Equilibrium distribution of shapes for linear and star macromolecules

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Abstract
We investigate the equilibrium distribution of three-dimensional shapes adopted by isolated linear and star-shaped macromolecules, both with and without intramolecular interactions, using an implementation of the Monte Carlo method suitable for macromolecules with branches. We compute the joint probability distribution function for two quantities which together characterise invariant features of the normalised radius of gyration tensor associated with the shape (rather than size) of macromolecular configurations. Amongst other things, knowledge of this distribution function allows us to compute the expectation values ⟨Δ0⟩ and ⟨S0⟩ introduced by Aronovitz and Nelson (J. Phys. France 47 (1986) 1445) to characterise the extent and nature of anisotropy of typical shapes drawn from the ensemble of macromolecular configurations. We also compute a third expectation value ⟨Σ0⟩ which isolates the nature of the anisotropy from its extent. Furthermore, our simulation permits a comparison of ⟨Δ0⟩ and ⟨S0⟩ with the less natural (but analytically tractable) alternative quantities Δ, the asphericity examined by Rudnick and Gaspari (J. Phys. A 19 (1986) L191) and by Aronovitz and Nelson, and S, examined by Aronovitz and Nelson, which have enhanced sensititvity to larger configurations and therefore convolve shape information with size information. It is found that although Δ and S do provide some characterisation of anisotropy, they differ considerably from the natural measures ⟨Δ0⟩ and ⟨S0⟩. In particular, if Δ and S are regarded as approximations to Δ0⟩ and ⟨S0⟩ then, for both linear and branched macromolecules, they severely underestimate the increase (or overestimate the decrease) in extent and prolateness of anisotropy due to intramolecular interactions

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