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ZENO -- A PROGRAM FOR COMPUTING HYDRODYNAMIC PROPERTIES
OF MACROMOLECULES.
Marc L. Mansfield
Department of Chemistry
Stevens Institute of Technology
Hoboken, New Jersey 07030
OUTLINE
I. INTRODUCTION
II. THE PROGRAM MODELS ARBITRARY SHAPES AS UNIONS OF SIMPLE
BODY ELEMENTS
III. COMPILING AND INVOKING THE PROGRAM
IV. GRAMMATICAL RULES FOR THE BODY FILE
V. DESCRIPTION OF THE INTEGRATIONS
VI. DESCRIPTION OF THE OUTPUT DATA
APPENDIX A. Examples.
APPENDIX B. Internal dictionary (words recognized by the grammar of the body file).
APPENDIX C. Error estimates and propagation.
APPENDIX D. Random numbers.
REFERENCES AND NOTES
I. INTRODUCTION
This document describes procedures for using the program zeno, which computes various
shape functionals (e.g., certain electrostatic and hydrodynamic properties) of
macromolecules of arbitrary shape. The fortran code is stored in the file zeno.f. You
must create a text file beforehand with the complete body specification, which will be
referred to hereafter as the “body file,” and the program returns its results in a second text
file, referred to as the “zeno file,” or the “report file.” Each shape is identified to the
program by a character string of at most 25 characters, which will be represented in this
document by the symbol <identifier>. The body file has the name <identifier>.bod, and
the zeno file has the name <identifier>.zno.
The program performs as many as three different numerical integrations on the body.
(You only request the integrations that are desired. You need not request all three.)
The three integrations are:
1. The “Zeno” computation, a numerical path integration technique that solves Laplace’s
equation for two separate boundary value problems; an isolated, charged conductor, and a
conductor in an external electric field. NZ brownian paths are initiated from a sphere of
radius RL, the launch sphere, which completely encloses the body. You specify the value
of NZ, but RL is determined internally from the specification of the body. This
computation determines the electrostatic capacity or capacitance, C, and the nine
1
components of the electrostatic polarizability tensor,   . Because of analogies between
the hydrodynamic and the electrostatic boundary value problems, these quantities then
permit the program to estimate the hydrodynamic radius, Rh, and the intrinsic viscosity,
 .
2. The “interior” computation, a Monte Carlo integration over the interior of the body.
You specify a large number Ni. The program begins generating points at random inside
the launch sphere, and continues until 2Ni points are found that also lie inside the body.
The volume, V, of the body is obtained as the volume of the launch sphere times the ratio
of successful points to trial points. The mean-square radius of gyration of the interior of
the body, Rgi2 , is obtained as one half of the mean-square distance between successive
pairs of interior points.
3. The “surface” computation, a Monte Carlo integration over the surface of the body.
You specify a large number Ns. The program generates 2Ns points distributed randomly
over the surface, and uses these to compute the surface area, A, the Kirkwood radius, RK,
defined as the harmonic mean distance between arbitrary pairs of surface points, and the
mean-square radius of gyration of the surface of the body, Rgs2 , defined as one half the
mean square distance between arbitrary pairs of surface points.
The raw values obtained from these integrations are then used to compute a number of
derived quantities.
II. THE PROGRAM MODELS AN ARBITRARY SHAPE AS A UNION OF
SIMPLE BODY ELEMENTS
The body is modeled as the union of simple, component body elements. Currently, eight
different types of elements are recognized. In this document, we use the terms “open
cylinder” and “closed cylinder,” to refer to cylinders with and without ends. (Think of a
tin can: A closed cylinder is the can with the ends intact, an open cylinder is the can with
both ends cut off.) Table 1 defines the body elements currently in use.
TABLE 1. SUMMARY OF BODY ELEMENT TYPES
BODY
ELEMENT
sphere
triangle
disk
cylinder
DEFINITION
Set of points within a given distance of a given point.
Three points constitute the vertices of a triangle, the triangle
element is the set of points in the plane defined by the vertices
and in the interior of the resulting triangle.
The set of points in a plane within a given distance of a given
point.
The locus of points generated by rotating a line segment about
an axis to which it is parallel, an open cylinder.
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SHAPE
TYPE
A
B
B
B
torus
lens
ellipsoid
cube
The locus of points generated by rotating a circle about an axis
outside the circle.
The set of points formed as the intersections of two distinct
spheres with different centers; the spheres need not have the
same radius.
The set of points inside an arbitrarily oriented and translated
ellipsoid; the three axes can all be distinct.
This body element is only defined for cubes whose edges are
parallel to the Cartesian axes; arbitrary orientations can always
be achieved with 12 triangles (2 for each face of the cube).
A
A
A
A
The body elements are of two types. The first type (type A) includes the elements that
have three-dimensional interiors; or that consist of surfaces enclosing a region of threedimensional space: spheres, tori, lenses, ellipsoids and cubes. The second type (type B)
includes the elements that are two-dimensional surfaces: triangles, disks, and cylinders.
In all cases, the overall shape is taken to be the union of some collection of these body
elements. This permits considerable power in defining shapes. Two important examples
are first, defining a molecule as the region of space occupied by a set of overlapping
spheres, and second, defining an arbitrary surface with a grid of triangles. (E.g., a
geodesic dome.) The complete set of body elements used in any given specification,
along with their positions and orientations, are specified in the body file. See below for
the grammatical rules that must be followed in setting up the body file.
This program can consider three major shape classifications. The first are those shapes
that have three-dimensional interiors, such as spheres, or a set of overlapping spheres.
The second are those shapes that are two-dimensional surfaces but that are embedded in
three-dimensional space, such as an open cylinder. The third are those shapes that are
only two-dimensional, that exist in a plane, such as a square. (There are also hybrid
shapes, for example, the union of a sphere and a square.) Certain shape functionals, such
as the electrostatic capacity, can be defined for all three classifications, and the zeno
integration works successfully on all three. However, the interior integration can only be
performed for shapes of the first classification, for only in this case is the interior defined.
This creates a problem, because it is not always easy to program the computer to
determine which of the three classifications we might have. For example, we can
represent any surface as a grid of triangles, but it probably requires evaluation of certain
topological invariants to determine whether or not the surface is in the first or the second
class – and this evaluation is beyond the scope of this program. Therefore, the following
rules apply:
(1) If the body contains any shape elements of type B, an interior integration will not be
performed, even if one is requested.
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(2) It follows that the volume of the body can be determined only if it contains body
elements of type A. It is defined as the total volume occupied by points inside the launch
sphere that are also inside at least one of the body elements.
(3) The surface area of the body will be defined as the area contributed by all points on
the surface of each body element (of either type) that are not in the interior of any other
body element of type A.
(Users should realize that Rule 3 introduces vagueness into the definition of the surface
area, but only for some kinds of bodies. Imagine constructing a body as the union of a
square and a sphere, but representing both as grids of triangles. Then by Rule 3, points
on the square but inside the sphere will contribute to the surface area as calculated by this
program.)
III. COMPILING AND INVOKING THE PROGRAM
The program was developed using the Linux f77 compiler. Therefore, it should compile
effortlessly with either of the two Linux compilers f77 or f90. The following Linux
commands can be used:
f77 zeno.f –o zeno
or
f90 zeno.f –o zeno
The above f77 or f90 command prepares an executable file of name zeno. You then issue
the following command to invoke the program:
./zeno <identifier> <action-code-1> <action-code-2> <action-code-3>
The four strings <identifier>, <action-code-1>, etc., are accessed by the program via a
call to the intrinsic fortran subroutine getarg.
The string <identifier> is the same identifier name discussed above. The full body
specification must be prepared beforehand and saved in the “body” file, which should be
named <identifier>.bod. Section IV describes in detail the grammatical rules required for
the body file. Output goes to a file with the name <identifier>.zno.
Each action code is a string with three parts: a one-character prefix, a set of digits in the
middle, and optionally a suffix. The prefix is one of the following three characters:
z
i
s
do a zeno integration on the body
do an interior integration on the body
do a surface integration on the body
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Allowed suffixes are any of the three characters:
t = thousand
m = million
b = billion
but these suffixes are optional. A few examples demonstrate the proper format of the
action codes:
z100t
i1b
s5000000
requests the zeno integration with Nz = 100 thousand.
requests the interior integration with Ni = 1 billion.
requests the surface integration with Ns = 5000000.
You can specify as few as zero (which will result in no action being taken by the
program) or as many as three action codes.
For example, the following invocation of the zeno program:
./zeno spheroid z1m i1m s1m
directs the program to get data on the body from the file spheroid.bod, to perform the
zeno, interior, and surface integrations on the body, each with one million steps, and to
put the results in a file named spheroid.zno.
As another example, this invocation:
./zeno w85 i100t
directs the program to get data on the body from the file w85.bod, to perform only the
interior integration with 100 thousand steps, and to put the results in a file named
w85.zno.
IV. GRAMMATICAL RULES FOR THE BODY FILE
You use the “body file” to give the full specification of the body to the program. The
name of the body file is <identifier>.bod, where <identifier> represents the indentifier
string provided during the invocation of the program. As mentioned above, the body is
set up as a union of simple body elements. Allowable elements are listed in Table 1.
The data in the body file consist of a series of commands, which in turn consist of a series
of character strings. The strings are delimited by spaces or by carriage returns. A single
line is 80 characters or less, so do not put more than 80 characters between carriage
returns.
5
The first string of a command is its “predicate,” and identifies the type of command. The
remaining strings in each command are “modifiers” of the predicate. The modifiers to
each predicate come in a specific order following that predicate, and each predicate
requires a specific number of modifiers. There are no punctuation marks flagging the end
of one command or the beginning of another. The command is defined as a valid
predicate followed by the correct number of modifiers, which are then followed by the
predicate of the next command. A single command, i.e., a predicate with its modifiers,
can be spread across more than one line, and one command may end and another begin
on the same line. However, for ease in reading by humans, you will probably want to
design the body file with carriage returns between commands.
To process the file, the program looks at the first string on the file. This string must be a
valid predicate. If it is not, then the program aborts. Then, the program takes the next n
strings, where n is the number of modifiers required for this particular predicate. The
program also aborts if it has trouble interpreting any of the modifiers. Assuming these n
strings are interpreted successfully, then the program repeats, reading the next predicate
and its modifiers, etc., until it encounters the end of the file.
The strings are of two types, “numeric” strings, or simply “numbers,” and “alphabetic”
strings, or simply “words.” A valid “numeric string” or “number” is any character string
that can be interpreted by the fortran internal-read, free-format command:
read(string,*) value
(This converts the numeric string into a floating point number.)
A valid “alphabetic string” or “word,” is one of the fifty or so words found in the
program’s “internal dictionary.” All these words are given in this section, and also
summarized in Appendix A. The program knows whether to expect a word or a number
based on the position of the string relative to the beginning of the command.
Any line with an asterisk in column 1 is interpreted as a comment, and is skipped over by
the program. Blank lines can also be inserted for readability; these are also skipped over.
Table 2 summarizes each command-type, giving the valid predicate or predicates, the
valid modifiers, and the action of the command. The commands either add a bodyelement to the growing body (ADD commands) or set the value of some variable
(SPECIFY commands). Most of the commands have several synonymous predicates,
e.g., the four strings “SPHERE,” “sphere,” “S,” and “s” are all valid predicates for the
ADD- SPHERE command. The order of the modifiers as given below must be followed
in the body file.
The order of the commands is not important. Body elements can be added in any order,
and the SPECIFY and ADD commands can be interspersed.
See Appendix A for examples of valid body files.
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Table 2. VALID BODY FILE COMMANDS
ADD-SPHERE COMMAND
Valid predicates: SPHERE, sphere, S, s
Modifiers: Four numbers: cx c y cz r
Action: Adds a sphere to the list of body elements. The sphere is centered at the point
cx , c y , cz  and has radius r .
ADD-TRIANGLE COMMAND
Valid predicates: TRIANGLE, triangle, T, t
Modifiers: Nine numbers: v1x v1 y v1z v2 x v2 y v2 z v3 x v3 y v3 z
Action: Adds a triangle to the list of body elements. The three vertices of the triangle are
the points v1x , v1 y , v1z  , v2 x , v2 y , v2 z  , v3 x , v3 y , v3 z .
ADD-DISK COMMAND
Valid predicates: DISK, disk, D, d
Modifiers: Seven numbers: cx c y cz nx n y nz r
Action: Adds a circular disk to the list of body elements. The disk is centered at the
point cx , c y , cz  . The vector nx , n y , nz  specifies the direction normal to the disk, and
need not be normalized. The disk has radius r .
ADD-CYLINDER COMMAND
Valid predicates: CYLINDER, cylinder
Modifiers: Eight numbers: cx c y cz nx n y nz r L
Action: Adds an open cylinder to the list of body elements. The cylinder has center
cx , c y , cz  . The vector nx , ny , nz  specifies the principle axis of the cylinder, and need
not be normalized. The cylinder has radius r and length L .
(NOTE: You can add a closed cylinder as one open cylinder and two disks.)
ADD-TORUS COMMAND
Valid predicates: TORUS, torus, TO, to
Modifiers: Eight numbers: cx c y cz nx n y nz r1 r2
Action: Adds a torus to the list of body elements. The torus has center cx , c y , cz  . The
vector nx , n y , nz  specifies the principle axis of the torus, and need not be normalized.
The two radii r1 and r2 are defined such that if the torus were in a reference frame with
the center at the origin and the principle axis in the z-direction, it would be formed by
2
revolving the circle x  r1   y 2  r22 about the z-axis.
7
Table 2, cont’d. VALID BODY FILE COMMANDS
ADD-LENS COMMAND
Valid predicates: LENS, lens
Modifiers: Eight numbers: cx c y cz d x d y d z rc rd
Action: Adds a lens to the list of body elements. A lens is defined as the intersection of
two spheres, one sphere centered at cx , c y , cz  and having radius rc , the other centered at
d , d , d  and having radius r .
x
y
z
d
ADD-ELLIPSOID COMMAND
Valid predicates: ELLIPSOID, ellipsoid, E, e
Modifiers: Twelve numbers: cx c y cz n1x n1 y n1z n2 x n2 y n2 z a b c
Action: Adds an ellipsoid to the list of body elements. The center is at cx , c y , cz  , one
axis is parallel to the vector n1x , n1 y , n1z  , another to the vector n2 x , n2 y , n2 z  . These two
axis vectors need not be normalized; they will be automatically normalized by the
program. However, they should be orthogonal. The third axis is determined by the
program as the cross-product of these two. Then a is the semiaxis along the direction
n1x , n1 y , n1z , b is the semiaxis along the direction n2 x , n2 y , n2 z  , and c is the semiaxis
along the third direction.
ADD-CUBE COMMAND
Valid predicates: CUBE, cube
Modifiers: Four numbers: cx c y cz s
Action: Adds a cube to the list of body elements. The cube is defined as the locus of
points x, y, z  satisfying cx  x  cx  s , c y  y  c y  s , cz  z  cz  s . Note,
therefore, that this version only adds cubes that are aligned parallel to the Cartesian axes.
To add cubes with other orientations, use 12 triangles, two on each face.
SPECIFY-SKIN-THICKNESS COMMAND
Valid predicates: ST, st
Modifiers: One number: 
Action: Sets the value of the skin-thickness parameter,  .
Note: This command is optional. A value of  is only needed if the zeno integration is
to be performed. If you omit this command,  defaults to the value RL  106 .
8
Table 2, cont’d. VALID BODY FILE COMMANDS
SPECIFY-LENGTH-UNITS COMMAND
Valid predicates: UNITS, units
Modifiers: This command takes a single word as a modifier. It does not take number
modifiers. Only one of the following five strings will be accepted as the modifier:
m (meters)
cm (centimeters)
nm (nanometers)
A (Ångstrom units)
L (generic or unspecified length units)
Action: Sets the length unit for quantities found in the body file. All coordinates, radii,
lengths, etc., must always be given in the same length units, and you use this command to
specify the units.
Note: This command is optional. If not given, the length unit defaults to L (generic or
unspecified units).
SPECIFY-TEMPERATURE COMMAND
Valid predicates: TEMP, temp
Modifiers: The predicate must be followed by two modifiers. The first is a number, the
second gives the temperature units. Valid temperature unit codes:
C (Celsius)
K (Kelvin)
Action: Specifies the temperature.
Note: This command is optional. It needs to be present if you want the program to
compute the diffusion coefficient from the Stokes-Einstein formula.
SPECIFY-MASS COMMAND
Valid predicates: MASS, mass
Modifiers: The predicate must be followed by two modifiers. The first is a number, and
the second gives the mass units. Valid mass unit codes:
Da
(Daltons)
kDa
(kilodaltons)
g
(grams)
kg
(kilograms)
Action: Specifies the mass of the molecule.
Note: This command is optional. It needs to be present if you want the program to
compute the intrinsic viscosity in conventional units.
9
Table 2, cont’d. VALID BODY FILE COMMANDS
SPECIFY-SOLVENT-VISCOSITY COMMAND
Valid predicates: VISCOSITY, viscosity
Modifiers: The predicate must be followed by two modifiers. The first is a number, the
second gives the viscosity units. Valid viscosity unit codes:
p
(poise)
cp
(centipose)
Note: This command is optional. If you want the program to compute the diffusion
coefficient by the Stokes-Einstein formula, it will need the solvent viscosity. There are
two options: (1) this command, or (2) the SPECIFY-SOLVENT command in
conjunction with the SPECIFY-TEMPERATURE command.
SPECIFY-SOLVENT COMMAND
Valid predicates: SOLVENT, solvent
Modifiers: The predicate takes only one modifier. At present, the modifier must be one
of the two codes:
water
WATER
Note: This command is optional. If you want the program to compute the diffusion
coefficient by the Stokes-Einstein formula, it will need the solvent viscosity. There are
two options: (1) the SPECIFY-SOLVENT-VISCOSITY command, or (2) this command
in conjunction with the SPECIFY-TEMPERATURE command. At some later date, it
may be possible to add other solvents to the list.
V. DESCRIPTION OF THE INTEGRATIONS
A. The Zeno Integration.
The zeno integration is a numerical path integration. It simultaneously solves two
separate boundary value problems in electrostatics, the charge distribution on, first, a
charged conductor, and second, on a grounded conductor in a uniform external field. A
flowchart summarizing the computation is given in Ref. [7]. Three quantities are
required as input: The integration size, Nz, which you set through the action-code during
program invocation, the launch radius, RL, which is determined automatically by the
program from the body specification, and the skin thickness, ε, which is either set in the
body file, or, by default, is set equal to RL  106 . If you set the skin thickness yourself,
we recommend values five to six orders of magnitude smaller than the lateral dimensions
of the body.
Outputs are the electrostatic capacity, C, and the nine components of the electrostatic
polarizability tensor,   .
10
B. The Interior Integration.
This is a Monte Carlo integration over the total volume of the object. Two quantities are
required as input: The integration size, Ni, which you set through the action-code during
program invocation, and the launch radius, RL, which is determined automatically by the
program from the body specification. The calculation is performed by generating points
at random inside the launch sphere, and discarding all those that do not also lie inside the
body. This continues until 2 Ni points have been found inside the body.
Outputs are: first, the volume, V, of the object, which is set equal to 4 / 3RL3 f i , where
f i is the fraction of points found inside the body; and second, the square radius of
gyration accumulated by points over the interior, which has the definition:
Rgi2 
where
 dr
1
1
dr1  dr2 r122
2 
2V V
V
denotes integration over the volume of the body. This integral is evaluated
V
internally by averaging the square-distance between pairs of points for a total of Ni pairs.
The integration will be skipped if any of the body elements are of type B, as explained
above.
C. The Surface Integration
This is a Monte Carlo integration over the total surface area of the object. One quantity is
required as input, the integration size, Ns, which you set through the action-code during
program invocation. The calculation is performed by generating points at random over
the surface of each body element (with each body element weighted according to its own
surface area), and discarding all those that lie inside some other body element. This
continues until 2 Ns points have been located on the surface of the body.
Outputs are: first, the surface area, A, of the object, which is set equal to A0 f s , where f s
is the fraction of points retained and where A0 is the total combined surface area of all
the body elements; second, the square radius of gyration accumulated by points over the
surface, which has the definition:
Rgs2 
where
 dr
1
1
dr1  dr2 r122
2 A2 A
A
denotes integration over the surface of the body of the body, and third, the
A
Kirkwood radius, or harmonic mean distance between arbitrary surface points:
RK1 
1
1
dr1  dr2
2 
r12
A S
S
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These integrals are evaluated internally by averaging over pairs of points for a total of Ns
pairs.
D. Computation times.
Of the three integration procedures, the zeno is generally slower than the other two for
comparable values of NZ, NI, or NS, although the precise timing depends on the shape.
The times for each of the three integrations are generally linear in NZ, NI, and NS,
respectively. The time for a zeno integration is also linear in the number of body
elements. When the body elements are spheres, and for Pentium III processors, an
estimate of the time for a zeno integration is 2.3  108 N Z minutes per body element.[7]
(This is notable because finite element computations are cubic in the number of body
elements.) The ellipsoid body elements are somewhat slower than the others.


VI. DESCRIPTION OF THE OUTPUT DATA
Results of the computation are reported in the “zeno file,” a text file created by the
program. The name of the zeno file is <identifier>.zno, where <identifier> represents the
name-string introduced above. Whether or not a result is reported in the zeno file
depends, obviously, on whether or not the requisite computation was performed and
whether or not other requisite variables were set. To codify the rules followed by the
program in reporting a quantity, let us first define several Boolean variables:
TABLE 3. BOOLEAN VARIABLES CONTROLLING DATA OUTPUT.
BOOLEAN
VARIABLE
BL
BK
BZ
BS
BI
BT
BM
BV
BU
DEFINITION
The launch radius, RL , was successfully determined.
The skin thickness, ε, was set.
The zeno integration finished successfully.
The surface integration finished successfully.
The interior integration finished successfully.
The temperature was set.
The mass was set.
The solvent viscosity was set or determined.
Specific length units, rather than the generic unit “L,” were set.
The following table summarizes all quantities that are reported in the zeno file. It
includes the Boolean truth-function that determines whether or not the program displays
the quantity, and in some cases, a brief definition, the formula by which the quantity is
computed, and appropriate literature references.
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TABLE 4. SUMMARY OF OUTPUT DATA
Launch radius, RL
Displayed if: BL
Definition: Radius of a sphere centered at the origin which encloses the body.
Skin thickness, ε
Displayed if: BK
During the zeno integration, paths approaching to within this distance are considered to
have made first-passage onto the surface.
Temperature, T
Displayed if: BT
Mass, m
Displayed if: BM
Solvent viscosity, η
Displayed if: BV
The viscosity is either supplied in the body file, or else it is computed from the
temperature, using formulas given on page F49 of the CRC Handbook of Chemistry and
Physics, 55th edition.
Zeno Monte Carlo steps, NZ
Displayed if: BZ
The number of independent paths employed in the zeno integration.
Capacitance, C
Displayed if: BZ
The proportionality between total charge and electrostatic potential for a charged,
conducting body. It has the units of length and provides one measure of the size of the
body. Also an excellent approximation to the hydrodynamic radius.
Polarizability,  
Displayed if: BZ
The tensor giving the proportionality between induced dipole moment and external field
for a polarized conducting body. All nine components are determined in the zeno
integration.
Surface Monte Carlo steps, Ns
Displayed if: BS
The number of independent pairs of points sampled over the surface during the surface
integration.
Kirkwood radius, RK
Displayed if: BS
1
1
, harmonic mean distance between arbitrary pairs of surface
RK1  2  dr1  dr2
r12
A S
S
points. Also used, via the Kirkwood double sum formula, to approximate the
hydrodynamic radius and capacity. See Ref. [10] for an appraisal of the accuracy of this
approximation.
Surface area, A
Displayed if: BS
13
TABLE 4, cont’d. SUMMARY OF OUTPUT DATA
Surface radius of gyration, Rgs
Displayed if: BS
Radius of gyration as contributed only by the surface points.
Russell radius, RRus
Displayed if: BS
1/ 2
 A
RRus  
 , an approximation to the hydrodynamic radius and capacitance, most
 4 
accurate for nearly spherical ellipsoids. See Refs. [9] and [10].
Rayleigh radius, RRay
Displayed if: BS
1/ 2
 2  A 
RRay     , an approximation to the hydrodynamic radius and capacitance, most
    
accurate for two-dimensional, nearly circular bodies. See Refs. [9] and [10].
Interior Monte Carlo steps, Ni
Displayed if: BI
The number of independent pairs of points sampled through the interior during the
interior integration.
Volume, V
Displayed if: BI
Interior radius of gyration, Rgi
Displayed if: BI
Radius of gyration as contributed by the interior points.
Hydrodynamic radius, Rh
Displayed if: BZ
Rh  q1C , where q1 = 1. The radius of a hypothetical sphere having the same diffusion
coefficient as the molecule in question. This formula results from the electrostatichydrodynamic analogy and is valid to within 1% or so. In our opinion, this is the best
approximation for Rh
Trace of polarizability tensor, Tr  
Displayed if: BZ
Tr    11   22  33
Hydrodynamic volume, Vh
Displayed if: BZ
q Tr  
Vh  2
, where q2 = 0.79. The volume of a hypothetical sphere having the same
3
intrinsic viscosity as the molecule in question. Like the hydrodynamic radius, this
formula results from the electrostatic-hydrodynamic analogy and is valid to within about
5%. In our opinion, this is the best approximation for Vh.
14
TABLE 4, cont’d. SUMMARY OF OUTPUT DATA
13
 3V 
C0  

 4 
Displayed if: BI
The capacitance of a hypothetical sphere having the same volume as the molecule in
question.
  
 
3V
Displayed if: BZ  BI
Diagonal elements of polarizability tensor normalized by the volume.
   22

   11
;   33   33
3V
3V
Displayed if: BZ  BI
Only significant for bodies with rotational symmetry about the z-axis, in which case this
represents a decomposition of σ into components parallel and perpendicular to the
rotation axis.
Tr  

  11   22   33     
3V
Displayed if: BZ  BI
Trace of the polarizability tensor normalized by the volume. This is a convenient shape
functional.
Intrinsic viscosity (volume-normalized),  V
Displayed if: BZ  BI
 V  Vh  q2Tr   , where q2 = 0.79. Intrinsic viscosity in terms of volume fraction.
V
3V
Approximation good to about 5%.
C Rgi , Rgi Rh , C C0 , V Vh
Displayed if: BZ  BI
Various dimensionless shape functionals.
C RK , C Rgs , Rgs Rh , RRus C , RRay C
Displayed if: BZ  BS
Various dimensionless shape functionals.
Sphericity
Displayed if: BI  BS
A
, a dimensionless shape functional, quantifies departure from sphericity.
13
36V 2


15
TABLE 4, cont’d. SUMMARY OF OUTPUT DATA
Intrinsic viscosity (mass-normalized),  M
Displayed if: BZ  BM
 M  Vh  q2Tr   , where q2 = 0.79. Intrinsic viscosity defined in terms of mass
m
3m
concentration (conventional units). Approximation good to about 5%.
Diffusion coefficient, D
Displayed if: BZ  BT  BV  BU
kT
D
. Stokes-Einstein relation for the diffusion coefficient.
6Rh
16
Appendix A. EXAMPLES
Example 1. A cube. Tables A.1 and A.2 display the body and zeno files, respectively, for
the computation performed on a cube.
Table A.1 box.bod
cube -1 -1 -1
2
Table A.2 box.zno
Body name: box
Number of body elements:
1
==================================================
JOB SUMMARY:
Actions
Checked if
Monte Carlo
requested
completed
size
-------------------------------------------------z
*
1000000
s
*
1000000
i
*
1000000
==================================================
launch radius . . . . . . . .
1.73205
L
skin thickness . . . . . . . .
0.173205E-05
L
zeno m.c. steps . . . . . . .
1000000
capacitance, C . . . . . . . . 1.3230(7)
L
polarizability 11 . . . . . . 2.916(8)E+01
L^3
polarizability 12 . . . . . . -7(8)E-02
L^3
polarizability 13 . . . . . . 1.0(7)E-01
L^3
polarizability 21 . . . . . . 5(9)E-02
L^3
polarizability 22 . . . . . . 2.923(7)E+01
L^3
polarizability 23 . . . . . . 1.0(8)E-01
L^3
polarizability 31 . . . . . . 9(7)E-02
L^3
polarizability 32 . . . . . . -5(7)E-02
L^3
polarizability 33 . . . . . . 2.923(5)E+01
L^3
surface m.c. steps . . . . . .
1000000
RK . . . . . . . . . . . . . . 1.294(3)
L
surface area . . . . . . . . . 2.40000(0)E+01
L^2
Rg (surface) . . . . . . . . . 1.2916(3)
L
R(Russell) . . . . . . . . . . 1.38198(0)
L
R(Rayleigh) . . . . . . . . . 1.75959(0)
L
interior m.c. steps . . . . .
1000000
volume . . . . . . . . . . . . 7.993(4)
L^3
Rg (interior) . . . . . . . . 9.996(4)E-01
L
Rh . . . . . . . . . . . . . . 1.32(1)
L
Tr(alpha) . . . . . . . . . . 8.76(1)E+01
L^3
Vh . . . . . . . . . . . . . . 2.3(1)E+01
L^3
C0 . . . . . . . . . . . . . . 1.2403(2)
L
sig 11 . . . . . . . . . . . . 1.216(3)
17
sig 22 . . . .
sig 33 . . . .
sig(normal) .
sig(parallel)
sigma . . . .
[eta](V) . . .
C/Rg(int) . .
Rg(int)/Rh . .
C/C0 . . . . .
V/Vh . . . . .
C/RK . . . . .
C/Rg(surf) . .
Rg(surf)/Rh .
R(Russell)/C .
R(Rayleigh)/C
sphericity . .
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18
1.219(3)
1.219(2)
2.435(5)
1.219(2)
3.654(5)
2.9(1)
1.3235(8)
7.56(8)E-01
1.0666(6)
3.5(2)E-01
1.023(3)
1.0243(6)
9.8(1)E-01
1.0446(5)
1.3300(7)
1.2414(5)
Example 2. Five overlapping spheres. Tables A.3 and A.4 display the body and zeno
files for a body constructed from five overlapping spheres. It displays some of the
freedom you have in formatting the body file, including use of comments, blanks, and
extending commands across more than one line.
Table A.3 some.spheres.bod
* This body consists of 5 spheres
* Blank lines are OK:
* This line inserts a sphere of radius 1 at the origin
SPHERE 0 0 0 1
* The next line inserts a sphere of radius 2 tangent
* to the first sphere with center on the x axis
S 3 0 0 2
* Carriage returns are permissible during the
*specification of any one element:
sphere -3 0 0
2
*
*
You can also run different elements together on
the same line
S 1 1 1 1
s -1 -1 -1 1
* This command establishes nanometers as the length unit
units nm
Table A.4 some.spheres.zno
Body name: some.spheres
Number of body elements:
5
==================================================
JOB SUMMARY:
Actions
Checked if
Monte Carlo
requested
completed
size
-------------------------------------------------z
*
1000000
i
*
1000000
s
*
1000000
==================================================
launch radius . . . . . . . .
5.00000
nm
skin thickness . . . . . . . .
0.500000E-05
nm
zeno m.c. steps . . . . . . .
1000000
capacitance, C . . . . . . . . 3.101(2)
nm
polarizability 11 . . . . . . 7.98(3)E+02
nm^3
polarizability 12 . . . . . . 7(3)
nm^3
19
polarizability 13 .
polarizability 21 .
polarizability 22 .
polarizability 23 .
polarizability 31 .
polarizability 32 .
polarizability 33 .
surface m.c. steps .
RK . . . . . . . . .
surface area . . . .
Rg (surface) . . . .
R(Russell) . . . . .
R(Rayleigh) . . . .
interior m.c. steps
volume . . . . . . .
Rg (interior) . . .
Rh . . . . . . . . .
Tr(alpha) . . . . .
Vh . . . . . . . . .
C0 . . . . . . . . .
sig 11 . . . . . . .
sig 22 . . . . . . .
sig 33 . . . . . . .
sig(normal) . . . .
sig(parallel) . . .
sigma . . . . . . .
[eta](V) . . . . . .
C/Rg(int) . . . . .
Rg(int)/Rh . . . . .
C/C0 . . . . . . . .
V/Vh . . . . . . . .
C/RK . . . . . . . .
C/Rg(surf) . . . . .
Rg(surf)/Rh . . . .
R(Russell)/C . . . .
R(Rayleigh)/C . . .
sphericity . . . . .
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20
5(2)
5(1)
2.167(9)E+02
4.5(9)
4.6(9)
1.55(9)E+01
2.179(9)E+02
1000000
2.944(5)
1.2588(3)E+02
3.316(2)
3.1650(4)
4.0298(5)
1000000
7.824(5)E+01
3.181(1)
3.10(3)
1.233(3)E+03
3.2(2)E+02
2.6533(6)
3.40(1)
9.23(4)E-01
9.28(4)E-01
4.32(1)
9.28(4)E-01
5.25(1)
4.1(2)
9.751(9)E-01
1.03(1)
1.169(1)
2.4(1)E-01
1.054(2)
9.352(9)E-01
1.07(1)
1.0205(8)
1.299(1)
1.4230(7)
nm^3
nm^3
nm^3
nm^3
nm^3
nm^3
nm^3
nm
nm^2
nm
nm
nm
nm^3
nm
nm
nm^3
nm^3
nm
Example 3. The myoglobin molecule. Tables A.5 and A.6 display the application of
these techniques to a protein molecule, myoglobin. The structure of the molecule was
taken from the Protein Data Bank, entry code 1a6m. This protein consists of 151 amino
acids, and was modeled with overlapping spheres, one sphere centered at each alphacarbon. Each sphere has radius 5 Å. This example demonstrates use of the SPECIFYMASS, SPECIFY-TEMPERATURE, and SPECIFY-SOLVENT commands in the body
file. Since these variables were specified, the program was able to compute the massnormalized intrinsic viscosity and the diffusivity, neither of which appear in the previous
zeno files. (For the sake of brevity, over 140 lines have been omitted from the body file.)
Table A.5 1a6m.5.bod
s
-3.526
15.758
s
-0.689
14.190
s
-1.487
12.495
s
0.324
13.366
.
.
.
s
-0.870
33.550
s
-1.223
31.968
s
1.894
29.853
ST 0.001
mass 16747 Da
temp 20 C
solvent water
units A
14.900
16.862
20.143
23.335
5.000
5.000
5.000
5.000
-1.190
-4.522
-4.417
5.000
5.000
5.000
Table A.6 1a6m.5.zno
Body name: 1a6m.5
Number of body elements:
151
==================================================
JOB SUMMARY:
Actions
Checked if
Monte Carlo
requested
completed
size
-------------------------------------------------z
*
100000
s
*
100000
i
*
100000
==================================================
launch radius . . . . . . . .
47.3601
A
skin thickness . . . . . . . .
0.100000E-02
A
temperature . . . . . . . . . 2.932(5)E+02
K
mass . . . . . . . . . . . . . 1.67470(5)E+04
Da
solvent viscosity (computed) . 1.002(1)
cp
zeno m.c. steps . . . . . . .
100000
capacitance, C . . . . . . . . 2.071(6)E+01
A
polarizability 11 . . . . . . 1.29(2)E+05
A^3
21
polarizability 12 .
polarizability 13 .
polarizability 21 .
polarizability 22 .
polarizability 23 .
polarizability 31 .
polarizability 32 .
polarizability 33 .
surface m.c. steps .
RK . . . . . . . . .
surface area . . . .
Rg (surface) . . . .
R(Russell) . . . . .
R(Rayleigh) . . . .
interior m.c. steps
volume . . . . . . .
Rg (interior) . . .
Rh . . . . . . . . .
Tr(alpha) . . . . .
Vh . . . . . . . . .
C0 . . . . . . . . .
sig 11 . . . . . . .
sig 22 . . . . . . .
sig 33 . . . . . . .
sig(normal) . . . .
sig(parallel) . . .
sigma . . . . . . .
[eta](V) . . . . . .
C/Rg(int) . . . . .
Rg(int)/Rh . . . . .
C/C0 . . . . . . . .
V/Vh . . . . . . . .
C/RK . . . . . . . .
C/Rg(surf) . . . . .
Rg(surf)/Rh . . . .
R(Russell)/C . . . .
R(Rayleigh)/C . . .
sphericity . . . . .
[eta](M) . . . . . .
D . . . . . . . . .
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22
-2(3)E+03
-1.7(2)E+04
-7(2)E+03
1.01(2)E+05
-1.7(2)E+04
-1.3(2)E+04
-2.6(2)E+04
1.12(2)E+05
100000
1.957(6)E+01
7.18(2)E+03
1.930(2)E+01
2.391(2)E+01
3.044(3)E+01
100000
2.957(6)E+04
1.646(2)E+01
2.07(2)E+01
3.42(4)E+05
9.0(5)E+04
1.918(1)E+01
1.46(3)
1.14(2)
1.26(3)
2.59(4)
1.26(3)
3.85(5)
3.0(2)
1.258(4)
7.95(8)E-01
1.079(3)
3.3(2)E-01
1.058(4)
1.073(3)
9.3(1)E-01
1.154(4)
1.470(5)
1.553(4)
3.2(2)
1.03(1)E-06
A^3
A^3
A^3
A^3
A^3
A^3
A^3
A^3
A
A^2
A
A
A
A^3
A
A
A^3
A^3
A
cm^3/g
cm^2/s
Appendix B. SUMMARY OF WORDS RECOGNIZED BY THE GRAMMAR OF
THE BODY FILE.
Table B1 gives the “dictionary” for the grammar file. All the synonyms of any one word
are listed together. In column 2, “P” indicates predicate, “M” indicates modifier.
Table B1.
WORD
A
C
cm
cp
cube, CUBE
cylinder,
CYLINDER
d, D, disk, DISK
Da
e, E, ellipsoid,
ELLIPSOID
g
K
kDa
kg
L
lens, LENS
m
mass, MASS
nm
p
s, S, sphere,
SPHERE
solvent,
SOLVENT
st, ST
t, T, triangle,
TRIANGLE
temp, TEMP
to, TO, torus,
TORUS
units, UNITS
viscosity,
VISCOSITY
water, WATER
TYPE
MEANING
M
M
M
M
P
P
Ångstrom units
Celcius
centimeters
centipoise
COMMAND IN WHICH THIS
WORD IS FOUND
SPECIFY-LENGTH-UNITS
SPECIFY-TEMPERATURE
SPECIFY-LENGTH-UNITS
SPECIFY-SOLVENT-VISCOSITY
ADD-CUBE
ADD-CYLINDER
Daltons
ADD-DISK
SPECIFY-MASS
ADD-ELLIPSOID
P
M
P
M
M
M
M
M
P
M
P
M
M
P
grams
Kelvins
kilodaltons
kilograms
Generic length unit
meters
nanometers
poise
SPECIFY-MASS
SPECIFY-TEMPERATURE
SPECIFY-MASS
SPECIFY-MASS
SPECIFY-LENGTH-UNITS
ADD-LENS
SPECIFY-LENGTH-UNITS
SPECIFY-MASS
SPECIFY-LENGTH-UNITS
SPECIFY-SOLVENT-VISCOSITY
ADD-SPHERE
P
SPECIFY-SOLVENT
P
P
SPECIFY-SKIN-THICKNESS
ADD-TRIANGLE
P
P
SPECIFY-TEMPERATURE
ADD-TORUS
P
P
SPECIFY-LENGTH-UNITS
SPECIFY-SOLVENT-VISCOSITY
M
SPECIFY-SOLVENT
23
Appendix C. UNCERTAINTY ESTIMATES AND PROPAGATION.
Like any Monte Carlo integration, the results display sampling error. The program
estimates the sampling error in the integrations, and propagates the errors through
subsequent computations. Error estimation, propagation, and reporting by the program
are explained here.
Significant figures in the input data.
The input quantities set in the body file, namely temperature, mass, and solvent viscosity,
are assumed to contain experimental error. All digits displayed in a SPECIFY command,
including trailing zeros, will be considered significant by the program. In other words,
the two commands
“MASS 20000 Da”
and
“MASS 2.00E4 Da”
will be interpreted as m  20000  0.5 Da , and m  20000  50 Da , respectively.
These uncertainties will then be propagated through subsequent computations as
explained below.
Sampling errors in the integrations.
The following quantities are results of a Monte Carlo integration, and therefore display
sampling error:
Capacitance, C
Polarizability,  
Kirkwood radius, RK
Surface area, A
Square surface radius of gyration, Rgs2
Volume, V
Square interior radius of gyration, Rgi2
To estimate the integration error, each integral is performed 20 times independently,
using an integration size of N/20. The final value is taken as the mean of these 20
independent integrations, while the sampling error is taken as the standard deviation
divided by 20 .
Uncertainties resulting from the electrostatic-hydrodynamic analogy.
Two formulas, first given above,
Rh  q1C and  V 
24
Vh q2Tr  

V
3V
result from analogies between electrostatic and hydrodynamic boundary value problems.
But the analogies are only approximate, so that q1 and q2 are not constant; rather they
vary from shape to shape. The variation is small, however, and so using standard values
of the two coefficients lets us use the results of an electrostatic calculation to approximate
hydrodynamic properties. The current version of the program uses the values:
q1  1.00  0.01 and q2  0.79  0.04
These uncertainties are propagated through subsequent calculations. This means that no
matter how many Monte Carlo steps are used in the zeno integration, hydrodynamic
properties directly related to these two coefficients will never appear with more than 2 or
3 significant figures.
Propagation of uncertainties.
All other quantities are computed from the above values. Suppose that the computation
of a variable y in terms of several variables xj is represented in the following functional
form:
y  f x1, x2 ,
Furthermore, let y and x j represent uncertainties in y and xj, respectively. Then, to
estimate y , the program uses
2


y    j  f  x j 2
 x j 
2
Final display of error estimates.
As explained in the preceding paragraphs, uncertainty estimates are calculated for all
quantities. The final results are always rounded, with the uncertainty in the final digit
enclosed in parentheses. For example, the string 1.03(1)E-06 represents the range of
numbers (1.03  0.01)  106 .
25
Appendix D. Random numbers.
The program uses the random number generator ran2 published in Press, Teukolsky,
Vetterling, and Flannery, Numerical Recipes in Fortran 77, 2nd edition, Cambridge
University Press (1992).
The program uses the Linux date command to generate the seed for the random numbers.
Therefore, after execution, you will find the file <identifier>.dfl in you directory. It
contains the Linux date stamp at the time of program initiation. It is perfectly safe to
delete this file.
26
References:
[1] Derivation of the analogy between capacitance and hydrodynamic radius: Hubbard
and Douglas, “Hydrodynamic friction of arbitrarily shaped Brownian particles,” Physical
Review E, 47, R2983-R2986 (1993).
[2] Derivation of the path integral technique for the capacitance: Zhou, Szabo, Douglas,
and Hubbard, “A Brownian dynamics algorithm for calculating the hydrodynamic
friction and the electrostatic capacitance of an arbitrarily shaped object,” J. Chem. Phys.,
100, 3821-3826 (1994).
[3] Test of the analogy between capacitance and hydrodynamic radius: Douglas, Zhou,
and Hubbard, “Hydrodynamic friction and the capacitance of arbitrarily shaped objects,”
Physical Review E, 49, 5319-5331 (1994).
[4] Derivation and testing of the analogy between polarizability and intrinsic viscosity:
Douglas and Garboczi, “Intrinsic viscosity and the polarizability of particles having a
wide range of shapes,” Adv. Chem. Phys., 91, 85-153 (1995), and Garboczi and Douglas,
Physical Review E, 53, 6169-80 (1996).
[5] The derivation of the path integral formulation for the polarizability and its
application to a number of different shapes: Mansfield, Douglas, and Garbozci, “Intrinsic
viscosity and the electrical polarizability of arbitrarily shaped objects,” Physical Review
E, 64, 061401 (2001).
[6] Application zeno algorithm to flexible polymer models: Mansfield and Douglas,
“Numerical path-integration calculation of transport properties of star polymers and
theta-DLA aggregates,” Condensed Matter Physics, 5, 249 (2002).
[7] Application of the zeno algorithm to proteins: Kang, Mansfield, and Douglas,
“Numerical path integration technique for the calculation of transport properties of
proteins,” Physical Review E, 69, 031918 (2004). This reference also gives a flowchart
for the zeno algorithm.
[8] A good introduction to the use of path integral techniques in solving boundary value
problems: Douglas and Friedman, “Coping with complex boundaries,” IMA Series on
Mathematics and its Applications, Vol. 67 (Springer, New York, 1995), p. 166-185.
[9] The quantities RRus and RRay are discussed in an appendix of Douglas and Freed,
“Competition between hydrodynamic screening (‘draining’) and excluded volume
interactions in an isolated polymer chain,” Macromolecules, 27, 6088-6099 (1994). This
also contains references to the original literature.
[10] An an appraisal of how accurately RK, RRus, and RRay represent the hydrodynamic
radius: Mansfield and Douglas, “Accuracy of several approximate formulas for the
hydrodynamic radius and the diffusion coefficient,” in preparation. .
27
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