At low pulsing frequency, there are few such frequencies. At high pulsing frequency, there are many more such slowly relaxing terms present. It is these slowly relaxing terms that give rise to the characteristic increase in signal observed in a CPMG experiment. PARP inhibitor An expression for the effective transverse relaxation rate of the ground state ensemble is sought: equation(1) R2,eff=-1TrellnIG(Trel)IG(0)where Trel is the total time of the concatenated CPMG elements and IG specifies the signal intensity from the observed ground state at the specified times. In order to calculate the relevant signal intensities a
kinetic model for the exchange process and types of magnetisation present need to be specified. The simplest and most widely encountered kinetic scheme is the two-site case for in-phase magnetisation. Here, a ground state and an excited state undergo the conformational rearrangement G⇄kEGkGEE. In this scheme, the exchange rate kEX = kEG + kGE and the fractional populations of the excited (PE ) and ground (PG ) states are given by kGE /kEX and kEG /kEX respectively. The CPMG experiment consists of a number of free precession elements interspersed with 180° pulses. To evaluate Cyclopamine their combined effect, how magnetisation evolves in the absence of pulses needs first
to be calculated. This is accomplished most conveniently using the shift basis (I + = Ix + iIy and I − = Ix − iIy ) using a modified Bloch–McConnell equation : equation(2) ddtIG+IE+=R+IG+IE+where E and G denote the magnetisation on the excited and ground states, respectively. The evolution matrix is: equation(3) R+=-kGE-R2GkEGkGE-kEG-R2E-iΔωR 2G and R 2E specify the intrinsic RG7420 relaxation of the ground and excited states respectively, and Δω is the chemical shift difference between the ground and excited states in rad s−1. The solution for Eq. (2) is: equation(4) I(t)=eR+tI(0)=OI(0)I(t)=eR+tI(0)=OI(0)where I (0) are I (t ) specify the magnetisation on the ground and excited states at time zero and t respectively. Initially the system
is in equilibrium, and so I(0)†=(PG,PE)I(0)†=(PG,PE) where †† indicates a transpose. The derivation of I(t) first requires the well known matrix O (Eq. (17)) that determines how magnetisation evolves during free precession . In the shift basis, the effect of a 180° on-resonance ideal pulse switches magnetisation on I+ terms to I−, leading magnetisation to evolve according to the complex conjugate of R+ (Eq. (3)), (R+)*. Following a 180° pulse therefore, magnetisation will evolve according to the matrix O*. By applying Eq. (4) iteratively, taking the complex conjugate where appropriate, an expression that represents the entire CPMG experiment can be built. This, when used with Eq. (1) enables us to derive an expression for R2,eff. The matrix M that represents the CPMG experiment will enable us to evaluate I(t) = MI(0).