3.1 Thermodynamics and dynamics of low complexity periodic motifs vs random heteropolymers

Before delving into material properties encoded by energy landscapes of stickers and spacers, we first analyze how patterns of stickers and spacers encode the thermodynamics of phase equilibrium. All the simulated chains have the same length , mimicking the length scales of LCDs commonly found in condensate-forming proteins. We systematically vary the periodicity of the sticker motifs (where ) while keeping the total “sticker energy” conserved. Despite having constant sticker energy, the critical temperatures exhibit significant variation with sticker periodicity (Fig 3A). For shorter period sequences, i.e., , the critical temperature is . For longer periodic sequences like , the critical temperature increases to . In the case of much longer period , T_c is almost twice as high as that of the sequence arrangements, (Fig 3A).

As the extreme case, we examine the fully randomized arrangement of sticker and spacer motifs denoted as . Intriguingly, regardless of the permutation of the periodically distributed chain, the phase diagram of the randomly arranged sequences manifests approximately the same critical temperatures, denoted as , (Fig 3A). Notably, randomizing the spatial arrangement of the sticker and spacer motifs of results in a halving of the critical temperature from to . Observing the phase diagram, one notices that periodically arranged stickers demonstrate a greater propensity for phase separation than randomly distributed stickers. Moreover, phases will likely be more stable at higher temperatures if the periodicity broadens. Augmenting the periodicity of attractive beads within our model resembles increasing the number of stickers, akin to folded binding domains, in a protein structure [17,30]. Experiments have demonstrated that mutations at sticker sites, rendering them functionally inert, reduce the tendency of proteins to undergo phase separation [31], a phenomenon analogous to our findings.

Structure factor: Here, we explore the correlation between sticker spacer arrangements and the emergence of local structural order. We observe an increase in the thickness of the sticker layer or domain size with elongated periodic chains, resulting in a shift of scattering peaks towards lower q-values (see in (Fig 3B)). In randomly distributed chains, S(q) decays as q^-3, without anomalies in the peak, while in periodically distributed chains, one peak is significantly higher, indicating local structures of sticker domains. The reciprocal of the peak position in Fourier space corresponds to the thickness layer (r ≈ 1 . 2σ) in , which increases longer periodicity for (r ≈ 2 . 5σ), and further increases for , the shift to even lower values (r ≈ 5σ), as indicated in (Fig 3B). The S(q) reveals the condensates’ local architecture, which, as we shall see in subsequent sections, influences nonequilibrium rheological and viscoelastic properties. An increase in sticky domain size suggests the creation of folded sticky domains, enhancing stability and promoting a more viscoelastic behavior. Conversely, randomly distributed chains exhibit fluidic behavior due to the absence of local structure in the condensed phase. Equilibrium contact map calculations of condensates support the local structures in S(q) peaks. Specifically, the contact maps illustrate spatial proximity () between motifs i and j within a biomolecular condensate (Fig A in S1 Text). In periodically distributed chains, well-formed domains of sticker motif contacts are evident, while consistent contact between spacers is lacking. Conversely, randomly distributed chains exhibit no consistent contact regions capable of forming sticker domains. The radial distribution function g(r) between sticky particles are shown in (Fig E in S1 Text), where the short-distance behavior is nearly identical irrespective of the sequence patterning.

Mean squared displacement: We use the mean squared displacement (MSD) to elucidate the subdiffusive and arrested dynamics of condensates arising from different sticker-spacer patterning. The MSD is defined as , where D is related to the diffusion constant and α is the scaling exponent. Our analysis reveals subdiffusive behavior, with MSD scaling as to , driven by the formation of sticky domains and caging dynamics (Fig 3C). The MSD exhibits a scaling for randomly organized sequences, reflecting slowed molecular motion due to topological entanglements and localized interactions. Similar subdiffusion behavior has been reported in protein droplets, where the viscoelastic nature of the droplet results in significant differences in protein diffusivity between its interior and interface [32]. This subdiffusive scaling is consistent across different periodicities of sticker molecules, even when the stickers are randomly distributed along the chain (Fig 3C). As the degree of periodicity broadens, sticky clusters grow, leading to spatiotemporally arrested condensed phases that exhibit progressively slower dynamics, as shown by the blue curves in Fig 3C.

Notably, the non-monotonic trend among the systems arises from the combined effects of system dynamics and spatial organization. For , the sticky domains are less stable and highly dynamic, resulting in behavior similar to randomly arranged sequences at longer times. In contrast, shows an initial scaling akin to random chains but deviates and decreases over time, following a power law. The behavior of is particularly intriguing as it exhibits two distinct scaling regimes. Due to its long spacer regions, initially behaves like random chains for shorter distances. However, as the spacer dynamics exceed the spacer mesh size, the system becomes increasingly arrested, causing the curve to surpass those of lower values. This anomaly highlights the interplay between sticker periodicity and spacer fluctuations, which significantly influences the long-time dynamics of these systems. These observations underscore the complex relationship between sequence organization, dynamic arrest, and molecular diffusion in condensed phases.

Viscoelasticity: Recent microrheology experiments have demonstrated that the material properties of biomolecular condensates can vary widely, ranging from intricate Maxwell fluids to glassy or Kelvin-Voigt viscoelastic gels, depending on their composition and maturation times [21,33–35]. Here, we elucidate how the energy landscape of sequences composed of stickers and spacers encodes the diverse viscoelastic properties observed in experiments. We find that randomly distributed stickers (Fig 3D) show multiple relaxation times seen by fitting the complex modulus G to the generalized Maxwell model . The complex modulus G goes to zero for all three randomly distributed sticker cases at long t, revealing Maxwell fluid behavior. Maxwell-like behavior persists also for sticker-spacer systems with short periodicity (Fig 3D). However, broadening the period of sticker domains ( & ) changes the material properties of condensate from Maxwell fluid to Kelvin-Voigt elastic solid behavior.

The complex modulus for & gets nearly saturated at long time scales (Fig 3D). The Fourier transform of & shows that the elastic modulus is independent of imposed deformation frequency ω at long times (Fig 3E) a signature of Kelvin-Voigt solid nature of the condensed phase. As sticker domain size increases, entangled sticker segments fold back to make a strongly elastic-dominated viscoelastic condensed phase, where of is an order of magnitude larger relative to its randomly distributed sequence. Note that, even though the elastic dominance differs for different arrangements of the sticker and spacers, viscous response, , collapses on the same curve for all systems. The scaling behavior of Maxwell fluid in the viscous dominated long time scale regime indicates that as ω → 0, exhibits and . Thus, one can think of the aging of biomolecular condensates as rearranging the sticker residue for a long time, which makes the systems fluidic to an elastic-dominated Kelvin-Voigt nature [19]. Our simulation reveals a strong correlation between the sticker profile and bulk viscosity of condensates (Fig 3F), linked across scales using Rouse theory [36]. The periodic arrangement of the stickers results in an order of magnitude more viscous (Fig 3F) phase compared to their random counterparts. This is consistent with its thermodynamically more stable phases and structurally larger sticky domains formed by chains with periodically arranged stickers.

3.2 Dynamical impact of local randomness in periodic stickers and spacers motifs

Having discussed the two extreme limits of periodic and random sticker patterns, we now turn to examine two intermediate cases of the disorder in stickers and spacers denoted as and (Fig 4A and 4B) respectively. We use the 25-period residue sequence as the reference for comparison. In the case of , we fix the spacers defined by the lowest ε interaction strengths and randomize sticky domains defined by the higher values of ε. In case , the sticker period is fixed, but the strengths of residues in the sticker domain alternate between higher and lower values sequentially (Fig 4A). This corresponds to typical RGRG, YGYG motifs in many phase-separating proteins [1,37,38].

Simulations show that one can have very distinct bulk equilibrium structures and material properties by simply tuning the degree of order in sticker and spacer domains (Fig 4B). Chains with periodically arranged ordered sticky regions self-assembled make micelle-like architecture where the bulk is heterogeneously distributed with the sticker types (Fig 4A and 4B). A disordered sticker arrangement makes a structure-less, unstable, homogeneous melt (Fig 4B). In the intermediate states, & , there are also structures present; however, they are not as prominent as (Fig 4A and 4B).

The phase diagram of these different sticker arrangements (Fig 4C) shows that extreme ordered and random arrangement of stickers possess the most stable and least stable phases, respectively. The sequences lie in the intermediate stability regime. To structurally distinguish the architecture of these condensates, we calculate the structure factor S(q). An anomalous peak in the case of indicates the existence of local structure in the condensed phase. The peak position at q refers to the size of the sticker micelle-like local structures (Fig 4D).

The prominent peak is lowered in the case of , suggesting the local structure is fading out. In the case of a fully randomized sequence , one ends up with a homogeneously distributed melt system. The emergence of structural order is also reflected in subdiffusive dynamics. The exponent of the MSD decays with periodically arranged stickers (Fig 4E).

In terms of viscoelastic properties quantified by (Fig 4F), we find that local structural order leads to elevated elastic modulus (Fig 4F). We plotted viscosity over time, whereas the saturation value as , describes the total viscosity of the system. In condensates with periodically arranged stickers, condensates form the most viscous systems, whereas it is lowest for (Fig 4G).

One can also look into the contact time autocorrelation function, C(t), which tracks the dynamics of contact formation and offers insights into the viscoelasticity [39]. Short timescales show rapid decay in C(t), suggesting fluid-like behavior with frequent contact disruptions and reforms. On longer timescales, C(t) indicates a shift toward a gel-like arrested state, where sustained interactions enhance viscosity and structural rigidity, mainly due to sticker-sticker interactions (Fig D in S1 Text). Periodic sticker arrangements result in longer correlation times in C(t), while higher temperatures accelerate correlation decay, indicating reduced viscoelastic stabilization from sticker interactions (Fig D in S1 Text).

3.3 Comparison with experiments

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Fig 5. Comparison of the experimental results of reported in [21] and [40] in (A) with our model of periodic and random sequences in (B) as a function of periodicity of the sequence.

Both experimental and simulation results show an increase in viscoelasticity as the periodicity of the arranged sequences increases.

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We have reported that viscoelasticity grows as a function of the periodicity of stickers and that mutations in sticker regions have the most dramatic impact on material properties. In this section, we show that these results fall within the expected range of experimentally characterized material properties despite the limited data available in the literature on the viscoelastic properties of protein condensates with varying sticker and spacer residues [21,40] (sequences are listed in the SI section D). We define periodicity for a real protein sequence to facilitate comparison with our energy landscape predictions. For this, we use a correlation length scale where the maximum energy strengths as defined by hydropathy index [41] , decays to half-width total maxima along the chain,  ⟨ κ ⟩ . We plot the inverse loss tangent, 1 ∕ tan ⁡  δ, or the ratio of , as a function of , where the chain length is N_p and is the energy amplitude of the i-th residue, and is the maximum energy among the sequence residues. Note that most of the experiments have been conducted with changing residues; therefore, along with the periodicity  ⟨ κ ⟩ , we also incorporate the unitless interaction energy parameter λ. A similar increasing trend in the elastic modulus with increasing periodicity is observed in our simulation data (Fig 5B). It is consistent with the limited experimental data from [21] (Fig 5A). An alternative parametrization of these sequences using the Wasserstein distance, , reveals similar behavior, as shown in SI (Fig F in S1 Text). This analysis underscores the predictive capability of the sequence arrangements, with the Wasserstein distance serving as a key metric to quantify deviations from randomness. Observing the viscoelastic properties of such predictive sequence patterning would provide valuable insights into the spatial molecular grammar and its relationship to material properties.

3.4 Orientational order and correlations

A semi-flexible homopolymer chain’s persistence length () exhibits resistance to bending, which characterizes the mechanical and viscoelastic properties of constitutive bulk systems. The angle between a tangent vector at position i and another tangent vector at a distance along the polymer contour, s, determines the correlation , indicating the extent (e.g., persistence length, ) over which correlations along the contour approach zero. Here,  ⟨ ⋯ ⟩  is taken over all possible configurations of the polymer and ensemble of the multiple chains of specific bead type. Typically, the homopolymer with bending rigidity (or persistence length ) reveals viscoelastic properties. As a reference, we note that tangent correlation (exponential decay) along the chain contour is spatially invariant in the case of homopolymer, as every bead is identical.

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Fig 6. Tangent-tangent correlation in condensates was calculated for the periodically distributed (A–E) and randomly distributed (F) stickers.

(A) For , periodic sticky regions aggregate; however, the chains show uniform exponential decay correlation. (B) shows anomalous exponential decay in , indicating a folded structure. (C) displays diverse position-dependent residue characteristics and long-range order; spacer residues show exponential decay, while spool-like folded domains show non-exponential decay and negative correlation. (D) also shows anomalous exponential decay, and (E) has distinct exponential decay lengths. (F) A collapsed exponential decay in the tangent correlation of randomly distributed chains (, , ) exhibits worm-like chain behavior with a persistence length of .

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In our model, even though the chain is not homopolymeric, we find a single exponential decay tangent correlation for random sequences as depicted in Fig 6D–6F. Interestingly, sticky residues exhibit different correlation characteristics in the case of periodic sticker-spacer motifs. Due to the formation of clusters among sticky residues, there is strong anti-correlation behavior [42–44]. The orientational ordering in periodically arranged sticker systems reveals that certain beads show non-monotonous decay in tangent correlation. For spacer beads with repulsive interactions, tangent correlation decays exponentially. One sees long-distance ordering as a function of sticker interaction strength, which also leads to negative correlations, dictating that chains are folding back. The strength of interaction among sticker residues does not solely dictate the arrangement of folded sections (Fig 6). Still, it is also influenced by the spatial distribution of the sticker beads (Fig 6B and 6C).

The tangent correlation for short periodic sticker () systems exponentially decays, revealing non folded sticky patch region (Fig 6A). Therefore, these systems do not form tightly connected sticky domains and exhibit viscous-dominated rheological behavior under applied deformation. Consequently, they are also less stable at high temperatures, leading to a lower critical temperature () in the phase diagram than systems with broadly distributed periodic sticker beads. Systematic increase in periodicity of the sticker in Fig 6B and 6C, sticker residues with have a higher tendency to fold back compared to spacers . Those have . The radius of curvature of the coiled folded sticky regions can be extracted from the minimum correlation distance. Spacer residues exhibit the expected exponential decay characteristics (Fig 6 C); however, with an incremental increase in sticker strength, residues tend to accumulate within a sticky domain in a folded configuration.

Periodically arranged sticker molecules show arrested dynamics, whereas the randomly distributed stickers are more fluid. We calculate the self-van Hove function to differentiate the spatial-temporal characteristics that resemble the distinct sub-diffusive nature (different exponent) of the dynamics. The self-part of the van Hove function (where N is the total number of beads, δ is Dirac delta and ensemble average  ⟨ . . . ⟩  over multiple trajectories), captures the probability distribution of particle displacements over a time interval t [45]. The approximation of the self-van Hove function is Gaussian [46]. Unlike the Gaussian observed in structurally simple fluids [46], our analysis reveals nuanced characteristics, particularly in randomly distributed frustrated sticker-spacer condensed phase. Frustration arises because the system cannot simultaneously minimize all interaction energies due to geometric or spatial constraints. In this phase, exponential decay tails signify anomalous behavior [44,47,48]. In the case of periodically arranged systems, the self-van Hove function is Gaussian in nature, in early (dashed line Gaussian fit) and as late as (Fig 7A, 7B and 7C). Chains , with shuffled stickers, remain Gaussian. However, chains, with a sticker periodicity of 25 and alternating sticker-spacer arrangements, deviate from Gaussian behavior at long times. On the other hand, even though initially randomly arranged stickers show Gaussian nature, for a long time, they fail to exhibit Gaussian behavior (Fig 7F). The frustration leads to a disordered structure within the condensed phase, significantly affecting the dynamical behavior, as evident from the deviations of the Gaussian behavior. Therefore, we conclude that sticker beads are localized in the case of periodic stickers, as shown in Fig 7A–7E, compared to the randomly placed sticker-spacer arrangement. This temporal evolution of provides invaluable insights into the evolving microenvironment surrounding each sticky core, revealing structural rearrangements within the cluster [45].

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Fig 7. The self part of the van Hove distribution function of the sticker residues, , are shown with the color scheme indicating the temporal evolution: dark colors represent the earlier distribution (t ~ 10τ), whereas light colors depict the late-time behavior ().

All the curves correspond to distributions taken at the same time intervals. Gaussian fits are shown for the early time (t ~ 10τ, blue dashed line) and late time (, red solid line). (A–C) represent chains with periodic sticker patterns having periods 10, 25, and 100, which remain Gaussian at long times. (D) shows chains with a sticker periodicity of 25 and shuffled stickers, which also remain Gaussian at long time. However, (E) depicts chains with a sticker periodicity of 25 and alternating sticker-spacer arrangements, which deviate from Gaussian behavior. (F) corresponds to R₂₅ (other shows similar behavior), chains with randomly arranged beads, which fail to exhibit Gaussian behavior at long times, indicating the lack of spatial organization in the system.

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Fig 8. Biomolecular condensate phases of at different number densities ρ (by varying the box volume ) creates a range of architectures, from a network fluid structure to clusters of micelles.

(G) Orthogonal radius of gyration of sticky clusters , (H) average number of motifs in a single cluster, , and (I) the average number of sticky clusters in bulk, , as a function of box dimensions ρ are shown here. The vertical lines separate the percolated single and multiple clusters in the bulk phase.

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Fig 9. The first row displays snapshots of sticker clusters for four different chain lengths N: (A) 25, (B) 50, (C) 100, and (D) 200, with individual clusters depicted in various colors.

In the second row, corresponding systems for each chain length illustrate how cluster shapes evolve across different density regimes, transitioning from spherical to elongated micelles. The third row presents the probability distribution of the radius of gyration, , for the sticker clusters.

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3.5 Density dependent condensate architectures:

The energy landscapes of stickers encode the final equilibrium structural and dynamical properties and impact the time scales and percolation characteristics during the formation of condensates in the dilute biomolecular regime. We carried out simulations in the dilute regime observing a network-like fluid structure [29], whereas, in the dense phase, sticky domains form percolated membrane-like structures (Fig B in S1 Text). In the intermediate state, where chain densities are low, spherical micelles form, and at higher densities, micelles are no longer spherical but elongated along an axis. As density increases, these elongated tube-like micelles form large branch micelles. Above a critical density, these branched micelles connect and create a single percolated porous foam-shaped membrane-like architecture.

To quantitatively characterize the shapes of the sticky clusters, we identify the clusters and calculate the gyration tensor of the cluster as a function of chain concentrations ρ. The principal components of the gyration tensor, (i = x , y , z, denoted by three distinct markers), are plotted as a function of ρ, for chain lengths (Fig 8G). The vertical line separates the percolation and disperse clusters. Interestingly, all the orthogonal components of are equivalent at lower densities. However, degenerates as density increases, indicating that the three principal components , and which are equal for spherical clusters, become unequal. This deviation suggests that the clusters lose their spherical symmetry and transition to an elongated shape. This behavior persists across different chain lengths, as shown in Fig 8G.

The average number of particles in each cluster as a function of ρ (Fig 8H) for different shows similar behavior. It is also interesting to examine the number of clusters in the system as a function of chain density, which linearly increases with density (Fig 8I and Fig C in S1 Text), irrespective of the chain length. The critical length scale here is the size of the cluster or the width of the membrane-like percolated surface shape, which can be quantified by identifying the anomalous peak in S(q).

To explore the morphology of clusters we use shape characteristics such as asphericity, and nature of asphericity, where is the gyration tensor [49–52]. The parameter , ranging from 0 to 2, characterizes cluster deviation from spherical to elongated shapes, with extreme values representing spheres and rigid rods, respectively. Whereas on the x–axis provides a quantitative measure of shape anisotropy, it delineates the distinction between oblate and prolate shapes. We report distinct micelle-like sticky cluster shapes (Fig 9) for two different scenarios: i) four different chain lengths (Fig 9A–9D) ii) different chain densities ρ. Notably, the red line delineates the structural diversity within clusters between accessible and inaccessible closed shapes. We vary the density ρ for each chain length , indicating that spherical sticky clusters form at lower densities (ρ ≤ 0 . 2). In contrast, the sticky clusters form elongated micellar structures at higher densities (Fig 9). The histogram of the radius of gyration of these sticky clusters further supports this observation (Fig 9). Under dilute conditions, equilibrium clusters are smaller, exhibit more fluid-like behavior, and are loosely connected within the network (Fig D in S1 Text). Conversely, the cluster node size increases at higher densities, and the sticky nodes are tightly connected, resulting in a more elastic network.