1. Introduction 2. NEB/STRING 3. ADMD 4. Numerical...

The 11th KIAS Protein Folding Winter School, HIGH1 Resort, February 6-10, 2012

1. Introduction 2. NEB/STRING 3. ADMD 4. Numerical Experiments


Hamilton’s principle Lagrange’s equation

{q(0)}

{q(P)}

{q(0)}

{q(P)}

Action optimization

Known two conformations

Reactants

Products

solution


q(0)

q(t)

Hamilton’s principle Lagrange’s equation

Complexity 3Natom (P-1)

S2

S1 S3

Saddle?

Microscopic

reversibility

Fast sine transformation {q} {a}

http://gams.nist.gov/serve.cgi/Module/SLATEC/SINT/7394/

O(P2) O(P log P)

The nature of the stationary points depends on time step.


Phys. Rev. 159, 98 (1967)

aka Onsager-Machlup

“path quality”measure


The Verlet algorithm Phys. Rev. 159, 98

(1967)

Elber et al. (1996)

+

aka Onsager-Machlup


Adv. Chem. Phys. 126, 93 (2003)


improved path quality : new object function

Phys. Rev. Lett. 87, 10832 (2001)

Fictitious T

Target energy

Phys. Rev. B 68, 064303 (2003)

Chem. Phys. Lett. 105, 9299 (1996)

Onsager-Machlup

Passerone-Parrinello

Phys. Rev. Lett. 90, 065501 (2003)

J. Chem. Phys. 120, 4672 (2004)


Q representation


example

C60 molecule formation process simulation

N=60, P=150, 6 fs = Time increment, 3N(P-1)=26820

Atomic unit = 0.529177 Angstrom



Phys. Rev. B 68, 064303 (2003)


ADMD vs NEB

systematic barrier estimation : target E Phys. Rev. B 68, 064303 (2003)


Numerical Optimization, Nocedal and Wrights, Springer, 1999


Complexity 3N(P-1),

N = the number of atoms,

P = the number of steps

Typical distribution of

I : 1, 2, 3,…., N

j : 1, 2, 3,….,P-1

HP-36


N=504, P=2000

3N(P-1)=3022488


Parallel ADMD start

end

j=0,1,2,…,P

Vj,{F}j

V0,{F}0 V1,{F}1 V3,{F}3 V2,{F}2 VP,{F}P …

{R}j configurations

Object function/gradient Relaxation {R}j

Converged?

No

j=0 and j=P

Yes

,…,

speedup

{Communication}/{CPU} ratio

Time consuming part

LBFGS

Energy/force


subroutine multiv

! Written by In-Ho Lee, KRISS, August 10 (2007)

USE action, ONLY : qq,force_ac,vofqj,np_ac,natom_ac,isequence,isymbol,iattyp,ii12

IMPLICIT NONE

include 'mpif.h'

integer n1,n2,j,jstart,jfinish

real*8, allocatable :: exqq(:,:,:),exforce(:,:,:),exvofqj(:)

integer myid,nproc,ierr,kount,iroot

!

call MPI_COMM_RANK( MPI_COMM_WORLD, myid, ierr )

call MPI_COMM_SIZE( MPI_COMM_WORLD, nproc, ierr )

if(nproc > np_ac+1) then

write(6,*) 'you have too many CPUs nproc, np+1',nproc,np_ac+1

call MPI_FINALIZE(ierr)

stop

endif

if(myid == 0)then ! -----{ PROCESS ID = 0

vofqj=0.0d0 ; force_ac=0.0d0

endif ! -----} PROCESS ID = 0

!

if(nproc == 1)then

do j=0,np_ac

call xtinker(qq(1,1,j),force_ac(1,1,j),vofqj(j),natom_ac,isequence,isymbol,iattyp,ii12)

enddo

return

endif

!

allocate(exqq(natom_ac,3,0:np_ac), exforce(natom_ac,3,0:np_ac), exvofqj(0:np_ac))

if(myid == 0)then ! -----{ PROCESS ID = 0

exqq=qq

endif ! -----} PROCESS ID = 0

iroot=0 ; kount=3*natom_ac*(np_ac+1)

call MPI_BCAST(exqq, kount,MPI_REAL8, iroot,MPI_COMM_WORLD,ierr)

n1=0 ; n2=np_ac

call equal_load(n1,n2,nproc,myid,jstart,jfinish)

!

exforce=0.0d0 ; exvofqj=0.0d0

do j=jstart,jfinish

call xtinker(exqq(1,1,j),exforce(1,1,j),exvofqj(j),natom_ac,isequence,isymbol,iattyp,ii12)

enddo

iroot=0 ; kount=3*natom_ac*(np_ac+1)

call MPI_REDUCE(exforce,force_ac,kount,MPI_DOUBLE_PRECISION,MPI_SUM,iroot,MPI_COMM_WORLD,ierr)

iroot=0 ; kount=(np_ac+1)

call MPI_REDUCE(exvofqj,vofqj,kount,MPI_DOUBLE_PRECISION,MPI_SUM,iroot,MPI_COMM_WORLD,ierr)

!

deallocate(exforce, exvofqj, exqq)

return

end


subroutine equal_load(n1,n2,nproc,myid,istart,ifinish)

! Written by In-Ho Lee, KRISS, September (2006)

implicit none

integer nproc,myid,istart,ifinish,n1,n2

integer iw1,iw2

iw1=(n2-n1+1)/nproc ; iw2=mod(n2-n1+1,nproc)

istart=myid*iw1+n1+min(myid,iw2)

ifinish=istart+iw1-1 ; if(iw2 > myid) ifinish=ifinish+1

! print*, n1,n2,myid,nproc,istart,ifinish

if(n2 < istart) ifinish=istart-1

return

end


http://en.wikipedia.org/wiki/File:Protein_backbone_PhiPsiOmega_drawing.jpg http://en.wikipedia.org/wiki/Ramachandran_plot

CNCC

NCCN


Conformational isomerization

480 fs

N=22

JKPS 46, 601 (2005).


Conformational isomerization

690 fs 690 fs

N=28

JKPS 46, 601 (2005).


Hammond postulate "If two states, as for example, a transition state and an unstable intermediate, occur consecutively during a reaction process and have nearly the same energy content, their interconversion will involve only a small reorganization of the molecular structures."

G. S. Hammond

That is, along the reaction coordinate, species with similar energies also have similar structures.

activation energy: the minimum energy necessary to form an activated complex in a reaction.

The Bell–Evans–Polanyi principle states that highly exothermic chemical reactions have a low activation energy.


Exp. analysis

• (9,0)-capsule : C114 (C42+18x4) : 0.704 nm

• (5,5)-capsule : C120 (C40+20x4) : 0.678 nm

• kBT=800 oC ~ 0.09 eV vs SW ~ 6 eV

Chem. Phys. Lett. 384, 320 (2004)

Phys. Rev. B 69, 73402 (2004)

Rep. Prog. Phys. 62, 1181 (1999)

~0.8 eV (th. & exp.)

Migration energy : inside SWCNT

6 atoms

J. Phys. Chem. 96, 3574 (1992)

exchange

A beam of 13C ions


Science 272, 87 (1996)

autocatalysis

Chem. Phys. Lett. 351, 178 (2002)

Ea= 6.2 eV4.0 eV2.3 eV

kBT= 800 oC ~ 0.09 eV


Microscopic reversibility

C. H. Xu, C. Z. Wang, C. T. Chan, and K. M. Ho, J. Phys.: Condens. Matter 4, 6047 (1992) Tight-binding

ADMD simulation

Appl. Phys. Lett. 88, 011913 (2006)


Dynamic pathway models

C60 precursor models Potential-energy fluctuations

Simulated annealing

J. Phys. Chem. 98, 1810 (1994)

J. Chem. Phys. 120, 4672 (2004)


C60 formation ADMD simulation

Chain/hexagon

Tangled polycyclic I

J. Chem. Phys. 120, 4672 (2004)


C60 activation energy, formation rate

Phys. Rev. Lett 72, 2418 (1994)

Tight-binding C. H. Xu, C. Z. Wang, C. T. Chan, and K. M. Ho, J. Phys.: Condens. Matter 4, 6047 (1992)

C-C: 3.6 eV concerted motions

J. Phys. Chem. 98, 1810 (1994)

J. Chem. Phys. 120, 4672 (2004)


C60 fusion

nanotubes C60

Double walled fused C60

Chem. Phys. Lett. 315, 31 (1999)

~800 oC

Chem. Phys. Lett. 384, 320 (2004)


C60 fusion ADMD simulation

C60 + C60 (5,5)-capsule

Phys. Rev. Lett. 90, 065501 (2003)


An extra carbon atom

C60 fusion capsule (5,5) chirality

Two extra carbon atoms

catalyst

Appl. Phys. Lett. 88, 011913 (2006)


Single/double exchange

C60 + C60 + C C120 + C C60 + C60 + C + C C120 + C + C

Appl. Phys. Lett. 88, 011913 (2006)


Partial density of state

Appl. Phys. Lett. 88, 011913 (2006)


Potential energy profile along pathway

C60 + C60 + C C120 + C

C60 + C60 + C + C C120 + C + C

Appl. Phys. Lett. 88, 011913 (2006)


J. Am. Chem. Soc. 129, 8954 (2007)


Alignment of vacancies

Phys. Rev. B 79, 174105 (2009)

The 11th KIAS Protein Folding Winter School, HIGH1 Resort, February 6-10, 2012 Phys. Rev. B 74, 195409 (2006)


„Y‟ junction


„T‟ junction


Screw dislocation in Cu

New cross slip mechanism (ADMD, empirical potential model)

Lower activation energy process = Fleischer +Friedel-Escaig mechanism

Appl. Phys. Lett. 88, 201908 (2006)

The activation energy is around 2.14 eV that

is the lowest energy barrier ever reported by atomistic

simulations.

The 11th KIAS Protein Folding Winter School, HIGH1 Resort, February 6-10, 2012 Phys. Rev. B 76, 245408 (2007) Phys. Rev. B 76, 245407 (2007)

Transition-pathway models of atomic diffusion on fcc metal surfaces. I. Flat surfaces

Transition-pathway models of atomic diffusion on fcc metal surfaces. II. Stepped surfaces


The two most important secondary structure motifs

N=112

N=247

Chem. Phys. Lett. 412, 307 (2005)


helix formation

Acetyl-(Ala)10-N-methyl amide Generalized Born solvation model

N=112


helix formation

Generalized Born solvation model J. Mol. Biol. 267, 963 (1997)

Chem. Phys. Lett. 412, 307 (2005)


b-hairpin formation

Residues 41-56 of protein G Generalized Born solvation model

N=247

Turn forms first

Chem. Phys. Lett. 412, 307 (2005)


b-hairpin formation

Generalized Born solvation model Nature 390, 196 (1997)

Turn forms first

At the central part : the first native H bond

Chem. Phys. Lett. 412, 307 (2005)

The 11th KIAS Protein Folding Winter School, HIGH1 Resort, February 6-10, 2012 J. Comput. Chem. 31, 57 (2009)

HP-36


HP-36

3N(P − 1) = 3 × 596 × 1999 = 3,574,212 Amber94, GB/SA 36.28 fs

J. Comput. Chem. 31, 57 (2009)


J. Comput. Chem. 31, 57 (2009)


Principal component analysis

Principal component analysis is used to extract a small number of the most contributing elements (principal

components) of the spatial correlation between residue pairs. M defined as Mij = (xi − )(xj − ),

where x1, x2, . . . , x3 Nα are the Cartesian coordinates of the Nα Cα

atoms. The average . . . is over all structural frames from the ADMD trajectory (i.e., P + 1 = 2001). By

diagonalizing M, 3Nα eigenvectors with their corresponding eigenvalues are obtained.

J. Comput. Chem. 31, 57 (2009)


Eigenvalues As Percentages Cumul. Percentages

----------- -------------- ------------------

0.55364246E+02 51.2632 51.2632

0.16711985E+02 15.4741 66.7372

0.10515863E+02 9.7369 76.4741

0.77891107E+01 7.2121 83.6863

0.52260976E+01 4.8390 88.5253

0.33910422E+01 3.1399 91.6651

0.17537344E+01 1.6238 93.2889

0.12711619E+01 1.1770 94.4660

0.12134945E+01 1.1236 95.5895

0.92323029E+00 0.8548 96.4444

0.64476693E+00 0.5970 97.0414

0.56307554E+00 0.5214 97.5628

0.46476060E+00 0.4303 97.9931

0.27635869E+00 0.2559 98.2490

0.26253435E+00 0.2431 98.4921

0.21947116E+00 0.2032 98.6953

0.18363801E+00 0.1700 98.8653

0.14950135E+00 0.1384 99.0038

0.13822177E+00 0.1280 99.1317

0.11565286E+00 0.1071 99.2388

0.11010055E+00 0.1019 99.3408

0.89341208E-01 0.0827 99.4235


Principal component analysis

J. Comput. Chem. 31, 57 (2009)


Reaction coordinate(s) ADMD simulation

Reaction coordinate(s) Markov state model

free energy profile

time scale


Predicting the weather

• very simple model (sunny-rainy model) • sunny, rainy, K=2, 2x2 matrix P • sunny sunny : rainy = 0.9 : 0.1 • rainy sunny : rainy = 0.5 : 0.5

http://en.wikipedia.org/wiki/Examples_of_Markov_chains

Markov model


Steady state of the weather

program steady_state_test implicit none integer k real*8 pp(2,2),vec1(2),vec2(2) pp(1,1)=0.9d0 ; pp(1,2)=0.1d0 pp(2,1)=0.5d0 ; pp(2,2)=0.5d0 vec1(1)=1.d0 ; vec1(2)=0.d0 do k=1,1000 vec2=matmul(vec1,pp) vec1=vec2 end do write(6,*) vec2 stop end 0.8333333333333345 0.1666666666666669 FORTRAN STOP

In conclusion, in the long term, 83% of days are sunny.

Stationary distribution

vec2(1)*pp(1,2)-vec2(2)*pp(2,1)=0 Detailed balance


J. Chem. Phys. 129, 064107 (2008).

Markov State Model: example

Free energy profile and characteristic time scale

A. C. Pan and B. Roux


Transition-count matrix

• Mij= Mij+1, state i; 0 state j; , lag time • time-consumming part : many (relatively) short MD simulations : parallel job

• Mij/(sumj Mij) Pij : finite sampling

• Posterior_distribution (P | M) ~ Likelihood(M | P) x Prior_distribution (P)

• Likelihood(M | P) ~ Pij M

ij

• Prior_distribution (P) ~ Pij b

ij-1

• Posterior_distribution (P | M) ~ Pij

ij-1, a Dirichlet distribution

• ij=bij+Mij • bij= ½ or bij= 1 or bij= 1/K, [ KxK matrices, M, P, , b]

• Dirichlet distribution, ~ Pij

ij-1

• Bayesian Inference : error analysis

W. C. Swope , J. W. Pitera, and F. Suits, J. Phys. Chem. B 108, 6571 (2004).


Transition matrix

• If the state space is finite, the transition probability distribution can be represented by a matrix, called the transition matrix, with the ij element of P equal to Pij= Pr(Xn+1=j| Xn=i). p = pP, p: left-eigenvector, eigenvalue =1 probability of the state i : pi : sumi pi = 1

• a left-eigenvector of the transition matrix associated with the eigenvalue 1 • a right-eigenvector of the transition matrix associated with the eigenvalue 1

• single eigenvalue=1 for left-eigenvector and right-eigenvector • a time-homogenous chain: Markov chain, a stationary distribution • Perron-Frobenius theorem • other eigenvalues, |eigenvalue| < 1

• (relaxation time scale)s = - /ln(ls), largest eigenvalue of P matrix less than 1 • l1,l2,l3,l4,l5,....,lK-1,lK=1, (in ascending order), lK-1=ls

• Pij >=0

• sumj Pij =1. (stochasticity) : sum of each row is unity • pi Pij = pj Pji (detailed balance) : p = pP • sumi piPij= sumi pjPji =pj sumi Pji=pj


0.15 microsecond

41x41 matrix

Langevin dynamics, 300 K, 91 ps-1


Energ

y

Reaction coordinate

Weighted-Ensemble

bins

Free energy surface

Time variations

weight(bin,time)

Transition probability : P(t)

Weighted-ensemble Brownian dynamics simulations for protein association reactions Biophysical Journal, Volume 70, Issue 1, Pages 97-110

ADMD-pathway


Example

Potential energy profile

Population changes


1l2y

32 replicas/bin, 300 K, 91 ps-1

states pico-second

popula

tion

Popula

tion a

t final sta

te (

t)


Crooks‟ fluctuation theorem

The Crooks equation is an equation in statistical mechanics that relates the work done on a system

during a non-equilibrium transformation to the free energy difference

between the final and the initial state of the transformation.

WAB, i.e. the work done on the system during the forward transformation (from A to B)

ΔF = F(B) − F(A), i.e. the Helmholtz free energy difference between the state A and B

http://incredible.egloos.com/4646410


ADMD + PCA

ADMD + Markov state model

(stochastic model)

ADMD + Replica-exchange molecular dynamics

(thermodynamics)

ADMD + Weighted-ensemble

(kinetics)

ADMD + Crooks‟ fluctuation theorem


summary

• Rare event simulations

• Minimum energy path, transition state

• NEB, string

• Action-derived molecular dynamics reaction coordinates

• PCA, MSM, REMD, weighted ensemble

• Crooks’ fluctuation theorem


1. Introduction 2. NEB/STRING 3. ADMD 4. Numerical Experiments


/home/ihlee/tutorial_string_admd/potential_plot

=============================================================

vi plot_pot.f90

pgf90 plot_pot.f90 energy?.f90

./a.out

ls -ltra

gnuplot

>set pm3d

>set palette rgbformulae 33,32,10

>splot 'fort.7' with pm3d

Remember that it is an interactive mode.

The followings are the gnuplot commands for getting a color eps file.

gnuplot> set pm3d

gnuplot> set palette rgbformulae 33,32,10

gnuplot> splot 'fort.7' with pm3d

gnuplot> set term postscript eps enhance color

Terminal type set to 'postscript'

Options are 'eps enhanced color colortext \

dashed dashlength 1.0 linewidth 1.0 defaultplex \

palfuncparam 2000,0.003 \

butt "Helvetica" 14'

gnuplot> set output "test.eps"

gnuplot> replot

gnuplot> quit

[ihlee@helix source]$ gv test.eps

=============================================================

gnuplot> set pm3d map

gnuplot> set size square

gnuplot> splot 'fort.7'

gnuplot> set palette rgbformulae 33,32,10


gnuplot> set xrange[-25: 25]

gnuplot> set yrange[-25: 25]


gnuplot> set term postscript eps enhanced color

Terminal type set to 'postscript'

Options are 'eps enhanced color colortext \

dashed dashlength 1.0 linewidth 1.0 defaultplex \

palfuncparam 2000,0.003 \

butt "Helvetica" 14'

gnuplot> set output "test1.eps"

gnuplot> replot

gnuplot> quit

[ihlee@helix potential_plot]$ gv test1.eps


/home/ihlee/tutorial_string_admd/string_m1/source

====================================================

make clean

make

make a2ps

gv main.ps

or evince energy.ps

ls -ltra ../*.x

====================================================

cp include.f init.f inout.f ../

cd ../

pgf90 init.f inout.f

./a.out

ls -ltra

data0 will be present. This file will be an input file for the program string1.x.

./string1.x

ls -ltra

data1 will be present.

xmgrace data0 data1


====================================================

cmpl= pgf90

segl= pgf90

#OPT = -fast

OPT =

FFLAGS = -c ${OPT}

LIBS =

FILES= main.o dcopy.o energy.o force.o inout.o intpol.o mvper.o mvtan.o tangent.o

/home/ihlee/tutorial_string_admd/string_m1/string1.x: $(FILES)

$(segl) -o ../string1.x $(FILES) $(LIBS)

main.o:main.f

$(cmpl) $(FFLAGS) main.f

dcopy.o:dcopy.f

$(cmpl) $(FFLAGS) dcopy.f

energy.o:energy.f

$(cmpl) $(FFLAGS) energy.f

force.o:force.f

$(cmpl) $(FFLAGS) force.f

inout.o:inout.f

$(cmpl) $(FFLAGS) inout.f

intpol.o:intpol.f

$(cmpl) $(FFLAGS) intpol.f

mvper.o:mvper.f

$(cmpl) $(FFLAGS) mvper.f

mvtan.o:mvtan.f

$(cmpl) $(FFLAGS) mvtan.f

tangent.o:tangent.f

$(cmpl) $(FFLAGS) tangent.f

clean:

rm -f *.x *.o *.mod *.M core*

touch:

touch *.f90 *.i makefile ; chmod 600 *.f90 *.i makefile ; ls -l *.f90 *.i makefile *.f

rmo:

rm -f *.o *.mod *.M core*

lsl:

ls -l *.f90 makefile *.i

a2ps:

a2ps --prologue=color -o main.ps main.f ; a2ps --prologue=color -o energy.ps energy.f ; a2ps --prologue=color -o force.ps

force.f ; a2ps --prologue=color -o inout.ps inout.f ; a2ps --prologue=color -o mvper.ps mvper.f ; a2ps --prologue=color -o

mvtan.ps mvtan.f ; a2ps --prologue=color -o tangent.ps tangent.f ; a2ps --prologue=color -o intpol.ps intpol.f


subroutine lbfgs_driv(theta,posi,grad,nn)

! Written by In-Ho Lee, KRISS, January 31 (2003)

implicit none

integer nn

real*8 theta,posi(nn),grad(nn)

integer n_local,m,msave

real*8, allocatable :: diag(:),w(:)

real*8 eps,xtol,gtol,t1,t2,stpmin,stpmax

integer iprint(2),iflag,icall,mp,lp,j,nwork

logical diagco

real*8, allocatable :: gmatwork(:,:)

integer itrelax

!

! The driver for LBFGS must always declare LB2 as EXTERNAL

external lb2

common /lb3/mp,lp,gtol,stpmin,stpmax

!

itrelax=1000

n_local=nn

m=5 ; msave=7 ; nwork=n_local*(2*msave+1)+2*msave

allocate(diag(n_local),w(nwork))

iprint(1)= 1 ; iprint(2)= 0

!

! We do not wish to provide the diagonal matrices Hk0, and

! therefore set DIAGCO to FALSE.

diagco= .false. ; eps= 1.0d-4 ; xtol= 1.0d-16 ; icall=0 ; iflag=0

!

!

allocate(gmatwork(nn/3,nn/3))

20 continue

call lj(posi,grad,theta,nn/3,gmatwork)

!

write(6,'(e14.7)') theta

call lbfgsac(n_local,m,posi,theta,grad,diagco,diag,iprint,eps,xtol,w,iflag)

if(iflag.le.0) go to 50

icall=icall + 1

! We allow at most 2000 evaluations of Function and Gradient

if(icall > itrelax) go to 50

go to 20

50 continue

deallocate(gmatwork)

deallocate(diag,w)

return

end

Local minimization LBFGS : function, gradient


implicit none

integer natoms,nn

real*8 elj

real*8, allocatable :: x(:),v(:)

real*8, allocatable :: g(:,:)

integer i

real*8 tmpx,tmpy,tmpz,obj

natoms=7

allocate(x(3*natoms),v(3*natoms))

allocate(g(natoms,natoms))

do i=1,natoms

x(3*(i-1)+1)=0.3*i

x(3*(i-1)+2)=0.3*i**2+1.

x(3*(i-1)+3)=0.d0

enddo

i=1

x(3*(i-1)+1)=-1.0d0

x(3*(i-1)+2)=0.0d0

x(3*(i-1)+3)=0.0d0

i=2

x(3*(i-1)+1)=1.0d0

x(3*(i-1)+2)=0.0d0

x(3*(i-1)+3)=0.0d0

i=3

x(3*(i-1)+1)=0.0d0

x(3*(i-1)+2)=-1.0d0

x(3*(i-1)+3)=0.0d0

i=4

x(3*(i-1)+1)=0.0d0

x(3*(i-1)+2)=1.0d0

x(3*(i-1)+3)=0.0d0

i=5

x(3*(i-1)+1)=1.0d0

x(3*(i-1)+2)=1.0d0

x(3*(i-1)+3)=0.0d0

i=6

x(3*(i-1)+1)=-1.0d0

x(3*(i-1)+2)=-1.0d0

x(3*(i-1)+3)=0.0d0

i=7

x(3*(i-1)+1)=0.0d0

x(3*(i-1)+2)=0.0d0

x(3*(i-1)+3)=0.0d0

tmpx=0.d0 ; tmpy=0.d0 ; tmpz=0.d0

do i=1,natoms

tmpx=tmpx+x(3*(i-1)+1)

tmpy=tmpy+x(3*(i-1)+2)

tmpz=tmpz+x(3*(i-1)+3)

enddo

tmpx=tmpx/float(natoms) ; tmpy=tmpy/float(natoms) ; tmpz=tmpz/float(natoms)

do i=1,natoms

x(3*(i-1)+1)=x(3*(i-1)+1)-tmpx

x(3*(i-1)+2)=x(3*(i-1)+2)-tmpy

x(3*(i-1)+3)=x(3*(i-1)+3)-tmpz

enddo

call lj(x,v,elj,natoms,g)

write(6,*) natoms

write(6,*) elj

do i=1,natoms

write(6,'(a2,2x,3f18.8)') 'Ar',x(3*(i-1)+1),x(3*(i-1)+2),x(3*(i-1)+3)

enddo

write(6,*)

do i=1,natoms

write(6,'(3f18.8)') v(3*(i-1)+1),v(3*(i-1)+2),v(3*(i-1)+3)

enddo

nn=3*natoms

call lbfgs_driv(obj,x,v,nn)

elj=obj

tmpx=0.d0 ; tmpy=0.d0 ; tmpz=0.d0

do i=1,natoms

tmpx=tmpx+x(3*(i-1)+1)

tmpy=tmpy+x(3*(i-1)+2)

tmpz=tmpz+x(3*(i-1)+3)

enddo

tmpx=tmpx/float(natoms) ; tmpy=tmpy/float(natoms) ; tmpz=tmpz/float(natoms)

do i=1,natoms

x(3*(i-1)+1)=x(3*(i-1)+1)-tmpx

x(3*(i-1)+2)=x(3*(i-1)+2)-tmpy

x(3*(i-1)+3)=x(3*(i-1)+3)-tmpz

enddo

write(6,*) natoms

write(6,*) elj

do i=1,natoms

write(6,'(a2,2x,3f18.8)') 'Ar',x(3*(i-1)+1),x(3*(i-1)+2),x(3*(i-1)+3)

enddo

write(6,*)

do i=1,natoms

write(6,'(3f18.8)') v(3*(i-1)+1),v(3*(i-1)+2),v(3*(i-1)+3)

enddo

deallocate(x,v)

deallocate(g)

stop

end

Local minimization: function, gradient


subroutine lj(x,v,elj,natoms,g)

! Written by In-Ho Lee, KRISS, December 22 (2011); based on Wales's GMIN programs in fortran 77.

implicit none

integer natoms

real*8 x(3*natoms),v(3*natoms),elj,g(natoms,natoms)

integer j1, j2, j3, j4

real*8 r6, dummyx, dummyy, dummyz, xmul2, dummy, dist

elj=0.0d0

do j1=1,natoms

j3=3*j1

g(j1,j1)=0.0d0

do j2=j1+1,natoms

j4=3*j2

dist=(x(j3-2)-x(j4-2))**2+(x(j3-1)-x(j4-1))**2+(x(j3)-x(j4))**2

dist=1.0d0/dist

r6=dist**3

dummy=r6*(r6-1.0d0)

elj=elj+dummy

dist=dist*r6

g(j2,j1)=-24.0d0*(2.0d0*r6-1.0d0)*dist

g(j1,j2)=g(j2,j1)

enddo

enddo

elj=elj*4.0d0

do j1=1,natoms

j3=3*j1

dummyx=0.0d0

dummyy=0.0d0

dummyz=0.0d0

do j4=1,natoms

j2=3*j4

xmul2=g(j4,j1)

dummyx=dummyx+xmul2*(x(j3-2)-x(j2-2))

dummyy=dummyy+xmul2*(x(j3-1)-x(j2-1))

dummyz=dummyz+xmul2*(x(j3) -x(j2))

enddo

v(j3-2)=dummyx

v(j3-1)=dummyy

v(j3)=dummyz

enddo

end

LJ potential : energy, gradient


make clean

make

cd ../work1

../string_m_lj.x

../string_m_lj.x

xmgrace fort.34

cp fort.8 test.xyz

xmakemol -f test.xyz

control --> frames --> start

control --> frames --> make anim

ls *.xpm -ltra

convert lj7.1.xpm lj7_001.jpg



.....

Use Windows Live Movie Maker


program string_driver

! Written by In-Ho Lee, KRISS, December 22 (2011); based on W. Ren's string programs in fortran 77.

USE string, ONLY : initial,final,mvtan,mvper,print_intermediate

implicit none

real*8 dt,error

integer ic

dt=0.0004d0

dt=1.0d-7

call initial(dt)

!

ic=0

100 continue

ic=ic+1

if(mod(ic,10).eq.1) then

call mvtan

endif

call mvper(dt,error)


write(*,*) 'ic=',ic, ' error =',error

endif


call print_intermediate

endif

if(error.lt.1.e-6) then

goto 101

else

if(ic > 1000000)then

write(6,*) 'ic=',ic, ' error =',error

goto 101

endif

endif

goto 100

101 continue

call final

!

stop

end program string_driver


subroutine rmarin(ij,kl)

! This subroutine and the next function generate random numbers. See

! the comments for SA for more information. The only changes from the

! orginal code is that (1) the test to make sure that RMARIN runs first

! was taken out since SA assures that this is done (this test didn't

! compile under IBM's VS Fortran) and (2) typing ivec as integer was

! taken out since ivec isn't used. With these exceptions, all following

! lines are original.

! This is the initialization routine for the random number generator

! RANMAR()

! NOTE: The seed variables can have values between: 0


Simulation details

No extended atoms, no constraints on bonds &

angles

AMBER94 (http://dasher.wustl.edu/tinker)

atomistic simulation in implicit solvents

Generalized Born solvation model

TINKER 3.8

TINKER 4.2

TINKER 5.0

TINKER 6.0


.seq

.key

.xyz

.pdb

protein.x

newton.x

pdbxyz.x

xyzpdb.x

minimize.x

superpose.x


MPI/FORTRAN 90

RS/6000 SP: Practical MPI Programming An IBM Redbooks publication

Yukiya Aoyama and Jun Nakano

http://www.redbooks.ibm.com/abstracts/sg245380.html

포트란 및 MPI 레퍼런스, 인터넷


FORTRAN 90

Fortran 90

A Conversion Course for Fortran 77 Programmers (The University of Manchester)

The University of Liverpool

Fortran 90 course notes

두 개의 포트란 90 레퍼런스, 인터넷


http://incredible.egloos.com


References • Allen and Tildesley, Computer simulation of liquids • Haile, Molecular dynamics simulation • Frenkel and Smit, Understanding molecular simulation • Landau and Binder, A guide to Monte Carlo simulations in

statistical physics • Rapaport, The art of molecular dynamics simulation • Newman and Barkema, Monte Carlo methods in statistical

physics • http://dasher.wustl.edu/tinker • http://www.ccp5.ac.uk/ • http://www.expasy.org/swissmod/course/course-index.htm

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