DPL94

The Davis et al., 1994 2-site off-resonance fast exchange relaxation dispersion model for R1rho-type data. It extends the M61 model to off-resonance data, hence it collapses to this model for on-resonance data. The model is labelled as DPL94 in relax.

Equation

[math] \mathrm{R}_{1\rho}= \mathrm{R}_1\cos^2\theta + \left( \mathrm{R}_{1\rho}{´} + \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + \omega_\textrm{e}^2} \right) \sin^2\theta [/math]

Parameters

The DPL94 model has the parameters {$R_{1\rho}'$, $...$, $\Phi_{ex}$, $k_{ex}$}.

Essentials

It is essential to read in $R_{1}$ values before starting a calculation:
Note, R1 should be provided in rad/s.

relax_data.read(ri_id='R1', ri_type='R1', frq=cdp.spectrometer_frq_list[0], file='R1_values.txt', mol_name_col=1, res_num_col=2, res_name_col=3, spin_num_col=4, spin_name_col=5, data_col=6, error_col=7)

Where the data could be stored like

# mol_name    res_num    res_name    spin_num    spin_name    value   error   
None               13           L        None            N 1.323940 0.146870
None               15           R        None            N 1.344280 0.140560
None               16           T        None            N 1.715140 0.136510

Parameter name space in relax

At time of writing (March 2014) the parameters in relax was stored as:

# Load the outcome from an analysis
state.load(state="results.bz2", dir="results/final")

# import spin functions
from pipe_control.mol_res_spin import return_spin, spin_loop

# Alias one spin
s13 = return_spin(":13@N")

# See attributes
dir(s13)

# See parameters
print(s13.params)
['r2', 'phi_ex', 'kex']

# Print parameters
print(s13.r2)
{'R1rho - 799.77739910 MHz': xx.yy}
print(s13.phi_ex)
print(s13.kex)

# See Ri data (ri_type:  The relaxation data type, i.e. 'R1', 'R2', 'NOE', or 'R2eff'. )
print(s13.ri_data)
{'R1': 1.3239399999999999}

# Print all spin id
for curspin, mol_name, res_num, res_name, spin_id in spin_loop(full_info=True, return_id=True, skip_desel=False):
    if curspin.select == False:
        print(mol_name, res_num, res_name, spin_id)
    else:
        print(mol_name, res_num, res_name, spin_id, curspin.r2, curspin.phi_ex, curspin.kex)

Please see the summary of the model parameters here.

Which means:

$R_{1\rho}'$ = spin.r2 (Fitted)
$R_{1\rho}$ = spin.r2eff (Back calculated)
$\Phi_{ex}$ = spin.phi_ex (Fitted)
$k_{ex}$ = spin.kex (Fitted)
$R_{1}$ = spin.ri_data['R1'] (Loaded)

Please also see this thread: http://thread.gmane.org/gmane.science.nmr.relax.devel/5164

Equation - re-writed forms

Discussed in: http://thread.gmane.org/gmane.science.nmr.relax.devel/5207

Evenäs, J., Malmendal, A. & Akke, M. (2001). Dynamics of the transition between open and closed conformations in a calmodulin C-terminal domain mutant. Structure 9, 185–195 http://dx.doi.org/10.1016/S0969-2126(01)00575-5
Kempf, J.G. & Loria, J.P. (2004). Measurement of intermediate exchange phenomena. Methods Mol. Biol. 278, 185–231 http://dx.doi.org/10.1385/1-59259-809-9:185
Palmer, A.G. & Massi, F. (2006). Characterization of the dynamics of biomacromolecules using rotating-frame spin relaxation NMR spectroscopy. Chem. Rev. 106, 1700–1719 http://dx.doi.org/10.1021/cr0404287
Palmer, A.G., Kroenke, C.D. & Loria, J.P. (2001). Nuclear magnetic resonance methods for quantifying microsecond-to-millisecond motions in biological macromolecules. Meth. Enzymol. 339 http://dx.doi.org/10.1016/S0076-6879(01)39315-1
Francesca Massi, Michael J. Grey, Arthur G. Palmer III* (2005). Microsecond timescale backbone conformational dynamics in ubiquitin studied with NMR R1ρ relaxation experiments, Protein science http://dx.doi.org/10.1110/ps.041139505
Trott, O. and Palmer, 3rd, A. G. (2002). R1rho relaxation outside of the fast-exchange limit. J. Magn. Reson., 154(1), 157–160. (http://dx.doi.org/10.1006/jmre.2001.2466).

Different graphs.

The R1rho: R2 or R2eff as function of effective field in rotating frame: w_eff

Discussion:
It is clear that there is no real name for the pseudo-parameter. It looks like that $R_{\text{eff}}$ was Art's original way of denoting this and that he has now changed to $R_{2}$ instead.
But if one look at the reference for the TP02 dispersion model TP02, one will see yet another notation:

Here $R_{2}$ does not contain the $R_{ex}$ contribution. Also, $R_{\text{eff}}$ is absent of $R_{ex}$.
But in Art's Protein Science paper (Ref [5]), the definition $R_{2} = R^{0}_2 + R_{ex}$ is used. The MP05 model reference also does not use $R_{\text{eff}}$ MP05.

The $R_{\text{eff}}$ parameter name is confusing and it seems to have been dropped from 2005 onwards. The $R_{\text{eff}}$ name appears to be specific to Art Palmer's group and as he himself has dropped it, then it would be best to avoid it too.

Ref [2], Equation 27. Here the calculated value is noted as: R_eff: $R_{\text{eff}} = R^{0}_2 + R_{ex} = R_{1\rho}' + R_{ex} = R_{1\rho} / \sin^2(\theta) - R_1 / \tan^2(\theta)$
Ref [3], Equation 20. Figure 11+16, would be the reference. Here the calculated value is noted as: R_2: $R_{2} = R_{1\rho} / \sin^2(\theta) - R_1 / \tan^2(\theta)$.
Ref [4], Equation 43. $R_{\text{eff}} = R_{1\rho} / \sin^2(\theta) - R_1 / \tan^2(\theta)$
Ref [5], Material and Methods, page 740. Figure 4 would be the wished graphs. Here the calculated value is noted as: R_2: $R_{2} = R^{0}_2 + R_{ex}$

The following suggestions for the definition of the pseudo-parameters, which can be extracted, is then

$R_2 = R^{0}_2 + R_{ex} = R_{1\rho}' + R_{ex} = R_{1\rho} / \sin^2(\theta) - R_1 / \tan^2(\theta) = \frac{R_{1\rho} - R_1\cos^2(\theta)}{\sin^2(\theta)}$

Reference

The reference for the DPL94 model is:

Davis, D., Perlman, M., and London, R. (1994). Direct measurements of the dissociation-rate constant for inhibitor-enzyme complexes via the T1rho and T2 (CPMG) methods. J. Magn. Reson., 104(3), 266–275. (10.1006/jmrb.1994.1084)

Related models

The DPL94 model is simply the extension of the M61 model for off-resonance data.

Links

The implementation of the DPL94 model in relax can be seen in the: