# 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

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

## Parameters

The DPL94 model has the parameters {R', ..., Φex, kex}.

## Essentials

 Note  R1 should be provided in rad/s, the SI default unit for this relaxation rate.

It is essential to read in R1 values before starting a calculation:

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

Which means:

1. R' = spin.r2 (Fitted)
2. R = spin.r2eff (Back calculated)
3. Φex = spin.phi_ex (Fitted)
4. kex = spin.kex (Fitted)
5. R1 = spin.ri_data['R1'] (Loaded)

## Equation - re-written forms

• Evenäs, J., Malmendal, A. and Akke, M. (2001). Dynamics of the transition between open and closed conformations in a calmodulin C-terminal domain mutant. Structure, 9(3), 185-195. (DOI: 10.1016/S0969-2126(01)00575-5)
• Kempf, J. G. and Loria, J. P. (2004). Measurement of intermediate exchange phenomena. Methods Mol. Biol., 278, 185-231. (DOI: 10.1385/1-59259-809-9:185)
• Massi, F., Grey, M. J., Palmer, 3rd, A. G. (2005). Microsecond timescale backbone conformational dynamics in ubiquitin studied with NMR R1ρ relaxation experiments Protein science, 14(3), 735-742. (DOI: 10.1110/ps.041139505)
• Palmer, 3rd, A. G., Kroenke, C. D., and Loria, J. P. (2001). Nuclear magnetic resonance methods for quantifying microsecond-to-millisecond motions in biological macromolecules. Methods Enzymol., 339, 204-238. (DOI: 10.1016/S0076-6879(01)39315-1)
• Palmer, 3rd, A. G. and Massi, F. (2006). Characterization of the dynamics of biomacromolecules using rotating-frame spin relaxation NMR spectroscopy. Chem. Rev., 106(5), 1700-1719. (DOI: 10.1021/cr0404287)
• Trott, O. and Palmer, 3rd, A. G. (2002). R1rho relaxation outside of the fast-exchange limit. J. Magn. Reson., 154(1), 157-160. (DOI: 10.1006/jmre.2001.2466)

Different graphs.

## The R1ρ: R2 or R2,eff as function of effective field in rotating frame: ωe

### Discussion

It is clear that there is no real name for the pseudo-parameter. It looks like that Reff 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, 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 R2 = R20 + Rex is used. The MP05 model reference also does not use Reff.

The Reff parameter name is confusing and it seems to have been dropped from 2005 onwards. The Reff 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. G., Perlman, M. E., and London, R. E. (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. (DOI: 10.1006/jmrb.1994.1084)

## Related models

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