Intrinsically disordered proteins play important roles throughout biology, yet our understanding of the relationship between their sequences, structural properties, and functions remains incomplete. The dynamic nature of these proteins, however, makes them difficult to characterize structurally. Many disordered proteins can attain both compact and expanded conformations, and the level of expansion may be regulated and important for function. Experimentally, the level of compaction and shape is often determined either by small-angle x-ray scattering experiments or pulsed-field-gradient NMR diffusion measurements, which provide ensemble-averaged estimates of the radius of gyration and hydrodynamic radius, respectively. Often, these experiments are interpreted using molecular simulations or are used to validate them. We here provide, to our knowledge, a new and efficient method to calculate the hydrodynamic radius of a disordered protein chain from a model of its structural ensemble. In particular, starting from basic concepts in polymer physics, we derive a relationship between the radius of gyration of a structure and its hydrodynamic ratio, which in turn can be used, for example, to compare a simulated ensemble of conformations to NMR diffusion measurements. The relationship may also be valuable when using NMR diffusion measurements to restrain molecular simulations.
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