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Chapter 3 Parallel Computing of Resonance Raman Intensities Using a Transputer Array
E.J. Karjalainen (Editor), Scientific Computingand Automation (Europe) 1990 1990 Elsevier Science Publishers B.V., Amsterdam
21
CHAPTER 3
Parallel Computing of Resonance Raman Intensities Using a Transputer Array R.G. Efremov Shemyakin Institute of Bioorganic Chemistry, USSR Academy of Sciences, ul. MiklukhoMaklaya 16/10, 117871 Moscow, V-437,USSR
1. Introduction Recent progress in optical spectroscopy of biological molecules is closely connected to the development of the computcr power. The computational approaches have a great importance in solving the problems of spectral processing, interpretation and simulation. For example, computing the resonance Raman (RR) intensities of large molecules provides detailed information about the equilibrium geometry and dynamics of the resonant excited electronic state. The traditional approach to calculating the RR cross section involves a summation over all the vibrational levels of the resonant electronic state. Such sum-over-state algorithm performs a direct search for the complete set of excited state parameters that gcncratc the best fit to the expcrimcntal RR data. Usually, when there are only a fcw RR active modcs, the sum-over-state method is much more efficient compared with alternative approaches [I]. But in gcncral, using a standard sequential sum-over-state procedure can be computationally intractable for large (especially, biological) moleculcs because the algorithm scts up calculations the Raman cross sections of N vibrational modes of the molcculc and for each mode there are N logically nested loops. Therefore, such a technique demands powerful computer resourccs. Thc problem becomes processing-intensive espccially if a hard optimization is required-in a case when the initial estimates of excitcd slate paramctcrs are not known exactly. An altcrnativc approach to solving this task is based on the fact that the intensities of all modcs can bc calculalcd indepcndcntly at difkrent parallcl processors. The transputcr providcs an idcal unit for inexpensive, high power parallcl computers which can perform the algorithm in rcasonablc time and for real moleculcs of biological interest. The transputcr TSOO manufactured by INMOS [2] combines a fast 32 bit RISC (rcduccd instruction sct) proccssor (10 MIPS), fast static mcinory (4 Kb of on-chip RAM), a 61-bit floating point coproccssor (which can opcrate concurrcnlly with thc
22
ccntral proccssor) and four vcry fast (20 Mbit/s) bidirectional scrial communication links on onc chip. Transputcrs usc thc links for synchronous point-to-point communications with cach othcr. Thc links can bc switchcd so as to crcate any nctwork configuration. Some applications of the transputer arrays in the fields of computational chemistry, physics and biophysics havc bccn rcccntly dcscribcd [3-53. Thcsc studics clcarly dcmonstratcd the cfficicncy of the transputer architecture. It was shown that cvcn IBM AT or microVAX computcrs cquippcd with a transputer boards can be useful for such a proccssing-intcnsive problems like Monte Carlo [3], direct SCF [4]and biomolccular cncrgy [5] calculations. This work prcscnts a mcthod of parallel sum-over-state computing of rcsonancc Raman and absorption cross scctions using a transputer array. Wc havc implcmcntcd this approach to dircct modcling of thc cxpcrimcntal absorption spectrum of adcnosinc triphosphatc (ATP) and RR cxcilation profilc of ATP in watcr solution. As a rcsult, the sct of cxcitcd statc paramctcrs of ATP that provide the bcst fit to the expcrimcntal data has bccn obtaincd.
2. Computing resonance Raman and absorption cross scctions Thc rcsonancc Raman cross section in the Condon approximation can bc writtcn as thc convcntional vibronic sum over stiltcs [I]:
6.
i+f
=
c,
I,
< f l v > < vli> M E , E L C E v - E i + E 0 - E L -ir
Iz
Hcrc I i >, I v > and IS> arc thc initial, intermediate, and final vibrational states; ~i and E , arc thc cncrgics (in cm-1) of thc statcs I i > and I v >; M is the clcctronic transition Icngth; E, and EL are Lhc cncrgics of thc incidcnt and scattered photons; EO is thc cncrgy scparalion bclwccn the lowcst vibrational lcvcls of the ground and excited clcctronic statcs (zcro-zcro cncrgy); is the homogeneous linewidth (in cm-I), and C,, is a constant. The corrcsponding cxprcssion for thc absorption cross section is:
a,(E L ) =
c,
2
I< vI i >I 2
r
M ELE v X
(Ev-
Ei
+ Eo-
E L )2 +
12
Thc absorption cross scction OA (A2/molccule) is rclatcd to thc molar absorptivity E (M-l cm-I) by 0 ,= 2.303 10I9 €IN,, where iVA is Avogadro’s numbcr. In thc simplcst approach thc vibrational frequencics and normal coortiinatcs arc idcntical in thc ground and cxcitcd statcs, and a system of nmod vibrational modcs can bc
23
trcatcd as a collection of N indcpcndcnt pairs of harmonic oscillators with frequencies a k (k = 1, ..., nrnod). Therefore, multidimensional Frank-Condon factors can be written as the products of onc-dimcnsional overlaps: nmod
nmd
v > n < f i l v i>
and
i =1
E,-
.ci =
C
;=1
h n , ( v i - ij)
So, thc cquations (1) and (2) for fundamental resonance Raman and absorption cross scclions are given morc cxplicitly as follows:
O,(RL)=
c, M 2 E L rCC
(4)
v1 v 2
Hcrc the Raman active modc has subscript 1. One-dimcnsional Franck-Condon factors can bc calculatcd with a recursive rclation [6]. If there were no changes in vibrational frcqucncics (Q) in thc cxcitcd clcctronic state then the factors for each mode vk can be shown to bc only a function of its displacement dk in the excited state. In equations ( 3 ) and (4) nrnod vibrational modcs are included in the summation over quantum numbcrs vi = 0, 1, 2, ... of the uppcr clcctronic state for which the product of Franck-Condon factors excceds a cutoff lcvcl (wc uscd -104-105 times the magnitude of the zero-zero transition). Raman and absorption cross sections are also dependent on the environment effects bccausc in the condcnscd phase diffcrent scatterers may be either in different initial quantum states or in slightly diffcrcnt surroundings. Such phenomena lead to inhomogencous broadcning of RR excitation profiles and absorption spectra. The vibrational modcs which are active in resonance Raman don't undergo large displaccmcnts and frcqucncy shifts upon excitation. Thcrcforc, little error results from neglccting thcrmal effccts [l]. Variations in the local cnvironmcnt can be taken into account if a Gaussian distribution of zcro-zcro cncrgics with standard dcviation 0 (in cm-1) is proposcd to describe the sitc broadcning cffccts [7]:
24
where
3. Hardware and software environment The hardware used in the present work is shown in Figure 1. It consists of five NMOS transputers T800-25, each with 1 Mb RAM (the transputer array) of which one node (root transputer) is attached to a host personal computer AT 386-25 running under MS-DOS, A file server program (3L, Ltd.) placed on the PC controls the access of the transputer to the disk, the screen, the keyboard etc. The transputer processor has four N M O S links, to connect it with other transputers. Each link has two channels, one in each direction. They provide synchronized, unidirectional communication. The hardware configuration as well as logical interconnections between the processors and tasks were described by a configuration language included in 3L Parallel FORTRAN package [8]. As it can be seen from Figure 1, the server task placed on the host PC is not directly connected to the application program. The filter task is interposed
. . DATA
RES.
110 SERVER
...
I
#
’
COMPUTER
4
RES.
ROOTT1
DATA
t
RES.
TASK 4
T4 Figure 1. Hardware configuration of the transputer array.
RES.
T2
T5
25
between them. It runs in parallel with the server program and the application task and passes on messages traveling in both directions. Such a configuration file has the form: PROCESSOR Host PROCESSOR Root PROCESSOR T1
..........
PROCESSOR T4 WIRE ? Host[O] Root[O] WIRE ? Root[l] T I [O] WIRE ? Root[2] T2[0] WIRE ? Root[3] T3[0] WIRE ? T I [ l ] T4[0] TASK Afserver INS=l OUTS=l TASK Filter INS=2 OUTS=2 DATA=lOK
TASK I N S 2 O U T S 2 DATA=...K TASK I N S 1 O U T S 1 DATA=...K TASK
................ PLACE
The program for RR intcnsitics calculation was written in 3L Parallel FORTRAN.
4. Algorithm The general flowchart of the program is shown in Figure 2. The main algorithm placed on the root processor performs all input/output instructions and starts the nonlinear
26
INPUT DATA 1. starting values of the parameters to the optimized:d(i) (i=,... ,nrnod),pJ2. other parameters necessary for cross sections (resonance Raman or absorption) calculations 3. parameters of optimization process 4. I
exp
-experimental Rarnan (i=l ,...,N) cross sections
THE BODY OF OPTIMIZATION ALGORITHM (Dickinson's random search and/or Davidon-Fletcher-Powellmethods)
1
INTERMEDIATE VALUES OF d ( i ) , W
1
MESSAGE OF d(i),@,r
I
TO SLAVE TRANSPUTERS
CALCULATIONS OF CROSS SECTIONS OF NMOD VIBRATIONAL MODES ON ROOT AND SLAVE TRANSPUTERS
OPTIMAL VALUES OF d(i),8,T
h
TERMINATION MESSAGE TO SLAVE TRANSPUTERS
I
ROOT TRANSPUTER
CALCULATION OF U-FUNCTION ON ROOT TRANSPUTERS
I
Figurc 2. Gcncral flowchart of thc parallcl FOKTKAN program for calculation rcsonance Ranian and absorption cross scctions.
21
optimization with a Dickinson’s random search or/and Davidon-Fletcher-Powell gradicnt mcthods [9]. For random numbcr gcncration with the 32-bit processor wc have uscd the function RAN4 [lo]. The optimization function to be minimized was written in the form: for rcsonancc Raman:
for absorption:
,y ( d ,
0,r)=
U ( d , 0, r)=
2
nmod
C ( 17’- I$) i =1
N
i =1
1
( 17 - Il?)
I
2
whcrc d = dj 0’ = 1, ..., nmod)-the displacement of the excited state potential cncrgy curvc along thcjth normal coordinate; I pxP and correspond to cxpcrimcnlal and calculatcd Raman or absorption cross scctions, nmod is a number of the vibrational modcs (in this work nmod = 4 or 5), N is a numbcr of points in digitized absorption spectrum. According to Equation (5) we calculated inhomogeneous broadening as a finite sum (usually 50-100 stcps) ovcr a Gaussian distribution of zero-zero energies with standard dcviation 0. An avcragc value of Eg and a set of l f x P (i = 1, ..., nmod) for ATP were estimated from the analysis of high-resolution absorption spectra as well as detailed resonance Raman cxcitation profile (R.G. Efrcmov and A.V. Feofanov, unpublished data). At thc first stcp of computational scheme the excited state parameters ( d , 0,r> which givc the bcst fit to thc absorption spectrum wcrc found using the optimization-function U&(d, 0, 0. Thcn the refinement of the data obtained was performed to achieve the minimal discrcpancy between the experimental and calculated RR cross sections. Bcforc the calculation of optimization function at fixed values of d(i) (i = 1, ..., nmod), 0 and I-, thc main task scnds the messages containing the parameters d , 0,r to the “computational” tasks placcd on the slave processors T1 (for mode v2), T2 (for mode v3), ctc. and also pcrforms thc samc algorithm for the first vibrational modc ( y ) . So, thc data arc dividcd bctwccn fivc proccssors working in parallel. Onc of thc most important considcrations is that each Lransputcr performs a similar amount of work and all thc rcsourccs arc utilizcd at this stcp. Whcn the cross scctions (Raman or absorption) arc calculatcd, thc msks immcdiatcly scnd thc rcsults obtained to thc main module and thc ncxt stcp of optimization bcgins. It should bc notcd that all thc proccsscs arc synchronizcd, so, it’s impossiblc for thc main program to start the next iteration bcforc it rcccivcs thc mcssagcs from the slave transputcrs. Each Lransputcr operates indcpendcntly of the othcrs. Thc proccsscs on differcnt Lransputcrs synchronizc only if necessary, i.e., if thcre is a data transfcr bctwccn the tasks.
5. Speed and efficiency of parallel code Thc timing and cfficicncy of the parallcl implementation for calculation thc FU7 cross scclions offour and fivc vibrational modcs (nnzod = 4 or nmod = 5 ) are givcn in Table 1. The
28
TABLE 1 CPU times for computing the resonance Raman cross sections of 4 and 5 vibrational modes with a PC and PC-based transputer array. Number of modes
PC
CPU time, sec
T800 transputer array
Speed-up coeff.
Sequential code obtained with MS-FORTRAN4.10 compiler
Parallel code obtained with 3L Parallel FORTRAN compiler
4
1,352
7
193
5
20,610
101
204
actual speed-up obtained for this simulation using four and five transputers when comparing the elapsed time on a heavily used PC AT 386-25 are approximately 193 and 204 times, correspondently.Potentially greater reductions are available with increasing number of processors for molecules which contain more than five normal modes revealed in resonance Raman spectra. However, as the number of vibrational modes and transputers in the array is increased, the time for communication between transputers may become important. Thus, the efficiency will decrease as the communication time becomes more dominant. For the tasks dividing the data bctwcen differentprocessors each of which performs the same calculation it was shown that the elapsed time drops linearly when the number of processors is less than 10 [Goodfellow JM [lo]]. Therefore, it is reasonably to suggest that for molecules with a large number of intensive Raman active modes (nmod > 20) the computational efficiency of thc transputer farm will not be so impressive and elapsed time will be almost independent of the number of proccssors.
6. Application to analysis of absorption and resonance Raman spectra of ATP We have followed the procedure described above to determine the excited state displacements (&)), homogeneous width (I)as well as inhomogeneous standard deviation (@ for 5 vibrational modes of ATP. Experimental absorption spectrum and calculated fit generated from the parameters estimated in the result of sum-over-statemodeling are shown in Figure 3. It is clearly seen that the experimental profile and the simulated one are in good agreement. There are only small discrepancies on the high-energy edge of thc absorption. At the next step wc uscd the parameters obtained and experimental resonance Raman spccua of ATP excited with different laser lines to calculate the refined set of the
29
6
Energy, cm -1
x10
Figure 3. Experimental (solid line) and calculated (dashed line) absorption spectra, for ATP. Calculated curve was generated using the parameters found by means of parallel computing procedure (di for 1302, 1333, 1482, 1510 and 1582 cm-1 vibrational modes are: 0.14, 1.13,0.55, 0.42 and 0.37; 8 = 750 cm-1; 725 cm-1; E c 37594 cm-1).
r=
Energy, ern-'
x10'
Figure 4. Experimental (solid line) and calculated (dashed line) resonance Raman excitation profiles of 1333 cm-1 mode of ATP. Calculated curve was generated using a set of best-fit excited state parameters (see Fig. 3). Experimental profile was taken from: Efremov RG and Feofanov AV, unpublished data.
30
paramctcrs which provide the best fit to resonance Raman absolute cross sections. The final bcst-fit resonance Raman cxcilation profile of ATP is shown in Figure 4. Thc calculatcd contour corrcctly rcproduccs thc main features of the expcrimcntal one. It is intcrcsting to note that the bcst-fit paramctcrs are very similar to those obtained from the analysis of absorption spcctrum. The details of cxpcrimental and calculated resonance Raman profilcs will be published in the nearest future.
7. Conclusion The use of five T-800 transputers connected in a farm allowcd the calculation of absorbance and resonance Raman cross sections of ATP molccule with a sum-over-slates tcchniquc. Large increascs in spccd wcrc obtained using FORTRAN code and with a simple approach to parallclism. The method proposed was shown to be crficient in scarching for the sct of cxcitcd statc parameters which give the best fit to the spcctral data-thc spcctra gcncratcd with such paramctcrs are in good agrccmcnt with thc cxpcrimcnlal oncs. Rcccntly wc hnvc succccdcd in obtaining the UV-RR spcctra of a functional complcx oC ATP with thc mcmbranc-bound protein Na+,K+-ATPasc[ 111. Thc substrate binding in thc cnzymc’s active sitc has bccn shown to bc accompanicd with significant changcs in thc clcctronic-vibrational structure of the adcnine ring of ATP. Thc computing prcscntcd in this study is an indispcnsablc step in solving thc problcm of intcrprcmtion of Lhe spcctral changcs induced by thc ATP binding in the activc sitc of Na+,K+-ATPasc.
References Mycrs AR, Mathics RA. In: Spiro TG, Ed. Biological Applicalions of Raman Spectroscopy. Vol. 2. New York: Wilcy, 1987, 1-58. 2. Homcwood M. May D, Shcphcrd D, Shcphcrd K.IEEE Micro 1987; 7: 10. 3. Gorrod MJ, Coc MJ, Ycarworth M. Nuclear Instrum and Methods in Physics Res 1089; A281: 156. 4. LVcdig U, Rurkhartlt A, v. Sclmcring HG. 2 Phys D 1989; 13: 377. 5. Goodfellow JM, Vovcllc F. Eur Biophys J 1989; 17: 167. 6. hlanncback C. Physica 1951; 17: 1001. 7. Mycrs AB,H a m s KA, Mathics RA. J Chem Phys 1983; 79: 603. 8. Parallel FORYRAN Uscr Guide 2.0. 3L Ltd., I’ccl Housc, Ladywell, Livingston EH53 6AG, S c o h x i , 1988. 9. Siddal JN. “OPTISEP” Designer’s Optimizalion Subroutines (ME’I71IDSNIREPl). Faculty of Enginccring, McMastcr Univcrsity, Hamilton, Ontario, Canada, 1971. 10. Prcss WH, Flanncry BP, Tcukolsky SA, Vcttcrling WT. hrumerical Rccipes. The Art of ScicnliJic Compuling. Carnbridgc: Cambridge University I’rcss, 1986,215. 11. Efrcmov KG, Fcofanov AV, Dzhandzliugazyan KN, Modyanov NN,Nabicv IR. FEOS LcIt 1990; 260: 257. 1.
21
CHAPTER 3
Parallel Computing of Resonance Raman Intensities Using a Transputer Array R.G. Efremov Shemyakin Institute of Bioorganic Chemistry, USSR Academy of Sciences, ul. MiklukhoMaklaya 16/10, 117871 Moscow, V-437,USSR
1. Introduction Recent progress in optical spectroscopy of biological molecules is closely connected to the development of the computcr power. The computational approaches have a great importance in solving the problems of spectral processing, interpretation and simulation. For example, computing the resonance Raman (RR) intensities of large molecules provides detailed information about the equilibrium geometry and dynamics of the resonant excited electronic state. The traditional approach to calculating the RR cross section involves a summation over all the vibrational levels of the resonant electronic state. Such sum-over-state algorithm performs a direct search for the complete set of excited state parameters that gcncratc the best fit to the expcrimcntal RR data. Usually, when there are only a fcw RR active modcs, the sum-over-state method is much more efficient compared with alternative approaches [I]. But in gcncral, using a standard sequential sum-over-state procedure can be computationally intractable for large (especially, biological) moleculcs because the algorithm scts up calculations the Raman cross sections of N vibrational modes of the molcculc and for each mode there are N logically nested loops. Therefore, such a technique demands powerful computer resourccs. Thc problem becomes processing-intensive espccially if a hard optimization is required-in a case when the initial estimates of excitcd slate paramctcrs are not known exactly. An altcrnativc approach to solving this task is based on the fact that the intensities of all modcs can bc calculalcd indepcndcntly at difkrent parallcl processors. The transputcr providcs an idcal unit for inexpensive, high power parallcl computers which can perform the algorithm in rcasonablc time and for real moleculcs of biological interest. The transputcr TSOO manufactured by INMOS [2] combines a fast 32 bit RISC (rcduccd instruction sct) proccssor (10 MIPS), fast static mcinory (4 Kb of on-chip RAM), a 61-bit floating point coproccssor (which can opcrate concurrcnlly with thc
22
ccntral proccssor) and four vcry fast (20 Mbit/s) bidirectional scrial communication links on onc chip. Transputcrs usc thc links for synchronous point-to-point communications with cach othcr. Thc links can bc switchcd so as to crcate any nctwork configuration. Some applications of the transputer arrays in the fields of computational chemistry, physics and biophysics havc bccn rcccntly dcscribcd [3-53. Thcsc studics clcarly dcmonstratcd the cfficicncy of the transputer architecture. It was shown that cvcn IBM AT or microVAX computcrs cquippcd with a transputer boards can be useful for such a proccssing-intcnsive problems like Monte Carlo [3], direct SCF [4]and biomolccular cncrgy [5] calculations. This work prcscnts a mcthod of parallel sum-over-state computing of rcsonancc Raman and absorption cross scctions using a transputer array. Wc havc implcmcntcd this approach to dircct modcling of thc cxpcrimcntal absorption spectrum of adcnosinc triphosphatc (ATP) and RR cxcilation profilc of ATP in watcr solution. As a rcsult, the sct of cxcitcd statc paramctcrs of ATP that provide the bcst fit to the expcrimcntal data has bccn obtaincd.
2. Computing resonance Raman and absorption cross scctions Thc rcsonancc Raman cross section in the Condon approximation can bc writtcn as thc convcntional vibronic sum over stiltcs [I]:
6.
i+f
=
c,
I,
< f l v > < vli> M E , E L C E v - E i + E 0 - E L -ir
Iz
Hcrc I i >, I v > and IS> arc thc initial, intermediate, and final vibrational states; ~i and E , arc thc cncrgics (in cm-1) of thc statcs I i > and I v >; M is the clcctronic transition Icngth; E, and EL are Lhc cncrgics of thc incidcnt and scattered photons; EO is thc cncrgy scparalion bclwccn the lowcst vibrational lcvcls of the ground and excited clcctronic statcs (zcro-zcro cncrgy); is the homogeneous linewidth (in cm-I), and C,, is a constant. The corrcsponding cxprcssion for thc absorption cross section is:
a,(E L ) =
c,
2
I< vI i >I 2
r
M ELE v X
(Ev-
Ei
+ Eo-
E L )2 +
12
Thc absorption cross scction OA (A2/molccule) is rclatcd to thc molar absorptivity E (M-l cm-I) by 0 ,= 2.303 10I9 €IN,, where iVA is Avogadro’s numbcr. In thc simplcst approach thc vibrational frequencics and normal coortiinatcs arc idcntical in thc ground and cxcitcd statcs, and a system of nmod vibrational modcs can bc
23
trcatcd as a collection of N indcpcndcnt pairs of harmonic oscillators with frequencies a k (k = 1, ..., nrnod). Therefore, multidimensional Frank-Condon factors can be written as the products of onc-dimcnsional overlaps: nmod
nmd
v > n < f i l v i>
and
i =1
E,-
.ci =
C
;=1
h n , ( v i - ij)
So, thc cquations (1) and (2) for fundamental resonance Raman and absorption cross scclions are given morc cxplicitly as follows:
O,(RL)=
c, M 2 E L rCC
(4)
v1 v 2
Hcrc the Raman active modc has subscript 1. One-dimcnsional Franck-Condon factors can bc calculatcd with a recursive rclation [6]. If there were no changes in vibrational frcqucncics (Q) in thc cxcitcd clcctronic state then the factors for each mode vk can be shown to bc only a function of its displacement dk in the excited state. In equations ( 3 ) and (4) nrnod vibrational modcs are included in the summation over quantum numbcrs vi = 0, 1, 2, ... of the uppcr clcctronic state for which the product of Franck-Condon factors excceds a cutoff lcvcl (wc uscd -104-105 times the magnitude of the zero-zero transition). Raman and absorption cross sections are also dependent on the environment effects bccausc in the condcnscd phase diffcrent scatterers may be either in different initial quantum states or in slightly diffcrcnt surroundings. Such phenomena lead to inhomogencous broadcning of RR excitation profiles and absorption spectra. The vibrational modcs which are active in resonance Raman don't undergo large displaccmcnts and frcqucncy shifts upon excitation. Thcrcforc, little error results from neglccting thcrmal effccts [l]. Variations in the local cnvironmcnt can be taken into account if a Gaussian distribution of zcro-zcro cncrgics with standard dcviation 0 (in cm-1) is proposcd to describe the sitc broadcning cffccts [7]:
24
where
3. Hardware and software environment The hardware used in the present work is shown in Figure 1. It consists of five NMOS transputers T800-25, each with 1 Mb RAM (the transputer array) of which one node (root transputer) is attached to a host personal computer AT 386-25 running under MS-DOS, A file server program (3L, Ltd.) placed on the PC controls the access of the transputer to the disk, the screen, the keyboard etc. The transputer processor has four N M O S links, to connect it with other transputers. Each link has two channels, one in each direction. They provide synchronized, unidirectional communication. The hardware configuration as well as logical interconnections between the processors and tasks were described by a configuration language included in 3L Parallel FORTRAN package [8]. As it can be seen from Figure 1, the server task placed on the host PC is not directly connected to the application program. The filter task is interposed
. . DATA
RES.
110 SERVER
...
I
#
’
COMPUTER
4
RES.
ROOTT1
DATA
t
RES.
TASK 4
T4 Figure 1. Hardware configuration of the transputer array.
RES.
T2
T5
25
between them. It runs in parallel with the server program and the application task and passes on messages traveling in both directions. Such a configuration file has the form: PROCESSOR Host PROCESSOR Root PROCESSOR T1
..........
PROCESSOR T4 WIRE ? Host[O] Root[O] WIRE ? Root[l] T I [O] WIRE ? Root[2] T2[0] WIRE ? Root[3] T3[0] WIRE ? T I [ l ] T4[0] TASK Afserver INS=l OUTS=l TASK Filter INS=2 OUTS=2 DATA=lOK
TASK
................ PLACE
The program for RR intcnsitics calculation was written in 3L Parallel FORTRAN.
4. Algorithm The general flowchart of the program is shown in Figure 2. The main algorithm placed on the root processor performs all input/output instructions and starts the nonlinear
26
INPUT DATA 1. starting values of the parameters to the optimized:d(i) (i=,... ,nrnod),pJ2. other parameters necessary for cross sections (resonance Raman or absorption) calculations 3. parameters of optimization process 4. I
exp
-experimental Rarnan (i=l ,...,N) cross sections
THE BODY OF OPTIMIZATION ALGORITHM (Dickinson's random search and/or Davidon-Fletcher-Powellmethods)
1
INTERMEDIATE VALUES OF d ( i ) , W
1
MESSAGE OF d(i),@,r
I
TO SLAVE TRANSPUTERS
CALCULATIONS OF CROSS SECTIONS OF NMOD VIBRATIONAL MODES ON ROOT AND SLAVE TRANSPUTERS
OPTIMAL VALUES OF d(i),8,T
h
TERMINATION MESSAGE TO SLAVE TRANSPUTERS
I
ROOT TRANSPUTER
CALCULATION OF U-FUNCTION ON ROOT TRANSPUTERS
I
Figurc 2. Gcncral flowchart of thc parallcl FOKTKAN program for calculation rcsonance Ranian and absorption cross scctions.
21
optimization with a Dickinson’s random search or/and Davidon-Fletcher-Powell gradicnt mcthods [9]. For random numbcr gcncration with the 32-bit processor wc have uscd the function RAN4 [lo]. The optimization function to be minimized was written in the form: for rcsonancc Raman:
for absorption:
,y ( d ,
0,r)=
U ( d , 0, r)=
2
nmod
C ( 17’- I$) i =1
N
i =1
1
( 17 - Il?)
I
2
whcrc d = dj 0’ = 1, ..., nmod)-the displacement of the excited state potential cncrgy curvc along thcjth normal coordinate; I pxP and correspond to cxpcrimcnlal and calculatcd Raman or absorption cross scctions, nmod is a number of the vibrational modcs (in this work nmod = 4 or 5), N is a numbcr of points in digitized absorption spectrum. According to Equation (5) we calculated inhomogeneous broadening as a finite sum (usually 50-100 stcps) ovcr a Gaussian distribution of zero-zero energies with standard dcviation 0. An avcragc value of Eg and a set of l f x P (i = 1, ..., nmod) for ATP were estimated from the analysis of high-resolution absorption spectra as well as detailed resonance Raman cxcitation profile (R.G. Efrcmov and A.V. Feofanov, unpublished data). At thc first stcp of computational scheme the excited state parameters ( d , 0,r> which givc the bcst fit to thc absorption spectrum wcrc found using the optimization-function U&(d, 0, 0. Thcn the refinement of the data obtained was performed to achieve the minimal discrcpancy between the experimental and calculated RR cross sections. Bcforc the calculation of optimization function at fixed values of d(i) (i = 1, ..., nmod), 0 and I-, thc main task scnds the messages containing the parameters d , 0,r to the “computational” tasks placcd on the slave processors T1 (for mode v2), T2 (for mode v3), ctc. and also pcrforms thc samc algorithm for the first vibrational modc ( y ) . So, thc data arc dividcd bctwccn fivc proccssors working in parallel. Onc of thc most important considcrations is that each Lransputcr performs a similar amount of work and all thc rcsourccs arc utilizcd at this stcp. Whcn the cross scctions (Raman or absorption) arc calculatcd, thc msks immcdiatcly scnd thc rcsults obtained to thc main module and thc ncxt stcp of optimization bcgins. It should bc notcd that all thc proccsscs arc synchronizcd, so, it’s impossiblc for thc main program to start the next iteration bcforc it rcccivcs thc mcssagcs from the slave transputcrs. Each Lransputcr operates indcpendcntly of the othcrs. Thc proccsscs on differcnt Lransputcrs synchronizc only if necessary, i.e., if thcre is a data transfcr bctwccn the tasks.
5. Speed and efficiency of parallel code Thc timing and cfficicncy of the parallcl implementation for calculation thc FU7 cross scclions offour and fivc vibrational modcs (nnzod = 4 or nmod = 5 ) are givcn in Table 1. The
28
TABLE 1 CPU times for computing the resonance Raman cross sections of 4 and 5 vibrational modes with a PC and PC-based transputer array. Number of modes
PC
CPU time, sec
T800 transputer array
Speed-up coeff.
Sequential code obtained with MS-FORTRAN4.10 compiler
Parallel code obtained with 3L Parallel FORTRAN compiler
4
1,352
7
193
5
20,610
101
204
actual speed-up obtained for this simulation using four and five transputers when comparing the elapsed time on a heavily used PC AT 386-25 are approximately 193 and 204 times, correspondently.Potentially greater reductions are available with increasing number of processors for molecules which contain more than five normal modes revealed in resonance Raman spectra. However, as the number of vibrational modes and transputers in the array is increased, the time for communication between transputers may become important. Thus, the efficiency will decrease as the communication time becomes more dominant. For the tasks dividing the data bctwcen differentprocessors each of which performs the same calculation it was shown that the elapsed time drops linearly when the number of processors is less than 10 [Goodfellow JM [lo]]. Therefore, it is reasonably to suggest that for molecules with a large number of intensive Raman active modes (nmod > 20) the computational efficiency of thc transputer farm will not be so impressive and elapsed time will be almost independent of the number of proccssors.
6. Application to analysis of absorption and resonance Raman spectra of ATP We have followed the procedure described above to determine the excited state displacements (&)), homogeneous width (I)as well as inhomogeneous standard deviation (@ for 5 vibrational modes of ATP. Experimental absorption spectrum and calculated fit generated from the parameters estimated in the result of sum-over-statemodeling are shown in Figure 3. It is clearly seen that the experimental profile and the simulated one are in good agreement. There are only small discrepancies on the high-energy edge of thc absorption. At the next step wc uscd the parameters obtained and experimental resonance Raman spccua of ATP excited with different laser lines to calculate the refined set of the
29
6
Energy, cm -1
x10
Figure 3. Experimental (solid line) and calculated (dashed line) absorption spectra, for ATP. Calculated curve was generated using the parameters found by means of parallel computing procedure (di for 1302, 1333, 1482, 1510 and 1582 cm-1 vibrational modes are: 0.14, 1.13,0.55, 0.42 and 0.37; 8 = 750 cm-1; 725 cm-1; E c 37594 cm-1).
r=
Energy, ern-'
x10'
Figure 4. Experimental (solid line) and calculated (dashed line) resonance Raman excitation profiles of 1333 cm-1 mode of ATP. Calculated curve was generated using a set of best-fit excited state parameters (see Fig. 3). Experimental profile was taken from: Efremov RG and Feofanov AV, unpublished data.
30
paramctcrs which provide the best fit to resonance Raman absolute cross sections. The final bcst-fit resonance Raman cxcilation profile of ATP is shown in Figure 4. Thc calculatcd contour corrcctly rcproduccs thc main features of the expcrimcntal one. It is intcrcsting to note that the bcst-fit paramctcrs are very similar to those obtained from the analysis of absorption spcctrum. The details of cxpcrimental and calculated resonance Raman profilcs will be published in the nearest future.
7. Conclusion The use of five T-800 transputers connected in a farm allowcd the calculation of absorbance and resonance Raman cross sections of ATP molccule with a sum-over-slates tcchniquc. Large increascs in spccd wcrc obtained using FORTRAN code and with a simple approach to parallclism. The method proposed was shown to be crficient in scarching for the sct of cxcitcd statc parameters which give the best fit to the spcctral data-thc spcctra gcncratcd with such paramctcrs are in good agrccmcnt with thc cxpcrimcnlal oncs. Rcccntly wc hnvc succccdcd in obtaining the UV-RR spcctra of a functional complcx oC ATP with thc mcmbranc-bound protein Na+,K+-ATPasc[ 111. Thc substrate binding in thc cnzymc’s active sitc has bccn shown to bc accompanicd with significant changcs in thc clcctronic-vibrational structure of the adcnine ring of ATP. Thc computing prcscntcd in this study is an indispcnsablc step in solving thc problcm of intcrprcmtion of Lhe spcctral changcs induced by thc ATP binding in the activc sitc of Na+,K+-ATPasc.
References Mycrs AR, Mathics RA. In: Spiro TG, Ed. Biological Applicalions of Raman Spectroscopy. Vol. 2. New York: Wilcy, 1987, 1-58. 2. Homcwood M. May D, Shcphcrd D, Shcphcrd K.IEEE Micro 1987; 7: 10. 3. Gorrod MJ, Coc MJ, Ycarworth M. Nuclear Instrum and Methods in Physics Res 1089; A281: 156. 4. LVcdig U, Rurkhartlt A, v. Sclmcring HG. 2 Phys D 1989; 13: 377. 5. Goodfellow JM, Vovcllc F. Eur Biophys J 1989; 17: 167. 6. hlanncback C. Physica 1951; 17: 1001. 7. Mycrs AB,H a m s KA, Mathics RA. J Chem Phys 1983; 79: 603. 8. Parallel FORYRAN Uscr Guide 2.0. 3L Ltd., I’ccl Housc, Ladywell, Livingston EH53 6AG, S c o h x i , 1988. 9. Siddal JN. “OPTISEP” Designer’s Optimizalion Subroutines (ME’I71IDSNIREPl). Faculty of Enginccring, McMastcr Univcrsity, Hamilton, Ontario, Canada, 1971. 10. Prcss WH, Flanncry BP, Tcukolsky SA, Vcttcrling WT. hrumerical Rccipes. The Art of ScicnliJic Compuling. Carnbridgc: Cambridge University I’rcss, 1986,215. 11. Efrcmov KG, Fcofanov AV, Dzhandzliugazyan KN, Modyanov NN,Nabicv IR. FEOS LcIt 1990; 260: 257. 1.