Generalized cylinders

Generally, cylinders that are broad and squat in contour are for low-pressure service, such as the propane tanks used on automobiles or with campers. Those that are tall and thin are generally used for high-pressure containment, such as for oxygen, hydrogen, or nitrogen. 

The central construct of the expert system DOCENT which we are developing is its capacity to represent a macromolecule by "generalized cylinders", and to permit manipulation of the cylinders directly, instead of adjusting the molecular internal coordinates or space-fixed axes. The inverse problem, to recover reasonable values of the underdetermined atomic coordinates from the disposition of the generalized cylinders, is posed. 

Brooks achieves a representation of classes of objects by an unusual data structure we will call this a "Brooks Data Structure" — BDS. We can adapt the BDS to a LISP-coded specification of a macromolecule as follows the macromolecule is to be described by a TREE, the ROOT of which is the coarsest characterization of the molecule. For a macrocycle, the ROOT would be the great ring for a branched chain it would be the longest backbone. This ROOT would be a list, the basic data structure of LISP, with each element in the list a data set defining a generalized cylinder. 

Figure 1. Examples of generalized cylinders, which are defined by a space curve spine, a cross sectional form, and a sweeping rule.

Z, - = Z, + H, where H is the height of a riser. Their top ends define tne spine of the generalized cylinder representing the bannister. 

The steps are also constrained they are planar, arranged to fit the bottom cuts of the newel posts. Their exact shape depends on the shape of the wall, which we preseume is smooth and vertical. It need not be flat any curvature can be represented by a section of a generalized cylinder. 

Even in the atomic-scale graphical representation of macromolecules it is easy to recognize helical segments. In our display we represent any helical portion of a molecule by a single rod. Properties of the rod would include the number of turns and the number and nature of atoms or residues involved in the local helical structure. Other types of secondary structure lend themselves to representation by generalized cylinders, but have not been incorporated into our code, which is in an early stage of development. 

Deducing Atomic-Scale Geometry from Generalized Cylinders... 

The generalized cylinder representation of a macromolecule is a much sparser description of the system than is the full set of Cartesian or internal coordinates which define the position of every atom. The generalized cylinder representation leaves the molecular structure seriously underdetermined. It is eventually necessary to recover an estimate of all the atomic coordinates for more detailed study. Here is where DOCENT takes on some of the features of an expert system, which must make plausible judgments without complete information. 

It is evident that the "solid" models developed by Carson for purposes of vivid display are sequences of simple generalized cylinders, with elliptical cross sections and sweeping rules locally simple, but capable of encompassing very general cahin structures. The smooth appearance of the ribbon diagram is desirable in most displays, but require a large number of local cylinders. We expect that a cruder representation, with a bare minimum of distinct cylinders, will be adequate to our purposes. Our priority is the direct manipulation of cylinders, which is easier if there are only a few. 

But if that geometric fit defines the parameters of generalized cylinders, and if we can infer something of the atomic coordinates of chains from those parameters as we suggest above, the possibility exists that we can deduce characteristics of the primary structure which would produce the tertiary structure represented by the set of cylinders which take on the desired shape. 

There are considerable advantages, of simplicity and power, to the large-scale and approximate representation of macromolecules by "generalized cylinders". Gross features of the structure are made easier to grasp visually. But more significant, major alterations to the... 

The constraint,/ + Hl/ X = const, defines a three-dimensional manifold M, a generalized cylinder. = 0 is an invariant manifold of the isokinetic equations, with solutions therefore confined for all time to one or the other half-space (f i < 0 or 1 > 0). It is enough to sample one or the other of the two half spaces. To show that the solutions do so, we need to demonstrate that an appropriate Hbrmander condition holds. Define/ andg as the vector fields... 

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