**Keywords:** *computational model, computer network, visualization,
virtual reality*.

The well-known models of molecules with different information for a
chemist are derived from the molecular structure hypothesis: a) The 3-dimensional
ball-and-stick model with balls for the atomic nuclei, sticks for the atomic
bonds and their angles, b) its 2-dimensional representation as structural
formula, and c) its 1-dimensional representation as linguistic name which
can be derived from the structural formula. Graphic models are applications
of mathematical *graph theory* which is a part of combinatorial topology.
This mathematical theory became fundamental to chemistry, when in the midst
of the last century the molecular structures of chemical substances were
discovered (Mainzer 1997b).

Van’t Hoff’s *stereochemistry* regarding the three-dimensional
structure of molecules must initially have appeared to be a highly speculative
idea with a certain proximity to platonic forms. Kekulé may have
been particularly adept at three-dimensional visualization as a result
of his prior study of architecture. Simultaneously with stereochemistry,
geometry and algebra were also undergoing fruitful development. Van’t Hoff’s
success in experimental explanation and prediction made his geometry and
algebra of the molecule soon a method accepted by chemists. However, it
lacked any definitive physical justification. At this stage of development,
stereochemistry remains a successful heuristic approach which meets chemists’
need for a means by which they can visualize their structural analyses.

From an *experimental point of view,* the shape of molecules can
be illustrated by an outer envelope of their electronic charge distributions.
These representations are similar to the pictures of atoms which we can
obtain today experimentally by the scattering of electrons in super microscopes
or from the scanning tunneling electron microscope. It is the distribution
of charge that scatters the X-rays or electrons in these experiments. Thus,
it is the distribution of charge that determines the form of molecular
matter in the 3-dimensional space.

Mathematical methods of *differential topology *enable us to identify
atoms in terms of the morphology of the charge distribution. The charge
density D(r) is a scalar field over 3-dimensional space with a definite
value at each point. Positions of extrema in the charge density with maxima,
minima, or saddles, where the first derivative of D(r) vanish, can be studied
in the associated gradient vector field Ñ D(r).
Whether an extremum is a maximum or a minimum is determined by the sign
of the second derivative or curvature at this point. The gradient vector
field makes visible the molecular graph with a set of lines linking certain
pairs of nuclei in the charge distribution.

Local maxima in an electronic charge distribution are found only at the positions of the nuclei. This is an observation based on experimental results obtained from X-ray diffraction and on theoretical calculations on a large number of molecular systems. Thus, a nucleus seems to have the special role of an attractor in the gradient vector field of the charge density. In short: The topology of the measurable charge density defines the corresponding molecular structure.

The molecular graph is the network of bond paths linking pairs of neighboring
nuclear attractors. An atom, free or bound, is defined as the union of
an attractor and its basin. Atoms, bonds, and structure are topological
consequences of a measurable molecular charge distribution. In a next step,
it is necessary to demonstrate that the *topological atom *and its
properties have a basis in *quantum mechanics.* Topological atoms
and bonds have a meaning in the real 3-dimensional space. However, this
structure is not reflected in the properties of the abstract infinite-dimensional
Hilbert-space of the molecular state function. The state function contains
all the information that can be known about a quantum system of nuclei
and electrons. From an operational point of view, there is too much and
redundant information in the state function because of the instinguishableness
of the electrons and because of the symmetry of their interactions. Some
of that is unnecessary as a result of the two-body nature of the Coulomb
interaction. Thus, there is a reduction of information in passing from
the state function in the infinite-dimensional Hilbert space to the charge
distribution function in the real 3-dimensional space. On the other hand,
we thus get a description of the molecular structure in the observable
and measurable space.

*Quantum chemistry *uses several mathematical procedures of approximation
to achieve this kind of reduction. A well-known approximation is the Born-Oppenheimer
procedure that allows a distinction of the electronic and nuclear mass
of a molecule. We get the nuclear structure of a molecule that is represented
by its structural formula. In order to distinguish the electrons as quasi-classical
objects in orbitals, the Hartree-Fock method is sometimes an appropriate
approximation for the electronic state function. *The coincidence of
the topological and quantum definitions of an atom in a molecular structure
*means
that the topological atom is an open quantum subsystem of the molecular
quantum system, free to exchange charge and momentum with its environment
across boundaries which are defined in the three-dimensional real space.

Topological forms can be characterized by *topological index numbers*
correlated to certain chemical properties (*e.g.* boiling point, toxicity;
*cf.*
Bonchev 1983). Examples are hydrocarbon molecules with nearly the same
volume but different topological structures and properties. An example
of a topological index is the *Wiener index.* If the molecular structure
is represented in a topological graph with atoms as nodes and bonds as
edges, the Wiener index is the sum of all shortest connections between
the atoms along the edges by counting the number of edges. The Wiener index
correlates the molecular structure with many properties of certain chemical
substances, *e.g.*, boiling point, viscosity, and refractive index.

The *bonding index* refers to the topological structure of a molecule
by ranking the atomic modes and their connection edges. The rank of an
edge is the product of the ranks of its two modes. The bonding index of
a molecule is the sum of the ranks of all molecular edges. The bonding
index can be computed for molecular fragments (*e.g.* path, cluster,
or closed ring). They can be correlated to chemical properties of certain
medicaments, the toxicity, smell, or taste of new substances.

The number of *double bond equivalents *corresponds to the number
of independent rings and double bonds in a molecule. A special topological
index results if it is multiplied with the relative frequency of open chains
and closed rings. That index can be correlated to the amount of soot produced
by burning hydrocarbon. Polycyclic aromatic hydrocarbon compounds contain
certain topological regions which are involved in chemical reactions with
cancerous effects. They can be characterized by topological indices, too.

A challenge to modern chemistry is the development of *computer-aided
molecular design* and *artificial intelligence* (Brandt & Ugi
1989, Mainzer 1992). In the 60s, the application of *knowledge based
expert systems* started with the DENDRAL program in chemistry. It automatically
searches chemical structural formulas according to a given molecular formula
and the corresponding mass spectrogram. In this case, the research strategy
of a chemist tries to generate topologically possible molecular structures
and to test or select the chemically possible ones. Mathematically, the
research strategy is performed by a recursive algorithm (‘British Museum
algorithm’) in a LISP-program (Mainzer 1995).

In the 80s, there was a boom of programs producing molecular models
by CAMD (*computer aided molecular design) *methods*. *A simpleexample
is a program using a method to draw 2D structures of organic molecules,
including ring systems and stereoisomers, which can automatically be converted
into 3D models. The automatic process uses an advanced distance geometry
algorithm. Another program generates and displays molecular volumes for
one or more molecules, and makes a range of comparisons between the volumes.
It can also generate volumes from the output of a dynamic calculation and
from a systematic search file.

The *complex shape of macromolecules *dramatically effects the
electrostatic field and can be crucial to their functions. The program
calculates these electrostatic properties and visualizes complex structures.
As a result, the researcher can predict the electrostatic effects and screen
compounds before experimentation. The program uses a finite difference
algorithm to solve the Poisson-Boltzmann-equation. There are also CAMD-programs
for simulating the *molecular dynamics* by trajectories in 3D-models.
These programs incorporate a broad spectrum of molecular mechanics and
dynamics methodologies. By using an empirical force field as foundation,
minimum energy confirmations as well as families of structures and dynamic
trajectories of molecular systems can be computed. The program can help
develop and refine working hypotheses as well as guide experimental directions.

In general, complex CAMD-programs consist of several *modules *combining
more and more activities of a researcher. There are the following standard
modules: *Viewer*, for viewing and comparing molecules, contours,
and other graphic objects; *Builder, *for constructing new molecules
from molecular fragments or atoms; *Docking*, for calculating the
interaction between two molecules using an combination of van der Waals
energy and/or Coulomb energy. Optional modules of specific interest are,
*e.g.*,
*Biopolymer, *for building and modifying proteins, peptides, and nucleic
acids; *Analysis,* for analyzing trajectory data, conformational data
*etc*. The program should enable chemists to design drugs, chemicals,
and materials. The goal is to help scientists comprehend the amount of
information produced by theory-based models, and to focus the research
in a more productive manner.

The World Wide Web started with the 2D format of HTML (hypertext markup language). But the interactive 3D world of VRML is the future of the web. The speed of realization only depends on various technical aspects like bandwidth and CPU requirements. The VRML 1.0 specification of 1995 was a means of creating static 3D worlds. The extension of VRML 2.0 provides enhanced 3D worlds, interaction, animation, and prototyping. Complex molecular structures, which had no chance of visualization before, can now be interactively experienced and designed. Further on, the Internet programming system Java is a flexible environment for the integration of VRML into chemical teleworking. Collaborative work on 3D molecular structures can be realized by research groups spread over the World Wide Web.

*Object-oriented programming* of VRML corresponds to the structure
of complex molecules consisting of atomic building blocks and their bonds.
From a methodological point of view, object-oriented programming relies
on basic ideas of *mathematical systems theory.* According to the
complex systems approach, any system can be separated from the external
environment by some real or fictitious system boundary. The features of
the system boundary determine the exchange of material, energy, and information
between the system and its environment, corresponding to the input and
output transfer in information systems. Further on, any system is defined
as a structure of related subsystems. Stepwise decomposition of the system
leads to smaller subsystems on different hierarchical levels of details
until the level of elementary subsystems, which are regarded as not being
further decomposable. An elementary subsystem can be an abstract or a real
material entity.

These modeling entities support the development and the handling of
mathematical models. Composite modeling objects at a higher degree of complexity
are derived by selection and aggregation of the predefined elementary modeling
objects. In *object-oriented programming,* an object refers to a *data
structure* that is used to mimic the conceptual entities of the application
area to be modeled. Different types of modeling objects can be represented
by object classes. A molecular structure with atomic elements and bonds
corresponds to a data structure with __i__nheritance relationships between
*classes*
and their *subclasses*. ‘Inheritance’ means that manipulations of
object classes (*e.g.,* turning a molecular structure) are transmitted
to their subclasses (*e.g.*, the atoms of a molecule). The similarities
among the modeling objects can be utilized to develop class taxonomies
of modeling objects.

Object-oriented modeling of molecular structures provides new tools
of *computer-assisted problem solving* in chemistry. An example is
*protein
fold recognition in biochemistry* (Fig. 1): Given a virtual library
of folds representative of the database of experimentally solved structures
and a query sequence, the tool identifies that fold among the representatives
which is most plausible, *i.e.* most similar to the predicted structure,
for the sequence in question. This is done by computing sequence-structure
alignments of the query sequence with each of the representatives and then
ranking the latter according to the alignment score as an approximation
of sequence-structure computability. The alignments with the top-ranked
structures define detailed mappings of sequence to structure positions
which immediately lead to rough structure models for the query sequence.

The exhaustive comparison results in a relation of sequence similarities
within the genome. The visualization of such a relation is the *genome
sequence similarity graph*. A graph is a network of vertices connected
by edges. Each vertex of the graph represents a DNA block. An edge connecting
two vertices represents a similarity relationship between two blocks. Each
vertex contains information about its position on a distinct chromosome.
Edges are labeled according to their similarity score values. Each vertex
contains associated information about known genetic elements identified
by other sequence analysis methods. In the framework of the *object-oriented
programming language Java*, the Genomebrowser and the visualization
of the genome sequence similarity graphs support powerful services of interactive
biochemical research.

Chemistry does not only explore static structures, but also dynamic
processes like chemical reactions as applications of kinetic equations.
They correspond to the *‘dynamical view’ *of applied mathematics.
The prediction or determination of chemical events and properties need
sophisticated computational procedures of numerical mathematics, approximation,
and algorithmic theory. Typical examples are *ab initio* computations
in quantum chemistry. These applications correspond to, what I call, the
*‘numerical
view*’ of applied mathematics. Today, the numerical procedures become
more and more efficient by the increasing capacities of computer technology,
for example, the power of massively parallel computers.

However, chemistry is not only interested in numerical procedures, but
also in the construction of 3D geometric models and the derivation of linguistic
terms such as chemical formulas. In the past, these activities have been
already assisted and even simulated by knowledge based expert systems,
3D computer aided molecular design (CAMD) programs, and computer aided
knowledge processing of AI-programs. This aspect of modern computer mathematics
is called the *‘program view’, *which I have demanded for modern philosophy
of science (Mainzer 1995, p. 705). With the development of object-oriented
programming languages, virtual chemical structures can be designed and
explored in computer networks by worldwide distributed research groups.

There is a clear tendency of research in all natural sciences that the
traditional experiment in the laboratory is assisted by *computer experiments.*
They are not only supplementary visualization. An example is the dendritic
growth of materials. For this kind of diffusion limit aggregation (DLA),
there is no analytical theory, but a direct computer simulation. Diffusion
processes are mathematically considered as random walks of particles. The
growth of DLA-clusters is simulated by an algorithm that can easily be
translated into an appropriate programming language (Mainzer 1999, p. 114).
Algorithms and programs of DLA-processes can be tremendously accelerated
by high-speed computers. Thus, the fractal dimensions of even large dendritic
clusters are computable. Typical structures of dendritic growth can be
observed and classified under varying conditions of complex experiments,
providing fruitful hints on lab experiments and industrial design of new
materials. For example, consider the chemistry of polymerization. Long
chains of identical monomers play an enormous role in technical applications
of materials science, but also in living organisms with complex DNA structures.
Mathematically, the growth of polymeric chains seems to be simulated by
random walks, again. However, random walks may cross themselves, polymeric
chains may not. Thus, polymeric chains are examples of self-avoiding-walk
(SAW)-processes. There are no differential equations for their computation.
In general, there is only the possibility of direct computer experiments
or lab experiments.

However, experiments in chemical laboratories spend time, materials,
and money. In the age of accelerating innovation cycles and increasing
costs in technology and industry, computer experiments will help to select
and decide on future tendencies of research. Their programs provide strategies
to refine scientific conceptions and to focus the research in a productive
manner. They help to prevent and to select less productive, expensive or
even dangerous experiments in the laboratory. But, of course, research
cannot do without lab experiments. Finally, computer networks enable worldwide
collaboration of chemists on virtual objects of research. From an epistemic
point of view, computational models and virtual reality are an essential
enlargement of human imagination and recognition, opening new avenues of
research. However, virtual reality in chemistry does not compete with the
wet reality of chemical substances in nature. It is a software tool of
modeling, no more and no less.

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Mainzer, K.: 1999, *Computernetze und virtuelle Realität*,
Springer, Berlin.