  Liquid-liquid phase separation (LLPS) of proteins regulated by small
molecules, such as Adenosine triphophate (ATP), ssDNA and RNA, has been
observed. However, the physical mechanism of this regulation remains largely
unknown. Here we develop a minimal multi-scale model to quantitatively study
the influence of ATP on LLPS of Fused in Sarcoma (FUS). Based on Flory-Huggins
theory, we explicitly include multivalent interactions instead of treating all
interactions via simple Flory parameters. We find the nonlinear shift of phase
diagrams with the increase of ATP concentration, which agrees with the first
enhancement and then inhibition of LLPS observed in the experiment. The reason
lies in the positive and negative cooperation between ATP-FUS and FUS-FUS
interactions, where the existence of the ATP bridging plays a key role.
Furthermore, we systematically study the phase behavior in a larger parameter
space, and introduce a quantity to reflect different regulation degrees. The
present model provides a clear physical picture relating microscopic
interactions to macroscopic phase diagrams, as well as a good example for
adapting a classical theory in polymer physics to understanding the new
phenomena in intracellular phase separation.
