Analysis and design of disordered polypeptides with optimized sequence patterning properties

Abstract

Intrinsically disordered proteins (IDPs) exhibit phase separation behavior that is closely linked to their degree of single-chain compaction, which in turn is governed by both amino acid composition and sequence patterning. Existing metrics such as sequence charge decoration (SCD) and sequence hydropathy decoration (SHD) describe these effects but are largely limited to describing differences between sequences of similar length and overall composition. In this work, we present a shuffle-based normalization scheme for SCD and SHD, enabling comparison of sequence patterning between very different IDP sequences. Leveraging this normalization scheme toward design space, we develop a Monte Carlo, based sequence design algorithm that generates novel IDPs with desired patterning features. Our design framework is further strengthened by incorporating additional metrics such as sequence aromatic decoration (SAD), compositional RMSD, and a previously developed sequence based ΔG predictor. We validate our approach through coarse-grained MD simulations, showing that the designed sequences exhibit tunable phase behavior. This strategy lays the groundwork for rational design of IDPs for biomedical and biotechnology applications, as well as basic biophysical research.

Author summary

Intrinsically disordered proteins behave similar to polymers in solution, having no defined structure. Their behavior is dictated by the collection of shapes the protein adopts, known as it’s “conformational ensemble” which is tuned by its amino acid sequence, and the solution environment. In this work, we have developed parameters to describe the patterning of charged and hydrophobic amino acids within these protein sequences, which are predictive of their ability to phase separate and form dense liquid-like droplets in solution. Importantly, the parameters we develop are motivated by physics and can be applied across a large number of amino acid sequences rapidly. This will enable researchers to rapidly predict the behavior of large libraries of protein sequences. We have additionally developed a software to design randomized amino acid sequences with desired amino acid composition, and patterning properties. Finally, we have tested our design scheme and parameters by running simulations of designed IDP sequences and quantified each of their ability to phase separate.

Introduction

Biomolecular condensates are specialized cellular compartments that form without membranes, helping cells organize important molecules through a process called liquid-liquid phase separation (LLPS) [1]. Unlike traditional organelles, condensates form membraneless organelles, compartments which can assemble and disassemble in response to changing conditions, and protein properties [2]. They play essential roles in organizing and regulating biological processes, such as DNA replication and repair, transcription, chromatin remodeling, RNA metabolism, ribosome biogenesis, protein quality control, cell division, intercellular adhesion, and signal transduction [3–9]. Disruptions in condensate dynamics have been linked to neurodegenerative diseases, cancer, and viral infections, with aberrant signaling and assembly potentially contributing to pathological states and pathways such as uncontrolled proliferation, and aggregation [10–13]. Given their broad functional importance, understanding how biomolecular condensates form and operate is vital for unraveling their roles in both normal physiology and disease [14].

A major factor influencing condensate formation is the distinct behavior of intrinsically disordered proteins (IDPs), which lack a stable three-dimensional structure and instead fluctuate among a diverse ensemble of conformations [15]. This conformational ensemble is governed by the IDP’s amino acid sequence and composition, which in turn determines how it interacts with other molecules and contributes to phase separation and condensate dynamics [16, 17]. Compositional features such as net charge, mean hydropathy, and charge density offer valuable insights into the conformational preferences and phase behavior of IDPs [16, 18]. Beyond composition, the arrangement of charged or aromatic residues within the amino acid sequence further modulates chain compaction and phase separation propensity [19–22]. Understanding these principles opens the door to rational sequence design, not only to map how sequence governs behavior, but also to engineer IDPs for specific functions or subcellular localizations [23–26]. Moreover, short synthetic sequences can serve as minimal models for LLPS, enabling mechanistic studies under controlled conditions [27].

Computational and theoretical studies have shown that the degree of single-chain compaction in soluble IDPs strongly correlates with their phase separation propensity—more compact chains generally phase separate more readily [21, 28–30]. This compaction is influenced by overall protein composition as well as the patterning of particular amino acid chemistries within the sequence. For example, blocky distributions of like-charged or hydrophobic residues promote intra-chain attractions, leading to collapse, and in turn, enhance intermolecular interactions that drive LLPS [20, 21, 31]. Quantitative metrics such as sequence hydropathy decoration (SHD) and sequence charge decoration (SCD) capture these effects and help predict phase behavior [24, 32, 33]. Our goal here is to refine the definitions of various patterning parameters and utilize them in design of unique IDP sequences.

In this work, we address limitations in how sequence-based patterning parameters—such as sequence charge decoration (SCD) and sequence hydropathy decoration (SHD), are currently defined and applied. While these metrics capture how charged and hydrophobic residues are arranged along a sequence, it is difficult to compare the degree of “blockiness” between sequences that are of different length or composition. Thus, studies applying SCD and SHD are typically limited to mutagenesis or randomization of individual sequences. To provide a more generalized metric for sequence “blockiness”, we first introduce a refined, shuffle-based normalization scheme for SCD and SHD that better contextualizes the influence of sequence patterning and how blocky different sequences are. We then develop a Monte Carlo design algorithm that leverages these improved metrics to generate novel IDP sequences with desired patterning characteristics. To enhance the robustness of this design framework, we later incorporate additional parameters including sequence aromatic decoration (SAD), compositional RMSD, and a sequence-based ΔG predictor [34]. Finally, we validate the designed sequences across diverse protein sequences using HPS-Urry simulations, demonstrating the algorithm’s ability to generate proteins with tunable phase behavior, laying the groundwork for applications such as biomolecular condensate-based drug delivery, and determination of compositional control within condensates [2, 35].

Results

Generalized parameters to quantify patterning within a fixed composition

Previous work has shown that the critical temperature of IDP phase separation can be roughly predicted by degree of collapse as a single chain [21, 29, 30, 36], generally demonstrating that IDPs are more collapsed will be more capable of phase separating, while those that are more extended, will be incapable of phase separation on their own. It’s also been demonstrated that the patterning of charged and hydrophobic amino acids is predictive of IDP chain dimensions [19, 20, 32], and phase separation [21, 37, 38] In this work, we focus specifically on the patterning parameters sequence hydropathy decoration (SHD) and sequence charge decoration (SCD). SHD is calculated from the scaled hydropathy values and sequence separation of all residues in an amino acid sequence:

$$SHD=\frac{1}{N}\sum _i^{N-1}\sum _{j>i}^N\frac{\left({\lambda }_i+{\lambda }_j\right)}{j-i}$$ (1)

where N is the sequence length, ${\lambda }_i$ is the scaled hydropathy value of residue i according to the Urry hydropathy scale [39] (S1 Table). SCD is calculated from the relative charge and sequence separation of formally charged residues in an amino acid sequence:

$$SCD=\frac{1}{N}\sum _i^{N-1}\sum _{j>i}^N{q}_i{q}_j{(j-i)}^{1/2}$$ (2)

where N is the sequence length and q_i is the formal charge of residue i.

Both parameters are demonstrated to be very effective at predicting IDP conformations [32, 40] as well as phase separation and materials properties [33, 38]. We note, however, that there is a limitation to using such metrics since they effectively describe the degree of patterning, but only within sequences of identical length and composition. When used for sequences having drastically different composition or size, we observe widely varying values of SCD and SHD. To demonstrate this disconnect, we consider seven disordered sequences of different sizes and compositions from biology (S2 Table). To evaluate the effect of sequence composition on range of accessible SCD and SHD values, we take an amino acid sequence, and shuffle it 1 million times, producing only sequences with identical composition, but random ordering of amino acids. This can be considered as a partial ensemble of constant-composition sequences, from which we can approximate the underlying probability density of patterning parameters, and “blockiness” of sequences. From these sequence ensembles, we calculate SCD and SHD of and plot the probability distributions (Fig. 1A,B). We note that for each individual sequence, the probability distributions have no overlap in SHD, and only moderate overlap in SCD, making direct comparison of SHD or SCD values between different proteins challenging to interpret. For instance, the randomized sequence of FUS LC that is the least “blocky” has a greater SHD value than the most “blocky” randomized sequence of hnRNPA2 LC. Thus, we note the limitation of SCD and SHD for quantifying blockiness of disparate sequences, particularly those with differing size and composition, and the need for a universal normalization scheme.

Fig 1.

The probability density of patterning parameter values based on real IDP sequences that have been shuffled 1 million times each. A) SHD values of shuffled IDP sequences each follow normal distributions with different mean values, and relatively small spread. B) SCD values from shuffled sequences show more spread and overlap than SHD (FUS and hnRNPF shown in inset since they have very few charged residues), C) collapsed SHD_norm and D) SCD_norm values match well with the standard normal distribution, with SCD_norm having an extended tail toward negative numbers.

$$SH{D}_{norm}=\frac{SHD-{\mu }_{SHD}}{{\sigma }_{SHD}}$$ (3)

$$SC{D}_{norm}=\frac{SCD-{\mu }_{SCD}}{{\sigma }_{SCD}}$$ (4)

We find that since the probability distributions are roughly gaussian in shape, this functions to collapse each to the standard normal distribution (Fig. 1C,D). This normalization serves to identify the significance of a value within each parameter’s respective fixed-composition distributions, giving a sense of whether this protein sequence is well-mixed, or segregated for something of its average composition. For example, the SCD_norm of WT LAF-1 RGG is on the right arm of the probability distribution, and that of WT α-synuclein is far to the left of the main density of the distribution. This indicates that the charged residues of LAF-1 RGG and α-synuclein, are more evenly distributed and more blocky than would be expected if randomly arranged, respectively.

We then ask the question whether we can bypass the need for shuffling sequences to obtain the probability density of accessible SCD and SHD values. Toward this goal of approximating ${\mu }_{SHD}$ and ${\sigma }_{SHD}$ for a given composition, we must first identify descriptors that don’t change upon shuffling, being only composition-dependent and not sequence-dependent. Starting with SHD, we find that the shuffled ensemble probability distribution can be predicted by just three composition-based parameters, i.e. the mean hydropathy (commonly denoted $<\lambda >$, but referred to as ${\mu }_{\lambda }$ here), standard deviation of hydropathy (${\sigma }_{\lambda }$) of all residues in the sequence, and chain length (N). To test the predictability, we designed 48 mock sequences ranging in size from 50 to 200 residues, with ${\mu }_{\lambda }$ between 0.4 tp 0.7 , and with ${\sigma }_{\lambda }$ values within the realm of possibility for the given ${\mu }_{\lambda }$. For example, a polyglycine sequence will have ${\mu }_{\lambda }\sim 0.5$ and ${\sigma }_{\lambda }=0$, while a sequence comprising half glutamate and half tryptophan will have roughly the same ${\mu }_{\lambda }$, but will have a larger ${\sigma }_{\lambda }$ value, ~ 0.5. Taking these 48 sequences, we shuffled each one million times, and calculated the ${\mu }_{SHD}$ and ${\sigma }_{SHD}$ values to provide real distributions which can then be predicted by the composition-based parameters (Fig. 2).

Fig 2.

Predicting probability density of SHD from composition-dependent parameters. A) The average SHD for a constant-composition ensemble depends on the average hydropathy and length of the sequence. B) The standard deviation of SHD values for constant-composition ensemble depends on standard deviation of hydropathy and length.

We first compare ${\mu }_{SHD}$ with ${\mu }_{\lambda }$ and find that there is a positive linear correlation, having a slope that is related to the length of the sequence (Fig. 2A). We then compare ${\sigma }_{SHD}$ with with ${\sigma }_{\lambda }$ and find another linear correlation, where the slope is dependent on N (Fig. 2B). Notably, ${\mu }_{SHD}$ is larger for longer sequences, indicating that the value of SHD increases for longer and more hydrophobic sequences (Fig. 2A). In contrast, ${\sigma }_{SHD}$ is smaller with increasing N, indicating smaller fluctuations occurring for greater sampling (Figure 2B). Intriguingly, we find that the average hydropathy is not predictive of ${\sigma }_{SHD}$ on its own, and the standard deviation of hydropathy is not predictive ${\mu }_{SHD}$ (S1 Fig). Using this data set, we derive a simple empirical formula for both the mean and standard deviation of possible SHD values just based on sequence composition within the bounds of the tested sequences,

$${\mu }_{SHD}=2.761{\mu }_{\lambda }\cdot N^{0.2354}$$ (5)

$${\sigma }_{SHD}=0.6500{\sigma }_{\lambda }\cdot N^{-0.4592}$$ (6)

This enables prediction of the probability distribution of SHD values for any amino acid sequence without the need for thousands or millions of shuffles, and provides a useful normalization scheme for the SHD patterning parameter.

We repeated this process for the SCD parameter, aiming to identify composition-dependent but sequence-independent features that could be used to estimate ${\mu }_{SCD}$ and ${\sigma }_{SCD}$. We similarly generated a library of synthetic sequences with varied charge compositions, having varied values of fraction of charged residues (FCR), net charge per residue (NCPR), and chain length, (N).

We observed that both the mean (${\mu }_{SCD}$) and standard deviation (${\sigma }_{SCD}$) of the SCD distribution showed approximately linear relationships with FCR (Fig. 3A,B), indicating that FCR alone provides a useful predictor in these cases. On the other hand, the relationships between SCD statistics and NCPR were non-linear (Fig. 2C,D), suggesting more complex dependencies when considering net charge effects. Using these trends, we determined empirical equations that combine FCR, NCPR, and N to estimate the mean and standard deviation of the SCD distribution for a given sequence composition:

$${\mu }_{SCD}=0.2555\cdot N^{1.5092}\cdot \left(NCP{R}^2-0.0054\cdot FCR\right)$$ (7)

$${\sigma }_{SCD}=\left(0.0531\cdot N^{1.1216}\right)\cdot FCR\cdot \left(NCP{R}^2-1.3640\cdot NCP{R}^4+N^{-0.3991}\right)$$ (8)

Fig 3.

Prediction of SCD distribution from sequence-independent parameters. Each point represents statistics calculated from 1 million shuffled sequences per composition, with solid lines indicating empirical fits. A-B) Mean and standard deviation of SCD vs. FCR for sequences with net charge of zero. C-D) Mean and standard deviation of SCD vs. NCPR for sequences of size N=100. Inset in panel C highlights smaller influence of FCR on average SCD.

In addition to charge and hydropathy patterning, there is also known to be an influence of aromatic residue patterning on IDP conformations and phase separation [22]. We thus develop a third patterning parameter, focused specifically on aromatic residues within the amino acid sequence, termed sequence aromaticity decoration (SAD) and following a similar functional form to SHD:

$$SAD=\sum _i^{N-1}\sum _{j>i}^N\frac{\left(a_i\cdot a_j\right)}{j-i}$$ (9)

where $a_i$ and $a_j$ is the aromaticity of residue i and j, where aromatic residues are assigned a value of 1, and non-aromatic residues are assigned a value of 0. We find that upon shuffling of random sequences, the SAD also follows a roughly gaussian distribution, and can be predicted using empirical equations as well:

$${\mu }_{SAD}=1.6188\cdot {\mu }_A^{2.0390}\cdot N^{0.2110}$$ (10)

$${\sigma }_{SAD}=1.4177\cdot {\sigma }_A^{2.4598}\cdot N^{-0.4404}$$ (11)

where ${\mu }_{SAD}$ and ${\sigma }_{SAD}$ are mean and standard deviation of aromaticity in the sequence. Intriguingly, both mean and standard deviation of the SAD distribution are dependent on both mean and standard deviation of the aromaticity in the amino acid sequence (${\mu }_a$ and $\left.{\sigma }_a\right)$). We find this is due to the fact that aromaticity is a binary in the sequence, and thus, the mean and standard deviation of aromaticity in a set composition are related to each other through the relation of random binary data, i.e. $\sigma =\sqrt{\mu (1-\mu )}$, since ${\mu }_A$ is constrained to be between 0 and 1. SAD will serve as a useful design parameter as it should be able to help design sequences with well-dispersed aromatic residues, and reducing the likelihood of having blocks of aromatic residues that can lead to aggregation or more solid or gel-like behavior of condensates [22].

Sequence Monte Carlo to design polypeptides with desirable properties

Contextualization and normalization of patterning parameters across all compositions and sizes offers great potential to accelerate design of sequences with multiple optimized physical properties. We have effectively created a coordinate space, within which values of these patterning parameters are equally weighted regardless of differing sequence compositions and lengths. Toward this goal, we leverage a Monte Carlo sampling algorithm to design new sequences with desirable properties. This includes quickly formulating sequences with targeted values of patterning parameters, as well as identical-composition of existing sequences with tuned propensity to phase separate.

We first define a loss function as a Monte Carlo energy function by inputting “target” values for a set of parameters, namely the aforementioned patterning parameters, and measuring the deviation of a designed sequence from each target value, and scaled by a weighting parameter.

$$E_{seq}=\sum _{i\in [SCD,SHD,\dots ]}{W}_i\left(abs\left(i-i_{\mathit{target}}\right)\right)$$ (12)

In order to apply the normalization to each term’s distance, the weight is set to $1/{\sigma }_i$. This effectively makes the change in the MC energy equivalent to the sum of the change of $SC{D}_{\mathit{norm}}$ and $SH{D}_{\mathit{norm}}$. As a result, weighing each term with respect to the previously derived standard deviation functions sets each parameter to have roughly equivalent importance in the overall design and optimization of an IDP sequence. This approach can also incorporate other design parameters, such as aromatic patterning, and deviation from a set sequence composition (Fig. 4A–C). We can also incorporate direct predictions of phase separation capability, such as the ΔG predicted from sequence through the algorithm developed by von Büllow et al. [34]. We implement each of these and discuss further in the following sections. Notably, the process of sequence design is modular and can be made to incorporate even more design parameters than discussed in this work.

Fig 4.

Design parameters and moveset for sequence Monte Carlo algorithm. A) SCD is related to the degree of segregation of charged residues, where well-mixed polyampholytes are close to zero, blocky polyampholytes have large negative SCD, and sequences with a significant net charge have a positive SCD. B) SHD tends to be higher for sequences having hydrophobic groups more localized toward the middle of the sequence. Typically, IDP sequences are not so extreme, so SHD functions to quantify blockiness of hydrophobic residues primarily, and is somewhat influenced by proximity of hydrophobic groups toward the edges of the sequence. C) Composition RMSD is a simple metric of how similar the current composition is from that of the base composition. D) Three types of moves are attempted for the MC code, mutation moves change a single residue’s type, swap moves swap the positions of two random residues, and shuffle moves randomize a part of the sequence.

To carry out sequence design, we use a Metropolis Monte Carlo approach where perturbations are made to the sequence, the new sequence “energy” is calculated, and moves are accepted or rejected based on the Metropolis Criterion [41]. To perturb the sequence, we introduce three types of “moves”, namely mutations, swaps and partial sequence shuffles (Fig. 4D). The MC moves result in changes in the loss function (ΔE) which is then used to accept or reject the move based on the Metropolis criterion

$$P_{\mathit{accept}}=\mathit{min}\left(e^{-\mathrm{\Delta }{E}_{\mathit{seq}}/T},1\right)$$ (13)

where T is a scaling parameter that determines the likelihood of accepting uphill moves to higher energies which may aid in escaping local minima in the energy function. For our study, we have set this to be 10⁻² as an intermediate value, allowing some exploration, but overall migration toward the optimized design parameters. This process will repeat until all parameters are within an interval of their desired values, or until the maximum set number of desired cycles has been reached.

As a trial run, we use LAF-1 RGG as an input, and begin to design a target sequence with each design parameter set to several standard deviations away from the initial value, with the exception of composition, which is set to stay as close to the original sequence composition as possible. SCD and SHD optimization was particularly fast, reaching target values around 200 moves (Fig. 5A,B). Alternatively, SAD and ΔG predictor were able to reach their target values within an attempted 2500 Monte Carlo moves (Fig. 5C,D). The composition of the sequence was allowed to fluctuate, but only did so moderately (Fig. 5E), staying very close to the original sequence composition. Consequently, the total loss function reached a minimum at ~ 2500 moves (Fig. 5F).

Fig 5.

(A-E) Trajectory of sequence MC converging to desirable properties of SCD, SHD, SAD, ΔG, and Composition RMSD, respectively. (F) Total loss progression over 5000 Monte Carlo moves.

To evaluate the success of the normalization scheme for patterning parameters, we plotted the energy decomposition for additional design runs for LAF-1 RGG, setting target values equal to initial values to minimize drift throughout the course of the simulation (S2 Fig). Within the same move type, comparing the mean of each graph gives an idea of the difference in average energy contributions from all parameters. As such, we would expect a successful normalization scheme to keep all contributions within a reasonable distance of one another. We find that for SCD, SHD, and SAD, each move type only shows variations within an order of magnitude in energy fluctuations S2 Fig, indicating successful normalization of these parameters and having each parameter equally likely to be optimized in the design process.

Efficient and guided design of low SCD sequences

With the identification of significant changes in phase-behavior between protein sequences of fixed-composition, a commonly explored area of interest is the relation between patterning parameters and a protein’s propensity for phase separation. In a previous study, we investigated the relationship between the phase-behavior of fixed-composition LAF-1 RGG variants at a wide-range of SCD values [38]. By performing 10 million random shuffles to the wild type sequence with an SCD of 0.6, the variant with the lowest attained SCD only reached −7.3. To recreate this effort using our Monte Carlo approach, we utilize LAF-1 RGG as a reference sequence to which further variants will be designed through shuffling. To keep composition from changing, we disabled mutation moves for this set of sequence designs. The shuffling moves attempted differ from the original study in that they are applied to only segments of the sequence, and their contribution to SCD is evaluated, such that changes that increase SCD are not accepted. This causes the sequence to walk a directed path toward the desired properties. We repeat this process 10 times for LAF-1 RGG variants with three different target SCDs of −5, −10, and −15, which averaged 180, 370, and 800 cycles, respectively, to accomplish (Fig. 6A–C). Each design task, two of which extended significantly past the attainable sequences from the aforementioned paper, was attained in fewer than 1000 shuffle moves. Notably, the calculation of SCD and SHD for millions of sequences can be time consuming, and also scales poorly with increasing chain length, making it even less feasible for longer IDP designs. Although the Monte Carlo algorithm relies on the same shuffling and calculation methodologies, its motivated search was able to reach more significant extremes of LAF-1 RGG’s SCD sequence ensemble in a fraction of the iterations. As such, this tool provides an efficient means of designing polypeptide sequences that lie far outside the range of easily-accessible sequences.

Fig 6.

(A-C) 15 Monte Carlo runs starting with LAF-1 RGG and attempting to create variants with an SCD of −5 (A), −10 (B), and −15 (C). (D) Coexistence curve of LAF-1 RGG WT along with the 3 previously created variants.

We then tested these designed sequences using coarse-grained MD simulations in the HPS-Urry model with slab geometry [42]. We simulated the WT LAF-1 RGG as well as three designed variants with increasingly negative values of SCD, finding that more negative values of SCD resulted in upward shifts of the binodal phase envelope, indicating stronger self-association and greater phase separation propensity (Fig. 6D). We also determine the critical temperature for these variants (Table 1) and find that the observed increase in critical temperature is comparable to that reported for shuffled sequences designed in previous work [38]. The designed sequences are listed in S3 Table.

Table 1.

SCD, SHD, SAD, and T_c values for LAF-1 RGG wild type and variants.

Protein name	SCD	SHD	SAD	Tc(K)
LAF-1 RGG WT	0.020	5.339	0.019	282.6
LAF-1 V1	−5.000	5.327	0.031	302.6
LAF-1 V2	−9.997	5.328	0.020	307.7
LAF-1 V3	−15.002	5.349	0.028	319.9

Adding new features to sequence optimization

Next, we incorporate an IDP’s predicted transfer free energy (ΔG) that directly quantifies the degree to which a sequence will phase separate, being related to the log of the IDP’s partition coefficient. To do this, we utilize the predictive algorithm from von Büllow et al. which predicts ΔG of phase separation directly from amino acid sequence [34]. We note that this predictive algorithm was trained on a separate dataset utilizing a different hydropathy scale from what we use to calculate SHD. This may be seen as an advantage however, since SHD and ΔG will have less redundancy in the design space.

We also introduce composition root-mean-square deviation (compRMSD) as a novel parameter that quantifies how much a designed sequence deviates from a reference sequence composition while allowing for Mutation moves.

$$compRMSD=\sqrt{\frac{1}{20}\sum _{i=1}^{20}{\left(f_i-f_{\mathit{target}},i\right)}^2}$$ (14)

where ${\text{f}}_i$ is the fraction of residues in the sequence that are amino acid i. Unlike fixed-composition approaches, enabling mutation moves, and restratining overall composition drift through compRMSD enables controlled variation, making it particularly valuable for designing IDPs that retain similar composition to the template sequences, but may differ somewhat in size, or total length. The weight of the compRMSD component can also be adjusted to modify the importance of preserving sequence composition.

By integrating these two parameters into our Monte Carlo sequence design approach, we designed two variants of the FUS LC wild-type (WT) protein: Variant 1 (V1) and Variant 2 (V2). As shown in the phase diagrams (Figure 7), both variants exhibit a lower ΔG compared to WT, correlating with an increased propensity for phase separation. These results demonstrate how incorporating additional predictive metrics, such as transfer free energy, allows for the rational design of IDP sequences with enhanced phase behavior.

Fig 7.

Phase diagrams of a series of FUS-like IDP sequences.

In summary, the designed FUS LC WT variants maintain compositions that are reasonably close to the wild type, with compRMSD values of 0.008 and 0.012 for variants V1 and V2, respectively (Table 2). By keeping compositional changes minimal, we ensure that the observed differences in predicted phase behavior primarily arise from altered residue patterning rather than shifts in overall amino acid content. The designed sequences are listed in S3 Table.

Table 2.

SCD, SHD, SAD, compRMSD, and ΔG values for FUS LC WT wild type and variants.

Protein name	SCD	SHD	SAD	compRMSD	ΔG (kB T)
FUS LC WT	0.039	5.922	0.065	0	−5.9
FUS WT V1	−0.121	5.977	0.101	0.008	−7.9
FUS WT V2	0.000	6.187	0.178	0.012	−10.1

Leveraging design strategy for short peptides to mimic larger IDPs

It is well-established that shorter chain lengths, particularly molecules with lower valence, are less prone to phase separation [28, 43–46]. However, recent work has shown that even short peptide sequences are capable of phase separation, usually owing to significant charge or aromatic content [27, 47, 48]. Short peptides have several advantages for experimental and computational research, including that simulations will converge more quickly and likely require fewer computing resources. We employed this method to design peptides that replicate the phase separation properties of FG Nucleoporin 153 (Nup153), a 1475-residue protein which is a component of the nuclear pore complex, and implicated in binding to HIV-1 viral capsid proteins [49]. To demonstrate an extreme example of reducing protein sequence size, we have designed several miniature versions of Nup153, or mini-NUP sequences of length 30 (Table 3).

Table 3.

SCD, SHD, SAD, compRMSD, and ΔG values for designed mini-NUP sequences. Values in parentheses are normalized values.

Sequence	SCD	SHD	SAD	compRMSD	ΔG (kB T)
mini-NUP seq1	−0.80(−8.18)	3.89(−1.98)	0.23(−2.45)	0.044	−6.39
mini-NUP seq2	−0.72(−7.39)	3.97(−2.05)	0.32(−2.12)	0.047	−7.66
mini-NUP seq3	−0.25(−4.71)	4.13(−3.66)	0.32(−2.11)	0.049	−7.84

Since we have shortened the Nup153 sequence by a factor of 50, we expect that keeping identical composition will significantly reduce the phase separation propensity to a degree that it will not phase separate at accessible experimental conditions. We thus, generated multiple variants of the peptide to have increasing likelihood of phase separation. To achieve the desired values of SHD, ΔG, and other sequence parameters, we increased the number of aromatic residues by introducing an additional 10% tyrosine and phenylalanine into the original FG Nucleoporin, Nup153. To quantify the compositional changes, we computed the compRMSD, which captures the difference in amino acid composition between the original sequence with added aromatic residues and the final designed sequences.

We conducted MD simulations of these peptide sequences to determine the phase separation propensity of each and find that we achieve a range of phase behaviors for each sequence (Fig. 8). The sequences of designed Nup153-like peptides are included in S3 Table.

Fig 8.

Phase diagrams of a series of 30-residue peptide sequences based on the composition of Nup153.

Methods

Simulations of designed IDP sequences were conducted using OpenMM simulation software [50] using the HPS-Urry coarse-grained model [42]. Each IDP was simulated at a range of temperatures where both a dense phase and a dilute phase were present to gather sufficient statistics to calculate the binodal coexistence curve. We ran simulations for at least 1 μs per temperature case using Langevin dynamics. For all trajectories, the largest cluster was centered in the simulation box using an autocorrelation function of the instantaneous density profile of each simulation frame [51]. The critical point of each sequence was determined using a fitting function, assuming the universality of the 3D Ising model as in previous work [45].

Force Field and Interaction Potentials

The HPS-Urry coarse-grained model was employed, where each amino acid is represented by a single bead located at the position of its $C_{\alpha }$ atom. The total interaction energy ($U_{\text{total}}$) in this model is a sum of bonded and non-bonded terms:

$$U_{\text{total}}=\sum _{i=1}^{N-1}{k}_b{\left(r_{i,i+1}-r_0\right)}^2+\sum _{i,j}{\varphi }_{ij}^{\text{HPS}}+\sum _{i,j}{\varphi }_{ij}^{\text{elec}}$$

The first term models the bonded interactions between adjacent residues as a harmonic potential with a spring constant k_b = 20 kJ/Ų and an equilibrium bond length of r₀ = 3.82 Å. The non-bonded interactions are composed of hydrophobic (${\varphi }_{ij}^{\mathrm{H}\mathrm{P}\mathrm{S}}$) and electrostatic (${\varphi }_{ij}^{\text{elec}}$) terms.

Hydrophobic Interactions

These interactions are described by a modified Ashbaugh-Hatch potential [52], which is based on the Lennard-Jones (LJ) potential:

$${\varphi }_{ij}(r)=\left\{\begin{array}{ll}{\varphi }_{ij}^{\mathrm{L}\mathrm{J}}(r)+\left(1-{\lambda }_{ij}\right)ϵ,& \text{for}r\le 2^{1/6}{\sigma }_{ij}\\ {\lambda }_{ij}{\varphi }_{ij}^{\mathrm{L}\mathrm{J}}(r),& \text{for}r>2^{1/6}{\sigma }_{ij}\end{array}\right.$$

where the standard LJ potential is defined as:

$${\varphi }_{ij}^{\mathrm{L}\mathrm{J}}(r)=4ϵ\left[{\left(\frac{{\sigma }_{ij}}{r}\right)}^{12}-{\left(\frac{{\sigma }_{ij}}{r}\right)}^6\right]$$

The energy scale is set by $ϵ=0.2$ kcal/mol. The residue-specific parameters for size (${\sigma }_i$) and hydropathy/stickiness (${\lambda }_i$) are combined using an arithmetic mean: ${\sigma }_{ij}=\left({\sigma }_i+{\sigma }_j\right)/2$ and ${\lambda }_{ij}=\left({\lambda }_i+{\lambda }_j\right)/2-\mathrm{\Delta }$, where $\mathrm{\Delta }=0.08$.

Electrostatic Interactions

Electrostatic forces are modeled using a Debye-Hückel potential to account for charge screening in solution [53]:

$${\varphi }_{ij}^{\mathrm{e}\mathrm{l}}(r)=\frac{q_i{q}_j}{4\pi Dr}{e}^{-\kappa r}$$

Here, $q_i$ and $q_j$ are the charges of the interacting residues. The dielectric constant of the solvent (D) was set to 80 to represent water, and the inverse Debye screening length (κ) was set to 1 nm⁻¹, corresponding to a monovalent salt concentration of approximately 100 mM.

Conclusion

By defining trends between composition-specific terms and patterning parameters, we have successfully established a method of approximating the properties of a given IDP’s constant-composition ensemble, and thus derived a normalization scheme for the common patterning parameters, SCD and SHD. Using this normalization scheme, we then leverage this into a method of comparing changes between different patterning parameters, putting them on an equal playing field in the design space. We test this out with an IDP sequence design Monte Carlo algorithm which can simulataneously optimize multiple design critera. Existing literature linking patterning parameters and LLPS further allows this Monte Carlo to represent a means of tuning the phase behavior of a given IDP by creating variants. We first demonstrate that our code can design extreme outliers of specific patterning parameters such as super charge-segregated LAF-1 RGG variants. We apply this method to design variants of FUS that are more prone to LLPS than the wild-type, focusing primarily on predicted ΔG of phase separation. We also design mini-NUP variants that maintain similar composition to the base sequence, Nup153, while scaling the importance of perfect compositional similarity to enable design of more “sticky” sequences more capable of phase separation, offsetting the significantly reduced size of the sequence.

Looking forward, this work can be supplemented by inclusion of multiple other parameters for optimization, such as: secondary structure predictors, disorder predictors or any other sequence-based parameter. Furthermore, as the link between patterning parameters of an IDP and the properties of its resultant condensate continue to be elucidated, this work could incorporate condensate properties into its analysis as opposed to just a general change to phase separation propensity. For example, as SAD seems to indicate a more gel-like condensate, this work could be improved to give users better control over designed condensate properties by giving additional guidance into what parameters of SAD may correlate with liquid-like and gel-like condensates.

Supplementary Material

Supplement 1

S1 Fig. Correlation of different predictive parameters with ensemble-calculated mean and standard deviation of SCD.

S2 Fig. Energy decomposition of five parameters with respect to mutations, swaps, and shuffles separately. These also include unaccepted Monte Carlo moves. The range of values observed for the three patterning parameters are within an a few-fold difference for each move type, indicating success in the normalization scheme for application to MC sequence design.

S1 Table. Urry normalized hydropathy values used for each amino acid.

S2 Table. List of amino acid sequences for real IDPs used in initial shuffling study.

S3 Table. List of designed sequences used in this work in MD simulations of phase separation.

S1 Appendix. Supporting information document with 2 figures, and 3 tables.

Data availability statement

All data used to generate figures in this work will be deposited in a repository on Zenodo following acceptance of this manuscript. We will also publish the source code for the Monte Carlo sequence design software in the same Zenodo repository upon acceptance of this manuscript. Before publication, we will include reference numbers and doi for the Zenodo repository in the final version of the article. Additional data such as raw simulation trajectories will be available upon reasonable request.