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Derivation of segmental motional models from experimental spin relaxation rates
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Solvent-dependent segmental dynamics in intrinsically disordered proteins

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The fitted relaxation rates are given by the following functions$R1=110(μ0ℏγHγN4πrN−H3)2(J(ωH−ωN)+3J(ωN)+6J(ωH+ωN))+215ωN2(σ∥−σ⊥)2J(ωN)$(1)$R2=120(μ0ℏγHγN4πrN−H3)2(4J(0)+J(ωH−ωN)+3J(ωN)+6J(ωH+ωN)+6J(ωH))+145ωN2(σ∥−σ⊥)2(4J(0)+3J(ωN))$(2)$σNH=110(μ0ℏγHγN4πrN−H3)2(6J(ωH+ωN)−J(ωH−ωN))$(3)$ηxy=115P2(cosθ)(μ0ℏγHγN4πrN−H3)(σ∥−σ⊥)ωN(4J(0)+3J(ωN))$(4)$ηz=115P2(cosθ)(μ0ℏγHγN4πrN−H3)(σ∥−σ⊥)ωN(6J(ωN))$(5)

J(ω) is the spectral density function at frequency ω, and θ is the angle between the N-H dipole-dipole interaction and the principal axis of the chemical shift anisotropy (CSA) tensor (assumed axially symmetric with anisotropy σ − σ= −172 ppm). rNH is the NH internuclear distance (assumed to be 1.02 Å); γH and γN are the gyromagnetic ratio of 1H and 15N nuclei, respectively; μ0 is the permittivity of free space; and ℏ is Planck’s constant.

The spectral density function was modeled by the sum of Lorentzian functions$J(ω)=∑kAkτk1+ω2τk2$(6)where ∑kAk = 1.

The model-free analysis optimizes A2, A3, τ1, τ2, τ3, and θ using a nonlinear least-squares fitting approach by minimizing the following function$χi2=∑n=15∑m=1N{(Rn,expm−Rn,calcm)σn,expm}2$(7)

Segments were analyzed by constraining one of the correlation times that is common to the segment to adopt the same value for each residue, while the other two correlation times and the amplitudes were all optimized simultaneously. The optimization was carried out by summing Eq. 7 over all of the residues in the given segment.

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