Determinant algorithm
In order to eliminate the �fixed
Z-coordinate� effect as well as optimise the processing of stereochemical
structures during registration in the databases the authors designed and
implemented a new effective algorithm for handling of geometry at stereochemical
centres. The algorithm, due to its mathematical representation, is described
here as the determinant algorithm.
The starting point of the reasoning
while designing the algorithm was the obvious assumption that stereoisomers with
their variety of spatial distribution of differentiated ligands around an
asymmetric centre can be described � mathematically �using the notion of the
space orientation (sign of space). For a 3D space, the space orientation is
defined by equation (1): 
The sign of space is equivalent to
the sign of a determinant constructed by unit vectors defining the space itself.
It can be easily proved that equation (1) � when traversing from the vector
notation to the notation based on four points defining space � can be converted
into the fourth grade determinant (2):
The determinant equals zero only for
all four points lying on the same plane or for at least two identical points.
Relating this equation with the four ligands at stereocentres one can deduce
that transformation of configuration can be only achieved for situation when all
ligands are located on the same plane. Additionally one can see that in order to
create space the points have to be differentiated. Besides, one can see that
permutation of two rows of the determinant (2) causes the change of its sign.
The determinant (2), when extended at the third column attains the much more
practical and convenient representation (3):
This representation allows one to
determine the sign of the determinants (3) [and thus (1) and (2)] without
setting any value for the Z-coordinate. For summands having the same sign, the
sign of (3) is independent of Z values (Z1, Z2 and Z3)
and thus no matter if Z is zero, negative or positive the sign of the sum (3)
stays the same. In other words, addition of only positive numbers results in a
positive number just as adding only negative numbers leads to a negative result.
Besides, the addition of zero to anything does not change the result at all. The
conclusion is obvious: if summands with different signs occur
in (3), then the structure
represented by (3) is unambiguous. It can be proven that the resulting value of
(3) depends on particular values of Z and in particular it becomes zero (the
case of a flat structure). The determinant (3), when related to a 2D space can
be interpreted as 2D space orientation (sign of space) or as the sign of
triangle (4):
The determinant (4) is very
convenient for manual as well as algorithmic analysis while determining its
sign. The manual method (based on graphic representation) is similar to the CIP
procedure and consists in moving along sequences of vertexes of the triangle
(5). The clockwise direction of the movement corresponds to a negative sign
while the counter clockwise one is related to a positive sign of (4).
The form (4) of the triangle
determinant is not very �computer friendly�. Its grade can however be reduced
giving the following form:
Formula (6) is exceptionally simple
to program and can be particularly fast if designed for use with the CPU
registers storage technique [18]. It is particularly important for registering
very big databases with millions of compounds some of them having multiple
stereochemical centres.
Using the determinant algorithm for manual (non
algorithmic) assignment of stereochemical centres
If the representation of direct
neighbourhood of asymmetric centres in a structure is deformed then usually the
stereochemistry of such structures, even during manual assignment by a chemist,
is declared as undefined. Direct discussions with nomenclature specialists
professionally analysing such structures on an everyday basis for systematic
naming purposes [18] before storage in the Beilstein File proved that deformed
structure are in most of the cases rejected and declared as �stereochemically
ambiguous flat� structures.
It seems that application of
equation (3) in the course of the manual stereocentre assignment directly from
the graphical representation of a structure is fully possible and should prove
to be faster and reliable. In practice it is limited, for a given asymmetric
centre, to ranking - using the CIP rules � the ligands a, b ,c, d ,
selecting the ligand connected to the asymmetric centre with a stereochemical
bond and then determining the sign of triangle that can be created by the other
ligands. The sign of the triangle is then multiplied by (+1) or (-1) depending
on the sign of the chosen Z-coordinate. The sign of the summand in (3) is also
considered. Negative values of (3) correspond to the R (rectus)
descriptor while a positive one corresponds to S (sinister). The
result should be fully independent of the selection of the stereochemical
bonding (if more than one such a bonding present). Should selection of different
stereochemical bonds lead to different signs of (3), then the asymmetric centre
cannot be unambiguously defined. The application of determinant method in the
manual stereocentre assignment process can be exemplified using the following
real case scenario. The non-steric occurrence of the structure chosen for the
example below comes from the Beilstein File where it is registered under the
Beilstein Registry Number (BRN): 1900185. Its computer generated (by the AutoNom
program) systematic name is 1-aminoethanol. CAS registered this
structure under the CAS Registry Number (CRN): 75-39-8
Example:
Let us assume the
CIP ranking for all ligands at a given stereocentre atom of a structure located
ligand c as lying above the plane of the graph. Thus the sign of triangle set by
the vertices a, b, and d should be calculated and the route along the vertices a
through b and then to d should be analysed. If traversing from a via b to d goes
in the clockwise direction then the sign of the third grade determinant is
negative. Let us assume for this example that this is the case. Since the c
ligand is connected to the asymmetric centre by a steric bond then the sign at
the third summand of sum (3) is negative and while ligand c is above the plane
then the Z-coordinate is positive. Thus minus multiplied by plus and then once
more by minus gives minus and consequently the correct descriptor of the
stereocentre is R (rectus).
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