CERTAIN ASPECTS of nature that lie beyond the direct grasp of the human senses can be represented usefully by physical models, such as the constructions made with balls and sticks to depict the architecture of crystals and molecules. The information conveyed by a model of this kind depends on the meaning that is assigned to its parts. A ball can represent the position of an atomic nucleus or its surface can define the boundary of an atom. The sticks can symbolize the forces that bind atoms together or the direction in which the forces act.

Models of simple molecules can be made inexpensively from kits. The more interesting structures, however, can include hundreds of atoms and can be quite expensive if they are made with commercial parts. A construction technique that enables amateurs to assemble models of giant molecules with inexpensive materials has been developed by Richard E. Goodman, who teaches biology at California State College in San Bernardino, Calif. He has described the technique formally in The American Biology Teacher (Volume 32, Number 1, January, 1970). His following discussion is partly based on that article.

"Models of three basic kinds have been devised to illustrate molecular structures [see illustration at right]. On employs both balls and sticks. Another consists of sticks without the balls. The third model has balls without sticks. Each type is uniquely capable of illustrating a selected aspect of molecular structures.

"The ball-and-stick model is conventionally used to represent the structure of crystals. The center of each ball symbolizes the location in space of the atomic nucleus. The balls are supported by slender rods in the form of a lattice that matches the configuration of the crystal. The internal structure of the model, including the planes in which the atoms lie, is clearly visible and can be examined in detail. Ball-and-stick models accurately represent the angles at which atoms are bonded to one another. On the other hand, such models do not show the relative sizes of the atoms or the volume of space they occupy.

"Constructions made exclusively of sticks are known as skeletal models. They are valuable for depicting the dimensions of molecules, particularly those that include atoms of carbon. A straight stick symbolizes a chemical bond between two atoms; its free ends mark the locations of the nuclei. The length of the stick is proportional to the internuclear distance. Molecules are represented by two or more sticks joined at points representing the nuclei of the atoms. The angles at which the sticks are joined conform to the directions through which the atom's bonding forces act.

The skeletal model of a methane molecule, for example, consists of four sticks that meet to form tetrahedral angles of about 110 degrees. The length of the sticks is proportional to the internuclear distance between a hydrogen atom and an atom of carbon. The size of a stick model scaled in this way is proportional to the size of the molecule the model represents. These models also represent accurately the possible spatial configurations of a molecule and therefore can be used in research. For example, skeletal models can be assembled to predict the probable size and spatial configuration of a proposed molecule to be synthesized in the laboratory.

"Constructions made exclusively of balls are called space-filling models. The surface of a ball represents the boundary of a free atom in space. (Such an atom is not attached to neighboring atoms by bonding forces.) The radius of the unbonded ball is a measure of the effective size of the atom and is known as the van der Waals radius.

"In models of this kind the bonding forces and the angles through which they act are symbolized by cutting away part of the spherical surface to create a facet, or flat surface. For example, a sphere that represents an atom of bonded hydrogen has one facet, which symbolizes the single bond that hydrogen can make with a neighboring atom. A sphere representing an atom of carbon has four facets. Straight lines projected at right angles through the centers of the facets would meet in the center of the ball to form tetrahedral angles of 110 degrees. The depth of the facets varies inversely with the covalent radius of the atom: the distance between the nucleus and the boundary of a bonded atom as measured in the direction of the bonding force. The covalent radius is always shorter than the van der Waals radius.

"Space-filling models are customarily used to illustrate the structure of biological compounds because they present the viewer with a clear picture of the shape and volume of the molecule: its space-filling property. Moreover, they display precise bond angles. Their dimensions can be conveniently scaled to conform to both the interatomic distances and the van der Waals radii.

"Various materials are available from which models can be made inexpensively. The plastics industry manufactures spheres in sizes that range from a fraction of an inch to several inches. Some workers prefer to mold their own spheres from epoxy and polyester resins or plaster. Others use methacrylate spheres, which are also available commercially.

"Material for rods and sticks can range from copper tubing and wooden dowel stock to plastic soda straws and chenille pipe cleaners. It takes little ingenuity to devise jigs for cutting facets on balls at accurate bond angles. Great accuracy is not required, however, in models used for demonstrations or exhibits because small errors in bond angles, covalent radii and van der Waals radii are not serious or even noticeable, particularly in models of giant molecules.

"To make my own space-filling models I use balls of expanded polystyrene and join them with connectors made from chenille. The balls could be cemented together, of course, but this would prevent atomic groups from rotating around their bonds and would preclude the possibility of demonstrating alternate structural configurations with the model. Balls of expanded polystyrene are inexpensive. At the time I bought my supply they were priced at about $11 per 1,000 by the Plasteel Corporation (26970 Princeton, Inkster, Mich. 48141). This firm also markets chenille and a coloring kit at reasonable prices.

"I wanted to make several dozen models, including an assortment of giant molecules such as deoxyribonucleic acid, or DNA. To conserve time and effort I devised a few shortcuts that involved the sacrifice of some accuracy in exchange for the desired economies. The resulting models are not of research quality, but they are more than adequate for making demonstrations.

"I used spheres of 1/2-inch diameter to represent hydrogen atoms and spheres of 3/4-inch diameter to represent all other atoms. In addition to carbon the atoms that occur frequently in biological compounds include hydrogen (van der Waals radius 1.2 angstroms), nitrogen (1.5), oxygen (1.4), phosphorus (1.9) and sulfur (1.85). The ratio of the diameters of the small spheres to the large spheres is .67 whereas the ratio of the van der Waals radius of hydrogen to that of the other atoms ranges from .63 to .86. The maximum error introduced by using 3/4inch balls for all atoms except hydrogen is therefore not serious. The scale of approximately .22 inch per angstrom allows the construction of giant-molecule models of convenient size.

"As the first step in building a model I make the required atoms by sandpapering flat surfaces on the styrofoam balls to depths that represent the covalent radii at the correct bond angles. The angle between a pair of facets is supplementary to the desired bond angle. For example, an atom of oxygen unites two atoms of hydrogen at a bond angle of 104 degrees to form a molecule of water [see left]. The supplementary angle between the facets is equal to 180 degrees minus 104 degrees, or 76 degrees.

"To make the facets I use a 3/8-inch electric drill fitted with a disk sander. The depth to which the plastic is sanded and the bond angles between facets are judged by eye. Balls that represent hydrogen atoms are held in contact with the sandpaper by needle-nosed pliers. The larger balls are held by hand.

"Atoms frequently used in models of biological compounds are listed in the accompanying table [right] together 10 with the required number of facets and the covalent radii. Bond angles of the tetrahedral carbon atom should be about 110 degrees, of the dihedral oxygen atom about 104 and of the sulfur atom about 102.

"Atoms other than those of hydrogen must be labeled in some way because they are all made with balls of the same size. I identify the various chemical elements by means of a color code that is in common use: white for hydrogen and phosphorus, black for carbon, red for oxygen, green or blue for nitrogen and yellow for sulfur. To apply the color I push a toothpick into the ball and immerse the plastic in an alcohol-soluble dye. I then invert the toothpick and stick the free end into a slab of styrofoam to support the ball in air until it dries.

"To assemble the model I push a piece of chenille into the center of a facet to a depth of about a quarter of an inch. With wire cutters I snip the chenille at a point about a quarter of an inch from the facet. The mating facet of the ball to be joined is pushed into the projecting chenille. The fibers of the chenille fold smoothly against the wire spine as the connector passes into the plastic but act as barbs when they are pulled in the other direction. The bond is surprisingly strong, but the assembly can be pulled apart without damaging the plastic.

"I have illustrated the structures of many compounds with the models. Sequences of models have also been constructed to depict enzymatic reactions and even entire metabolic pathways. Models of polypeptides, or chains of amino acid units, are easy to construct. The low density of expanded plastic and the small size of the spheres result in models of giant molecules that are easy to handle, store and transport. A model of DNA consisting of 25 base pairs was assembled from its constituent atoms in a single evening. The resulting structure, which shows the celebrated double helix, was supported by scaffolding fabricated with D-Stix, a construction toy."

IN A few of its several forms the interferometer serves as a convenient instrument for observing and recording local variations in the density of gas, such as disturbances caused in air by the heat of an open flame or the flow of air around an object in a small wind tunnel. Essentially the instrument splits a source of light into two beams of parallel rays, recombines the beams at a distant point and focuses the converging rays on a screen. The optical parts of the instrument, which consist of mirrors and lenses, can be adjusted so that light waves in one beam of parallel rays fall slightly out of step with those the other beam. When the beams are recombined and projected on the screen, the crests of the waves will coincide in some areas of the screen to form bright patterns. In other areas the crests of the waves in one beam will coincide with the troughs of the waves m the other beam; the screen appears dark in these areas.

When the instrument is appropriately adjusted, the screen displays a gridlike pattern, being crossed by fringes, or alternately dark and light bands, that are uniformly spaced and parallel. If either of the split beams traverses a gas of nonuniform density in an instrument so adjusted, the form of the grid is altered to a pattern that corresponds to the density variations. Interferometers of conventional design must be made with parts of good optical quality because the projected image displays imperfections in the lenses and mirrors as well as non- uniformities in the density of a gas. Instruments of this quality are priced -beyond the reach of most casual experimenters. Chris F. Bathurst of Invercargill in New Zealand has devised a scheme for making a useful interferometer from scraps of ordinary plate glass and with lenses of the kind used in inexpensive magnifying glasses. He discusses the construction as follows:

"My instrument is a variation of the cyclic interferometer, so called because the test and reference beams of light traverse a triangular path in opposite directions. The unusual feature of the instrument is a lens that is inserted in the base of the triangular path. The lens can be adjusted to minimize the effect of imperfections in the optical parts.

"Rays of light from a conventional source are focused into a converging cone that falls on a beam splitter in the form of a partially silvered mirror [see illustration at right], The apex of the transmitted cone passes through the test section of the instrument-the vessel that contains the gas specimen. The cone is deliberately located in a region of the vessel that is known to contain stable gas.

"The rays proceed through the test section as a diverging cone and are reflected by a mirror to the lens in the base of the triangular path, as shown by the colored rays in the illustration. The lens bends the rays into a parallel bundle that proceeds to the second mirror, which reflects the bundle to the beam splitter, thus closing the triangular path. Half of the light is reflected to the source by the beam splitter and is lost. The beam splitter transmits the remaining half to a field lens that focuses the rays at the viewing position. Essentially these rays function as a reference beam with which rays that are disturbed in the test chamber are compared.

"Converging rays from the source that are reflected by the beam splitter traverse the triangular path in the opposite direction, as shown by the black rays in the illustration. Following this reflection they impinge on the second mirror and are reflected to the lens in the base of the triangular path. The lens transforms the diverging cone into a beam of parallel rays. After subsequent reflection by the first mirror the beam floods the full area of the test section with parallel rays. The direction and the velocity of the rays will be altered more or less by local variations of density in the gas.

"Rays that emerge from the test section return to the beam splitter. Half of the light proceeds through the beam splitter to the source and is lost. The remaining rays are reflected by the beam splitter, interfere with rays of the reference beam and proceed through the third lens to the focal plane at the viewing position. If the gas in the test section is of uniform density, the instrument can be adjusted to project a pattern of fringes that are reasonably parallel and straight even though the surfaces of the optical elements are imperfect. The effects of optical imperfections can be reduced about 80 percent. The residual imperfections in the fringe pattern of the instrument described here are about seven fringes in a field of view three inches square.

"None of the lenses I used is of high quality. All are symmetrical. Lens 1 (focal length 10 inches, diameter four inches) is from a magnifying glass. Lens 2 (focal length 42 inches, diameter four inches) and lens 3 (focal length 25 inches, diameter four inches) were made by a local craftsman with a machine that is normally used for grinding and polishing spectacle lenses. Lens 4 (focal length 5 1/2inches, diameter 1 1/4inches) is of relatively small diameter and reasonably good optical quality. Most of the light passes through it close to its axis. Hence the lens introduces little distortion.

"The green filter (Wratten No. 64) restricts the light in the image to monochrome. I use it for isolating the 5,461angstrom line emitted by a mercury lamp. The filter can be omitted if a laser is used as the light source.

"All mirrors and the windows of the test section were made of ordinary plate glass 3/8 inch thick. The reflecting surfaces are films of aluminum deposited in a vacuum. A series of partially coated beam splitters was made to determine the optimum thickness of the aluminum in terms of reflection. I found that the thickness of the film is not crucial. The metallized coating must be on the surface of the beam splitter that is nearest the test section.

"The mirrors are supported at three points by mechanical mounts that were improvised from available materials. Small pieces of cardboard separate the metal clamps from the glass. The windows of the test chamber are sealed with gaskets of cork. Two micrometer screws, which function as adjustments, rotate the mirror mounts about the horizontal and vertical axes in the plane of the mirror.

"In addition to these adjustments the mounting of lens 2 must be adjustable in both directions parallel to the optical axis and at a right angle to the axis. I did not fit this mounting with screws for making translational adjustments, although they would be convenient. The base of the instrument is made of 1/4-inch steel channel sections and measures three inches in width by 1-1/2 inches in height. Photographs of fringe patterns were made with a 35-millimeter camera.

"The adjustment procedure is somewhat tedious, as it is with most interferometers. My instrument can be aligned and adjusted in from 10 to 30 minutes, depending on the experience of the operator. After the mountings of the optical elements have been assembled to the base remove the beam splitter and lenses 2 and 3. Put a rectangle of white cardboard at the position normally occupied by lens 3. Move lens 1 toward the light source to the point at which the lens projects a light beam of substantially constant width. Align the mirrors to the position at which the beam illuminates the screen.

"Adjust lens 1 and mirrors 1 and 2 to focus a sharp image of the light source on the center of the screen. Replace the beam splitter. Two images of the source should appear on the screen. Adjust the beam splitter to the position at which the two images coincide exactly.

"Move lens 1 toward the light source. A pattern of light and dark fringes should appear on the screen. If fringes do not appear, return to the previous step and again attempt to adjust the beam splitter to the exact point at which the images of the source coincide.

"Look for the fringes. Repeat the procedure until they appear. The operation calls for patience until you acquire experience. After the fringes appear adjust the beam splitter to the position at which you observe the minimum number of fringes.

"Replace lens 2 and move lens 1 toward the beam splitter to the point at which the apex of the cone of converging light rays lies inside the test section. The location of the apex within the test section can be altered by shifting lens 1 up and down or sideways. Avoid regions where the apex would interfere with the phenomenon under study.

"Next, move lens 2 along its axis to the position at which fringes fill the field of view. If the fringes do not appear, return to the critical step in which the images of the light source were made to coincide by manipulating the beam splitter. When the field of view eventually fills with fringes, manipulate lens 2 and the mirror mounts to remove as many fringes as possible. Adjust either of the mirrors to the position at which a desired pattern of fringes, such as a grid, appears in the field of view.

"Finally, transfer the image to the focal plane of the instrument by replacing lens 3, the field lens. Adjust the position of aperture near the camera to intercept the halo of stray light that surrounds the converging rays. The position of the focal plane of lens 4 can be located with a screen of white cardboard.

"To make sharp photographs of fringes that arise from phenomena in the test section place the film of the camera in the focal plane of the instrument. I have used light sources of three kinds. A frosted incandescent lamp can be located behind lens 1 for adjusting the camera to the position of sharpest focus with respect to phenomena in the test section. All other adjustments have been made with a 250-watt, high-pressure mercury arc lamp. Photographic exposures are made by replacing the mercury lamp with a xenon flash lamp of the kind available from dealers in photographic supplies."

Bibliography

THE ARCHITECTURE OF MOLECULES. Linus Pauling and Roger Hayward. W. H. Freeman and Company Publishers, 1964.

BIOCHEMISTRY: THE MOLECULAR BASIS OF CELL STRUCTURE AND FUNCTION. Albert L. LeEninger. Worth Publishers, Inc., 1970.

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