*
Molecular surfaces are widely used for characterizing
molecules and displaying and quantifying their interaction properties.
Here we consider molecular surfaces defined as iso-contours of a function
(a sum of exponential functions centred on each atom) that approximately
represents electron density. The smoothness is advantageous for surface
mapping of molecular properties (e.g. electrostatic potential). By varying
parameters, these surfaces can be constructed to represent the van der
Waals or solvent accessible surface of a molecule with any accuracy.
*

*
We describe numerical algorithms to operate on the
analytically defined surfaces. Two applications are considered. (1) We
define and locate extremal points of molecular properties on the surfaces.
The extremal points provide a compact representation of a property on a
surface obviating the necessity to compute values of the property on an
array of surface points as is usually done. (2) A molecular surface patch
or interface is projected onto a flat surface (by introducing curvilinear
coordinates) with approximate conservation of area for analysis purposes.
Applications to studies of protein-protein interactions are described.
*

- Introduction
- Distance to the surface
- Operation on the surface
- Finding potential extrema
- Projections of molecular surfaces and interfaces
- Example of interface analysis
- Accuracy of area values
- The program
- Conclusions

The surface of a molecule bears information about how it interacts with other
molecules and its solvent. As the surface of a molecule is not a quantity for
which a unique physico-chemical definition exists, several definitions have
been introduced. Those most commonly used are:

The analytical molecular surface is smoother than the van der Waals surface and this facilitiates its analysis. However, its primary importance is not its smoothness but that it gives a different description of the molecular interior from the van der Waals surface, since the cavities inside the molecule which are not accessible to the solvent probe appear to belong to the molecule's interior.

The properties of a molecule are usually displayed on its surface
by assigning values to points on the surface -
all but a minority of which will be on the surface of
a single atom and thus possess the properties of that atom.
The points can be displayed on a graphical device
or used for computational analysis. The analysis of surface properties
is usually based on visual inspection for which points are coloured or
assigned sizes corresponding to the value of a property.
Solid rendering, such as exemplified by the GRASP program of Nicholls, can be used to enhance visual quality.
There are several algorithms allowing automation of surface analysis.
We mention two by way of example: a) The fully automated detection
of clusters of surface points with like properties in order to
specify hydrophobic patches on protein surfaces by
Lijnzaad *et al*. b) Location of knobs and
holes on a protein surface using a geometric hashing algorithm applied to
coordinates derived from a dot representation of the molecular surface
by Fischer
*et al*.
A reasonably accurate representation of a molecular surface
requires 10-30 points per ,
generating 100,000-300,000 points per surface for medium-sized proteins.
While the scanning of all surface points is fast,
considerable computational effort is necessary to establish the connectivity
(neighbourhood) of the points. Operations with dot surfaces
can, however, be made extremely efficient by avoiding time
consuming distance checks
(see Eisenhaber et al ).

An alternative description of a molecular surface, which is analogous to the
van der Waals surface, may be derived from a Gaussian description of the
molecule (Duncan and Olson).
It is defined using the approximate electron density distribution:

,

representing the contribution from each atom* i*
of the molecule with different weighting factors
and taking into account the different sizes
of the atoms.
This form of representation is rather arbitrary and exponential functions
such as:

,

would give a more correct description of the asymptotic behavior of
electron density.
Advantage of Gaussian surface description is that it may describe the
fuzziness of molecular surfaces due to high frequency atomic vibrations
(see Agishtein).

The volumetric properties of the Gaussian molecular representation have been compared to those of other surfaces by Duncan and Olson and Grant and Pickup. They found that a Gaussian representation could reproduce area and volume quantitites computed using a hard-sphere model. The Gaussian representation is exploited to locate specific regions on molecular surfaces in the SURFNET program of Laskowski. Physically, the Gaussian representation provides a much more natural and realistic description of the shape of molecules than the atomic hard sphere representation. Another advantage is its continuity which allows straight-forward analytical estimation of the derivatives of surface dependent properties, in contrast to the computational difficulties of dealing with the discontinuous hard sphere representation.

In this study, we use a related, exponential, density function to
construct molecular surfaces. It differs from previous uses of
Gaussian functions in the following respects:

a) The function is constructed to define molecular surfaces
rather than the molecular interior.
We intentionally study the surfaces directly, since quantization
and numerical operations on the 2D surface require considerably
fewer operations and memory than operations on the
3D density function itself.

b) Surfaces similar to the solvent accessible surface of the
molecule can be derived as well as those corresponding to its
van der Waals surface. The solvent accessible surface is
easier to treat analytically because of its relative simplicity.

c) The parameters of the density function are assigned to
define the degree of smoothing of a hard-sphere surface.
These parameters are related to electron density parameters to the
extent that the hard-sphere model reflects the electron density.

The starting point for this study is the analytical definition of the
surface approximating the solvent accessible or van der Waals surface.
The surface is defined implicitly using the functional of the distance to it
(simply, distance=0).
Using this functional, one can make a numerically fast and stable
projection to the surface on which one can place a set of approximately
equally spaced
points with obvious connectivity.
The latter may be used to build a grid of
quasi-curvilinear coordinates on the surface.
We describe two important applications of these surfaces.

Let us define an exponential parametric function
*g(***r***,A;d)*
associated with atom
A,
whose center is positioned at
and which has a radius
.
The parametric function depends on coordinate
**r** and adjustable parameter
*d* :

.

While other functional forms may be equally appropriate, we will only consider the above exponential function. A simple manipulation gives the exact value of the distance to the surface of the atom A, :,

which is independent of the parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1a)

Then, the distance functional is:This definition is formally equivalent to an isocontour value of the sum in (1a), but is easier to handle numerically.

At any point, Depending on which radii
are ascribed to the atoms, one obtains the distances to different surfaces:
*a)* if all
are atomic van der Waals radii, then
*=0*
defines an approximation to the van der Waals surface;
*b)* if all
are van der Waals radii of atoms incremented by a solvent probe radius,
then the solvent accessible surface is approximated.

The choice of the value of the parameter
*d*
should be related to the characteristic interatomic distances.
At any given point on the surface,
the relative contributions to the sum in
equation (1b) arising from the
the closest atom and the next closest atom can be compared.
Considering a typical interatomic distance of 1.5 Å between
these atoms, the contribution to the sum from the second atom will be
0.05 and 0.22 times that of the closest atom for values of
*d* of 0.5 and 1.0 Å, respectively.
Consequently, considering only the two closest atoms, the
*=0*
surface will follow the van der Waals or solvent accessible
surface with a deviation of only 0.025 Å at *
d=*0.5 Å or 0.2 Å at
* d=* 1.0 Å.
If the geometrical positions of the atoms is different that
from described above, or more atoms are considered,
the distortion may be much greater.
Figures 01-04 illustrate the dependence of the computed surfaces on the
parameter *d* for the
human growth factor hormone.

Figure 01: Contours on the solvent accessible surface of
human growth factor (hGH)
computed with *d *=1.0 Å.
There are 6911 points at the surface. The solvent
accessible surface area, computed from the hard-sphere
representation is 9785 .

Figure 02: Contours on the solvent accessible surface of
human growth factor (hGH)
computed with *d *=0.5 Å.
There are 7497 points at the surface.

Figure 03:
Planar section through the same molecule,
hGH,
showing contours corresponding to van der Waals surfaces at
*d *=1.0, 0.50 and 0.25 Å.
The contour with *d *=0.5Å
essentially shows all the details of the molecular surface except the cavities
within the molecule. At *d*<0.25 Å
the contours are not of closed form, making continuous tracing
of the surface impossible.

Figure 04 shows contours on the same cross-section plane as in
Figure 03 for solvent accessible surfaces computed with four values of
*d*. Note that surfaces computed with
*d*<0.25 Å are almost
indistinguishable.

The surface of molecule *M*
is thus defined by
** =0**.
To define the interface between two molecules,

=

where the distances and are the distances to the surfaces of the first and second molecules, respectively. The parameters in the definition of the distance functional (1b) can be modified to obtain different interfaces defined as equidistant to molecular surfaces using atomic radii or defined as equidistant to the closest atoms by setting all radii to zero.

The performance of some of the operations to be described below
depends on the value assigned to the parameter
*d*,
defining the closeness of the computed surface to
a hard-sphere surface. Except for the first two, which are performable
at any *d*, mapping of the surface can usually
(for proteins) be done only at *d* >0.5 Å
when the surface is smooth enough to keep the mapping operations stable.
Improvement of the algorithms may, however, enable mapping of the
more complicated surfaces obtained at smaller *d*.

a. Placing a point on the surface

The surface is defined implicitly and is smooth. The basic operation for moving from any given point to the point on the surface is projection to the surface along the gradient of the distance function (1b). Projection is performed iteratively with a step size equal to the distance to the surface.

b. Motions along the surface

Motions along the surface from any point can be done using the surface tangents. The choice of the functional (1b) makes the higher order derivatives small, so the tangent movements can be corrected by a few iterations (typically 1-2, when the deviations from the surface are less than 1 Å), when the step size of the tangent motion is within 1-3Å. Consequently, the move along the surface costs several computations of sums like that in (1b) (the functional itself and its 3 first derivatives are computed at each iteration). Computation of the sum (1b) scales as the number of atoms: the same number of motions wll cost only twice as much when the molecule size is doubled.

c. Mapping part of a surface

Locally, the molecular surface resembles a Euclidean rectangle (being topologically equivalent to it). One can define pseudo-Euclidean coordinates on an interesting part of the surface. It is natural to introduce polar coordinates, starting from some specific point on the surface. This can be achieved by growing up rings of points starting from a given center. A variety of algorithms can be used. We introduce the basal distance D to define the distance between the points on the surface. Each ring of points is constructed at this distance from the previous one and then projected onto the surface. The points on each newly constructed ring are then checked, one after the other, to see if the distance to the next point is less than 0.75D, in which case this next point is eliminated, or larger than 1.7D, in which case one more point is added. An example of a set of points resulting from applying this procedure is shown in Figure 05. This method is similar to that used by Bacon and Moult, who introduced web coordinates on molecular surface patches by fitting the precomputed surface points by B-splines and constructing a self-growing web. We use the ring construction algorithm to map the interfaces between two molecules. If D=1 Å, the algorithm places ca. 1 point per 1.225 of the surface.

Figure 05: Example of a set of points resulting from the mapping procedure.

d. Scanning the whole molecular surface

To represent the molecular surface as a whole, one needs a representative set of points on the surface that cover the entire surface as uniformly as possible. The simplest solution is to project every atom center in the molecule onto the closest point on the surface. A drawback of this procedure is that the set of points on the surface cannot be made more dense unless a check over all surface point pairs is done because the points are not ordered according to proximity. It is just such computationally intensive searching that we wish to avoid. An alternative procedure is to use the algorithm described above in section (c) to map the entire molecular surface. For relatively spherical molecular surfaces, the growing rings may eventually contract to a point on the other side of the molecular surface from that at which growing was started, thus defining the spherical coordinates. These mapped coordinates may be used to locate every point on the surface. Making a more dense representation is straightforward since the set of points is ordered. Examples are presented in Figures 01 and 02 where this algorithm was used to cover the solvent accessible surface of the molecule by a sequence of rings.

Potentials representing diverse properties, for example, electrostatic potential, lipophilicity, hydrophobicity, or the curvature of the surface itself, can be studied. In order to locate the potential's extrema on the surface, the formalism of Lagrange multipliers is used and the equations are solved by Newton's method (see, for example, Korn and Korn). For this, it is necessary to compute the first and second derivatives of the potential and the distance functional. These can be computed analytically if the potential and its gradient are smooth.

Figure 06 shows an example of the characterization of a potential
on a molecule's surface by its extremal points.
The circles are drawn to show the sizes (derived from the second
derivative) of the minima and maxima along
the surface tangent plane.
Saddle points are shown as lines along the two main directions,
defining the steepest increase and decrease of the potential.
Surprisingly, the potential at the (smoothed with
*d* = 1 Å)
1.4 Å probe accessible surface of the molecule
is not extremely complicated, having only 50-100 extremal points in all. These
points can be used as a minimal representation to restore the potential
and to compare molecules. For instance, many molecular properties correlate with
the properties of its electrostatic potential as described, for example, by
Murray *et al* and
Richard.

Figure 06: Minima, maxima and saddle points of the electrostatic potential on the smoothed solvent accessible surface (shown by yellow contours) of hGH. Red circles shows the position and extension of minima (which, in this example, all have a negative value of the potential), blue circles show the maxima, and green crosses show the saddle points of the potential on the surface.

Surfaces are usually defined as a set of points that is visualized with the aid of graphical programs and analysed by eye. The automation of the procedure would not only save time spent in analysis, but would also avoid possible errors resulting from the subjective nature of the manual analysis procedure. The projection of two different surfaces onto one simple surface is necessary for the comparison of these surfaces (see review by Masek ). A natural solution is to project the surface onto a plane where it can be quantified in a straightforward manner. However, simple projection of the entire molecular surface encounters a topological problem (met already in mapping the surface of the earth), since a closed surface can at best be projected onto a sphere rather than a plane without destroying the connectivities between points. A beautiful solution to this problem for small molecules was suggested by Gasteiger et al, who used Kohonen maps to project the surface onto a torus conserving neighbourhood.

For projecting the surface onto a sphere, the gnomonic projection (see Chau and Dean) provides the simplest solution. The surface is placed within the sphere and then projected along the radials of the sphere. The feasibility of approximately conserving distances between points on the surface upon projection is highly dependent on the relative orientations of the sphere and surface and distance conservation will not be uniform over the surface. Moreover, two points will often be projected onto one, resulting in loss of information. A projection that avoids overlaps is the spherical harmonics representation developed by Duncan and Olson in which a special procedure eliminates overlaps resulting from the gnomonic projection.

The solution proposed in this work is based on the analytical definition of molecular surfaces and the algorithm of growing rings, which builds up quasi-polar coordinates on the surface. This construction may overlap onto itself, causing the duplicate projection of some regions of the surface. The duplication problem is inevitable as can be appreciated from imagining covering an irregular surface with a piece of paper. The duplicate covering can obviously not be completely avoided except in the case of a planar surface. However, most of the duplications can be avoided by weakening the requirement of distance conservation, in an analogous way to covering the surface with a piece of stretch film rather than paper. Depending on the curvature of the surface, this will cause different degrees of distortion. On the other hand, overlaps caused by topological differences between the original and projection surfaces should be handled either by projecting onto an appropriate simple surface or by additional description, for example, specifying the equivalence of opposite borders of a rectangle to which a torus has been projected.

Once the surface has been covered by a set of rings, the *
m*-th point of the
*n*-th ring can be projected onto a
point *(x,y)* on the plane using the
following formulae:

As a result various properties on the surface or interface are transferred to an appropriate Euclidian rectangle.

Figures 01 and 02 show surfaces mapped avoiding duplications. A correction of the procedure described in section (d) above avoids local sources of duplication, so that the growing rings eventually evolve to a point on the other side of the molecule. The ring numbering can serve as a latitude and the length parameter along each ring as a longitude. This provides a means by which to automatically introduce spherical coordinates for a molecular surface.

By way of illustration, we provide a new view (in Figures 07-13) of the hGH-hGH receptor interfaces analysed by Clackson and Wells. The available crystal structure consists of the ternary complex of one hGH molecule bound to two identical receptors: hGHR1 and hGHR2. Binding at the first interface, hGH-hGHR1, is high affinity while that at the second interface, hGH-hGHR2, is low affinity. The buried areas on the hGH-hGHR1 and hGH-hGHR2 interfaces are 1300 and 900 , respectively. The two binding surfaces of hGH differ in sequence while the binding surfaces of the two receptors are similar (but not identical) both spatially and in terms of the residues involved.

The pictures below are the links to stereo images of: a) the ribbon representation of the complex; b) the complex together with 2 interfaces; c) hGHR2 and hGH and interface; d) hGHR1 and hGH and interface.

For the mapped interfaces shown in Figures 07-13, the areas in the maps approximately equal the areas on the interface in 3D. The scale is such that each map has dimensions of about 60 Å by 60 Å, and the tick marks are positioned at approximately every 1 Å. The contour plots of 2 dimensional functions on a reactangle are generated using the program XFarbe.Figure 07: Mapping displaying the interprotein distances (in Å) for the hGH-hGHR2 (left) and hGH-hGHR1 (right) interfaces. The distance is computed at every interface point as a sum of the distances to the closest atoms of the two proteins. The latter distances are derived from the distance functional (1b) with atomic radii set to zero.

Figure 08: Mapping of the residues of hGHR1 (left) and hGH (right) onto the hGH-hGHR1 interface coloured by their number. Only the residues projecting onto more than 17 points are coloured.

Figure 09: As figure 08 but for the hGH-hGHR2 interface.

Figure 10: Mapping of the hydrophobicity properties (following Eisenberg and McLachlan) of the closest atoms of hGHR1 (left) and hGH (right) onto the hGH-hGHR1 interface. Blue regions are hydrophobic, red ones are hydrophilic.

Figure 11: As figure 10 but for the hGH-hGHR2 interface.

Figure 12: Mapping of the crystallographic temperature factor distribution (in of the closest atoms of hGHR1 (left) and hGH (right) onto the hGH-hGHR1 interface. Red regions are more mobile with temperature factors above 30 ; blue regions are more rigid with temperature factors below 25

Figure 13: As figure 12 but for the hGH-hGHR2 interface.

Figure 14: Mapping of the electrostatic potential (in kcal/mole/e) from hGHR1 (left) and hGH (right) on the hGH-hGHR1 interface.

Figure 15: As figure 14 but for the hGH-hGHR2 interface.

Interfacial cavities and crevices can be identified from Figure 07. Those for which the interprotein distance is greater than about 5 Å are large enough to accommodate water molecules. As can be seen from Figures 08 and 09, the interfaces consist of residues from discontiguous parts of the sequences of the proteins. However, there are some continuous stretches of residues at the interfaces, notably, the participation of the D helix and C-terminus of hGH (residues 155-190) in interactions with the hGHR1 is readily seen in Figure 08 (right) as a more ordered region of residues with close numbers. The interface to HGHR1 is, however, not centred on this helix but inbetween two separate secondary structure patterns. On hGHR1, there are three residues that contribute large surface areas to the interface. Tryptophans 104 and 169 contact part of hGH's D helix (residues 155-184) and AB loop (residues 35-71, especially the N-terminal part of the minihelix 2, built of residues 64-70). Asparagine 218 interacts with part of hGH's A helix (residues 9-34). Experimentally it has been shown that the high affinity of the hGH:hGHR1 complex is due to the contribution of a "hot spot" on the hGHR1 receptor consisting of a small number of hydrophobic residues, including W104 and W169 (Clackson and Wells). The hydrophobic patch at the interface created by these residues is clearly visible in Figure 10. It is the largest continuous hydrophobic patch on the interface. It satisfies the description of the binding epitope as consisting of a hydrophobic center surrounded by polar moieties. An equivalent patch, due also to W104 and W169, is observed on the hGHR2 in the second (lower affinity) interface, but it differs in shape and in the arrangement of the polar residues around it. The second patch on the hGHR1 interface (formed mainly by N218 of hGHR1) also has a large buried area. However, it contributes much less to the binding energy, as revealed by mutational analysis (see Clackson and Wells ). It can be seen from Figure 12, that this patch has less hydrophobic complimentarity, being composed of hydrophilic residues of hGHR1. Moreover, it is rather mobile unlike the first patch. It is interesting that it is the mobility rather than hydrophobicity that clearly distinguishs the first patch from the second (see Figure 12 vs Figure 10) implying that the interactions between less mobile residues define the binding free energy of these two proteins. The residues forming the second patch of the interface to hGHR1 have much smaller interfacial areas on the interface to hGHR2.

The electrostatic potentials of the two proteins show a high degree of complimentarity at the interface, as can be seen from Figures 14 and 15, where regions of positive (negative) potential from one protein match regions of negative (positive) potential from the other. The electrostatic potentials were computed with UHBD. The potentials of the hormone at the two binding interfaces are similar. This is seen by comparing the right sides of Figures 14 and 15 taking into account the different transformations applied during projection: in both cases, the potential of the hormone consists of patches in the sequence: negative-positive-negative-positive (from the left to the right).

Figure 16 shows the similarity index for the potentials from hGHR1 and hGH plotted on every interface point, as well as a simple product of the two potentials. To compute the potentials used in Figure 16, the polar hydrogen atoms were simply added to the crystal structure with the HBUILD option in QUANTA. As can be seen from Figure 16, the complimentarity of the two potentials (given by negative values of the similarity index) is not uniformly distributed over the interface. There are at least 4 regions where the potentials from the 2 molecules are in conflict having the same sign and having considerable absolute values (see the right-side panel of Figure 16). To evaluate whether these result from assignment of poor hydrogen atom positions, we energy minimized the hydrogen atoms (fixing all heavy atoms) with QUANTA/CHARMM for 150 steps of steepest descent minimization. Figure 17 shows the same properties as Figure 16 for the resulting conformation.

Figure 16: Similarity index
(known as Hodgkin index) as a function of the
interface point (left) and the product (right)
of the two molecular potentials for the hGH-hGHR1 interface.
Namely, given the potentials from
molecule 1, , and molecule 2,
,
at the point **r**
of the interface, the function shown on the left-side is
and on the
right: .
There are 4 clickable regions
(A, B
,C and D)
for which 3D stereo images available.

Figure 17: As Figure 16 but for the energy minimized conformation of the hGH-hGHR1 complex.

As a result of minimization, one highly non-complementary patch (B) disappeared. The other non-complementarity patches, however, remain. The conformational changes due to minimization are rather small, as can be seen by clicking on the marked patches of non-complimentarity in Figure 16 - they have links to 3D stereo images of conformations before (coloured by atom type) and after (coloured green) minimization. The receptor (hGHR1) is always drawn on the left side of the interface (yellow). Non-complimentarity in region B is removed by a conformational change in Threonine 175 of hGH, the added hydroxyl hydrogen atom of which was in obvious conflict with the side chain atoms of Arginine 43 of hGHR1. The small non-complimentarity in region A is not affected by minimization. Non-complimentarity in region D is reduced, but this is not only due to minimization: we also interchanged the NE and OE atoms of the side chain of Glutamine 46 of hGH, which, as a result, has a favourable electrostatic interaction with Glutamate 120 and Threonine 77 of the hGHR1. The interactions in region C are very complicated and remained essentially unoptimized after minimization. It is interesting that there is a correlation between electrostatically non-complimentary regions, mobile regions and the regions having a small contribution to binding.

In summary, this example shows the different features contributing to binding. The "hot spot" is a particularly rigid (blue spot in Figure 12 ) hydrophobic (blue spot in Figure 10 and white spot on the right-side panel of Figure 17 ) region of high shape complimentarity (red spot in the right side panel of Figure 07 ) involving 2 Tryptophans ( Figure 08 ) surrounded by polar residues in highly complimentary positions (best seen in Figure 17 ).

There is approximate area conservation during the mapping procedure. When the reference distance for ring generation is D=1Å, approximately one point is placed per 1.225 . However the distribution of the points is not uniform. Projection of the surface rings to the rings of the polar coordinates on a planar rectangle further distorts the areas.

We repeated the mapping for the two interfaces for hGH binding to hGHR1 and hGHR2 using different starting points. In both cases, the interface remains the same since its definition is independent of mapping. Distortions occur when the surface is projected onto the planar square and these distortions are related to the position of the interface with respect to the center/starting point. There is no strict correlation between the distance to the chosen center and the distortions, although the area around the center will always be less distorted. Mapping with different centers is a good test of area conserving properties. Figure 18 shows the correlation between the areas assigned to every residue of both proteins at both interfaces during two different mappings. Deviations are within either the absolute limit of points or the relative limit of These deviations depend mainly on the surface curvature which defines the degree of distortion.

The distortions above are introduced by the projection procedure. However, the placing of the points on the interface or surface is also subject to error. These distortions can be quantified by relating the accessibility of each residue to the number of surface points generated on the analytical surface which are closer to this residue than any other (see Figure 19). There are two sources for deviations. The first is caused by the deviation of the analytical smooth surface from the hard-sphere surface used to compute residue accessibilities. The second is defined by the algorithm generating the points on the surface, which is designed to generate non-intersecting rings that evolve to a single point when mapping of the surface is complete. As can be seen in Figure 19, although the upper estimate of the error for the surface shown in Figure 02 is 22 , for the majority of residues the number of surface points per residue correlates much better (with a scaling factor of 1.25) with the residue accessibilities. The scaling factor arises because the surface was constructed so that there is on average 1 point per 1.225 .

The illustrations above are generated by a 4-step protocol using the ADS (Analytically Defined molecular Surfaces) program package. In the first step, rings of points are constructed on the interface using a program which maps the surface of the molecule. In the second step, the required properties are assigned to the points on the surface. In the third step, the surface rings are projected onto a rectangle, producing data files to be used in the fourth step, which is to display the properties and plot the pictures. Output data files are readable by the X-window oriented program XFarbe of Preusser, which is used to make a contour representation of the functions on a rectangle. Computer times necessary to perform the manipulations (which have not yet been optimized for speed) are within 2-5 minutes on an SGI Indy RC 4600 workstation for proteins having ca 1500 atoms.

The source codes (in fortran 77) of the ADS program package are currently under development and it is intended that they will be made available upon request.

**Check the latest status of ADS development at ADS home page.**

We have introduced an analytical definition of smooth molecular surfaces as isocontours of the sum of exponential functions centered on atoms. Depending on the smoothing parameter, the analytically defined surfaces can approximate van der Waals or solvent accessible surfaces of a molecule with any desired accuracy. We constructed a distance functional to make the implicit definition of the surface easy to use numerically. Moreover, the interface between two molecules may be defined using this definition of distance to the molecules.

We exploited the main property of these surfaces, their smoothness,
in order
* a) * to couple potential functions to a surface and locate extremal
points of the potential on the surface by gradient methods;
* b) * to define pseudo-Euclidean (on part of the surface) or
pseudo-spherical (on the whole surface) coordinates on the surface;
* c) * to define pseudo-Euclidean coordinates on the interface between
two molecules. These representations are useful to visualize and study
molecular surfaces and interfaces.

The main advantage of these representations is that their complexity or simplicity can be tuned by the user. This attribute is especially important for analysing the many properties of protein surfaces and interfaces.

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