Dispersion model summary

Except for `R2eff' and `No Rex', all models can be fit to clusterings of spins, or spin blocks. The models are described in more detail below and summarised in Table 11.1. The parameters of the models and of relaxation dispersion in general are given in Table 17.5.

R₁ parameter optimisation

For a number of models, the off-resonance R₁ value can be optimised. Normally the off-resonance models will use fixed experimental R₁ values for optimisation. However if the experimental values are not loaded, then the R₁ values will be automatically optimised. For finer control of this optimisation behaviour, see the relax_disp.r1_fit user function (page

). The models which support off-resonance R₁ fitting include:

In the future, support for off-resonance effects in the CPMG experiments is planned (see section 11.10 on page

**Table 11.1:** The dispersion models supported by relax.

Model name	Solution	Sites	Parameters	Restrictions	Reference


Experiment independent

R2eff	-	-	{R_2eff,^...}	Fixed relaxation time period	-
R2eff	-	-	{R_2eff, I₀,^...}	Variable relaxation time period	-
No Rex	Closed	1	{R₂⁰,^...}	-	-

CPMG-type

LM63	Analytic	2	{R₂⁰,..., Φ_ex, k_ex}	Fast exchange	Luz and Meiboom (1963)
LM63 3-site	Analytic	3	{R₂⁰,..., Φ_{ex, B}, k_B, Φ_{ex, C}, k_C}	Fast exchange, p_A > p_B and	Luz and Meiboom (1963)
				p_A > p_C
CR72	Analytic	2	{R₂⁰,..., p_A, Δω, k_ex}	p_A > p_B, not very slow exchange	Carver and Richards (1972)
CR72 full	Analytic	2	{R_2A⁰, R_2B⁰,..., p_A, Δω, k_ex}	p_A > p_B, not very slow exchange	Carver and Richards (1972)
IT99	Analytic	2	{R₂⁰,..., p_A, Δω, τ_ex}	p_A $\gg$ p_B	Ishima and Torchia (1999)
TSMFK01	Analytic	2	{R_2A⁰,..., Δω, k_AB}	p_A $\gg$ p_B slow exchange	Tollinger et al. (2001)
B14	Analytic	2	{R₂⁰,..., p_A, Δω, k_ex}	p_A > p_B,	Baldwin (2014)
B14 full	Analytic	2	{R_2A⁰, R_2B⁰,..., p_A, Δω, k_ex}	p_A > p_B,	Baldwin (2014)
NS CPMG 2-site expanded	Numeric	2	{R₂⁰,..., p_A, Δω, k_ex}	p_A > p_B	Tollinger et al. (2001)
NS CPMG 2-site 3D	Numeric	2	{R₂⁰,..., p_A, Δω, k_ex}	p_A > p_B	-
NS CPMG 2-site 3D full	Numeric	2	{R_2A⁰, R_2B⁰,..., p_A, Δω, k_ex}	p_A > p_B	-
NS CPMG 2-site star	Numeric	2	{R₂⁰,..., p_A, Δω, k_ex}	p_A > p_B	-
NS CPMG 2-site star full	Numeric	2	{R_2A⁰, R_2B⁰,..., p_A, Δω, k_ex}	p_A > p_B	-
MMQ CPMG-type

MMQ CR72	Analytic	2	{R₂⁰,..., p_A, Δω, Δω^H, k_ex}	p_A > p_B	Korzhnev et al. (2004a)
NS MMQ 2-site	Numeric	2	{R₂⁰,..., p_A, Δω, Δω^H, k_ex}	p_A > p_B	Korzhnev et al. (2005a)
NS MMQ 3-site linear	Numeric	3	{R₂⁰,..., p_A, p_B, Δω_AB, Δω_BC,	p_A > p_B and p_B > p_C	Korzhnev et al. (2005a)
			Δω^H_AB, Δω^H_BC, k_ex^AB, k_ex^BC}
NS MMQ 3-site	Numeric	3	{R₂⁰,..., p_A, p_B, Δω_AB, Δω_BC,	p_A > p_B and p_B > p_C	Korzhnev et al. (2005a)
			Δω^H_AB, Δω^H_BC, k_ex^AB, k_ex^BC, k_ex^AC}

R_1ρ-type

M61	Analytic	2	{R_1ρ',..., Φ_ex, k_ex}	Fast exchange, on-resonance,	Meiboom (1961)
				R₁ = R₂
DPL94	Analytic	2	{R_1ρ',..., Φ_ex, k_ex}	Fast exchange	Davis et al. (1994)
M61 skew	Analytic	2	{R_1ρ',..., p_A, Δω, k_ex}	p_A $\gg$ p_B, on-resonance	Meiboom (1961)
TP02	Analytic	2	{R_1ρ',..., p_A, Δω, k_ex}	p_A $\gg$ p_B, not fast exchange	Trott and Palmer (2002)
TAP03	Analytic	2	{R_1ρ',..., p_A, Δω, k_ex}	Weak condition of p_A $\gg$ p_B	Trott et al. (2003)
TP04^11.1	Analytic	N	{R_1ρ',..., p₁,..., p_N, $\overline{{\omega}}$ , k₁₂,... k_1N}	One site dominant	Trott and Palmer (2004)
MP05	Analytic	2	{R_1ρ',..., p_A, Δω, k_ex}	p_A > p_B	Miloushev and Palmer (2005)
BK13	Analytic	2	{R₂⁰,..., p_A, Δω, k_ex}	p_A > p_B,	Baldwin and Kay (2013)
BK13 full	Analytic	2	{R_2A⁰, R_2B⁰,..., p_A, Δω, k_ex}	p_A > p_B,	Baldwin and Kay (2013)
NS R1rho 2-site	Numeric	2	{R_1ρ',..., p_A, Δω, k_ex}	p_A > p_B	-
NS R1rho 3-site linear	Numeric	3	{R_1ρ',..., p_A, p_B, Δω_AB, Δω_BC,	p_A > p_B and p_A > p_C	-
			k_ex^AB, k_ex^BC}
NS R1rho 3-site	Numeric	3	{R_1ρ',..., p_A, p_B, Δω_AB, Δω_BC,	p_A > p_B and p_A > p_C	-
			k_ex^AB, k_ex^BC, k_ex^AC}

**Table 11.2:** The parameters of relaxation dispersion.

Parameter	Equation	Description	Units

ν_CPMG	1/(4τ_CPMG)	CPMG frequency	Hz
τ_CPMG	1/(4ν_CPMG)	Delay between CPMG π pulses	s
T_relax	-	The relaxation delay period	s
I₀	-	Reference peak intensity when T_relax is zero	-
I₁	-	Peak intensity for a given ν_CPMG or spin-lock field strength ω₁	-
R₂⁰	-	R₂ relaxation rate in the absence of exchange	rad.s^-1
R_2A⁰	-	R₂ relaxation rate for state A in the absence of exchange	rad.s^-1
R_2B⁰	-	R₂ relaxation rate for state B in the absence of exchange	rad.s^-1
R_1ρ'	-	R_1ρ relaxation rate in the absence of exchange	rad.s^-1
$\overline{{\Omega}}$	$\overline{{\omega}}$ - ω_rf	The average resonance offset in the rotating frame	rad.s^-1
Ω_A	ω_A - ω_rf	The resonance offset in the rotating frame for state A	rad.s^-1
Ω_B	ω_B - ω_rf	The resonance offset in the rotating frame for state B	rad.s^-1
Ω_C	ω_C - ω_rf	The resonance offset in the rotating frame for state C	rad.s^-1
ω_A	-	The Larmor frequency of the spin in state A	rad.s^-1
ω_B	-	The Larmor frequency of the spin in state B	rad.s^-1
ω_C	-	The Larmor frequency of the spin in state C	rad.s^-1
ω^H_A	-	The proton Larmor frequency of the spin in state A (for MMQ data)	rad.s^-1
ω^H_B	-	The proton Larmor frequency of the spin in state B (for MMQ data)	rad.s^-1
ω^H_C	-	The proton Larmor frequency of the spin in state C (for MMQ data)	rad.s^-1
$\overline{{\omega}}$	p_Aω_A + p_Bω_B	The population averaged Larmor frequency of the spin	rad.s^-1
ω₁	-	Spin-lock field strength, i.e. the amplitude of the rf field	rad.s^-1
ω_e	$\sqrt{{\overline{\Omega}^2 + \omega_1 ^2}}$	Effective field in the rotating frame	rad.s^-1
ω_rf	-	Spin-lock offset, i.e. the frequency of the rf field	rad.s^-1
θ	arctan $\left(\vphantom{ \frac{\omega_1 }{\overline{\Omega}} }\right.$ ${\frac{{\omega_1 }}{{\overline{\Omega}}}}$ $\left.\vphantom{ \frac{\omega_1 }{\overline{\Omega}} }\right)$	Rotating frame tilt angle	rad
k_AB	p_Bk_ex	The forward exchange rate from state A to state B (2-site)	rad.s^-1
k_BA	p_Ak_ex	The reverse exchange rate from state B to state A (2-site)	rad.s^-1
k_AB	p_Bk_ex^AB	The forward exchange rate from state A to state B (3-site)	rad.s^-1
k_BA	p_Ak_ex^AB	The reverse exchange rate from state B to state A (3-site)	rad.s^-1
k_BC	p_Ck_ex^BC	The forward exchange rate from state B to state C (3-site)	rad.s^-1
k_CB	p_Bk_ex^BC	The reverse exchange rate from state C to state B (3-site)	rad.s^-1
k_AC	p_Ck_ex^AC	The forward exchange rate from state A to state C (3-site)	rad.s^-1
k_CA	p_Ak_ex^AC	The reverse exchange rate from state C to state A (3-site)	rad.s^-1
k_ex	1/τ_ex	Chemical exchange rate constant	rad.s^-1
k_ex^AB	k_AB + k_BA	Chemical exchange rate constant between sites A and B	rad.s^-1
k_ex^BC	k_BC + k_CB	Chemical exchange rate constant between sites B and C	rad.s^-1
k_ex^AC	k_AC + k_CA	Chemical exchange rate constant between sites A and C	rad.s^-1
k_B	$\approx$ k_ex^AB	Approximate chemical exchange rate constant between sites A and B	rad.s^-1
k_C	$\approx$ k_ex^AC	Approximate chemical exchange rate constant between sites A and C	rad.s^-1
τ_ex	1/k_ex	Time of exchange	s.rad^-1
p_A	-	Population of state A	-
p_B	1 - p_A	Population of state B (2-site)	-
p_B	1 - p_A - p_C	Population of state B (3-site)	-
p_C	1 - p_A - p_B	Population of state C (3-site)	-
Δω	ω_B - ω_A	Chemical shift difference between sites A and B (2-site)	rad.s^-1 (stored as ppm)
Δω_AB	ω_B - ω_A	Chemical shift difference between sites A and B (3-site)	rad.s^-1 (stored as ppm)
Δω_BC	ω_C - ω_B	Chemical shift difference between sites B and C (3-site)	rad.s^-1 (stored as ppm)
Δω_AC	Δω_AB + Δω_BC	Chemical shift difference between sites A and C (3-site)	rad.s^-1 (stored as ppm)
Δω^H	ω^H_B - ω^H_A	Proton chemical shift difference between sites A and B (2-site)	rad.s^-1 (stored as ppm)
Δω^H_AB	ω^H_B - ω^H_A	Proton chemical shift difference between sites A and B (3-site)	rad.s^-1 (stored as ppm)
Δω^H_BC	ω^H_C - ω^H_B	Proton chemical shift difference between sites B and C (3-site)	rad.s^-1 (stored as ppm)
Δω^H_AC	Δω^H_AB + Δω^H_BC	Proton chemical shift difference between sites A and C (3-site)	rad.s^-1 (stored as ppm)
Φ_ex	p_Ap_BΔω²	Fast exchange factor	rad².s^-2 (stored as ppm²)
Φ_{ex, B}	See 11.20a on page	Fast exchange factor between sites A and B	rad².s^-2 (stored as ppm²)
Φ_{ex, C}	See 11.20b on page	Fast exchange factor between sites A and C	rad².s^-2 (stored as ppm²)

Dispersion model summary

R1 parameter optimisation

R₁ parameter optimisation