theoretical and computational 
 modeling has provided many insights 
 into the remarkable ability of proteins to rapidly fold from unstructured 
 coils into their native , functional conformations . especially 
 for small structured proteins , entire folding processes can be investigated 
 via atomistic , equilibrium molecular dynamics ( md ) simulations , where 
 the ability to sample multiple folding events ( with s simulations ) 
 with a transferable force field is an important milestone in algorithm 
 development and hardware parallelization . when multiple folding events are observed , the underlying 
 kinetics and conformational features that promote structural transitions 
 and the eventual attainment of the native state can be statistically 
 compared . such studies reveal important characteristics of the underlying 
 free energy landscape ( fel ) , the high - dimensional surface of hills 
 and valleys that govern the likelihood of structural transitions and 
 the occupancy probabilities of energetically coherent states , called 
 conformational substates . for structured proteins , 
 the fel has been conceptualized as a funnel with a low - energy native ensemble at its global minimum , where near - native 
 intermediates are kinetic neighbors , and a nonnative ensemble composed of freely interconverting conformers at some further reaction 
 distance . while the majority of protein functions 
 are accomplished via the native ensemble , quantifying the structural 
 and kinetic characteristics of the nonnative ensemble can aid calibration 
 of coarse - grained polypeptide models and improve our understanding 
 of folding initiation pathways , protein misfolding , protein aggregation , and synergistic folding ( i.e. , folding in tandem with a binding 
 partner ) . although nonnative ensembles 
 recapitulate several properties of 
 idealized random - coil models , they have also been 
 shown to deviate from polymeric predictions in important ways . substantial 
 secondary structure can accrue in the nonnative ensemble , and these nucleation locations 
 have been implicated as consistent waypoints in folding pathways . larsen et al . likewise 
 showed that for transition pathways specifically , secondary structure 
 accumulates before native contacts are formed , a temporal bias that 
 is inconsistent with an idealized nonnative ensemble . from the kinetic 
 perspective , another surprise is that the nonnative ensemble can be 
 modeled as a hub - like transition map , where interchange between unfolded peptide geometries is mediated 
 preferentially via the native ( hub ) ensemble instead of by direct 
 routes ( see ref ( 29 ) for a contrasting interpretation ) . the minimally frustrated model of protein folding harmonizes some 
 of these observations by recognizing that folding is energetically 
 downhill and will thus avoid the enthalpic frustration of nonnative 
 structure formation . analogously , we can ask whether the nonnative ensemble is minimally 
 frustrated in a kinetic sense . does folding proceed sequentially from unfolded to folded substates or are there 
 off - pathway kinetic inhibitors populating the fel ? computational 
 studies have invoked cartesian , angular , topological , 
 subspace - projection , or other structural descriptors to identify nonnative 
 conformational states , but assessing 
 their impact on folding rates requires additional analysis . to quantify 
 the kinetic contribution to folding of specific conformational substates , 
 we present here a methodology that ( 1 ) permits comparisons of kinetic 
 inhibition across multiple folding events and between multiple proteins 
 and ( 2 ) can query any proposed structural parameters that may impact 
 folding kinetics . instead of only specifying the existence of kinetic 
 traps , hubs , or preferential pathways in md trajectories , we quantify 
 the overall kinetic burden , or kinetic frustration , that structural deformations ( secondary structure , tertiary structure , 
 standard rmsd - to - native , or others ) effect in protein simulations . in the rest of this introduction we give an overview of our model s 
 assumptions , justification for a topological definition of kinetic 
 frustration , and primary results . we invoke a kinetic modeling 
 of md simulation data where a simulation 
 trajectory and its conformers are represented as ( 1 ) sets of clusters , 
 or conformational substates , whose kinetically indistinguishable members 
 share conformational features , and ( 2 ) transitions , which capture 
 the observed jumps between substates . such a network of substates 
 ( nodes ) and edges , when constructed with an appropriate lag - time between 
 sampled trajectory snapshots and clustering criteria , satisfies the 
 properties of a markov state model ( msm ) . these models are guided by the motivation to equate conformational 
 transitions with probability flow , enabling multistep transition pathways 
 to be associated with a probability and expected duration even if 
 the path itself was never observed within contiguous trajectory frames . 
 by representing a protein s fel as an evolving finite markov 
 chain , msms permit computation of the stationary distribution , the 
 unique set of substate probabilities that is stable over time . it 
 is then possible to calculate the expected time for any substate to 
 transition to the assigned native ensemble , that is , the mean first 
 passage time ( mfpt ) or transit time . mfpt 
 values express temporal expectations for random walks along the weighted 
 edges of the conformational network . they are robust and can be compared to diffusional models of folding and nanosecond laser t - jump experiments . whereas mfpts necessarily are a function of two specified end points , 
 our concern is only with those transition paths that terminate at 
 the native ensemble , a convention implicit throughout this study and 
 indicated by the subscript of mfpt values , nat . can these values tell us which substates are responsible for accelerating 
 or hindering folding ? not directly , but that exact information is 
 revealed when substates are theoretically removed from the transition 
 network and the change in nat values among the rest 
 of the nonnative ensemble compared . kinetic frustration , quantified 
 in frustration scores , fnat , captures 
 these changes and quantifies the degree to which a particular conformational 
 substate state inhibits or facilitates transitions to the native state . the terms inhibit and facilitate summarize a substate s topological neighborhood with respect 
 to the native ensemble : a substate that facilitates folding is highly 
 connected to native or native - like substates , whereas a folding inhibitor 
 promotes transitions to non - native regions of the fel . mfpts 
 and frustration scores , fnat , are 
 therefore related but , importantly , distill different information . nat values reflect expected transit times given 
 a network structure and designated native ensemble , whereas frustration 
 scores quantify the impact on the network given the substate of interest . nodes that have equal transit times need not share frustration scores , 
 for example ( see figures 1 and 2b ) . moreover , because a demarcated native state is inherent 
 to their definition , frustration scores go beyond a quantification 
 of local topology , such as average neighbor connectivity , which treats equally links leading toward or 
 away from the native state ( for a given nonnative conformational substate ) . that is , frustration scores are sensitive to the native state location , 
 whereas local connectivities are not ( cf . supporting 
 information ( si ) figure s6 ) , an advantage that warrants the 
 more computationally demanding node by node perturbation approach 
 presented here . frustration scores are thus individually computed 
 for all nodes in the nonnative ensemble by observing changes in nat values when each is removed from the network model ( figure 1 ) , a process akin to 
 the eigenvalue estimation problem in matrix perturbation theory . observations obtained by altering the transition 
 network in this way provide a quantitative framework for understanding 
 its unperturbed behavior . specifically , each frustration score fnat  i can be understood as the mean percentage 
 change in all transit times from all possible paths as a result of 
 node i , a kinetic interpretation lacking from many 
 local topology metrics . substates with fnat > 0 are labeled facilitators since folding rates would 
 decrease ( i.e. , nat increase ) in their absence ; 
 states with fnat < 0 are inhibitors , 
 or kinetic traps , in that folding rates would increase ( nat decrease ) if they were to be removed from the conformational 
 landscape . we thus invoke the concept of kinetic frustration because 
 mfpt values alone can not elucidate these causal relationships . ( a ) each conformational substate 
 in the nonnative ensemble ( 1 ... knn ) is 
 colored according to frustration scores , fnat ; substate diameters indicate the stationary probability . transition 
 probabilities are shown along observed transitions where values above 
 the transition path always denote left  right transitions 
 and values beneath the arrow refer to moving right  left . ( b ) computed mfpts , nat values , for each nonnative 
 substate to reach the native ensemble . each nonnative substate is removed 
 from the transition matrix ( states i = 1 ... 4 , top 
 to bottom ) and transit times for all remaining knn  1 substates are compared with unperturbed values 
 ( left panels : black bars , unperturbed ; red , gray , or blue bars , perturbed ) . relative changes in transit times ( wedges , right panels ) are averaged 
 over remaining substates to yield fnat ( dashed lines ) . these frustration scores are then depicted by the 
 color scale on the original intact network ( a ) . substates 1 and 2 
 have identical mfpts , whereas fnat values indicate substate 2 is a facilitator and increases folding 
 rates from all other substates by an average of 25% , while substate 
 1 is kinetically neutral . substate 4 is a kinetic trap , slowing all 
 transit times by 30% on average in the unperturbed network . ( a ) nodes 
 indicate conformational substates determined with rmsd clustering ; node diameters are proportional to substate 
 probability . white diamonds indicate 
 the substate containing the conformation closest to the experimental 
 structure . frustration scores , fnat , are denoted by the color spectrum , 
 centered at fnat = 0 . positive 
 scores , blue , indicate substates that facilitate transition to the 
 native ensemble ; negative scores , red , indicate kinetic traps . ( b ) 
 comparison of frustration scores , fnat , and transit times , nat ( s ) , for nonnative 
 substates in part a. color values correspond to frustration scores 
 as in part a. within our simulation 
 data set of 10 fast - folding proteins , substantial 
 kinetic traps were observed for four proteins ( trp , bbl , pb , and hmdm ) 
 whereas kinetic inhibition was chiefly absent in the nonnative ensembles 
 of ww , pg , and a3d . the largest frustration scores ( i.e. , most extreme 
 facilitators ) were observed in the ww and pg simulations , but none 
 of the 10 globular proteins examined ( table 1 ) presented a single facilitator , an ultimate gatekeeper substate , 
 to the native ensemble . as shown in figure 2 , kinetic traps were unequally distributed throughout the nonnative 
 ensembles . the transition networks , or transition maps , did display 
 unique topological features , and we were able to ask to what relative 
 degree secondary structure , tertiary structure , and nonnativeness 
 ( standard rmsd - to - native ) are associated with positive or negative 
 kinetic frustration . we chose these structural parameters because 
 of their broad interpretability and popularity for monitoring folding 
 progress but emphasize that the approach is compatible 
 with any geometric or energetic feature that can be computed for all 
 trajectory frames . folding is a conformationally heterogeneous 
 process , but the recognized prevalence 
 of preferred folding 
 routes and transition pathways highlights the need for tools linking specific 
 nonnative substates to folding kinetics . quantifying these relationships 
 is a legitimate aim in its own right , but our findings relate to the 
 wider problem of predicting the kinetic impact of direct perturbations 
 to protein systems . mutations , small molecule ligands , or solvent 
 conditions that modulate the populations of conformational substates 
 can influence folding rates or folding routes , and quantifying 
 any such changes therefore has applications to pathway inhibition , 
 aggregation - based diseases , and protein engineering . a summary 
 of the proteins and 
 simulations studied , adapted from lindorff - larsen et al . data columns indicate sequence length ( nres ) , total aggregated simulation duration ( ttotal ) , number of conformational substates ( and 
 any substates excised during transition matrix mle ) ( k ) , protein data bank accession code ( residue indices ) , simulation 
 temperature , number of folding events ( nf ) , number of unfolding events ( nf ) , and 
 the native - ensemble rmsd cutoff ( rnc ) . we applied our analysis to 10 proteins 
 within a large simulation data set generated by d. e. shaw research 
 as reported in lindorff - larsen et al . and 
 analyzed further elsewhere . aggregate simulations of the 
 10 proteins comprise 5.1 ms of total sampling where each protein undergoes 
 at least 10 folding and unfolding events ( table 1 ) ; details for the simulations and folding event classification are 
 contained in the original reference . snapshots were recorded every 200 ps . in this study , 
 multiple simulations for the same protein , if present , were concatenated . to identify conformational 
 substates for each protein , we performed hierarchical clustering with 
 msmbuilder2 . trajectories were first 
 subsampled to obtain snapshots every 50 ns based on implied time scale 
 plots ( si figure s1 ) , then clustered into 
 substates using root mean squared distance ( rmsd ) and ward s 
 algorithm . the number of substates , k ( see table 1 ) , is a heuristic user 
 parameter that was selected to be approximately equal to ( simulation 
 frames)/10 . this parameter has been shown 
 to have little dependence on peptide length , nres . the k values 
 chosen here correspond closely to those in ref ( 29 ) . the transition probability 
 matrix p was then approximated using the msmbuilder2 
 maximum likelihood estimation ( mle ) routine , and substates not included 
 in the estimated matrix ( i.e. , those separate from the primary connected 
 component ) were excised from subsequent analysis . connected singletons 
 ( substates with a single member ) were retained , however , and constituted 
 0% of total conformers for bba , bbl , pb , and trp and 0.810% 
 for a3d , hmdm , lamda , pg , vhp , and ww . distributions of cluster sizes 
 ( number of member conformers ) and widths ( defined as mean pairwise 
 rmsd of any two substate members ) are given in the supporting information ( figures s4 and s5 ) . our network folding model 
 requires a demarcated native state to function as a kinetic end point , 
 that is , a theoretical absorbing state where folding is defined as complete . selecting the largest conformational 
 substate , the substate closest to the 
 pdb - deposited coordinates , or a hard rmsd threshold is too restrictive , 
 excluding many substates with  native - like properties 
 and artificially increasing theoretical nat values . instead , we chose to designate a native 
 ensemble , or a set of conformational substates that interconvert 
 more frequently with each other than with outside substates . such 
 a graph property is captured by an algorithm called modularity optimization , and is particularly suited for this classification 
 task in that it reflects and adapts to the actual network topology , 
 unlike an rmsd threshold . modularity optimization proceeds by initially 
 designating each substate as its own ensemble and then iteratively 
 combining them until only highly intraconnected ensembles remain , 
 at which point modularity is maximized . for a transition network , 
 modularity is defined aswhere cij is the number of transitions 
 between substates i and j , ki is the total number of transitions 
 to substate i , kj is the number of 
 transitions to substate j , 2 m is 
 the total transition count in the network , and (si , sj ) = 1 when substates i and j reside in the same ensemble s and 0 otherwise ( elsewhere lnn or ln denote 
 total edges among nodes in the nonnative or native ensembles , respectively ) . other formulations for optimizing modularity are possible , including 
 normalized cut and conductance criteria . maximizing w yields multiple ensembles for each 
 of 10 analyzed transition networks , but only one per network will 
 be defined as the native ensemble . within each candidate ensemble , 
 five random conformers were sampled from every constituent substate , 
 and the aggregate number of conformers within a cutoff rnc to the pdb - deposited native structure 
 were counted and compared with similar counts from the remaining candidate 
 ensembles . the ensemble with the most conformers under the cutoff 
 was designated to be the native ensemble , and all its substates , not 
 only those under the cutoff , were included ( all substates not in the 
 native ensemble were defined to be in the nonnative ensemble ) . the 
 cutoff itself , rnc , was determined by 
 identifying the rmsd value such that 5% of total trajectory frames 
 were within rnc  from the pdb - deposited 
 crystal structure ( table 1 ) . for 8 of 10 proteins , 
 100% of the frames with rmsd < rnc were 
 found in the native ensemble ; the same values were 82% for bba and 
 93% for bbl . substates containing the snapshot nearest the experimental 
 native structure were always contained in the assigned native ensemble 
 ( figure 2 , white diamonds ) . the number of substates 
 assigned to the native and nonnative ensembles was kn and knn , respectively ( table 1 ) . although the nonnative ensembles were partitioned 
 variously during iterations of modularity optimization , the constituent 
 substates of the ( eventually defined ) native ensemble were in fact 
 identically preserved through all iterations for all proteins . modularity optimization 
 operates exclusively on transitions counts ; however , we observed that 
 secondary structure also separated cleanly between the native and 
 nonnative ensembles as a result of this classification ( si figure s2 ) . the identified native ensembles 
 are shown in green in the network representations ( figure 2 ) . for purposes of computing rmsd , a native 
 conformation was defined to be the trajectory snapshot nearest 
 the theoretical mean structure of the entire native ensemble ( figure 2 , black diamonds , and figure 6 , tube representations ) . the proportion of native contacts present in any 
 trajectory conformer , q , is a useful reaction coordinate 
 for monitoring folding 
 progress or modeling energy barriers . it is a degenerate quantity in that many distinct 
 conformers could map to an identical q value . we defined native contacts as those residue pairs whose separation 
 ( cc ) was less than 10  for at least 65% of the conformers 
 within the native ensemble ( si figure s3 ) . native contacts separated by fewer than 7 amino acids in the primary 
 structure were excluded . we denote the percentage of native contacts 
 as qn(t ) and the percentage 
 of nonnative contacts as qnn(t ) for some time t. we quantified the presence 
 of secondary structure in each trajectory frame using protein secondary 
 element assignment ( p - sea ) , which labels every residue as in either 
 an unstructured coil ,  helix ,  sheet , or  other 
 configuration ( si 
 figure s2 ) . ideal sequence of native secondary 
 structure assignment was defined as the residue - wise assignment most 
 common within the native ensemble and termed the structure sequence . the presence of native secondary structure throughout the simulation 
 was quantified by dividing the number of native - like p - sea assignments 
 by the total number of  sheet and  helix assignments 
 within the structure sequence . nonnative secondary structure , 
 which captures the percentage of  and  secondary structure 
 assignment that is unlike that found in the structure sequence , is 
 denoted hnn(t ) . having determined the set 
 of substates defining the native state , we next derived the expected 
 mean first passage time of each nonnative substate to the native ensemble 
 as put forward in ref ( 48 ) ( alternative algorithms for computing transit times are given in 
 torchala et al . ) . first , we estimated 
 the symmetric transition probability matrix p from 
 the clustering results using the msmbuilder2 mle method to guarantee 
 detailed balance . this matrix carries 
 jump probabilities for the embedded discrete markov chain , but can also be expressed as a rate matrix   that 
 approximates the continuous time transition 
 rates . for each nonnative substate i , we modify   to 
 have zero transition rates to all substates 
 previously connected to i. we then compute the formal 
 matrix exponentiation e ) we then ask how 
 these mean first 
 passage times to the native ensemble , or transit times , change in 
 response to network perturbation , that is , the removal of a substate 
 in the nonnative ensemble . to that end , we remove a substate i in the nonnative ensemble from the network and then observe 
 the percentage change between unperturbed ( jn ) and perturbed ( jn * ) transit times , in both cases 
 for all nonnative substates m  [ 1 ... knn  i ] ( see figure 1 ) . that is , where the bar thus indicates 
 the average percentage 
 change in transit time over all knn1 
 substates in the nonnative ensemble , and the multiplicative factor 
 allows frustration scores to be interpreted as percentages . any isolates resulting from removing node i were 
 discarded when computing fnat  i , but 
 this was rare ( ( isolates)/(knn ) < 0.01 
 for all proteins except lamda , ( isolates)/(knn ) < 0.021 ) . frustration scores fnat quantify the kinetic impact , positive or negative , 
 for each nonnative substate i , expressed as percentages 
 in figure 3 and table 2 . states with positive frustration scores are termed facilitators , those with negative frustration scores are termed inhibitors or kinetic traps . all analysis subsequent to clustering 
 was performed in matlab . due to the matrix 
 exponentiation , complexity of fnat computation is o(n ) , and runtimes were between 4 min ( pb , knn = 167 ) and 20h ( pg , knn = 2248 ) on a 12-core cluster . because the 
 substates of the native ensemble can not be associated with fnat values , the number of data points 
 corresponds to the number of substates in the nonnative ensemble , knn ( table 1 ) . values 
 less than zero indicate a kinetic trap , those above zero indicate 
 a substate that facilitates transition to the native ensemble . central 
 red marks indicate the median ; box edges are the 25th and 75th percentiles . substate transition matrices 
 are usually very sparse , especially in the nonnative ensemble ( table 2 ) . most transitions are forbidden due to the involved 
 steric clashes , backbone geometry restrictions , and repulsive electrostatics . graph - based visualizations of conformation space thus have interpretive 
 value in conveying only the transitions that do take place as well 
 as the relative sizes of conformational states . we used gephi to represent each protein s 
 transition network ( figure 2 ) . network layouts 
 were optimized using the force atlas algorithm , first allowing and 
 then penalizing node overlap , in both cases with an internode repulsion 
 strength of 200 . edge weights were scaled according to the transition 
 matrix , specifically 1000 , but 
 are not differentiated visually in 
 the figure . the repulsion force acts between all nodes , whereas node 
 attraction is relative to connecting edge weight , so unconnected nodes 
 feel zero direct attraction . node diameters reflect their relative 
 populations , but the smallest node is shown no smaller than /30 the size of the largest node for clarity . conformational landscapes 
 of the 10 proteins can be conveniently 
 depicted as networks of nodes and edges that illustrate folding properties 
 of each peptide . the native ensembles as defined in methods are colored green , though the maps layouts 
 themselves were created without preknowledge of native or nonnative 
 substate assignments . columns 17 : parenthetical 
 values denote properties of the native ensemble , all others to the 
 nonnative ensemble . the range of frustration scores is given in the 
 first two columns . median substate size refers to the number of trajectory 
 snapshots clustered into each conformational substate . the number of substates 
 classified by modularity optimization as being in the nonnative ( native ) 
 ensemble is denoted by knn(kn ) . maximum modularity value , w , for 
 the modularity optimization algorithm utilized ( see methods ) is also given . the last column shows the ratio between 
 ( 1 ) total transition edges connecting nonnative and native ensembles 
 ( lnnn ) and ( 2 ) the total number 
 of edges in the nonnative ensemble ( lnn ) , expressed as a percentage . many general phenomenological 
 aspects of protein folding are visible in these abstractions in addition 
 to features that distinguish the folding behavior of specific polypeptides . the prevalence of large substates within the highly interconnected 
 native ensemble ( see table 2 ) , for example , 
 reflects the loss of entropy a folding peptide experiences upon attaining 
 the energetically favorable folded conformation ( cluster size and 
 widths given in si figures s4 and s5 ) . second , folding facilitators ( blue substates ) are as expected mostly 
 proximal to the native ensemble due to being conformationally very 
 native - like . the topological isolation 
 of the native ensemble , especially for trp , ww , pg , a3d , and lamda , 
 suggests modularity optimization effectively classifies the folded 
 and unfolded ensembles without invoking any protein - specific parameters . a particularly evident separation between the two ensembles characterized 
 the transition maps of ww , pg , and a3d , all of which had less than 
 one percent of nonnative edges connecting the nonnative and native 
 ensembles ( range for all proteins : 0.520.9% , see lnnn / lnn in table 2 ) . higher values of this measure indicate less homogeneous 
 folding pathways , most evident in hmdm 
 and pb transition maps . large kinetic traps , transition bottlenecks , 
 and facilitators , among other topological motifs , are unequally prominent 
 among the networks . several of the maps depict a nonnative ensemble 
 of freely interconverting structures that form no apparent energetically 
 coherent substates ( min fnat values 
 for ww , pg , and a3d are 0.6,0.7 , and 0.6 , 
 respectively ) , whereas substantial kinetic traps characterize the 
 nonnative ensembles of trp , bbl , pb , and hmdm ( min fnat = 1.8,1.6,4.2 , and 1.7 , 
 table 2 ) . we also 
 compared nat and fnat values to corresponding quantities computed for phantom ( i.e. , synthesized ) 
 networks in si figure s7 . the comparison 
 reveals that the degree distribution inherent to the transition network 
 of each peptide is sufficient input for approximating nat and fnat in generated 
 networks . however , a more quantitative analysis is necessary 
 to reveal the 
 specific conformational features , or structural parameters , that are 
 responsible for the frustration scores unique to each protein s 
 unfolded ensemble . we focus primarily on properties of kinetic traps 
 because facilitators inherently border a native / nonnative delineation 
 that is convenient but imposed ; any conformational differences between 
 facilitators and native substates are likely to be subtle with regard 
 to the structural parameters used here . we first address clustering properties that could be thought to 
 cause negative outlier fnat values 
 and then discuss the structural features that indeed correlate with 
 kinetic frustration . as 
 shown in figure 3 , most substates within the 
 nonnative ensemble are kinetically neutral ; their individual presence 
 in the fel has little impact on the expected transit times of other 
 substates . importantly , these substates need not be small , or have 
 few constituent conformers . while substates with substantial positive 
 or negative frustration scores tend to be above average in size , the 
 converse is not true ( si figure s4 ) . that 
 kinetic traps , as identified through fnat , must contain a substantial number of conformers reflects 
 our intuition that kinetic traps represent local minima with stabilizing 
 intramolecular interactions in the nonnative fel . substate size ( i.e. , number of members ) as a descriptive 
 trait can be contrasted with substate width ( i.e. , the mean pairwise 
 rmsd of any two of its members ) , which provides an approximation of 
 local entropy . if kinetic traps presented increasing substate width 
 as fnat values became more extreme , 
 we could conclude that fnat values 
 are actually artifacts of the clustering step . in this scenario larger 
 and larger peripheral regions of the configurational state space are 
 unfairly grouped together during clustering , resulting in artificially 
 exaggerated fnat values . we observed , 
 in contrast , that kinetic traps display decreasing width values , suggesting 
 they represent genuine local energy minima ( si 
 figure s5 ) . having discussed that clusters in 
 our networks are well - formed , we next investigate conformational causes 
 of this observed topology . we 
 selected five structural parameters that share the desirable properties 
 of normalizability and interpretability , and we computed them for 
 all substates ( all nonnative and native trajectory frames ) in the 
 transition networks . definitions for native contacts , qn , nonnative contacts , qnn , native secondary structure hn , and 
 nonnative secondary structure hnn are 
 given in methods , and our fifth structural 
 parameter was standard rmsd ( against the native conformation ) . figure 4 illustrates the relationship between fnat and fractional contacts ( qnn or qn ) for all nonnative substates within 
 the conformational landscape of lamda . the relationship observed confirms 
 our expectation that kinetic traps display more nonnative contacts 
 than the nonnative ensemble in general . the presence of interresidue 
 contacts , both native and nonnative , is normalized against the corresponding 
 quantity observed in the native ensemble . mean values for these features 
 are shown with dashed horizontal lines , gray for the entire nonnative 
 ensemble , green for the entire native ensemble . for lamda , we conclude 
 that the enrichment of nonnative contacts among kinetically frustrated 
 substates is one hypothesis for the appearance of outlying red substates 
 in lamda s transition network ( figure 2 , white arrow ) . average intrasubstate qnn ( c ) and qn ( d ) values for lamda are 
 plotted against frustration scores , fnat , showing that nonnative contacts are associated with kinetic 
 frustration . structural parameter values , qn and qnn included , are normalized against 
 the average corresponding values within the native ensemble . marker 
 widths are scaled according to substate populations , and error bars 
 indicate one standard deviation . dashed lines show normalized average 
 values for the nonnative ( gray ) and native ( green ) ensembles . cumulative 
 sums , ( a ) sqnn = ( qnn  qnn ) and ( b ) sqn = ( qn  qn ) , ( see main text ) show the propensity 
 of structural features to be more associated with negative or positive fnat values . when integrated these curves 
 yield the bias values ,  , that allow quantitative comparison 
 between proteins ( figure 5 and si table s1 ) . color values along the curve correspond 
 to substate color values in figure 2 . in normalizing qn and qnn against their respective 
 values in the native ensemble , 
 we can evaluate their correlative relationship to frustration scores 
 and then compare among protein systems . we thus quantify whether a 
 feature is more enriched among kinetic traps or facilitators with 
 a bias valuefor any feature f with mean 
 value f ( within the nonnative ensemble ) , where f indexes the ascendingly sorted fnat values . bias values convey whether feature f is more enriched for negative or positive fnat values ( figure 4a and b ) , where 
 the negative sign allows us to compare with standard linear correlation , 
 which we performed with the addition of weighted substate size ( si table s1 ) . values 
 near zero indicate the structural parameter is not strongly associated 
 with kinetic frustration . we performed permutation tests to check the 
 statistical significance of these bias values ( si table s1 ) . as we would expect , native secondary structure 
 and native contacts frequently have positive bias values ( trp qn is the only statistically significant 
 exception ) , indicating that facilitator substates contain many native - like 
 structural features , whereas kinetic traps do not . significant nonnative 
 secondary structure bias values were observed for bba , pg , a3d , and 
 lamda ( hnn = 0.13,0.19,0.12 , 
 and 0.10 respectively ) , and significant nonnative contacts 
 bias values were observed for trp , bba , bbl , pg , a3d , lamda ( qnn = 0.33,0.13,0.18,0.15,0.13 , 
 and 0.28 , respectively ) . because rmsd - to - native is so commonly 
 invoked as a distance metric for how far a simulation has progressed , 
 we also computed bias values for rmsd , which were negative and statistically 
 significant for all proteins except trp , vhp , and pb . especially for 
 bba , ww , bbl , hmdm , and a3d , nonspecific structural deformity , the 
 characteristic summarized by rmsd , appears more associated with kinetic 
 frustration than the specific structural features tested . vhp and 
 pb simulations did not present statistically significant bias values 
 for any structural parameters , perhaps due to modularity optimization 
 defining the native ensemble too inclusively for these networks ( see si figure s2 ) . negative 
 values indicate the structural feature is strongly associated with 
 kinetic frustration , that is , slowing transition to the native state 
 for that protein . positive values indicate the feature is associated 
 with states that facilitate attainment of the native state . the rmsd 
 distance from the native conformation has the largest negative bias 
 value for bba , ww , bbl , hmdm , and a3d . nonnative contacts have the 
 largest negative biases for trp , pb , and lamda . nonnative secondary 
 structure , hnn , is the most biased structural 
 parameter only for pg . some bias values close to zero are not statistically 
 significant ( si table s1 ) , indicating the 
 structural parameter has little kinetic impact on folding for that 
 protein . conformational ensembles 
 consisting of snapshots from the most kinetically frustrated substates 
 are shown in figure 6 , rendered with pymol . the 5% of frames 
 that were members of substates with the most negative frustration 
 scores were aligned to their collective mean structure and represented 
 as ensembles ( topological context shown in si 
 figure s8 ) . then the native conformation ( see methods ) was added , aligned , and shown in a thicker tube representation . these ensembles illustrate properties of the nonnative ensemble that 
 were suggested by the transition networks in figure 2 . the diffuse nonnative ensembles of ww , pg , and a3d , for 
 example , have few stabilizing nonnative interactions , so even their 
 most kinetically frustrated states appear almost completely unstructured 
 ( figure 6a ) . in contrast , the nonnative transition 
 maps with more topological isolation among inhibitor substates , especially 
 bbl and pg , show much more homogeneity in the respective structural 
 ensembles . bbl s bias for qnn was 
 0.18 , suggesting that the nonnative tertiary structure apparent 
 in the ensemble is responsible for the cluster of kinetic traps evident 
 in the network ( figure 2 black arrow ) . the 
 kinetic trap ensemble for trp shows that the peptide can get conformationally although 
 the nonnative configuration in the ensemble and the superimposed native 
 state have very different backbone geometries , the relative compactness 
 of the former may explain why trp presented a statistically significant 
 negative qn , a property 
 not observed for any other peptide . peptides with simple contact topologies 
 have been shown in lattice models to allow more interplay between 
 native and nonnative contacts , consistent 
 with our findings on trp . that stabilizing interactions generally 
 may be responsible for kinetic traps is suggested by figure 6b , which shows that kinetic traps commonly have 
 smaller widths ( lower average pairwise rmsd of constituent members ) 
 than the nonnative ensemble . ( a ) representative structures of kinetic traps , shown in topological 
 context in si figure s8 . red , substates classified 
 as kinetic traps ; gray , all nonnative substates ; green , all native 
 substates . to compare 
 the folding properties of 10 protein sequences , we have 
 exploited both quantitative and interpretive aspects of network models 
 of protein folding . our definition of kinetic frustration is grounded 
 in graph theoretic principles while being consistent with qualitative 
 definitions of kinetic features , such as kinetic traps . the method 
 thus allows direct comparison between temporal folding behaviors and 
 conformational features , the latter summarized by five standard structural 
 metrics that were normalized against their prevalence in the native 
 ensemble . while nonnative intramolecular interactions or nonnative 
 secondary structure formation have been recognized as contributing 
 factors to misfolding or folding rate reductions , we quantified the influence of these structural malformations through 
 a normative process that requires no prior domain knowledge of the 
 protein of interest . specifically , folding for trp , vhp , pb , pg , and 
 lamda was most kinetically frustrated by deformations other than that 
 characterized by rmsd , suggestive of stabilizing forces that trap 
 a folding protein in a semistructured but nonnative conformation . these details were resolvable because we chose to perturb individual 
 substates rather than substate collections within the transition networks . we additionally observed that phantom networks 
 constructed by mimicking gross topological attributes of the observed 
 networks mostly reproduced emergent kinetic properties ( nat and fnat ) ( si figure s7 ) . if transition networks directly 
 reflect the kinetic barriers , traps , and pathways caused by conformational 
 fluctuations , as argued , then further topological properties can hopefully 
 be linked to more nuanced categories of structural deformation . certainly , 
 subjective concepts such as misfolded intermediates , unstructured 
 intermediates , and kinetic traps , often invoked in the literature 
 of misfolding pathologies , can especially benefit from this type of quantification since simulations 
 are increasingly sampling the distant or rare fel regions where these 
 events occur .
experiments 
 and atomistic simulations of polypeptides have revealed 
 structural intermediates that promote or inhibit conformational transitions 
 to the native state during folding . 
 we invoke a concept of  kinetic 
 frustration  to quantify the prevalence and impact of these 
 behaviors on folding rates within a large set of atomistic simulation 
 data for 10 fast - folding proteins , where each protein s conformational 
 space is represented as a markov state model of conformational transitions . 
 our graph theoretic approach addresses what conformational features 
 correlate with folding inhibition and therefore permits comparison 
 among features within a single protein network and also more generally 
 between proteins . 
 nonnative contacts and nonnative secondary structure 
 formation can thus be quantitatively implicated in inhibiting folding 
 for several of the tested peptides .