Sequence-encoded Spatiotemporal Dependence of Viscoelasticity of Protein Condensates Using Computational Microrheology

Many biomolecular condensates act as viscoelastic complex fluids with distinct cellular functions. Deciphering the viscoelastic behavior of biomolecular condensates can provide insights into their spatiotemporal organization and physiological roles within cells. Though the re is significant interest in defining the role of condensate dynamics and rheology in physiological functions, the quantification of their time-dependent viscoelastic properties is limited and mostly done through experimental rheological methods. Here, we demonstrate that a computational passive probe microrheology technique, coupled with continuum mechanics, can accurately characterize the linear viscoelasticity of condensates formed by intrinsically disordered proteins (IDPs) . Using a transferable coarse-grained protein model, we first provide a physical basis for choosing optimal values that define the attributes of the probe particle, namely its size and interaction strength with the residues in an IDP chain. W e show that the technique captures the sequence- dependent viscoelasticity of heteropolymeric IDPs that differ either in sequence charge patterning or sequence hydrophobicity. We also illustrate the technique’s potential in quantifying the spatial dependence of viscoelasticity in heterogeneous IDP condensates. The computational microrheology technique has important implications for investigating the time-dependent rheology of complex biomolecular architectures, resulting in the sequence-rheology- function relationship for condensates. .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 2

Characterizing the spatiotemporal evolution of biomolecular condensates is crucial for understanding their role in modulating cellular biochemistry 1-3 and how they transform into pathological aggregates.4-6 Liquid-liquid phase separation through multivalent interactions in a protein sequence can drive the formation of these condensates .7-12 Liquid-like (viscous) behavior of these cellular compartments is thought to define their functional landscape, by enabling extreme dynamics13 and efficient transport of biomolecules that can aid in biochemical processes. 14 However, recent investigations have demonstrated the loss of condensate liquidity over time, yielding dominant elastic behavior that may eventually promote the formation of solid fibrillar states.15-17 Thus, it is critical to accurately quantify the viscoelastic spectrum of biomolecular condensates to establish how the protein sequence governs their (dys)functional paradigm.18 Transitions in the material states of protein condensates can occur due to a multitude of factors, e.g., post-translational modifications that alter sequence charge patterning or mutations that alter sequence hydrophobicity.19-21 Such sequence alterations commonly occur in intrinsically disordered proteins (IDPs) or regions (IDRs), which are prevalent in biocondensates, thereby causing a speed-up or slowdown in their dynamics. Deciphering the sequence-encoded molecular interactions of IDPs that dictate prominent dynamical changes in conjunction with the measurements of viscoelasticity are essential for establishing the sequence-rheology-function relationship of condensate biology. S imulations can serve as a computational lens into the molecular interactions of condensates and a tool for accurately measuring their rheolog ical properties. For example, the viscosity and viscoelasticity of IDP condensates can be quantified using equilibrium molecular dynamics (MD) simulations along with the Green-Kubo (GK) relation and nonequilibrium MD simulations.22-26 While these techniques are useful for measuring the bulk rheology of single -component systems, they suffer from the inability to capture the spatial variations in viscoelasticity seen in multi-component in vitro and in vivo condensates. Knowledge of the spatial dependence of viscoelastic properties within the condensates can provide important insights for studying how the partitioning and transport of small drug molecules into condensates depends on their local environment.27 Particle tracking experimental microrheology is widely used for investigating the time- dependent bulk viscoelasticity and viscosity of in vitro phase -separated droplets displaying .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 3 different material characteristics ranging from liquid-like to solid-like behaviors.28-33 It is a highly sought-after method because it requires only small volumes of biological samples and enabl es a label-free approach where the biomolecules need not be fluorescently tagged.34, 35 In addition, the technique has the potential to quantify the local viscoelasticity of heterogen eous condensate systems.34 The technique relies on connecting the probe particle motion in a complex fluid system and its microscopic viscoelastic properties through continuum mechanics. 36, 37 A computational analogue of the technique via MD simulations has been shown to yield quantitatively accurate viscoelastic modulus for the homopolymer and colloidal systems,38-41 but a rigorous implementation of it remain s untested for the heteropolymeric protein condensates. The success of the computational microrheology technique relies on carefully choosing the parameters for the probe particle, namely its size and interactions with the medium of interest, such that they follow continuum mechanics assumptions. This is also a primary concern in experimental microrheology where the nonspecific interactions between probe beads and protein molecules need to be prevented for reliable measurements.35 Establishing a physical basis for choosing the attributes of the probe is important because of the unexplored questions regarding the technique’s capability in two following aspects: ( 1) capturing the sequence-dependent viscoelasticity of condensates formed by heteropolymeric IDPs and (2) quantifying the spatial variations in viscoelasticity found in heterogeneous condensates formed by a pair of heteropolymeric IDPs. To this end, we systematically demonstrate in this work that computational microrheology is a robust technique for studying the time-dependent rheological properties of protein condensates.

To investigate the s equence-dependent viscoelasticity of IDP condensates using computational microrheology, we employed coarse-grained (CG) model sequences formed by an equal number of negatively charged glutamic acid (E) and positively charged lysine (K) with chain length 𝑁𝑁= 250 residues. Recently, we used similar IDP sequences to establish the sequence - dependent material properties, namely diffusion coefficient 𝐷𝐷 and viscosity 𝜂𝜂 of condensates formed by charge-rich model and naturally occurring IDPs.24 We chose three E−K sequences that varied in the sequence charge patterning, with a high degree of charge segregation quantified using a high value of normalized sequence charge decoration (nSCD ∈ [0,1]) parameter42-44 (see Table 𝐒𝐒𝟏𝟏 for the amino acid sequences). Specifically, we used variants with nSCD values of 0.067, .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 4 0.468, and 1.000, respectively. We performed computational passive probe microrheology via MD simulations in a cubic simulation box at the preferred dense phase concentrations of the E−K variants (see Methods). We found that the dense phase concentration 𝜌𝜌 increased with increasing degree of charge segregation (i.e., increasing nSCD), highlighting stronger effective interactions between the oppositely charged residues (Fig. 𝐒𝐒𝟏𝟏). A probe particle of bare mass 𝑚𝑚bare, modeled as a rough sphere for ensuring no-slip boundary conditions,45 was dispersed in the dense phase of the E−K variants (Fig. 𝟏𝟏𝐚𝐚). All the simulations were carried out via a transferable hydropathy scale (HPS) model 46, 47 at a constant temperature of 𝑇𝑇= 300 K (see Methods for model and simulation details). The friction encountered by the probe particle during its Brownian motion can be related to the linear viscoelastic properties (elastic 𝐺𝐺′ and viscous 𝐺𝐺′′ modulus) of the dense phase of a condensate using the inertial generalized Stokes-Einstein relation (IGSER)48, 49 𝐺𝐺∗(𝜔𝜔) = 𝑖𝑖𝑖𝑖𝑍𝑍∗(𝑖𝑖) 6𝜋𝜋𝑅𝑅h + 𝑚𝑚eff𝑖𝑖2 6𝜋𝜋𝑅𝑅h + 𝑅𝑅h2𝑖𝑖2 2 �� 𝜌𝜌2 + 2𝜌𝜌 3𝜋𝜋𝑅𝑅h3 � 𝑍𝑍∗(𝑖𝑖) 𝑖𝑖𝑖𝑖− 𝑚𝑚eff� − 𝜌𝜌�, (1) where 𝐺𝐺∗ = 𝐺𝐺′ + 𝑖𝑖𝐺𝐺′′ is the dynamic modulus of the medium , 𝑍𝑍∗ is the frequency -dependent friction experienced by the probe through its interactions with the medium, 𝑅𝑅h is the hydrodynamic radius of the probe, and 𝑚𝑚eff is the effective mass of the probe particle. As per continuum mechanics, 𝑚𝑚eff = 𝑚𝑚bare + 𝑚𝑚add, where 𝑚𝑚add = 2 3 𝜋𝜋𝑅𝑅h 3𝜌𝜌 is the added mass from the medium.50 For IGSER to accurately capture the viscoelastic properties of IDP condensates, two important characteristics of the probe emerge, namely its size (i.e., 𝑅𝑅h) relative to the relevant length scale of the IDP dense phase and its interaction s with the IDP chains. This prompted us to do a systematic assessment of the attributes of the probe particle for the successful implementation of computational microrheology for protein condensates. Probe particle s ize is a critical determinant for the computational microrheology of IDP condensates Given that the length of our model E−K sequences is representative of naturally occurring IDPs, we first asked whether there is an appropriate size of the probe particle, which can be rationalized in terms of the relevant length scale such as the mesh size within IDP condensates .51 To investigate this aspect, we varied the probe sizes, ranging from bare radius 𝑅𝑅b = 2.5 Å (similar .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 5 to the radius of the residue beads in the HPS model) to 𝑅𝑅b = 20 Å (similar to the single -chain radius of gyration 𝑅𝑅𝑔𝑔 of the investigated E −K variants). For our continuum mechanics analysis, we used the hydrodynamic radius 𝑅𝑅h of the probe particle, defined as the location of the first peak in the radial distribution function (RDF) between the probe and the protein residues within the dense phase of the condensates (Fig. 𝐒𝐒𝟐𝟐). This choice of definition for 𝑅𝑅h was previously shown to recover the Stokes frictional force and torque for the rough spherical particle moving through a complex fluid as well as yield accurate 𝐺𝐺′ and 𝐺𝐺′′ values of homopolymers with increasing 𝑁𝑁.38, 39, 45 For all sizes of the probe, its interaction with the protein residues was modeled via a modified Lennard-Jones potential with interaction strength 𝜀𝜀HPS = 0.2 kcal/mol, the strongest possible van der Waals interaction between a pair of residue beads in our HPS model. In our passive rheology simulations, w e tracked the Brownian displacement of the center of mass of the probe of different sizes in the dense phase of the E −K sequences (insets of Figs. 𝟏𝟏𝐛𝐛, 𝐒𝐒𝐒𝐒𝐚𝐚, and 𝐒𝐒𝐒𝐒𝐚𝐚). U sing the displacement data, w e then computed the probe ’s mean square displacement MS D(𝑡𝑡) = 〈[𝒓𝒓(𝑡𝑡) − 𝒓𝒓(0)]2〉, (2) wh ere 𝒓𝒓 is its position at time 𝑡𝑡 (Figs. 𝟏𝟏𝐛𝐛, 𝐒𝐒𝐒𝐒𝐚𝐚, and 𝐒𝐒𝐒𝐒𝐚𝐚). W e found that probes of all sizes 𝑅𝑅h exhibited a ballistic motion (MSD ∝ 𝑡𝑡2) at short times, but only the smallest probe ( 𝑅𝑅h = 6.2 Å) showed a diffusive motion ( MSD ∝ 𝑡𝑡) at long times. We observed a sub -diffusive behavior at intermediate times, which became increasingly prominent until long times with increasing 𝑅𝑅h, indicating that larger probes move slower in the condensate within the simulation duration of the E−K sequences investigated. Further, MSD at intermediate to long times drastically decreased for smaller 𝑅𝑅h, followed by a gradual decrease for larger 𝑅𝑅h, highlighting high friction exerted on large probe particles. In fact, the friction 𝑍𝑍∗ that appears in IGSER can be related to the MSD of the probe as 𝑍𝑍∗(𝜔𝜔) = 2𝑑𝑑𝑘𝑘B𝑇𝑇 (𝑖𝑖𝑖𝑖)2MSD(𝑖𝑖), (3) where 𝑑𝑑 is the number of dimensions in which the particle is tracked, 𝑘𝑘B is the Boltzmann constant and MSD(𝜔𝜔) is the one -sided Fourier-transformed MSD. For obtaining MSD in the frequency domain, we used a previously developed analytical fitting procedure by expressing MSD in terms .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 6 Fig. 𝟏𝟏. (a) Simulation snapshot of the dense phase of a condensate formed by a select E−K sequence (nSCD = 1) of chain length 𝑁𝑁= 250 with a spherical probe particle (magenta) of hydrodynamic radius 𝑅𝑅h embedded in it. (b) Mean square displacement MSD(𝑡𝑡) of the center of mass of the probe particle for different 𝑅𝑅h in the dense phase of the E −K sequence with nSCD = 1. The dashed lines are the fits based on the Baumgaertel-Schausberger-Winter-like power law spectrum to the MSD data. The inset shows probe’s displacement in the three -dimensional Cartesian coordinates for select 𝑅𝑅h values. (c) Elastic 𝐺𝐺′ (dashed line) and viscous 𝐺𝐺′′ (solid line) modulus for the dense phase of the E−K sequence with nSCD = 1 as a function of 𝑅𝑅h, with circles corresponding to the crossover frequency. The inset shows the relaxation time 𝜏𝜏, computed as the inverse of crossover frequency, with varying 𝑅𝑅h. (d) Viscosity 𝜂𝜂, normalized by that obtained based on the Green-Kubo (GK) relation 𝜂𝜂GK, as a function of normalized probe particle size 𝑅𝑅h/𝜉𝜉, where 𝜉𝜉 is the correlation length, for three different nSCD sequences. The two shaded regions delineate regions where 𝑅𝑅h is smaller or larger than 𝜉𝜉. The black dashed line corresponds to 𝜂𝜂= 𝜂𝜂GK. .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 7 of a Baumgaertel -Schausberger-Winter-like power law spectrum (see Methods; Figs. 𝟏𝟏𝐛𝐛, 𝐒𝐒𝐒𝐒 𝐚𝐚, and 𝐒𝐒𝐒𝐒 𝐚𝐚).39, 52 This method is usually superior to the use of an approximate Fourier transform expression used for the polynomial fit to MSD (𝑡𝑡) in experimental microrheology,37 which often leads to poor estimates when the slope of MSD varies rapidly. Ne xt, we used IGSER (Eqs. 1 and 3) to obtain the elastic 𝐺𝐺′ and viscous 𝐺𝐺′′ modulus of the dense phase of E −K sequences (Figs. 𝟏𝟏𝐜𝐜, 𝐒𝐒𝐒𝐒 𝐛𝐛, an d 𝐒𝐒𝐒𝐒 𝐛𝐛). Consistent with the trends in the probe motion, we found that 𝐺𝐺′ and 𝐺𝐺′′ were significantly lower as well as the terminal regime (i.e., 𝐺𝐺′ ∝ 𝜔𝜔2 and 𝐺𝐺′′ ∝ 𝜔𝜔1) was observed at high frequencies for smaller probe particles (i.e., 𝑅𝑅h ≤ 8.2 Å) as compared to the larger ones for all E−K sequences. For 𝑅𝑅h > 8.2 Å, the onset of terminal viscous characteristics occurred at similar frequency values, as indicated by similar crossover frequencies below which 𝐺𝐺′′ > 𝐺𝐺′. This was also evident by further looking at the similar relaxation times 𝜏𝜏, computed as the inverse of crossover frequency, for larger probe particles in the dense phase of E −K variants (insets of Figs. 𝟏𝟏𝐜𝐜, 𝐒𝐒𝐒𝐒 𝐛𝐛, an d 𝐒𝐒𝐒𝐒 𝐛𝐛). Our findings indicate that for probe sizes beyond a certain threshold value (i.e., 𝑅𝑅h > 8.2 Å), the probe motion yields a viscoelastic spectrum of the dense phase of E −K sequences that exhibit similar characteristics (i.e., similar 𝜏𝜏). G iven that the motion of the probe is intricately tied to the dense environment of the IDP condensates, we next investigated whether the similar viscoelastic modulus yielded based on the probe motion of certain sizes is because of them being larger than the mesh size53 within the dense phase of E−K sequences. Note that IGSER requires the probe particle to be large enough to see the medium as a continuum. Following previous work, 51 we estimated the correlation length 𝜉𝜉 from the overlap concentration 𝜌𝜌∗, which is related to the polymer mesh size in a dense phase system. We found the values of 𝜉𝜉 to be in the range of 5.24 Å to 8.32 Å for the E−K sequences. We also computed the viscosity of the dense phase of E −K sequences from the microrheology simulations as 𝜂𝜂= 𝐺𝐺′′/𝜔𝜔 in the terminal region as well as from the GK relation that yields macroscopic viscosity 𝜂𝜂GK (see Methods). When we normalized 𝜂𝜂 by 𝜂𝜂GK, we found that probes larger than the correlation length (i.e., 𝑅𝑅h/𝜉𝜉 ≳1.5) yielded macroscopic viscosity (Fig. 𝟏𝟏𝟏𝟏). This finding highlights that probes larger than 𝜉𝜉 feel the ‘true’ macroscopic friction present within the dense phase. However, given that very large probes sample the dense phase of the condensates much slower (insets of Figs. 𝟏𝟏𝐛𝐛, 𝐒𝐒𝐒𝐒 𝐚𝐚, an d 𝐒𝐒𝐒𝐒 𝐚𝐚) , we concluded that the smallest probe size that .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 8 Fig. 𝟐𝟐. (a) Velocity 𝑣𝑣𝑥𝑥 profile of the residues of protein chains with nSCD = 1 around the probe particle translating with a velocity 𝑣𝑣𝑥𝑥,probe for differen t probe-protein interaction strengths 𝜀𝜀, normalized by the strongest possible interaction strength 𝜀𝜀= 0.20 kcal/mol in the HPS model. (b) Mean square displacement MSD(𝑡𝑡) of the center of mass of the probe particle for different 𝜀𝜀/𝜀𝜀HPS in the dense phase of the E −K sequence with nSCD = 1. The dashed lines are the fits based on Baumgaertel-Schausberger-Winter-like power law spectrum to the MSD data. The inset shows probe’s displacement in the three -dimensional Cartesian coordinates for select 𝜀𝜀/𝜀𝜀HPS values. (c) Elastic 𝐺𝐺′ (dashed line) and viscous 𝐺𝐺′′ (solid line) modulus for the dense phase of the E−K sequence with nSCD = 1 as a function of 𝜀𝜀/𝜀𝜀HPS, with circles corresponding to the crossover frequency. The inset shows the relaxation time 𝜏𝜏 with varying 𝜀𝜀/𝜀𝜀HPS. (d) Viscosity 𝜂𝜂, normalized by that obtained based on the Green-Kubo (GK) relation 𝜂𝜂GK as a function of 𝜀𝜀/𝜀𝜀HPS for three different nSCD sequences. The black dashed line corresponds to 𝜂𝜂= 𝜂𝜂GK. .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 9 satisfied the criteria 𝑅𝑅h/𝜉𝜉 ≈1.5 (i.e., 𝑅𝑅h = 10.6 Å) would be a computationally efficient choice to obtain reliable viscoelastic measurements of any IDP condensates investigated using the HPS model and other analogous CG models.54-57 Confluence of probe particle size and its interaction strength with IDPs correctly captures condensate rheology T he friction experienced by the probe particle depends on its interaction with the IDP chains constituting the condensates. Having established a suitable probe size for computational microrheology, we next investigated whether the strength of probe -protein interactions ha d a significant effect on the viscoelastic modulus of the dense phase of IDP condensates. An important requirement imposed by IGSER is the need for a no-slip boundary condition at the probe particle surface.39, 40 To identify the interaction strengths 𝜀𝜀 that would satisfy the condition, we varied it in the range of 𝜀𝜀/𝜀𝜀HPS = 0 (purely repulsive interactions) to 𝜀𝜀/𝜀𝜀HPS = 4 for a probe particle size of 𝑅𝑅h = 10.6 Å. We then computed the velocity 𝑣𝑣𝑥𝑥 profile of the protein residues around the probe particle that was moving at a pre- defined translational velocity 𝑣𝑣𝑥𝑥,probe (Fig. 𝟐𝟐 𝐚𝐚) . We found that the velocity of the protein residues adjacent to the probe surface became increasingly similar to that of 𝑣𝑣𝑥𝑥,probe with increasing 𝜀𝜀/𝜀𝜀HPS, indicating that sufficient attractive interactions prevent a slip at its surface. In line with this observation, we observed that only probe particles with 𝜀𝜀/𝜀𝜀HPS < 0.75 exhibited very high displacements as well as a diffusive behavior within the course of the simulations of E−K sequences (Figs. 𝟐𝟐 𝐛𝐛, 𝐒𝐒𝐒𝐒 𝐚𝐚, and 𝐒𝐒𝐒𝐒 𝐚𝐚) . To quantify the variations in the probe’s MSD for different 𝜀𝜀/𝜀𝜀HPS in terms in viscoelasticity, we measured 𝐺𝐺′ and 𝐺𝐺′′ of the dense phase of E−K variants via IGSER (Figs. 𝟐𝟐 𝐜𝐜, 𝐒𝐒𝐒𝐒 𝐛𝐛, and 𝐒𝐒𝐒𝐒 𝐛𝐛). Fo r probes with 𝜀𝜀/𝜀𝜀HPS < 0.75, we found that 𝐺𝐺′ and 𝐺𝐺′′ were significantly lower and did not show a crossover between them as compared to the higher 𝜀𝜀/𝜀𝜀HPS values. When sufficient attractive interactions ( 𝜀𝜀/𝜀𝜀HPS ≥ 0.75) exist between the probe and the protein residues , similar relaxation timescales delineating dominant viscous and elastic behaviors were observed irrespective of 𝜀𝜀/𝜀𝜀HPS for all sequences (Figs. 𝟐𝟐 𝐜𝐜, 𝐒𝐒𝐒𝐒 𝐛𝐛, an d 𝐒𝐒𝐒𝐒 𝐛𝐛). H owever, for 𝜀𝜀/𝜀𝜀HPS = 4, the modulus values were much higher as compared to 𝜀𝜀/𝜀𝜀HPS = 1. This is further highlighted in the normalized viscosity 𝜂𝜂/𝜂𝜂GK values, which revealed an optimal window for interaction strengths (1 ≤ 𝜀𝜀/𝜀𝜀HPS ≤ 1.5) that yielded the macroscopic viscosity ( Figs. 𝟐𝟐 𝟏𝟏). N ote that, for 𝜀𝜀/𝜀𝜀HPS ≳ .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 10 1.75, we observed the IDP chains getting strongly adsorbed to the probe particle surface (Fig. 𝐒𝐒𝐒𝐒). This is a common concern in experimental microrheology as well, in which, for example, polystyrene beads are often passivated with polyethylene glycol, to ensure negligible chemical interactions with the biomolecules as well as to prevent the beads from getting constrained within the droplet.35, 58, 59 From these findings, we concluded that an attractive strength (𝜀𝜀/𝜀𝜀HPS = 1) that is low enough, but a value that resides in the optimal window would be ideal for correctly capturing the time-dependent rheology of IDP condensates computationally. Computational microrheology reveals the sequence -encoded time-dependent viscoelasticity of heteropolymeric IDP condensates T hrough the continuum analysis of the motion of a single probe particle in our computational microrheology simulations, we have identified suitable probe particle size and its interaction strength with the IDP chains that would yield accurate macroscopic dense phase viscosities of the E−K sequences. However, the microrheology experiments are often performed with multiple probe particle beads within the same droplet, and the average motion of all particles is used for quantifying the condensate viscoelasticity.13, 28 This approach is efficient as it reduces the uncertainties in the viscoelastic measurements, which may arise from tracking only a single particle. Though the continuum mechanics expressions are for a single particle, which necessitates that no hydrodynamic interactions exist between the probe particles, we next asked the extent to which multiple probes in an IDP dense phase system would influence their motion, which can alter the corresponding viscoelastic measurements. For this purpose, we performed the passive rheology simulation with 𝑛𝑛= 2 to 48 probes in the dense phase of E−K sequences (Fig. 𝐒𝐒𝐒𝐒). Surprisingly, we found that the probe’s average MSD was nearly the same at all times with varying 𝑛𝑛, but the statistical noise in the MSD profile has significantly reduced for systems with multiple particles. To demonstrate the ability of passive rheology simulations to capture the changes in viscoelasticity with increasing nSCD, we again obtained the viscoelastic modulus 𝐺𝐺 ′ and 𝐺𝐺′′ of the dense phase of E−K sequences based on the average MSD of 𝑛𝑛= 8 probes ( Fig. 𝐒𝐒𝐚𝐚). The modulus values obtained from the microrheology simulations using multiple probes are in quantitative agreement with those obtained based on a single probe particle (Fig. 𝐒𝐒𝐒𝐒). Further, we also observed a semi- quantitative agreement, yet similar trends in the modulus values obtained from the microrheology simulations and based on the Green -Kubo relation for all E −K sequences, even though the latter .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 11

suffers from noise at long times (see Methods; Figs. 𝐒𝐒 𝟏𝟏 𝐒𝐒 and 𝐒𝐒 𝟏𝟏𝟏𝟏). O ur microrheology simulations revealed that the modulus 𝐺𝐺′ and 𝐺𝐺′′, relaxation time 𝜏𝜏, and viscosity 𝜂𝜂 were higher with increasing nSCD, highlighting that charge segregation slows down dynamics , thereby resulting in longer timescales for displaying the terminal flow behavior (Fig. 𝐒𝐒𝐚𝐚). Finally, we also tracked the probe motion in the dense phase of E−K sequences of 𝑁𝑁= 50, but with the same nSCD values as those used for 𝑁𝑁= 250 (Fig. 𝐒𝐒 𝟏𝟏𝟐𝟐𝐚𝐚) . We found that 𝐺𝐺′ and 𝐺𝐺′′ did not show a cross-over (i.e., no dominant elastic response in the entire frequency space) for the shorter sequences, indicating negligible entanglement effects between the chains in these systems (Fig. 𝐒𝐒 𝟏𝟏𝟐𝟐𝐛𝐛). O ur findings indicate that condensates formed by longer chains exhibit Maxwell fluid-like behavior over the entire frequency range investigated, which is in agreement with recent experimental28 and computational studies.60, 61 T he dynamics and rheology of IDP condensates can be modulated not only through sequence charge patterning but also via alterations to sequence hydrophobicity. We next investigated the ability of the passive rheology technique coupled with our CG model to capture Fig. 𝐒𝐒. (a) Elastic 𝐺𝐺′ (dashed line) and viscous 𝐺𝐺′′ (solid line) modulus for three E−K sequences with different nSCD, obtained based on multiple (𝑛𝑛= 8) probes in each dense phase system. (b) Elastic 𝐺𝐺′ (dashed line) and viscous 𝐺𝐺′′ (solid line) modulus for A1-LCD WT and its three variants, obtained based on multiple (𝑛𝑛= 6) probes in each dense phase system. The circles in (a) and (b) correspond to the sequence-specific crossover frequencies. The insets in (a) and (b) shows the changes in relaxation time 𝜏𝜏 and viscosity 𝜂𝜂 with changing nSCD and with mutational changes in the A1-LCD WT sequence, respectively. .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 12 the sequence alterations affecting the hydrophobic character. To this end, we tracked multiple (𝑛𝑛= 6) probe particles in a naturally occurring IDP A1 -LCD wildtype (WT) sequence and three of its variants, each with 𝑁𝑁= 137, with different aromatic residue (tyrosine Y, tryptophan W, phenylalanine F) identities, namely allY, allW, and allF (see Table 𝐒𝐒𝟐𝟐 for the amino acid sequences). Again, we ensured that the average probe MSD computed based on 𝑛𝑛= 6 particles was similar to the MSD of a single particle in these systems , except for long times where the statistics for the MSD based on multiple probes w ere better as compared to a single probe MSD (Fig. 𝐒𝐒 𝟏𝟏 𝐒𝐒). The values of 𝐺𝐺′ and 𝐺𝐺′′, 𝜏𝜏, and 𝜂𝜂 obtained using IGSER increased in the following order: allW > allY ≈ WT > allF (Fig. 𝐒𝐒𝐛𝐛). T his finding is in agreement with the recent experimental microrheology measurements on the same set of IDP sequences. 62 Taken together, we concluded that the computational microrheology technique coupled with continuum mechanics and CG models can accurately quantify the sequence- encoded time-dependent viscoelasticity of IDP condensates. Com putational microrheology unmasks the spatial variations in the viscoelasticity of heterogeneous IDP condensates I ntracellular biomolecular condensates are multi -component in nature, often exhibiting complex molecular architectures with spatial heterogeneities. 63 We next asked whether computational microrheology can be used to quantify the spatial variations in the viscoelasticity of heterogeneous condensates. It is known that IDPs with large differences in nSCD values demix in such a way that a highly charge-segregated sequence forms the condensate core and the well- mixed sequence forms a shell around the core.64 Specifically, we formed such a condensate using a pair of E−K sequences with nSCD = 0.067 and nSCD = 1.000, respectively (Fig. 𝐒𝐒𝐚𝐚). A lso, the maximum densities of the two E−K sequences in the heterogeneous mixture were the same as that we observed in the bulk condensates simulated in a cubic geometry (Fig. 𝐒𝐒𝐚𝐚). To characterize the local viscoelasticity within this heterogeneous condensate, we spatially restrained 𝑛𝑛 = 3 probes along the 𝑧𝑧 direction encompassing distinct local environments: one among nSCD= 1.000 chains, one among nSCD = 0.067 chains, and one at the interface of the two nSCD sequences (see Methods). We found that the two-dimensional probe particle displacements were different in each of the three regions at intermediate to long times, with the probe’s among nSCD = 0.067 chains and nSCD = 1.000 chains exhibiting the highest and lowest mobilities, respectively (Fig. .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 13 Fig. 𝐒𝐒. (a) Simulation snapshot of a heterogeneous condensate formed by two different E−K sequences with nSCD = 0.067 and nSCD = 1.000. Three probe particles are spatially restrained in three regions of the heterogeneous condensate: one among nSCD = 1.000 chains, one among nSCD = 0.067 chains, and one at the interface of the two nSCD sequences. Also shown are the density profiles of the heterogeneous condensate. (b) Mean square displacement MSD(𝑡𝑡) of the probe particle in three different regions of the heterogeneous condensate. The dashed lines are the fits based on Baumgaertel-Schausberger-Winter-like power law spectrum to the MSD data. The inset shows probe’s displacement in the two-dimensional Cartesian coordinates in the chosen regions within the condensate. (c) Elastic 𝐺𝐺′ (open symbols) and viscous 𝐺𝐺′′ (closed symbols) modulus sampled from three different regions of the heterogeneous condensate , with circles corresponding to the location - specific crossover frequency. The inset shows the relaxation time 𝜏𝜏 in the chosen regions within the condensate. (d) Viscosity 𝜂𝜂 in three regions of the heterogeneous condensate compared with those obtained from the bulk simulations of the E−K sequences. .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 14 𝐒𝐒𝐛𝐛) . This was evident in the elastic 𝐺𝐺′ and viscous 𝐺𝐺′′ modulus obtained by using the two- dimensional probe particle displacement information in IGSER (Fig. 𝐒𝐒𝐜𝐜) . Specifically, the values of modulus and the corresponding relaxation times 𝜏𝜏 were lower for the shell region around the condensate core as compared to the core itself, while the values at the interface between the core and the shell regions displayed intermediate viscoelasticity. Interestingly, we also found that viscosity 𝜂𝜂 values at the condensate core and shell regions, obtained based on 𝐺𝐺′′ in the terminal region, were quantitatively similar to that obtained from the bulk condensates ( Fig. 𝐒𝐒𝟏𝟏). T hese observations further highlight the potential of computational microrheology for revealing the location-dependent viscoelasticity of complex heterogeneous condensates, which can gain insights about the small molecules that partition into such environments27 as well as help define the role of condensate interfaces that have been found to exhibit distinct conformational characteristics as compared to the core of the condensates.65, 66

Knowledge of the molecular interactions that govern the viscoelastic transitions would allow for establishing how protein sequence dictates the spatiotemporal evolution of condensates. Using the HPS model , we demonstrate that a computational passive probe microrheology technique, in which the probe particle motion is analyzed via continuum mechanics, can accurately quantify the sequence-dependent viscoelasticity of IDP condensates. We do so by first rationalizing our choice for the two important attributes of the probe particle, namely its size and interaction strength with the IDPs. We found that a probe with a hydrodynamic radius 𝑅𝑅h being slightly greater than the correlation length 𝜉𝜉 within the condensates (𝑅𝑅h ≈ 1.5𝜉𝜉) and its interactions with the IDPs being optimally strong that it prevents slip (𝜀𝜀/𝜀𝜀HPS = 1) is sufficient to accurately quantify the viscosity and viscoelasticity of IDP condensates. By tracking the motion of the probe particle and converting its displacement information into viscoelastic modulus using IGSER, we found that the measurements from microrheology simulations are in quantitative agreement with those obtained based on the Green -Kubo relation. Further, the elastic and loss modulus, relaxation time, and terminal viscosity increased with increasing degree of charge segregation , exhibiting Maxwell fluidic nature for the E −K sequences with identical sequence composition. This observation highlighted that computational microrheology captures the time-dependent rheological transitions with sequence alterations that result in pronounced electrostatic interactions between the .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 15 oppositely charged residues. Further, we have show n that the microrheology simulations can capture changes in viscoelasticity with changes in the composition of aromatic residues in the naturally occurring A1-LCD WT sequence through mutations. Taken together, we conclude that computational microrheology is a robust technique for characterizing the sequence-encoded time- dependent viscoelasticity of heteropolymeric protein condensates. Microrheology experiments are often performed with multiple probe particles within the same droplet and the average displacement of the particle is used to characterize condensate viscoelasticity. We also showed that microrheology simulations performed with multiple probe particles yield accurate viscoelastic modulus values of the dense phase of IDP sequences investigated in this work. This suggests that a single simulation with multiple probes can lead to improved statistical accuracy of the probe’s displacement at long times. Consequently, condensate viscoelasticity can be sampled based on a single simulation trajectory as opposed to other conventional techniques such as the MD simulations along with the Green -Kubo (GK) relation and the NEMD simulations . In the method involving G K relation , the values of shear stress relaxation modulus, used for obtaining the viscoelastic modulus, are typically prone to noise at long times because of which very long MD simulations are required.22 In the NEMD method , viscoelastic modulus can only be obtained by applying an oscillatory shear strain on the system at a specific frequency and is prohibitive of accessing low frequency (long time) viscoelasticity, rendering it computationally expensive.26, 38 Further, the use of multiple probe particles can allow for the accurate characterization of viscoelasticity in biomolecular condensates with inherent heterogeneities due to physical aging. 62, 67 Finally, we spatially restrained multiple probe particles at distinct locations within a heterogeneous IDP condensate formed by a pair of E−K sequences with vastly different charge patterning for quantifying local viscoelasticity. By tracking each of the particle’s displacements in unrestrained directions, we demonstrated the technique’s ability to accurately quantify the viscoelasticity that depends on the local environment within such condensates. This finding highlights that passive rheology can be used for accurately sampling the spatial variations in viscoelasticity that are prevalent in intracellular condensates . Knowledge of the spatial dependence of viscoelasticity can provide important insights for designing drug- delivery nanoparticles with targeted partitioning and transport within condensates. 68 We believe that the computational microrheology technique can be an easy-to-use technique for establishing the sequence determinants of condensate viscoelasticity for a wide range of protein condensates. .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 16

Hydropathy scale (HPS) model We used our transferable CG model based on the hydropathy scale to computationally investigate the IDP sequences.46, 47 This CG framework has been widely used in deciphering the sequence-dependent conformations and phase separation of a wide range of IDPs.44, 54, 69-74 In this framework, we represent the IDPs as fully flexible polymeric chains with a single bead per residue representation. Interactions between bonded residues occurred via the harmonic potential 𝑈𝑈b(𝑟𝑟) = 𝑘𝑘b 2 (𝑟𝑟 − 𝑟𝑟0)2, (4) with distance 𝑟𝑟 be tween residues, spring constant , and equilibrium bond length set to 𝑘𝑘b = 20 kcal �mol Å2�⁄ and 𝑟𝑟0 = 3.8 Å, respectively . The van der Waals interactions between nonbonded residues i and j were modeled using the modified Lennard- Jones (LJ) potential 75, 76 based on the average hydropathy 𝜆𝜆= �𝜆𝜆𝑖𝑖+ 𝜆𝜆𝑗𝑗�/2 𝑈𝑈vdW(𝑟𝑟) = �𝑈𝑈LJ(𝑟𝑟) + (1 − 𝜆𝜆)𝜀𝜀, 𝑟𝑟 ≤21 6⁄ 𝜎𝜎 𝜆𝜆𝑈𝑈LJ(𝑟𝑟), otherwise , (5) where 𝑈𝑈LJ is the standard LJ potential 𝑈𝑈LJ(𝑟𝑟) = 4𝜀𝜀 �� 𝜎𝜎 𝑟𝑟� 12 − � 𝜎𝜎 𝑟𝑟� 6 �. (6) The parameters of LJ potential are the average diameter 𝜎𝜎= � 𝜎𝜎𝑖𝑖+ 𝜎𝜎𝑗𝑗�/2, and the interaction strength 𝜀𝜀= 𝜀𝜀HPS = 0.2 kcal mol⁄ . We used 𝜆𝜆 values based on the Kapcha-Rossky scale77 for the E−K sequence variants and those based on the Urry scale 78 for the A 1-LCD sequence variants. We used the Urry scale for A 1-LCD as it is known to capture the changes in the phase behavior of natural proteins upon mutations of arginine to lysine and tyrosine to phenylalanine.47 The values of 𝑈𝑈vdW and its forces were truncated to zero at a distance of 4 𝜎𝜎. Finally, the nonbonded charged residues interacted through a Coulombic potential with Debye-Hü ckel electrostatic screening79 𝑈𝑈e(𝑟𝑟) = 𝑞𝑞𝑖𝑖𝑞𝑞𝑗𝑗 4𝜋𝜋𝜖𝜖r𝜖𝜖0𝑟𝑟𝑒𝑒−𝑟𝑟/ℓ, (7) with vacuum permittivity 𝜖𝜖0, relative permittivity 𝜖𝜖r = 80, and Debye screening length ℓ = .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 17 10 Å. The choices for 𝜖𝜖r and ℓ were made to represent an aqueous solution with a physiological salt concentration of ~100 mM. The values of 𝑈𝑈e and its forces were truncated to zero at a distance of 35 Å. Model of the probe particle and its interactions with IDPs We modeled the probe particle by carving out a spherical region from a face-centered cubic (FCC) crystal lattice structure of the LJ beads (𝜎𝜎= 1.5 Å and mass 𝑚𝑚= 100 g/mol), with a lattice spacing (i.e. distance between corner atoms) of 2.12 Å. This value was chosen because it ensures that the corner and the face atoms in the FCC lattice are at a distance of 1.5 Å (i.e., just touching each other ). The spherical shape of the probe particle was maintained by connecting the neighboring LJ particles , constituting the corner and the face atoms , using stiff harmonic bonds with a spring constant 𝑘𝑘b = 250 kcal �mol Å2�⁄ . We ensured that 𝑅𝑅g of the probe particle was nearly identical to the expected value for a spherical particle 𝑅𝑅g = � 3𝑅𝑅b 2/5 as well as its relative shape anisotropy 𝜅𝜅2 was nearly zero (𝜅𝜅2 = 0 for a sphere) during the simulations (Fig. 𝐒𝐒 𝟏𝟏 𝐒𝐒).53 The probe particle beads interacted with all protein residues in an IDP chain via the modified LJ potential 𝑈𝑈vdW (Eqs. 5 and 6), in which the values of 𝜆𝜆 were varied between the values of 0 and 4 to control the interaction strength 𝜀𝜀. Specifically, the values of 𝜀𝜀 were in the range of 0 kcal/mol to 0.8 kcal/mol. Again, we truncated 𝑈𝑈vdW and its forces were to zero at a distance of 4 𝜎𝜎. We note that the interactions between the beads constituting the probe particle were modeled using a purely repulsive potential, which corresponds to 𝜆𝜆= 0 in Eq. 5. Microrheology simulation details We simulated the IDP sequences in a cubic simulation box at a constant pressure of 𝑃𝑃= 0 atm for a duration of 0.2 𝜇𝜇𝜇𝜇. At the end of this equilibration run, the IDPs reached their preferred sequence-dependent dense phase concentration 𝜌𝜌. We then performed the Langevin dynamics (LD) simulations in the canonical ensemble for a total duration of 1 𝜇𝜇𝜇𝜇. For these simulations , a damping factor of 𝑡𝑡damp = 1 ns was used to set the friction coefficient of a residue in the chain as well as a bead constituting the probe particle to 𝑓𝑓= 𝑚𝑚 𝑡𝑡damp⁄ . For characterizing the spatial dependence of viscoelasticity, we performed LD simulations .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 18 of a heterogeneous condensate formed by a pair of E −K sequences in a slab geometry (225 Å × 225 Å × 1687.5 Å) for a duration of 1 𝜇𝜇𝜇𝜇. The friction coefficient for the IDP residues and the probe particle beads were the same as those used in the bulk dense phase simulations. Three probe particles were restrained via a harmonic potential with 𝑘𝑘b = 20 kcal �mol Å2�⁄ at specific locations along the 𝑧𝑧 direction of the heterogeneous condensate via the restrain functionality (i.e., restrain.plane) within azplugins.80 For comparison with the microrheology simulations, we performed equilibrium MD simulations of the E −K sequence variants in the absence of a probe particle to compute their viscosity and viscoelasticity using the Green-Kubo relation 𝜂𝜂= ∫ 𝐺𝐺(𝑡𝑡) 𝑑𝑑𝑡𝑡 ∞ 0 , (8) where 𝐺𝐺(𝑡𝑡) is the shear stress relaxation modulus. We measured 𝐺𝐺(𝑡𝑡) (Fig. 𝐒𝐒𝟏𝟏𝐒𝐒) ba sed on the autocorrelation of the pressure tensor components 𝑃𝑃𝑎𝑎𝑎𝑎 22, 81 𝐺𝐺(𝑡𝑡) = 𝑉𝑉 5𝑘𝑘B𝑇𝑇�〈𝑃𝑃𝑥𝑥𝑥𝑥(0)𝑃𝑃𝑥𝑥𝑥𝑥(𝑡𝑡)〉 + 〈𝑃𝑃𝑥𝑥𝑥𝑥(0)𝑃𝑃𝑥𝑥𝑥𝑥(𝑡𝑡)〉 + 〈𝑃𝑃𝑥𝑥𝑥𝑥(0)𝑃𝑃𝑥𝑥𝑥𝑥(𝑡𝑡)〉�+ 𝑉𝑉 30𝑘𝑘B𝑇𝑇�〈𝑁𝑁𝑥𝑥𝑥𝑥(0)𝑁𝑁𝑥𝑥𝑥𝑥(𝑡𝑡)〉 + 〈𝑁𝑁𝑥𝑥𝑥𝑥(0)𝑁𝑁𝑥𝑥𝑥𝑥(𝑡𝑡)〉 + 〈𝑁𝑁𝑥𝑥𝑥𝑥(0)𝑁𝑁𝑥𝑥𝑥𝑥(𝑡𝑡)〉�, (9) where 𝑉𝑉 is the volume of the simulation box and 𝑁𝑁𝑎𝑎𝑎𝑎= 𝑃𝑃𝑎𝑎𝑎𝑎− 𝑃𝑃𝑎𝑎𝑎𝑎 is the normal stress difference. For computing viscosity 𝜂𝜂, we followed the approach of Tejedor et al.22 by fitting the smooth 𝐺𝐺(𝑡𝑡) profile at long times to a series of Maxwell modes (𝐺𝐺𝑖𝑖exp (−𝑡𝑡/𝜏𝜏𝑖𝑖) with 𝑖𝑖= 1…. 4) equidistant in logarithmic time.53 We obtained 𝜂𝜂 by summing up the values from numerical integration at short times and analytical integration based on the fits to Maxwell modes at long times. For computing elastic 𝐺𝐺′ and loss 𝐺𝐺′′ modulus, we Fourier transformed 𝐺𝐺(𝑡𝑡) using the RepTate software.82 All the simulations were performed with periodic boundary conditions applied to all three Cartesian directions. The simulations were performed with a timestep of 10 fs using HOOMD- blue (version 2.9.3)83 with features extended using azplugins (version 0.10.1).80 Continuum analysis of the probe motion In microrheology, the viscoelastic modulus ( elastic 𝐺𝐺′ and loss 𝐺𝐺′′ modulus) is estimated from the probe particle’s motion, which is considered to obey the generalized Langevin equation .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 19 (GLE) 𝑚𝑚eff 𝑑𝑑2𝒓𝒓(𝑡𝑡) 𝑑𝑑 𝑡𝑡2 = − ∫ 𝑍𝑍 𝑡𝑡 −∞ (𝑡𝑡 − 𝑡𝑡′) 𝑑𝑑𝒓𝒓�𝑡𝑡′� 𝑑𝑑𝑡𝑡′ 𝑑𝑑𝑡𝑡′ + 𝑭𝑭B(𝑡𝑡) + 𝑭𝑭ex(𝑡𝑡), (10) where 𝑍𝑍(𝑡𝑡) is the time-dependent friction, 𝑭𝑭B is the Brownian force on the probe particle, and 𝑭𝑭ex is the external force on the probe particle, which is zero in the case of our passive rheology simulations. Because of the low time scales inherent in the MD simulations, inertia plays an important role in accurately quantifying the modulus values of protein condensates. When the inertial terms are included, the GLE in the frequency domain takes the form 38, 39, 48 𝑍𝑍∗(𝜔𝜔) = 6𝜋𝜋𝑅𝑅h𝐺𝐺∗(𝑖𝑖) 𝑖𝑖 𝑖𝑖 + 6𝜋𝜋𝑅𝑅h 2�𝜌𝜌𝐺𝐺∗(𝜔𝜔) + 𝑖𝑖 𝜔𝜔𝑚𝑚eff, (11) where the terms on the right side correspond to the generalized Stokes drag, the Basset force arising from the medium inertia, and the effective probe particle inertial force, respectively. In experimental microrheology, the generalized Stokes drag alone is sufficient to obtain the viscoelastic modulus of complex fluid systems. On rearranging Eq. 11, we obtained IGSER as given in Eq. 1. Analytical expression describing the probe’s displacement data We followed the approach originally proposed by Karim et al. 38, 39, which we discuss here for the sake of completeness. The probe’s mean square displacement can be described through the Baumgaertel-Schausberger-Winter-like power law spectrum MSD(𝑡𝑡) = ∫ ℎ(𝜏𝜏) 𝜏𝜏𝑓𝑓1(𝑡𝑡, 𝜏𝜏)𝑑𝑑 𝜏𝜏+ 𝑔𝑔0𝑓𝑓0(𝑡𝑡, 𝜏𝜏max) ∞ 0 , (12) with 𝑓𝑓1, 𝑓𝑓0, 𝜏𝜏 being the first function for capturing the short-time ballistic behavior (MSD = 𝐶𝐶𝑡𝑡2, where 𝐶𝐶 is the ballistic coefficient), the second function for capturing the long- time diffusive behavior (MSD = 6𝐷𝐷 𝑡𝑡, where 𝐷𝐷 is the diffusion coefficient), and characteristic time representing the changes in MSD, respectively. Further, the spectrum ℎ(𝜏𝜏) can be written as ℎ(𝜏𝜏) = ∑ 𝑔𝑔𝑗𝑗𝜏𝜏𝛼𝛼𝑗𝑗𝐻𝐻�𝜏𝜏𝑗𝑗− 𝜏𝜏�𝐻𝐻(𝜏𝜏 − 𝜏𝜏𝑗𝑗−1)𝑛𝑛max 𝑗𝑗=1 , (13) where 𝑛𝑛max is the number of j modes, with each mode having a relaxation time 𝜏𝜏𝑗𝑗 (𝜏𝜏𝑗𝑗> 𝜏𝜏𝑗𝑗−1) and exponent 𝛼𝛼𝑗𝑗. 𝐻𝐻(𝜏𝜏) is the Heaviside step function. The function in the first term is .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 20 𝑓𝑓1(𝑡𝑡, 𝜏𝜏) = 1 − �1 + 𝑡𝑡 𝜏𝜏� 𝑒𝑒−𝑡𝑡 𝜏𝜏. (14) The function in the second term is 𝑓𝑓0(𝑡𝑡, 𝜏𝜏max) = 𝑒𝑒 − 𝑡𝑡 𝜏𝜏max − 1 + 𝑡𝑡 𝜏𝜏max , (15) which is weighted by the constant 𝑔𝑔0 = ∑ 𝑔𝑔𝑗𝑗(𝜏𝜏𝑗𝑗 𝛼𝛼𝑗𝑗 − 𝜏𝜏𝑗𝑗−1 𝛼𝛼𝑗𝑗 )/𝑛𝑛max 𝑗𝑗=1 𝛼𝛼𝑗𝑗, ensuring that the first term in Eq. ( 12) equals the second term when 𝑡𝑡 = 𝜏𝜏max. The other weighted term s include 𝑔𝑔𝑗𝑗= 𝑔𝑔1 ∏ 𝜏𝜏𝑘𝑘 𝛼𝛼𝑘𝑘−𝛼𝛼𝑘𝑘−1𝑗𝑗−1 𝑘𝑘=1 for 2 ≤ 𝑗𝑗 ≤ 𝑛𝑛max. The value of 𝑔𝑔1 can be obtained from 𝐶𝐶= 𝑔𝑔1 2 ∑ �∏ 𝜏𝜏𝑘𝑘 𝛼𝛼𝑘𝑘−𝛼𝛼𝑘𝑘−1𝑗𝑗−1 𝑘𝑘=1 � 𝜏𝜏𝑗𝑗 𝛼𝛼𝑗𝑗−2 −𝜏𝜏𝑗𝑗−1 𝛼𝛼𝑗𝑗−2 𝛼𝛼𝑗𝑗−2 + 𝑔𝑔0 𝜏𝜏max2 𝑛𝑛max 𝑗𝑗=1 . Given that 𝜏𝜏max is usually large, the second term in the expression for 𝐶𝐶 is negligible, and once the slope of the ballistic regime is known, 𝑔𝑔1 and other weighted terms can be readily computed. The integration of Eq. ( 12) with respect to 𝜏𝜏 then gives rise to MSD(𝑡𝑡) = ∑ 𝑔𝑔𝑗𝑗�𝜏𝜏𝛼𝛼𝑗𝑗� 1 𝛼𝛼𝑗𝑗 − 𝑒𝑒 𝑡𝑡 𝜏𝜏+ �𝛼𝛼𝑗𝑗− 1�𝐸𝐸1+𝛼𝛼𝑗𝑗� 𝑡𝑡 𝜏𝜏��� 𝜏𝜏=𝜏𝜏𝑗𝑗−1 𝜏𝜏=𝜏𝜏𝑗𝑗 +𝑔𝑔0 �𝑒𝑒 − 𝑡𝑡 𝜏𝜏max − 1 + 𝑡𝑡 𝜏𝜏max � , 𝑛𝑛max 𝑗𝑗=1 (16) where 𝐸𝐸1+𝛼𝛼𝑗𝑗� 𝑡𝑡 𝜏𝜏� = ∫ 𝑥𝑥−1−𝛼𝛼𝑒𝑒−𝑥𝑥𝑡𝑡/𝜏𝜏𝑑𝑑𝑥𝑥 ∞ 1 is the exponential integral function. The Fourier transform of Eq. (16) gives the real and imaginary parts of the MSD in the frequency domain as MSD′(𝜔𝜔) = −2 ∑ 𝑔𝑔𝑗𝑗 1+𝛼𝛼𝑗𝑗 �𝜏𝜏1+𝛼𝛼𝑗𝑗 . 2𝐹𝐹1 �2, 1+𝛼𝛼𝑗𝑗 2 ; 3+𝛼𝛼𝑗𝑗 2 , −𝜏𝜏2𝜔𝜔2�� 𝜏𝜏=𝜏𝜏𝑗𝑗−1 𝜏𝜏=𝜏𝜏𝑗𝑗 −𝑛𝑛max 𝑗𝑗=1 𝑔𝑔0 𝜏𝜏max𝑖𝑖2�1+𝜏𝜏max2 𝑖𝑖2�, (17) MSD′′(𝜔𝜔) = − 1 𝑖𝑖∑ 𝑔𝑔𝑗𝑗 𝛼𝛼𝑗𝑗(2+𝛼𝛼𝑗𝑗) �𝜏𝜏𝛼𝛼𝑗𝑗�2 + 𝛼𝛼𝑗𝑗− 𝛼𝛼𝑗𝑗𝜏𝜏2𝜔𝜔2.2𝐹𝐹1 �1, 2+𝛼𝛼𝑗𝑗 2 ; 4+𝛼𝛼𝑗𝑗 2 , −𝜏𝜏2𝜔𝜔2� −𝑛𝑛max 𝑗𝑗=1 2𝛼𝛼𝑗𝑗𝜏𝜏2𝜔𝜔2 . 2𝐹𝐹1 �1, 2+𝛼𝛼𝑗𝑗 2 ; 4+𝛼𝛼𝑗𝑗 2 , −𝜏𝜏2𝜔𝜔2��� 𝜏𝜏=𝜏𝜏𝑗𝑗−1 𝜏𝜏=𝜏𝜏𝑗𝑗 − 𝑔𝑔0 𝑖𝑖�1+𝜏𝜏max2 𝑖𝑖2�, (18) where . 2𝐹𝐹1 is the hypergeometric function and [𝑥𝑥(𝜏𝜏)]𝜏𝜏=𝜏𝜏𝑗𝑗−1 𝜏𝜏=𝜏𝜏𝑗𝑗 = 𝑥𝑥�𝜏𝜏𝑗𝑗−1� − 𝑥𝑥(𝜏𝜏𝑗𝑗) for an arbitrary function 𝑥𝑥(𝜏𝜏). The parameters �𝛼𝛼1, … , 𝛼𝛼max,𝜏𝜏0, 𝜏𝜏1, … , 𝜏𝜏max� are obtained based on the fit to the probe’s MSD data. We fixed 𝜏𝜏𝑗𝑗 values and numerically sought 𝛼𝛼𝑗𝑗 values that minimize 𝜒𝜒2 of MSD, which were then used to obtain the MSD values in the frequency domain (Eqs. 17 and 18). .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 16, 2024. ; https://doi.org/10.1101/2024.08.13.607792doi: bioRxiv preprint 21 Acknowledgments This material is based on the work supported by the National Institute of General Medical Science of the National Institutes of Health under grant s R01GM136917 and R 35GM153388, and the Welch Foundation under grant A -2113-20220331. We thank Prof. Michael P. Howard (Auburn University) for bringing to our attention about the feature s available within azplugins to restrain the particles at specific locations in our simulations. We also thank Prof. Benjamin Schuster (Rutgers University) and Dr. Avijeet Kulshrestha (Texas A&M University) for their helpful comments on the manuscript. The authors acknowledge the Texas A&M High Performance Research Computing (HPRC) for providing computational resources that have contributed to the

reported in this research article.

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