Hi Andy,
One part you missed is Troels' correction to your CarverRichards() function:
"""
@@ -109,7 +110,7 @@
keg=(1-pb)*kex
kge=pb*kex
- zeta=2*dw*(-deltaR2+keg-kge) #zeta=g1
+ 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)
"""
This makes the CR result much closer to your new model:
Maximum error over grid (s-1):
CarverRichards (before): 4989.8296812
CarverRichards (after): 8.22388524626
For reference, these values are simply the maximum of all
back-calculated R2eff deviations to the numeric R2eff values over a
large set of different dispersion parameters (the Grid). As an aside,
a distribution bar chart over all R2eff deviations would be a much
better visualisation of the deviations in this Grid. A value of
delta_R2eff = 8 rad.s^-1 seems much more reasonable to me and is what
I would expect from playing with many different data sets and models
in relax.
If this change is correct, which I believe it is as it restores the
original CR equations, then this change makes the code identical to
relax:
fact = r20a - r20b - k_BA + k_AB
zeta = 2.0*dw * fact
identical to the funCarverdR.m file in the sim_all software from
Nikolai and Martin:
dR=Ra-Rb;
eta=2*dw*(dR+kex*(pb-pa));
and identical to the cpmg_fit software from Dmitry Korzhnev:
zeta=2*DW*(R2A-(R2B-I*DWH)-PA*KEX+(1-PA)*KEX);
Ignore the I*DWH part as this is the multiple quantum data extension
of the CR72 formula that Dmitry derived. The only other software
which has R20A != R20B is Glove from Peter Wright, but I don't have
that source code. Though they are likely to have implemented the
original equations too.
This change would however not affect figure 1 whereby R20A == R20B, as
zeta is squared. I don't know if this correction would affect any
part of your paper or supplement. But it is significant and it does
affect the statements you made earlier on the mailing list that the
original Carver and Richards equations were incorrect.
Regards,
Edward
On 6 May 2014 09:25, Andrew Baldwin <[email protected]> wrote:
> Hiya Troels,
>
> Very commendable tenacity with the fluorescence result :) Good stuff indeed!
>
> 1) Definition of alpha- :
>
> That's a fair point. I agree. Alpha- is later squared so either:
>
> zeta(B14) = 2.0*dw (Delta R2 + k_eg - k_ge)
> zeta(B14) = 2.0*dw (-Delta R2 - k_eg + k_ge)
>
> Will work equally well. Which you use depends on how you choose to
> parametrise the Eigenvalues. h1 can be plus or minus depending on your mood.
> So my mood says keeping the positive deltaR2 makes the equation prettier.
>
> 2) You might need to give a bit more thought to the numerical
> implementation. Notice that I add the factor of R2g back in, during the
> function Numdisp. Turns out you can factor anything out from the diagonal as
> long as you put it back in at the end, hence the form of the matrix in the
> code I sent.
>
> You can prove this by looking at the Eigenvalues, and how they propagate
> when you take things like matrix exponentials. In the same vein, notice that
> In the paper I subtract R2G from both f00 and f11. I add it back in again
> later in the constants out the front. This is following from the lead of
> Nikolai in his reconstructing invisible states paper. Implicitly looking at
> deltaR2s shows clearly where differences in relaxation have effects, so this
> brings out the physics.
>
> So look at the 'error' reported in the outputs. When you made the Troels
> version of the numerical code, the error for the Baldwin formula was 10s-1.
> This is actually the exact value you set R2G to. So while you added in R2G
> in the evolution matrix (which is not wrong to do), you didn't then remove
> it in numdisp, so your solution is out by one unit of R2G.
>
> Looking at the outputs, the code I sent for CR had the carver richard's
> formula set in the wrong 'minus' configuration, added during testing while
> writing the previous email. This was certainly my foobar. However, note that
> the maximum 'error' reported by the code was 4989 s-1 when you ran the code
> for the first time. In my previous email:
>
>
>> with the alpha- sign made consistent with the Carver Richard's paper gives
>> an error of 4989 s-1 in the same test
>
>
> That should have set off alarm bells for you. Setting either:
>
> zeta=2*dw*(deltaR2+keg-kge) #zeta=g1
> zeta=2*dw*(-deltaR2-keg+kge) #zeta=g1
>
> In the CR function as you note (and as you do in relax) is good. If you
> subtract the 'Baldwin error' from the 'CR error' you get 8. 8 s-1 is the
> biggest error incurred using CR in this parameter range (as I stated in the
> last email). So it's fair to call the CR equation an approximation. Fig 1
> speaketh the truth. Calculate it yourself first using your code before
> wielding the accusing stick!
>
> So I'm deducting one half pint from your score for not checking the results
> from your modified numerical code more carefully.
>
> And now for beer counting:
> 1) Appendix 1 – recipe for exact calculation of R2,eff
> is: h1 = 2.0*dw (R2B - R2A + k_BA - k_AB) = - CR72 (But it have no
> influence, but please keep history! Change it back to CR72. :-))
> same for h2.
>
>
> Pint for the typo in keg/kge. That's a genuine foobar in the manu - the
> equations should match code i've written, and I know the code is right
> because i've matched it to simulation. I'll keep my Eigenvalue definitions
> though.
>
>
> 2) zeta typo in eq 70. z=zeta
>
> Agreed! +1.
>
>
> 3) Watch alpha_ in eq 70. Get Delta R straight
>
> See 1. No extra pint here.
>
> 4) Comments in CODE:
> http://svn.gna.org/viewcvs/*checkout*/relax/trunk/lib/dispersion/b14.py?revision=HEAD
> g1=2*dw*(deltaR2+keg-kge) #same as carver richards zeta
> That is not true. That is minus Zeta. But it does not matter, since order
> **2.
>
> See 1. No extra pint here either.
>
> 5) Did you actually implement ExchangeMat2x2 wrong??
> If yes, theeeeeen, no wonder figure 1 shows what it does. :-)
> What is up for the deltaOmegas story???
>
>
> -1/2 pint for an over-reached conclusion from your modified code.
>
> So current score = 1.5
>
>
> Meiboom challenge: I'd be interested in a proof for how Carver Richard's
> reduces to Meiboom. I'd pay a pint for that. To get you started, note that
> in the fast exchange limit, E0 and E2 can be simplified using the fast
> exchange limits in supp section 1.
>
> Best,
>
> Andy.
>
>
>
> On 05/05/2014 15:27, Troels Emtekær Linnet wrote:
>
> Hi Andy.
>
> Heh, the funniest part with the fluorescence was to battle with the
> moderators of wikipedia to change 9000 to 9 in R06.
> http://en.wikipedia.org/wiki/F%C3%B6rster_resonance_energy_transfer#Theoretical_basis
> That was a brawl, to convince them that the bible was wrong. But
> referring to the old dusty library book of Forster (Ref 14) made the
> deal.
>
> Back to the beer.
>
> You are swapping two things in your code, according to CR72.
>
> If I write up CR72: (Directly from cr72.py)
> #######
> fact = r20a - r20b - k_BA + k_AB
> zeta = 2.0*dw * fact
>
> So in total:
> %%%%%
> zeta(CR72) = 2.0*dw (R2A - R2B + k_AB - k_BA)
> %%%%
>
>
> Then we take your code:
> #######
> deltaR2=R2e-R2g
> g1=2*dw*(deltaR2+keg-kge) #same as carver richards zeta
>
> Change notation, to make it easier to see:
> %%%%%
> zeta(B14) = 2.0*dw (R2B - R2A + k_BA - k_AB)
> %%%%
> But huh, is that not just:
>
> zeta(B14) = - zeta(CR72)
> = 2.0*dw ( - R2A + R2B - k_AB + k_BA)
> = 2.0*dw ( R2B - R2A + k_BA - k_AB)
>
> Well, that is interesting.
> So your g1 is the negative of carver richards zeta.
>
> Fortunately we are lucky, that zeta is to the power **2.
> So that leaves it out.
>
> How about psi?
> It boils down to that
> fact(B14) = - fact(CR72) <=>
> alpha_ (B14) = - alpha_ (CR72)
>
> But alpha_ is also to the power **2. Phew...
>
> So what is up.
> If we look at: (compareTroells.py)
> https://gna.org/support/download.php?file_id=20636
>
> ##############
> 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
> #############
>
> Well here we have something !
> zeta = 2.0 * dw ( R2A - R2B - k_AB + k_BA )
>
> So this is not about how you make the difference of ( R2A - R2B )
> it is the swapping of + k_AB - k_BA << - k_AB + k_BA
> kex(1 - 2pa) << kex( -1 + 2pa )
>
> You have implemented CR72 wrong ?
>
> Hmmmm. What is up?
>
> You get a difference between CR72, and numerical.
> You cannot get CR72 to work, unless you modify alpha_.
> You invert the signs of forward / backward exchange rate.
>
> You say that numerical is true, so CR72 is wrong.
>
> Lets inspect numerical.
>
> If we look at: (compareTroells.py)
> https://gna.org/support/download.php?file_id=20636
> Now we look up the numerical solution.
> #####
> NumDisp(kex,pb,dw,ncyc,Trelax,R2g,R2e,outfile,verb='n'):
> array=CPMG2x2(dw,(R2e-R2g),kex,pb,Trelax,ncyc)
> CPMG2x2(dOmega,DeltaR2,kex,pb,Trel,ncyc):
> L=ExchangeMat2x2(dOmega,DeltaR2,kex,pb)
> ExchangeMat2x2(dOmega,DeltaR2,kex,pb):
> L = mat(zeros((2, 2),dtype=complex))
> L[1,1] -= DeltaR2
>
> Hmmm.
> According to your paper at equation 3.
>
> R+ =
> [[ -kAB - R2A] [kBA] ]
> [ kAB] [ -kBA - R2B - i dw] ]
>
> But I see that it gets populated like
> R+ =
> [[ -kAB ] [kBA] ]
> [ kAB] [ -kBA - R2B + R2A - i dw] ]
>
> Has this something to do with it:
>
> ????
>
> So I took the liberty to change your code, hoping the Theory is right.
>
> ############# Normal
> Baldwin 1.183188 1.0
> Numerical 58.556647 49.4905687008
> Meiboom 0.391873 0.331200958766
> Carver 0.693676 0.586277075156
> compareTroells_w_Troels_mod.py:260: RuntimeWarning: invalid value
> encountered in log
> R2eff=(1/Tc)*numpy.log(intensity0/intensity); # we did not
> factor out (Ka+Kb)/2 here
> Nikolai 3.434938 2.90312105938
> NikolaiLong 585.485493 494.837247335
>
> Maximum error over grid (s-1):
> Meiboom: 93015865.9296
> Baldwin: 1.49539403083e-09
> CarverRichards: 4989.8296812
> Nikolai dougle (9): 204.761587913
> Nikolai long double (23): 2.0692254415912185547869e-7
>
>
> ############ Modifying Meiboom to 4 NCYC
> Maximum error over grid (s-1):
> Meiboom: 22664952.3388
> Baldwin: 1.49539403083e-09
> CarverRichards: 4989.8296812
>
> ########### Modifying ExchangeMat2x2 to your paper version.
> Meiboom: 22664942.3388
> Baldwin: 10.0000000012
> CarverRichards: 4979.8296812
>
> ########## Modify CR72 zeta=2*dw*(-deltaR2-keg+kge)
> Baldwin 1.216869 1.0
> Numerical 60.302727 49.555644034
> Meiboom 0.397395 0.326571718073
> Carver 0.700189 0.575402118059
> compareTroells_w_Troels_mod.py:261: RuntimeWarning: invalid value
> encountered in log
> R2eff=(1/Tc)*numpy.log(intensity0/intensity); # we did not
> factor out (Ka+Kb)/2 here
> Nikolai 3.569577 2.93341107383
> NikolaiLong 591.459824 486.050531323
> Maximum error over grid (s-1):
> Meiboom: 22664942.3388
> Baldwin: 10.0000000012
> CarverRichards: 18.2238852463
> Nikolai dougle (9): 214.761587913
> Nikolai long double (23): 10.000000207062814176945
>
> WOOOOOWWWW
>
> The change I made was:
>
> ##############################################
> 37c37
> < def ExchangeMat2x2(dOmega,DeltaR2,kex,pb):
> ---
>
> def ExchangeMat2x2(dOmega,R2e,R2g,kex,pb):
>
> 47c47
> < L[1,1] -= DeltaR2
> ---
>
> L[1,1] -= R2e
>
> 49a50
>
> L[0, 0] -= R2g
>
> 64,65c65,66
> < def CPMG2x2(dOmega,DeltaR2,kex,pb,Trel,ncyc):
> < L=ExchangeMat2x2(dOmega,DeltaR2,kex,pb)
> ---
>
> def CPMG2x2(dOmega,R2e,R2g,kex,pb,Trel,ncyc):
> L=ExchangeMat2x2(dOmega,R2e,R2g,kex,pb)
>
> 86,87c87,88
> < tcp=1/(2*nu_cpmg)
> <
> R2eff=(1-pb)*R2g+pb*R2e+R2Fast*(1.-(2./(kex*tcp))*numpy.tanh(kex*tcp/2.0))
> ---
>
> tcp=1/(4*nu_cpmg)
>
> R2eff=(1-pb)*R2g+pb*R2e+R2Fast*(1.-(4./(kex*tcp))*numpy.tanh(kex*tcp/4.0))
>
> 112c113
> < zeta=2*dw*(-deltaR2+keg-kge) #zeta=g1
> ---
>
> zeta=2*dw*(-deltaR2-keg+kge) #zeta=g1
>
> 145c146
> < array=CPMG2x2(dw,(R2e-R2g),kex,pb,Trelax,ncyc)
> ---
>
> array=CPMG2x2(dw,R2e,R2g,kex,pb,Trelax,ncyc)
>
> 546c547
> < #tom,jarr=TestDisp(ncyc,Trelax,R2g,'NikolaiLong',tim)
> ---
>
> tom,jarr=TestDisp(ncyc,Trelax,R2g,'NikolaiLong',tim)
>
> 560c561
> < print 'Nikolai long double (18):
> ',numpy.max(numpy.abs(narr[:,:,1]-larr[:,:,1]))
> ---
>
> #print 'Nikolai long double (18):
> ',numpy.max(numpy.abs(narr[:,:,1]-larr[:,:,1]))
>
> [tlinnet@tomat ~/Downloads]$ diff compareTroells.py
> compareTroells_w_Troels_mod.py
> 37c37
> < def ExchangeMat2x2(dOmega,DeltaR2,kex,pb):
> ---
>
> def ExchangeMat2x2(dOmega,R2e,R2g,kex,pb):
>
> 47c47
> < L[1,1] -= DeltaR2
> ---
>
> L[1,1] -= R2e
>
> 49a50
>
> L[0, 0] -= R2g
>
> 64,65c65,66
> < def CPMG2x2(dOmega,DeltaR2,kex,pb,Trel,ncyc):
> < L=ExchangeMat2x2(dOmega,DeltaR2,kex,pb)
> ---
>
> def CPMG2x2(dOmega,R2e,R2g,kex,pb,Trel,ncyc):
> L=ExchangeMat2x2(dOmega,R2e,R2g,kex,pb)
>
> 86,87c87,88
> < tcp=1/(2*nu_cpmg)
> <
> R2eff=(1-pb)*R2g+pb*R2e+R2Fast*(1.-(2./(kex*tcp))*numpy.tanh(kex*tcp/2.0))
> ---
>
> tcp=1/(4*nu_cpmg)
>
> R2eff=(1-pb)*R2g+pb*R2e+R2Fast*(1.-(4./(kex*tcp))*numpy.tanh(kex*tcp/4.0))
>
> 112c113
> < zeta=2*dw*(-deltaR2+keg-kge) #zeta=g1
> ---
>
> zeta=2*dw*(-deltaR2-keg+kge) #zeta=g1
>
> 145c146
> < array=CPMG2x2(dw,(R2e-R2g),kex,pb,Trelax,ncyc)
> ---
>
> array=CPMG2x2(dw,R2e,R2g,kex,pb,Trelax,ncyc)
>
> 546c547
> < #tom,jarr=TestDisp(ncyc,Trelax,R2g,'NikolaiLong',tim)
> ---
>
> tom,jarr=TestDisp(ncyc,Trelax,R2g,'NikolaiLong',tim)
>
> 560c561
> < print 'Nikolai long double (18):
> ',numpy.max(numpy.abs(narr[:,:,1]-larr[:,:,1]))
> ---
>
> #print 'Nikolai long double (18):
> ',numpy.max(numpy.abs(narr[:,:,1]-larr[:,:,1]))
>
> [tlinnet@tomat ~/Downloads]$ diff compareTroells.py
> compareTroells_w_Troels_mod.py
> 37c37
> < def ExchangeMat2x2(dOmega,DeltaR2,kex,pb):
> ---
>
> def ExchangeMat2x2(dOmega,R2e,R2g,kex,pb):
>
> 47c47
> < L[1,1] -= DeltaR2
> ---
>
> L[1,1] -= R2e
>
> 49a50
>
> L[0, 0] -= R2g
>
> 64,65c65,66
> < def CPMG2x2(dOmega,DeltaR2,kex,pb,Trel,ncyc):
> < L=ExchangeMat2x2(dOmega,DeltaR2,kex,pb)
> ---
>
> def CPMG2x2(dOmega,R2e,R2g,kex,pb,Trel,ncyc):
> L=ExchangeMat2x2(dOmega,R2e,R2g,kex,pb)
>
> 86,87c87,88
> < tcp=1/(2*nu_cpmg)
> <
> R2eff=(1-pb)*R2g+pb*R2e+R2Fast*(1.-(2./(kex*tcp))*numpy.tanh(kex*tcp/2.0))
> ---
>
> tcp=1/(4*nu_cpmg)
>
> R2eff=(1-pb)*R2g+pb*R2e+R2Fast*(1.-(4./(kex*tcp))*numpy.tanh(kex*tcp/4.0))
>
> 112c113
> < zeta=2*dw*(-deltaR2+keg-kge) #zeta=g1
> ---
>
> zeta=2*dw*(-deltaR2-keg+kge) #zeta=g1
>
> 145c146
> < array=CPMG2x2(dw,(R2e-R2g),kex,pb,Trelax,ncyc)
> ---
>
> array=CPMG2x2(dw,R2e,R2g,kex,pb,Trelax,ncyc)
>
> 546c547
> < #tom,jarr=TestDisp(ncyc,Trelax,R2g,'NikolaiLong',tim)
> ---
>
> tom,jarr=TestDisp(ncyc,Trelax,R2g,'NikolaiLong',tim)
>
> 560c561
> < print 'Nikolai long double (18):
> ',numpy.max(numpy.abs(narr[:,:,1]-larr[:,:,1]))
> ---
>
> #print 'Nikolai long double (18):
> ',numpy.max(numpy.abs(narr[:,:,1]-larr[:,:,1]))
>
> ####################################
>
> File can be downloaded from:
> https://gna.org/support/download.php?file_id=20642
>
>
>
> What have we (maybe) learned?????
>
> 1) Baldwin has created a code, which is as good as the numerical of
> Nicolai. Thanks !!! Wuhuuu !
> 2) CR72 is still valid though. Doing well.
>
> And now for beer counting:
> 1) Appendix 1 – recipe for exact calculation of R2,eff
> is: h1 = 2.0*dw (R2B - R2A + k_BA - k_AB) = - CR72 (But it have no
> influence, but please keep history! Change it back to CR72. :-))
> same for h2.
>
> 2) zeta typo in eq 70. z=zeta
>
> 3) Watch alpha_ in eq 70. Get Delta R straight
>
> 4) Comments in CODE:
> http://svn.gna.org/viewcvs/*checkout*/relax/trunk/lib/dispersion/b14.py?revision=HEAD
> g1=2*dw*(deltaR2+keg-kge) #same as carver richards zeta
> That is not true. That is minus Zeta. But it does not matter, since order
> **2.
>
> g2=(deltaR2+keg-kge)**2.0+4*keg*kge-dw**2 #same as carver richards psi
> Not totally correct, but since deltaR2 is swapped, it gets minus
> alpha, and goes to order 2.
>
> 5) Did you actually implement ExchangeMat2x2 wrong??
> If yes, theeeeeen, no wonder figure 1 shows what it does. :-)
> What is up for the deltaOmegas story???
>
> This is my 2 cents. :-)
>
> Let me hear if my hours have been wasted.
>
> Best
> Troels
>
>
> 2014-05-05 11:00 GMT+02:00 Andrew Baldwin <[email protected]>:
>
> I had to struggle long to find out the Forster distance calculation in
>
> Lakowicz was wrong. The bible of fluorescence...
>
>
> Very good stuff :) Most would not take the time to check :) The finest
> biophysics really does demand both first rate biology and first rate
> physics.
>
> I don't think anyone has ever used the CR equation to analyse data outside
> of the limit where R2A=R2B, so it's very unlikely that this has error has
> had any real consequence. Note also that in general, it's actually much
> easier to code up a numerical solution - ie once you've typed up an
> evolution matrix, all you have left to do is write some lines taking matrix
> exponentials. Whereas with an equation, not only are you making implicit
> approximations that aren't necessarily obvious, there are many, many, more
> lines to type, thus increasing the probability of a foobar. So numerical
> solutions for spin dynamics = no mistakes, and also easy to code! This has
> been the approach in the Kay group, arguably cultivated by Dimitri and
> Nikolai, and then pushed hard more recently by Flemming, Pramodh and
> Guillaume.
>
> This paper was truly groundbreaking (in my view):
> Journal of biomolecular NMR. 2000 18(1):49-63
>
> On this note, there are a few suggestions that Flemming and I are putting
> together for another thread sensibly started by Edward a few days ago for
> analysing R1rho and CPMG data. I think that with just a few extra tweaks,
> the backend to your code can be as powerful and as versatile as your front
> end, for millisecond data analysis. I agree that it really is in the
> interests of the field to have people able to perform the experiments and
> get information out of their data as efficiently (and as rigorously) as
> possible.
>
> This means more citations for those that made the experiments, and the
> community can keep showing more examples where Xray people have
> inadvertently missed functionally relevant conformers :)
>
> Best,
>
> Andy.
>
>
>
>
>
>
> On 04/05/2014 23:09, Troels Emtekær Linnet wrote:
>
> Well Andrew.
>
> Your paper popped out of the Mail alerts. :-)
>
> My supervisor Kaare Teilum guided my attention to it, and asked me to
> go through it to check.
>
> I discussed it with a lab colleague, and we were thrived.
>
> The math derivations still kills me, but I thought that the essential
> results was alarming.
> I mean, half of our lab do CPMG or R1rho.
> Any changes to those equations gets our attention.
>
> The "worst" problem we have in our lab, is the ability to analyse data.
> With relax, we now have a GUI, which make sure that at least the
> Bachelor Students still survive.
>
> And what else can I then do than get dirty with your code, to test if
> it is right?
>
> If you suspicion about the CR72 is correct, then I really really
> wonder why not errata article is out there?
>
> This situation reminds me of my master thesis.
> I had to struggle long to find out the Forster distance calculation in
> Lakowicz was wrong. The bible of fluorescence...
>
> Jesus....
>
> but thanks for the beer!
>
> Best
> Your biggest fan. ;-)
>
>
> 2014-05-04 22:59 GMT+02:00 Andrew Baldwin <[email protected]>:
>
> 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 <[email protected]>:
>
> 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.
>
>
>
>
>
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