Progress of Theoretical Physics Supplement No. 191, 2011
135
GPU Based Molecular Dynamics Simulations of Polymer Rings
in Concentrated Solution: Structure and Scaling
Daniel Reith,1,∗) Leonid Mirny2,∗∗) and Peter Virnau1,∗∗∗)
1 Institut
für Physik, Johannes Gutenberg-Universtität Mainz,
Staudingerweg 7, 55128 Mainz, Germany
2 Harvard-MIT Division of Health Sciences and Technology,
and Department of Physics, Massachusetts Institute of Technology,
Cambridge, MA, USA
(Received May 8, 2011)
We report on equilibrium properties of a concentrated solution of non-concatenated ring
polymers by Molecular dynamics simulations using HooMD-blue, a fast implementation on
graphics processor units (GPUs). We are able to identify the intermediate scaling regime for
the radius of gyration Rg ∝ N 0.4 as well as indication for a crossover to Rg ∝ N 1/3 for rings
with chain length N in our fully flexible off-lattice polymer model. This crossover takes place
between a ring size of 2500 and 7500 monomers for monomer density ρ = 0.5. Our results are
in agreement with recent studies for lattice and stiff off-lattice models and show once again
that this scaling is not model dependent at all. Furthermore this study demonstrates that
GPUs are suited to tackle large scale problems in computational polymer science. To this
end we present performance measurements relating the computing performance of a single
NVidias GTX580 consumer graphic card to conventional state of the art supercomputers.
§1.
Introduction
The dynamics in a polymer melt is driven by the reptation mechanism.1) Thereby
a polymer chain moves along its contour in a snake-like motion because every other
motion is hindered by entanglements imposed by neighboring chains. The lack of
this mechanism in ring polymers changes the static and dynamic behavior dramatically.2)–22) Furthermore, it is impossible to change self-entanglements (knots) of a
ring polymer without breaking bonds. The typical restriction to unknotted loops in
this context corresponds to a constriction of phase space to a specific topological subclass. Knots also provide a direct measure of entanglements in single polymers,23)–25)
akin to interchain concepts like the primitive path in melts26) and can serve as a
starting point for the understanding of DNA in viral capsids,27), 28) proteins29)–34)
and synthetic polymers.23), 35) Again, the presence or absence of such entanglements
influences both static and dynamic properties.
While in a melt of open chains the radius of gyration in principle scales like a
random walk (Rg ∝ N 1/2 ) in the asymptotic limit, there are three different scaling
regimes8), 9), 13), 18), 19), 21), 36) in a melt or concentrated solution of non-concatenated,
unknotted rings. For very small polymer rings, Rg ∝ N 1/2 , which is followed by an
∗)
∗∗)
∗∗∗)
E-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
136
D. Reith, L. Mirny and P. Virnau
intermediate region with Rg ∝ N 2/5 and finally a crossover to Rg ∝ N 1/3 . Computer
simulations of lattice models first showed evidence of this transition around 10 years
ago13) and recently a first off-lattice simulation of a semiflexible polymer model
confirmed these scaling laws.21), 22) In this manuscript we investigate a fully flexible
off-lattice model. While the intermediate 2/5 region is consistent with a Flory-like
free energy minimization in which each topological constraint is penalized by kB T ,
the asymptotic 1/3 scaling is consistent with the fractal globule concept introduced
by Grosberg et al. in the late 1980s.37) Grosberg also concluded that DNA has to
be organized in exactly this structure in order to remain functional.38)
In the interphase (between two cell divisions) chromosomes are localized in socalled chromosome territories in the eukaryotic cells,39)–41) which can be visualized
with experimental techniques like Fluorescence In Situ Hybridization (FISH).42)
New experimental techniques like HiC43) allow for the visualization of the threedimensional architecture of whole genomes by constructing spatial proximity maps
with a resolution down to 1 megabase. With this spatial proximity maps it is possible to calculate the probabilities for two loci being in contact, which are separated
by a distance s along the genome. Several studies try to explain the development
of these chromosome territories in the context of polymer physics.44)–49) While linear chromosomes may simply not have enough time to fully equilibrate as argued in
Ref. 45), an analogy can be drawn to melts of rings which segregate due to topological
constraints.
In the following we report on simulations of fully flexible coarse-grained ring
polymers in the semi-dilute regime. We see indication for a crossover from the
intermediate ν = 2/5 to the asymptotic ν = 1/3 scaling in the Flory exponent and
analyse structures in both regimes. All calculations were notably done on low-budget
graphics processor units (GPUs) the performance of which we compare to state of
the art supercomputers in the second part of the paper.
§2. Model and simulation techniques
Our model system consists of simple flexible Lennard-Jones + FENE homopolymers23), 50), 51) with cut and shifted Lennard-Jones interactions between all monomers
VLJ =
12 6
− σr +
4 σr
127
16384
0,
,
if r < rcut
otherwise
,
(2.1)
and FENE interactions between adjacent beads:
r 2 VF EN E = −33.75 ln 1 −
.
1.5σ
(2.2)
Polymer Rings in Concentrated Solutions
137
Table I. Parameters of the simulated systems. All boxes are of cubic size L × L × L. We used
periodic boundary conditions in all three directions.
type
rings
chain lengthN number of chains Nc number of monomers Ntotal side length L
250
4000
100000
58.48
500
2000
100000
58.48
1000
108
108000
60.
2500
100
250000
79.37
5000
100
500000
100.
7500
100
750000
114.472
10000
100
1000000
125.992
open chains
1000
1000
1000000
125.992
√
Unless noted, rcut = 2 6 2. The single chain simulations took place at T =
4 /kB , a temperature significantly above the θ-temperature TΘ = 3.2 /kB (for
a chain of infinite length52) ), to ensure sufficient interpenetration. Note that at
these high temperatures, the attractive part of the Lennard-Jones potential could in
principle be omitted to further speed up the computations. The molecular dynamics
simulations were performed using the GPU accelerated code HooMD blue revision
357453) in the NVT ensemble with a timestep of dt = 0.002 for the production and
dt = 0.004 for the√equilibration runs. We used the included DPD-thermostat54)
with rcut,DP D = 2 6 2 and γ = 0.5. Note that all calculations were done using the
GPU’s single-precision arithmetic. All simulations started from globular single loop
configurations, which were placed in the simulation box. For N = 5000, 7500 and
10000 we prepared additional starting configurations from the final configurations
for N = 2500 and N = 5000 to check for equilibrium. (Volume and loops were scaled
up from N = 2500 and N = 5000 and additional particles were placed in between
neighboring particles.) The potentials of Eqs. (2.1) and (2.2) were originally designed
to prevent bond crossings. However, the probability of crossing a bond is small but
non-zero and does depend on time step and details of the thermostat. To address
this concern, we performed a primitive path analysis26) of the final configuration for
N = 5000, which ran for more than three months. Unfortunately, we did discover a
single link between two rings. A knot analysis,55) however, did not reveal any knots.
Hence we do not believe that this single entanglement (which probably occurred in
the initial relaxation phase) significantly alters our results. Further details on the
specific systems can be found in Table I. A crucial parameter of the model is the
entanglement length. This length can at least for open chains be estimated according
to Ref. 26) via an empirical relation between packing and entanglement length
Ne = p3
4
.
5 × 0.00226
(2.3)
The packing length is defined as p = N Re−2 /ρ, where Re2 is the squared end-to-end
vector, N the number of monomers per chain and ρ the overall monomer density.
Putting in the numbers for the open chain simulation we obtain a value of roughly
Ne ≈ 400. Therefore, the biggest rings of size N = 10000 considered correspond
approximately to 25Ne of the corresponding linear polymer at the same density.
138
D. Reith, L. Mirny and P. Virnau
Fig. 1. Squared radius of gyration Rg2 over time t for different ring sizes. N = 7500 and N = 10000
are not fully equilibrated, yet, and only the last few configurations were used for analysis. The
three time series on the right-hand side of the graph are runs starting from alternative starting
configurations as described in the main text.
Fig. 2. Squared radius of gyration Rg2 as a function of chain length N in double logarithmic
representation. The two straight lines ∝ N 0.8 and ∝ N 2/3 are guides to the eyes only to illustrate
the intermediate and the asymptotic scaling regime. The additional squares at N = 5000, 7500
and 10000 denote the averages from runs with alternative initial configurations. Note that
N = 7500 and N = 10000 are not fully equilibrated, yet.
§3.
Results and discussion
One of the central static properties of concentrated solutions or melts of polymers
is the radius of gyration of the polymer which measures the size of a single chain.
Figure 1 shows the squared radius of gyration Rg2 as a function of time to determine
if chains are equilibrated. At each time the average is calculated over all rings in
the simulation box. While the systems appear to be equilibrated for ring sizes up
to 5000 monomers, the plateau is not reached yet for ring sizes of 7500 and 10000
monomers. Our control runs from our “blown-up” configurations (to the right of
Fig. 1) yield the same results for N = 5000 and similar results for the not yet
fully equilibrated sizes N = 7500 and N = 10000. The scaling of the radius of
gyration Rg is expected to behave like Rg ∝ N 0.4 for the intermediate and like
Polymer Rings in Concentrated Solutions
139
Fig. 3. Self density ρS of a ring as a function of the radial distance d to its center of mass.
Rg ∝ N 1/3 for the asymptotic regime. We observe the intermediate scaling regime
between ring sizes of N = 500 and N = 2500 monomers (Fig. 2). A fit yields an
exponent ν = 0.403 using ring sizes N = 500, 1000, 2500. At ring size N = 5000
monomers the scaling starts to deviate from the ν = 0.4 scaling. Ring sizes of
N = 7500 and N = 10000 are not fully equilibrated, yet, but indicate a crossover
to ν = 1/3. Following Vettorel et al.19) the onset of the asymptotic regime can be
estimated by an argument using the entanglement length Ne of linear chains at the
same density. Topological constraints start to play an important role in melts or
concentrated solutions of open chains on a length scale larger than the entanglement
length Ne . On such a length scale ‘tube defects’ become entropically
√ unfavorable if
the strands leaking out of the chain’s tube are longer than Le = b Ne , where b is
the bond length. Vettorel et al. concluded that large rings favor collapsed states,
because rings cannot reptate out of this situation. Relating the core size rc of such a
collapsed states to Le and assuming rc > Le they expect to observe collapsed states
with ring sizes
4π 3 4π 3
ρLe =
ρb (Ne )3/2 .
(3.1)
N > Nc =
3
3
This would yield Nc ≈ 15000 for our model, assuming that rings squeeze each other
out completely and that at the center of mass of each ring there is no overlap with
the surrounding rings at all. Halverson et al.21) did not see this complete squeeze out
in their simulations and we observe a similar behavior for the case of a fully flexible
model at a lower density. Furthermore they estimated the self-density ρS ≈ 0.23 (for
their largest system) at a melt density of ρ = 0.85 for their data. In other words in
their semi-flexible model at melt density roughly 75 percent of the monomers in the
vicinity of the center of mass have to be attributed to other chains. Figure 3 shows
that we see a similar behavior for our fully flexible model. In our case the self-density
converges at roughly ρ = 0.22 at the center of mass (55 percent of monomers in the
vicinity of the center of mass belong to neighboring chains). Halverson et al.21) again
proposed to include this observation in the arguments of Vettorel et al. and modify
Eq. (3.1) such that the overall density ρ is replaced by the self-density ρS . Hence,
we end up with a new estimate of the critical ring size of Ñc ≈ 6700.
140
D. Reith, L. Mirny and P. Virnau
Fig. 4. (left): Contact probability P (s) as a function of the distance along the chain contour s for
different ring sizes and a concentrated solution of open chains of chain length N = 1000.
(right): Radius of gyration Rg (s)2 of a segment of size s for different ring sizes and a concentrated solution of open chains of chain length N = 1000.
In Fig. 4 we report on internal properties of the rings. The contact probability
of two monomers over the separation of these two monomers along the chain is
plotted in double logarithmic representation (Fig. 4 (left)) and defined
as follows:
√
6
Two monomers are in contact if their distance is smaller than dmin = 2· 2. The data
for open chains of length N = 1000 is also included for comparison in the plot. The
open chain contact probability P (s) ∝ s−3/2 scales nicely with an exponent of −3/2
over two orders of magnitude over the range from 10 to almost 1000 as predicted by
theory and simulations.56), 57) The contact probability of two monomers on a ring,
however, differs already at length scales larger than 10 monomers. Halverson et al.21)
found evidence, that the structure of non-concatenated rings in a melt is consistent
with the model of the fractal globule37) with fractal dimension df = 3. For such
structures the contact probability scales with exponent −1 as recently demonstrated
in Refs. 43) and 57). Theoretically, the fractal globule is only expected for ring
sizes N Ne .37) Unfortunately we are not able to clearly observe the scaling with
exponent γ = −1 in our data, probably because the rings are still too small compared
to the linear entanglement length of Ne ≈ 400. For our largest system (N = 10000,
which is not yet fully equilibrated) γ = −1 scaling starts to emerge for N > 1000.
Right next to the contact probability, Fig. 4 (right) shows the radius of gyration of
sub-segments s as an additional measure for the internal structure of a ring. Again,
data for open chains of length N = 1000 is plotted for comparison. The sub-segment
radius of gyration for open chains in concentrated solutions is expected to show two
different scaling regimes: Rg2 (s) ∝ s2ν with ν ≈ 3/5 if s is smaller than the number
of monomers in a blob and ν = 0.5 if s is larger. For our data, this crossover takes
place at roughly s = 30. The scaling of the sub-segment radius of gyration of the
rings does not exhibit the same behavior. While starting with ν ≈ 3/5, the subsegment radius of gyration appears to grow at a smaller rate for rings than for open
chains. For larger sub-segments we are not able to observe a clear scaling law in
our data. If the rings have a similar structure as a fractal globule, the sub-segment
radius of gyration Rg (s) should scale with ν = 1/3 because a fractale globule is
Polymer Rings in Concentrated Solutions
141
a self-similar structure. As for the contact probability, a ring size of roughly 25Ne
is probably not sufficient to reach the N Ne limit and to observe the expected
scaling.
§4. Performance
Modern Graphics Processing units (GPUs) are able to perform massively parallel scientific computations at low cost, but only few algorithms from the field of
statistical physics have been proven to run efficiently on such devices. Besides calculations for the Ising-model58), 59) and analysis of time series60) a lot of effort has
focussed on porting Molecular dynamics simulations to GPUs.53), 61), 62) In the meantime traditional Molecular dynamics packages for parallel CPUs such as Gromacs,63)
Lammps,64) Espresso65) and NAMD66), 67) have also announced GPU support. In this
section we would like to show that GPUs can be used to simulate systems, which
originally relied on the usage of supercomputing clusters. As a test case we studied a concentrated solution of non-concatenated rings and their scaling behavior on
traditional supercomputers and compare those results with single GPUs.
The system used for the benchmark is almost identical to the biggest system
considered in Ref. 21), a Kremer-Grest (KG) based model.50) With this model, the
authors demonstrated for the first time the discussed scaling relations for an offlattice model. As it also includes a chain stiffness and as the densities are higher
(ρ = 0.85), the largest systems (N = 1600) actually corresponds to 57 entanglement lengths according to Eq. (2.3) and is effectively larger than the biggest system
used by us. Hence, the system is also better suited for identifying scaling relations.
The nonbonded interaction is modeled with the help of a Weeks-Chandler-Anderson
(WCA) potential,
very similar to the one described in Eq. (2.1), but with a smaller
√
6
cutoff rcut = 2 and shifted in such a way, that VW CA (rcut ) = 0 holds. The bonded
interactions are described with the help of the FENE potential in Eq. (2.2). However
the model considered in Ref. 21) is a semi-flexible and not a fully-flexible model as
discussed in the previous section. To this end we introduce a chain stiffness by an
additional angular potential
1
Vangular = k (θ − θ0 )2
2
(4.1)
with k = 1.5 and θ0 = π. Note that this is not exactly identical to the angular potential of Ref. 21) but rather its Taylor expansion up to second order. The
benchmark system consists of 200 rings of size 1600 monomers with a total number
of 320000 monomers in the system. They are packed into a cubic simulation box
with side length 72.207σ with periodic boundary conditions in all three dimensions.
This packing corresponds to a monomer density of ρ = 0.85. Figure 5 shows an
equilibrated configuration of the system.
We compare HooMD-blue revision 357453) running on different types of GPUs
with Gromacs 4.x63) running on two state of the art parallel computing clusters at
the JSC Jülich: An Intel Xeon cluster (JUROPA) and a Blue Gene P supercomputer
(JUGENE). To this end 100000 NVE integration steps with a timestep dt = 0.01
142
D. Reith, L. Mirny and P. Virnau
Fig. 5. Benchmark system. 200 semiflexible rings of length N = 1600 at a monomer density
ρ = 0.85. Each ring is colored differently.
Table II. Details on the benchmarked platforms.
name
Juropa
JUGENE
GPU
GTX580
Tesla M2070
CPU
Intel [email protected]
PowerPC 450@850Mhz
MD code
Gromacs
Gromacs
HooMD-blue
HooMD-blue
Version
4.5.3
4.0.7
r3547
r3547
were undertaken in each run to obtain a reasonable estimate for the speed of the
integration. Note that both HooMD and Gromacs use single precision floating point
operations. Of course, single precision is not particularly well-suited for NVE simulations. Therefore, these runs should only be considered for benchmark purposes.
Table II provides further details on the platforms. Figure 6 shows the speed of the
NVE-integration over the number of used CPU cores. No performance loss through
communication between the different CPU cores implies a perfect scaling and corresponds to a straight line with slope one in the double logarithmic scale of the plot.
Gromacs measures the duration of the integration in hours per nanosecond (ns).
In these units dt = 0.1 ps. HooMD-blue gauges its speed in timesteps per second.
The HOOMD-blue units can be converted to the ones of GROMACS by multiplying
their reciprocal value by 1000/(dt × 60 × 60). Hence the speed in units of ns/hour
is plotted over the number of CPU cores in Fig 6. This plot shows that the specific
systems scale rather nicely except for the biggest number of CPU cores taken into
account on JUROPA as well as JUGENE. GPUs are massive parallel computing
devices, but HOOMD-blue r3574 is not capable to run on more than one device in
parallel. Therefore, the runtime of HOOMD-blue r3574 on different types of GPUs is
denoted by different horizontal lines in the plot. The intersections of these horizontal
lines with the scaling lines from JUROPA and JUGENE denote the number of CPU
cores needed to obtain the same computing power from the respective cluster. In our
Polymer Rings in Concentrated Solutions
143
Fig. 6. Reciprocal time (speed) over the number of CPU cores on the JUROPA and JUGENE
supercomputers used for integrating the benchmark system with Gromacs as described in the
main text. The horizontal lines are the speed of integration with HOOMD-blue on different
GPUs. A GTX580 roughly corresponds to 75 2.93 Ghz JUROPA cores and 850 BlueGene P
cores on JUGENE.
test runs, a GTX 580 consumer card roughly corresponds to 75 2.93 Ghz JUROPA
cores and 850 Blue Gene P cores on JUGENE. The corresponding speed-ups for the
professional Tesla card are somewhat lower due to lower clock speeds and a lower
number of compute cores. The exact speed up factors should, however, be taken
with a grain of salt and rather serve as a guideline.
§5.
Conclusion
The aim of this paper is twofold. First we demonstrate that GPUs can be used
to tackle state of the art problems in the field of computational polymer physics. To
this end we investigate the structure and scaling behavior of non-concatenated ring
polymers in concentrated solution with a fully flexible off-lattice model by Molecular
dynamics. Although our model and the parameters differ from models considered
in the literature before, our results confirm once again that structural properties
and scaling do not depend on a specific model. We find indication for a crossover
to ν = 1/3-scaling in the radius of gyrations in the framework of a fully flexible
off-lattice model. Furthermore we are able to give some additional evidence that
the self-density of a ring at its center of mass does not converge towards the system
density in a fully flexible model. The analysis of the contact probability and the
sub-segment radius of gyrations indicate, that even the longest chains considered in
this study are not yet long enough to observe the scaling that is predicted for fractal
globules.
In the second part of the paper we compare the speed of Molecular dynamics
integration on a GPU using HooMD-blue to corresponding integrations on massively
parallel computing clusters using Gromacs. We demonstrate that GPUs are an efficient and economic alternative to massive parallel clusters for Molecular dynamics
integration of coarse-grained systems consisting of 100000 to 1000000 particles. Systems much bigger than 1 million particles cannot be simulated yet because of memory
144
D. Reith, L. Mirny and P. Virnau
limitations, and an efficient implementation of Molecular dynamics parallelized over
tens of GPUs is still missing. We found that a high end consumer graphic card (which
at the time of writing costs less than 400 ) yields the same number of Molecular
dynamics integration timesteps as 75 Xeon processor cores and roughly 850 Blue
Gene P processor cores. These numbers should however only serve as a guideline.
Acknowledgements
We would like to thank A. Stasiak, K. Binder, H.-P. Hsu, A. Tröster, K. Koschke
and M. Imakaev for fruitful discussions on the topic and J. Ahrens for technical
support. We are also grateful to K. Kremer for sending preprints of Refs. 21) and
22). This work received financial support from the Deutsche Forschungsgemeinschaft
(SFB 625-A17). Computing time on the JUROPA supercomputer at the NIC Jülich
and the GPU cluster of the Center for Computational Sciences Mainz at the ZDV
Mainz are also gratefully acknowledged.
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