A three-dimensional building procedure for computer modelling of microbiological structures* W . J. Perkins
P. Polihroniadis
E . A . Piper
P. S m a r t
The National Institute for Medical Research, Mill Hill, London NW7, England
A b s t r a c t - - O n e method for determining the three-dimensional shape of microbiological structures is to build a model incorporating available data and ideas, that may be manipulated for comparison with the diverse projections observed in related electron micrographs. Data in the form of dimensions may be obtainable from the e/ectronmicrographs whilst other biological information may give some clues to the structure. As physical models are difficult to manipulate and cannot easily be compared with the electronmicrographs, a method was developed for building up a model on to a computer display, using a few simple instructions, that can then be orientated into any position for comparison with related electronmicrographs. Keywords--lnteractive computing. Displays, Modelling, Biological structure, Electron micrographs
Introduction THE shape of a biological structure is important in providing information as to its function. Highresolution electron microscopy provides a means for visualising the shape of a structure from the diverse projections obtained, but until the orientation of these projections can be precisely defined, reconstruction in three dimensions will be difficult. Specified orientations may be obtained by successively tilting a specimen by defined angles and taking a series of micrographs, but, for biological specimens, only a limited number of views may be considered because of radiation damage to the specimen. The number of views required for three-dimensional reconstruction is determined by the resolution required, but may be reduced if the study is restricted to highly symmetrical structures. Each view may then be transformed and reconstructed in a computer to provide an indication of its three-dimensional shape (DE ROSIERand KLUG, 1968; CROWTHER et al., 1970a; CROWTHERet al., 1970b; GORDON and BENDER, 1970; GILBERT, 1972). Symmetrical structures are also amenable to simulation by an equivalent geometric shape whose projections may be compared with electron micrographs of the structure (FINCH and KLUG, 1966; CASPER, 1966; FINCH and KLUG, 1967; KLUG and FINCH, 1968). To avoid the problem of specimen damage, and to be able to study less symmetrical structures, an alternative approach was adopted in which ideas and available data about a biological structure were able to be incorporated into a model that could be built up in three dimensions on to a computer display and then manipulated for comparison of its different orientations with electronmicrographs of the structure. ~ First received 18th A p r i l and in final form 27th June 1975
274
Procedure The basis of the technique was to use a convenient building brick or cell that could be made up into different sub-units, with these in turn being incorporated into a final structure. Three-dimensional display was achieved by drawing in three orthogonal planes, displayed in three quadrants of the display screen (Fig. la). As complex structures could take some time to draw and there was no apparent need for continuous rotations, at least during the building stage, a Tektronic 611 storage oscilloscope was chosen in preference to a refresh-type display with limited persistence. The computer used was a Honeywell DDP516 with 16K of 16 bit store and a 3 '6M word disc. For biomedical staff to be able to build and manipulate their models in an interactive manner, the program was written so that options could be called by simple instructions, the complete set being shown in Table 1. A sphere was chosen as a basic cell, as this could be represented by a circle in any orientation, the radius being normally fixed at 0" 5 ram. A cell could be brought into any desired position on the screen by using the BRING instruction BR, followed by the x, y, z co-ordinates of the centre of the sphere. Cells could then be joined together by the JOIN instruction JN, followed by the cell number N to which the new cell was to be joined, then the two angles of rotation about N, ~b and ~,, in the screen y and x axes, respectively (Fig. lb). A SHOW instruction sH was used to either label selected cells or to print out their co-ordinates, and errors could be erased with a DELETE instruction DE, specifying the lower and upper limits of the cells to be removed or with a zero zero to delete all cells. When a sub-unit was completed it could be made into a structure by means of a MAKE STRUCTURE
Medical and Biological Engineering
May 1976
Table I Building instructions Function
Instructions A celt into position x, )1, z A cell to cell N at angles ~ (about y) and (about z) The number of cell N, or all cells if N is zero. Type co-ordinates in region N1 to N2 Cells from M to N, or all cells if M is zero As drawn if N is zero, with CG at centre of display if not. Take next available structure number if M is zero; otherwise M as structure number Number N into position x, y, z rotated by 0 (about x) and ~ (about z) Angles O and ~ are set for quadrant N; set all quadrants if N = 5 Add ~ and 0 in each quadrant Rotate structure about the appropriate axis through given angles Added of N dots/square inch. 250 if N is zero Distribution to new bin values (default 5, 15, 2 ~ ) An established structure from disc file An established structure to a disc file Of image Program
BRING JOIN SHOW DELETE MAKE STRUCTURE
BRING STRUCTURE AXES VIEW ROTATE STRUCTU RES BACKGROUND CHANGE DENSITY READ OUTPUT STOP DRAWING TERMINATE
BR/X./Y./Z./ JN/-N/~./~./ SH/--N/-N1-N2/ DE/--M--N/ MST/M/-N/
B RSI-NIX.IY.IZ.IO.I~r.I AX/N/~5./O./
Vll~.lO.i RTX/0./RTY/~b./RTZ/~./ BN/--N/ CB/--N/--N/--N/ RE/(filename)/ OU/(filename)/ Space bar EXIT
Display options: establish by setting sense switches followed by/ SS1 SS2 SS3 SS4 SS5
= = = = =
plotter density central display--responds with VIEW = (N/) scale factor--responds with FACTOR = (N./) 4th view
instruction MST, which then numbered and stored the structure, either as drawn or with its centre of gravity at the centre of the display area. Structures could then be joined together by bringing them as stored into desired positions in the viewing area with a BRING STRUCTUREinstruction BRS, followed by the structure number, x, y, z co-ordinates and the angles of rotations 0, ~ about the x and z axes, respectively. To save drawing time while building up structures, the drawing routine could be halted by a Teletype interrupt using a space bar character. To indicate density, each circle could be replaced by a distribution of dots whose positions were randomly distributed within the circle area in order to show overlapping spheres. This type of display was selected by a sense switch. Other sense switch Normal Display. 1
2
3
4
1= Frontal plane,Y-X 2=Side plane, Y-Z 3=Top plane, Z-X 4 = Blank
. YA ~ ,
Screen Quadrants
Z~
Axes
X
Fig. 1 Display planes Medical and Biological Engineering
M a y 1976
options allowed for the use of a Calcomp plotter, a central display of any of the different individual views, scaling factor and a fourth view. For remote operation, when the sense switch panel was not available, software instructions provided an alternative. When a structure had been established it could be stored on disc with an OUTPUTinstruction ou and given a file name. Typing a READinstruction RE would call for a file name, following which the complete structure would be displayed. The next requirement was to be able to manipulate these structures into different positions for subjective comparisons with the electronmicrographs; there was little point at this stage in seeking the best fit. The fourth quadrant could be brought in for display by setting sense switch 5, and other orientations selected by changing the angles for each quadrant by the AXES instruction AX, giving the respective angles 0 and ~ for each quadrant. To rotate all four quadrants by the same amount, the vi instruction could be used. The AXESinstruction was suitable for observing successive rotation about one axis, but rotations around more than one axis could be better appreciated by leaving the axes set for the orthcgonal planes and rotating the structure by a specified angle about a given axis using the RoTate x, RoTate Y and RoTate z instructions. 275
Sense Switche$
I'
I
Ic m""d/
Plotter Density
~i
I
/from T T Y J ~' I I | I ,
I
i
l r~
. ~ t.entral Display Scale Factor ~ 4th.View
/ .eoo /
P~gr,Trl
.I.I.,. 1 I
~
, . I.
I
IDEI SrI.EICRIRTxIRTVlRTzlaNIBRIB.slsHI I I
I
I
I
I
I__.___I
I I
I
L.
l
REraseScreenl edraw Axes]
I
l Dis ,lay
Fig. 2 Schematic flowchart of system
The shape of a structure seen in an electronmicrograph may be affected by the fact that less dense fringes could be submerged in the background. This could be produced by applying a random uniform background to the whole display using a ~N instruction which supplied a preset level unless a new value was typed in.
Density indication Initially the representation of sphere was achieved by positioning 60 dots within the area of the circle projection by taking two random numbers from a random-number generator: the first, setting the distance from the centre (0-5 ram), the second, the angle of rotation (O-2n). This gave a distribution with a greater concentration near the centre, which was adequate for representing a single sphere but reached a saturation level at about four overlapping spheres. Although the sphere was convenient as a building brick it did not provide an even representation of depth in more complex models, for which an equal density distribution was required. To achieve this, a system of three bins formed by equally spaced concentric circles produced areas in a ratio of 1 : 3 : 5, into which dots were added in the same ratio. Better space filling could have been obtained by extending the radius of the outer band to 7 mm, then to limit in x and y to 5 ram, producing a square distribution, but the edges of the structures were
276
then too sharply defined by comparison with the electronmicrographs. The distribution which by subjective comparison with electronmicrographs gave the best relationship had three bins in the ratio of I : 3 : 4 with the outer radius extending to 6 ram. It was then required to produce a density gradient in the model similar to that in the electronmicrograph of the structure. F o r the storage display oscilloscope only fixed sites are available for dots. As the available sites are taken up, the probability of further dots being accepted is reduced, and the dots accepted for each overlapping cell will decrease in an exponential form. As the dots are applied in groups of N for each cell it was assumed that the number accepted for each overlapping cell was successively decreased by some fraction ~t. A more rigorous analysis would demand that the probability of acceptance of each new dot be considered in turn rather than each group, but the approximation does not materially affect the numbers accepted in this case, giving a slightly higher value for the initial layer. The fraction ct, of N accepted at each layer depends upon the ratio of available sites to total sites: o~n -
M - a n _ _ _
1
M
where M = maximum number of dots per cell (total
Medical and Biological Engineering
May 1976
sites) S. = number of sites occupied at the nth layer. The number of dots accepted at any layer n is An = Ng.. F o r the first layer all are accepted:
Then S,, = M(1-0c") = M 1 -
At = N~ ~ = N
M-N
n--1
and An = Notn = No~"-l
S.=
i=.
i=.
~=~
i=t
Y,, A , = N ~
(l
0e"-l=N -
-
=")
(1-~)
For the second layer n = 2
~z = :t -
M-SI M
SI=N
and M-N M
In our case the software for drawing on the storage oscilloscope had been scaled by a factor of F in X and Y so that the actual drawing area Ca = /r(rF) 2. The maximum density of dots for the display screen, D, = 386 dots/cm 2. Then M = D, Co giving a value to M of 170 for r = 5 m m and F = 0-75. For the selected ratio of 1 : 3 : 4, N must equal 81 (F1NCH and KLUG, 1967), where I is an integer. ce determines the density resolution, a higher value enabling more layers to be resolved. This demands a lower value of N, but this is limited by the need to represent a circle area by a minimum number of dots. The distribution selected had three bins in the ratio of 5 : 15 : 20 dots and gave a value to 9 of 0.76. Using a test pattern this arrangement gave a subjective discrimination between the 6th and 7th layers, above which the increase in the number of dots became less than 5 % for each layer, which was the effective saturation level for density indication. Program
Fig. 3a Electron micrograph--glutamine synthetase
Fig. 3b Basic model--glutamine synthetase
Medical and Biological Engineering
organisation
The main program was written in Fortran IV
Fig. 3c Density model-~glutamine synthetase
May 1976
277
and used approximately 9K of store with an additional 4K for data storage. The program was originally developed as a core-based program with established structures held on paper tape, but when a disc was obtained, these were transferred to disc. The temporary substructures required in the building up of a main structure were also transferred by creating temporary disc files with a number allocated to each. The x, y and z co-ordinates of each cell to be displayed were held in an array from which the drawing routine could produce an image of up to 650 cells. When retrieving a structure, the file is transferred into the array following the points already held on the storage oscilloscope, and only the additional structure is drawn. When displaying an image it was useful to be able to stop the drawing at any point; so this feature was included by using a Teletype interrupt out of the drawing routine and back ready for the next command. Standard plotting software produced a short line for drawing on to the storage oscilloscope, which reduced the number of points for density representation; so a subroutine was included for drawing a spot. Speeding up of the dot plotting can be achieved by using integer SIN and cos routines for the positioning of the dot.
There is no further manipulation on these points, and therefore no accumulation of error; and the accuracy of a single 16 bit word is sufficient when plotting on the display scope. To reduce unnecessary interaction, a number of normal values were preset in the program, such as orthogonal axes, bin densities and background level, and these values could be overwritten when required. The circles were represented by an 18-sided polygon with the respective SIN and cos values for each point held in a look-up table. For each command that altered the display, the appropriate processing was performed on the array of points before transferring to the drawing routine (Fig. 2). This was organised in a loop, drawing each quadrant with a different origin and selected rotational angles, so that any one quadrant could be selected for a central display or a 4th quadrant added by changing the loop delimiters.
Application The technique was first tested by comparison with a simple symmetrical structure, glutamine synthetase, that had already been established as two rings of six subunits with three stable positions marked A, B and C in the micrograph of Fig. 3a (VALENTINE and SHAPIRO, 1968). The computer model was built up as shown in Fig. 3b to show the orthogonal orientations, which equate to the three projections observed, and density indication is incorporated in Fig. 3c; in this example using a spherical distribution. A computer model was then built up of a more complex but still symmetrical structure, the Adenovirus, in which the protein subunits were represented as spheres on an icosahedral surface (PERE1RAet al.,
Fig. 4a Basic model~Nonamer
Fig. 4b Electronmicrograph--Nonamer 278
Fig. 5 Model in orthogonal planes--Adenov/rus Medical and Biological Engineering
May 1976
Fig, 6 Electronmicrograph--Adenovirus
Fig. 7 Electronmicrograph--Adenovirus hexons
Fig. 8a Basic model, orthogonal planes--Adenovirus hexon Medical and Biological Engineering
May 1976
1974). First a triangular face was constructed of cells (Fig. 4a), some of which were deleted to leave the equivalent of a hexon nonamer in the electronmicrograph of Fig. 4b. Two such faces ABC and BCD were connected to form a unit structure with point A on the vertical axis. Reproducing this structure at 72 ~ intervals around the vertical axis provides a half icosahedron which can then be combined with its mirror images by 180~ rotation about the horizontal axis, to obtain the complete structure. Fig. 5 shows the orthogonal plane views of the Adenovirus model, incorporating density indication and background, which may be compared with the electronmicrograph of the Adenovirus in Fig. 6. Having established that models could be built of known symmetrical structures that could be related to the electronmicrographs, the next step was to use the method for studying structures that were not known. For this, the individual hexons forming the Adenovirus nonamers were chosen. When combined, these appeared to have a stable state as in Fig. 4b, but, when isolated, other orientations could be observed (Fig. 7). The hexon is assumed to consist of three polypeptide subunits and estimates of their dimensions and configuration, provided by M. V. NERMUT, were incorporated into the various models that were tried. The model was first built up in the three orthogonal planes (Fig. 8a), then density, using spatial distribution and background were added (Fig. 8b). A quick assessment of the model was obtained by changing the axes, to show the effect of rotation about the horizontal axis (Fig. 9a) and the vertical axis (Fig. 9b). At this stage a model would either be rejected or refined; then, for a more thorough assessment, a model would be manipulated
Fig. 8b Density model, orthogonal planes--Adenovirus hexon 279
about all axes to see if a match could be obtained with selected micrographs. However to achieve a precise match, the effect of negative strain at different orientations will need to be incorporated into the computer model and this is now being investigated. Some projections of the present hexon model, obtained by rotation about three axes, are shown in Fig. 10 and may be compared with the electronmicrographs of Fig. 7.
of microbiological structures by reconstruction from their electronmicrographs necessitates an involvement of biological research workers with mathematics and computing at a reasonably high level.
Conclusions
The determination of the three-dimensional shape
Fig. I 0 Selected projections o f hexon model
Fig. 9a Successive rotation of hexon model about horizontal axis
Fig. 9b Successive rotation of hexon model about vertical axis
280
Many biologists are understandably reluctant to devote too much time and effort in this direction, which is perhaps best left to those with a natural interest in combining the relevant disciplines, and prefer to construct physical models which undoubtedly help them to formulate new ideas but are obviously limited for more complex structures and when a more precise assessment is needed. The information and ideas that led to a physical model can be incorporated into a computer model that may be manipulated as easily, with the added advantage of providing a better assessment of comparison with electronmicrographs and the possibility of more extensive calculation on a model. It is not necessary for biologists transferring their models into a computer to become expert in computing, provided programs are developed which allow them to interact with the computation by means of simple instructions and to be able to put in their ideas and observe their effect (PERKINSand HAMMOND, 1975). References CASPER, D. L. D. (1966) An analogue for negative staining. J. Mol. Biol. 15, 365. CROWTHER,R. A., AMos, L. A., FINCH,J. T., DE ROS~ER, D. J. and KLUG, A. (1970) Three dimensional reconstructions of spherical viruses by Fourier synthesis from electron micrographs. Nature 226, 421. CROWTHER,R. A., DE ROSIER,D. J. and KLUG,A. (1970) The reconstruction of a three dimensional structure from projections and its application to electron microscopy. Proc. Roy. Soc. 'A' 317, 319. DE ROSIER,D. J. and KLUG,A. (1968) Reconstruction of
Medical and Biological Engineering
May 1976
three dimensional structures from electron micrographs. Nature 217, 130-134. FINCH, J. T. and KLUG, A. (1966) Arrangements of protein subunits and the distribution of nucleic acid in turnip yellow mosaic virus. J. Mol. Biol. 15, 344-364. FINCH, J. T. and KLUG, A. (1967) Structure of Broad Bean Mottle Virus I. Analysis of electron micrographs and comparison with turnip yellow mosaic virus and its top component. J. Mol. Biol. 24, 289. GILBERT, P. (1972) Iterative method for the three dimensional reconstruction of an object from projections. J. Theor. Biol. 36, 105. GORDON,R. and BENOER,R. (1970) Algebraic reconstruction techniques (ART) for three dimensional electron
microscopy and X-ray photography. J. Theor. Biol. 29, 471. KLUG, A. and F~NCH, J. T. (1968) Structure of viruses of the Papilloma--Polyoma Type. Analysis of Tilting experiments in the electron microscope. J. Mol. Biol. 31, 1-12. NERMtJT, M. V. and PERKINS, W. J. (to be published). PEREmA, N. S., WRIGLEY,N. G. and PERmNS, W. J. (1974) In vitro reconstruction, hexon bonding and handedness of incomplete adenovirus capsid. J. Mol. Biol. 85, 617. PERKINS,W. J. and HAMMOND,B. J. (1975) Computeraided thought. Nature 256, 171. VALENTINE,R. C. and SHAPIRO,B. M. (1968) Regulation of Glutamine synthetase. Biochemistry 7, 2143.
Procddure de construction t~ trois dimensions d'un module ordinateur des structures micro-biologiques Sommaire--II existe une m~thode pour d6terminer la forme 5. trois dimensions des structures microbiologiques, notamment la construction d'un module, incorporant les donn6es et id6es disponibles, qu'on peut manipuler 5. des fins de comparaison avec les diff6rentes projections observ6es dans les micrographes 61ectroniques associ6s. On peut obtenir les donn6es sous forme de dimensions dans les micrographes 61ectroniques, et la structure sera sugg6r6e par d'autres informations biologiques. Etant donn6 que les modules physiques sont difficiles 5. manoeuvrer et 5. comparer avec les micrographes 61ectroniques, on a 6labor6 une m6thode pour bfitir un modele sur un dispositif d'affichage ordinateur (en suivant quelques consignes tr~s simples) qu'on peut orienter dans n'importe quelle position m3cessaire pour fair des comparaisons avec les micrographes 61ectroniques associ6s.
Dreidimensionales bauverfahren zur Computermodellierung von mikrobiologischen Strukturen Zusammenfassung~Ein Verfahren zur Bestimmung der dreidimensionalen Form yon mikrobiologischen Strukturen wurde in einem Modell verk6rpert, das aus verfiigbaren Daten und Ideen besteht, die zum Vergleich mit den verschiedenen, in verwandten elektronenmikrographischen Aufnahmen beobachteten Projektionen manipuliert werden k6nnen. Daten in Form von Abmessungen sind aus elektronenmikrographischen Aufnahmen erhNtlich, w~ihrend andere biologische Informationen gewisse Aufschltisse tiber die Struktur vermitteln. Da physikalische Modelle nur schwer zu manipulieren sind und sich nicht ohne weiteres mit den elektronenmikrographischen Aufnahmen vergleichen lassen, wurde ein Verfahren zum Aufbau eines Modells auf einer Computeranzeige entwickelt, bei dem einige einfache Befehle verwendet werden, die zum Vergleich mit verwandten elektronenmikrographischen Aufnahmen in jede Position gebracht werden k6nnen.
Medical and Biological Engineering
May 1976
281
P. Polihroniadis
E . A . Piper
P. S m a r t
The National Institute for Medical Research, Mill Hill, London NW7, England
A b s t r a c t - - O n e method for determining the three-dimensional shape of microbiological structures is to build a model incorporating available data and ideas, that may be manipulated for comparison with the diverse projections observed in related electron micrographs. Data in the form of dimensions may be obtainable from the e/ectronmicrographs whilst other biological information may give some clues to the structure. As physical models are difficult to manipulate and cannot easily be compared with the electronmicrographs, a method was developed for building up a model on to a computer display, using a few simple instructions, that can then be orientated into any position for comparison with related electronmicrographs. Keywords--lnteractive computing. Displays, Modelling, Biological structure, Electron micrographs
Introduction THE shape of a biological structure is important in providing information as to its function. Highresolution electron microscopy provides a means for visualising the shape of a structure from the diverse projections obtained, but until the orientation of these projections can be precisely defined, reconstruction in three dimensions will be difficult. Specified orientations may be obtained by successively tilting a specimen by defined angles and taking a series of micrographs, but, for biological specimens, only a limited number of views may be considered because of radiation damage to the specimen. The number of views required for three-dimensional reconstruction is determined by the resolution required, but may be reduced if the study is restricted to highly symmetrical structures. Each view may then be transformed and reconstructed in a computer to provide an indication of its three-dimensional shape (DE ROSIERand KLUG, 1968; CROWTHER et al., 1970a; CROWTHERet al., 1970b; GORDON and BENDER, 1970; GILBERT, 1972). Symmetrical structures are also amenable to simulation by an equivalent geometric shape whose projections may be compared with electron micrographs of the structure (FINCH and KLUG, 1966; CASPER, 1966; FINCH and KLUG, 1967; KLUG and FINCH, 1968). To avoid the problem of specimen damage, and to be able to study less symmetrical structures, an alternative approach was adopted in which ideas and available data about a biological structure were able to be incorporated into a model that could be built up in three dimensions on to a computer display and then manipulated for comparison of its different orientations with electronmicrographs of the structure. ~ First received 18th A p r i l and in final form 27th June 1975
274
Procedure The basis of the technique was to use a convenient building brick or cell that could be made up into different sub-units, with these in turn being incorporated into a final structure. Three-dimensional display was achieved by drawing in three orthogonal planes, displayed in three quadrants of the display screen (Fig. la). As complex structures could take some time to draw and there was no apparent need for continuous rotations, at least during the building stage, a Tektronic 611 storage oscilloscope was chosen in preference to a refresh-type display with limited persistence. The computer used was a Honeywell DDP516 with 16K of 16 bit store and a 3 '6M word disc. For biomedical staff to be able to build and manipulate their models in an interactive manner, the program was written so that options could be called by simple instructions, the complete set being shown in Table 1. A sphere was chosen as a basic cell, as this could be represented by a circle in any orientation, the radius being normally fixed at 0" 5 ram. A cell could be brought into any desired position on the screen by using the BRING instruction BR, followed by the x, y, z co-ordinates of the centre of the sphere. Cells could then be joined together by the JOIN instruction JN, followed by the cell number N to which the new cell was to be joined, then the two angles of rotation about N, ~b and ~,, in the screen y and x axes, respectively (Fig. lb). A SHOW instruction sH was used to either label selected cells or to print out their co-ordinates, and errors could be erased with a DELETE instruction DE, specifying the lower and upper limits of the cells to be removed or with a zero zero to delete all cells. When a sub-unit was completed it could be made into a structure by means of a MAKE STRUCTURE
Medical and Biological Engineering
May 1976
Table I Building instructions Function
Instructions A celt into position x, )1, z A cell to cell N at angles ~ (about y) and (about z) The number of cell N, or all cells if N is zero. Type co-ordinates in region N1 to N2 Cells from M to N, or all cells if M is zero As drawn if N is zero, with CG at centre of display if not. Take next available structure number if M is zero; otherwise M as structure number Number N into position x, y, z rotated by 0 (about x) and ~ (about z) Angles O and ~ are set for quadrant N; set all quadrants if N = 5 Add ~ and 0 in each quadrant Rotate structure about the appropriate axis through given angles Added of N dots/square inch. 250 if N is zero Distribution to new bin values (default 5, 15, 2 ~ ) An established structure from disc file An established structure to a disc file Of image Program
BRING JOIN SHOW DELETE MAKE STRUCTURE
BRING STRUCTURE AXES VIEW ROTATE STRUCTU RES BACKGROUND CHANGE DENSITY READ OUTPUT STOP DRAWING TERMINATE
BR/X./Y./Z./ JN/-N/~./~./ SH/--N/-N1-N2/ DE/--M--N/ MST/M/-N/
B RSI-NIX.IY.IZ.IO.I~r.I AX/N/~5./O./
Vll~.lO.i RTX/0./RTY/~b./RTZ/~./ BN/--N/ CB/--N/--N/--N/ RE/(filename)/ OU/(filename)/ Space bar EXIT
Display options: establish by setting sense switches followed by/ SS1 SS2 SS3 SS4 SS5
= = = = =
plotter density central display--responds with VIEW = (N/) scale factor--responds with FACTOR = (N./) 4th view
instruction MST, which then numbered and stored the structure, either as drawn or with its centre of gravity at the centre of the display area. Structures could then be joined together by bringing them as stored into desired positions in the viewing area with a BRING STRUCTUREinstruction BRS, followed by the structure number, x, y, z co-ordinates and the angles of rotations 0, ~ about the x and z axes, respectively. To save drawing time while building up structures, the drawing routine could be halted by a Teletype interrupt using a space bar character. To indicate density, each circle could be replaced by a distribution of dots whose positions were randomly distributed within the circle area in order to show overlapping spheres. This type of display was selected by a sense switch. Other sense switch Normal Display. 1
2
3
4
1= Frontal plane,Y-X 2=Side plane, Y-Z 3=Top plane, Z-X 4 = Blank
. YA ~ ,
Screen Quadrants
Z~
Axes
X
Fig. 1 Display planes Medical and Biological Engineering
M a y 1976
options allowed for the use of a Calcomp plotter, a central display of any of the different individual views, scaling factor and a fourth view. For remote operation, when the sense switch panel was not available, software instructions provided an alternative. When a structure had been established it could be stored on disc with an OUTPUTinstruction ou and given a file name. Typing a READinstruction RE would call for a file name, following which the complete structure would be displayed. The next requirement was to be able to manipulate these structures into different positions for subjective comparisons with the electronmicrographs; there was little point at this stage in seeking the best fit. The fourth quadrant could be brought in for display by setting sense switch 5, and other orientations selected by changing the angles for each quadrant by the AXES instruction AX, giving the respective angles 0 and ~ for each quadrant. To rotate all four quadrants by the same amount, the vi instruction could be used. The AXESinstruction was suitable for observing successive rotation about one axis, but rotations around more than one axis could be better appreciated by leaving the axes set for the orthcgonal planes and rotating the structure by a specified angle about a given axis using the RoTate x, RoTate Y and RoTate z instructions. 275
Sense Switche$
I'
I
Ic m""d/
Plotter Density
~i
I
/from T T Y J ~' I I | I ,
I
i
l r~
. ~ t.entral Display Scale Factor ~ 4th.View
/ .eoo /
P~gr,Trl
.I.I.,. 1 I
~
, . I.
I
IDEI SrI.EICRIRTxIRTVlRTzlaNIBRIB.slsHI I I
I
I
I
I
I__.___I
I I
I
L.
l
REraseScreenl edraw Axes]
I
l Dis ,lay
Fig. 2 Schematic flowchart of system
The shape of a structure seen in an electronmicrograph may be affected by the fact that less dense fringes could be submerged in the background. This could be produced by applying a random uniform background to the whole display using a ~N instruction which supplied a preset level unless a new value was typed in.
Density indication Initially the representation of sphere was achieved by positioning 60 dots within the area of the circle projection by taking two random numbers from a random-number generator: the first, setting the distance from the centre (0-5 ram), the second, the angle of rotation (O-2n). This gave a distribution with a greater concentration near the centre, which was adequate for representing a single sphere but reached a saturation level at about four overlapping spheres. Although the sphere was convenient as a building brick it did not provide an even representation of depth in more complex models, for which an equal density distribution was required. To achieve this, a system of three bins formed by equally spaced concentric circles produced areas in a ratio of 1 : 3 : 5, into which dots were added in the same ratio. Better space filling could have been obtained by extending the radius of the outer band to 7 mm, then to limit in x and y to 5 ram, producing a square distribution, but the edges of the structures were
276
then too sharply defined by comparison with the electronmicrographs. The distribution which by subjective comparison with electronmicrographs gave the best relationship had three bins in the ratio of I : 3 : 4 with the outer radius extending to 6 ram. It was then required to produce a density gradient in the model similar to that in the electronmicrograph of the structure. F o r the storage display oscilloscope only fixed sites are available for dots. As the available sites are taken up, the probability of further dots being accepted is reduced, and the dots accepted for each overlapping cell will decrease in an exponential form. As the dots are applied in groups of N for each cell it was assumed that the number accepted for each overlapping cell was successively decreased by some fraction ~t. A more rigorous analysis would demand that the probability of acceptance of each new dot be considered in turn rather than each group, but the approximation does not materially affect the numbers accepted in this case, giving a slightly higher value for the initial layer. The fraction ct, of N accepted at each layer depends upon the ratio of available sites to total sites: o~n -
M - a n _ _ _
1
M
where M = maximum number of dots per cell (total
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sites) S. = number of sites occupied at the nth layer. The number of dots accepted at any layer n is An = Ng.. F o r the first layer all are accepted:
Then S,, = M(1-0c") = M 1 -
At = N~ ~ = N
M-N
n--1
and An = Notn = No~"-l
S.=
i=.
i=.
~=~
i=t
Y,, A , = N ~
(l
0e"-l=N -
-
=")
(1-~)
For the second layer n = 2
~z = :t -
M-SI M
SI=N
and M-N M
In our case the software for drawing on the storage oscilloscope had been scaled by a factor of F in X and Y so that the actual drawing area Ca = /r(rF) 2. The maximum density of dots for the display screen, D, = 386 dots/cm 2. Then M = D, Co giving a value to M of 170 for r = 5 m m and F = 0-75. For the selected ratio of 1 : 3 : 4, N must equal 81 (F1NCH and KLUG, 1967), where I is an integer. ce determines the density resolution, a higher value enabling more layers to be resolved. This demands a lower value of N, but this is limited by the need to represent a circle area by a minimum number of dots. The distribution selected had three bins in the ratio of 5 : 15 : 20 dots and gave a value to 9 of 0.76. Using a test pattern this arrangement gave a subjective discrimination between the 6th and 7th layers, above which the increase in the number of dots became less than 5 % for each layer, which was the effective saturation level for density indication. Program
Fig. 3a Electron micrograph--glutamine synthetase
Fig. 3b Basic model--glutamine synthetase
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organisation
The main program was written in Fortran IV
Fig. 3c Density model-~glutamine synthetase
May 1976
277
and used approximately 9K of store with an additional 4K for data storage. The program was originally developed as a core-based program with established structures held on paper tape, but when a disc was obtained, these were transferred to disc. The temporary substructures required in the building up of a main structure were also transferred by creating temporary disc files with a number allocated to each. The x, y and z co-ordinates of each cell to be displayed were held in an array from which the drawing routine could produce an image of up to 650 cells. When retrieving a structure, the file is transferred into the array following the points already held on the storage oscilloscope, and only the additional structure is drawn. When displaying an image it was useful to be able to stop the drawing at any point; so this feature was included by using a Teletype interrupt out of the drawing routine and back ready for the next command. Standard plotting software produced a short line for drawing on to the storage oscilloscope, which reduced the number of points for density representation; so a subroutine was included for drawing a spot. Speeding up of the dot plotting can be achieved by using integer SIN and cos routines for the positioning of the dot.
There is no further manipulation on these points, and therefore no accumulation of error; and the accuracy of a single 16 bit word is sufficient when plotting on the display scope. To reduce unnecessary interaction, a number of normal values were preset in the program, such as orthogonal axes, bin densities and background level, and these values could be overwritten when required. The circles were represented by an 18-sided polygon with the respective SIN and cos values for each point held in a look-up table. For each command that altered the display, the appropriate processing was performed on the array of points before transferring to the drawing routine (Fig. 2). This was organised in a loop, drawing each quadrant with a different origin and selected rotational angles, so that any one quadrant could be selected for a central display or a 4th quadrant added by changing the loop delimiters.
Application The technique was first tested by comparison with a simple symmetrical structure, glutamine synthetase, that had already been established as two rings of six subunits with three stable positions marked A, B and C in the micrograph of Fig. 3a (VALENTINE and SHAPIRO, 1968). The computer model was built up as shown in Fig. 3b to show the orthogonal orientations, which equate to the three projections observed, and density indication is incorporated in Fig. 3c; in this example using a spherical distribution. A computer model was then built up of a more complex but still symmetrical structure, the Adenovirus, in which the protein subunits were represented as spheres on an icosahedral surface (PERE1RAet al.,
Fig. 4a Basic model~Nonamer
Fig. 4b Electronmicrograph--Nonamer 278
Fig. 5 Model in orthogonal planes--Adenov/rus Medical and Biological Engineering
May 1976
Fig, 6 Electronmicrograph--Adenovirus
Fig. 7 Electronmicrograph--Adenovirus hexons
Fig. 8a Basic model, orthogonal planes--Adenovirus hexon Medical and Biological Engineering
May 1976
1974). First a triangular face was constructed of cells (Fig. 4a), some of which were deleted to leave the equivalent of a hexon nonamer in the electronmicrograph of Fig. 4b. Two such faces ABC and BCD were connected to form a unit structure with point A on the vertical axis. Reproducing this structure at 72 ~ intervals around the vertical axis provides a half icosahedron which can then be combined with its mirror images by 180~ rotation about the horizontal axis, to obtain the complete structure. Fig. 5 shows the orthogonal plane views of the Adenovirus model, incorporating density indication and background, which may be compared with the electronmicrograph of the Adenovirus in Fig. 6. Having established that models could be built of known symmetrical structures that could be related to the electronmicrographs, the next step was to use the method for studying structures that were not known. For this, the individual hexons forming the Adenovirus nonamers were chosen. When combined, these appeared to have a stable state as in Fig. 4b, but, when isolated, other orientations could be observed (Fig. 7). The hexon is assumed to consist of three polypeptide subunits and estimates of their dimensions and configuration, provided by M. V. NERMUT, were incorporated into the various models that were tried. The model was first built up in the three orthogonal planes (Fig. 8a), then density, using spatial distribution and background were added (Fig. 8b). A quick assessment of the model was obtained by changing the axes, to show the effect of rotation about the horizontal axis (Fig. 9a) and the vertical axis (Fig. 9b). At this stage a model would either be rejected or refined; then, for a more thorough assessment, a model would be manipulated
Fig. 8b Density model, orthogonal planes--Adenovirus hexon 279
about all axes to see if a match could be obtained with selected micrographs. However to achieve a precise match, the effect of negative strain at different orientations will need to be incorporated into the computer model and this is now being investigated. Some projections of the present hexon model, obtained by rotation about three axes, are shown in Fig. 10 and may be compared with the electronmicrographs of Fig. 7.
of microbiological structures by reconstruction from their electronmicrographs necessitates an involvement of biological research workers with mathematics and computing at a reasonably high level.
Conclusions
The determination of the three-dimensional shape
Fig. I 0 Selected projections o f hexon model
Fig. 9a Successive rotation of hexon model about horizontal axis
Fig. 9b Successive rotation of hexon model about vertical axis
280
Many biologists are understandably reluctant to devote too much time and effort in this direction, which is perhaps best left to those with a natural interest in combining the relevant disciplines, and prefer to construct physical models which undoubtedly help them to formulate new ideas but are obviously limited for more complex structures and when a more precise assessment is needed. The information and ideas that led to a physical model can be incorporated into a computer model that may be manipulated as easily, with the added advantage of providing a better assessment of comparison with electronmicrographs and the possibility of more extensive calculation on a model. It is not necessary for biologists transferring their models into a computer to become expert in computing, provided programs are developed which allow them to interact with the computation by means of simple instructions and to be able to put in their ideas and observe their effect (PERKINSand HAMMOND, 1975). References CASPER, D. L. D. (1966) An analogue for negative staining. J. Mol. Biol. 15, 365. CROWTHER,R. A., AMos, L. A., FINCH,J. T., DE ROS~ER, D. J. and KLUG, A. (1970) Three dimensional reconstructions of spherical viruses by Fourier synthesis from electron micrographs. Nature 226, 421. CROWTHER,R. A., DE ROSIER,D. J. and KLUG,A. (1970) The reconstruction of a three dimensional structure from projections and its application to electron microscopy. Proc. Roy. Soc. 'A' 317, 319. DE ROSIER,D. J. and KLUG,A. (1968) Reconstruction of
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three dimensional structures from electron micrographs. Nature 217, 130-134. FINCH, J. T. and KLUG, A. (1966) Arrangements of protein subunits and the distribution of nucleic acid in turnip yellow mosaic virus. J. Mol. Biol. 15, 344-364. FINCH, J. T. and KLUG, A. (1967) Structure of Broad Bean Mottle Virus I. Analysis of electron micrographs and comparison with turnip yellow mosaic virus and its top component. J. Mol. Biol. 24, 289. GILBERT, P. (1972) Iterative method for the three dimensional reconstruction of an object from projections. J. Theor. Biol. 36, 105. GORDON,R. and BENOER,R. (1970) Algebraic reconstruction techniques (ART) for three dimensional electron
microscopy and X-ray photography. J. Theor. Biol. 29, 471. KLUG, A. and F~NCH, J. T. (1968) Structure of viruses of the Papilloma--Polyoma Type. Analysis of Tilting experiments in the electron microscope. J. Mol. Biol. 31, 1-12. NERMtJT, M. V. and PERKINS, W. J. (to be published). PEREmA, N. S., WRIGLEY,N. G. and PERmNS, W. J. (1974) In vitro reconstruction, hexon bonding and handedness of incomplete adenovirus capsid. J. Mol. Biol. 85, 617. PERKINS,W. J. and HAMMOND,B. J. (1975) Computeraided thought. Nature 256, 171. VALENTINE,R. C. and SHAPIRO,B. M. (1968) Regulation of Glutamine synthetase. Biochemistry 7, 2143.
Procddure de construction t~ trois dimensions d'un module ordinateur des structures micro-biologiques Sommaire--II existe une m~thode pour d6terminer la forme 5. trois dimensions des structures microbiologiques, notamment la construction d'un module, incorporant les donn6es et id6es disponibles, qu'on peut manipuler 5. des fins de comparaison avec les diff6rentes projections observ6es dans les micrographes 61ectroniques associ6s. On peut obtenir les donn6es sous forme de dimensions dans les micrographes 61ectroniques, et la structure sera sugg6r6e par d'autres informations biologiques. Etant donn6 que les modules physiques sont difficiles 5. manoeuvrer et 5. comparer avec les micrographes 61ectroniques, on a 6labor6 une m6thode pour bfitir un modele sur un dispositif d'affichage ordinateur (en suivant quelques consignes tr~s simples) qu'on peut orienter dans n'importe quelle position m3cessaire pour fair des comparaisons avec les micrographes 61ectroniques associ6s.
Dreidimensionales bauverfahren zur Computermodellierung von mikrobiologischen Strukturen Zusammenfassung~Ein Verfahren zur Bestimmung der dreidimensionalen Form yon mikrobiologischen Strukturen wurde in einem Modell verk6rpert, das aus verfiigbaren Daten und Ideen besteht, die zum Vergleich mit den verschiedenen, in verwandten elektronenmikrographischen Aufnahmen beobachteten Projektionen manipuliert werden k6nnen. Daten in Form von Abmessungen sind aus elektronenmikrographischen Aufnahmen erhNtlich, w~ihrend andere biologische Informationen gewisse Aufschltisse tiber die Struktur vermitteln. Da physikalische Modelle nur schwer zu manipulieren sind und sich nicht ohne weiteres mit den elektronenmikrographischen Aufnahmen vergleichen lassen, wurde ein Verfahren zum Aufbau eines Modells auf einer Computeranzeige entwickelt, bei dem einige einfache Befehle verwendet werden, die zum Vergleich mit verwandten elektronenmikrographischen Aufnahmen in jede Position gebracht werden k6nnen.
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281
