# Protein–Polymer Mixtures in the Colloid Limit: Aggregation, Sedimentation
and Crystallization

Rui Cheng  HH Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK
Bristol Centre for Functional Nanomaterials, University of Bristol, Tyndall
Avenue, Bristol, BS8 1TL, UK  Jingwen Li  HH Wills Physics Laboratory, Tyndall
Avenue, Bristol, BS8 1TL, UK  Bristol Centre for Functional Nanomaterials,
University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK  School of
Chemistry, University of Bristol, Cantock’s Close, Bristol, BS8 1TS, UK.
Ioatzin Ríos de Anda  HH Wills Physics Laboratory, Tyndall Avenue, Bristol,
BS8 1TL, UK  School of Mathematics, University Walk, Bristol, BS8 1TW, UK.
Thomas W. C. Taylor  HH Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8
1TL, UK  Bristol Centre for Functional Nanomaterials, University of Bristol,
Tyndall Avenue, Bristol, BS8 1TL, UK  Malcolm A. Faers  Bayer AG, 40789,
Monheim am Rhein, Germany.  J. L. Ross Anderson  School of Biochemistry,
University of Bristol, Bristol, BS8 1TD, UK.  School of Cellular and Molecular
Medicine, University Walk, Bristol, BS8 1TD, UK.  BrisSynBio Synthethic
Biology Research Centre, Life Sciences Building, Tyndall Avenue, Bristol, BS8
1TQ, UK.  Annela M. Seddon  HH Wills Physics Laboratory, Tyndall Avenue,
Bristol, BS8 1TL, UK  Bristol Centre for Functional Nanomaterials, University
of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK  C. Patrick Royall
<EMAIL_ADDRESS>HH Wills Physics Laboratory, Tyndall Avenue, Bristol,
BS8 1TL, UK  Bristol Centre for Functional Nanomaterials, University of
Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK  School of Chemistry, University
of Bristol, Cantock’s Close, Bristol, BS8 1TS, UK.  Gulliver UMR CNRS 7083,
ESPCI Paris, Université PSL, 75005 Paris, France.

###### Abstract

While proteins have been treated as particles with a spherically symmetric
interaction, of course in reality the situation is rather more complex. A
simple step towards higher complexity is to treat the proteins as
non–spherical particles and that is the approach we pursue here. We
investigate the phase behavior of enhanced green fluorescent protein (eGFP)
under the addition of a non–adsorbing polymer, polyethylene glycol (PEG). From
small angle x-ray scattering we infer that the eGFP undergoes dimerization and
we treat the dimers as spherocylinders with aspect ratio $L/D-1=1.05$. Despite
the complex nature of the proteins, we find that the phase behaviour is
similar to that of hard spherocylinders with ideal polymer depletant,
exhibiting aggregation and, in a small region of the phase diagram,
crystallization. By comparing our measurements of the onset of aggregation
with predictions for hard colloids and ideal polymers [S.V. Savenko and M.
Dijkstra, J. Chem. Phys 124, 234902 (2006) and F. lo Verso et al., Phys. Rev.
E 73, 061407 (2006)] we find good agreement, which suggests that the eGFP
proteins are consistent with hard spherocylinders and ideal polymer.

††preprint: AIP/123-QED

##  I Introduction

Protein aggregation, and crystallization behavior has important consequences
in determining their structure, functions and understanding pivotal challenges
ranging from condensation diseases Schneider, Rasband, and Eliceiri (2012);
Morris, Watzky, and Finke (2009); Cohen  _et al._ (2013) to the development of
new materials.Bai, Luo, and Liu (2016) Controlling their assembly into states
in which their functionality can be exploited is crucial to fully realise
their potential. Predictions can be made based on phase diagrams, with
detailed conditions such as protein concentration, temperature, pH,
etc.Alberti, Gladfelter, and Mittag (2019); Lai, King, and Yeates (2012);
Dumetz  _et al._ (2008)

Crystallization is one of the most complex and least understood topics in
biology. Protein crystallization is fundamental to obtain protein structure
and to get insights into protein function. However, crystallization protocols
are mainly based on trial and error assays, with a lack of standardized
approaches. In fact, in average only 0.04% of crystallization experiments
yields good quality crystals. Fusco and Charbonneau (2016) This is due in part
to inherent protein shape and surface complexity as well as the dependence of
protein-protein interactions on combinations of pH, temperature and
precipitants (salts and polymers).Hui and Edwards (2003); Huang, Teng, and Niu
(1999); Bhamidi, Varanasi, and Schall (2005); Fusco and Charbonneau (2016);
Glaser and Glotzer (2019); Nicolai and Durand (2013)

Recent developments include an improved understanding of the role of clusters
in protein crystallization,Yamazaki  _et al._ (2017) the effect of polymers in
inducing crystallization,McPherson (1976); Tanaka and Ataka (2002); Tardieu
_et al._ (2002) the effects of salts on crystallization Tardieu  _et al._
(2002); Yamanaka  _et al._ (2011); Velev, Kaler, and Lenhoff (1998) and
crystal growth rate,Schmit and Dill (2012) and the role of temperature in the
protein phase diagram.Astier and Veesler (2008); Grouazel  _et al._ (2006) An
emphasis has been placed on the role of entropy in contact–contact
interactions in proteins.Cieślik and Derewenda (2009) Perhaps unsurprisingly,
fine manipulation of protein interactions is necessary for self-assembly, Kim
_et al._ (2015); Mandell and Kortemme (2009) and if this can be successfully
achieved and coupled with protein engineering, it is possible to manipulate
proteins to enable new paths of self–assembly.Suzuki  _et al._ (2016); Zhang
_et al._ (2020); Bai, Luo, and Liu (2016)

By contrast, the field of _Soft Matter_ often operates at rather larger
lengthscales than the supramolecular lengthscale of proteins. Yet phenomena
exhibited by soft materials have been applied to proteins, with some degree of
success.McManus  _et al._ (2016); Fusco and Charbonneau (2016); Stradner and
Schurtenberger (2020); Hamley and Castelletto (2007) Among these is the
concept inspired by colloidal systems of effective interactions between the
protein molecules that can be altered by other components in the system such
as added salts and polymers. Dorsaz  _et al._ (2012); Zhang  _et al._ (2012);
Zhang (2017); Piazza (2004) In this way a soft matter perspective offers some
insights to understand and quantify protein interactions and their equilibrium
phase diagrams by simplified models, which might provide a systematic way to
improve protein crystallization. McManus  _et al._ (2016); Fusco and
Charbonneau (2016); Stradner and Schurtenberger (2020); Dorsaz  _et al._
(2012); Hamley and Castelletto (2007) Indeed, a parameter that has been used
to relate protein phase behavior Tessier  _et al._ (2002); Piazza (2004);
George and Wilson (1994a); Elcock and McCammon (2001); Wentzel and Gunton
(2008) to that of colloidal suspensions Noro and Frenkel (2000); Lu  _et al._
(2008); Royall (2018) is the second viral coefficient, which can be calculated
experimentally from osmotic pressure,Elcock and McCammon (2001) static light
scattering, dynamic light scattering, SAXS or SANSZhang  _et al._ (2008);
Foffi  _et al._ (2014); Kim, Dumont, and Gruebele (2008); Huang, Yao, and
Olsen (2019). While proteins often aggregate at high concentration, some do
not and indeed interesting glassy behaviour reminiscent of hard sphere
colloids has been seen for concentrated solutions of eye lens
$\alpha$-crystallin Foffi  _et al._ (2014), which opens the potential for
further analogies with colloidal systems.

Examples of the insights gained from this approach of comparing proteins to
colloidal systems include the prediction of enhanced nucleation rates in the
vicinity of a (metastable) critical point, Ten Wolde and Frenkel (1997);
Pellicane, Costa, and Caccamo (2004) which have been realized using colloids
with a short–ranged attractionSavage and Dinsmore (2009); Taylor, Evans, and
Royall (2012) and gelationCardinaux  _et al._ (2007); Kulkarni, Dixit, and
Zukoski (2003); Zhang  _et al._ (2008) and so–called liquid–liquid phase
separation. Broide  _et al._ (1991) It is also possible to control the pathway
of crystallization by manipulating the interactions.Whitelam (2010) Analogies
have also been made between proteins and colloids with so–called mermaid
(short–range repulsion, long–range attraction) interactions, through the
discovery of finite–sized clusters, Groenewold and Kegel (2001); Stradner  _et
al._ (2004); Sedgwick, Egelhaaf, and Poon (2004); Campbell  _et al._ (2005);
Sciortino, Tartaglia, and Zaccarelli (2005); Malins  _et al._ (2011); Klix,
Royall, and Tanaka (2010); van Gruijthuijsen  _et al._ (2018) although the
existence of the protein clusters has been questioned. Shukla  _et al._ (2008)
Meanwhile the colloid systems have been shown to exhibit much more complex
behavior than was originally supposed, through a fundamental breakdown in
spherical symmetry in the electrostatic repulsions. Klix  _et al._ (2013);
Royall (2018) Moreover simple isotropic models applied to other colloidal
systems Likos (2001); Velev, Kaler, and Lenhoff (1998) often fall short and
fail to fully characterize key protein phenomenology due to anisotropic shape
and a non-uniform surface charge and hydrophobic/hydrophilic pattern in
proteins.Elcock and McCammon (2001); Fusco and Charbonneau (2016); Baaden and
Marrink (2013) This anisotropy is responsible for the directional and
localized protein interactions that yields non-closed packed crystal
structures as well as directed self-assembly.Liu, Kumar, and Douglas (2009)
This more complex behavior can be captured and reproduced to some extent via
_patchy particle_ models, where the simulated particles include angular
surface directionality of attractive short-range interactions.Glaser and
Glotzer (2019); Fusco  _et al._ (2014); Doye and Poon (2006) By changing the
number, size and specificity of said patches, the system can be optimised to
fully describe the protein behavior. Fusco  _et al._ (2014); Altan  _et al._
(2018); James, Quinn, and McManus (2015); Li  _et al._ (2012); Gnan,
Sciortino, and Zaccarelli (2019); Cai and Sweeney (2018); Zhang  _et al._
(2020)

In colloidal systems, an effective attraction between the particles can be
induced by adding non–adsorbing polymers.Asakura and Oosawa (1954); Poon
(2002) This can lead to gelationPoon (2002); Zaccarelli (2007); Royall  _et
al._ (2021); Asherie, Lomakin, and Benedek (1996) and enhanced crystallization
rates.Savage and Dinsmore (2009); Taylor, Evans, and Royall (2012) Although
they are often smaller than colloids, polymers are typically rather larger
than proteins,Vivares  _et al._ (2002); Zhang (2017) leading to the concept of
the _protein limit_ ,Bolhuis, Meijer, and Louis (2003); Mutch, van
Duijneveldt, and Eastoe (2007); Mutch  _et al._ (2008) where the polymers are
so much larger than the proteins that the relevant lengthscale is the _intra–_
polymer persistence length, rather than the polymer radius of gyration that is
typically considered in the case of colloid–polymer mixtures. However, here we
consider a scenario more akin to colloid–polymer mixtures, where the polymer
radius of gyration $R_{g}$ is smaller than or comparable to the protein
radius.

Polymer–induced protein precipitation has been investigated via volume
exclusion interactions i.e. depletion (for high molecular weight polymer).
Atha and Ingham (1981); Hui and Edwards (2003) However, some polymers can also
interact with positively charged amino acids (lysine, arginine and
histidine),Hašek (2006) and/or through hydrophobic chemical interactions (for
example with -CH2OCH2\- groups)Shkel, Knowles, and Record (2015) both present
on the protein surface. Additionally, there can be a preferential formation of
hydrogen bonds between the polymer and water, which in turn enhances protein-
protein interactions.Durbin and Feher (1996) These scenarios can lead to more
complex interactions than the non-adsorbing polymer-protein depletion picture.

Here, we consider mixtures of _enhanced_ Green Fluorescent Protein (eGFP) and
poly–ethylene–glycol (PEG). eGFP readily undergoes dimerizationKim  _et al._
(2015) such that the proteins resemble short rods. By comparing our results to
the extensive colloid–polymer literature,Poon (2002) we treat the proteins as
spherocylinders (with dimensions deduced from x–ray scattering), in particular
as mixtures of hard spherocylinders and ideal polymer Bolhuis  _et al._
(1997); Savenko and Dijkstra (2006). In this way, a we consider a model of the
protein–polymer mixture where the only level at which the complexity of the
system is treated is via a simplified form for the anisotropy of the protein
dimers, namely a spherocylinder. We thus treat the protein dimers as hard
spherocylinders and the polymers as ideal polymers.

Here we follow the literature on spherocylinder–polymer mixtures Bolhuis  _et
al._ (1997); Savenko and Dijkstra (2006) and express the aspect ratio as
$L/D-1$ in which the aspect ratio of spheres is then zero where $L$ is the
spherocylinder length and $D$ is the diameter. We interpolate the predictions
of polymer _fugacity_ required for polymer–induced demixing, between
spherocylinders with aspect ratio $L/D-1=5$Savenko and Dijkstra (2006) and
spheres Lo Verso  _et al._ (2006). Remarkably, given the simplicity of the
model, we find good agreement for the geometric parameters of our system with
$L/D-1=1.05$.

This paper is organised as follows. In section II we describe the methods of
protein preparation, characterization, as well as estimation of their
interactions as spherocylinders. Section III.1 consists of phase behavior,
including aggregation and crystallization, in salt-screened and salt-free
mixtures. We then compare the onset of aggregation with theory in Section
III.2, where the polymer radius of gyration at different molecular weights are
fitted by interpolating predictions from hard colloids and ideal polymers.
Finally, a discussion of our findings presented in Section IV.

##  II Methodology

###  II.1 Estimation of Interactions Between Proteins

Our system is governed by two control parameters: the protein concentration
and the polymer concentration. In the context of treating the system in the
spirit of a colloid–polymer mixture, we consider the proteins as
spherocylinders and thus the volume fraction

$\phi_{\mathrm{eGFP}}=\rho_{\mathrm{eGFP}}\left(\frac{\pi}{6}D^{3}+\frac{\pi}{4}D^{2}(L-D)\right)$
(1)

where $\rho_{\mathrm{eGFP}}$ is the protein number density. We determine the
diameter $D$ and length $L$ from x–ray scattering and compare our results to
literature values (see section II.3). Our choice of spherocylinders is
motivated by the literature on colloid-polymer mixtures, for which phase
diagrams for hard spherocylinders plus ideal polymer are available.Bolhuis
_et al._ (1997); Savenko and Dijkstra (2006); Lekkerkerker and Tuinier (2011)

Our second control parameter, polymer concentration, is expressed as the
polymer fugacity $z_{\mathrm{pol}}$. We make the significant assumption that
the polymers can be treated as an ideal gas (of polymers), and then the
fugacity is equal to the polymer number density in a reservoir
$z_{\mathrm{pol}}=\rho_{\mathrm{pol}}^{\mathrm{res}}$ in thermodynamic
equilibrium with the system.

To compare with predictions from theory and computer simulation, we use the
fraction of available volume $\alpha$ to estimate the reservoir polymer number
density from that in the experimental system $\rho_{p}^{\mathrm{exp}}$, viz
$\rho_{p}^{\mathrm{exp}}=\alpha\rho_{p}^{\mathrm{res}}$. We use the free-
volume approximation for $\alpha$.Lekkerkerker and Tuinier (2011)

$\begin{array}[]{c}\alpha=\left(1-\phi_{\mathrm{eGFP}}\right)\exp\left[-\left\\{A\left(\frac{\phi_{\mathrm{eGFP}}}{1-\phi_{\mathrm{eGFP}}}\right)+B\left(\frac{\phi_{\mathrm{eGFP}}}{1-\phi_{\mathrm{eGFP}}}\right)^{2}+C\left(\frac{\phi_{\mathrm{eGFP}}}{1-\phi_{\mathrm{eGFP}}}\right)^{3}\right\\}\right]\vspace{2ex}\\\
A=\frac{6\gamma}{3\gamma-1}q+\frac{3(\gamma+1)}{3\gamma-1}q^{2}+\frac{2}{3\gamma-1}q^{3}\vspace{2ex}\\\
B=\frac{1}{2}\left(\frac{6\gamma}{3\gamma-1}\right)^{2}q^{2}+\left(\frac{6}{3\gamma-1}+\frac{6(\gamma-1)^{2}}{3\gamma-1}\right)q^{3}\vspace{2ex}\\\
C=\frac{2}{3\gamma-1}\left(\frac{12\gamma(2\gamma-1)}{(3\gamma-1)^{2}}+\frac{12\gamma(\gamma-1)^{2}}{(3\gamma-1)^{2}}\right)q^{3}\end{array}$
(2)

where $\gamma=L/D$ is the length-to-diameter ratio of spherocylinders, and $q$
is the polymer-protein size ratio, of $2R_{g}/D$, $R_{g}$ is the radius of
gyration of polymers. Below we compare the phase behavior we obtain for our
system with literature values for spherocylinder–polymer mixtures.Bolhuis  _et
al._ (1997); Savenko and Dijkstra (2006); Lekkerkerker and Tuinier (2011)

Our proteins carry an electrostatic charge, which we determine below (section
II.3). To estimate the electrostatic interactions we used a screened Coulomb
(Yukawa) potential. Here, it is convenient to treat the proteins as spheres.
We shall see below that although they are not spherical, the electrostatic
interactions turn out to be so weak that we believe that to a large extent
they can be neglected. Therefore we merely estimate their strength with a
spherically–symmetric approximation.

$\beta u_{\mathrm{yuk}}(r)=\left\\{\begin{array}[]{ll}\infty&\text{ for
}r<D\\\ \beta\epsilon_{\mathrm{yuk}}\frac{\exp(-\kappa(r-D))}{r/D}&\text{ for
}r\geq D\end{array}\right.$ (3)

where the contact potential,

$\beta\epsilon_{\mathrm{yuk}}=\frac{\mathscr{Z}^{2}}{(1+\kappa
D/2)^{2}}\frac{\lambda_{B}}{D}$ (4)

$\kappa$ is the inverse Debye length,

$\kappa^{2}=4\pi\lambda_{\mathrm{B}}\sum_{i}\rho^{\mathrm{ion}}_{\mathrm{i}}(Z_{i}^{\mathrm{ion}})^{2}$
(5)

with $\mathscr{Z}$ the charge number, $\lambda_{\mathrm{B}}$ is the Bjerrum
length, $\rho_{i}^{\mathrm{ion}}$ is the number density of the $i$th ionic
species, $\mathscr{Z}_{i}^{\mathrm{ion}}$ is the valency of the $i$th ionic
species. For the system with no added salt, the ionic strength
$I=\sum_{i}\rho^{\mathrm{ion}}_{\mathrm{i}}(\mathscr{Z}_{i}^{\mathrm{ion}})^{2}$
was evaluated as the sum of the ion contributions of the weak dissociation of
25 mM HEPES (pKa = 7.66) and the counter ions contribution assuming charge
neutrality. Thus, by varying protein concentrations we obtained a range of $I$
= 1–4 mM. For the system where 10 mM NaCl was added, we included to the sum
the ions contribution from this salt dissociation, giving a range of $I$ =
15–18 mM. Further details of Yukawa potential are listed in Table 1.

It is important to highlight, as pointed out by Roosen-Runge and
collaborators,Roosen-Runge  _et al._ (2013) that in addition to assuming a
spherical shape for the proteins, also an isotropic distribution of ions on
their surfaces is assumed. This is not the case for eGFP dimers, thus the
charges calculated should only be considered as effective charges suitable to
describe the phenomena observed in our experiments. However, the magnitude of
the charge that we determine is sufficiently small that within the DLVO
treatment we employ, the electrostatic interactions prove to be very weak, so
we believe that at the level of this analysis, a spherical approximation is
reasonable.

###  II.2 Protein Expression and Purification

Cellular Culture for the Expression of eGFP. A mini-culture of competent
_Escherichia coli_ BL21 (DE3) previously transformed with the DNA plasmid-
pET45b(+)-eGFP was prepared by inoculating 100 mL of lysogeny broth (LB) and
the antibiotic carbenicillin (50 $\mu$g/mL) with an isolated _E. coli._
colony. The culture was left to grow overnight (16 h) at 37∘ C and 180 rpm. 2
mL of this culture inoculated to a 1 L of LB containing the same antibiotic
and which was left to grow under the same previous conditions. The optical
density (OD600nm) was monitored until a value of 0.5–0.6 was reached. Then,
the production of eGFP was induced by adding 1 mM of Isopropyl
$\beta$-D-1-thiogalactopyranoside (IPTG). After 1 h of induction time, the
temperature was changed to 28∘ C and was incubated overnight. The cell culture
was centrifuged at 4500 g for 15 min at 4∘ C. The supernantant obtained was
discarded and the pellet was resuspended in a _lysis buffer_ (20 mM imidazole,
300 mM NaCl and 50 mM potassium phosphate at pH 8.0) and stored at -20∘ C.Tang
_et al._ (2018)

Purification and concentration of eGFP. Cell pellets were thawed and kept on
ice, sonicated for 3 cycles of 30 seconds (Soniprep 150 plus MSE) and
centrifuged at 18000 rpm (Sorvall SS34 rotor) at 4∘ C for 30 min. The
supernatant was recovered and filtered through a 0.22 $\mu$m syringe filter
(Millipore) and injected to a Ni-NTA (nickel-nitrilotriacetic acid) agarose
column (Qiagen) connected to an ÄKTA START purification system (GE
Healthcare), previously equilibrated with the lysis buffer mentioned. The
bound eGFP was washed with the same lysis buffer to elute the rest of unbound
proteins and eGFP was later eluted with a linear gradient (0–100%) of a 500 mM
imidazole, 300 mM NaCl, 50 mM potassium phosphate buffer at pH 8.

The recovered proteins were then further purified through size exclusion
chromatography to eliminate aggregates and unfolded protein. A single peak
corresponding to a single protein size was collected. eGFP was concentrated to
$\sim$3 mL in a 25 mM Tris-Base 150 mM NaCl buffer at pH 7.4. The proteins
were applied to a HiLoad Superdex 75 16/600 size exclusion column using ÄKTA
START purification system (GE Healthcare) pre-equilibrated with the same
buffer. Protein elution was monitored at 280 nm.

Purified eGFP was filtered through a 0.22 $\mu$m syringe filter (Millipore)
and concentrated using protein 30 kDa concentrators (ThermoFisher Scientific)
at 5000 rpm and 4∘ C for the time required to reach the desired volume. The
protein concentration was determined by measuring the absorbance at
$\lambda_{\mathrm{eGFP}}$ = 488 nm with a molar extinction coefficient
$\epsilon_{\mathrm{eGFP}}$ = 56000 M-1cm-1. Kaishima  _et al._ (2016)

Sample preparation. From small-angle X-ray scattering, the purified eGFP
showed to form dimers with a height of 8.2 nm and length of 4 nm, as shown in
Fig. 1. Thus we treated the eGFP molecules as spherocylinders, which are made
of two hemispheres of diameter $D=4$ nm, length $L=8.2$ nm. We changed the
protein buffer to 25 mM HEPES at pH = 7.4. A separate buffer solution with 0
or 200 mM NaCl was used as a stock solution to adjust the final protein and
salt concentrations.

We carried out experiments with two different polymer sizes, in particular,
polyethylene glycol (PEG, Polymer Laboratories) with molecular weights $M_{w}$
of 620 and 2000. The polymer radius of gyration $R_{g}$ was estimated from
polymer scaling with an empirical prefactor
$R_{g}^{\mathrm{scale}}=0.020M_{\mathrm{w}}^{0.58}$,George and Wilson (1994b)
leading to $R_{g}^{\mathrm{scale}}$ of 0.83 and 1.64 nm, and a polymer-protein
size ratio ($q^{\mathrm{scale}}=2R^{\mathrm{scale}}_{g}/D$) of 0.42 and 0.82,
for the small and large polymers respectively.

Figure 1:  Determining protein form factor with small angle X-ray scattering
(SAXS). (a) SAXS scattering intensity with a cylinder model fitting (line) for
10 mg/mL protein solutions of eGFP. (b) Illustration of the dimer structure of
eGFP and the spherocylinder representation used for data analysis, showing the
diameter $D$ and the length $L$. (c) Schematic of eGPF depletion zone when
polymers are present.

For each sample, we first mixed different volumes of the protein stock
solution (106.4 mg/mL) with different volumes of the HEPES buffer with and
without salt to complete a fixed volume of 5 $\mu$L, giving a range of protein
concentrations of 0.7–30 mg/mL and a constant NaCl concentration of 10 mM (for
the samples with salt). To induce effective attractions between the protein
molecules, we added different amounts of PEG by weight at room temperature
such that we obtained a polymer concentration between 0–0.8 gcm-3 (fugacity
$\sim 1$–$50$). The samples were thoroughly shaken for 5 seconds by a touch-
vortexer, immediatly imbibed to inside-diameter of 0.5 mm capillaries (CM
Science), and sealed with optical adhesive (Norland Optical no 81). Within 5
minutes, the different phases obtained were characterised through laser
scanning confocal microscopy (Leica SP8) at an excitation wavelength of 488 nm
and emission wavelength of 509 nm.

###  II.3 Characterisation

SAXS analysis To characterise the size and shape of the expressed and purified
eGFP, we performed SAXS measurements on 25 $\mu$L of 10 mg/mL eGFP in 25 mM
Tris-Base 150 mM NaCl buffer at pH 7.4 on a SAXSLAB Ganesha 300XL instrument.
Samples were loaded into 1.5 mm borosilicate glass capillaries (Capillary Tube
Supplies UK) and sealed with optical adhesive under UV light (Norland 81). The
wavevector $k$ range was of 0.006–0.30 Å-1. Background corrections were
carried out with both an empty cell and a cell with the buffer only. The
obtained data were fitted using the SasView version 4.0 software
package.Doucet  _et al._ (2017)

The results are shown in Fig. 1a. The scattering intensity, _I(k)_ is given by
the product of the form factor $P(k)$ and the static structure factor $S(k)$
via

$I(k)=\phi_{\mathrm{eGFP}}V_{\mathrm{eGFP}}(\Delta\rho_{\mathrm{scat}})^{2}P(k)S(k)$
(6)

$V_{\mathrm{eGFP}}$ is the volume of a protein dimer (Eq. 1) and
$\Delta\rho_{\mathrm{scat}}$ is the difference in scattering length density
between the proteins and its solvent.Zhang  _et al._ (2007); Wolf  _et al._
(2014); Singh  _et al._ (2019) The scattering data was successfully fitted by
a cylindrical form factorArpino, Rizkallah, and Jones (2012) with a diameter
of 4.0 $\pm$ 0.02 nm and a length of 8.2 $\pm$ 0.08 nm (see full parameters in
Appendix A 2). These dimensions are consistent with dimers of eGFP as
illustrated in Fig. 1b. These results are in agreement with previous work on
eGFP, where it was found that the protein exists in dimers.Myatt  _et al._
(2017)

Electrophoretic Mobility Measurement. We performed electrophoretic mobility
$\mu_{e}$ measurements on 1 mL protein solutions of 2 mg/mL at 20∘ C in a NaCl
10 mM solution using a Zetasizer Nano ZS (Malvern, UK) at a detector angle of
13∘ and a 4 mW 633 nm laser beam to determine the charge of eGFP following
Roosen-Runge, _et al_.Roosen-Runge  _et al._ (2013) Care was taken in order to
have the same pH with the buffer used in phase diagram determination. By using
electrophoretic light scattering (ELS) via phase analysis light scattering
(M3-PALS), the electrophoretic mobility $\mu_{e}$ of eGFP was determined as an
external electric field is applied.

From this we obtained the zeta potential $\zeta$ for a spherical particle with
diameter _D_ using

$\mu_{e}=\dfrac{2\epsilon\zeta f(\kappa D/2)}{3\eta}$ (7)

where $\epsilon_{r}$ and $\eta$ are the dielectric constant and the viscosity
of the solvent, respectively, and $f(\kappa D/2)$ is the Henry function
evaluated at $\kappa D/2$. The relation between surface charge density
$\sigma$ and the reduced zeta potential _$\tilde{\zeta}$ = ($e\zeta$)/(2kBT)_
is:

$\frac{\sigma
e}{k_{B}T}=2\epsilon_{r}\kappa\left[\sinh^{2}\left(\frac{\tilde{\zeta}}{2}\right)+\frac{4}{\kappa
D}\tanh^{2}\left(\frac{\tilde{\zeta}}{4}\right)+\frac{32}{(\kappa
D)^{2}}\ln\left(\cosh\left(\frac{\tilde{\zeta}}{4}\right)\right)\right]^{1/2}$
(8)

Finally we can obtain the total charge using $\mathscr{Z}e=\pi D^{2}\sigma$,
where $\mathscr{Z}$ is the charge number and $e$ is the elementary charge. The
zeta potential value measured was $\zeta$ = -7.02 mV, which corresponds to a
charge number of $\mathscr{Z}$ = 1.16. We list the parameters for Yukawa
potential in Table 1.

##  III Results

We divide our results as follows. First a phase diagram is presented for the
eGFP-PEG620 system, showing the fluid-aggregation transition in section III.1.
We increased the polymer molecular weight to obtain a larger size ratio (using
PEG2000), investigating the effects of polymer size on phase boundary. To
check any residual effects of protein charges, the comparison between salt-
screened and salt-free system is discussed. In section III.2, we consider a
spherocylinder-sphere system of $L/D-1=1.05$. The polymer radius of gyration
is fitted by interpolating between theoretical and computer simulation
predictions. Finally the protein crystallization, formed through depletion
attractions with polymer, is discussed in section IV.

###  III.1 Phase Behavior

Salt–screened system. The phase diagram with different states as determined
from images from confocal microscopy for the eGFP – PEG620 (small polymer)
system with 10 mM of added NaCl salt is shown in Fig. 2. The phase diagram is
presented in the plane of protein volume fraction ($\phi_{\mathrm{eGFP}}$) and
polymer fugacity $z_{\mathrm{pol}}$. The phase boundaries are determined by
the average between the fluid and aggregated phase points. Note that in
depletion systems, aggregation and gelation are identified with the liquid–gas
phase boundary. Lu  _et al._ (2008); Royall (2018); Royall  _et al._ (2021)
Thus while these are non–equilibrium states, comparison with equilibrium phase
behaviour is nevertheless highly informative. For lower protein volume
fraction below we tested, a dotted line is drawn based on the intuition from
literature.Poon (2002) The smaller the concentration of proteins, the higher
the polymer concentration needed for phase separation. As noted above, the
protein volume fraction is estimated by assuming that the eGFP molecules are
spherocylinders of aspect ratio $L/D-1=1.05$. The polymer fugacity is obtained
from the polymer number density. The protein dimensions determined from SAXS
(section II.3) and estimated polymer size gave a size ratio
$q=(2R_{g}^{\mathrm{scale}})/D\sim 0.4$.

Figure 2: Phase diagram of the eGFP – PEG system with 10 mM NaCl. Here the
phase behavior is shown in the protein volume fraction – polymer fugacity
($\phi_{\mathrm{eGFP}}-z_{\mathrm{pol}}$) plane, coupled with their
concentrations. Fluids, aggregating systems and crystals are denoted by blue
circles, red triangles, and green diamonds respectively. Confocal microscopy
images of different states, along with a schematic representation of the
system behavior are shown in (a)-(d) as follows: (a) fluids, (b) crystals, (c)
aggregation, (d) denser sediments. The scale bars indicate 10 $\mu$m.

As a function of polymer concentration, we first encounter protein solutions
where the eGFP appears stable and exhibits no observable aggregation, but
instead there is a uniform fluorescent intensity, as the protein dimers are
far below the resolution of the microscope (Fig. 2a). Upon increasing the
polymer concentration, we see aggregation for polymer fugacity
$z_{\mathrm{pol}}=30.0\pm 1.0$ (which corresponds to a protein volume fraction
around 0.002), shown in Fig. 2c. Now the polymer concentration here is rather
high, indeed the polymer volume fraction
$\phi_{\mathrm{pol}}=\rho_{\mathrm{pol}}\pi R_{g}^{3}/6$ is of order unity. We
return to this point below in section IV.

As the protein concentration is increased, the polymer concentration required
for aggregation decreases. Upon further increase in polymer concentration,
protein aggregates form quickly and become large enough that considerable
quantities sediment to the bottom of the sample where a denser sediment builds
up (Fig. 2d). This is reminiscent of aggregation and sedimentation behavior in
colloidal systems.Piazza (2014) In a small region of the phase diagram, we
encounter protein crystallization, indicated as green diamonds (see region
denoted as “X”) in Fig. 2. We note that there is some lack of smoothness in
the phase boundary. Such fluctuations in phase boundaries we well–known in
soft matter systems (see. e.g. Poon, Weeks, and Royall (2012)) and we leave
this for further investigation.

Protein crystallization has been related to near–critical behavior.Ten Wolde
and Frenkel (1997) Here, although the regime of crystallization occurs near
the aggregation line (which, by itself might link it to criticality ten Wolde,
Ruiz-Montero, and Frenkel (1996); Taylor, Evans, and Royall (2012)) the
protein volume fraction is vastly lower than any critical isochore that would
be expected to occur for this system. Indeed, the volume fraction of the
critical isochore for spherical colloids plus polymers with size ratio $q\sim
0.4$ is estimated to be at least $\phi_{c}\gtrsim 0.25$,Lo Verso  _et al._
(2006) so it is hard to imagine that critical fluctuations are important here.
The lengths of the crystallites that we find are in the range of $\sim$4–80
$\mu m$. Fig. 5b is pure crystal, while Figs. 5a and c show aggregates which
we presume to be amorphous.

Figure 3:  A Phase diagram of eGFP with PEG2000 and 10 mM NaCl. Fluids and
aggregating systems are denoted by blue circles and red triangles,
respectively.  Figure 4: Phase diagram of the eGFP – PEG620 system with no
added salt. Fluids and aggregating systems are denoted by blue circles and red
triangles, respectively.

Salt–free system. To investigate the effect of the (weak) electrostatic
interactions between the proteins, we determine the phase behavior in the
absence of added salt shown in Fig. 4. We find a boundary for aggregation
estimated at $z_{\mathrm{pol}}=30.9\pm 1.9$ for a rather lower volume fraction
of 0.002, which is almost indistinguishable behavior to the case with added
salt (Fig. 4) at the same protein volume fraction. This is quite consistent
with the soft matter inspired analysis of treating the proteins as hard
spherocylinders. However we do not encounter any crystallization behavior here
and return to this in the discussion below.

Effects of PEG molecular weight. So far, we have discussed the system with the
smaller polymer (PEG620), we now switch to the larger polymer. We chose
PEG2000 here because its size is comparable to that of the protein. We
therefore expect normal depletion behaviour, as described by the
Asakura–Oosawa model, unlike the protein limit $q\gg 1$.Bolhuis, Meijer, and
Louis (2003); Mutch, van Duijneveldt, and Eastoe (2007); Mutch  _et al._
(2008) The phase diagram for the eGFP–PEG2000 system is shown in Fig. 3. The
aggregation shows at a much lower fugacity, $z_{\mathrm{pol}}=1.20\pm 0.04$,
compared with the smaller polymer at same protein volume fraction of 0.02.
This is qualitatively consistent with the literature,Poon (2002); Lekkerkerker
_et al._ (1992); Atha and Ingham (1981) that the larger the polymer, the lower
fugacity is needed for phase separation. Below we provide a more quantitative
comparison.

Figure 5:  Crystallization close to the demixing phase boundary, see Fig. 2b for location on the phase diagram. Upon increasing the polymer volume fraction, crystals were found around $z_{\mathrm{pol}}=32.5$ and $\phi_{\mathrm{eGFP}}$ = 0.0011. Bars = 10 $\mu$m. Table 1: Effects of adding salts on Yukawa potential NaCl Concentration (mM) |  $\kappa^{-1}$ (nm) | $\kappa D$ | $Z_{e}$ | $\beta\epsilon_{\mathrm{yuk}}$
---|---|---|---|---
0 | 4.65 | 1.51 | 0.95 | 0.0320
10 | 2.48 | 2.82 | 1.31 | 0.0322
Figure 6:  The interpolation from the data presented by Savenko and Dijkstra
Savenko and Dijkstra (2006) for $L/D-1=5$ (purple triangles) and Lo Verso et
al.Lo Verso  _et al._ (2006) for $L/D-1=0$ (green circles), green and purple
lines are fitted by power functions. The blue line is the interpolation, and
the pink and black stars are crossovers denote matching for PEG620 and
PEG2000, respectively.

###  III.2 Comparison with theory

In order to make a comparison with theoretical and computer simulation
predictions, we interpolate between phase boundaries determined for spheres
$(L/D-1=0)$Lo Verso  _et al._ (2006) and spherocylinders of a larger aspect
ratio than those we consider here $(L/D-1=5)$ (Fig. 6).Savenko and Dijkstra
(2006) It is important to note what the exact phase _is_. In the case of
sphere–polymer mixtures, upon adding polymer at low colloid volume fraction,
the first phase transition that is encountered (for $q\gtrsim 0.3$) is the
(colloidal) liquid–gas demixing.Lekkerkerker  _et al._ (1992); Poon (2002);
Dijkstra, Brader, and Evans (1999); Lekkerkerker and Tuinier (2011) In the
case of spherocylinders with aspect ratio $L/D-1=5$ it is fluid–crystal
coexistence.Bolhuis  _et al._ (1997); Savenko and Dijkstra (2006)
Nevertheless, for spheres at $q\approx 0.4-0.5$ the liquid–gas and
fluid–crystal phase boundaries occur at quite similar values of the polymer
fugacity and so here we neglect the difference. We are, in any case unaware of
any computation of the phase diagram for our parameters, and note that the
free volume theory of Lekkerkeker _et al._ Lekkerkerker  _et al._ (1992) is
not highly accurate for these parameters.Lo Verso  _et al._ (2006)

We fit the data for spheresLo Verso  _et al._ (2006) and
spherocylindersSavenko and Dijkstra (2006) by a power law
$z_{\mathrm{pol}}=a-bq^{c}$ at a low value of protein volume fraction
$\phi_{\mathrm{eGFP}}\sim 0.02$ with different $q$. Here $a$, $b$ and $c$ are
fitted constants. The interpolation is done linearly, by
$z_{\mathrm{pol}}^{(0)}-1.05(z_{\mathrm{pol}}^{(0)}-z_{\mathrm{pol}}^{(5)})/5$,
where $z_{\mathrm{pol}}^{(0)}$ is for spheres, Lo Verso  _et al._ (2006) and
$z_{\mathrm{pol}}^{(5)}$ is for spherocylinders ($L/D-1=5$). Savenko and
Dijkstra (2006) Our interpolation is shown in Fig. 6 where we plot the fitted
phase boundaries for fitting data and our interpolation. We interpolate to
obtain values of $q$ that are consistent with our measured fugacity for
demixing $z_{\mathrm{pol}}=8.63$ (smaller polymer) and $z_{\mathrm{pol}}=1.20$
(larger polymer). We have in addition some uncertainty in determining the size
ratio $q$. As noted above, our estimate for the polymer radius of gyration
$R_{g}^{\mathrm{scale}}$ relied on polymer scaling, which may not be accurate
for such small polymers. Moreover there are a variety of other assumptions,
such as polymer ideality, rigidity, which have been addressed in more refined
theoretical treatments.Lekkerkerker and Tuinier (2011); Fleer and Tuinier
(2008) We therefore accept some adjustment in our fitted values and take
$q^{\mathrm{fit}}=0.59$ for smaller polymer and $q^{\mathrm{fit}}=0.90$ for
larger polymer, which agree well with our data.

For larger polymers the fitted polymer radius of gyration
$R_{g}^{\mathrm{fit}}$ of 1.80 nm falls close to the one from empirical
equation of $R_{g}^{\mathrm{scale}}=1.64$ nm (see section II.2). It is worth
noting that there is an fitted increase from the size ratio from scaling
$q^{\mathrm{scale}}=0.42$ to the fitted size ratio of $q^{\mathrm{fit}}=0.59$
for the smaller polymer (PEG620). Now we consider the assumption that polymers
are ideal as in the standard AO model as with $R_{g}=1.2$ nm, these are _very_
small polymers to treat as ideal. Asakura and Oosawa (1954) Dijkstra _et al._
Dijkstra, Brader, and Evans (1999) compared additive hard-spheres with ideal
polymers using thermodynamic perturbation theory, they found that for small q
and polymer packing fraction $\phi_{p}$, the phase separation is very similar
between two models. Here we have $q=0.59$, and under these conditions of
larger depletants, the behaviour of spherical colloids plus ideal polymer and
spherical colloids plus hard sphere depletant is rather different, at least at
the level of the effective interactions between the larger spheres.Royall,
Louis, and Tanaka (2007) While we cannot rule out that the polymers may
exhibit significant deviations from ideality, given that the phase behavior we
find is similar to that of hard spherocylinders and ideal polymer, we note
that at the level of our analysis the polymers appear more likely to be
behaving in a manner similar to a polymer depletant rather than hard spheres.

##  IV Discussion

We have seen that the model fluorescent protein–polymer system can, rather
surprisingly, be treated in the spirit of a colloid–polymer mixture where the
only additional complexity is an approximate treatment of the anisotropy of
the protein dimers. This is notable, and a simple depletion picture of hard
spherocylinders with non–absorbing ideal polymers is consistent with our
observation. Furthermore, we observe no aggregation for eGFP in the absence of
polymer at least to 500 mg/mL, corresponding to a volume fraction of 0.48. At
this volume fraction, the protein solution becomes very viscous, consistent
with previous work which found glassy behavior reminiscent of colloidal
systems in concentrated eye lens $\alpha$-crystallin.Foffi  _et al._ (2014)
Furthermore, we found that upon dilution aggregated protein solutions
re–dissolved, behavior which is compatible with weak, depletion–driven
aggregation.

The crystallization behavior in our system re-emphasises that protein crystals
can be produced through addition of polymer, as noted previously.Arpino,
Rizkallah, and Jones (2012) This is significant because the process is
apparently immediate without a fine–tuning of the system. We focus on the low
volume fraction regime in this work, and we note that crystals only appeared
in a limited region in our phase diagram and then only in the system with
smaller polymer and added salt, not in the case of the larger polymer or
without added salt. At first sight, it may seem surprising that we find above
(section II.1) that the electrostatic interactions are very weak in our
system, with or without salt. It is important to highlight that the
isoelectric point (pI) of the monomeric unit of the eGFP (obtained from its
amino acid sequence) is 5.8, Pep (2015) which is close to the pH 7 used in the
experiments. This might explain the small values found for the surface charge.

We now enquire as to why not adding salt suppresses the crystallization. The
observation of crystallization only in a very limited region of polymer
concentration (i.e. attraction strength) is consistent with previous work with
(spherical) colloids and polymer mixtures,Poon (2002); Royall and Malins
(2012); Taylor, Evans, and Royall (2012) and has been interpreted in terms of
fluctuation–dissipation theorem violation.Klotsa and Jack (2011) Additionally,
it has been observed that acidic proteins are more likely to crystallize when
the pH of the solution is 0–2.5 units above their pI. Kirkwood  _et al._
(2015) Our experiments fall within such a range. Thus, only a small amount of
salt would be required to overcome small electrostatic repulsions under these
favourable conditions. What is perhaps more notable is the limited range of
protein concentration in which we see crystallization and the failure of the
salt–free system to crystallize. It is quite possible that the region of the
phase diagram in which crystallization occurs is so small is somehow related
to more complex behavior than that which we treat here. For example, Fusco _et
al._ showed the importance of contacts in the crystallization behavior of
rubredoxin. Fusco  _et al._ (2014) We speculate that a decrease in the
electrostatic repulsions only needs to occur around or in these regions to
promote crystal formation, leading to only small amounts of salt required to
yield a crystal, in contrast for example with isotropic systems. Finally,
salts can also affect the hydrophobic protein-protein interactions by
increasing the surface tension.Durbin and Feher (1996) These interactions have
shown to be relevant in the formation of a crystal phase and protein
solubility,Prevost  _et al._ (1991); Quinn, James, and McManus (2019) which
cannot be discarded in the present study.

Nevertheless, the crystallization that we observe is compatible with the
spherocylinder–polymer phase behavior ($L/D-1=5$), Savenko and Dijkstra
(2006); Bolhuis  _et al._ (1997). It would be most interesting to determine
the phase diagram for hard spherocylinders of aspect ratio $L/D-1=1.05$ plus
polymer, but for now we conclude that our finding of protein crystallization
is not inconsistent with some of the literature for hard particle – polymer
mixtures.Savenko and Dijkstra (2006); Bolhuis  _et al._ (1997); Lekkerkerker
_et al._ (1992); Poon (2002); Lekkerkerker and Tuinier (2011)

The polymer volume fractions at which we find aggregation are rather high, of
order unity. It is important to enquire whether one can still apply the
concept of polymer–induced depletion under these conditions. Accurate computer
simulations in which the polymer chain segments predict that for the polymer
fractions that we consider here, only small deviations of ideal Asakura–Oosawa
behavior are expected.Louis  _et al._ (2002) While we have treated our eGFP as
spherocylinders, and this work refers to spherical particles, we are unaware
of similar work which pertains to anisotropic particles and thus, in absence
of evidence to the contrary, presume that a simple depletion picture remains
reasonably accurate at these polymer concentrations.

While we have suggested that it is possible to account for the behavior of our
system by treating the eGFP as hard spherocylinders in a solution of ideal
polymers, we can be confident that the situation in reality is much more
complex. In addition to an enhancement of hydrophobic interactions from salt
addition discussed above, due to the amphiphilic nature of PEG, additional
hydrophobic Curtis and Lue (2006) and chemical Shkel, Knowles, and Record
(2015) interactions (via PEG -CH2OCH2\- groups) between PEG and proteins might
also contribute to this phenomenon. Furthermore, PEG molecules can also
enhance aggregation and crystallization via effective repulsion since PEG
might preferentially form hydrogen bonds with water compared to the
proteins.Durbin and Feher (1996) Finally, we have determined electrostatic
interactions between eGFP dimers to be weak, if we only consider the _net_
charge. Of course this is a very significant approximation. Monomeric eGFP has
a number of charging groups, e.g. 32 acidic residues and so a more
sophisticated approach which takes this into account may prove valuable. Such
an approach as that noted above for rubredoxin Fusco  _et al._ (2014) would be
most interesting to pursue here.

In short, further work is needed to explore throughout the metastable region
and then predictions can be validated using the depletion theory. Moreover the
properties of those crystals formed at this low protein concentration and by
purely depletion interactions, are certainly worth investigating in future
research.

##  V Conclusion

We studied the phase behavior of a model system of fluorescent proteins and
polymers (eGFP-PEG) in the “colloid limit” where the polymer depletant is
smaller than or comparable in size to the protein. A phase behavior of
fluid–aggregation was observed for two polymer sizes, i.e. two polymer–protein
size ratios). In addition to a small region of the phase diagram of a system
with added salt (NaCl) and small polymers where protein crystallization
occurred. At high polymer concentration, protein aggregates were large enough
to sediment on the timescale of the experiment and form a sediment whose
structure is reminiscent of a gel. In the absence of polymer, solutions of
eGFP are stable at least to a concentration of 500 mg/ml (volume fraction at
0.48). This suggests that the eGFP dimers interact rather weakly and that
approximating them as hard particles may be reasonable.

Based on the shape of eGFP dimers as deduced from small angle x-ray
scattering, we treat them as hard spherocylinders with aspect radio
$L/D-1=1.05$. In the case of the small polymer (PEG 620), the aggregation
boundary of polymer fugacity around protein volume fraction of 0.002, was
found almost indistinguishable, between $30.0\pm 1.0$ for salt-screened system
and $30.9\pm 1.9$ for salt-free system. For the larger polymer (PEG2000)
aggregation was found at a polymer fugacity of 1.20. Consistent with DLVO
theory for colloids, the effects of electrostatic interactions between the
proteins were found to be weak. Intriguingly, in the case of no added salt,
and also in the case of no added polymer, we observed no protein
crystallization. Due to the uncertainty of the polymer radius of gyration, we
interpolated the fugacity for the aggregation phase boundary from existing
literature, between $L/D-1=0$ for spheres–polymer mixtures Lo Verso  _et al._
(2006) and $L/D-1=5$ for spherocylinder–polymer mixtures Savenko and Dijkstra
(2006) and fitted a polymer radius of gyration of 1.1 nm for PEG620. Compared
with the empirical estimation of 0.83 nm, this somewhat larger size may be
related to some non–ideality in the polymers Lekkerkerker and Tuinier (2011)
(we note that polymer scaling theory is expected to break down for such small
polymers in any case). The smaller difference for larger polymer (PEG2000) is
consistent with this.

The behavior we observed is consistent with the depletion picture of hard
spherocylinders and ideal polymers. But in reality the system is rather more
complex. At our level of analysis and observation, we cannot exclude the
possibility that other interactions drive the phenomena that we observe, for
example hydration effects, hydrophobic or electrostatic “patches”.
Nevertheless, the fact that in the absence of polymer, the eGFP solution
exhibits no aggregation to such high concentrations, at that the aggregates
re–dissolve upon dilution gives us some cautious optimism that the behavior we
observe may be driven by such simple interactions as the excluded volume
effects of polymer–induced depletion.

###### Acknowledgements.

We would like to thank John Russo and Mike Allen for helpful discussions;
Richard Stenner for protein expression and purification; and Angélique
Coutable-Pennarun for assistance with zeta potential measurements. This work
was financially supported by Bristol Centre for Functional Nanomaterials
(BCFN), Chinese Scholarship Council (CSC), and Bayer AG. IRdA was funded by
the Philip Leverhulme Prize 2018 awarded by the Leverhulme Trust. IRdA, JLRA
and CPR were funded by the Leverhulme Trust grant “Unifying Protein Design and
Assembly of Soft Matter for New Materials”. RC, IRdA and CPR gratefully
acknowledge the ERC Grant agreement no. 617266 NANOPRS for financial support
and Engineering and Physical Sciences Research Council (EP/H022333/1). The
Ganesha X-ray scattering apparatus used for this research was purchased under
EPSRC Grant Atoms to Applications (EP/K035746/1). This work benefitted from
the SasView software, originally developed by the DANSE project under NSF
award DMR-0520547.

Data availability statement The data that support the findings of this study
are available from the corresponding author upon reasonable request.

##  Appendix A Geometric Parameters of eGFP Determined with SAXS

Table 2: Parameters obtained from SASView Doucet  _et al._ (2017) fitting using a cylinder form factor Protein |  Radius (Å) |  Error Radius (Å) |  Length (Å) |  Error Length (Å) |  Fitting $\chi^{2}$
---|---|---|---|---|---
eGFP |  20.5 |  0.08 |  82.3 |  0.7 |  1.19

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