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Random polyampholytes (PAs) contain positively and negatively chargedmonomers that are distributed randomly along the polymer chain. The interactionbetween charges is assumed to be given by the Debye-Huckel potential. We showthat the size of the PA is determined by an interplay between electrostaticinteractions, giving rise to the polyelectrolyte (PE) effect due to net chargeper monomer ($\sigma$), and an effective attractive PA interaction due tocharge fluctuations, $\delta \sigma$. The interplay between these terms givesrise to non-monotonic dependence of the radius of gyration, $R_g$ on theinverse Debye length, $\kappa$ when PA effects are important (${\frac{\delta\sigma}{\sigma}} > 1$). In the opposite limit, $R_g$ decreases monotonicallywith increasing $\kappa$. Simulations of PA chains, using a charged bead-springmodel, further corroborates our theoretical predictions. The simulationsunambiguously show that conformational heterogeneity manifests itself amongsequences that have identical PA parameters. A clear implication is that thephases of PA sequences, and by inference IDPs, cannot be determined using onlythe bare PA parameters ($\sigma$ and $\delta \sigma$). The theory is used tocalculate the changes in $R_g$ on $N$, the number of residues for a set ofIntrinsically Disordered Proteins (IDPs). For a certain class of IDPs, with $N$between 24 to 441, the size grows as $R_g \sim N^{0.6}$, which agrees with datafrom Small Angle X-ray Scattering (SAXS) experiments.

Charge fluctuation effects on the shape of flexible polyampholytes with applications to Intrinsically disordered proteins