Random crosslinks generate anomalous scaling of dynamic modulus of biomolecular condensates

Random crosslinks generate anomalous scaling of dynamic modulus of biomolecular condensates | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Random crosslinks generate anomalous scaling of dynamic modulus of biomolecular condensates Bohan Lyu , View ORCID Profile Jie Lin doi: https://doi.org/10.1101/2025.11.05.686888 Bohan Lyu 1 Peking-Tsinghua Center for Life Sciences, Peking University , Beijing, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site Jie Lin 1 Peking-Tsinghua Center for Life Sciences, Peking University , Beijing, China 2 Center for Quantitative Biology, Peking University , Beijing, China Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Jie Lin For correspondence: linjie{at}pku.edu.cn Abstract Full Text Info/History Metrics Supplementary material Preview PDF Abstract Biomolecular condensates are viscoelastic, and their mechanical properties are intimately related to their biological functions. However, the connection between microscopic networks formed by intermolecular crosslinks and viscoelasticity is still elusive. Here, we model biomolecular condensates as random crosslinked polymer solutions to elucidate how random connectivity fundamentally alters their viscoelasticity. We decompose the entire solution into multiple tree networks and demonstrate that for networks with size n , their spectra of relaxation rates λ exhibit a power-law scaling p n ( λ ) ∼ λ −1 / 3 with a lower cutoff λ min ∼ n −3 / 2 . By integrating all networks, we show that for the entire solution, random crosslinks generate an abundance of soft modes involving multiple linear polymers with a flat spectrum of relaxation rates. The soft modes cause anomalous linear frequency scaling of the dynamic modulus, in particular, they significantly boost the low-frequency storage modulus relative to uncrosslinked systems. Our predictions agree quantitatively with the experimental data from distinct biomolecular condensates. Introduction Biomolecular condensates, also known as membraneless organelles, play crucial roles in cellular organizations and functions, e.g., by concentrating specific proteins and nucleic acids into dynamic compartments such as P granules [ 1 , 2 ], stress granules [ 3 , 4 ], transcriptional condensates [ 5 – 7 ], and many others [ 8 – 10 ]. Notably, many experiments have demonstrated that biomolecular condensates are viscoelastic, exhibiting both liquid-like and solid-like behaviors, which are quantified by the storage G ′ and loss moduli G ′′ [ 11 – 15 ]. Importantly, the material properties of condensates also significantly influence their biological functions and may even lead to diseases [ 16 – 20 ]. Experiments and simulations have highlighted the central role of intermolecular interactions in shaping the mechanical properties, morphologies, and aging behaviors of biomolecular condensates [ 13 , 15 , 21 – 27 ]. Importantly, biomolecular condensates often involve multivalent proteins, e.g., via intrinsically disordered regions (IDRs) or sticky patches, and may form networks, giving rise to emergent viscoelastic behaviors distinct from those of linear polymers [ 28 – 30 ]. Recent computational studies have also shown that accounting for crosslinks between different proteins is critical for matching numerical and experimental data on the dynamic modulus [ 31 , 32 ]. Moreover, recent experiments have revealed anomalous linear scalings in the low-frequency regime of storage and loss modulus: G ′ ( ω ) ∼ ω and G ′′ ( ω ) ∼ ω where ω is the frequency [ 11 , 12 , 32 ], distinct from the predictions of Maxwell model [ 33 ]. Classical models of viscoelasticity of polymer solutions, including Rouse and Zimm theories, provide foundational frameworks for understanding the rheology of linear polymer solutions, but fall short in capturing the complexity inherent in biomolecular condensates [ 33 – 36 ]. How inter-molecular crosslinks and network formation at the molecular scale affect the macroscopic viscoelastic behavior of biomolecular condensates and the frequency scaling of the dynamic modulus, remains poorly understood. In this work, we model biomolecular condensates as random crosslinked polymer solutions and elucidate how random connectivity profoundly transforms their viscoelastic behaviors. We generalize the Rouse model to study a polymer solution generated by adding random crosslinks between linear polymers, resulting in a solution composed of tree networks whose sizes follow a power-law distribution near gelation. We investigate the eigenvalue spectra of the connectivity matrices for these networks, which we call the relaxation spectra, since they determine the condensate’s viscoelastic responses. Intriguingly, we show that adding crosslinks between linear polymers generates eigenvalues that are bounded between the unperturbed eigenvalues of the original Rouse chain, and it is the eigenvalues between 0 and the smallest eigenvalue of the Rouse chain, denoted λ 1 , that sets the long-time rheology of the condensate. For a given tree network with n proteins, we show that the corresponding relaxation spectrum exhibits a power-law scaling, p n ( λ ) ∼ λ − α for small eigenvalues with a lower cutoff λ min ( n ) ∼ n − β , where α ≈ 1 / 3 and β ≈ 3 / 2. Notably, we prove that for an arbitrarily connected network, its viscosity is precisely proportional to its mean-square radius of gyration. Using this relationship, we predict αβ = 1 / 2, in perfect agreement with our numerical calculations. By combining all tree networks, we further reveal that the entire polymer solution exhibits a flat relaxation spectrum, reflecting the presence of an abundance of soft collective modes involving multiple chains. Remarkably, these soft modes lead to anomalous linear scaling in both the storage and loss moduli (with a logarithmic correction) at low frequency; in particular, they significantly enhance the low-frequency storage modulus relative to that of uncrosslinked polymer solutions. Importantly, we compare our predictions with recent experimental data on the storage and loss moduli of diverse biomolecular condensates and find excellent agreement [ 11 , 12 , 32 ], strongly suggesting that the soft collective modes generated by random crosslinking underlie the universal long-time-scale viscoelastic behavior of biomolecular condensates. Random crosslinked polymer solutions We model biomolecular condensates as polymer solutions in which random crosslinks are added between long linear chains, which can be generated via sticky amino acids such as tyrosine and arginine or sticky patches at a protein’s surface [ 21 , 37 – 39 ] ( Figure 1a ). In this scenario, the mean-field model is an excellent approximation [ 33 ]: any pair of chains has an equal probability of being crosslinked. Below the gelation point, one can coarse-grain the system as an Erdős-Rényi random graph where each node represents a polymer chain and each edge represents a crosslink; the degree distribution follows a Poisson distribution P ( k ) = c k e − c /k ! where c is the average degree, i.e., the average number of crosslinks per chain[ 40 , 41 ] ( Figure 1b ). Download figure Open in new tab FIG. 1. (a) We model biomolecular condensates as a crosslinked polymer solution where linear polymer chains, e.g, proteins, are connected via random crosslinks (red lines). (b) In this schematic, the network consists of n = 5 chains, each with m = 9 monomers. The total number of monomers is N = nm = 45. We demonstrate that the viscosity generated by this network, η , is proportional to its mean-square radius of gyration and also related to the viscosity of the corresponding coarse-grained network, η G , with 5 nodes, each representing one original chain. Here, η R is the viscosity of the corresponding uncrosslinked system. For a solution with N p chains, the number of crosslinks is N p c/ 2. In the thermodynamic limit ( N → ∞) below gelation threshold ( c < 1), the solution contains virtually no closed loops, resulting in a collection of tree networks. The probability distribution for a randomly selected chain to belong to a tree of size n is given by (Supplemental Material) For n ≫ 1, P ( n ) ∝ n −3 / 2 e − n/n ∗ , where n ∗ ∝ (1 − c ) −2 . General properties of networks We generalize the Rouse model to an arbitrary network configuration, where the position of bead i in three dimensions follows an overdamped dynamics Here, ζ is the friction coefficient, and k is the spring constant, which is related to the Kuhn length b by k = 3 k B T/b 2 where k B is the Boltzmann constant and T is the temperature. For simplicity, we use the same spring constant for polymer backbones and crosslinks, which does not affect our main conclusions. The random force ξ i is generated by thermal fluctuation and satisfies the fluctuation-dissipation theorem (FDT) [ 42 ]. The matrix L ij is a symmetric matrix that describes the connectivity between beads with eigenvalues λ p [ 31 , 35 , 43 ]. The relaxation modulus G ( t ) (i.e., the shear stress relaxation given a unit strain) of a network is equivalent to that of a parallel connection of multiple Maxwell fluids with their relaxation rates given by the eigenvalues of the connectivity matrix (see the detailed derivations in Supplemental Material) [ 44 ]: where V is the system volume. One should note Eq. (3) applies to the whole solution as well, which contains numerous disconnected networks; each of them contributes a zero eigenvalue due to translational invariance and does not contribute to the relaxation modulus. From Eq. (3) , we compute the zero-shear viscosity as To simplify the notation, we set the time unit as ζ/k , the length unit as the Kuhn length b , the viscosity unit as k B Tζ/ 2 V k , and the stress unit as k B T/V . In the following, all variables are dimensionless unless otherwise mentioned, e.g., η = ∑ p 1 /λ p . We also compute the frequency-dependent storage modulus G ′ ( ω ) and loss modulus G ′′ ( ω ) as where τ p = 1 / (2 λ p ) is the relaxation time of mode p . Relaxation spectra of tree networks In the Erdős-Rényi random graph, varying the mean degree c only alters the relative abundance of trees of different sizes within the entire forest, leaving the conditional probabilities of different topological structures for trees of a given size n unchanged. Consequently, we only need to study the relaxation spectra of trees of a given size and sum the spectra of trees of different sizes to obtain the full spectrum of the entire polymer solution. In the absence of any crosslinks, the network is composed of n disconnected linear chains. In this case, the spectrum has m distinct modes, each with n -fold degeneracy [ 33 , 36 , 45 ]: where q = 0, 1, …, m − 1. In the following, we denote the eigenmode of the uncrosslinked system with eigenvalue λ q as the q -th Rouse mode. Due to the eigenvalue interlacing theorem [ 46 ], adding a crosslink reduces the degeneracy of each mode by one and introduces a new eigenvalue in the gap between λ q and λ q +1 . As more crosslinks are added, more eigenvalues escape from the reference value λ q and enter the gap. Therefore, we predict that for a tree network with n − 1 crosslinks, the q -th mode retains exactly one eigenvalue while each inter-mode gap contains n − 1 eigenvalues. To verify our prediction, we generate fixed-size tree structures with random topologies and compute the eigenvalue distributions averaged over all sampled topologies ( Figure 2a ). Indeed, the perturbed eigenvalues reside inside the gap between λ q and λ q +1 (see the example for q = 0 in Figure 2a inset). One should note that the overall shapes of the cumulative distributions of eigenvalues for the tree networks resemble that of a single Rouse chain, λ = 4 sin 2 ( πQ ( λ ) / 2) due to the confinement of the perturbed eigenvalues (the dashed line in Figure 2a ). Therefore, the relaxation spectrum G ( t ) of a tree network for a time scale shorter than the Rouse time 1 /λ 1 , where λ 1 is the smallest eigenvalue for a linear Rouse chain, should approach that of a single Rouse chain. Download figure Open in new tab FIG. 2. Eigenvalues for fixed-size tree networks. (a) The cumulative distributions of eigenvalues for different tree sizes n . The black dashed curve represents the theoretical prediction λ = 4 sin 2 ( πQ/ 2) for the corresponding uncrosslinked system. The red vertical dashed line indicates the position of the first mode λ 1 . The inset shows the cumulative distribution between 0 and the first mode. (b) The probability density of eigenvalues in the interval (0, λ 1 ). (c) Scaling of the lower cutoff λ min with the tree size n . The error bars show the geometric standard deviation across 100 different topologies. In panels a, b, and c, each linear chain has m = 10 monomers. (d) The cumulative distributions for tree networks of size n = 20 under different values of m . The inset shows the distribution of eigenvalues between 0 and λ 1 after normalization by λ 1 , which converges to a limit for sufficiently large m . We remark that coarse-graining the number of monomers in a linear chain should not change the physics when m is sufficiently large. As m changes, the friction coefficient and the spring constant scale as ζ ∼ m −1 and k ∼ m : coarse-graining several Kuhn monomers into one monomer makes the effective spring constant smaller and the friction coefficient felt by one monomer larger [ 33 ]. Meanwhile, the relaxation modulus in physical units beyond the Rouse time, where µ i are the n −1 eigenvalues in (0, λ 1 ), must remain invariant under changes in the coarse-graining level of m , leading to the relation µ i ∼ ζ/k ∼ m −2 ∼ λ 1 . Consequently, the normalized eigenvalues µ i /λ 1 must be independent of m . We confirm the scaling collapse numerically for fixed n = 20, where we find that mQ 20 ( λ ) vs. λ/λ 1 ( m ) converges to a universal curve once m ≳ 10 ( Figure 2d ). Therefore, we use m = 10 in the following simulations, as it suffices to reach the asymptotic limit. The above analysis suggests the following scaling form for the probability distribution of eigenvalues between 0 and λ 1 , p n ( λ ) = f n ( λ/λ 1 ) /λ 1 , which converges to a stable distribution as n increases and exhibits a power-law scaling for small λ ( Figure 2a inset, b): with a lower cutoff, x min ( n ) ∼ n − β ( Figure 2c ). In the following, we show that the two exponents α and β are related. Relation between viscosity and radius of gyration As a short diversion, we introduce a relationship between the viscosity generated by an arbitrary polymer network (which does not need to be a tree) in a solution and its mean-square radius of gyration: ( Figure 1b , see the detailed proof in Supplemental Material). Here, N is the total number of monomers in the network. To our surprise, we could not find any previous papers mentioning this identity, despite its simplicity. Using the above relationship, we further show that for a particular tree of size n , where each node is a linear chain with m beads, the viscosity increment relative to the uncrosslinked state is given by Here, η R ( n ) is the viscosity of the uncrosslinked Rouse chains, which scales as η R ( n ) ∼ m 2 n and η G ( n ) is the viscosity of the coarse-grained network of size n , where each node is a single monomer, which can be approximated by (Supplemental Material). Meanwhile, for a tree of size n , its incremental viscosity relative to the uncrosslinked state is dominated by the slow modes between 0 and λ 1 : Therefore, we predict that which agrees perfectly with our numerical calculations where we find α ≈ 1 / 3 and β ≈ 3 / 2 ( Figure 2b, c ). From tree networks to the entire polymer solution We next connect the spectra of tree networks to that of the entire solution. Given a mean degree c , the number of eigenvalues in the inter-mode gap is N p c/ 2, which is the number of crosslinks ( Figure 3a ). To obtain the probability distribution between 0 and λ 1 for the entire solution, we only need to integrate all possible tree sizes: where n min ( λ ) is the minimum tree size to observe the target λ ( Figure 2c ), which satisfies n min ( λ ) ∼ ( λ/λ 1 ) −1 /β ( Figure 2d ), and n ∗ ∝ (1 − c ) −2 . Here, we have used the relationship αβ = 1 / 2. Surprisingly, the integration over tree sizes yields a flat relaxation spectrum for the entire solution independent of λ , with a lower cutoff λ min happens when the smallest tree capable of contributing to a given mode is comparable to the characteristic tree size n ∗ , i.e., n min ( λ min ) ≈ n ∗ . This gives: Download figure Open in new tab FIG. 3. Eigenvalues for the entire polymer solution. (a) The cumulative distributions of eigenvalues for different mean degrees c . The black dashed curve represents the theoretical prediction for the corresponding uncrosslinked system. The vertical dashed line indicates the position of the first Rouse mode λ 1 . (b) The cumulative distribution between 0 and λ 1 . We highlight the linear increase of Q ( λ ) as λ approaches zero, which means a constant probability distribution. (c) The probability density of eigenvalues in the interval (0, λ 1 ), which becomes constant for small λ as c approaches 1. (d) Scaling of the lower cutoff λ min of the entire polymer solution with 1 − c . For each data point, λ min is the geometric mean of the 100 eigenvalues (10 smallest eigenvalues from each of 10 independent realizations), with the error bar indicating the geometric standard deviation. In this figure, N p = 2 × 10 5 and m = 10. Our predictions agree perfectly with our numerical calculations ( Figures 3c , d), and also with previous calculations based on replica methods for random connectivity matrices [ 47 , 48 ]. Viscoelasticity of the crosslinked polymer solution Given the flat spectrum between 0 and λ 1 near the gelation threshold at c = 1, the relaxation modulus for becomes , exhibiting distinct behaviors for different time scales: Here, the first crossover from the characteristic 1 / 2 decay of Rouse chains [ 34 ] to the inverse decay occurs at . The storage and loss moduli also exhibit time scale-dependent behaviors (see the detailed derivations in Supplemental Material): We remark that the linear frequency scaling of G ′ ( ω ) and G ′′ ( ω ) (with a logarithmic correction) comes from the constant relaxation spectrum between λ min and λ 1 : the abundance of soft modes involving multiple chains. We confirm our predictions numerically ( Figure 4 ). From the low-frequency scaling of the loss modulus, we also obtain the scaling of viscosity vs. the distance to gelation threshold: η = lim ω →0 G ′′ ( ω ) /ω ∼ ln(1 − c ), in agreement with previous works [ 44 ]. In the Supplemental Material, we also include a more systematic derivation of the viscosity over the full range 0 < c < 1, using the relationship between viscosity and mean-square radius of gyration. In contrast, the enhancement of storage modulus due to crosslinks at low frequency ω < λ 1 is much more significant than the loss modulus (comparing Figure 4b and c ); in particular, increasing the chain length m and the average degree c boosts the storage modulus much faster than the loss modulus [ Eqs. (14 , 15 )]. To understand the boost of storage modulus, it is convenient to consider the low frequency limit ω ≪ λ min where , while clearly, the storage modulus is more sensitive to the abundant soft modes. Download figure Open in new tab FIG. 4. Viscoelasticity of the random crosslinked polymer solution computed with N p = 2 × 10 5 and m = 10, averaged over 30 independent realizations. (a) The relaxation modulus G ( t ) as a function of time t . (b) The storage modulus G ′ ( ω ) as a function of angular frequency ω . (c) The loss modulus G ′′ ( ω ) as a function of angular frequency ω . (d) G ′′ ( ω ) /ω as a function of angular frequency ω . Comparison with experimental data We validate our predictions by comparing the theoretically predicted dynamic modulus with experimental data obtained from diverse protein condensates [ 11 , 12 , 32 ]. We rescale the experimental data by the crossing point ( ω 0 , G 0 ) where G ′ ( ω 0 ) ≈ G ′′ ( ω 0 ) ≈ G 0 to plot them on the same plot. Surprisingly, our theoretical curves with the same set of fixed parameters, m = 5 and c = 0.95, align with all the experimental data, even though the underlying components of these condensates are entirely different. We note that the results are not sensitive to the exact value of c as long as it is close to 1. In particular, our model successfully captures the linear scaling of G ′ and G ′′ at low frequency. The universality of this anomalous scaling across diverse condensates suggests that the soft modes involving multiple chains in a random crosslinked polymer solution are general stress-relaxation mechanisms. Discussion In this work, we study a minimal model of biomolecular condensates as crosslinked polymer solutions. Since the entire solution can be decomposed into tree networks, we first investigate the relaxation spectra of tree networks, and demonstrate that each gap between the eigenvalues of the uncrosslinked system contains n − 1 eigenvalues, where n is the tree size. Furthermore, the spectrum for the slow modes between 0 and λ 1 exhibits the scaling form p n ( λ ) ∼ 1 /λ 1 f n ( λ/λ 1 ) where f n ( x ) becomes independent of the coarse-graining level m for sufficiently large m . In particular, for small x, f n ( x ) ∼ x − α with a lower cutoff x min ∼ n − β , where α ≈ 1 / 3 and β ≈ 3 / 2. We also establish an exact relationship between the zero-shear viscosity and mean-square radius of gyration for arbitrary Rouse networks, (in physical units), revealing a fundamental connection between molecular architecture and macroscopic rheology. Using this relation, we further uncover an exact relationship between the two scaling exponents: αβ = 1 / 2. Going from single trees to the entire solution, we prove that the relaxation spectrum of the whole solution must be flat near the gelation threshold, with a lower cutoff λ min ∼ (1 − c ) 2 β . The abundance of soft modes leads to the anomalous scaling of the storage and loss moduli: G ′ ( ω ) ∼ ω and G ′′ ( ω ) ∼ ω ln( λ 1 /ω ) in the frequency regime λ min ≪ ω ≪ λ 1 . In particular, the storage modulus is significantly higher in the random crosslinked system than in the uncrosslinked system at low frequency. Remarkably, our theoretical predictions show excellent quantitative agreement with experimental data from diverse biomolecular condensate systems, including binary-mixture condensates, sequence-variant condensates, and aging protein condensates ( Figure 5 ). The model successfully captures the characteristic linear frequency scaling of the storage modulus at low frequency, providing a mechanistic explanation for the enhanced elasticity observed in biomolecular condensates. Our results establish a foundation for understanding the rheology of biomolecular condensates as complex polymer networks and provide predictive capabilities for designing condensates with tailored viscoelastic properties. Download figure Open in new tab FIG. 5. Comparison between theoretical predictions and experimental data. (a) Data from multiple studies: (1)-(4) condensates formed by binary mixture [ 12 ]; (5)-(8) condensates formed by different variants of the intrinsically disordered prion-like low-complexity domain of the RNA-binding protein hnRNPA1 [ 32 ]. (b) condensates formed by protein PGL-3 at different waiting times after sample formation [ 11 ]. The solid lines are the theoretical predictions (blue for G ′ , yellow for G ′′ ) with fixed parameters m = 5 and c = 0.95. Data are normalized by the crossing point where G ′ ( ω 0 ) ≈ G ′′ ( ω 0 ) ≈ G 0 . We thank Huan-Xiang Zhou for discussions related to this work. The research was funded by the National Natural Science Foundation of China (Grant No. 12474190), the National Key Research and Development Program of China (2024YFA0919600), and grants from the Peking-Tsinghua Center for Life Sciences. Funder Information Declared National Natural Science Foundation of China , 12474190 National Key Research and Development Program of China , 2024YFA0919600 Peking-Tsinghua Center for Life Sciences References [1]. ↵ Clifford P Brangwynne , Christian R Eckmann , David S Courson , Agata Rybarska , Carsten Hoege , Jöbin Gharakhani , Frank Jülicher , and Anthony A Hyman . Germline p granules are liquid droplets that localize by controlled dissolution/condensation . 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