stabilization of helices compared to other , more compact conformers , a contribution that should indeed make a difference in actual proteins as well . this contribution is intrinsic to the helix and should therefore act largely independently , not entangled with environment - dependent terms such as a solvent entropy . interestingly , an essential role of low - frequency modes is also actively debated in other areas of protein science , related to their function . beginning with the terms that shape the potential - energy surface ( pes ) , known reasons for helix stability include @xcite ( i ) their efficient hydrogen - bond ( h - bond ) network and increasing h - bond cooperativity with helix length @xcite , ( ii ) suitably bonded and/or electrostatically favorable termination , for instance the lysh@xmath1 termination considered here @xcite , and ( iii ) remarkably , a rather specific favorable contribution of van der waals ( vdw ) interactions for @xmath3-helices @xcite . clearly , the peptide chain length plays a role : too short chains have too few and too weak hydrogen bonds for helices to compensate the cost of strain in the backbone . in practice , environment effects will necessarily influence helix stability @xcite . in an aqueous medium , the hydrophobic effect will be prominent @xcite . in water - poor conditions , like membranes , helices are frequently observed @xcite . finally , in vacuum conditions , the longer members ( @xmath48 ) of the polyalanine based peptide series studied here assume helical structure in experiment @xcite . these helices are stable _ in vacuo _ even up to extreme temperatures ( not expected in solution ) @xcite , or after soft landing on a surface @xcite . the potential energy surface also shapes entropy , and thus the effect of temperature @xmath5 . with increasing @xmath5 , the conformational entropy of the backbone will favor an unfolded state @xcite ( so - called `` random coil '' ) , while at low @xmath5 helices may also compete with other , enthalpically more stable conformers . for instance , gas - phase ion mobility spectrometry ( ims ) by jarrold and coworkers @xcite showed that the ac - ala@xmath6-gly@xmath7-ala@xmath6h@xmath1 polypeptide is helical at @xmath5=400 k but globular at room temperature . a similar structural change was observed in experiments involving multiply protonated polyalanine in the gas - phase in ref . @xcite . empirical force - field based simulations by ma and coworkers @xcite of more than 60 small peptides indicate that the _ vibrational _ entropy ( harmonic approximation ) could stabilize @xmath3-helices or @xmath8-hairpins over competing low - temperature conformers . very recently , plowright and coworkers @xcite used density - functional theory ( dft ) including dispersion contributions ( the b97-d @xcite exchange - correlation functional ) to suggest that , for a small neutral four - residue peptide , @xmath8-sheets and conformers containing 3@xmath9 helical loops are stabilized by the harmonic vibrational entropy at finite temperatures .    in the present work , we provide independent , unambiguous , and quantitative computational evidence that the vibrational entropy acts to stabilize helical conformers with increasing temperature over more compact , enthalpically competitive structures . the reason , in short , is traced to the softer low - frequency modes of helices , which are also reflected in the dynamics ( anharmonic case ) . we focus on polyalanine - based peptides , since alanine is known to have a high helix propensity both in solution @xcite and _ in vacuo _ @xcite . for ac - ala@xmath0-lysh@xmath1 ( @xmath2=4 - 20 ) in the gas phase , ims @xcite and first - principles calculations compared to experimental vibrational spectroscopy at room temperature @xcite suggest a cross - over from non - helical to helical preferred conformers as a function of polyalanine chain length . for @xmath2=5 there is a competition between different conformers , while the @xmath2=10 and @xmath2=15 conformers are found to be firmly in the helical range @xcite . in a previous publication @xcite we have quantified , from first principles , the contributions from electrostatics , h - bond cooperativity , and van der waals interactions on the stability of unsolvated polylalanine - based helices against unfolding . this class of systems is thus an ideal testing ground to clarify the structural competition of non - helical ( compact ) and helical conformers as a function of chain length also toward the opposite temperature limit ( low temperature , folded state ) . in the following , we address conformational preference of ac - ala@xmath0-lysh@xmath1 _ in vacuo _ for @xmath2=4 - 8 , i.e. , the length range in which the helical preference at room temperature develops . our work is based on an exhaustive prediction of low - energy conformers using dft and the pbe @xcite exchange - correlation functional , corrected to account for long - range vdw interactions @xcite ( here called pbe+vdw ) . this level of theory treats accurately , and without system - specific empirical parameters , critical length - dependent contributions such as h - bond cooperativity @xcite and vdw interactions @xcite , including their effect on vibrational frequencies and in _ ab initio _ molecular dynamics . our conformational search strategy , used to find structure candidates of ac - ala@xmath0lysh@xmath1 , @xmath2=4 - 8 , consists of two steps . for a detailed discussion of the search strategy we use here , we refer the reader to ref . @xcite . both the lys residue and the c - terminal cooh are considered protonated throughout , as is known to be the gas - phase preference when the n - terminus is capped.@xcite    in the first step , we begin by an extensive , unconstrained force field ( opls - aa @xcite ) basin - hopping search ( using the tinker package @xcite ) , aiming to simply `` list '' as many different conformers as possible . the same strategy was employed in ref . @xcite , enumerating a huge number of candidate structures : at least 10@xmath10 conformers for each of the molecules in question . our particular choice of the opls - aa force field is not motivated by any other reason than that an input structure `` generator '' for dft was needed . in particular , care was taken that small changes to the parameters in the opls - aa part of the search do not affect the structure manifold considered in the dft step below.@xcite    in the second step , a wide range of conformers suggested by the force field is fully relaxed using dft with the pbe+vdw exchange - correlation functional in all - electron total energy calculations ( fhi - aims program package @xcite ) . we employ essentially converged numeric atom - centered basis sets and other numerical settings @xcite for the relaxations . in detail , the pbe+vdw part of our searches covers 1068 , 1000 , 800 , 800 , and 820 conformers for @xmath2=4 , 5 , 6 , 7 , 8 , respectively . for @xmath2=6 , 7 , and 8 , these numbers are chosen to ensure that at least all conformers identified in the lowest 7 kcal / mol ( @xmath11 0.3 ev ) of the force field step are included in the pbe+vdw relaxations . for @xmath2=4 and 5 , we explored the limits of our search strategy @xcite . when comparing locally relaxed minima in pbe+vdw that were obtained by starting from an opls - aa local pes minimum structure , we observe a correlation of the relative energy hierarchies , but with a large scatter ( max . around 50 mev per residue ) . therefore , a large number of conformers must be considered for post - relaxation in pbe+vdw . the comparison also reveals structure - specific force field errors that might otherwise go unnoticed . for example , we observe that opls - aa systematically overestimates the energy of 3@xmath9-helical structures _ in vacuo_. to exclude any unwanted impact of the overestimation of 3@xmath9-helices , we performed additional constrained basin - hopping searches in which we forced certain h - bonds of the molecule to remain 3@xmath9-helical , again followed by individual , unconstrained pbe+vdw relaxations . the pbe+vdw relaxed conformers were sorted into `` families '' according to their h - bond pattern . we define an h - bond to be present when an o acceptor atom is closer than 2.5  to a h donor atom . within each h - bond family thus defined , small conformational variations are still possible , e.g. , by slight bends of the backbone atoms or different rotamers of the lysh@xmath1 side chain . typically , the lowest - energy pbe+vdw conformer is found among the family members arising within 5 kcal / mol ( 0.2 ev ) of the lowest - energy force field conformer . harmonic vibrational frequencies and intensities were computed from finite differences . the accuracy of the vibrational frequencies calculated in this manner is estimated by analyzing the rotational and translational modes . these modes should be at zero frequency , and thus the deviation observed gives a limit for the accuracy of the rest of the frequencies . we have not observed deviations of more than 2 @xmath12 in any calculations . harmonic free energies were calculated in the harmonic oscillator / rigid body approximation . vibrational density of states beyond the harmonic approximation were calculated from the fourier transform of the velocity time autocorrelation function , taken from _ ab initio _ molecular dynamics trajectories . for all molecules containing @xmath2>8 alanine residues and for the li@xmath1 terminated model structures discussed in this work , no extensive conformational search was performed . these peptides are structure models used specifically for a computer experiment to determine the development of low - frequency vibrational modes with increasing helix length for two different terminations . their geometries are fully relaxed pbe+vdw structures . for additional details about these calculations we refer the reader to the supporting information . [ fig : fig1 ] summarizes the energetic ordering of the lowest - energy ( pbe+vdw ) h - bond families for @xmath2=48 , found employing our search strategy . only the energy of the lowest energy structure belonging to each family is reported , and families are included up to 0.12 ev ( @xmath11 3 kcal / mol ) of the lowest identified minimum of the pbe+vdw pes . @xmath3-helical conformers are highlighted in red . the 3@xmath9-helical conformer for @xmath2=4 is highlighted in blue . we define purely @xmath3- ( or purely 3@xmath9- ) helical conformers as those where , counting from the n - terminus , all the backbone co groups at residues @xmath13 are either connected to nh groups at residues @xmath14 ( or @xmath15 ) or to the lysh@xmath1 side chain ( usually the final three or four co groups at the c terminus ) . coordinates and a more detailed analysis of all the geometries shown in fig . [ fig : fig1 ] are given in the supporting information . the conformer associated with the lowest - energy pes minimum for @xmath2=4 is rather small , connecting almost all its backbone co groups to the lysh@xmath1 termination . the remaining h - bond at the n - terminus is bifurcated , comprising an @xmath3- and a 3@xmath9-helical h - bond . this conformer could therefore be classified as the smallest possible @xmath3-helical prototype in this series . in contrast , the lowest - energy pes minima for @xmath2=5 and 6 are _ not _ simple helices . each contains an `` inverted '' h - bond where one co group points to the n - terminus and its connecting nh group points to the c - terminus , producing more compact structures . in fig . [ fig : fig1 ] , these bonds are highlighted in yellow and pointed to by an arrow . for @xmath2=5 , we have previously denoted this conformer as `` g-1 '' @xcite . for @xmath2=7 and 8 , the lowest - energy pes minima correspond to @xmath3-helices . in each case , they are closely followed by a conformer that we characterize as compact / globular ( families 2 of @xmath2=7 and 8 , with an energy separation of 20 and 60 mev respectively ) . thus , we already observe a cross - over with peptide length to @xmath3-helical lowest - energy minima of the pes at @xmath2=7 . however , based on the energy hierarchies of the structures of the local pes minima alone , one would not expect a purely helical ensemble of conformers at room temperature at @xmath2=7 or 8 . simple boltzmann factors would indicate a mix of structure candidates . yet , the experimental work by kohtani and jarrold @xcite does indicate a complete room - temperature structural cross - over at @xmath2=8 at the latest , albeit based on a completely different line of reasoning ( water adsorption behavior of `` helical '' versus `` globular '' conformers ) .    for the low - energy conformers in fig . [ fig : fig1 ] , we also compute and show in the same figure the impact of the vibrational free energy at room temperature ( @xmath5=300  k ) in the harmonic approximation . remarkably , the relative stability of the @xmath3-helices is systematically enhanced by the vibrational free energy contribution for _ all _ @xmath2 . for @xmath2=5 and 6 the @xmath3-helical conformers move down in energy with respect to the ( non - helical ) lowest energy pes minimum . for @xmath2=7 and 8 the @xmath3-helices now become the isolated minima . in detail , we observe : + - for @xmath2=8 , the energetic interval between the @xmath3-helical lowest energy conformer and the nearest globular one ( red shaded areas ) now amounts to 0.14 ev . the additional family 1a at 0.13 ev is another @xmath3-helix with a slightly modified terminating h - bond network . with the vibrational free energy included , the energy hierarchy is thus consistent with the experimental claim that @xmath3-helices dominate over all other possible conformers in gas - phase experiments for @xmath2=8 @xcite . + - for @xmath2=7 , the same qualitative picture emerges . here , the next remaining conformer ( family 3 ) at 300  k is a mixed 3@xmath9/@xmath3-helix . the competing compact conformers ( representatives of family 2 and 2a ) are significantly destabilized by @xmath16(@xmath5 ) . + - for @xmath2=6 , the @xmath3-helix emerges as the room - temperature minimum free energy conformer , but the competing non - helical pes minimum , family 1 , remains close ( 50 mev ) . + - for @xmath2=5 , the difference between the @xmath3-helix and family 1 ( g-1 ) decreases by 60 mev at 300  k , compared to the pes minimum , but family 1 remains overall more favorable . + in essence , there is an uniform stabilization of helical over more compact ( globular ) structures as the temperature increases . the stabilization tendency increases with peptide length , confirming quantitatively and systematically from first principles the related observations in refs . we next demonstrate that it is indeed the _ vibrational entropy _ term which is critical , and then pinpoint the physical reason among the low - frequency vibrational modes .      in fig . [ fig : fig3 ] we analyze the individual quantities composing the vibrational free energy differences between : families 1 and 2 of @xmath2=4 , families 1 and the @xmath3-helical conformers of @xmath2=5 and 6 , as well as families 1 ( @xmath3-helical ) and 2 for @xmath2=7 and 8 . the energy terms plotted are the pbe+vdw pes minimum energy , the harmonic internal energy ( containing the zero - point energy ) @xmath17 , and the entropy term @xmath18 . they are reported in fig . [ fig : fig3 ] as energy differences , taking the non - helical conformer of each @xmath2 as the reference , such that negative slopes correspond to a stabilization of the @xmath3-helices . upon inspection of fig . [ fig : fig3 ] , we observe a monotonic stabilization of all helical conformers with increasing @xmath5 . while for the shortest molecule ( @xmath2=4 ) there is hardly any observable effect , the stabilization trend is enhanced with increasing length . for @xmath2=6 , we predict a cross - over of the lowest - energy structures at @xmath19 150 k. it is clear that among the individual contributions to the vibrational part of the free energy @xmath20 , the entropy term @xmath21 is indeed always the most important helix - favoring term . the zero - point - energy ( @xmath22 at @xmath5=0 ) , is also favorable , but on a smaller scale . = 4 ; families 1 ( g-1 ) and 2 ( @xmath3 ) of @xmath2=5 ; families 1 (  g-1 " like ) and 3 ( @xmath3 ) of @xmath2=6 ; families 1 ( @xmath3 ) and 2 ( compact ) of @xmath2=7 ; and families 1 ( @xmath3 ) and 2 ( compact ) of @xmath2=8 . in each case , the non-@xmath3-helical conformer was taken as the reference ( i.e. negative slopes mean @xmath3-helix stabilization ) . the pbe+vdw energy differences are plotted with a yellow dash - dash - dot line , the harmonic internal energy @xmath23 as black dash - dot lines , the entropy term @xmath24 as red dashed lines , and the sum @xmath16 as green full lines . [ fig : fig3],scaledwidth=60.0% ]    the observed stabilization of the @xmath3-helical conformers can be understood by analyzing the lowest - energy vibrational modes of different conformations . for simple helices , these modes have been estimated and analyzed before @xcite . the important point here is that we have available a direct comparison against many other , non - helical structure candidates . these modes dominate in the harmonic free energy expression at room @xmath5 , so that conformers with lower - frequency modes will effectively be stabilized over others . in table [ tab : all - first - modes ] , the frequencies ( in @xmath12 ) corresponding to the first normal modes of all conformers shown in fig . [ fig : fig1 ] are reported . the lowest - energy @xmath3-helices , marked in red in table [ tab : all - first - modes ] , show first vibrational normal modes between 8  @xmath12 and 13  @xmath12 . the same is true for families 1a of @xmath2=7 and 8 , which only differ in details of the termination . in contrast , the conformers that have first vibrational modes of at least 20  @xmath12 are all compact non - helical , for example : 28 @xmath12 for family 2 of @xmath2=7 , and 22 @xmath12 for family 1 ( g-1 motif ) of @xmath2=5 . the g-1 motif has the first vibrational mode lying around 20  @xmath12 for @xmath2=5 , 6 , and 7 ( marked with a * * * symbol in table [ tab : all - first - modes ] ) .    of course , the energy destabilization of the compact conformers with respect to the helices will depend not only on the first vibrational mode but also on the overall distribution of low - frequency modes . to illustrate this point , we show the frequencies of vibration lying between 0 and 50 @xmath12 for families 1 ( @xmath3 ) and 2 ( globular ) of @xmath2=7 and 8 in fig . [ fig : fig4](a ) and ( b ) . the @xmath3-helices have lower first vibrational normal modes , _ and _ have slightly higher ( one or two modes ) densities of modes in this region than the globular conformers . .position of the lowest vibrational mode , in @xmath12 , for the lowest energy conformers of each family shown in fig . [ fig : fig1 ] for @xmath2=4 - 8 . we use red characters to indicate the @xmath3-helical conformers , a * * * symbol for the g-1-like conformers of @xmath2=5 and 6 , and blue characters for the 3@xmath9-helical conformer . the numerical accuracy of all computed frequencies is 2 @xmath12 or better ( see supplementary material ) . [ tab : all - first - modes ] [ cols="^,^,^,^,^,^",options="header " , ]     here , a comment regarding structures that are more elongated than @xmath3-helices is in order . it is customary to compare the stability of @xmath3-helices with 3@xmath9 helices and @xmath8-sheets or fully extended structures ( e.g. , in references @xcite and many others ) . following our rationale above , the more extended the structure is , the softer the low vibrational modes will be . this indeed happens for the most extreme case , the fully extended structure ( fes ) , where we find the first vibrational mode to lie around only 2 @xmath12 ( calculated for @xmath2=8 and 15 ) . in fact , entropically stabilized @xmath8-sheets in _ neutral _ polyalanine in the gas - phase have been suggested in ref . for 3@xmath9 helices , in our own structure searches for @xmath2=4 - 8 , we find the first vibrational mode to lie always very close to their @xmath3-helical counterpart . accordingly , it is the enthalpic energy difference that favors @xmath3-helices specifically over 3@xmath9 for all molecules studied . for infinite periodic structures , calculation of phonons and vibrational free energies for @xmath3- , 3@xmath9- , and @xmath25-helices , and the fes in ref . @xcite corroborate our results . there , all helices are destabilized with respect to the fes at 300 k. the @xmath25-helix , which is the most compact among the helices studied in ref . @xcite , is most destabilized by the vibrational entropy term .     for ( a ) family 1 ( @xmath3 ) and family 2 ( globular ) of ac - ala@xmath7-lysh@xmath1 and ( b ) family 1 ( @xmath3 ) and family 2 ( globular ) of ac - ala@xmath26-lysh@xmath1 . ( c ) vibrational density of states taken from the velocity autocorrelation function of 21ps of microcanonical aimd at @xmath27 300 k , for family 1 ( @xmath3 ) and family 2 ( globular ) of ac - ala@xmath26-lysh@xmath1 . light colored bars in ( c ) correspond to the harmonic normal frequencies of vibration [ same as ( b ) ] . [ fig : fig4],scaledwidth=50.0% ]      we next show that the observations discussed in the last paragraph should hold also in a fully anharmonic picture . even at relatively low temperature , where the structure still stays generally constant , we expect first the inherent anharmonicity of the local , nearly harmonic pes , then the lowest - barrier transitions between basins ( side chain rotations ) , and then transitions between locally different backbone conformations and h - bond networks @xcite to contribute . unfortunately , a direct calculation of these terms ( e.g. , by thermodynamic integration ) is computationally prohibitive in dft . we can , however , use explicit _ ab initio _ molecular dynamics simulations to gain some insights . in fig . [ fig : fig4](c ) , we compare the fourier - transformed velocity autocorrelation functions of the family 1 ( helical ) and family 2 ( compact non - helical ) conformers of @xmath2=8 , extracted from explicit microcanonical _ ab initio _ molecular dynamics simulations ( 21  ps total time , 1  fs time step , initially thermalized to approximately room temperature ) . the fourier transform of the velocity autocorrelation function corresponds to a vibrational density of states ( vdos ) . at @xmath5=0 and for classical nuclei , the autocorrelation function should reflect only the harmonic eigenmodes of fig . [ fig : fig4](b ) . compared to these modes , the onset of the calculated vdos at @xmath5@xmath11300  k is noticeably shifted towards lower frequencies for both conformers , but the onset frequency for the helix is still significantly lower than for the non - helical structure . thus , the lower vibrational frequencies of the helix are also carried over to the full ( anharmonic ) motion of the conformers . in addition , the integral over the vdos up to 50 @xmath12 is 4% larger for the @xmath3-helical family 1 than for the compact family 2 , i.e. , the general downshift of frequencies in this region is also preserved . regarding the overall local structure stability during the aimd simulation , we observe that the h - bond pattern of family 2 ( compact non - helical ) stays essentially the same throughout the entire simulation ( see figure s.1 in the supporting information ) . in contrast , family 1 displays local structural fluctuations in the helical part , occasionally forming short - lived 3@xmath9-like h - bond connections . similar fluctuations also occur in simulations of longer helices ( e.g. , ac - ala@xmath28-lysh@xmath1 in ref . it seems plausible that the overall greater `` floppiness '' of the helix compared to the more compact non - helical h - bond network adds another favorable entropic contribution at room temperature . this direct observation that we make has been guessed as the motive for the loss in entropy observed in @xmath3-helical formation , compared to a ppii helical structure in ref . @xcite . according to our reasoning , the ppii structure , being less compact than the @xmath3-helices should indeed show more fluctuations , just like @xmath3-helices do if compared to more compact conformers . finally , we show that , especially for the short peptides considered here , there are two independent aspects to the stabilization of helical conformers : length and termination . to isolate their role , we examine the character of the lowest - frequency modes as a function of peptide length ( also for longer helices ) for two different terminations . the first is the lysh@xmath1-terminated series , which is the main subject of this work . we contrast this series with li@xmath1-terminated polyalanine molecules ala@xmath0-li@xmath1 , a much more rigid termination as we shall see . for the latter , we assume @xmath3-helices for all @xmath2 ( fully relaxed in dft - pbe+vdw ) , which are at least locally stable when li@xmath1 is placed in contact with the last three residues at the c terminus . in fig . [ fig : termination_deform](a ) , we compare the position of the first and second vibrational modes of these @xmath3-helical geometries of ala@xmath0-li@xmath1 , @xmath2=5 - 15 , and 20 , with the @xmath3-helical geometries of ac - ala@xmath0-lysh@xmath1 , @xmath2=5 - 15 , and 19 . for both peptide series , there is a monotonic decrease of the first and second vibrational frequencies for @xmath297 , but the starting point is much higher for the li@xmath30 termination ( 26 @xmath12 at @xmath2=7 ) than for the lysh@xmath30 termination . a softening trend of the respective modes in neutral polyalanine helices with increasing length has also been observed in ref . @xcite . in order to characterize this vibrational mode in more detail , fig . [ fig : termination_deform](b ) visualizes the displacement of the backbone atoms of ac - ala@xmath31-lysh@xmath1 when deforming this molecule along the first vibrational mode . fig . [ fig : termination_deform](c ) shows the relative length changes of the hydrogen bonds in the structure upon deformation along this mode . subfigures ( d ) and ( e ) show the equivalent data for ala@xmath32-li@xmath1 . for both molecules , the vibration spans the helical part of the structures . for the lysh@xmath1 terminated molecule , we show that the actual lysh@xmath30 termination ( connected to the last four co residues ) is clearly involved in the vibration . in contrast , the hypothetical li@xmath1 charged termination constrains especially the c terminus to be much more rigid , as evidenced by the almost zero change of all li - o distances . here , the n - terminus is much more involved in the lowest - frequency modes . the same trends are observed for the smaller molecules in both series . movies containing 3d visualizations of the first vibrational modes for helical and compact structures are contained in the supporting information . we thus conclude two points : + ( i ) helices are entropically favored by allowing delocalized , soft low - frequency modes that we do not observe in competing , more compact conformers of the same lysh@xmath1 termination ( evidenced by table [ tab : all - first - modes ] ) . + ( ii ) for short conformers , the already electrostatically favorable lysh@xmath1 termination @xcite is additionally helpful by allowing soft , delocalized modes to include the termination also for short conformers , in contrast to the hypothetical , much harder charged termination by li@xmath1 . for long enough helices these softer low frequency modes should exist regardless of the termination . in summary , we show from first principles , quantitatively , and for a particularly well studied series of polyalanine peptides how helices emerge with length and temperature as the leading structural pattern from a vast array of possible competing conformers . the crossover to helical stability with length is already apparent based on local structural minima of the potential - energy surface alone , due to the critical role of h - bond networks including their cooperativity @xcite as well as that of vdw terms @xcite . in addition , the contribution from softer low - frequency vibrational modes acts to stabilize helices with increasing temperature over their more compact competition . the specific experimental claim @xcite of exclusively helical conformers at and above @xmath2=8 is thus explained by _ both _ enthalpic and entropic effects acting together at finite temperature . _ ab initio _ molecular dynamics simulations corresponding to approximately room temperature suggest that these trends are further strengthened by anharmonic effects . the emergence of room - temperature helix stability with length in ac - ala@xmath0-lysh@xmath1 is thus the result of a subtle balance of enthalpic and entropic terms . in a non - vacuum environment , further terms would obviously contribute , but we expect the fact that helices , in general , allow locally softer vibrational modes to hold . regarding the overall ubiquity of the helical motif in folded peptides and proteins , at least , we here show that low - frequency modes will be a significant quantitative contribution . the authors would like to acknowledge dr . carsten baldauf for numerous helpful discussions and suggestions about the figures and paper in general , including the schematic backbone vibration visualization in fig . [ fig : termination_deform ] . additional computational details , xyz geometries of conformers discussed in this paper , detailed information about the structures discussed in this paper , and movies illustrating the first vibrational modes of helices and compact conformers . rossi , m. _ ab initio study of alanine - based polypeptide secondary - structure motifs in the gas phase _ , ph.d . thesis ( fritz - haber - institut der max - planck - gesellschaft and technische universitt berlin ) , * 2011 * , http://opus.kobv.de / tuberlin / volltexte/2012/3429/.
helices are a key folding motif in protein structure . 
 the question which factors determine helix stability for a given polypeptide or protein is an ongoing challenge . 
 here we use van der waals corrected density - functional theory to address a part of this question in a bottom - up approach . we show how intrinsic helical structure is stabilized with length and temperature for a series of experimentally well studied unsolvated alanine based polypeptides , ac - ala@xmath0-lysh@xmath1 . by exploring extensively the conformational space of these molecules 
 , we find that helices emerge as the preferred structure in the length range @xmath2=4 - 8 not just due to enthalpic factors ( hydrogen bonds and their cooperativity , van der waals dispersion interactions , electrostatics ) , but importantly also by a vibrational entropic stabilization over competing conformers at room temperature . 
 the stabilization is shown to be due to softer low - frequency vibrational modes in helical conformers than in more compact ones . 
 this observation is corroborated by including anharmonic effects explicitly through _ ab initio _ molecular dynamics , and generalized by testing different terminations and considering larger helical peptide models .