MASARYK UNIVERSITY
FACULTY OF INFORMATICS
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Calculation of dissociation constants
using triangulation methods
BACHELOR’S THESIS
Jaroslav Ol’ha
Brno, spring 2012
Declaration
Hereby I declare, that this paper is my original authorial work, which I have worked out
by my own. All sources, references and literature used or excerpted during elaboration
of this work are properly cited and listed in complete reference to the due source.
Advisor: RNDr. Radka Svobodov´a Vaˇrekov´a, Ph.D.
ii
Acknowledgement
I would like to thank RNDr. Radka Svobodov´a Vaˇrekov´a, Ph.D. for coming up with an
interesting thesis that turned out to be quite fun to do and for helping me understand
it and get through it. I’m also grateful to my parents for giving me the opportunity to
study. And to Joss Whedon for making Firefly. Also to Sim. You know who you are.
iii
Abstract
One of the great challenges of present-day chemoinformatics research is prediction of
dissociation constants (pKa) of molecules. The values of pKa are very interesting for various
fields of research, particularly for the area of drug development - for instance, the
value of a molecule’s pKa is one of the criteria that allow us to discard an unfit molecule
from a list of potential drug molecules. However, the experimental measuring of pKa is
difficult and computational methods for acquiring pKa are nontrivial. An efficient alternative
is approximation of dissociation constant using the values of partial charges of
atoms within a molecule. This can be done using various methods of computer science
and mathematics. One such method - using Delaunay triangulation - has been designed
and implemented in this thesis. Part of the thesis is an application demonstrating the
effectiveness of this method, which has shown to be considerable. The method has been
tested on several datasets of molecules containing phenols and/or benzoic acids, and in
most cases it has been quite accurate at predicting their pKa. Its results have also been
compared to the results of a method using linear regression over a similar dataset.
iv
Keywords
Dissociation constant, Delaunay triangulation, phenol, benzoic acid, approximation.
v
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Molecule in a computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.2 Descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Atomic charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Dissociation constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3.1 Dissociation constant of phenols and benzoic acids . . . . . . . . . 5
2.3.2 Prediction of dissociation constant . . . . . . . . . . . . . . . . . . . 6
2.4 Delaunay triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Methods of evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.1 Microsoft Visual Studio 2010 . . . . . . . . . . . . . . . . . . . . . . 10
3.1.2 MIConvexHull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.3 Excel Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.2 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.4 Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.5 Programming language . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.6 Classes and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.1 Descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.2 Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.3 Weighted average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1 Phenols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Benzoic acids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 All molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Comparison with linear regression . . . . . . . . . . . . . . . . . . . . . . . 22
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
vi
6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.1 Contents of the attached CD . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
vii
Chapter 1
Introduction
The demands of pharmaceutical industry, combined with an enormous increase of available
information on structures of organic molecules, have recently led to the emergence
of a field of science called chemoinformatics [1] [2]. This field sets out to solve chemical
problems using algorithmic approaches of computer science and methods of computational
chemistry and molecular modelling [3].
One of the great challenges of chemoinformatics is prediction of dissociation constants
of molecules (pKa) [4]. The values of dissociation constants are very interesting
for chemical, biological and environmental research, because pKa determines important
physically-chemical properties of molecules [5] [6]. There is a special emphasis on
pKa in the pharmaceutical industry, particularly in drug development - the value of a
molecule’s pKa is one of the criteria that allow us to eliminate unfit molecules from a
selection of molecules proposed for drug research. That is why pKa is also used in the
so-called ’virtual screening’ [3], which involves searching through large databases in
order to find molecules viable for pharmacy. Experimental measuring of pKa is difficult
and the computational methods for acquiring pKa are nontrivial [4]. Fortunately, there
is an effective alternative - prediction of pKa based on partial charges of atoms within
a molecule. There are several ways to perform this kind of calculation - be it by using
linear regression, database methods or neural network. An approach that seems very
interesting is one that would use Delaunay triangulation [7] - and the implementation
of such approach is precisely what lies at the heart of this thesis. The goals of this thesis
are:
• Study up on topics concerning the computerization of molecules and algorithms
for computing pKa.
• Design and implement software tools for prediction of dissociation constants.
• Test the created tools on real data of organic molecules.
• Compare the accuracy of the method that uses Delaunay triangulation with a
method that uses linear regression.
1
Chapter 2
Theory
This thesis combines certain areas of chemistry, informatics and mathematics – this
chapter sets out to explain the main principles behind all of these. The first section will
focus on the interface between chemistry and informatics – namely how a molecule
can be stored and processed in a computer to allow for the use of computer science in
chemistry.
The next two sections are dedicated to the chemistry behind the thesis – namely the
atomic charges of atoms, the dissociation constant of a molecule and a few other related
phenomena. It is assumed here that the reader already has a certain understanding of
the most basic physical and chemical terms – such as molecule or charge (presumably
not too big a leap of faith), and only those terms and principles which an average computer
scientist might be unfamiliar with will be explained here.
The fourth section takes on the mathematical principles used in this thesis. Of all the
theoretical sections, this is probably the one with the greatest overall importance for the
thesis. The main focus of this section is the description of Delaunay triangulation and
the definition of a few other geometrical terms.
The last section will briefly mention the statistical methods used to evaluate the results
of our approximation.
2.1 Molecule in a computer
In order for computer science to be used in chemistry, there must first be a way to
convert chemical structures and chemical activity to a form that is comprehensible to
a computer – from the ’analog’ system of the natural world to the digital system of
virtual reality. This conversion will not be covered extensively here, but at least a few
approaches to working with molecules on a computer should be mentioned.
2.1.1 Graph theory
Since molecules are, on the most basic structural level, just groups of atoms held together
by bonds that connect pairs of atoms, the graph theory seems like a perfect fit
for describing molecules. We just need to treat atoms as though they were vertices and
bonds as though they were edges of a graph. We can then easily store the molecule on
a computer (for example using adjacency matrix or adjacency list) and perform all the
2
2. THEORY
operations that we could perform with a regular graph. Of course, this does not account
for the finer and more complex aspects of molecular structure, but no digitalization can
ever be perfect.
2.1.2 Descriptors
Another way of working with a molecule on a computer are descriptors. Descriptors are
the quantifiable properties of a molecule – used precisely for the purpose of digitalizing
molecules and allowing for further calculations. A molecule has many qualities that can
be used as descriptors – molecular charge, molecular weight, the number of atoms...
even structural motifs that can not be quantified numerically can be used as binary
descriptors (for example by assigning 1 to the descriptor if the particular motif is present
and 0 if it is not).
Although in this thesis descriptors will only be used to allow for calculations involving
a molecule’s qualities, they can also be used to identify a molecule with certain properties
among a large number of molecules through a technique called ’fingerprinting’.
This is nothing more than a suitable grouping of several descriptors – resulting in an
array of values that gives us the relevant information about a molecule in a very simple
form. Such molecules can then easily be searched through with the right search criteria.
Although this may sound like a good way of storing a molecule on a computer, it is
not - the main point of a fingerprint is that it only contains the values that are relevant
in a given situation. Creating a fingerprint that would describe any molecule uniquely
in some sort of standardized form would be complicated, inefficient and perhaps even
impossible. Fingerprints are generally used as a tool for finding relevant similarities
between molecules.
2.2 Atomic charges
Atomic charges are very important for this thesis - exploiting their influence on the
dissociation constant of a molecule is what allows us to approximate the constant in
this way. Although there are other ways of approximating the dissociation constant
(some of which will be mentioned later), the atomic charges are ideal for building an
n-dimensional model since there are multiple atomic charges useful for approximation
in each molecule.
The partial atomic charges (referred to simply as ’atomic charges’ throughout the rest
of this thesis) are those charges of a molecule that are caused by the distribution of electrons
between bonded atoms. Normally, an atom has a neutral charge if it has an equal
number of protons (positively charged particles) and electrons (negatively charged particles).
The protons always remain inside an atom and can not be shared with other
atoms. The electrons, however, are shared by two atoms whenever they share a bond this
bond is formed by two electrons; one from each atom. Each of these atoms has a certain
electronegativity, which determines how strongly it draws electrons. If one of the
3
2. THEORY
atoms in a pair has a higher electronegativity than the other, it draws the other atom’s
electron closer to itself. The electron then contributes to the atom’s charge, making it
slightly negative (since its electrons now ’outweigh’ its protons). The other (less electronegative)
atom, however, loses some portion of the electron’s negative charge, which
makes it slightly positive. But the charge created in this way is always only partial - the
more electronegative atom never gains the whole electron (at least not without the help
of external forces).
The partial charge also propagates through the molecule - the less electronegative
atom that lost a portion of its electron’s charge is now positive, and therefore drawing
its neighbors’ electrons more strongly than before (if it has any neigbors, of course).
Although it may not seem like it, it is quite difficult to determine the partial atomic
charges of atoms in a molecule. What may be a trivial task in the world of macrophysics
becomes a lot more complicated in the quantum world of atoms and molecules - there
is currently no way to determine the partial atomic charges directly by experiment, and
they can only be acquired through the methods of quantum mechanics.
2.3 Dissociation constant
In the most basic terms (which should be sufficient for the purposes of this thesis), the
dissociation constant (pKa) of a molecule describes the willingness of the molecule to
lose one of its hydrogen atoms in a certain environment. The constant is specific to
molecules – a molecule will always have the same dissociation constant under normal
conditions (which is what makes it a constant), but it is different between two different
molecules. The constant determines a molecule’s acidity - a lower pKa value means
that the molecule is more prone to losing a hydrogen atom and therefore more acidic.
Knowing how a molecule will interact with other molecules in the environment that
contain hydrogen atoms is critically important, especially when working with living
organisms – and the dissociation constant allows us to quantify this interaction. It is
calculated as a negative common logarithm of Ka
pKa = −log10Ka, (2.1)
which is, in turn, discovered by experiment. In simple (and slightly incorrect) terms,
Ka represents the amount of molecules (of the same compound) that dissociate spontaneously
relative to the amount of molecules that do not. In the following equation (2.2),
dissociated molecules are represented as hydrogen atom H+
and the rest of the acid A−
(the bond between the hydrogen and the acid has been broken), while non-dissociated
molecules are HA (the bond between the acid and the hydrogen atom persists). The
square brackets represent concentration (for example, [H+
] means ’concentration of
H+
’).
Ka =
[H+
].[A−
]
[HA]
(2.2)
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2. THEORY
2.3.1 Dissociation constant of phenols and benzoic acids
Any molecule containing a hydrogen atom has a certain pKa, but in this thesis the focus
is on phenols and benzoic acids. Both phenols and benzoic acids contain a chemical
structure known as benzene core. This structure consists of six carbon atoms arranged
in a hexagon, each of them also bonded with a hydrogen atom. When drawing benzene,
the carbon and hydrogen atoms are usually not explicitly denoted, but it is understood
that they are there.
Figure 2.1: A molecule of benzene, in complete form (left) and in simplified form (right).
In addition to this core, phenols contain a hydroxyl group, consisting of one hydrogen
and one oxygen atom – the group is bonded with the benzene core via the oxygen
atom which bonds with one of the carbon atoms of the core in place of its hydrogen.
Instead of the hydroxyl group, benzoic acids contain a carboxyl group – consisting of
a carbon bonded with a hydroxyl group (via oxygen) and another oxygen atom (by a
double bond). This group bonds with the benzene core via this carbon – which is, again,
connected to one of the six benzene carbons in place of its hydrogen.
Figure 2.2: Derivatives of benzene - phenol (on the left) and benzoic acid (on the right).
The orange line indicates the dissociating hydrogen.
5
2. THEORY
Adding various other atoms and structures to this basic structure produces various
phenols and benzoic acids. Although these structures will always contain multiple hydrogen
atoms, the one in the hydroxylic or carboxylic group (for phenols and benzoic
acids, respectively) is the one most likely to be dissociated and therefore the one we will
be most interested in.
When the hydrogen atom is dissociated, the partial charges of all atoms within the
molecule change. Therefore, if we calculate an atom’s partial charges in a dissociated
molecule as well as in a non-dissociated one, we get two different descriptors. This
principle will be used later in this thesis.
It should also be mentioned that at some point we will be distinguishing between a
few basic kinds of benzoic acids. Many kinds of substituents can be added to the basic
structure in place of its hydrogen atoms to create various benzoic acids. We will make a
distinction according to where these substituents bond with the main structure (shown
in figure 2.3). If they bond with one of the carbons next to the carboxylic group, they are
called ortho benzoic acids. If they bond with one of the carbon atoms that are two bonds
away from the carboxylic group, they are called meta benzoic acids. And finally, if they
bond with the carbon atom directly opposite the carboxylic group, they are called para
benzoic acids.
Figure 2.3: Three types of benzoic acids (left to right) - ortho, meta and para. X represents
a substituent.
2.3.2 Prediction of dissociation constant
There are quite a few ways of approximating a molecule’s dissociation constant without
actually having to perform the experiments needed to determine it precisely [8].
The dissociation constant of a molecule is affected by the atomic charges of atoms
within the molecule. As discussed earlier, all the atomic charges are molecular descrip-
6
2. THEORY
tors that can be quantified, translated to a computer and further worked with. If there is
a correlation between two properties of a molecule, we can use one of them to approximate
the other. As a rule of thumb, the better the correlation, the better the approximation.
The beauty of using descriptors for approximation is that we can use more of
them at once. Adding good descriptors adds information and improves the approximation,
but adding descriptors with weak correlation can actually decrease the accuracy of
the approximation. Choosing the right descriptors is therefore one of the challenges of
building a good approximation model.
The partial charges on atoms are only some of the descriptors acquired through
calculations of quantum mechanics. Some of the others are also correlated with the
molecule’s pKa - one of the most prominent being the free energy of a molecule (used by
methods utilizing the so-called ’linear free-energy relationship’ or LFER). Other descriptors
derived from quantum mechanics include electrostatic potential or polarizability.
There are also methods that do not use descriptors - simulations that try to calculate
the dissociation constant directly, from scratch (’ab initio’ methods), again using quantum
mechanics. These do not rely on correlated structural properties and are much more
accurate, but also demand a lot of computational capacity.
Quantum mechanics is not always required, however - some methods search through
databases of molecules trying to find the most similar ones and approximate using their
dissociation constants.
2.4 Delaunay triangulation
In geometry, the term ’triangulation’ refers to a certain transformation of a set of vertices
to a connected graph (without changing the positions of the vertices and without any
of the edges crossing), in such a way that no more edges can be added without adding
additional vertices or creating double edges. This creates simplices in the graph (in two
dimensions these simplices are triangles, hence the name). From now on, these simplices
will be referred to as ’triangulation cells’, except when discussing a two-dimensional
triangulation where it is easier to simply call them ’triangles’. Let us also define the
term ’triangulation border’, which will be used to refer to the hyperplane shared by two
neighboring cells. Again, this term will only be used when discussing n-dimensional
space; in 2D it will be called simply ’edge’.
Delaunay triangulation in particular has a few unique properties that all non-Delaunay
triangulations lack – for the sake of simplicity, these properties will be described in a
two-dimensional space, but they are analogous in a higher dimension. Firstly, a Delaunay
triangulation maximizes the angles inside the cells. This means that of all possible
triangulations of a two-dimensional set of vertices, the Delaunay triangulation will have
the ’least skinny’ triangles. Secondly, the circumcircle of any triangle will not contain another
vertex inside. The two properties are evidently related – a ’skinny’ triangle’s circumcircle
would obviously be too big to avoid other vertices. From now on, we can say
that a set of connected vertices satisfies the ’Delaunay condition’ if none of its circumcir-
7
2. THEORY
cles contains another vertex. There is at least one Delaunay triangulation for every set
of vertices (unless the vertices form a hyperplane and cannot generate the whole space),
and for most sets of vertices it is unique.
There are many ways to create a Delaunay triangulation from a set of unconnected
vertices. Probably the easiest one to grasp is the technique of ’flipping’ edges. Let us consider
a set of four points. There are two ways this set can be triangulated – shown in the
figure below. Unless the vertices form a perfect square, only one of these triangulations
satisfies the Delaunay condition. The ’wrong’ triangulation can be made angularly optimal
by simply ’flipping’ the diagonal edge to create the other possible triangulation. In a
larger graph, if all such subsets (of four vertices forming two triangles sharing an edge)
satisfy the Delaunay condition, then so does the whole graph. Any random triangulation
can therefore be turned into Delaunay triangulation simply (albeit inefficiently) by
’flipping’ all non-Delaunay pairs of triangles until the whole graph meets the Delaunay
condition [9].
Figure 2.4: Two possible triangulations of a set of four points - only the one on the right
satisfies the Delaunay condition.
2.5 Methods of evaluation
To determine the accuracy of approximations, three statistical methods will be used:
Coefficient of determination R2
is defined as the square of Pearson correlation coefficient
r, which is in turn defined as:
r =
n
i=1(Xi − ¯X)(Yi − ¯Y )
n
i=1(Xi − ¯X)2 n
i=1(Yi − ¯Y )2
, (2.3)
where ¯X and ¯Y represent the average values in datasets X and Y. R2
can reach values
between (and including) 0 and 1 - with higher values suggesting a better correlation
between the two datasets (so in our case the higher the better).
Root-mean-square deviation (RMSD) is another method of determining the similarity
of two datasets X and Y. It is exactly what the name suggests - individual deviations
between sets X and Y are squared, averaged and then rooted:
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2. THEORY
RMSD =
n
i=1(Xi − Yi)2
n
(2.4)
Since a good approximation is, naturally, supposed to produce the smallest possible
deviations, we will be aiming for a lower RMSD as a sign of a better approximation.
The average deviation of two datasets (occasionally referred to as AD in this thesis)
is the simplest of these metrics - it is simply the average of all the values’ absolute
deviations from the other dataset’s value:
AD =
n
i=1 |Xi − Yi|
n
. (2.5)
Again, like with RMSD - smaller the deviations, better the approximation.
All of these methods are designed to test similarities between two datasets - in our
case, the set of pKa values approximated by our application and the set of actual pKa
values.
9
Chapter 3
Application
At the core of this thesis is a computer application – one that tests how successfully
the Delaunay triangulation can approximate a property of a molecule using structural
descriptors and helps the user decide which methods and descriptors are the most useful
when applying this mathematical tool. In its current form, the application does just
that – but with a few minor alterations it is possible to use it for actual approximation
or plug it into a larger program. This chapter has two sections – firstly, there is a short
description of each of the software tools I used to create this application. Secondly, there
is a significantly longer section explaining how the actual application works – how it is
used, what outputs it produces and, most importantly, how it produces the results that
will be summarised in the next chapter.
3.1 Methods
Rather than an in-depth analysis, this section is more of an acknowledgement of the
tools that were instrumental in completing this thesis but were not created by me.
3.1.1 Microsoft Visual Studio 2010
MS Visual Studio [10] is a popular IDE for work with C#. For work on this thesis I used
the Ultimate Edition of the environment – not because of its special features, but because
as a student of the Faculty of informatics I have a free license for its use.
3.1.2 MIConvexHull
As has already been mentioned several times in this thesis, this application uses triangulation
to transform a set of molecules into a model that allows for a good approximation.
This triangulation is performed by an external library – MIConvexHull [11]. This library
takes a set of vertices and returns a set of cells. The molecules have to satisfy certain expected
properties by implementing the library’s vertex interface – the molecules have
to be converted to vertices (and will often be referred to as such in the mathematical
sections of this thesis). The returned cells are C# objects representing the simplices of
the triangulation – triangles in 2D, tetrahedrons in 3D and so on. Each of these objects
10
3. APPLICATION
contains a list of vertices forming the cell – we can therefore think of the triangulation
we receive as a ’list of lists’ (a list of cells, each containing a list of vertices).
One further thing should be mentioned, though - originally, the library did not compute
my triangulation at all because of a tiny flaw in the library which I had to find
and fix myself in order for the triangulation to work. That means that my application
is running with a slightly altered version of MIConvexHull, and if someone were to
download the original version and use it with the same data, they would need to make
the same adjustment again in order to make it calculate any triangulation.
In the FindInitialPoints method of the ConvexHullInternal class, a variable called
maximum (on line 609) was initialized with value 0.0001, which was supposed to be an
arbitrary minimal value of a simplex volume (the size of a triangulation cell) - meant to
be overwritten by the first ’actual’ value that would be greater. However, since the data
used for this thesis contains values that are very close to each other, even the greatest
simplex volumes are smaller than 0.0001. Seeing as the Double type does actually have
an infinitesimal number (Double.MinValue), I used that instead.
3.1.3 Excel Library
MS Visual Studio 2010 does have a built-in library for reading and writing .xls files.
However, this library can only be used on a machine with Microsoft Office already installed.
In order to make the application more portable (and because I do not own MS
Office), an external library was required. I chose ExcelLibrary because it is open source
and fairly easy to use [12].
3.2 Implementation
From the user’s viewpoint, the application looks quite limiting - it was, indeed, not
designed for a wide use. It has no user interface and is launched from the command
prompt, with the option of adding launch parameters to change the methods used by
the application. The main purpose of the application was to provide the needed results
for demonstration and evaluation of Delaunay triangulation as a method for prediction
of dissociation constants of molecules, with little regard paid to its further use or user
friendliness.
The application may be ugly and unfriendly; its source code, however, is not. The
possibility that the method may prove useful in the future has been accounted for and
rules like basic decomposition and frequent use of comments have not been neglected
in case the classes and methods were to be used again in another application.
3.2.1 Functionality
To test out the application’s approximation methods, we have a dataset of molecules
with known atomic charges and known pKa. The application uses the obvious way to
11
3. APPLICATION
test the approximation - it selects a molecule, builds a model using all the remaining
molecules and then tries to approximate this molecule using the model (pretending it
does not know the molecule’s actual pKa value). After this process has been repeated
with every molecule in the dataset, statistical methods are applied to determine how
accurate the approximations were.
3.2.2 Input
The molecules with known pKa are loaded from several .xls and .csv files that were
provided to test the application. Each of these files contains over a hundred phenols or
benzoic acids - their names, pKa values and charges on their main atoms (atoms of their
basic structure). The application searches for these files in its current directory.
3.2.3 Output
The output of the application is twofold. Firstly, statistical results of the approximation
are printed to the command prompt from which the application was launched – namely
the root-mean-square deviation (RMSD), the coefficient of determination (R2
), and the
average deviation. Secondly, an .xls file is created, containing three columns for each
approximated molecule – its name, its real pKa value, and the approximated pKa value.
The file is named according to the method used for the approximation, so as to avoid
overwriting the results if another method is used afterwards.
3.2.4 Execution
The application is launched from the command prompt, with launch parameters determining
what method will be used to approximate, and what molecules will be approximated.
The possible launch parameters are listed in the next section, and also in the
’readme’ file enclosed with the application. By default, when no parameters are entered,
the approximation is performed on all molecules using two-dimensional triangulation
(with hydrogen and oxygen charges as descriptors) and weighted average.
3.2.5 Programming language
As mentioned in the previous section of this chapter, I chose C# to create the application
part of the thesis. C# is a programming language developed by Microsoft, very similar
to the more popular Java from Oracle Corporation. This language allowed for easy work
with the MIConvexHull library (which is also written in C#), and it is also a personal
preference of mine. It has several unique features, which have gone largely unused by
me.
12
3. APPLICATION
3.2.6 Classes and methods
Aside from the external tools mentioned in the previous section (ExcelLibrary and MIConvexHull),
the application contains several original classes. This will be a brief summary
of these classes and their methods.
Directing the whole lifetime of the application is, naturally, the Main method in class
Program. The application’s lifetime is as follows: Program parses launch parameters
from the command line and sets its variables to appropriate values. It then asks ReaderXLS
and ReaderCSV classes to parse the input files and return the dataset in form of a
list of instances of Molecule class. Once Program class has these molecules, it goes into
a cycle wherein it always takes one molecule from the dataset and tries to approximate
it with the others - using methods from class Approximate. Once it has the approximations,
it uses methods from Calculator class to calculate statistical properties of the
whole approximation (R2
, RMSD and average deviation) - these values are printed to
the console. Finally, the approximations are written to an .xls file using class Writer.
Approximation is the most important class - as the name suggests, all the logic behind
the approximation (with or without triangulation) is here. The only methods called
from the outside are ApproximateWithTriangulation and ApproximateWithoutTriangulation.
These methods take a molecule and a list of other molecules and return an approximation
of the single molecule’s pKa based on the molecules from the list. The fact
that they only approximate one molecule at a time allows this class to be exported to
another application, one that may wish to approximate a (truly) unknown pKa using
known values. ApproximateWithTriangulation method requires the input molecules in
form of instances of Molecule class, then transforms them to instances of Vertex class
to pass them on to MIConvexHull library. Vertex implements the library’s IVertex interface,
but also contains a method that allows a vertex to compute its distance to another
vertex (needed for weighted average).
All the other methods in Approximation serve only to aid the two approximation
methods; they are mainly mathematical and geometrical operations and their names are
almost always self-explanatory (for example FindNormalVector or WeightedAverage).
3.3 Approximation
The approximation performed by this application basically consists of three steps. First,
the choice of structural properties that will be used to build a model (described in the
first part of this section). Then, the assembly of a model using these properties and
location of the right molecules for the approximation in this model (shown in the second
part). And finally, the actual approximation using the molecules we found (described in
the third part of the section).
13
3. APPLICATION
3.3.1 Descriptors
In the theoretical part of this thesis, I established that the choice of descriptors is critical
for a good model of approximation. The application therefore allows the user to choose
from several descriptors in order to see their effectiveness in approximating the dissociation
constant. The number of descriptors used to build a model determines its dimension
– two descriptors will make the model two-dimensional and three descriptors will
make it three-dimensional. Therefore the user is not required to input the dimension of
the model.
Although there are many descriptors we could theoretically use, our focus will be
on five of them:
• Partial charge on hydrogen atom of the hydroxyl group.
• Partial charge on oxygen atom of the hydroxyl group.
• Partial charge on dissociated oxygen atom (this is the same oxygen atom, in a
dissociated molecule).
• Partial charge on the nearest carbon atom (the one that is bonded with the hydroxyl
or carboxyl group).
• Partial charge on the other oxygen atom from the carboxylic group (the one that
is not bonded with hydrogen) - this is only applicable to benzoic acids.
By default, only hydrogen and oxygen atoms are used for the approximation. If the
user wishes to add a third descriptor to make the model three-dimensional, they can
do so by adding parameter -c to add the charge of the nearest carbon atom among the
descriptors, parameter -o to add the charge of the dissociated oxygen, or parameter -o2
to add the charge of the other oxygen (this will only approximate benzoic acids since
phenols do not have this atom).
3.3.2 Triangulation
In chapter 2, we have discussed what Delaunay triangulation is and how it can be
achieved. Earlier in this chapter, it was described how the external triangulation library
is interacted with. Now that we have an optimal triangulation of vertices with known
pKa, it is time to approximate the remaining vertex. Our best bet would be to place the
unknown vertex in the triangulation, find out which cell it lies in, and average the cell’s
vertices.
But how do we determine which cell the unknown vertex lies in? Remember – the
only information we have are the coordinates of each vertex of every triangulation cell,
and the coordinates of the tested vertex.
14
3. APPLICATION
Two dimensions
As one might expect, geometry has the answer. For the sake of simplicity, let us consider
a two-dimensional model at first and then move on to a solution for three dimensions.
Probably the easiest way to find if a vertex lies inside a triangle formed by three
other vertices is to check the position of the tested point relative to each edge of the
triangle. Supposing the edges are AB, BC and CA (that is, they are directed and they
form a cycle), a point that would lie within the triangle would have to be either to the
right (if points A, B and C are ordered clockwise) or to the left (if they are ordered
counter-clockwise) of all three edges (figure 3.1).
Figure 3.1: A triangle, its vertices ordered clockwise. Any point that lies within the triangle
must be to the right of each edge.
Determining where a point lies relative to a straight line defined by two other points
is nothing more than a matter of calculating a simple determinant


1 px py
1 qx qy
1 rx ry


where p and q are two different points from the line (in our case two vertices of a triangle)
and r is the point to be determined. If the result is a positive number, the point lies
to the left of the line; analogically, a negative result indicates a position to the right of
the line, and a result of 0 means that the point lies directly on the line. Unfortunately,
this means that to find the triangle a point lies in, we have to check every edge of every
triangle in our triangulation until we find the one that satisfies our criterium.
A minor improvement can be made by disregarding all triangles that lie completely
to the right, to the left, above or below the tested point (that is, all three of their vertices
have a lower/higher x or y coordinate). This means that we do not need to compute the
determinant for the vast majority of triangles, but we still need to check every vertex of
every triangle to see if we can discard them this way.
15
3. APPLICATION
Three dimensions
In 3D, ’left’ and ’right’ lose meaning for us, and we need a better way to determine
whether a point lies to the correct side of the triangulation border. For this, we can
use the normal vector of the border. In three dimensions, the triangulation cells are
tetrahedrons and triangulation borders are triangles. The normal vector of a plane is a
vector that is perpendicular to it, and it is calculated as the cross product of two vectors
generating the plane. Once we have the vector, we can create a line that is perpendicular
to the border plane and runs through our tested vertex, and test the vertex against the
intersection of the line and the border plane. Since the line connecting the tested vertex
and the intersection has the direction of the normal vector, we can reach the tested vertex
from the intersection either by going ’along’ the vector (multiplying it by a positive
parameter) or ’against’ the vector (multiplying it by a negative parameter). The signum
of the parameter therefore shows us which side (positive or negative) the vertex lies on
- as shown in figure 3.2.
Figure 3.2: The cross product of two vectors, in our case two edges of a triangle. By
multiplying the vector by a positive number, we reach points lying ’along’ the vector
(in this picture ’above’ the triangle), and vice versa.
But how do we determine whether it is the correct side of the border, if we can call
neither of these sides ’left’ or ’right’? Well, we still have the fourth vertex to help us out
- that is, of course, the vertex that is part of the tetrahedron, but not part of the triangulation
border we are testing against. If we perform the same procedure with this vertex
as we did with the unknown vertex – that is, we create a line that is perpendicular to
the triangulation border and runs through the fourth vertex, and test the vertex against
the intersection of the line and the border – we will once again get a parameter that will
be either positive or negative.
Now we have two parameters, one for the unknown vertex and the other for the
fourth vertex of the tetrahedron. The size of these two parameters tells us how far the
vertices are from the triangulation border. However, finding out whether the tested ver-
16
3. APPLICATION
tex is closer to the border than the fourth vertex is of little use to us – it is, after all,
possible for the tested vertex to be closer to the border than the fourth vertex (even in
the same general direction), and yet still be outside the tetrahedron. The only thing of
importance is the signum of each parameter. If these are the same (both parameters are
positive or both are negative), we can say that they lie to the same side of the triangulation
border. That means that the tested vertex is safely on the ’inner’ side of this
particular border. If we perform this procedure on each border of the tetrahedron, and
in each case the vertex lies to the same side of the border as the remaining (fourth)
vertex, then the unknown vertex is surely inside the tetrahedron.
This solution would not work in a four-dimensional space, since normal vector is
undefined (and therefore can not be calculated) in four dimensions.
By default, the option to triangulate is turned on. It can be turned off using launch
parameter -nt. In that case the application will simply approximate using the nearest
vertices in the surrounding space (three in case of a 2D model, four in case of 3D).
3.3.3 Weighted average
The pKa values of the molecules that contribute to the approximation can be weighted
by distance – that is, how close the charges on the molecules are to the charges on the
tested molecule. We can calculate this distance exactly as we would calculate the distance
of two points in space, if we think of a molecule’s charge values as the coordinates
of a point:
d = (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 (3.1)
where x1, y1 and z1 are the coordinates of the first vertex (or properties of the first
molecule in our case), and x2, y2 and z2 are the coordinates of the second vertex (properties
of the other molecule).
Naturally, the closer a molecule is to our tested molecule, the more weight it should
be assigned – therefore the weighting will be inversed (each pKa value will be divided
by the distance rather than multiplied). The application also offers the option to use
the regular average (all vertices are treated equally regardless of their distance to the
tested vertex). By default, however, the approximation is performed using the weighted
average (since it has consistently proven to be more accurate). This can be changed using
launch parameter -nw.
17
Chapter 4
Results
Finally, we need to evaluate the results yielded by the application of Delaunay triangulation
to the problem of approximation of dissociation constants of molecules. This will
be done mostly in form of comprehensive tables and short comments on the results. Besides
the results of our triangulating method, the tables will also contain the results of a
simple method without triangulation. Since the viability of the approach needs not necessarily
be the same for all types of molecules, a distinction will be made between the
results of approximating phenols (section 1) and the results of approximating benzoic
acids (section 2) - just before these results are merged for a grand evaluation (section
3). To offer some perspective, some of these results will then be compared to the results
of another model created as part of another bachelor’s thesis, which used linear
regression for approximation of the same data (section 4). The results were achieved on
data provided by Tom´aˇs Bouchal, who used them as part of his bachelor’s thesis [13] namely
molecules from the database of NCI (National Cancer Institute) [14], pKa values
obtained from PhysProp database [15] and partial charges of atoms calculated with
a program called Gaussian (using method HF/6-31G* MPA) [16].
4.1 Phenols
First, the results for various phenols 4.1. The first three columns describe the method
used to achieve the results. There is no ’dimension’ column - it would be redundant
since the dimension of a model is always equal to the number of descriptors used.
Hydrogen, oxygen and dissociated oxygen seem to be the best descriptors. Note that
the results of triangulation are almost universally better than the results of the simpler,
non-triangulating method and that using weighted average always improves the approximation
when compared with regular average.
4.2 Benzoic acids
When evaluating the results for benzoic acids, we should make a distinction between
ortho benzoic acids and the rest (meta and para). For fairly complicated reasons, ortho,
meta and para forms of the same benzoic acids do not have the same acidity. The ortho
form of a benzoic acid will always be more acidic than its meta and para forms. Also,
18
4. RESULTS
Table 4.1: Results of approximation of dissociation constant of phenol molecules. The
best results are highlighted.
Descriptors Weighted average Triangulation R2
RMSD AD
H, O NO NO 0.9045 0.6824 0.4652
H, O YES NO 0.9117 0.6603 0.4509
H, O, C NO NO 0.9159 0.6432 0.4415
H, O, C YES NO 0.9295 0.5983 0.3998
H, O, O−
NO NO 0.8651 0.8402 0.5653
H, O, O−
YES NO 0.8701 0.8177 0.5347
H, O NO YES 0.9347 0.5206 0.3894
H, O YES YES 0.9394 0.4946 0.3633
H, O, C NO YES 0.9043 0.5774 0.4414
H, O, C YES YES 0.9420 0.4354 0.3222
H, O, O−
NO YES 0.9362 0.4887 0.3843
H, O, O−
YES YES 0.9626 0.3789 0.2784
while the dissociation constants of meta and para forms will usually be similar, the pKa
of ortho form might be quite different.
The results will be presented in three tables - first only for ortho benzoic acids, then
only for meta and para benzoic acids 4.2 and finally for all benzoic acids together 4.3.
Please note that O2 represents the second oxygen of the carboxyl group, not an oxygen
molecule consisting of two atoms.
You can see that the results of approximation of ortho benzoic acids are abysmal.
This suggests that the acidity of ortho benzoic acids is less correlated with partial atomic
charges than it is for phenols or other benzoic acids. That actually agrees with the chemistry
behind this - due to the so-called ortho effect, when a substituent is in the ortho
position, it interferes with the carboxyl group and affects its hydrogen atom’s willingness
to dissociate - lessening the influence of the partial charges.
Apparently, triangulation does not work very well with benzoic acids, since even
the simple non-triangulating method (using nearest neighbors in space) gives us better
results.
Strangely enough, when looking for the best set of descriptors, you may notice that
the best descriptors are different for each set of molecules. Still, the best ones appear
to be H, O and O2 - trailing only a little in the first two result tables and dominating
the third. Also note that the weighted average still keeps providing consistently better
results than the regular average.
19
4. RESULTS
Table 4.2: Results of approximation of benzoic acids - first for ortho benzoic acids, then
for meta and para benzoic acids. The best results for each dataset are highlighted.
Descriptors Weighted average Triangulation R2
RMSD AD
H, O NO NO 0.7402 0.4933 0.4062
H, O YES NO 0.7620 0.4671 0.3910
H, O, C NO NO 0.6139 0.6043 0.5090
H, O, C YES NO 0.7018 0.5408 0.4438
H, O, O−
NO NO 0.6886 0.5832 0.4469
H, O, O−
YES NO 0.7473 0.4990 0.3921
H, O, O2 NO NO 0.7116 0.5294 0.4121
H, O, O2 YES NO 0.7219 0.5129 0.3928
H, O NO YES 04988 0.4196 0.3483
H, O YES YES 0.6599 0.3254 0.2788
H, O, C NO YES 0.3516 0.5090 0.4353
H, O, C YES YES 0.4640 0.4372 0.3640
H, O, O−
NO YES 0.5132 0.3957 0.3185
H, O, O−
YES YES 0.7077 0.2923 0.2444
H, O, O2 NO YES 0.4716 0.5806 0.4495
H, O, O2 YES YES 0.6613 0.3482 0.2421
Descriptors Weighted average Triangulation R2
RMSD AD
H, O NO NO 0.8891 0.1826 0.1353
H, O YES NO 0.9083 0.1524 0.1107
H, O, C NO NO 0.8815 0.2055 0.1513
H, O, C YES NO 0.9113 0.1583 0.1247
H, O, O−
NO NO 0.8574 0.2176 0.1630
H, O, O−
YES NO 0.8650 0.2071 0.1543
H, O, O2 NO NO 0.8839 0.1849 0.1373
H, O, O2 YES NO 0.8897 0.1746 0.1287
H, O NO YES 0.7267 0.1845 0.1491
H, O YES YES 0.7373 0.1818 0.1441
H, O, C NO YES 0.6655 0.1768 0.1485
H, O, C YES YES 0.6899 0.1741 0.1392
H, O, O−
NO YES 0.7088 0.1677 0.1393
H, O, O−
YES YES 0.7390 0.1617 0.1286
H, O, O2 NO YES 0.8727 0.1292 0.1008
H, O, O2 YES YES 0.8477 0.1260 0.0869
20
4. RESULTS
Table 4.3: Results of approximation of all benzoic acids together. The best results are
highlighted.
Descriptors Weighted average Triangulation R2
RMSD AD
H, O NO NO 0.3912 0.7131 0.5205
H, O YES NO 0.5040 0.6350 0.4653
H, O, C NO NO 0.3875 0.7069 0.5354
H, O, C YES NO 0.4617 0.6650 0.5005
H, O, O−
NO NO 0.5396 0.6175 0.4741
H, O, O−
YES NO 0.5557 0.6045 0.4598
H, O, O2 NO NO 0.8128 0.4107 0.2864
H, O, O2 YES NO 0.8273 0.3897 0.2581
H, O NO YES 0.2145 0.6584 0.5267
H, O YES YES 0.3849 0.5446 0.4072
H, O, C NO YES 0.3886 0.5285 0.4412
H, O, C YES YES 0.4691 0.4816 0.3793
H, O, O−
NO YES 0.1845 0.6877 0.5705
H, O, O−
YES YES 0.4422 0.5335 0.4198
H, O, O2 NO YES 0.7172 0.4630 0.3135
H, O, O2 YES YES 0.8249 0.3025 0.1893
4.3 All molecules
Now let us see what happens when we pool all phenols and benzoic acids together
and build one big common model. Intuitively, it is hard to guess which way the results
will sway. They could become more accurate, since our model will have more values
to approximate from. But then again, they could also become less accurate as we will
use benzoic acids to approximate phenols and phenols to approximate benzoic acids.
It all seems to come down to how similar phenols and benzoic acids are in terms of
dissociation constant and its relationship with partial charges. The results are presented
in table 4.4.
The results are quite positive - in fact, they are better than any results achieved by
only approximating one type of molecules. This shows us two things: firstly, that the
method’s accuracy improves considerably with larger datasets, and secondly, that phenols
and benzoic acids can be used to approximate each other’s dissociation constants a
phenol and a benzoic acid with similar partial charges on atoms will also have similar
pKa.
The best descriptors are H, O and O−
, by a wide margin. The results of using the
weighted average are, once again, significantly better than those of the regular average.
21
4. RESULTS
Table 4.4: Results of approximation of all the molecules together. The best results are
highlighted.
Descriptors Weighted average Triangulation R2
RMSD AD
H, O NO NO 0.9394 0.7065 0.4867
H, O YES NO 0.9461 0.6678 0.4596
H, O, C NO NO 0.9405 0.7004 0.4795
H, O, C YES NO 0.9477 0.6600 0.4401
H, O, O−
NO NO 0.9301 0.7754 0.5359
H, O, O−
YES NO 0.9334 0.7556 0.5105
H, O NO YES 0.9177 0.8127 0.5310
H, O YES YES 0.9671 0.5123 0.3637
H, O, C NO YES 0.8962 0.8946 0.6593
H, O, C YES YES 0.9463 0.6516 0.4310
H, O, O−
NO YES 0.9488 0.6216 0.4702
H, O, O−
YES YES 0.9755 0.4287 0.3219
4.4 Comparison with linear regression
A benchmark set for the methods used in this thesis is linear regression. The results
should be compared with the results achieved by Michal Kudera in his bachelor’s thesis
[17]. This is not as straightforward a task as it may seem - there are considerable
differences between the two approximations and, as a result, there are multiple ways to
approach this comparison.
Firstly, the method used in my application can not approximate all of the molecules
in a dataset using triangulation - regardless of implementation or the dataset’s size. The
method relies on taking a molecule from a dataset, building the triangulation from the
rest of the molecules, and then placing the molecule inside the triangulation. But if the
molecule is part of the dataset’s convex hull, then it will be outside the new triangulation
and therefore will not be inside any triangle.
This can be solved by approximating the remaining molecules (the ones in the convex
hull) using the ’plain’ method - simply averaging the nearest neighbors in space.
When we are trying to evaluate triangulation, is it better to omit the results triangulation
can not give us, or to mix in another method entirely? Both are shown below.
There is also another thing. The application using linear regression was run on a
similar, but smaller dataset than mine. It was, in fact, a subset of my dataset. It is therefore
reasonable to only use that subset when comparing the two methods (even though
the subset is quite small and my approximation loses accuracy with smaller datasets).
But it offers a third way to compare the results - build the model using all the molecules
from my dataset, but only compare the results of the molecules from the shared sub-
22
4. RESULTS
Figure 4.1: An attempt to approximate a molecule from the convex hull.
set. This removes the problem of molecules from the convex hull, because now there
will be other molecules on the convex hull - ones that do not need to be approximated.
However, since it uses extra information that was not available to the application using
linear regression, it is not completely fair.
The three tables below represent the three approaches to results - omitting the molecules
that are on the convex hull, approximating the molecules from the convex hull with another
(less accurate) method, or using the extra values to help build the model. Note
that there are no results for benzoic acids, because the small dataset used in the linear
regression application only contains phenols.
Table 4.5: Results gained when omitting the molecules from the convex hull.
Linear regression 0.9732
Triangulation using O, H 0.9855
Triangulation using O, H, C 0.8764
Triangulation using O, H, O−
0.9733
Table 4.6: Results gained when approximating the molecules from the convex hull with
another method.
Linear regression 0.9732
Triangulation using O, H 0.9094
Triangulation using O, H, C 0.8106
Triangulation using O, H, O−
0.9293
It is easy to see that when we try to incorporate another method to help with the
approximation, the accuracy drops sharply. But even when using all available values
to build the model, the approximations of this small dataset are still less accurate with
23
4. RESULTS
Table 4.7: Results gained when using a larger dataset to build the model.
Linear regression 0.9732
Triangulation using O, H 0.9439
Triangulation using O, H, C 0.9499
Triangulation using O, H, O−
0.9703
triangulation than they would have been with linear regression. However, the results
are definitely comparable and linear regression still has a serious limitation - it can only
be used on homogenous sets of molecules (e.g. only phenols or only benzoic acids).
Not only is our triangulation method capable of approximating heterogenous sets of
molecules (as shown earlier in this chapter), its results can actually be improved by
bringing in different types of molecules into the approximation. The fact that the results
of a method that can be used on any set of molecules are comparable to a much
more specialised method - even when compared over a homogenous dataset - makes
triangulation a very interesting and viable option when trying to predict a molecule’s
dissociation constant.
24
Chapter 5
Conclusion
One of the great challenges of present-day chemoinformatics research is prediction of
dissociation constants (pKa) of molecules [3]. The values of pKa are very interesting
for chemical, biological and environmental research. A particular attention is paid to
pKa in the pharmaceutical industry, especially in the area of drug development. The
value of a molecule’s pKa is one of the criteria that allow us to discard a molecule from
a list of potential drug molecules. The experimental measuring of pKa is difficult and
computational methods for acquiring pKa are nontrivial. An efficient alternative is approximation
of dissociation constant using the values of partial charges on atoms within
a molecule.
An interesting way to do this is by using Delaunay triangulation, and that was the
focus of this thesis. First, I designed and implemented a software tool capable of reading
charges and dissociation constants of molecules from a file and predicting pKa of
other molecules. For these predictions, my application uses Delaunay triangulation and
direct calculation based on pKa of the molecules with the most similar atomic charges.
This tool allows various types of charges to be included in the calculation and offers
its user the option to work with either weighted or regular average of the neighboring
molecule’s properties.
The software was then tested on several datasets of molecules - one containing phenol
molecules, one containing ortho benzoic acids, one containing meta and para benzoic
acids and a few combinations of the three. The results show that triangulation
methods are very successful in majority of cases (R2
> 0.9). The results of these methods
improve with larger datasets. H, O and O−
have shown to be the best descriptors, and
using weighted average is better than using regular average.
Compared to methods of linear regression, triangulation gives us slightly worse results.
On the other hand, methods using linear regresion can only be used on homogenous
sets of molecules, which is very limiting for their application. That is why triangulation
is showing to be a very promising method for predicting pKa in any given set of
molecules.
The software tools developed and the results gathered as part of this thesis will be
used at the National Centre for Biomolecular Research. They will be used to predict pKa
and their results will serve to test the effectiveness of the above-described methods for
predicting pKa.
25
Chapter 6
Appendices
6.1 Contents of the attached CD
The attached CD contains the application - both as a C# solution and as a compiled
executable file. It also contains all the files needed to run it (testing sets of molecules and
ExcelLibrary.dll) and a short Readme file describing the launch parameters available to
the user.
26
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