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"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c
:link(lws,http://lammps.sandia.gov)
:link(ld,Manual.html)
:link(lc,Commands_all.html)
:line
fix nve/dot command :h3
[Syntax:]
fix ID group-ID nve/dot :pre
ID, group-ID are documented in "fix"_fix.html command :ulb,l
nve/dot = style name of this fix command :l
:ule
[Examples:]
fix 1 all nve/dot :pre
[Description:]
Apply a rigid-body integrator as described in "(Davidchack)"_#Davidchack1
to a group of atoms, but without Langevin dynamics.
This command performs Molecular dynamics (MD)
via a velocity-Verlet algorithm and an evolution operator that rotates
the quaternion degrees of freedom, similar to the scheme outlined in "(Miller)"_#Miller1.
This command is the equivalent of the "fix nve/dotc/langevin"_fix_nve_dotc_langevin.html
without damping and noise and can be used to determine the stability range
in a NVE ensemble prior to using the Langevin-type DOTC-integrator
(see also "fix nve/dotc/langevin"_fix_nve_dotc_langevin.html).
The command is equivalent to the "fix nve"_fix_nve.html.
The particles are always considered to have a finite size.
An example input file can be found in /examples/USER/cgdna/examples/duplex1/.
Further details of the implementation and stability of the integrator are contained in "(Henrich)"_#Henrich3.
The preprint version of the article can be found "here"_PDF/USER-CGDNA.pdf.
:line
[Restrictions:]
These pair styles can only be used if LAMMPS was built with the
"USER-CGDNA"_Package_details.html#PKG-USER-CGDNA package and the MOLECULE and ASPHERE package.
See the "Build package"_Build_package.html doc page for more info.
[Related commands:]
"fix nve/dotc/langevin"_fix_nve_dotc_langevin.html, "fix nve"_fix_nve.html
[Default:] none
:line
:link(Davidchack1)
[(Davidchack)] R.L Davidchack, T.E. Ouldridge, and M.V. Tretyakov. J. Chem. Phys. 142, 144114 (2015).
:link(Miller1)
[(Miller)] T. F. Miller III, M. Eleftheriou, P. Pattnaik, A. Ndirango, G. J. Martyna, J. Chem. Phys., 116, 8649-8659 (2002).
:link(Henrich3)
[(Henrich)] O. Henrich, Y. A. Gutierrez-Fosado, T. Curk, T. E. Ouldridge, Eur. Phys. J. E 41, 57 (2018).

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"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c
:link(lws,http://lammps.sandia.gov)
:link(ld,Manual.html)
:link(lc,Commands_all.html)
:line
fix nve/dotc/langevin command :h3
[Syntax:]
fix ID group-ID nve/dotc/langevin Tstart Tstop damp seed keyword value :pre
ID, group-ID are documented in "fix"_fix.html command :ulb,l
nve/dotc/langevin = style name of this fix command :l
Tstart,Tstop = desired temperature at start/end of run (temperature units) :l
damp = damping parameter (time units) :l
seed = random number seed to use for white noise (positive integer) :l
keyword = {angmom} :l
{angmom} value = factor
factor = do thermostat rotational degrees of freedom via the angular momentum and apply numeric scale factor as discussed below :pre
:ule
[Examples:]
fix 1 all nve/dotc/langevin 1.0 1.0 0.03 457145 angmom 10
fix 1 all nve/dotc/langevin 0.1 0.1 78.9375 457145 angmom 10 :pre
[Description:]
Apply a rigid-body Langevin-type integrator of the kind "Langevin C"
as described in "(Davidchack)"_#Davidchack2
to a group of atoms, which models an interaction with an implicit background
solvent. This command performs Brownian dynamics (BD)
via a technique that splits the integration into a deterministic Hamiltonian
part and the Ornstein-Uhlenbeck process for noise and damping.
The quaternion degrees of freedom are updated though an evolution
operator which performs a rotation in quaternion space, preserves
the quaternion norm and is akin to "(Miller)"_#Miller2.
In terms of syntax this command has been closely modelled on the
"fix langevin"_fix_langevin.html and its {angmom} option. But it combines
the "fix nve"_fix_nve.html and the "fix langevin"_fix_langevin.html in
one single command. The main feature is improved stability
over the standard integrator, permitting slightly larger timestep sizes.
NOTE: Unlike the "fix langevin"_fix_langevin.html this command performs
also time integration of the translational and quaternion degrees of freedom.
The total force on each atom will have the form:
F = Fc + Ff + Fr
Ff = - (m / damp) v
Fr is proportional to sqrt(Kb T m / (dt damp)) :pre
Fc is the conservative force computed via the usual inter-particle
interactions ("pair_style"_pair_style.html,
"bond_style"_bond_style.html, etc).
The Ff and Fr terms are implicitly taken into account by this fix
on a per-particle basis.
Ff is a frictional drag or viscous damping term proportional to the
particle's velocity. The proportionality constant for each atom is
computed as m/damp, where m is the mass of the particle and damp is
the damping factor specified by the user.
Fr is a force due to solvent atoms at a temperature T randomly bumping
into the particle. As derived from the fluctuation/dissipation
theorem, its magnitude as shown above is proportional to sqrt(Kb T m /
dt damp), where Kb is the Boltzmann constant, T is the desired
temperature, m is the mass of the particle, dt is the timestep size,
and damp is the damping factor. Random numbers are used to randomize
the direction and magnitude of this force as described in
"(Dunweg)"_#Dunweg3, where a uniform random number is used (instead of
a Gaussian random number) for speed.
:line
{Tstart} and {Tstop} have to be constant values, i.e. they cannot
be variables. If used together with the oxDNA force field for
coarse-grained simulation of DNA please note that T = 0.1 in oxDNA units
corresponds to T = 300 K.
The {damp} parameter is specified in time units and determines how
rapidly the temperature is relaxed. For example, a value of 0.03
means to relax the temperature in a timespan of (roughly) 0.03 time
units tau (see the "units"_units.html command).
The damp factor can be thought of as inversely related to the
viscosity of the solvent, i.e. a small relaxation time implies a
hi-viscosity solvent and vice versa. See the discussion about gamma
and viscosity in the documentation for the "fix
viscous"_fix_viscous.html command for more details.
Note that the value 78.9375 in the second example above corresponds
to a diffusion constant, which is about an order of magnitude larger
than realistic ones. This has been used to sample configurations faster
in Brownian dynamics simulations.
The random # {seed} must be a positive integer. A Marsaglia random
number generator is used. Each processor uses the input seed to
generate its own unique seed and its own stream of random numbers.
Thus the dynamics of the system will not be identical on two runs on
different numbers of processors.
The keyword/value option has to be used in the following way:
This fix has to be used together with the {angmom} keyword. The
particles are always considered to have a finite size.
The keyword {angmom} enables thermostatting of the rotational degrees of
freedom in addition to the usual translational degrees of freedom.
The scale factor after the {angmom} keyword gives the ratio of the rotational to
the translational friction coefficient.
An example input file can be found in /examples/USER/cgdna/examples/duplex2/.
Further details of the implementation and stability of the integrators are contained in "(Henrich)"_#Henrich4.
The preprint version of the article can be found "here"_PDF/USER-CGDNA.pdf.
:line
[Restrictions:]
These pair styles can only be used if LAMMPS was built with the
"USER-CGDNA"_Package_details.html#PKG-USER-CGDNA package and the MOLECULE and ASPHERE package.
See the "Build package"_Build_package.html doc page for more info.
[Related commands:]
"fix nve"_fix_nve.html, "fix langevin"_fix_langevin.html, "fix nve/dot"_fix_nve_dot.html, "bond_style oxdna/fene"_bond_oxdna.html, "bond_style oxdna2/fene"_bond_oxdna.html, "pair_style oxdna/excv"_pair_oxdna.html, "pair_style oxdna2/excv"_pair_oxdna2.html
[Default:] none
:line
:link(Davidchack2)
[(Davidchack)] R.L Davidchack, T.E. Ouldridge, M.V. Tretyakov. J. Chem. Phys. 142, 144114 (2015).
:link(Miller2)
[(Miller)] T. F. Miller III, M. Eleftheriou, P. Pattnaik, A. Ndirango, G. J. Martyna, J. Chem. Phys., 116, 8649-8659 (2002).
:link(Dunweg3)
[(Dunweg)] B. Dunweg, W. Paul, Int. J. Mod. Phys. C, 2, 817-27 (1991).
:link(Henrich4)
[(Henrich)] O. Henrich, Y. A. Gutierrez-Fosado, T. Curk, T. E. Ouldridge, Eur. Phys. J. E 41, 57 (2018).

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"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c
:link(lws,http://lammps.sandia.gov)
:link(ld,Manual.html)
:link(lc,Commands_all.html)
:line
pair_style oxdna/excv command :h3
pair_style oxdna/stk command :h3
pair_style oxdna/hbond command :h3
pair_style oxdna/xstk command :h3
pair_style oxdna/coaxstk command :h3
[Syntax:]
pair_style style1 :pre
pair_coeff * * style2 args :pre
style1 = {hybrid/overlay oxdna/excv oxdna/stk oxdna/hbond oxdna/xstk oxdna/coaxstk} :ul
style2 = {oxdna/excv} or {oxdna/stk} or {oxdna/hbond} or {oxdna/xstk} or {oxdna/coaxstk}
args = list of arguments for these particular styles :ul
{oxdna/stk} args = seq T xi kappa 6.0 0.4 0.9 0.32 0.75 1.3 0 0.8 0.9 0 0.95 0.9 0 0.95 2.0 0.65 2.0 0.65
seq = seqav (for average sequence stacking strength) or seqdep (for sequence-dependent stacking strength)
T = temperature (oxDNA units, 0.1 = 300 K)
xi = temperature-independent coefficient in stacking strength
kappa = coefficient of linear temperature dependence in stacking strength
{oxdna/hbond} args = seq eps 8.0 0.4 0.75 0.34 0.7 1.5 0 0.7 1.5 0 0.7 1.5 0 0.7 0.46 3.141592653589793 0.7 4.0 1.5707963267948966 0.45 4.0 1.5707963267948966 0.45
seq = seqav (for average sequence base-pairing strength) or seqdep (for sequence-dependent base-pairing strength)
eps = 1.077 (between base pairs A-T and C-G) or 0 (all other pairs) :pre
[Examples:]
pair_style hybrid/overlay oxdna/excv oxdna/stk oxdna/hbond oxdna/xstk oxdna/coaxstk
pair_coeff * * oxdna/excv 2.0 0.7 0.675 2.0 0.515 0.5 2.0 0.33 0.32
pair_coeff * * oxdna/stk seqdep 0.1 1.3448 2.6568 6.0 0.4 0.9 0.32 0.75 1.3 0 0.8 0.9 0 0.95 0.9 0 0.95 2.0 0.65 2.0 0.65
pair_coeff * * oxdna/hbond seqdep 0.0 8.0 0.4 0.75 0.34 0.7 1.5 0 0.7 1.5 0 0.7 1.5 0 0.7 0.46 3.141592653589793 0.7 4.0 1.5707963267948966 0.45 4.0 1.5707963267948966 0.45
pair_coeff 1 4 oxdna/hbond seqdep 1.077 8.0 0.4 0.75 0.34 0.7 1.5 0 0.7 1.5 0 0.7 1.5 0 0.7 0.46 3.141592653589793 0.7 4.0 1.5707963267948966 0.45 4.0 1.5707963267948966 0.45
pair_coeff 2 3 oxdna/hbond seqdep 1.077 8.0 0.4 0.75 0.34 0.7 1.5 0 0.7 1.5 0 0.7 1.5 0 0.7 0.46 3.141592653589793 0.7 4.0 1.5707963267948966 0.45 4.0 1.5707963267948966 0.45
pair_coeff * * oxdna/xstk 47.5 0.575 0.675 0.495 0.655 2.25 0.791592653589793 0.58 1.7 1.0 0.68 1.7 1.0 0.68 1.5 0 0.65 1.7 0.875 0.68 1.7 0.875 0.68
pair_coeff * * oxdna/coaxstk 46.0 0.4 0.6 0.22 0.58 2.0 2.541592653589793 0.65 1.3 0 0.8 0.9 0 0.95 0.9 0 0.95 2.0 -0.65 2.0 -0.65 :pre
[Description:]
The {oxdna} pair styles compute the pairwise-additive parts of the oxDNA force field
for coarse-grained modelling of DNA. The effective interaction between the nucleotides consists of potentials for the
excluded volume interaction {oxdna/excv}, the stacking {oxdna/stk}, cross-stacking {oxdna/xstk}
and coaxial stacking interaction {oxdna/coaxstk} as well
as the hydrogen-bonding interaction {oxdna/hbond} between complementary pairs of nucleotides on
opposite strands. Average sequence or sequence-dependent stacking and base-pairing strengths
are supported "(Sulc)"_#Sulc1. Quasi-unique base-pairing between nucleotides can be achieved by using
more complementary pairs of atom types like 5-8 and 6-7, 9-12 and 10-11, 13-16 and 14-15, etc.
This prevents the hybridization of in principle complementary bases within Ntypes/4 bases
up and down along the backbone.
The exact functional form of the pair styles is rather complex.
The individual potentials consist of products of modulation factors,
which themselves are constructed from a number of more basic potentials
(Morse, Lennard-Jones, harmonic angle and distance) as well as quadratic smoothing and modulation terms.
We refer to "(Ouldridge-DPhil)"_#Ouldridge-DPhil1 and "(Ouldridge)"_#Ouldridge1
for a detailed description of the oxDNA force field.
NOTE: These pair styles have to be used together with the related oxDNA bond style
{oxdna/fene} for the connectivity of the phosphate backbone (see also documentation of
"bond_style oxdna/fene"_bond_oxdna.html). Most of the coefficients
in the above example have to be kept fixed and cannot be changed without reparameterizing the entire model.
Exceptions are the first four coefficients after {oxdna/stk} (seq=seqdep, T=0.1, xi=1.3448 and kappa=2.6568 in the above example)
and the first coefficient after {oxdna/hbond} (seq=seqdep in the above example).
When using a Langevin thermostat, e.g. through "fix langevin"_fix_langevin.html
or "fix nve/dotc/langevin"_fix_nve_dotc_langevin.html
the temperature coefficients have to be matched to the one used in the fix.
Example input and data files for DNA duplexes can be found in examples/USER/cgdna/examples/oxDNA/ and /oxDNA2/.
A simple python setup tool which creates single straight or helical DNA strands,
DNA duplexes or arrays of DNA duplexes can be found in examples/USER/cgdna/util/.
Please cite "(Henrich)"_#Henrich1 and the relevant oxDNA articles in any publication that uses this implementation.
The article contains more information on the model, the structure of the input file, the setup tool
and the performance of the LAMMPS-implementation of oxDNA.
The preprint version of the article can be found "here"_PDF/USER-CGDNA.pdf.
:line
[Restrictions:]
These pair styles can only be used if LAMMPS was built with the
USER-CGDNA package and the MOLECULE and ASPHERE package. See the
"Build package"_Build_package.html doc page for more info.
[Related commands:]
"bond_style oxdna/fene"_bond_oxdna.html, "fix nve/dotc/langevin"_fix_nve_dotc_langevin.html, "pair_coeff"_pair_coeff.html,
"bond_style oxdna2/fene"_bond_oxdna.html, "pair_style oxdna2/excv"_pair_oxdna2.html
[Default:] none
:line
:link(Henrich1)
[(Henrich)] O. Henrich, Y. A. Gutierrez-Fosado, T. Curk, T. E. Ouldridge, Eur. Phys. J. E 41, 57 (2018).
:link(Sulc1)
[(Sulc)] P. Sulc, F. Romano, T.E. Ouldridge, L. Rovigatti, J.P.K. Doye, A.A. Louis, J. Chem. Phys. 137, 135101 (2012).
:link(Ouldridge-DPhil1)
[(Ouldrigde-DPhil)] T.E. Ouldridge, Coarse-grained modelling of DNA and DNA self-assembly, DPhil. University of Oxford (2011).
:link(Ouldridge1)
[(Ouldridge)] T.E. Ouldridge, A.A. Louis, J.P.K. Doye, J. Chem. Phys. 134, 085101 (2011).

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"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c
:link(lws,http://lammps.sandia.gov)
:link(ld,Manual.html)
:link(lc,Commands_all.html)
:line
pair_style oxdna2/excv command :h3
pair_style oxdna2/stk command :h3
pair_style oxdna2/hbond command :h3
pair_style oxdna2/xstk command :h3
pair_style oxdna2/coaxstk command :h3
pair_style oxdna2/dh command :h3
[Syntax:]
pair_style style1 :pre
pair_coeff * * style2 args :pre
style1 = {hybrid/overlay oxdna2/excv oxdna2/stk oxdna2/hbond oxdna2/xstk oxdna2/coaxstk oxdna2/dh} :ul
style2 = {oxdna2/excv} or {oxdna2/stk} or {oxdna2/hbond} or {oxdna2/xstk} or {oxdna2/coaxstk} or {oxdna2/dh}
args = list of arguments for these particular styles :ul
{oxdna2/stk} args = seq T xi kappa 6.0 0.4 0.9 0.32 0.75 1.3 0 0.8 0.9 0 0.95 0.9 0 0.95 2.0 0.65 2.0 0.65
seq = seqav (for average sequence stacking strength) or seqdep (for sequence-dependent stacking strength)
T = temperature (oxDNA units, 0.1 = 300 K)
xi = temperature-independent coefficient in stacking strength
kappa = coefficient of linear temperature dependence in stacking strength
{oxdna2/hbond} args = seq eps 8.0 0.4 0.75 0.34 0.7 1.5 0 0.7 1.5 0 0.7 1.5 0 0.7 0.46 3.141592653589793 0.7 4.0 1.5707963267948966 0.45 4.0 1.5707963267948966 0.45
seq = seqav (for average sequence base-pairing strength) or seqdep (for sequence-dependent base-pairing strength)
eps = 1.0678 (between base pairs A-T and C-G) or 0 (all other pairs)
{oxdna2/dh} args = T rhos qeff
T = temperature (oxDNA units, 0.1 = 300 K)
rhos = salt concentration (mole per litre)
qeff = effective charge (elementary charges) :pre
[Examples:]
pair_style hybrid/overlay oxdna2/excv oxdna2/stk oxdna2/hbond oxdna2/xstk oxdna2/coaxstk oxdna2/dh
pair_coeff * * oxdna2/excv 2.0 0.7 0.675 2.0 0.515 0.5 2.0 0.33 0.32
pair_coeff * * oxdna2/stk seqdep 0.1 1.3523 2.6717 6.0 0.4 0.9 0.32 0.75 1.3 0 0.8 0.9 0 0.95 0.9 0 0.95 2.0 0.65 2.0 0.65
pair_coeff * * oxdna2/hbond seqdep 0.0 8.0 0.4 0.75 0.34 0.7 1.5 0 0.7 1.5 0 0.7 1.5 0 0.7 0.46 3.141592653589793 0.7 4.0 1.5707963267948966 0.45 4.0 1.5707963267948966 0.45
pair_coeff 1 4 oxdna2/hbond seqdep 1.0678 8.0 0.4 0.75 0.34 0.7 1.5 0 0.7 1.5 0 0.7 1.5 0 0.7 0.46 3.141592653589793 0.7 4.0 1.5707963267948966 0.45 4.0 1.5707963267948966 0.45
pair_coeff 2 3 oxdna2/hbond seqdep 1.0678 8.0 0.4 0.75 0.34 0.7 1.5 0 0.7 1.5 0 0.7 1.5 0 0.7 0.46 3.141592653589793 0.7 4.0 1.5707963267948966 0.45 4.0 1.5707963267948966 0.45
pair_coeff * * oxdna2/xstk 47.5 0.575 0.675 0.495 0.655 2.25 0.791592653589793 0.58 1.7 1.0 0.68 1.7 1.0 0.68 1.5 0 0.65 1.7 0.875 0.68 1.7 0.875 0.68
pair_coeff * * oxdna2/coaxstk 58.5 0.4 0.6 0.22 0.58 2.0 2.891592653589793 0.65 1.3 0 0.8 0.9 0 0.95 0.9 0 0.95 40.0 3.116592653589793
pair_coeff * * oxdna2/dh 0.1 1.0 0.815 :pre
[Description:]
The {oxdna2} pair styles compute the pairwise-additive parts of the oxDNA force field
for coarse-grained modelling of DNA. The effective interaction between the nucleotides consists of potentials for the
excluded volume interaction {oxdna2/excv}, the stacking {oxdna2/stk}, cross-stacking {oxdna2/xstk}
and coaxial stacking interaction {oxdna2/coaxstk}, electrostatic Debye-Hueckel interaction {oxdna2/dh}
as well as the hydrogen-bonding interaction {oxdna2/hbond} between complementary pairs of nucleotides on
opposite strands. Average sequence or sequence-dependent stacking and base-pairing strengths
are supported "(Sulc)"_#Sulc2. Quasi-unique base-pairing between nucleotides can be achieved by using
more complementary pairs of atom types like 5-8 and 6-7, 9-12 and 10-11, 13-16 and 14-15, etc.
This prevents the hybridization of in principle complementary bases within Ntypes/4 bases
up and down along the backbone.
The exact functional form of the pair styles is rather complex.
The individual potentials consist of products of modulation factors,
which themselves are constructed from a number of more basic potentials
(Morse, Lennard-Jones, harmonic angle and distance) as well as quadratic smoothing and modulation terms.
We refer to "(Snodin)"_#Snodin and the original oxDNA publications "(Ouldridge-DPhil)"_#Ouldridge-DPhil2
and "(Ouldridge)"_#Ouldridge2 for a detailed description of the oxDNA2 force field.
NOTE: These pair styles have to be used together with the related oxDNA2 bond style
{oxdna2/fene} for the connectivity of the phosphate backbone (see also documentation of
"bond_style oxdna2/fene"_bond_oxdna.html). Most of the coefficients
in the above example have to be kept fixed and cannot be changed without reparameterizing the entire model.
Exceptions are the first four coefficients after {oxdna2/stk} (seq=seqdep, T=0.1, xi=1.3523 and kappa=2.6717 in the above example),
the first coefficient after {oxdna2/hbond} (seq=seqdep in the above example) and the three coefficients
after {oxdna2/dh} (T=0.1, rhos=1.0, qeff=0.815 in the above example). When using a Langevin thermostat
e.g. through "fix langevin"_fix_langevin.html or "fix nve/dotc/langevin"_fix_nve_dotc_langevin.html
the temperature coefficients have to be matched to the one used in the fix.
Example input and data files for DNA duplexes can be found in examples/USER/cgdna/examples/oxDNA/ and /oxDNA2/.
A simple python setup tool which creates single straight or helical DNA strands,
DNA duplexes or arrays of DNA duplexes can be found in examples/USER/cgdna/util/.
Please cite "(Henrich)"_#Henrich and the relevant oxDNA articles in any publication that uses this implementation.
The article contains more information on the model, the structure of the input file, the setup tool
and the performance of the LAMMPS-implementation of oxDNA.
The preprint version of the article can be found "here"_PDF/USER-CGDNA.pdf.
:line
[Restrictions:]
These pair styles can only be used if LAMMPS was built with the
USER-CGDNA package and the MOLECULE and ASPHERE package. See the
"Build package"_Build_package.html doc page for more info.
[Related commands:]
"bond_style oxdna2/fene"_bond_oxdna.html, "fix nve/dotc/langevin"_fix_nve_dotc_langevin.html, "pair_coeff"_pair_coeff.html,
"bond_style oxdna/fene"_bond_oxdna.html, "pair_style oxdna/excv"_pair_oxdna.html
[Default:] none
:line
:link(Henrich)
[(Henrich)] O. Henrich, Y. A. Gutierrez-Fosado, T. Curk, T. E. Ouldridge, Eur. Phys. J. E 41, 57 (2018).
:link(Sulc2)
[(Sulc)] P. Sulc, F. Romano, T.E. Ouldridge, L. Rovigatti, J.P.K. Doye, A.A. Louis, J. Chem. Phys. 137, 135101 (2012).
:link(Snodin)
[(Snodin)] B.E. Snodin, F. Randisi, M. Mosayebi, et al., J. Chem. Phys. 142, 234901 (2015).
:link(Ouldridge-DPhil2)
[(Ouldrigde-DPhil)] T.E. Ouldridge, Coarse-grained modelling of DNA and DNA self-assembly, DPhil. University of Oxford (2011).
:link(Ouldridge2)
[(Ouldridge)] T.E. Ouldridge, A.A. Louis, J.P.K. Doye, J. Chem. Phys. 134, 085101 (2011).