JOURNAL OF CHEMICAL PHYSICS
VOLUME 121, NUMBER 18
8 NOVEMBER 2004
Translational friction coefficients for cylinders of arbitrary axial ratios
estimated by Monte Carlo simulation
Steen Hansena)
Department of Natural Sciences, The Royal Veterinary and Agricultural University, Thorvaldsensvej 40,
DK-1871 FRB C, Denmark
共Received 27 April 2004; accepted 11 August 2004兲
Translational friction coefficients for cylinders of arbitrary axial ratios 共including disks兲 are
calculated using Monte Carlo simulation and an approximate description of the hydrodynamic
interaction. The calculations indicate that the approximate description is exact for ellipsoids and this
result is generalized to include cylinders, which possess the same symmetry as ellipsoids. From the
result an approximate formula for the translational friction coefficient of cylinders is calculated
which is compared to results from other sources. © 2004 American Institute of Physics.
关DOI: 10.1063/1.1803533兴
I. INTRODUCTION
lated using Monte Carlo simulation of the chord distribution
for the cylinder and an approximate description of the hydrodynamic interaction. As a result of these calculations an approximate formula for the friction coefficients of cylinders as
a function of their axial ratios is given for axial ratios between 0.01 and 100. This formula is tested against the approximate formulas of Tirado and Garcı́a de la Torre, Ortega
and Garcı́a de la Torre as well as against specific calculations
using the program HYDROSUB.10 The comparisons indicate
that the suggested method is valid.
Various methods for calculation of the hydrodynamic
properties of rigid macromolecules have been tested since
Stokes derived his formula for the translational friction coefficient of a sphere in 1847. For prolate and oblate ellipsoids
of revolution analytical expressions for the friction coefficients were derived by Perrin.1 For molecules of more complex shapes a modeling procedure where the molecule of
interest was approximated by combinations of spheres of
equal sizes 共beads兲 was used by Kirkwood.2,3 The modeling
procedure of Kirkwood was extended by Bloomfield to include beads of unequal sizes.4 Garcı́a de la Torre and Bloomfield refined the method by introducing an exact description
of the hydrodynamic interaction of the beads.5 The work of
Garcı́a de la Torre and Bloomfield was the basis for the successful program suite HYDRO of Garcı́a de la Torre and
various co-workers.6 –10 These bead modeling methods are
all special cases of the boundary element 共BE兲 method developed by Youngren and Acrivos where the surface of the
molecule is approximated by a number of platelets.11,12 A
more detailed review of the many methods for hydrodynamic
modeling is given by Garcı́a de la Torre, Huertas, and
Carrasco.8
Apart for the Perrin formulas for ellipsoids of revolution,
formulas relating the geometry of macromolecules to their
translational friction coefficients proved difficult to obtain.
For short cylinders Broersma calculated an approximate formula for the translational friction coefficient.13,14 This result
was later modified by Tirado and Garcı́a de la Torre for cylinders having axial ratios 共length/diameter兲 between 2 and
20.15,16 Allison compared calculations using the BE method
for cylinders of similar axial ratios as those of Tirado and
Garcı́a de la Torre and found good agreement between the
methods.17 Recently Ortega and Garcı́a de la Torre repeated
their calculations using bead modeling, extending the axial
ratios down to 0.1.18
The present paper argues that translational friction coefficients for cylinders of arbitrary axial ratios can be calcu-
II. THEORY
A. Hydrodynamic modeling
Most modeling procedures approximate the molecule of
interest by a combination of small spheres 共beads兲 either as a
space-filling model or as a shell model, where all beads are
positioned at the surface of the molecule. For a single sphere
of radius ␴ i the friction coefficient ␨ i is given by Stokes’ law
␨ i ⫽6 ␲ ␩ 0 ␴ i ,
where ␩ 0 is the viscosity of the solvent. The hydrodynamic
properties of the molecule are calculated using the hydrodynamic interaction tensor, which describes the interaction of
the beads from which the molecule is composed. This tensor
was originally formulated by Oseen as
冋
T共 Ri j 兲 ⫽ 共 8 ␲ ␩ 0 R i j 兲 ⫺1 I⫹
Ri j Ri j
R 2i j
册
共2兲
,
where I is the unit tensor, Ri j is the vector connecting beads
i and j, and R i j is the length of the vector. Various modifications of the Oseen tensor have been suggested, but Eq. 共2兲
has been shown to give an adequate description of the hydrodynamic interaction for the shell models, which are used
in the present manuscript.19 Writing Ti j ⫽T(Ri j ) the components of the interaction tensor can be written as
冋
共 Ti j 兲 ␣␤ ⫽ 共 8 ␲ ␩ 0 R i j 兲 ⫺1 ␦ ␣␤ ⫹
共 ␣ , ␤ ⫽x,y,z 兲 .
a兲
Electronic mail: [email protected]
0021-9606/2004/121(18)/9111/5/$22.00
共1兲
9111
共 R i j 兲 ␣共 R i j 兲 ␤
R 2i j
册
,
共3兲
© 2004 American Institute of Physics
9112
J. Chem. Phys., Vol. 121, No. 18, 8 November 2004
Steen Hansen
兺
兺 ⫽n 2 冕
i⫽ j j⫽1 R i j
n
Defining tensors Bi j by
Bi j ⫽Ti j for i⫽ j,
共4兲
Bii ⫽ 共 1/␨ i 兲 I,
n
兺 Bi j Cj ⫽I
j⫽1
共5兲
共 i⫽1, . . . ,n 兲
the friction tensor ⌶ for a molecule modeled by n beads can
be written as
n
⌶⫽
兺
j⫽1
共6兲
Cj ,
where it is seen that the tensor C j is the effective friction
tensor of bead j. The friction coefficient f t for the molecule
is calculated from the inverted friction tensor
f t ⫽3/Tr共 ⌶
⫺1
共7兲
兲
and the translational diffusion coefficient D t is given by D t
⫽kT/ f t , where T is the temperature of the solvent and k is
Boltzmann’s constant. Also the hydrodynamic radius R H for
a molecule is determined by the friction coefficient according to
f t ⫽6 ␲ ␩ 0 R H .
冓冔
g共 l 兲
1
dl⫽n 2
l
l
共11兲
denoting averaging with respect to g by 具•典. Inserting this in
the equation for the hydrodynamic radius one obtains,
and tensors C j by the equations
n
1
R H⫽
冓
冔
冓冔
n␴
n␴
1
⫽
→
␴ 2 1
1
l
1⫹ n
1⫹n ␴
n
l
l
冓冔
⫺1
as n→⬁.
共12兲
It may be noted that the chord length distribution g(l) is
not related in any simple way to the chord length distribution
G(l) which may be obtained from solution scattering experiments by indirect Fourier transformation.22 However, for the
simple case of a sphere the chord length distributions G(l)
and g(l) are identical and
G 共 l 兲 ⫽g 共 l 兲 ⫽
2l
.
D2
共13兲
The results of the present paper are obtained by Monte Carlo
simulation of the shell of the molecule placing random points
in a thin shell on the surface. These points are used to estimate the chord length distribution g(l) from which the hydrodynamic radius of the ellipsoid can be calculated using
Eqs. 共11兲 and 共12兲.
共8兲
III. RESULTS
B. Approximate methods
A. Ellipsoids of revolution
Knowing the hydrodynamic interaction tensors for a
given model, it is in principle quite simple to calculate the
corresponding friction coefficient from Eqs. 共5兲–共7兲. However for models consisting of more than 2000–3000 beads
this may still be a time consuming task and various approximations have been tested over the years. The errors and relations of some of the approximate methods have been estimated by Garcı́a de la Torre et al.20
By truncation of a series expansion for the friction tensor
an approximate expression for the friction coefficient of a
molecule consisting of n beads can be calculated:2,3
f t⫽
n
兺 i⫽1
␨i
n
1⫹ 共 6 ␲ ␩ 0 兺 i⫽1
␨ i 兲 ⫺1 兺 ni⫽ j 兺 nj⫽1 ␨ i ␨ j R ⫺1
ij
.
共9兲
The truncation of the series expansion leading to Eq. 共9兲 is
often referred to as ‘‘approximate hydrodynamics.’’ It should
be noted that for certain applications Eq. 共9兲 may lead to
relatively large errors.21 From Eqs. 共9兲 and 共8兲 the hydrodynamic radius can be found for a molecule ‘‘covered’’ by n
beads 共a shell model兲 all having identical radii ␴,
R H⫽
n␴
␴
1⫹ 共 兺 ni⫽ j 兺 nj⫽1 R ⫺1
ij 兲
n
.
共10兲
In the limit of infinitely small beads R i j is a chord for the
molecule. Denoting the distribution of chords given by the
distance between two arbitrary points on the surface of the
molecule by g(l), the double sum in Eq. 共10兲 can be calculated using g(l) according to
The Perrin formulas1 for the translational diffusion coefficients of prolate and oblate ellipsoids of revolution have
provided a good test of the various modeling procedures
which have estimated the hydrodynamic behavior of molecules. For comparison of calculations of translational friction coefficients it may be useful to quote the ratio between
the hydrodynamic radius R H of molecule and the 共hydrodynamic兲 radius R S of a sphere having the same volume. In
terms of the ratio R H /R S the Perrin formulas can be written
冑p 2 ⫺1
RH
⫽ 1/3
R S p ln共 p⫹ 冑p 2 ⫺1 兲
p⫽a/b⬎1 prolate,
共14兲
冑p 2 ⫺1
RH
⫽ 2/3
R S p arctan共 冑p 2 ⫺1 兲
p⫽b/a⬎1 oblate.
共15兲
Frequently the ratio of the translational friction coefficient f t
to the friction coefficient f 0 of a sphere having the same
volume is used instead of R H /R S . Due to Eq. 共8兲 f t / f 0
⫽R H /R S . The translational friction coefficient 共or equivalently the hydrodynamic radius兲 for a molecule is primarily
dependant upon the volume of the molecule. This is apparent
from Fig. 1 showing R H /R S for prolate and oblate ellipsoids
of axial ratios p⫽a/b between 0.01 and 100.
Using a shell model with approximate hydrodynamics
for calculation of the translational friction coefficients for
ellipsoids Carrasco and Garcı́a de la Torre found deviations
from the Perrin formulas of about 2%.23 Carrasco and Garcı́a
de la Torre concluded that it was likely that the shell models
predicted the correct translational diffusion coefficient even
J. Chem. Phys., Vol. 121, No. 18, 8 November 2004
FIG. 1. R H /R S calculated using the Perrin formulas for ellipsoids of various
axial ratios.
with approximate hydrodynamics. These findings are corroborated in the following, where it is argued that Monte
Carlo 共shell-兲modeling of the ellipsoid may give arbitrarily
small deviations from the Perrin formulas.
The deviation between the result of the Monte Carlo
simulation and the Perrin formulas is shown in Fig. 2 using
simulations of 2500, 5000, and 10 000 points for 96 axial
ratios p in the interval 关0.01:100兴 共a simulation using 10 000
points uses about 1 sec of CPU time on a 2 GHz processor兲.
Apparently the estimate converges towards the correct values
as the number of points used for the simulation is increased.
To improve the precision it is more efficient to repeat the
calculations, which will also provide the calculations with
error estimates.
This has been done for the calculations shown in Fig. 3.
The figure shows the deviation of the result of the Monte
Carlo simulation from the Perrin formulas as the number of
calculations is increased. The relative deviation between the
Perrin formula and the Monte Carlo simulation is reduced to
about 0.01% for 100 repetitions each using 10 000 points.
The error can be reduced even further by introducing a thinner shell for the simulation and/or increasing the number of
repetitions of the simulation at the cost of an increase in the
CPU time. Doing this for specific axial ratios of an ellipsoid
consistently resulted in deviations from the Perrin formulas
which were less than the error estimates provided by the
simulations.
FIG. 2. Deviation between simulations of ellipsoids of revolution using
2500 共⫻兲, 5000 共⫹兲, and 10 000 共dots兲 points and Perrin formulas.
Translational friction coefficients for cylinders
9113
FIG. 3. Error estimate for the calculations. 共⫻兲 shows the standard deviation
calculated from ten repetitions using 2500 points, 共⫹兲 10 repetitions using
5000 points, and 共dots兲 10 repetitions using 10 000 points. 共Square兲 shows
the deviation between the Perrin formulas and the calculated R H /R S of 100
repetitions using 10 000 points.
B. Cylinders
As a cylinder possesses the same symmetry as an ellipsoid it is likely that the approximation which appears to be
valid for ellipsoids may also be valid for cylinders. For cylinders of axial ratios between 0.01 and 100 R H /R S the result
of simulations using 100 repetitions of 10 000 points is
shown in Fig. 4.
Using the error estimates for each axial ratio the R H /R S
curve is fitted using a polynomial in ln p, where p⫽L/d is
the axial ratio of the cylinder having length L and diameter
d. Writing x⫽ln p the following expression was found to
give a good fit to the simulation in all of the interval
关0.01:100兴:
RH
⫽1.0304⫹0.0193x⫹0.062 29x 2 ⫹0.004 76x 3
RS
⫹0.001 66x 4 ⫹2.66⫻10⫺6 x 7 .
共16兲
Tirado and Garcı́a de la Torre calculated a formula for
the frictional coefficient of cylinders of axial ratios in the
interval 关2:20兴.15,16 In terms of the ratio R H /R S the formula
of Tirado and Garcı́a de la Torre can be written,
FIG. 4. R H /R S for cylinders of various axial ratios. Error bars: Monte Carlo
simulation using 100 repetitions of 10 000 points. Full line: fitted function in
Eq. 共16兲.
9114
J. Chem. Phys., Vol. 121, No. 18, 8 November 2004
Steen Hansen
FIG. 5. R H /R S for cylinders of various axial ratios. Full line: Result of
Monte Carlo simulation using 100 repetitions of 10 000 points. Dashed line
in the interval 关2:20兴: Eq. 共17兲 from Tirado and Garcı́a de la Torre 共Ref. 15兲.
Dotted line: Eq. 共18兲 from Ortega and Garcı́a de la Torre 共Ref. 18兲. Squares:
calculations by HYDROSUB. Insert shows: Full line: Relative difference
between the formula of the Monte Carlo simulation and Ortega and Garcı́a
de la Torre. Dotted line: Relative difference between the formula of Tirado
and Garcı́a de la Torre and the formula of Ortega and Garcı́a de la Torre.
Dashed line: Relative difference between the formula of the Monte Carlo
simulation and Tirado and Garcı́a de la Torre.
R H 共 2/3p 2 兲 1/3
⫽
and ␥ ⫽0.312⫹0.565/p⫺0.100/p 2 .
RS
ln p⫹ ␥
共17兲
These calculations were repeated by Ortega and Garcı́a de la
Torre and extended to include cylinders of axial rations in the
interval 关0.1:20兴 leading to the approximate formula18
RH
⫽1.009⫹0.013 95x⫹0.0770x 2 ⫹0.006 040x 3 ,
RS
共18兲
again writing x⫽ln p.
The results of Eqs. 共16兲, 共17兲, as well as 共18兲 are shown
in Fig. 5. The differences between the various methods of
calculation are also shown as insert in the figure.
As a further test of the validity of the Monte Carlo
method this author used the programs HYDROPIX and
HYDROSUB for calculation of R H /R S for some selected
axial ratios.9,10 The programs gave similar results and for
HYDROSUB some results are shown in Fig. 5.
IV. DISCUSSION
For ellipsoids of revolution further calculations show
that it is possible to make the deviation between the Perrin
formulas and the result of the Monte Carlo simulation arbitrarily small and no systematic deviation was found. Although calculations of this kind by their nature cannot give
absolute proof, they do provide strong indication that the
suggested procedure is able to reproduce the correct value of
the hydrodynamic radius for an ellipsoid of revolution.
Assuming the Monte Carlo simulation to be correct for
ellipsoids, the explanation for this must be found in some
special features of the Oseen interaction tensors which reflect
the geometry of an ellipsoid.
As the geometrical characteristics of a cylinder are similar to those of an ellipsoid it appears very likely that the
interaction tensors of a cylinder should behave like the tensors of an ellipsoid for the calculation of the friction tensor.
Previously Freire and Garcı́a de la Torre have evaluated
some conditions for the chord length distribution for which
the so called ‘‘diagonal approximation’’ 共where the interaction tensors are approximated by their diagonal values兲 is
identical to the Kirkwood approximation in Eq. 共9兲, but apparently their reasoning cannot be transferred directly to the
problem of the present paper.24
The indication that the approximate method suggested
here is correct for ellipsoids of revolution and cylinders may
be related to the fact that both of these geometric shapes are
‘‘symmetric tops’’ 共having two of the moments of inertia
identical兲. For symmetric top molecules the friction tensor is
diagonal,15 but axial symmetry may not be sufficient for the
approximation to be valid. In a comparison of different computational procedures Carrasco and Garcı́a de la Torre mention that shell modeling with approximate hydrodynamics
seems to predict the translational diffusion coefficient of ellipsoids correctly.23 However this does not seem to be the
case for the oligomeric structures tested in their paper, and
Carrasco and Garcia de la Torre speculate that the reason for
this might be that ‘‘the sphere and the ellipsoid are simple
convex bodies, whereas the oligomeric arrays are geometrically more complex, with holes, convex, and concave parts,
etc.’’
Calculation of the friction tensor by the Jacobi or the
Gauss–Seidel method involves the double sum of the intern
action tensors 兺 i⫽1
兺 nj⫽1 Ti j . 5 By symmetry this sum is a
diagonal tensor for cylinders and ellipsoids for n→⬁. It is
n
noted that all of the tensors 兺 i⫽1
兺 nj⫽1 Bi j , 兺 nj⫽1 C j , and
n
兺 j⫽1 Bi j C j in Eqs. 共4兲–共6兲 are diagonal. This may have implications for the expansion of the 共diagonal兲 friction tensor,
which leads to Eq. 共9兲 for the limit of infinitely many beads.
For cylinders within the range of axial ratios between 2
and 20 a maximum difference of about 4% is found between
the formula of Tirado and Garcı́a de la Torre and the formula
of Ortega and Garcı́a de la Torre 共Fig. 5, insert兲. The maximum difference between the Monte Carlo simulation and the
formula of Ortega and Garcı́a de la Torre within the range of
axial ratios between 0.1 and 20 is about 2% and found for an
axial ratio close to one.
Compared to the approximative formulas the program
HYDROSUB as used by the author appears to give a small
overestimation of R H . This overestimation may appear
somewhat strange as the approximative formula of Ortega
and Garcı́a de la Torre was based upon calculations done by
HYDROSUB. However the calculation of HYDROSUB involves an extrapolation to zero bead size requiring the user
to chose parameters for the extrapolation which may lead to
small variations in the result. The extrapolation covers a relatively wide range of bead sizes and Ortega and Garcı́a de la
Torre comment on this in their paper wondering whether a
quadratic or a linear approximation is more appropriate. Ortega and Garcı́a de la Torre found that the extrapolation led
to variations of ‘‘a few percent’’ in the calculated
properties.18
Another reason for variations in the results of HYDROSUB is that present version of HYDROSUB is restricted to a
relatively small number of beads to simulate the cylinder or
disc 共to avoid excessively large CPU times兲. Especially for
J. Chem. Phys., Vol. 121, No. 18, 8 November 2004
Translational friction coefficients for cylinders
9115
From the precision found in the calculation for ellipsoids
R H /R S for cylinders is expected to be reproduced to within
0.1% by the approximative Eq. 共16兲. Consequently Eq. 共16兲
covers a wider range of axial ratios than the corresponding
Eq. 共18兲 and it is likely to be more precise.
V. CONCLUSION
It has been argued that the translational friction coefficients of cylinders may be calculated using Monte Carlo
simulation of the chord distribution for the cylinder and an
approximate description of the hydrodynamic interaction.
Similar calculations for ellipsoids as well as comparisons
with previous formulas and calculations for cylinders indicate that the suggested method is valid. An approximate formula for the hydrodynamic radius of cylinders of axial ratios
between 0.01 and 100 is given.
F. Perrin, J. Phys. Radium 7, 1 共1936兲.
J. G. Kirkwood, Recl. Trav. Chim. Pays-Bas 68, 649 共1949兲.
J. G. Kirkwood, J. Polym. Sci. 12, 1 共1954兲.
4
V. A. Bloomfield, W. O. Dalton, and K. E. Van Holde, Biopolymers 5, 135
共1967兲.
5
J. Garcı́a de la Torre and V. A. Bloomfield, Biopolymers 16, 1747 共1977兲.
6
J. Garcı́a de la Torre and V. A. Bloomfield, Q. Rev. Biophys. 14, 81
共1981兲.
7
J. Garcı́a de la Torre, S. Navarro, M. C. L. Martı́nez, F. G. Diaz, and J. L.
Cascales, Biophys. J. 67, 530 共1994兲.
8
J. Garcı́a de la Torre, M. L. Huertas, and B. Carrasco, Biophys. J. 78, 719
共2000兲.
9
J. Garcı́a de la Torre, Biophys. Chem. 94, 265 共2001兲.
10
J. Garcı́a de la Torre and B. Carrasco, Biopolymers 63, 163 共2002兲.
11
G. K. Youngren and A. Acrivos, J. Fluid Mech. 69, 377 共1975兲.
12
S. A. Allison, Macromolecules 31, 4464 共1998兲.
13
S. Broersma, J. Chem. Phys. 32, 1626 共1960兲.
14
S. Broersma, J. Chem. Phys. 32, 1632 共1960兲.
15
M. M. Tirado and J. Garcı́a de la Torre, J. Chem. Phys. 71, 2581 共1979兲.
16
M. M. Tirado, M. C. L. Martı́nez, and J. Garcı́a de la Torre, J. Chem. Phys.
81, 2047 共1984兲.
17
S. A. Allison, Macromolecules 32, 5304 共1999兲.
18
A. Ortega and J. Garcı́a de la Torre, J. Chem. Phys. 119, 9914 共2003兲.
19
B. Carrasco and J. Garcı́a de la Torre, J. Chem. Phys. 111, 4817 共1999兲.
20
J. Garcı́a de la Torre, M. C. Lopez, M. M. Tirado, and J. J. Freire, Macromolecules 16, 1121 共1983兲.
21
R. Zwanzig, J. Chem. Phys. 45, 1858 共1966兲.
22
W. Gille, Eur. Phys. J. B 17, 371 共2000兲.
23
B. Carrasco and J. Garcı́a de la Torre, Biophys. J. 75, 3044 共1999兲.
24
J. J. Freire and J. Garcı́a de la Torre, Macromolecules 16, 331 共1983兲.
1
2
3
FIG. 6. Cylinder and disc with axial ratios of 8 and 1/8 respectively modeled by approximately 2000 beads. The figures were generated using
RasMol.
large or small axial ratios, the used 2000 beads may not be
sufficient as is apparent from Fig. 6 and furthermore, the
beads are positioned at the surface of the cylinder in a regular pattern, which may also lead to artifacts. The Monte
Carlo simulation does not suffer from these limitations, but
approximates the shell of the cylinder using 106 arbitrarily
positioned points.