fiber diffraction.pdf

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Some Macromolecules form Fibers rather than Crystals

Many biological macromolecules will not or cannot crystallize. However, an important group of fibrousmacromolecules such as DNA or many of the components of the cytoskeleton form orientated fibers in whichthe axes of the long polymeric structures are parallel to each other. Often, as in the case of muscle fibers, theorientation is intrinsic; sometimes the long molecules can be induced to form orientated fibers by pullingthem from a gel with tweezers, sometimes by flowing a gel through a capillary tube, or even by subjectingthem to intense magnetic fields.The experimental set-up is rather simple: the orientated fiber is placed in a collimated x-ray beam at rightangles to the beam and the "fiber diffraction pattern" is recorded on a film placed a few cm away from thefibre.

Fibers show helical symmetry rather than the three-dimensional symmetry taken on by crystals. By analysingthe diffraction from orientated fibers one can deduce the helical symmetry of the molecule and in favourablecases one can deduce the structure. In general this is done by constructing a model of the fiber (as in DNA)and then calculating the expected diffraction pattern. By comparing the calculated and observed diffractionpatterns one eventually arrives at a better model.

Fiber diffraction patterns fall into two main classes: crystalline and non-crystalline.

In the crystalline case (e.g. A-form of DNA) The long fibrous molecules pack to form long thin micro-crystals which share a common axis (usually referred to as the c-axis). The micro-crystals are randomlyarranged around this axis. The resulting diffraction pattern (on left of figure h1) is

equvalent to taking one long crystal and spinning it about its axis during the x-ray exposure. All Braggreflexions are registered at one time. The reflexions are grouped along "layer-lines" which arise from therepeating structure along the c-axis. However, particularly at high resolution the Bragg reflexions tend to fall

on top of each other. If the Bragg reflexions could be separated andmeasured out to high resolution then the standard methods we havedescribed for crystals could be used. Unfortunately this is never thecase and model building must be employed - as is generally the

case for non-crystalline fibers.

In non-crystalline fibers (e.g. B-form of DNA) the long fibrousmolecules are arranged parallel to each other but each moleculetakes on a random orientation around the c-axis. The resultingdiffraction pattern (right of figure h1) is also based on layer-lines,which reflect the periodic repeat of the fibrous molecule. Theintensity along the layer-lines is continuous and can be calculatedvia a "Fourier-Bessel Transform" of the repeating structure of the

diffraction http://homes.mpimf-heidelberg.mpg.de/~holmes/fibre/bran

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fibrous molecule (the Fourier-Bessel Transform replaces theFourier transform used in standard crystallography - The Fourier-Bessel transform arises because of the cylindrical symmetry).

Fig h1 Diffraction from the A and B forms of DNA

Diffraction from a helix

Calculating the x-ray diffraction pattern from a helix was of central significance in the development ofmolecular biology. It was first described by Francis Crick in his doctoral thesis. He wished to understand thediffraction to be expected from an [alpha]-helix. However, the theory was very quickly applied todetermining the structure of DNA.

Crick showed that the diffraction from a helix occurs along a series of equidistant lines rather than the Braggspots one obtains from a three dimensional crystal. These lines (known as layer-lines) are at right angles tothe axis of the fiber and the scattering along each layer-line is made up fromBessel functions. In helicaldiffraction Bessel functions take the place of sines and cosines one uses for crystals: Bessel functions (writtenJ

n(x), where n is called the order and x the argument) are the form that waves take in situations of cylindrical

symmetry (e.g. the waves you get if you throw a pebble into the middle of a pond). Bessel was a Germanastronomer who calculated accurately the orbits of the planets. Fourier used Bessel functions to calculate theflow of heat in cylindrical objects. Bessel functions characteristically begin with a strong peak and thenoscillate like a damped sine wave asx increases. The position of the first strong peak depends on the ordernof the Bessel function. A Bessel function of order zero begins in the middle of the pattern, a Bessel function

of order 5 has its first peak at aboutx = 7, a Bessel function of order 10 does everything roughly twice as farout.


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(Fig h2 plots of J0(x) to J

15(x))

Crick showed that for a continuous helix the order of Besselfunction n occuring on a certain layer line is the same as the layerline numberl (counted from the middle of the diffraction pattern).In Fig h3 we show a continous helix and its diffraction pattern.Because the order of Bessel function increases with layer linenumber so does the position of the first strong peak. which thenform the characteristic "helix cross". The position of the firststrong peak is also inversely proportional to the radius of the helix.The spacing of the layer-lines is reciprocal to the pitch (P) of thehelix. There is a reciprocal relationship between the layer lineseparation and the pitch- small separation large P, large separationsmall P.

Fig h3: A continuous helix and its diffraction pattern

However, real helicies are not continuous, rather they consist of repeating groups of atoms or molecules.The

symmetry of a discontinuous helix can be defined in a number of ways: the most general is to quote how farone goes along the axis from one repeating subunit to the next in the macromolecule (the rise per residuep)and by what angle you turn ([phi]) between one subunit and the next. This is enough to define a helix.Directly derivable from these parameters is the pitch P. The main effect of shifting from a continous to a

discontinuous helix is to introduce new helix crosses with theirorigins diplaced up and down the axis (meridian) of the diffractionpattern by a distance 1/p. The diffraction pattern of a discontinoushelix (with 10 subunits in one turn) is shown in Fig h4.Note thatthe layer-lines can be grouped into two kinds: those which are


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strong on the meridian of the fiber diffraction pattern (meridional)and those which have no intensity on the meridian(non-meridional). For a simple helix which repeats in one turn thefundamental layer line repeat is 1/P. The distance out along themeridian of thefirst meridional layer-line (not counting the origin)gives 1/p.

h4 A discontinuous helix and its diffraction pattern

Helix Selection Rule

For more complicated helices which repeat after two or more turns n and l are related by the helix selectionrule

l = tn + um

The selection rule is an integer equation which makes use of an alternative definition of helix symmetry:there are t subunits in u turns of a repeat. m can take all positive and negative integer values. As an example,for an [alpha] helix t=5 and u=18 i,e, there are 18 subunits arranged on 5 turns per repeat. Solutions to theselection rule tell you which Bessel functions will turn up on which layer lines. Bessel functions with verylarge orders can be forgotten since they will occur so far out in the diffraction pattern (at such highresolution) that they will not be visible. Converseley, if one can figure out which Bessel functions turn up onwhich layer lines one knows the symmetry. The effects helical symmetry are in fact very useful for ananalysis of fiber diffraction patterns. Without helical symmetry all Bessel functions would turn up on all layerlines, which would be a mess. The helical symmetry limits the allowed Bessel to one or two per layer-linewhich renders such problems tractable.

How this applies to DNA

DNA-B form is a simple helix which repeats in one turn. It has 10 basepairs per turn so that the angle turned ber base [phi] is 36. The spacingbetween the bases is 3.4 (i.e.p = 3.4 ) and the pitch is ten timesgreater (i.e. P = 34. For low order layer lines the order of the Besselfunction n which occurs on the l'th layer line is l. Because of their massthe phosphate groups are the dominant scatterers in a nucleic acids.Thephosphate oxygens show up as prominent white spheres in the atomicmodel (Fig h5). The spacing of the layer lines in Fig h1 corresponds to34 - the helix repeats in 34. In this case this is also the pitch P.Knowing this and Francis Crick's formula for the scattering of a helix,Jim Watson was able to glean the radius of the phosphate groups from thelook of the helix cross in Rosalind Franklin's fiber diffraction patterns ofDNA . Furthermore, the strong meridional reflexion (see Fig h1) whichhas a Bragg spacing of 3.4, must correspond to 1/p, (i.e. the spacingbetween bases was 3.4). These pieces of information went a long waytowards defining the essential parameters of the Watson-Crick model.


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Fig h5 A space filling model of DNA-B form (W. Fuller)

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Fourier Transform in cylindrical coordinates

In 2-D we have

For comparion, transforms of a slit, cylinder and sphere

Object : Transform

1D (slit)

:

2D (cylinder)

:

3D (sphere)

:

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Diffraction from a continous helix (Cochran et al. 1952)

A uniform helix (a helical wire ) is defined by the parametric equations

Cochran, W., F. H. C. Crick and V. Vand (1952). The structure of synthetic polypeptides. I. The transform of

atoms on a helix.Acta Cryst. 5: 581-586.

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Diffraction from a discontinous helix

This can be derived by multiplying the continous helix of pitch P by a set of planes with spacingp (the planes

are at right angles to the helix axis). This operation produces a set of points (e.g. atoms) of vertical spacing p

along the helix of pitch P.

The transform can be generated by convoluting (folding) the transforms of the two functions. The process of

convolution reduces to setting down the transform of the continuous helix with its origin at each of the points(0,0,0), (0,0,+1/p), (0,0,-1/p), (0,0,+2/p), (0,0,.-2/p) etc. and taking the sum. The transform of the first

function is non-zero for

Z = n/P

and the transform of the set of planes is non-zero for

Z=m/p

so that the transform of the discontinuous helix is non-zero on planes

Z= n/P + m/p A

where n andm arepositive or negative integers.

On such planes the transform takes the value

If the pitch P and the rise per subunit p have a common repeat c then the transform of the disontinuous helix

can be written

where l andn are related by the integer equation

l = tn + um B

l = layer line number

c= repeat distance along the helix-axis

and the helix has

u subunits in c

tturns in c

m may take any positive or negative integer value

The integer equation B is equivalent to the convolution in A and is a useful form. B is known as thehelix

selection rule.


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Impressum


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Diffraction from a helical array of molecules

From above, for one atom we have

This formula is readily generalized to many atoms by inserting the appropriate changes of origin for each

atom and summing over all atoms. If the coordinates of the atoms in a molecule are

and the scattering factor isfj

we obtain

where n takes all values on layer line l. allowed by the helix selection rule. Using the terminology of (Klug et

al. 1958) it is often convenient to group all contributions from one Bessel function n on layer-line l into a

function

the Fourier Transform F is now

Klug, A., F. H. C. Crick and H. W. Wyckoff (1958). Diffraction by helical structures.Acta Cryst. 11:

199-213.


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Cochran, W., F. H. C. Crick and V. Vand (1952). "The structure of synthetic polypeptides. I. The transform ofatoms on a helix."Acta Cryst.5: 581-586.

Cohen, C. and D. A. D. Parry (1990). "a-Helical coiled coils and bundles: how to design an a-helical protein."

Proteins7: 1-15.

Crick, F. H. C. (1953). "The fourier transform of a coiled-coil."Acta Cryst.6: 685-689.

Crick, F. H. C. (1953). "The packing of a-helices: simple coiled-coils."Acta Cryst.6: 689-697.

DeRosier, D. J. and P. B. Moore (1970). "Reconstruction of three-dimensional images from electronmicrographs of structures with helical symmetry."J. Mol. Biol.52: 355-369.

Franklin, R. E. and R. G. Gossling (1953). "Molecular structure of nucleic acids: molecular configuration ofsodium thymonucleate."Nature171: 740-741.

Holmes, K. C. (1995). "Solving the structure of macromolecular complexes with the help of x-ray fiberdiffracxtion diagrams."J. Struct. Biol.115: 151-158.

Holmes, K. C. and J. Barrington Leigh (1974). "The effect of disorientation on the intensity distribution ofnon-crystalline fibres I . Theory."Acta Crystallograph.A 30: 635-638.

Holmes, K. C., D. Popp, W. Gebhard and W. Kabsch (1990). "Atomic model of the actin filament."Nature347: 44-49.

Huxley, H. E., R. M. Simmons, A. R. Faruqi, M. Kress, J. Bordas and M. H. J. Koch (1981).

"Millisecond Time-Resolved changes in X-ray Reflections from Contracting Muscle during RapidMechanical Transients, Recorded using Synchrotron Radiation." Proc. Natl. Acad. Sci. USA78:2297- 2301.

Klug, A., F. H. C. Crick and H. W. Wyckoff (1958). "Diffraction by helical structures."Acta Cryst.11:199-213.

Langridge, R., M. D.A., W. E. Seeds, H. R. Wilson, C. W. Hooper, M. H. F. Wilkins and L. D. Hamilton(1960). "The molecular configuration of deoxyribonucleic acid: II. Molecular models and their Fouriertransforms."J. Mol. Biol.2: 38-64.

Lorenz, M., K. J. V. Poole, D. Popp, G. Rosenbaum and K. C. Holmes (1995). "An atomic model of theunregulated thin filament obtained by x-ray fiber diffraction on orientated actin-tropomyosin gels."J. Mol.Biol.246: 108-119.

Namba, K., R. Pattanayek and G. Stubbs (1989). "Visualisation of protein-nucleic acid interactions in a virus.Refined structure of intact tobacco mosaic virus at 2.9 resolution by x-ray fiber diffraction." J. Mol. Biol.208: 307-325.

Namba, K. and G. Stubbs (1985). "Solving the phasing problem in fiber diffraction. Application to tobacco


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mosaic virus at 3.6 resolution."Acta Cryst.A41: 252-262.

Namba, K. and G. Stubbs (1986). "Structure of tobacco mosaic virus at 3.6 resolution:: implications forassembly." Science231: 1401-1406.

Rayment, I., H. M. Holden, M. Whittaker, C. B. Yohn, M. Lorenz, K. C. Holmes and R. A. Milligan (1993)."Structure of the Actin-Myosin Complex and its Implications for Muscle Contraction." Science261:58-65.

Reedy, M. K., K. C. Holmes and R. T. Tregear (1965). "Induced changes in the orientation of the cross-bridges of glycerinated insect flight muscle."Nature (Lond)207: 1276- 1280.

Stubbs, G. and C. Stauffacher (1981). "Structure of the RNA in tobacco mosaic virus."J. Mol. Biol.152:387-396.

Stubbs, G., S. Warren and K. C. Holmes (1977). "Structure of RNA and RNA binding-site in tobacco mosaicvirus from 4 map calculated from x-ray fibre diagrams."Nature267: 216-221.

Stubbs, G. J. and R. Diamond (1975). "Phase problem for cylindrically averaged diffraction patterns. Solutionby isomorphous replacement and application to tobacco mosaic virus."Acta Cryst.A31: 709-718.

Watson, J. D. and F. H. C. Crick (1953). "Genetic implications of the structure of desoxyribonucleic acid."Nature171: 964-967.

Watson, J. D. and F. H. C. Crick (1953). "Molecular structure of nucleic acids: A structure for deoxyribosenucleic acid."Nature171: 737-738.

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