cyclic peptides have a 
 number of properties that make them useful 
 for biomedical applications . the constraints of cyclization give them 
 a smaller accessible conformational space than acyclic peptides , which 
 leads to a smaller loss of entropy when they bind to a receptor .
they are also very stable because they are not 
 broken down by exopeptidases , which digest peptides by removing residues 
 from the end of the peptide chain .
cyclic 
 peptides of all sizes are biologically active , starting from cyclic 
 dipeptides , which are known as diketopiperazines .
interesting cyclic tetrapeptides include the antitumor 
 agent trapoxin , the antimalarial apicidin , and the phytotoxic tentoxin .
there are also many examples of biologically active cyclic 
 peptides containing five , six , or more peptide groups .
understanding the 
 energy landscapes of cyclic peptides will account 
 for their conformational dynamics and provide some insight into their 
 biological activity .
small cyclic peptides have very different conformational 
 behavior to acyclic peptides , most significantly with respect to cis / trans 
 isomerization of the peptide groups .
cyclic dipeptides have both peptide 
 bonds in the cis conformation because this is the only configuration 
 that allows for closure of the ring .
all of the known experimental 
 structures for cyclic tripeptides have all - cis conformations , but 
 ab initio calculations on cyclic triglycine show that the all - cis 
 and trans  cis  cis isomers are close in energy . many cyclic tetrapeptides exhibit interesting 
 conformational dynamics with slow interconversion of several structures 
 and competition between the cis and trans isomers of the peptide groups . as the size of the ring increases , the cis / trans 
 ratio in cyclic peptides
the conformations of cyclic peptides have 
 been explored with a 
 variety of computational techniques .
the most stable conformers of 
 several cyclic tetrapeptides have been located either by systematic or monte carlo searches . 

ab initio methods have been used to study the pathways for conversion 
 between a small number of minima in cyclic tri- and hexapeptides .
many larger cyclic 
 peptides , comprising up to ten residues , have also been studied with 
 molecular dynamics . other methods to generate cyclic peptide conformers 
 include dihedral angle sampling , distance 
 geometry methods , and the nccyp method , which uses a combination of coarse - grained and 
 all atom models to generate the conformers of large cysteine - rich 
 cyclic peptides .
this is a small enough conformational space for discrete path 
 sampling to sample all of the physically relevant minima and 
 transition states . in this study , we present a detailed 
 analysis of the energy landscape of cyclo-[gly4 ] and compare 
 this to some larger and less strained cyclic peptides as well as an 
 acyclic peptide .
we then study a number of peptides where some of 
 the glycine residues are replaced by the l- and d - isomers of alanine , to study the effect of side chains on the backbone 
 of the peptide without the additional expense of large flexible side 
 chains .
we also consider substitution by proline , in which the cis 
 and trans isomers are much closer in energy than in other amino acids and which has been shown to promote structural 
 features like -turns .
we examine the energy landscapes 
 of several cyclic tetrapeptides , 
 the simplest of which is cyclo-[gly4 ] .
we compare this 
 energy landscape with the larger cyclic peptides cyclo-[gly5 ] and cyclo-[gly6 ] and the methyl - capped acegly3nme , which contains four peptide groups and is the acyclic peptide 
 that most closely resembles cyclo-[gly4 ] .
we have constructed 
 energy landscapes for all of the cyclic tetrapeptides where one or 
 more of the glycine residues have been replaced by alanine residues 
 ( cyclo-[alagly3 ] , cyclo-[(alagly)2 ] , cyclo-[ala2gly2 ] , cyclo-[ala3gly ] , and cyclo-[ala4 ] ) .
we also study the conformations of cyclic peptides containing 
 both d- and l - peptides by looking at cyclo-[d - alagly l - alagly ] and cyclo-[(d - ala l - ala)2 ] .
many biologically active tetrapeptides contain 
 at least one proline residue , and we 
 study cyclo-[gly3pro ] , cyclo-[(glypro)2 ] , and 
 the larger cyclic peptides cyclo-[gly4pro ] and cyclo-[gly5pro ] .
the initial minima for 
 the discrete path sampling 
 calculations were located with the basin - hopping algorithm as implemented in gmin .
initially , pairs of minima for connection were selected with the 
 missing connection algorithm .
transition 
 states were located using the doubly nudged elastic band method with interpolation between end points using 
 a cartesian coordinate interpolation scheme and optimized by hybrid eigenvector following in optim .
later , pairs of minima for connection were selected 
 using the untrap method to remove artificial frustration . these networks of stationary points are visualized 
 as disconnectivity graphs .
we present only the 
 most important disconnectivity graphs here , with the graphs for all 
 the other cyclic peptides discussed in this paper available as supporting information .
some of the cyclic peptides 
 studied here are highly symmetrical , with many symmetry equivalent 
 minima and transition states . in the disconnectivity graphs for these 
 compounds , all of the symmetry equivalent
isomers are collected together . 
 in the most symmetrical compounds this gives a much smaller number 
 of stationary points .
for example , cyclo-[alagly3 ] has 
 369 minima accessible via transition states less than 30 kcal mol above the global minimum , but cyclo-[gly4 ] has just 54 symmetry unique minima .
construction of the database 
 of stationary points for a typical unsymmetrical cyclic tetrapeptide , 
 such as cyclo-[alagly3 ] , requires about 24 h walltime on 
 four cores of an intel xeon e5405 cpu with a clock speed of 2.0 ghz .
topology files for cyclic peptides 
 prepared using the amber leap program give small energy differences between structures that should be 
 degenerate .
this problem was resolved by reordering the atoms defining 
 the improper torsion angles at the point of ring closure .
cyclic tetrapeptides are strained molecules , 
 and we must check that the amber force field accurately generates 
 the relative energies of the stationary points . the smallest cyclic 
 tetrapeptide , cyclo-[gly4 ] , was chosen for higher level 
 calculations because it is the least computationally demanding in 
 terms of the number of stationary points and the size of each calculation . 

the energies of all stationary points on the cyclo-[gly4 ] landscape were re - evaluated by single - point density functional 
 theory ( dft ) calculations at the b3lyp/6 - 31 g * level as implemented 
 in nwchem . additionally , the structures 
 of key minima from the amber potential energy surface were reoptimized 
 at the b3lyp/6 - 31 g * level .
. a cyclic tetrapeptide can be described by up 
 to four different sequences because the starting point for the sequence 
 in a cyclic peptide is arbitrary .
we assign the first position in 
 the sequence to the amino acid that is first alphabetically .
when 
 labeling conformers , the plane of the ring is defined by the mean 
 plane of the four -carbon atoms . the molecule is oriented with 
 the ring running clockwise , and the peptide groups are labeled as 
 up ( u ) or down ( d ) from the position of the peptide oxygen relative 
 to the plane .
each minimum is labeled by the sequence of cis / trans 
 and up / down isomers ( e.g. , ctct - uudd ) .
in the gas phase , 
 both the b3lyp and amber calculations agree on the s4 symmetrical 
 tttt - dudu conformer as the global minimum ( figure 1a ) of cyclo-[gly4 ] .
the energies of all the minima 
 relative to the global minimum show a good correlation between both 
 methods ( figure 2a ) .
however , the energy separations 
 in amber are slightly lower than those calculated with b3lyp .
the 
 agreement is also good for the transition states up to 30 kcal mol above the global minimum .
reoptimization of the tttt - udud 
 conformer at the b3lyp/6 - 31 g * level leads to no structural change 
 except for a small increase in the pyramidalization of the nitrogen 
 atoms in the peptide bonds .
relative energies of the stationary points on the cyclo-[gly4 ] energy landscape calculated at the b3lyp/6 - 31 g * and amber / ff03 
 levels .
points in red are minima , and points in blue are transition 
 states . in the aqueous phase ,
the agreement between the 
 amber and b3lyp 
 energies is less good but still acceptable ( figure 2b ) .
the two methods disagree on the ordering of the most stable 
 structures . in both cases ,
however , amber prefers the 
 c2 symmetrical uuuu structure , with two of the peptide 
 groups almost perpendicular to the plane of the ring and the other 
 two tilted outward by 15 ( figure 1b ) . 
 with the amber force field ,
the uuuu conformer is a higher - order saddle 
 point on the potential energy surface in vacuo .
breaking the symmetry 
 of this structure followed by minimization leads to the duuu isomer . 

the uuuu isomer is a minimum on the b3lyp gas - phase potential energy 
 surface , but it lies 29 kcal mol above the dudu 
 global minimum .
the relative energies 
 of the cis- and trans - peptides and the barriers between them are strongly 
 dependent on the size of the cyclic peptide ring ( table 1 ) . in the acyclic peptide ,
the most stable isomer containing 
 a cis - peptide is 4.9 kcal mol above the all - trans 
 global minimum and separated from it by a barrier of 21.5 kcal mol ( figure 3 ) . in the unstrained 
 cyclic hexapeptide cyclo-[gly6 ] , the energies and barriers 
 for the trans  cis rearrangement are almost identical to those 
 for the acyclic peptide . reducing the size of the ring to cyclo-[gly5 ] gives a smaller difference of 3.5 kcal mol between the trans and cis isomers and a much smaller barrier to 
 their interconversion of 16.8 kcal mol . in the 
 cyclic tetrapeptide cyclo-[gly4 ] the energy required to 
 introduce a cis - peptide and the barriers to trans 
. no conformation 
 of the twelve - membered ring can satisfy all of the preferred values 
 of its component bond angles and torsions , and this strain is responsible 
 for lowering the barriers to cis / trans isomerization .
the energies 
 and barriers associated with introducing two or more cis bonds into 
 cyclo-[gly4 ] are also much lower than in the acyclic peptide . 
 when two cis - peptides are present , the ctct arrangement is more stable 
 than cctt .
disconnectivity graph showing the energy landscape of ace - gly3-nme in water including the 841 minima and 5786 transition 
 states accessible via transition states lower than 30 kcal mol from the global minimum .
minima are colored by the 
 number of trans - peptide groups from orange ( 1 ) to purple ( 4 ) .
disconnectivity graph showing the energy landscape of 
 cyclo-[gly4 ] in water including the 54 minima and 255 transition 
 states 
 accessible via transition states lower than 30 kcal mol from the global minimum .
minima are colored by the number of trans - peptide 
 groups from red ( 0 ) to purple ( 4 ) .
all energies are in 
 kcal mol and calculated with the amber ff03 force 
 field in water . in the acyclic peptide , the barriers to rotation of 
 the  
 and  torsion angles in the peptide backbone tend to be less 
 than 5 kcal mol ( figure 3 ) . these correspond to the transitions between the up and down isomers 
 in the cyclic peptides , which require concerted motion of several 
 torsional angles and vary over a much wider range of energies . in 
 the tttt arrangement of cyclo-[gly4 ] , these barriers are 
 all smaller than 5 kcal mol . however , in conformers 
 with at least one cis - peptide arrangement many of the barriers to 
 rotation of the single bonds are much larger . in the case of the ctct 

isomers some of these barriers are within 12 kcal mol of the cis / trans barriers .
these larger barriers 
 occur because up / down isomerization of a cis - peptide requires substantial 
 deformation of the rest of the molecule .
the global minimum structure for acegly3nme has a hydrogen 
 bond between the two peptide groups at the ends of the chain ( figure 3 ) .
the global minimum of the cyclic hexapeptide 
 ( figure 5b ) and many of the other low - lying 
 structures contain two transannular hydrogen bonds .
however , the most 
 stable structure with no hydrogen bonds ( figure 5c ) is only 0.3 kcal mol less stable then the 
 global minimum .
a single hydrogen bond is present in all of the low - lying 
 structures in cyclo-[gly5 ] . in cyclic tetrapeptides ,
the 
 constraints of the ring make it difficult to form hydrogen bonds without 
 introducing strain into the peptide backbone . in the aqueous phase
, 
 the tttt - uuuu global minimum contains no hydrogen bonds , and hydrogen 
 bonding only makes a small contribution to the other tttt structures . 

as will be discussed in the next section , hydrogen bonding becomes 
 more important to the tttt structures in less polar solvents .
if a 
 single cis - peptide bond is introduced , the two peptide groups on either 
 side of this are well aligned to form a hydrogen bond ( figure 1c ) . in the ctct structures ,
hydrogen bonding becomes 
 impossible because the two cis - peptides point outward in the plane 
 of the ring while the two trans - peptides have to lie perpendicular 
 to the plane of the ring ( figure 1d ) .
selected conformers 
 of cyclo-[gly5 ] and cyclo-[gly6 ] optimized with 
 the amber ff03 potential in water . in a low dielectric medium , 
 the dudu conformer of cyclo-[gly4 ] is the most stable by 
 a significant margin ( figure 6 ) .
the dipoles 
 of the four peptide groups are aligned so that this conformer has 
 no dipole moment .
changing the polarity of the solvent distorts this 
 structure due to changes in the hydrogen bonding . in the gas phase 

each peptide group is hydrogen bonded to the peptide groups at positions i1 and i+1 , but these hydrogen 
 bonds are rather bent with n  h  o angles of 134 
 and an h 
the 
 hydrogen bonding becomes much weaker , and the ring relaxes to place 
 the h and o atoms 2.8  apart with an n 
o angle 
 of 116. increasing the polarity of the solvent stabilizes the 
 polar uuuu isomer , and it is the global minimum for values of r > 15 .
the duuu and dduu isomers both 
 have 
 small dipole moments and so are stabilized by increasing the polarity 
 of the solvent but not to the same extent as the uuuu isomer . in nonpolar 
 solvents , each of these isomers splits into two minima stabilized 
 by different patterns of hydrogen bonds .
the h and c nmr spectra of cyclo-[gly4 ] have been recorded 
 in trifluoroacetic acid ( r = 8.4 ) and show that all four peptide groups are equivalent .
the proposed structure was tttt - dudu , which 
 is consistent with the calculations presented here .
relative energies of 
 the up / down isomers of tttt cyclo-[gly4 ] as a function 
 of the solvent dielectric constant .
the lines 
 represent the uuuu ( red ) , duuu ( green ) , dduu ( blue ) , and dudu ( pink ) 
 conformations . where multiple structures have the same u / d configuration 
 only the lowest is shown .
the relative stabilities of the conformers can 
 be understood in 
 terms of the components of the amber energy ( table 2 ) .
the tttt - udud isomer has the lowest electrostatic energy 
 because of a favorable alignment of the dipoles of the four peptide 
 groups .
if the tttt - udud isomer is moved from polar to nonpolar conditions , 
 the structure becomes distorted by the shortening of the hydrogen 
 bonds .
this gives a more favorable electrostatic component of the 
 energy at the expense of worse steric and strain components .
the tttt - uuuu 
 isomer has a high electrostatic energy because the four dipoles are 
 almost parallel .
however , solvation in water stabilizes conformers 
 with a large dipole moment , such as tttt - uuuu over those with no dipole . 

the tttt - uuuu conformer has the lowest strain energy , and introduction 
 of cis - peptide bonds leads to increased strain .
the components for the gas - phase 
 tttt - dudu isomer are included for comparison .
the strain energy includes 
 the bond stretching , angle and torsion terms in the potential .
if one of the glycine residues in cyclo-[gly4 ] is replaced with an alanine residue to make cyclo-[alagly3 ] , then the four peptide groups are nonequivalent , which gives 
 a much larger number of stationary points ( figure 7 ) .
the global minimum has a tttt - dddd conformation similar 
 to that seen in cyclo-[gly4 ] .
the relative energies of 
 minima containing cis - peptide bonds are similar to those seen in cyclo-[gly4 ] ( table 3 ) as are the barriers to 
 cis / trans transitions ( figure 7 ) . if a second 
 alanine residue is introduced to make cyclo-[(alagly)2 ] 
 or [ ala2gly2 ] , the tttt - dddd conformer is still 
 the global minimum ( table 3 ) .
the relative 
 energies of the lowest cis isomers and the barriers linking them to 
 the global minimum are similar to those seen in cyclo-[gly4 ] . however , some of the barriers to up  down isomerization 
 are larger than those for cis  trans isomerization .
disconnectivity 
 graph showing the energy landscape of cyclo-[alagly3 ] in 
 water including the 369 minima and 2708 transition states 
 accessible via transition states lower than 30 kcal mol from the global minimum .
minima are colored by the number of trans - peptide 
 groups from red ( 0 ) to purple ( 4 ) .
if a third alanine residue is added , the relative 
 energies of the 
 tttt and ctct isomers are unchanged , but the cttt isomer becomes less 
 stable .
this trend continues with cyclo-[ala4 ] , where the 
 most stable cttt and ctct conformers are of similar energy .
the electrostatic 
 energy contributions from the interactions of the peptide dipole moments 
 are important in determining the relative energies of the structures 
 ( table 4 ) , with the polar dddd structure the 
 most stable in water and the nonpolar udud structure the most stable 
 in vacuo .
nmr spectra of cyclo-[ala4 ] show a mixture of 
 several stable conformations . in water , 
 four conformers are observed , with three of these merging at higher 
 temperatures .
it is possible that the three signals that merge correspond 
 to the tttt isomer and two cttt isomers , because the downhill barriers 
 separating them are relatively small ( figure 8) .
the ctct isomers are separated by larger downhill barriers and 
 could be the source of the signals that do not merge at high temperatures . 
 in cdcl3 , only the tttt and ctct isomers are observed , 
 with none of the cttt isomer .
the disconnectivity graphs here show 
 the potential energy surface , but free energy disconnectivity graphs 
 may be more appropriate for studying the relative populations of each 
 conformer .
disconnectivity graph showing the energy landscape of 
 cyclo-[ala4 ] in water including the 67 minima and 199 transition 
 states 
 accessible via transition states lower than 30 kcal mol from the global minimum .
minima are colored by the number of trans - peptide 
 groups from orange ( 1 ) to purple ( 4 ) .
the strain energy includes the 
 bond stretching , angle and torsion terms in the potential .
the introduction of d - amino acid residues 
 leads to an 
 increase in the amount of strain .
the most stable conformer of cyclo-[d - alagly - l - alagly ] is 2.3 kcal mol less stable than the most stable conformer of cyclo-[(alagly)2 ] .
this allows the ring to adopt a chairlike structure , with one of the 
 methyl groups axial to the ring .
the most stable tttt structure is 
 tttt - dudd , which is 0.9 kcal mol higher in energy . 

the tttt - dddd isomer , which is the global minimum in water for all 
 the previously discussed peptides , is 1.7 kcal mol above the global minimum .
this conformer places one of the alanine 
 methyl groups close to two of the peptide oxygen atoms below the plane 
 of the ring , which accounts for its destabilization .
the global 
 minimum of cyclo-[(d - ala - l - ala)2 ] is 
 1.7 kcal mol less stable than the 
 global minimum of cyclo-[ala4 ] .
the most stable conformers 
 with tttt , cttt , and ctct arrangements are all close in energy ( figure 9 ) .
the tttt - uuuu isomer is 5.7 kcal mol above the global minimum . due to steric clashes , conformers containing 
 three or four
cis - peptides are less stable in this compound than in 
 all other cyclic tetrapeptides ( table 3 ) .
disconnectivity 
 graph showing the energy landscape of cyclo-[(d - ala - l - ala)2 ] in water including the 116 
 minima and 524 transition states accessible via transition states 
 lower than 30 kcal mol from the global minimum . 

minima are colored by the number of trans - peptide groups from orange 
 ( 1 ) to purple ( 4 ) . in the proline containing cyclic tetrapeptides
, there are groups 
 of minima with the same pattern of cis / trans and up / down configurations 
 separated by barriers of 23 kcal mol . 

these minima correspond to distortions of the five - membered rings 
 in the proline residues . due to the conformational restriction imposed 
 by the proline ring
, simple up / down isomerizations of the amide group 
 attached to this ring do not occur .
the global minimum structure 
 for cyclo-[gly3pro ] has the peptide group in the proline 
 residue in the cis conformation ( figure 10 ) . 

if the proline group is in the trans conformation the ring can not 
 be closed without at least one of the other peptide groups being in 
 an unfavorable conformation . in this cyclic tetrapeptide
, the proline 
 residue strongly favors a cis conformation , with several ctct conformers 
 below the most stable one containing a trans proline .
however , the 
 barriers to cis  trans isomerization of the proline residue 
 are smaller than those for the glycine residues .
disconnectivity graph 
 showing the energy landscape of cyclo-[gly3pro ] in water 
 including the 358 minima and 2472 transition 
 states accessible via transition states lower than 30 kcal mol from the global minimum .
minima are colored by the 
 number of trans - peptide groups from red ( 0 ) to purple ( 4 ) . as the size of the ring in the cyclic peptide increases , 

the trans 
 isomer of the proline residue becomes more stable ( table 5 ) .
the relative energies of the conformers containing cis 
 and trans proline groups in cyclo-[gly5pro ] are very close . 

this trend is similar to that seen for the glycine residues in cyclic 
 polyglicines of different sizes .
trans isomerization of the peptide bond preceding 
 the proline ring does not vary much with the size of the cyclic peptide 
 ring . the global minimum for cyclo-[(glypro)2 ]
the lowest all - trans isomer is 5.0 kcal mol above the global minimum and slightly above one ccct 
 conformer ( table 3 ) .
the predicted preference 
 for a ctct structure is consistent with the available nmr spectra and in agreement with the known crystal structureof cyclo-[(glypro)2 ] .
all energies are in 
 kcal mol as calculated with the amber ff03 force 
 field in water and are relative to the global minimum for that cyclic 
 peptide .
the energy landscapes of 
 small cyclic peptides are very different 
 from those of larger cyclic peptides and acyclic peptides . as the 
 size of the ring decreases ,
isomers containing cis - peptide groups 
 become more stable , and the barriers to trans 
cis isomerization 
 become smaller . in cyclo-[gly4 ] , the simplest cyclic tetrapeptide , 
 the global minimum is all - trans , and the energy of the molecule increases 
 when the number of cis - peptide bonds increases . substituting one or 
 two of these glycine residues with alanine gives a much larger number 
 of minima due to the lower symmetry of these molecules , but the energy 
 differences and barriers between these minima are similar to those 
 seen in cyclo-[gly4 ] . introducing more alanine residues 
 leads to higher barriers and destabilization of some minima due to 
 steric crowding .
the peptide bonds preceding proline groups have a 
 much smaller preference for the trans conformation in cyclic hexapeptides 
 and adopt the cis conformation in smaller systems .
solvation 
 has a substantial effect on the energy landscapes of 
 cyclic tetrapeptides . in nonpolar solvents conformers with no net 
 dipole moment , such as the udud isomers ,
are the most stable . as the 
 polarity of the solvent increases , isomers with large dipole moments 

are stabilized and become competitive with , or more stable than , the 
 nonpolar udud conformers . due to the small ring size , the structures 
 including hydrogen bonds have strained geometries .
such structures 
 are only stable in nonpolar conditions where the strength of the hydrogen 
 bond overcomes this strain . we have presented the potential 
 energy landscapes of several cyclic 
 tetrapeptides . in the future
, we must consider free energy landscapes to obtain a full picture of the conformational dynamics of these 
 molecules . here
, we have only considered cyclic tetrapeptides comprising 
 four types of amino acid residue .
it is likely that other natural 
 or unnatural amino acids will influence the conformations of cyclic 
 peptides in different ways , and these will also be the focus of future 
 studies .
 cyclic tetrapeptides are an important class of biologically 
 active 
 molecules that exhibit interesting conformational dynamics , with slow 
 interconversion of several different structures . 
 we present calculations 
 on their energy landscapes using discrete path sampling . in acyclic 
 peptides and large cyclic peptides , isomers containing 
 cis - peptide 
 groups are much less stable than the all - trans isomers and separated 
 from them by large barriers . 
 strain in small cyclic peptides causes 
 the cis and trans isomers to be closer in energy and separated by 
 much lower barriers . 
 if d - amino acids or proline residues 
 are introduced , isomers containing cis - peptides become more stable 
 than the all - trans structures . 
 we also show that changing the polarity 
 of the solvent has a significant effect on the energy landscapes of 
 cyclic tetrapeptides , causing changes in the orientations of the peptide 
 groups and in the degree of intramolecular hydrogen bonding . 
Introduction
Methods
Results and Discussion
Conclusions