Direct Methods for Solving Macromolecular Structures Ed. by S. Fortier Kluwer Academic Publishes, The Netherlands, 1998, pp

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1 Drect Metods for Solvng Macromolecular Structures Ed. by S. Forter Kluwer Academc Publses, Te Neterlands, 998, pp SAYRE EQUATION, TANGENT FORMULA AND SAYTAN FAN HAI-FU Insttute of Pyscs, Cnese Academy of Scences Bejng 00080, P. R. Cna. Notaton V te volume of a unt cell r real-space postonal vector defned wtn a unt cell ρ (r) electron densty functon wt r as argument recprocal-lattce vector, wc corresponds to te dffracton ndex kl F te structure factor wt as argument; te Fourer transform of ρ (r) F sq te Fourer transform of ρ (r) f j te scatterng factor of te j t atom n te unt cell f sq j te scatterng factor of te j t squared atom n te unt cell F te magntude of F ϕ Φ 3 E E σ n te pase of F = ϕ +ϕ +ϕ, te tree-pase structure nvarant te normalsed structure factor correspondng to F te magntude of E = Σ j (Z j ) n, Z j s te atomc number of te j t atom n te unt cell, n s an nteger. Sayre s equaton Te Sayre equaton [] s an exact equaton lnkng structure factors. It olds under te followng condtons: ) postvty; ) atomcty; ) equal-atom structure. Gven a crystal structure represented by ρ (r), we can construct a squared structure expressed as - -

2 ρ (r) = ρ (r) ρ (r). () Accordng to te convoluton teorem, te Fourer transform of () yelds were F sq = F' F V N ' ', () F = f j exp( π r j ). (3) j= Snce F sq s te Fourer transform of ρ (r), accordng to () te squared structure can be determned troug te convoluton of structure factors F. Now f we can fnd te relatonsp between ρ (r) and ρ (r), ten te F sq n () can be converted to F leadng to an equaton lnkng structure factors. If te frst two condtons mentoned above are satsfed,.e. we ave ) ρ (r) 0, ) te electron denstes of dfferent atoms do not overlap, ten ρ (r) and ρ (r) wll ave te same number of maxma (atoms) stuated at te same postons. We can wrte F sq N sq f exp( j ) j j= = π r. (4) If te trd condton s also satsfed, we ave F / F sq = f / f sq and ence F f = FF f sq V ' ' '. (5) Equaton (5) s Sayre s equaton. An mportant outcome of Sayre s paper, and two oter papers publsed alongsde tat of Sayre by Cocran [] and Zacarasen [3], was te relatonsp between te sgns of structure factors n centrosymmetrc case: S S S, (6) were means probably equals. Ts can be seen from (5); f F s a large structure factor ten t s more lkely tan not tat a large product on te rgt and sde wll ave te same sgn (pase). Te probablty for (6) to be true was gven by Woolfson [4] and more generally by Cocran and Woolfson [5] - -

3 σ3 Ps ( ) = + tan σ EE E, ' 3 / ' '. (7) Durng te early days of drect metods, te sgn relatonsps were used n varous ways to solve centrosymmetrc structures or centrosymmetrc projectons of noncentrosymmetrc structures [3, 6, 7]. A very successful tecnque of applyng Sayre s equaton n ab nto pasng s te SAYTAN metod, wc wll be descrbed later n ts paper. On te oter and, Sayre s equaton and ts varatons ave also been successfully used n pase extenson and refnement for a wde varety of structures, from protens to aperodc crystals [8]. It sould be notced tat, wle te tree condtons mentoned above are necessary for dervng te Sayre equaton, tey are not satsfed exactly n practce. It s useful to know wat would appen wen one or more condtons does not old. In teory, any volaton of te tree condtons would lead to te collapse of equaton (5). However even n ts case equaton () s stll vald. Consequently results of applyng Sayre s equaton would tend to ρ (r) rater tan ρ (r). Te problem s: to wat extent wll ρ (r) and ρ (r) resemble eac oter? For example, n neutron dffracton te scatterng factor of some elements s negatve leadng to negatve atoms n te densty functon ρ (r). Wen Sayre s equaton s used wt neutron dffracton data, te result, approxmately ρ (r), wll dffer from ρ (r) manly n tat te negatve atoms are canged to postve ones. In anoter case, f te crystal contans eavy atoms togeter wt lgt atoms, te map resultng from Sayre s equaton wll ave eavy atoms eaver and lgt atoms lgter tan tose n ρ (r). Hence n many cases Sayre s equaton may stll be applcable even wen te tree condtons are not completely fulflled. 3. Cocran s dstrbuton Te tree-pase relatonsp and te probablty dstrbuton of were gven by Cocran [9] were ϕ + ϕ + ϕ 0 (modulo π). (8) Φ 3 = ϕ + ϕ + ϕ ' ', (9) [ πi κ, '] ( κ, ' Φ3) P( Φ ) = ( ) exp cos 3 0, (0) - 3 -

4 κ for non-equal-atom structures and /, ' ' ' = σ 3 σ 3 E E E () / κ, ' = ' ' N E E E () for equal-atom structures. Te dervaton of (0) s based on te central lmt teorem: Gven a set of n ndependent random varables, x, wt means < x > and varances σ, te functon y = n = a x (3) as a probablty dstrbuton tat tends, as n becomes large, to a normal (Gaussan) dstrbuton ( y < y > ) Py ( ) = exp / σy ( π) σ y (4) wt mean < y > = a < x > n = and varance σ y n = aσ. = It as been sown by Ktagorodsk [0] tat, takng cos(π.r j ) as x and takng F as y, for a crystal n space group P contanng more tan 0 atoms n te unt cell, te probablty dstrbuton of F tends, to a good approxmaton, to a Gaussan dstrbuton. Now let te trgonometrc factors cos sn [ π ( ) r ] cos sn ( ) cos sn j π rj [ π ( ') rj ] ', were cos sn means sn or cos, be te ndependent random varables x and te product E- E E - be te functon y. Assumng te probablty dstrbuton of bot te real and te magnary component of y tend to a Gaussan dstrbuton, and assumng te ampltudes E-, E and E - are known, we would fnally obtan Cocran s dstrbuton (0)

5 4. Te tangent formula From equatons (9) and (0), f tere are more tan one par of known pases, ϕ and ϕ, assocated wt te same ϕ, ten te total probablty dstrbuton for ϕ wll be P( ϕ ) = P' ( Φ ) = N exp κ, ' cosφ ' were N s a normalsng factor. 3 3, (5) ' Let and α snβ = κ, sn (ϕ + ϕ ) (6) α cosβ = κ, cos (ϕ + ϕ ) (7) (5) becomes [ I0 ] P( ϕ ) = π ( α) exp[ αcos( ϕ β)]. (8) By maxmsng P(ϕ ) n (8) we ave ϕ = β. Ten from (6) and (7) we obtan wt tanϕ = ' ' κ sn( ϕ + ϕ ), ' ' ' κ cos( ϕ + ϕ ), ' ' ' α = κ ϕ + ϕ κ ϕ ϕ + +, ' sn( ' ' ), ' cos( ' ' ) ' '. (9) (0) ndcatng te relablty of te estmaton of ϕ. Ts s te tangent formula ntroduced by Karle and Hauptman [], wc s te most wdely used formula n drect metods. Te form of te tangent formula gven ere dffers a lttle from, but s equvalent to, tat of te orgnal one. It s easly obtaned from Sayre s equaton a formula smlar to (9) but wt qute dfferent meanng. By splttng equaton (5) nto te real and te magnary parts and by dvdng te magnary part wt te real part, t follows / - 5 -

6 tanϕ = ' ' F F sn( ϕ + ϕ ) ' ' ' F F ' cos( ϕ + ϕ ) ' ' ' '. () Equaton () may be regarded as te angular porton of Sayre s equaton. It dffers from (9) n tat te summaton n () sould nclude all avalable terms, wle tat n (9) may just nclude a few or even only one term of. Besdes, () s an exact equaton, wle (9) gves te most probable value of ϕ. Te tangent formula s muc easer to use for ab nto pasng. A storcal breaktroug on te applcaton of drect metods was made by Karle and Karle [] wen tey solved te non-centrosymmetrc crystal structure of L-argnne dydrate by te symbolc-addton procedure usng te tangent formula. A few years later a systematc procedure to use te tangent formula and a computer program MULTAN (MULtsoluton TANgent-formula metod) [3] were ntroduced by Woolfson and s colleagues. Te development and applcaton of te MULTAN and related procedures led to te domnaton of drect metods n solvng small molecular structures. 4. SAYTAN Te basc dea bend SAYTAN s to use not only te relatonsp among pases but also tat among ampltudes mpled n Sayre s equaton. Te plosopy s tat a good set of pases sould satsfy a system of Sayre equatons. E K = E' E g ' ', (0) were g s te scatterng factor for squared atoms and K s an overall scalng constant, wc allows for te fact tat only structure factors wt large magntude are ncluded on te rgt-and sde. Te dervaton started wt te followng resdual for a system of Sayre equatons: R= g E K E ' E ' '. () As a condton tat R sould be a mnmum t s necessary tat R = 0 for all ϕ - 6 -

7 and ts leads to te Sayre-equaton tangent formula [4] pase of ( g g " g ") E E K E E E. () ϕ = + + " " " ' " ' " " ' As s seen te Sayre-equaton tangent formula ncludes bot trplet and quartet terms. A dstnctve feature of te Sayre-equaton tangent formula s tat t can use te nformaton from Sayre equatons for wc te values of E are small, deally zero. As s well known n powder-metod crystal-structure analyss, a good structure model sould satsfy te weakest reflectons as well as te strongest. Te Sayre-equaton tangent formula tends to develop pase sets, wc satsfy te smallest magntudes as well as te largest. Snce t uses extra nformaton SAYTAN s more effectve tan MULTAN, eter gvng a soluton n fewer trals or gvng a soluton were MULTAN would not. Reference. Sayre, D. (95) Te squarng metod: a new metod for pase determnaton, Acta Cryst. 5, Cocran, W. (95) A relaton between te sgns of structure factors, Acta Cryst. 5, Zacarasen, W. H. (95) A new analytcal metod for solvng complex crystal structures, Acta Cryst. 5, Woolfson, M. M. (954) Te statstcal teory of sgn relatonsps, Acta Cryst. 7, Cocran, W. and Woolfson, M. M. (955) Te teory of sgn relatons between structure factors, Acta Cryst. 8, Woolfson, M. M. (957) An effcent process for solvng crystal structures by sgn relatonsps, Acta Cryst. 0, Grant, D. F., Howells, R. G. and Rogers, D. (957) A metod for te systematc applcaton of sgn relatons, Acta Cryst. 0, Woolfson, M. M. and Fan, H. F. (995) Pyscal and Non-pyscal Metods of Solvng Crystal Structures, Cambrdge Unversty Press, Cambrdge, pp ; Cocran, W. (955) Relatons between te pase of structure factors, Acta Cryst. 8, Ktagorodsk, A. I. (957) Teory of structure analyss, Academc Press, USSR, Moscow. (n Russan). Karle, J. and Hauptman, H. (956) A teory of pase determnaton for te four types of noncentrosymmetrc space groups P, P, 3P, 3P, Acta Cryst. 9, Karle, I. L. and Karle, J. (964) An applcaton of te symbolc addton metod to te structure of L- argnne dydrate, Acta Cryst. 7, German, G. and Woolfson, M. M. (968) On te applcaton of pase relatonsps to complex structures, Acta Cryst. B4, Debaerdemaeker, T., Tate, C. and Woolfson, M. M. (988) On te applcaton of pase relatonsps to complex structures. XXVI. Developments of te Sayre-equaton tangent formula, Acta Cryst. A44,