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GRID2016
Institute of Mathematical Problems of Biology RAS the Branch of Keldysh Institute of Applied Mathematics
of Russian Academy of Sciences
Comparison of CPU and GPU performance
for the charge-transfer simulation
in the 1D molecular chains
Nadezhda Fialko, I. Likhachev, E. Sobolev
GRID2016
Outline
• I. Preliminary – biopolymer DNA
• II. Semi-classical Holstein model
• III. Numerical experiments
• IV. Comparison of CPU and GPU
performance
•
Conclusions
GRID2016 I.1
Biopolymer DNA
Bacis element of DNA –
nucleobase –
cytosine (C), guanine (G),
adenine (A), and thymine (T)
Base pairing rules:
A with T, and C with G
DNA is a long polymer consist of repeating units.
Diameter of DNA helix ~ 20 Å
GRID2016 I.2
DNA properties are interesting
for nanobioelectronics
Self-assembly properties of a double helix DNA may be
used to constract various structures
Biophysical experiments on charge transfer in DNA have
demonstrated strong dependence of conducting
property on the type of nucleotide sequence
N.C. Seeman. Nanotechnology and the Double Helix. Scientific American, a
Division of Nature America, Inc. (2004)
A.J. Turberfield. Molecular Machinery from DNA: Synthetic Biology from the
Bottom up. Biophysical Journal 106, 23A (2014)
H. Dietz. DNA nanotechnology: Nanoscale cable tacking. Nature
Nanotechnology 10, 829–830 (2015)
E.M. Roller, L.K. Khorashad, M. Fedoruk et al. DNA-assembled nanoparticle
rings exhibit electric and magnetic resonances at visible frequencies. Nano
letters 15 (2015) 1368-1373
GRID2016 II.1
Semi-classical model of system
«DNA + charge»
Nucleotide pair as a site
Sites move in a plane
perpendicular to the DNA
helix direction
Excess electron or hole (quantum
particle) migrates along chain of
classical sites.
GRID2016 II.2
Hamiltonian averaged by state
n = 1, …, N
bn (t) is the probability amplitude describing the
charge evolution on the site n.
un(t) is intrasite oscillations about the mass center
GRID2016 II.3
Equations of motion
variables: bn – probability amplitude of finding the charge on
the n-th site ,
un – displacement of n-th site from its equilibrium,
n = 1, …N,
T –temperature of thermostat
GRID2016 II.4
Model + temperature
Chain of sites is under the influence of temperature
Each sample initially has its own displacements and
velocities of sites and is influenced by background
temperature noise
Example: disrupting of polaron state
GRID2016 III.1
Computer simulations
One sample is trajectory of the system with its own random
sequence and from its initial data
e.g., displacements and velocities of sites correspond to
temperature prescribed
And for quantum subsystem,e.g., at one site in the center
of the chain |bn(t=0)| = 1, other |bk| = 0
GRID2016 III.2
Task for HPC
To calculate large number of samples
on purpose
to find averaged by ensemble timedependence functions.
Than estimate charge mobility, total
energy of the system, heat capacity,
etc.
GRID2016 III.3
Examples
System becomes to thermodynamically equilibrium state.
Dimer, sets 250 samples
T=20 K
The time of attaining the thermal
equilibrium t ~ 100 000
Etot (t ) ~ 1.97
T=300 K
The time of attaining the thermal
equilibrium t < 10 000
Etot (t ) ~ 0.88
GRID2016 III.4
Set of samples.
MPI
Natural parallelism with the calculation of each sample
(the dynamics of the charge distribution from the different
initial conditions and with different values of the random
force) on a single node, using MPI to collect data
at the master node, than  averaging by ensemble
the time-dependences.
The effectiveness is ~ 100%
GRID2016 III.5
Individual sample.
openMP
dbn
i
 nn 1bn 1  nnbn  nn 1bn 1  unbn
dt
d 2 un
dun
2
2



u




|
b
|
Z n (t )
n
n
2
dt
dt
Zn (t )  0
Zn (t )Zk (s)  nk (t  s)
n = 1,…N.
Equations include explicitly only the nearest neighbors.
We can "divide" the chain into several fragments, which
are integrated in one step independently on different
cores.
GRID2016 III.6
openMP, 2o2s1G - method
At the stages of integration step calculations of the
fragments are overlapped.
Synchronization is executed once, at the end of the step.
GRID2016 IV.1
openMP, tests for short chains
Speed-up T1/Tp dependence on number of sites.
T1 – machine time of sequentional version,
Tp – duration of the task, parallelized on p threads.
Tests performed on 4-core processors
(Intel Xeon E5520, 2.3GHz )
GRID2016 IV.2
openMP, tests for long chains
Speed-up T1/Tp dependence on number of sites.
T1 – machine time of sequentional version,
Tp – duration of the task, parallelized on p threads.
For sequences of 1000 sites and more T1/T4 ~ 3
GRID2016 IV.3
tests GPU / CPU
Not enough memory
Red line – Tesla 2090
Black line – a sample at a CPU core
GRID2016
Conclusions
MPI or openMP –
choice depends on task
For fixed number of nodes in the cluster,
equal machine time
MPI
to calculate more
short trajectories of
K samples
openMP
to calculate 3 time
longer trajectories of
K/4 samples
It seems that the use of GPU is not profitable.
GRID2016
We are grateful to
Joint Supercomputer Center
of the Russian Academy of Sciences
The work is supported by
Russian Foundation for Basic Research,
grants 15-07-06426, 16-07-00305,
14-07-00894
and Russian Science Foundation,
project 16-11-10163.
GRID2016
Thank you for your attention!
GRID2016 I.2
DNA properties
Self-assembly properties of
a double helix DNA may
be used to constract
various structures
(С.Dekker at al., Physics World, 2001)
Biophysical experiments on charge
transfer in DNA have demonstrated
strong dependence of this transfer
on the type of nucleotide sequence
J.K. Barton et. al. (1993)
Science 262, 1025-1029
NAA′16 III.5
Results
“Charge part” of total energy
Eq  Etot (t )  2 kBT
NAA′16 IV.1
Estimation of Free energy F
(1)
2 d
Etot (T )  T
dT
F
 
T 
 F (a) T Etot ( x)

 

dx  T  F (T )
2
x
a
 a

value of F(a) at temperature a0 ?
(2)
F  Etot (T )  TS
Assume: for big T
S?
S  Sclass  S q
Sclass
 T

  N  ln  1
  
Choose T=a, find F(a) using (2), integrate (1) from a to 0 with Etot
Calculate F(T) using (2) with Sq
NAA′16 IV.2
Comparison of results
AA dimer
The red circles indicate
the results of calculations
by the (2) ,
black curves - using
integral (1) with
F(a=700) for AA
and
F(a=1000) for GG
GG dimer
NAA′16 IV.3
Comparison of results
The red circles indicate
the results of calculations
by the (2) ,
black curves - using (1)
with F(a=1600) for TT
TT dimer
Results diverge at low
temperatures
NAA′16
Thank you for your attention!
NAA′16 IV.1
Partition function and averaging
 1

*
Z  Const   exp   E (bn , bn , un , vn )  db1
 T


where
du1
duN
dvN
2
1

2
*
E (t )   bnbn*1   vn2 
u


u
b
b
n nnn
2
2
 = R2NS2N : un (;+), vn (;+) , n(bnbn*) = 1
For systems {I} (,,) and {II} (,С,С)
variable substitution Un = Cun  dun = (1/C) dUn and domain of
integration un has not changed, so
N
Z{I}
1
   Z{II}
C 
and
E{I}  E{II} , R{I}  R{II} ,...
NAA′16 IV.1
Partition function and averaging
 1

*
Z  Const   exp   E (bn , bn , un , vn )  db1
 T


where
du1
duN
dvN
2
1

2
*
E (t )   bnbn*1   vn2 
u


u
b
b
n nnn
2
2
 = R2NS2N : un (;+), vn (;+) , n(bnbn*) = 1
For systems {I} (,,) and {II} (,С,С)
variable substitution Un = Cun  dun = (1/C) dUn and domain of
integration un has not changed, so
N
Z{I}
1
   Z{II}
C 
and
E{I}  E{II} , R{I}  R{II} ,...
NAA′16
Tests for Holstein parameters ~~1.
Evolution to the thermodynamical equilibrium
T = 200
T = 150
Dynamics of R(t), calculated from different initial data. The bottom curve
represents calculation from the initial ‘polaron distribution’, the state with the
lowest energy Emin. The upper curve – from uniform initial distribution of the
charge over all the sites (which, without regard for temperature, corresponds to
the state with the highest energy Emax).
NAA′16
Stationary solution, T=0
ibn  (bn1  bn1 )  unbn ,
0  2un   | bn |2
 un  

2
|
b
|
n
2
bn = rn exp(iWt)
W rn = ( rn1 + rn+1)  (/)2 rn3
n =1,…,N
2
r
 n =1
Systems with parameters (,,){I} and (,С,С){II} have
the same steady-state solutions {r1,…,rN,W} (but un{I}  un{II} )
and the same energy values
NAA′16
Tests for T  0
10-site G..G fragment, =1.276, T=100K, averaging over 50 samples
 = 0.01,  = 0.02 (DNA)
 = 0.1,  = 0.2
 = 0.5,  = 1.
Time of evolution to thermodynamic equilibrium can be
significantly reduced by changing the parameters with /=const.
Thermodynamic averaging function
F (T )



F  exp( E (b, b* , u, v) / T )dbdb*dudv


exp( E (b, b* , u, v) / T )dbdb*dudv
For systems {I} (,,) and {II} (,С,С)
this variable substitution leads to equality
E{I}  E{II}
2o2s1G* - integration method
Zn – random variable with Gauss distribution (0;1)
[*H.S. Greenside, E. Helfand (1981) The Bell System Tech. J.
60, 1927]
Calculation of  E , small T
All coefficients are of the order of 1, averaging by 500 samples
nonequilibrium initial data
of quantum subsystem, at
the centre of chain
|bn(t=0)| = 1, other |bk| = 0
1
1

Echain = N  2 u 2  v 2 
2
2

N  kBT
E  Echain
1
1 2
2
  n un    n un2   n bnbn*1   n unbnbn*
2
2
Calculation of  E , small T, polaron at t=0
E  Echain 
1
1 2
2
2
*
*
u


u


b
b


u
b
b
 n 2  n n  n n n1  n n n n
2 n
Estimation of “thermodynamic”
parameters
1
1 2
2
T = 0 : E   n un    n un2   n bnbn*1   n unbnbn*
2
2
T0:
d F
E  T
 
dT  T 
2
T
F (T)  E0  T 
0
E0 = E (T=0)
E ( x)  E0
x
2
dx
Calculation of  E , small T
All coefficients are of the order of 1, averaging by 500 samples
Nonequilibrium initial data
of quantum subsystem, at
the centre of chain
|bn(t=0)| = 1, other |bk| = 0
For classical subsystem thermodynamical
equilibrium distribution
at t=0
Next: numerical integration
E ( x)  E0
F (T)  E0  T 
dx
2
x
0
T
E( x)  E  Echain
Theoretical line Echain + E0 , Echain = N  kBT
and results of computer experiments (marking squares )