The determination of the helical screw angle of a helical particle from its diffraction pattern

STRUCTURAL

INFORMATION

FROM

PHAGE

T4 TAIL

271

King, J. & Litemmli, U. K. (1973). J. Mol. Biol. 75, 315-337. King, J. & Mykolajewycz, N. (1973). J. Mol. Biol. 75, 339.-358. Klug, A., Crick, F. H. C. & Wycoff, H. W. (1958). Actu Crystullogr. 11, 199-213. Krimm, S. & Anderson, T. F. (1967). J. &foZ. Biol. 27, 197-202. Moody, M. F. (1967a). J. Mol. Biol. 25, 167-200. Moody, M. F. (1967b). J. Mol. Biol. 25, 201.-208. Moody, M. F. (1971). First Euro. Biophys. Congr., Baden, Austria (Broda, E., Locker, A. & Springer-Lederer, H., eds), pp. 543-546. Smith, P. R. & Aebi, U. (1974). J. Phys. A.: Math. iI’/lcl. Gen. 7, 1627-1633. Smith. P. K., Prters, T. M. & Bates, R. H. T. (1973). J. Phys. A.: Math. Nucl. Gen. 6, 361~-382. ten Eyck, L. (1973). Acta Crystallogr. sect. A, 29, 183-191. To, C. M.. Kellenberger, E. & Eisenstark, A. (1969). .J. Mol. Biol. 46, 493-511. van Holde, K. E. (1975). In The Proteins (Neurath. H. & Hill, R. L., eds) 3rd edn, vol. 1, pp. 225-291, Academic Press, New York. Venyaminov, S. Yu., Rodikova, L. P., Metlina, A. J,. & Poghzov. B. F. (1975). J. Mol. Biol. 98, 657-664. Williams, R. C. $ Fisher, H. LI’. (1970). J. ~Vol. Biol. 52, 121-123.

APPENDIX

The Determination

of the Helical Screw Angle of a Helical Particle from its Diffraction Pattern P. R. SMITH AND U. AEBJ

Theory A discrete helix (Cochran et al.? 1952: Klug et al., 1958) can be thought of as a stack of identical annuli or axially repeating units of axial thickness, p, where each annulus is rotated by a constant angle, Sz. with respect to the one beneath it. In general, the continuous mass distribution, /\, of any discrete helix can be defined by the following two equations (Crowther & Klug. Notes on Fourier transforms for EMBO advanced study course on image processing of electron micrographs. Cambridge, 1973; Smith & Aobi. 1974). For all r,O.z: X(rJ3,z) = h(rJ + 52. 3 + p)

(AlI

h(r,e,z) z 0 for all r>a.

(A.?

(r,B,z) are cylindrical polar co-ordinates for radius r. azimuth 8 and distance along the helix axis, z, respectively. a is the helix outer radius and Sz is called the helical screw angle. p is the axial repeating distance and is defined as the smallest non-zero positive value such that a value for J2 can be found for which equation (Al) above is satisfied. The two quantities Q and p describe the basic symmetry of a discrete helix. In addition, the annuli which constitute the basic “building bricks” of the discrete helix may possess N-fold axial rotational symmetry and dyad axes perpendicular to the helix axis (Klug et al., 1958). In the case of N-fold axial rotational symmetry of the annuli D may adopt any of an infinite discrete set of values:

“72

1’. H. SML’I’H

.-1X1) U. AEl3l

and equation (Al) will still be satisfied for all J2,. /s is any integer and L$ is the smallest numerical value which Q can adopt,. Cochran et al. (1952) have shown t,hat the three-dimensional Fourier transform of a discrete helix is characterized by being only non-zero on layer planes. The general expression for the alt’itude, 2, of any of these layer planes given in terms of the helical screw angle Q and the axial repeating distance 1~: (A41 where n is the order of the angular harmonic sampled on the layer plane and m is an integer (Smith & Aebi, 1974). An electron micrograph of a helical particle lying on a support grid shows a projection of the density of the particle to a good approximation. A subsequent optical diffraction pattern of the particle image therefore records the intensities of the three-dimensional Fourier transform on a near-axial two-dimensional central section. The layer planes in t’he three-dimensional transform will be sampled on layerlines in the diffraction pattern and their a,ltitudes as measured in the diffraction pattern will be proportional bo t’he altitudes of the layer planes in the three-dimensional transform, independent of moderate tilts of the helix out of the plane perpendicular t’o the projection (beam) direction. This proportionality allows equation (A4) to be used to obtain the screw angle of the helix, Q, from the measured altitudes of the layerlines. The vital step in this procedure is to index the diffraction pattern so that values for n and m are available for each layerline. The procedure for doing this is given by Klug et al. (1958, see Figs 1 and 2) and by Moody (1971). n is the order of the angular harmonic sampled on the layerline and m is the label of the “branch”, as drawn on the (n,Z) plot (Klug et al., 1958, Fig. 2), to which the layerline belongs. Since the measurement of layerline altitudes involves measurement errors, a leastsquares procedure can be used to find an estimate for Q which minimizes their effects. An appropriate residual for minimization is : R = 2 (Zi - an, -

tmJ2wi,

i

(A5)

where the 2, values are the measured layerline altitudes, n, is the order of the angular harmonic sampled on the ith layerline and m, is the label of the “branch” of the (n,E) plot. wI is a weight reflecting the relative accuracy of the altitude measurement if this is known. The summation extends over all measured layerline altitudes. The values of a and b are chosen which minimize R and the screw angle is then given by

In addition, the absolute from b using the formula

value of the axial repeating

distance lo can be calculated

D ’ = At’. b . cosw ’

(A7)

where D/I is the micrograph magnification, D is the diffractometer constant and w is the angle of tilt of the particle out of the plane perpendicular to the beam direction,

UETERMIh-ATIOX

OF HELIC.41,

S(‘REW

ANGLE

273

estimated using the method proposed by DeRosier & Moore (1970). Both essential helical parameters, Q and p, can therefore be estimated in an optimum fashion. Once the helical screw angle has been found, it is sometimes convenient to approximate the exact selection rule, given in equation (A4) above, to obtain an approximate integer selection rule following the example given by Cochran ef al. (1952) (eqn 5). f Q This is done by finding a suitable rational approximation. -. t*o -. and setting ‘11 277

(*W where 1 is an integer. integer

selection

Substituting

52

1

- for Z and f for G in eqn (A4) above, allows the U UP

rule 1 = t*n -+ u*n/.

(A9)

to be derived. This definition follows Klug et al. (1958) (eqn 2), DeRosier & Moore screw angle (1970) (eqn 2) and Smith & Aebi (1974) (eqn 2.17). The approximate D approx is then

+.” J-2B!mPOX Finally it is important to point out that approximate int’eger selection rule :

(AlO)

U’

there is an alternative

1 = q * n’ -1 v . m,

definition

of the

(*411)

where l,q,n’,r and m are integers and the angular harmonic, n, sampled on the layerline, I, is given by n=N.n’ (see for example Finch & Klug, 1971). It is most important to realize that if this definition has been used, the screw angle of the helix is not, in general, given by the expression (2rq/r). Equation (AlO) above can only be used when the integer selection rule has been obtained following the definitions given for the terms in equation (AS), not those for equation (Al 1).

An example The indexing of the diffraction pattern of an intact T4 phage tail is well known and the approximate integer selection rule (corresponding to the basic helix), 1 = -2n + 7m, can be used to locate the approximate positions of the layerlines in the diffraction pattern and to assign them (n,m) pairs as defined above. It is assumed that the layerlines are “shifted” from the positions the approximate selection rule would predict because the helical screw angle 52 of the sheath is slightly different from the approxi4rr mate value of --7- (-102+357”). Table Al gives the measured layerline altitudes obtained by measuring the original optical diffraction pattern of particle A, described in the main text, together with the layerline numbers and the appropriate (n,m) pairs given by the approximate selection rule. The values of a and b which minimized R in equation (A4) above were then calculated and the set of altitudes corresponding to the optimum a and b were calculated. These “optimum” layerline altitudes and the differences between them and the measured altitudes are also listed in Table Al. Weights (wi values) of 1 were used for all measured layerlines.

n, (=Q-,) = -102.387” QIr (. $2,) = 17-613” rJn.s. of (Z,,,, -&d

5 7

0

0,039

-6

6 -12 12

2

3 4

n

A4ngular harmonic

1

Layerline number

mm

-1 1

-s 7 4

2

112

Branch lebel: helical family I

6-81 9-55

3.94 5-60

2.87

2 meas

6.76 9.58

3.95 5.62

2.81

z C3.k

Calculated layerline altitude (mm)

TABLE Al indexation and altitudes

Measured layerline altitude (mm)

Layerline

- 0~05 0.03

0.01 0.02

-0.06

6

1 1

0 1 0

711

0.061* ().O(j+

~-0.063” 0.124” -0.125’

A’

DETERMINATION

OF HELICAL

SCREW

ANGLE

275

The values of m given in “helical family (I)” column correspond to the indexing of the (n,Z) plot of the tail leading to the approximate selection rule 1 = -2n + 7m. There are an infmite number of other valid indexings of the (n,Z) plot, however, each giving a different choice of m values and each leading to another of the 0, values in equation (A3) above. The positions of the “optimum” layerline altitudes are independent of these different indexings of course. The set “helicalfamily (II)” corresponds to the value of Q, = 17.613” and this corresponds to the set of so-called “coarse” helices seen on the extended T2 sheath after freeze-etching (Bayer & Remsen, 1970). Measurement accuracy was checked by repeated measurements of the same diffraction pattern and was estimated to be (&0.05 mm) and essentially independent of the layerline measured. The deviations of the calculated optimum layerline altitudes and the measured altitudes lie within this measurement error. Stability of the deter52 a mination of the ratio - = G against measurement error was tested by adding a b ( > fixed error of +O*l

mm to each of the optimized

layerline

altitudes

and recomputing

the screw angle. The resulting deviations are tabulated in Table Al as a function of the layerline measurement dist.urbed. We would estimate that our calculations of Q are correct to within

O-1”.

REFERENCES Bayer, M. E. & Remsen, C. C. (1970). L’i’irology, 40, 703-718. Cochran, W., Crick, F. H. C. &I Vand, V. (1952). Acta Crystallogr. 5, 581-586. DeRosier, D. J. & Moore, P. B. (1970). J. Mol. Biol. 52, 355-369. Finch, J. T. & Klug, A. (1971). Phil. Trans. Roy. Sot. London, ser. B, 261, 211-219. Klug, A., Crick, F. H. C. & Wycoff, H. W. (1958). Acta Crystallogr. 11, 199-213. Moody, M. F. (1971). Phil. Tram. Roy. Sot. London, ser. B, 261, 173-179. Smith, P. R. & Aebi, U. (1974). J. Phys. A.: Math. Xd. Qen. 7, 1627-1633.