mathematical expressions for the jensen - shannon divergence (  ) and s - score are given by equations 2 and 3 , respectively : 
 ( 2)2(wit(m),wip(m))=s(wit(m)+wip(m)2)-12[s(wit(m))+s(wip(m ) ) ] 
 ( 3)s(wit(m),wip(m))=12i=1nwit(m)-wip(m) in which { 
 wit(m ) } and { 
 wip(m ) } represent the population weights for the i bin in ensemble t and p , respectively for a given bin size , m. s(wi)=wi(m)log2wi(m ) ) in equation 2 is the information entropy . and s vary between 0 and 1 for maximum and minimum similarity , and are equal to zero if and only if 
 { wit(m)}={wip(m)}. equations 2 and 3 show that for non - overlapping regions in two distributions , defined as cases in which 
 { wit(m)}=0;{wip(m)}0 or 
 { wit(m)}0;{wip(m)}=0 , the contribution to  and s is independent of the extent of structural similarity . in the sas approach , experimental rdcs are used to guide construction of an ensemble by selecting n conformations from a conformational pool that minimize the following  function , 
 ( 4)2=i=1l(dicalc - diexp)2/l in which l is the total number of rdcs used in sas , 
 dicalc and 
 diexp are calculated and experimentally measured rdcs , respectively . in our implementation of sas , first an initial ensemble of n conformations is randomly selected from the pool . then at each step ( k ) of the selection procedure one conformation in the ensemble is randomly chosen and replaced by a conformation randomly selected from the rest of the pool . the change from step k to k+1 (k ) ; if (k+1 )  (k ) with a probability p = exp(((k)(k+1))/t ) , where t is an effective temperature that is linearly decreased using a simulated - annealing scheme . the initial effective temperature is set to sufficiently high so that > 99% of the conformations can be replaced and slowly decreased until the acceptance probability is smaller than 10 . at each effective temperature , a matlab script ( available from authors upon request ) was used to implement this sas - based ensemble construction . the capability of rdcs to reconstruct inter - helical ensembles using the sas approach was investigated using synthetic rdc data , using up to five rdc data sets corresponding to five perfectly orthogonal alignment tensors . in these simulations , a given conformation is represented using three inter - helical euler angles ( h , h , h ) describing the relative orientation of the two idealized a - form helices representing the tar helices connected by a trinucleotide bulge ( fig . the conformational pool necessary for the sas selection was generated by using the corresponding topologically allowed space . this space corresponds to all possible inter - helical orientations that satisfy basic steric and connectivity restraints imposed by the bulge . the pool was generated using a 5 resolution grid ( i.e. each conformation differs from its closest neighbor by a 5 change in one of the three euler angles ) . for a trinucleotide bulge , the pool represents ~10% of the total possible inter - helical orientations . a target ensemble containing five distinct conformations ( n=5 ) five orthogonal alignment tensors arbitrarily fixed on the reference helix were then generated using the gram - schmidt procedure . for each of the five alignment tensors , all possible one bond ch rdc were computed for the target ensemble . for each alignment tensor , the rdcs for the five conformations were averaged and error - corrupted assuming 2hz rdc uncertainty . the sas approach was then implemented to select an ensemble of n=5 distinct conformations using one , two , three , four and five sets of input rdcs to guide selection . the target and the predicted ensemble were then compared using similarity measurements including  , s - score ,  and bhattacharyya distance at various bin sizes as described below . the same process was repeated 50 times and the similarity between target and predicted ensembles were averaged over these 50 comparisons at each bin size . standard deviation is calculated to estimate the variation in repeated simulations using different numbers of input alignments . the standard deviation is similar across the groups , and small compared to the observed differences between them ( fig . ensembles determined using one , two , three and four rdc data sets in the sas selection were used to predict a fifth rdc data set that was not used in the selection . the resultant rmsd between the rdcs for this fifth data set and values back - calculated from the predicted ensemble was then computed . the cartesian distance in the euler space , ( ( hahb ) + ( hahb ) + ( hahb ) ) between two sets of euler angles a and b defining two distinct inter - helical orientations does not provide a measure of structural similarity between the two conformations . first , there are inherent degeneracies ( h=h+180 , h=h , h=h+180 ; h=h180 , h=h , h=h180 ; h=h+180 , h=h , h=h180 ; h=h180 , h=h , h=h+180 ) that map several sets of distinct inter - helical euler angles to the same conformation . this problem was overcome by using a restricted grid of euler angles devoid of any degeneracy . second , even after taking into account the above degeneracy , the cartesian distance between two sets of euler angles does not provide a faithful measurement of structural similarity . for example , the cartesian distances between ( 0 , 0 , 0 ) and ( 5 , 5 , 5 ) is ~9 in the euler space whereas the two conformations differ by single axis rotation with amplitude ~11. likewise , the conformations ( 5 , 5 , 0 ) and ( 170 , 10 , 170 ) differ by a cartesian distance of ~237 but the two conformations differ by a single axis rotation with amplitude ~25. more generally , the cartesian distance between euler angles can be smaller than , equal to or larger than the actual difference between two conformations . therefore we used the amplitude of single axis rotation to bin inter - helical orientations together and measure similarity between ensembles ( see below ) . the binning grid points are constructed by picking a binning origin , defined by minimum value of each of the three euler angle in the two ensembles upon comparison , and then incrementing each euler angles by an amount defined by the bin size to cover the entire non - degenerate 3d euler space . changing in the binning origin has minimal effects on the resulting analysis ( data not shown ) . next , the amplitude of a single axis rotation (  ) connecting a given conformation in the ensemble defined by euler angles ( h1 , h1 , h1 ) and a point on the grid ( h2 , h2 , h2 ) is computed , 
 ( 5)r(h1,h1,h1)=o(x , y , z,)r(h2,h2,h2 ) in which o(x , y , z ,  ) represents a single axis rotation about a unit vector ( x , y , z ) with amplitude (  ) . o(x , y , z ,  ) can also be expressed by a 3 by 3 matrix in terms of x , y , z and  and the rotation amplitude  is given by , 
 ( 7)=arccos(o11+o22+o33 - 12 ) in which o11 , o22 and o33 are the three diagonal elements of o(x , y , z ,  ) . in this manner , the amplitude of the single axis rotation connecting a given conformation in an ensemble to every grid point is computed , and the conformation is binned to the grid point that leads to the minimum single axis rotation amplitude . the population of each grid point is then calculated to be the number of conformations binned divided by the total number of conformations in the ensemble . in our case , binning of the target and the predicted ensemble led to two population distributions on the same binning grid for a given bin size , and the value of  between the two ensembles at the given bin size is then calculated using equation 2 . this analysis was preformed using a matlab script that is available from the authors upon request . an in - house perl script was used to compute inter - helical angles ( h , h , h ) describing the relative orientation of two a - form helices . all intra- and inter - base - pair parameters were computed using curves+   and all the local torsion angles defining the sugar and backbone geometry were computed using an in - house c script . the resulting inter - helical orientations defined by three euler angles were binned and analyzed as described above . distributions of base - pair parameters , sugar and backbone torsion angles were directly binned to a binning grid ranging between 5 and 360 with variable increments defined by the bin size . the value of  was calculated at each given bin size for each parameter / angle distribution using equation 2 and the values of  are calculated using equation 1 for distributions of inter - helical orientation , base - pair parameter , sugar , and backbone torsion angles . mathematical expressions for the jensen - shannon divergence (  ) and s - score are given by equations 2 and 3 , respectively : 
 ( 2)2(wit(m),wip(m))=s(wit(m)+wip(m)2)-12[s(wit(m))+s(wip(m ) ) ] 
 ( 3)s(wit(m),wip(m))=12i=1nwit(m)-wip(m) in which { 
 wit(m ) } and { 
 wip(m ) } represent the population weights for the i bin in ensemble t and p , respectively for a given bin size , m. s(wi)=wi(m)log2wi(m ) ) in equation 2 is the information entropy . and s vary between 0 and 1 for maximum and minimum similarity , and are equal to zero if and only if 
 { wit(m)}={wip(m)}. equations 2 and 3 show that for non - overlapping regions in two distributions , defined as cases in which 
 { wit(m)}=0;{wip(m)}0 or 
 { wit(m)}0;{wip(m)}=0 , the contribution to  and s is independent of the extent of structural similarity . in the sas approach , experimental rdcs are used to guide construction of an ensemble by selecting n conformations from a conformational pool that minimize the following  function , 
 ( 4)2=i=1l(dicalc - diexp)2/l in which l is the total number of rdcs used in sas , 
 dicalc and 
 diexp are calculated and experimentally measured rdcs , respectively . in our implementation of sas , first an initial ensemble of n conformations is randomly selected from the pool . then at each step ( k ) of the selection procedure one conformation in the ensemble is randomly chosen and replaced by a conformation randomly selected from the rest of the pool . the change from step k to k+1 (k ) ; if (k+1 )  (k ) with a probability p = exp(((k)(k+1))/t ) , where t is an effective temperature that is linearly decreased using a simulated - annealing scheme . the initial effective temperature is set to sufficiently high so that > 99% of the conformations can be replaced and slowly decreased until the acceptance probability is smaller than 10 . at each effective temperature , a matlab script ( available from authors upon request ) was used to implement this sas - based ensemble construction . the capability of rdcs to reconstruct inter - helical ensembles using the sas approach was investigated using synthetic rdc data , using up to five rdc data sets corresponding to five perfectly orthogonal alignment tensors . in these simulations , a given conformation is represented using three inter - helical euler angles ( h , h , h ) describing the relative orientation of the two idealized a - form helices representing the tar helices connected by a trinucleotide bulge ( fig . the conformational pool necessary for the sas selection was generated by using the corresponding topologically allowed space . this space corresponds to all possible inter - helical orientations that satisfy basic steric and connectivity restraints imposed by the bulge . the pool was generated using a 5 resolution grid ( i.e. each conformation differs from its closest neighbor by a 5 change in one of the three euler angles ) . for a trinucleotide bulge , the pool represents ~10% of the total possible inter - helical orientations . a target ensemble containing five distinct conformations ( n=5 ) five orthogonal alignment tensors arbitrarily fixed on the reference helix were then generated using the gram - schmidt procedure . for each of the five alignment tensors , all possible one bond ch rdc were computed for the target ensemble . for each alignment tensor , the rdcs for the five conformations were averaged and error - corrupted assuming 2hz rdc uncertainty . the sas approach was then implemented to select an ensemble of n=5 distinct conformations using one , two , three , four and five sets of input rdcs to guide selection . the target and the predicted ensemble were then compared using similarity measurements including  , s - score ,  and bhattacharyya distance at various bin sizes as described below . the same process was repeated 50 times and the similarity between target and predicted ensembles were averaged over these 50 comparisons at each bin size . standard deviation is calculated to estimate the variation in repeated simulations using different numbers of input alignments . the standard deviation is similar across the groups , and small compared to the observed differences between them ( fig . for the rdc cross validation analysis , ensembles determined using one , two , three and four rdc data sets in the sas selection were used to predict a fifth rdc data set that was not used in the selection . the resultant rmsd between the rdcs for this fifth data set and values back - calculated from the predicted ensemble was then computed . the cartesian distance in the euler space , ( ( hahb ) + ( hahb ) + ( hahb ) ) between two sets of euler angles a and b defining two distinct inter - helical orientations does not provide a measure of structural similarity between the two conformations . first , there are inherent degeneracies ( h=h+180 , h=h , h=h+180 ; h=h180 , h=h , h=h180 ; h=h+180 , h=h , h=h180 ; h=h180 , h=h , h=h+180 ) that map several sets of distinct inter - helical euler angles to the same conformation . this problem was overcome by using a restricted grid of euler angles devoid of any degeneracy . second , even after taking into account the above degeneracy , the cartesian distance between two sets of euler angles does not provide a faithful measurement of structural similarity . for example , the cartesian distances between ( 0 , 0 , 0 ) and ( 5 , 5 , 5 ) is ~9 in the euler space whereas the two conformations differ by single axis rotation with amplitude ~11. likewise , the conformations ( 5 , 5 , 0 ) and ( 170 , 10 , 170 ) differ by a cartesian distance of ~237 but the two conformations differ by a single axis rotation with amplitude ~25. more generally , the cartesian distance between euler angles can be smaller than , equal to or larger than the actual difference between two conformations . therefore we used the amplitude of single axis rotation to bin inter - helical orientations together and measure similarity between ensembles ( see below ) . the binning grid points are constructed by picking a binning origin , defined by minimum value of each of the three euler angle in the two ensembles upon comparison , and then incrementing each euler angles by an amount defined by the bin size to cover the entire non - degenerate 3d euler space . changing in the binning origin has minimal effects on the resulting analysis ( data not shown ) . next , the amplitude of a single axis rotation (  ) connecting a given conformation in the ensemble defined by euler angles ( h1 , h1 , h1 ) and a point on the grid ( h2 , h2 , h2 ) is computed , 
 ( 5)r(h1,h1,h1)=o(x , y , z,)r(h2,h2,h2 ) in which o(x , y , z ,  ) represents a single axis rotation about a unit vector ( x , y , z ) with amplitude (  ) . o(x , y , z ,  ) can also be expressed by a 3 by 3 matrix in terms of x , y , z and  and the rotation amplitude  is given by , 
 ( 7)=arccos(o11+o22+o33 - 12 ) in which o11 , o22 and o33 are the three diagonal elements of o(x , y , z ,  ) . in this manner , the amplitude of the single axis rotation connecting a given conformation in an ensemble to every grid point is computed , and the conformation is binned to the grid point that leads to the minimum single axis rotation amplitude . the population of each grid point is then calculated to be the number of conformations binned divided by the total number of conformations in the ensemble . in our case , binning of the target and the predicted ensemble led to two population distributions on the same binning grid for a given bin size , and the value of  between the two ensembles at the given bin size is then calculated using equation 2 . this analysis was preformed using a matlab script that is available from the authors upon request . an in - house perl script was used to compute inter - helical angles ( h , h , h ) describing the relative orientation of two a - form helices . all intra- and inter - base - pair parameters were computed using curves+   and all the local torsion angles defining the sugar and backbone geometry were computed using an in - house c script . the resulting inter - helical orientations defined by three euler angles were binned and analyzed as described above . distributions of base - pair parameters , sugar and backbone torsion angles were directly binned to a binning grid ranging between 5 and 360 with variable increments defined by the bin size . the value of  was calculated at each given bin size for each parameter / angle distribution using equation 2 and the values of  are calculated using equation 1 for distributions of inter - helical orientation , base - pair parameter , sugar , and backbone torsion angles .
methods for comparing ensembles of biomolecules assess the population overlap between distributions but fail to fully quantify structural similarity . 
 we present a simple and general approach for quantifying population overlap and structural similarity between ensembles . 
 this approach captures improvements in the quality of ensembles determined using increasing input experimental data that go undetected using conventional methods and reveals unexpected similarities between rna ensembles determined using nmr and molecular dynamics simulations .