A New Analytical Method for Computing

Solvent-Accessible Surface Area of Macromolecules

J Comput Chem 26: 334–343, 2005

By SHURA HAYRYAN, CHIN-KUN HU, JAROSLAV SKRˇ IVA′ NEK, EDIK HAYRYAN, IMRICH POKORNY

THE QUESTION

• Biological macromolecules’ most important properties are related to their interactions with the surrounding water molecules

• Exact determination of the protein–solvent interactions remains a hard problem

• One of the approaches is the atomic solvation parameters method proposed by Eisenberg and McLachlan1 in 1986

the atomic solvation parameters method assumed that:

1. the solvation energy of the atoms or atomic groups is proportional to the area of the part of the atomic surface exposed to the solvent

2. The total solvation energy of the whole protein is then the sum of the individual contributions from all atoms

• Ai is the conformation-dependent solvent-accessible surface area

• is the atomic solvation parameter of the atomic group i, which can be determined from experiments on the model compounds. Determining the parameters is the question left to the experimental biologists

• So calculating Ai is THE QUESTION for theoretical biologists

TO DEFINE THE QUESTION

two definitions of Ai :1. solvent accessible surface: the locus of the

center of probe sphere (solvent molecule) when it rolls on the van der Waals surface of the molecule without penetrating any atom

2. molecular surface, which consists of two parts: the part of the van der Waals surface to which the probe can contact immediately, and the inward looking part of the probe surface when it is in contact with more than one atom simultaneously

• The bold continuous line is the molecular surface. The bold dashed line shows the solvent accessible surface. P is the probe sphere and represents the water molecule.

• here, we only calculate the solvent accessible surface, i.e. the dashed blue line above

• Computation of the surface area for a proteins or other macromolecules, especially in the Monte Carlo (MC) or molecular dynamics (MD) simulations, has always been a difficult and time consuming work

• this paper presents a new analytical method for computing solvent-accessible surface area of macromolecules against the coordinates of the atoms

THE METHOD

describe the protein molecule as a set of N spherical atoms with ( xi, yi, zi) being the Cartesian coordinates of their centers and ri being the radii (it is assumed that ri equals the actual radius of the atom plus the radius of the probe sphere）

every point on the surface of the sphere can be projected onto the plane from its “top point” (TP)

The equation above shows how x,y,z can be shown in terms of t,s

Consider a case with 2 spheres’ intersection

The intersection of any two spheres is mapped into a circle on the tangent plane except the case when the TP is located on the intersection circle

All points of the i-th sphere that are not located inside the j-th sphere satisfy

From the equations above, after mapping on the tangent plane, we can obtain

a,b,c,d are fixed once the locations and the radii of the 2 spheres are fixed

So , what can we find from this equation geographically?

There are 6 different mapping results depending on the mutual position of the i-th and j-th spheres

6 mapping results• 1. An empty set. The i-th sphere is fully

inside the j-th sphere and does not contact it at any point

6 mapping results

• 2. A single point. The i-th sphere is fully inside the j-th sphere and contacts it at one single point.

6 mapping results

• 3. Interior of a circle .The i-th and j-th spheres overlap, and the TP on the i-th sphere is inside the j-th sphere.

6 mapping results

• 4. Half plane. The TP belongs to the intersection circle.

6 mapping results

• 5. Exterior of a circle. The TP is outside of the j-th sphere.

6 mapping results

• 6. The whole plane. The i-th and j-th spheres do not overlap at all.

Calculation of the Area

x,y,z can be expressed in terms of s,t, thus we can express the area Ai

• the plane region represents the image of the spherical region

This equation shows that the region represents one of the sets 1–6 described above or the intersection of several such sets

Improvement of accuracy

• This can be done by rotating the whole molecule or only its relevant part by a certain angle to bring a more convenient point of the i-th sphere to the top.

• This possibility allows us to reduce the problem to only two cases

Improvement of accuracy

(1) (t, s) is a finite region bounded by circle arcs

(2) (t, s) represents the whole plane without some finite regions bounded by circular arcs

an example of 3 spheres intersect with the target sphere

the white area is the wanted area

another example of 3 spheres intersect with the target sphere

the white area is the wanted area

• After using Green’s formula and some math skills, we turn the area integral into line integral of closed loops. Thus, we obtain

where

• Here all arcs together form the boundary of and are oriented positively with respect to . In general, each arc is a part of a circle.

• is a circular arc on the plane formed by the projection of the intersection circle of the i-th and the j-th spheres

• can be labeled by the local polar coordinates through the following equations

• After some math skills, we integral the function

Here,

So, we have

where

Summary: Description of the Algorithm for Surface Calculations

• STEP 1

the algorithm reads the coordinates and radii of the atoms, the algorithm generates a list of spheres that contains the labels of the spheres reaching the i-th sphere

Summary: Description of the Algorithm for Surface Calculations

• STEP 2

Project the area to the plane. By using the formulas we introduced before, find the corresponding coordinates of the necessary pointes

Summary: Description of the Algorithm for Surface Calculations

• STEP 3

Put all the values we got in step 2 into the following 2 functions, and the output will be our final result

Summary: Description of the Algorithm for Surface Calculations

• The 2 functions are

Tests for Accuracy and Performance

Tests for Accuracy and Performance

Concluding Remarks and Further Work

• The method is tested for accuracy and has shown good agreement with the published data as well as with the direct numerical calculations for artificial small systems.

• This is the first version of the ACCAR routine, which was mainly written for checking the accuracy of the method, and hence, complete attention was not paid to its efficiency. So, efficiency can be improved in the future work

A QUESTION ABOUT ZHOU TAO’S SEMINAR

• The data we are using in brenda is incomplete. Thus, completing the database is important for further research

• The network zhou tao was working on is one that consists of 2 interacting network, i.e. the users’ and the products’.

• His aim was to find, using the algorithm he described, the candidate product for the target user.

A QUESTION ABOUT ZHOU TAO’S SEMINAR

• Here in our network, it also consists of 2 interacting network, i.e. the metabolic network and the regulatory network.

• Using the same algorithm, can we find the candidate enzymes that a target compound may control?

• When dealing with the compounds with few degrees, dividing the network into several sub-network may be helpful