Power Law Rheology of Folded Protein Hydrogels

Anders Aufderhorst-Roberts1,2 Sophie Cussons3,4 David J. Brockwell,3,4 Lorna
Dougan2,3∗

1 Centre for Materials Physics, Department of Physics,
University of Durham, Durham, DH1 3LE, UK
2 School of Physics and Astronomy,
University of Leeds, Leeds, LS2 9JT, UK
3 Astbury Centre for Structural Molecular Biology,
University of Leeds, Leeds, UK
4 School of Molecular and Cellular Biology, Faculty of Biological Sciences,
University of Leeds, Leeds, LS2 9JT, UK

<EMAIL_ADDRESS>
## Abstract

Folded protein hydrogels are prime candidates as tuneable biomaterials but it
is unclear to what extent their mechanical properties have mesoscopic, as
opposed to molecular origins. To address this, we probe hydrogels of the
muscle-derived protein $I27_{5}$, using a multimodal rheology approach. Across
multiple protocols, the hydrogels consistently exhibit power-law
viscoelasticity in the linear viscoelastic regime with an exponent
$\beta=0.03$, suggesting a dense fractal meso-structure, with predicted
fractal dimension $d_{f}=2.48$. In the nonlinear viscoelastic regime, the
hydrogel undergoes stiffening and energy dissipation, indicating simultaneous
alignment and unfolding of the folded proteins. Remarkably, this behaviour is
highly reversible, as the value of $\beta$, $d_{f}$ and the viscoelastic
moduli return to their equilibrium value, even after multiple cycles of
deformation. This highlights a previously unrevealed diversity of viscoelastic
properties that originate on the mesoscopic scale. These considerations are
likely to be key to controlling the viscoelasticity of folded protein
hydrogels.

## Introduction

Hydrogels are hydrated three-dimensional networks with significant potential
in applications that involve the repair and replacement of biological
tissue.[1] A desired aspect of their design is that they mimic the material
properties of the tissue that they are designed to replace. The ability to
emulate biomechanical properties such as toughness, strain-stiffening and
mechanical memory is therefore an important scientific challenge, particularly
for applications involving cell growth and proliferation.[2] The design
challenges to recreate these properties are significant since materials must
be sufficiently soft to allow cell proliferation and sufficiently stiff to
protect the material from large and sudden deformations.[3] In living matter,
this trade-off is enabled by the unique mechanical properties of biopolymers
and their ability to increase their stiffness when deformed and to accommodate
large strains through force induced unfolding.[4] Discovering approaches that
can recreate these properties in synthetic materials remains highly
challenging.

An increasingly promising approach to address this challenge is to construct
chemically crosslinked hydrogels from folded proteins.[5, 6, 7, 8] In nature,
protein domains have precise structures which change their conformation under
precise chemical and mechanical conditions. When engineered as polyproteins,
their relatively small molecular size ($\sim$10-100kDa) distinguishes them
from conventional biopolymers, however w hen crosslinked into hydrogels,
polyproteins exhibit impressive mechanical properties such as high
elasticity[9] and resilience[6]. These properties can be controlled through
chemical stimuli[10, 11], which suggests they may be used in applications
involving shape change and shape programming.[12, 7, 13] Their use is further
advantageous since the molecular mechanics of folded protein domains have been
extensively interrogated using force spectroscopy[14, 15, 16] and can be
correlated to bulk viscoelasticity .[17, 18]

All hydrogels are intrinsically cross-length scale materials; however, most
studies of folded protein hydrogels have focused only on the molecular scale
mechanics of the protein and on the bulk properties of the resulting hydrogel.
Recent findings from our group have shown that structural parameters at the
meso-scale may also be of importance. These include the intermediate
structural hierarchy including cross-linking,[19, 20, 21] which can be used to
modulate the network growth and the fractal dimensions and force distribution
of percolating clusters. [22, 23] This rich hierarchy of structural properties
indicates that polyprotein hydrogels are likely to exhibit complex
viscoelastic behaviours, which should manifest themselves in their mechanical
response. Relating this mechanical response to the underlying structure can be
achieved through shear rheology, since the time-dependent viscoelasticity of
the hydrogel directly reflects the length scale of the hydrogel structure.[24]
However, most studies of polyprotein hydrogels have instead used dynamic
mechanical analysis,(DMA) which is instructive for quantifying bulk elastic
properties such as toughness and fracture stress but less effective in
quantifying viscoelastic and structural properties. In comparison to shear
rheology, DMA may also induce volumetric sample changes at high strains, which
may frustrate accurate interpretation of the data.[25] However, a multi-modal
rheological characterisation of polyprotein hydrogels that relates time-
dependent viscoelasticity to underlying structure has yet to be carried out.

In this study, we seek to bridge the gap between the molecular and bulk
mechanics of folded protein hydrogels, using shear rheological tests to probe
their meso-scale mechanics. We construct hydrogels using $I27_{5}$
polyproteins, so called because they comprise five concatenated $I27$
immunoglobulin domains of the muscle protein titin. Titin is an elastomeric
component of the sarcomeres of striated muscle, where its $I27$ domains are
able to unfold and refold to prevent overstretching.[26] Reflecting this
biological role, the $I27$ protein possesses mechanical stability whereas
other domains in titin extend with low resistance. It also possesses a high
degree of mechanical reversibility,[27, 28] as well as being suitable for
crosslinking into a hydrogel using photochemical approaches.[29] This makes it
a putative robust and tough polyprotein and an excellent candidate for this
study. We probe $I27_{5}$ hydrogels through frequency-dependent small
amplitude oscillatory shear, stress relaxation and strain ramp experiments, to
probe the viscoelastic properties at a range of timescales and within
different regimes of deformation. By combining these different modes of
rheology, we present a comprehensive rheological analysis of $I27_{5}$
hydrogels that infers their meso-scale structure, reveals how this structure
relates to viscoelasticity and explores how this viscoelasticity determines
the nonlinear and fracture mechanics of the hydrogel.

## Methods

#### Polyprotein Purification

The assembly of $I27_{5}$ polyproteins was performed using a PCR-based Golden
Gate protocol with a modified pET14b, with all original BsaI sites removed, as
the destination expression vector.[30] After verification of the DNA sequence,
the resulting vector was transformed into BLR (DE3) pLysS E.coli cells. 2 ml
LB starter culture was used to inoculate 0.5 L autoinduction medium. 10
$\times$ 0.5 L cell cultures were incubated at 28∘C, 200 rpm for 24 h, for
protein expression and then harvested and lysed. The protein was loaded on to
2 $\times$ 5 ml HisTrap FF columns (Cytiva) for Ni2+ affinity chromatography.
The column was then equilibrated in wash buffer (20 mM Tris, 300 mM NaCl, 10
mM imidazole, pH 8), before the protein was eluted with elution buffer (20 mM
Tris, 300 mM NaCl, 500 mM imidazole, pH 8), in a ratio of 1:3 to wash buffer.
The protein was further purified using a 5ml HiTrap DEAE Sepharose FF Ion
Exchange chromatography column (Cytiva) for ion exchange purification. A
gradient elution of 0-100$\%$ elution buffer (25 mM Sodium phosphate pH 7.4,
0-500 mM NaCl) was performed over 300 ml. The protein was then purified by
size exclusion chromatography (HiLoad 26/600 Superdex 75 pg, Cytiva) and
eluted in 25 mM Sodium Phosphate, pH 7.4. The purified protein was dialysed
into MilliQ water and freeze-dried for storage at -20∘C.

#### Hydrogel Preparation

$I27_{5}$ was re-suspended in 25 mM sodium phosphate, pH = 7.4 and placed on a
rotary wheel for a minimum of 1 h to enable complete dissolution of protein.
The suspensions were then centrifuged for 10 minutes at 5000 RCF, retaining
the supernatant to ensure that any insoluble protein was removed. Protein
concentration was measured by absorption at 280 nm, using an extinction
coefficient of 43824M-1cm-1. The final suspension was then diluted and mixed
with an assembly buffer of Tris(2,2’-bipyridyl)-dichlororuthenium(II)
hexahydrate (Ru(BiPy)3) and sodium persulfate (NaPS) directly before use, with
final concentrations of 0.38 mM $I27_{5}$, 100 $\mu$M Ru(BiPy)3 and 10 mM
NaPS. Sample gelation was initiated photochemically through 60 seconds of in-
situ illumination with a 460 nm light emitting diode as previously
described[31]. The illumination intensity was calibrated to 40 mW.cm-2 at 452
nm.

#### Shear Rheology

Rheology measurements were performed using a stress-controlled rheometer
(Kinexus Malvern Pro) equipped with a 10 mm steel plate and plate geometry.
The lower plate of the rheometer was replaced with a custom-built photodiode
module which allowed the photochemical crosslinking to be carried out in-situ.
To prevent the evaporation, low viscosity (5 cSt) mineral oil was applied to
the air-sample interface. All measurements were performed at 25 ∘C and 2 hours
after crosslinking. For all measurements, rheology data was captured over at
least 3 measurements on independently prepared samples.

#### Small Amplitude Oscillatory Measurements

All small amplitude oscillatory measurements were carried out at a strain
amplitude of $1\%$ \- comfortably within the linear viscoelastic regime for
all samples studied. Time dependent measurements were carried out at an
angular frequency of 6.28 s${}^{-}1$ unless otherwise stated. At high
frequencies, artifacts from instrument inertia were avoided by discounting all
data points for which the phase angle exceeded 150∘.

#### Cyclic Loading Measurements

Cyclic loading experiments were performed to probe the mechanical response of
the polyprotein hydrogel to large deformations and to examine the effect of
multiple deformations on the hydrogel mechanics. Following crosslinking, the
sample was subjected to a shear strain pulse of loading rate 0.01 s-1. After
each pulse, the sample was left to recover for 23 min, during which time a
single oscillatory frequency sweep was carried out to assess any permanent
changes to material properties. The process was then repeated with
incrementally increasing strain pulse magnitudes.

#### Stress relaxation

Stress relaxation experiments were carried out by applying a step strain to
the sample for 10 mins. The resulting relaxation in stress over time was
fitted to a range of different constitutive models using the Rheology Open
Source (RHEOS) software.[32, 33] A constant applied strain was typically
attained within 2-3 s and any data from before this point were discarded from
the analysis.

#### Circular Dichroism

Circular dichroism (CD) measurements were performed on an Applied Photophysics
Chirascan CD spectropolarimeter using 0.01 mm path length cuvettes at a
concentration of 0.5 mM. Samples were gelled inside the cuvette through
photochemical crosslinking using identical buffers and the same protocol to
that used in shear rheology experiments. Samples were also probed at higher
dilution (5 $\mu$M), in which case the protein was probed in a phosphate
buffer without crosslinking agents (25 mM sodium phosphate, pH = 7.4) in a 1mm
path length cuvette. Each measurement was averaged over 1 s and any data
points for which the photomultiplier tube voltage (HV) exceeded 600 V were not
included in the analysis. A minimum of 4 independently prepared samples were
used for each measurement to ensure reproducibility. Standard deviations did
not exceed 10$\%$ of the mean value.

### Results

#### I27 Protein Structure is Retained following Crosslinking

Figure 1: (a) The polyprotein used in this study comprises five concatenated
I27 domains inter-spaced with flexible ’linker’ regions. Illumination of
polyprotein solutions with 490 nm blue light in the presence of crosslinking
reagents leads to gelation through the formation of dityrosine bonds. (b) A
sharp increase in G’ (triangles) and G” (squares) over a timescale of seconds
indicates rapid crosslinking. (c) Within the linear viscoelastic regime, after
2 hours both G’ and G” exhibit a weak power law dependence on frequency (d)
Circular dichroism spectra show a $\beta$-sheet peak at 211 nm, with a slight
peak shift and decrease in intensity after gelation. (e) Analytical fitting
spectra using the CONTINLL reference set confirms the structure is
predominantly $\beta$-sheet, before and after gelation and indicates reduction
in the fraction of folded protein of approximately 12$\%$, taking place at
equivalent timescales to the decrease in G’/G”. Error bars in (b)-(d) (shaded
regions) represent averaged measurements.

The $I27_{5}$ polyprotein comprises five sequential repeats of the $I27$ as
shown in figure 1 (a). The I27 polyproteins are assembled into hydrogels using
the well-known ruthenium crosslinking strategy, [34] which causes tyrosine
residues on the surface of the polyproteins to radicalise and form dityrosine
linkages under illumination of visible light and in the presence of sodium
persulfate. Each polyprotein has five native tyrosine sites which are retained
in the polyprotein design. The polyprotein also contains a hexahistidine
repeat at its N-terminus and a dicysteine residue at its C-terminus, both of
which are known to crosslink under the same conditions, albeit at lower
efficiency.[35] Therefore the polyprotein can be said to possess 7 potential
crosslinking sites. The sodium persulfate that drives the crosslinking
reaction is present at an approximately 4$\times$ stochiometric excess to the
potential crosslink sites therefore we may assume a high crosslinking
efficiency. Recent computational work from our group has predicted that the
coordination efficiency of pentameric polyproteins with a similar number of
crosslinks to $I27_{5}$ is close to unity,[21] therefore without making a
priori assumptions about the meso-scale structure, it can already be assumed
that the network is densely crosslinked, with each polyprotein being
permanently crosslinked to multiple neighbouring polyprotein molecules.

To probe this crosslinking process, we carry out small amplitude oscillatory
shear measurements, probing the storage (G′) and loss (G′′) moduli of the
sample during in situ illumination at time $t=t_{0}$. As plotted in figure
1(b) there exists a short lag time before a sharp increase is observed in both
G′ and G′′. In a recent study of globular bovine serum albumin (BSA) hydrogels
we showed that this lag time[31] is strongly indicative of a two-step
nucleation and growth process, in which diffusion-limited pre-gel clusters
mature towards eventual percolation. Although the intensity of light and the
stochiometric ratio are similar to this previous study, the observed lag time
is significantly shorter ($\sim$ 3 s) for $I27_{5}$ than for BSA ($\sim 30$
s).

The value of G’ and G” both increase sharply following the lag phase, reaching
a peak of $1.7\pm 0.6\times 10^{3}$ Pa after approximately 30 s, before
undergoing a steady decrease and reaching a final G’ of $1.0\pm 0.3\times
10^{3}$ Pa, after approximately 1 hr. To examine the possible molecular
origins of this decrease in moduli, we carry out CD spectroscopy across each
stage of the crosslinking process as shown in figure 1(d). Before crosslinking
the mean residue ellipticity spectrum indicates an anti-parallel $\beta$-sheet
structure which, in I27, is typically characterised by a negative band at 211
nm [36]. A slight positive band can also be seen close to 230 nm, which
reflects exciton coupling between aromatic residues[37, 38]. Compared with the
spectra of dilute uncrosslinked $I27_{5}$, there is a close alignment ($<10\%$
difference in peak at 211 nm) indicating that the protein remains largely
stable at the increased concentrations used for gelation, while the magnitude
of the 211 nm band ($5.4\pm 0.2\times 10^{3}$ deg. cm2. dmol-1) is comparable
to previous studies of $I27_{5}$. Following photochemical crosslinking, the
peak intensity reduces and shifts to a lower wavelength, which may indicate a
decrease in the folded protein fraction. Assuming that the 211 nm peak
intensity is directly proportional to the fraction of folded protein, we
estimate a reduction in folded fraction of $I27_{5}$ of 7$\%$. The 230 nm peak
is also reduced, presumably due to the conversion of the aromatic tyrosine
residues into dityrosine residues during crosslinking. Overall, this indicates
a slight reduction in the folded fraction of $I27_{5}$. To investigate this
further, we deconvoluted the CD spectra using the CONTINLL parameter set in
Dichroweb[39], using protein reference set 4 (optimised for 190 - 240 nm). As
shown in figure 1(e), table 1, and figure S3 there is a slight reduction in
$\beta$-sheet fraction and $\beta$-turn fraction and an increase in the
proportion of the protein that is unordered. Taking this as being indicative
of protein unfolding under gelation, we estimate a decrease in the fraction of
folded protein of approximately 12$\%$, which takes place in the 2 hours
following gelation, the same time over which a decrease in G’ and G” is
observed. A similar reduction in folded fraction that coincides with a
decrease in G’ and G” has previously been observed for other folded protein
hydrogels studied in our group[31, 22, 23], which suggests that it is a
general feature of folded protein hydrogels, where it is caused by forced
unfolding of weak protein domains on the molecular-scale as the hydrogel
network forms.

|  $\beta$-Strands |  $\beta$-Turns |  Unordered
---|---|---|---
Dilute (5 $\mu$M) |  0.39 |  0.23 |  0.34
Pre-Crosslink |  0.38 |  0.24 |  0.34
Post-Crosslink |  0.35 |  0.24 |  0.34
Post-Gel, 2 hrs |  0.30 |  0.20 |  0.42
Table 1:  Secondary structure composition of I275 polyprotein hydrogels.
$\alpha$-helix content (not shown) is the result of deconvolution errors.

#### Frequency Sweeps Indicate a Fractal Network Structure

Rheological frequency sweeps (figure 1(c)) show that the equilibrated
hydrogels are predominantly elastic in nature, with a storage modulus G’ that
is over an order of magnitude higher than the loss modulus G” across all
probed frequencies. Frequency sweeps reflect the distribution in timescales of
the underlying relaxation mechanisms in the hydrogel. For our I275 hydrogels
we find that the frequency dependence of G’ and G” are well-described by weak
power law relationships (solid lines in figure 1(c)). The physical origins of
this power law viscoelasticity in hydrogels remain a matter of debate[40] but
they are commonly associated with fractal structure in a variety of soft and
biological materials[40, 41, 42, 24, 43, 44, 45, 46, 47, 48, 49]. In recent
work from our group, we have shown that fractal structures are also present in
folded protein hydrogels[23, 22] and that these structures arise from the
diffusion- and rate-limited crosslinking of constituent folded proteins[31]
and in situ protein unfolding.[23, 22]

The emergence of a power law relaxation spectrum can be rationalised by
considering that a fractal hydrogel network has a self-similar structure and
consequentially will exhibit self-similar relaxation dynamics. [44] Assuming
that the excluded volume interactions are screened, this allows the fractal
dimension $d_{f}$ of the hydrogel network to be directly calculated using the
following previously derived[24] relation:

$\beta=\frac{3(5-2d_{f})}{2(5-d_{f})}$ (1)

where $G^{\prime}\sim\omega^{\beta}$ and $0<\beta<1$, which correspond to
$1.25<d_{f}<2.5$. The above relation was originally devised for gels at the
percolation threshold but has also been shown to be broadly applicable at
later timescales,[40] as used here. A simple least squares fit to figure 1(c)
reveals a weak power law ($\beta$ =0.02). We also independently verify the
value of $\beta$ using the Kramers-Kronig relation:
$G^{\prime\prime}/G^{\prime}=tan(\beta\times\pi/2)$ which is a consequence of
the interdependence between G” and G’.[50] We find $\beta=0.022\pm 0.002$
(figure S2, supplementary), in excellent agreement with the power law fit.
Taking $\beta=0.02$, equation (1) indicates a fractal dimension of 2.48. In
previous work we have carried out structural characterisation of $I27_{5}$
hydrogels[29], which revealed dense nanoscale clusters of polyproteins with a
mean diameter of 20 nm, surrounded by sparser inter-cluster regions, which may
be either folded[23] or unfolded[22], depending on the constituent protein.
These nanoscale clusters have been shown to be fractal in nature with values
of $d_{f}$ across a broad range 2.1-2.7.[23] How then should we compare the
value of $d_{f}$ of the nanoscale clusters to those extracted from time-
dependent rheology data? The rheological response of folded protein hydrogels
is likely[31] to depend not on the structure of the nanoscale clusters but
rather on the connectivity of inter-cluster regions. This is analogous to
colloidal gels, in which the hydrogel elasticity is a property that emerges
from the density of connections between colloidal clusters.[51] Our analysis
is therefore reminiscent of other soft matter systems and intriguingly points
towards the possibility of a fractal structure that is length-scale dependent.
Although we cannot infer the precise length scale of the inter-cluster regions
through rheology, recent work from our group has measured this length scale
for MBP hydrogels as being approximately 100 nm.[23]

#### Hydrogel Structure is Responsive to Deformation

So far, we have identified that the relaxation spectrum of an $I27_{5}$
hydrogels can be functionally fitted to a power law model, with exponent
$\beta$. To validate this, we independently carry out a series of stress
relaxation experiments, in which a step strain is applied to the sample and
the sample’s shear stress decays over time. This approach avoids the changes
in sample strain that are inherent to creep measurements while also probing a
longer range of timescales than oscillatory frequency sweeps. We probe the
extent of the linear viscoelastic regime through a shear strain ramp (figure
S4), which confirms that the linear viscoelastic regime corresponds to
$\gamma$ $<$ 0.1, with only a slightly nonlinear response at higher strains
($\gamma$ $<$ 0.3).

Figure 2: (a) Relaxation data are fitted using a fractional model, which
represents an arbitrarily large network of springs and dashpots, using just
two parameters $c_{\beta}$ and $\beta$. (b) A step strain $\gamma$ at time 0
is applied to an $I27_{5}$ hydrogel and the normalised stress response
$\sigma$/$\sigma_{0}$ is probed. The model (grey line) provides an excellent
fit across a range of values of $\gamma$.

While the exponent $\beta$ is easily extracted from the frequency sweeps,
extracting it from relaxation data requires a mechanical constitutive
modelling approach. Here, elements of Hookean elasticity (springs) and
Newtonian viscosity (dashpots) are combined in an empirical manner to
approximate the viscoelastic response. In many materials, a finite number of
such elements are sufficient,[52] however for power law viscoelasticity the
spectrum of relaxation times is broad and hence the approach requires a large
number of fitting parameters making it unsuitable.[53] To address this, we use
a so-called fractional rheological model, which captures power law rheology
using a minimal number of fitting parameters. This is achieved by reducing an
arbitrary large number of springs and dashpots (figure 2 (a)) to a single
mathematical element commonly known as a springpot model [54] with two fitting
parameters $\beta$ and $c_{\beta}$. The parameter $c_{\beta}$, is often
referred to as a quasi-property [53] since its SI units (Pa.sβ) contain a
power law exponent. From a practical perspective $c_{\beta}$ can most simply
be conceptualised as referring to the ‘firmness’ or ‘strength’ of the material
[55]. The springpot model describes stress relaxation $\sigma(t)$ following an
applied step strain $\gamma$ over a measurement timescale $t$ as:

$\sigma(t)=c_{\beta}\frac{d^{\beta}\gamma}{dt^{\beta}}$ (2)

Here $\beta$ has the same physical meaning as previously defined. The elegance
of the above relation is reflected in the fact that it reduces to the equation
for a Hookean spring when $\beta=0$ and to a Newtonian dashpot when $\beta=1$.
The underlying physical basis for this model is supported by the fact that
when fractal soft materials are embedded with passive particles, the particle
motion is sub-diffusive and anomalous ($<\Delta x^{2}>\sim
t^{\beta},\beta<1$).[56, 57] This sub-diffusive behaviour arises from the
multi-scale structure of the material which also directly leads to power law
stress relaxation. [58] As such, the fractional approach has come to be
established as the most predictive model for power-law viscoelastic
materials.[54, 59, 60, 40]

As shown in figure 2(b), equation 2 provides an excellent fit to the stress
relaxation data, both within the linear viscoelastic regime ($\gamma$ $<$ 0.1)
and even for the nonlinear regime (0.1 $<\gamma<$0.3 ). For step strains that
exceed this strain (figure S5) we consistently observe an instantaneous
decrease in stress during the measurement, suggesting that the sample
undergoes irreversible fracture. The fitted parameters $\beta$ and $c_{\beta}$
are shown in figure 3.

The fractional model provides an excellent fit to the relaxation data. We also
attempted to fit several different variations of the two-component fractional
model, incorporating different combinations of springs, dashpots and
springpots arranged in series and in parallel in attempt to improve our model.
The use of either fractional Maxwell model (two springpots in series, figures
S6[61]) or a modified power law model (springpot combined with a dashpot and
spring, figure S7 [33]) provide a poor fit to the data as both predict a
plateau in $\sigma(t)$ at long timescales. A fractional Kelvin Voigt model
(two springpots in parallel, figure S8) does provide a close fit to the data,
however the additional parameters are degenerate to fitting.

Within the linear viscoelastic regime, the value of $\beta$ is independent of
applied strain, as expected, and has an average measured value of 0.03 $\pm$
0.01. This is in excellent agreement with the frequency spectra
($\beta=0.02$), demonstrating that the power-law model accurately predicts the
viscoelastic material properties of the $I27_{5}$ hydrogels, which are likely
to originate from a cross length-scale fractal structure. Interestingly, the
springpot model appears to provide an excellent fit to the relaxation data at
strains that exceed the linear viscoelastic regime. The reasons for this are
not immediately clear since the fractional model that we employ is only
strictly valid for linear viscoelastic materials, [62] although attempts have
been made to extend it to nonlinear behaviour.[47, 63] It may be that the
deviations from linear viscoelasticity for $0.1<\gamma<0.3$ (figure S4) are
sufficiently minor so as not to invalidate the model. We find that $c_{\beta}$
increases with strain, which suggests a strengthening of the hydrogel network.
This, accompanied by an increase in $\beta$ over the same range of applied
strain, suggests progressive changes in network structure as the strain is
increased. $\beta$ reaches a maximum of $0.065\pm 10^{-5}$ for $\gamma$, which
using the equation (1), would correspond to a slight decrease in fractal
dimension from $d_{f}=2.47$ to $d_{f}=2.44$. Assuming that our model is indeed
valid at higher strains, we propose that this change may reflect the
progressive unfolding of the constituent $I27$ domains as the applied strain
is increased, leading to changes in the hydrogel network structure. At the
same time, the increase in $c_{\beta}$ would imply a strengthening or
reinforcement of the network at increased strain. One explanation for this is
that the polyprotein chains stiffen, by aligning with the direction of shear
(leading to an increase in $c_{\beta}$) while simultaneously unfolding
(leading to an increase in $\beta$).

Figure 3: Fractional fit parameters extracted from applying equation 2 to data
in shown in figure 2. The shaded region indicates the linear viscoelastic
regime. Dashed lines are guides for the eye. Error bars represent fits over
multiple measurements

Comparison can be made to a previous study by our group on crosslinked BSA
hydrogels. In that study, the addition of a reducing agent resulted in
breakage of covalent bonds within the BSA protein, resulting in a more force
labile protein.[22] Upon cross-linking, in situ protein unfolding was found to
define the network architecture and mechanics, with regions of clusters of
folded protein interconnected by unfolded protein. The study also noted an
increase in the power law exponent when the disulphide bonds that prevent
unfolding of the protein were broken through chemical reduction. Remarkably,
the power law exponents before and after covalent bond breaking ($0.027\pm
0.002$ and $0.061\pm 0.001$, respectively) are almost identical to the
exponents at low and high strains measured here ($0.03\pm 0.01$ and $0.065\pm
10^{-5}$, respectively), providing further support for our hypothesis that the
structural changes implied by our data arise from the force induced unfolding
of individual protein domains.

#### Nonlinear Mechanics Indicate Reversible Alignment and Unfolding of
Polyproteins

How do the structural properties of $I27_{5}$ hydrogels that we have
encountered, change under deformation? Our relaxation data show intriguing
clues that suggest network strengthening and unfolding under applied strain.
To probe this independently we carry out nonlinear cyclic deformation
experiments[64, 65]. As depicted in figure 4(a) and figure S9, the $I27_{5}$
polyprotein network is loaded and unloaded up to a threshold strain
$\gamma_{max}$ through a triangular strain pulse at a constant shear rate and
the magnitude of $\gamma_{max}$ is increased with each subsequent shear pulse.

Figure 4: The bulk response of polyprotein networks under large deformation
reveals insights into the mechanisms of energy dissipation. A series of
triangular strain ($\gamma$) pulses of increasing magnitude are applied (a).
For $\gamma<0.4$ each stress ($\sigma$) curve overlays closely with its
predecessor (b) indicating that the hydrogel’s mechanical response is fully
reversible. Increasing applied strain correlates closely with energy
dissipation (c) and strain stiffening (d), which are macroscopic
manifestations of the unfolding of individual protein domains and the
stretching of polyproteins, respectively. This is shown diagrammatically (e).
At $\gamma=0.4$ the energy dissipated increases sharply and subsequent loading
curves no longer overlay, indicating the irreversible fracture of the
hydrogel.

Within the linear viscoelastic regime, ($\gamma_{max}<0.1$) the loading and
unloading curves (figure 4 (a), solid and dashed lines respectively) appear to
overlay closely suggesting a reversible and purely elastic response. However
as $\gamma_{max}$ increases, a deviation between the loading and unloading
curves begins to emerge. Calculating the energy loss in each cycle (4(c)) from
the ratio of the areas beneath the loading and unloading curves (see methods),
shows that inelastic energy dissipated during loading steadily increases with
$\gamma_{max}$. Furthermore, each subsequent loading curve appears to overlay
with its predecessor, indicating that this inelastic dissipation does not
bring about any permanent changes to the network stiffness. This complete
reversibility of the network is unusual as most polymer networks that exhibit
reversibility, such as filled rubbers[66] and biological tissues also exhibit
a well-defined residual cyclic softening. By contrast, we observe an almost
perfect overlay between loading curves for strains up to $\gamma=0.4$. The
profile of the loading curves is also revealing of changes in hydrogel
stiffness during deformation. For $\gamma>0.1$, the stress $\sigma$ required
to increase the strain becomes progressively higher, indicating strain-
stiffening.[67] This can be most clearly seen in the evolution of the
differential elastic modulus $K^{\prime}$ with respect to strain (figure
4(d)), which increases approximately by a factor of 2 before the onset of
rupture. As with the loading curves, the relationship between $K^{\prime}$ and
$\gamma$ overlays closely across all curves, confirming that no softening
takes place between loading cycles. Plotting $K^{\prime}$ with respect to
$\sigma$, reveals that the stiffening follows a power law relationship of
$K^{\prime}\sim\sigma^{0.6}$ (figure S10). Such power law stiffening is a
general feature of semiflexible networks for which the distance between
crosslink nodes is similar to the polymer persistence length and arises from
the entropic resistance of thermal undulations against stretching.[67] To
assess whether our system can be considered to be semiflexible we carry out an
analysis of the polyprotein dimensions (see supplementary information) which
reveals that the distance between the crosslink sites is $5.44$ nm. To
calculate the persistence length of the polyproteins we draw on our previous
computational calculations for a range of polyprotein dimensions. [19] The
most important parameter here is the ratio between the length of the linker
domain and the radius of the globular domains, which for our dimensional
analysis is $\approx 0.34$. This corresponds[19] to a persistence length in
the range of $\approx$ 7-14 nm. Therefore, the persistence length and
crosslink length are of the same order of magnitude and the polyprotein can be
considered as semiflexible. For semiflexible networks it is expected that
$K^{\prime}\sim\sigma^{1.5}$[68] although exponents as low as $0.5$[69] have
been observed elsewhere and are thought to arise from inelastic effects such
as crosslink unbinding or axial stretch.

For our system, strain stiffening appears to occur concurrently with energy
dissipation, which agrees with the stiffening and dissipation hypothesis that
emerges from our relaxation data. From the strain ramp data, it seems likely
that strain stiffening occurs due to the stretching of the constituent
semiflexible polyproteins and that these polyproteins undergo force-induced
unfolding concurrently (figure 4 (e)).

#### Fracture of Polyprotein Hydrogels is Driven by Structural Adaption

If the unfolding and refolding of $I27_{5}$ in the hydrogel is indeed
reversible then how does the fracture, as observed from strains exceeding 0.4,
take place? To investigate this further, we carry out oscillatory frequency
sweep measurements after each of the triangular strain pulse measurements.
Each frequency sweep therefore represents the linear viscoelastic properties
of the hydrogel after each cycle of deformation.

Figure 5: The linear viscoelastic properties of the $I27_{5}$ hydrogels after
strain pulse deformation cycles of magnitude $\gamma_{max}$ reveal the
fracture properties of the hydrogel network (a). The elastic modulus remains
unchanged below a threshold strain of $\gamma_{max}=0.4$, after which it
steadily decreases, indicating network fracture (b). The power law exponent
$\beta$, which represents the fractal structure of the hydrogel increases
after the fracture point, indicating structural changes post-fracture (c). The
negative stress $\sigma_{0}$ needed to return the network to zero strain after
each deformation cycle steadily increases before fracture, suggesting a build-
up of tension after each deformation cycle (d). Meanwhile the network
dissipation (tan$\delta$ = G”/G’) remains relatively low over all values of
$\gamma_{max}$, indicating that bond rupture rather than protein unfolding is
the underlying cause of the fracture of the hydrogel network above a threshold
strain of $\gamma_{max}$=0.4.

The profiles of the individual frequency sweeps of the hydrogels remain highly
reproducible at pre-fracture strains, (figure S12) which is consistent with
our previous observation that the deformations are fully reversible. At higher
strains, the linear elastic modulus $G^{\prime}$ begins to decrease (figure 5
(a)). Over the same strain range, the power law exponent of the frequency
sweeps $\beta$ increases,(figure 5 (b)) indicating permanent structural
change, even after the removal of strain. At this final stage, the value
$\beta$ reaches $0.064\pm 0.008$. Remarkably, this is identical to the maximum
value of $\beta$ measured through stress relaxation experiments (Figure 3).

The fracture of biopolymer networks is a complex phenomenon that is
challenging to model.[70] In network-forming systems, fracture typically
occurs suddenly and is preceded by a dynamic microscopic changes[71]. The
eventual mechanism of fracture can originate from the dissociation of
noncovalent bonds, the permanent rupture of covalent bonds, or both
concurrently.[72] We expect that covalent bond rupture is the most likely
cause in our system since the hydrogels are covalently crosslinked.
Furthermore, the high energy dissipation at $\gamma_{max}<0.4$ (figure 4)
implies protein unfolding occurs pre-rupture. This observation is consistent
with the unfolding forces of an $I27$ domain ($\sim$ 0.2 nN[27]), which are
lower than the typical rupture forces of a covalent bond ($\sim$ 1-2 nN[73] at
a pulling speed of 10 nN.s-1). One possible mechanism for fracture is that as
the polyprotein approaches a fully unfolded state, the tension that was
previously directed to domain unfolding instead is applied to the unfolded
polypeptide chain leading ultimately to irreversible peptide bond rupture. It
is intuitive that such rupture would occur at the weakest bond in the chain,
in this case a peptide bond in either the unfolded $I27$ domains or the linker
region. Further evidence for the mechanism of rupture can be found by re-
examining the strain pulse data in figure 4. Specifically, the stress required
to return the hydrogel to zero strain after each pulse is denoted $\sigma_{0}$
(shown diagrammatically in figure S11). This represents the residual stress
after each deformation cycle. $\sigma_{0}$ is initially close to 0 and becomes
more negative after each cycle until the onset of fracture (figure 5(c)). Post
fracture, $\sigma_{0}$ no longer becomes more negative, which may reflect the
loss of tension in the network as the polyprotein chains rupture. Finally, the
value of tan$\delta$ (figure 5 (d)), which quantifies the relative viscous
dissipation of the network in the form of the ratio between the elastic and
viscous moduli, remains relatively low ($\approx 0.2$, at the highest values
of $\gamma_{max}$). Given that the chains are highly dissipative at high
strains (figure 4 (d)) and given the theoretical range of tan$\delta$ (0-1), a
much higher dissipation would be expected if the polyprotein chains remained
intact but unfolded. It is therefore likely that the network fracture is
determined by the reversible unfolding of the protein domains which, at
significantly high tension leads to eventual bond rupture as shown
diagrammatically in figure 5 (e).

### Discussion

In summary, we have applied a multi-modal rheology approach to interrogate the
viscoelastic properties of folded protein hydrogels. Using these approaches,
we highlight several important and previously unappreciated aspects of their
material properties.

Firstly, we determine that the viscoelasticity of $I27_{5}$ hydrogels can be
accurately modelled by a simple two-parameter fractional viscoelastic
constitutive model. This model manifests itself as a power law in both the
frequency response (figures 1 (c) and S12) and in the time-dependent stress
relaxation response (figure 2). From a practical perspective, this indicates
that hydrogel’s stress relaxation is governed not by a single characteristic
timescale but rather by a broad spectrum of timescales. This is perhaps
surprising as it suggests that the viscoelastic response of the hydrogel, at
least within the linear viscoelastic regime, is not governed primarily by the
narrow and well-defined unfolding timescale of the $I27$ protein domains when
stretched axially.[15] Instead the architecture of the hydrogel appears to
result in a more complex spectrum of relaxation timescales, which likely
arises from the orientation and arrangements of the protein domains. There are
several possible mechanisms to explain this behaviour. Firstly, it could be
argued that the formation of the hydrogel network causes partial unfolding,
however we show a reduction of the fraction of folded protein of just 12$\%$
(figure 1). A more likely explanation is that the fractal structure of the
hydrogel[23, 22, 31] leads to a broad spectrum of relaxation modes, which
reflects the broad spectrum of length scales within the network.[74] One
additional contributing factor could be the random alignment of the
polyprotein chains within the hydrogel network in relation to the direction of
shear deformation. In principle this could lead to changes in the relaxation
timescales as the mechanical stiffness of folded proteins is known to depend
on their pulling geometry[75]. The broad spectrum of relaxation times we
observe here appears to be common in polymeric systems[76], cells[77, 33] and
tissues[78]. Here, we show that this behaviour also extends to folded protein
hydrogels.

Using a fractional viscoelastic model, we show that the power law exponent
$\beta$ is remarkably consistent between the oscillatory shear and stress
relaxation experiments used in this study. Therefore, we may say that power
law viscoelasticity is not an artifact of any particular technique but is
instead an intrinsic property of the hydrogel and is likely to reflect the
hydrogel’s fractal structure. A consistent value of $\beta$ is also measured
after multiple cycles of deformation, indicating that this fractal structure
is robust to perturbation. More surprisingly, our model also closely fits the
relaxation response at larger deformations in the nonlinear viscoelastic
regime (figure 3(a)). While our model is strictly valid only for linear
viscoelasticity, we note that the fitted parameter $\beta$ in this regime is
close to the value of $\beta$ as probed by oscillatory shear after the
hydrogel fracture. We speculate that this may be indicative of permanent
changes to the hydrogel’s structure at large deformations. Modelling this
behaviour using more advanced constitutive modelling that accounts directly
for nonlinear effects[47, 63] would an interesting topic of further study.

Within this nonlinear regime we observe that the hydrogel undergoes stiffening
while subjected to shear strains, which is manifested both in relaxation data
(figure 3 (b)) and in strain ramp data (figure 4 (d)). In line with
predictions,[19] this is consistent with the semiflexible nature of the
polyprotein chain, a property that arises directly from the dimensions of the
folded proteins. Recreating strain stiffening in synthetic hydrogels is a
growing area of research[79] and although the degree of stiffening is
relatively low for this system (a twofold increase in modulus before fracture)
it is likely that a modification of the polyprotein dimensions could tune and
enhance this behaviour further.

This strain-induced stiffening occurs at low strains, which corresponds to the
onset of nonlinearity in stress relaxation experiments (figure 2) and to an
observed increase in energy dissipation (figure 4 (d)). Our relaxation data
indicates an increase in the power law exponent $\beta$ during deformation but
notably there is no permanent changes to the hydrogel’s viscoelasticity or
$\beta$ (figure 5), suggesting that the network structure is not significantly
altered by repeated deformation. This stands in direct contrast to
conventional rubber materials, in which the viscoelastic response from each
subsequent deformation cycle is directly determined by the strain history of
the sample.[66] It is likely that this behaviour is a consequence of the $I27$
domain’s propensity to re-fold upon removal of stress, since the same
behaviour has been observed for hydrogels constructed from the similarly
reversible GB1 domains.[9] Nevertheless, our analysis of the repeated loading
curves indicates that some stress accumulates in the network after each
deformation cycle (figure 5(d)), which likely contributes to its eventual
fracture.

Finally, we wish to re-emphasise that the weak power-law rheology that we
observe here is also commonly observed in biological tissue. Indeed, slow
internal relaxation processes appear to be intrinsic to a broad range of
living materials [63]. The mechanical similarities between biological tissues
and folded protein hydrogels are apparent and may provide further support for
their potential applications in replacing and repairing living tissue.

In summary, we have shown that the viscoelastic response of $I27_{5}$ folded
protein hydrogels is governed not only by the molecular mechanics of the
folded protein domains, as has commonly been assumed, but also by their meso-
scale structure. Within the linear viscoelastic regime, the protein domains
remain folded, and the response follows a characteristic power-law rheological
behaviour. This is well described by a fractional viscoelastic model and most
likely arises from the hydrogel’s fractal structure. At higher strains,
nonlinear viscoelasticity is observed in the form of strain stiffening, which
is accompanied by an increase in energy dissipation as the polyprotein chains
align and unfold. This unfolding shows an excellent degree of reversibility,
which affirms the ability of the constituent $I27$ domains to return to their
folded confirmation upon removal of stress. Overall, we show that folded
protein hydrogels exhibit a rich diversity of viscoelastic properties which
reflect the hierarchy of length-scales, from their energy-dissipative
properties (at the length scale of individual proteins) to their stiffening
(at the length scale of polyprotein chains) and their linear viscoelasticity
and fracture properties (at the length scale of the hydrogel network). We
expect that these insights will be instructive in the continuing efforts to
modify and control the mechanics of folded protein hydrogels as well as for
their envisaged applications as biomaterials.

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Acknowledgments: We thank David Head, Ben Hanson, and Matt Hughes for helpful
discussions. Funding: We gratefully acknowledge funding from the Engineering
and Physical Sciences Research Council with grant EP/P02288X/1. The CD
spectropolarimeter was funded by the Wellcome Trust with grant WT094232.
Author contributions: A.A-R, L.D and D.J.B conceived the experiments. A.A-R
performed the experiments, analysed the data and wrote the manuscript with
contributions from all other authors. S.C expressed and purified proteins.
L.D. is the principal investigator and corresponding author. Competing
interests: The authors declare that they have no competing interests. Data and
materials availability: All data needed to evaluate the conclusions in the
paper are present in the paper and/or the Supplementary Materials.

Supplementary Materials for
Power Law Rheology of Folded Protein Hydrogels
Anders Aufderhorst-Roberts, Sophie Cussons, David J. Brockwell, Lorna Dougan∗
∗Corresponding author. Email<EMAIL_ADDRESS>
This PDF file includes:
Supplementary Text
Figs. S1 to S12

### Supplementary Text

#### Analysis of Polyprotein Dimensions

The polyprotein structure comprises a concatenated series of five $I27$
protein domains as shown in figure S1. Linkers comprising 4-6 amino acids were
inserted between domains to decrease inter-domain interactions. A
hexahistidine tag is found at the N-terminus and a dicysteine group is found
at the C-terminus.

Dimensions and distances are determined using the ’distance’ tool in the
molecular visualization software PyMOL. The individual $I27$ domains each have
an N-C end-to-end length of $l_{domain}$ = 4.66 nm. Each of the four linker
domains contain between 4-6 amino acids with no intrinsic secondary structure.
Given that the average linker length is 5 residues, that the average length of
an amino acid is 0.35 nm, and assuming that the linker adopts a freely-
jointed-chain structure, the total average length of the linker $l_{link}$ is:
$l_{link}=$0.35 nm$\times\sqrt{5}=$0.78 nm.

From this we may estimate the following:

  1. 1.

The mean distance between tyrosine crosslink sites is
$l_{domain}+l_{link}=$5.44 nm

  2. 2.

The ratio between the linker length and polyprotein radius as
$\frac{2l_{link}}{l_{domain}}=0.34$.

Figure S1: Secondary structure (a) and protein sequence (b) of the $I27_{5}$
polyprotein used in this study. The molecule comprises five mutated $I27$
domains concatenated with unstructured linker regions (light blue). The
secondary structure is predominantly composed of $\beta$-sheets (rectangles)
and contains 5 tyrosine residues, (ringed by purple circles) which act as
crosslink sites Figure S2: Power law exponent $\beta$ as verified
independently using the Kramers-Kronig relation of G” and G’. For
$\omega>1~{}\mathrm{rad}~{}s^{-1}$, $\beta$ is shown to plateau to an average
value of $0.022\pm 0.002$, in excellent agreement with power laws fitted to
$G^{\prime}$ in figure 1. Error bars (shaded regions) represent averaged
measurements. Figure S3: Best fits (solid red lines) to the mean residue
ellipticity of $I27_{5}$ hydrogels (black circles) at dilute concentrations (5
$\mu$M) and at each stage of the gelation process. Fits are carried using the
CONTINLL parameter set in Dichroweb. Error bars (shaded regions) represent
averaged measurements across different samples.

4

Figure S4: Representative stress-strain Curve of an $I27_{5}$ hydrogel,
captured at a shear strain rate of 0.01 s-1 (black dotted line). Excellent
agreement with a linear viscoelastic response (red dashed line) is observed
for strain $\gamma<0.1$.  Figure S5: Representative stress relaxation curve at
high step strain ($\gamma$ = 0.35). The relaxation curve is discontinuous,
indicating that the sample undergoes irreversible fracture.  Figure S6: Best
fit of a fractional maxwell model (a) to stress relaxation data at a range of
different applied strains. Model fits are shown as dashed lines. Figure S7:
Best fit of a modified Fractional viscoelastic model comprising a fractional
springpot element in series with a spring and in parallel with a dashpot (a).
The fit is applied to stress relaxation data at a range of different applied
strains (b)-(g). Model fits are shown as dashed lines. Figure S8: Best fit of
a fractional Kelvin Voigt model comprising two springpot elements in parallel.
(a). The fit is applied to stress relaxation data at a range of different
applied strains (b)-(g). While the model fits the data closely, the parameters
$c_{\alpha}$ and $\alpha$ are within the error margins of parameters
$c_{\beta}$ and $\beta$ (h)-(i) suggesting that additional parameters cause
the fitting to become degenerate. Error bars represent averaged measurements
across different samples. Model fits are shown as dashed lines.  Figure S9:
Representative cyclic loading data showing measured stress with respect to
applied strain. Solid lines are loading curves, dashed lines are unloading
curves. Colours indicate different loading cycles. Repeat measurements show
largely identical nonlinear response, with only slight differences in the
network fracture point, denoted by a sudden decrease in stress.  Figure S10:
Differential elastic modulus K’ with respect to the sample stress, extracted
from multiple loading cycles (a). Colours and regimes of mechanical response
(1-3) are equivalent to figure 4. Once K’ is normalised to account for slight
differences in the linear modulus $G_{0}$, data from all repeat measurements
superimpose perfectly on to a single curve (b). The hydrogel stiffens as
$K^{\prime}\sim\sigma^{0.6}$. Figure S11: Definition of the remodelling
parameter $\sigma_{0}$. As the network is progressively remodelled after
repeated loading cycles, a negative stress $\sigma_{0}$ required to return the
network to zero strain. An increase in $\sigma_{0}$ indicates an increase in
network remodelling.  Figure S12: Representative frequency sweeps of the
$I27_{5}$ hydrogels following the application of triangular strain pulses of
increasing magnitude. Each frequency sweep probes the linear viscoelastic
properties of the hydrogel after deformation. The spectra remain remarkably
consistent at applied strains $\gamma$ below approximately 0.4, at which point
the network undergoes irreversible fracture