Re: [sr #3154] Implementation of Baldwin (2014) B14 model - 2-site exact solution model for all time scalesRe: [sr #3154] Implementation of Baldwin (2014) B14 model - 2-site exact solution model for all time scales|
Dear Troels,
The existence of typos is exceedingly probable! The editors have added more errors in the transcrpition, so it'll be curious to see how accurate the final draft will be. Your comments come just in time for alterations. So I'll pay a pint of beer for every identified error. I didn't use that particular supplementary section to directly calculate anything, so this is the place in the manu where errors are most likely to occur. I do note the irony that this is probably one of few bits in the article that people are likely to read. 1) Yep - that's a typo. alpha- should definitely be KEG-KGE (score 1 pint). 2) Yep! zs should be zetas (score 1 pint). 3) This one is more interesting. Carver and Richard's didn't define a deltaR2. To my knowledge, I think Nikolai was the first to do that in his 2002 paper on reconstructing invisible states. So here, I think you'll find that my definition is the correct one. There is a typo in the Carver Richard's equation. I've just done a brief lit review, and as far as I can see, this mistake has propagated into every transcription of the equation since. I should note that in the manu in this section. Likely this has gone un-noticed as differences in R2 are difficult to measure by CPMG, instead R2A=R2B being assumed. Though thanks to Flemming, it is now possible to break this assumption: J Am Chem Soc. 2009 Sep 9;131(35):12745-54. doi: 10.1021/ja903897e. Also, in the Ishima and Torchia paper looking at the effects of relaxation differences: Journal of biomolecular NMR. 2006 34(4):209-19 I understand their derivations started from Nikolai's formula, which is exact, so the error wouldn't come to light there either, necessarily. To prove the point try this: numerically evaluate R2,effs over a big grid of parameters, and compare say Nikolai's formula to your implementation of Carver Richard's with a +deltaR2 and a -deltaR2 in the alpha- parameter. Which agrees better? You'll see that dR2=R2E-R2G being positive wins. The attached should also help. It's some test code I used for the paper that has been briefly sanitised. In it, the numerical result, Carver Richards, Meiboom, Nikolai's and my formula are directly compared for speed (roughly) and precision. The code added for Nikolai's formula at higher precision was contributed by Nikolai after the two of us discussed some very odd results that can come from his algorithm. Turns out evaluating his algorithm at double precision (roughly 9 decimal places, default in python and matlab) is insuffficient. About 23 decimal places is necessary for exactness. This is what is discussed in the supplementary info in the recent paper. In short, on my laptop: formula / relative speed / biggest single point error in s-1 over grid Mine 1 7.2E-10 Meiboom 0.3 9E7 Carver Richards 0.53 8 Nikolai (double precision 9dp) 3.1 204 Nikolai (long double 18dp) 420 3E-2 Nikolai (23 dp) 420 2E-7 Numerical 118 0 The error in Carver Richard's with the alpha- sign made consistent with the Carver Richard's paper gives an error of 4989 s-1 in the same test. Ie, it's a bad idea. Note that the precise calculations in python at arbitrary precision are slow not because of the algorithm, but because of the mpmath module, which deals with C code via strings. Accurate but slow. This code has a small amount of additional overhead that shouldn't strictly be included in the time. I've no doubt also that algorithms here could be sped up further by giving them a bit of love. Those prefaces aside, I'm surprised that your benchmarking last Friday showed Nikolai's to be so much slower than mine? In my hands (attached) at double precision, Nikolai's formula is only 3x slower than mine using numpy. Though this is not adequate precision for mistake free calculations. Perhaps more physically compelling, to get my equation to reduce to Carver Richard's, alpha- has to be defined as I show it. Note that the origin of this parameter is its role in the Eigenvalue of the 2x2 matrix. This is the one advantage of a derivation - you can see where the parameters come from and get a feel for what their meaning is. So this typo, and the fact that Carver and Richards inadvertently change their definition of tau_cpmg half way through the paper, are both errors in the original manuscript. Though given that proof reading and type-setting in 1972 would have been very very painful, that there are this few errors is still actually quite remarkable. I notice in your CR72 code, you are currently using the incorrect definition, so I would suggest you change that. So overall, I think you're still ahead, so I'm calling that 2 pints to Troels. I would actually be very grateful if you can bring up any other inconsistencies in the paper! Also from an editorial perspective, please feel free to note if there are parts of the paper that were particularly unclear when reading. If you let me know, there's still time to improve it. Thusfar, I think you are the third person to read this paper (the other two being the reviewers who were forced to read it), and you're precisely its target demographic! Best, Andy. On 03/05/2014 13:54, Troels Emtekær Linnet wrote: Dear Andrew Baldwin. I am in the process of implementing your code in relax. I am getting myself into a terrible mess by trying to compare equations in papers, code and conversions. But I hope you maybe have a little time to clear something up? I am looking at the current version of your paper: dx.doi.org/10.1016/j.jmr.2014.02.023 which still is still "early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form." 1) Equations are different ? In Appendix 1 – recipe for exact calculation of R2,eff h1 = 2 dw (dR2 + kEG - kGE) But when I compare to: Supplementary Section 4. Relation to Carver Richards equation. Equation 70. h1 = zeta = 2 dw alpha_ = 2 dw (dR2 + kGE - kEG) Is there a typo here? The GE and EG has been swapped. 2) Missing zeta in equation 70 There is missing \zeta instead of z in equation 70, which is probably a typesetting error. 3) Definition of " Delta R2" is opposite convention? Near equation 10, 11 you define: delta R2 = R2e - R2g if we let e=B and g=A, then delta R2 = R2B - R2A And in equation 70: alpha_ = delta R2 + kGE - kEG but i CR original work, A8, alpha_ = R2A - R2B + kA - kB = - delta R2 + kGE - kEG So, that doesn't match? Sorry if these questions are trivial. But I hope you can help clear them up for me. :-) Best Troels 2014-05-01 10:07 GMT+02:00 Andrew Baldwin <andrew.baldwin@xxxxxxxxxxxxx>:The Carver and Richards code in relax is fast enough, though Troels might have an interest in squeezing out a little more speed. Though it would require different value checking to avoid NaNs, divide by zeros, trig function overflows (which don't occur for math.sqrt), etc. Any changes would have to be heavily documented in the manual, wiki, etc. The prove that the two definitions are equivalent is relatively straightforward. The trig-to-exp command in the brilliant (and free) symbolic program sage might prove helpful in doing that. For the tau_c, tau_cp, tau_cpmg, etc. definitions, comparing the relax code to your script will ensure that the correct definition is used. That's how I've made sure that all other dispersion models are correctly handled - simple comparison to other software. I'm only aware of two definitions though: tau_cpmg = 1 / (4 nu_cpmg) tau_cpmg = 1 / (2 nu_cpmg) tau_cpmg = 1/(nu_cpmg). Carver and Richard's switch definitions half way through the paper somewhere. What is the third? Internally in relax, the 1 / (4 nu_cpmg) definition is used. But the user inputs nu_cpmg. This avoids the user making this mistake as nu_cpmg only has one definition. I've seen people use nu_cpmg defined as the pulse frequency. It's just an error that students make when things aren't clear. I've often seen brave student from a lab that has never tried CPMG before do this. Without someone to tell them that this is wrong, it's not obvious to them that they've made a mistake. I agree with you that this is solved with good documentation. You guys are free to use my code (I don't mind the gnu license) or of course implement from scratch as needed. Cheers! For a valid copyright licensing agreement, you'll need text something along the lines of: "I agree to licence my contributions to the code in the file http://gna.org/support/download.php?file_id=20615 attached to http://gna.org/support/?3154 under the GNU General Public Licence, version three or higher." Feel free to copy and paste. No problem: "I agree to licence my contributions to the code in the file http://gna.org/support/download.php?file_id=20615 attached to http://gna.org/support/?3154 under the GNU General Public Licence, version three or higher." I'd like to note again though that anyone using this formula to fit data, though exact in the case of 2site exchange/inphase magnetisation, evaluated at double floating point precision should not be doing so! Neglecting scalar coupling/off resonance/spin flips/the details of the specific pulse sequence used will lead to avoidable foobars. I do see value in this, as described in the paper, as being a feeder for initial conditions for a more fancy implemenation. But beyond that, 'tis a bad idea. Ideally this should appear in big red letters in your (very nice!) gui when used. In relax, we allow the user to do anything though, via the auto-analysis (hence the GUI), we direct the user along the best path. The default is to use the CR72 and 'NS CPMG 2-site expanded' models (Carver & Richards, and Martin Tollinger and Nikolai Skrynnikov's Maple expansion numeric model). We use the CR72 model result as the starting point for optimisation of the numeric models, allowing a huge speed up in the analysis. The user can also choose to not use the CR72 model results for the final model selection - for determining dispersion curve significance. Here's the supp from my CPMG formula paper (attached). Look at the last figure. Maybe relevant. Nikolai's algorithm blows up sometimes when you evaluate to double float point precision (as you will when you have a version in python or matlab). The advantage of Nicolai's formula, or mine is that they won't fail when Pb starts to creep above a per cent or so. Using the simple formula as a seed for the more complex on is a good idea. The most recent versions of CATIA have something analogous. As for the scalar coupling and spin flips, I am unaware of any dispersion software that handles this. CATIA. Also cpmg_fig I believe. In fact, I think we may have had this discussion before? https://plus.google.com/s/cpmg%20glove If you consider experimental validation a reasonable justification for inclusion of the effects then you might find this interesting: Spin flips are also quite relevant to NH/N (and in fact any spin system). The supplementary section of Flemming and Pramodh go into it here for NH/N http://www.pnas.org/content/105/33/11766.full.pdf And this: JACS (2010) 132: 10992-5 Figure 2: r2 0.97, rmsd 8.0 ppb (no spin flips) r2 0.99, rmsd 5.7 ppb (spin flips). The improvements are larger than experimental uncertainties. When we design these experiments and test them, we need to think about the details. This is in part because Lewis beats us if we don't. You can imagine that it comes as a surprise then when we see people neglecting this. In short, the parameters you extract from fitting data will suffer if the details are not there. In the case of spin flips, the bigger the protein, the bigger the effect. In your code, you have the opportunity to do things properly. This leaves the details behind the scenes, so the naive user doesn't have to think about them. And only Flemming's CATIA handles the CPMG off-resonance effects. This is all explained in the relax manual. I have asked the authors of most dispersion software about this too, just to be sure. I don't know how much of an effect these have though. But one day they may be implemented in relax as well, and then the user can perform the comparison themselves and see if all these claims hold. Myint, W. & Ishima, R. Chemical exchange effects during refocusing pulses in constant-time CPMG relaxation dispersion experiments. J Biomol Nmr 45, 207-216, (2009). also: Bain, A. D., Kumar Anand, C. & Nie, Z. Exact solution of the CPMG pulse sequence with phase variation down the echo train: application to R(2) measurements. J Magn Reson 209, 183- 194, (2011). Or just simulate the off-resonance effects yourself to see what happens. For NH you see the effects clearly for glycines and side chains, if the carrier is in the centre around 120ppm. The problem generally gets worse the higher field you go though this of course depends when you bought your probe. You seem to imply that these effects are almost mythical. I assure you that they come out happily from the Bloch equations. Out of curiosity, from a philosophical perspective, I wonder if you'd agree with this statement: "the expected deviations due to approximations in a model should be lower than the experimental errors on each datapoint." Anyway, the warnings about analytic versers numeric are described in the manual. But your new model which adds a new factor to the CR72 model, just as Dimitry Korzhnev's cpmg_fit software does for his multiple quantum extension of the equation (from his 2004 and 2005 papers), sounds like it removes the major differences between the analytic and numeric results anyway. In any case, I have never seen an analytic result which is more than 10% different in parameter values (kex specifically) compared to the numeric results. I am constantly on the lookout for a real or synthetic data set to add to relax to prove this wrong though. I think there's a misunderstanding in what I mean by numerical modelling. For the 2x2 matrix (basis: I+, ground and excited) from the Bloch McConnell equation, you can manipulate this to get an exact solution. Nikolai's approach does this, though his algorithm can trip up sometimes for relatively exotic parameters when you use doubles (see attached). My formula also does this. I agree with you entirely that in this instance, numerically solving the equations via many matrix exponentials is irrelevant as you'll get an identical result to using a formula. My central thesis here though is that to get an accurate picture of the experiment you need more physics. This means a bigger basis. To have off resonance effects, you need a minimum 6x6 (Ix,Iy,Iz, ground and excited). To include scalar coupling and off resonance, you need a 12x12 (Ix,Iy,Iz,IxHz,IyHz,IzHz, ground and excited). Including R1 means another element and so on. The methyl group, for example, means you need 24. So when we use the term numerical treatment, we generally mean a calculation in a larger basis, as is necessary to take into account the appropriate spin physics. There aren't formulas to deal with these situations. In the 6x6 case for example, you need 6 Eigenvalues, which is going to make life very rough for someone brave enough to attempt a close form solution. The Palmer and Trott trick used in 2002 for R1rho is a smart way of ducking the problem of having 6 Eigenvalues, but for CPMG unfortunately you need several Eigenvalues, not just the smallest. The 2x2 matrix that Nikolai and Martin, Carver Richard's and I analyse does not include scalar coupling, as magnetisation is held in-phase (in addition to neglecting all the other stuff detailed above). So it is a good representation for describing the continuous wave in-phase experiments introduced here (neglecting relaxation effects and off resonance): Vallurupalli, P.; Hansen, D. F.; Stollar, E. J.; Meirovitch, E.; Kay, L. E. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 18473–18477. and here: Baldwin, A. J.; Hansen, D. F.; Vallurupalli, P.; Kay, L. E. J. Am. Chem. Soc. 2009, 131, 11939–48. But I think are the only two where this formula is directly applicable. Only if you have explicit decoupling during the CPMG period do you satisfy this condition. So this is not the case for all other experiments and certainly not true for those used most commonly. Anyhow. Best of luck with the software. I would recommend that you consider implementing these effects and have a look at some of the references. The physics are fairly complex, but the implementations are relatively straightforward and amount to taking many matrix exponentials. If you do this, I think you'd end up with a solution that really is world-leading. As it stands though, in your position, I would worry that on occasion, users will end up getting slightly wrong parameters out from your code by neglecting these effects. If a user trusts this code then, in turn, they might lead themselves to dodgy biological conclusions. For the time being, I'll stick to forcing my students to code things up themselves. All best wishes, Andy. |
#!/usr/bin/python
######################################################################
# Script to simulate different forms of the CPMG intensity,
# and to compare approaches
#
# Started
# A.Baldwin 20th October 2011
# Simplified 30th Jan 2014
#
# NikolaiLong function coded up using mpmath module. This was kindly
# contributed by Nikolai. The python implementation is going via
# strings and so is hell-of-slow. It does give the user full control
# of precision. Note mp.mp.dps>=23 required for numerical precision
# 18 is long double precision and gives acceptable errors.
# 9 is double precision, and can give foobars.
# March 2014.
#
# Copyright (c) A.Baldwin.
# University of Oxford.
# Please do not circulate without permission.
#
import numpy,sys
from math import cos,sin,atan2,log10
from datetime import datetime,timedelta
from scipy import mat,zeros,log
from scipy.linalg import expm
import mpmath as mp
mp.mp.dps=23
mp.mp.dps=18
#######################################################################
#Numerically take eigenvalues and propagate the 2x2 matrix
#
#setup 2x2 matrix
def ExchangeMat2x2(dOmega,DeltaR2,kex,pb):
#rates in s-1, domage in rad s-1, dfrq in MHz
k_ab=pb*kex
k_ba=(1-pb)*kex
L = mat(zeros((2, 2),dtype=complex))
#add chemical shift evolution
L[0,0]=complex(0,0.0)
L[1,1]=complex(0,-dOmega)
#add intrinsic relaxation
L[0,0] -= 0.0
L[1,1] -= DeltaR2
#add exchange
L[0, 0] -= k_ab
L[1, 0] += k_ab
L[1, 1] -= k_ba
L[0, 1] += k_ba
return L
def make_free_precess_propagator(L, time):
return mat(expm(L * time))
def Make2x2Equil(pb):
I0=mat(zeros((2, 1),dtype=complex))
I0[0,0]=complex((1-pb),0.0)
I0[1,0]=complex(pb,0.0)
return I0
def CPMG2x2(dOmega,DeltaR2,kex,pb,Trel,ncyc):
L=ExchangeMat2x2(dOmega,DeltaR2,kex,pb)
R2eff=[]
nu_cpmg=(ncyc)/Trel
for i in range(len(ncyc)):
I0=Make2x2Equil(pb)
if(i==0):
Ifin=I0
tcp=Trel/(4*ncyc[i])
LP=make_free_precess_propagator(L,tcp) #make the tau propagator
LPncyc=(LP*LP.conjugate()*LP.conjugate()*LP)**ncyc[i] #make the
(tau/180/tau/tau/180/tau)*ncyc propagator
Ifin=LPncyc*I0
R2eff.append((nu_cpmg[i],(1/Trel)*log((1-pb)/Ifin[0].real)))
return R2eff
#######################################################################
#Meiboom formula
def Meiboom(kex,pb,dw,ncyc,Trelax,R2g,R2e,outfile,verb='y'):
R2Fast=pb*(1-pb)*dw**2.0/kex
nu_cpmg=ncyc/Trelax
tcp=1/(2*nu_cpmg)
R2eff=(1-pb)*R2g+pb*R2e+R2Fast*(1.-(2./(kex*tcp))*numpy.tanh(kex*tcp/2.0))
if(verb=='n'):
return
else:
array=[]
for i in range(len(ncyc)):
array.append((nu_cpmg[i],R2eff[i]))
if(outfile!='Null'):
outy=open(outfile,'w')
for i in range(len(array)):
outy.write('%f\t%f\n' % (array[i][0],array[i][1]))
outy.close()
return array
#######################################################################
#Carver Richard's equation
def CarverRichards(kex,pb,dw,ncyc,Trelax,R2g,R2e,outfile,verb='n'):
deltaR2=R2e-R2g
keg=(1-pb)*kex
kge=pb*kex
zeta=2*dw*(-deltaR2+keg-kge) #zeta=g1
psi=(deltaR2+keg-kge)**2.0+4*keg*kge-dw**2 #psi=g2
NpFac=numpy.sqrt( psi+numpy.sqrt(psi**2.0+zeta**2.0))*2./numpy.sqrt(2.0)
NmFac=numpy.sqrt(-psi+numpy.sqrt(psi**2.0+zeta**2.0))*2./numpy.sqrt(2.0)
Dp=0.5*( 1.+(psi+2.*dw**2.0)/(numpy.sqrt(psi**2.0+zeta**2.0)))
Dm=0.5*(-1.+(psi+2.*dw**2.0)/(numpy.sqrt(psi**2.0+zeta**2.0)))
nu_cpmg=ncyc/Trelax
tcp=1/(4*nu_cpmg)
Np=NpFac*tcp
Nm=NmFac*tcp
R2eff=(R2g+R2e+kex)/2.0-(1/(4*tcp))*(numpy.arccosh(Dp*numpy.cosh(Np)-Dm*numpy.cos(Nm)))
if(verb=='n'):
return
else:
array=[]
for i in range(len(ncyc)):
array.append((nu_cpmg[i],R2eff[i]))
if(outfile!='Null'):
outy=open(outfile,'w')
for i in range(len(array)):
outy.write('%f\t%f\n' % (array[i][0],array[i][1]))
outy.close()
return array
#######################################################################
#numerical solution
def NumDisp(kex,pb,dw,ncyc,Trelax,R2g,R2e,outfile,verb='n'):
array=CPMG2x2(dw,(R2e-R2g),kex,pb,Trelax,ncyc)
if(verb=='n'):
return
else:
#add on R2g to R2eff
arrayNew=[]
for i in range(len(array)):
arrayNew.append((array[i][0],array[i][1]+R2g))
array=arrayNew
if(outfile!='Null'):
outy=open(outfile,'w')
for i in range(len(array)):
outy.write('%f\t%f\n' % (array[i][0],array[i][1]))
outy.close()
return array
#######################################################################
#coding up nikolai's formula
def Nikolai(kex,pb,dw,ncyc,Trelax,R2g,R2e,outfile,verb='n'):
#rename variables
Tc=Trelax;
Ra=R2g
Rb=R2e
dR=Ra-Rb;
pa=1-pb;
Ka=kex*pb;
Kb=kex*pa;
nu_cpmg=ncyc/Trelax; # get cpmg frequencies
tcp=1./(2*nu_cpmg); # tcp is the interval between consequitive 180
pulses
N=ncyc*2; # ncyc*2 (no need for round with these variables)
#off we go.
Rinf=(Ra+Rb)/2;
t3 = complex(0,1)/2;
t4 = t3*dw;
t5 = Ka/2;
t6 = Ra/2;
t7 = Kb/2;
t8 = Rb/2;
t9 = -2*(complex(0,1));
t14 = 2*(complex(0,1));
t20 = 2*Kb*Ka;
t22 = 2*Ka*Rb;
t24 = 2*Ra*Kb;
t26 = 2*Ra*Rb;
t28 = 2*Kb*Rb;
t30 = 2*Ka*Ra;
t31 = Rb**2;
t32 = Kb**2;
t33 = Ka**2;
t34 = Ra**2;
t35 = dw**2;
t36 =
t9*dw*Kb+t9*Rb*dw+t14*Ka*dw+t14*Ra*dw+t20-t22-t24-t26+t28+t30+t31+t32+t33+t34-t35;
t37 = numpy.sqrt(t36);
t38 = t37/2;
t42 = numpy.exp((t4-t5-t6-t7-t8+t38)*tcp/2);
t43 = 1/t37;
t49 = numpy.exp((t4-t5-t6-t7-t8-t38)*tcp/2);
t52 = t42*t43*Ka-t49*t43*Ka;
t61 =
t14*Rb*dw+t20-t22-t24-t26+t28+t30+t31+t32+t33+t34+t9*Ka*dw+t9*Ra*dw-t35+t14*dw*Kb;
t62 = numpy.sqrt(t61);
t63 = t62/2;
t64 = t4-t5-t6+t7+t8+t63;
t65 = -complex(0,1)/2;
t66 = t65*dw;
t69 = numpy.exp((t66-t5-t6-t7-t8+t63)*tcp);
t71 = 1/t62;
t73 = t4-t5-t6+t7+t8-t63;
t76 = numpy.exp((t66-t5-t6-t7-t8-t63)*tcp);
t79 = t64*t69*t71-t73*t76*t71;
t81 = complex(0,1);
t82 = t81*dw;
t85 = t42*(t82+Ka+Ra-Kb-Rb+t37)*t43;
t88 = t49*(t82+Ka+Ra-Kb-Rb-t37)*t43;
t89 = t85-t88;
t94 = t69*t71*Ka-t76*t71*Ka;
t96 = t52*t79+t89*t94/2;
t97 = t66-t5-t6+t7+t8+t38;
t98 = 1/Ka;
t101 = t66-t5-t6+t7+t8-t38;
t104 = t97*t98*t85-t101*t98*t88;
t105 = t96*t104/2;
t109 = t69*(t82-Ka-Ra+Kb+Rb-t62)*t71;
t114 = t76*(t82-Ka-Ra+Kb+Rb+t62)*t71;
t116 = -t64*t98*t109+t73*t98*t114;
t118 = -t109+t114;
t120 = t52*t116/2+t89*t118/4;
t121 = t120*t89/2;
t126 = t97*t42*t43-t101*t49*t43;
t129 = t126*t79+t104*t94/2;
t130 = t129*t126;
t133 = t126*t116/2+t104*t118/4;
t134 = t133*t52;
t135 = t105+t121;
t136 = t135**2;
t137 = t130+t134;
t140 = t137**2;
t146 = t96*t126+t120*t52;
t150 = numpy.sqrt(t136-2*t137*t135+t140+4*(t129*t104/2+t133*t89/2)*t146);
t151 = t105+t121-t130-t134-t150;
t153 = N/2;
t154 = (t105/2+t121/2+t130/2+t134/2+t150/2)**t153;
t156 = 1./t150;
t158 = t105+t121-t130-t134+t150;
t160 = (t105/2+t121/2+t130/2+t134/2-t150/2)**t153;
t165 = 1./t146;
t177 =
1/(Ka+Kb)*((-t151*t154*t156/2+t158*t160*t156/2)*Kb+(-t151*t165*t154*t158*t156/2+t158*t165*t160*t151*t156/2)*Ka/2);
intensity0 = pa ; # pA
intensity = numpy.real(t177);
R2eff=(1/Tc)*numpy.log(intensity0/intensity); # we did not factor
out (Ka+Kb)/2 here
if(verb=='n'):
return
else:
array=[]
for i in range(len(ncyc)):
array.append((nu_cpmg[i],R2eff[i]))
if(outfile!='Null'):
outy=open(outfile,'w')
for i in range(len(array)):
outy.write('%f\t%f\n' % (array[i][0],array[i][1]))
outy.close()
return array
def NikolaiLong(kex,pb,dw,ncyc,Trel,R2g,R2e,outfile,verb='n'):
#print 'Running test for exactness'
dfrq=mp.mpf('200.0') #spectrometer frequency of nuclei (MHz)
kex=mp.mpf(str(kex))
pb=mp.mpf(str(pb))
dw=mp.mpf(str(dw))
#rename variables
Tc=mp.mpf(str(Trel));
Ra=R2g
Rb=R2e
dR=Ra-Rb;
pa=1-pb;
Ka=kex*pb;
Kb=kex*pa;
R2eff=[]
nu_cpmg=[]
for i in range(len(ncyc)):
ncycV=mp.mpf(str(ncyc[i])) #number of cpmg cycles
nu_cpmgV=mp.mpf(str(ncycV))/Trel; # get cpmg frequencies
tcp=1/(2*nu_cpmgV); # tcp is the interval between consequitive
180 pulses
N=ncycV*2; # ncyc*2 (no need for round with these
variables)
nu_cpmg.append(nu_cpmgV.real) #off we go.
t3 = mp.mpc(0,1)/2;
t4 = t3*dw;
t5 = Ka/2;
t6 = Ra/2;
t7 = Kb/2;
t8 = Rb/2;
t9 = -2*(mp.mpc(0,1));
t14 = 2*(mp.mpc(0,1));
t20 = 2*Kb*Ka;
t22 = 2*Ka*Rb;
t24 = 2*Ra*Kb;
t26 = 2*Ra*Rb;
t28 = 2*Kb*Rb;
t30 = 2*Ka*Ra;
t31 = Rb**2;
t32 = Kb**2;
t33 = Ka**2;
t34 = Ra**2;
t35 = dw**2;
t36 =
t9*dw*Kb+t9*Rb*dw+t14*Ka*dw+t14*Ra*dw+t20-t22-t24-t26+t28+t30+t31+t32+t33+t34-t35;
t37 = mp.sqrt(t36);
t38 = t37/2;
t42 = mp.exp((t4-t5-t6-t7-t8+t38)*tcp/2);
t43 = 1/t37;
t49 = mp.exp((t4-t5-t6-t7-t8-t38)*tcp/2);
t52 = t42*t43*Ka-t49*t43*Ka;
t61 =
t14*Rb*dw+t20-t22-t24-t26+t28+t30+t31+t32+t33+t34+t9*Ka*dw+t9*Ra*dw-t35+t14*dw*Kb;
t62 = mp.sqrt(t61);
t63 = t62/2;
t64 = t4-t5-t6+t7+t8+t63;
t65 = -1*mp.mpc(0,1)/2;
t66 = t65*dw;
t69 = mp.exp((t66-t5-t6-t7-t8+t63)*tcp);
t71 = 1/t62;
t73 = t4-t5-t6+t7+t8-t63;
t76 = mp.exp((t66-t5-t6-t7-t8-t63)*tcp);
t79 = t64*t69*t71-t73*t76*t71;
t81 = mp.mpc(0,1);
t82 = t81*dw;
t85 = t42*(t82+Ka+Ra-Kb-Rb+t37)*t43;
t88 = t49*(t82+Ka+Ra-Kb-Rb-t37)*t43;
t89 = t85-t88;
t94 = t69*t71*Ka-t76*t71*Ka;
t96 = t52*t79+t89*t94/2;
t97 = t66-t5-t6+t7+t8+t38;
t98 = 1/Ka;
t101 = t66-t5-t6+t7+t8-t38;
t104 = t97*t98*t85-t101*t98*t88;
t105 = t96*t104/2;
t109 = t69*(t82-Ka-Ra+Kb+Rb-t62)*t71;
t114 = t76*(t82-Ka-Ra+Kb+Rb+t62)*t71;
t116 = -t64*t98*t109+t73*t98*t114;
t118 = -t109+t114;
t120 = t52*t116/2+t89*t118/4;
t121 = t120*t89/2;
t126 = t97*t42*t43-t101*t49*t43;
t129 = t126*t79+t104*t94/2;
t130 = t129*t126;
t133 = t126*t116/2+t104*t118/4;
t134 = t133*t52;
t135 = t105+t121;
t136 = t135**2;
t137 = t130+t134;
t140 = t137**2;
t146 = t96*t126+t120*t52;
t150 = mp.sqrt(t136-2*t137*t135+t140+4*(t129*t104/2+t133*t89/2)*t146);
t151 = t105+t121-t130-t134-t150;
t153 = N/2;
t154 = (t105/2+t121/2+t130/2+t134/2+t150/2)**t153;
t156 = 1/t150;
t158 = t105+t121-t130-t134+t150;
t160 = (t105/2+t121/2+t130/2+t134/2-t150/2)**t153;
t165 = 1/t146;
t177 =
1/(Ka+Kb)*((-t151*t154*t156/2+t158*t160*t156/2)*Kb+(-t151*t165*t154*t158*t156/2+t158*t165*t160*t151*t156/2)*Ka/2);
intensity0 = mp.mpf(str(pa)); # pA
intensity = t177.real;
R2eff.append( ((1/Tc)*mp.log(intensity0/intensity))); # we did
not factor out (Ka+Kb)/2 here
#print 'Precision of calculations mp.dps: ', mp.dps
#print 'R2eff: ', R2eff
# R2eff.append(2.)
array=[]
for i in range(len(ncyc)):
array.append((nu_cpmg[i],R2eff[i]))
if(outfile!='Null'):
outy=open(outfile,'w')
for i in range(len(array)):
outy.write('%f\t%f\n' % (array[i][0],array[i][1]))
outy.close()
return array
############################################################################
#numpy approximate formula
def Baldwin(kex,pb,dw,ncyc,Trel,R2g,R2e,outfile,verb='n'):
pa=(1-pb)
keg=kex*(1-pb)
kge=kex*pb
deltaR2=R2e-R2g
nu_cpmg=ncyc/Trel
tcp=Trel/(4.0*ncyc) #time for one free precession element
#########################################################################
#get the real and imaginary components of the exchange induced shift
g1=2*dw*(deltaR2+keg-kge) #same as carver richards zeta
g2=(deltaR2+keg-kge)**2.0+4*keg*kge-dw**2 #same as carver richards psi
g3=cos(0.5*atan2(g1,g2))*(g1**2.0+g2**2.0)**(1/4.0) #trig faster than
square roots
g4=sin(0.5*atan2(g1,g2))*(g1**2.0+g2**2.0)**(1/4.0) #trig faster than
square roots
#########################################################################
#time independent factors
N=complex(kge+g3-kge,g4) #N=oG+oE
NNc=(g3**2.+g4**2.)
f0=(dw**2.+g3**2.)/(NNc) #f0
f2=(dw**2.-g4**2.)/(NNc) #f2
#t1=(-dw+g4)*(complex(-dw,-g3))/(NNc) #t1
t2=(dw+g4)*(complex(dw,-g3))/(NNc) #t2
t1pt2=complex(2*dw**2.,-g1)/(NNc) #t1+t2
oGt2=complex((deltaR2+keg-kge-g3),(dw-g4))*t2 #-2*oG*t2
Rpre=(R2g+R2e+kex)/2.0 #-1/Trel*log(LpreDyn)
#do calc in numpy
E0= 2.0*tcp*g3 #derived from relaxation #E0=-2.0*tcp*(f00R-f11R)
E2= 2.0*tcp*g4 #derived from chemical shifts
#E2=complex(0,-2.0*tcp*(f00I-f11I))
E1=(complex(g3,-g4))*tcp #mixed term (complex) (E0-iE2)/2
ex0b=(f0*numpy.cosh(E0)-f2*numpy.cos(E2)) #real
ex0c=(f0*numpy.sinh(E0)-f2*numpy.sin(E2)*complex(0,1.)) #complex
ex1c=(numpy.sinh(E1)) #complex
v3=numpy.sqrt(ex0b**2.-1) #exact result for v2v3
y=numpy.power((ex0b-v3)/(ex0b+v3),ncyc)
v2pPdN=(( complex(deltaR2+kex,dw) )*ex0c+(-oGt2-kge*t1pt2)*2*ex1c)
#off diagonal common factor. sinh fuctions
Tog=(((1+y)/2+(1-y)/(2*v3)*(v2pPdN)/N))
Minty=Rpre-ncyc/(Trel)*numpy.arccosh((ex0b).real)-1/Trel*numpy.log((Tog.real))
#estimate R2eff
if(verb=='n'):
return
else:
array=[]
for i in range(len(ncyc)):
array.append((nu_cpmg[i],Minty[i]))
if(outfile!='Null'):
outy=open(outfile,'w')
for i in range(len(array)):
outy.write('%f\t%f\n' % (array[i][0],array[i][1]))
outy.close()
return array
#######################################################################
#Loop over a bunch of exchange parameters
def TestDisp(ncyc,Trel,R2g,flg,tim):
kexMax=1.0 #in s-1
kexMin=5000.0
dwMax=0.1 #in ppm
dwMin=1000.0
pbMax=0.001
pbMin=0.1
dRMin=1.
dRMax=1E4
Grid=10
now1=datetime.now()
rar=[]
for k in range(Grid):
for i in range(Grid):
for j in range(Grid):
for l in range(Grid):
kex=kexMin*10**(+(1.0*i/(Grid-1.0))*log10(kexMax/kexMin))
pb= pbMin*10**(+(1.0*j/(Grid-1.0))*log10(pbMax/pbMin))
dw= dwMin*10**(+(1.0*k/(Grid-1.0))*log10(dwMax/dwMin))
R2e= dRMin*10**(+(1.0*l/(Grid-1.0))*log10(dRMax/dRMin))
if(flg=='Carver'):
arr=CarverRichards(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trel,R2g,R2e,'Null',verb='y')
if(flg=='Baldwin'):
arr=Baldwin(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trel,R2g,R2e,'Null',verb='y')
if(flg=='Numerical'):
arr=NumDisp(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trel,R2g,R2e,'Null',verb='y')
if(flg=='Nikolai'):
arr=Nikolai(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trel,R2g,R2e,'Null',verb='y')
if(flg=='NikolaiLong'):
arr=NikolaiLong(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trel,R2g,R2e,'Null',verb='y')
if(flg=='Meiboom'):
arr=Meiboom(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trel,R2g,R2e,'Null',verb='y')
rar.append(arr)
now2=datetime.now()
val=timedelta.total_seconds(now2-now1)
if(flg=='Baldwin'):
tim=timedelta.total_seconds(now2-now1)
print flg,val,val/tim
return val,numpy.array(rar)
def ppm_to_rads(ppm,dfrq):
return ppm*2*numpy.pi*dfrq
if __name__ == "__main__": #run this if this file is run as standalone
print 'Relative times of the various approaches:'
dfrq=200. #spectrometer frequency of nucleci (MHz)
ncyc=numpy.array((2,4,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40))
#number of cpmg cycles
R2g=10. #relaxation rate of ground
R2e=10. #relaxation rate of excited
Trelax=0.02 #total time of CPMG block
tim,barr=TestDisp(ncyc,Trelax,R2g,'Baldwin',0.)
tom,narr=TestDisp(ncyc,Trelax,R2g,'Numerical',tim)
tom,marr=TestDisp(ncyc,Trelax,R2g,'Meiboom',tim)
tom,carr=TestDisp(ncyc,Trelax,R2g,'Carver',tim)
tom,sarr=TestDisp(ncyc,Trelax,R2g,'Nikolai',tim)
mp.mp.dps=18
#tom,larr=TestDisp(ncyc,Trelax,R2g,'NikolaiLong',tim)
mp.mp.dps=23
#tom,jarr=TestDisp(ncyc,Trelax,R2g,'NikolaiLong',tim)
print 'Maximum error over grid (s-1):'
print 'Meiboom: ',numpy.max(numpy.abs(narr[:,:,1]-marr[:,:,1]))
print 'Baldwin: ',numpy.max(numpy.abs(narr[:,:,1]-barr[:,:,1]))
print 'CarverRichards: ',numpy.max(numpy.abs(narr[:,:,1]-carr[:,:,1]))
#remove nan's from Nikolai's double output.
test=numpy.abs(narr[:,:,1]-sarr[:,:,1])
for i in range(len(test)):
for j in range(len(test[i])):
if(test[i][j]!=test[i][j]):
test[i][j]=0.
print 'Nikolai dougle (9): ',numpy.max(test)
print 'Nikolai long double (18):
',numpy.max(numpy.abs(narr[:,:,1]-larr[:,:,1]))
print 'Nikolai long double (23):
',numpy.max(numpy.abs(narr[:,:,1]-jarr[:,:,1]))
"""
print
print 'Short test for exactness'
kex=1.
pb=0.1
dw=1000
ncyc=numpy.array((2,4,6,8,10,12,14,16,20,24,28,32,36,40)) #number of cpmg
cycles
Narr=NumDisp(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trelax,R2g,R2e,'Null',verb='y')
Carr=CarverRichards(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trelax,R2g,R2e,'Null',verb='y')
Aarr=Baldwin(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trelax,R2g,R2e,'Null',verb='y')
Warr=Nikolai(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trelax,R2g,R2e,'Null',verb='y')
Marr=Meiboom(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trelax,R2g,R2e,'Null')
larr=NikolaiLong(kex,pb,ppm_to_rads(dw,dfrq),ncyc,Trelax,R2g,R2e,'Null')
col=len(ncyc)-1 #comparing ncyc=40
print 'R2eff for ncyc=40:'
print 'Numerical result :',Narr[col][1]
print 'Nikolai result : ',Warr[col][1]
print 'CarverRichards : ',Carr[col][1]
print 'Baldwin result : ',Aarr[col][1]
print
print 'Errors for ncyc=40, R2(calc)-R2(numerical)'
print 'Nikolai Error : ',Warr[col][1]-Narr[col][1]
print 'Baldwin Error : ',Aarr[col][1]-Narr[col][1]
print 'CarverRichards err: ',Carr[col][1]-Narr[col][1]
print 'To compare directly with matlab code:'
print 'Python implementation of Nikolais formula gives (nu_cpmg,R2eff):'
for i in range(len(Warr)):
print Warr[i]
print Marr
for i in range(len(Aarr)):
print larr[i]
"""