understanding the mechanisms that lead 
 to the biological function 
 of metalloproteins requires detailed knowledge of the metal - site structure . so far , metal - site structures of metalloproteins have been studied 
 experimentally mainly by techniques such as x - ray crystallography , 
 x - ray absorption spectroscopy , extended x - ray absorption fine 
 structure , and epr . these techniques provide information about the geometric 
 and electronic structure of the metal - site in the crystal phase or 
 in frozen liquids . however , proteins normally function in solution 
 where the flexibility and dynamics of the proteins often are essential 
 for their biological activity . characterization of both the dynamics 
 and the structure of the metal - site in aqueous solution at the atomic 
 level is therefore important for the elucidation of functional mechanisms 
 of metalloproteins . paramagnetic nuclear magnetic resonance 
 ( nmr ) spectroscopy is a 
 unique tool for providing information about metal - site structure and 
 dynamics of paramagnetic metalloproteins such as iron  sulfur 
 proteins , copper proteins , heme proteins , and other 
 metalloproteins . thus , the nuclear spins act as 
 structural probes yielding information about the geometric and electronic 
 structures of the metal - site in solution through their interaction 
 with the unpaired electron spin . the chemical shift of nuclei 
 close to a paramagnetic metal - site 
 is affected strongly by the hyperfine fermi contact shift . the fermi contact shift arises from a through - bond 
 electron  nuclear scalar coupling and is proportional to the 
 unpaired electron spin density at the nuclear position . therefore , 
 it provides information about the delocalization of the unpaired electrons 
 onto the ligand nuclei . theoretical quantum chemical calculations , 
 experimental nmr , and endor studies have shown that the fermi contact 
 shift depends on the geometry of the protein in the immediate vicinity 
 of the metal - site . therefore , it also gives information about 
 the structure and structural changes close to the metal - site . the 
 nuclear longitudinal paramagnetic relaxation enhancement ( pre ) is 
 complementary to the fermi contact shift . it stems from a through - space 
 electron  nucleus dipole interaction , which depends on the geometric 
 position of the nucleus relative to the total unpaired electron spin 
 distribution of the metal - site . the pre can therefore provide long - range 
 distance information in proteins and information about spatial distribution of the electron spin 
 in metalloproteins . despite the large and versatile amount of information provided 
 by the nuclear electron interactions , the use of paramagnetic 
 nmr in the determination of the structure and dynamics of metal - sites 
 of metalloproteins has so far been limited . this is mainly due to 
 the fact that a proper description of the unpaired electron spin density 
 in proteins requires a large basis set , which makes restrained dynamics 
 simulations and accurate structure determinations computationally 
 highly demanding . here , we show how a natural bond orbital ( nbo ) description of the unpaired electron spin density 
 of metalloproteins 
 can alleviate this problem . first , we show that the majority of the 
 unpaired electron spin density of metalloproteins can be modeled by 
 a small number of nbos that reflect the chemical bonding environment 
 of the protein . second , we show how accurate nmr parameters can be 
 calculated from a small and transferable basis set that is based on 
 the nbos . for h nuclei , the pre can be calculated accurately 
 from the metal  ligand nbos , as for example the pre of h nuclei of the iron sulfur protein rubredoxin that 
 can be calculated using the nbos that describe the fe  s bond . 
 for heavy atoms , such as c and n , local 
 nbos that are centered on the heavy atom paramagnetic 
 interactions in both proteins have been studied extensively , both 
 experimentally by nmr and theoretically by density functional theory 
 ( dft ) calculations . moreover , these two metalloproeins are very 
 different both with respect to symmetry , structure , and electronic 
 spin . accurate paramagnetic nmr parameters are calculated using the 
 small nbo basis set for both plastocyanin and rubredoxin , which suggests 
 that the approach developed below is applicable to a wide range of 
 metalloproteins . overall , combining the nbo description of the electron 
 spin with experimental nmr parameters allows an accurate characterization 
 of the dynamics and structures of metalloproteins in solution , as 
 shown below . the longitudinal 
 paramagnetic relaxation enhancement ( pre ) contains contributions from 
 the fermi contact relaxation , the curie spin relaxation , and the dipolar relaxation . the fermi 
 contact contribution to longitudinal paramagnetic relaxation is in 
 general small even for hyperfine shifted signals . specifically , the contact contribution is less than 0.5% 
 of the dipolar relaxation for all of the nuclei used in this study and is therefore neglected . the experimentally measured longitudinal 
 pre of the nuclei depends on the unpaired electron spin density as 
 follows:1where ge is the electron g - factor , b is the bohr magneton , s is the spin quantum number 
 of the unpaired electron , 0 is the magnetic permeability 
 of free space , i is the gyromagnetic ratio of the 
 nucleus , c,1 is the correlation time of the electron nucleus 
 dipolar interaction , r is the rotational correlation 
 time of the protein , and sz is the thermal average of the electron spin magnetization . the effect of curie spin relaxation , described by the term modulated by r in eq 1 , can often be neglected . for example , for the high - spin 
 fe(iii ) protein rubredoxin ( blue copper protein plastocyanin ) studied 
 here , the contribution to r1p from curie 
 spin relaxation is about 2.5% ( 0.03% ) . even in studies where curie 
 spin relaxation can not be neglected , the longitudinal relaxation enhancement 
 is proportional to reff6 , where  is the proportionality 
 constant . moreover , cross - correlated relaxation between curie - spin 
 relaxation ( nsz ) and dipole relaxation with a directly bond 
 nuclei ( nh ) can be neglected if ( 1 ) h or n relaxes 
 faster than the scalar coupling between h and n or ( 2 ) the relaxation 
 rate of n is measured on a coherence that is in - phase with respect 
 to h. both criteria are fulfilled here , and cross - correlated relaxation 
 is therefore neglected . the effective nucleus electron distance 
 is given by the unpaired spin distribution as follows:23where 2(r ) = ry2(r/r ) are the spatial components of the 
 electron nucleus 
 dipolar operator centered at the nucleus at the position r , and y2(r/r ) ,  = 2, ... ,2 , are the spherical harmonics . furthermore , f2(r ) is the matrix 
 representation of the dipolar operator in the basis of the spin density , 
 and p is the spin - only fock  dirac density matrix . experimentally , the longitudinal pre of a given nucleus is obtained 
 as the difference between the relaxation rate of the nucleus in the 
 paramagnetic and the corresponding diamagnetic species . for example , 
 for the nuclei of the blue copper protein plastocyanin , a longitudinal 
 pre is obtained as r1p = r1(cu(ii ) ) r1(cu(i ) ) , r1(cu(ii ) ) and r1(cu(i ) ) being the longitudinal relaxation rates measured in the paramagnetic 
 copper(ii ) protein , and in the copper(i ) analogue diamagnetic protein , 
 respectively . subsequently , the effective distance , reff , can be calculated from the pre once the correlation 
 time , c,1 , or the proportionality constant ,  , 
 is known . overall , the effective distance depends on the distribution 
 of the total unpaired electron spin in the molecule and therefore 
 provides information on the overall electron spin distribution via 
 (r ) = (r ) (r ) , as well as 
 information on the position of the nucleus relative to the distribution 
 via 2(r r ) . the unpaired 
 electron spin density causing the paramagnetic relaxation enhancement 
 and given in eq 3 is located either in the metal  ligand orbitals 
 or in the local orbitals of the relaxing nucleus . even though only 
 a small fraction of the spin density is in the local orbitals of the 
 relaxing nucleus , this density is spatially very close to the nucleus 
 and can therefore give a significant contribution to the relaxation . it is our goal to estimate the spin density of the local orbitals 
 from the hyperfine chemical shift of heavy atoms and subsequently 
  correct  the experimentally derived relaxation enhancements 
 such that these enhancements report exclusively on the position of 
 the nucleus relative to the metal  ligand center . the 
 chemical shift observed for nuclei of a paramagnetic protein consists 
 of contributions from the hyperfine interaction , hyp , and a diamagnetic contribution . the hyperfine shift , hyp , has in general two contributions , that is , the fermi contact 
 shift , con , and the dipolar ( through space ) pseudocontact 
 shift , pcs:4for nuclei in the immediate vicinity of a 
 paramagnetic metal ion , the fermi contact shift is usually much larger 
 than the pseudocontact shift . we find that 
 pcs/hyp < 0.085 for all of 
 the nitrogen nuclei of rubredoxin included in the analysis below with 
 the average pcs/hyp 
 = 0.025  0.025 . an upper limit for pcs was 
 calculated from the susceptibility tensor of fe(iii ) rubredoxin ( ax = 5.3  10 cm , rh = 2.1  10 cm ) , and hyp was obtained experimentally . the fermi contact shift , con , relates to the 
 unpaired electron spin density through:5where k is the boltzmann 
 constant , t is the absolute temperature , and (r ) is the matrix representation of the dirac - delta 
 function centered at the nuclear position . the nuclear position , r , is given in the coordinate frame of { |i}i , which 
 is the basis set in which the electron spin density is described . from eq 5 , it is seen that the peak position in an nmr spectrum 
 of hyperfine - shifted nuclei is a direct measure of the electron spin 
 density at the position of the nucleus , if the pseudocontact shift 
 pcs is much smaller than con or 
 if pcs can be calculated . below , we use the fermi 
 contact shift to calculate the spin density of the local orbitals 
 to take the relaxation effect of the local orbitals into account . in general , the contribution to the pre from the local orbital can 
 be neglected , when the pseudocontact shift is similar to or larger 
 than the fermi contact shift . for the nuclei of the ligand binding 
 residues of rubredoxin , the fermi contact shift is obtained as con = fe(iii )  dia , where fe(iii ) is the chemical shift in the paramagnetic 
 high - spin fe(iii ) protein and dia is the chemical shift in the analogue diamagnetic protein , 
 which in this case was estimated from the random coil value . natural bond orbital ( nbo ) analysis is based on local block eigenvectors 
 of the one - particle density matrix and has emerged as a technique 
 for studying hybridization and covalency effects in polyatomic wave 
 functions . in short , the nbo between two atoms a and b is derived 
 by first transforming the one - particle density matrix to the natural 
 atomic orbital basis and subsequently only considering the part of 
 the density matrix , pab , that involves natural 
 atomic orbitals centered at a or b. the matrix pab is then depleted for orbitals such as core orbitals and 
 lone pairs that are only centered at one of the two atoms . an eigenvector 
 of the depleted pab density matrix with an 
 occupation number ( eigenvalues ) over a certain threshold constitutes 
 a pre - nbo ( pnbo ) . the calculated eigenvectors can be used to transform 
 the original set of basis functions ( often gaussian type atomic orbitals ) 
 into the basis set of pnbos , core orbitals , and lone - pairs . the set 
 of nbos corresponds closely to the picture of localized bonds and 
 lone pairs , and ab initio wave functions transformed into pnbos are 
 found to be in good agreement with the lewis structure concepts of 
 bond hybridization . the nbos are the 
 set of orbitals obtained by orthogonalizing the two - center pnbos . the pnbo basis set will serve as a starting point for a robust 
 and transferable modeling of the unpaired electron spin density in 
 metalloproteins . whenever possible , the two - center preorthogonalized 
 natural bond orbitals ( pnbo ) will be used instead of the fully orthogonalized 
 nbos . the genuine two - center pnbos are easily transferred from one 
 molecule to another because they are linear combinations of atomic 
 orbitals centered only at the two nuclei that are connected with a 
 bond . thus , as shown below , the metal  ligand pnbo description 
 of a metal - site model complex can be transferred from the small model 
 complex to a full metalloprotein and thereby used to describe the 
 unpaired electron spin density in the protein . most of the unpaired 
 electron spin density of a paramagnetic metalloprotein occupies metal 
 and ligand atomic orbitals . intuitively , when the unpaired electron 
 spin density is described in the pnbo basis set of orbitals , the metal ligand 
 antibond orbitals and metal and ligand lone - pairs contain the majority 
 of the unpaired electron spin density . to verify that the metal ligand 
 pnbos and lone - pairs alone form a good basis for the unpaired electron 
 spin density in metalloproteins , we consider two small model complexes : 
 ( 1 ) the spin 5/2 iron  sulfur model complex , fe(sch3)4 , 
 and ( 2 ) the spin 1/2 copper model complex , cu(im)2(sch3)s(ch3)2 + . the structures of the models were based on 
 x - ray crystal structures of the iron sulfur protein rubredoxin 
 ( pdb code 4rxn(51 ) ) and the blue copper protein plastocyanin 
 ( pdb code 1plc(52 ) ) ; these model complexes are shown in 
 figure 1 . the two small model complexes used to 
 study the unpaired electron 
 spin density of rubredoxin and plastocyanin : ( a ) the fe(sch3)4 model 
 complex , which is used as a model of rubredoxin , and ( b ) cu(sch3)(im)2s(ch3)2 + , which is a model of plastocyanin . below , we calculate the spin occupation numbers 
 for the pnbos of 
 the two small model complexes and demonstrate ( 1 ) that the majority 
 of the unpaired electron spin density occupies metal ligand 
 pnbos and lone - pairs and ( 2 ) that only very few pnbos have significant 
 occupancy . finally , we use the pnbos that are highly spin - occupied 
 to calculate nmr parameters . electron densities ,  and  , of the two small model complexes were obtained by quantum chemical 
 calculations at the ub3lyp/6 - 31 g * level . subsequently , an nbo analysis 
 was performed , and the fock - dirac spin 
 density matrix in the atomic orbital ( ao ) basis , pao , was transformed to the two - center pnbo basis ( -spin ) 
 by the transformation matrices provided by the nbo analysis program 
 as implemented in gaussian 03 and described in material 
 and methods . gross occupation numbers , ni , for the unpaired electron spin density 
 of the individual basis functions , ni = ( s1p)ii , were calculated in the bases  
 = ao and  = pnbo , respectively . here , s is the overlap matrix of the basis set in question , while p is the fock dirac spin density 
 matrix in the same basis . the gross spin occupation numbers , ni of the unpaired electron spin , ( s1p)ii , 
 for the fe(sch3)4 model complex in two different basis 
 sets : ( a ) the two - center pnbo basis and ( b ) the atomic orbital basis , pao ( in the 6 - 31 g * basis ) . sigma antibond orbitals 
 are indicated as  * , lone pairs as lp , and rydberg orbitals 
 as ry . only a few orbitals have significant spin occupation in the 
 pnbo basis , and the occupied orbitals are strongly correlated to the 
 chemical bonding environment . the gross spin occupation numbers of the unpaired spin , ni = ( s1p)ii , 
 for the cu(sch3)(im)2s(ch3)2 + model complex 
 in two different basis sets : ( a ) the two - center pnbo basis and ( b ) 
 the atomic orbital basis , pao ( in the 6 - 31 g * 
 basis ) . sigma and bond orbitals are shown as  and  , 
 respectively , while sigma and pi antibond orbitals are shown as  * 
 and  * , respectively , and lp indicates a lone pair . only a few 
 orbitals in the pnbo basis are occupied by unpaired electron spin . figures 2 and 3 show 
 the gross spin occupation numbers , ni = ( s1p)ii , of the unpaired electron spin density 
 in the ao and the pnbo bases and for the two model complexes . as it 
 appears from the figures , only a small number of the two - center pnbo 
 basis functions are significantly spin - populated ; that is , the unpaired 
 electron spin density can be described , to a good approximation , by 
 a few basis functions , which relate to the chemical bonding environment . the gross occupation numbers for the unpaired electron spin of 
 the metal  ligand pnbos are summarized for the two model complexes 
 in table 1 . the two - center metal ligand 
 pnbos account for approximately 90% of the total unpaired electron 
 spin density for both the rubredoxin and the plastocyanin complex . also , primarily valence antibond orbitals (  * ,  * ) and 
 lone pairs have significant population of unpaired electron spin , 
 whereas only a small amount of the unpaired electron spin occupies 
 the high - energy rydberg ( ry ) orbitals . the accumulative gross 
 spin density . the pnbos 
 number 54 , 56 , 58 , 60 , 
 61 , and 62 of the fe(sch3)4 model complex correspond to  
 antibonds between the iron and the four sulfur atoms . the mixed  and  antibond 
 orbitals number 261 and 262 can be transformed into  and  
 antibond orbitals with spin occupancies of 1.1  10 and 0.83 , respectively . the same transformation of 
 the corresponding bond orbitals , that is , 11 and 12 , results in  
 and  bond orbitals with spin occupancies of 3.7  10 and 0.040 , respectively ( see text ) . the populations 
 of these transformed orbitals are shown in the last four rows of columns 58 . from figures 2 and 3 , it is seen that the unpaired electron spin population 
 of the bond 
 orbitals is small as compared to the population of the antibond natural 
 orbitals . thus , the tight connection between the chemical bonding 
 environment and the pnbos is evident from the types of pnbos that 
 are populated , because one would expect that the bonding orbital will 
 be fully occupied with one  and one  electron , while 
 the antibonding orbital is only partly occupied . for the rubredoxin 
 model complex , the spin - populated pnbos fall in two groups ; four correspond 
 to fe  s  antibond (  * ) , while six pnbos correspond 
 to  fe  s antibond formed by lone pairs at the iron and 
 the sulfur atoms . for the plastocyanin model complex , the two 
 most spin - populated 
 pnbos correspond to a mixture of cu  s  and  antibond 
 orbitals with spin occupancy of 0.55 and 0.28 , respectively . the submatrix 
 of the spin density matrix that corresponds to the two cu  s 
 orbitals can be transformed into two orbitals with  and  
 antibond character . in this , the gross 
 spin occupation of the  antibond orbital is 0.83 , while the 
 occupation of the corresponding  antibond orbital is 1.1 
  10 . thus , the unpaired electron spin density 
 occupies only the antibond  orbital and does not occupy the 
  * orbital . this is in agreement with the uv  vis spectra 
 of plastocyanin , which show that the unpaired electron spin predominantly 
 is in the  antibond cu the calculations 
 above show that the unpaired electron spin density 
 can be modeled by the pnbos that reflect the chemical bonding environment . 
 in figure 4 , the -lumo accounts 
 for the total spin density of the plastocyanin model complex and is 
 shown in figure 4h . it is evident that most 
 of the -lumo spin density ( given by the 286 atomic basis functions ) 
 can be represented accurately by a linear combination of the three 
 pnbos sketched for plastocyanin in the figure . the nitrogen lone - pairs 
 shown in figure 4f and g are very similar in 
 shape , which is expected due to the high transferability of the pnbos . considering the large difference between the two model complexes 
 studied here , the metal  ligand pnbos are likely , in general , 
 to account for the majority of the unpaired electron spin density 
 of metalloproteins . this implies that properties that depend on the 
 overall spin distribution can be calculated accurately from the few 
 metal  ligand pnbos instead of using the full basis set , thus 
 speeding up calculations of such parameters significantly . below , 
 we will show that the pre of protons , which primarily depend on the 
 overall unpaired electron distribution , can be calculated accurately 
 from the metal  ligand pnbos . schematic representation of the pnbos 
 of figures 2 and 3 with 
 high spin occupation numbers . ( a ) lone pair at the sulfur , pnbo number 54 in figure 2 ; ( b ) lone pair at the ferric , pnbo number 61 in figure 2 ; ( c ) lone pair at the ferric , pnbo number 62 in 
 figures 2 and 3 ; ( d ) 
 the  * antibond orbital between the sulfur and the ferric ion , 
 pnbo number 176 in figure 2 ; ( e ) the  * 
 orbital obtained from pnbo numbers 261 and 262 in figure 3 as described in the text ; ( f ) lone pair at the nitrogen , 
 number 68 in figure 3 ; ( g ) lone pair at the 
 nitrogen , number 75 in figure 3 ; and ( h ) the 
 -lumo of the plastocyanin model complex ( 286 basis functions ) . 
 for blue copper proteins , as indicated , the -lumo can 
 be obtained by a linear combination of the pnbos shown in ( e ) , ( f ) , 
 and ( g ) . the density at the protons of sch3 in ( h ) corresponds to the pnbo 
 265 in figure 3 , which is a c  h antibond 
 orbital . the longitudinal 
 dipolar pre , eq 1 , of protons is primarily caused by the unpaired 
 electron spin density located in the metal  ligand orbitals . although unpaired electron spin may also occupy local orbitals of 
 the proton ( see figure 4h ) , these orbitals 
 are predominantly s - type orbitals that do not contribute to dipolar 
 relaxation because they are spherical symmetric . overall , the longitudinal 
 paramagnetic relaxation of protons in metalloproteins is the simplest 
 case to analyze in terms of unpaired electron spin density and can , 
 intuitively , be calculated using the metal  ligand pnbos shown 
 in table 1 . therefore , paramagnetic proton 
 relaxation is our starting point in the search for a general , small , 
 and transferable basis for the unpaired electron spin density , which 
 allows accurate calculations of paramagnetic nmr parameters . two large model complexes are considered to evaluate in detail the 
 dependence of the pre of protons on the unpaired electron spin density . the structures of these model complex are shown in figure s1 . as compared to the model complexes shown in figure 1 , the larger model complexes include all of the 
 amino acids in the vicinity of the metal - site and the hydrogen bonds 
 that stabilize the metal charge . thus , a 144-atom complex was used 
 to model the metal - site environment of plastocyanin , while a 104-atom complex was used to model the metal - site 
 environment of rubredoxin ( see materials and methods ) . dft calculations and nbo 
 analyses of the spin density form the basis for the evaluation of 
 the pre . the large model complexes serve as  true  models 
 for the proteins , and the nmr parameters calculated from the full 
 dft spin densities of the complexes serve as  true  
 values . moreover , a comparison of the derived nmr parameters with 
 the experimentally measured parameters serves as a test of the use 
 of metal  ligand pnbos for calculation of nmr parameters . in figure 5 , we compare pres calculated 
 from different models of the unpaired electron spin density using 
 the corresponding effective distances ( eqs 1 and 2 ) of the nuclei 
 in the two proteins rubredoxin and plastocyanin as proxies for the 
 pres . the expected  true  values of the pres are those 
 calculated from the total unpaired electron spin density obtained 
 from a dft calculation ( 1380 and 1223 basis function for rubredoxin 
 and plastocyanin , respectively ) . thus , the effective distances , reff , dft , calculated by integrating the contributions 
 from the total unpaired electron spin density , (r ) (r ) , obtained from the dft calculation are compared to the effective 
 distances calculated from the unpaired electron spin density of the 
 metal  ligand pnbos . the latter distances were obtained from 
 the spin density matrix , ppnbo , corresponding 
 to the metal  ligand pnbos and lone - pairs of table 1 , using eq 3 . in addition , the effective distances , reff , dft , are compared to geometric metal proton 
 distances ( euclidean distances ) , rpd = 
 rh  me , obtained from 
 the x - ray structure ( pdb code 1plc(52 ) and pdb code 4rxn(51 ) ) . finally , figure 5b compares the 
 effective distances calculated from the experimental r1p for the protons of spinach plastocyanin using eq 1 
 and a correlation time c,1 = ( r1e , cu(ii ) + r1 ) = 
 ( 4.1  0.6 )  10 s at 18.8 t , with 
 the pnbo and the geometric distances . well for the longitudinal pre of all of the protons 
 in both plastocyanin and rubredoxin . it is also noteworthy that , in 
 the case of plastocyanin , the effective distances calculated from 
 the four pnbos of table 1 agree extremely well 
 with the experimentally measured distances , while discrepancies of 
 up to 1  are seen for the point - dipole approximation ( rpd vs reff , exp = 
 ( r1p , exp/ ) ) . the main reason for the breakdown of the point - dipole approximation 
 is the highly anisotropic electron spin delocalization in plastocyanin , 
 where about 40% of the unpaired electron spin is located at the strongly 
 bound cysteine sulfur ; yet the pnbos account 
 for this delocalization . for rubredoxin , the effective distances 
 derived on the basis of 
 the point - dipole description deviate only about 0.5  from those 
 derived from the dft density ( figure 5c ) this 
 is because the spin delocalization in rubredoxin is smaller and more 
 isotropic than in plastocyanin . thus , only 25% of the unpaired electron 
 spin is located at the ligand sulfur atoms in rubredoxin , and the 
 sulfur ligands are arranged in a symmetrical tetrahedron around the 
 ferric ion . still , the pnbo description of the unpaired electron spin 
 provides a significantly better agreement with the full dft description 
 than the point - dipole description . an important property of 
 the metal  ligand pnbos is the transferability ; 
 that is , the metal  ligand pnbos of the large complexes are 
 very similar to the metal  ligand pnbos calculated for the small 
 complexes . as a comparison of the transferability , we have transferred 
 the metal  ligand pnbos from the small model complexes onto 
 the large model complexes and subsequently calculated effective proton a comparison of the resulting effective 
 distances , which is detailed in supporting information , shows that effective proton electron distances that are 
 within 0.25  are obtained by using the pnbos from a small complex . in conclusion , the metal  ligand pnbos electron density allows 
 accurate calculation of the longitudinal pre for all protons of the 
 two metalloproteins , rubredoxin and plastocyanin . thus , effective 
 distances are obtained that are within 0.25  ( 0.084  rmsd ) 
 of the full dft description for plastocyanin and within 0.10  
 ( 0.068  rmsd ) for rubredoxin . this implies that the longitudinal 
 pres of all protons in a metalloprotein can be included as constraints 
 in a molecular dynamics simulation , provided the unpaired electron 
 is described by the metal  ligand pnbos or by a more sophisticated 
 model . evaluation of the effective distances , reff , between the protons and the unpaired electron spin of ( a ) the 144-atom 
 model complex of plastocyanin , and ( c ) 
 the 104-atom model complex of rubredoxin . in ( a ) and ( c ) , the distances calculated from the total unpaired 
 electron spin density obtained from the dft calculation , reff , dft , are compared to the corresponding effective distances 
 calculated from the metal  ligand pnbos of table 1 , ( red  ) reff , pnbo , and 
 the geometric distances obtained from the x - ray structures of the 
 model complexes ( blue  ) , rpd = 
 rh  me. in ( b ) , the 
 experimentally derived effective distances , reff , exp , calculated from the experimentally measured r1p of spinach plastocyanin using eq 1 and a 
 correlation time c,1 = ( r1e , cu(ii ) + r1 ) = 
 ( 4.1  0.6 )  10 s , are compared to 
 distances calculated from the pnbo density , ( red  ) reff , pnbo , and with the geometric distances , rpd , of spinach plastocyanin ( blue  ) . the paramagnetic relaxation enhancement of heavy atoms such as c and n is more complicated to interpret in 
 terms of unpaired electron spin density and geometric distances than 
 the relaxation of h nuclei discussed above . even so , the n data can often provide complementary information in addition 
 to that obtained from the h relaxation . the smaller gyromagnetic 
 ratio of n as compared to h results in sharper n nmr signals that can give information about paramagnetic 
 metal - sites in cases where proton data are unavailable due to extensive 
 line broadening and signal overlap . thus , pres of heavy atoms are 
 particularly valuable in studies of high - spin metalloproteins , where 
 the pres and line broadenings are large and the nmr signals of protons 
 in the vicinity of the metal below , it is demonstrated how structural information can 
 be derived from the relaxation rates and the chemical shifts of n nuclei using nbos to describe the electron spin density . in contrast to the proton pres ( figure 5 ) , the amide n nuclei pres can not be calculated from 
 the metal  ligand orbitals alone , as shown for the rubredoxin 
 model in figure 6a . this is in accordance with 
 previous observations that the relaxation of n nuclei 
 is affected also by unpaired electron spin density in the local 2p 
 orbitals . in general , a very 
 small amount of the total spin density resides in the local n orbitals ( less than 0.03% per amide nitrogen of the rubredoxin 
 complex ) . yet , these small spin densities have substantial contributions 
 to the paramagnetic relaxation because the local spin density is located 
 in the immediate vicinity of the n nucleus . therefore , 
 to analyze n pres of n nuclei in the immediate 
 vicinity of the metal - site , the contribution from the local orbitals 
 must be included . in figure 6b , the effective 
 distances , reff , dft = r1p1/6/ , calculated from the full dft density are 
 compared to effective 
 distances calculated from a density that includes contributions from 
 eight local orbitals as well as from the metal  ligand orbitals . 
 for nonproline amide nitrogens , the eight local nbos are lnbos = { nbo(n  h ) , 
 nbo*(n  h ) , nbo(n  c ) , nbo*(n  c ) , nbo(n  co ) , nbo*(n  co ) , lp(n ) , cr(n)}. 
 the local antibonding nbos , nbo * , are shown in figure 7 . for other nitrogen atoms ( proline , histidine side - chain , 
 etc . ) , the set of eight local orbitals consists of the lone - pair ( lp ) , 
 the core orbital ( cr ) , and the nbo and nbo * to the three neighboring 
 atoms . figure 6b shows a good agreement between 
 the calculated distances , which implies that the pre of n can be derived from a small set of spin densities and basis functions 
 that consists of the metal  ligand pnbos and a few local orbitals . a similar comparison for the plastocyanin model ( figure s3 ) leads to the same conclusion . overall , it is clear 
 from the two widely different model complexes studied here that the 
 unpaired electron spin density , which causes the relaxation of the n nuclei , must include the local n orbitals 
 in addition to the metal  ligand orbitals . n distances , reff , obtained from the paramagnetic relaxation 
 enhancement of n nuclei of the 104-atom rubredoxin complex 
 calculated from ( a ) the fe  s pnbos specified in table 1 and ( b ) the fe  s pnbos and eight local n orbitals . in both cases , the distances are compared to 
 the effective distances , reff , dft , calculated 
 from the full dft spin density ( y - axis ) taken as 
 the  true  density . two experimental paramagnetic quantities can be obtained for each 
 nucleus near a paramagnetic site , the pre , which translates into the 
 effective distance , reff , and the fermi 
 contact shift that reports on the spin density at the position of 
 the nucleus . for protons , it is common to use the experimental relaxation 
 parameters ( r1p , reff ) to derive either geometric distance constraints or information about the electronic structure . for n , the situation is more complicated as shown above 
 because the n paramagnetic relaxation depends on the 
 electron spin density in both the metal  ligand orbitals and 
 the local orbitals . representation of local n orbitals , ( a ) the 
 antibonding 
 n  c natural bond orbital , ( b ) the combined 
 core ( cr ) and lone - pair ( lp ) orbital ( see text ; eq 6 ) , and ( c ) the 
 antibonding n  h natural bond orbital . as shown below , by using the n pre 
 and the n fermi contact shift in combination , information 
 on the geometric 
 structure and geometric ( metal - site)n distances 
 can be obtained . in particular , we seek a single local n orbital , local , and its spin density , local , that together with the pre permit the calculation of 
 ( metal - site)n distance restraints . to that end , 
 we first show that the fermi contact shift can be calculated from 
 the density of the local natural bond orbitals , lnbos , combined with 
 the metal  ligand pnbos . in particular , we show later that all 
 of the local spin density can be projected onto one local orbital 
 that allows distance constraints to be obtained from the pres . in figure 8 , the fermi contact shift for n nuclei of plastocyanin and rubredoxin is calculated using 
 two different spin densities ; : the full dft density ( x - axis ) and the density formed by the lnbos and the metal the dft shifts 
 was obtained from eq 5 by summation over the complete dft spin density . as above , we use the dft shift as the  true  
 experimental shift . the nbo{me  ligand , local } shifts ( y - axis ) were obtained by including only 
 the metal  ligand pnbos and the lnbos . the good correlations 
 of figure 8 show that the small set of orbitals 
 consisting of the lnbos and the metal  ligand pnbos together 
 with their spin densities describe accurately the unpaired electron 
 spin density at the nitrogen position . conversely , the spin density 
 of the local orbitals can be obtained if the fermi contact shift is 
 known experimentally . fermi contact shift for n nuclei of ( a ) the 
 144 atom 
 plastocyanin model complex and ( b ) the 104 atom rubredoxin model complex 
 calculated from the full dft density ( x - axis ) and 
 from the electron spin density formed by the metal  ligand nbos 
 and eight local n orbitals . the excellent correlations 
 show that the experimentally observed fermi contact shifts ( represented 
 by dft , see text ) can be calculated from a small 
 number of orbitals , that is , the metal  ligand nbos and a few 
 local orbitals . previous 
 studies have shown that the natural bond orbitals resemble the chemical 
 bonding environment . also , as shown above 
 and illustrated in figure 6b , figure s3b , and figure 8 , the unpaired 
 electron spin density of a paramagnetic metal - site can be represented 
 accurately by a few ( mainly antibond ) natural bond orbitals . it is 
 therefore tempting to hypothesize that the unpaired electron spin 
 density of the local n orbitals can be described by one 
 antibonding local nbo , local , that relates to the 
 spin - polarization / spin - delocalization pathway from the metal - site 
 to the n nucleus . many of the amide protons of 
 rubredoxin are hydrogen bonded to one of the ligand sulfurs including 
 the amide protons of val8 , cys9 , tyr11 , leu41 , cys42 , and val44 . h ) 
 antibonding orbital is the main carrier of the local unpaired electron 
 spin in these cases . similarly , the amide nitrogen of the ligand - residue 
 cys84 of plastocyanin is only three bonds from the sulfur that carries 
 approximately 40% of the unpaired electron spin density . a very likely 
 spin - polarization pathway is therefore through the -carbon 
 ( cu  s c 
 c n ) in which case the nbo*(n c ) is the main carrier of electron spin density into 
 the local nitrogen orbitals . for some nitrogens , there could be multiple 
 options for the spin - polarization path , for example , his37 n of plastocyanin or the cys42 n of rubredoxin . for his37 
 n , one could imagine a spin - polarization path both 
 through c or through c , while 
 for cys42 n of rubredoxin , the spin - polarization can come 
 from the hydrogen bond or through the c. in all 
 cases , we find that the dominant spin - polarization path is the one 
 that involves the lowest number of bonds between the nucleus and the 
 metal - site , thus , c n for 
 his37 and h n for cys42 . a key property of 
 the local n orbitals , local , is the 
 ratio of p - orbitals ( only relaxation effect ) and s - orbitals ( only 
 fermi contact shift ) . for residues where a distinct spin polarization 
 pathway can be identified , the inherent property of the nbos ensures 
 that the ratio of p - orbitals and s - orbitals is correct . in these cases , the local nbo will  bridge  the fermi contact shift 
 and the local relaxation effect . for n , where no distinct 
  through - bond  spin polarization pathway can be identified , 
 we propose that the unpaired spin density migrates to the n nucleus through its lone - pair ( lp ) , which , in turn , polarizes an 
 s - orbital to give a fermi contact shift . thus , we propose that if no  through - bond  spin polarization 
 pathway can be identified , which will often be the case when the n nuclei that are not hydrogen - bonded to the metal - site or 
 in a ligand residue , the local unpaired electron spin density is given 
 by a mixture of the lone - pair ( lp ; p - orbitals ) and a core - orbital 
 ( cr ; s - orbitals):6where  is a coefficient that describes 
 the mixing between the cr and the lp orbitals . moreover , it is shown 
 in the supporting information that  
 of eq 6 is to a very good approximation 86. this value of  
 is valid for both plastocyanin and rubredoxin that have very different 
 metal - sites , that is , different paramagnetic metal ion , different 
 spin , and different coordination sphere . therefore , we propose that 
 the angle   86 is general applicable for calculating n relaxation of other paramagnetic systems . knowing the local 
 orbital and its spin density allows us to calculate the effective 
 ( metal - site)n distances according to eq s6:7where m is the metal ligand 
 pnbos , local is the local n orbital 
 that accounts for both the fermi contact shift and the pre , while 
 local is the spin - density of the local orbital , 
 and m is the metal  ligand spin - density ; m  1 . we have tested the 
 derived nbo approach by investigating how well the experimentally 
 measured pres and fermi contact shift agree with geometric structure 
 information obtained from the crystal structure of rubredoxin ( 5rxn ) . figure 9 compares the effective distances , reff(r,local,con , exp ) , calculated from the position , r , 
 of the n nuclei relative to the fe s4 site of rubredoxin and the experimentally measured fermi contact 
 shift , con , exp , using eq 7 , with the effective distances , r1p , exp1/6 , obtained from the experimentally measured paramagnetic relaxation 
 enhancements . the fermi contact shifts , con , exp , 
 were those measured previously from n labeled rubredoxin . the local spin densities , local , were derived from the experimental shifts , con , exp , as described in the supporting information . for residues with a through - bond spin - polarization pathway the 
 local orbital , local , was derived from that nbo , 
 while a well - defined mixture of a lone - pair and a core orbital , eq 
 6 , and  = 86 was used for residues without a spin - polarization 
 pathway . the excellent correlation between reff(r,local,con , exp ) and r1p , exp1/6 in figure 9 corresponding to a straight line with slope  ( see eq 1 ) shows that distance information with an rmsd of 0.15 
  can be obtained from n longitudinal pres , provided 
 that the fermi contact shift is obtained . in contrast , the correlation 
 of r1p , exp1/6 with distances obtained from the 
 point - dipole approximation ( the geometric distance , rpd , between n and fe ) is quite low with a 
 pearson s correlation coefficient of r = 0.40 , which clearly demonstrates the improvement achieved 
 by using the scheme detailed above . n distances , reff(r,local,con , exp ) , calculated 
 from the metal  ligand pnbos , the experimentally measured fermi 
 contact shift , con , and the position , r , in the crystal structure . n distances calculated within the 
 point dipole approximation ; that is , the distance is given by the 
 geometric distance between the nitrogen and the iron , rpd , and compared to the longitudinal pres . vertical bars 
 represent uncertainties derived from the uncertainties of the experimentally 
 measured pres . the red ( blue ) dashed line is a best fit line , y = ax , to the reff(rxray,local,con , exp ) ( rpd ) correlations with r1p , exp1/6 . structural information is now directly available 
 from eq 7 via r , because the spin density local ( calculated from con , exp ) and the form 
 of the 
 local orbital local can be obtained for all n nuclei . the allowed positions for a n nucleus 
 are those r that solve eq 7 , with reff6(r,local,con , exp ) = r1p , exp/. this is illustrated in figure 10a for n of pro40 of rubredoxin , where 
 the red surface represents the positions r that solve 
 eq 7 (  = 3000 ms  for high - spin rubredoxin ) . each point 
 on the surface , that is , for each r of reff(r,local,con , exp ) , took approximately 25 s to calculate 
 on a standard linux computer . for comparison the solution of the simple 
 point - dipole approximation , rpd6 = rfe  n = r1p thus the allowed positions , r , obtained from the solution of eq 7 are in excellent agreement with 
 the crystal structure because the surface goes through the n of pro40 . in contrast , positions derived from the point - dipole 
 approximation shown in figure 10b are at variance 
 with the crystal structure , because the surface does not pass through 
 the n. the distance between the  geometric surface  
 and the nitrogen in figure 10b is 0.9  . surface 
 representation of allowed positions for n 
 of pro40 of rubredoxin , ( a ) derived from the experimental fermi contact 
 shift , the local orbital of eq 6 and the experimentally measured longitudinal 
 relaxation rate , r1p , exp , using eq 7 . ( b ) allowed position derived using the point - dipole approximation , 
 rfe n = r1p , exp/. clearly , the surface 
 derived using the nbo approach described above is in excellent agreement 
 with the crystal structure , while the surface derived using the point - dipole 
 approximation is at variance with the crystal structure . all taken together , the unpaired electron spin 
 density of the local 
 orbitals of the n nuclei can to a good approximation 
 be described by one local orbital , local , where 
 the density local is derived from the fermi contact 
 shift . thus , the metal  ligand pnbos together with local and con , exp can be combined with the 
 experimental pre to derive allowed positions of the nitrogen . subsequently , 
 the experimental longitudinal relaxation rate and fermi contact shift 
 can be used as constraints in structure determination of paramagnetic 
 metalloproteins . restrained molecular dynamics simulations 
 has become an important tool for determining allowed conformations 
 of protein structures and for sampling protein dynamics . these simulations 
 and structure optimizations have previously been restrained with experimental 
 data , such as chemical shifts , order parameters , residual dipole couplings , proton pseudocontact shifts and proton pres , and has provided valuable information about protein structure and 
 dynamics . here , we include the pre and the fermi contact shift of 
 the n of rubredoxin as restraints in a molecular dynamics 
 simulation and a structure optimization using the nbo approach described 
 above . as detailed in the supporting information , we generated starting structures that were sufficiently different 
 from the crystal structures to test our approach . these initial structures 
 were generated by a molecular dynamics simulation at 373 k for 10 
 ns , and only the five metal ligand atoms , fes4 , 
 were restrained by a set of distance restraints obtained from the 
 crystal structure . the backbone rmsd of residues 611 and 3944 
 that surround the metal - site to the crystal structure was approximately 
 1.5  for these starting structures . only the rmsd of residues 
 611 and 3944 are considered here , because these are 
 the only residues where paramagnetic nmr data are available . to check 
 the ability of the md force field to generate structures that are 
 in agreement with the crystal structure , we performed a 500 ns simulation 
 at 298 k where only the fes4 atoms were restrained ( figure s5 ) . during the 500 ns , the rmsd of residues 
 611 and 3944 stayed at approximately 1.5  , thus 
 showing that the md force field on its own is not capable of bringing 
 the structure to a state that is in agreement with the crystal structure . consequently , we need to restrain the simulation with experimental 
 data to generate structures that are in agreement with the crystal 
 structure . the paramagnetic nmr parameters of rubredoxin were 
 included using 
 eq 7 to calculate effective ( metal - site)n distances 
 that were converted to distance restraints ( see the supporting information ) . the effective distances were approximated 
 with distance restraints and were updated every 100 ps of the simulation . thus , every 100 ps of the simulation , the metal  ligand pnbos 
 and the local n orbitals were transferred from the initial 
 dft calculation of the model complex to the current structure of the 
 simulation . the effective distance , reff(r,local,con , exp ) , was calculated at three positions along the fe n vector using eq 7 as shown in the supporting 
 information . finally , a distance restraint was created at the 
 distance where the experimental effective distance , ( r1p , exp/ ) , agrees with the 
 calculated ( see figure s6 ) . the structure 
 of the residues around the metal - site approaches 
 that of the crystal structure ( rmsd of residues 611 and 3944 
 is 0.55  0.1  ) , figure 11 , when 
 the molecular dynamics simulation is run at 298 k and is restrained 
 by the paramagnetic restraints . moreover , when the temperature of 
 the simulation is lowered to 50 k , the structures becomes even more 
 similar to the crystal structure ( rmsd of residues 611 and 
 3944 is 0.40  0.03  ; average three runs ) . this 
 shows that a well - defined and accurate structure is obtained when 
 the paramagnetic nmr parameters are included as restraints in simulations 
 that use the nbo approach described above . to test the strength of 
 the nbo approach as compared to the traditional point - dipole approximation , 
 we performed two molecular dynamics simulations where the n pres were included using the point - dipole approximation ( blue squares 
 of figure 9 ) . during the two 10 ns simulations 
 at 298 k , the rmsd of residues 611 and 3944 to the 
 crystal structure was 0.65  0.05  ( figure s7 ) . when the temperature was lowered to 50 k , the 
 rmsd dropped to 0.63  0.02  ( average of two ) . the n pres and fermi contact shifts of residues 611 
 and 3944 were included as geometric restraints in the simulations . ( a ) backbone rmsd calculated for residues 611 and 3944 
 between the crystal structure and the structure during the molecular 
 dynamics simulation . during the 373 k simulation , the rmsd reaches 
 ca . 0.6  within the first few nanoseconds , cooling to 298 k 
 results in a slightly better agreement between the structures and 
 the crystal structure ( rmsd  0.55  ) , while cooling to 
 50 k results in an excellent agreement ( inset ) . ( b ) the structures 
 at 10 , 11 , ... , 19 ns ( green ) overlaid with the crystal structure 
 ( magenta ) . residues where paramagnetic nmr data are available ( residues 
 611 and 3944 ) are shown with bright colors , whereas 
 other residues are shown with pale colors . ( c ) the final structure , at 22 ns , obtained after cooling to 
 50 k and overlaid with the crystal structure ( 5rxn ) of rubredoxin . in summary , including the n pre and 
 fermi contact 
 shift as restraints in a molecular dynamics simulation ensures that 
 the structure samples states that are in accordance with the crystal 
 structure . moreover , the obtained structures are in excellent agreement 
 ( rmsd  0.40  ) with the crystal structures after lowering 
 the temperature of the simulation . restraining the molecular dynamics 
 simulation with the n pres in the point - dipole approximation 
 improves the simulation as compared to the unbiased simulation ; yet , 
 the backbone rmsd of the residues in the vicinity of the metal - site 
 is significantly larger than that of the simulation restrained with 
 the nbo approach . we have shown that the natural bond orbitals 
 ( nbos ) form a simple 
 and natural basis for the unpaired electron spin density of paramagnetic 
 metalloproteins , which marks a new area of the use of nbos to describe 
 electronic structures . the majority of the unpaired electron spin 
 density can be described by a small number of metal ligand 
 nbos that are closely related to the chemical bonding environment 
 of the metal - site . thus , for the model complexes of the blue copper 
 protein plastocyanin and the iron  sulfur protein rubredoxin , 
 about 90% of the unpaired electron spin density is described by the 
 two - center metal the nuclear 
 paramagnetic relaxation rates of protons in these metalloproteins 
 are calculated accurately by the two center metal ligand pnbos , 
 even for protons within a few chemical bonds from the metal - site . this holds , because the paramagnetic relaxation of the protons is 
 affected by the delocalization of the unpaired spin onto the ligand 
 atoms , but is unaffected by unpaired electron spin density in local 
 orbitals . thus , the effect on the proton relaxation caused by the 
 electron spin delocalization can be accounted for by the metal ligand 
 natural bond orbitals alone . the paramagnetic relaxation of n nuclei in metalloproteins 
 is affected also by unpaired electron spin in the local orbitals . we have shown that also the paramagnetic relaxation and fermi contact 
 shift of the n nuclei can be described within the nbo 
 formalism , despite a more complex dependence of the relaxation on 
 the unpaired electron . a local orbital , local , that 
 relates to the spin - polarization / spin - delocalization pathway , together 
 with the metal  ligand pnbos , can account for both the nuclear 
 relaxation and the fermi contact shift of the n nuclei . conversely , the spin density of the local orbital can be determined 
 from the observed fermi contact shift and subsequently used to calculate 
 the contribution to the pre from the local orbitals . our method allows 
 pres of n to be directly interpreted in terms of the 
 position of the nucleus relative to the metal treatment 
 of metal ions and in particular divalent metal ions is 
 quite challenging for molecular dynamics force fields . describing 
 the metal ion in a metalloprotein by a quantum mechanics calculation 
 and the rest of the protein with a molecular mechanics force field 
 ( qm - mm method ) allows simulations of metalloproteins ; however , these 
 simulations are in general slow , and only a few nanoseconds can by 
 obtained in best - case scenarios . the hybrid nbo approach that we developed 
 allows paramagnetic pres and fermi contact shifts of n nuclei to be included as restraints in molecular dynamics simulations . we show that structures that are in good agreement with the crystal 
 structure of rubredoxin are obtained when molecular dynamics simulations 
 of rubredoxin are restrained with paramagnetic nmr parameters using 
 the nbo approach . the nbo approach , which allows paramagnetic nmr 
 data of n nuclei to be used as restraints in molecular 
 dynamics simulations , adds to a growing list of experimental restraints 
 in molecular dynamics simulations that are significantly impacting 
 the utility of simulations to describe dynamics and allowed conformations 
 of proteins . plastocyanin from spinach prepared 
 and purified as described previously was kindly 
 supplied by jens ulstrup and hans e. m. christensen , the technical 
 university of denmark . the nmr experiments on plastocyanin 
 were performed at 298.2 k and a h frequency of 800 mhz 
 using a 1.5 mm fully oxidized spinach plastocyanin sample . the signal 
 eliminating relaxation filter ( serf ) experiment was used for measuring the relaxation rates of the fast 
 relaxing contact shifted signals , as described previously . the sweep width used in the serf experiments 
 was 160 khz , and the specific serf parameters were r1 = 50 s , r1s = 2 s , ta = 90 ms , and c = 1 . all n resonances 
 of oxidized rubredoxin were assigned experimentally by reference to 
 chemically synthesized rubredoxin samples that contain n labeling of only one type of amino acid or one specific amino acid . longitudinal relaxation measurements 
 of the hyperfine - shifted n signals of oxidized rubredoxin 
 were collected on a bruker dmx-500 nmr spectrometer by direct observation 
 of the n spectrum and were 
 recorded at 283 k. the relaxation decays were monitored by an inversion the structure of the 
 fe(sch3)4 model complex shown in figure 1a was constructed from the crystal structure of oxidized c. pasteurianum rubredoxin ( pdb code 4rxn(51 ) ) . the density functional calculations were performed with 
 the gaussian 03 program using the spin - unrestricted 
 b3lyp hybrid density functional approach . the basis set used for all atoms was the 6 - 31 g * basis ( 196 basis functions ) , which was implemented with the following 
 keyword in the gaussian input :  # p ub3lyp/6 - 31 g * scf = tight 
 prop=(efg , epr )  . the positions of the hydrogen atoms were optimized 
 by minimizing the total dft energy , while the positions of the ferric 
 iron and the four sulfur atoms were fixed . transformation matrices from the atomic orbital basis to the pnho , 
 to the pnbo , and to the nbo basis were obtained with the nbo keylist 
  $ nbo archive plot $ end  in the gaussian input . here , 
 the nbo keyword plot  provides the transformation matrices 
 from the atomic orbital basis to the following bases : pnao , nao , pnho , 
 nho , pnbo , nbo , pnlmo , nlmo , and mo . alternatively , the transformation 
 matrices from the atomic orbital basis to the pnho , to the pnbo , and 
 to the nbo basis can be provided by the nbo keylist the structure of the 104-atom model 
 complex was identical to the model complex used previously to study the hyperfine fermi contact shifts and paramagnetic relaxation 
 of oxidized rubredoxin . thus , the model complex was constructed from 
 the x - ray structure of oxidized c. pasteurianum rubredoxin ( pdb code 4rxn(51 ) ) and consisted of the 
 ferric iron and the two hexapeptide chains , cys6-thr7-val8-cys9-gly10-tyr11 
 and cys39-pro40-leu41-cys42-gly43-val44 . the residues thr7 , val8 , 
 and leu41 were converted into glycine residues , formyl groups were 
 added to the n - terminus , while the c - terminal residues tyr11 and val44 
 were truncated to n - methyl groups . the density functional 
 calculations were performed with the gaussian 03 program using the spin - unrestricted b3lyp hybrid density 
 functional approach , which allows the 
 effects of spin polarization and electron correlation to be included 
 for large model systems . the basis set used for all 104 atoms in the 
 model complex was the 6 - 311 g * * basis ( 1380 
 basis functions ) . the structure of the 
 cu(im)2(sch3)s(ch3)2 + model complex shown in figure 1b was constructed from the crystal structure of 
 plastocyanin from poplar leaves ( pdb code 1plc(52 ) ) . the density 
 functional calculations were performed with the gaussian 03 program using the spin - unrestricted b3lyp hybrid density 
 functional approach , and the basis set 
 used for all atoms was the 6 - 31 g * basis ( 286 basis 
 functions ) . the positions of the hydrogen atoms were optimized by 
 minimizing the total dft energy , while the positions of the other 
 atoms were fixed . the 144-atom model complex of 
 plastocyanin was constructed from the crystal structure of plastocyanin 
 from poplar leaves ( pdb code 1plc(52 ) ) . the model consisted 
 of the cupric ion and two peptide chains , pro36-his37-asn38 and tyr83-cys84-ser85-pro86-his87-gln88-gly89-ala90-gly91-met92-val93 . the model was further truncated by substituting p36 and y83 by formyl 
 groups and n38 and v93 by n - methyl groups . although 
 ser85 and gln88 were substituted by alanine residues to reduce computational 
 time , all charge stabilizing atoms were included in the model . the 
 density functional calculations were performed with the program gaussian 
 03 revision c.02 using the spin - unrestricted 
 b3lyp hybrid density functional approach . the basis set used for the sulfur and copper atoms was triple- 
 basis tzvp , while the 6 - 31 g * basis was used for all other atoms ( 1223 basis functions ) . the positions 
 of the hydrogen atoms , the carbonyl oxygen atoms , and the imidazole 
 ring of h37 were optimized in the scf calculation . the nbo analysis 
 was performed as described for the fe(sch3)4 model complex 
 above . the dft calculations on the two protein model complexes were 
 carried out using an apple g5 xserve cluster with 34 processors at 
 the university of copenhagen , and two sgi altix 3300 servers each 
 with 12 intel itanium 2 processors at the university of wisconsin  madison . the effective distances 
 were calculated from the matrix representation of the dipole operator 
 and the fock the spin density matrix in the atomic orbital basis was obtained 
 as a part of the output from the nbo analysis . the matrix representation 
 of the dipole operator in the atomic orbital basis , f2 , was 
 calculated by in - house software . here , the dipole tensor was calculated 
 in the irreducible tensor representation , 2(r ) = ry2(r/r ) and8where i(r ) and j(r ) are atomic orbitals and the integral is over the complete volume 
 ( v ) . the contribution to the integral from the singularity , r  r , was accounted for by a 
 dirac - delta function , as described previously . all md simulations were performed using the gromacs 
 4.5.4 package with standard parameters 
 and the amber99sb - ildn force field . the 
 simulations were initiated from the crystal structure of rubredoxin , 5rxn , where protons were 
 added . the protein was placed in a box ( 150 nm ) of explicit 
 tip3p water , and periodic boundary conditions were applied . the structure 
 was first energy minimized with 1000 steps of steepest decent gradient 
 minimization , and subsequently the water and the proton positions 
 were equilibrated by a 100 ps simulation at 298 k with positional 
 restraints applied to all heavy atoms . for all of the simulations 
 thereafter , the iron and sulfur atoms were restrained to the conformation 
 of the crystal structure with a distance matrix restraint with force 
 constant of 10 kj / mol / nm . first , a 20 
 ns simulations at 373 k was carried out to create structures that 
 are different from the crystal structure and structures on which we 
 could test the performance of the paramagnetic nmr restraints for 
 improving the structure . the total backbone rmsd after 20 ns to the 
 crystal structure was 2.3  , and the backbone rmsd of the residues 
 around the metal - site ( residue 611 and residue 3944 ) 
 was 1.5  . the structures obtained after the 20 ns run serve 
 as the starting point for the restrained molecular dynamics simulations 
 and structure optimization .
a natural bond orbital ( nbo ) analysis of unpaired electron 
 spin 
 density in metalloproteins is presented , which allows a fast and robust 
 calculation of paramagnetic nmr parameters . 
 approximately 90% of the 
 unpaired electron spin density occupies metal  ligand nbos , 
 allowing the majority of the density to be modeled by only a few nbos 
 that reflect the chemical bonding environment . 
 we show that the paramagnetic 
 relaxation rate of protons can be calculated accurately using only 
 the metal  ligand nbos and that these rates are in good agreement 
 with corresponding rates measured experimentally . 
 this holds , in particular , 
 for protons of ligand residues where the point - dipole approximation 
 breaks down . to describe the paramagnetic relaxation of heavy nuclei , 
 also the electron spin density in the local orbitals 
 must be taken 
 into account . 
 geometric distance restraints for 15n can 
 be derived from the paramagnetic relaxation enhancement and the fermi 
 contact shift when local nbos are included in the analysis . 
 thus , 
 the nbo approach allows us to include experimental paramagnetic nmr 
 parameters of 15n nuclei as restraints in a structure optimization 
 protocol . 
 we performed a molecular dynamics simulation and structure 
 determination of oxidized rubredoxin using the experimentally obtained 
 paramagnetic nmr parameters of 15n . 
 the corresponding structures 
 obtained are in good agreement with the crystal structure of rubredoxin . 
 thus , the nbo approach allows an accurate description of the geometric 
 structure and the dynamics of metalloproteins , when nmr parameters 
 are available of nuclei in the immediate vicinity of the metal - site .