key: cord-0292271-jpo6pji8
authors: Shillcock, Julian C.; Lagisquet, Clément; Alexandre, Jérémy; Vuillon, Laurent; Ipsen, John H.
title: Model biomolecular condensates have heterogeneous structure quantitatively dependent on the interaction profile of their constituent macromolecules
date: 2022-03-27
journal: bioRxiv
DOI: 10.1101/2022.03.25.485792
sha: 199c7e9bbce93ca786fcca8490c2ae362580100c
doc_id: 292271
cord_uid: jpo6pji8

Biomolecular condensates play numerous roles in cells by selectively concentrating client proteins while excluding others. These functions are likely to be sensitive to the spatial organization of the scaffold proteins forming the condensate. We use coarse-grained molecular simulations to show that model intrinsically-disordered proteins phase separate into a heterogeneous, structured fluid characterized by a well-defined length scale. The proteins are modelled as semi-flexible polymers with punctate, multifunctional binding sites in good solvent conditions. Their dense phase is highly solvated with a spatial structure that is more sensitive to the separation of the binding sites than their affinity. We introduce graph theoretic measures to show that the proteins are heterogeneously distributed throughout the dense phase, an effect that increases with increasing binding site number, and exhibit multi-timescale dynamics. The simulations predict that the structure of the dense phase is modulated by the location and affinity of binding sites distant from the termini of the proteins, while sites near the termini more strongly affect its phase behaviour. The relations uncovered between the arrangement of weak interaction sites on disordered proteins and the material properties of their dense phase can be experimentally tested to give insight into the biophysical properties and rational design of biomolecular condensates.

The phase separation of intrinsically-disordered proteins (IDP) into biomolecular condensates has taken centre stage in cellular physiology.(1) Biomolecular condensates (BC) appear in numerous locations in cells where they carry out many biochemical functions.(2, 3) They modulate enzymatic activity,(4-7), a function that has been reproduced with rationally designed peptides; (8) (9) (10) buffer protein concentration, (11) reduce noise in gene expression, (12) and regulate cell migration. (13) Phase separation of IDPs is crucial for the healthy functioning of neuronal synapses in the presynaptic axon (14, 15) and postsynaptic dendrite. (16, 17) However, dysfunctional phase separation underlies many pathological processes.

Many neurodegenerative diseases involve aberrant phase transitions of IDPs, (18) and interfering with such transitions presages new routes for therapeutic advances in neurology. (19) Viral replication occurs within phase-separated inclusion bodies, (20) and proteins responsible for packing the RNA of SARS-Cov2 also undergo phase separation. (21) These processes depend on the ease with which enzymes and reactants can diffuse, fluctuate, and interact within. (22, 23) A better understanding of how the molecular architecture of scaffold IDPs influences the material properties of BCs would help elucidate the physical mechanisms underlying their functions and guide the identification of targets for drugs, (24) opening up new approaches to attack cancer and other diseases. (25) (26) (27) (28) Although many biomolecular condensates are fluid,(29) they do not behave as simple liquids. A slow transition from the fluid phase to a rigid, fibrillous phase has been observed in vivo and in vitro for IDPs including Huntingtin (30, 31) , FUS, (32, 33) and alpha synuclein (18) . Chromatin has been shown to undergo phase separation in vitro, (34) and to be mechanically restructured in a process that depends on its viscoelastic properties. (35, 36) Such experiments show that recapitulating the dynamics of condensates does not require the complex environment of a living cell. (23) It has been conjectured that reversible phase separation is indicative of cellular health while irreversible rigidification of BCs marks a cell's transition into disease states. (29) Empirically classifying biomolecular condensates on a spectrum from fluid (healthy) to rigid (disease) motivates us to learn how their material properties arise from their constituent IDPs. (37) (38) (39) But although an enormous amount of experimental data is accessible in online databases, (40) (41) (42) via web-based interfaces (https://forti.shinyapps.io/mlos/), (43) there is no mechanistic understanding of how IDP molecular architecture controls the structure of their dense phase.

Computational modelling has been extensively used to connect molecular details such as multivalency and hydrophobicity to IDP phase behaviour in model systems. (44) (45) (46) (47) Exquisite detail on residue-residue interactions (48) and the conformational dynamics of IDPs within condensates is revealed by Atomistic Molecular dynamics (aaMD) simulations, (23, 49) but these are limited to nearmolecular system sizes and short times. Coarse-grained simulation techniques simplify the atomic details of proteins in order to simulate thousands of molecules. (50) (51) (52) (53) (54) (55) A minimal model of the phase separation of multivalent proteins is the patchy particle scheme in which proteins are treated as spheres with attractive sites on their surface. (56, 57) These models, however, ignore conformational fluctuations that may be important for flexible proteins. Other coarse-grained models represent an IDP as a linear polymer of monomers (or beads) held together by bonds (or springs). A bead represents one or more amino acids and the models differ chiefly in which atomic details are kept. (37, 44, 45, 50, 53, 58) The translational and conformational degrees of freedom of the IDPs are therefore retained whilst their enthalpic interactions are simplified to increase the accessible length and time scales.

Recent reviews describe how modeling is able to reveal physical principles underlying phase separation of IDPs. (51, 59) In this work, we go beyond establishing the phase boundaries of a model IDP in aqueous solution, and quantitatively study its dense phase. An IDP is represented as a semi-flexible polymer with multiple, punctate, attractive sites immersed in a good solvent. (4, 60) Our results are thus distinct from previous applications of similar stickers and spacers models, in which the polymers are in a poor solvent and possess a large fraction of hydrophobic monomers. (37, 38, 45, 61, 62) Although motivated by the study of IDPs, our model is also applicable to the phase separation of uncharged, associative polymers. (63) We use the coarse-grained simulation technique of dissipative particle dynamics, (64, 65) which is suitable for simulating (uncharged) IDPs because of their polymeric nature and transient, non-specific interactions. (66) We find that the dense phase has a low (~ mM) internal concentration and approximately 70% solvent by volume, both results in good agreement with experiments on the uncharged IDP FUS. (33, 67) The IDPs form a structured fluid network in which their binding sites transiently meet at junctions. The length scale between the junctions is much larger than the monomer size, and varies more strongly with the binding site separation than with their affinity. It is reminiscent of the diffraction peaks observed in SANS experiments on the dense phase of the multivalent nucleolar protein NPM1. (68) The spatial heterogeneity is revealed further using graph theoretic measures, (69, 70) which have been applied to metabolic networks,(71) allosteric pathways, (72) and the importance of residue mutations in proteins. (73) We map the junctions to the nodes of a graph, and place edges between nodes spanned by at least one polymer. The local clustering coefficient quantifies the crowding of polymers around the nodes. (74, 75) Thirdly, the dense phase evolves on multiple timescales, indicating a complex internal dynamics of the IDPs.(23) Finally, our model predicts that the location of attractive domains on IDPs is critical for their phase separation. When the binding sites at the polymer endcaps are disabled, the dense phase dissolves.

We use the Dissipative Particle Dynamics simulation technique (DPD) to study the phase behaviour of a series of model IDPs. The source code for the simulations is available on Github.(76) DPD is a coarse-grained, explicit-solvent simulation technique invented to study the hydrodynamic behaviour of complex fluids. (64, 65) It has since been applied to many soft matter systems including amphiphilic membranes, (77) (78) (79) vesicle fusion, (80, 81) and domain formation in vesicles (82), among many others.(83) It is highly suited to simulations of (uncharged) IDPs because the interactions are weak, and it has been shown that polymers in DPD exhibit self-avoiding walk scaling.(84) DPD is able to follow the evolution of fluid systems over large length and time scales by grouping atoms or atomic groups into beads, and replacing complex, interatomic potentials by effective forces that are softer and short-ranged. This reduces the number of degrees of freedom being integrated, and allows a larger integration time step in the equations of motion.

Beads in DPD have mass and interact via three non-bonded interactions that are soft, short-ranged (vanish beyond a fixed length-scale ! ), pairwise additive, and conserve linear momentum. A conservative force gives each bead an identity such as hydrophilic or hydrophobic:

for "# < ! . The maximum value of the force is "# ; = − is the relative position vector from bead j to bead i, "# is its magnitude, and . is the unit vector directed from bead to bead . The other two non-bonded forces constitute a thermostat that ensures the equilibrium states of the simulation are Boltzmann distributed.(65) The dissipative force is:

where "# is the strength of the dissipative force and is the relative velocity between beads and . This force destroys relative momentum between interacting particles.

The random force is:

where + is the system temperature and "# is a symmetric, uniform, unit random variable that is sampled for each pair of interacting beads and satisfies "# = #" ,

. This force creates relative momentum between pairs of interacting particles and . The factor 1 √ ⁄ is required in the random force so that the discretized form of the Langevin equation is well defined. (65) Molecules are constructed by tying beads together using Hookean springs with potential energy:

where the spring constant, ( , and unstretched length, ! may be different for each bead type pair but are here fixed at the values ( = 128 + ! ( ⁄ and ! = 0.5 ! . The semi-flexible nature of IDPs is represented by a chain bending potential applied to the angle θ defined by adjacent backbone bead triples (BBB):

with parameters 0 = 5 + and ! = 0. All bonded and non-bonded interaction parameters are given in Table 1 . For further details of the force field and DPD method applied to IDPs, the reader is referred to previous work. (47) An IDP is represented as a linear polymer of hydrophilic backbone beads (B)

interspersed by short segments of hydrophilic sticky beads (see Figure 1 ). Each segment contains 4 beads of type F that adopt approximately spherical binding sites. ⁄ and ! respectively). Note that E represents the end-cap beads and F the internal binding site beads for computational and visualisation purposes, but these bead types have identical interactions except when one or the other is turned off as described in the text. NA = not applicable. A three-body bending potential is applied to adjacent trips of backbone beads (BBB) with parameters 0 = 5 + and ! = 0.

The number, location and affinity of the binding sites are the main parameters of the model. We use the nomenclature nBm (or nIm) to identify the structure of a polymer in the text, where n is the total number of binding sites (internal plus endcaps), and m is the number of backbone beads between adjacent binding sites (referred to as the gap between binding sites.) The letter B indicates that endcaps are present in the polymer and the letter I indicates that only internal binding sites are present. Selected binding sites can be turned on or off thereby changing the effective separation of the remaining active sites in a manner analogous to PTMs on a protein revealing/occluding specific interaction motifs. A typical phase separated droplet is shown in Figure 1a together with its equivalent graph in Figure 1b , which is further discussed in Section E. We refer to the dense phase equivalently as a droplet, 

We first explore how the phase behaviour of the model IDPs depends on the number and strength (or affinity) of their binding sites. These sites represent residue motifs with weak, uncharged, attractive interactions. (4, 60) Because the model IDPs are hydrophilic, the only driving force for phase separation comes from the attraction between these sites. The conservative interaction between beads of type , is characterized in DPD by a parameter "# . We parametrize the binding site attraction by a dimensionless parameter that is defined in terms of the DPD conservative force parameters for the binding site/solvent beads ( 12 = 32 ) and binding site/binding site beads ( 11 = 33 ) via = ( 12 − 11 ) 12 ⁄ . The value = 0 corresponds to no attraction, and ~ 1 is a strong attraction between binding site beads.

Although representing an IDP as a semi-flexible polymer with discrete binding sites is a great simplification, the resulting model still possesses a large parameter space.

At a minimum, the polymer length, bending stiffness, concentration, and the number, location and affinity of the binding sites must be specified. After a preliminary exploration, we focused on B48 polymers with 6 binding sites separated by 6, 8, or 10 backbone beads (referred to hereafter as 6B6, 6B8, 6B10 respectively). The top row of Figure 2 shows the dense phase of 6B6 polymers for three values of the affinity decreasing from left to right. The droplet shows little visible change in cohesion over this range. By contrast, the bottom row shows that a droplet composed of 6B10 polymers begins to dissolve for the same change in affinity. The stability of the dense phase is clearly sensitive to the affinity and separation of the binding sites: more widely-separated binding sites require a higher affinity to drive their phase separation.

We establish a correspondence between the simulated droplets and experiments by calculating the concentration of the dense phase. Although it is difficult to measure the volume of arbitrarily-shaped droplets, their fluidity usually results in their being approximately spherical (cp. Fig 2) , which enables their volume to be estimated from their radius of gyration. This is obtained from the coordinates of the polymers' binding site beads under the assumption (verified by visual inspection of snapshots) that they are uniformly distributed throughout the droplet's volume. with the same affinities.

In equilibrium, the dense phase concentration C is: Tomasso et al., (87) and Marsh and Forman-Kay(88) have empirically fit the hydrodynamic radius E of a wide range of uncharged IDPs with the formula:

where N is the number of residues in the proteins, and the exponent is close to the Flory exponent of 1/2 for ideal chains. (89) In order to relate the hydrodynamic radius of an IDP to its radius of gyration one has to adopt a particular polymer model, e.g.,

treating it as an ideal chain or self-avoiding walk (SAW interesting to note that these relations imply that an IDP's hydrodynamic radius is smaller than its radius of gyration, which is the opposite of a uniformly-dense sphere, for which the relationship is @ ( = 0 G E ( . This implies that a fluctuating polymer in the dilute phase diffuses faster than a sphere with the same radius of gyration.

FUS-LC contains 163 residues and its hydrodynamic radius is predicted to be 2.9 nm from Eq. 2. Its radius of gyration is @ = 4.36 using the relation for ideal chains or 4.61 nm for a SAW. The value of ! is then fixed by:

Combining all the numerical factors together with 1000/0.6 to convert the concentration from polymers/nm 3 into mM in Eq. 1 allows the dense phase concentration to be determined from the formula 4.06

where we have adopted the SAW model for the polymer. farther apart on the polymers, which is intuitively expected. But it is largely independent of the affinity, apart from a small divergence (less than the size of a single bead) for the 6B10 polymers, which suggests that the network is loosened by decreasing affinity (cp. bottom row of Fig. 2 ). This result implies that the porosity of a biomolecular condensate is insensitive to small changes in the strength of the interactions between the constituent IDPs but increases with increasing separation of their binding domains (cp. the top row of Fig. 2 ). Panel 3c shows that the mean junction mass increases systematically (albeit slowly) with increasing binding site affinity and decreasing separation. Combined with Panel 3b, we find that increasing the affinity while keeping their separation unchanged does not change the structure of the dense phase, but only packs more polymers between existing junctions.

We have verified that the dense phase is in equilibrium with the surrounding dilute phase by two methods. It is clear from panels 3b and 3c that the junction separation and mass are independent of the total polymer concentration in the range studied, which supports the droplets being in thermodynamic equilibrium. We also simulated systems in a larger box (64 ! ) 0 to confirm that the dense phase structure is near the origin counts binding sites present at the same junction, and is uninteresting.

The first interesting peak in the RDF quantifies the separation of junctions that are spanned by polymers. Notably, it does not shift significantly when the binding site affinity is reduced, but increases as expected when their separation is increased. The second peak also moves to larger distances but the peaks are less pronounced for the weaker affinity than the stronger. The near coincidence of the peak heights for the solid and dashed curves of the same colour shows that the mean junction separation shown in Figure 3 is an average over junctions formed by endcaps and internal binding sites. The peak heights are proportional to the number of binding sites (4 internal and 2 endcaps), so dashed curves are always above the solid curves for the same affinity and spacing. to their interaction with the water beads, which sets their affinity = 0. Note that the binding site beads are still present in the polymer, so its length is unaltered. Figure 5 shows typical droplets for these cases, in which the dense phase concentration Although results are shown for a single affinity ( = 0.84), we expect from Figure 3 that disabling internal sites for other affinities would be similar unless the combination of binding site separation and affinity were such as to render the droplet already close to the phase boundary, e.g., 6B10 with = 0.76. Comparing Figures 3 and 6 shows that droplet's characteristic length scale is more sensitive to the spacing of the binding sites than their affinity, while the junction mass is sensitive to both.

Although the model BC is an equilibrium thermodynamic phase, there is considerable variability in the spatial distribution of the mass. Figure 4 shows that the dense phase has a structure that extends far beyond the monomer size and is not smeared out over time in spite of its fluidity. The regulation of biochemical reactions has been identified as a key function of BCs, and the spatial organisation of scaffold proteins within a BC should be expected to influence the diffusion and reactivity of client molecules. (3, 91) Previous simulations of telechelic polymers has shown that the distribution of the number of polymers that bind at junctions throughout the dense phase is broad. (47) In this section we quantify the spatial heterogeneity of the dense phase using measures from graph theory.

The dense phase is mapped to a graph by identifying a Node with each junction where polymer binding sites meet, and connecting two nodes by an Edge when the corresponding junctions are spanned by at least one polymer. For the purpose of constructing the graph, an edge is assigned to two nodes if two adjacent binding sites on a polymer connect two junctions. Only polymers with at least two binding sites connecting them to the dense phase are included in the graph to avoid skewing its properties with polymers that dangle off into the dilute phase. The graph is recalculated each time the simulation is sampled to generate a time series of graphs that represent the droplet's state from which equilibrium values of observables can be calculated.

All properties of the graph we use are calculated from its adjacency matrix "# . This is the square matrix whose dimension is equal to the number of vertices in the graph, and whose elements "# = 1, if the vertices , are connected by an edge, and 0 if not.

Diagonal elements are zero because we ignore self-loops, which represent polymers multiply bound to the same junction. The number of neighbours of a node , referred to as its degree, is the sum " = ∑ "# # of the elements in row . 

where " is its degree. The mean clustering coefficient (CC) is the average of " taken over all nodes in the graph. It measures the local connectivity of the nodes, taking its largest value of unity when all the neighbours of a node are also connected to each other (see Supplementary Material Section 1, Figures S1 -S6 , for the CC of some simple graphs). The CC just defined is the unweighted clustering coefficient. We also define the weighted clustering coefficient (wCC), (75, 93) which is calculated using Eq.

8 with a modified adjacency matrix whose elements are "# = if there are edges between nodes , . The weighted clustering coefficient contains information about the number of polymers that connect two junctions, and provides a better measure of the mass distribution in the droplet than the unweighted CC that reflects only its connectivity. S13 ). We find that that both the number, separation, and affinity of the binding sites influence the clustering coefficient. The wCC in Figure 7b shows that the average connectivity of junctions increases sharply with increasing binding sites per polymer and its variance increases particularly for polymers with 5 and 6 sites. Polymers with multiple binding sites form regions with a high local connection density as well as regions with a low density.

These results are averaged over samples taken from simulations of 600,000 timesteps (after discarding 10 6 steps) indicating that they persist in equilibrium. We point out here that the wCC is calculated for each node in the graph and normalized by dividing by the quantity " ( " − 1), which is the maximum number of triangles the node could make, and then averaged over all nodes. The number of triangles a node actually makes is usually much smaller than this maximum (cp. Fig. 7a) , which is why the ordinate of Fig. 7b is not large. But its vertical extent reflects the large variation in the number of polymers spanning the junctions. The wCC of the random graph is not shown because it has edges that span the whole graph, which renders it not comparable to locally-connected droplets. Although moving internal binding sites closer together along the IDPs systematically reduces the separation of the junctions in the BC (cp. Fig. 3b ) and increases the mass distribution (cp. Fig. 3c ), it remains highly heterogeneous.

We hypothesize that a cell may regulate the structure of a BC via post-translational modifications to scaffold proteins that occlude or expose interaction sites. To explore the prediction of our model in this case, we have compared the CC for the networks composed of 6B6 polymers with all six sites active and when two internal binding sites are disabled. These correspond to the networks shown in the middle column of Figure 5 . Figure 8 shows that the CC for the network is largely independent of concentration for the four points A -D, but decreases when two internal binding sites are disabled (cp. "A" and "A two off", etc.) 

where is the number of nodes in the graph, " ( ) is the number of polymer binding sites present on node at time t, and " ( , + ) is the number of descendent nodes at time + that contain at least one binding site that was bound to node at time .

Angle brackets indicate an average over all starting times t. The fluidity is normalized so that (0) = 0 because each node is then its own descendent, " ( , ) = 1. We note that this definition implies that if a polymer has more than one of its binding sites bound to a node and any of them move to a different node, they contribute to the fluidity measure. That is, the fluidity tracks the motion of the binding sites not the whole polymer. We also note that if a binding site leaves a junction and rebinds to the same one within the sampling time of the simulations (50,000 time steps) this is not reflected in the fluidity measure. It would appear if the sampling rate were increased.

The function ( ) measures the number of descendent nodes into which each node splits during a time interval , averaged over the whole graph and normalized so that if a node is its own descendent it contributes 0, and if it totally splits up it contributes 1. It is similar to the auto-correlation function of the number of binding sites present on a node with the important difference that it tracks not only the number of binding sites per node but also their identity: if a binding site leaves a node and a different one joins, the fluidity measure reflects this even though the number of binding sites on the node has not changed. We introduce this definition because binding sites fluctuating between nodes represents a motion that will likely impact the behaviour of other molecules within it, e.g., by permitting the diffusion of objects larger than the mean junction separation because the network can rearrange around them.

The fluidity of the dense phase of telechelic 2B16 polymers is shown in Figure 10a for a range of binding site affinities. These polymers form box-spanning networks for the lowest two affinities = 0.76, 0.74, and phase separated droplets for higher values. (47) The fluidity function for weak affinity increases steeply over short times revealing that the polymers easily rearrange their binding configuration as they diffuse within the dense phase. As the affinity increases, the droplet dynamics slows down.

The low-affinity networks rearrange their structure after a few hundred thousand timesteps, whereas the high affinity polymers require millions of steps, which means these droplets evolve slowly on the time scale of the simulations. Table 2 shows the timescale for the dynamic rearrangement of the networks shown in Figure 10 .

The curves for affinities above = 0.8 are well fitted by a single exponential in Figure   10b , indicating that their dynamics is dominated by a characteristic timescale. The fluidity of droplets composed of polymers with weaker affinity cannot be described Adding more binding sites might be expected to reduce the dense phase fluidity, but this is not observed. Figure 11 shows that droplets with multiple binding sites are more fluid than telechelic polymers ( Figure 10 ) with the same affinity = 0.84, and exhibit fluctuations on shorter timescales. Figure 11a shows that the dynamics, as measured by the fluidity ( ), slows down with increasing affinity as intuitively expected, but that the fast motion of the polymers is reduced with increasing binding site separation for stronger affinities (dashed 6B6 curves are above solid 6B10 curves). We attribute this effect to the increasing difficulty of the binding sites on a fluctuating polymer to move to a nearby junction as their separation along the polymer backbone increases (cp. junction separation in Figure 3 ). Figure 11b shows that the fluidity of droplets containing multi-binding site polymers cannot be described by single exponential decay for any of the affinities studied. Their internal dynamics therefore possesses at least two distinct time-scales. The departure of the curves for 6B10 polymers with the lowest affinity and concentration reflects the small size and instability of the dense phase. 

The Flory-Huggins (FH) theory of polymer phase separation (89) has been widely used to construct and interpret simulation studies of IDP phase behaviour. (37, 38, 45, 61, 96) This theory assigns all monomers in a polymer an energetic interaction with the solvent that is quantified by a parameter χ (for heteropolymers this is an effective parameter). When χ is negative, the entropy of mixing combines with the favourable solvation energy to keep the single-phase mixture stable. This corresponds to the Figure 7) . We have constructed a novel measure (Eq. 9) to quantify the dynamics of polymer fluctuations within the dense phase. It reflects the reversible binding/unbinding of the IDPs, and shows that they exhibit multi-exponential decay in which both fast and slow relaxation processes are important ( Figure 11 ). The relaxation dynamics we find is almost independent of IDP concentration and is not accompanied by stiffening of the network structure in the presence of more (weak) internal binding sites.

Fluid FUS-LC droplets have been observed to undergo an irreversible transition to a rigid fibrous state over several hours in vitro. (32, 109) , and passive rheology experiments show that BCs undergo ageing and exhibit glassy behaviour. (110) However, condensates with material properties that range from liquid to solid also 

The 

The Supporting Information is available free of charge via the internet, and includes the following: 

121) as well as transport and mobility networks, which typically involve shortest-path and trajectory prediction on time-varying graphs.(122, 123) In conclusion, the predicted relations between the model IDPs and the structure of their condensed phase will, if confirmed in experiments, help the design of synthetic BCs with tunable properties.(124-126) Combined with the ability to modify the residue sequence of IDPs,(60) and online tools to predict the effects of point mutations on local disorder and hydropathy of IDPs

Conceptualization, Methodology, Software, Validation, Formal analysis, Data curation, Visualization, Writing -original draft, Writing -review and editing

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Software, Validation, Formal analysis, Resources, Writing -review and editing, Funding acquisition

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Solutions of Associative Polymers

Entropic networks in colloidal, polymeric and amphiphilic systems

Lattice cluster theory of associating polymers. I. Solutions of linear telechelic polymer chains

Collapse Transitions of Proteins and the Interplay Among Backbone, Sidechain, and Solvent Interactions

Phase Diagram of Solutions of Associative Polymers

A Hybrid Monte Carlo Self-Consistent Field Model of Physical Gels of Telechelic Polymers

Structural Analysis of Telechelic Polymer Solution Using Dissipative Particle Dynamics Simulations

Structure of Model Telechelic Polymer Melts by Computer Simulation

Structured Fluids

Solutions of Associative Polymers

A guide to regulation of the formation of biomolecular condensates

The prion-like domain of Fused in Sarcoma is phosphorylated by multiple kinases affecting liquid-and solid-liquid phase transitions

ALS/FTD Mutation-Induced Phase Transition of FUS Liquid Droplets and Reversible Hydrogels into Irreversible Hydrogels Impairs RNP Granule Function

Protein condensates as aging Maxwell fluids

Organization and Function of Non-dynamic Biomolecular Condensates

α-Synuclein aggregation nucleates through liquid-liquid phase separation

A condensate-hardening drug blocks RSV replication in vivo

Friend or foe-Post-translational modifications as regulators of phase separation and RNP granule dynamics

A predictive coarse-grained model for position-specific effects of post-translational modifications

Quantifying Dynamics in Phase-Separated Condensates Using Fluorescence Recovery after Photobleaching

A time-varying group sparse additive model for genome-wide association studies of dynamic complex traits

Stream graphs and link streams for the modeling of interactions over time

Predicting interactions between Individuals with structural and dynamical information

Effects of time-dependent diffusion behaviors on the rumor spreading in social networks

A new time-dependent shortest path algorithm for multimodal transportation network

LOKI: Long Term and Key Intentions for Trajectory Prediction

Probing and engineering liquid-phase organelles

Phase Separation in Biology & Disease: The next chapter

Designer Condensates: A Toolkit for the Biomolecular Architect

Contiguously hydrophobic sequences are functionally significant throughout the human exome

Designer membraneless organelles sequester native factors for control of cell behavior

Therapeutics -how to treat phase separation-associated diseases

VMD -Visual Molecular Dynamics