Liquid-liquid microphase separation leads to formation of membraneless organelles
LLiquid-liquid microphase separation leads toformation of membraneless organelles
Srivastav Ranganathan and Eugene Shakhnovich ∗ Department of Chemistry and Chemical Biology
E-mail: [email protected]
Abstract
Proteins and nucleic acids can spontaneously self-assemble into membraneless droplet-like compartments, both in vitro and in vivo . A key component of these dropletsare multi-valent proteins that possess several adhesive domains with specific interac-tion partners (whose number determines total valency of the protein) separated bydisordered regions. Here, using multi-scale simulations we show that such proteinsself-organize into micro-phase separated droplets of various sizes as opposed to theFlory-like macro-phase separated equilibrium state of homopolymers or equilibriumphysical gels. We show that the micro-phase separated state is a dynamic outcome ofthe interplay between two competing processes: a diffusion-limited encounter betweenproteins, and the dynamics within small clusters that results in exhaustion of availablevalencies whereby all specifically interacting domains find their interacting partnerswithin smaller clusters, leading to arrested phase separation. We first model thesemulti-valent chains as bead-spring polymers with multiple adhesive domains separatedby semi-flexible linkers and use Langevin Dynamics (LD) to assess how key timescalesdepend on the molecular properties of associating polymers. Using the time-scales fromLD simulations, we develop a coarse-grained kinetic model to study this phenomenonat longer times. Consistent with LD simulations, the macro-phase separated state a r X i v : . [ q - b i o . B M ] J a n as only observed at high concentrations and large interaction valencies. Further, inthe regime where cluster sizes approach macro-phase separation, the condensed phasebecomes dynamically solid-like, suggesting that it might no longer be biologically func-tional. Therefore, the micro-phase separated state could be a hallmark of functionaldroplets formed by proteins with the sticker-spacer architecture. Significance Statement
Membraneless organells (MO) are ubiquitous in healthy living cells, with an altered statein disease. Their formation is likened to liquid-liquid phase separation (LLPS) betweenMO-forming proteins. However most models of LLPS predict complete macrophase separa-tion while in reality MOs are small droplets of various sizes, which are malleable to rapidmorphological changes. Here we present a microscopic multiscale theoretical study of ther-modynamics and kinetics of formation of MO. We show that MOs are long-living dynamicstructures formed as a result of arrested macrophase separation. Our study provides a di-rect link beween the molecular properies of MO-forming proteins and the morphology anddynamics of MO paving a path to rational design and control of MO.
Introduction
Biomolecular phase transitions are widespread in living systems. Transitions that result inirreversible, solid-like assemblies such as amyloid fibrils are a hallmark of disease while thoselike cytoskeletal filaments play a functional role. The third self-assembled state, in additionto the soluble and the solid-like state, is the loosely associated droplet phase held togetherby several weak, transient interactions. The transient nature of these interactions makesthese self-assemblies reversible and thereby a potential strategy for temporally regulated sub-cellular organisation. Several examples of spatiotemporally regulated, droplet-like objectswithin the cell have been discovered in the past few decades, composed of different types2f proteins often co-localised with nucleic acids.
The importance of these membrane-lesscompartments is potentially two-fold: (i) localising biochemical reactions within the cell,and (ii) sequestering biomolecules to regulate their activity.
Examples of cytoplasmicmembrane-less organelles include P bodies, germ granules and stress granules (SGs). How-ever, aberrant granule dynamics and a transition from a liquid-like to a more solid-like stateare often hallmarks of degenerative diseases.
Solid-like RNP aggregates have been reportedas cytoplasmic inclusions and as nuclear RNP aggregates in degenerative diseases. Inves-tigating the physical principles governing the formation of these high-density phases is vitalto understand the subcellular organisation and the conditions leading to disease.Several experimental studies have highlighted the ’multi-valent’ nature of the constituentproteins in membrane-less organelles . In other words, these proteins carry multipleassociative or ’adhesive’ domains.
Multivalency could be achieved by several different ar-chitectures, the simplest being a linear sequence of folded domains that are connected bylinker regions that are flexible and unstructured. Li et al., employed such a linear multivalent2-component model system (SH3 and PRM domains threaded together by flexible regions)to show that liquid-like droplet formation can result from just two multivalent interactingcomponents (repeats of the same domain). Another intriguing feature of condensate pro-teins is the presence of intrinsically disordered regions or low complexity sequences that linkfolded domains together. Therefore, a combination of these motifs and different architec-tures could form a basis for different types of phase-separated structures within the cell.Due to the heterogeneous composition of droplets and complex set of factors dictating theirgrowth, theoretical and computational models of varying resolutions have previously beenemployed to study this problem.
However, much of the focus has been on identifying theself-assembled state at equilibrium. In particular, Flory theory of phase separation in poly-mer solution has been employed to describe self-assembly of membrane-less organelles. TheFlory theory which is applicable to solutions of homopolymers predicts two phases mixedand macrophase separated. However, in reality membrane-less organelles are droplets of fi-3ite size akin to microphase separated entities in variance with simple Flory-type theoreticalpredictions. Despite significant experimental and computational efforts, a key question re-mains unanswered – what are the physical mechanisms that govern the formation of tunablemicrophase separated droplets rather than complete macrophase separation? The multi-component nature of membrane-less organelles is an important feature that distinguishesthem from simple homopolymers that undergo phase separation. How do droplet size distri-butions depend on the intrinsic characteristics of the biopolymers? What are the key featuresinherent to the architecture of these self-assembling molecules that make their assemblies tun-able? Since the equilibrium states in such systems are either mixed or macrophase separated,the kinetic factors must play a crucial role in the observed microphase separation outcome. However, a detailed mechanistic understanding of the process of formation and growth ofthese droplets that leads to the microphase separated state has been lacking.In this work, we address these questions using multi-scale coarse-grained models. Inthe first part of the paper, we probe the physical determinants of intracellular microphaseseparation using microscopic model of multivalent polymers composed of a linear chain ofadhesive domains separated by semi-flexible linkers. The results of our Langevin dynamics(LD) simulations for this model shed light on the early stages of droplet growth and themechanism of arrested phase-separation. Next, using the LD simulations as the basis toidentify vital timescales for condensate growth, we explore the phenomenon at biologicallyrelevant timescales using a phenomenological kinetic model. Broadly, we explore liquid-liquidmicrophase separation (LLmPS) on multiple scales from mesoscale to macroscopic. To thatend, detailed LD simulations explore microscopic mechanisms that determine key events dic-tating the early stages of growth which serve as the basis for developing the phenomenologicalkinetic model. Overall, using these two approaches, we provide a mechanistic explanationfor the predominance of the microphase separated state for multi-valent heteropolymers.4igure 1: Model. Schematic of the polymer model for studying phase-separation by multi-valent biopolymers 5 odel
Despite the complexity of the intracellular space, experiments suggest that in vitro
LLmPScan be achieved even using simple two-component systems. The tractability of simplermodels makes them powerful tools to investigate the role of individual factors in modulatingdroplet formation and growth. Here, we perform LD simulations (see Methods for detail)to understand the process of self-association between two types of polymer chains composedof specific interaction sites (red and yellow beads in Fig.1). These adhesive sites are linkedtogether by non-specifically interacting linkers (blue beads in Fig.1). The red and yellowbeads on these chains mimic complementary domains on different chains that can participatein a maximum of one specific interaction (between yellow and red beads). For our simulations,we consider 400 such semi-flexible polymer chains (200 of each type) in a cubic box withperiodic boundary conditions). Each chain in the simulation box is composed of 5 specificinteraction sites that are linked together by non-specific linker regions that are 35 beadslong. (blue beads in Fig.1). This linker length was based on previous theoretical studiesof phase-separating proteins. The specificity of the functional interactions is modelledby imposing a valency of 1 between different complementary specific bead types (red andyellow beads) via a bonding vector attached to each bead. Valency 1 means that eachspecific interaction site can only be involved in one such interaction. The total valency ofan individual polymer chain ( λ ) is the number of adhesive sites that are part of a singlechain. Bond formation (modelled using stochastically forming harmonic springs) occurswith a probability ( P form ) if two complementary beads are within a defined interactioncutoff distance ( r c ). We employ conventional Langevin dynamics simulations to study theself-assembly of the model biopolymers, wherein the size of the specific interaction sites(diameter of 20 ˚ A ) is roughly four times that of the linker beads which represent individualamino acid residues (diameter of 4.2 ˚ A ). This difference in sizes is to mimic a folded adhesivedomain – SH3 domain (diameter of 20 ˚ A , PDB ID:1SHG,) that is often involved in liquid-liquid phase-separation. The folded adhesive domains are modeled at a lower resolution6one bead per domain of 60 amino acids) than linkers which are more dynamic and hencemodeled with a 1 bead per amino acid resolution.
Results
Self-assembly of multi-valent polymers with inter-chain interactions between complementarydomains is a simple model system for studying phase-separation by intracellular polymers.In the current study, we employ this model to discover microscopic factors driving phaseseparation. In the first part of this study, we present results of self-assembly driven byirreversible (highly stable) functional interactions and identify two key timescales that de-fine cluster growth and their size distributions. In the latter part of this paper, we builda coarse-grained kinetic model demonstrating the tunability of this phenomenon. In thefollowing subsections, we use the term specific interactions for the finite valency interactionsbetween functional domains and non-specific interactions to refer to the inter-linker isotropicinteractions.
Micro-phase separation – the most likely outcome at low concen-trations and irreversible functional interactions
Multivalent polymers with adhesive domains separated by flexible linkers can potentially self-assemble through two types of interactions, a) the finite number of adhesive contacts betweenthe functional domains (yellow and red beads in Fig.1), and b) the non-specific, isotropicinteractions between the linker regions (blue beads in Fig 1). We first performed controlsimulations with specific interactions turned off, where we varied free monomer concentration C mono , from 10 to 200 M for a weak non-specific interaction strength of (cid:15) ns = 0.1 kcal/mol.In these control simulations, we observe no phase-separation in the whole range of C mono (Fig.2A, purple curve). Therefore, in this regime of (cid:15) ns and C mono , at the simulation timescaleof 16 s, the polymer assemblies do not reach large sizes. Not surprisingly, this result suggests7igure 2: Cluster Sizes in Langevin Dynamics simulations. A) The single largest clusteras a function of free monomer concentration (in µ M). The largest cluster size is shown asa fraction of the total number of monomers in the simulation box. The smooth curves areplotted as a guide to the eye, using the cspline curve fitting. The vertical bars representthe standard error. B) Cluster size distributions for increasing free monomer concentrations.The linker stiffness for the self-assembling polymer chains in this plot is 2 kcal/mol whilethe strength of inter-linker interaction is 0.1 kcal/mol (per pair of interacting beads). Themean and distributions of the single largest cluster sizes were computed using 500 differentconfigurations from 5 independent simulation runts of 16 µ s. The simulations were performedfor an instantaneous bond formation assumption ( P form = 1).8hat the assembly driven by non-specific interactions (linker-driven) alone is achieved onlyat stronger non-specific interactions and/or high free monomer concentrations. However,intracellular microphase separation often results in the enrichment of biomolecules withinthe self-assembled phase, at relatively low bulk concentrations. Under such conditions,specific interactions which are fewer in number become critical determinants of microphaseseparation. In our LD simulations, we employ polymer chains comprised of 5 univalentadhesive domains and 4 linker regions totalling 145 beads bringing total valency of 5 foreach polymer. For the sake of simplicity, we begin with a situation where these bonds, onceformed, do not break for the rest of the simulation. Although an idealized construct withrespect to biological polymers, these simulations can aid in our understanding of cluster sizedistributions and arrest of cluster growth when the spontaneously assembled clusters areunable to undergo further reorganization.In Fig.2 we show the single largest cluster size, for varying free monomer concentrations.The polymer-chains, for the data plotted in Fig.2, have flexible linkers ( κ =2 kcal/mol, seeModel and Methods for defintion of bending rigidity) with weak inter-linker interactions( (cid:15) ns = 0.1 kcal/mol). As evident from Fig.2, the polymer chains can assemble into largerclusters in the presence of functional interactions (black curve, Fig.2A), as opposed to thecontrol simulations where inter-linker interactions alone drive self-assembly (purple curve,Fig.2A). Simulations with highly stable functional interactions suggest that, for low andintermediate concentrations ( C mono <
100 uM) these polymer chains would never achieveaggregate sizes that approach the number of chains in the system (fraction of monomersin largest cluster → µ M to 100 µ M results in macrophase separation - a system-spanning aggregate with cluster sizes approaching the total number of monomers. This isfurther evident from the size-distribution of the single-largest cluster computed using 500different configurations from 5 independent simulation runs of 16 µ s (Fig.2B). The largestcluster approaches the total number of monomers, only at large C mono . However, the cellular9oncentration of phase-separating proteins is often in the nanomolar to the low micromolarrange . Our LD simulations suggest that for irreversible specific interactions macro-phaseseparation can only be observed at large non-physiological concentrations.Figure 3: Tracking cluster formation at early timescales. A) and B) show the temporalevolution of specific contacts for a free monomer concentration of 10, and 50 µ M, respectively.For a low concentration of 10 µ M, there is an initial decrease in the monomer population(purple curve) which is concomitant with an increase in the dimer population. A negligiblefraction of the clusters is in the form of large-mers (size > = 10, orange curve) at these lowconcentration since the available valencies for growth are consumed by the smaller aggregatedspecies. An increase in concentration from 10 µ M to 50 µ M results in an increase in thelarge-mer (orange curve, 50 µ M) population as the monomer fraction decreases during thesimulation. A higher free monomer concentration allows the larger clusters to grow due toconsumption of free monomers (with unsatisfied valencies) before they get converted intosmaller clusters (dimers, trimers) with satisfied valencies. C) Time evolution of availablevalencies within the single largest cluster and outside the single largest cluster, for a C mono of 50 µ M, (cid:15) ns of 0.1 kcal/mol and linker bending rigidity of 2 kcal/mol. D) A schematic figureshowing the possible mechanisms of cluster growth and arrest and the competing timescalesthat could punctuate the process. 10 xhaustion of free valencies results in kinetically arrested droplets. The size of the largest cluster shows a dependence on concentration (Fig.2), for weak inter-linker interactions ( (cid:15) ns = 0.1 kcal/mol). While the macro-phase separated state could be apotential equilibrium outcome, for stable functional interactions, such a state is not observedin our simulations except for very high concentrations ( > = 100 µ M in Fig.2). Therefore, iden-tifying the timescales which are vital for cluster growth could reveal the cause of arrestedmacro-phase separation. In Fig.3A and B, we show the time evolution of the individualspecies at different monomer concentrations (10 µ M and 50 µ M). As seen from Fig.3A andB, for irreversible functional interactions, the monomer fraction continues to monotonicallydecrease during the simulations with the fraction of other competing species increasing con-comitantly. However, at low concentrations (Fig.3A, 10 µ M), the monomer fraction curveshows a cross-over with the dimer curve (Fig.3A, green curve) while higher order clusters donot appear at simulation tmescales.. This result suggests that the spontaneous formation oflarge assemblies held together by functional interactions is contingent upon two timescales.The first is the diffusion-limited timescale that governs initial dimerisation and the subse-quent growth of these smaller clusters. The second, competing timescale is the one whereall functional valencies get exhausted within the smaller initial clusters. At an intermediateconcentration of 50 µ M, (Fig.3B) we observe that the fraction of large aggregates ( > P form ,results in larger cluster sizes for smaller free monomer concentrations, suggesting that thedynamicity of bond formation can alter the cluster size distributions significantly (Supple-mentary Fig. S1). These results suggest that the ability to form a large, macro-cluster islimited by the exhaustion of free valencies within smaller sized clusters, thereby arrestingtheir growth (Fig.3D).Figure 4: The characterisation of time-scales associated with the growth of a cluster using adimerisation model. A) The mean first passage time for the first specific interaction betweena pair of polymer chains as a function of the polymer concentration (represented as thepolymer number density, N mono / L , where N mono and L are, the number of chains the lengthof the box, respectively). B) The mean first passage time for a pair of polymer chains toexhaust all the available valencies within the dimer. The shaded region in B) shows that forflexible linkers, this timescale for the exhaustion of valencies is of the order of the reptationtime for an equivalent polymer, given by T rep . T rep = L ∗ π ∗ D bead , where L and D bread refer tothe contour length and the diffusion coefficient of the individual monomer beads within thepolymer chain. Identifying the vital timescales determining cluster growth
In the limit of irreversible functional interactions, the process of phase separation gets ar-rested due to kinetically trapped clusters which do not participate in further cluster growthdue to lack of available valencies (Fig.3D). As discussed above, two critical timescales wouldthen dictate the growth of clusters: i) the timescale for two chains to meet and form the firstfunctional interaction, and ii) the time it takes for the polymer chains within an assembly12o exhaust all valencies within the cluster before new chains join in. We explore the scal-ing behaviour of these two timescales using the primary unit of any self-assembly process,the dimer. Using LD simulations, we first compute the mean first passage times for twopolymer chains to form their first functional interaction. Given the diffusion-limited nature,this timescale scales with the concentration of free monomers in the system (Fig.4A). Thisdiffusion limited timescale dictates the encounter probability of the two chains. Once thefirst bond is formed, resulting in an ’active dimer’ (one that has unsatisfied valencies), thecluster can only grow to larger sizes as long as the dimer remains active. Therefore, thetime taken by the dimer to exhaust all its valencies becomes a vital second timescale. InFig.4B, we plot the mean first passage times for a dimer to exhaust all its valencies once thefirst bond is formed. For flexible linkers ( κ < T exhaust − valency ), is roughly of the order of the reptation time of flexiblepolymer chains ( T rep In Fig.4B) that are 145 beads long. The reptation time, i.e. the timetaken for the polymer chain to ’scan’ its entire contour length, is given by, T rep = L ∗ π ∗ D bead , (1)where L and D bead refer to the contour length and the diffusion coefficient of the individualmonomer beads within the polymer chain. Also, D bead = k B T πηa . For an amino acid, D bead is ofthe order of 9 ∗ − m /s . For a polymer of contour length L = 145 beads in length (eachbead representing an amino acid), this timescale is of the order of 0.5 µ s. As seen from theright panel in Fig.4, the flexible polymer chains indeed exhaust their valencies at timescalescomparable to the reptation time for an equivalent flexible polymer chain. As the linkerregion becomes more rigid, this timescale becomes slower, resulting in the dimer remaining‘active’ for much longer. A slower reorganisation time and a faster diffusion encounter timefavours the cluster growing into larger sizes. Conversely, a faster reorganisation time that isof the order of the meeting timescales results in a higher likelihood of the clusters getting13ocked at smaller sizes. For stable functional interactions, the interplay between these twotimescales would limit the size of the largest cluster.Figure 5: Linkers as modulators of self-assembly propensity. A) The size of the largest clusterfor flexible linker regions( κ = 2 kcal/mol) with varying inter-linker interaction strength(black curve, (cid:15) ns = 0.1 kcal/mol and purple curve, (cid:15) ns = 0.5 kcal/mol). Sticky inter-linkerinteractions result in smaller cluster sizes, also corroborated by the collapsed nature of theseaggregates in B). Mean Radius of gyration for the individual-polymer chains ( Rg Mono ) withina self-assembled cluster as a function of increased inter-linker interactions. C) The meanreorganisation times ( T exhaust − valency , as a function of linker-stiffness, for different values ofinter-linker interaction strengths. Linker regions as modulators of self-assembly propensity
The spatial separation of specifically interacting domains (with finite valencies) and the non-specific linker regions is an architecture that could be highly amenable to being tuned forphase-separation propensities. In this context, we further extend the findings from our LDsimulations to probe how the microscopic properties of the linker region could modulate theextent of phase-separation. In Fig.5, we demonstrate how the linker properties could be usefulmodulators of cluster sizes, without any alteration of the nature of the specific functionalinteractions. Further, to model an unstructured linker, we consider a scenario where thelinkers participate in inter-linker interactions alone. Strong inter-linker interactions increasedoccupied valencies (Supplementary Fig.S2), for all the concentrations under study, whileresulting in much smaller equilibrium cluster sizes (Fig.5A) during the timescales accessedby our simulations. Further, the mean radius of gyration ( Rg Mono ) for the monomers within14hese clusters shows a transition with an increase in inter-linker interaction strength ( (cid:15) ns ),with a sharp decrease in Rg Mono for an increase in (cid:15) ns from 0.3 kcal/mol to 0.5 kcal/molindicating an onset of linker-driven coil-globule transitions for the polymers. The assemblies that ensue at these strong values of (cid:15) ns are more compact and condensedakin to homopolymer globules. Intramolecular compaction of polymers due to nonspecificinter-linker interactions brings specific domains closer in space leading to a higher likelihoodof the exhaustion of specific interaction valencies within small assemblies. In Fig.5C, weshow how the inter-linker interaction strength can influence the time it takes to exhaust spe-cific interaction valencies within a dimeric cluster ( T exhaust − valency ). For weaker inter-linkerattraction ( (cid:15) ns < T exhaust − valency fordimers with (cid:15) ns < (cid:15) ns = 0.5kcal/mol exhaust their valencies almost an order of magnitude faster than their 0.1 kcal/molcounterparts (Fig.5C). Upon exhaustion of these specific interaction valencies, these clusterscan only grow via inter-linker interactions, a phenomenon that could be less dynamic (Supple-mentary Fig S3) and tunable than the functional interaction driven cluster growth. It must,however, be noted that the observation of further coalescence of clusters formed by stickyinter-linker interactions was limited by the timescales accessible to the LD simulations. Anyalteration to the stickiness of the linker can shift the mechanism of assembly, and thereby re-sult in altered kinetics of cluster growth by modulating the T exhaust − valency . Our dimerisationsimulations show that a second mechanism of slowing down the T exhaust − valency timescale is byaltering the flexibility of the linker region (increasing linker stiffness in Fig.5C). Primarily thestiffer linkers model spacer regions that prefer more ‘open’ configurations (because of say, ahigh local density charges). This is corroborated by a shift in the cluster size distribution to-wards larger sizes, for the polymer chains with less flexible linkers (Supplementary Fig.S4).The linker region can thereby serve as a modulator of microphase-separation propensity.15urther, we probed how the linker region influences the density profiles of these clusters. InSupplementary Fig.S4, we show the probability distributions for cluster densities normalizedby the bulk densities, as a measure of the degree of enrichment within the self-assembledstate. For weak inter-linker interactions, we find a 10-fold enrichment of monomers withinthe assemblies (Supplementary Fig.S4, purple curve). On the other hand, a sticky linkerresults in an almost 100-fold enrichment within the collapsed globule-like clusters (Fig.5Band Supplementary Fig.S5). These values are consistent with experimental findings of a ≈ Therefore, a variation inthe intrinsic properties of the linker, with no modifications to the functional region can beused as a handle to tune the degree of enrichment within the condensed phase. This lendsmodularity and functionality to these condensates, with the linker regions controlling thedegree of enrichment within a condensate.Figure 6: A schematic figure detailing the different rates in our phenomenological kineticmodel simulated using the Gillespie algorithm. The particles on the lattice can diffusefreely (when there are no neighbouring particles) with a rate k diff . In the presence of aneighbouring particle, a non-specifically interacting monomer can diffuse away with a rate k diff (cid:15) ns . Neighbouring particles can also form specific interactions (with fixed valency λ )at a rate k bond or break an existing interaction with a rate k break . Clusters could diffuse at arate that is scaled by their sizes. (cid:15) ns and (cid:15) sp refer to the strength of non-specific and specificinteractions, respectively. 16 henomenological kinetic simulations predict microphase separa-tion at biologically relevant timescales The LD simulations help us identify the initial events that mark phase-separation by multi-valent polymer chains assembling via finite-valency, specific interactions. However, the modelis limited by its ability to access longer, biologically relevant timescales at which dropletstypically form and grow in living cells. Also, while the assumption of non-transient interac-tions is a useful simplification to identify conditions arresting cluster growth, bio-molecularinteractions are often transient in nature. Hence, to probe whether micro or macrophaseseparation becomes kinetically favored for reversible functional interactions, we employ acoarse-grained approach wherein the whole polymer chain from the bead-spring model (witha fixed valency) is represented as an individual particle on a 2D-lattice. In our simulations,the lattice is populated by N such multi-valent particles (at varying densities) that diffusefreely, at a rate k diff , and particle collisions per unit time is proportional to k diff * φ . Here, φ is the bulk number density of particles on the lattice. In our MC study, we vary the bulkdensity of the lattice in a range of 0.01 to 0.1 (see Methods section for rationale). When twosuch particles occupy neighbouring sites on the lattice, they interact non-specifically withan interaction strength of (cid:15) ns , a parameter that is analogous to the inter-linker interactionsin our LD simulations. Additionally, two neighbouring particles with unfulfilled valenciescan form a specific bond (with finite valencies per lattice particle) with a rate of k bond .The valency per particle ( λ ) here can be utilized to form bonds with one to four potentialneighbors, with each pair being part of one, or more than one specific interactions betweenthemselves. However, unlike the irreversible specific interactions in our LD simulations, thespecific bonds in the lattice model can break with a rate k break = k bond *exp(- (cid:15) sp ), where (cid:15) sp isthe strength of each specific bond. Additionally, clusters can diffuse with a scaled diffusionrate that is inversely proportional to the cluster size. It must be noted that the timescalefor the first bond formation in the LD simulations is an outcome of two phenomenologicalrates in this model, k diff and k bond . The second timescale, T exhaust − valency , is a timescale17hat depends on the k bond and k break parameters in this model. The details of the simulationtechnique and the various rate processes can be found in the Methods section and describedschematically in Fig.6.Using kinetic Monte Carlo simulations, we explore the cluster formation (at times reach-ing a physiologically relevant scale of hours) by varying parameters such as a) bulk density ofparticles on the lattice ( φ ), (2) rate of bond formation ( k bond ), (c) valency per interacting par-ticle ( λ ), and (d) the strength of specific interactions ( (cid:15) sp ). With the assumption that, for lowconcentrations, the rate of free diffusion k diff is the fundamental timescale limiting clustergrowth, we first explore the relationship between the rate of specific bond formation ( k bond ),and k diff (Fig.7A and Supplementary Fig.S6). It must be noted that the LD simulationsemployed the assumption that bond formation, upon the two functional domains coming incontact, is an instantaneous event. Here, we show that for values of k bond / k diff →
0, thereis no phase-separation. As the bond formation rate approaches that of free diffusion cor-reponding to instant bond formation in LD simulations –, we encountered phase-separatedstates in our simulations. However, the system largely favors the micro-phase separatedstate (bluish-red regions in the phase diagram) for low and intermediate densities. Themacro-phase separated state is only observed for higher densities. The macro-phase sepa-rated regime in this phase diagram is, however, absent at low valency λ of 3 (SupplementaryFig.S6). Further, this phase diagram (Fig.7A) also establishes that for the value of non-specific interaction strength ( (cid:15) ns = 0.35 kT, see methods section for rationale) used here thecluster formation is driven by the finite-valency specific interactions (absence of phase sep-aration for k bond / k diff → λ =0).The (cid:15) ns - φ phase diagram shows that non-specific interaction-driven cluster formation occursat only high values of (cid:15) ns (Supplementary Fig.S7B). Therefore, microphase separation is con-tingent on the bonding rate being of the same order as the free-diffusion rate ( k bond → k diff )establishing the validity of the instantaneous bonding assumption in the LD simulations.18igure 7: Kinetic Monte Carlo Simulations. A.) Phase diagram highlighting the differentphases (microphase (Micro-PS) or macrophase separated (Macro-PS), and the non-phaseseparated state (No PS) ) encountered upon varying the ratio of k bond / k diff , and the bulkdensity of monomers ( φ ) within the box. The assembling particles have a valency ( λ ) of 5in these simulations. B.) Phase diagram highlighting the different phases encountered uponvarying valency and bulk density as the phase parameters. The macro-phase separated stateis only encountered for larger valency particles at higher densities. This phase diagram wascomputed for k bond / k diff = 1 and (cid:15) ns of 0.35 kT. C.) The difference in the densities of thelargest cluster, for clusters that are driven by specific (solid purple curve) and non-specific(solid black curve) interactions. The non-specific interaction driven clusters are denser thantheir specific-interaction driven counterparts. The cluster sizes, however, for both these casesare comparable (dashed curves). k bond / k diff = 1, and φ =0.04, for this plot. (cid:15) ns was set to0.35 kT. D) The fraction of monomers in the largest cluster as a function of epsilon, for k diff = k bond and φ =0.04. The single largest cluster sizes in all sub-panels of this Figure wascomputed for a simulation timescale of 2 hrs (with the fundamental timsecale of diffusionbeing set to k diff = 1 s-1). 19t is the ratio of k bond / k diff , and not the absolute magnitudes that is a vital parameter forthese simulations. Hence, in all the kinetic Monte carlo (KMC) simulations we set the valueof k diff to 1 s − and vary the ratio k bond / k diff to tune phase separation. It must be notedthat, unless mentioned otherwise, the results from the kMC simulations presented here arefor a weak non-specific interaction strength of 0.35kT. All simulations were performed for atimescale of 2 hours (actual time). As proof of convergence of these simulations, we com-pare results at the end of 2 hrs to those at longer simulation timescale and show that thereis negligible difference in cluster sizes (Supplementary Fig.S8). Further, we systematicallyprobed the effect of specific interaction valency on the extent of phase separation. Fig.7Bshows phase diagram with λ and φ as the phase parameters. For smaller λ and low φ , thesingle largest cluster sizes do not approach the macrophase separated limit (blue and blackregions in Fig.7B). This suggests that the critical densities for macrophase separation forlower valency particles would be extremely large, making it an improbable scenario at low in vivo concentrations. Detailed versions of these phase diagrams can be found in Supple-mentary Fig.S5. This phase-diagram is consistent with in vitro experiments involving SH3and PRM chains with varying valencies, with higher valency molecules displaying a lowercritical concentration for phase separation. Another interesting feature that differentiatesthe clusters stabilized by specific interactions are relatively lower densities of these clusters,as opposed to their counterparts stabilized by non-specific interactions (Fig.7C). While theassemblies formed by specific interactions ( λ =5, (cid:15) ns = 0.35 kT; purple curve in Fig.7C) areof comparable sizes to those driven by non-specific interactions, their densities are almost2-fold lower. This is consistent with the spatial density profiles for strong and weakly inter-acting linkers in our LD simulations (Supplementary Fig.S4). In addition to the number ofspecific interactions, a vital parameter that would determine the droplet sizes is the strengthof these specific interactions ( (cid:15) sp ). As evident from Fig.7D, for specific interactions that areextremely weak ( (cid:15) sp < <
50% of available monomers)20s lower for higher valency particles ( λ = 5 curve in Fig.7D). Interestingly, the SH3-PRMinteraction strength is reported to be of the order of 2 kT. A more detailed (cid:15) sp - k bond phase diagram can be found in Supplementary Fig.S9. Overall, our systematic study of clus-ter sizes suggests that the propensity to phase separate at biologically relevant timescalescould be tuned via different parameters offering the cell several handles to modulate sizesand morphologies of droplets. Interestingly, a macro-phase separated state exists only ina very narrow window of parameters in the limit of weak non-specific interactions, unlikeaggregation processes where isotropic interactions always result in macrophase separation atlonger timescales. Tunability of exchange times in microphase-separation
In the KMC simulations so far, we discuss the manner in which different phase parameterscould shape droplet size distributions. However, the functionality of a condensate hingesnot only on the ability of biomolecules to assemble into larger clusters but also to exchangewith the surrounding medium at biologically relevant timescales. These exchange timescalesare also a measure of the material properties of the droplets themselves.
Therefore asystematic understanding of the dependence of molecular exchange times on intrinsic andextrinsic parameters is crucial to get a grasp of the tunability of intracellular self-organisationdriven by finite-valency specific interactions. In this context, we probed the extent to whichthe exchange times could be tuned by modulating the intrinsic features of the self-assemblingunits. Here, we define monomer exchange times as the mean first passage time for a monomerto go from having four neighbours to being completely free. To compute first passage times,we kept track of exchange events from across 300 simulation trajectories of 10 hours each.Our simulations suggest that, for a given valency, a slight increase in interaction strength ( (cid:15) sp )within a narrow window could result in a dramatic increase in phase-separation (Fig.7D).This raises an interesting questionis there an optimal range for these phase parametersthat promotes phase separation as well as maintains the dynamicity of the clusters? In21igure 8: Effect of model parameters on the exchange times between monomers and theaggregates. The parameter values used in panels A and B are φ =0.04, k bond / k diff = 1. A)Mean first passage time for the monomers to go from the buried state (with 4 neighbours)to the free state (with no neighbours) in response to varying values of specific interactionstrength. The two curves show the trends for species with different interaction valencies.The dashed solid lines refer to value of (cid:15) sp beyond which the largest cluster consumes 50% ormore of available monomers. B) The state of the system, for variation in (cid:15) sp and λ suggeststhat the system displays remarkable malleability in dynamicity and size distributions. Weak (cid:15) sp and a low λ results in no phase separation (shaded black). For higher values of boththese parameters, the system can access the macrophase separated state (shaded yellow),however with a dramatic slowdown in exchange times. For an intermediate range, the systemshows microphase separation, with either slow (shaded red) or fast-exchange (shaded blue)dynamics. C) The experimentally determined molecular exchange times for molecules ofvarying interaction valencies. D) The extent of recovery after a photobeaching experiment,for interacting species with varying valencies. The data in panel C) and D) has been obtainedfrom a study by Xing et al. The solid curves in C and D are added to guide the eye22his context, we first computed the mean first passage times for monomer exchange uponsystematic variation of (cid:15) sp (Fig.8A). Interestingly, as with cluster sizes, a slight increase in (cid:15) sp results in a dramatic increase in monomer exchange times. For particles with λ =5, a slightincrease in (cid:15) sp from 2 kT to 2.5 kT, there is a four-fold slowdown in exchange times indicatingdramatic malleability of dynamicity of these assemblies. This shift from the fast to slowexchange dynamics is less abrupt in case of particles with λ =3 suggesting that an interplaybetween λ and (cid:15) sp could tune the droplets for desired exchange properties. We furthervaried these two parameters ( λ and (cid:15) sp ) systematically to probe their effect on cluster sizes(Supplementary Figure.S10A) and molecular exchange times (Supplementary Figure.S10B).As expected, for weak (cid:15) sp and a low λ there is no phase separation (black region in Fig.8B).For an intermediate regime in this phase-space, the system is predominantly in a micro-phase separated state, with either slow (red region in Fig.8B) or fast (blue region in Fig.8B)molecular-exchange times. However, macrophase separation is only observed for really large λ and (cid:15) sp , with a dramatic slowdown in exchange times (shaded yellow region in Fig.8Band Supplementary Figure.S10B ), suggesting that these assemblies might be biologicallynon-functional. Valency is, therefore, a key determinant of how frequently a molecule getsexchanged between the droplet and the free medium. Similar observations have been madeexperimentally by Xing et al. (Fig 8C and D) with regards to several condensate proteinsfeaturing different valencies, with low valency species showing exchange times that are ordersof magnitude faster than the higher valency ones. Given that functional droplets are tunedfor liquid-like behaviour, microphase separation is, therefore, the most likely outcome fordynamically exchanging droplets. 23 iscussion
Micro-phase separation: a potential signature of multivalent het-eropolymers
Membrane-less organelles are heterogeneous pools of biomolecules which localize a high den-sity of proteins and nucleic acids.
An interesting feature of several complex monomers that constitute these droplets is ’multivalency’ or multiple repeats of adhesive domains.
These adhesive domains can bind to complementary domains on other chains, thereby facili-tating phase separation. In this study, we model this phenomenon as that of self-associativepolymers that possess folded domains (represented as idealized spheres in ) separated by flex-ible linker regions. Recent computational studies have employed similar models to charac-terize the equilibrium state of these polymer systems, notably the coarse-grained simulationsby Harmon et al. and Choi et al. These studies employ the ’sticker and spacer’ model tounderstand the phase behaviour of these linear multivalent polymers, largely focusing onthe nature of the phase-separated state at equilibrium. Using lattice-polymer Monte carlosimulations, they establish the role of intrinsically disordered linker regions as moleculardeterminants that dictate the equilibrium state of a system of associative polymers that in-teract via non-covalent interactions. Crucially, these works focus on the cross-linked gel-likenature of the macrophase-separated state of these polymers. However, equilibrium latticesimulations and classical Flory-like theories do not address the dynamics of cluster growthresulting in the observed peculiar micro-phase separated nature of multiple biomolecularcondensates, as opposed to a single macro-phase separated state.
This raises a numberof interesting questions is microphase separation a signature of multi-valent protein assem-blies? How does the physics of associative polymers multivalent proteins differ from those ofsimple, homopolymer chains? In this work, we provide a kinetic analysis of these questionsusing a combination of coarse-grained LD simulations to probe the early stages of clustergrowth, and a reaction-diffusion model to probe the problem at longer timescales.24 xhaustion of specific interaction valencies – a barrier to macro-phase separation
First, we studied self-assembly by multivalent polymers whose adhesive domains interact viastable, ‘non-transient’ interactions to understand the early events in the growth process. Weobserved that, except for extremely high concentrations where there is a system spanningnetwork, the most feasible scenario at smaller concentrations is that of micro-droplets witha concentration-dependent distribution of droplet sizes. The temporal evolution of aggre-gated species suggests that at lower concentrations, the available interaction valencies getconsumed within smaller-sized assemblies, making them inert for further cluster growth.Our results suggest that two critical timescales decide whether a cluster continues to growfurther – a) concentration-dependent timescale of two chains encountering each other andforming the initial functional interaction, and, b) the exhaustion of valencies within a smallcluster, a timescale dependent on intrinsic features of the polymer (Fig.4). Crucially, thesetwo timescales are sensitive to subtle modifications in the self-assembling polymer chain(Fig.5). Therefore, by tuning these two time-scales, the cell can modulate the degree ofphase separation. Modifications to the linker can also result in altered densities of theself-assembled state, with a 10-100 fold enrichment in molecular concentrations within thedroplets (Supplementary Fig.S5), consistent with the experimentally reported degrees ofenrichment within condensates .
Overall, the simulations involving non-transient inter-actions help us establish an understanding of the essential physical mechanisms determiningmicrophase separation in membrane-less organelles, with the finite nature of the specific in-teractions driving the peculiar phenomenon. Although these findings were for an artificialassumption of extremely stable functional interactions, the highly cross-linked nature of thephase-separated state might result in these early micro-phase droplets not being able totransition to the macro-phase separated state even for reversible, transient interactions.25 ridging the gap between the early and biologically relevant timescales To reach the time scales relevant to biology we employed a coarse-grained kinetic model whereeach polymer (from the LD simulations) is represented as a diffusing reacting centre on a2D lattice which can interact either non-specifically (mimicking inter-linker interactions) orspecifically with neighbouring centres. The difference between the two types of interactionsis that the number of specific interactions that each centre can make is limited by its valency.In an extension of the LD model, specific interactions are stable yet reversible and can formand break with rates dictated by detailed balance. Consistent with our LD simulations, ourkinetic Monte-Carlo simulations of the phenomenological model reveal that for timescalesrelevant to biology, macro phase-separation is a phenomenon that is observed in a very nar-row regime of parameters in the kinetic phase-space (Fig.7). Further, the phenomenon ofexhausted adhesive valencies is even more prominent for species with lower valency (feweradhesive domains in the prototypical polymer), as evident from much smaller sizes of thelargest cluster after an hour-long (actual time) simulation run. The lattice-diffusion model,despite its minimal nature, captures the well-known relationship between molecular valencyand critical concentration for phase separation (Fig.7B). An interplay between the valencyof the generalized polymer and the strength of interactions can also alter the exchange timesof molecules with the bulk medium dramatically. Interestingly, a slight shift in either thesevalencies or interaction strengths could result in a change in exchange rates by orders ofmagnitude. Further, for regions of the parameter space ( (cid:15) sp and λ ) that favor macro-phaseseparation we observe a dramatic slow-down in molecular exchange times. In other words,for parameters that result in a fast-exchanging condensed state, micro-phase separation isthe most favored outcome (Fig.8B and Supplementary Fig.S10). Such a discontinuity makesthese systems extremely sensitive to mutations that might cause a shift in dynamics andeventual loss of function of these droplets. Also, such shifts could also make these sys-tems extremely responsive to non-equilibrium processes such as RNA-processing, side-chainmodifications (acetylation, methylation) that are often attributed to modulating condensate26ynamics. The differential exchange times in response to variation of interaction param-eters in our model lends further support to the scaffold-client model. The scaffolds that areslower exchange species with higher valencies acting could, therefore, recruit faster exchangeclients with lower valencies. The valencies and strength of interactions could have thusevolved to achieve exchange times that ensure the functionality of spatial segregation vialiquid-liquid phase separation. A multi-domain architecture such as the sticker-spacer archi-tecture allows for separation of two functions with the folded domains (conserved) performingfunctional role while the spacer regions being modified over time to tune the propensity tophase separate and also the material nature of the condensate. Overall, our multi-scalestudy shows that the block co-polymer like organization of these multi-valent proteins, withfinite specific interactions driving phase separation could manifest itself in the micro-phaseseparated droplet state. A switch in the driving force for self-assembly, from the specific tonon-specific interactions via sticky linkers, could not only alter the kinetics of assembly butalso have implications in disorders associated with aberrant phase separation.
Methods
Langevin Dynamics simulations
Force Field.
The polymer chains in the box are modelled using the following interactions. Adjacent beadson the polymer chain are connected via harmonic springs through the potential function, E stretching = k s M − X i =1 ( | ~r i − ~r i +1 | − r ) , (2) ~r i and ~r i +1 refer to the adjacent i th and ( i + 1) th bead positions, respectively. Here, r isthe equilibrium bond length and k s represents the spring constant. To model semi-flexibilityof the polymer chain, any two neighbouring bonds within the linker regions of the polymer27hains interact via a simple cosine bending potential E bending = κ M − X i =1 (1 − cos θ i ) , (3)where θ i describes the angle between i th and ( i +1) th bond while κ is the energetic cost forbending. The non-bonded isotropic interactions between linker beads and linker-functionalinteractions were modelled using the Lennard-Jones (LJ) potential, E nb = 4 (cid:15) X i The dynamics of these coarse-grained polymers was simulated using the LAMMPS moleculardynamics package. In these simulations, the simulator solves for the Newtonss equationsof motion in presence of a viscous drag (modeling effect of the solvent) and a Langevin ther-mostat (modeling random collisions by solvent particles). The simulations were performedunder the NVT ensemble, at a a temperature of 310 K. The mass of linker beads was set to110 Da while the mass of the idealized functional domains (red and yellow beads in Fig.1)was set to 7000 Da that is approximately equal to the mass of the SH3 domain. The sizeof the linker beads was set at 4.2 A (of the same order as amino acids) while that of thefunctional domains was set at 20 A ( size of a folded SH3 domain ). The viscous drag wasimplemented via the damping coefficient, γ = m/ πηa . Here, m is the mass of an individualbead, ‘ η is the dynamic viscosity of water and a is the size of the bead. An integrationtime step of 15 fs was used in our simulations, and the viscosity of the surrounding mediumwas set at the viscosity of water. Similar values of these parameters have been previouslyemployed for coarse-grained Langevin dynamics simulations of proteins. Kinetic Monte carlo simulations To assess biologically relevant time-scales, we develop a phenomenological kinetic modelwherein the individual multi-valent polymer chains are modelled as diffusing particles withfixed valencies. Each particle in our lattice Monte carlo simulations is a coarse-grained repre-sentation of the bead-spring polymer chains in the LD simulations, with an effective valencythat is a simulation variable. Particles occupying adjacent sites on the lattice experiencea weak, non-specific interaction (of strength (cid:15) ns ) that is isotropic in nature. This isotropicattractive force is analogous to the inter-linker interactions in the LD simulations. A func-tional bond can stochastically form between any pair of neighbouring particles, providedboth the particles possess unsatisfied valencies. The rate of bond formation between a pairof particles with unsatisfied valencies is k bond . On the other hand, an existing functional29ond can break with a rate k break that is equal to k bond exp − ( (cid:15) sp ) in magnitude. (cid:15) sp is thestrength of each functional interaction. Assuming that there are unoccupied neighbouringsites, the monomers can diffuse on the lattice in any of the four directions (in 2D) with arate k diff . Additionally, the entire cluster that any given monomer is a part of can diffusein either of the 4 directions with a scaled diffusion rate that is inversely proportional to thesize of the cluster. A particle that is part of a cluster can diffuse away from the clusterwith a rate k diff exp − ( P (cid:15) sp + (cid:15) ns )). Here, P (cid:15) sp + (cid:15) ns is the magnitude of net interactionsthat any particle is involved in. These rates are schematically described in Fig.6. Usingthese set of phenomenological rates, we allow the system to evolve using the exact stochasticsimulation method or the Gillespie kinetic Monte Carlo algorithm ]. This algorithm haspreviously been used to model biological processes as diverse as gene regulation and cy-toskeletal filament growth. In this approach, a set of ’N’ rates is initiated for any givencurrent configuration (set of coordinates of each particle and their valencies). In other words,10 potential events are initiated for each particle on the lattice. Namely, the 10 events perparticle are a) 4 diffusion events, b) 4 cluster diffusion events, c) bond formation and d)bond breakage events. For any given configuration, all or only a subset of these events couldbe possible. We then advance the state of the system by executing one reaction at a time.The probability of each event is equal to its rate, r ji / P P r ji . Here i and j refer to the iden-tity of the monomer and the event type, respectively. Given these set of propensities, wechoose the event to be executed by drawing a uniformly distributed random number and bycomparing how the random number compares with the event propensities. The simulationtime is advanced using the following expression, δt = − (1 /r total ) ln ( z ), where z is a uniformrandom number. This δt is based on the assumption that the waiting times between any twoevents are exponentially distributed. The above algorithm has to be iterated several timessuch that each reaction has been fired multiple times, suggesting the system has reachedsteady-state behaviour. 30 ationale behind phase parameters in KMC simulations: Two key phase parameters in our KMC simulations are the valency per interacting particleand the bulk density of particles on the lattice. We performed simulations for valencies rang-ing from 3 to 6, typical of multivalent proteins enriched in membrane-less organelles. Unlikethe non-specific which can only be one per neighbor, thereby a maximum of 4 per particle fora 2D-lattice, there could be more than one specific interactions between a pair of neighboringparticles, as long as both the participating members have unsatisfied valencies. The bulkdensity of particles is defined as, φ = N mono / L , where N mono and L are the number of par-ticles and lattice size, respectively). In our kMC simulations, we varied φ within the rangeof 0.01 to 0.1, a range that is consistent with the analogous parameter for LD simulations,the occupied volume fractions within the LD simulation box ( φ LD = N mono .4/3 πR mono ). Forinstance, a free monomer concentration ( C mono ) of 10 uM corresponds to a φ LD of 0.008which increases to 0.17 for 200 uM. Further, the phase diagrams for KMC simulations wereprimarily computed for a weak non-specific (cid:15) ns of 0.35 kT, chosen in order to focus on specificinteraction-driven cluster growth. In the kMC simulations, (cid:15) ns refers to the net non-specificinteraction for a pair of monomers as opposed to a pair of interacting linker beads in theLD simulations. It must, however, be noted that the interaction strength for a pair of in-teracting chains is not merely additive (with the length of the polymer chain). A potentialexplanation for this could be found in earlier studies wherein it has been argued that smallersegments ( length of 7-10 amino acids) within a long, solvated polymer chain (like IDPs)could behave like independent units referred to as blobs. Since the degree of coarse-grainingemployed in KMC is of the order of one particle per polymer chain, it lacks the microscopicdegrees of freedom of the bead-spring polymer chain (in LD simulations). Hence we do notemploy a higher non-specific interaction strength in our phase plot computations. As proofof principle, we show the mean pairwise interaction energies for dimers from the LD simula-tions, for a weak inter-linker interaction strength of 0.1 kcal/mol (Supplementary Fig.S7A)and a corresponding phase-diagram for non-specifically driven interactions (Supplementary31ig.S7B). All the mean cluster sizes in the MC simulations were computed over 300 indepen-dent kinetic Monte carlo trajectories. 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A polymer physics perspective ondriving forces and mechanisms for protein aggregation. Archives of Biochemistry andBiophysics , , 132–141. 36 UPPLEMENTARY INFORMATION:Liquid-liquid microphase separation leads toformation of membraneless organelles Srivastav Ranganathan, ∗ Eugene Shakhnovich ∗ a r X i v : . [ q - b i o . B M ] J a n max_avg.log’ u 1:($2/400)’../k_2_p_0p02/max_avg.log’ u 1:($2/400)’../k_2_p_0p5/max_avg.log’ u 1:($2/400)’../k_2_p_0p3/max_avg.log’ u 1:($2/400)’../k_2/max_avg_k2_0p1.log’ u 1:($3/400)’max_avg.log’ u 1:($2/400)’../k_2_p_0p02/max_avg.log’ u 1:($2/400)’../k_2_p_0p5/max_avg.log’ u 1:($2/400)’../k_2_p_0p3/max_avg.log’ u 1:($2/400)’../k_2/max_avg_k2_0p1.log’ u 1:($3/400) P form 40 80 120 160 200C mono, uM L a r g e s t C l u s t e r ( F r a c t i o n o f M o n o m e r s ) Figure S1. The size of the largest cluster, for different values of bondformation probability, P form . A lower P form results in a slower arrest of theclusters, and thereby results in increased cluster sizes for smaller free monomerconcentration, C mono . Figure S2. Fraction of valencies utilized as a function of increasinginter-linker interaction strength, for A) 10 µ M and B) 50 µ M free monomerconcentration. 2 A Lag Time (ns)0.5 kcal/mol0.3 kcal/mol0.1 kcal/mol D=7.5 A2/nsD=3.1 A2/ns Figure S3. The mean-squared displacements of the center of masses of theconstituent polymer chains as a function of increasing lag-times, in the absenceof functional interaction. The three curves show the lagtimes for varyingvalues of isotropic interaction strength between the linker residues.3 P ( C l u s t e r ) Largest Cluster (Fraction of Total Monomers) κ =2 kcal/molFlexible Linker κ =5 kcal/molRigid Linker Figure S4. Comparision between the size distributions of the largest cluster,for stiff ( κ =2 kcal/mol) versus flexible ( κ =5 kcal/mol) linker regions. The freemonomer concentration used for this plot was 50 µ M and an a weak interlinkerinteraction strength of (cid:15) n = 0.1 kcal/mol was used.4 ε ns 0.1 kcal/mol ε ns 0.5 kcal/mol ε ns 0.35 kcal/mol0 20 40 60 80 100 120 14000.050.10.150.20.250.3 φ ' (Local Density/Bulk Density) P ( φ ' ) Figure S5. The probability of finding clusters with varying densities(normalized by the bulk densities) for different values of inter-linkerinteractions. As the inter-linker interactions increase, the degree of enrichmentcan go from 10-fold to 100-fold. 5 igure S6. Detailed phase diagrams for A) and B) φ - k b ond , C) φ - λ as thephase parameters. The cluster sizes were computed at the end of a simulationrun of 2 hours (actual time), setting the rate of diffusion k d if f to 1 s − . D)The bonding rate k bond was varied to identify the relationship between k bond and k d if f . 6 ns, kT0 0.5 1 1.5 2 2.5 30.010.030.050.070.09 D e n s i t y () L a r g e s t C l u s t e r ( F r a c t i o n M o n o m e r s ) ’e_dimer_v_k.dat’ u 1:($2/0.6) Linker bending rigidity, kcal/mol0.60.811.2 E d i m e r , k T A B Figure S7. A) The mean pair-wise interaction energy for 100 different dimericstructures (from the LD simulations), for an inter-linker interaction strength of0.1 kcal/mol, for different values of linker bending rigidity. B) The (cid:15) ns - φ phasediagram (for λ =0) showing no phase separation for low values of isotropicinteraction strength. However, for values of (cid:15) ns > .01 0.02 0.03 0.04 0.05 0.06 0.070.050.10.150.20.250.30.35 Density L a r g e s t C l u s t e r ( F r a c t i o n o f a v a il a b l e M o n o m e r s ) Density L a r g e s t C l u s t e r ( F r a c t i o n o f a v a il a b l e M o n o m e r s ) B u l k D e n s i t y ( φ ) Valency ( λ )0.010.030.050.070.09 Valency ( λ ) A λ = 3 λ = 5 BC D 10 hrs Figure S8. Convergence of phase diagrams. A) and B) shows the fraction ofmonomers in the largest cluster for 1 and 10 hours of actual time, for valencyof 3 and 5, respecively. C) and D). The λ - φ phase diagram at the end of 2 and10 hours of simulation time, respectively, showing very little difference.8 igure S9. Detailed phase diagrams for A) and B) (cid:15) sp - k bond as the phaseparameters. The bulk density of particles was set to 0.04, an intermediatedensity identified from the previous phase diagrams with density as a phaseparameter. The cluster sizes were computed at the end of a simulation run of2 hours (actual time), setting the rate of diffusion k d if f to 1 s − . ε s p , k T ε s p , k T Valency2 3 4 5 612340 123407 Valency2 3 4 5 6 7 < F P T r e l e a s e > , s e c o n d s L a r g e s t C l u s t e r ( F r a c t i o n o f M o n o m e r s ) A B Figure S10. Mean cluster sizes for variation in (cid:15) sp and λ , for a bulk densityof 0.04, and a k bond / k diff ratio of 1. B) Mean first passage times for a particleto exchange between a cluster and the bulk. The parameter values are same asin panel A. 9 .20.40.60.80 0.02 0.04 0.06 0.08Bulk Density ( φ ) F r a c t i o n o f m o n o m e r s i n l a r g e s t c l u s t e r λ = 5, ε ns = 0.35 kT λ = 0, ε ns = 3 kT λ = 4, ε ns = 0.35 kT λ = 3, ε ns = 0.35 kT λ = 1, ε ns = 0.35 kT λ = 0, ε ns = 0.35 kT φ ) D e n s i t y w i t h i n l a r g e s t c l u s t e r ( φ ) λ = 3, ε ns = 0.35 kT λ = 5, ε ns = 0.35 kT λ = 0, ε ns = 3 kT A B Figure S11. Cluster formation driven by isotropic versus specificinteractions. A) Comparison of cluster sizes (as a fraction of total monomers)for different scenarios. The black curve shows cluster sizes for assembly drivenby strong non-specific interactions alone ( λ =0 and (cid:15) ns =3 kT). For a scenarioinvolving weak isotropic interactions ( (cid:15) ns = 0.35 kT), the curves approachthat of the isotropic interactions for higher valencies ( λ → (cid:15) ns = 3kT, λλ