Introduction

The incommensurate crystal phases have a need of expansion of crystallography beyond Euclidean crystallography which is already superficial with the superspace approaches and lattice symmetry is also restored by this in a higher dimensional space only. Since scaling does not leave the Euclidean distance invariant, therefore, it also requires a suitable non-Euclidian extension by the discovery of quasi-crystals with scaling invariant diffraction peak position. In the golden mean ratio, the one-dimensional Bonacci chain is characterized by the two interval sequences i.e., L and S where,

L/S = ɽ = (1+51/2)/2

The invariants in relation to the scale transformation are,

S → L and L → L + S

A two-dimensional invertible integral matrix is expressible by this transformation, and a square lattice of invariant is left by the square of it is a hyperbolic rotation in the superspace. The Fibonacci scale transformation justifies the necessity to extend Euclidean crystallography and for including hyperbolic rotations, therefore, is of fundamental importance for the structural properties of a Fibonacci chain. Multimetric crystallography which has been adopted in this study reflects the appearance of circular and hyperbolic rotations within the same group. Normal crystals can also be associated and applied by the multimetric crystallography. Particularly, multimetrical space group left invariant to the atomic structure of ice. The structural features in snow crystals can be interpreted by its point group of infinite order. The hexagonal scaled forms which is much similar within the snow crystals, reveals the possible relevance of an analogous multimetrical point group for six-fold helical nucleic acid molecules, but this process has taken place by the microscopic molecular level. In the involved molecules, if a number of atomic positions in the asymmetric unit of the Euclidean line symmetry group are considered as modulo translations along the helical axis, it can be linked by the elements of this extended point group. The opportunities for crystallographic approaches is provided by this observation which is applied to molecular structures particularly, in biomacromolecules having a given axial rotation symmetry. It is important to know before considering the technical details that crystallography always describes structural relations of a Euclidean system. It is still the matter of question that the present approaches are physically relevant.

Model and real structures

Hereby considering the model structures, X is termed as positions given as a discrete set of points, x is in the three-dimensional Euclidean space:

X = {x|x ϵ Ɛ(3)} (1)

In the same way, with the correspondence of ρ between model and real positions, the real structure Y is given,

ρ: X0→ Y0 for ρ(X0) = Y0 C Y, X0 = ρ-1 (Y0) C X (2)

Having the following properties:

 In given upper bound Ɛ, the model structure is an approximation of the real one.

|x - y| ≤ Ɛ for y = ρ(x);

 Here definition domain of ρ is a subset X0 of the model structure X

 The mapping ρ is a monomorphic as it is injective:

Ρ(x) ≠ ρ(xʹ) denotes x ≠ xʹ equally

 A subset Y0 of the whole structure Y is the image of ρ.

Crystallographic structures and groups

An interplay between geometry, number theory, and algebra, on which crystallography is based on and all the three aspects are very significant and necessary for the extension. The Euclidian group is the covariance group of the system, as the law of Euclidian geometry is followed by the crystallographic structure. The laws of the system are left invariant by a co-variance group, whereas, the system is left invariant by an invariance group. The sets of rational integers, the indices, can labeled the equivalent positions, due to the underlying lattices and indexed positions have nontrivial invariance group (Janner, 2001).

Crystallography and Drug Design

There are many techniques available through which, determination of the structure of molecule at atomic resolution can be done, among which, X-ray crystallography is one of them. Functions and structure are intimately related with each other. For rational drug design and structure-based functional studies, there are some prerequisites like accurate knowledge of molecular structures are needed. An unambiguous, accurate, and reliable 3-dimensional structural parameters are provided by the results from X-ray crystallographic studies at a time even sometime, before the completion of characterizations. An determined by the X-ray crystallography method. In a biological system, the absolute configuration is a critical property and alteration in the response of the biological system may be caused by any change in it (Deschamps, 2005). The determination of the 3D structure of several macromolecules have been done which enables the structure-based drug design for the treatment of various diseases, therefore structure plays an important role in healthcare. Structure-based designed molecules cannot be used as drugs but various other properties such as bioavailability and solubility are needed to be optimized. The properties and abilities of passing through the membrane for reaching up to target site, recognizing and binding specifically with their target to control its function is needed to possess by drugs. The drug remains likely to be accepted by the body and also must be chemically synthesized and cost-effective. The examples of how crystal structures help in the drug development are discussed below

HIV Drugs

AIDS (Acquired Immunodeficiency Syndrome) is the disease which is previously unknown and came to know in the year 1980s, which was found to be caused by a virus called as HIV (Human Immunodeficiency Virus). The disease causes severe infections and also destroys the immune system of the patient with no medication available. A few years later, treatment of this dangerous disease is made available but, before that person suffering from this deadly disease did not escape from death. Aspartyl protease is the first structure to be determined and is essential for the replication and maturity of the virus. Crystallography, binding energy measurements, and computations are carried out widely at each stage of the development. Viral protease (HIV PR) is given in combination with the reverse transcriptase, which inhibits another enzyme of the virus, in a treatment with cocktail drugs.

Hypertension

Hypertension leads to the cardiovascular diseases hence it is a major health concern worldwide. There are two enzymes which convert enzymes into the renin-angiotensin system of the reaction pathways are renin and angiotensin- I regulating the blood pressure in the body and are probed as drug targets. One drug aliskiren has made for the market after many years. A zinc-dependent dipeptidyl carboxypeptidase named as ACE its inhibitors were designed before its structure determination based on the reported structures of carboxypeptidase-A and thermolysin. The insights into the hydrolytic mechanism of the enzyme are provided by the structures of ACE and also showed the basis for first-generation drugs binding. The new improve drugs stay longer in the body and have a better tendency to penetrate the tissue.