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Order-Disorder phenomena in myelinated nerve sheaths
d. Mol. Biol. (1990) 215, 373-384
O r d e r - D i s o r d e r P h e n o m e n a in M y e l i n a t e d N e r v e Sheaths I. A Physical Model and Its Parametrization:
Exact and Approximate Determination of the Parameters Vittorio Luzzatit Centre de G~ndtique Moldculaire, Laboratoire Propre du C N R S Associg d l'Universit~ Pierre at Marie Curie 91198 Gif-sur- Yvette, France and Leonardo Mateu Centro de Bioflsica y Bioquimica, I VIC Apdo 21827, Caracas 1020-A, Venezuela (Received 24 November 1989; accepted 8 June 1990) An algorithm is developed for the analysis of the X-ray scattering spectra of lamellar systems, by reference to a precise physical model. The model consists of identical planar lamellae (the motif), all parallel and stacked in a one-dimensional crystal with four types of defect: stacking disorder, finite size of the crystallites, and presence of diffuse and blank scattering. In addition, the spectra are distorted by collimation aberrations. In order to evaluate the effects of these distortions, the following assumptions are made: (1) beyond some point Slimit the intensity curve can be expressed as a function of a (small),number of parameters; (2) the blank scattering, restricted to very small angles, can be identified and eliminated; and (3) the diffuse scattering is entirely defined by the values of idif~(h/D) at the lattice sh = hiD (h is a positive integer _
rod outer segments (ROS) are the best examples. These organelles consist of identical and intrinsically asymmetric membranes, associated in pairs of opposite polarity; the membranes are all (quasi) parallel and the distances between the membranes of each pair and between pairs of membranes are both (quasi) constant throughout the stack. In MyS the membranes are wrapped spirally around the cylindrical nerve fibers, in ROS they form closed disks. Usually, the number of membranes per stack is large, as is the diameter of tile disks in ROS and
(a) The problem In Nature, a few highly ordered membranecontaining systems exist that behave like onedimensional crystals in X-ray and neutronscattering experiments: myelin sheaths (MyS:~) and t Author to whom correspondence should be sent. Abbreviations used: MyS, myelin sheaths; ROS, rod outer segments. 0022-2836/90[ 190373-12 $03.00
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(~ 1990 AcademicPress Limited
374
V. Luzzati and L. Mateu
tim diameter of tim nerve fibers in MyS, so tlmt the diffraction spectra, like those of one-dimensional crystals, may be reduced to a row of fairly shar I) reflections, all equally spaced. The repeat distances arc in the range 100 to 400 A (1 A = 0-1 nm); reflections up to the 15th order are commonly observed. Extensive studies of the structure of MyS and ROS lmve been carried out using X-ray and neutronscattering teclmiques (see Yaeger et al., 1980; Nelander & Blaurock, 1978; Kirsclmer el al., 1984; Worthington, 1986, and references therein). Besides, MyS and ROS are such excellent scatterers that kinetic experiments have been performed and the structure correlated with physiologically significant I)arameters, especially since position sensitive detectors and high-flux neutron and X-ray sources have come into use. In tile case of ROS, tim effects of light (Chabre & Cavaggioni, 1975) and in tile case of MyS, tile effects of action potential proi)agation (Mor,'in & Mateu, 1983), of anesthesia (Mateu & Mor~in, 1986) and of exI)erimental allergic neuritis (Vonasek et al., 1987) Imvc been studied. Tile one-dimensional order, though, is not perfect: tile reflections are not quite, nor equally, sharp and some background scattering is always present in tim spectra. In order to rid the data of noise and distortions one must distinguish reflections from background, dissociate the overlapping reflections and integrate the intensity of each reflection; these steps usually involve, to some degree, ill-defined operations, especially ira the regions of tile spectra in which tile signal-to-noise ratio is low (Yaeger, 1975; Nelan(ler & Blaurock, 1978). This problem is even more serious for kinetic experiments: tim analysis, ira tiffs case, hinges upon small differences between spectra recorded in tile shortest possible time. A significant improvement is achieved by analyzing tim spectra i)y reference to a precise i)hysical model. In tile ease of myelin, previous authors ( s e e Blaurock & Nelander, 1976; Worthington, 1986; Inouye el al., 1989, art(1 referenccs therein) assumed tlmt tim real structure is derived from an ideal one-dimensional crystal via six types Of distortion: erystallite size, namely the finite number of lamcllae present in each coherent stack; stacking (or lattice) disorder, i.e. variations of tile repeat distance; positional disorder, i.e. variations in the separation of tim two membranes comprising the elementary pair; blank scattering, due to tile instrument and to chemical impurities in the samI)le; diffuse scattering, related to structural imperfections within the lamellar system; collimation aberrations, due to tim finite width of tile incident beam. From a practical standpoint, traditional analyses of tile X-ray spectra involve, as a rule, tim following steps. Firstly, identify the noise urtderlying tim signal; usually the lower envelope of the intensity curve, drawn by baud, is ascribed to diffuse scattering (see, for examI)le, Fig. l0 of Nelander & Blaurock, 1978). Secondly, subtract the noise from tile signal, dissociate (if required) tile overlapping reflections and draw tile intensity curve associated
with each lattice point. Thirdly, integrate tile intensity of each reflection. Fourtldy, ascribe tile integrated intensities to tile reflections of an ideal onedimensional crystal, search for tile signs of tile corresponding structure factors and determine tile electron density profiles. In addition, in a few cases tile shape of the reflections has been analyzed in terms of a model involving crystallite size and stacking disorder, and the values of tile parameters used to specify tile model have been determined (Blaurock & Nelander, 1978; Inouye el al., 1989). Tile procedures used for those purposes call for a number of comments. Some of tile operations, for example, hand drawing of tile curve corrcsi)onding to tile "noise" (that curve plays a critical role in the analysis of disorder) and separation of overlapping reflections, are neither defined in quite objective terms nor totally exempt from groping. Moreover, the goals are not always consistent with the model setting the framework for tim analysis; for example, the very notion of " m o t i f " (and of its structure factor and electron density profile) is problematic ira the presence of positional disorder (Worthington, 1986); also, the operation of integrating tim intensit), deserves some theoretical justification. Finally, the results are often presented as a faithful representation of tile system, overlooking tim fact that tim analysis hinges upon models whose relevance may well be open to question. The awareness of these limitations, and also tim urgent need to analyze with tractable speed and sufficient accuracy the large wealth of data recorded in myelin X-ray scattering experiments performed with position-sensitive detectors (by contrast with photographic films, used in most previous studies of myelin) and under a variety of physiological and pathological conditions (PadrSn el al., 1980; PadrSn & Mateu, 1982; Mor,4n & Mateu, 1983; Mateu & Mor,-in, 1986; Vonasek el al., 1987) prompted us to try to overcome some of the drawbacks and to devise a routine involving clearly and objectively defined operations, which are easy to implement with a computer. The approach is concei)tually the same as the one that we adopted in the mathematical treatment of solution scattering experiments and used in tile analysis of the information content of solution and crystal scattering studies (Taupin & Luzzati, 1982; Luzzati & Taui)in, 1984, 1986a,b; see also .Moore, 1980). That i)roccdure involves the following steps. (1) Introduce a model and Sl)ccify tim degrees of freedom of the I)roblem (i.e. define the mathematical parameters that are necessary and sufficient to take into account tile I)hysical phenomenon) by reference to that model. (2) Express the exI)erimental observations (usually counts at a finite number of clmnnels) as functions of the degrees of freedom. (3) Determine the values of the degrees of freedom, whenever their numl)er is smaller than or equal to tim number of tim independent experimental observations. (4) Analyze the errors.
Order-Disorder in Myelinated Nerve Sheaths. I
(b) The model and the algorithm We tackle in this paper steps (1), (2) and (3), above. The model is defined by introducing the following distortions into an ideal one-dimensional crystal: crystallite size, each coherent stack contains a finite number of motifs; stacking disorder, we assume t h a t the motifs, all identical, are stacked with variable spacing; blank scattering, due to the instrument and to chemical impurities in the sample, and limited to very small angles; diffnse scattering, which we parametrize via the assumption that its Fourier transform has the finite support - - D / 2 < r < D / 2 ; collimation aberrations, due to the finite width of the incident beam. The model is similar to other models used previously (Schwartz et al., 1975; B l a u r o c k & Nelander, 1976; Worthington, 1986; Inouye el al., 1989). The main differences bear on t h e parametrization of diffuse scattering and on positional disorder. For the sake of simplicity, we develop the mathematical treatment described in the following section, disregarding positional disorder; we later take it into account (see accompanying paper, Mateu et al., 1990) with the restriction t h a t positional and stacking disorder are independent of each other. This is an essential precaution if the structural notion of "motif '" is to be preserved. Tile degrees of freedom of the model are the parameters specifying the lattice and its disorder: namely, tile average repeat distance D and its vari2 ante ao, the average number of motifs per cr 3 stallife ( N ) , and the sets {i~,(h/D)} and {imot~r(k/2D)}, which define, respectively, the continuous diffnse scattering and the intensity curves of one motif. A few additional parameters are involved in the definition of the asymptotic trend, of the blank seattering antl of the angular spread of the sheaths' orientation (see section 2 of tile accompanying paper). We elaborate a simple and fast, but not strictly rigorous, algorithm that operates on the raw d a t a and yields the values of the whole set of parameters with a minimum of intermediate manipulations. More precisely, the manual interventions are restricted to the determination of the asymptotic trend of tile experimental curves and of tile repeat distance D, and to the elimination of the blank scattering. The main virtues of the algorithm are to eliminate m a n y of the intermediate, and questionable, operations (hand-drawing of the diffuse scattering curve, discrimination of the overlapping reflections, integration of the intensities) to yield at once the value of all tile parameters specifying structure and disorder, and especially to determine the continuous intensity fnnction corresponding to tile motif. Naturally, tile validity of our approach (and of all previous approaches to the problem) is limited by the validity of the model. The model will be shown to be consistent with all the experimental observations; the question of its relevance to the particular
375
system under investigation is more difficult to answer. One way to approach tile problem would be to carry out a Z2 test; this involves elaborate operations that we leave for the future (see section l(a), step (4)). (c) Layout of the two papers The present and the accompanying paper (hereinafter called papers I and II, respectively) apply to the ideal case of perfectly oriented fibers. The experimental problems are discussed in paper II, section 2(a): angular spread of the sheaths' orientation; collimation aberrations including electronic distortions and drifts; determination of the average repeat distance D; elimination of the blank scattering surrounding the incident beam; determination of the asymptotic trend of the intensity curve; operational definition, from s = 0 to infinity, of the experimental curve i*r involved in all the subsequent calculations. In paper I we introduce the model that sets the framework for the mathematical analysis and we derive an algorithm, involving straightforward operations of tile experimental points i*xp(si), which leads to tile determination of all tile parameters specifying tile structure and tile disorder. An additional parameter, introduced in paper II, is the fraction a~.... of myelin lamellae that are not tightly packed in the sheaths. Once the complete intensity curve of tile motif is known, and since the structure is centrosymmetrie, it should in principle be possible to determine the electron density profile. The presence of positional disorder complicates tile problem. This point is discussed in paper II, section 2(d). The very notion of electron density profile is shown to become problematic when positional disorder is present; two tests are proposed to select the "best" map. The algorithm, whose derivation involves a few aI)proximations, is tested in section 3, point (4) using simulated experimental curves. In paper II, the algorithm is applied to X-ray scattering experiments performed on rat sciatic nerves, both native and swollen in non-isotonic solvents, and also as a function of the age of the animal (from 4 days old to adult). All the results are consistent with the model, which sets the framework of the mathematical analysis. The swelling experiments show that in native myelin the stacking disorder involves the cytoplasmic space, the external space in swollen myelin. The analysis of myelogenesis shows t h a t the number of membranes per crystallite varies dramatically with the age of the animal but that the structure of the individual membranes of the sheaths is practically invariant. 2. M a t h e m a t i c a l M o d e l a n d Analysis o f the D a t a
(a) Notation r, position in real space (in A). s, position in reciprocal space; s = (2 sin 0)/2, 20 is
the scattering angle, ). tile wavelength (in A).
V. Luzzati and L. Mateu
376
D, a~, mean repeat in real space and its variance. v = 1/(N), where ( N ) is the average number of lamellae per crystallite. W, positive integer specifying the repeat WD of a virtual periodic function (see section (b)(iv)). n, h (and k), integers specifying positions in real or in reciprocal space, in units WD and (WD)-t:
r, = nWD, sh =h(WD) -1.
I
Pdl(r)/~ f I I
X, specifies position in real space, in units WD. FT, operator expressing Fourier transformation. *, operator expressing convolution, namely:
f(z) * g(x) = ~f(y) g(x--y) dy. p(r) and i(s), (with various sub- and superscripts) a pair of reciprocal functions: p(r)FTi(s). 5(r-R), Dirae delta function, centered at r = R. f(x, ylA, B . . . ) , ad hoc expression of a function f(x, y, A, B . . . ) in which x, y play the role of variables, A, B . . . t h a t of parameters. fl(r--RlL), step function of width L, centered at the point R:
fl(r--RIL) = 1, if - L / 2 < _ r - R < _ L / 2 fl(r--R[L) = 0 otherwise.
(1)
Theorem 1, [p:(r) * pb(r)] FT [i:(s) ib(s)].
(2)
Theorem 2,
[A
]
]~ ngn (r-nO)
AAA,i, AAA
g4(r+40) 9~(r+20) g o ( r ) gz(t-20) g4(r-4O) gs(r+SO)gl(r+O) gl(r--O) gS(r-SO) i
-40-30-29-'0
0
i~ 2'0 i9 ;O r
Figure 1. Generation of the function P*od.l(r) = pi.c(r) [Pdiff(r) + Pmotu(r)*Pli..,.t.o.(r)] (eqn (17a)), assuming that the incident beam is infinitely narrow (namely pi.r v = 0 and ao/D=00495. Note that the support of the function Pmo.r(r) is - D < r < D, that of Pdlff(r) is --D]2 < r < D]2. The lower curve represents the function ~.g,,(r-nD) (eqn (23b)), which, in this case, takes the form of a set of Gaussian functions, go(r), represented by a vertical arrow, is a delta function centered at r = 0.
istep(s]L, R), Fourier transform of pstep(r]L, R):
Theorem 3,
i(S--So) FT [p(r) exp ( - 2~irso)].
(4)
P(rID), periodic function, of repeat D and motif Prnotif(r):
P(rID) = ~
Pmotif(r--nD)
I
+L/2
[cos 2n(r+R)s
J - L/:Z + cos 2rr(r-- R)s] dr/2 = cos 2rcRs sin (uLs)/us.
,0, I(s]D), Fourier transform of P(r[D); it consists (see eqns (2) and (3)) of a set of delta functions of position h]D and content (l[D)i(h]D): (6)
Support, of a function g(r) is the range of r outside of which 9(r) is identically zero. According to Shannon's (1949) theorem, if - L / 2 <_r< L/2 is the support of g(r), then its Fourier transform ~(s) is entirely defined by the values it takes at the points
sh = h/I,:
(b) The model of the disordered structure We refer to a one-dimensional crystal of repeat D: the electron density and the structure factor of its motif are Pmotif(r) and F(s), Pmotif(r) and imotif(8) are the autocorrelation and the intensity functions of one isolated motif. The functions Pcry(r[D) and Ic,r(s]D) corresponding to the ideal crystal take the form (see eqns (5) and (6) and Fig. 1):
Pc'Y(rlD) = Pm~
* [,,=~-oo6(r--riD) 1'
I.y(4D) = {imo.f(s)(l/D)6(s-h/D)}.
V(s)=L ~. v(h/L)sin[r~(Ls-h)]/n(Ls--h).
(9)
INT(x), integer part of a real number x.
?1= - o o
I(slD) = {i(s)(l/D)5(s-h/D)}.
istcp(s[L, R) =
(7)
h= -~o
pst=p(r[L, R), pair of (half) step functions (eqn (1)) centered at the points R and - R :
p~t~,(r[L, R) = (l/2)[p(r--R[l.,) + p(r+ R[L)]. (8)
(1O) (11)
Thus, the function I.y(s) is entirely defined by the set {i,..otif(h[D)}. The support of the electron density map Pmo.f(r) is --D]2 < r < D]2; the support of the autocorrelation function Pmotif(r) is twice as large, - - D < r < D . Therefore, the function Pmotif(r) is entirely defined by the set {F(h]D)}; on the other
Order-Disorder in Myelinated Nerve Sheaths. I hand, the functions Pmotif(r) and imotif(S) (and also Pmotif(r) when it is centrosymmetric; Hosemann & Bagchi, 1962) are entirely defined by the set {imotif(k/2D)}, whose number of elements is twice the number of reflections of the crystal. As discussed in section 1, above, and in agreement with previous workers (Blaurock & Nelander, 1976, and references therein), we assume that the real structure derives from the ideal crystal via five types of distortion: finite crystallite size, stacking disorder, blank scattering, diffuse scattering, collimation aberrations. The finite size of the crystallites has the effect of multiplying each term of equation (10) by a function psiz~(n[v), which has the property (Blaurock & Nelander, 1976): [Psizc(n]V)]nwO "-~ Psize(0) (1 -- [n[v).
(12a)
For the sake of convenience and by way of approximation we adopt the form: pslz~(n[v) = exp (-Inlv)
(12b)
(setting Psiz~(O)= 1 has no consequence, since the analysis of this section is independent of scale) (we anticipate, see paper II, section 2(b)(iii), that a constant term will be added at the origin of p~i~(n) to take into account the presence of a loosely packed membrane component, called loose myelin). With regard to stacking disorder we adopt a random walk model (Blaurock & Nelander, 1976): namely, we assume that the motifs are all identical, that the distance from one motif to the next fluctuates around the average value D and that these fluctuations are uncorrelated. Under these conditions, and applying the central limit approximation, each delta flmction of equation (10) is replaced by a Gaussian function centered at point nD:
l(r--nnln, a~) ,.~ (2nln[a~)-! x exp [-- (r-- nD)Z/(2lnla~)]. Note that this model involves the assumption:
(13)
following
Assumption (1): over the bordering region D / 2 > r > D / 2 - e , with'~ of the order of aD, the electron density is constant and equal to the average electron density over the unit cell. (This condition implies that /motif(0)=0; in fact theory shows and simulations confirm that the final result is barely sensitive on the precise values of /motlf(0) and of p(D]2), at least as long as >> 1 and av<
= ~
p~i~o(nlv)l(r-nDln, a.),
]1~ --o0
isi~r ~,~ck(slD, av, v)
= ~
psi~(n]v)2(sln, aD)exp(--2~inDs)
(14a)
= ~ n=
377
exp[-[n[(v+2n2a~s2)--2ninDs],
(14b)
--O0
2(sln, a9) = exp (-2n2[nla~s :) FT l(r[n, at)). (14e) Equation (14b) represents a geometric progression; a trivial transformation yields a more condensed expression (Guinier, 1964): isi.... tack(S) = (1 --e-2a)/(1 +e-2a--2 e -~ cos b), (15a) a = v-k27t2r 2, (15b)
b = 2~zDs.
(15c)
The convolution of equation (14a) by Pmotif(r), equivalent, in reciprocal space, to the multiplication of equation (14b) by imofif(8), takes account of the electron density distribution of the motif. Background scattering, arising from air and windows, electronic noise, sample constituents that do not belong to the quasi-crystalline component, and structural disorder of myelin, is also present and must be taken into account. It is convenient to decompose the background scattering into two components: /back(S) = /blank(S) -{- idiff(8 ).
(16)
The diffuse scattering idiff(8) is a smooth function that, according to a widespread (and loosely defined) procedure, is drawn by hand to fit the presumed minima of the spectra. The criterion of smoothness implicitly underlying this procedure can be expressed more formally (and more rigorously) by the assumption that the support of Pdiff(r) is --D/2 <__r <_1)/2. This assumption entails the interpolation rule expressed by equation (7), therefore, the entire c u r v e idiff(8) can be expressed as a function of the set {iditf(h/D)}h. The term iblank(S) is aimed at removing the sharper bump centered at the origin,, whose shape makes exception to the interpolation rule, and also at extrapolating the data to the origin (see Fig. 2 of paper II). This decomposition must be viewed as an empirical procedure; the possible sources of background scattering are indeed so multifarious that it would be hazardous to try to correlate the terms of equation (16) with precise physical properties of the sample. In reciprocal space the effect of collimation aberrations is to convolute the whole of the intensity curve, with all the distortions discussed above, by the intensity distribution of the incident beam, iinc(s); the real space counterpart is the multiplication by Pine(r). Thefinal effect of these distortions is expressed by the following equations (see eqns (14) and (15)): p*modet(r) = Pmodel(r)--Pbhnk(r) = Pinc(r)[Pdiff(r) "{-Pmofif(r) * Psi .... tack(r)],
(17a)
i'model(S) = imodel(S) -- iblank(S) = iinc(S ) * [idiff(8)+iraotif(S)isize, stack(S)]. (17b)
Note that equations (17) are functions of the parameters D, at), v, {imot,f(k/2D)}, {idir,(h/D)}.
V. Luzzati and L. Mateu
378
and to infinity are discussed in paper II, section 2(b)(ii). We also assume t h a t a few of tim reflections stand out sharp and strong, so that the mean repeat (D) can be determined: we thus remove D from the unknowns. We then proceed to elaborate an algorithm according to the following procedure.
W=I
X=3/2
AAAt - 4D
- 319 - 2D
-D
tAAA .....A . . . .
(r
N
.____/!/
D
2D
3D
4D 9
.
nD
Figure 2. Generation of the functions Pmodd.st,p(r) * (eqn (24)) for W= 1 and ( X + t ) integer. The drawings represent the functions ~ , g d r - n D ) and (hatched) pstcp(r): the product of the 2 functions is also hatched. The motif of the periodic fimction I model,step(r) is represented on the right side of the Figure; the repeat is D. )*
(c) Deleiminations of the parameters of the model Tile result of an experiment is the measure of the intensity of the incident and of the scattered beams, ii,r and i=~p(j), respectively, at a finite n u m b e r of channels j. As discussed above, the blank scattering must be removed from the experimental curve (see eqn (17b)): i*exp(Sj) = iexp(Sj) -- iblank(Sj). (18) We discuss this operation in paper II, section 2(b)(iii)..We suppose t h a t tile incident beam is symmetric and regular and that its sampling is sufficiently dense so that tile function ii~r ) can be interpolated at any point s. Besides, as often happens in tile analysis of scattering phenomena (Taupin & Luzzati, 1982), the mathematical problems are not manageable unless the intensity scattered beyond some point Slimit can be expressed as a fimction of a finite (and small) number of parameters, M. We also assume t h a t beyond 81imit the flmction imotif(s) is negligibly small. I f the conditions above are all fulfilled, then the data consist of the values of the intensity i*._(s,) at a finite set of J k' J points si and of the curve h,r (see section 3). To each clement of the set {i*.p(Sj)}, one equation (17b) corresponds: i*~p(Sj) =
f_
P=,p, step(riD, XD) = [p**p(r)p~t,p(rlD, XD)] * L=~_oo 5 ( r - - n D ) ]
(20)
The reciprocal space counterpart is (see cqn (9)):
I*~p, step(BID, XD) = {I*~p(h/D]D, XD)(I/D)5(s-hlD)},
(21a)
where:
I%(h/DID, XD) = F, Asjq(sj~h, D, X)i%(sA, (21b) .l=l Asi = sj+ 1 --sp
(21c)
q(s, h, D, X) = i~t~p(h/D-- s) + i~t~p(h/D+ s) = D[cos 27:X(h- Ds) sin n(h- Ds)/n(h- Ds) + cos 2uX(h + Ds) sin u(h + Ds)/n(h + Ds)]. (21d) Thirdly, we carry out the operations above on the function Pmodct(r) * (see eqn (17a)). The result is a periodic function of repeat D. Its motif (see Fig. 1):
P*od~L~t=p(rlD, XD, aD, v) Pmodr v)p~tcp(r]D, XD) I
--
$
-
(22a)
is a fimction of r, whose support, like t h a t of
pstcp(r[XD, D), is split into two halves, D ( - X - - 1/2) <_r <_D ( - - X + 1/2) and D ( X - 1/2) <_r <_D(X + 1/2).
[imo.f(s)i~i .... t~k(S) c~
+idiff(s)liinc(Sj--s)ds
(i) Construction of a virtual periodic function Operating in real space, we firstly cut out of the flmction p*xp(r) a pair of sections of width D, centered at the points + X D (Fig. 2); this operation is formally equivalent to multiplying pc*xp(r) by the function Pstep(rlD, XD) (see eqn (8)). The result, expressed ill space s, is the convolution of i**p(S) by istcp(slD, XD) (see eqn (9)). Secondly, we construct a periodic fimction P*w.st=p(rlD, XD) whose repeat is D and whose motif is tile product pc*xp(r) Pstcp(rlD, XD):
(19)
of tile unknowns D, go, v, {imotif(l['/2D)}, {idiff(h/D)}. Tile munbers of elements of tile sets {imotif(k/2D)} and {iditf(h/D)} are, respectively, 2lNT(sjlmitD ) and INT(811mltD); thus the total number of unknowns is 3INT(SlimitD ) + 3 + ~1/. I f this number is smaller than the number J of experimental points, then the most probable values of the unknowns and the associated error matrix can be determined (Taui)in & Luzzati, 1982). We leave for the future a rigorous mathematical treatment of the problem; we devise below a simple, but approximate, solution, which we illustrate in paper II, section 3. 9 We suppose that tile fimction i*exp(Sj) is known from s = 0 to infinity (the extrapolations to s = 0
The convolution by the lattice brings the two halves on top of each other and adds them. I t is thus possible to use an equivalent expression (see eqn (10)):
Pmodel. step(r[D, XD, ao, v) = {(l/2)[P*modr
v)+P*odc,(--rla.,
x fl(r--XDID)}*
s tl ~
The definition of Pmodr * (17a):
.
Pmod,t(r]aD, V)= pi~r
5(r-riD). -
v)] (22b)
oo
v) is given in equation
{Pdit-f(r) +/)motif(r)
Order-Disorder in Myelinated Nerve Sheaths. I (ii) Simplifying assumptions We introduce the following simplifications and approximations.
379
When X + I is an integer, and ouly in this case (see Fig. 2), tile same terms gn(r--nD) are involved ill tile two sides of equation (25) and tile inequality becomes a quasi-equality:
Assumption (2): tile incident beam is sufficiently narrow in reciprocal space so t h a t tim fimction Pine(r) can be assumed to be constant and equal to plne(XD) over tile range D(X- 89 <_r < (X +89 Assumption (3): tile flmctions l(rln, ao) are narrow: more precisely, na~ <
P*odr step(rlD, XD, at), v)
In this case, and for r > D (namely, beyond tlm support of pdlff(r)) tim expression of Pmod,l(rlcrD, V) (eqn (22e)) can be simplified:
Tile reciprocal space counterpart of equation (26a) is (see cqns (14), (21a) and (23b):
{P*odel(rlao, v)}~u~Pmotif(r) * ~ n=
g.(r--nD),
~oo
(23a)
gn(r-- nD) = pinr
exp (-- nv)(2nlnia2 ) -89 xexp[-(r-nD)2[(2lnla~)].
(23b)
A set of functions gn(r--nD) is represented in Figure 1. The multiplication [P*odel(rlaD, v)p~tcp(rlD, XD)] and tile subsequent convolution with tile lattice generate tile periodic function (see eqn (22) and Fig. 2):
Pmotif(r) * {(89189 + gx+ t( r - [X + 89
9 ~ n=
~(r--nD).
(26a)
-oo
{l*oael(h/nlD, XD, no, V)}x~ x .~ imotif(h/D)H2(h/D]D, X, a D, v), (26b) H2(sID, X, no, v) = (89
X - k , no, v)
+ HI(slD, X +89 no, v)], = HI(slD, X, no, v)cosh (v/2 + r?a2os2), (26e) Ha(slD, X, Up, V) F T g~(z) = pin~(XD)exp[--X(v+2t~za~s2)],
(26d)
z = r - XD.
(26e)
=
Equations (26) apply to tile points X _> I. At tim point X = 89 and normalizing Pine(0)----I, equation (26b) takes tile form (see Figs 1 and 2):
Prouder, step(rlD, XD, at), v) ~{[Pmot,f(r'*n=~_~oyn(r--nD']Pstep(rlXD, D) }
l*odel(h[DlD, D/2, up, v) .,~ (2){Zmotlf)(h[D) x [1 +Hx(h/DID, 1, no, v)] 9
9 9
+idlff(h/D)}.
The support of Pmotlf(r) is --D<_r<_D and each gn(r-nD) is an even and narrow function of r-riD: as a consequence only tim terms [g.(r--nD)]:_ 1<_.~x+1 contribute to the function
[P motif(r) * n~=~ogn(r)] ' over tim range (X--k)D<_r<(X+ 89 On the other hand, and over tl)e same range of r, only tile terms [gn(r--nD)]x_89189 are involved in tim multiplication
[pstep(rlXD, D)n=~r162g.(r)]. As a rule, therefore, the sequence of operations of equation (24) cannot be inverted.
(27)
(iii) Partial solution of the problem A comparison of the pair of functions I*odet(h/DlD, XD, up, v) (eqn (26)) and I*,~,(h/DID, XD) (eqn (21)) may yield an approximate and a partial solution of tim problem. For that purpose we apply tile following equation:
{l*,p(h/DID, XD)}xzx = imotif(h/D)H2(h/DID, X, up, v) (28) at the points X = 3/2, 5/2 . . . . . and thus determine the values of ao, v and {imotle(h/D)} using the algorithms discussed in Appendix. We subsequently determine the values of {iditt(h[D)} using equation (27):
id~rt(h[n) = 21*,p(h[nln, D[2) --imotif(h/D)[1 +HI(h/DID, 1, no, v)]. It may also be noted that for X = 3/2, 5/2 . . . . ratio of tile fimctions I*,,p(hl/DlD, XD) relevant pair of lattice points h I and hz (see eqns (28)
(29) tim to a and
(26d)): 9 ~
I*~p(hx/DID, XD)[I*p(hz/DID, XD) = [im~176 2 x exp [-2rc2Xa~(h~-h2)l D ] x [cosh (v[2 + rt2a~h~/DZ)[ cosh (v/2 + rc2aZhZ,In2)].
6(r-riD) Cp~o~idr) 9
9 n=-D~6(r-nD
I} .
(25)
(30)
is also a fimction of tile parameters o-D and {i~otif(h/D)}h, barely dependent on v if v<< 1.
V. Luzzati and L. Mateu
380
aOx(r-2nO)
aox(, -tz~+~lo)
I**(k] WDI WD, X WD)
t
= ~ As~q(sj, k, WD, X)ie**p(si). (32c) j=t
1
The value of the function I**(k/WD]WD, XWD) can be computed a t any point {k, WD, X}. F o r ' t h e sake of simplicity, we restrict the explicit t r e a t m e n t to the case W = 2 ; a generalization to W > 2 is straightforward. Using the same arguments as in section (c){ii), above, we single out the values of X for which the edges of the function pstep(r[WD, XWD) coincide with the lattice points r.=nD, namely X = 0 , 1/2, 1,3/2 . . . . . As illustrated in Figure 3, the m o t i f of the periodic function PmoaeL**step(rl2D, X2D, a., v) consists of two functions, GOx(r) and GDx(r), centered, respectively, at r = 0 and r = D :
I
W=2 X=I
AAAt !,AAA ...... ~
A i
gz (r- 2 ~ t
-[~ (r-Iv,+ I Iv)+g3 t~-Iz,,Hlo)}/2
1I I
1I
AA-A/ !AAA Ai W=2 x =3 / 2
I
{gzlr-~.nOl+g3[r --ZnO)}/2
-4D-SO-20
--'O
O
~ / -
2D ~O 4D
r
. . . . -/ 7 I /
Pmotif(r) * {[GOx(r) + GDx(r + D)]
2aO (?-n§
Figure 3. The same as Fig. 2, for W = 2 and 2X integer. In this case the repeat of the periodic function is 2D, its motif consists of 2 peaks, one [GO(r)[ centered at the origin r = 0, the other [GD(r-D)] centered at the point r = D; the 2 peaks are linear combinations of the functions g~(r) (see eqn (34)).
Therefore, the analysis of the ratio [I*,p(h~/DID, XD)/I*,p(h2/DID, XD)]
the incident beam is not known. (iv) Completing the solution 9Once the values of the parameters {imotir(h]D)}, {io~fr(h/D)}, aD and v are known, the elements of the set {imotir(k/2D)} with k odd are still required for the full determination of the function imotif(s). A generalization of the same procedure may yield the value of the missing parameters. Since the function idlft(s) is entirely defined by the set {ioiff(h/D)} (see eqn (7)) its contribution to both leap(S) and p,,p(S) can be eliminated: "** ) = iexp(S ) -- iblank(8 ) -- idiff(8), Zexp(S
(31a)
Pexp(r) = pexp(r)--Pbtank(r)--Pdiff(r).
(31b)
B y analogy with section (e)(i), above, we generate a virtual periodic function whose repeat is WD (W is a positive integer), whose motif is the product of p**(r) by a pair of step functions centered at the points +_XWD and of width WD (see Fig. 3 and eqns (20), (21) and (24)):
9 ~ n=
6(r-nWD),
IVD)] (32a)
~oo
Ie,p. step(k/WD[ WD, X WD)
n=
(33)
-oo
GOx(r) and GDx(r) are linear combinations of the functions g,(r) (eqn (23b)), which, according to the parity of X, take the form:
GOx(r) = (cos~X)2g2x(r) +(sinuX)2[g2x+x(r)+g2x_t(r)]/2, GDx(r ) = (sin zcX)2g2x(r) 9 +(cosxX)2[g2x+l(r)+gzx_l(r)]]2.
(34a) (34b)
The F T of P~odcl, ~tep(r[2D, X2D, ao, v) yields the expression of I**odel.st,p(k/2D[2D,X2D, aD, v) (see eqn (26d)):
l**oael(k[2Dl2D, X2D) ,~ imotif(k/2DlX)H3(k]2DlD, X, ao, v) .~. I*~(k/2D[2D, X2D), H3(k]2DID, X, a D, v) = cos 2ukX{HI(k/2D[D, 2X, aD, V) + 89COSnk[H l (k/2DID, 2X + l, aD, V) +II~(k/2DID, 12X-- II, aD, v)]}.
(32b)
(35a)
(355)
Thus, one value of imotif(k]2DlX ) can be determined at each point {k,X}x=o,89 a.... :
i~ot,f(k]2D[X) = I*~(k]2Dl2D, X2D)[ H3(k]2D[D, X, aD, V),
(36)
and the value of imotif(k]2D) determined b y averaging the points imotif(k[2DIX) over X (see Appendix).
3. Verifications and Practical Considerations The application of the algorithms to myelin data is described in paper II (Mateu et al., 1990) and will be documented in other forthcoming papers. Our purpose in this section is to verify the algorithm by treating a variety of simulated "experimental" curves i*m(s). The expression of i*m(S) is: i~m(# ) = is~m*(8) + idiff(S),
= {I**(k/WDI WD, X WD)
x (I/WD)g(s-k/WD)},
9 ~: 6(r-n2D)}.
**
versus X may yield the values of aD and of {i:otif(h/D)}, even when the intensity distribution of
Pe*x~,step(r[ IVD, X WD) = [p**(r)pst,p(r[WD, X
Pmoact,**st,p(rl2D, X2D, aD, V)
i**(s) = [i,i .... tack(slD, ao, V)imo.f(S)] * ii.~
(37a)
(37b)
Order-Disorder in Myelinated Nerve Sheaths. I
I 9
9
.~ 0.5
-I0
-5
o J (channel)
10
5
I
~0.5
1
I
!
!
!
o
I
I
rid
Figure 4. iinr is the intensity distribution of the virtual incident beam used in the simulation, defined at equally spaced points jAs, with As = 7-6018 x 10 -5/l -t, sample to detector distance 330mm, 2 = 1-54A, D = 175-92 A. pi.c(r) is the FT of ii.o(s).
381
We adopt for idiff(s) one of the curves experimentally observed with myelin (see paper II), which we compute by Shannon's (1949) interpolation of the points sh=h/D (see section 2(b)). The function isizc.,t~,ck(slD,aD,V ) is defined by equation 05); we adopt for imotif(s) the intensity curve corresponding to the electron density profile of myelin reported by Mateu & Mor~n (1986, their Fig. 5). The intensity distribution of the incident beam (see Fig. 4) is typical of our experimental conditions. The function i*m(s) is generated at approximately 1300 channels, with a step As = 7"6018 x l0 -5 A-1, D = 175"92 A and for several sets of values {aD, V} chosen within an experimentally relevant range. A few examples are represented in Figure 5. No random noise is added to the curves, since our purpose is to test the algorithms and the approximations (see Assumptions (1) to (3)). The application of equation (21) to i*~(s) and the division by pinc(XD) yields a set of points {I*~(h, X)/pinc(XD) }. The points corresponding to the strongest and sharpest reflections (h = 2, 3, 4, 5; see Fig. 6) are used to determine ao and v; these values are subsequently involved in the determination of the parameters imom(h/D) and of idlff(h/D ) (see section 2(e)(iii)). The points I$*(k/2DI2D, 2DX) and the values of a o and v are also used to determine imotif(k/2D ) (see section 2(c)(iv)). Tables 1 and 2 report the values of the para-
Table 1
Input and output values of i,.oa~(s) a o (A):
In
Out
In
Out
In
Out
lfl
Out
1-993 110"5
3 25
2.993 25"7
5 75
4"991 81"I
8 5
7"990 5-03
W=2
W= 1
W=2
W= 1
W=2
W=I
W=2
(N):
2 100
k = 2Ds
Model
W= 1
1 2 3 4 5 6 7 8 9 l0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
48 1 128 606 723 147 133 773 729 193 2 9 2 0 8 21 9 0 0 12 56 50 3 13 27 7
1 619 153 758 185 ll 1 24 0 14 54 12 6
61 1 181 620 516 154 188 758 594 185 26 II 2 l 6 24 l0 0 --6 15 60 54 --7 12 27 9
l 619 154 758 185 ll l 24 1 15 55 13 8
54 1 146 620 652 154 151 758 669 186 '14 ll 3 1 9 24 12 1 --1 15 03 55 2 15 30 13
1 620 154 758 186 II 2 25 2 18 60 23 29
62 1 154 620 684 154 145 758 698 186 11 12 3 2 13 26 14 3 6 19 67 62 16 26 42 34
I 621 156 760 190 14 7 37
49 l 143 621 734 156 139 760 718 190 l0 15 l0 7 21 37
?
Results of applying the algorithms to t h e simulated experimental curves i*m(S) (see eqn (37)) corresponding to the input values of t h e a r a m e t e r s {imotlr(k/2D), i a i . ( h / D ) , at), v}. The o u t p u t values are obtained by applying equations (28) (for It' = 1) and (35) (for W = 2) to t~m(s). D = 170 A. Note the excellent agreement, especially for W = 1.
V. Luzzati and L. Mateu
382
^
oo 15
P
=,
j.o
V "- I / I 0 0
30
.~ ~0
i
A 15
50
o-o =3~ V = 1/25
I
0
50
oo 15
o'0--5~ V = 1/75
I A
o? 15
30
O-o = 8 ~ V = 1/5
I 0"05
X-'
0
0.05
o
$
$
F i g u r e 5. Left frames. A few examples of simulated "experimental" curves ,i*m(s ) (eqn (37)) recorded at equally spaced channels (As = 7"6018 x 10 -5 .~-t). The curves correspond to different values of a D and v, and to tile input functions idi,(s) and i=otit(s) of Tables 1 and 2 (see also below). Right frames. Values of tlle ftmetion ,imotlr(k/2D): (O) l i ' = l (eqn (28)); ( O ) I 1 ' = 2 (eqn (35)) (see also Table l). The continuous curves are obtained by Shannon's (1949) interpolation. Tile uppermost frame reports the input curve Si~o,r(S) used in the simulation; that curve corresponds to the electron density profile reported by Mateu & MorSn (1986, their Fig. 5). Note the very close agreement of all the curves Simo,ir(S). With regard to S/~itf(s), the input and the output curves are indistinguishable at the scale of tile drawing (see Table 2).
Order-Disorder in Myelinated Nerve Sheaths. I I
>X
0
50 ~ I000~ * ~
5 0
5 0
5
.~ I00
*i ~
i h=
I 3
~o=2
=,= I/100
5 1/75
1/25
8~ I/5
Figure 6. A few points I~.(/dDID, XD)/Pin~(XD) (see eqns (26)) obtained by the application of algorithm (2]) to the curves i*~(s) (see Fig, 5) at the lattice points h = 2, 3, 4, 5 and for X = 1/2, 3/2 . . . . . The- straight lines represent the function imo.r(h/D)exp[-X(v+2~2a~s2)] (see eqns (28) and (26d)) corresponding to the output values of {imo.r(k[2D), idifr(h/D), aD, v} reported in Table l (W = ]) and in Table 2. Note the excellent fit of the points to the straight lines.
meters ao, v, {imot~e(h/D)}, {imoti,(k/2D)}, {idir,(h/D)} used in the calculation of i*m(S) (the input) and the corresponding values obtained from tire analysis of tile curves i*m(S) (tile output). The results (see also Figs 5 and 6) bear on several aspects of the mathematical treatment and of its implementation: (1) The l)oints {[l*m(h/D,X)/pi.r excellent agreement with the curve:
383
(3) The points {imo,~f(h/D)}w=t and {imo,,(k/2)}w=2 are very close to those used for the determination of i*m(S) (Table 1); as a consequence, the output function imotlf(s) obtained by Shannon's (1949) interpolation of tire points {imot~f(k/2D)} is ahnost identical with the input function. This is a most interesting novelty of this work. (4) Tile algorithm accurately discriminates the "noise" idlrf(S) from tile "signal" imotlf(S) (Fig. 5 and Table 2). Tire overall conclusion is that the algorithm provides an efficient and accurate tool for solving the problems defined in Introduction: namely, discriminate the diffirse scattering from the signal; determine tim parameters ao and v, which specify the stacking disorder and the finite size of the crystallites, respectively; and draw the complete intensity curve of a single motif. Naturally, this section is aimed at testing the algorithms, and tells nothing about the relevance of the mathematical model to tile physical structure of any particular system. This problem will be treated in paper II. As a final comment we point out that the mathematical procedure is based upon simple geometric operations performed in real space. In addition, the algorithm is simple and easy to implement in a computer. In our hands, and using a personal computer, the complete analysis of a typical X-ray scattering experiment takes less than one hour; simple hardware and software imtn'ovements would easily cut tiffs time down to minutes.
are in Appendix
[imotif(h/D) exp [-- X(v + 2nZaghZ/DZ)]
Determination of the parameters
(Fig. 6). (2) The output values of a D and v agree almost perfectly with those used in the calculation of i~]m(S) (Table 1).
Tile applieation of equation (21) to tile experimental c u r v e i*exp(Sj) yields tile values of the flmetion l*,p(h/DlD, XD) at a set of points {h, X}. These are the data of the problem: tile unknowns are ao, v, {imot,t(k/eD)}, {idifr(h/D)}. We solve the problem as follows. Firstly, we determine the values of a and v; we analyze, for that purpose, a subset of sharp and strong reflections (for example, h = 2, 3, 4, 5 in tile case of Figs 5 and 6), for X = 3 / 2 , 5 / 2 , . . . . Tile relevant equation is (28):
Table 2
Input and output values of iaisf(S ) ao (A): (N):
h= Ds
Model
0 1 2 3 4 5 6 7 8 9 l0 II 12 13
365 775 1013 1085 960 808 355 217 195 181 207 342 413 358
2 100
3 25
5 75
8 5
365 776 1024 1079 961 786 330 217 190 182 212 355 408 347
356 776 1023 1079 961 785 330 217 190 181 211 354 407 345
365 776 1023 1078 959 784 330 216 188 179 207 348 396 323
365 776 1021 1076 955 780 326 210 176 152
The 1st eohmm reports the input values used in the calculation of i%(s). The other values are those obtained by applying eqn (29) to i*m(s).
I*~p(h/DlD, XD)x~ t = imotif(h/D)Pine(XD )
x exp [-- X(v + 2~2o'~ h2[D2)] x eosh (v/2 + rc2a~h2/D2).
(38)
We solve tlfis equation by an iterative i)roeedure. Elementary transformations yield: Yp(X, h) = I*~p(h/D]D, XD)/[p,,r x cosh (vp_l/2+uza~ p_lh2[DZ)] (39a) lg YX.h.p = ah.p-X(bp+%h:),
(39b)
ah. p = lg [imotlf(h/D)p ],
(39c)
bp = vp, o_~_2 ,r~z 9 Cp = --1~ OD, pl.lJ
(39d) (39e)
V. Luzzati and L. Mateu
384
The suffix p specifics the cycle of the iteration: in each cycle p, bp_ l and cp_ t are treated as constants, {{ah.p}h, bp, %} as unknowns. The system is solved by least squares, minimizing the quantity: Z2 = ~ ~ [lg Yx.h.t,-ah.p+X(bp+cph2)]2[(a2,,r) p, X
h
(40a)
This work was supported in part by an exchange grant CNRS-CONICIT and research grants from the Association Fran~aise contre les Myopathies, the Association Fran~aise pour la Recherche M6dicale and CONICIT (Sl:1413). The authors are grateful to Dr Alain Henaut for the generous loan of computing equipment, to Dr Salvino Ciccariello for valuable suggestions and to Dr Rodo[fo Vargas for a critical reading of the manuscript.
where (a~gr)p is the variance of lg Yx, h,p. Assuming t h a t at. << l*,p the variance takes the form: ajgr(x,h,p) ,,, a (x,h,p)[ y2 (x,
h,p)
~
at.L (x.h)/(Ir@
z
(X, h).
(40b)_
We also assume that the variance of I*,p is independent of X and h. We set the partial derivatives of X2 equal to zero, we eliminate the variables {imoar(h/D)} and thus obtain two equations of the unknowns aD, p, vp; the final values of ao and v are obtained when convergence is achieved. Once ao and v are known, imotit(h/D) can be determined for any value of h b y minimizing the quant i t y (see eqns (28) and (39a):
X2 = ~ [I*,p(h/DlD, XD)--imot~r(h/n) X
x H2(h/DID, X, ao, v)]2[al,] -2,
(41)
with respect to i~otir(h]D). Assuming again t h a t the variance of l*~p is independent of I*,p and of X, the result is:
imotir(h]D) = [ ~ I:,v(h/DID, XD) • HI(h[DID, X, ao, v)I ...I
)t Icosh (v/2 + u2a2oh2/D 2) I._
x ~. It~(hlnln, X , ~ , v)
]-'
9 (42)
X
The values of the {Qi~dh[D)} are determined using equation (28):
iaift(h/D) = 21*r
D / 2 ) - imotlf(h/D)
[1 +Ht(h]DID, 1, aD, v)]}.
(43)
Finally, the parameter i~otif(k[2D ) is determined for each value of k by applying the algorithm above to equations (34):
[-_ =
x2D)
x H3(k/2DID, X, av, v)/ ...1
•
H~(k]2D]D, X, al), v)
1'
.(44)
References Blaurock, A. E. & Nelander, J. C. (1976). J. Mol. Biol. 103, 421--431. Chabre, M. & Cavaggioni, A. (1975). Biochim. Biophys. Acta, 389, 336-343. Guinier, A. (1964). Thdorie et Technique de la Radiocrlstallographie, Dunod, Paris. Hosemann, R. & Bagchi, S. N, (1962). Direct Analysis of Diffraction by Matter, North-Holland, Amsterdam. Inouye, H., Karthigasan, J. & Kirschner, D. A. (1989). Biophys. J. 56, 129-137. Kirschner, D. A., Ganser, A. L. & Caspar, D. L. D. (1984). In Myelin (Morell, P., ed.), pp. 51-95, Plenum Publishing Co., New York. Luzzati, V. & Taupin, D. (1984). J. Appl. Crystallogr. 17, 273-285. Luzzati, V. & Taupin, D. (1986a). J. Appl. Crystallogr. 19, 39-50. Luzzati, V. & Taupin, D. (1986b). J. Appl. Crystallogr. 19, 51-60. Mateu, L. & Mor~n, O. (1986). Biochim. Biophys. Acta, 862, 525-534. Mateu, L., Luzzati, V., Vargas, R., Vonasek, E. & Borgo, M. (1990). J. Mol. Biol. 215, 385-402. Moore, P. B. (1980). J. Appl. Crystallogr. 13, 168-175. Mor~in, O. & Mateu, L. (1983). Nature (London), 304, 344-345. Nelander, J. C. & Blaurock, A. E. (1978). J. Mol. Biol. 118, 497-532. Padr6n, R. & Mateu, L. (1982). Biophys. J. 39, 183-188. Padr6n, R., Mateu, L. & Requena, J. (1980). Biochim. Biophys. Acta, 602, 221-233. Schwartz, S., Cain, J. E., Dratz, E. A. & Blasie, K. (1975). Biophys. J. 15, 1201-1233. Shannon, C. E. (1949). Proc. Inst. Radio Enff. N.Y. 37, 10-21. Taupin, D. & Luzzati, V. (1982). J. Appl. Crystallogr. 15, 289-300. Vonasek, E., Mor~in, O. & Mateu, L. (1987). J. Neurocytol. 16, 105-114. Worthington, C. R. (1986). Biophys. J. 49, 98-101. Yaeger, M. J. {1975). In Neutron Scattering for the Analysis of Biological Structures (Schoenborn, B. P., ed.), pp. VII-77-106, Brookhaven National Laboratory, Upton, New York. Yaeger, M. J., Schoenborn, B. P., Engelman, D., Moore, P. & Stryer, L. (1980). J. Mol. Biol. 137, 315-348.
Edited by A. Klug
O r d e r - D i s o r d e r P h e n o m e n a in M y e l i n a t e d N e r v e Sheaths I. A Physical Model and Its Parametrization:
Exact and Approximate Determination of the Parameters Vittorio Luzzatit Centre de G~ndtique Moldculaire, Laboratoire Propre du C N R S Associg d l'Universit~ Pierre at Marie Curie 91198 Gif-sur- Yvette, France and Leonardo Mateu Centro de Bioflsica y Bioquimica, I VIC Apdo 21827, Caracas 1020-A, Venezuela (Received 24 November 1989; accepted 8 June 1990) An algorithm is developed for the analysis of the X-ray scattering spectra of lamellar systems, by reference to a precise physical model. The model consists of identical planar lamellae (the motif), all parallel and stacked in a one-dimensional crystal with four types of defect: stacking disorder, finite size of the crystallites, and presence of diffuse and blank scattering. In addition, the spectra are distorted by collimation aberrations. In order to evaluate the effects of these distortions, the following assumptions are made: (1) beyond some point Slimit the intensity curve can be expressed as a function of a (small),number of parameters; (2) the blank scattering, restricted to very small angles, can be identified and eliminated; and (3) the diffuse scattering is entirely defined by the values of idif~(h/D) at the lattice sh = hiD (h is a positive integer _
rod outer segments (ROS) are the best examples. These organelles consist of identical and intrinsically asymmetric membranes, associated in pairs of opposite polarity; the membranes are all (quasi) parallel and the distances between the membranes of each pair and between pairs of membranes are both (quasi) constant throughout the stack. In MyS the membranes are wrapped spirally around the cylindrical nerve fibers, in ROS they form closed disks. Usually, the number of membranes per stack is large, as is the diameter of tile disks in ROS and
(a) The problem In Nature, a few highly ordered membranecontaining systems exist that behave like onedimensional crystals in X-ray and neutronscattering experiments: myelin sheaths (MyS:~) and t Author to whom correspondence should be sent. Abbreviations used: MyS, myelin sheaths; ROS, rod outer segments. 0022-2836/90[ 190373-12 $03.00
373
(~ 1990 AcademicPress Limited
374
V. Luzzati and L. Mateu
tim diameter of tim nerve fibers in MyS, so tlmt the diffraction spectra, like those of one-dimensional crystals, may be reduced to a row of fairly shar I) reflections, all equally spaced. The repeat distances arc in the range 100 to 400 A (1 A = 0-1 nm); reflections up to the 15th order are commonly observed. Extensive studies of the structure of MyS and ROS lmve been carried out using X-ray and neutronscattering teclmiques (see Yaeger et al., 1980; Nelander & Blaurock, 1978; Kirsclmer el al., 1984; Worthington, 1986, and references therein). Besides, MyS and ROS are such excellent scatterers that kinetic experiments have been performed and the structure correlated with physiologically significant I)arameters, especially since position sensitive detectors and high-flux neutron and X-ray sources have come into use. In tile case of ROS, tim effects of light (Chabre & Cavaggioni, 1975) and in tile case of MyS, tile effects of action potential proi)agation (Mor,'in & Mateu, 1983), of anesthesia (Mateu & Mor~in, 1986) and of exI)erimental allergic neuritis (Vonasek et al., 1987) Imvc been studied. Tile one-dimensional order, though, is not perfect: tile reflections are not quite, nor equally, sharp and some background scattering is always present in tim spectra. In order to rid the data of noise and distortions one must distinguish reflections from background, dissociate the overlapping reflections and integrate the intensity of each reflection; these steps usually involve, to some degree, ill-defined operations, especially ira the regions of tile spectra in which tile signal-to-noise ratio is low (Yaeger, 1975; Nelan(ler & Blaurock, 1978). This problem is even more serious for kinetic experiments: tim analysis, ira tiffs case, hinges upon small differences between spectra recorded in tile shortest possible time. A significant improvement is achieved by analyzing tim spectra i)y reference to a precise i)hysical model. In tile ease of myelin, previous authors ( s e e Blaurock & Nelander, 1976; Worthington, 1986; Inouye el al., 1989, art(1 referenccs therein) assumed tlmt tim real structure is derived from an ideal one-dimensional crystal via six types Of distortion: erystallite size, namely the finite number of lamcllae present in each coherent stack; stacking (or lattice) disorder, i.e. variations of tile repeat distance; positional disorder, i.e. variations in the separation of tim two membranes comprising the elementary pair; blank scattering, due to tile instrument and to chemical impurities in the samI)le; diffuse scattering, related to structural imperfections within the lamellar system; collimation aberrations, due to tim finite width of tile incident beam. From a practical standpoint, traditional analyses of tile X-ray spectra involve, as a rule, tim following steps. Firstly, identify the noise urtderlying tim signal; usually the lower envelope of the intensity curve, drawn by baud, is ascribed to diffuse scattering (see, for examI)le, Fig. l0 of Nelander & Blaurock, 1978). Secondly, subtract the noise from tile signal, dissociate (if required) tile overlapping reflections and draw tile intensity curve associated
with each lattice point. Thirdly, integrate tile intensity of each reflection. Fourtldy, ascribe tile integrated intensities to tile reflections of an ideal onedimensional crystal, search for tile signs of tile corresponding structure factors and determine tile electron density profiles. In addition, in a few cases tile shape of the reflections has been analyzed in terms of a model involving crystallite size and stacking disorder, and the values of tile parameters used to specify tile model have been determined (Blaurock & Nelander, 1978; Inouye el al., 1989). Tile procedures used for those purposes call for a number of comments. Some of tile operations, for example, hand drawing of tile curve corrcsi)onding to tile "noise" (that curve plays a critical role in the analysis of disorder) and separation of overlapping reflections, are neither defined in quite objective terms nor totally exempt from groping. Moreover, the goals are not always consistent with the model setting the framework for tim analysis; for example, the very notion of " m o t i f " (and of its structure factor and electron density profile) is problematic ira the presence of positional disorder (Worthington, 1986); also, the operation of integrating tim intensit), deserves some theoretical justification. Finally, the results are often presented as a faithful representation of tile system, overlooking tim fact that tim analysis hinges upon models whose relevance may well be open to question. The awareness of these limitations, and also tim urgent need to analyze with tractable speed and sufficient accuracy the large wealth of data recorded in myelin X-ray scattering experiments performed with position-sensitive detectors (by contrast with photographic films, used in most previous studies of myelin) and under a variety of physiological and pathological conditions (PadrSn el al., 1980; PadrSn & Mateu, 1982; Mor,4n & Mateu, 1983; Mateu & Mor,-in, 1986; Vonasek el al., 1987) prompted us to try to overcome some of the drawbacks and to devise a routine involving clearly and objectively defined operations, which are easy to implement with a computer. The approach is concei)tually the same as the one that we adopted in the mathematical treatment of solution scattering experiments and used in tile analysis of the information content of solution and crystal scattering studies (Taupin & Luzzati, 1982; Luzzati & Taui)in, 1984, 1986a,b; see also .Moore, 1980). That i)roccdure involves the following steps. (1) Introduce a model and Sl)ccify tim degrees of freedom of the I)roblem (i.e. define the mathematical parameters that are necessary and sufficient to take into account tile I)hysical phenomenon) by reference to that model. (2) Express the exI)erimental observations (usually counts at a finite number of clmnnels) as functions of the degrees of freedom. (3) Determine the values of the degrees of freedom, whenever their numl)er is smaller than or equal to tim number of tim independent experimental observations. (4) Analyze the errors.
Order-Disorder in Myelinated Nerve Sheaths. I
(b) The model and the algorithm We tackle in this paper steps (1), (2) and (3), above. The model is defined by introducing the following distortions into an ideal one-dimensional crystal: crystallite size, each coherent stack contains a finite number of motifs; stacking disorder, we assume t h a t the motifs, all identical, are stacked with variable spacing; blank scattering, due to the instrument and to chemical impurities in the sample, and limited to very small angles; diffnse scattering, which we parametrize via the assumption that its Fourier transform has the finite support - - D / 2 < r < D / 2 ; collimation aberrations, due to the finite width of the incident beam. The model is similar to other models used previously (Schwartz et al., 1975; B l a u r o c k & Nelander, 1976; Worthington, 1986; Inouye el al., 1989). The main differences bear on t h e parametrization of diffuse scattering and on positional disorder. For the sake of simplicity, we develop the mathematical treatment described in the following section, disregarding positional disorder; we later take it into account (see accompanying paper, Mateu et al., 1990) with the restriction t h a t positional and stacking disorder are independent of each other. This is an essential precaution if the structural notion of "motif '" is to be preserved. Tile degrees of freedom of the model are the parameters specifying the lattice and its disorder: namely, tile average repeat distance D and its vari2 ante ao, the average number of motifs per cr 3 stallife ( N ) , and the sets {i~,(h/D)} and {imot~r(k/2D)}, which define, respectively, the continuous diffnse scattering and the intensity curves of one motif. A few additional parameters are involved in the definition of the asymptotic trend, of the blank seattering antl of the angular spread of the sheaths' orientation (see section 2 of tile accompanying paper). We elaborate a simple and fast, but not strictly rigorous, algorithm that operates on the raw d a t a and yields the values of the whole set of parameters with a minimum of intermediate manipulations. More precisely, the manual interventions are restricted to the determination of the asymptotic trend of tile experimental curves and of tile repeat distance D, and to the elimination of the blank scattering. The main virtues of the algorithm are to eliminate m a n y of the intermediate, and questionable, operations (hand-drawing of the diffuse scattering curve, discrimination of the overlapping reflections, integration of the intensities) to yield at once the value of all tile parameters specifying structure and disorder, and especially to determine the continuous intensity fnnction corresponding to tile motif. Naturally, tile validity of our approach (and of all previous approaches to the problem) is limited by the validity of the model. The model will be shown to be consistent with all the experimental observations; the question of its relevance to the particular
375
system under investigation is more difficult to answer. One way to approach tile problem would be to carry out a Z2 test; this involves elaborate operations that we leave for the future (see section l(a), step (4)). (c) Layout of the two papers The present and the accompanying paper (hereinafter called papers I and II, respectively) apply to the ideal case of perfectly oriented fibers. The experimental problems are discussed in paper II, section 2(a): angular spread of the sheaths' orientation; collimation aberrations including electronic distortions and drifts; determination of the average repeat distance D; elimination of the blank scattering surrounding the incident beam; determination of the asymptotic trend of the intensity curve; operational definition, from s = 0 to infinity, of the experimental curve i*r involved in all the subsequent calculations. In paper I we introduce the model that sets the framework for the mathematical analysis and we derive an algorithm, involving straightforward operations of tile experimental points i*xp(si), which leads to tile determination of all tile parameters specifying tile structure and tile disorder. An additional parameter, introduced in paper II, is the fraction a~.... of myelin lamellae that are not tightly packed in the sheaths. Once the complete intensity curve of tile motif is known, and since the structure is centrosymmetrie, it should in principle be possible to determine the electron density profile. The presence of positional disorder complicates tile problem. This point is discussed in paper II, section 2(d). The very notion of electron density profile is shown to become problematic when positional disorder is present; two tests are proposed to select the "best" map. The algorithm, whose derivation involves a few aI)proximations, is tested in section 3, point (4) using simulated experimental curves. In paper II, the algorithm is applied to X-ray scattering experiments performed on rat sciatic nerves, both native and swollen in non-isotonic solvents, and also as a function of the age of the animal (from 4 days old to adult). All the results are consistent with the model, which sets the framework of the mathematical analysis. The swelling experiments show that in native myelin the stacking disorder involves the cytoplasmic space, the external space in swollen myelin. The analysis of myelogenesis shows t h a t the number of membranes per crystallite varies dramatically with the age of the animal but that the structure of the individual membranes of the sheaths is practically invariant. 2. M a t h e m a t i c a l M o d e l a n d Analysis o f the D a t a
(a) Notation r, position in real space (in A). s, position in reciprocal space; s = (2 sin 0)/2, 20 is
the scattering angle, ). tile wavelength (in A).
V. Luzzati and L. Mateu
376
D, a~, mean repeat in real space and its variance. v = 1/(N), where ( N ) is the average number of lamellae per crystallite. W, positive integer specifying the repeat WD of a virtual periodic function (see section (b)(iv)). n, h (and k), integers specifying positions in real or in reciprocal space, in units WD and (WD)-t:
r, = nWD, sh =h(WD) -1.
I
Pdl(r)/~ f I I
X, specifies position in real space, in units WD. FT, operator expressing Fourier transformation. *, operator expressing convolution, namely:
f(z) * g(x) = ~f(y) g(x--y) dy. p(r) and i(s), (with various sub- and superscripts) a pair of reciprocal functions: p(r)FTi(s). 5(r-R), Dirae delta function, centered at r = R. f(x, ylA, B . . . ) , ad hoc expression of a function f(x, y, A, B . . . ) in which x, y play the role of variables, A, B . . . t h a t of parameters. fl(r--RlL), step function of width L, centered at the point R:
fl(r--RIL) = 1, if - L / 2 < _ r - R < _ L / 2 fl(r--R[L) = 0 otherwise.
(1)
Theorem 1, [p:(r) * pb(r)] FT [i:(s) ib(s)].
(2)
Theorem 2,
[A
]
]~ ngn (r-nO)
AAA,i, AAA
g4(r+40) 9~(r+20) g o ( r ) gz(t-20) g4(r-4O) gs(r+SO)gl(r+O) gl(r--O) gS(r-SO) i
-40-30-29-'0
0
i~ 2'0 i9 ;O r
Figure 1. Generation of the function P*od.l(r) = pi.c(r) [Pdiff(r) + Pmotu(r)*Pli..,.t.o.(r)] (eqn (17a)), assuming that the incident beam is infinitely narrow (namely pi.r v = 0 and ao/D=00495. Note that the support of the function Pmo.r(r) is - D < r < D, that of Pdlff(r) is --D]2 < r < D]2. The lower curve represents the function ~.g,,(r-nD) (eqn (23b)), which, in this case, takes the form of a set of Gaussian functions, go(r), represented by a vertical arrow, is a delta function centered at r = 0.
istep(s]L, R), Fourier transform of pstep(r]L, R):
Theorem 3,
i(S--So) FT [p(r) exp ( - 2~irso)].
(4)
P(rID), periodic function, of repeat D and motif Prnotif(r):
P(rID) = ~
Pmotif(r--nD)
I
+L/2
[cos 2n(r+R)s
J - L/:Z + cos 2rr(r-- R)s] dr/2 = cos 2rcRs sin (uLs)/us.
,0, I(s]D), Fourier transform of P(r[D); it consists (see eqns (2) and (3)) of a set of delta functions of position h]D and content (l[D)i(h]D): (6)
Support, of a function g(r) is the range of r outside of which 9(r) is identically zero. According to Shannon's (1949) theorem, if - L / 2 <_r< L/2 is the support of g(r), then its Fourier transform ~(s) is entirely defined by the values it takes at the points
sh = h/I,:
(b) The model of the disordered structure We refer to a one-dimensional crystal of repeat D: the electron density and the structure factor of its motif are Pmotif(r) and F(s), Pmotif(r) and imotif(8) are the autocorrelation and the intensity functions of one isolated motif. The functions Pcry(r[D) and Ic,r(s]D) corresponding to the ideal crystal take the form (see eqns (5) and (6) and Fig. 1):
Pc'Y(rlD) = Pm~
* [,,=~-oo6(r--riD) 1'
I.y(4D) = {imo.f(s)(l/D)6(s-h/D)}.
V(s)=L ~. v(h/L)sin[r~(Ls-h)]/n(Ls--h).
(9)
INT(x), integer part of a real number x.
?1= - o o
I(slD) = {i(s)(l/D)5(s-h/D)}.
istcp(s[L, R) =
(7)
h= -~o
pst=p(r[L, R), pair of (half) step functions (eqn (1)) centered at the points R and - R :
p~t~,(r[L, R) = (l/2)[p(r--R[l.,) + p(r+ R[L)]. (8)
(1O) (11)
Thus, the function I.y(s) is entirely defined by the set {i,..otif(h[D)}. The support of the electron density map Pmo.f(r) is --D]2 < r < D]2; the support of the autocorrelation function Pmotif(r) is twice as large, - - D < r < D . Therefore, the function Pmotif(r) is entirely defined by the set {F(h]D)}; on the other
Order-Disorder in Myelinated Nerve Sheaths. I hand, the functions Pmotif(r) and imotif(S) (and also Pmotif(r) when it is centrosymmetric; Hosemann & Bagchi, 1962) are entirely defined by the set {imotif(k/2D)}, whose number of elements is twice the number of reflections of the crystal. As discussed in section 1, above, and in agreement with previous workers (Blaurock & Nelander, 1976, and references therein), we assume that the real structure derives from the ideal crystal via five types of distortion: finite crystallite size, stacking disorder, blank scattering, diffuse scattering, collimation aberrations. The finite size of the crystallites has the effect of multiplying each term of equation (10) by a function psiz~(n[v), which has the property (Blaurock & Nelander, 1976): [Psizc(n]V)]nwO "-~ Psize(0) (1 -- [n[v).
(12a)
For the sake of convenience and by way of approximation we adopt the form: pslz~(n[v) = exp (-Inlv)
(12b)
(setting Psiz~(O)= 1 has no consequence, since the analysis of this section is independent of scale) (we anticipate, see paper II, section 2(b)(iii), that a constant term will be added at the origin of p~i~(n) to take into account the presence of a loosely packed membrane component, called loose myelin). With regard to stacking disorder we adopt a random walk model (Blaurock & Nelander, 1976): namely, we assume that the motifs are all identical, that the distance from one motif to the next fluctuates around the average value D and that these fluctuations are uncorrelated. Under these conditions, and applying the central limit approximation, each delta flmction of equation (10) is replaced by a Gaussian function centered at point nD:
l(r--nnln, a~) ,.~ (2nln[a~)-! x exp [-- (r-- nD)Z/(2lnla~)]. Note that this model involves the assumption:
(13)
following
Assumption (1): over the bordering region D / 2 > r > D / 2 - e , with'~ of the order of aD, the electron density is constant and equal to the average electron density over the unit cell. (This condition implies that /motif(0)=0; in fact theory shows and simulations confirm that the final result is barely sensitive on the precise values of /motlf(0) and of p(D]2), at least as long as
= ~
p~i~o(nlv)l(r-nDln, a.),
]1~ --o0
isi~r ~,~ck(slD, av, v)
= ~
psi~(n]v)2(sln, aD)exp(--2~inDs)
(14a)
= ~ n=
377
exp[-[n[(v+2n2a~s2)--2ninDs],
(14b)
--O0
2(sln, a9) = exp (-2n2[nla~s :) FT l(r[n, at)). (14e) Equation (14b) represents a geometric progression; a trivial transformation yields a more condensed expression (Guinier, 1964): isi.... tack(S) = (1 --e-2a)/(1 +e-2a--2 e -~ cos b), (15a) a = v-k27t2r 2, (15b)
b = 2~zDs.
(15c)
The convolution of equation (14a) by Pmotif(r), equivalent, in reciprocal space, to the multiplication of equation (14b) by imofif(8), takes account of the electron density distribution of the motif. Background scattering, arising from air and windows, electronic noise, sample constituents that do not belong to the quasi-crystalline component, and structural disorder of myelin, is also present and must be taken into account. It is convenient to decompose the background scattering into two components: /back(S) = /blank(S) -{- idiff(8 ).
(16)
The diffuse scattering idiff(8) is a smooth function that, according to a widespread (and loosely defined) procedure, is drawn by hand to fit the presumed minima of the spectra. The criterion of smoothness implicitly underlying this procedure can be expressed more formally (and more rigorously) by the assumption that the support of Pdiff(r) is --D/2 <__r <_1)/2. This assumption entails the interpolation rule expressed by equation (7), therefore, the entire c u r v e idiff(8) can be expressed as a function of the set {iditf(h/D)}h. The term iblank(S) is aimed at removing the sharper bump centered at the origin,, whose shape makes exception to the interpolation rule, and also at extrapolating the data to the origin (see Fig. 2 of paper II). This decomposition must be viewed as an empirical procedure; the possible sources of background scattering are indeed so multifarious that it would be hazardous to try to correlate the terms of equation (16) with precise physical properties of the sample. In reciprocal space the effect of collimation aberrations is to convolute the whole of the intensity curve, with all the distortions discussed above, by the intensity distribution of the incident beam, iinc(s); the real space counterpart is the multiplication by Pine(r). Thefinal effect of these distortions is expressed by the following equations (see eqns (14) and (15)): p*modet(r) = Pmodel(r)--Pbhnk(r) = Pinc(r)[Pdiff(r) "{-Pmofif(r) * Psi .... tack(r)],
(17a)
i'model(S) = imodel(S) -- iblank(S) = iinc(S ) * [idiff(8)+iraotif(S)isize, stack(S)]. (17b)
Note that equations (17) are functions of the parameters D, at), v, {imot,f(k/2D)}, {idir,(h/D)}.
V. Luzzati and L. Mateu
378
and to infinity are discussed in paper II, section 2(b)(ii). We also assume t h a t a few of tim reflections stand out sharp and strong, so that the mean repeat (D) can be determined: we thus remove D from the unknowns. We then proceed to elaborate an algorithm according to the following procedure.
W=I
X=3/2
AAAt - 4D
- 319 - 2D
-D
tAAA .....A . . . .
(r
N
.____/!/
D
2D
3D
4D 9
.
nD
Figure 2. Generation of the functions Pmodd.st,p(r) * (eqn (24)) for W= 1 and ( X + t ) integer. The drawings represent the functions ~ , g d r - n D ) and (hatched) pstcp(r): the product of the 2 functions is also hatched. The motif of the periodic fimction I model,step(r) is represented on the right side of the Figure; the repeat is D. )*
(c) Deleiminations of the parameters of the model Tile result of an experiment is the measure of the intensity of the incident and of the scattered beams, ii,r and i=~p(j), respectively, at a finite n u m b e r of channels j. As discussed above, the blank scattering must be removed from the experimental curve (see eqn (17b)): i*exp(Sj) = iexp(Sj) -- iblank(Sj). (18) We discuss this operation in paper II, section 2(b)(iii)..We suppose t h a t tile incident beam is symmetric and regular and that its sampling is sufficiently dense so that tile function ii~r ) can be interpolated at any point s. Besides, as often happens in tile analysis of scattering phenomena (Taupin & Luzzati, 1982), the mathematical problems are not manageable unless the intensity scattered beyond some point Slimit can be expressed as a fimction of a finite (and small) number of parameters, M. We also assume t h a t beyond 81imit the flmction imotif(s) is negligibly small. I f the conditions above are all fulfilled, then the data consist of the values of the intensity i*._(s,) at a finite set of J k' J points si and of the curve h,r (see section 3). To each clement of the set {i*.p(Sj)}, one equation (17b) corresponds: i*~p(Sj) =
f_
P=,p, step(riD, XD) = [p**p(r)p~t,p(rlD, XD)] * L=~_oo 5 ( r - - n D ) ]
(20)
The reciprocal space counterpart is (see cqn (9)):
I*~p, step(BID, XD) = {I*~p(h/D]D, XD)(I/D)5(s-hlD)},
(21a)
where:
I%(h/DID, XD) = F, Asjq(sj~h, D, X)i%(sA, (21b) .l=l Asi = sj+ 1 --sp
(21c)
q(s, h, D, X) = i~t~p(h/D-- s) + i~t~p(h/D+ s) = D[cos 27:X(h- Ds) sin n(h- Ds)/n(h- Ds) + cos 2uX(h + Ds) sin u(h + Ds)/n(h + Ds)]. (21d) Thirdly, we carry out the operations above on the function Pmodct(r) * (see eqn (17a)). The result is a periodic function of repeat D. Its motif (see Fig. 1):
P*od~L~t=p(rlD, XD, aD, v) Pmodr v)p~tcp(r]D, XD) I
--
$
-
(22a)
is a fimction of r, whose support, like t h a t of
pstcp(r[XD, D), is split into two halves, D ( - X - - 1/2) <_r <_D ( - - X + 1/2) and D ( X - 1/2) <_r <_D(X + 1/2).
[imo.f(s)i~i .... t~k(S) c~
+idiff(s)liinc(Sj--s)ds
(i) Construction of a virtual periodic function Operating in real space, we firstly cut out of the flmction p*xp(r) a pair of sections of width D, centered at the points + X D (Fig. 2); this operation is formally equivalent to multiplying pc*xp(r) by the function Pstep(rlD, XD) (see eqn (8)). The result, expressed ill space s, is the convolution of i**p(S) by istcp(slD, XD) (see eqn (9)). Secondly, we construct a periodic fimction P*w.st=p(rlD, XD) whose repeat is D and whose motif is tile product pc*xp(r) Pstcp(rlD, XD):
(19)
of tile unknowns D, go, v, {imotif(l['/2D)}, {idiff(h/D)}. Tile munbers of elements of tile sets {imotif(k/2D)} and {iditf(h/D)} are, respectively, 2lNT(sjlmitD ) and INT(811mltD); thus the total number of unknowns is 3INT(SlimitD ) + 3 + ~1/. I f this number is smaller than the number J of experimental points, then the most probable values of the unknowns and the associated error matrix can be determined (Taui)in & Luzzati, 1982). We leave for the future a rigorous mathematical treatment of the problem; we devise below a simple, but approximate, solution, which we illustrate in paper II, section 3. 9 We suppose that tile fimction i*exp(Sj) is known from s = 0 to infinity (the extrapolations to s = 0
The convolution by the lattice brings the two halves on top of each other and adds them. I t is thus possible to use an equivalent expression (see eqn (10)):
Pmodel. step(r[D, XD, ao, v) = {(l/2)[P*modr
v)+P*odc,(--rla.,
x fl(r--XDID)}*
s tl ~
The definition of Pmodr * (17a):
.
Pmod,t(r]aD, V)= pi~r
5(r-riD). -
v)] (22b)
oo
v) is given in equation
{Pdit-f(r) +/)motif(r)
Order-Disorder in Myelinated Nerve Sheaths. I (ii) Simplifying assumptions We introduce the following simplifications and approximations.
379
When X + I is an integer, and ouly in this case (see Fig. 2), tile same terms gn(r--nD) are involved ill tile two sides of equation (25) and tile inequality becomes a quasi-equality:
Assumption (2): tile incident beam is sufficiently narrow in reciprocal space so t h a t tim fimction Pine(r) can be assumed to be constant and equal to plne(XD) over tile range D(X- 89 <_r < (X +89 Assumption (3): tile flmctions l(rln, ao) are narrow: more precisely, na~ <
P*odr step(rlD, XD, at), v)
In this case, and for r > D (namely, beyond tlm support of pdlff(r)) tim expression of Pmod,l(rlcrD, V) (eqn (22e)) can be simplified:
Tile reciprocal space counterpart of equation (26a) is (see cqns (14), (21a) and (23b):
{P*odel(rlao, v)}~u~Pmotif(r) * ~ n=
g.(r--nD),
~oo
(23a)
gn(r-- nD) = pinr
exp (-- nv)(2nlnia2 ) -89 xexp[-(r-nD)2[(2lnla~)].
(23b)
A set of functions gn(r--nD) is represented in Figure 1. The multiplication [P*odel(rlaD, v)p~tcp(rlD, XD)] and tile subsequent convolution with tile lattice generate tile periodic function (see eqn (22) and Fig. 2):
Pmotif(r) * {(89189 + gx+ t( r - [X + 89
9 ~ n=
~(r--nD).
(26a)
-oo
{l*oael(h/nlD, XD, no, V)}x~ x .~ imotif(h/D)H2(h/D]D, X, a D, v), (26b) H2(sID, X, no, v) = (89
X - k , no, v)
+ HI(slD, X +89 no, v)], = HI(slD, X, no, v)cosh (v/2 + r?a2os2), (26e) Ha(slD, X, Up, V) F T g~(z) = pin~(XD)exp[--X(v+2t~za~s2)],
(26d)
z = r - XD.
(26e)
=
Equations (26) apply to tile points X _> I. At tim point X = 89 and normalizing Pine(0)----I, equation (26b) takes tile form (see Figs 1 and 2):
Prouder, step(rlD, XD, at), v) ~{[Pmot,f(r'*n=~_~oyn(r--nD']Pstep(rlXD, D) }
l*odel(h[DlD, D/2, up, v) .,~ (2){Zmotlf)(h[D) x [1 +Hx(h/DID, 1, no, v)] 9
9 9
+idlff(h/D)}.
The support of Pmotlf(r) is --D<_r<_D and each gn(r-nD) is an even and narrow function of r-riD: as a consequence only tim terms [g.(r--nD)]:_ 1<_.~x+1 contribute to the function
[P motif(r) * n~=~ogn(r)] ' over tim range (X--k)D<_r<(X+ 89 On the other hand, and over tl)e same range of r, only tile terms [gn(r--nD)]x_89189 are involved in tim multiplication
[pstep(rlXD, D)n=~r162g.(r)]. As a rule, therefore, the sequence of operations of equation (24) cannot be inverted.
(27)
(iii) Partial solution of the problem A comparison of the pair of functions I*odet(h/DlD, XD, up, v) (eqn (26)) and I*,~,(h/DID, XD) (eqn (21)) may yield an approximate and a partial solution of tim problem. For that purpose we apply tile following equation:
{l*,p(h/DID, XD)}xzx = imotif(h/D)H2(h/DID, X, up, v) (28) at the points X = 3/2, 5/2 . . . . . and thus determine the values of ao, v and {imotle(h/D)} using the algorithms discussed in Appendix. We subsequently determine the values of {iditt(h[D)} using equation (27):
id~rt(h[n) = 21*,p(h[nln, D[2) --imotif(h/D)[1 +HI(h/DID, 1, no, v)]. It may also be noted that for X = 3/2, 5/2 . . . . ratio of tile fimctions I*,,p(hl/DlD, XD) relevant pair of lattice points h I and hz (see eqns (28)
(29) tim to a and
(26d)): 9 ~
I*~p(hx/DID, XD)[I*p(hz/DID, XD) = [im~176 2 x exp [-2rc2Xa~(h~-h2)l D ] x [cosh (v[2 + rt2a~h~/DZ)[ cosh (v/2 + rc2aZhZ,In2)].
6(r-riD) Cp~o~idr) 9
9 n=-D~6(r-nD
I} .
(25)
(30)
is also a fimction of tile parameters o-D and {i~otif(h/D)}h, barely dependent on v if v<< 1.
V. Luzzati and L. Mateu
380
aOx(r-2nO)
aox(, -tz~+~lo)
I**(k] WDI WD, X WD)
t
= ~ As~q(sj, k, WD, X)ie**p(si). (32c) j=t
1
The value of the function I**(k/WD]WD, XWD) can be computed a t any point {k, WD, X}. F o r ' t h e sake of simplicity, we restrict the explicit t r e a t m e n t to the case W = 2 ; a generalization to W > 2 is straightforward. Using the same arguments as in section (c){ii), above, we single out the values of X for which the edges of the function pstep(r[WD, XWD) coincide with the lattice points r.=nD, namely X = 0 , 1/2, 1,3/2 . . . . . As illustrated in Figure 3, the m o t i f of the periodic function PmoaeL**step(rl2D, X2D, a., v) consists of two functions, GOx(r) and GDx(r), centered, respectively, at r = 0 and r = D :
I
W=2 X=I
AAAt !,AAA ...... ~
A i
gz (r- 2 ~ t
-[~ (r-Iv,+ I Iv)+g3 t~-Iz,,Hlo)}/2
1I I
1I
AA-A/ !AAA Ai W=2 x =3 / 2
I
{gzlr-~.nOl+g3[r --ZnO)}/2
-4D-SO-20
--'O
O
~ / -
2D ~O 4D
r
. . . . -/ 7 I /
Pmotif(r) * {[GOx(r) + GDx(r + D)]
2aO (?-n§
Figure 3. The same as Fig. 2, for W = 2 and 2X integer. In this case the repeat of the periodic function is 2D, its motif consists of 2 peaks, one [GO(r)[ centered at the origin r = 0, the other [GD(r-D)] centered at the point r = D; the 2 peaks are linear combinations of the functions g~(r) (see eqn (34)).
Therefore, the analysis of the ratio [I*,p(h~/DID, XD)/I*,p(h2/DID, XD)]
the incident beam is not known. (iv) Completing the solution 9Once the values of the parameters {imotir(h]D)}, {io~fr(h/D)}, aD and v are known, the elements of the set {imotir(k/2D)} with k odd are still required for the full determination of the function imotif(s). A generalization of the same procedure may yield the value of the missing parameters. Since the function idlft(s) is entirely defined by the set {ioiff(h/D)} (see eqn (7)) its contribution to both leap(S) and p,,p(S) can be eliminated: "** ) = iexp(S ) -- iblank(8 ) -- idiff(8), Zexp(S
(31a)
Pexp(r) = pexp(r)--Pbtank(r)--Pdiff(r).
(31b)
B y analogy with section (e)(i), above, we generate a virtual periodic function whose repeat is WD (W is a positive integer), whose motif is the product of p**(r) by a pair of step functions centered at the points +_XWD and of width WD (see Fig. 3 and eqns (20), (21) and (24)):
9 ~ n=
6(r-nWD),
IVD)] (32a)
~oo
Ie,p. step(k/WD[ WD, X WD)
n=
(33)
-oo
GOx(r) and GDx(r) are linear combinations of the functions g,(r) (eqn (23b)), which, according to the parity of X, take the form:
GOx(r) = (cos~X)2g2x(r) +(sinuX)2[g2x+x(r)+g2x_t(r)]/2, GDx(r ) = (sin zcX)2g2x(r) 9 +(cosxX)2[g2x+l(r)+gzx_l(r)]]2.
(34a) (34b)
The F T of P~odcl, ~tep(r[2D, X2D, ao, v) yields the expression of I**odel.st,p(k/2D[2D,X2D, aD, v) (see eqn (26d)):
l**oael(k[2Dl2D, X2D) ,~ imotif(k/2DlX)H3(k]2DlD, X, ao, v) .~. I*~(k/2D[2D, X2D), H3(k]2DID, X, a D, v) = cos 2ukX{HI(k/2D[D, 2X, aD, V) + 89COSnk[H l (k/2DID, 2X + l, aD, V) +II~(k/2DID, 12X-- II, aD, v)]}.
(32b)
(35a)
(355)
Thus, one value of imotif(k]2DlX ) can be determined at each point {k,X}x=o,89 a.... :
i~ot,f(k]2D[X) = I*~(k]2Dl2D, X2D)[ H3(k]2D[D, X, aD, V),
(36)
and the value of imotif(k]2D) determined b y averaging the points imotif(k[2DIX) over X (see Appendix).
3. Verifications and Practical Considerations The application of the algorithms to myelin data is described in paper II (Mateu et al., 1990) and will be documented in other forthcoming papers. Our purpose in this section is to verify the algorithm by treating a variety of simulated "experimental" curves i*m(s). The expression of i*m(S) is: i~m(# ) = is~m*(8) + idiff(S),
= {I**(k/WDI WD, X WD)
x (I/WD)g(s-k/WD)},
9 ~: 6(r-n2D)}.
**
versus X may yield the values of aD and of {i:otif(h/D)}, even when the intensity distribution of
Pe*x~,step(r[ IVD, X WD) = [p**(r)pst,p(r[WD, X
Pmoact,**st,p(rl2D, X2D, aD, V)
i**(s) = [i,i .... tack(slD, ao, V)imo.f(S)] * ii.~
(37a)
(37b)
Order-Disorder in Myelinated Nerve Sheaths. I
I 9
9
.~ 0.5
-I0
-5
o J (channel)
10
5
I
~0.5
1
I
!
!
!
o
I
I
rid
Figure 4. iinr is the intensity distribution of the virtual incident beam used in the simulation, defined at equally spaced points jAs, with As = 7-6018 x 10 -5/l -t, sample to detector distance 330mm, 2 = 1-54A, D = 175-92 A. pi.c(r) is the FT of ii.o(s).
381
We adopt for idiff(s) one of the curves experimentally observed with myelin (see paper II), which we compute by Shannon's (1949) interpolation of the points sh=h/D (see section 2(b)). The function isizc.,t~,ck(slD,aD,V ) is defined by equation 05); we adopt for imotif(s) the intensity curve corresponding to the electron density profile of myelin reported by Mateu & Mor~n (1986, their Fig. 5). The intensity distribution of the incident beam (see Fig. 4) is typical of our experimental conditions. The function i*m(s) is generated at approximately 1300 channels, with a step As = 7"6018 x l0 -5 A-1, D = 175"92 A and for several sets of values {aD, V} chosen within an experimentally relevant range. A few examples are represented in Figure 5. No random noise is added to the curves, since our purpose is to test the algorithms and the approximations (see Assumptions (1) to (3)). The application of equation (21) to i*~(s) and the division by pinc(XD) yields a set of points {I*~(h, X)/pinc(XD) }. The points corresponding to the strongest and sharpest reflections (h = 2, 3, 4, 5; see Fig. 6) are used to determine ao and v; these values are subsequently involved in the determination of the parameters imom(h/D) and of idlff(h/D ) (see section 2(e)(iii)). The points I$*(k/2DI2D, 2DX) and the values of a o and v are also used to determine imotif(k/2D ) (see section 2(c)(iv)). Tables 1 and 2 report the values of the para-
Table 1
Input and output values of i,.oa~(s) a o (A):
In
Out
In
Out
In
Out
lfl
Out
1-993 110"5
3 25
2.993 25"7
5 75
4"991 81"I
8 5
7"990 5-03
W=2
W= 1
W=2
W= 1
W=2
W=I
W=2
(N):
2 100
k = 2Ds
Model
W= 1
1 2 3 4 5 6 7 8 9 l0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
48 1 128 606 723 147 133 773 729 193 2 9 2 0 8 21 9 0 0 12 56 50 3 13 27 7
1 619 153 758 185 ll 1 24 0 14 54 12 6
61 1 181 620 516 154 188 758 594 185 26 II 2 l 6 24 l0 0 --6 15 60 54 --7 12 27 9
l 619 154 758 185 ll l 24 1 15 55 13 8
54 1 146 620 652 154 151 758 669 186 '14 ll 3 1 9 24 12 1 --1 15 03 55 2 15 30 13
1 620 154 758 186 II 2 25 2 18 60 23 29
62 1 154 620 684 154 145 758 698 186 11 12 3 2 13 26 14 3 6 19 67 62 16 26 42 34
I 621 156 760 190 14 7 37
49 l 143 621 734 156 139 760 718 190 l0 15 l0 7 21 37
?
Results of applying the algorithms to t h e simulated experimental curves i*m(S) (see eqn (37)) corresponding to the input values of t h e a r a m e t e r s {imotlr(k/2D), i a i . ( h / D ) , at), v}. The o u t p u t values are obtained by applying equations (28) (for It' = 1) and (35) (for W = 2) to t~m(s). D = 170 A. Note the excellent agreement, especially for W = 1.
V. Luzzati and L. Mateu
382
^
oo 15
P
=,
j.o
V "- I / I 0 0
30
.~ ~0
i
A 15
50
o-o =3~ V = 1/25
I
0
50
oo 15
o'0--5~ V = 1/75
I A
o? 15
30
O-o = 8 ~ V = 1/5
I 0"05
X-'
0
0.05
o
$
$
F i g u r e 5. Left frames. A few examples of simulated "experimental" curves ,i*m(s ) (eqn (37)) recorded at equally spaced channels (As = 7"6018 x 10 -5 .~-t). The curves correspond to different values of a D and v, and to tile input functions idi,(s) and i=otit(s) of Tables 1 and 2 (see also below). Right frames. Values of tlle ftmetion ,imotlr(k/2D): (O) l i ' = l (eqn (28)); ( O ) I 1 ' = 2 (eqn (35)) (see also Table l). The continuous curves are obtained by Shannon's (1949) interpolation. Tile uppermost frame reports the input curve Si~o,r(S) used in the simulation; that curve corresponds to the electron density profile reported by Mateu & MorSn (1986, their Fig. 5). Note the very close agreement of all the curves Simo,ir(S). With regard to S/~itf(s), the input and the output curves are indistinguishable at the scale of tile drawing (see Table 2).
Order-Disorder in Myelinated Nerve Sheaths. I I
>X
0
50 ~ I000~ * ~
5 0
5 0
5
.~ I00
*i ~
i h=
I 3
~o=2
=,= I/100
5 1/75
1/25
8~ I/5
Figure 6. A few points I~.(/dDID, XD)/Pin~(XD) (see eqns (26)) obtained by the application of algorithm (2]) to the curves i*~(s) (see Fig, 5) at the lattice points h = 2, 3, 4, 5 and for X = 1/2, 3/2 . . . . . The- straight lines represent the function imo.r(h/D)exp[-X(v+2~2a~s2)] (see eqns (28) and (26d)) corresponding to the output values of {imo.r(k[2D), idifr(h/D), aD, v} reported in Table l (W = ]) and in Table 2. Note the excellent fit of the points to the straight lines.
meters ao, v, {imot~e(h/D)}, {imoti,(k/2D)}, {idir,(h/D)} used in the calculation of i*m(S) (the input) and the corresponding values obtained from tire analysis of tile curves i*m(S) (tile output). The results (see also Figs 5 and 6) bear on several aspects of the mathematical treatment and of its implementation: (1) The l)oints {[l*m(h/D,X)/pi.r excellent agreement with the curve:
383
(3) The points {imo,~f(h/D)}w=t and {imo,,(k/2)}w=2 are very close to those used for the determination of i*m(S) (Table 1); as a consequence, the output function imotlf(s) obtained by Shannon's (1949) interpolation of tire points {imot~f(k/2D)} is ahnost identical with the input function. This is a most interesting novelty of this work. (4) Tile algorithm accurately discriminates the "noise" idlrf(S) from tile "signal" imotlf(S) (Fig. 5 and Table 2). Tire overall conclusion is that the algorithm provides an efficient and accurate tool for solving the problems defined in Introduction: namely, discriminate the diffirse scattering from the signal; determine tim parameters ao and v, which specify the stacking disorder and the finite size of the crystallites, respectively; and draw the complete intensity curve of a single motif. Naturally, this section is aimed at testing the algorithms, and tells nothing about the relevance of the mathematical model to tile physical structure of any particular system. This problem will be treated in paper II. As a final comment we point out that the mathematical procedure is based upon simple geometric operations performed in real space. In addition, the algorithm is simple and easy to implement in a computer. In our hands, and using a personal computer, the complete analysis of a typical X-ray scattering experiment takes less than one hour; simple hardware and software imtn'ovements would easily cut tiffs time down to minutes.
are in Appendix
[imotif(h/D) exp [-- X(v + 2nZaghZ/DZ)]
Determination of the parameters
(Fig. 6). (2) The output values of a D and v agree almost perfectly with those used in the calculation of i~]m(S) (Table 1).
Tile applieation of equation (21) to tile experimental c u r v e i*exp(Sj) yields tile values of the flmetion l*,p(h/DlD, XD) at a set of points {h, X}. These are the data of the problem: tile unknowns are ao, v, {imot,t(k/eD)}, {idifr(h/D)}. We solve the problem as follows. Firstly, we determine the values of a and v; we analyze, for that purpose, a subset of sharp and strong reflections (for example, h = 2, 3, 4, 5 in tile case of Figs 5 and 6), for X = 3 / 2 , 5 / 2 , . . . . Tile relevant equation is (28):
Table 2
Input and output values of iaisf(S ) ao (A): (N):
h= Ds
Model
0 1 2 3 4 5 6 7 8 9 l0 II 12 13
365 775 1013 1085 960 808 355 217 195 181 207 342 413 358
2 100
3 25
5 75
8 5
365 776 1024 1079 961 786 330 217 190 182 212 355 408 347
356 776 1023 1079 961 785 330 217 190 181 211 354 407 345
365 776 1023 1078 959 784 330 216 188 179 207 348 396 323
365 776 1021 1076 955 780 326 210 176 152
The 1st eohmm reports the input values used in the calculation of i%(s). The other values are those obtained by applying eqn (29) to i*m(s).
I*~p(h/DlD, XD)x~ t = imotif(h/D)Pine(XD )
x exp [-- X(v + 2~2o'~ h2[D2)] x eosh (v/2 + rc2a~h2/D2).
(38)
We solve tlfis equation by an iterative i)roeedure. Elementary transformations yield: Yp(X, h) = I*~p(h/D]D, XD)/[p,,r x cosh (vp_l/2+uza~ p_lh2[DZ)] (39a) lg YX.h.p = ah.p-X(bp+%h:),
(39b)
ah. p = lg [imotlf(h/D)p ],
(39c)
bp = vp, o_~_2 ,r~z 9 Cp = --1~ OD, pl.lJ
(39d) (39e)
V. Luzzati and L. Mateu
384
The suffix p specifics the cycle of the iteration: in each cycle p, bp_ l and cp_ t are treated as constants, {{ah.p}h, bp, %} as unknowns. The system is solved by least squares, minimizing the quantity: Z2 = ~ ~ [lg Yx.h.t,-ah.p+X(bp+cph2)]2[(a2,,r) p, X
h
(40a)
This work was supported in part by an exchange grant CNRS-CONICIT and research grants from the Association Fran~aise contre les Myopathies, the Association Fran~aise pour la Recherche M6dicale and CONICIT (Sl:1413). The authors are grateful to Dr Alain Henaut for the generous loan of computing equipment, to Dr Salvino Ciccariello for valuable suggestions and to Dr Rodo[fo Vargas for a critical reading of the manuscript.
where (a~gr)p is the variance of lg Yx, h,p. Assuming t h a t at. << l*,p the variance takes the form: ajgr(x,h,p) ,,, a (x,h,p)[ y2 (x,
h,p)
~
at.L (x.h)/(Ir@
z
(X, h).
(40b)_
We also assume that the variance of I*,p is independent of X and h. We set the partial derivatives of X2 equal to zero, we eliminate the variables {imoar(h/D)} and thus obtain two equations of the unknowns aD, p, vp; the final values of ao and v are obtained when convergence is achieved. Once ao and v are known, imotit(h/D) can be determined for any value of h b y minimizing the quant i t y (see eqns (28) and (39a):
X2 = ~ [I*,p(h/DlD, XD)--imot~r(h/n) X
x H2(h/DID, X, ao, v)]2[al,] -2,
(41)
with respect to i~otir(h]D). Assuming again t h a t the variance of l*~p is independent of I*,p and of X, the result is:
imotir(h]D) = [ ~ I:,v(h/DID, XD) • HI(h[DID, X, ao, v)I ...I
)t Icosh (v/2 + u2a2oh2/D 2) I._
x ~. It~(hlnln, X , ~ , v)
]-'
9 (42)
X
The values of the {Qi~dh[D)} are determined using equation (28):
iaift(h/D) = 21*r
D / 2 ) - imotlf(h/D)
[1 +Ht(h]DID, 1, aD, v)]}.
(43)
Finally, the parameter i~otif(k[2D ) is determined for each value of k by applying the algorithm above to equations (34):
[-_ =
x2D)
x H3(k/2DID, X, av, v)/ ...1
•
H~(k]2D]D, X, al), v)
1'
.(44)
References Blaurock, A. E. & Nelander, J. C. (1976). J. Mol. Biol. 103, 421--431. Chabre, M. & Cavaggioni, A. (1975). Biochim. Biophys. Acta, 389, 336-343. Guinier, A. (1964). Thdorie et Technique de la Radiocrlstallographie, Dunod, Paris. Hosemann, R. & Bagchi, S. N, (1962). Direct Analysis of Diffraction by Matter, North-Holland, Amsterdam. Inouye, H., Karthigasan, J. & Kirschner, D. A. (1989). Biophys. J. 56, 129-137. Kirschner, D. A., Ganser, A. L. & Caspar, D. L. D. (1984). In Myelin (Morell, P., ed.), pp. 51-95, Plenum Publishing Co., New York. Luzzati, V. & Taupin, D. (1984). J. Appl. Crystallogr. 17, 273-285. Luzzati, V. & Taupin, D. (1986a). J. Appl. Crystallogr. 19, 39-50. Luzzati, V. & Taupin, D. (1986b). J. Appl. Crystallogr. 19, 51-60. Mateu, L. & Mor~n, O. (1986). Biochim. Biophys. Acta, 862, 525-534. Mateu, L., Luzzati, V., Vargas, R., Vonasek, E. & Borgo, M. (1990). J. Mol. Biol. 215, 385-402. Moore, P. B. (1980). J. Appl. Crystallogr. 13, 168-175. Mor~in, O. & Mateu, L. (1983). Nature (London), 304, 344-345. Nelander, J. C. & Blaurock, A. E. (1978). J. Mol. Biol. 118, 497-532. Padr6n, R. & Mateu, L. (1982). Biophys. J. 39, 183-188. Padr6n, R., Mateu, L. & Requena, J. (1980). Biochim. Biophys. Acta, 602, 221-233. Schwartz, S., Cain, J. E., Dratz, E. A. & Blasie, K. (1975). Biophys. J. 15, 1201-1233. Shannon, C. E. (1949). Proc. Inst. Radio Enff. N.Y. 37, 10-21. Taupin, D. & Luzzati, V. (1982). J. Appl. Crystallogr. 15, 289-300. Vonasek, E., Mor~in, O. & Mateu, L. (1987). J. Neurocytol. 16, 105-114. Worthington, C. R. (1986). Biophys. J. 49, 98-101. Yaeger, M. J. {1975). In Neutron Scattering for the Analysis of Biological Structures (Schoenborn, B. P., ed.), pp. VII-77-106, Brookhaven National Laboratory, Upton, New York. Yaeger, M. J., Schoenborn, B. P., Engelman, D., Moore, P. & Stryer, L. (1980). J. Mol. Biol. 137, 315-348.
Edited by A. Klug