Direct computations of viscoelastic moduli of biomolecular condensates

Abstract

Biomolecular condensates are viscoelastic materials defined by time-dependent, sequence-specific complex shear moduli. Here, we show that viscoelastic moduli can be computed directly using a generalization of the Rouse model that leverages information regarding intra- and inter-chain contacts, which we extract from equilibrium configurations of lattice-based Metropolis Monte Carlo (MMC) simulations of phase separation. The key ingredient of the generalized Rouse model is a graph Laplacian that we compute from equilibrium MMC simulations. We compute two flavors of graph Laplacians, one based on a single-chain graph that accounts only for intra-chain contacts, and the other referred to as a collective graph that accounts for inter-chain interactions. Calculations based on the single-chain graph systematically overestimate the storage and loss moduli, whereas calculations based on the collective graph reproduce the measured moduli with greater fidelity. However, in the long time, low-frequency domain, a mixture of the two graphs proves to be most accurate. In line with the theory of Rouse and contrary to recent assertions, we find that a continuous distribution of relaxation times exists in condensates. The single crossover frequency between dominantly elastic vs dominantly viscous behaviors does not imply a single relaxation time. Instead, it is influenced by the totality of the relaxation modes. Hence, our analysis affirms that viscoelastic fluid-like condensates are best described as generalized Maxwell fluids. Finally, we show that the complex shear moduli can be used to solve an inverse problem to obtain the relaxation time spectra that underlie the dynamics within condensates. This is of practical importance given advancements in passive and active microrheology measurements of condensate viscoelasticity.

FIG. 1.

Illustration, for completeness, of the well-established and well-known Rouse theory for an ideal chain in solution. (a) Bead-and-spring model for a single chain (N = 7). The beads are Kuhn monomers, each of size b. (b) The central object of our adaptation of the Rouse theory is the graph Laplacian, which is shown here for an ideal chain of seven beads. (c) The spectrum of relaxation times, τ_p, for a 137-mer chain is calculated from the eigenvalues of the graph Laplacian. A fit of the dimensionless relaxation times to [1/(6π²)](N/p)² agrees well with the slowest relaxation times. The mode corresponding to τ_N−1 describes the relaxation time of a single monomer, whereas τ₁ describes the relaxation time of the entire chain. (d) Plots of the computed relaxation moduli for an ideal 137-mer. (e) The storage modulus (G′) and the loss modulus (G″) for an ideal chain are plotted against angular frequency. The scaling ∼ ω^1/2 in the frequency range, 1/τ₁ ≪ ω ≪ 1/τ_N−1, is observed, with G″ converging to G′. (f) Viscosities are calculated using iωη* = G′ + iG″. The quantities here are normalized to the zero-shear viscosity, η₀, and the polymer-mediated viscosity of the solvent, η_s. In (c)–(f), we plot the dimensionless quantities.

FIG. 2.

Assessments of how non-bonded contacts impact computed moduli of an ideal chain. (a) Starting with the 137-mer ideal chain, we randomly add contacts by increasing the off-diagonal density of the graph Laplacian starting with the ideal chain at ρ = 0.0146. We compute the dynamical moduli (solid lines) and compare the results to the ideal chain (dashed lines) for (b) ρ = 0.015, (c) ρ = 0.017, and (d) ρ = 0.019. The intermediate region in the dynamical moduli disappears as random contacts are added.

FIG. 3.

The single-chain graph and comparisons of computed and measured moduli. (a) Snapshot of a single chain extracted from MMC simulations of the dense phase of A1-LCD: WT^+NLS. Each residue is a Kuhn monomer. (b) The graph Laplacian accounts for both bonded and non-bonded intra-chain contacts. The inset indicates a region of the graph Laplacian showing the types of intra-chain contacts that form in dense phases. (c) The corresponding dynamical moduli show an intermediate response scaling as ∼ω^1/2. The Rouse time is longer than the Rouse time, τ₁, for the ideal chain due to internal friction imparted by the non-bonded interactions. (d) Comparisons to measurements following a rescaling to match the measured crossover frequency show that the single-chain graph overestimates the experimental storage and loss moduli. The computed moduli were analyzed over 30 snapshots across three replicates by averaging the pth relaxation times over all chains in the dense phase.

FIG. 4.

In the collective graph, the entire polymer network is treated as a single chain. (a) We analyze dense phases from LaSSI simulations of WT^+NLS. (b) Chains in the dense phase have a high likelihood of forming inter-chain interactions. (c) A coarse-grained model of the dense-phase network was developed by treating each chain as a node. An undirected edge between nodes indicates that at least one pair of residues between the chains forms a contact. (d) The graph Laplacian corresponding to the dense phase in panel (a). The insets indicate regions with different sparsities.

FIG. 5.

Performance of the collective graph, the computed relaxation modulus, and the mixture of two graphs. (a) The storage and loss moduli computed using the collective graph are plotted alongside the experimental values for WT^+NLS. (b) The relaxation modulus shown here is computed using both the single-chain and collective graphs. The collective graph leads to a shortening of the relaxation times as the network is less stiff and more compliant. In (a) and (b), 30 snapshots were analyzed over three replicates. The error bands indicate the standard deviation. (c) A linear combination of the moduli computed using the single-chain and collective graphs reproduces the experimental values with a coefficient of $\approx 0.88$ for the collective graph.

FIG. 6.

A continuous spectrum of relaxation times indicates that a generalized Maxwell model best describes dense phases of PLCDs. (a) We calculated the spectrum for the WT^+NLS from the experimentally measured moduli in Fig. 5(a) following a nonlinear regularization strategy. As a consistency check, we solve the forward problem exactly from the knowledge of $h\left(\tau \right)$, thereby recovering the storage and loss moduli. (b) We also compute the spectrum from the experimentally measured moduli for the allY variant following the same approach. The arrow in each panel indicates the peak in the relaxation time spectrum corresponding to the crossover in the dynamical moduli.