Random crosslinks generate anomalous scaling of dynamic moduli of biomolecular condensates

Biomolecular condensates are viscoelastic, and their mechanical properties are intimately related to their biological functions. However, the connection between microscopic networks formed by intermolecular crosslinks and viscoelasticity is still elusive. Here, we model biomolecular condensates as random crosslinked polymer solutions to elucidate how random connectivity fundamentally alters their viscoelasticity. We decompose the entire condensate into multiple clusters without loops and demonstrate that for clusters with size n, their eigenvalue distributions exhibit a power-law scaling pn({lambda}) [~]{lambda}-1/3 with a lower cutoff{lambda} min [~]n-3/2. By integrating all clusters, we show that for the entire condensate, random crosslinks generate abundant slow modes involving multiple linear polymers with a constant eigenvalue distribution. The slow modes cause anomalous linear frequency scaling of the dynamic moduli; in particular, they significantly boost the low-frequency storage modulus relative to uncrosslinked systems. Our model rationalizes the anomalous scaling of the dynamic moduli observed in multiple biomolecular condensates.