] Author to whom the correspondence should be addressed

# Fine structures of Intrinsically Disordered Proteins

Swarnadeep Seth  Brandon Stine  Aniket Bhattacharya  [
<EMAIL_ADDRESS>Department of Physics, University of Central
Florida, Orlando, Florida 32816-2385, USA

###### Abstract

ABSTRACT

We report simulation studies of 33 single intrinsically disordered proteins
(IDPs) using coarse-grained (CG) bead-spring models where interactions among
different amino acids are introduced through a hydropathy matrix and
additional screened Coulomb interaction for the charged amino acid beads. Our
simulation studies of two different hydropathy scales (HPS1, HPS2) [Dignon et
al., PLOS Comp. Biology, 14, 2018, Tesei et al. PNAS, 118, 2021] and the
comparison with the existing experimental data indicates an optimal
interaction parameter $\epsilon=0.1$ kcal/mol and $0.2$ kcal/mol for the HPS1
and HPS2 hydropathy scales. We use these best-fit parameters to investigate
both the universal aspects as well as the fine structures of the individual
IDPs by introducing additional characteristics. (i) First, we investigate the
polymer specific scaling relations of the IDPs in comparison to the universal
scaling relations [Bair et al., J. Chem. Phys. 158, 204902 (2023)] for the
homopolymers. By studying the scaled end-to-end distances $\langle
R_{N}^{2}\rangle/(2L\ell_{p})$ and the scaled transverse fluctuations
$\tilde{l}^{2}_{\perp}=\sqrt{\langle{l_{\perp}^{2}}\rangle}/{L}$ we
demonstrate that IDPs are broadly characterized with a Flory exponent of
$\nu\simeq 0.56$ with the conclusion that conformations of the IDPs
interpolate between Gaussian and self-avoiding random walk (SAW) chains. Then
we introduce (ii) Wilson charge index ($\mathcal{W}$) that captures the
essential features of charge interactions and distribution in the sequence
space and (iii) a skewness index ($\mathcal{S}$) that captures the finer shape
variation of the gyration radii distributions as a function of the net charge
per residue and charge asymmetry parameter. Finally, our study of the (iv)
variation of $\langle R_{g}\rangle$ as a function of salt concentration
provides another important metric to bring out finer characteristics of the
IDPs which may carry relevant information for the origin of life.

##  I Introduction

Intrinsically disordered proteins (IDPs) are characterized by a low proportion
of hydrophobic residues and a high content of polar and charged amino acids
which make them distinct from those which fold [1]-[7]. IDPs lack well defined
three-dimensional structures and do not participate in forming
$\alpha$-helices or $\beta$-strands, and other secondary or tertiary
structures. Since their discovery almost three decades ago the number of IDPs
has been growing at a steady rate [8]. It is now known that almost 30% of the
proteins are either IDPs or folded proteins have intrinsically disordered
regions (IDR) [9] which play crucial roles in numerous biological processes,
such as regulating signaling pathways, helping in molecular recognition, in
initiating protein-protein interactions, and serve as molecular switches [10,
11]. The conformal flexibility of IDPs helps mediate interactions with binding
partners to form components of macromolecular complexes [12, 13]. The
flexibility and often faster dynamics allow IDPs to bind to multiple different
proteins [11, 14]. The IDP complexes [12] have also been realized to play a
central to the pathology of several degenerative diseases: $\alpha$-synuclein
(Parkinson’s disease), tau (Alzheimer’s disease), and IAPP (Type II Diabetes)
[15, 16]. IDPs form membrane-less intra-cellular compartments, have been
identified to be the key drivers of liquid–liquid phase separation in the cell
[17, 18, 19], and therefore, have generated tremendous interest in the
underlying phase separation in IDPs, formation of IDP complexes and the role
of charge separation in such systems using well established concepts of
polymer physics [12, 20, 21, 22]

Evidently the study of IDPs in the last two decades has been an active area in
various branches of science. Despite tremendous growth and interest in
studying IDPs - the discovery of new IDPs and their fast dynamics made it
difficult to study experimentally using small angle X-Ray scattering (SAXS)
[23], single molecule Förster resonance energy transfer (smFRET) [22, 24, 25,
26] and solution nuclear magnetic resonance (sNMR) [27] which have produced
conflicting results. The conformational information, such as end-to-end
distances and gyration radii, are also available for a limited number of IDPs.
Thus, an integrative structural biology approach that combines experimental
techniques [28], using NMR spectroscopy SAXS, smFRET, combined with
computational methods seem to be a practical and feasible approach to unravel
the conformational properties and interactions of IDPs, shedding light on
their structural ensembles.

Historically, computer simulation studies of CG models of polymers have played
an important role as a stand-alone discipline between theory and experiments
successfully predicting conformational and dynamic properties of neutral and
charged polymers [29]. Similar studies have been generalized for the IDPs
taking into account different sizes, charges, and hydropathy indices of the 20
different CG amino acid beads [30, 31, 32, 33, 34, 35]. Data driven approaches
aided by Machine Learning algorithms have helped to build hydropathy indices
for numerical studies of IDPs [32, 36]. In this article we enlarge the scope
of validity of a subset of these models (HPS and M3) by comparing simulation
results obtained for a large number of IDPs using these two models with
experimental data [37, 38, 6, 39, 28, 40] and with the results contained using
other CG models [33, 34].

##  II Coarse grained models of IDPs

One of the hallmarks of IDPs is their characterization using the Uversky plot
[1], where it has been shown that when the mean net absolute charge $\langle
Q\rangle$ of a polypeptide chain at neutral pH is plotted against the mean
side chain hydropathy $\langle H\rangle$, measured using the Kyte-Doolittle
[41] hydrophobicity scale, a boundary line

$\langle Q\rangle=2.785\langle H\rangle-1.151$ (1)

separates the compact (natively folded or globular) and expanded (coil-like or
pre-molten globular) conformations [1, 42]. Habchi et al. [42] improved Eqn. 1
but the basic observation remains the same. Evidently, a larger charge and a
smaller hydropathy ensure the extended structure of the IDPs. The simplicity
has been appealing to build CG models of IDPs based on hydropathy, where the
standard bead-spring model of a homopolymer has been generalized to
incorporate the relative well depth between any two amino acids through a
hydropathy matrix. [30]. Mittal and coworkers [31, 43] have used this HPS
model to compare the gyration radii for several IDPs and found a reasonably
good agreement. A slightly different version has been used by Tesei et al.
[32]. Unlike the HPS models [30, 31, 32] where hydropathy is introduced
directly, other implicit solvent CG models have been used to study various
properties of IDPs. Pappu and coworkers developed a software called ABSINTH
(Assembly of Biomolecules Studied by an Implicit, Novel, and Tunable
Hamiltonian) to study phase transitions in IDPs [33]. Thirumalai and coworkers
used another CG model called SOP-IDP (self-organized polymer (SOP) coarse-
grained model for IDPs) with a finer level of granularity where, except for
Glycine and Alanine, the rest of the amino acid residues are represented using
a backbone bead and a side-chain (SC) bead [34]. All these models are
computationally more efficient compared to the models with explicit solvent
molecules and hence can be used to study macromolecular condensates of IDPs
leading to liquid-liquid phase separation [17, 18, 19, 44, 45, 46, 48, 47] in
membraneless organelles.

Since their discovery, progressively more IDPs are being cataloged [8].
Compared to the models for the folded proteins, the CG models of the IDPs are
relatively new. Due to their flexibility and faster dynamics, the experimental
studies of the IDPs are relatively limited and often very difficult to
interpret. Thus, studies of several CG models with convergence to the
experimental results is an important aspect of developing a better
understanding of the fundamental physics of the IDPs which share properties of
the polyelectrolytes and polyampholytes, but exhibit very different sequence-
specific behaviors. In this article we critically examine the parameters of
two hydropathy models of the IDPs by enlarging the scope of previous work
[31]-[32]. The gyration radii obtained from the simulation then have been
reanalyzed with reference to the universal properties of the homopolymer
model. Another important aspect of this work is that we have introduced new
physically motivated metrics to analyze these fine structures of the IDPs.
Finally, we provide specific examples in the result section, that IDPs
clustered around different regions of the Charge-Hydropthay space may exhibit
markedly different characteristics.

Figure 1: (a) Coarse-grained model of a portion of IDP containing 9 amino acid
residues with one letter code EGKDRCVWR. Each amino acid residue is
represented by the corresponding three letter code bead of different diameter
$\sigma$ (not to the scale) listed on the table. (b) The table lists the one-
letter and three-letter codes of the amino acids in the $1^{st}$ and $2^{nd}$
columns. The mass, charge, and diameter of the amino acid residues are
tabulated in the $3^{rd}$, $4^{th}$, and $5^{th}$ columns respectively.
Finally, the HPS1 and HPS2 hydropathy scales are shown in $6^{th}$ and
$7^{th}$ columns.

##  III Simulation Model

The CG amino acid residues interact among themselves by a modified Van der
Waals interaction potential, first introduced by Ashbaugh and Hatch [30] given
by

$\displaystyle
U_{VdW}\left(r_{ij}\right)=\begin{cases}U_{LJ}\left(r_{ij}\right)+(1-\lambda_{ij})\epsilon_{ij},&\text{$r_{j}\leq
2^{\frac{1}{6}}\sigma_{ij}$}\\\
\lambda_{ij}U_{LJ}\left(r_{ij}\right),&\text{otherwise}\end{cases}$ (2)

where $U_{LJ}$ is the Lennard-Jones (LJ) potential,

$\displaystyle
U_{LJ}\left(r_{ij}\right)=4\epsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right],$
(3)

$r_{ij}=\left|\vec{r}_{i}-\vec{r}_{j}\right|$ and
$\sigma_{ij}=\frac{1}{2}\left(\sigma_{i}+\sigma_{j}\right)$ are the distance
and the effective diameter between the amino acid beads of diameter
$\sigma_{i}$ and $\sigma_{j}$ with indices $i$ and $j$ positioned at
$\vec{r}_{i}$ and $\vec{r}_{j}$ respectively. The strength of the van der Waal
interaction interaction
$\epsilon_{ij}=\sqrt{\epsilon_{i}\epsilon_{j}}\cong\epsilon$ chosen to be the
same, and $\lambda_{ij}=\frac{1}{2}\left(\lambda_{i}+\lambda_{j}\right)$ is
the average hydropathy factor between any two amino acids with indices $i$ and
$j$. A harmonic bond potential

$U_{b}\left(r_{ij}\right)=\frac{k_{b}}{2}\left(\frac{r_{ij}-r^{0}_{ij}}{\sigma_{ij}}\right)^{2}.$
(4)

acts between two consecutive amino acid residues $i$ and $j=i\pm 1$. The
spring constant $k_{b}$ = 8033 kJ/(mol$\cdot$nm2) = 1920 kcal/(mol$\cdot$nm2),
and the equilibrium bond length is $r^{0}_{ij}=r_{0}=0.38$ nm.

A screened-Coulomb (SC) interaction acts between any two charged amino acids

$U_{SC}\left(r_{\alpha\beta}\right)=\frac{q_{\alpha}q_{\beta}e^{2}}{4\pi\epsilon_{0}\epsilon_{r}}\left(\frac{e^{-\kappa
r_{\alpha\beta}}}{r_{\alpha\beta}}\right)$ (5)

where the indices $\alpha$ and $\beta$ refer to the subset of the indices $i$
and $j$ for the charged amino acids, $\epsilon_{r}$ is the dielectric constant
of water, and $\kappa$ is the Debye screening length [49]. The inverse Debye
length $\kappa^{-1}$ is dependent on the ionic concentration (I) and expressed
as

$\displaystyle\kappa^{-1}$ $\displaystyle=\sqrt{8\pi l_{B}IN_{A}\times
10^{-24}}$ (6)

where $N_{A}$ is the Avogadro’s number and $l_{B}$ is the Bjerrum length,

$\displaystyle l_{B}=\frac{e^{2}}{4\pi\epsilon_{0}\epsilon_{r}k_{B}T}.$ (7)

At higher temperatures, the dielectric constant typically decreases, which
affects the strength of the electrostatic interactions. If the dielectric
constant does not account for temperature effects, the electrostatic
interactions may be overestimated, leading to unrealistic protein
conformations or interactions. Hence, we implement the temperature-dependent
dielectric constant of water as expressed by the empirical relation [50]

$\displaystyle\epsilon_{r}(T)=\frac{5321}{T}+233.76-0.9297T$
$\displaystyle+1.147\times 10^{-3}T^{2}-8.292\times 10^{-7}T^{3}.$ (8)

##  IV Hydropathy scales

Historically many hydropathy scales have been introduces to model the
properties of amino acids and provide a quantitative measure of the
hydrophilicity or hydrophobicity of amino acids based on their propensity to
reside in a water-soluble or water-insoluble environment [41, 51, 52, 53, 54].
Each of these scales assigns a numerical value to each amino acid, reflecting
its hydrophobic or hydrophilic nature. The scores obtained from hydropathy
scales are useful in predicting protein structure and function. Recently,
specific hydropathy scales are employed to study formation of condensates and
liquid–liquid phase separation (LLPS) behavior in IDPs. Dignon et al. proposed
HPS [31] hydropathy scale where Proline and Phenylalanine are considered to be
the most hydrophobic with $\lambda^{HPS}=1$ and Arginine is the least
hydrophobic with $\lambda^{HPS}=0$. All the amino acids’ hydropathy are scaled
to fit in the range. Later, Tesei et al. used the Bayesian parameter-learning
procedure to further optimize the hydropathy values and showed M3 [32]
hydropathy scale performs better to produce radius of gyration values closer
to the experiments.

###  IV.1 HPS1 Scale

Dignon et al. proposed HPS [31] hydropathy values which are listed in the
$6^{th}$ column of Fig. 1(b). The original HPS model uses the constant
dielectric constant of water $\epsilon_{r}=80$ for simulating the IDPs.
However, in some cases, such as, for CoINT (row 13 in Table-I), experiments
are performed at a lower temperature. To account these temperature changes we
used the analytic expression of temperature-dependent dielectric constant (see
Eq. 8) which makes our model slightly different from the original HPS model.
However, the amino acid hydropathy scale remains identical. We denote it as
the HPS1 scale in our simulation.

###  IV.2 HPS2 Scale

Tesei et al. used the Bayesian parameter-learning procedure to further
optimize the hydropathy values and showed M3 [32] hydropathy scale performs
better to produce radius of gyration values closer to the experiments. In the
original M3 model the charges of the end amino acids of an IDP are modified
and the charge of the Histidine residue is tuned as a function of the pH.
However, in our simulation we do not alter the end charges and the charge of
the Histidine residue to make a consistent comparison of the results with the
other existing models. We denote this hydropathy scale of the amino acids as
HPS2 and are shown in $7^{th}$ column of Fig. 1(b).

Table 1: Comparisons of gyration values from experimental (column 11) and
simulation studies obtained using HPS1 (column 12) and HPS2 (column 14)
models. The symbols are explained in the main text. Experimental values listed
in column 11 correspond to the IDP references in the 2nd column obtained from
Baul et al. [34]($\circ$), Dignon et al. [31]($\square$), Tesei et al.
[32]($\Diamond$), and Dannenhoﬀer-Lafage and Best [36]($\triangle$). The 12th
and 14th columns represent the simulation radius of gyration results obtained
by the HPS1 and the HPS2 hydropathy scale for $\epsilon=0.1$ kcal/mol, and
$0.2$ kcal/mol respectively producing least MSE (see Eqn. 9, and Fig. 2) with
the corresponding percentage errors in the 13th and the 15th column.

##  V Results

We studied 33 different IDPs with varying numbers of amino acids (N = 24 -
283) with both net positive and net negative charges (see the $6^{th}$ column
of Table- 1). All these IDPs have been studied earlier by different CG models
[31, 34, 32]. However, some of the IDPs such as CspTm, Protein-L, hCyp, R15,
and R17 are either actually intrinsically disordered regions (IDRs) of the
folded proteins or unfolded state of normally folded proteins [26, 22]. Within
the range of our studied IDPs, An16 is a polyelectrolyte (PE) containing only
six positively charged Histidine residues, SIC1 contains six positively
charged Lysine and five positively charged Arginine residues, Nucleoporin153
which contains only uncharged residues, and the rest of the 30 IDPs are
polyampholytes (PAs). The table is sorted according to their net charge $Q$
($6^{th}$ column) from highly positive (in red) to highly negative (in dark
green) values. The first row of the table lists K32 that contains 14 $\%$
Lysine (+1) and 4 $\%$ Aspartic acid (-1) which makes it highly positive. On
the other hand, the bottom row is ProTa-N that contains 20 $\%$ Glutamic acid
(-1), 17 $\%$ Aspartic acid (-1) and 8 $\%$ Lysine (+1), which makes it highly
negative. We assign unique letter codes for each of the IDPs, Greek letters
$\alpha$-$\rho$ in the ascending order starting from highly positively charged
IDPs and in descending order with the alphabets starting from the negatively
charged IDPs. The second to sixteen columns of Table-I list the names of the
IDPs, the length (N), the ionic concentration (I) in mM, the temperature
(T(K)) at Kelvin scale, total charge $Q$, number of charged beads (Nq), the
absolute and net charge per unit length $q_{\rm abs}$ and $q_{\rm net}$ (Eqn.
21), the charge asymmetry parameter $f^{*}$ (Eqn. 18), $R_{g}^{\rm expt}$,
$R_{g}^{\rm HPS1}$, $\Delta R_{g}^{\rm HPS1}$, $R_{g}^{\rm HPS2}$, $\Delta
R_{g}^{\rm HPS2}$ in units of nm, and the Wilson parameter $\mathcal{W}$ (Fig.
4) respectively.

Figure 2: The comparison of $R_{g}^{sim}$ with $R_{g}^{expt}$ for two
different hydropathy models - the $1^{st}$ row shows the simulation results
with HPS1 hydropathy scale ([31]) and the HPS2 hydropathy scale ([32]) results
are shown in the $2^{nd}$ row for different values of LJ interaction strengths
$\epsilon$ in kcal/mol unit. The black dashed line serves as a guide to show
the highest positive correlation between the experiment and simulation results
with a unit slope, and the deviations are calculated as mean square errors,
shown as MSE, in each subplot.

The aforementioned two hydropathy scales HPS1 [31] and HPS2 [32] are used to
study 33 IDPs using the coarse grained simulation method described in Sec. II.
The gyration radii $R_{g}^{HPS1}$ and $R_{g}^{HPS2}$ are obtained from
simulation using hydropathy scales HPS1 and HPS2 as a function of $\epsilon$
(in kcal/mol) are then used to calculate the mean square error (MSE)

$\displaystyle
MSE=\frac{1}{N_{tot}}\sum_{i=1}^{N_{tot}}\left[R_{g}^{expt}(i)-R_{g}^{k}(i)\right]^{2}$
(9)

to compare the deviation from the experimental gyration radii $R_{g}^{expt}$.
Here $N_{tot}$ is the number of different IDPs studied, and $k$ refers to
either HPS1 or HPS2. In Fig. 2 we show the scatter plots of the experimental
vs. gyration radii for five different values of $\epsilon$
($\epsilon=0.08-0.13$ kcal/mol for the HPS1 model and $\epsilon=0.1-0.3$
kcal/mol for the HPS2 model) and conclude that $\epsilon$ = 0.1 kcal/mol and
0.2 kcal/mol has the lowest errors (MSE = 0.23 and 0.26) for the HPS1 and HPS2
models respectively. The gyration values from these models are nearly
identical as shown in the inset (left) of Fig. 3(a). In this paper we use HPS1
hydropathy scale with $\epsilon$ = 0.1 kcal/mol to further analyze the
properties of the IDPs. However, we have carried out the same simulation
studies using the HPS2 scale using $\epsilon=0.2$ kcal/mol (not shown in the
manuscript) and the results are nearly identical to those from HPS1 model.
Both the scales show $\approx 14$% error (Table-I, Columns 13 & 15) for the
set used here.

###  V.1 Universal Scaling Properties of the IDPs

Despite the fact the IDPs are mostly described as polyampholytes (PAs) or
polyelectrolytes (PEs) [55], a fraction of experimental and theoretical
studies using the HPS model by Dignon et al. describe IDPs as Gaussian chains
[43], while in a recent publication, Thirumalai and coworkers using a two-bead
CG model calculated the RMS $R_{g}\equiv\sqrt{\langle R_{g}^{2}\rangle}$ and
the end-to-end distance $R_{N}\equiv\sqrt{\langle R_{N}^{2}\rangle}$, and
concluded that globally the IDPs are described not as the Gaussian chains, but
described as fully flexible excluded volume (EV) self-avoiding-walk (SAW)
chains that obey the Flory scaling $R_{g}=aL^{0.59}$ in three dimensions (3D),
where $L$ is the contour length of the IDP [35]. We investigate this point
further to find out to what extent the properties of the IDPs are universal.
From theoretical arguments following Schaefer et al. [56] and Nakanishi [57]
it is established that a proper description of a semi-flexible EV chain
characterized by a contour length $L$ and a persistence length $\ell_{p}$ in
$d$ spatial dimensions is given by

$\sqrt{\langle R_{N}^{2}\rangle}\simeq
b_{l}^{\frac{d-2}{d+2}}N^{\frac{3}{d+2}}\ell_{p}^{\frac{1}{d+2}}=b_{l}^{\frac{d+1}{d+2}}\left(\frac{L}{b_{l}}\right)^{\nu}\ell_{p}^{\frac{1}{d+2}}.$
(10)

Here, $N$ is the number of monomers of the chain so that length of the IDP is
$L=(N-1)b_{l}\simeq Nb_{l}$ (for $N\gg 1$), $b_{l}$ is the bond length between
two neighboring monomers, and the mean-field Flory exponent $\nu=3/(d+2)$ in
2D = 0.75 and in 3D = 0.60 ($\approx 0.588$ actual) respectively. This EV
chain accurately describes the limit $L/\ell_{p}>>1$ and supersedes the Worm-
like-chain model [58]

$\frac{\langle
R_{N}^{2}\rangle}{L^{2}}=\frac{2\ell_{p}}{L}\left(1-\frac{\ell_{p}}{L}[1-\exp(-L/\ell_{p})]\right).$
(11)

which does not take into account the EV effect and hence saturates to $\langle
R_{N}^{2}\rangle=2L\ell_{p})$ even when $L/\ell_{p}>>1$. In a previous
publication we have shown that scaled end-to-end distance $\langle
R_{N}^{2}\rangle/(2L\ell_{p})$ and the scaled transverse fluctuation
$\sqrt{\langle{l_{\perp}^{2}}\rangle}/{L}$ as a function of $L/\ell_{p}$
collapse onto the same master curve [59, 60] for all ratios of $L/\ell_{p}$
spanning rod to Gaussian and the EV limit. We would like to discuss our
simulation findings for the IDPs in the context of these universal scaling
plots (Fig. 3).

Fig. 3(a) summarizes the simulation results from HPS1 (magenta solid circles)
and HPS2 (blue solid circles) models. For comparison we have also included the
experimental points (green circles). The red dashed line corresponds to
$\langle R_{g}\rangle=0.38L^{0.56}$ (in nm shown explicitly as a power-law at
the inset (bottom-right)). Please note that the prefactor 0.38 nm is exactly
the same as the average bond-length obtained from the simulation. The dashed
line (black) $\langle R_{g}\rangle\approx L^{0.588}$ corresponds to the
excluded volume chain in 3D.

To get a clearer perspective we have calculated the length of the IDPs and
compared the scaled end-to-end distances $\langle R_{n}^{2}\rangle/2L\ell_{p}$
in units of dimensionless length $L/\ell_{p}$ [59, 60] (blue circles) in Fig.
3(b). These data points will serve as a guide and help readers to visualize
the deviation of the scaling properties of the IDPs in reference to the
universal master plot for the self-avoiding chain. The data points for the
IDPs show that while H5T5, CspTm and ProTaC lie in the Gaussian regime, the
rest of the IDPs are in between Gaussian (WLC) and self-avoiding-walks in 3D
but closer to being represented as self-avoiding-walks in 3D, consistent with
the conclusion from Fig. 3(a). It is worth mentioning that a slightly lower
exponent ($0.56<0.588$) may be attributed partly due to the finite size effect
for the chain lengths $24\leq N\leq 283$ used in this study. Our conclusion is
further strengthened by Fig. 3(c) inset plot, where we find that the scaled
transverse fluctuations collapse with a slope of $0.56-1=-0.44$. These results
possibly bridge and explain both the previous findings by Dignon et al. [43]
where they identified the Gaussian behavior of the IDPs and by Baul et al.
[34] where they described IDPs as self-avoiding chains. We will see in section
V-C that IDPs can be further classified in terms of their skewness indices
($\mathcal{S}$-index captures the overall shape). $\mathcal{S}$-index will
also affect an IDP’s global positioning with respect to the universal master
plot. For example, CspTm and ProTa-C having larger $\mathcal{S}$-index lie on
the WLC line. This finer distinctions make IDPs more interesting. One can use
the universal curve to understand why they are more Gaussian or behave more
like a self-avoiding chain. In the next two sections we will introduce such
characteristics to further classify IDPs.

Figure 3: (a) Log-log plot of IDPs radius of gyration as a function of chain
length $L$ obtained from HPS1 (magenta circles) and HPS2 (blue circles)
models, and from the experiments green circles). The red dashed line shows the
best fit of the simulation data. (top) The inset (top-left) show the
comparison of the gyration radii from HPS1 and HPS2 models and the black
dashed line indicates line of unit slope. The other inset (bottom-right) shows
the simulation gyration radii $\sqrt{\langle R_{g}^{2}\rangle}$ as a function
of $L^{0.56}$ and the red dashed line corresponds to $\sqrt{\langle
R_{g}^{2}\rangle}=0.38L^{0.56}$ (b) Log-log plot of the scaled end-to-end
distances, $\langle R_{N}^{2}\rangle/2Ll_{p}$, as a function of $L/l_{p}$ for
homopolymer chains in $\circ$ for a variety of combinations of $L$ and
$\ell_{p}$ for different IDPs in colored symbols for the HPS1 model. The
dashed purple line in each figure shows the behavior of the WLC model [Eq.
(11)]. (c) Log-log plot of the scaled transverse fluctuation $\sqrt{\langle
l_{\perp}^{2}\rangle}/L$, as a function of $L/l_{p}$ with the symbols having
the same meaning as (b).

###  V.2 Wilson index $\cal{W}$ of the IDPs

Figure 4: The schematic diagram shows the derivation of Wilson charge
$\tilde{\mathcal{W}}(n)$ of a hypothetical fasta sequence ESRKRT of length 6.
The table shows the different $\tilde{\mathcal{W}}(n)=0.75\rm{-}0.333$ for
different window lengths $n=2\rm{-}6$ averages.

Now that we have demonstrated an approximate universal description of the IDPs
in terms of the HPS1 model, we want to demonstrate the sequence specific
features that make each IDP distinct and may exhibit very different behaviors
from their homopolymer counterparts. Evidently, the charges present along the
sequence play a crucial role in shaping the structure and dynamics of the
IDPs. Out of 20 amino acids, only five of them carry a charge. Specifically,
in the HPS1 model, both “R” and “K” have a $+1$ charge, “H” possesses a $+0.5$
charge, and “D” and “E” have charge of $-1$. However, the charges are randomly
distributed along the chain backbone and therefore, IDPs in general can be
classified either as PA or PE [21, 7]. Das and Pappu showed that weak
polyampholytes form globules, whereas for a strong polyampholyte the net
charge per residue and their linear distribution along the chain backbone
controls their conformational preferences [21]. FRET experiments have
explicitly demonstrated expansion of charged IDPs at low ionic concentration
[65]. Extensive research has been conducted to study this phenomenon in the
existing literature. Nonetheless, our study delves deeper to investigate the
positional implications of the amino acid sequence in terms of Wilson index
($\mathcal{W}$) as described below.

Unlike a homopolymer, an IDP can have varied degree of local stiffness and
flexibility resulting in the amino acids in different segments interacting
with neighboring sequences even if they are far apart in the sequence space.
To capture these potential dynamical interactions, we employ the concept of
Wilson Renormalization extensively used to study the spin systems [61]. This
renormalization approach allows us to analyze the sequence of charges and
their unfolding interactions, considering interactions up to the next nearest
neighbor. Fig. 4 illustrates a hypothetical example of a short IDP sequence
“ESRKRT” of length 6, showcasing the presence of a negative charge at the
beginning followed by three positive charges in the middle, and the remaining
amino acids being neutral. To initiate the averaging procedure, we select a
window of length $n$. The simplest case $\tilde{\mathcal{W}}(2)$, considers
sliding averages of window length n=2 and denotes the next neighbor
interactions. The window length for averaging can range from 2 to $N$, where
$N$ represents the number of amino acids in the IDP sequence. For the general
case of $\tilde{\mathcal{W}}(n)$, where n consecutive charges are averaged,
the process begins by sliding an averaging window from one end of the sequence
toward the other. After the first step of averaging, denoted as S1, we obtain
a new sequence of length $N-n+1$, and use the new sequence to carry on the
averaging procedure as,

Figure 5: The Wilson construction of charge distribution plotted against the
corresponding window length for (a) K32, (b) sNase (c) CoINT (d) SH4UD, (e)
An16, (f) R15, (g) CspTm, (h) FhuA, (i) ACTR, (j) $\alpha$-synuclein, (k) OPN,
and (l) ProTa-N, in the order of positive to negative net charge content of
the IDPs. $Q$ denotes the net charge of the IDP along with positive and
negative charges as $Q+$ and $Q-$ respectively. $\mathcal{W}$ denotes the area
under the Wilson constructed curve. The color shade magenta/green shows the
positive/negative intensity of the Wilson charge as a function of window
length.

$S0:\left[a_{1},a_{2},\dots,a_{N}\right]$ (12a)
$S1:\left[\frac{1}{n}\sum_{i=1}^{n}a_{i},\;\frac{1}{n}\sum_{i=2}^{n+1}a_{i},\dots,\;\frac{1}{n}\sum_{i=N-n+1}^{N}a_{i}\right]$
(12b) $SN:\tilde{\mathcal{W}}(n)=\frac{1}{n}\sum_{i=1}^{n}a_{i}.$ (12c)

We continue this procedure iteratively until we reach the final average value,
represented as $\tilde{\mathcal{W}}(n)$. If the length of the charge sequence
becomes less than the window length $n$ during the averaging process, we
terminate the procedure and calculate a global average to obtain
$\tilde{\mathcal{W}}(n)$.

These averaging procedure with varied window size $n\in[2,N]$ can effectively
capture the combination of charge interactions at different length scale. One
can show that $\tilde{\mathcal{W}}(2)$ consider binomial interactions among
the charges and expressed as

$\displaystyle\tilde{\mathcal{W}}(n=2)=\frac{1}{2^{N-1}}\sum_{m=0}^{N-1}\binom{N-1}{m}a_{m}.$
(13)

The higher-order window averaging considers interactions of varying
magnitudes, which can have an impact on determining dynamic conformations of
IDPs. In Fig. 5 we explore $\tilde{\mathcal{W}}(n)$ for 12 IDPs with different
total charges from highly positive (a) K32 (Q=22.5), (g) CspTm (Q=2.0) to
highly negatively charged IDP (l) ProTa-N (Q=-43.0) as a function of
normalized window length n/N. In the case of highly positively and negatively
charged IDPs, the Wilson curves consistently remain above or below the zero
line respectively. However, for IDPs with lower net charges, we sometimes
observe the Wilson curve crossing from negative to positive in the case of (f)
R15, (g) CspTm, and from positive to negative in the case of (j)
$\alpha$-synuclein, and (k) OPN. The area under the Wilson curve is denoted by
$\mathcal{W}=\sum_{n=2}^{N}\tilde{\mathcal{W}}(n)$ and listed in the 16th
column of Table-1. It is conceivable that when plotted in normalized unit
length scale, IDPs with similar Wilson charge $\tilde{\mathcal{W}}(n)$ will
behave the same way and thus, can be used as their fingerprints.

###  V.3 Charge patches and the local persistence length

Figure 6: The persistence length $l_{p}$ is shown as a function of the
normalized amino acid index for (a) K32, (b) An16, (c) Nucleoporin153, (d)
CspTm, (e) $\alpha$-synuclein, and (f) ProTa-N. In each case, the light pink
line represents the persistence length from Eqn. 14 while the magenta line
corresponds to the sliding average $\langle
l_{p}\rangle=\left(\sum_{i-2}^{i+2}l_{p}(i)\right)/5$. The positions of
charged residues along the chain backbone are indicated by filled blue
$\bullet$, cyan $\bullet$, and darkgreen $\bullet$ circles for $+1$, $+0.5$,
and for $-1.0$ charged residues respectively. The green dashed lines denote
the average persistence length in each case.

The presence of charge patches introduce varying degree of local stiffness
along the chain backbone. During the BD simulation use a discrete chain and
the persistence length is calculated from [58]

$\ell_{p}/\sigma=-\frac{1}{\ln\left(\cos\theta_{i}\right)},$ (14)

where $\theta_{i}$ is the angle between two bond vectors connecting the
$i^{th}$ bead to the $(i\pm 1)^{th}$ beads [60]. We have checked that for a
homopolymer chain this matches well with the continuum description of
persistence length [62]

$\ell_{p}/\sigma=\kappa/k_{B}T\quad{\rm(3D)}.$ (15)

IDPs with very similar net charges can have markedly different distribution of
charges. An IDP containing correlated charge patches will have increased chain
stiffness along that region that will affect its conformations and dynamics.
To demonstrate this, we calculate the local persistence length ($l_{p}$) in nm
along the chain using Eqn. 14 for a few IDPs shown in Fig. 6. For example,
CspTm has sparsely distributed charged residues with less net charge compared
to ProTa-N containing mostly negatively charged residues in patches, and we
observe increase in $l_{p}$ on those regions. The electrostatic repulsion
among the same charge residues makes the chain locally stiffer and possibly
has a deeper effect in their participation in biophysical processes.

###  V.4 Skewness index ($\mathcal{S}$-index) of the radius of gyration

The variation of the chain persistence length due to different charge species
along the chain backbone is manifested in the shapes of the corresponding
gyration radii that we measure in terms of the $\mathcal{S}$-index. The
$\mathcal{S}$-index is obtained by fitting the distribution
$P\left(\bar{R}_{g}\right)$ of the scaled gyration radii
$\bar{R}_{g}=\sqrt{R_{g}^{2}}/\sqrt{\langle R_{g}^{2}\rangle}$ with an
exponentially-modified Gaussian distribution (exponnorm) [63] function given
by

$f(x,K)=\frac{1}{2K}\exp\left(\frac{1}{2K^{2}}-\frac{x}{K}\right){\rm{erfc}}\left(-\frac{x-1/K}{\sqrt{2}}\right)$
(16)

where $x$ is a real number, $K>0$ and erfc is the complementary error
function. The $\mathcal{S}$-index is then obtained from the following equation

$\mathcal{S}=\frac{2K^{3}}{\left(1+K^{2}\right)^{3/2}},$ (17)

where $K$ corresponds to the shape parameter of the exponnorm distribution of
Eqn. 16. As shown below, the usefulness of the $\mathcal{S}$-index is
manifested in capturing the overall shape of the distributions of the the
gyration radii which then can be further analyzed as a function of the amino
acid compositions and their net charge content. We observe in Fig. 7 that the
shapes of the distribution of the gyration radii vary from being near Gaussian
($\mathcal{S}\rightarrow 0$) to an exponentially modified Gaussian
distribution (that exhibits a tail for larger value of $\mathcal{S}$).

Figure 7: $P\left(\bar{R}_{g}\right)$ for several IDPs as a function of the
$\mathcal{S}$-index. In each figure, the solid colored line denotes the
exponentially modified Gaussian fit for the $P\left(\bar{R}_{g}\right)$
histograms (please see text) with the solid lines for $\mathcal{S}\geq 0.1$
while the dotted line for the near Gaussian distribution ($\mathcal{S}\leq
0.1$). The corresponding $\mathcal{S}$-index are written under parentheses in
the legends. Figure 8: (a) $\mathcal{S}$-index as a function of $f^{*}$. The
solid blue line is an exponential fit. Light pink, green, and blue shaded
regions denote the high, moderate, and low skewed radius of gyration of the
IDPs. (b) $f^{*}$ as a function of net charge. The straight line is a linear
fit through the points.

Fig. 7 confirms that most of the IDPs have long exponential tails
($\cal{S}\geq$ 1.0) such as CspTm, ProTa-C, An16, FhuA, OPN, CoINT, SH4UD,
$\alpha$-synuclein, ACTR, sNase, R15, and K32. The highly charged IDP ProTa-N
and HST5 are observed to have almost perfect Gaussian distribution with
$\mathcal{S}\rightarrow 0$. Moreover, for the highly skewed distributions the
peaks shift towards the left that signifies the median is smaller than the
mean and there is a propensity of these IDPs to expand occasionally. This
skewness parameter can be utilized to further classify IDPs broadly into three
categories (three colored regions in Fig 8(a)) that characterize the
propensity of expansions.

We now study how the $\mathcal{S}$-index depends on the charge content of the
IDPs in more detail. In a previous study, Pappu et al. [20, 21] demonstrated
that radius of gyration depends on the charge asymmetry parameter

$f^{*}=\frac{(f^{+}-f^{-})^{2}}{f^{+}+f^{-}},$ (18)

where $f^{+}/f^{-}$ is the net positive/negative charge per residue of an IDP.
In Fig. 8(a) we plotted $\mathcal{S}$-index as a function of the $f^{*}$ and
based on the data it appears that there is an exponential decay of
$\mathcal{S}$-index, although a power law fit could not be excluded. However,
when plotted as a function of the net charge $q_{net}$ (column 9 of Table-I) a
simpler relation is obtained (Fig. 8(b)). The simulation data indicates that
$\mathcal{S}$-index decreases linearly with $q_{net}$. The linear fit which
can be simplified to

$\mathcal{S}\approx-2q_{net}+1$ (19)

can serve as a useful relation. Since $\mathcal{S}$ can not be negative one
observes a general relation

$\lim_{q_{net}\rightarrow 0.5}\mathcal{S}\rightarrow 0\;{\rm(Gaussian)}.$ (20)

From Fig. 7 (last two rows of the right column) one observes that the shapes
of the IDPs Prota-N and HST5 with $q_{net}=0.384$ and 0.354 are almost
Gaussian with $\mathcal{S}=0.09$ and $0$ respectively. This general conclusion
will get tested when data for more IDPs will be available and analyzed in
future.

###  V.5 Chain conformations and Ionic Concentration

Solvent conditions, such as pH, temperature, ionic strength, and the presence
of specific molecules, can significantly influence the conformations and hence
the behaviors of the IDPs, particularly in a cellular environment. The
robustness of the IDPs under external conditions can also be associated with
the

Figure 9: (a) Characterization of IDPs using their net charge per residue
$q_{net}$ as a function of the absolute charge per residue $q_{abs}$. The two
dotted black horizontal and vertical lines $q_{net}=0.13$, and $q_{abs}=0.275$
respectively, further identify the IDPs in terms of their placement into one
of the four quadrants (I - IV). (b) $\sqrt{R_{g}^{2}/R_{g}^{2}(0)}-1$ as a
function of ionic concentration $I$. Separate colors and in each quadrant with
different symbols for each IDP clearly identifies different dependencies of
the IDPs on the salt concentration. $P\left(\bar{R}_{g}\right)$s for p53 (c),
ProTa-N (d), K16 (e), Nucleoporin153 (f), hCyp (g), and K25 (h) are shown for
three different ionic concentrations 10 mM (cyan), 100 mM (green) and 300 mM
(magenta) respectively. The black lines denote the exponentially modified
Gaussian fit of the histograms and the corresponding $\mathcal{S}$-indices are
denoted on the legends.

origin of life. Previous experimental studies [64, 65, 66] and simulation
studies using CG models [67, 68] have revealed conformational changes and
salt-induced phase transition and looked at the liquid-liquid phase
transitions in IDPs. IDPs are described either as PEs or PAs with varying
amounts of net charge [20]. Thus, it is conceivable that screening will affect
the conformational aspects in a significant way. Intuitively one can
understand the behavior by using the idea of screening. The IDPs that are PE,
an increase in salt concentration will screen the net charge reducing the
electrostatic repulsion and hence by and large, all the PEs with a net
positive or negative charge will have reduced gyration radii as a function of
increased screening. The case of PAs is a bit more subtle depending on an
IDP’s not only the net charge per residue $q_{net}$, but the fraction of the
residues that are charged $q_{abs}$ as defined below.

$q_{net}=\frac{||Q_{+}|-|Q_{-}||}{N}\quad{\rm and}\quad
q_{abs}=\frac{|Q_{+}|+|Q_{-}|}{N}$ (21)

Here, $Q_{+}$, $Q_{-}$, and $N$ represent the total positive and negative
charges, and the number of amino acids in the IDP. In literature, $q_{net}$
and $q_{abs}$ are previously denoted as NCPR and FCR respectively [21]. For
the PA the loss/gain in electrostatic energy and entropy ultimately controls
the show. We have made an extensive study of the dependence of gyration radii
of the 33 IDPs (listed in Table- 1) on salt concentration under physiological
conditions ranging from 0-300 mM shown in Fig. 9. The IDPs can be placed on
any one of the four quadrants (I, II, III, and IV) of $(q_{abs},q_{net})$ to
study their dependency on salt concentration (Fig. 9(a)). Based on the values
of $(q_{abs},q_{net})$, two decision boundary lines $q_{abs}=0.275$ and
$q_{net}=0.13$ place the IDPs into four subclasses. Fig. 9(a) displays scatter
plots of the 33 IDPs classified into four quadrants, represented by blue,
green, orange, and red symbols corresponding to the I, II, III, and IV
quadrants, respectively. For each quadrant, a plot of the saturation values at
each concentration is used to plot $\sqrt{R_{g}^{2}/R_{g}^{2}(0)}-1$ as a
function of ionic concentration, where $R_{g}(0)$ corresponds to the radius of
gyration under ion-free conditions. Quadrant-I and IV are easy to understand.
A strong PA, such as Prota-N lies in the quadrant-I as expected. Other PAs
(HST5, OPN, ProTa-C) with large $q_{net}$ and $q_{abs}$ belong here. In this
case, as the salt concentration increases, charge screening occurs, leading to
a decrease in their gyration radii. This condition holds when only one type of
charged residue is abundant in number. On the other hand, in quadrant IV,
$q_{abs}$ is high, but $q_{net}$ is low, corresponding to a situation where
there is a higher number of charged residues, yet they are almost equal in
numbers. As both types of charges are present, at low salt concentration, the
attraction between opposite charges reduces their radius of gyration due to
electrostatic interactions. However, with increasing salt concentration, the
charge screening effect comes into play, and the strength of electrostatic
attraction among the oppositely charged residues decreases. Consequently, we
observe a swelling of the IDPs, leading to a higher radius of gyration. Twelve
IDPs belong to this category. In quadrant II and III, the $q_{abs}$ value is
low, indicating a low content of charged residues. In quadrant II, we find
that p53 is the only IDP out of the 33 that falls into this category but has a
high value of $q_{net}$. This pathological case is characterized by having
only 17 negative charge residues (GLU and ASP) and two positively charged
residues (ARG and LYS). Due to the charge screening effect mostly on the
negatively charged residues, it can be inferred that the radius of gyration
will decrease and that is indeed true as observed from the plot. In quadrant
III, the $q_{net}$ is low corresponding to IDPs that have less net charge per
residue. We find 15 IDPs belong to this category. In this case gyration radii
of hCyp, FhuA, and K10 increase while gyration radii of K18, K17, K32, k27 and
K16 decrease as a function of salt concentration. On the other hand, the
remaining 5 IDPs, namely Protein-L, SH4UD, An16, Nucleoporin153, and CoINT are
robust to the variation of ionic concentration as they are mostly low charge
containing IDPs. The segregation of IDPs into four subsections unravels
insights about their responses to salt concentration and provides a framework
to classify other unknown proteins based on how they will behave under a wide
range of salt solutions.

We further studied the accompanying variation in the shape of the distribution
of the gyration radii by monitoring the $\mathcal{S}$-index as progressively
more screening is introduced for the reason discussed in section D. Some
examples are shown Fig. 9(c)-(h). The skewness of the distributions for K25
for 10 mM, 100 mM, and 300 mM ionic concentration changes from 1.73, 1.62, and
1.06 respectively, and they span a larger conformational space. On the other
hand, for ProTa-N while the gyration radii decrease at higher salt
concentrations implying they become more compact without altering the
distribution shape. We also observe that gyration radii for a few IDPs
(Protein-L, SH4UD, An16, Nucleoporin153, and CoINT) in Fig. 9(a) remain
unaffected within the low salt limit of our study. With the change in the salt
concentration, our simulation studies show that the degree of alteration in
the shape of IDPs is different. The shape deformation is drastic in K25
compared to ProTa-N.

###  V.6 Summary and Outlook

In conclusion, we used two different CG models (HPS1 & HPS2) to study both
universal and fine structures of 33 IDPs and compared our results with
available experimental data as well as simulation results for the same IDPs
using other CG models. Our systematic studies of IDPs with fairly disparate
levels of absolute and net charge ($q_{abs}$, $q_{net}$), and net hydropathy
add many interesting characteristics to the prior studies using similar
models. For the HPS1 model use of a larger set of IDPs and new experimental
data [36] converge on the interaction parameter $\epsilon\approx 0.1$
kcal/mol, as opposed to 0.2 kcal/mol if we use the original set of Dignon et
al. [31] (please refer to the supplementary section). For the HPS2 model, our
optimized parameter 0.2 kcal/mol using 33 IDPs concurs with that of Tesei et
al. [32, 69] and seem to be more robust compared to the HPS1 model. Analysis
of the results using the expanded dataset establishes a more robust and
reliable framework for studying IDPs in bulk. The experimental results
converge well with the simulation results for the optimum $\epsilon$ for both
HPS1 and HPS2 models.

A natural question that has been addressed in the community that if sequence
specificity makes every IDP distinct from each other, or they share some
universal characteristics of homopolymers described by Flory’s theory. We have
been able to address both issues. A comparison of the scaled end-to-end
distance and transverse fluctuations in reference to our recently established
universal results [59], we observe that IDPs studied here interpolates from
being Gaussian chains to self-avoiding chains to a different degree depending
upon their amino acid composition.

We then study in detail how the absolute and net charge per residue
($q_{abs}$, $q_{net}$) manifest themselves in finer characteristics of the
IDPs. We come up with several new metrics that reveal the uniqueness of each
IDP, yet leaves room for making further classification of IDPs in different
categories. The first one is the Wilson index $\mathcal{W}$ that on a
normalized unit length scale demonstrates the uniqueness of each IDP and hence
can be used as their fingerprints (e.g. We believe that using the
$\mathcal{W}$ index of the newly discovered IDPs in future will allow us to
predict their behaviors with those of similar $\mathcal{W}$ index.

Likewise, we demonstrate how the charge patches control the local stiffness
and hence the overall conformations of the IDPs that we further characterize
by introducing a skewness index $\mathcal{S}$. We first demonstrate that the
distribution of the scaled gyration radii for highly charged IDPs with
$\mathcal{S}\rightarrow 0$ are near-perfect Gaussians and that moderately
charged IDPs with a larger value of the $\mathcal{S}\rightarrow 1.0-1.5$ are
characterized by exponential tails. This observation immediately leads us to
the observation of a simple linear dependence of $\mathcal{S}$ on $q_{net}$ as
approximated in Eqn. 19. We related $\mathcal{S}$ parameter to the net charge
charge asymmetry parameter $f^{*}$ introduced by Das and Pappu [21] that we
observe a stronger exponential dependence. We believe Fig. 8 along with Eqn.
19 can serve as a good reference to understand properties of the individual
IDPs.

An important classification of the IDPs emerges from the study of salt
dependence of the conformations of the IDPs. We find that IDPs exhibit very
different characteristics and can be broadly placed in four different region
in the ($q_{abs},q_{net}$) space.

We conclude with some comments which may promote further studies to perfect
the CG models. We and many others used isotropic radius of gyration as the
sole physical quantity for comparison as this is the mostly available from the
experiments. The CG model can be refined by introducing other quantities. For
example, Wohl et al. [68] studied salting-out effect on the liquid–liquid
phase separation (LLPS) of IDPs by introducing a salt-dependent term into the
hydropathy used in the HPS1 model. Maity et al. [67] introduced Molecular
Transfer Model to study the Salt-Induced Transitions. In the low
concentrations of salts ($\leq$ 1M) IDP conformations are affected by the
degree of screening of electrostatic interactions of the charged residues and
are independent of the specific salt identity which is likely the regime that
we have studied. However, at high concentrations, salts affect IDP
conformations through salt-specific Hofmeister effects [70, 71]. Thus, our
studies will be useful in refining the existing hydropathy models for a wider
range of parameter space.

We note that some of the IDPs remain unaffected with the variation of salt
concentration and thus can be compared with the behavior of other simpler
amino acids identified and studied in the context of the origin of life [72].
Recently Tesei et al. developed an efficient model to generate conformational
ensembles of IDRs, and reported conformational properties 28,058 IDRs from
sequence only [73]. We believe that some of the new ideas introduced here can
be tested for a much larger set and will open up several exciting avenues for
future research to obtain a deeper understanding of the unique properties and
behavior of IDPs.

##  VI Supplementary Materials

A comparison of the gyration radii data from our simulation based on HPS1
hydropathy scale with the experimental data for a smaller subset of IDPs [31]
is included in the supplementary materials.

##  VII Acknowledgments

All computations were carried out using STOKES High-Performance Computing
Cluster at UCF. We thank Prof. Kresten Lindorff-Larsen and Dr. Giulio Tesei
from University of Copenhagen for discussions, comparison of simulation
results, and bringing reference [36] to our attention. We sincerely thank both
the referees for their critical comments on the manuscript. Simulation results
were obtained using HOOMD-blue [74].

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