circular dichroism of chromaffin granule proteins in situ ... · circular dichroism (cd) and...

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Proc. Nat. Acad. Sci. USA Vol. 70, No. 12, Part I, pp. 3458-3462, December 1973 Circular Dichroism of Chromaffin Granule Proteins In Situ: Analysis of Turbidity Effects and Protein Conformation (catecholamine secretory vesicles/chromogranins/Mie scattering/coated sphere/membrane proteins) KURT ROSENHECK* AND ALLAN S. SCHNEIDERt * Polymer Department, Weizmann Institute of Science, Rehovot, Israel; and t Laboratory of Neurobiology, National Institute of Mental Health, 36/1D02 Bethesda, Maryland, 20014 Communicated by Francis 0. Schmitt, July 20, 1973 ABSTRACT The circular dichroism spectra for pro- teins in situ in catecholamine secretory vesicles (chro- maffin granules) is presented together with an analysis of protein conformation and turbidity effects on the spec- tra. The calculational analysis has resolved scattering and absorption effects in the turbid suspension spectra using a coated-sphere scattering model which allows for dif- ferent materials in its shell and core. The intrinsic con- formation of the proteins in situ was estimated by an itera- tive procedure with various trial protein conformations, for the chromaffin granules both intact and after release of their contents. The resulting average secondary struc- tures (within about 10%) are: (25% a helix, 15% ,6 struc- ture) for the membrane proteins and (15% a helix, 5% # structure) for the soluble contents. The protein confor- mation did not change with osmotic release of the gran- ule's contents. Consequently, if chromogranins are in- volved in a catecholamine storage complex, this is not re- flected in any detectable change in their average secondary structure. Circular dichroism (CD) and optical rotatory dispersion (ORD) have proven very useful in monitoring protein and nucleic-acid conformations in solution. While the solution work has laid important groundwork for the structural in- terpretation of the spectra, it is the study of biopolymers in situ in functioning biological complexes that is of ultimate interest for elucidation of cellular processes. In the last few years a beginning has been made in this direction, primarily for turbid suspensions of biological membranes (1). The membrane CD and ORD spectra contain characteristic dis- tortions in the shape and magnitude of the spectral bands, and these were experimentally shown to be caused by the scattering properties and particulate nature of the membrane suspension (1-4). Similar results were found for turbid sus- pensions of hemoglobin within erythrocytes (1) and for poly- peptide aggregates (3). Thus, in order to interpret the spectra of such turbid suspensions, the scattering contributions had to be resolved. A theoretical analysis of the circular polariza- tion dependence of scattered light followed (5) and formed the basis for a calculational method of analyzing distorted spectra (6). Numerical computations using this approach have appeared (7, 8) and have successfully accounted for the CD and ORD spectra of suspensions of erythrocyte ghosts as well as polypeptide aggregates. While this work did not attempt a quantitative determination of membrane-protein secondary structure, it demonstrated that scattering distortions in op- tical activity spectra could be resolved by calculation. Most recently, the importance of scattering effects on the CD spectra of viruses has been experimentally demonstrated with instrumentation modified to collect a large part of the scat- tered light (9). In the present work we analyze the CD spectra and protein conformation of both the membrane and internal soluble proteins of intact catecholamine secretory vesicles known as chromaffin granules. These are subcellular organelles which synthesize, store, and release the catecholamines, epinephrine and norepinephrine (10). They can be detected in electron micrographs (11) of the adrenal medulla as round, membrane- bound, electron-dense vesicles of a diameter of about 2000 A. They exhibit in vitro, under proper conditions, functional properties mirroring their natural role in the organism, such as uptake or release (12) of catecholamines. The internal, soluble proteins constitute about 78% of the total proteins (13). The main component, chromogranin A, has been postulated to be involved in a storage complex with catecholamine, and possibly also with ATP, in the intact granule (12). Isolated chromogranin A in dilute aqueous solution shows the hydrodynamic and optical properties of a rather extended random coil (14). No study of its conforma- tion in the presence of the other soluble granule contents at the very high concentrations prevalent in the intact organelle has been done, although this would obviously be relevant to its postulated function. Part of the present work is devoted to this problem. The other part concerns protein conformation in situ in the chromaffin granule membrane. Several impor- tant aspects of chromaffin granule function are likely to reside in its membrane, such as uptake of catecholamine (10, 12), as well as release by exocytosis (15). A quantitative estimate of the conformation of chromaffin granule proteins in situ is made by an iterative procedure with various trial protein secondary structures and their corre- sponding solution spectra. The latter are used with a coated- sphere scattering model (16) to obtain by calculation scatter- ing CD spectra for comparison with experimental turbid suspension spectra. The Mie-type scattering functions used for the coated sphere contain the full dependence on the particle's complex optical properties, and are applicable to scattering particles with different substances in the shell and core. In what follows we first briefly outline the calculational method used and then present both experimental and cor- rected (by calculation) CD spectra for both the intact chro- 3458 Abbreviations: CD, circular dichroism; ORD, optical rotatory dispersion. Downloaded by guest on March 26, 2020

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Page 1: Circular Dichroism of Chromaffin Granule Proteins In Situ ... · Circular dichroism (CD) and optical rotatory dispersion (ORD) have proven very useful in monitoring protein and nucleic-acid

Proc. Nat. Acad. Sci. USAVol. 70, No. 12, Part I, pp. 3458-3462, December 1973

Circular Dichroism of Chromaffin Granule Proteins In Situ: Analysis ofTurbidity Effects and Protein Conformation

(catecholamine secretory vesicles/chromogranins/Mie scattering/coated sphere/membrane proteins)

KURT ROSENHECK* AND ALLAN S. SCHNEIDERt

* Polymer Department, Weizmann Institute of Science, Rehovot, Israel; and t Laboratory of Neurobiology, National Institute ofMental Health, 36/1D02 Bethesda, Maryland, 20014

Communicated by Francis 0. Schmitt, July 20, 1973

ABSTRACT The circular dichroism spectra for pro-teins in situ in catecholamine secretory vesicles (chro-maffin granules) is presented together with an analysisof protein conformation and turbidity effects on the spec-tra. The calculational analysis has resolved scattering andabsorption effects in the turbid suspension spectra usinga coated-sphere scattering model which allows for dif-ferent materials in its shell and core. The intrinsic con-formation ofthe proteins in situ was estimated by an itera-tive procedure with various trial protein conformations,for the chromaffin granules both intact and after releaseof their contents. The resulting average secondary struc-tures (within about 10%) are: (25% a helix, 15% ,6 struc-ture) for the membrane proteins and (15% a helix, 5% #structure) for the soluble contents. The protein confor-mation did not change with osmotic release of the gran-ule's contents. Consequently, if chromogranins are in-volved in a catecholamine storage complex, this is not re-flected in any detectable change in their average secondarystructure.

Circular dichroism (CD) and optical rotatory dispersion(ORD) have proven very useful in monitoring protein andnucleic-acid conformations in solution. While the solutionwork has laid important groundwork for the structural in-terpretation of the spectra, it is the study of biopolymers insitu in functioning biological complexes that is of ultimateinterest for elucidation of cellular processes. In the last fewyears a beginning has been made in this direction, primarilyfor turbid suspensions of biological membranes (1). Themembrane CD and ORD spectra contain characteristic dis-tortions in the shape and magnitude of the spectral bands,and these were experimentally shown to be caused by thescattering properties and particulate nature of the membranesuspension (1-4). Similar results were found for turbid sus-pensions of hemoglobin within erythrocytes (1) and for poly-peptide aggregates (3). Thus, in order to interpret the spectraof such turbid suspensions, the scattering contributions hadto be resolved. A theoretical analysis of the circular polariza-tion dependence of scattered light followed (5) and formedthe basis for a calculational method of analyzing distortedspectra (6). Numerical computations using this approach haveappeared (7, 8) and have successfully accounted for the CDand ORD spectra of suspensions of erythrocyte ghosts as wellas polypeptide aggregates. While this work did not attempt aquantitative determination of membrane-protein secondarystructure, it demonstrated that scattering distortions in op-

tical activity spectra could be resolved by calculation. Mostrecently, the importance of scattering effects on the CDspectra of viruses has been experimentally demonstrated withinstrumentation modified to collect a large part of the scat-tered light (9).

In the present work we analyze the CD spectra and proteinconformation of both the membrane and internal solubleproteins of intact catecholamine secretory vesicles known aschromaffin granules. These are subcellular organelles whichsynthesize, store, and release the catecholamines, epinephrineand norepinephrine (10). They can be detected in electronmicrographs (11) of the adrenal medulla as round, membrane-bound, electron-dense vesicles of a diameter of about 2000 A.They exhibit in vitro, under proper conditions, functionalproperties mirroring their natural role in the organism, suchas uptake or release (12) of catecholamines.The internal, soluble proteins constitute about 78% of the

total proteins (13). The main component, chromogranin A,has been postulated to be involved in a storage complex withcatecholamine, and possibly also with ATP, in the intactgranule (12). Isolated chromogranin A in dilute aqueoussolution shows the hydrodynamic and optical properties of arather extended random coil (14). No study of its conforma-tion in the presence of the other soluble granule contents atthe very high concentrations prevalent in the intact organellehas been done, although this would obviously be relevant toits postulated function. Part of the present work is devoted tothis problem. The other part concerns protein conformationin situ in the chromaffin granule membrane. Several impor-tant aspects of chromaffin granule function are likely to residein its membrane, such as uptake of catecholamine (10, 12), aswell as release by exocytosis (15).A quantitative estimate of the conformation of chromaffin

granule proteins in situ is made by an iterative procedure withvarious trial protein secondary structures and their corre-sponding solution spectra. The latter are used with a coated-sphere scattering model (16) to obtain by calculation scatter-ing CD spectra for comparison with experimental turbidsuspension spectra. The Mie-type scattering functions usedfor the coated sphere contain the full dependence on theparticle's complex optical properties, and are applicable toscattering particles with different substances in the shell andcore.

In what follows we first briefly outline the calculationalmethod used and then present both experimental and cor-rected (by calculation) CD spectra for both the intact chro-

3458

Abbreviations: CD, circular dichroism; ORD, optical rotatorydispersion.

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Conformation of Chromaffin Granule Proteins 3459

maffin granule and its membrane. The distorted suspensionspectra are resolved into scattering and absorptive contribu-tions, and the origins of the distortions are made clear. Theconformation of the internal soluble proteins and the mem-brane proteins are determined both before and after hormonerelease, and the biological implications are discussed.

MATERIALS AND METHODS

Experimental. Intact chromaffin granules were prepared byisotonic density gradient centrifugation (17) in sucrose-Ficoll-D20. This procedure yields granules that are essentially freeof other subcellular organelles and soluble contaminants. Forspectroscopic measurement, the granules were suspended in0.27 M sucrose, containing 10mM Tris buffer (pH 7.4) atconcentrations giving optical densities of 1.0-1.5 at the wave-length of the absorption maximum, near 198 nm.Membranes were prepared by osmotic lysis of the intact

granules in 10 M Tris buffer (pH 7.4) followed by three tofour washings in the same medium. After each washing themembranes were sedimented at 24,000 X g for 20 min. Mem-branes were also prepared by use of hypertonic gradients (18).Membranes prepared by these two methods had identicalspectroscopic properties. Suspensions for spectroscopic mea-surement were prepared in 10 mM Tris buffer (pH 7.4). Thegranule and membrane preparations were analyzed for cat-echolamine content (19) and Mg++-stimulated adenosinetriphosphatase activity (13), and both gave normal values.Absorption spectra were recorded on a Cary 15 far-ultra-

violet spectrophotometer, with 1-mm pathlength silica cells.Particle sedimentation during the spectral recording waschecked for and found to be absent. The errors in the opticaldensity measurement, due to intense small-angle scattering(20), were estimated by reducing the acceptance half angle ofthe photomultiplier to about one degree with the aid of adouble-aperture attachment, one aperture being placed rightafter the absorption cell and the other in front of the collectinglens. The difference in optical density between this mode andthe normal light detection geometry was less than 10% andtherefore neglected. CD spectra were recorded on a Cary 60spectropolarimeter equipped with CD attachment, with 1- or0.5-mm pathlength cells. The potential sources of error due tointense small-angle scattering were checked and again foundto be negligible.Membrane preparations were sonicated, where indicated

below, in a Raytheon Sonic Oscillator model DF 101 for 5min at 40.

Calculation Methods. The calculational methods outlinedbelow are aimed at the following problem: given a distortedexperimental CD spectrum for a turbid suspension, how do weobtain the intrinsic CD and thus the secondary structure ofthe macromolecules composing the scattering particle? Forour membrane-bound secretory vesicles (the chromaffingranule) we have a measured spectrum for a turbid suspensionof intact granules and wish to know the intrinsic spectra andconformation of both the membrane proteins and internalsoluble proteins. Starting from an intrinsic solution spectrum,it is possible to directly calculate a turbid suspension spec-trum, but no reverse procedure exists. Our general approachtherefore is to first assume a trial solution spectrum cor-responding to a given protein conformation. We then calculatethe turbid suspension spectrum, and compare this with themeasured suspension spectra. The calculation is repeated

with various protein conformations and intrinsic solutionspectra until the best fit to the experimental suspensionspectrum is achieved. The trial solution CD and ORD spectrawill then be a good estimate of the intrinsic optical activity ofproteins in situ. In order to be sure that the corrected proteinspectrum is unique, the calculated spectra are tested for sen-sitivity to various trial protein conformations. This procedurewill also give an estimate of the precision with which differentprotein conformations can be distinguished in the calculatedsuspension spectra.The suspension CD is calculated (5, 6) as:

CDsusp = - Iml/22VX [A (m p) - A (mTp) ] [1]where N is the particle concentration, X the wavelength, andA(mr) is the scattering amplitude which is a function ofparticle size, wavelength, and complex refractive index, mp =nP - inp'. The subscripts I and r on m indicate refractiveindices for left and right circular polarization, and the sub-scripts susp and p refer to suspension and intrinsic particle(solution) properties, respectively. The suspension CD is re-lated to the intrinsic optical activity of the molecules makingup the scattering particle, CD. and ORD., by noting thelatter's relation to left and right particle refractive indexes:

nlp= h + k-1ORDp; nrp = Av k-kORDp; and

p= /2(nlp + nr7), [2]

and

nl= ' + k-'CD.; n7p' = hp' k-'CDp; and

Vp' = '/2(n lp' + np'), [3]where the wave vector, k = 27r/X, and np and ii' are the realand imaginary parts of the average particle refractive index.The calculated suspension CD (Eq. 1) may easily be sepa-

rated into absorption and scattering contributions: CD8u,, =CDabs + CDscat (6). The former, CDabs, is what would bemeasured by a spectropolarimeter that has been modified tocollect the scattered light (9). Such instrumentation can atbest only eliminate the differential scattering effects, CDs8at,but not the absorption flattening distortions which are aninherent property of absorbing particles. Consequently, ex-perimental spectra corrected by such instrumentation maystill be highly distorted relative to the true intrinsic spectra ofthe macromolecules in situ. Indeed, as we will show below forour 2000-A diameter vesicles, the dominant effect on the CDspectrum will be a flattening distortion which would still bepresent in an experimental spectrum that had the differentialscatter removed.

Scattering Model. Since chromaffin granules are spherical,membrane-bound, filled vesicles, the appropriate scatteringmodel will be large, absorbing, coated spheres. The Mie-typescattering functions used in the present computations for thismodel have been described by Aden and Kerker (16). Thenumerical computations were performed with the equivalentexpressions of Pilat (21) on the Golem computer (19 significantfigures) at the Weizmann Institute. The computed resultswere checked against values for coated spheres with absorbingcore (22) and absorbing coat (21), as well as nonabsorbingdielectric coated spheres (23). Agreement was achieved in allcases.The coated-sphere model is illustrated in Fig. 1, where r

signifies the outer radius, t the membrane (shell) thickness,

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3460 Biochemistry: Rosenheck and Schneider

FIG. 1. Concentric sphere scattering model with relevantparameters.

and ml and m2 the complex refractive indexes (relative to thatof the medium) of the core (soluble contents) and shell (mem-brane), respectively. By letting ml equal the medium re-fractive index, we obtain a limiting case of the more generalcoated-sphere model, namely the hollow shell. In applyingthe general coated-sphere scattering functions to our intactchromaffin granule, we first determine the optical properties(M2, CD2, ORD2) of the empty membrane, using the hollow-shell limiting case. These are then used in the coated-spherecalculation for the intact granule.The input parameters that define the coated-sphere model

of the chromaffin granule were determined as indicated below.The radius, r, was taken from published values (24) as 1100 A.Various membrane thicknesses (t = 70-120 A), were triedaround the generally accepted value of 100 A, and found tohave an insignificant effect on the calculated spectra providedthe refractive index was accordingly adjusted for the associ-ated change in membrane density. A value of 100 A for t wasthus chosen. The imaginary part of particle refractive index(n1' and n2') is linearly related to absorbance. It was deter-mined for the vesicle contents by measuring the absorptionspectra of the supernatant of lysed vesicles and adjusting forthe concentration within the vesicle. Similarly, n2' for themembrane was determined from measured absorption spectrafor a sonicated preparation of chromaffin-granule membranesthat had been prepared by osmotic lysis. The real part of theparticle refractive index is a more difficult parameter to deter-mine, and no experimental values exist in the ultravioletregion for proteins, lipids, or membranes. Previous scatteringcalculations for biological membranes (7, 8) use an assumedconstant value of real particle refractive index, neglectingdispersion. In the present work the dispersion of nli and n2 isaccounted for in the following way: Mie optical densities werecalculated from trial values for nli and n2 at each wavelengthand compared with experimental suspension OD spectra. Thebest fit was determined with either a nonlinear least-squaresprogram (25) or a quadratic interpolation method (Shkoller,B., personal communication), and the corresponding trialvalues of ni and n2 were taken as an estimate of the refractiveindexes of the soluble contents and membrane, respectively.This procedure was tested in the visible region of the spectrumwhere our estimated refractive indexes could be comparedwith experimental values for membranes (26) and proteins andlipoproteins (27). Reasonable agreement was achieved. The

range of relative refractive index dispersion thus estimatedin the ultraviolet was n2 = 1.07-1.11 for the membrane andni = 1.02-1.05 for the soluble contents of the granule.The trial values for the intrinsic CD and ORD of the gran-

ule proteins were chosen by assuming a given protein secon-dary structure and determining the corresponding CD andORD from the protein reference spectra of Yang and co-workers (28). These reference spectra are derived from theCD and ORD of five proteins with known x-ray structures.Protein conformational estimates resulting from these refer-ence spectra should be accurate within about 10% of the truesecondary structure. In addition to the protein contributionto the intrinsic optical activity, that of the catecholaminesand ATP were also considered. Epinephrine, which is by farthe major catecholamine in the system, has two negativebands in the spectral region of our measurements. One is at203 nm with a molar ellipticity of 7200 deg cm2 dmol-1 andthe other is at 228 nm with a molar ellipticity of 2400 degcm2 dmol-1. Because the concentration of ATP is low, relativeto that of the protein, its optical activity was neglected.The equations used in our computations assume indepen-

X(nm)FIG. 2. Calculated turbid suspension CD for various input

trial protein conformations of (a) membrane.s and (b) intactgranules. The experimental suspension CD is shown by 0.[0]is given in deg cm2 dmol-1. (a) in %: (...) a = 40, j8 = 10;

- ) a = 30,1 = 20; (- -) a = 30,,6 = 10; (--) a = 20,= 20; (---) a = 20,13 = 10; (---) a = 10,13 = 10, (b)

in %: (--)a = 25,1, = 5; (-.-)a = 15,1 = 5; (.**)a =10,,B = 15; (-- )a = 10,1 = 5; (---)a = 0,1 = 5.

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Conformation of Chromaffin Granule Proteins 3461

dent, discrete scattering particles and no significant multiplescattering.

RESULTS

In Fig. 2 we demonstrate the sensitivity of the calculated CDto trial protein conformations for suspensions of chromaffin-granule membranes and the intact granule. The trial mem-brane-protein conformations varied from 10-40% a helix and10-20% , structure, while the trial conformations of thesoluble proteins in the intact granule ranged from 0-25% ahelix and 5-15% , structure. When the conformation of theinternal soluble proteins was varied, the membrane-proteinstructure was held constant at its previously estimated valueof 25% a helix and 15% ,B structure. The experimental pointsshown in Fig. 2 are the measured CD spectra for the granulemembranes and for the intact granule. The calculated spectracan readily distinguish among trial protein conformationswithin about 10% variation in secondary structure and, thus,should yield a unique determination within this range. Otherregions of secondary structure were explored and did notproduce calculated suspension spectra in agreement withexperiment.A comparison is shown in Fig. 3 between the distorted sus-

pension CD spectra and corrected intrinsic spectra of thechromaffin-granule membranes. The intrinsic membranespectrum for a protein conformation of 25% a helix and 15%, structure yields a calculated scattering spectrum that re-produces the essential features of the experimental suspensionspectrum.

Fig. 4 illustrates the CD results for the intact chromaffingranule. As for the membrane, excellent agreement is achievedbetween calculated and experimental spectra of turbid sus-pensions. The corrected intrinsic spectrum of the granulecorresponds to an internal protein conformation of 15% ahelix and 5% p structure and a membrane protein conforma-tion of 25% a helix and 15% , structure. Also shown in Fig.4 is the measured CD spectrum of lysed granules, whichclosely approximates the corrected intrinsic spectrum. Theobvious implication of this result is that no significant proteinconformational changes occur during release of the granulecontents. In order to ascertain whether the protein conforma-tions obtained were unique (within about 10%) we also triedvarying the membrane-protein conformation while holdingthe core protein fixed at 15% a helix and 5% , structure. Noother regions of membrane-protein secondary structure were

A(nm)

FIG. 3. Chromaffin granule membranes. Calculated (-- -)and experimental (i) suspension CD and corrected intrinsic CD(--) for a protein conformation of 25% a helix and 15% ,structure.

5

-10-

20 210 22 230 240

A (nm)

FIG. 4. Intact chromaffin granules. Calculated ( )and experimental (D) suspension CD, experimental CD of agranule lysate (4), and corrected intrinsic CD ( --) for an in-ternal protein conformation of 15% a helix and 5% ,8 structureand a membrane-protein conformation of 25% ca helix and 15% ,structure.

found to give reasonable values of calculated spectra otherthan those already mentioned.The separation of scattering and absorption contributions

to the suspension spectra of the intact granule is shown inFig. 5. It is evident that the differential scattering contribu-tion to the CD is considerably smaller than the absorptivecontribution. Consequently most of the distortion in the CDspectra of intact chromaffin granules will arise from the ab-sorption flattening effect previously mentioned. This resultis not surprising in light of the relatively high absorption ofeach filled granule (optical density of about 0.6 near the 198-nm absorption peak).

DISCUSSIONWe have shown how the CD spectrum for proteins in situ inthe membrane and contents of catecholamine secretoryvesicles (chromaffin granules) can be interpreted. The rel-atively low amount of ordered structure of the internalproteins is consistent with previous studies of chromograninA in dilute solution (14). No significant change in proteinconformation occurred during release of the granule contents.Consequently, if chromogranin A is involved in a catechol-amine storage complex, this is not reflected by a detectablechange in its secondary structure. It may be noted that theclearly distinguiihable CD spectra (Fig. 4) for intact com-

A (nm)

FIG. 5. Intact chromaffin granules. Calculated differentialscattering (---) and absorptive (-- ) contribution to the sus-pension CD (- -). Solid curve, as in legend of Fig. 4.

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3462 Biochemistry: Rosenheck and Schneider

pared with lysed chromaffin granules offers a simple and sen-sitive way of monitoring release.

There are several aspects of the present calculationalanalysis of scattering optical activity spectra that have notpreviously been demonstrated. One is that we have takeninto account refractive index dispersion in the ultravioletregion of the spectrum. Another is our use of an iterativecalculational procedure with various trial protein conforma-tions, unlike earlier work which used fragmentation (1-4)and detergent (7) solubilization methods to obtain a non-scattering solution spectrum. Finally, our use of the generalcoated-sphere model with Mie-type scattering functionspermits an extension of optical activity studies to scatteringparticles having different materials on their surface and in-side their core. This model can thus serve as a prototype forCD scattering corrections for a large number of interestingbiological particles. The coat of the sphere can represent themembranes of prokaryotic cells and subcellular organelles,the coat of spherical viruses, or the cell wall of bacteria byassigning the appropriate values of shell thickness (t) radius(r), and complex refractive index (M2). Similarly, the contentsof various biological particles would be specified by the ap-propriate value of mi.The good agreement achieved between calculated and ex-

perimental suspension spectra (as well as between the cor-responding trial solution spectra and the experimental lysedgranule spectra) provides an encouraging demonstration thata reasonable structural interpretation of the optical activityof macromolecules in situ in complex biological systems cannow be made.

The work reported here was initiated at the Polymer Depart-ment of the Weizmann Institute which was headed by the lateProf. Aharon Katchalsky. It was his encouragement, enthusiasm,and warm expansive personality that inspired our work. K.R.thanks Prof. Francis 0. Schmitt and his coworkers for the hos-pitality and excitement they provided while he was a ResidentScientist at the Neurosciences Research Program, Boston, andwas first exposed to molecular problems in neurobiology. Wethank Dr. B. Shkoller of the Department for Applied Mathe-matics, Weizmann Institute of Science, who wrote the computerprograms used in this work. Present address for A.S.S.: SloanKettering Institute for Cancer Research and Cornell UniversityGraduate School of Medical Sciences, 410 East 68th Street, NewYork, N. Y. 10021.

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