Intro

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)

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_eff was Art's original way of denoting this and that he has now changed to R2 instead.
But if one look at the reference for the TP02 dispersion model TP02, one will see yet another notation:

Here R2 does not contain the Rex contribution. Also, Reff is absent of Rex.
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 Reff MP05.

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: $R_{2} = R^{0}_2 + R_{ex}$

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

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:

DPL94