[1]Roy Beck

1]School of Physics and Astronomy, the center for Nanoscience and
Nanotechnology, and the center of Physics and Chemistry of Living Systems, Tel
Aviv University, Tel Aviv, Israel

2]Materials Department, Biomolecular Sciences and Engineering Program, and
Physics Department, University of California, Santa Barbara, USA

# From isolated polyelectrolytes to star-like assemblies:
The role of sequence heterogeneity on the statistical structure of the
intrinsically disordered Neurofilament-low tail domain

Mathar Kravikass  Gil Koren  Omar A. Saleh<EMAIL_ADDRESS>[  [

###### Abstract

Intrinsically disordered proteins (IDPs) are a subset of proteins that lack
stable secondary structure. Given their polymeric nature, previous mean-field
approximations have been used to describe the statistical structure of IDPs.
However, the amino-acid sequence heterogeneity and complex intermolecular
interaction network have significantly impeded the ability to get proper
approximations. One such case is the intrinsically disordered tail domain of
Neurofilament low (NFLt), which comprises a 50 residue-long uncharged domain
followed by a 96 residue-long negatively charged domain. Here, we measure two
NFLt variants to identify the impact of the NFLt two main subdomains on its
complex interactions and statistical structure. Using synchrotron small-angle
x-ray scattering, we find that the uncharged domain of the NFLt induces
attractive interactions that cause it to self-assemble into star-like polymer
brushes. On the other hand, when the uncharged domain is truncated, the
remaining charged N-terminal domains remain isolated in solution with typical
polyelectrolyte characteristics. We further discuss how competing long- and
short-ranged interactions within the polymer brushes dominate their ensemble
structure, and, in turn, their implications on previously observed phenomena
in NFL native and diseased states.

###### keywords:

Intrinsically disordered proteins, Small angle X-ray scattering,
Neurofilament, Polymer physics, Polymer Brushes

## Introduction

Intrinsically disordered proteins (IDPs) are a subset of proteins that,
instead of forming a rigid singular structure, fluctuate between different
conformations in their native form [1, 2]. Nonetheless, IDPs serve significant
biological functions and account for about 44% of the human genome [3]. The
lack of fixed structure provides IDPs many advantages in regulatory systems in
which they often play a crucial role in mediating protein interaction [4, 5].
These roles often come into play from intrinsically disordered regions (IDRs)
of folded proteins interacting with other IDRs. For example, in the
Neurofilament proteins, tails emanating from the self–assembled filament
backbone domains bind together and form a network of filaments [6, 7, 8, 9,
10].

The ensemble statistics of IDPs stem from their sequence composition and the
surrounding solution [2]. For example, previous studies showed that IDPs
comprising mostly negatively charged amino acids (polyelectrolytes) are
locally stretched due to electrostatic repulsion between the monomers [11].
Moreover, different properties, such as hydrophobicity, were shown to be
linked with local IDP domain collapse [12]. The complex interactions that
arise from sequence heterogeneity allow IDPs to form specific complexes
without losing their disordered properties [13]. For example, Khatun et al.
recently showed how, under limited conditions, the human amylin protein
self–assembles into fractal structures [14].

As IDPs are disordered chains, polymer theories are prime candidates to relate
the measured structural statistics to known models, which can help link the
sequence composition of the IDP to its conformations [15, 16, 17, 18].
Specifically, polymer scaling theories allow us to derive the statistical
structure of IDPs given sequence–derived parameters, such as charge density
and hydrophobicity [11, 19, 12, 20, 21]. However, due to the heterogeneity of
the IDP primary structure (i.e., the amino acid sequence), some systems showed
contradictions with the behavior theorized by standard heterogeneous polymer
physics [22, 23, 17, 24, 19].

The unique biological properties of IDPs have given rise to numerous attempts
to use them as building blocks for self–assembled structures [25]. For
example, IDPs were proposed as brush–like surface modifiers, due to their
enhanced structural plasticity to environmental conditions [26, 27]. Another
example of an IDP brush system is the Neurofilament (NF) protein system [6,
28, 29], described as interacting bottle–brushes. NF subunit proteins form
mature filaments with protruding disordered C–terminus IDR known as ‘tails.’
NF tails were shown to mediate NF network formation and act as shock
absorbents in high–stress conditions [29]. Moreover, NF aggregations are known
to accumulate alongside other proteins in several neurodegenerative diseases,
such as Alzheimer’s, Parkinson’s, etc. [30].

The NF low disordered tail domain (NFLt) sequence can be divided into two
unique regions: an uncharged region (residues 1–50) starting from its N
terminal and a negatively charged region (residues 51–146). The NFLt can be
described as a polyelectrolyte with a net charge per residue (NCPR) of -0.24.
Furthermore, the statistical structures of segments within the NFLt are
influenced by the amount, type, and disperse of the charged amino acid within
a segment [22]. Nonetheless, other structural constraints, particularly
long–range contacts, impact the local statistical structures. Additionally,
NFLt was shown to have glassy dynamics with the response to tension [31]. Such
dynamics were associated with multiple weakly interacting domains and
structural heterogeneity.

In this paper, we revisit NFLt as a model system for charged IDP and focus on
the contribution of its neutral and hydrophobic N–terminal domain. We will
show that increased salt concentration causes NFLt to form star–like brushes
with increased aggregation number ($Z$). We will further quantify the
competition between hydrophobic attraction and electrostatic and steric
repulsion in the formation of the structures of NFLt.

## Results

To study the N–terminal domain contribution to the structure of NFLt, we
designed two variants and measured them at various buffer conditions. The
first construct is the entire 146 residues of the NFLt chain, which we term as
WT (NCPR = -0.24), and the second is isolating the 104 negatively charged
residues from the C-terminal of NFLt (NCPR = -0.33), termed as
$\rm{\Delta}$N42. We expressed the variants in E-coli and purified it up to
96% (see methods).

We assessed the variants in solution using small–angle X–ray scattering
(SAXS), a technique extensively used to characterize the statistical
structures of IDPs [32]. From the raw SAXS data, measured at various
salinities, we can already find high structural differences between the two
variants (Fig. 1a). Dominantly at the low wave-vector ($q$) region, the WT
variant scattering ($I$) rises with added NaCl salt. Such an increase at low
$q$ implies high molecular mass particles due to aggregation of the WT
variant.

In contrast, $\rm{\Delta}$N42 shows a separated Gaussian polymer profile
(Figs. 1a, S1), nearly insensitive to total salinity ($C_{s}=20-520$ mM).
Similarly, the data presented in Kratky format ($qI^{2}$ vs. $q$, Fig. 1a)
shows the $\rm{\Delta}$N42 has the signature of a disordered polymer. In
contrast, the WT variant, in particular at high salinity, has a combination of
a collapse domain (the peaks from below $q=0.25\rm{nm}^{-1}$) and a disordered
polymeric structure (the scattering rise at higher $q$ Fig. 1a).

Figure 1: SAXS measurements of WT and $\rm{\Delta}$N42 at different salinity
($C_{s}$). a. For increasing $C_{s}$, the WT variant shows increased small
angle scattering, a signature for aggregation. In contrast, $\rm{\Delta}$N42
remains structurally intrinsically disordered as $C_{s}$ vary. Data points are
shifted for clarity. Lines are form-factor fittings, as described in the text.
b. Normalized Kratky plot of the same SAXS measurements. The $\rm{\Delta}$N42
variant remains disordered and unchanged with salinity, while the WT variant
shows a hump at low $q$, typical for a collapse region. With increasing
$C_{s}$, the hump at the lower $q$ range becomes a sharper peak accompanied by
a scattering rise at the higher $q$ range. Such behavior indicates that the
aggregation coexists with the WT variant’s highly dynamic and disordered
regions. Both variants shown are at the highest measured concentration (Table
S1, S3). WT measurements are in 20 mM Tris pH 8.0 with 0, 150, 250, and 500 mM
added NaCl (from bottom to top). Likewise, for $\rm{\Delta}$N42, measurements
are in 20 mM Tris pH 8.0 with 0 and 150 mM added NaCl (bottom to top).

Being completely disordered, $\rm{\Delta}$N42 lacks a stable structure and can
be described using a statistical ensemble of polymeric conformations [33]
were:

$\begin{split}I(q)=I_{0}\exp\\{-\frac{1}{3}(qR_{\rm G})^{2}+\\\
0.0479(\nu-0.212)(qR_{\rm G})^{4}\\}.\end{split}$ (1)

Here, $I_{0}$ is the scattering at $q=0$, $\nu$ is Flory scaling exponent, and
$R_{\rm G}$ is the radius of gyration defined by:

$R_{\rm
G}=\sqrt{\frac{\gamma(\gamma+1)}{2(\gamma+2\nu)(\gamma+2\nu+1)}}bN^{\nu},$ (2)

where $\gamma=1.615$ and $b=0.55\rm{nm}$ (see [33]) and the analysis is viable
up to $qR_{\rm G}\sim 2$ (Fig. S2, S3). In all $\rm{\Delta}$N42 cases, the
scattering profile fits Eq. 1 and with $\nu$ ranging between 0.63–0.69
depending on the buffer salinity (Table S1). In ‘infinite dilution’ conditions
(zero polymer concentration), we find $\nu$ to decrease monotonically from
0.73 to 0.62 with added salt (Table S2).

Given the noticeable aggregation for the WT variant, alternative form factors
were considered to match the scattering profiles (lines in Fig. 1). The
absence of structural motifs at high $q$ values ($q>0.3\,\,\rm{nm}^{-1}$)
indicates a disordered nature for WT at shorter length scales. Conversely, in
the lower $q$ region ($q<0.3\,\,\rm{nm}^{-1}$), the scattering suggests stable
structural motifs or a larger molecular weight particles. Such SAXS resembles
that of self–assembled decorated spherical micelles [34]. Variations of
micelle models are shown to fit the data (Figs. 1, S4-S6). Sufficiently low
aggregation number and core size distil the description of the spherical
micelle into a ‘star–like’ brush. Alternative attempts to fit the scattering
profiles to other form factors models, including vesicles and lamellar, were
unsuccessful.

For the star–like model, the aggregated variants form a small spherical core
of volume $V_{\rm core}$ made out of $n\cdot Z$ monomers (comparison with
different cores described in [35] and in Fig. S4), where $n$ denotes the
peptide length per polypeptide within the core, and $Z$ is the aggregation
number, i.e. the number of polypeptides per ‘star.’ The remainder of the WT
variant then protrudes from the core as the star polymer brush (Figs. 2a,
S4-S6).

Figure 2: a. Schematic of the system’s structure variation with salinity
($C_{s}$). While $\rm{\Delta}$N42 remains disordered and segregated, the WT
variant aggregates to a star–like polymer with a higher aggregation number at
higher $C_{s}$. b–e. Structural parameters for WT (blue symbols) and
$\rm{\Delta}$N42 (black symbols) variants extracted from fitting the SAXS
data. Full and hollow circles represent the spherical and cylindrical core
fitted parameters, respectively. d. In all cases, the brush heights ($h$) are
much larger than the corresponding grafting length ($\rho$), indicative of a
brush regime. e. The structurally intrinsically disordered $\rm{\Delta}$N42
variant compacts with higher $C_{s}$ values and remains more compacted from
the projected brushes for the WT variant. All values are the extrapolated
‘zero concentration’ fitting parameters (see Fig. S7)

The star–like scattering form factor is described as a combination of four
terms [34]: the self–correlation term of the core $F_{\rm c}$, the self-
correlation term of the tails $F_{\rm t}$, the cross-correlation term of the
core and the tails $S_{\rm ct}$ and the cross-correlation term of the tails
$S_{\rm tt}$:

$\begin{split}F_{\text{total}}(q)=Z^{2}\beta_{\rm c}^{2}F_{\rm
c}(q)+Z\beta_{\rm t}^{2}F_{\rm t}(q)+\\\ 2Z^{2}\beta_{\rm c}\beta_{\rm
t}S_{\rm ct}(q)+Z(Z-1)\beta_{\rm t}^{2}S_{\rm tt}(q).\end{split}$ (3)

Here, $\beta_{\rm c}$ and $\beta_{\rm t}$ are the excess scattering length of
the core and the tails, respectively. From fitting the scattering data, we
extracted the height of the tails $h=2R_{\rm G}$, the aggregation number $Z$,
and the relevant core’s parameters (e.g., core radius $R$ for a spherical
core, cylinder radius $R$ and length $L$ for a cylindrical core [35]),
schematically illustrated in Fig. 2a. All fitting parameters are found in
Table S3.

To avoid misinterpretation and to minimize intermolecular interaction effects,
we present the fitting results at the ‘infinitely diluted regime’ by
extrapolating the relevant parameters measured at various protein
concentrations to that at zero protein concentration (Fig. S7, Table S4). The
parameters are mostly independent of the concentration unless explicitly
mentioned.

At low salinity (20mM), the aggregation number for the WT variant is of a
dimer ($Z\approx 2$), and the core’s shape is that of a cylinder (with a
radius $R=0.89$ nm and length $L=1.19$ nm). At higher salt conditions (170-520
mM), the form factor fits spherical core aggregates with increasingly higher
$Z$’s (Fig. 2a).

Given the relatively small core volume ($V_{\rm core}\approx 1-2\rm{nm}^{3}$,
Fig. 2c), it is crucial to evaluate the ‘grafting’ distance between
neighboring chains, $\rho$, on the core surface ($S=4\pi R^{2}=Z\rho^{2}$) and
the brush extension, $h$, outside the core. As shown in Fig. 2d, in all cases,
$h/\rho\gg 1$ indicates a ‘brush regime’ where neighboring chains repel each
other while extending the tail’s height [36].

The repulsion between the grafted tail is further emphasized when comparing
$h/2$ for WT to the equivalent $\rm{\Delta}$N42 length-scale ($R_{\rm G}$),
showing a significant extension for WT (Fig. 2e). We notice that the WT tail’s
length ($h$) increases at low salt (during the transitions from a dimer to a
trimer), followed by a steady mild decrease as the $C_{s}$, and following $Z$
increase. Similar compactness with increasing $C_{s}$ is shown for
$\rm{\Delta}$N42 and is expected for polyelectrolyte due to the reduction in
electrostatic repulsion [37]. To better compare the statistical structure of
two variants of disordered regions, we followed the polymeric scaling notation
$\nu$ that quantifies the compactness of the chain. For $\rm{\Delta}$N42, we
extracted $\nu$ from Eqs. 1 and 2 and found a significant decrease in its
value as 50 mM of NaCl is added to the 20 mM Tris buffer (Fig. 3a). The
following monotonic decline is in line with polyelectrolytic models and
electrostatic screening effects [38], shown in a solid red line in Fig. 3a.
Interestingly, previous measurements of segments within the NFLt charged
domain were shown to have similar $\nu$ values as in $\rm{\Delta}$N42 .
However, the same decline in salinity was not observed (Fig. 3a) [22].

Figure 3: Deduced structural parameters from the SAXS data fitting. a. Flory
exponent ($\nu$) of WT tails and $\rm{\Delta}$N42 variants showing extended
disordered scaling. The red line refers to the theoretical brush model [39],
and the blue line refers to the theoretical polyelectrolyte [38].
$\rm{\Delta}$N42 shows a decrease in the protein extension due to the decline
in intermolecular electrostatic repulsion (see also Fig. 4). WT shows an
increase in the extension when shifting from a dimer to a trimer, followed by
a slight decline with a further increase in salinity. In gray, average $\nu$
is obtained from measuring separate NFLt segments with an NCPR of -0.3 to -0.6
[22]. b. The core (aggregated) peptide length per polypeptide as a function of
salinity. At high salinity, each polypeptide aggregates via 2–3 amino acids
that form the star–like polymer core. Both panels’ values are the extrapolated
‘zero concentration’ parameters (supplementary Fig. S8).

For the WT variant, the scaling factor ($\nu$) of the ‘star–like polymer’
brushes is extracted from Eq. 2. Here, we use $R_{\rm G}=h/2$, where $h$ is
obtained from Eq. 3. For $C_{s}=20$ mM, we find a $\nu$ is of similar scale as
for $\rm{\Delta}$N42 . This similarity can be attributed to the nature of the
dimer, where the intermolecular electrostatic interactions dominate the
expansion of each of the two tails. As $C_{s}$ increases by $150$ mM, $\nu$
exhibits a considerable increase, presumably due to neighboring tail
repulsion. Above $C_{s}=170$ mM, $\nu$ shows a weak decrease. We attribute
this weak decline to the salt–brush regime of polyelectrolyte brushes [39]
shown in solid blue in Fig. 3a. In this regime, $h\propto C_{s}^{-1/3}$, and
subsequently $\nu\propto-\frac{1}{3}log(C_{s})$.

We note that the cores of the star–like polymers are relatively small and that
each polypeptide aggregates through only a few, most likely hydrophobic, amino
acids. From the tabulated amino-acid partial volume, $\langle\phi_{aa}\rangle$
[40], we evaluated the average number of amino-acids per polypeptide inside
the core $n=V_{\rm core}/(\langle\phi_{aa}\rangle\cdot Z)$.

In Fig. 3a, we indeed see that the most significant change occurs at the low
salt regime, where $n$ drops from an average of 7 to 3 amino acids
($C_{s}=20,170$ mM, respectively). We presume this behavior to be due to a
salting–in effect, as is known to occur within globular proteins [41] and were
recently alluded to impact IDPs [42]. The following trend is a further
decrease in $n$, albeit much weaker, which results in a final average $n$ of
about two as the salinity reaches $C_{s}=520$ mM.

Last, in Fig. 4, we quantify the intermolecular interactions by evaluating the
second virial coefficient, $A_{2}$, using a Zimm analysis [43] (Table S5).
Here, $A_{2}$ describes the deviation of the statistical ensemble from an
ideal gas. In agreement with our previous data, we find that the
inter–molecular interactions of $\rm{\Delta}$N42 change from repulsive
($A_{2}>0$) to weakly attractive ($A_{2}\leq 0$) as the salinity increases. In
contrast, for WT, $A_{2}$ changes from a nearly neutral state of
intermolecular interactions (i.e., ideal gas regime) to mildly attractive
($A_{2}<0$). These findings are reflected in the dependency of the variant
Flory coefficient $\nu$ in concentration. While at the lowest salinity,
$\rm{\Delta}$N42 is shown to expand as protein concentration is decreased, for
higher salinities and for the WT measurements, $\nu$ remain primarily
unchanged (Fig. S8a).

Combining our results for both variants, we find an exemplary role of
long–range electrostatic interactions tuning the statistical structure of
IDPs. Without the uncharged N-terminal domain, the NFLt exhibited significant
change as the electrostatic interactions were screened, causing them to
condense further. In contrast, the presence of the uncharged domain incurred
aggregation of the proteins, bringing the tails much closer to each other. The
increase in proximity was reflected in a significant increase in the expansion
compared to the truncated variant, which exhibited a much weaker contraction
with salinity.

Figure 4: The osmotic second virial coefficient $A_{2}$ as a function of the
two variants’ salinity ($C_{s}$). $\rm{\Delta}$N42 intermolecular interactions
transition from repulsive to attractive as $C_{s}$ increases. WT changes from
a nearly neutral state of intermolecular interactions to attractive. Inset: A
demonstration (WT variant, 20 mM Tris and 500 mM NaCl pH 8.0) for the Zimm
analysis used to extract $A_{2}$ from SAXS data measured at various protein
concentrations ($C$). Values shown in the graph are in mg/ml units. The dashed
lines show the extrapolation from the measured data (colored lines) to the
fitted $q\rightarrow 0$ and $C\rightarrow 0$ yellow lines, where $\alpha=0.01$
is an arbitrary constant used in this analysis.

## Discussion and Conclusions

We investigated the effects of structural heterogeneity on the interactions of
NFLt, an IDP model system. For NFLt, the N-terminal region consisting of the
first $\sim 50$ residues is hydrophobic and charge neutral, while the
remaining chain is highly charged. We found that the sequence heterogeneity
differentiates between the structures of the entire WT NFLt and a variant
lacking the N-terminal domain. In particular, the WT variant self-assembles
into star-like structures while the $\rm{\Delta}$N42 one remains isolated in
all measured cases.

Since $\rm{\Delta}$N42 can be attributed as a charged polymer, weakly
attractive interactions take center stage as the electrostatic repulsion
diminishes with charge screening (Fig. 4). These interactions could be
attributed to monomer-monomer attractions that arise from the sequence
heterogeneity of the IDP, such as weak hydrophobic attraction from scattered
hydropathic sites [22, 29, 28, 44, 45, 46].

For the WT variant, the intermolecular interactions started from a near-
neutral state and transitioned to weakly attractive. However, as the WT
measurements describe self-assembling complexes, the interpretation of these
results differs from $\rm{\Delta}$N42. As such, we interpret the
intermolecular interactions as the ‘aggregation propensity,’ the protein
complex’s growing ability. The aggregation propensity grows as the
attractivity between the complex and the other polypeptides in the solution
increases. This behavior can be observed when examining the responsiveness of
the aggregation number $Z$ to protein concentration $C$ (Fig. S7). In the
lowest measured screening, $Z$ dependency on protein concentration was
minimal. As we increase the screening effects, this dependency becomes more
substantial. This characterization is also found in folded proteins, where
inter-molecular interactions were shown to indicate aggregation propensity
[47].

In our previous study [22], Flory exponents ($\nu$) of shorter segments from
the same NFLt were measured independently and in the context of the whole NFLt
using SAXS and time-resolved Förster resonance energy transfer (trFRET).
There, regardless of the peptide sequence, in the context of the entire NFLt,
the segments’ structural statistics were more expanded (i.e., with larger
$\nu$ values) than when measured independently. Similarly, these short
segments measured with SAXS have smaller $\nu$ values (i.e., with a compacted
statistical structure) than those of measured here for $\rm{\Delta}$N42 in all
salt conditions (Fig. 3a, grey symbols).

The expansion of segments in the context of a longer chain corroborates that
long-range contacts contribute to the overall disordered ensemble [22].
Interestingly, at $C_{s}=520$ mM salinity, we found similar $\nu$ values of
the $\rm{\Delta}$N42 and the previous short segment measurements, indicating a
comparable expansion. We suggest that at higher salinities, the significance
of electrostatic long-range contacts diminishes, aligning the expansion
‘scaling laws’ regardless of the chain length. Importantly, comparisons
between our $\rm{\Delta}$N42 variant results (and not to the WT variant) to
the previous segments’ measurements are more suitable as the chains did not
aggregate in those cases.

Compared to $\rm{\Delta}$N42, WT exhibits a mild contraction in salt,
resembling the behavior of the ‘salt–brush’ regime observed in polyelectrolyte
brushes, as demonstrated in Fig. 3. Similar salt–brush behavior was previously
observed in Neurofilament high tail domain brushes grafted onto a substrate
[26], and in a recent polyelectrolytic brush scaling theory [48]. In the salt-
brush regime, Pincus showed that brush mechanics resemble neutral brushes,
determined by steric inter-chain interactions [49]. In this interpretation,
the effective excluded volume per monomer enlarges and is proportional to
$1/\kappa_{\rm s}^{2}$, where $\kappa_{\rm s}$ is the Debye length attributed
to the added salt. Consequently, we suggest that the heightened charge
screening in the WT solution allows steric interactions between brushes to
play a more significant role in determining the brush ensemble. Additionally,
we deduce that the increased prevalence of steric repulsion counteracts the
attractive forces responsible for aggregation, thereby preventing brush
collapse.

The NFLt contraction aligns with previous studies of native NFL hydrogel
networks [29, 28]. At high osmotic pressure, the NFL network showed weak
responsiveness to salinity higher than $C_{s}=100$ mM, in agreement with
theory [48]. With the observed salt–brush behavior for WT, we suggest that
weak salt response in NFL hydrogels coincides with the increase in steric
repulsion shown for the star-like structures (Fig. 3a, blue line).

Additionally, our measurements show that the hydrophobic N-terminal regime of
the NFLt domain aggregates. This result is consistent with the findings of
Morgan et al. [31], where single-molecule pulling experiments were performed
on WT NFLt, and slow aging effects were observed, likely due to collapse (and
potential aggregation) of the neutral domain. Indeed, follow–up studies by
Truong et al. [50] used single-molecule stretching to show that added
denaturant led to a swelling of the chain (increased $\nu$), demonstrating
that the WT chain has hydrophobic aggregation that can be disrupted by the
denaturant. These observations suggest that at higher salt, the loss of
repulsion may lead to attractive hydrophobic interactions growing more
prominent in the NFL network. However, the steric repulsion from the remaining
NFL tail may shield such an unwanted effect. Nonetheless, such effects may
grow more prominent as the native filament assembly is disrupted.

In summary, we showed how the sequence composition of the NFLt IDP caused
structural deviation from a disordered polyelectrolyte to a self–assembled
star–like polymer brush. Together with the self–regulatory properties of the
brushes, such behavior can be exploited to design structures that can resist
specific environmental conditions. Additionally, our results showed possible
implications on NFL aggregates that could shed light on the underlying
correlations between the complex structure and the conditions driving it.
While IDPs resemble polymers in many aspects, as we showed here, it is
critical to assess their sequence to distinguish where and how to use the
appropriate theoretical arguments to describe their statistical properties and
structure.

## Methods

Protein purification Protein purification followed Koren et al. [22]. Variant
$\rm{\Delta}$N42, included two cysteine residues at the C- and N terminals.
After purification, $\rm{\Delta}$N42 variants were first reduced by 20 mM
2-Mercaptoethanol. Next, 2-Mercaptoethanol was dialysed out with 1 L of 50 mM
HEPES at pH 7.2. To block the cysteine sulfhydryl group, we reacted
$\rm{\Delta}$N42 variants with 2-Iodoacetamide at a molar ratio of 1:20. At
the reaction, the variants’ concentrations were $\sim$2 mg/ml. The reaction
solution was kept under dark and slow stirring for 5 hr and stopped by adding
50 mM 2-Mercaptoethanol followed by overnight dialysis against 1 L of 20 mM
Tris at pH 8.0 with 0.1% 2-Mercaptoethanol. Final purity was $>$95% as
determined by SDS-PAGE (Fig. S9).

SAXS measurement and analysis Protein samples were dialyzed overnight in the
appropriate solution and measured with a Nanodrop 2000 spectrophotometer
(Thermo Scientific) for concentration determination. Buffers were prepared
with 1 mM of TCEP to reduce radiation damage and 0.2% of Sodium Azide to
impair sample infection. The samples were prepared in a final concentration of
2 mg/ml, measured in a series of 4 dilutions. Preliminary measurements were
measured at Tel-Aviv University with a Xenocs GeniX Low Divergence CuK$\alpha$
radiation source setup with scatterless slits [51] and a Pilatus 300K
detector. All samples were measured at three synchrotron facilities: beamline
B21, Diamond Light Source, Didcot, UK [52], beamline P12, EMBL, DESY, Hamburg,
Germany [53], and beamline BM 29 ESRF, Grenoble, France [54]. Measurements at
ESRF were done using a robotic sample changer [55].

Integrated SAXS data was obtained from the beamline pipeline and 2D
integration using the ”pyFAI” Python library [56]. Extended Guinier analyses
for the $\rm{\Delta}$N42 variant were done with the ”curve_fit” function from
the ”Scipy” Python library [57]. To extract $R_{g}$ and $\nu$, extended
Guinier analysis was conducted for $0.7<qR_{g}<2$. Error calculation was done
from the covariance of the fitting.

Model fittings for the WT variant were done using the ”lmfit” Python library
[58] using the model described in [34, 35]. Due to the complexity of the
model, cylindrical core fittings were done by binning the data in 100
logarithmic bins to reduce computation time. Within the same model, core
parametres (cylinder radius $R$ and cylinder length $L$) were set constant, to
offset fitting errors. Initial values of $R$ and $L$ were calculated with the
highest measured concentration. Physical boundary conditions were imposed on
the fitting, and scattering length (SL) values were set to be unchanged by the
fitting process. SL values of both the core and the tail domains were
determined by tabulated values of amino acid SLD in 100% H2O [59] (Table S3).
Fitting parameter error evaluation was done by finding the covariant of the
returning fitting parameters. Error calculation of the volume was done using:
$\frac{dV}{V}=\sqrt{3\left(\frac{dR}{R}\right)^{2}}$. In addition, $\nu$
values of WT were found by a recursive search of the corresponding tail height
$h/2$ over Eq. 2. Errors of $\nu$ were then found by assuming a simple case of
$R_{g}=bN^{\nu}$, from which:
$d\nu\sim\frac{\ln{(1+dR/R)}}{\ln{N}}\sim\ln{(N)}^{-1}\frac{dR}{R}$

Zimm analysis Zimm analysis was performed as described in [43]. Data
normalization was done by first determining $I_{0}$ by fitting a linear curve
over the Guinier plot ($\ln{I(q)}$ vs $q^{2}$). Normalized $1/I(q)$ linear
fitting was done starting with the earliest possible data point until a
deviation from the linear behavior occurs. Data points were then binned for
visual clarity without impacting the result.

Brush model fitting Brush height model as described in [39] was fitted with a
prefactor $c=0.33$ to match data. Resulting heights were converted to $\nu$ by
$h=bN^{\nu}$ where $b=0.38$ nm and $N=146$. To accommodate for the change in
grafting density, a linear curve was fitted to the grafting density’s change
in salinity and was used to obtain a continuous plot.

Polyelectrolye fitting The fitting model was used as described in [38] with a
pre-factor $c=1.24$ to match data.

## Acknowledgments

R.B. and O.A.S. dedicate this article to Fyl Pincus, for his continuous
leadership and friendship over the years. His past works on charged polymer
brushes, and polymers’ scaling laws, inspired much research in the field,
including this work. The synchrotron SAXS data were collected at beamline P12,
operated by EMBL Hamburg at the PETRA III storage ring (DESY, Hamburg,
Germany), at beamline B21, operated by Diamond Light Source (Didcot, UK), and
at beamline BM29, operated by ESRF (Grenoble, France). We would like to thank
Cy M. Jefferies (DESY), Katsuaki Inoue (DLS), and Mark Tully (ESRF) for their
assistance in using the beamlines. This work has been supported by the NSF
(MCB-2113302), the NSF–BSF program (2020787), the Israel Science Foundation
(1454/20), and by iNEXT-Discovery (15410), funded by the Horizon 2020 program
of the European Commission. We also acknowledge the fruitful discussion and
help from Yacov Kantor, Uri Raviv, and Sagi Meir.

### Statements and Declarations

Conflicting interests The authors claim no conflicting interests.
Data availability The raw SAXS data is available in the Small–Angle Scattering
Biological Data Bank (SASBDB) with the identifier: XXXX.
Author contribution M.K., G.K., and R.B. designed the project. M.K. conducted
experiments and analysis with G.K.’s and O.A.S.’s assistance. M.K., G.K.,
R.B., and O.A.S. wrote the paper.

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## Supplementary Information

From isolated polyelectrolyte to star-like assemblies:
the role of sequence heterogeneity on the statistical structure of the
intrinsically disordered Neurofilament-low tail domain
Mathar Kravikass, Gil Koren, Omar Saleh, Roy Beck
Corresponding<EMAIL_ADDRESS>

Contents:
Table S1-S5,
Figure S1-S9

Figure S1: $\rm{\Delta}$N42 SAXS measurements Gaussian form factor fitting for all salinity concentrations $C_{s}$. $r_{G}$ used for the Gaussian form factor is as obtained by the Extended Guinier analysis (see table S1). Figure S2: $\rm{\Delta}$N42 SAXS measurement with corresponding extended Guinier curve. Bottom: Deviation from fit $\sigma_{fit}=(Y_{data}-Y_{fit})/\sigma_{data}$ Dashed line represents the maximum analysis point $qr_{G}=2$ from which deviation starts. Displayed data: $20$mM Tris pH$8.0$ in $1.1$mg/ml. Figure S3: $\rm{\Delta}$N42 SAXS measurements with corresponding extended Guinier curves. Dashed lines represent the maximum analysis point $qr_{G}=2$ from which deviation starts. Protein concentrations were offset for clarity, with the lowest (blue) being of the highest concentration. $C_{s}$ (mM) | C (mg/ml) |  $r_{G}$ (nm) | $\nu$ |  $I_{0}$ (cm-1)
---|---|---|---|---
20 | 1.1 | 4.23 $\pm$ 0.05 | 0.642 $\pm$ 0.002 | 0.0228
20 | 0.8 | 4.56 $\pm$ 0.07 | 0.660 $\pm$ 0.66 | 0.0254
20 | 0.6 | 5.11 $\pm$ 0.13 | 0.689 $\pm$ 0.007 | 0.027
70 | 1 | 4.41 $\pm$ 0.12 | 0.652 $\pm$ 0.007 | 0.0259
70 | 0.5 | 4.53 $\pm$ 0.22 | 0.659 $\pm$ 0.012 | 0.0257
70 | 0.3 | 4.99 $\pm$ 0.46 | 0.683 $\pm$ 0.023 | 0.032
170 | 1.5 | 4.51 $\pm$ 0.05 | 0.657 $\pm$ 0.003 | 0.19
170 | 0.8 | 4.59 $\pm$ 0.11 | 0.662 $\pm$ 0.006 | 0.18
170 | 0.3 | 4.56 $\pm$ 0.27 | 0.661 $\pm$ 0.015 | 0.18
270 | 1 | 4.43 $\pm$ 0.06 | 0.653 $\pm$ 0.003 | 0.026
270 | 0.5 | 4.45 $\pm$ 0.1 | 0.654 $\pm$ 0.006 | 0.026
270 | 0.3 | 4.64 $\pm$ 0.16 | 0.664 $\pm$ 0.009 | 0.028
520 | 1.5 | 4.14 $\pm$ 0.02 | 0.636 $\pm$ 0.001 | 0.026
520 | 0.78 | 4.00 $\pm$ 0.08 | 0.628 $\pm$ 0.005 | 0.023
520 | 0.38 | 4.05 $\pm$ 0.35 | 0.630 $\pm$ 0.02 | 0.023
Table S1: $\rm{\Delta}$N42 Extended guinier analyis data. Analysis parametres (radius of gyration $r_{G}$, scaling exponent $\nu$, and scattering intensity at $q=0$ ($I_{0}$)) obtained for different salt concentrations ($C_{s}$) and protein concentrations ($C$). $C_{s}$ (mM) |  $r_{G}$ (nm) | $\nu$
---|---|---
20 | 5.76 $\pm$ 0.31 | 0.729 $\pm$ 0.015
70 | 4.84 $\pm$ 0.27 | 0.677 $\pm$ 0.015
170 | 4.71 $\pm$ 0.04 | 0.669 $\pm$ 0.003
270 | 4.61 $\pm$ 0.16 | 0.663 $\pm$ 0.009
520 | 3.88 $\pm$ 0.04 | 0.620 $\pm$ 0.003
Table S2: Zero concentration extended Guinier analysis data. Analysis
parameters (radius of gyration $r_{G}$ and scaling exponent $\nu$) were
extrapolated to zero protein concentration at various salt concentrations
($C_{s}$). Figure S4: SAXS measurements of WT and its fitting to different
form factors. Both form factors are of the same model but use a different
core: Spherical or Ellipsoidal. Spherical core fitting yields a core radius of
$R=0.66\pm 0.016$ nm, and the ellipsoidal core yields a core radius of
$R=1.335\pm 0.23$ nm and a secondary radius of $\epsilon R$ where
$\epsilon=0.153\pm 0.08$. Both fittings yield close values of aggregation
number Z ($3.046\pm 0.04$ for spherical and $3.562\pm 0.07$ for ellipsoidal)
and tail height $h/2$ ($9.838\pm 0.04$ nm for spherical and $9.584\pm 0.12$
for ellipsoidal). Below: Fitting error
$\sigma_{fit}=(Y_{fit}-Y_{data})/\sigma_{data}$. Both curves show similar
error profiles. The spherical model proved best to describe the model due to
its simplicity. Displayed data: WT in $20$ mM Tris pH=8.0, and $170$ mM NaCl
at a concentration of $1.3$ mg/ml.  Figure S5: SAXS measurements and spherical
form factor fitting for all salinity concentrations ($C_{s}$). The$C_{s}=20$
mM data fit is to a cylindrical core. Dashed lines represent the Gaussian form
factor of the structure tails. Protein concentrations were offset for clarity,
with the lowest (blue) being the highest concentration. Figure S6: WT SAXS
measurement with cylindrical fitting. Measurements at the highest
concentration of $C=2.68$ mg/ml, in a $20$ mM Tris buffer at pH=$8.0$. To
alleviate fitting inconsistencies, consequent fittings of measurements with
lower protein concentrations in the same buffer were done using the obtained
core parameters: Core radius $R=0.89\pm 0.03$ nm, and core length $L=1.19\pm
0.09$ nm.

$C_{s}$ | $C$ | $h/2$ | $\nu$ | $Z$ | $n$ | $R$ | $L$ | $V$ | $\beta_{t}$ | $\beta_{c}$
---|---|---|---|---|---|---|---|---|---|---
(mM) | (mg/ml) | (nm) |  |  |  | (nm) | (nm) | (nm3) | (103nm) | (103nm)
20 | 2.68 | 9.16±0.15 | 0.786±0.0033 | 1.60±0.03 | 10.11±0.711 | 0.89±0.028 | 1.19±0.09 | 2.26±0.027 | 0.227 | 3.826
20 | 1.8 | 8.20±0.13 | 0.758±0.0032 | 1.83±0.04 | 8.52±0.059 | 0.89 | 1.19 | 2.26 | 0.195 | 3.558
20 | 1 | 8.57±0.17 | 0.768±0.0041 | 1.87±0.05 | 8.12±0.060 | 0.89 | 1.19 | 2.26 | 0.195 | 3.558
20 | 0.5 | 8.27±0.16 | 0.759±0.0040 | 1.91±0.05 | 7.78±0.057 | 0.89 | 1.19 | 2.07±0.006 | 0.17 | 3.881
170 | 1.3 | 9.96±0.03 | 0.796±0.0006 | 3.34±0.02 | 2.52±0.009 | 0.66±0.005 | X | 1.18±0.007 | 0.039 | 4.014
170 | 0.73 | 10.11±0.05 | 0.799±0.0009 | 3.27±0.03 | 2.32±0.014 | 0.63±0.007 | X | 1.06±0.010 | 0.039 | 4.014
170 | 0.57 | 10.37±0.08 | 0.806±0.0015 | 2.13±0.02 | 3.10±0.037 | 0.60±0.006 | X | 0.93±0.008 | 0.074 | 3.98
170 | 0.24 | 10.78±0.10 | 0.815±0.0019 | 2.71±0.05 | 3.23±0.025 | 0.66±0.011 | X | 1.23±0.017 | 0.074 | 3.98
270 | 2 | 9.40±0.02 | 0.781±0.0003 | 5.37±0.02 | 1.89±0.007 | 0.70±0.004 | X | 1.42±0.008 | 0.018 | 4.03
270 | 1.5 | 9.43±0.02 | 0.782±0.0004 | 5.02±0.02 | 1.93±0.009 | 0.69±0.005 | X | 1.36±0.009 | 0.018 | 4.03
270 | 1.5 | 9.43±0.02 | 0.782±0.0004 | 5.02±0.02 | 1.93±0.009 | 0.69±0.005 | X | 1.36±0.009 | 0.018 | 4.03
270 | 0.69 | 9.85±0.04 | 0.793±0.0009 | 4.05±0.04 | 2.70±0.016 | 0.71±0.009 | X | 1.53±0.016 | 0.039 | 4.014
370 | 2.3 | 9.36±0.01 | 0.780±0.0003 | 6.68±0.03 | 1.48±0.008 | 0.69±0.005 | X | 1.38±0.009 | 0.018 | 4.036
370 | 1.5 | 9.44±0.02 | 0.782±0.0003 | 6.43±0.03 | 1.65±0.009 | 0.71±0.006 | X | 1.49±0.010 | 0.018 | 4.036
370 | 0.96 | 9.53±0.02 | 0.785±0.0004 | 5.85±0.03 | 2.08±0.010 | 0.74±0.006 | X | 1.70±0.012 | 0.039 | 4.014
370 | 0.6 | 9.81±0.03 | 0.792±0.0007 | 5.33±0.05 | 2.13±0.017 | 0.72±0.010 | X | 1.59±0.019 | 0.039 | 4.014
520 | 2.5 | 9.57±0.01 | 0.786±0.0003 | 8.15±0.04 | 1.82±0.008 | 0.79±0.005 | X | 2.07±0.012 | 0.018 | 4.036
520 | 1.19 | 9.28±0.02 | 0.778±0.0004 | 6.66±0.04 | 1.57±0.010 | 0.70±0.006 | X | 1.47±0.012 | 0.018 | 4.036
520 | 0.66 | 8.90±0.03 | 0.769±0.0006 | 4.05±0.02 | 2.03±0.012 | 0.65±0.007 | X | 1.15±0.010 | 0.039 | 4.014
520 | 0.45 | 9.82±0.03 | 0.792±0.0005 | 6.36±0.06 | 1.94±0.016 | 0.74±0.010 | X | 1.72±0.020 | 0.039 | 4.014

Table S3: WT spherical and cylindrical fitting analysis data. Analysis
parameters (brush height ($h$), scaling exponent ($\nu$), aggregation number
($Z$), core peptide length ($n$), core radius ($R$), cylindrical core length
($L$), core volume ($V$), tail scattering length ($\beta_{t}$) and core
scattering length ($\beta_{c}$)) obtained for different salt concentrations
($C_{s}$) and protein concentrations ($C$). Cylinder length $L$ values are
only relevant to $C_{s}=20$mM where a cylindrical core fit was used. For the
cylindrical core, the same values of $L$ and $R$ were used for all
concentrations to alleviate fitting errors (see Methods).

$C_{s}$ | $h/2$ | $\nu$ | $Z$ | $n$ | $R$ | $L$ | $V$
---|---|---|---|---|---|---|---
(mM) | (nm) |  |  |  | (nm) | (nm) | (nm3)
20 | 8.01±0.46 | 0.751±0.013 | 2.03±0.08 | 7.08±0.39 | 0.89 | 1.19 | 2.26
170 | 10.60±0.33 | 0.811±0.008 | 2.83±0.31 | 3.17±0.55 | 0.64±0.03 | X | 1.10±0.16
270 | 9.84±0.21 | 0.793±0.006 | 3.52±0.22 | 3.06±0.30 | 0.70±0.03 | X | 1.49±0.20
370 | 9.73±0.12 | 0.790±0.003 | 5.19±0.29 | 2.39±0.12 | 0.76±0.02 | X | 1.67±0.01
520 | 9.47±0.44 | 0.783±0.011 | 5.67±0.35 | 1.82±0.29 | 0.68±0.06 | X | 1.28±0.38

Table S4: Zero concentration WT spherical and cylindrical fitting analysis data. Analysis parametres (brush height ($h$), scaling exponent ($\nu$), aggregation number ($Z$), core peptide length ($n$), core radius ($R$), cylindrical core length ($L$) and core volume ($V$)) were extrapolated to zero protein concentration at various salt concentrations ($C_{s}$). Cylinder length $L$ values are only relevant to $C_{s}=20$mM where a cylindrical core was used. Figure S7: Structural parameters for WT (circles) and $\rm{\Delta}$N42 (triangles) variants extracted from fitting the SAXS data. Dashed lines demonstrate the linear fitting of the data used to obtained the zero concentration extrapolations. a. Aggregation number ($Z$) dependency on protein concentration ($C$) increases with increasing salt. b. Core volume $V_{s}$ against protein concentration ($C$). In $C_{s}=20$ mM, the $V_{s}$ values are constant due to fitting constraints (see Methods). c. In all cases, the tail heights ($h$) are much larger than the corresponding grafting length ($\rho$), indicative of a brush regime. d. The structurally intrinsically disordered $\rm{\Delta}$N42 variant compacts with higher $C_{s}$ values and remains more compacted from the projected tails for the WT variant. For the $\rm{\Delta}$N42 variant $r_{G}$ drastically changes as a function of the protein concentration ($C$). $C_{s}$ | $A_{2}^{WT}$ | $A_{2}^{\Delta N42}$
---|---|---
(mM) | (cm3mol/g${}^{2}\times 10^{3}$) | (cm3mol/g${}^{2}\times 10^{3}$)
20 | -0.295±1.346 | 13.264±0.466
70 | X | 3.978±1.248
170 | -2.072±2.091 | 0.169±1.544
270 | -3.328±0.508 | -1.152±0.756
370 | -2.020±0.563 | X
520 | -1.933±3.582 | -4.417±1.514
Table S5: Second virial coefficient $A_{2}$ values for both variants in salt
concentration $C_{s}$. Figure S8: a. Flory exponent ($\nu$) of WT tails and
$\rm{\Delta}$N42 variants as a function of the concentration. $\rm{\Delta}$N42
shows to change radically as a function of the concentration at the lowest
salinities. This effect is reduced as salinity concentration $Cs$ reaches
$170$mM. WT and the rest of $\rm{\Delta}$N42 $\nu$ data shows little change as
a function of the protein concentration.b. The core (aggregated) peptide
length per polypeptide as a function of the concentrations. The large drop
observed from $C_{s}=20$mM to $C_{s}=170$mM can be attributed to the shift
from a dimer to a trimer. Core peptide length difference diminishes with
increasing salinity, however the value still remain largely similar.  Figure
S9: a. SDS-PAGE Tris-Glycine 15% of both $\rm{\Delta}$N42 and NFLt (WT),
showing purity above 95%. White dashed lines indicate where image lanes were
edited closer for clarity. Both show a higher molecular weight reading in the
gel, which is common for IDPs.b-c. Deconvoluted ESI-TOF MS spectra of
$\rm{\Delta}$N42 and NFLt respectively. Theoretical molecular weight values
are 12423.57 and 16233.79 for $\rm{\Delta}$N42 and NFLt, respectively