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{\bf 18.417 Introduction to Computational Molecular Biology}\\
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\begin{center}
{\large \bf Computational Tools for Drug Discovery}\\
(Adapted from notes handed out in class.)
\end{center}

\section*{Introduction}


The sale of pharmaceuticals resulting from research in drug discovery
is a multi-billion dollar per year industry.  A single big-selling drug
(such as Lipitor, which is at the center of the recent controversial
Pfizer/Warner-Lambert deal) can bring revenues of \$4-5 billion per
year.  Bringing such a drug to market, however, is a slow and expensive
process that involves several major steps:

\begin{tabular}{lp{7.5cm}rr}
{\bf process} & {\bf description} & {\bf years} & {\bf candidates}\\
\hline \\
lead discovery & Identifying a molecule (here called a ligand)
that can inhibit some process important for disease, even if only weakly. & 1-2 & \\
lead optimization & Finding versions of this molecule that bind
more tightly to the target of the drug (here called the receptor). &
1-2 & $10^4$ \\
assays & {\em In vitro} and {\em in vivo} testing of efficacy, including
ability of the drug to reach its target & 1-2 & \\
toxicology & Testing if the drug harms living tissue. & 1-2 & $10^2$  \\
human safety & Testing if the drug harms humans. & 1 & 10  \\
human efficacy & Testing if the drug works in humans and determining dose. & 1-2 & \\
\hline\\
{\bf TOTAL} && 6-12 & 1 \\
\end{tabular}

Computational tools are mostly used for the first two stages (lead
discovery and optimization) and fall into two classes:

\begin{enumerate}
\item \label{known_inhibitors}
methods that use information about known inhibitors (no structural
 data available)
\item \label{known_structure}
methods that use information about the structure
of the biological target (structure-based drug design)
\end{enumerate}

A useful overview of both kinds of methods can be found in [1].

\section*{Tools that don't use structural information}
Methods of type 1 try
to identify features common to known inhibitors to use in screening for
new or improved drugs.  Two ways of doing this are briefly outlined
here.
\begin{description}
\item[Generation of a pharmacophore]
A phar\-ma\-co\-phore is a 3D arrange\-ment of func\-tion\-ally-im\-por\-tant parts of
a molecule.  Phar\-ma\-co\-phores can be identified by searching the possible
low-energy conformations of known inhibitors to find recurring spatial
relationships between important groups.

The conformational search can be quite large (since lots of drug
molecules have many degrees of freedom) and can be approached using
\begin{enumerate}
\item systematic search
\item distance geometry methods
\item clique-detection
\end{enumerate}

\item[Quantitative Structure-Activity Relationship (QSAR)]
A QSAR study tries to establish a link between the ability of a certain
molecule to perform its desired function and properties of that
molecule.

A typical example of such a relationship is given by the Hansch
equation:
$$ \log\left(\frac{1}{C}\right) = k_1\log P + k_2(\log P)2 + k_3\sigma + k_4 $$

$C$ = concentration required to give a certain response (lower $C$ = more
effective compound)

$P$ = measure of hydrophobicity of molecule
(partition coefficient between octanol/water)

$\sigma$ = Hammett parameter;
used in organic chemistry to describe the electronic properties of
molecules

This can also include terms to describe steric properties, molecular volume,
etc.

Generally a least squares fit of the parameters to the inhibition data
is performed to give the constants $k_i$.  Neural networks or genetic
algorithms can also be used, for non-linear dependence on $k_i$.

\end{description}

\section*{Structure-Based Drug Design}

Based on the structure of the biological target (usually a protein) either
bound or unbound to an inhibitor or substrate, an inhibitor can be
designed or improved upon.  This is much
harder if the structure is of the unbound complex.  There are two main
approaches:

\begin{description}
\item[Docking] Scan a database of known molecules for those likely to
bind well to the receptor.  Generate possible binding geometries and
evaluate these using some scoring function.
\item[De Novo Design]  Build a ``custom'' ligand for a given receptor.
\end{description}

Two types of de novo design are ``inside-out'' and ``outside-in''.
In each of these, the idea is to:

\begin{enumerate}
\item identify points on the receptor
where a ligand can form favorable interactions.
\item choose ligand
groups of atoms which are complementary to these sites.
\item connect
them into a molecular framework in an exhaustive, unbiased way.
\end{enumerate}

Favorable interaction sites are frequently found by placing multi-atom
fragments on a grid inside the binding site and evaluating their
energy.  Another method is to use molecular dynamics or Monte Carlo
search methods to place fragments in a site.

For inside-out approaches an initial good interaction is chosen as a
starting point, and the rest of the molecule is ``grown'' from this site,
seqentially choosing fragments from a library and connecting them using
rules.  GrowMol is one example of a successful inside-out method.

For outside-in approaches groups of atoms from the ligand are placed in
the binding site, and then a search is made for possible fragments which
can join them in an appropriate geometry.  LUDI and CAVEAT are examples
of outside-in programs.

\subsection*{Docking vs. Design} The key advantage of docking is that the molecules
being considered are taken from a database of molecules that already
exist and can therefore be immediately used in an assay.  Most
companies have proprietary databases of molecules that they have
on-hand.  The advantage in De Novo design is that new classes of
molecules can be discovered (in case there are not any good candidates in
the known databases).  However, these designed molecules
might be very difficult to make.

The main challenge for both methods is the size of the search space
that must be considered.  There are many possible different inhibitors
to choose from and each has many possible conformations in which it
might bind to the receptor.  Also, fast methods are needed since the goal
is to screen many possibilities (a typical database size is $10^5$, and this is
a small portion of all possible designed inhibitors).

\subsection*{Success?} Examples of the design of successful inhibitors using these
methods are still very rare (though there are now a few).  In fact,
even in cases where a ligand was successfully designed or predicted to
be active, structural studies have often shown that the molecule binds
to the target differently than predicted (oops!).  Nevertheless,
especially when coupled with experimental information in a feedback
loop, structure-aided design can be a valuable part of the
drug-discovery process.

\section*{Docking}

\subsection*{Two approaches used for docking.}

In docking there are three degrees of translational and three degrees of
rotational freedom between the receptor and the ligand.  As this is
already a very large space to sample, most docking methods to date use
a rigid body assumption in the docking process (that is, there is no
conformational flexibility of the ligand or receptor considered).  This
is not a very good assumption, but it is made so that the problem will
be somewhat tractable.

A docking method requires:  
\begin{enumerate}
\item a representation of the receptor and ligand
\item a method for generating potential binding geometries 
\item a way of scoring these geometries
\end{enumerate}

The method may also involve some refinement of the initially generated
conformations before or after scoring.

\section*{Method 1:  the DOCK program (Kuntz, UCSF)}

\subsection*{Molecular Representation}  Both the ligand and the potential
binding sites of the receptor are modeled using clusters of spheres. [2]

Begin by describing the surface of both ligand and receptor using a
Connolly surface.  This surface is generated by taking a spherical
probe of some radius $r$ (typically $r = 1.4-1.8$\AA) and rolling it over the
surface of the molecule.  All points where the probe touches the
molecule once are included on the surface, along with points which lie
on the surface of the probe when the probe touches the molecule in
more than 1
point.  The Connolly surface basically smoothes out invaginations in
the molecular surface.  Rather than a smooth analytical
surface, a grid of points on this surface is commonly used.

Given the Connolly surface grid, a series of spheres are generated such
that:
\begin{enumerate}
\item each sphere touches the Connolly surface at two grid points
with the sphere center on the normal from one of the points
\item
spheres lie outside the surface of the receptor, inside the surface of
the ligand
\end{enumerate}

The spheres thus describe the ligand (key) and the negative image of
the receptor (key hole).  The ligand and any potential binding sites on
the receptor will be defined in this way as clusters of overlapping
spheres.  In a perfectly-complementary complex, the centers of the
ligand and receptor spheres will be coincident.

\subsection*{Generating receptor/ligand matches [3]}

\subsubsection*{Size of space}
For $n$ receptor spheres and $m$ ligand spheres (typically
hundreds of each), there are $\frac{n!}{(n-m)!}$ ways to map the ligand points
onto the receptor.  However, it only takes a mapping of three points of the
ligand onto the receptor to define a rigid body transformation.  (A fourth
point is needed to discriminate two possible reflections through the plane of
the first three points).  Thus, we need to consider roughly $n^4m^4$ possible
configurations.  This number can be further reduced by breaking spheres
into pre-defined subgroups (e.g.~corresponding to potential binding
sites) and only allowing interactions where all points are within the
cluster.

\subsubsection*{Acceptable matches}
DOCK generates matches between receptor and ligand spheres such
that internal distances (intra-ligand and intra-receptor) are
equivalent, within a tolerance, between matched points.

GOAL:  find a set of matched pairs $\left\{(r_i,l_i)\right\}, r_i = $ receptor point; $l_i =$
ligand point such that $\forall i,j  \left|d(r_i,r_j) - d(l_i,l_j)\right| < \epsilon$ where
$d(x,y) = \left|x-y\right| = $ distance between points $x,y$.

\subsubsection*{Matching using a bipartite graph}

This method involves systematic generation of possible matches between
ligand and receptor.  The receptor and ligand are represented as
separate graphs, and edges connect matched pairs of
points.  Correspondences between edges are found as follows by
successive trial and backtracking (see Figure 1 of [3]).

\begin{description}
\item[LOOP 1] From the $n\times m$ possible pairs of $(r_i,l_j)$ points choose one:
$(r_1,l_1)$.  This is the seed edge.  (Loop over all possible seed edges.)

\item[LOOP 2]  For each potential second edge $(r_2,l_2)$ evaluate:
\begin{eqnarray*}
\left|d(r_1,r_2)-d(l_1,l_2)\right| & \ge & \epsilon \rightarrow \mbox{reject match pair, back up and choose again}\\
		   & < & \epsilon \rightarrow \mbox{accept match pair}
\end{eqnarray*}


\item[LOOP 3]  For each accepted $(r_2,l_2)$ choose each possible $(r_3,l_3)$ and
evaluate:
$$ \left|d(r_2,r_3)-d(l_2,l_3)\right| \ge \epsilon \mbox{ OR } \left|d(r_1,r_3)-d(l_1,l_3)\right| \ge \epsilon \rightarrow
\mbox{reject, choose again} $$
$$ \left|d(r_2,r_3)-d(l_2,l_3)\right| < \epsilon \mbox{ AND } \left|d(r_1,r_3)-d(l_1,l_3)\right| < 
\epsilon \rightarrow \mbox{accept match pair} $$
\end{description}

Continue until you have generated all possible sets of 3 matched pairs
which give a unique transformation (except for the chirality which can
be checked separately).

This is not particularly elegant or speedy.  Heuristics can be used to
speed it up, though they compromise the exhaustive search.  One such
heuristic is to use distance binning (pre-sort points by their distance
from the points used in the seed edge) so that only potential edges
compatible with the seed edge will be considered.  See Figure 7 of [4] for a
picture of how this works.  A better way to avoid the backtracking
required when bad matches are considered is to do...

\subsubsection*{Matching using a single docking graph}

In the graph used for this search the receptor and ligand points are
combined to give $n \times m$ nodes where each node represents a
potential match pair $(r_i,l_i)$.  Edges exist between nodes with
compatible distances.

Edge if $\left|d(r_i,r_j) - d(l_i,l_j)\right| < \epsilon$

The problem is then defined as finding all unique, completely connected
subgraphs containing at least 3 nodes.  This is related to clique
detection but subgraphs need not be maximal so the problem is not as
hard.

The graph is pre-computed as an Adjacency Matrix, $A$.  $A$ is an $n^2m^2$
Boolean matrix defined as:
\begin{eqnarray*}
			A_{ij} & = & 1 \mbox{ if } \left|d(r_i,r_j) - d(l_i,l_j)\right| < \epsilon\\
			Aij & =& 0 \mbox{ if } \left|d(r_i,r_j) - d(l_i,l_j)\right| \ge \epsilon
\end{eqnarray*}

The matrix $A$ is sparse (ca.~1\% occupied), so the whole matrix is
not stored.

With prior knowledge (stored in $A$) of what pairs of points match, no
backtracking need be done.  The match generation proceeds
to exhaustively generate all matches consistent
with the distance criteria (see Figures 2 and 3 of [3]).  
The search time is proportional to the
number of allowed solutions.  This method is approximately $2\times$ faster
than the previous method using heuristics (which was not exhaustive).
It requires about 10 to 1,000 seconds per molecule depending on what
tolerances are used.

After matches are generated:
\begin{enumerate}
\item The ligand is transformed according to
the unique transformation specified by the match.
\item The match is checked for steric clashes of points other than
match points.
\item The match may be scored using some potential function (e.g.
might include some surface complementarity term, electrostatic term,
polar/non-polar term, etc.).
\end{enumerate}

\section*{Method 2:  Geometric Hashing}
(Fischer, Lin, Wolfson, Nussinov, Tel Aviv
University and NCI Maryland)

Geometric Hashing is a method adopted from computer vision.  In vision
applications where the goal is to recognize objects in a scene, given a
database of known objects, you want to be able to: 
\begin{enumerate}
\item recognize translated/rotated objects.
\item recognize objects that are partially occluded or in the presence of non-database objects.
\item avoid having to retrieve every object in turn and compare it to every part of the
scene.
\end{enumerate}

There are similarities between this problem and the docking problem,
where you want: 
\begin{enumerate}
\item to recognize suitable ligands that are
translated/rotated relative to the receptor binding pocket.
\item to recognize good matches between part of a ligand and part of the
receptor.
\item to avoid having to retrieve every ligand in turn and
compare it to every part of the receptor.
\end{enumerate}

The idea is to store information about ligands in a way such that it is easy to
retrieve the ones appropriate for binding to a given receptor

\subsection*{Molecular Representation}
The representation used by Fischer {\em et
al.} is different than that used in DOCK.  The goal is to have an
accurate, sparse, uniquely-defined description of the molecular
surface. [5]

Again start with the Connolly surface.  The Connolly surface can be
broken up into three different kinds of faces:

\begin{enumerate}
\item convex, where a probe touches the surface once
\item concave - where a probe touches the surface three times
\item saddle-like - where probe touches surface twice
\end{enumerate}

The idea is to choose one point on the surface per face, which will be
called a critical point.  The three types of critical points are called
caps (for convex faces), pits (for concave faces) and belts (for
saddle-like faces).  Each critical point has associated with it a
surface normal and a size, which reflects how much of the surface it
covers.

The main advantages of this representation are that 
\begin{enumerate}
\item the shape of the
whole surface is described 
\item there are only a few points per atom
(typically six, but this is often pruned to less than 1 by various
considerations) 
\item the location of the points is independent of the
density of points chosen (not true for Connolly surface, for example)
\end{enumerate}


\subsection*{Generating receptor/ligand matches [6]}

This occurs in two phases.

\subsubsection*{Preprocessing of ligand or ligand database} This only needs to be done
once, as follows:

\begin{enumerate}
\item For every pair of {\em cap} critical points on the
ligand, $cp_{l1}$ and $cp_{l2}$,
generate a unique reference frame using the two points and the average of
their associated normals (given {\em m} ligand points there will be $m^2$ such
reference frames). 
\item In each reference frame in turn, for every other
cap critical point, $cp_{li}$,  that meets some distance criterion, record
the coordinates of this point in the chosen frame (gives (x,y,z)).
\item
Build a hash table to hold this information where the address is the
coordinates of the critical point and the entry is: (the identity of
the critical point, the ligand reference frame in which it is
described).  The table will have $m^2 \times m = m^3$ entries.
\end{enumerate}

Example:
$$
\begin{array}{cc}
\mbox{Address}         &      \mbox{Entry}\\
(x_a,y_a,z_a) & (cp_1, cp_{l1}, cp_{l2})\\
(x_b,y_b,z_b)     & (cp_2, cp_{l1}, cp_{l2})\\
(x_c,y_c,z_c)    & (cp_3, cp_{l1}, cp_{l2})\\
\vdots& \vdots\\
(x_{a'},y_{a'},z_{a'})   & (cp_1, cp_{l2}, cp_{l3})\\
(x_{b'},y_{b'},z_{b'})  & (cp_2, cp_{l2}, cp_{l3})\\
\vdots& \vdots\\
(x_{a''},y_{a''},z_{a''})     & (cp_1, cp_{li}, cp_{lj})\\
\vdots& \vdots\\
\end{array}
$$

Notice that each ligand critical point will occur many times in this
table, indexed by its coordinates in many different reference frames.

What this is doing is storing information about triples of
points found in the ligand in the hash table.  The idea is that when we
find a receptor that needs three ligand atoms in a certain orientation, we
will know where to look.

\subsubsection*{Recognizing Matches}

Process the receptor in a way similar to the ligand:

\begin{enumerate}
\item For every pair of {\em pit} critical points on the receptor, $cp_{r1}$ and $cp_{r2}$,
generate a unique reference frame using the 2 points and the average of their associated
normals.
\item In each reference frame in turn, for every other pit critical point, $cp_{ri}$,
that meets some distance criterion, note the coordinates of this point in the chosen
frame (gives $(x,y,z)$).
\item Look in the hash table under the $(x,y,z)$ entry and (after checking a few
geometry requirements) cast a vote for the ligand reference frame that is found there.
\item For each receptor frame, tally the votes from different ligand frames and use
the best as potential matches.
\end{enumerate}


For a given receptor reference frame, every ligand reference frame that
got $K$ votes comes from a ligand that has at least $K+2$ critical points
that can be superimposed on the receptor.  The two reference frames
(ligand and receptor) can be used to compute the transformation which
accomplishes the superposition.

After potential matches are found the ligand is transformed, and the match
is checked for steric clashes and can be scored, etc., as described for
DOCK above.


\subsubsection*{Advantages of Geometric Hashing}
\begin{enumerate}
\item pre-processing of database of ligands is done only once
\item relatively efficient lookup $O(m^3)$ for well-balanced hash table
\item inherently parallel
\end{enumerate}

A disadvantage (perhaps) is the possibility of collisions in the hash
table (more than one entry with same address slows down lookup).  This could
potentially be a problem since indices describe interatomic distances
which are not really random.

Geometric Hashing and DOCK both have the potential to include
requirements of chemical complementarity between matched points.
They are also both
translation/rotation invariant,
and do pretty well at reproducing the correct geometry of known
complexes when starting with the X-ray structure of the components.  However, they are bad at
picking the best solution out of the others generated, and they are bad
when they do not have the correct structures to dock together.  Since
conformational changes upon binding are common, this is a big problem.

The main challenge for docking methods in general these days is to
incorporate some amount of flexibility into the ligand and/or
receptor.  Methods examined for receptor flexibility include allowing
pre-determined hinge motions, performing sidechain repacking
calculations (using DEE and A*) during docking or docking to an
``average'' structure that is derived from several known conformations of
the receptor.  The most obvious way to include ligand flexibility is to
consider explicitly multiple conformations of each molecule being
docked.  Other possibilities include using some amount of molecular
dynamics or Monte Carlo search in the procedure, or using analogies to
robotics to describe articulated motions of docking ligands.

[1]  Leach, A. R. Molecular Modelling Principles and Applications, Addison Wesley
Longman (1996).

[2] Kuntz {\em et al. J. Mol. Bio. 161} (1982) 269-288.

[3]  Ewing & Kuntz, {\em J. Comp. Chem. 18} (1997) 1175-1189.

[4]  Shoichet, Bodian & Kuntz, {\em J. Comp. Chem. 13} (1992) 380-397.

[5]  Lin {\em et al. Proteins 18} (1994) 94-101.

[6]  Fischer {\em et al.  J. Mol. Bio. 248} (1995) 459-477.
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