Conformational Properties of Polymers at Droplet Interfaces as Model Systems for Disordered Proteins

Polymer models serve as useful tools for studying the formation and physical properties of biomolecular condensates. In recent years, the interface dividing the dense and dilute phases of condensates has been discovered to be closely related to their functionality, but the conformational preferences of the constituent proteins remain unclear. To elucidate this, we perform molecular simulations of a droplet formed by phase separation of homopolymers as a surrogate model for the prion-like low-complexity domains. By systematically analyzing the polymer conformations at different locations in the droplet, we find that the chains become compact at the droplet interface compared with the droplet interior. Further, segmental analysis revealed that the end sections of the chains are enriched at the interface to maximize conformational entropy and are more expanded than the middle sections of the chains. We find that the majority of chain segments lie tangential to the droplet surface, and only the chain ends tend to align perpendicular to the interface. These trends also hold for the natural proteins FUS LC and LAF-1 RGG, which exhibit more compact chain conformations at the interface compared to the droplet interior. Our findings provide important insights into the interfacial properties of biomolecular condensates and highlight the value of using simple polymer physics models to understand the underlying mechanisms.


Simulation Details
We have used a coarse-grained model, in which each monomer is represented by a spherical bead that is connected to neighboring beads via a harmonic spring: where  is the distance between two bonded beads,  b = 20 kcal/(molÅ 2 ) is the spring constant and  0 = 3.8 Å is the equilibrium length.The van der Waals interactions between nonbonded beads were modeling using a modified Lennard-Jones (LJ) potential 1  vdW ( where  LJ is the standard LJ potential with average hydropathy  = 0.865 and bead diameter  = 6.46 Å for the homopolymer systems.
The interaction strength was fixed to  = 0.2 kcal/mol in all simulations which, in conjunction with other model parameters, was previously found to reproduce the experimentally measured  g values of several IDPs.For computational efficiency, the pair potential  vdW and its associated forces were truncated to zero at a distance of 4 σ.
To have good statistics, droplet simulations were performed by placing 5000 chains (50 beads per chain) into a cubic box of edge length 1010 Å with periodic boundary conditions in all directions.We performed Langevin dynamics simulations at temperature  = 300K.The friction coefficient was set as   =    damp ⁄ , where   = 163.2g/mol and  damp = 1000 fs.Simulations were performed for 2.5 μs with a time step of 10 fs using HOOMD-blue 2 (ver.2.9.3) with features extended using azplugins 3 (ver.0.10.2).The simulation trajectories were saved every 1 ns.
Analysis was based on the last 1.5 μs.
Bulk system simulations were performed by placing 500 chains into a cubic box at a constant pressure of P = 0 atm for 0.5 μs.After the chains achieved their preferred bulk-phase concentration, the simulation was performed using Langevin dynamics at constant volume for 1 μs.All the simulations were implemented at a fixed temperature of T = 300 K.The analysis was based on the last 0.5 μs.Other simulation settings were the same with droplet simulation.

Average angle calculation for isotropic distribution of segments
The average angle of segments distributed without any preferred orientation can be calculated as where  is the angle between the segment-to-droplet COM vector and the segment end-to-end vector.

Relative shape anisotropy (𝜿 2 ) calculation for droplet:
By considering droplet as one long polymeric chain,  2 can be calculated using eigenvalues (λ) of the gyration tensor: where  i is the vector from the chain's mass center to residue i, and N is the total number of monomers in the droplet.

Supplementary Figures
Fig. S1.Relative shape anisotropy ( 2 ) of the droplets with respect to time for homopolymer, FUS LC, and LAF-1 RGG.The purple shaded area is the interface region.
Single chain simulations were performed by placing one chain into a cubic box of edge length 160 Å. Langevin dynamics simulations were performed for 1 μs at constant temperature  = 300 K.The analysis was based on the last 0.5 μs.Other simulation settings were the same with droplet simulation.Natural protein simulations were performed by placing 1500 chains into a cubic box of edge length 1000 Å with periodic boundary conditions in all directions.Langevin dynamics simulations were performed at temperature  = 300K for FUS LC and  = 260K for LAF-1 RGG.The hydropathy values (derived based on experimental transition temperatures) for the residues in natural proteins were set according to the HPS-Urry model4 .The friction coefficient was set as  =    damp ⁄, where   was the residue mass and  damp = 1000 fs.Simulations were performed for 1 μs and other simulation settings were the same with droplet forming simulation.

Fig. S2 .
Fig. S2.Distribution of radius of gyration ( g ).The black dashed line represents the  g distribution in the bulk system.The black solid line represents the  g distribution of a single chain.The color gradient from red to purple corresponds to the distribution of  g for chains with  COM ranging from 150 to 255 Å.

Fig. S4 .
Fig. S4.Orientation calculation.(a) Schematic of calculation method for angle (θ) and the minimum of θ ( min ).(b) θ of the segments of different length located at the middle of a chain as a function of  COM .(c) θ of the segments of different length located at the ends of a chain as function of  COM .

Fig. S6 .
Fig. S6.Concentration as a function of distance from the droplet's center of mass ( COM ) for (a) FUS LC and (b) LAF-1 RGG.The red line is the fitted curve for the simulation data (symbols).The purple shaded area is the interface region.