11
X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density ProfileVittorio Luzzati 1 *, Patrice Vachette 2 , Evelyne Benoit 3 and Gilles Charpentier 3 1 Centre de Ge ´ne ´tique Mole ´culaire, UPR 2167 CNRS associe ´e a ` l’Universite ´ Pierre et Marie Curie, 91198 Gif-sur-Yvette Cedex, France 2 Laboratoire pour l’Utilisation du Rayonnement Electromagne ´tique (LURE) associe ´ au CNRS, au CEA et au MR, Universite ´ Paris-Sud BP34, 91898 Orsay Cedex France 3 Laboratoire de Neurobiologie Cellulaire et Mole ´culaire UPR 9040 CNRS, 91198 Gif-sur-Yvette Cedex, France Synchrotron radiation X-ray scattering experiments were performed on unmyelinated pike olfactory nerves. The difference between the meridional and the equatorial traces of the 2-D spectra yielded the 1-D equatorial intensity of the macromolecular components oriented with respect to the nerve: axonal membranes, microtubules and other cytoskeletal filaments. These 1-D spectra display a diffuse band typical of bilayer membranes and, at small s, a few sharper bands reminiscent of microtubules. All the spectra merge at large s. The intensity of the axonal membrane was determined via a noise analysis of the nerve-dependent spectra, involving also the notion that the thickness of the membrane is finite. The shape of the intensity function indicated that the electron density profile is not centrosymmetric. The knowledge of intensity and thickness paved the way to the electron density profile via an ab initio solution of the phase problem. An iterative procedure was adopted: (i) choose the lattice D of a 1-D pseudo crystal, interpolate the intensity at the points s h Zh/D, adopt an arbitrary set of initial phases and compute the profile; (ii) determine the phases corresponding to this profile truncated by the thickness D/2; (iii) repeat the operation with the updated phases until a stable result is obtained. This iterative procedure was carried out for different D-values, starting in each case from randomly generated phases: stable results were obtained in less than 10,000 iterations. Most importantly, for D in the vicinity of 200 A ˚ , the overwhelming majority of the profiles were congruent with each other. These profiles were strongly asymmetric and otherwise typical of biological membranes. q 2004 Elsevier Ltd. All rights reserved. Keywords: axonal membranes; synchrotron radiation; X-ray scattering; phase problem *Corresponding author Introduction Excitable membranes, like other disorderly packed membrane systems, are uninviting objects for X-ray scattering analyses. Yet, a recent synchrotron radiation study of unmyelinated pike olfactory nerves raised some hope that the structure of these membranes might be brought within reach of an X-ray scattering approach. 1 We showed in that work that the scattering spectra of pike olfactory nerve display a diffuse band similar to the spectrum of isolated myelin membranes, accompanied by a few sharper bands reminiscent of microtubules. The analysis of the high-angle region (sO0.022 A ˚ K1 ) of the spectra, where the contribution of the non-membrane components is presumably negligible, showed that membrane thickness decreases as temperature increases, with a thermal expansion coefficient equal to K0.97( G0.19)!10 K3 8C K1 , a typical phenomenon of lipid-containing systems with 0022-2836/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. We owe more than can be said to Ray Kado’s unfailing interest for the structural aspects of electrophysiology. His untimely death put a cruel end to our collaboration. Let these papers be a tribute to his memory. Present address: P. Vachette, IBBMC, Ba ˆt.430, Universite ´ Paris-Sud, 91405 Orsay Cedex, France. Abbreviations used: SDF, small diameter filament; PON, pike olfactory nerve; MT, microtubule; FT, Fourier transform. E-mail address of the corresponding author: [email protected] doi:10.1016/j.jmb.2004.08.031 J. Mol. Biol. (2004) 343, 187–197

X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Embed Size (px)

Citation preview

Page 1: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

doi:10.1016/j.jmb.2004.08.031 J. Mol. Biol. (2004) 343, 187–197

X-ray Scattering Study of Pike Olfactory Nerve: Intensityof the Axonal Membrane, Solution of the Phase Problemand Electron Density Profile†

Vittorio Luzzati1*, Patrice Vachette2, Evelyne Benoit3 andGilles Charpentier3

1Centre de GenetiqueMoleculaire, UPR 2167 CNRSassociee a l’Universite Pierre etMarie Curie, 91198Gif-sur-Yvette Cedex, France

2Laboratoire pour l’Utilisationdu RayonnementElectromagnetique (LURE)associe au CNRS, au CEA et auMR, Universite Paris-SudBP34, 91898 Orsay CedexFrance

3Laboratoire de NeurobiologieCellulaire et MoleculaireUPR 9040 CNRS, 91198Gif-sur-Yvette Cedex, France

0022-2836/$ - see front matter q 2004 E

†We owemore than can be said tointerest for the structural aspects of euntimely death put a cruel end to othese papers be a tribute to his memPresent address: P. Vachette, IBBM

Universite Paris-Sud, 91405 Orsay CAbbreviations used: SDF, small d

PON, pike olfactory nerve; MT, mictransform.E-mail address of the correspond

[email protected]

Synchrotron radiation X-ray scattering experiments were performed onunmyelinated pike olfactory nerves. The difference between the meridionaland the equatorial traces of the 2-D spectra yielded the 1-D equatorialintensity of the macromolecular components oriented with respect to thenerve: axonal membranes, microtubules and other cytoskeletal filaments.These 1-D spectra display a diffuse band typical of bilayer membranes and,at small s, a few sharper bands reminiscent of microtubules. All the spectramerge at large s. The intensity of the axonal membrane was determined viaa noise analysis of the nerve-dependent spectra, involving also the notionthat the thickness of the membrane is finite. The shape of the intensityfunction indicated that the electron density profile is not centrosymmetric.The knowledge of intensity and thickness paved the way to the electrondensity profile via an ab initio solution of the phase problem. An iterativeprocedure was adopted: (i) choose the lattice D of a 1-D pseudo crystal,interpolate the intensity at the points shZh/D, adopt an arbitrary set ofinitial phases and compute the profile; (ii) determine the phasescorresponding to this profile truncated by the thickness D/2; (iii) repeatthe operation with the updated phases until a stable result is obtained. Thisiterative procedure was carried out for different D-values, starting in eachcase from randomly generated phases: stable results were obtained in lessthan 10,000 iterations. Most importantly, for D in the vicinity of 200 A, theoverwhelming majority of the profiles were congruent with each other.These profiles were strongly asymmetric and otherwise typical ofbiological membranes.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: axonal membranes; synchrotron radiation; X-ray scattering;phase problem

*Corresponding author

Introduction

Excitable membranes, like other disorderlypacked membrane systems, are uninviting objectsfor X-ray scattering analyses. Yet, a recent

lsevier Ltd. All rights reserve

Ray Kado’s unfailinglectrophysiology. Hisur collaboration. Letory.C, Bat.430,edex, France.iameter filament;rotubule; FT, Fourier

ing author:

synchrotron radiation study of unmyelinated pikeolfactory nerves raised some hope that the structureof these membranes might be brought within reachof an X-ray scattering approach.1

We showed in that work that the scatteringspectra of pike olfactory nerve display a diffuseband similar to the spectrum of isolated myelinmembranes, accompanied by a few sharper bandsreminiscent of microtubules. The analysis of thehigh-angle region (sO0.022 AK1) of the spectra,where the contribution of the non-membranecomponents is presumably negligible, showed thatmembrane thickness decreases as temperatureincreases, with a thermal expansion coefficientequal to K0.97(G0.19)!10K3 8CK1, a typicalphenomenon of lipid-containing systems with

d.

Page 2: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Figure 1. (From von Muralt et al., with permission.8)(a) Cross-section of an olfactory nerve. Bundles of tightlypacked nerve fibres are ensheathed by glial cells, g. Largergroups of bundles are separated by basement mem-branes, bm, with some collagen fibres, cf, and thinextensions of connective tissue cells; mitochondria, MI,are frequent. Magnification 53,000!. (b) Inset: highmagnification of cross-section of one nerve fibre. Thefibre is delimited by the axon membrane. In the axon aremicrotubules (MT), small-diameter fibers (MF) (SDF inthe text) and one larger tubule of axoplasmic reticulum(AR). Magnification 174,000!.

188 Electron Density Profile of an Axonal Membrane

hydrocarbon chains in a disordered conformation.2

Also, exposure to local anesthetics had the effect ofdecreasing axonal membrane thickness. Finally,membrane depolarisation, achieved by replacingNaCl with KCl in the extra-axonal medium,apparently induced a sizable fraction of themembranes to associate by apposition of theirouter faces.

Here, we approach the structure of these axonalmembranes in their native form. The endeavourinvolved two distinct steps. The first was to rid thespectra of the spurious non-axonal components andthereby identify the spectrum of the axonal mem-brane itself. This operation hinged upon threenotions; (i) the spectra used in this work aredifferences between equatorial and meridionaltraces of 2-D spectra and involve only the macro-molecular components oriented with respect to thenerve axis, presumably axonal membranes, micro-tubules and other cytoskeletal filaments;3 (ii) theintensity scattered by the axonal membranes is thesame in all the nerves, whereas the spuriousintensity varies from nerve to nerve; (iii) thethickness of axonal membranes is finite. Thesenotions paved the way to the elimination of thespurious intensity, to the use of Shannon’s theoremand eventually to a formulation of the problem inmanageable mathematical terms.

The second step was to determine the averageelectron density profile across the axonal mem-brane. The shape of the intensity function showedthat the profile is not centrosymmetric. We thus setabout finding a 1-D real function knowing themodulus, but not the phase of its Fourier transform.The issue hinged upon the notion that the thicknessof the membrane is finite and upon the knowledgeof the intensity function at all s values, and involvedan iterative procedure advocated by Stroud &Agard4 and by Miao et al.,5 which, starting fromrandomly generated phases, led to the electrondensity profile.

In the accompanying paper6 we discuss thestructural modulations induced in the axonalmembrane by depolarisation, osmotic stress and avariety of drugs, and we approach the elastic,thermodynamic and physiological properties of themembrane.

We have described previously1,7 some of thetechnical aspects of the work: e.g. fish handling,nerve dissection, electrophysiological controls,X-ray scattering experiments and data reduction. Itis worthwhile to stress that the electrophysiologicalactivity of all the nerves used in this work wastested before and after the X-ray scatteringexperiments.

Morphology of pike olfactory nerve

We draw from von Muralt et al.8 the morpho-logical description of pike olfactory nerve (Figure 1):“The nerve is about 1 mm in diameter and 50 mm to70 mm long. In cross-sections it is seen to be madeof up to about 1000 densely packed bundles, which

measure 15–20 mm in diameter; each of thesecomprises several thousands of non-myelinatednerve fibres. These nerve fibres exhibit a strikingregularity in size and shape. The majority have adiameter between 0.1 mm and 0.3 mm with a fewsmaller and a few larger fibres.. In regions wherefibres of uniform diameter are present, the packingpattern can result in quite regular hexagonalpatterns. Each axon is bounded by a denselystained cell membrane, showing the structure of abilayer about 80 A thick. In its axoplasm eachfibre contains, without exceptions, three elements.(a) 4–20 Microfilaments which are about 10 nmthick and often form more or less regularly spacedbundles; (b) 2–5 microtubules of circular cross-section, 25 nm in diameter, whose wall showsmoderate contrast and is about 4 nm thick; (c) oneor more smooth tubules of variable width andshape, formed by a typical densely stained bilayermembrane about 6 nm thick.

In longitudinal sections of the fibres allthese elements are also found. Microtubules and

Page 3: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Electron Density Profile of an Axonal Membrane 189

micro-filaments assume a straight course along thefibre and are very uniform in width; the membranetubules, however, form an axoplasmic reticulum.

The population of small axons (diameter0.1–0.5 mm) comprises 95% of all fibres.

The number of nerve fibres in the olfactory nerveis 1!107 for the small and 2!106 for the largerfibres. The nerve fibres comprise about 66% of thenerve volume, the glial cells some 22% whereas theintercellular gaps between densely packed nervefibres amounts to 8%.

In the context of the present study the amount ofaxonal membranes contained in the nerve is ofimportance. It was found, by estimation of thesurface density of these membranes, that 1 cm3 ofolfactory nerve contained 84,000 cm2 of axonalmembrane.”8

The cytoskeletal components called “micro-filaments” by von Muralt et al.8 do in factencompass a variety of filaments, mainly actin andintermediate filaments,3 whose diameters aresmaller than 60 A and 100 A, respectively. Weshall call small diameter filaments (SDF) all thefilaments whose diameter is not larger than 100 A.

Technical Aspects

Notation and abbreviations

PON, MTand SDF stand for pike olfactory nerve,microtubule and small diameter filament.

rax(r), Lax and Iax(s) are respectively electrondensity profile, thickness and intensity of the axonalmembrane; FT stands for Fourier transform.

Fourier transformation

r(r) is the average electron density over the planeparallel with the membrane, at the distance r fromits centre. The FTof r(r) is called the structure factorF(s), where s is the distance in diffraction space fromthe origin of the spectrum (sZ2(sinq)/l (in AK1),where 2q is the scattering angle and l thewavelength of the X-rays (in A)). The modulusand phase of F(s) are jF(s)j and f(s):

FðsÞ ¼

ðNKN

rðrÞexpð2pirsÞdr ¼ AðsÞ þ iBðsÞ (1a)

where:

AðsÞ ¼

ðNKN

rðrÞcosð2prsÞdr ¼ FðsÞcos½fðsÞ� (1b)

BðsÞ ¼

ðNKN

rðrÞsinð2prsÞdr ¼ FðsÞsin½fðsÞ� (1c)

½FðsÞ�2 Z IðsÞZA2ðsÞCB2ðsÞ (1d)

fðsÞZ tnK1½AðsÞ=BðsÞ� (1e)

By Fourier inversion:

rðrÞ ¼

ðNKN

½AðsÞcosð2prsÞ þ BðsÞsinð2prsÞ�ds (2)

When r(r) is centrosymmetric, r(r)Zr(Kr) andB(s)h0.

Shannon-expansible functions

A function g(r) (centrosymmetric for the sake ofthe argument) that vanishes uniformly outside afinite range KD/2%r%D/2 is defined fully by thevalues of its FT G(s) at the points shZh/D9:

GðsÞ ¼XNh¼0

GðhÞSðs; h;DÞ (3a)

Sðs; h;DÞhs0 Z sin½pðDsChÞ�=½pðDsChÞ�

Csin½pðDsKhÞ�=½pðDsKhÞ� (3b)

Sðs; 0;DÞZ sinðpDsÞ=ðpDsÞ (3c)

The function G(s) will then be said to be Shannon-expansible. One example of Shannon-expansiblefunction relevant to this work, is the intensity Iax(s)whose FT:

PaxðrÞ ¼

ðNKN

raxðRÞraxðrKRÞdR (4)

has a support D equal to twice the thickness Lax ofthe axonal membrane.It must be stressed that a function that is

Shannon-expansible for DZDa is Shannon-expansible for any D larger than Da.The main practical interest of Shannon-

expansible functions G(s) is to be entirely definedby the set {G(h/D)}. If, moreover, G(s) is supposed tobe negligibly small beyond sZhmax/D, then thefunction G(s) can be interpolated at any s-valuefrom the finite set {G(h/D)jhj%hmax}:

GðsjÞZXhmax

hZ0

Gðh=DÞSðsj; h;DÞ (5)

One equation (5) is available for each experimentalchannel sj. If the number of channels is equal to orlarger than hmax, then the system of equations canbe solved into the {G(h/D)}. As a rule, nevertheless,D is not known beforehand. Luzzati & Taupin10–12

have advocated a procedure to tackle this problem.The system of equation (5) is solved for differentvalues of D, using for example a least-squaresalgorithm that yields the most probable value andthe associated error of the {G(h/D)} and the value ofsome parameters, R2(D) and c2(D), assessing thequality of the least-squares approximation. Some ofthese (and other) parameters are expected to takeconstant values, as D is larger than a minimumDmin. The parameters are then plotted as a functionof D, and the D-threshold is sought. This procedureis documented in Results.

Page 4: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

190 Electron Density Profile of an Axonal Membrane

Error analysis

The standard error scount associated with count-ing statistics and its propagation through thecommon mathematical operations were estimatedas usual.

When dealing with N experimental values In(s),whose scount,n(s) is known, it is convenient todistinguish the experimental variance sexp,hIi

2 (s)from the intrinsic variance due to counting errors,scount,hIi2 (s):

hIðsÞiZM1ðsÞ=M0ðsÞ (6a)

s2exp;IðsÞZM2ðsÞ=M0ðsÞK ½M1ðsÞ=M0ðsÞ�

2 (6b)

s2count;IðsÞZN=M0ðsÞ (6c)

where:

MtðsÞZX

n

ItnðsÞ=s

2count;nðsÞ (6d)

s2exp;hIiðsÞZs2

exp;IðsÞ=N (6e)

s2count;hIiðsÞZs2

count;IðsÞ=N (6f)

The mathematical analysis often involved the least-squares solution of a system of N equations,13

whose output is the most probable value and thestandard error of the P unknowns {p1, p2,.,pP}, aswell as the statistical parameters:

c2 ZX

j

½ðyexpðsjÞKycalðsjÞÞ=sj�2=ðN KPÞ (7)

R2 Z 1K ðN KPÞc2Þ=X

j

½ðyexpðsjÞ

K hyexpðsjÞiÞ=sj�2 (8)

where yexp(sj) and sj are the data and their standarderrors, ycal(sj) is the calculated value of y(sj) (afunction of the unknowns {p}). If the statisticalproblem is posed correctly and the sj are assessedproperly, then c2 is expected to be equal to 1. Thevalue of R2 depends on the accuracy of the data, andis expected to increase as accuracy increases; R2

tends to 1 for infinitely accurate data and ideallycorrect solution of the problem. The assessment ofthe statistical weight of the data will be discussed indue place.

Figure 2. A few examples of native nerve X-rayscattering spectra. The code (for example 5a/00) identifiesthe nerve (5a) and the year of the experiments (00 for2000). The intensities are all drawn on the same relativescale. Note that the spectra are strongly nerve-dependentat small s and merge beyond sz0.024 AK1.

Results

The spectra used in this work were recorded infour campaigns: 1998, 1999, 2000 and 2001. Thechannel separation Ds was the same in eachcampaign but slightly different in the differentcampaigns. Moreover, the temperature was notquite the same: 15 8C in 1998, 20 8C in 1999 and 2001,20.5 8C in 2000. After a thermal corrections20 8C ¼ sTð1K0:97!10K3ðTKT20ÞÞ,

1 the spectra of

the 1998, 1999 and the 2000 campaigns were allinterpolated at the points s of the 2001 campaign, at20 8C.

The steep rise of the spectra in the vicinity of thebeam-stop prevented collecting reliable data belowsz0.004 AK1; besides, the spectra reach a weak andapparently constant value beyond sZ0.04 AK1.Therefore, the useful range of the data was0.004!s!0.04 AK1.

I(sj), the normalised intensity at the jth channel, isa function of the experimental counts i(sj):

IðsjÞ ¼ ½iðsjÞK iN�=½iav K iN� (9)

iN is the asymptotic value of i(sj) (more precisely theaverage from sZ0.040 AK1 to 0.045 AK1). iav is theaverage of i(sj) over the range 0.019!sj!0.026 AK1.The values of iN and of the normalisation factor½iavK iN� were further refined in the course ofthe analysis (see the accompanying paper6).

The data belonged to scattering experimentsperformed on 48 native nerves. At small s(!0.023 AK1) the spectra of different nerves dis-played conspicuous differences, whose amplitudedecreased with increasing s; beyond approximatelysZ0.024 AK1 all the normalised spectra merged(Figure 2). We ascribed the invariant part of thespectra (0.024!s%0.04 AK1) to the axonal mem-brane itself.1 The nerve-dependent component,observed below sZ0.024 AK1, yielded the infor-mation required to discriminate the axonal-membrane spectrum from the non-membranecomponents.

Page 5: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Figure 3. hInative(s)i is the average spectrum of 48 nativenerves. I#tðsÞ and I#jðsÞ are the functions used to takeaccount of the contribution of the non-axonal-membranecomponents of the spectra (equations (A6b) and (A6c)).

Electron Density Profile of an Axonal Membrane 191

Ridding the spectra of the non-membranecomponents: thickness and intensity of theaxonal membrane

The PON spectra consist of a predominant bandcharacteristic of bilayer membranes, accompaniedby a few sharper bands, whose position and relativeintensity are reminescent of MT.1

The 1-D spectra analysed in this work wereobtained by subtracting the meridional from theequatorial trace of 2-D spectra,1 an operation aimedat discriminating the spectra of the macromolecularcomponents oriented with respect to the nerve fromthe intensity originating from the unorientedcomponents. The predominant oriented com-ponents are presumably axonal membranes, MTand SDF (see Morphology of pike olfactory nerve,above). The concentration of the different com-ponents of the cytoskeleton will henceforth benormalised to axonal membrane area.

The discrimination of the axonal membranesignal from the noise originating from the cyto-skeleton is treated in Appendix A. We show therethat the noise takes the form of a linear combinationof two functions, I#tðsÞ and I#jðsÞ (see equations (A6a)and (A6b); and Figure 3). As for hInativeðsÞi (A8):

hInativeðsÞi ¼ IaxðsÞ þ ktI#tðsÞ þ kjI#jðsÞ (10)

Since, moreover, the thickness of the axonalmembrane is finite, then Iax(s) can be replaced byits Shannon expansion (equation (3a)–(3c)):

hInativeðsÞi ¼Xhmax

h¼0

Iaxðh=DÞSðD; s; hÞ þ ktI#tðsÞ

þ kjI#jðsÞ (11)

where kt, kj and the {Iax(h/D)} are the unknowns.Equation (11) acts as a filter against those com-ponents of the function !Inative(s)O that are linearcombinations of I#tðsÞ and I#jðsÞ and/or are notShannon-expansible (note that I#tðsÞ and I#jðsÞ arenot Shannon-expansible functions).One equation (11) is available for every channel.

Since in our experimental conditions the number ofchannels is 111, the system is amply redundant. Asdiscussed in Technical Aspects, the equations weresolved for different D-values. The weight wj of thedata was supposed to depend on counting errorsand on the errors affecting the functions I#tðsÞ andI#jðsÞ. We adopted wjZ1=½0:55s2

exp;IðsÞ�, the factor0.55 being chosen so that the minimum c2(D) isclose to 1 (see Figure 4).Some of the parameters are plotted in Figure 4 as

functions of D. In keeping with the expectation, theD-dependence is strong at small D and flattens outas D increases. The separation between the tworegimes is in the vicinity of DZ200 A.The parameters {Iax(h,D)} are plotted in Figure 5

as a function of sh (Zh/D) and for differentD-values. Note that beyond h/Dz0.01 AK1 thepoints barely depart from the continuous Iax(s)curve. At smaller h/D, the points are scattered morewidely, but still within error bars, as long at least asD%200 A.We thus infer that Dmax is smaller than 200 A, and

that the thickness Lax of the axonal membrane issmaller than 100 A. The function Iax(s) can becomputed by Shannon interpolation of the{Iax(h/200)}.

Electron density profile rax(r): ab initio approachto the phase problem

The subsequent step was to determine theelectron density profile rax(r) of the axonal mem-brane knowing the amplitude but not the phase ofits FT, the well-known phase problem of X-raycrystallography.As for crystals, the problem is complicated

further by the fact that the intensity is sampled atthe discrete points of a lattice whose dimensions areof the order of the inverse of the dimensions of themolecule.14 Sayre15 has shown that one possibleway out of the problem is to oversample theintensity.As for the axonal membrane, the intensity can be

sampled at any point s via Shannon’s interpolationof the {Iax(h/D)}. Should the profile be centro-symmetric, then determining the phases (0 or p inthis case) would be a straightforward operation. Butthe presence of a minimum (near sZ0.008 AK1,Figure 5) without Iax(s) approaching zero tellsagainst this hypothesis. Indeed, Aax(0) is

Page 6: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Figure 4. The D-dependence of several parameters determined by the least-squares solution of equation (11) fordifferent D-values. R2 and c2 are parameters assessing the quality of the least-squares solution; kt and kj are theparameters used to take account of the non-axonal membrane components (equation (A8)); I(0) is the intensity of thezero-order reflection; Z is the average figure of merit of the profiles obtained with different sets of randomly generatedinitial phases and for the same D-value (equation (13)). Note that at small D all the parameters display a strongD-dependence, and flatten out beyond a threshold located in the vicinity of DZ200 A.

192 Electron Density Profile of an Axonal Membrane

proportional to the buoyant density of axonalmembranes, known to be positive;16 should, more-over, rax(r) be centrosymmetric, then the sign ofAax(s) would change with increasing s, since r(0),which is proportional to the integral of Aax(s), isnegative (see Figure 3 of Luzzati et al.1).

Therefore, the problem is the following: knowingthe intensity Iax(s) and the thickness Lax of theaxonal membrane, determine its electron densityprofile rax(s) without making assumptions onsymmetry. The implementation in the phasingprocess proceeds as follows. Choose a length Dequal to or larger than twice the thickness of theaxonal membrane and compute the set of intensities{I(h/D)} (see under Shannon-expansible functions).The r-space equivalent of sampling the s-space atthe points h/D is to transform r(r) into the periodicfunction:

r#ðr;DÞ ¼XNr

m¼KN

ðr þ mDÞ (12)

Since, moreover, r(r) has the support LZD/2,

then r#(r,D) must vanish over the range KD/2!r!KL/2 and L/2!r!D/2.

We tackled the problem using the followingiterative approach:4,5 (i) choose one D-value andcompute the intensities {I(h/D}; (ii) generate aninitial phase-set {f0(h/D)}; (iii) compute r1(r), theFT of {F(h/D),f0(h/D)}; (iv) Fourier transform thefunction r1(r) truncated by the support L anddetermine the phases {f1(h/D)}; (v) repeat theoperation until the phase-set is stable. A figure ofmerit Z(n,D) was ascribed to each iteration n:

Zðn;DÞZX

h

½Iaxðh=DÞ

K Itðn; h=DÞ�2=X

h

½Iaxðh=DÞ�2 (13)

where It(n,h/D) is the intensity of the truncatedrn(r,D).

It must be stressed that I(s) is invariant withrespect to r-origin position and to the signs of both rand r(r).

From the practical viewpoint, we chose DZ

Page 7: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Figure 5. Intensities I(h,D) computed by applyingShannon’s algorithm to the data hInative(sj)i, for differentD-values. Also plotted are the average spectrum of thenative nerves and the axonal membrane intensitycomputed by Shannon’s interpolation of the {I(h,200)}.Note that beyond sz0.008 AK1 the points I(h,D) followclosely the function Iax(s). At smaller s, the points arescattered more widely, although within error bars, as longat least as D%200 A. The points I(0,D) are plotted also inFigure 4.

Electron Density Profile of an Axonal Membrane 193

200 A, determined the {I(h/D)}, set f(0)Z0 (namelyF(0) positive16), generated a random set of phases{f0(h/D)} and entered the iterative process. A stableresult was reached in a few thousand iterations. We

Figure 6. Electron density profiles r(r) obtained byapplying the iterative algorithm to the {I(h,200)} for tensets of randomly generated initial phase-sets. Note thatthe profiles are all congruent and that their positions areshifted slightly in the r-direction. The numbers inparentheses specify the number of overlapping profiles.

oriented r(r) arbitrarily with the higher of thepositive bumps on the positive r-side (Figure 6). Weperformed the operation for ten different sets ofrandomly generated initial phases.The ten profiles are plotted in Figure 6. All the

profiles are almost exactly congruent with eachother and somewhat scattered in the r-direction.The profiles display the central trough surroundedby two positive bumps characteristic of lipoproteinbilayers. Also, the profiles are highly asymmetric, asexpected from physiologically asymmetricmembranes and as observed in myelin.2

The profile eventually adopted was the averageof the ten profiles of Figure 6, shifted in ther-direction so as to bring the position of the minimain coincidence. The functions Aax(s), Bax(s) and Iax(s)

Figure 7. Electron density profile rax(r) of the axonalmembrane. rax(r) is the average of the ten profiles ofFigure 6 shifted in the r-direction so as to bring theirminima into coincidence. Iax(s), Aax(s) and Bax(s) are theintensity, the real and the imaginary parts of the FT ofrax(r), respectively.

Page 8: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Table 1. Intensity and structure factors of the axonalmembrane, for DZ200 A

h Iax (h) Aax (h) Bax (h)

0 0.5603 C0.7485 0.00001 0.3143 C0.4814 C0.28732 0.2695 K0.1529 C0.49613 0.8049 K0.6845 C0.57994 1.0584 K0.9021 C0.49465 0.8468 K0.8492 C0.35446 0.3999 K0.5937 C0.21777 0.0868 K0.2750 C0.1059

Intensity, real and imaginary components of the structure factorswere determined by Fourier-transformation of the average rax(r)of the profiles yielded by the iterative phasing process applied toten randomly-generated initial phase-sets for DZ200 A.

194 Electron Density Profile of an Axonal Membrane

(Figure 7 and Table 1) were obtained by Fouriertransformation of rax(r).

Other aspects of the present approach to thephase problem are discussed in Appendix B.

Discussion and Conclusions

It is timely to discuss some technical aspects ofthe data collection and reduction procedures. Asmentioned under Morphology of pike olfactorynerve and Results, the 1-D spectra used in this workinvolve the macromolecular components orientedwith respect to the nerve fibre, presumably axonalmembranes, MT and SDF. The small s limit of thespectra was 0.004 AK1. The spectra were assumedto vanish beyond sz0.04 AK1, although someevidence (see Appendix B) points at small butsignificant truncation distortions. A rewardingobservation validating both the experimental tech-niques and the data reduction procedures was thatthe spectra of pairs of nerves dissected from thesame animal were identical, in spite of conspicuousdifferences displayed by nerves from differentanimals (results not shown).

A critical step was the removal of the intensityscattered by non-axonal membrane components.We worked out a noise-filtering analysis basedupon the nerve-dependence of the spectra and thefinite thickness of the membrane. We ascribed theresidual spectrum to the axonal membrane itself.

The presence of a deep minimum in the intensitycurve, without the intensity approaching zero,suggested that the profile lacks mirror symmetry.The final step was to determine the electron densityprofile compatible with this intensity. The problemcould be tackled thanks to the finite thickness of themembrane and to the knowledge of its continuousintensity function. We adopted an iterative pro-cedure,5 which, starting from a set of randomlygenerated phases, led to a stable profile, inde-pendent of the initial phase set. The final profile wasstrongly asymmetric (Figure 7).

The departure from mirror symmetry deserves afew comments. This notion hinges upon theobservation that the intensity at the minimum of

Iax(s) is far from negligible. The presence of otheroriented macromolecular components of the nerveand/or of sharp distortions of the axonal mem-branes away from planar symmetry, two possiblesources of the intensity observed near sZ0.008 AK1,is inconsistent with the electron microscopeevidence (Figure 1) and with Iax(s) behaving as aShannon-expansible function (Figure 4). Therefore,the lack of mirror symmetry is a direct consequenceof the noise analysis. Note also that the overallshape of the asymmetric electron density profile,namely the relative position and height of themaxima and the minimum (Figure 7), are consistentwith other biological membranes. In this respect, itis worthwhile to stress that only in a handful ofbiological membranes, all myelins, have the elec-tron density profiles been determined convincingly,and that the profiles have been shown to beasymmetric to a degree that varies with the nervefrom which the myelin is extracted.2

Acknowledgements

Our work on pike olfactory nerve was initiated inclose collaboration with Leonardo Mateu. We aregrateful for his interest in the recent developments.We thank Cecile Merigoux for help with some of thecomputations, Dominique Bigot for skilledtechnical assistance, and the staff of PiscicultureVasseur (28480F-Beaumont-les-Autels) for theirfriendly and competent cooperation.

Appendix A: Assessing the contributionof the non-axonal-membranecomponent to the scattering spectra

The intensity scattered by the nth nerve is thesum of the spectrum of the axonal membrane and ofthe spectra of the various oriented filaments(microtubule (MT) and small-diameter fibre(SDF)). The fact that the spectra vary from nerveto nerve (see Figure 2 of the main text) shows thatthe concentration of the different cytoskeletalcomponents is different in the different nerves butdoes not tell whether their relative concentration isthe same in all the nerves. We proceed to show howto exploit the heterogeneous distribution of the non-axonal membrane components in working out amathematical filter against the noise generated bythe cytoskeletal components.

Without loss of generality, the differently orientedmacromolecular components, axonal membraneand cytoskeletal fibres, can be associated formallyin a number of bundles, each bundle containingdifferent proportions of the various elements. In thiscase, the intensity takes the form:

InðsÞZ I#axðsÞCtnsItðsÞCjnsIjðsÞC/ (A1)

where I#axðsÞ is the sum of the intensity of the axonal

Page 9: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Figure A1. Analysis of the moments mlðsÞ of the 48spectra of native nerves, for lZ4, 6, 8, 10 (Appendix A).Upper frame: s-dependent plot of the points lnf½mlðsÞ�

2=l

=½m2ðsÞ�g; lnðalÞ is the s-average at large s (0.015!s!0.025). Lower frame: s-dependent plots of the pointsðlk4Þlnf½mlðsÞ�

2=l=½m2ðsÞal�g, where lk4 is a factor adjustingthe lth curve to the points lZ4. Note how closely the fourcurves fit together.

Electron Density Profile of an Axonal Membrane 195

membrane and of the filaments possibly associatedwith it, tn, jn, etc. are the number of bundles of typet, j, etc. of the nth nerve per unit area of axonalmembrane. Iax(s), ItðsÞ, IjðsÞ, etc. are the intensity ofthe axonal membrane and of the filaments of eachbundle. ItðsÞ, IjðsÞ, etc. are s-dependent functions,identical in all the nerves, tn, jn etc. are nerve-dependent and s-independent parameters.

We strive to eliminate the non-axonal membraneterms from equation (A1). We rely for this purposeon the moments of the nerve-dependent distri-bution of the {In(s)}:

mlðsÞ ¼ h½DIðsÞ�li

¼ h½ðDtÞsItðsÞ þ ðDjÞsIjðsÞ þ/�li (A2a)

where hXi is the average of Xn over n and:

DInðsÞZ InðsÞK hInðsÞi (A2b)

Assuming that the bundles of type t are predomi-nant over the other non-axonal membrane bundles,namely that ðDtÞsItðsÞ[ ½ðDjÞsIjðsÞC.� (seeequation (A1)), and that tn, jn etc. are statisticallyindependent parameters, then the odd-order cross-terms of mlðsÞ are negligible (see equation (A2a))and:

m2ðsÞzhðDtÞ2i½sItðsÞ�2½1Cc2ðsÞ� (A3a)

mlðsÞzhðDtÞli½sItðsÞ�lf1C ½lðlK1Þ=2�

!½hðDtÞlK2ihðDtÞ2i=hðDtÞli�c2ðsÞg (A3b)

where:

c2ðsÞ ¼ hðDjÞ2i½sIjðsÞ�2=hðDtÞ2i½sItðsÞ�

2/1 (A3c)

Eliminating sItðsÞ from equations (A3a)–(A3c)yields:

½mlðsÞ�2=l=m2ðsÞzal½1Cblc

2ðsÞ� (A4a)

where:

al Z hðDtÞli2=l=hðDtÞ2i (A4b)

bl Z ½ðlK1ÞhðDtÞlK2ihðDtÞ2i=hðDtÞli�K1 (A4c)

If blc2ðsÞ is also smaller than 1, then:

lnf½mlðsÞ�2=l=m2ðsÞgzlnðalÞ þ blc

2ðsÞ (A4d)

The functions lnf½mlðsÞ�2=l for the 48 spectra of

native nerves and for lZ4, 6, 8, and 10 are plotted inFigure A1. The plot displays some remarkablefeatures: (i) at small s all the moments ares-dependent; consequently c2(s)s0 (see equation(A3c)) and equation (A2a) must contain at least twoterms; (ii) blc

2ðsÞ is always smaller than 1; (iii) thefunctions c2(s) relevant to the different moments allhave the very same form. These observationsvalidate the hypotheses that blc

2ðsÞ/1 and that tand j are statistically independent parameters andtell that the number of non-axonal membranebundles is larger than 1. For the sake of simplicity,

the non-axonal membrane terms of equation (A1)will henceforth be limited to t and j.The function tlðsÞ:

tlðsÞ ¼ lnf½mlðsÞ�2=l=½alm2ðsÞ�g ¼ blc

2ðsÞ (A5a)

is defined experimentally. Two other experi-mentally defined functions are available (seeequations (A1) and (A3a)):

hInðsÞiZ I#axðsÞC htisItðsÞC hjisIjðsÞ (A5b)

m2ðsÞzhðDtÞ2i½sItðsÞ�2 C hðDjÞ2i½sIjðsÞ�

2� (A5c)

Eliminating sItðsÞ] and sIjðsÞ from equations (A5a)–(A5c) yields:

hInðsÞi ¼ I#axðsÞ þ k#tI#tðsÞ þ k#jI#jðsÞ (A6a)

Page 10: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

196 Electron Density Profile of an Axonal Membrane

where:

I#tðsÞ ¼ fm2ðsÞ=½1þ tlðsÞ=bl�g1=2 (A6b)

I#jðsÞ ¼ I#tðsÞ½tlðs�1=2 (A6c)

k#t Z hti=½hðDtÞ2i�1=2 (A6d)

k#j Z hji=½blhðDjÞ2i�1=2 (A6e)

The values of al and bl can be computed easily forlZ4 (equation (A4a)–(A4d)):

a4 Z ½hðDtÞ4i�1=2=hðDtÞ2i (A7a)

b4 Z ½3=ða4Þ2�K1 (A7b)

Since a4Z1.41 (Figure A1), b4Z0.51.If, moreover, the intensity of the hypothetical

non-axonal membrane terms of I#axðsÞ also takes theform of a linear combination of I#tðsÞ and I#jðsÞ, then:

hInðsÞi ¼ IaxðsÞ þ ktI#tðsÞ þ kjI#jðsÞ (A8)

where kt and kj are s-independent parameters thatneed not be defined explicitly.

The implementation of equation (A8) is discussedin the main text.

Figure A2. A few samples of the four main types ofprofiles obtained by applying the iterative algorithm tothe sets of intensities interpolated at DZ140 A, 160 A,180 A, 200 A, 220 A, 240 A and 260 A (see Appendix B),and for randomly generated initial phase-sets. Note thatthe predominant type of profile is I (40 out of 60 profiles)and that it is also the one that best agrees with theproperties of lipoprotein bilayers.

A correction of counting errors

Every measurement In(s) is corrupted by acounting error 3n(s) whose !32(s)OZm2(s) isknown. If the distribution of the {3n(s)} is gaussian,then:

h32pðsÞiZm2pðsÞZ 1:3:5.ð2pK1Þ ½m2ðsÞ�p (A9)

If M2q(s) is the experimental (2q)th moment:

M2qðsÞZ h½IðsÞK hIðsÞi�2qi (A10)

and m2q(s) the 3-corrected moment, then:

M2qðsÞ ¼X2q

2p¼0

½2qC2p�m2ðqKpÞðsÞm2pðsÞ (A11a)

where 2qC2p is the binomial coefficient:

2qC2p Z ð2pÞ!=½ð2pK2qÞ!ð2qÞ!� (A11b)

Equation (A11a) and (A11b) was applied step bystep to the moments. The results were:

m2 ZM2 Km2 (A12a)

m4 ZM4 K6m2 m2 K3ðm2Þ2 (A12b)

m6 ZM6 K15m2m4 K45ðm2Þ2m2 K15ðm2Þ

3 (A12c)

m8 ZM8 K28m2m6 K210ðm2Þ2m4

K420ðm2Þ3m2 K105ðm2Þ

4 (A12d)

m10 ZM10 K45m2m8 K630ðm2Þ2m6

K3150ðm2Þ3m4 K4725m2ðm2Þ

4

K945ðm2Þ5 (A12e)

Appendix B: Some comments on theapproach to the phase problem

Miao et al.5,17 have tested the iterative approach tothe phase problem on a 2-D model. Since we are notaware of any application to real problems, we feel itappropriate to discuss our results in some detail.

It is worthwhile to stress that, in spite of itsextreme simplicity, a 1-D object involving but ahandful of reflections, our problem would bedifficult to tackle by any of the conventionalapproaches.

The results obtained for DZ200 A are presentedunder Results in the main text. Most rewardingly,the iterative algorithm applied to ten different setsof randomly generated phases led to one and thesame profile (Figure 6 of the main text).

The algorithm was applied also to otherD-values, starting always from randomly generatedinitial phase-sets. For DZ160 A and 180 A, all theprofiles were congruent with the profilesobtained for DZ200 A. We call these type I profiles.For DZ140 A, 220 A, 240 A and 260 A only some ofthe profiles were of type I: 4/5 for 140 A, 4/10 for220 A, 2/10 for 240 A, 5/10 for 260 A. The otherprofiles belonged to one of the types documented inFigure 9 of the main text. The type I profile ispredominant (40/60), and is in much better

Page 11: X-ray Scattering Study of Pike Olfactory Nerve: Intensity of the Axonal Membrane, Solution of the Phase Problem and Electron Density Profile

Electron Density Profile of an Axonal Membrane 197

agreement with a proteolipid bilayer than any ofthe other profiles. The final choice is thusunambiguous, but a few problems arise.

One is the relative r-shift of the profiles relevantto the same D-value and obtained with differentinitial phase-sets, an issue related to the thicknessLax of the membrane. As long as D is larger than2Lax the r-position of the profile is expected to bedistributed randomly within the range D–2Lax; allthe profiles should merge when DZ2Lax. Theresults in fact show that the profiles merge forD-values in between 140 A and 160 A. This obser-vation could have one of two possible explanations.One is that the electron density profile displays acompact core approximately 75 A thickaccompanied by tails of much lower electrondensity contrast, approximately 25 A thick. Theother explanation is that the profiles that emergefrom the analysis are in fact distorted by thetruncation of the data at sZ0.04 AK1. The secondexplanation is consistent with the presence in theprofiles of weak ripples whose pseudo-period isclose to 25 A (namely, to 1/0.04) and with the factthat the thickness of membranes from other nervesof the central system (frog, rat and mouse opticnerve myelins) is closer to 75 than to 100 A.17,2

The second problem is the occasional conver-gence of the iterative process to unsatisfactoryprofiles. One cause seems to be the intensity of thezero-order reflection (see Figures 4 and 5 of themain text). Indeed, the presence of unsatisfactoryprofiles for D O200 A coincides with a drop of I(0)and an increase of kj (Figure 4 of the main text),suggesting that these parameters are correlatedstatistically, a circumstance adverse to a reliableassessment of their value.

The hardly unexpected conclusion is that thephasing procedure is particularly sensitive to thedefinition of the molecular envelope (membranethickness in the present case) and to the intensity ofthe zeroth-order reflection.2,17,18

References

1. Luzzati, V., Mateu, L., Vachette, P., Benoit, E.,Charpentier, G. & Kado, R. (2000). Physical structureof the excitable membrane of unmyelinated axons:X-ray scattering study and electrophysiological prop-erties of pike olfactory nerve. J. Mol. Biol. 304, 69–80.

2. Luzzati, V., Mateu, L., Marquez, G. & Borgo, M.(1999). Structural and electrophysiological effects oflocal anesthetics and of low temperature on myeli-nated nerves: implication of the lipid chains in nerveexcitability. J. Mol. Biol. 286, 1389–1402.

3. Lee, M. K. & Cleveland, D. W. (1996). Neuronalintermediate filaments. Annu. Rev. Neurosci. 19,187–217.

4. Stroud, R. M. & Agard, D. A. (1979). Structuredetermination of asymmetric membrane profilesusing an iterative Fourier method. Biophys. J. 25,495–512.

5. Miao, J., Hodgson, K. O. & Sayre, D. (2001). Anapproach to three-dimensional structure of bio-molecules by using single-molecule diffractionimages. Proc. Natl Acad. Sci. USA, 98, 6641–6645.

6. Luzzati, V., Benoit, E., Charpentier, G. & Vachette, P.(2004). X-ray scattering study of pike olfactory: elastic,thermodynamic and physiological properties of theaxonal membrane. J. Mol. Biol. 343, 199–212.

7. Benoit, E., Charpentier, G., Mateu, L., Luzzati, V. &Kado, R. (2000). Electrophysiology of the olfactorynerve of pike Esox lucius: a pilot study on optimalexperimental conditions. Cybium Rev. Eur. Ictyol. 24,241–248.

8. von Muralt, A., Weibel, E. R. & Howarth, J. V. (1976).The optical spike. Structure of the olfactory nerve ofpike and rapid birefringence changes duringexcitation. Pflugers Arch. 367, 67–76.

9. Shannon, C. E. (1949). Communication in the presenceof noise. Proc. Inst. Radio Eng. N.Y. 37, 10–21.

10. Luzzati, V. (1980). Imaging processes and coherencein physics. In Lecture Notes in Physics (Schlenker, M.,Finck, M., Goedgebuer, J. P., Malgrange, C., Vienot,J. C. & Wade, R., eds), pp. 209–215, Springer, NewYork.

11. Taupin, D. & Luzzati, V. (1982). Information contentand retrieval in solution scattering studies. I. Degreesof freedom and data reduction. J. Appl. Crystallog. 15,289–300.

12. Taupin, D. (1988). Probabilities, Data Reduction andError Analysis in the Physical Sciences. Les Editions dePhysique, Les Ulis, France.

13. Press, W. H., Flannery, B. P., Teukolsky, S. A. &Vetterling, W. T. (1986). Numerical Recipes. The Art ofScientific Computing. Cambridge University Press,Cambridge, UK.

14. Sayre, D. (2002). X-ray crystallography: the past andpresent of the phase problem. Struct. Chem. 13, 81–95.

15. Sayre, D. (1952). Some implications of a theorem dueto Shannon. Acta Crystallog. 5, 843–846.

16. Chacko, G. K., Barnola, F. V. & Villegas, R. (1977).Phospholipid and fatty acid composition of axon andperiaxonal cell plasma membranes in lobster legnerve. J. Neurochem. 28, 445–447.

17. Miao, J., Chapman, H. N., Kirz, J., Sayre, D. &Hodgson, K. O. (2004). Taking X-ray diffraction tothe limit: macromolecular structures from femto-second X-ray pulses and diffraction microscopy ofcells with synchrotron radiation. Annu. Rev. Biophys.Biomol. Struct. 33, 157–176.

18. Marchesini, S., He, H., Chapman, H. N., Hau-Riege,S. P., Noy, A., Howells, M. R. et al. (2003). X-ray imagereconstruction from a diffraction pattern alone. Phys.Rev. B, 68, 140101(R).

Edited by Sir A. Klug

(Received 15 June 2004; received in revised form 29 July 2004; accepted 10 August 2004)