Chromatin Modeling — supplement 1
Supplementary material for Structure with Folding and Design, ??:??–??, 2000.
Daniel A. Beard and Tamar Schlick
Department of Chemistry and Courant Institute of Mathematical Sciences, New York University and Howard
Hughes Medical Institute, 251 Mercer Street, New York, New York 10012.
beard@biomath.nyu.edu
, schlick@nyu.edu
An explicit formulation of the configuration-dependent potential energy function and the Brownian Dynamics (BD) algorithm used to simulate the chromatin polymer is detailed. For the readers’ convenience, this document provides a complete description of our methodology, and thus some of the material is redundant with that found in the Material and Methods section of Beard & Schlick (Structure with Folding and Design
??:??–??, 2000).
The model structure is illustrated in Figure 1. Each core particle disk is connected to one or two linker DNA segments (see Figure 1, top panel). In Figure 1, denotes the position of the center of a core disk, while
, , and , denote positions of linker DNA beads. The orientation of the core disk is specified by a local coordinate system , , where the unit vectors and lie in the plane of the flat surface of the disk, and . The attatchment of the linker DNA to the core particle is illustrated in the lower panel of Figure 1. The linker DNA enters the core particle at the position
! #"%$& "('*)+$-,-.0/1
, and
$
/1 follows:
547698 nm,
;:<698
2A nm,
$
A local coordinate system , ,
>=<?#@
(see Table 1).
is also associated with each linker bead position, , and is used to calculate the local torsion on the linker beads. We define the sets core beads and the set of linker DNA beads. The vectors
BDC for
FHGIBDE and
BDE to be, respectively, the set of are directed in the direction of the linker DNA segment:
O
QPSR%T
O
-T for
GUBDE
(1)
V
[V
! #"W$
%! #"W$\
"(' )X$ !P
.Y/ Z
%"(' )X$ .Y/1 [Z for
GUBDC
]
Since the wrapped DNA does not make two full turns around the core disk, the unstressed trajectory of is not parallel to when the
F_^ ` bead is a disk. The trajectory of the wrapped DNA, is denoted
(see Figure 1, bottom panel), and is calculated as
! #"&$H.
"(' )H$
. Similarly, the trajectory of

Chromatin Modeling — supplement
Elements of Computational Model:
Core Disk r o a
+ i
θ
ο c i b i a
− i
θ
ο r i a i b
i+1 r
i+1 a
i+1 a DNA i c
i+1 r
i−1
Core Disk Geometry:
θ o a
+ i
β
i+1 r
i+2
Linker DNA Bead
Figure 1: Diagram of the subunits of our chromatin model.
The upper panel illustrates the components of the computational model: a core disk is located at position , and linker DNA beads are located at and
,
. The lower panel illus-
, trates the core disk geometry.
a i
−
2 the wrapped DNA, as it exits the core is denoted by . The local coordinate systems of the wrapped DNA entering and exiting the core particle are given by:
! #"%$ "(' )H$
<"(' )H$a.
! #"&$
(2)
&deWf dgeWf dgeWf c and
+b Hb
. For the core particle, we introduce the vectors (see
Figure 1), to represent the local coordinate system associated with the DNA segment connecting the core at
&de%f deWf dgeWf and the linker vertex . The calculation of is outlined next.
The bending and twisting terms in the potential energy function are expressed in terms of the Euler angles h [2], bSi bcj k
, that transform one local coordinate system to the next. In the following sections we will make use of the following sets of Euler angles:
For
GmBDE
, the angles coordinate system.
h bSi bcj k transform from the k to the
n

Chromatin Modeling — supplement
Table 1: Elastic and geometric parameters used in the dinucleosome model.
Parameter Description
}~
$
/1 r#‚Qƒ
†
FGUBDE
FGUB
Equilibrium segment length
Stretching constant
Twisting rigidity constant
Elastic bending persistence length of DNA
Angular separation between linker segments
Radius of wound DNA supercoil
Width of wound DNA supercoil
Hydrodynamic radius of linker bead (set
BDE
Hydrodynamic radius of core bead (set )
)
Excluded volume parameter
Core/linker excluded volume parameter
Core/core excluded volume parameter
Numerical time step for dynamics simulation
Value
3.0 nm
???sr2tauvR
{
6y?zO:D?
50 nm erg nm
=<?
4.8 nm
3.6 nm
1.5 nm
5 nm
0.001
3.0 nm
2.0 nm
2.0 ps
3
For
G‡BDC dinate system.
, the angles h bSi bcj transform from the to the coor-
For
FGˆBDC h bSi bcj
, the angles h bSi
&de%f transform bcj deWf transform deWf to to &deWf
.
de%f de%f ; and the angles
&de%f deWf deWf The additional DNA coordinate system associated with each core bead, h ciated Euler angles bSi bcj
, and the asso-
, are calculated based on the assumption that the core particle protein/DNA complex remains rigid and twisting of the DNA that is wrapped around the core does not contribute to the
&de%f mechanics of the model. Therefore . Calculation of the vector and the angle is straightforward:
V .Z
D‰Š .Y/ Š‹Œ de%f
(3) n [Vn%.Z
&.Y/1
N
! #" de%f
PŠ6
We calculate from: for for
|W
N deWf
O! #"
N deWf
O! #"
|W
PSR "(' )
1’‘
PSR "(' )
1’“
, where
! #"
•”
|– deWf
O! #"
"(' )
—+˜
Then we set to ensure that no torsion is introduced into the Euler rotation de%f deWf the Euler rotations are applied to to obtain and .
h
HbSi vbcj
(4)
(5)
(6)
. Then

Chromatin Modeling — supplement 4
The energy associated with a given polymer structure is estimated from the sum of several elastic energy terms, an electrostatic potential, and an excluded volume term [3]:
™vš ™H› ™vœ ™ž (7)
The first three terms represent elastic contributions from stretching, twisting, and bending, respectively. The terms and are used to model electrostatic and excluded volume interactions, respectively.
computational time step.
is written as:
™žš
!P
(8)
Ÿ
¡ where is the length of the segment connecting particle to particle
[O&.Y
%.Y/1
[V .Y/1
[O
Z
, calculated as: for
F[GUBDC for
GUBDC otherwise.
(9)
We use an equilibrium segment length of nm, which corresponds to roughly 9 bp per segment.
It has been shown for the Brownian dynamics simulation of linear and circular DNA [4] that a choice of
¢:D??#r užR
, where is Boltzmann’s constant and is the absolute temperature, results in standard deviations in segment length of around 10% of . The motions of the relatively large nucleosome beads in the present model tend to produce greater stretching and compressive forces on the linker DNA than are typically found in supercoiled DNA alone. This means that a greater value for the stretching rigidity
???#r užR constant is required to achieve comparable accuracy. A value of results in segment length deviations of less than 2% of the equilibrium segment length and a mean segment length equal to the equilibrium segment length for time steps of 2 ps.
The torsional rotation about the segment connecting particles and angles
&.
, and the torsional energy, , is calculated from:
:P is given by the sum of the Euler
¡
Nh
%.
(10) where is the torsional rigidity constant.

Chromatin Modeling — supplement 5
The bending energy, , is calculated from the set of angles denoting the deformation between the linker
DNA segments.
tZ¤£
•Ÿ
¡
_P
(11)
]¥¦_§
The bending constant, , can be calculated from:
}~ užR
(12) where }~ is the bending persistence length.
Representative charges are assigned to the linker beads as well as on the surface of the core disks. The assignment of charges in the model is based on a discrete -body potential which approximates the solution to the nonlinear Poisson-Boltzmann equation in the solvent surrounding the core particle. This procedure is detailed in Beard & Schlick (2000). A single charge,
©nE is assigned to each linker bead; and a set of charges
©<ª of the is assigned to each core disk, where the index refers to the n¬
^ ` charge on the
^ ` particle is denoted by
The charge on a linker bead is calculated as for
«­GUBDC
.
^ ` charge on the core disk. The position
, where is the effective linear charge density
©nE
¯® on DNA, calculated by fitting the tail of the Debye-H¨uckel potential for an infinitely-long cylinder to the
Gouy-Chapman potential in the far zone [5]. Values for and , the inverse Debye length for various levels of monovalent salt concentration, are listed in Table 2.
Table 2: Electrostatic parameters for DNA.
wX± [Molar]
0.01
0.02
0.03
0.04
0.05
[e/nm]
6²4
69=³
69=´:
476 :
[1/nm]
0.330
0.467
0.572
0.660
0.738
The electrostatic contribution to the potential is calculated as the superposition of the fundamental solution to the linearized Poisson-Boltzmann equation:
%¾D¿
%¾D¿
·¹Â µÃSÄ
%¾D¿
ÂQ·¹ÃcÄk µ
E½
©nE ©<ª ©<ª (13) n *Å Åk
¡ ¡ ¡
µ(¶<·¹¸´º
]» ¥<¦_¼
µ(¶·Á¸´º
]¥¦{¼ » ¥¦Q§
µS¶·Á¸7º
]» ¥<¦Q§
ªnÆ ªnÆ where denotes the inverse Debye length (salt-dependent), is the dielectric constant of the medium, and
¨ÇC is the number of point charges on each core disk. In equation (13) we have introduced the notation n ÈT z to denote the distance between the centers of the between the center of the
F{^ ` charge on particle and
«^ ` particles. Similarly, the distance is denoted by
*Å
¤T
;
«GUBDC ^ ` particle and the
^ `
ªnÆ

Chromatin Modeling — supplement 6 and the distance between the by
Åk
ÈT
.
^ ` charge on particle
FÉGOBDC and the
^ ` charge on particle
«ˆGOBDC is denoted
ªnÆ The first term in equation (13) is due to linker/linker interactions, and the summation is over all pairs of linker beads. The summation in the second term is over all linker/core pairs and in the last term over all core/core pairs.
It is necessary to include an excluded volume term in the calculation of the potential to avoid the overlap of positive and negative charges. We represent the excluded volume potential, , as the sum of Lennard-Jones potentials arising from linker/core pair interactions and from core/core pair interactions as:
{ r‚_ƒr
µS¶·
]¥¦_¼» ¥¦ §•ÊË
¡
Å
ªnÆ
*Å
ªDÆ
—ÉÌ-ÍÎ
{ r‚_ƒr
(14)
]»
µS¶·
¥<¦ §•ÊË
¡ ¡
Åk
ªnÆ
Åk
ªnÆ
—É̊ÍÎ
The terms in equation (14) have a shallow minimum of value kÏ of , where and r#‚Qƒ
‚Qƒ r2tauvR4 at a separation between charges are the parameters describing the Lennard-Jones interactions.
The excluded volume parameters (see Table 1) are chosen to ensure that particles do not overlap one another over the course of a simulation. It is not necessary to include a linker/linker excluded volume term because the electrostatic repulsion between DNA segments proves sufficient to prevent overlap.
The systematic force acting on the
F_^ ` particle is obtained from the gradient of the potential energy function taken with respect to the position of the particle: vÑÒ tˆ.
(15) or
(16)
Details of the calculations are given below.
The force due to the stretching potential is expressed as
NkÔ
ÐÓ
]-P
(17) where
ÖÕ g g
%dgeWf for
FG×BDE for
FG×BDC
.
(18)

Chromatin Modeling — supplement 7
A change in the position of the
F{^ `
Nh
&.
particle effects a change in the torsions
. The torsional force on the
^ ` particle can be expressed as:
Nh
]%
N]Ø
%.ZÙDg ]aVÙD]
]%DP
, Nh
] ]-P and
(19) where the vectors and
ÙD are given by
WÝ
·Þ
MÛÚ¹Ü
WÝ
·Þ
ß(à
ß(à
Ú¹Ü
²! #"
PXá ! #"
O"('*) de%f
O"(' )
, deWf for for
F[GUBDE
F[GUBDC and
For the end disks, Øé
·åæº
WÝ
·åæºkÞ
MãÚäÜ
WÝ
·]Þ
ÚäÜ
ß(à
ß(à
ê?
.
]
PXá ! #"
²! #" ] %.ç"(' ) deWf
.è"(' )
] *, de%f for
F[GˆBDE for
F[GˆBDC
.
(20)
(21)
The force due to the bending potential is expressed as:
Në g ].çìvOìv
(22) where the and
ìv are given as follows:
N
N
N
[
! #" _PSR
<! #"
&de%f
! #"
PSR
"('*)
#"('*)
PSR
QP
"('*) for for for
F\GUBDC
GUBDE
G×BDC
(23) and
ìv
]
N
] ! #"
]
N&de%f
]
N &deWf
<! #"
! #"
]ŒPSR
]
PSR
PSR
"('*)
#"-' )
"(' )
]ŒP
]
>?
.
for for for
F\G×B
GˆBDE
.
GUBDC
(24)
For the end disks,
The electrostatic force, , acting on bead is given by a sum of forces due to pairwise interactions of the form:
%¾
±kïgð
(25)
í\î
íî
:PÉò(ó
ÀŠñ
íî
í\î where
í\î represents the force on bead due to interaction of a charge (located on bead ) with charge
(located on another bead). The charges and are the charges at positions and , respectively. The

Chromatin Modeling — supplement 8 vector connecting position to position is given by
The total electrostatic force acting on a given bead is: and
íî
ÈT is the scalar distance.
(26)
¥
¥
í\î where the notation
õ÷GèF denotes that summation occurs over all charges that are associated with bead .
Similarly, the notation
GOF denotes the set of all charges not located on bead . Specifically, DNA beads have only one charge, positioned at the center of the bead while core beads have several charges located on the surface of the disk.
The excluded volume force, , acting on bead is given by:
{ú r#‚Qƒ<r
Å
¡
µŒø
¥¦Q§ ªnÆ
*Å
ªDÆ
—žû(Í
*Å
Å
ªnÆ
ªnÆýü
(27) for all
F\GUBDE and
{ú
‚Qƒ tau r#‚Qƒ<r
µŠø
¥¦_¼
¡
µŒø
¥¦Q§
¡ ¡
Åk
ªnÆ
Åk
ªnÆ
{ú
Åk
ªnÆ
—Éû-ÍÎ
Åk
Åk
ªnÆ
ªnÆ
—žû-ÍÎ
Åk
Åk Åk
ªnÆ
(28)
Æýü for all
F[GUB
. When is a linker bead (equation (27) the excluded volume force comes from the summation interactions between particle the charges on the surface of each core disk. When is a core disk (equation (28)), the force comes from all pairwise interactions with linker beads and with other core particles.
We express the systematic torque acting on bead by a vector of torques acting in each of the local coordinate directions, .
The torque on the linker DNA beads comes from the elastic twisting potential and acts only in the direction:
N
] ê?
ê?
(29) where is the torsional rotation about the segment connecting particle and .

Chromatin Modeling — supplement 9
The torque acting on core particles acts not only in the direction, but also in the directions of the and local coordinates. The total torque acting on a core particle can be written as the sum of several terms:
(30) where is the torque associated with contributions to the force which are not applied at the center of mass of the particle. Additional torques and are associated with the torsional and bending potentials.
The term is expressed in the fixed laboratory reference frame as follows:
(31)
A where the summation is over all contributions to the force acting on the particle
F•G
. The vector connects the center of the core particle to the location of the action of the force . For electrostatic and excluded volume force terms, is given by the position of the charge on the surface of the core. For elastic forces arising from linker stretching, bending, and twisting, the force is directed at
Z O/1
O .Y/ or , depending on whether the force arises from interaction with the proceeding or preceeding segment. The torque can be projected onto the local frame using:
(32)
The additional torque associated with the torsional potential is expressed in the laboratory frame as: dgeWf
] ]
P+á deWf
"('*) de%f
! #"
Nh
7.
ß(à
Nh
.
] ] ፠"-' ) ]ý.
! #" ]
âab
(33)
ß(à which is converted to the local frame using
(34)
The torque associated with the bending potential, expressed in the local frame, is
]%"(' )X$
N
]
|
_P
"('*) ]
"(' )
N deWf
|
%! #"%$
N
]
"(' )
|
N de%f
|
"(' )
]
]
N
]
|n
! #"W$\ ] "(' )X$
"-' )
(35)

Chromatin Modeling — supplement 10
For hydrodynamic purposes, core particles are treated as spheres, and the rotational frictional coefficients are be expressed as:
>8
(36) where is the hydrodynamic radius of the core particle. For the linker beads, is the rotational friction coefficient of DNA and can be expressed as:
54
(37)
:<6 where nm is the hydrodynamic radius of DNA. Since rotation occurs only about the axis for linker beads, and are effectively infinite.
The movements of the various components of the chromatin system are coupled to one another through the action of the viscous medium. This viscous coupling is approximated by incorporating the configurationdependent hydrodynamic friction tensor into the the dynamic equations, as outlined below.
Since the beads used to represent the core particles are larger than those used to represent the linker
DNA (see below), the hydrodynamic interaction tensor for nonidentical subunits, , given by Garcia and
Bloomfield [6] can be used to calculate the frictional interaction tensor. The diffusion tensor used in the
Brownian dynamics algorithm (see below) is proportional to :
(38)
For this -bead system, is the x<¨ x<¨ matrix written as: where each is a be calculated from [6]:
.
..
S
Œ
.
..
{
S
|D|D|
|D|D|
.
..
|D|D|!
Í#"
Ÿ\Ÿ matrix representing the interaction between the
F_^ ` and
«^ ` beads. Each
(39) can for (same bead),
³ &%('
(40) for
F*)
(different beads),
8
—X˜ where is the identity matrix, is the bead radius, and is the viscosity of the surrounding fluid.
We use a touching-bead model for the linker DNA, and thus set the linker bead hydrodynamic radius to
:<6 nm for
FXG BDE
. The core particle is treated as a spherical bead for hydrodynamic purposes.
Yao et al. (1990) measure the translational diffusion coefficient of a core particle (octamer and wrapped
DNA) to be about
6y?
nm for
F[GUB
69=0ç:D?
cm sec . This allows us to calculate an effective hydrodynamic radius of particle using the relation:
>r užR<³,
(41)

Chromatin Modeling — supplement 11
We remark that the volume of the spherical bead with this radius is nearly equal to the volume of a disk of radius 6 nm and width 5 nm.
The Brownian dynamics (BD) algorithm [7] requires obtaining the Cholesky factorization, where is a lower triangular matrix. The entries the matrix are given by:
.-/-
, kÏc
10
äg32
]
¡
10
42
¡
AA if if if .
,
,
(42)
An alternative to the Cholesky factorization of is to compute the random force vector as an expansion in terms of Chebyshev polynomials [8]. Our recent application of this alternative showed reduction in computational complexity (roughly from a cubic to quadratic dependence on system size), but the benefits are realized on much larger systems than used here [9].
A given conformation for an -bead system is specified by the set of local coordinate systems, , ,
, and the position vector , where the superscript denotes quantities associated with the difference is the collective position vector listing
G65
To simulation the dynamics of the system we use a modified BD method [7] (see below) which involves translating and rotating the particles in the system by discrete steps. Rotation of an orthogonal coordinate system at a constant angular velocity over a finite time step involves a transformation governed by the following system of linear differential equations:
98
98 g
98 W
(43) where are the three components of the angular velocity.
kÏc
We define and note that equation (43) has constant coefficients for a constant rate of angular rotation and has the exact solution:
N
P[<;
N
Pa;
†
P[;
N
P1=>8
P1=>8
P1=>8
(44) where is the finite time step and the rotation operators,
;@?
,
£BA
, and
£BC
, are defined as follows:
†
! #"
;@?
O! #" "(' )

Chromatin Modeling — supplement
O! #"
†
"(' )
"(' ) Z! #"
O! #" "(' )
! #"
12
Z! #" "(' ) O! #" "(' )
! #"
PŠ6
(45)
We modify the second-order BD algorithm of Iniesta and Garcia de la Torre [10] for translational and rotational motions as follows:
1. A first-order estimate of the rotational and translational configuration update is made. The rotational update is calculated by:
ED
¿HG
ED
¿JI
ED (46)
ED where , system , ,
ED variance
ED , denote finite rotations of the th particle about its local coordinate
. The random torques are chose from zero-mean Gaussian distributions with
¿HG ¿HG
·HL
·HM
îí
îí
·M
îí
(47) where
îí is the Kroneker delta. Using equations (44) and (45) we rotate the the local coordinate systems according to:
S»
O;
S»
O;
S»
7î …†
;
(48)
The asterisk superscript again denotes here the first-order estimate. The tilde notation is used to specify that these estimates of the coordinate axes are made based on the rotation step alone. A further modification (described below) of the local coordinate axes is associated with the translation step due to the constraint that the DNA beads rotate only about the axes.

Chromatin Modeling — supplement 13
2. The first-order translational update is given by:
S»
.4P
(49) where denotes a random move chosen from a Gaussian distribution with zero mean and covariance structure,
îí
(50)
Here is the configuration-dependent diffusion matrix calculated for the configuration and
îí is the Kroneker delta. The vector is the systematic force at the th time step.
3. To enforce the constraint that the linker DNA beads are free to rotate only about the axes, we recom-
S» pute the local coordinate systems of the particles after the position vector has been calculated.
Namely, for all linker beads, we calculate according to its definition in equation (1):
S» S» S» S»
O PSR%T Z for
GUBDE
S»
S»
S»
V
V
S»
! #"%$\
S»
! #"%$\
S»
"-' )H$
S»
"-' )H$
.Y/1
.Y/1
S»
S»
O
O
S»
S» for
GUBDC
S» S»
We then define translation step), the new displacements are calculated [4]:
(51)
. Since we require all rotations about to vanish (for the
S» S»
(52)
S» S» and then
RQ dicular to
S»
(see Figure 1):
. Then
S» is determined as the component of
SQ
S» perpen-
S» S» S» S»
S»
S» S» S» S»
(53)
Finally,
S» can be calculated from the cross product:
S» S» S»
(54)
S»
4. The first-order estimate of the configuration at time the local coordinate systems ing at the end of the
S» S» S»
:P
† (given by the position vector time step. These forces and torques, denoted by are used to construct an explicit second-order coordinate update: and
) yields an estimate of the forces and torques act-
S»
S» S» and
S»
The finite rotations are given by
†
S»
ED
PSR
¿HG
,
†
S»
PSR
ED
†
S»
PSR
ED (55)

Chromatin Modeling — supplement 14 and the the position update is given by:
S»
.3P
(56)
Again, the local coordinate systems of the DNA beads are rotated according to the procedure outlined in equations (51)-(54).
We chose reasonable parameter values from the literature wherever possible. Table 1 lists the elastic and geometric parameters used in the model. The electrostatic parameters discussed above are derived based on a detailed analysis of the solvated core particle [11]. A reasonably accepted value for the twisting energy
{ constant is
6y?U
:D?
erg nm [12], which corresponds to a twisting persistence length of about
75 nm. The bending rigidity constant is calculated from equation (12) based on the persistence length of
{ú
³æ69=ˆ5:D?
DNA which is set here to be 50 nm, which yields erg. This value is consistent with recent measures of the non-electrostatic contribution to the bending rigidity (e.g., the high-salt limit of the persistence length) [13].
Excluded volume parameters , , and r#‚Qƒ are the only free parameters used in the model. We have shown [11] that the Debye-H¨uckel electrostatic potential is accurate at distances of greater than 1 nm at wX±
?´6y?
the salt concentration of M. Thus setting the core/core excluded volume distance parameter nm, ensures that the potential remains fairly accurate and the charges are not allowed to overlap.
The linker/core distance parameter is set to nm, to account for the approximate radius of double helical DNA.
[1] K. Luger, A. W. Mader, R. K. Richmond, D. F. Sargent, T. J. Richmond, Nature 389, 251 (1997).
[2] H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980).
[3] T. Schlick, Curr. Opin. Struct. Biol. 5, 245 (1995).
[4] H. Jian, A. Vologodskii, T. Schlick, J. Comp. Phys. 136, 168 (1997).
[5] D. Stigter, Biopolymers 16, 1435 (1977).
[6] J. Garcia de la Torre and V. A. Bloomfield, Biopolymers 16, 1747 (1977).
[7] D. L. Ermak and J. A. McCammon, J. Chem. Phys. 69, 1352 (1978).
[8] M. Fixman, Macromolecules 19, 1204 (1986).
[9] T. Schlick, D. A. Beard, J. Huang, D. Strahs, X. Qian, IEEE Comp. Sci. Eng. (to appear) (2000).
[10] A. Iniesta and J. G. de la Torre, J. Chem. Phys. 92, 2015 (1990).
[11] D. A. Beard and T. Schlick, Biopolymers (in press) (2000).
[12] P. J. Heath, J. B. Clendenning, B. S. Fujimoto, J. M. Schurr, J. Mol. Biol. 260, 718 (1996).
[13] C. G. Baumann, S. B. Smith, V. A. Bloomfield, C. Bustamante, PNAS, USA 94, 6185 (1997).