**Implementation of CD
spectra fit **

The problem is quite straight forward, namely to read data, baseline, standard curves and some experimental details and list the results of the fit in a format which can be plotted easily using any spreadsheet program. The program checks for proper range of data, applies a baseline correction if needed, and attempts to solve the fit by a fast general LSQ method. If that procedure returns a negative coefficient, the program imposes positivity on the problem by iteratively hacking through all possible combinations and flags the best one. Final output is a listing of statistical parameters, data lists and a spline to the data, just for exercise.

First, the data, baseline, standard and header files are read and the overlapping wavelength range determined. After reading the files, finding the proper range of data, and calculating the conversion factors from sample ellipticity to mean residual ellipticity in [deg.cm2/(dmole.residue)] we put out the header:

c --- knock out header CDFI0200 rres=float(ires) CDFI0201 write (8,'(a)') ' -----------------------------------------------'CDFI0202 write (8,'(a)') ' CD DATA fitted by CD-FIT program (B.Rupp, 1996)'CDFI0203 write (8,'(a)') ' -----------------------------------------------'CDFI0204 write(8,'(a,a)') ' CD raw data file : ',datfil CDFI0205 write(8,'(a,a)') ' CD baseline file : ',basfil CDFI0206 write(8,'(a,a)') ' Sample header file : ',hedfil CDFI0207 write(8,'(a,a)') ' Ellipticity output file : ',outfil CDFI0208 write(8,'(a,a)') ' Results file : ',resfil CDFI0209 write(8,'(a,a/)') ' Standard data file : ',lysfil CDFI0210 write(8,'(i3,a,i3,a,i3,a)') npts,' data points from ', CDFI0211 & ilo,' to ',ihi,' nm' CDFI0212 write(8,*) '--------------------------------------------' CDFI0213 write(8,'(a,f10.2)') ' Molecular weight g/mol : ',mw CDFI0214 write(8,'(a,f10.6)') ' Sample concentration mg/cm3 : ',cconc CDFI0215 write(8,'(a,f10.8)') ' Sample concentration g/cm3 : ',conc CDFI0216 write(8,'(a,f10.8)') ' Sample concentration mol/cm3 : ',cmol CDFI0217 write(8,'(a,f10.6)') ' Cell path mm : ',ppath CDFI0218 write(8,'(a,f10.6)') ' Cell path cm : ',path CDFI0219 write(8,'(a,f10.6)') ' Number of residues : ',rres CDFI0220 CDFI0240 c --- in deg*cm2/dmol CDFI0241 dfact=mw/(10.0*conc*path) CDFI0242 write(8,'(a,f12.2)') ' Conversion factor [cm2/dmole] :', CDFI0244 & dfact CDFI0245 write(8,*) '---------------------------------------------' CDFI0246

We perform baseline subtraction and a final baseline correction
assuming that, based on the model we are using, the signal
between 245 and 250 nm should be zero.

c --- create the CD data from bas and knp and 18/pi CDFI0252 itest=0 CDFI0253 do i=ilo,ihi CDFI0254 cd(i)=(knp(i)-bas(i))/raddml CDFI0255 c --- mean residual ellipticty CDFI0256 elli(i)=cd(i)*dfact/ires CDFI0257 if(itest.eq.1) write(*,*) i,cd(i),elli(i) CDFI0258 end do CDFI0259 CDFI0260 c --- create zero shift only if sufficient data available CDFI0261 if (ihi.ge.250) then CDFI0262 iavg=ihi-245 CDFI0263 do j=250-iavg,250 CDFI0264 sumbas=sumbas+elli(j) CDFI0265 end do CDFI0266 sumbas=sumbas/(iavg+1) CDFI0267 do j=ilo,ihi CDFI0268 elli(j)=elli(j)-sumbas CDFI0269 end do CDFI0270 write (*,'(a,f5.0)') ' Zero shift applied : ',sumbas CDFI0271 write (8,'(1x,a,f5.0)') ' Zero shift applied : ',sumbas CDFI0272 else CDFI0273 sumbas=0.0 CDFI0274 write(*,*)'Cannot determine zero shift for given data range' CDFI0275 write(8,'(a)')'Cannot determine zero shift for data range' CDFI0276 end if CDFI0277

Now we are ready to set up the normal equations
by filling the coefficient matrix *A* with the
corresponding sums and call the Gauss-Jordan elimination to solve
the equations:

c --- calculate fit by LSQ procedure -----------------------------------CDFI0279 c CDFI0280 c A.x = b where A LSQ matrix CDFI0281 c j : data points (1=helix,2=sheet,3=coil) at lambda, sum is over CDFI0282 c all used lambda's, m=measured value at lamba CDFI0283 c A(i,j)=sum(i,j) 3x3 CDFI0284 c b(i)=sum(m*j) 1x3 CDFI0285 c CDFI0286 c ----------------------------------------------------------------------CDFI0287 a=0 CDFI0288 b=0 CDFI0289 x=0 CDFI0290 datapt(1,ilo:ihi)=helix(ilo:ihi) CDFI0291 datapt(2,ilo:ihi)=sheet(ilo:ihi) CDFI0292 datapt(3,ilo:ihi)=coil(ilo:ihi) CDFI0293 c --- fill normal equations CDFI0294 do i=1,3 CDFI0295 do k=ilo,ihi CDFI0296 b(i)=b(i)+elli(k)*datapt(i,k) CDFI0297 end do CDFI0298 do j=1,3 CDFI0299 do k=ilo,ihi CDFI0300 a(i,j)=a(i,j)+datapt(i,k)*datapt(j,k) CDFI0301 end do CDFI0302 end do CDFI0303 end do CDFI0304 call gaussj(a,3,3,b,1,1) CDFI0305

Now we have to make sure that the sum of the components are 100%
of the measured signal. As experimental errors as well as
simplified model assumptions usually give less than 100% for the
sum of the three components. If the result is really off, there
might be a problem with the data, or the fit by sum of ideal
helix, sheet and coil is questionable or inappropriate.

c --- check results and normalize CDFI0306 scfac=0 CDFI0307 ifit=0 CDFI0308 do i=1,3 CDFI0309 if (b(i).lt.0) then CDFI0310 ifit=1 CDFI0311 else CDFI0312 scfac=scfac+b(i) CDFI0313 end if CDFI0314 end do CDFI0315 b=b*100 CDFI0316 write(*,'(/a)')' Least squares percentage(s) and scale factor' CDFI0317 write(*,'(4f10.2)') b(1)/scfac,b(2)/scfac,b(3)/scfac,scfac CDFI0318 write(8,'(/a)')' Least squares percentage(s) and scale factor' CDFI0319 write(8,'(4f10.2)') b(1)/scfac,b(2)/scfac,b(3)/scfac,scfac CDFI0320

We need to check for another problem resulting from the
oversimplified model : One of the fit parameters can be negative
and the solution is physically not meaningful (see LSQ tutorial for other means to constrain the
solutions). We then resume a brute force calculation of every
possible combination of helix coil and sheet and check which one
is best.

Let's think about this a little : if we consider 2% accurate
enough for a CD fit (rest assured that this is a rather
optimistic assumption) we need to loop over 50*50*50 summations
of all datapoints, about 60 here. Compare the LSQ solution : all
that was needed was a 3 times loop over the datapoints to sum up
the LSQ matrix:. The iteration therefore needs about 42000 times
more operations than the LSQ refinement. Not an option for any
problem of complexity, but still feasable in this case.

if (ifit) then CDFI0322 write (*,'(a)')' Negative LSQ parameters - cannot solve by LSQ'CDFI0323 write (*,'(a/)')' Will resume with iterative procedure ' CDFI0324 write (8,'(a)')' Negative LSQ parameters - cannot solve by LSQ'CDFI0325 write (8,'(a/)')' Will resume with iterative procedure ' CDFI0326 c --- calculate fit by iteration --- CDFI0327 srs=1.0E10 CDFI0328 aa=0 CDFI0329 bb=0 CDFI0330 cc=0 CDFI0331 iprec=2 CDFI0332 write(*,'(a$)')' Fitting' CDFI0333 do i=0,100,iprec CDFI0334 write(*,'(a$)')'.' CDFI0335 do j=0,100,iprec CDFI0336 do k=0,100,iprec CDFI0337 srs2=0 CDFI0338 do m=ilo,ihi CDFI0339 difa=elli(m)-0.01*(i*helix(m)+j*sheet(m)+k*coil(m))CDFI0340 srs2=srs2+difa**2 CDFI0341 end do CDFI0342 srs2=sqrt(srs2) CDFI0343 if (srs2.lt.srs) then CDFI0344 srs=srs2 CDFI0345 aa=i CDFI0346 bb=j CDFI0347 cc=k CDFI0348 end if CDFI0349 end do CDFI0350 end do CDFI0351 end do CDFI0352 scfac=(aa+bb+cc)/100 CDFI0353 else CDFI0354 aa=b(1) CDFI0355 bb=b(2) CDFI0356 cc=b(3) CDFI0357 end if CDFI0358 write (*,'(/a)') ' ' CDFI0359

Finally, we compute the r.m.s.d. of the fit
and a R-value and print out the results,
statistics, and fitted datat.

c --- compute the data values and differences for best fit CDFI0361 sdif=0 CDFI0362 sobs=0 CDFI0363 srs=0 CDFI0364 do j=ilo,ihi CDFI0365 cdft(j)=0.01*(aa*helix(j)+bb*sheet(j)+cc*coil(j)) CDFI0366 dif(j)=elli(j)-cdft(j) CDFI0367 sdif=sdif+abs(dif(j)) CDFI0368 srs=srs+dif(j)**2 CDFI0369 sobs=sobs+abs(elli(j)) CDFI0370 end do CDFI0371 rfac=100*sdif/sobs CDFI0372 rmsd=sqrt(srs)/(ihi-ilo) CDFI0373 CDFI0374 c --- write out statistics CDFI0375 write(*,*)' rmsd helix sheet coil scale' CDFI0376 write(*,'(5f10.2)') rmsd,aa,bb,cc,scfac CDFI0377 write(8,*)' rmsd helix sheet coil scale' CDFI0378 write(8,'(5f10.2)') rmsd,aa,bb,cc,scfac CDFI0379 CDFI0380 write(*,*)' Rfac % helix sheet coil' CDFI0381 write(*,'(4f10.2)') rfac,aa/scfac,bb/scfac,cc/scfac CDFI0382 write(8,*)' Rfac % helix sheet coil' CDFI0383 write(8,'(4f10.2)') rfac,aa/scfac,bb/scfac,cc/scfac CDFI0384 CDFI0385 write(8,*) '--------------------------------------------------' CDFI0469 write(8,*) 'lambda data fit delta spline' CDFI0470 c --- write the data file -- CDFI0471 do j=ilo,ihi CDFI0472 write(6,'(f6.1,4(1x,f10.1))') wav(j),elli(j),cdft(j),dif(j), CDFI0473 & spln(j) CDFI0474 write(8,'(f6.1,4(1x,f10.1))') wav(j),elli(j),cdft(j),dif(j), CDFI0475 & spln(j) CDFI0476 end do CDFI0477

Note that the actual program also calculates a cubic spline for the experimental data. This is not needed and you can throw the spline out if you do not have the IMSL library available. The *.res file contains the whole output for viewing, the *.cdf file the data only in a format suitable for plotting with a spreadsheet or similar application.

**Back to Introduction**

**This World Wide Web site conceived and maintained by
Bernhard Rupp**

Last revised
Dezember 27, 2009 01:40