Two center multipole expansion method: application to
macromolecular systems
Ilia A. Solov’yov∗, Alexander V. Yakubovich*, Andrey V. Solov’yov*, and Walter Greiner
Frankfurt Institute for Advanced Studies,
Max von Laue Str. 1, 60438 Frankfurt am Main, Germany
We propose a new theoretical method for the calculation of the interaction energy
between macromolecular systems at large distances. The method provides a linear
scaling of the computing time with the system size and is considered as an alternative
to the well known fast multipole method. Its efficiency, accuracy and applicability
to macromolecular systems is analyzed and discussed in detail.
I. INTRODUCTION
In recent years, there has been much progress in simulating the structure and dynamics
of large molecules at the atomic level, which may include up to thousands and millions of
atoms [1, 2, 3, 4]. For example, amorphous polymers may have segments each with 10000
atoms [4] which associate to form partially crystalline lamellae, random coil regions, and
interfaces between these regions, each of which may contribute with special mechanical and
chemical properties to the system.
With increasing computer powers nowadays it became possible to study molecular systems
of enormous sizes which were not imaginable just several years ago. For example in [1] a
molecular dynamics simulations of the complete satellite tobacco mosaic virus was performed
which includes up to 1 million of atoms. In that paper the stability of the whole virion and
of the RNA core alone were demonstrated, and a pronounced instability was shown for the
capsid without the RNA.
The study of structure and dynamics of macromolecules often implies the calculation of
the potential energy surface for the system. The potential energy surface of a macromolecule
carries a lot of useful information about the system. For example from the potential energy
landscape it is possible to estimate the characteristic times for the conformational changes [5,
∗ On leave from the A.F. Ioffe Institute, St. Petersburg, Russia. E-mail: ilia@fias.uni-frankfurt.de
http://arxiv.org/abs/0704.1672v1
6, 7] and for fragmentation [8]. The potential energy surface of a macromolecular system can
be used for studying the thermodynamical processes in the system such as phase transitions
[9]. In proteins, the potential energy surface is related to one of the most intriguing problems
of protein physics: protein folding [9, 10, 11, 12, 13]. The rate constants for complex
biochemical reactions can also be established from the analysis of the potential energy surface
[14, 15].
The calculation of the potential energy surface and molecular dynamics simulations often
implies the evaluation of pairwise interactions. The direct method for evaluating these
potentials is proportional to ∼ N2, where N is the number of particles in the system.
This places a severe restraint on the treatable size of the system. During the last two
decades many different methods have been suggested which provide a linear dependence of
the computational cost with respect to N [16, 17, 18, 19, 20, 21]. The most widely used
algorithm of this kind is the fast multipole method (FMM) [17, 18, 19, 20, 21, 22, 23, 24].
The critical size of the system at which this method becomes computationally faster than the
exact method is accuracy dependent and is very sensitive to the slope in the N dependence
of the computational cost. In refs. [18, 20, 25] critical sizes ranging from N ≈ 300 to
N ≈ 30000 have been reported. Many discrepancies of the estimates in the critical size arise
from differences in the effort of optimizing the algorithm and the underlying code. However,
it is also important to optimize the methods themselves with respect to the required accuracy.
The FMM is based on the systematic organization of multipole representations of a local
charge distribution, so that each particle interacts with local expansions of the potential.
Originally FMM was introduced in [21] by Greengard and Rokhlin. Later, Greengard’s
method has been implemented in various forms. Schmidt and Lee [20] have produced a
version based upon the spherical multipoles for both periodic and nonperiodic systems.
Zhou and Johnson have implemented the FMM for use on parallel computers [26], while
Board et al have reported both serial and parallel versions of the FMM [25].
Ding et al introduced a version of the FMM that relies upon Cartesian rather than spher-
ical multipoles [18], which they applied to very large scale molecular dynamics calculations.
Additionally they modified Greengard’s definition of the nearest neighbors to increase the
proportion of interactions evaluated via local expansions. Shimada et al also developed a
cartesian based FMM program [27], primarily to treat periodic systems described by molec-
ular mechanics potentials. In both cases only low order multipoles were employed, since
high accuracy was not sought.
In the present paper we suggest a new method for calculating the interaction energy
between macromolecules. Our method also provides a linear scaling of the computational
costs with the size of the system and is based on the multipole expansion of the potential.
However, the underlying ideas are quite different from the FMM.
Assuming that atoms from different macromolecules interact via a pairwise Coulomb
potential, we expand the potential around the centers of the molecules and build a two
center multipole expansion using bipolar harmonics algebra. Finally, we obtain a general
expression which can be used for calculating the energy and forces between the fragments.
This approach is different from the one used in the FMM, where the so-called translational
operators were used to expand the potential around a shifted center. Note that the final
expression, which we suggest in our theory was not discussed before within the FMM. Similar
expressions were discussed since the earlier 50’s (see e.g. [28, 29, 30, 31]). In these papers
the two center multipole expansion was considered as a new form of Coulomb potential
expansion, but the expansion was never applied to the study of macromolecular systems.
We consider the interaction of macromolecules via Coulomb potential since this is the
only long-range interaction in macromolecules, which is important for the description of
the potential energy surface at large distances. Other interaction terms in macromolecular
systems are of the short-range type and become important when macromolecules get close
to each other [8]. At large distances these terms can be neglected.
In the present paper we show that the method based on the two center multipole ex-
pansion can be used for computing the interaction energy between complex macromolecular
systems. In section II we present the formalism which lies behind the two center multipole
expansion method. In subsection IIIA we analyze the behavior of the computation cost of
this method and establish the critical sizes of the system, when the two center multipole
expansion method demands less computer time than the exact energy calculation approach.
In subsection IIIB we compare the results of our calculation with the results obtained within
the framework of the FMM. In section IV we discuss the accuracy of the two center multipole
expansion method.
II. TWO CENTER MULTIPOLE EXPANSION METHOD
In this section we present the formalism, which underlies the two center multipole expan-
sion method, which will be further referred to as the TCM method.
Let us consider two multi atomic systems, which we will denote as A and B. The pairwise
Coulomb interaction energy of those systems can be written as follows:
∣R0 + r
, (1)
where NA and NB are the total number of atoms in systems A and B respectively, qi and
qj are the charges of atoms i and j from the system A and B respectively, R0 is the vector
interconnecting the center of system A with the center of system B, rAi and r
j are the
vectors describing the position of charges i and j with respect to the centers of the system
A and B respectively. The centers of both systems can be any suitable points of each of the
molecules. It is natural to define them as the centers of mass of the corresponding systems,
but in some cases another choice might be more convenient (see for example [8], where we
have applied the TCM method for studying fragmentation of alanine dipeptide).
Expression (1) can be expanded into a series of spherical harmonics. The expansion
depends on the vectors R0, r
i and r
j . In the present paper we consider the case when
|R0| >
∣ (2)
holds for all i and j. This particular case is important, because it describes well separated
charge distributions, and can be used for modeling the interaction between complex objects
at large distances. In this case the expansion of (1) reads as [32]:
∣R0 + r
= qiqj
2L+ 1
∣rBj − rAi
RL+10
Y ∗LM
YLM (ΘR0,ΦR0) (3)
According to [32] the function
j − rAi
Y ∗LM
can be expanded into series of bipolar harmonics:
j − rAi
Y ∗LM
4π(2L+ 1)!
l1,l2=0
l1+l2=L
(−1)l2
(2l1 + 1)!(2l2 + 1)!
Yl1(ΘrA
)⊗ Yl2(ΘrB
where a bipolar harmonic is defined as follows:
(Yl1(Θr1 ,Φr1) ⊗ Yl2(Θr2,Φr2))LM =
m1,m2
CLMl1m1l2m2Yl1m1(Θr1 ,Φr1)Yl2m2(Θr2,Φr2). (5)
Here CLMl1m1l2m2 are the Clebsch-Gordan coefficients, which can be transformed to the 3j-
symbol notation as follows:
CLMl1m1l2m2 = (−1)
l1−l2+M
2L+ 1
l1 l2 L
m1 m2 −M
 (6)
Using equations (4),(6) and (5) we can rewrite expansion (3) as follows:
∣R0 + r
= qiqj
l1,l2=0
l1+l2=L
l1,l2
m1=−l1
m2=−l2
(−1)l1+M
(4π)3(2L)!
(2l1 + 1)!(2l2 + 1)!
l1 l2 L
m1 m2 −M
RL+10
Yl1m1(ΘrAi ,Φr
)Yl2m2(ΘrBj ,Φr
)Y ∗LM (ΘR0,ΦR0) (7)
The multipole moments of systems A and B are defined as follows:
QAl1m1 =
2l1 + 1
Yl1m1(ΘrA
) (8)
QBl2m2 =
2l2 + 1
Yl2m2(ΘrB
Summing equation (7) over i and j, and accounting only for the first Lmax multipoles in
both systems, we obtain:
Umult =
l1,l2=0
l1+l2=L
l1,l2
m1=−l1
m2=−l2
(−1)l1+M
RL+10
4π(2L)!
(2l1)!(2l2)!
l1 l2 L
m1 m2 −M
QAl1m1Q
Y ∗LM (ΘR0 ,ΦR0) . (9)
This expression describes the electrostatic energy of the system in terms of a two center
multipole expansion. Note, that this expansion is only valid when the condition R0 > r
holds for all i and j, otherwise more sophisticated expansions have to be considered, which
is beyond the scope of the present paper.
Summation in equation (9) is performed over l1, l2 ∈ [0...Lmax]; m1 ∈ [−l1...l1]; m2 ∈
[−l2...l2], and the condition M = m1 +m2 holds. Lmax is the principal multipole number,
which determines the number of multipoles in the expansion.
III. COMPUTATIONAL EFFICIENCY
A. Comparison with direct Coulomb interaction method
In this section we discuss the computational efficiency of the TCM method. For this
purpose we have analyzed the time required for computing the Coulomb interaction energy
between two systems of charges and the time required for the energy calculation within the
framework of the TCM method for different system sizes, and for different values of the
principal multipole number.
For the study of the computational efficiency of the TCM method we have considered
the interaction between two systems (we denote them as A and B) of randomly distributed
charges, for which the condition eq. (2) holds. The charges in both systems were randomly
distributed within the spheres of radiiRA = 1.0·N1/3A and RB = 1.0·N
B respectively and the
distance between the centers of mass of the two systems was chosen as R0 = 3/2(RA +RB).
The computational time needed for the energy calculations is proportional to the number
of operations required. Thus, the time needed for the Coulomb energy calculation (CE
calculation) can be estimated as:
τCoul = αCoulNANB ∼ N2 (10)
where αCoul is a constant depending on the computer processor power and on the efficiency
of the code, NA ∼ NB ∼ N . From equation (10) it follows that the computational cost of
the CE calculation grows proportionally to the second power of the system size.
For large systems the TCM method becomes more efficient because it provides a linear
scaling with the system size. The time needed for the energy calculation reads as follows:
τmult(Lmax) = βN +
m1=−l1
m2=−l2
(NAτl1,m1 +NBτl2,m2) ≈ (11)
≈ αmultLmax(1 + Lmax)3N,
where the first term, βN , corresponds to the computer time needed for allocating arrays in
memory and tabulating the computationally expensive functions like cos(Φ) and exp(imΦ).
τl,m is the time needed for evaluation of the spherical harmonic at given l and m, and αmult
is a numerical coefficient, which depends on the processor power and on the efficiency of the
code. In general it is different from αCoul.
In Fig. 1 we present the dependencies of the computer time needed for the CE calculation
(squares) and for the computation of energy within the TCM method for different values of
the principal multipole number as a function of system size. This data was obtained on a
1.8 GHz 64-bit AMD Opteron-244 computer.
From Fig. 1 it is clear that the time needed for the CE calculation has a prominent
parabolic trend that is consistent with the analytical expression (10). The fitting expression
which describes this dependance is given in the first row of Tab. I. At large N the N2 term
becomes dominant and the other two terms can be neglected. Thus, αCoul ≈ 4.46 · 10−8
(sec).
The fitting expressions which describe the time needed for the energy computations within
the TCM method at different values of the principal multipole number are given in Tab. I,
rows 2-10. These expressions were obtained by fitting the data shown in Fig. 1. Note the
linear dependence on N . The numerical coefficient in all expressions correspond to the factor
αmultLmax(1 + Lmax)
3 in equation (11). The fitting expressions in Tab. I were obtained by
fitting of data obtained for systems with large number of particles (see Fig. 1). Therefore
these expressions are applicable when N ≫ 1.
From equations presented in Tab. I it is possible to determine the critical system sizes at
which the TCM method becomes less computer time demanding then the CE calculation.
FIG. 1: Time needed for energy calculation as a function of the system size.
The critical system sizes calculated for different principal multipole numbers are shown in
the third column of Tab. I. These sizes correspond to the intersection points of the parabola
describing the time needed for the CE calculation with the straight lines describing the
computational time needed for the TCM method. In Fig. 1 one can see six intersection
points for Lmax = 2− 7.
From equation (11), it follows that computation time of the energy within the framework
of the TCM method grows as the power of 4 with increasing Lmax. To stress this fact,
in Fig. 2 we present the dependencies of the computation time obtained within the TCM
method at different system sizes as a function of principal multipole number. All curves
shown in Fig. 2 can be perfectly fitted by the analytical expression (11). In the inset to
Fig. 2, we plot the dependence of the fitting coefficient αmult as a function of the system
size. From this plot it is seen that αmult varies only slightly for all system sizes considered,
being equal to (1.982± 0.015) · 10−7 (sec).
Thus, the expression for the time needed for the energy calculation within the framework
TABLE I: Fitting expressions for the computational time needed for the CE calculation and for the
energy computation within the TCM method at different values of the principal multipole number,
Lmax (second column). System sizes, for which the Coulomb energy calculation becomes more
computer time demanding at a given value of Lmax are shown in the third column.
Lmax τ(N) (sec.) Nmax
Coulomb 0.11736 − 0.0002N + 4.6768 · 10−8N2 -
2 −0.01986 + 3.0 · 10−5N 4223
3 −0.03159 + 5.0 · 10−5N 4662
4 −0.04714 + 1.0 · 10−4N 5809
5 −0.16054 + 2.1 · 10−4N 8026
6 −0.14710 + 3.7 · 10−4N 11704
7 −0.59675 + 7.4 · 10−4N 19308
8 −0.35383 + 10.9 · 10−4N 27212
9 −1.15856 + 1.9 · 10−3N 44286
10 −0.83688 + 2.71 · 10−3N 61892
of the TCM method reads as:
τmult(Lmax) ≈ 1.98 · 10−7Lmax(1 + Lmax)3N. (12)
Note, that αmult = 1.98 · 10−7 (sec) is larger than αCoul ≈ 4.46 · 10−8 (sec), since in one turn
of the TCM method more algebraic operations have to be done, than in one turn of the CE
calculation.
From the analysis performed in this section it is clear that the TCM method can give a
significant gain in the computation time. However, at larger principal multipole numbers
(Lmax = 8, 9, 10) this method can compete with the CE calculation only at system sizes
greater than 27000-61000 atoms. The accounting for higher multipoles is necessary if the
distance between two interacting systems becomes comparable to the size of the systems. In
the next section we discuss in detail the accuracy of the TCM method and identify situations
in which higher multipoles should be accounted for.
FIG. 2: Time needed for the calculation of energy of the systems of different sizes computed within
the framework of the TCM method as a function of the principal multipole number Lmax. In the
inset we plot αmult as a function of the system size.
B. Comparison with the fast multipole method
The fast multipole method (FMM) [21, 22, 23] is a well known method for calculating
the electrostatic energy in a multiparticle system, which provides a linear scaling of the
computing time with the system size. In order to stress the computational efficiency of the
TCM method in this section we compare the time required for the energy calculation within
the framework of the FMM and using the TCM method.
To perform such a comparison we used an adaptive FMM library, which has been im-
plemented for the Coulomb potential in three dimensions [24, 33]. We have generated two
random charge distributions of different size and calculated the interaction energies between
them as well as the required computation time using the FMM and the TCM methods. As
in the previous section the charges in both systems were randomly distributed within the
FIG. 3: Time needed for the calculation of the interaction energy between two systems as a function
the total number of particles calculated within the framework of the TCM method (triangles) and
within the framework of the FMM (squares). In the upper and lower insets we plot the relative
error of the FMM and of the TCM methods as a function of the system size respectively.
spheres of radii RA = 1.0 ·N1/3A and RB = 1.0 ·N
B respectively and the distance between
the center of mass of the two systems was chosen as R0 = 3/2(RA +RB).
In Fig. 3 we present the comparison of the computer time needed for the FMM calculation
(squares) and for the computation of energy within the TCMmethod (triangles) as a function
of system size. These data were obtained on an Intel(R) Xeon(TM) CPU 2.40GHz computer.
In the upper and lower insets of Fig. 3 we show the relative error of the FMM and of the
TCM methods as a function of the system size respectively, which is defined as follows:
ηmethod =
|Ucoul − Umethod|
|Ucoul|
· 100%. (13)
Here method indicates the FMM or the TCM methods. For comparing the efficiency of the
two methods we have considered different charge distributions within the size range of 100
to 10000 particles. Each point in Fig. 3 corresponds to a particular charge distribution.
For each system size ten different charge distributions were used. The time of the FMM
calculation depends on the charge distribution, as is clearly seen in Fig. 3. Note that for a
given system size the calculation time of the FMM can change by more than a factor of 5,
depending on the charge distribution (see points for N = 10000 in Fig. 3).
For all system sizes FMM requires some minimal computer time for calculating the energy
of the system, which increases with the growth of system size (see Fig. 3). The comparison
of the minimal FMM computation time with the computation time required for the TCM
method shows that the TCM method appears to be significantly faster than the FMM. For
N = 10000 FMM requires at least 2.15 seconds to compute the energy, while TCM method
requires 0.53 seconds, being approximately 4 times faster.
The results of the TCM method calculation shown in Fig. 3, were obtained for Lmax = 2.
The analysis of relative errors presented in the inset to Fig. 3 shows that with this principal
multipole number it is possible to calculate the energy between two systems with an error
of less than 1 % for almost arbitrary charge distribution. Note that for the same charge
distributions the error of the FMM is much more, being about 5 % in almost all of the
considered systems. This allows us to conclude that the TCM method is more efficient and
more accurate than the classical FMM.
It is important to mention that in the traditional implementation, FMM calculates the
total electrostatic energy of the system while TCM method was developed for studying
interaction energy between system fragments. It is possible to modify the FMM to study only
interaction energies between different parts of the system. However, the computation cost of
the modified FMM is expected to be higher than of the TCMmethod. This happens because,
within the framework of the modified FMM method, the field created by one fragment of
the system should be expanded in the multipole series and the interactions of the resulting
multipole moments with the charges from another fragment should be calculated. Thus the
computation cost of this method will be proportional to NA ·NB, where NA and NB are the
number of particles in two fragments, while the TCM method is proportional to NA +NB.
The computation cost of the modified version of the FMM depends quadratically on the
size of the system, because in this method the interacting fragments should be considered as
two independent cells, while traditional FMM uses a hierarchical subdivision of the whole
system into cells to achieve linear scaling.
So far we have considered only the interaction between two multi particle systems in
vacuo, and demonstrated the efficiency of the TCM method in this case, although the TCM
method can also be applied to the larger number of interacting systems. The study of
structure and dynamics of biomolecular systems consisting of several components (i.e an
ensemble of proteins, DNA, macromolecules in solution) is a separate topic, which is beyond
the scope of this paper and deserves a separate investigation.
IV. ACCURACY OF THE TCM METHOD. POTENTIAL ENERGY SURFACE
FOR PORCINE PANCREATIC TRYPSIN/SOYBEAN TRYPSIN INHIBITOR
COMPLEX.
We have calculated the interaction energy between two proteins within the framework of
the TCM method and compared it with the exact Coulomb energy value. On the basis of
this comparison we have concluded about the accuracy of the TCM method.
In the present paper we have studied the interaction energy between the porcine pan-
creatic trypsin and the soybean trypsin inhibitor proteins (Protein Data Bank (PDB) [36]
entry 1AVW [37]). Trypsins are digestive enzymes produced in the pancreas in the form
of inactive trypsinogens. They are then secreted into the small intestine, where they are
activated by another enzyme into trypsins. The resulting trypsins themselves activate more
trypsinogens (autocatalysis). Members of the trypsin family cleave proteins at the carboxyl
side (or ”C-terminus”) of the amino acids lysine and arginine. Porcine pancreatic trypsin is a
archetypal example. Its natural non-covalent inhibitor (porcine pancreatic trypsin inhibitor)
inhibits the enzyme’s activity in the pancreas, protecting it from self-digestion.
Trypsin is also inhibited non-covalently by the soybean trypsin inhibitor from the soya
bean plant, although this inhibitor is unrelated to the porcine pancreatic trypsin inhibitor
family of inhibitors. Although the biological function of the soybean trypsin inhibitor is
mostly unknown it is assumed to help defend the plant from insect attack by interfering
with the insect digestive system.
The structure of both proteins is shown in Fig. 4. The coordinate frame used for our
computations is marked in the figure. This coordinate frame is consistent with the standard
coordinate frame used in the PDB.
FIG. 4: Structure of the porcine pancreatic trypsin and soybean trypsin inhibitor with the coor-
dinate frame used for the energy computation. Figure has been rendered with help of the VMD
visualization package [38]
We use this particular example as a model system in order to demonstrate the possible
use of the TCM method. Therefore environmental effects are omitted and we consider only
the protein-protein interaction in vacuo. The porcine pancreatic trypsin and the soybean
trypsin inhibitor include 223 and 177 amino acids respectively. Both proteins include 5847
atoms. Thus for such system size the TCM method is faster than the CE calculation if
Lmax ≤ 4 (see Tab. I).
We have calculated the interaction energy between the porcine pancreatic trypsin and
soybean trypsin inhibitor as a function of distance between the centers of masses of the
fragments, ~R0, and the angle Θ, which is determined as the angle between the x-axis and
the vector ~R0 (see Fig. 4). We have assumed that the porcine pancreatic trypsin is fixed
in space at the center of the coordinate frame and have restricted ~R0 to the (xy)-plane. Of
course, the two degrees of freedom considered are not sufficient for a complete description
of the mutual interaction between the two systems. For this purpose at least six degrees
of freedom are needed. However for our example of the energy calculation of the porcine
pancreatic trypsin/soybean trypsin inhibitor complex within the framework of the TCM
method the two coordinates ~R0 and Θ are sufficient.
The interaction energy of the porcine pancreatic trypsin with the soybean trypsin in-
hibitor as the function of R0 and Θ calculated within the framework of the TCM method is
shown in Fig. 5. The Coulomb interaction energy between the two proteins is shown in the
top-left plot. In [8] it has been shown that the interaction energy between two well sepa-
rated biological fragments arises mainly due to the Coulomb forces. In the present paper we
consider R0 ∈ [58, 100] Å and Θ ∈ [0, 360]◦, at which condition (2) holds and both proteins
can be considered as separated. This means that the potential energy surface shown in the
top-left plot of Fig. 5 describes the interaction energy between the porcine pancreatic trypsin
and the soybean trypsin inhibitor on the level of accuracy of 90 % at least.
The top-left plot of Fig. 5 shows that one can select several characteristic regions on
the potential energy surface marked with numbers 1-4. The corresponding configurations
(states) of the system are shown in Fig. 6. The potential energy surface is determined by the
Coulomb interactions between atoms, thus at large distances it raises and asymptotically
approaches to zero. State 1 has the maximum energy within the considered part of the
potential energy surface because this state corresponds to the largest contact separation
distance between porcine pancreatic trypsin and the soybean trypsin inhibitor being equal
to 54.8 Å.
At smaller distances the potential energy decreases due to the attractive forces acting
between the two proteins. State 2 corresponds to the minimum on the potential energy sur-
face. It arises because a positively charged polar arginine (R125) from the porcine pancreatic
trypsin approaches the negatively charged site of the soybean trypsin inhibitor, which in-
cludes negatively charged polar amino acids glutamic acid (E549) and aspartic acid (D551)
(see state 2 in Fig. 6). The strong attraction between the amino acids leads to the formation
of a potential well on the potential energy surface. This observation is essential for dynamics
of the attachment process of two proteins, because it establishes the most probable angle at
which the proteins stick in the (xy)-plane of the considered coordinate frame (Θ = 192◦).
States 3 and 4 correspond to the saddle points on the potential energy surface and have
energies higher than state 2. They are formed because at these configurations two positively
FIG. 5: The interaction energies of the porcine pancreatic trypsin with the soybean trypsin inhibitor
calculated as the function of R0 and Θ (see Fig. 4) within the framework of the TCM method at
different values of the principal multipole number, Lmax. The principal multipole number is given
above the corresponding image. The result of the CE calculation is shown in the top left plot.
charged polar amino acids from the two proteins become closer in space providing a source
of a local repulsive force. In state 3 these amino acids are lysines (K145 and K665) (see
state 3 in Fig. 6), and in state 4 these are arginines (R62 and R563)(see state 4 in Fig. 6).
FIG. 6: Relative orientations of the porcine pancreatic trypsin and the soybean trypsin inhibitor,
corresponding to the selected points on the potential energy surface presented in Fig. 5. Below
each image we give the corresponding values of R0 and Θ. Some important amino acids are marked
according to their PDB id. Figure prepared with help of the VMD visualization package [38]
In the top-right plot of Fig. 5 we present the potential energy surface obtained within
the framework of the TCM method with Lmax = 2, i.e. with accounting for up to the
quadrupole-quadrupole interaction term in the multipole expansion (9). From the figure it
is seen that the TCM method describes correctly the major features of the potential energy
landscape (i.e. the position of the minimum and maximum as well as their relative energies).
However, the minor details of the landscape, such as the saddle points 3 and 4 (see top-left
plot of Fig. 5) are missed.
The relative error of the TCM method can be defined as follows:
η(Lmax)(R0,Θ) =
|Ucoul(R0,Θ)− ULmaxmult (R0,Θ)|
|Ucoul(R0,Θ)|
· 100%, (14)
where Ucoul(R0,Θ) and U
mult (R0,Θ) are the Coulomb energy and the energy calculated at
given values of R0 and Θ within the TCM method respectively. In the top-left plot of Fig. 7
we present the relative error calculated according to (14) for Lmax = 2. From this plot it
is clear that significant deviation from the exact result arise at Θ ∼ 50 − 60◦, 140 − 150◦,
245◦, 300− 310◦ and 350◦. The discrepancy at Θ ∼ 50− 60◦, Θ ∼ 300− 310◦ and Θ ∼ 350◦
arises because the saddle points 3 and 4, can not be described within the framework of TCM
method with Lmax = 2. The discrepancy at Θ ∼ 140 − 150◦ and Θ ∼ 245◦ is due to the
error in the calculation of the slopes of minimum 2 at R0 = 58 Å and Θ = 198
FIG. 7: Relative error of the interaction energies of the porcine pancreatic trypsin with the soybean
trypsin inhibitor calculated as the function of R0 and Θ within the framework of the TCM method
at different values of the principal multipole number, Lmax. The principal multipole number is
given above the corresponding image.
It is worth noting that the relative error of the TCM method with Lmax = 2 is less than 10
%. With increasing distance between the proteins, the relative error decreases, and becomes
less than 5 % at R0 ≥ 72 Å and less than 3 % at R0 ≥ 86 Å. This means that already at
Lmax = 2 the TCM method reproduces with a reasonable accuracy the essential features of
the potential energy landscape. This observation is very important, because TCM method
with Lmax = 2 requires less computer time then the CE calculation already at N = 4223
(see Tab. I). Thus, the TCM method can be used for the identification of major minima
and maxima on the potential energy surface of macromolecules and modeling dynamics of
complex molecular systems.
Accounting for higher multipoles in the multipole expansion (9) leads to a more accurate
calculation of the potential energy surface. In the second row of Fig. 5 we present the
potential energy surfaces obtained within the framework of the TCM method with Lmax = 4
and 6. From these plots it is seen that all minor details of the Coulomb potential energy
surface, such as the saddle points 3 and 4 are reproduced correctly. Figure 7 shows that the
TCM method with Lmax = 4 gives the maximal relative error of about 5 % at R0 = 58 Å
and Θ = 75◦, in the vicinity of the saddle point 3. The relative errors in the vicinity of the
saddle point 4 and minimum 2 are equal to 4 % and 1 % respectively. The error becomes
less then 1 % for all values of angle Θ at R0 ≥ 70 Å. For Lmax = 6, the largest relative error
is equal to 1.5 % at R0 = 58 Å and Θ = 340
◦ (saddle point 4), becoming less then 1 % at
R0 ≥ 61 Å.
By accounting for the higher multipoles in the multipole expansion (9) one can increase
the accuracy of the method. Thus, with Lmax = 8 and 10 it is possible to calculate the
potential energy surface with the error less then 1 % (see bottom row in Fig. 5 and Fig. 7).
Although the time needed for computing the potential energy surfaces with Lmax = 8 and
10 is larger than the time needed for computing the Coulomb energy directly (see Tab. I),
we present these surfaces in order to stress the convergence of the TCM method.
V. CONCLUSION
In the present paper we have proposed a new method for the calculation of the Coulomb
interaction energy between pairs of macromolecular objects. The suggested method provides
a linear scaling of the computational costs with the size of the system and is based on the
two center multipole expansion of the potential. Analyzing the dependence of the required
computer time on the system size, we have established the critical sizes at which our method
becomes more efficient than the direct calculation of the Coulomb energy.
The comparison of efficiency of the TCM method with the efficiency of FMM allows us to
conclude that the TCM method has proved to be faster and more accurate than the classical
The method based on the two center multipole expansion can be used for the efficient
computation of the interaction energy between complex macromolecular systems. To de-
termine that we have considered the interaction between two proteins: porcine pancreatic
trypsin and the soybean trypsin inhibitor. The accuracy of the method has been discussed
in detail. It has been shown that accounting of only four multipoles in both proteins gives
an error in the interaction energy less than 5 %.
The TCM method is especially useful for studying dynamics of rigid molecules, but it can
also be adopted for studying dynamics of flexible molecules. In this work we have developed
a method for the efficient calculation of the interaction energy between pairs of large multi
particle systems, e.g. macromolecules, being in vacuo. The investigation of biomolecular
systems consisting of several components (i.e complexes of proteins, DNA, macromolecules
in solution) and the extension of the TCM method for these cases deserves a separate
investigation. If a system of interest consists of several interacting molecules being placed in
a solution, one can use the TCM method to describe the interaction between the molecules
and then to take account of the solution as implicit solvent. This can be achieved using for
example the formalism of the Poisson-Bolzmann [34, 35], similar to how it was implemented
for the description of the antigen-antibody binding/unbinding process [14, 15]. The other
possibility is to split the whole system into boxes and account for the solvent explicitly by
calculating the interactions between the boxes and the molecules of interest. This can be
achieved by using the TCM method or a combination of the FMM and the TCM methods.
In this case the FMM can be used for the calculation of the resulting effective multipole
moment of the solvent, while the TCM method is much better suitable for the description of
the macromolecules energetics and dynamics. Note that all of the suggested methodologies
provide linear scaling of the computation time on the system size.
The results of this work can be utilized for the description of complex molecular systems
such as viruses, DNA, protein complexes, etc and their dynamics. Many dynamical features
and phenomena of these systems are caused by the electrostatic interaction between their
various fragments and thus the use of the two center multipole expansion method should
give a significant gain in their computation costs.
VI. ACKNOWLEDGEMENTS
This work is partially supported by the European Commission within the Network of
Excellence project EXCELL and by INTAS under the grant 03-51-6170. We are grateful to
Dr. Paul Gibbon for providing us with the FMM code. We thank Dr. Axel Arnold for his
help with compiling the programs as well as for many insightful discussions. We also thank
Dr. Elsa Henriques and Dr. Andrey Korol for many useful discussions. We are grateful
to Ms. Stephanie Lo for her help in proofreading of this manuscript. The possibility to
perform complex computer simulations at the Frankfurt Center for Scientific Computing is
also gratefully acknowledged.
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	Introduction
	Two center multipole expansion method
	Computational efficiency
	Comparison with direct Coulomb interaction method
	Comparison with the fast multipole method
	Accuracy of the TCM method. Potential energy surface for porcine pancreatic trypsin/soybean trypsin inhibitor complex.
	Conclusion
	Acknowledgements
	References

  We propose a new theoretical method for the calculation of the interaction
energy between macromolecular systems at large distances. The method provides a
linear scaling of the computing time with the system size and is considered as
an alternative to the well known fast multipole method. Its efficiency,
accuracy and applicability to macromolecular systems is analyzed and discussed
in detail.

Introduction
	Two center multipole expansion method
	Computational efficiency
	Comparison with direct Coulomb interaction method
	Comparison with the fast multipole method
	Accuracy of the TCM method. Potential energy surface for porcine pancreatic trypsin/soybean trypsin inhibitor complex.
	Conclusion
	Acknowledgements
	References
