# Temperature-Dependent Coarse-Grained Model for Simulations of Intrinsically Disordered Protein LCST and UCST Liquid-Liquid Phase Separations

Yingmin Jiang, Tâp Ha-Duong

► To cite this version:

Yingmin Jiang, Tâp Ha-Duong. Temperature-Dependent Coarse-Grained Model for Simulations of Intrinsically Disordered Protein LCST and UCST Liquid-Liquid Phase Separations. Journal of Chemical Theory and Computation, 2025, 21 (9), pp.4939-4952. 10.1021/acs.jctc.5c00212. hal-05373622

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Submitted on 19 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
A temperature-dependent coarse-grained model
for simulations of intrinsically disordered protein
LCST and UCST liquid-liquid phase separations

Yingmin Jiang and Tâp Ha-Duong*

Université Paris-Saclay, CNRS, BioCIS, Orsay, 91400, France

E-mail: tap.ha-duong@universite-paris-saclay.fr

Phone: +33 1 80 00 63 76

Abstract

Many intrinsically disordered proteins (IDPs) can undergo a liquid-liquid phase separation (LLPS) in water, depending on the solution conditions (temperature, pH, ionic strength). There are two types of LLPS that are controlled by temperature, those occurring above a lower critical solution temperature (LCST) and those occurring below an upper critical solution temperature (UCST). IDP coarse-grained (CG) models are particularly well appropriate for investigating the physical chemical factors that govern their LLPS and supramolecular organization. However, the development of CG models allowing simulations of both LCST and UCST behavior of temperature-sensitive IDPs is still in its infancy. In this context, we present here a novel temperature-dependent (TD) CG model for IDP simulations based on MARTINI 3 force field. The model was developed by modifying the Lennard-Jones potentials between the apolar or charged solute beads and water with a TD rescaling factor. It was parametrized to fit the TD potentials of mean force (PMF) between two apolar or two charged molecules computed using all-atom (AA) simulations. We show that the TD CG model is able to
reproduce the experimentally known LLPS of both LCST and UCST low-complexity
sequences, and to estimate phase transition temperatures comparable to experimental
measurements.

# 1. Introduction

Liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) is a physi-
cal phenomenon underlying the formation of biomolecular condensates in cell, also referred to
as membraneless organelles. ¹,² The latter, including stress granules, P-bodies, and nucleoli,
are involved in various cellular processes, particularly the spatiotemporal compartmentaliza-
tion and regulation of biochemical reactions,³ such as DNA transcription, RNA translation,
storage, or degradation. ⁴,⁵ Aberrant LLPS of various IDPs was associated to various human
pathologies, including cancers, neurodegenerative diseases, and infections, emphasizing the
importance of IDP LLPS in cell activities. ⁶⁻⁸

LLPS is finely regulated by environmental conditions, such as temperature, pH, ionic strength, or post-translational modifications (PTM). ¹,⁹⁻¹³ Among these, LLPS that are controlled by temperature are of particular interest as they participate in the molecular mechanisms of cell response to heat stress. For example, the RNA binding protein Pab1 is soluble in vivo and in vitro at 30 °C but forms condensates by LLPS when temperature increases to 46 °C, releasing RNA from the protein and permitting a higher level of translation. ³ In *Arabidopsis* plants, two proteins have recently been shown to undergo LLPS in response to temperature changes: the transcriptional repressor ELF3 reversibly forms liquid droplets when the temperature is raised from 17 to 35 °C, shifting the protein into an inactive state. ¹⁴ Conversely, the FRI protein, which activates the flowering locus C (FLC) promoter, forms nuclear condensates after cold exposure, which sequester FRI away from FLC, down-regulating its transcriptional activity. ¹⁵ Overall, these reports highlight the importance of IDP temperature-controlled LLPS in gene regulation in response to cold or heat stress.

Molecular dynamics (MD) simulations are a valuable technique for elucidating the struc-
ture-dynamics-function relationship of biomolecules.¹⁶ However, exhaustive exploration of conformational ensembles of IDP single chains by all-atom (AA) simulations remains extremely time-consuming, precluding their use to investigate the collective behavior of multiple chains, with rare exceptions.¹⁷⁻¹⁹ A merit of AA models with explicit solvent is that they can fairly capture the temperature dependence of nonbonded interactions, allowing to reproduce the temperature-dependent (TD) conformational ensembles of IDPs. Notably, several AA MD studies have shown that IDP single chains with LCST or UCST property are more compact when temperature increases or decreases, respectively, in good agreement with experimental data.²⁰⁻²³ However, although coil-to-globule transition temperatures of single-chain IDPs have been shown to correlate with their phase transition temperatures,²⁴ these parameters are not strictly coupled by physical principles, as collapse is a continuous rather than an emergent transition, which requires multi-chain simulations.

For simulating IDP multiple chains, more appropriate approaches are simplified or coarse-grained (CG) models, which reduce the number of degrees of freedom to speed up simulations of large biopolymer systems.²⁵⁻²⁷ In many effective CG models of IDPs, each amino acid is described by one bead and the solvent is implicit. Coulombic interactions between charged beads are screened by a Debye-Huckel term and short-range nonbonded interactions are modeled with effective potentials depending on the amino acid chemical properties. For example, several models were based on a residue hydrophobicity scale (HPS), such as that developed by Kapcha and Rossky (HPS-KR)²⁸⁻³¹ or the HPS-Urry scale.³²,³³ Most of these models were optimized to best reproduce IDP single chain radii of gyration and further applied to retrieve their phase diagram. Another approach is to derive statistical potentials from distributions of residue-residue contacts which can be either extracted from structural databases (knowledge-based),³⁴,³⁵ or computed from AA MD simulations (physics-based).³⁶,³⁷ Among the latter, the Mpipi model which accurately accounts for the π-π and π-cation interactions, was able to predict both IDP single chain radii of gyration and multi-chain LLPS behavior.³⁷

Among higher resolution CG models, MARTINI force fields are well known for their
transferability and efficiency in simulating the self-assembly of various polymer multi-chain systems.38 In MARTINI models, each polymer building block is described by a few beads representing elementary chemical moieties, and the solvent and ions are explicitly represented. Bead size and effective interactions were parametrized mainly using a top-down approach targeting experimental thermodynamic data, such as free energies of solvation or water-octanol transfer.39 Although more time-consuming than previous models with implicit solvent, simulations with MARTINI force fields were able to reproduce the phase separation of several IDP multi-chain systems.40–45

With regard to temperature-controlled LLPS, it should be noted that none of the previously mentioned CG models are TD. Yet, several studies showed that they were able to capture the UCST behavior of several IDPs.29–31,33,37,42 In implicit solvent CG simulations, this can be explained by the increase in chain entropy (as temperature increases), which compensates for the enthalpy cost of disrupting polar and/or electrostatic interactions between IDP chains, leading to their separation at high temperature. Even in MARTINI models with explicit solvent, the enthalpy-entropy balance in effective free energies may not be transferable to temperatures different from those at which the models were parametrized, potentially affecting TD properties of IDPs.39,46 More importantly, these temperature-independent models are unlikely to correctly reproduce the LCST behavior of IDPs, as solvent entropy is not well accounted for, making it impossible to properly simulate the TD hydrophobic interactions which are the main driving force of IDP aggregation at high temperature.

Very few simplified models of IDPs capable of reproducing their LCST behavior have been reported in the literature to date. Among them, AA simulations using TD versions of the ABSINTH implicit solvent model47 were able to reproduce the collapse or extension (with increasing temperature) of IDP single chains with LCST or UCST property, respectively.48,49 The crucial ingredient of these models is the solvation free energies of the elementary chemical moieties, which now depend on temperature. In a similar spirit, Dignon et al. modified their one-bead-per-residue CG model by rescaling the amino acid hydrophobicity scale with a
TD factor (HPS-T). More specifically, interactions of hydrophobic and aromatic residues are strengthened, while interactions involving polar and charged beads are weakened, as temperature increases (in the range of 270-370 K). Parametrized with a top-down approach to fit the TD extension or compaction of single-chain IDPs, the HPS-T model allows to successfully distinguish IDPs with LCST or UCST property.⁵⁰

Here we present an alternative TD CG model based on MARTINI 3 force field, which can be used to simulate the LLPS of both LCST and UCST IDPs and to determine phase transition temperatures comparable to experimental measurements. With higher resolutions than residue-level HPS models, MARTINI force fields can be used to study proteins with other types of molecules, including nucleic acids and small ligands.⁵¹ This versatility makes them particularly suitable for studying IDP condensates and their interactions with RNA⁴³ or drugs.⁴⁵ Our CG model was parametrized to fit the TD potentials of mean force (PMF) between two apolar chemical moieties and between two charged ones. We demonstrate that the TD CG model is able to reproduce (i) the temperature dependence of π-π and π-cation interactions, (ii) the TD compaction or extension of IDP single chains, and (iii) the phase separation of multiple chains of low complexity IDPs with experimentally known LCST or UCST property.⁵² Thanks to the transferable building block philosophy of MARTINI 3 model, we strongly believe that our TD CG model can be applied to natural IDPs to investigate their LLPS and condensate properties.

# 2. Methods

## 2.1. A TD CG model of IDPs

A straightforward way to model the temperature dependence of solute-solute nonbonded interactions is to empirically rescale the effective potentials between solute beads by a function of the temperature. For instance, this approach was applied in implicit solvent CG simulations of methanol liquid,⁵³ synthetic polymers,⁵⁴ and proteins,⁵⁵ using rescaling func-
tions as simple as $f(T) = (T/T_0)$ or $(T/T_0)^{1/2}$ where $T_0$ is the temperature at which the temperature-independent CG model was developed. As MARTINI 3 is an explicit solvent CG model, we rather applied this approach to the effective potentials between solute and solvent beads and adjusted the rescaling function to reproduce the temperature dependence of the binding free energies of two apolar small molecules and of two charged ones. More specifically, inspired by the aforementioned studies, we rescaled the Lennard-Jones (LJ) potentials between water (W type) and solute beads with two simple and tunable exponential functions of temperature, according to their apolar (C type) or charged (Q type) nature:

$$ \begin{aligned} \epsilon'_{C-W} &= \epsilon_{C-W} \times \left(\frac{T}{T_0}\right)^{\alpha} \\ \epsilon'_{Q-W} &= \epsilon_{Q-W} \times \left(\frac{T}{T_0}\right)^{\beta} \end{aligned} \qquad (1) $$

Here, $\epsilon'_{C-W}$ and $\epsilon'_{Q-W}$ denote the TD depth of the LJ potentials between C and W beads, and between Q and W beads, respectively. The parameters $\epsilon_{C-W}$ and $\epsilon_{Q-W}$ are taken from MARTINI 3 force field which were optimized at temperature $T_0 = 300$ K.³⁹ To illustrate the impact of the rescaling temperature functions, we plotted the SC3-W and Q5-W LJ potentials at four different temperatures. As displayed in Figure S1, the rescaling functions primarily impact the depth of the LJ potential minima, with negligible effect on the repulsive excluded-volume part of the potentials.

It could be noted that we selectively modified the LJ parameters of apolar and charged beads, without affecting the other ones, notably the polar N and P beads. Indeed, several studies highlighted that IDPs undergoing LLPS are generally enriched in charged and aromatic amino acids, and that charge-charge, π-cation, and π-π interactions are key driving forces in condensate formation.⁵⁶,⁵⁷ Hydrophobic interactions are also crucial in temperature-controlled LLPS, in particular for distinguishing LCST from UCST IDPs.⁵² On the other hand, polar interactions and hydrogen bonds can also influence IDP phase separation but more indirectly through the formation of secondary structures and fibrillar aggregates.⁵⁸,⁵⁹
For these reasons, we only rescaled the C-W and Q-W LJ potentials, as a first intention.

## 2.2. Bottom-up parametrization of the TD CG model

To determine the optimal values for the $\alpha$ and $\beta$ exponents of Equation 1, we computed the PMF in explicit water⁶⁰ of two octane (OCT) and two positively charged $\alpha,\beta$-diamino acid (DAA) molecules (Figure 1A) as a function of the distance $\zeta$ between the geometric centers of the molecules. Then, we tested various discrete values of the $\alpha$ and $\beta$ parameters in the interval [-2, 0] and [0, 2], respectively, and compared the TD PMF of CG models to that of AA one. In MARTINI 3 force field, OCT is represented by two beads of C2 type, while DAA is modeled using one SQ4p and one P2 bead. CG water beads of W type were used to solvate the solutes. The bond lengths and force constants for each CG molecule are indicated in Figure 1B. As reference AA models, the GAFF force field⁶¹ and SPC/E water model⁶² were used to describe the molecules and water, respectively. Initial three-dimensional structures of the molecules were generated by using MolView webserver⁶³ and their CG models were built using the CGbuilder tool.⁶⁴ To ensure the reliability of the CG models, we performed short MD simulations (50 ns) of the isolated molecules, and computed the bond length and solvent-accessible surface area (SASA) distributions, then compared the CG results to those of the AA models (Figure 1C). A notable difference was observed between AA and CG SASA. However, the difference does not exceed 10% (8% for OCT and 7% for DAA), which is considered acceptable when using the MARTINI 3 model for small molecules.⁵¹ Additional examinations of the Connolly surfaces of the two molecules revealed that the shapes and volumes of the AA models were satisfactorily captured by the CG models (Figure 1D).

The PMF were computed using the umbrella sampling technique⁶⁵ at four different temperatures (280, 310, 340, and 370 K). Two solute molecules were initially placed in a cubic box of 6 nm size and solvated with water molecules. 150 mM of sodium chloride were added to maintain the system neutrality. For each pair of solute molecules, a set of initial configu-
<figure><img src="image_2.png" /></figure>

<table><thead><tr><th></th><th>Bond type</th><th>Bond length (nm)</th><th>Force constant (kJ/mol/nm²)</th><th>AA SASA (nm²)</th><th>CG SASA (nm²)</th></tr></thead><tbody><tr><td>OCT</td><td>C2-C2</td><td>0.5</td><td>4000</td><td>3.6 ± 0.2</td><td>3.3 ± 0.2</td></tr><tr><td>DAA</td><td>SQ4p-P2</td><td>0.27</td><td>16000</td><td>2.7 ± 0.1</td><td>2.9 ± 0.1</td></tr></tbody></table>

Figure 1: (A) Mapping of the OCT and DAA molecules into MARTINI 3 model. (B) Pseudobond parameters of the two solutes. Solvent-accessible surface area (SASA) values computed with the AA and CG models are indicated in the two last columns. (C) Comparison of the bond length and SASA distributions between the AA (blue lines) and CG (orange lines) models. Vertical dashed lines indicate average values. (D) Superimposition of the AA (grey) and CG (blue) Connolly surfaces of the two molecules.
rations was created with a distance $\zeta$ between their geometric centers increasing from 0.53 to 2.53 nm with an increment of 0.1 nm. Then, this distance was restrained using a harmonic potential with a 1000 kJ/mol/nm$^2$ force constant.

AA simulations were composed of the following steps: First, the entire system was submitted to an energy minimization using the steepest descent method with a maximum force tolerance of 100 kJ/mol/nm$^2$. Next, two equilibrations of 1 ns each were performed in the NVT ensemble, using successively the Berendsen and Nose-Hoover thermostats. Finally, a 50 ns production was run in the NPT ensemble with the Nose-Hoover thermostat and the Parrinello-Rahman barostat. In CG simulations, a 100 step energy minimization was first performed, then the system was equilibrated during 100 ps using the Berendsen thermostat, followed by a production run of 200 ns with the V-rescale thermostat and the Parrinello-Rahman barostat. The GROMACS 2020.6 version$^{66}$ was utilized for all AA and CG simulations in this work.

The PMF of each solute pair was obtained as a function of the distance $\zeta$ between the solute geometric centers using the *gmx wham* tool.$^{67}$ It is noteworthy that, since constraints were applied to the distance between two solutes (but not to their position), each pair of solutes could rotate freely. The number of rotational configurations being proportional to the surface of the sphere of radius $\zeta$,68 the PMF were corrected by adding the term $RT\ln(4\pi\zeta^2)$ to eliminate the rotational entropy contribution, resulting in a flat PMF at a large separation distance.$^{42}$ The convergence of PMF calculations was assessed by computing average PMF over 5 sliding time windows of MD trajectories. As shown in Figures S2A and S2B, the AA and CG PMF averaged over the last 2 or 3 time windows are very similar, indicating that convergences have been achieved. Furthermore, the relevance of the difference between TD PMF is illustrated in plots of a CG PMF with error estimates as colored shading (Figure S2C).
2.3. Refinement of the TD CG model with multi-chain simulations

The α and β exponents of Equation 1 were further refined so that the TD CG model can
effectively reproduce both LCST and UCST behaviors of IDPs. To do this, we considered the
two low-complexity sequences (TPVAVG)₁₄ and (RGDSPYG)₁₂ which were experimentally
shown to have a LCST and an UCST property, respectively.⁵² From each protein sequence,
100 atomic three-dimensional structures were generated by using the DIPEND program.⁶⁹
Then, 10 structures with different radii of gyration were converted into MARTINI 3 CG
models using the Martinize2 script³⁹,⁷⁰ (Figure 2) and submitted to a 200 ns simulation
each. Finally, we randomly selected 30 chains from the trajectories for setting the multi-
chain systems.

<figure><img src="image_3.png" /><figcaption>Figure 2: Mapping of the (TPVAVG)₁₄ and (RGDSPYG)₁₂ repeat patterns into MARTINI 3 model. Green, grey, red, and yellow colors indicate apolar, polar, intermediate, and charged beads, respectively.</figcaption></figure>

For each low-complexity sequence, 30 CG chains were randomly dispersed in a cubic
simulation box whose size was such that the protein concentration was 1.0 mM. Then the
box was filled with W water beads and 150 mM of sodium chloride ions. Short-range non-
bonded interactions were computed with the Verlet cutoff scheme, with a threshold distance
of 1.1 nm. Long-range electrostatic interactions were treated using the reaction-field method,
with a dielectric constant set to 15. Each system was first submitted to an energy minimiza-
tion of 100 steps, then equilibrated for 1 ns with a 10 fs time step using the Berendsen
thermostat. Production simulations were performed for 7 µs with a 20 fs time step, using
the V-rescale thermostat and Parrinello-Raham barostat. In this refinement procedure, we
simulated each multi-chain system at a low (280 K) and a high (370 K) temperature, under
the pressure of 1 bar. Starting from separate chains, we visually examined their aggregated
state and compared it to the expected behavior, and tested different combinations of (α, β)
parameters previously optimized.

**2.4. Assessment of the TD CG model**

Once the TD CG model had been refined, it was assessed on three levels. First, we verified
the temperature dependence of the π-π and π-cation interactions in water. As a π conjugated system and cation, we considered the tyrosine side chain and the ammonium group,
respectively (Figure 4A and B). By using the same protocol as described in section 2.2, we
computed the PMF between two tyrosine side chains and between a tyrosine and an ammonium cation at the four temperatures 280, 310, 340, and 370 K. Then we compared the
yielded results obtained with the AA and TD CG force fields.

Next, we examined the compaction or extension with increasing temperature of a single
chain of two other low-complexity sequences that were not used in the model refinement
procedure: $(GVPGVA)_{14}$ and $(QYPSDGRG)_{10}$ which have known LCST and UCST property, respectively.⁵² For each IDP, an initial three-dimensional structure was generated with DIPEND,⁶⁹ converted into MARTINI 3 CG models using Martinize²⁷⁰ (Figure 5A), and solvated with W water beads and 150 mM of sodium chloride ions in a cubic box of 6 nm size.
Using the same simulation parameters as in section 2.3, each single chain was submitted to a production run of 1 µs, and its radius of gyration was calculated over time with the *gmx gyrate* tool.

Finally, for the four low-complexity sequences considered in this report, using the same
protocol as described above, we carried out MD simulations of their 30-chain system at two
protein concentrations (0.5 mM and 1.0 mM) and at different temperatures (270, 280, 300,
310, 340, 370 and 400 K) to check whether the TD CG model can reproduce their collective
LCST or UCST behavior. It could be noted that enhanced simulation techniques, such
as replica exchange MD, would be more efficient for sampling conformational ensembles of multi-chain systems. However, while such approaches are generally required to efficiently sample the dihedral angles of protein backbones to explore their propensity for secondary structures, they are less necessary in CG simulations with MARTINI models that describe the protein backbones with only one bead per amino acid. In addition, an interesting feature of the IDP LLPS is the fluidity and reversibility of phase separation with temperature change. However, it is not easy to follow these dynamic processes of IDP aggregation and dissolution using biased simulation techniques. For these reasons, we believe that standard MD CG simulations remain an appropriate technique for our study.

For each of the four multi-chain systems, we simulated the aggregation and dissolution processes over 7 µs as a function of the temperature. Trajectory frames were saved every 1 ps for subsequent analysis. It should be noted that, for calculating average physical quantities, only the last 5 µs of the trajectories were used. MD analyses were carried out using GROMACS tools and homemade python scripts. Notably, the aggregation state of the multi-chain systems was monitored using the *gmx clustsize* tool, which computes the number of aggregates and their size (in terms of number of chains) over time. The threshold distance used to classify two chains as part of the same aggregate was set to 0.6 nm.

# 3. Results and discussion

## 3.1. Parametrization of the TD CG model with PMF calculations

The temperature dependence of hydrophobic and electrostatic interactions is the main driving force of either LCST or UCST behavior of low-complexity IDPs. For this reason, we have endeavored to adjust the MARTINI 3 force field to reproduce the temperature dependence of PMF in water between apolar molecules and between charged ones. To achieve this, we rescaled the LJ potentials between W water bead and C type (apolar) or Q type (charged) beads according to Equation 1. Using AA and CG simulations, we then computed the PMF
between two OCT molecules and between two DAA ones at the four temperatures 280, 310, 340, and 370 K, with different $\alpha$ and $\beta$ exponents.

<figure><img src="image_4.png" /><figcaption>Figure 3: PMF of OCT-OCT and DAA-DAA molecule pairs computed with the MARTINI 3 ($\alpha$ = 0 and $\beta$ = 0), TD CG, and AA models. Blue, orange, green, and red lines indicate temperatures of 280, 310, 340, 370 K, respectively.</figcaption></figure>

As shown in Figure 3, the AA PMF between two apolar molecules in water are strengthened upon heating, which is a well-known feature of hydrophobic interactions linked to the increase in solvent entropy with temperature.⁷¹⁻⁷³ In contrast, the standard MARTINI 3 model ($\alpha$ = 0) gives an apolar-apolar PMF that exhibits a slightly opposite trend to that observed with the AA model, where interactions appear somewhat weaker at higher temperatures. However, by reducing the LJ interactions between apolar and water beads with negative values of the $\alpha$ exponent, it is possible to reverse the trend and restore the temperature dependence of the CG PMF in line with the AA model by using $\alpha$ values around -1.5.

Regarding electrostatic interactions, the AA PMF between the two positively charged molecules exhibits a shallow minimum at a distance of 0.6 nm and a small energy barrier before slowly decreasing towards zero with increasing intermolecular distance (Figure 3).
This free energy minimum is dependent on the temperature, with deeper interactions at low
than at high temperatures, due to greater kinetic energy of the molecules which destabilize
the electrostatic interactions and reduce the system enthalpy at high temperature.⁷³,⁷⁴ The
energy minimum at 0.6 nm is also observed in MARTINI 3 PMF ($\beta$ = 0), but it also
exhibits a deep energy minimum at 0.9 nm which is not seen in the AA PMF. To better
understand this discrepancy, we analyzed the radial distribution functions g(r) of the water
and ion particles around DAA molecules in the PMF simulations. As shown in Figure S3,
the density of chloride ions in the first solvation shell of DAA molecules is much higher in CG
than in AA simulations, probably due to the deeper LJ potential of the chloride bead in CG
model (1.18 kJ/mol) compared to the AA ion (0.49 kJ/mol). As a result, the electrostatic
repulsion between the positively charged DAAs is more attenuated in the CG than in the
AA model, especially as it is screened by a dielectric constant of 15 (CG) instead of 1 (AA).
Taken together, these factors stabilize a chloride-separated conformation of the DAA-DAA
pair in CG but not in AA simulations.

Importantly, we observed that the depth of the two energy minima in MARTINI 3 PMF is
independent of temperature, unlike that of the AA PMF. This indicates that the increase in
solvent entropy with temperature counterbalances the increase in the enthalpy contribution
which should destabilize the solute-solute interactions at high temperatures. To reduce this
overestimated entropy contribution from CG solvent, one possibility is to strengthen the LJ
potentials between charged beads and water with positive $\beta$ exponents. Indeed, as shown in
Figure 3, the temperature dependence of the first CG PMF energy minimum becomes more
obvious when $\beta$ varies from 0 to 1.5. For $\beta$ values around 0.7, the variation of the minimum
depth as a function of temperature is close to that observed in the AA PMF.

It could be noted that increasing solvent temperature should also decrease the order of
water dipoles around charged particles, which in turn should decrease the screening effect of
the solvent dielectric, leading to more intense charge-charge repulsion or attraction. This ef-
fect could be mimicked by scaling down the direct charge-charge interactions with increasing
temperature. However, the $\beta$ positive sign is rather related to the entropy of water release when two charges come to interact. Indeed, MARTINI water model is a simple LJ particle and has no rotational degrees of freedom. Consequently, increasing temperature generates exaggerated translational motions of the solvent beads, resulting in an overestimated increase in the solvent release entropy to the free energy of charge-charge interactions. On the other hand, with AA water models, increasing temperature also increases the rotational motions of water molecules, which should counterbalance the increase in the translational entropy of water. To rectify this overestimated entropy of water release in our CG model, we empirically deepened the charge-solvent LJ potentials.

Overall, the comparison between AA and CG PMF of apolar or charged molecule pairs confirms that the temperature dependence of hydrophobic and electrostatic interactions is not well reproduced by MARTINI 3 force field.³⁹ This could be explained by the reduced number of degrees of freedom in CG models, which induces an improper enthalpy-entropy balance in the free energy of solute-solute interactions,⁴⁶ making their temperature dependence inaccurate. To address these limitations, we decreased the strength of the LJ potential between apolar and water beads and enhanced that one between charged and water beads, using the rescaling Equation 1. With $\alpha$ and $\beta$ exponents around -1.5 and 0.7, respectively, we were able to recover the temperature dependence of the AA PMF of apolar-apolar and charge-charge interactions in water. This rescaled MARTINI 3 force field will be hereafter referred to as TD MARTINI 3 model.

## 3.2. Refinement of the TD CG model with multi-chain simulations

To evaluate the capability of the TD MARTINI 3 force field to reproduce the LCST or UCST phase behavior of low-complexity sequences, we conducted MD simulations of 30 chains of the (TPVAVG)₁₄ or (RGDSPYG)₁₂ IDP at low (280 K) and high (370 K) temperatures. To save computing time, we performed these preliminary multi-chain simulations at the rather high protein concentration of 1 mM. Starting from separated chains, we then visually
examined their aggregated state and compared it to the expected behavior. As shown in Table 1, when using the parameters $\alpha$ = -1.5 and any $\beta$ value between 0.5 and 1.0, the (TPVAVG)$_{14}$ sequence exhibited an aggregated state at low temperature, contrary to what is expected. This indicates that the PMF between apolar beads is still too deep at low temperature.

<table><caption>Table 1: Expected and simulated phase behavior of (TPVAVG)$_{14}$ and (RGDSPYG)$_{12}$ multiple chains (at protein concentration of 1 mM) at low and high temperatures using refined parameters of the TD MARTINI 3 model.</caption><thead><tr><th>Sequence</th><th colspan="2">(TPVAVG)<sub>14</sub></th><th colspan="2">(RGDSPYG)<sub>12</sub></th></tr><tr><th>Temperature</th><th>280 K</th><th>370 K</th><th>280 K</th><th>370 K</th></tr><tr><th>Expected behavior</th><th>Separated</th><th>Aggregated</th><th>Aggregated</th><th>Separated</th></tr></thead><tbody><tr><td>α=-1.5, β=0.5</td><td>Aggregated</td><td>Aggregated</td><td>Aggregated</td><td>Separated</td></tr><tr><td>α=-1.5, β=0.7</td><td>Aggregated</td><td>Aggregated</td><td>Aggregated</td><td>Separated</td></tr><tr><td>α=-1.5, β=1.0</td><td>Aggregated</td><td>Aggregated</td><td>Aggregated</td><td>Separated</td></tr><tr><td>α=-2.0, β=0.5</td><td>Separated</td><td>Aggregated</td><td>Aggregated</td><td>Aggregated</td></tr><tr><td>α=-2.0, β=0.7</td><td>Separated</td><td>Aggregated</td><td>Aggregated</td><td>Separated</td></tr><tr><td>α=-2.0, β=1.0</td><td>Aggregated</td><td>Aggregated</td><td>Aggregated</td><td>Separated</td></tr></tbody></table>

Thus, an attempt was made to destabilize the apolar-apolar interactions at low temperature by reducing $\alpha$ to -2. With $\beta$ = 0.5, the LCST property of (TPVAVG)$_{14}$ sequence could be successfully retrieved, but the (RGDSPYG)$_{12}$ sequence now aggregates at high temperature, contrary to expectations. This is probably due to the few apolar beads in the Arg, Tyr, and Pro side chains (Figure 2), whose hydrophobic interactions became too strong at high temperatures. By increasing $\beta$ above 0.7, deeper LJ potentials between charged beads and water destabilized electrostatic interactions at high temperature and led to the separation of (RGDSPYG)$_{12}$ sequence, reproducing its UCST property. However, for $\beta$ = 1, the (TPVAVG)$_{14}$ chains aggregate again at low temperature, probably due to the too favorable interactions between their charged terminal residues. Finally, the optimal combination of the $\alpha$ and $\beta$ parameters to reproduce both LCST and UCST of the two low-complexity sequences is $\alpha$ = -2 and $\beta$ = 0.7 (Table 1). We acknowledge that the optimization of the $\alpha$ and $\beta$ exponents is entirely empirical and not strictly rigorous, as it has not involved minimizing
an error function. We could have tested more $\alpha$ and $\beta$ values, for example, by systematically varying them with an increment of 0.1, but this would have required significantly more computational time. In this heuristic approach, it is thus challenging to estimate the standard errors of the optimal parameters.

### 3.3. Assessment of the TD CG model with $\pi-\pi$ and $\pi$-cation PMF

As $\pi-\pi$ and $\pi$-cation interactions play a dominant role in IDP LLPS,$^{37,56,57}$ we wondered whether the TD MARTINI 3 model could reproduce the temperature dependence of these interactions. To achieve this, we considered a phenolic molecule as a $\pi$ conjugated system and a guanidinium cation (Figure 4A and 4B), computed the $\pi-\pi$ and $\pi$-cation PMF in water at four different temperatures, and compared the AA versus CG results. As shown in Figure 4C, the AA PMF between the two $\pi$ conjugated cycles deepens with increasing temperature, similar to hydrophobic interactions. This trend can be clearly reproduced with the TD CG model, in contrast to the MARTINI 3 force field.

Regarding $\pi$-cation interactions, a slight temperature dependence can be observed in the AA PMF, with the attraction being stronger at low than at high temperature, consistent with electrostatic interactions (Figure 4D). This temperature dependence is reversed in the PMF computed with the standard MARTINI 3 model, again probably due to an overestimation of the entropy of desolvation of charged moieties. By strengthening the LJ potentials between charged beads and water ($\beta = 0.7$), the TD CG model can recover the correct temperature dependence of the $\pi$-cation AA PMF (Figure 4D). Interestingly, these PMF analyses help to clarify the contribution of $\pi-\pi$ and $\pi$-cation interactions to TD LLPS phenomena. The former would be determinant for the LCST property of IDPs, while the latter would be more involved in their UCST behavior.
<figure><img src="image_5.png" /><figcaption>Figure 4: (A) Mapping of a π conjugated cyclic molecule and (B) of a guanidinium cation into MARTINI 3 model. (C) PMF between two π conjugated molecules and (D) between a π conjugated cycle and a cation at different temperatures. Blue, orange, green, and red lines indicate temperatures of 280, 310, 340, 370 K, respectively.</figcaption></figure>
### 3.4. Assessment of the TD CG model with IDP radius of gyration

Several studies have linked the LLPS collective behavior of IDPs to the compaction or extension of their single chain in water.<sup>24,49,75</sup> We have thus assessed the ability of the TD CG model to account for the temperature dependence of the radius of gyration of four single chains, including two additional low-complexity sequences that were not taken into account in the model parametrization (Figure 5A). In simulations with the standard MARTINI 3 force field, the radius of gyration of all four IDPs, irrespective of their known LCST or UCST property, increases slightly with temperature (Figure 5B). This can be explained by the increase in the chain entropy which favors more disordered and extended conformations at high temperature. This observation may confirm why non-TD force fields are able to reproduce the UCST behavior of IDPs,<sup>29–31,33,37,42</sup> but would be incorrect for simulations of LCST IDPs.

In simulations with the TD CG model, the increase in radius of gyration (with temperature) of the two UCST sequences (QYPSDGRG)<sub>10</sub> and (RGDSPYG)<sub>12</sub> is more pronounced than with the standard MARTINI 3 force field, indicating that the TD CG model would remain able to simulate UCST phase separation. More importantly, in simulations with the TD MARTINI 3 model of the two known LCST sequences (GVPGVA)<sub>14</sub> and (TPVAVG)<sub>14</sub>, the radius of gyration decreases when temperature increases in line with what is expected, indicating that the TD CG model would also be able to simulate LCST behavior. It could be noted in Figure 5B that variations in the IDP radius of gyration as a function of temperature are rather smooth, with no clear discontinuity. To determine their phase transition temperature, it is necessary to use multi-chain simulations, which can additionally account for concentration effects, unlike single-chain simulations.

### 3.5. Assessment of the TD CG model with multi-chain simulations

Here, we assessed the ability of our TD MARTINI 3 model ($α = -2$ and $β = 0.7$) to reproduce the LLPS of both LCST and UCST IDPs using multi-chain MD simulations.
<figure><img src="image_6.png" /><figcaption>Figure 5: (A) Mapping of the $(GVPGVA)_{14}$ and $(QYPSDGRG)_{10}$ repeat patterns into MARTINI 3 model. Green, grey, red, and yellow colors indicate apolar, polar, intermediate, and charged beads, respectively. (B) Single-chain radius of gyration for each of the four studied low-complexity sequences. Orange and blue indicate average values computed with the standard MARTINI 3 and TD CG models, respectively.</figcaption></figure>
To do this, we first considered 30 chains of each IDP in a randomly dispersed state at a concentration of 0.5 mM, and monitored their aggregation state over 7 µs at six different temperatures. In a second step, we considered an aggregated state of each IDP (extracted from the end of a previous aggregation trajectory) and simulated its dissolution process during 7 µs as a function of the temperature.

<figure><img src="image_7.png" /><figcaption>Figure 6: Simulation with TD MARTINI 3 force field of the aggregation and dissolution processes of the (GVPGVA)₁₄ and (TPVAVG)₁₄ sequences (at the concentration of 0.5 mM) at 270, 280, 300, 310, 340, and 370 K. Blue and orange lines indicate the number of chains in the largest aggregates and the number of aggregates in the simulation box, respectively.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: Simulation with TD MARTINI 3 force field of the aggregation and dissolution processes of the (QYPSDGRG)₁₀ and (RGDSPYG)₁₂ sequences (at the concentration of 0.5 mM) at 280, 300, 310, 340, 370, and 400 K. Blue and orange lines indicate the number of chains in the largest aggregates and the number of aggregates in the simulation box, respectively.</figcaption></figure>
In the aggregation simulations of the (GVPGVA)₁₄ and (TPVAVG)₁₄ sequences, we observed that the 30 chains rapidly formed a single aggregate at temperature higher than 300 K, and remained dispersed at 270 and 280 K (Figure 6). In the dissolution simulations, both (GVPGVA)₁₄ and (TPVAVG)₁₄ aggregates remained stable during additional 7 µs at temperature above 300 K. At 270 and 280 K, the (GVPGVA)₁₄ chains of the aggregate split in less than 2 µs, confirming their dispersed state at low temperatures. In the (TPVAVG)₁₄ dissolution simulation, the aggregate could not be clearly dissolved at 270 and 280 K, but we could observe that some chains stripped from it (Figure 8), indicating a beginning of a probably very slow dissolution process at these low temperatures. From the above, the TD MARTINI 3 model is fairly able to reproduce the collective LCST behavior of two different low-complexity sequences.

<figure><img src="image_9.png" /><figcaption>Figure 8: First and last frames of the aggregation simulations at high temperature and of the dissolution simulations at low temperature for the (GVPGVA)₁₄ and (TPVAVG)₁₄ sequences.</figcaption></figure>

The same TD CG model can also be used to simulate the LLPS of UCST sequences. Indeed, starting from a dispersed state, the 30 chains self-assemble into few large aggregates (Figure 9) in simulations at temperatures below 310 K for (QYPSDGRG)₁₀ sequence and below 340 K for (RGDSPYG)₁₂, while they remain separated at high temperatures (Fig-
<figure><img src="image_10.png" /><figcaption>Figure 9: First and last frames of the dissolution simulations at high temperature and of the aggregation simulations at low temperature for the (QYPSDGRG)₁₀ and (RGDSPYG)₁₂ sequences.</figcaption></figure>

ure 7). This typical behavior of UCST sequences is confirmed in our dissolution simulations in which IDP aggregates are rapidly dissolved at high temperatures but remain stable below 310 K and 340 K for (QYPSDGRG)₁₀ and (RGDSPYG)₁₂ sequences, respectively. It could be noted that, similar to the slow dissolution process observed for (TPVAVG)₁₄ at low temperatures, the aggregation kinetics of the (RGDSPYG)₁₂ chains at 280 K is also very slow, as the simulations could not reach a single large aggregate after 7 µs (Figure 9). In contrast, at 300 and 310 K, we observe that two (RGDSPYG)₁₂ aggregates merged into a larger one at 6 and 4 µs, respectively (Figure 7).

In summary, the LLPS behavior of both LCST and UCST low-complexity sequences can be reproduced using multi-chain MD simulations with the TD MARTINI 3 model. This is the first successful modeling of LCST and UCST properties of IDPs using a MARTINI-based force field, going beyond many previous UCST protein studies. Based on simulations of the (QYPSDGRG)₁₀ and (RGDSPYG)₁₂ sequences, transition temperatures appear to be influenced by amino acid composition, warranting further investigation into the effects of IDP composition on their LLPS using our TD CG model.
3.6. Transition temperatures of low-complexity sequences

In contrast to single-chain simulations, a sharp transition with temperature could be observed
in multi-chain simulations, as expected for thermoresponsive IDPs. Thus, multi-chain simu-
lations at the previous six temperatures allow us to determine the LLPS critical temperature
of each low-complexity sequence to precisely delineate their dissolved and aggregated states.
To do this, we first plotted the average percentage of chains in the largest aggregate over the
last 5 µs of both aggregation and dissolution simulations, as a function of temperature. We
then fitted the obtained values with the sigmoidal function y(x) = a0/(1+exp(-a1(x-a2)))
characteristic of phase transitions. It should be noted that we have excluded data from simu-
lations at the lowest temperatures from these analyses, as equilibrium was not always reached
due to very slow kinetics. Finally, the fit parameters and their standard errors were deter-
mined using the scipy optimize tools from the SciPy Python library,^76 yielding the transition
or critical temperature T_c = a2. We carried out these analyses on the MD simulations of
the four low-complexity sequences at the protein concentration of 0.5 mM (Figures 6 and 7)
but also at 1.0 mM (Figures S4 and S5) to highlight an effect of IDP concentration on their
transition temperatures.

As displayed in Figure 10, the fitting functions for LCST IDPs exhibit S-shaped curves.
For (GVPGVA)₁₄ sequence, the transition temperatures computed at concentrations of 1.0
and 0.5 mM are 282 ± 1 K and 286 ± 1 K, respectively, while an experimental value of 318 K
was found for an IDP concentration of 0.05 mM.⁵² Similarly, for (TPVAVG)₁₄ sequence,
the transition temperature shifts from 275 ± 1 K to 280 ± 2 K when IDP concentration
decreases from 1 to 0.5 mM, while the experimental value is equal to 315 K at 0.05 mM.
Conversely, the fitting functions of UCST IDPs exhibit reverse S-shaped curves (Figure 10).
For (QYPSDGRG)₁₀ sequence, the transition temperature decreases from 336 ± 2 K to
333 ± 4 K as protein concentration decreases from 1 to 0.5 mM, while the experiments
yield a value of 311 K at 0.05 mM. For the (RGDSPYG)₁₂ sequence, the predicted transition
temperatures at concentrations of 1.0 and 0.5 mM are 361 ± 3 K and 355 ± 4 K, respectively,
<figure><img src="image_11.png" /><figcaption>Figure 10: Temperature dependence of the average percentages of chains in the largest aggregate in TD CG simulations for the four studied low-complexity sequences. Solid lines represent sigmoidal functions $y(x) = a0/(1 + \exp(-a1(x - a2)))$ used to fit data from simulations with IDP concentrations of 1.0 (orange) and 0.5 mM (blue). Fit parameters and their standard errors were obtained using the `scipy optimize tools.⁷⁶ Vertical dashed lines and colored bands indicate predicted transition temperatures ($T_c = a2$) and standard errors, respectively, for IDP concentrations of 1.0 (orange) and 0.5 mM (blue). Experimental values measured for IDP concentration of 0.05 mM⁵² are indicated with green dashed lines.</figcaption></figure>
while experimental value is 323 K at 0.05 mM.⁵²

Overall, we found that the LLPS transition temperatures of both LCST and UCST low-complexity sequences increase and decrease, respectively, as protein concentration decreases, in line with experimental observations on several proteins with LLPS property.⁵²,⁷⁷,⁷⁸ We expect that for LCST and UCST IDPs, the phase transition temperatures predicted by CG simulations will approach the experimental values as protein concentration approaches experimental conditions.

## 4. Conclusion

We propose here an important development of the MARTINI 3 force field to simulate both the LCST and UCST phase separation of IDPs, by modulating the solute apolar bead-water and charged bead-water LJ potentials with two specific TD factors (Equation 1). The TD CG model was parametrized in a bottom-up approach to best reproduce the temperature dependence of the free energy profiles (PMF) of two hydrophobic and of two charged moieties in water which were computed with an AA model (Figure 3). We further refined the TD CG model to reproduce the LCST and UCST collective behavior of (TPVAVG)₁₄ and (RGDSPYG)₁₂ multiple chains at low (280 K) and high (370 K) temperatures. Optimal model performance has been achieved with the parameters $α = -2$ and $β = 0.7$. It could be noted that the rescaling of the solute-solvent LJ potentials produced a temperature dependence of the solute-solute interaction free energies that is consistent with the top-down HPS-T model of Dignon *et al.*: hydrophobic and aromatic moieties have stronger interactions, while interactions involving charged beads are weakened as temperature increases.⁵⁰ However, our model does not rescale LJ potentials involving polar beads, unlike the HPS-T model. This suggests that polar residues play less important roles in distinguishing LCST from UCST low-complexity sequences.

The TD MARTINI 3 force field was applied to simulate the individual and collective dy-
namics behaviors of two other low-complexity sequences, (GVPGVA)₁₄ and (QYPSDGRG)₁₀,
with experimentally known LLPS property. Simulations were able to reproduce the com-
paction and extension upon heating of LCST and UCST single chains, respectively (Fig-
ure 5). The TD aggregated or dissolved states of multi-chain systems were also retrieved
using simulations with the TD MARTINI 3 model (Figures 6-9). Importantly, the simu-
lated aggregation and dissolution processes were reversible upon heating and cooling, in line
with experimental observations of the phase separation behavior of most LCST and UCST
IDPs.⁷⁹ It should be pointed out that, to reduce calculation times and computational costs,
the simulations presented here were carried out with protein concentrations of 0.5 and 1 mM,
which is much higher than those in experiments (0.05 mM) or those that are biologically
relevant. Therefore the computed transition temperatures were found lower and higher than
experimental values for LCST and UCST sequences, respectively. However, it is encourag-
ing to emphasize that these theoretical transition temperatures approach the experimental
values as protein concentrations decrease towards experimental conditions.

These studies significantly contribute to our understanding of IDP thermodynamics, and
shed light on the factors that govern their phase separation. For LCST IDPs, we know that
hydrophobic interactions are dominant driving forces behind their LLPS. Consequently, the
solvent release entropy of their aggregation (which drives the hydrophobic effect) is expected
to be greater at high than at low temperatures. To account for this phenomenon in our TD
CG model, we have weakened the LJ interactions between apolar and water beads (α < 0)
to accentuate solvent release from apolar residues as temperature increases. For UCST
IDPs, electrostatic interactions, particularly π-cation interactions, are key factors in their
aggregation at low temperatures and separation at high temperatures. Consequently, solute-
solute electrostatic repulsion must overcome the increase in solvent release entropy (which
always favors solute association) upon heating. To capture this phenomenon in our TD CG
model, we empirically strengthened the LJ potentials between the charged and water beads
(β > 0) to reduce solvent release from charged amino acids as temperature increases.
In essence, our research advances the field of IDP LLPS by providing a comprehensive framework for the theoretical study of their temperature dependence. Successful simulations of LLPS of low-complexity sequences with the TD MARTINI 3 force field clearly underscore the influence of their hydrophobic or charged amino acid composition on their LCST or UCST property. This opens up new avenues for studying the impact of temperature on LLPS behavior of natural IDPs, which typically have more complex amino acid motifs or sequences, and whose balance of apolar/charged residues can be modulated by post-translational modifications. Future research efforts could also focus on studying the impact of sequence mutations on IDP phase transitions, further improving our understanding of these biologically important molecules and their link to pathological dysfunctions. Ultimately, this knowledge of LLPS holds promise for the development of new therapies and biomaterials designed to take advantage of this unique property of IDPs.

## Acknowledgement

The authors acknowledge the financial support from the China Scholarship Council (Fellowship 202206790021). This work was supported by HPC resources from GENCI-IDRIS (Grants AD010710968R1 and AD010715692R1).

## Supporting Information Available

Additional figures showing examples of TD LJ potentials of the TD CG model, information about the convergence and error estimates of PMF profiles, radial distribution functions of water and ions around DAA molecule, and aggregation states of low-complexity sequences in simulations at high protein concentration.
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