Matt D G Hughes1, Benjamin S Hanson1,2, Sophie Cussons2,3, Najet Mahmoudi4, David J Brockwell2,3, Lorna Dougan1,2. 1. School of Physics and Astronomy, Faculty of Engineering and Physical Sciences, University of Leeds, Leeds LS2 9JT, U.K. 2. Astbury Centre for Structural Molecular Biology, University of Leeds, Leeds LS2 9JT, U.K. 3. School of Molecular and Cellular Biology, Faculty of Biological Sciences, University of Leeds, Leeds LS2 9JT, U.K. 4. ISIS Neutron and Muon Spallation Source, STFC Rutherford Appleton Laboratory, Oxfordshire OX11 0QX, U.K.
Abstract
Hierarchical assemblies of proteins exhibit a wide-range of material properties that are exploited both in nature and by artificially by humankind. However, little is understood about the importance of protein unfolding on the network assembly, severely limiting opportunities to utilize this nanoscale transition in the development of biomimetic and bioinspired materials. Here we control the force lability of a single protein building block, bovine serum albumin (BSA), and demonstrate that protein unfolding plays a critical role in defining the architecture and mechanics of a photochemically cross-linked native protein network. The internal nanoscale structure of BSA contains "molecular reinforcement" in the form of 17 covalent disulphide "nanostaples", preventing force-induced unfolding. Upon addition of reducing agents, these nanostaples are broken rendering the protein force labile. Employing a combination of circular dichroism (CD) spectroscopy, small-angle scattering (SAS), rheology, and modeling, we show that stapled protein forms reasonably homogeneous networks of cross-linked fractal-like clusters connected by an intercluster region of folded protein. Conversely, in situ protein unfolding results in more heterogeneous networks of denser fractal-like clusters connected by an intercluster region populated by unfolded protein. In addition, gelation-induced protein unfolding and cross-linking in the intercluster region changes the hydrogel mechanics, as measured by a 3-fold enhancement of the storage modulus, an increase in both the loss ratio and energy dissipation, and markedly different relaxation behavior. By controlling the protein's ability to unfold through nanoscale (un)stapling, we demonstrate the importance of in situ unfolding in defining both network architecture and mechanics, providing insight into fundamental hierarchical mechanics and a route to tune biomaterials for future applications.
Hierarchical assemblies of proteins exhibit a wide-range of material properties that are exploited both in nature and by artificially by humankind. However, little is understood about the importance of protein unfolding on the network assembly, severely limiting opportunities to utilize this nanoscale transition in the development of biomimetic and bioinspired materials. Here we control the force lability of a single protein building block, bovine serum albumin (BSA), and demonstrate that protein unfolding plays a critical role in defining the architecture and mechanics of a photochemically cross-linked native protein network. The internal nanoscale structure of BSA contains "molecular reinforcement" in the form of 17 covalent disulphide "nanostaples", preventing force-induced unfolding. Upon addition of reducing agents, these nanostaples are broken rendering the protein force labile. Employing a combination of circular dichroism (CD) spectroscopy, small-angle scattering (SAS), rheology, and modeling, we show that stapled protein forms reasonably homogeneous networks of cross-linked fractal-like clusters connected by an intercluster region of folded protein. Conversely, in situ protein unfolding results in more heterogeneous networks of denser fractal-like clusters connected by an intercluster region populated by unfolded protein. In addition, gelation-induced protein unfolding and cross-linking in the intercluster region changes the hydrogel mechanics, as measured by a 3-fold enhancement of the storage modulus, an increase in both the loss ratio and energy dissipation, and markedly different relaxation behavior. By controlling the protein's ability to unfold through nanoscale (un)stapling, we demonstrate the importance of in situ unfolding in defining both network architecture and mechanics, providing insight into fundamental hierarchical mechanics and a route to tune biomaterials for future applications.
Entities:
Keywords:
biomaterials; biomimetic and bioinspired materials; hierarchical biomechanics; protein hydrogels; protein unfolding
It remains
a fundamental challenge
to relate the properties of an individual nanoscale polymer building
block to the collective macroscale response of a network of such building
blocks.[1] The hierarchical structures present
in some biopolymer assemblies are crucial to the translation of properties
across length scales in living systems[2−4] and lead to a diverse
range of behavior including reversible softening under compression[5] and both stiffening[6] and negative normal stress under shear.[7] New insight would both further our understanding of biopolymer assemblies
ubiquitous in living systems and allow for the development of biomimetic
and bioinspired materials.[8−11] Recently, networks of folded globular proteins have
been demonstrated to exhibit exciting cross length-scale properties,[12−15] emerging due to the added complexity and functionality of the folded
building block. Since the initial demonstration of folded globular
proteins as suitable network building blocks,[15] protein-based hydrogels have emerged as a new class of biomaterial,
exhibiting rich properties such as mimicking the mechanical properties
of tissues,[15,16] forming highly elastic and stimuli-responsive
materials,[13,17,18] and dynamically regulating their properties and shape.[14,19,20] However, a complete understanding
of the translation of nanoscale properties to macroscale networks,
which will allow for the rational design of hydrogels with predictable
and tuneable properties, remains a fundamental challenge. In this
work we demonstrate the potential of the use of nanoscale staples
within proteins to define the network architecture and subsequent
mechanical response. We will show that manipulation of intraprotein
nanostaples provides control of protein mechanics and in situ protein unfolding, which is critical for the formation of the precise
structure and mechanics of the hydrogel network.Hydrogels are
hydroscopic networks formed from hydrophilic building
blocks swollen by relatively large volumes of water, and have become
a popular engineered biomaterial, as their high biocompatibility makes
them suitable for biomedical applications.[21] In order to design hydrogels for a specific purpose, it is necessary
to understand how to tune and tailor the mechanical properties of
the hydrogel to its application. The mechanical properties of hydrogel
network structures have previously been modulated using different
methods. One approach uses so-called “fillers” to occupy
the void space between the connected building blocks in the hydrogel
network, restricting movement of the overall network.[22,23] This approach has resulted in increases in the storage modulus from
6-fold[23] to 10-fold,[24] as well as emergent shear stiffening behavior.[25] An alternative method of hydrogel mechanical
reinforcement involves the inclusion of a secondary network in the
hydrogel (either permanent[26] or stimuli-responsive[27]) to act as a scaffold for the original network.
These “double-network” hydrogels[28,29] can be carefully designed to create networks which are “interwoven”,
resulting in gels with hybrid mechanical properties[30,31] such as high mechanical stability of the rigid network and the repetitive
elasticity of the flexible network.Both the filler method and
the double network method reinforce
and enhance the mechanical properties of the hydrogel network, but
both involve the alteration of the hydrogel at the network level rather
than on the nanoscale. In contrast, unstructured peptides or those
that form α-helices and β-strands have been utilized to
investigate how the interpeptide interactions of the network at the
molecular level affect the self-assembly of the fibrous microstructure
and subsequent mechanics of the hydrogel. Multiple investigations
have focused on tuning the structure and mechanics of the hydrogels
through precise control of the amino acid composition of these structurally
simple, short peptide chain building blocks.[32−35] These studies and others have
demonstrated that changing the hydrophobic interactions of the peptide
building blocks, can significantly shift the network morphology, increase
the storage modulus by 10–100-fold,[33,34] and result in interesting thermomechanical properties.[35]The recent studies on peptide-based hydrogels
highlight the critical
role of intermolecular interactions on the structure and mechanics
of the hydrogel. However, the use of peptides as hydrogel building
blocks significantly limits the range of biological functionality.
In the past decade, folded globular proteins have been utilized as
hydrogel building blocks due to their evolutionarily optimized and
highly specialized molecular functions, well defined structures, and
thermodynamic/mechanical stability.[36−38] The folded structures
of globular proteins, which display a range of mechanical strengths,
provide the opportunity to investigate the translation of mechanical
stability of individual building blocks to assemblies of building
blocks, all while retaining the inherent biological functionality
of the protein. While in-depth analyses of peptide hydrogels using
combined multi-experimental technique approaches are common in the
literature, in particular through the use of structural techniques
in conjunction with bulk mechanical characterization,[33,34,39] the same detail is lacking in
folded globular protein hydrogels.The additional functionality
offered by the mechanically robust
well-defined folded structures of proteins offers powerful opportunities
for new biomaterials. Recently several studies have studies have (i)
examined the interplay between the mechanical stability of the folded
protein building block and of the cross-linker;[40] (ii) investigated the role of rigidity and flexibility
in hydrogels by exploiting protein engineering to make constructs
containing both folded proteins and unstructured peptide chains, as
approximations of rigid rods and flexible chains respectively;[41] and (iii) determined that the increase of thermodynamic
and mechanical stability at the molecular level translates to increased
mechanical strength at the network level.[12,14] All of these studies exploit the nanoscale native state mechanics
of globular proteins. However, the importance of the nanoscale transition
from a rigid folded to a flexible unfolded state has yet to be explored.Here, we demonstrate that a transition in the nanoscale structure
of the building block via protein unfolding has a
defining role on network architecture and mechanics. In this work,
we present a combined experimental and modeling approach to show that in situ unfolding of bovine serum albumin (BSA) modulates
the BSA hydrogel network architecture, in particular the intercluster
region which demonstrably dominates the mechanical response of the
hydrogel network.
Results
BSA was selected as a model
protein to investigate the effects
of unfolding of the protein building block in situ on the properties of a cross-linked hydrogel network. BSA is an
ideal model globular protein as it contains at least four solvent
exposed tyrosine cross-linking residues (necessary for the formation
of gel network via photochemical cross-linking[42]) and, importantly, 17 structural disulphide
bonds. These intramolecular disulphide bonds effectively act as “staples”
holding the folded structure together (Figure a, b) and are prevalent in many protein families
that function in an extracellular environment including serum albumins,
defensins, and insulins. The covalent staples are mechanically robust
and capable of withstanding forces up to 2nN,[43,44] which is greatly in excess of the 20–100 pN thought to be
generated in the cross-linked protein network,[12,45] yet these bonds are rapidly removed by reducing agents such as dithiothreitol
(DTT) used in this study. BSA has disulphide bonds throughout its
structure (Figure a and b) suggesting that it is highly resistant to force-induced
unfolding; however, in the presence of DTT these staples (i.e., the intramolecular disulphide bonds that allow BSA
to resist force induced unfolding) break rendering the protein force
labile (i.e., readily unfolds under the application
of forces present in protein-based hydrogels during gelation). BSA
therefore presents the opportunity to determine the importance of
force lability of the protein building block on the hydrogel network
structure.
Figure 1
(a) Crystal structures (PDB code: 3v03) and (b) 2D topographs of BSA, where
disulphide bonded cysteine residues and tyrosine residues are colored
yellow and green, respectively. Normalized circular dichroism spectra
of BSA hydrogels in the (c) absence and (d) presence of DTT, before
gelation, immediately post gelation and one-hour post gelation. (e)
Normalized CD signal at 222nm of BSA (light red) and BSA:DTT (dark
red) hydrogels as a function of time post gelation, with previously
published data on a hydrogel composed of another globular protein
MBP (gray) added for reference. (f) Percentage of folded protein remaining
in each hydrogel system, pre-gelation (open), post-gelation (striped),
and post-relaxation (cross-hatched). Error bars taken from exponential
fit to the curves in part e.
(a) Crystal structures (PDB code: 3v03) and (b) 2D topographs of BSA, where
disulphide bonded cysteine residues and tyrosine residues are colored
yellow and green, respectively. Normalized circular dichroism spectra
of BSA hydrogels in the (c) absence and (d) presence of DTT, before
gelation, immediately post gelation and one-hour post gelation. (e)
Normalized CD signal at 222nm of BSA (light red) and BSA:DTT (dark
red) hydrogels as a function of time post gelation, with previously
published data on a hydrogel composed of another globular protein
MBP (gray) added for reference. (f) Percentage of folded protein remaining
in each hydrogel system, pre-gelation (open), post-gelation (striped),
and post-relaxation (cross-hatched). Error bars taken from exponential
fit to the curves in part e.In order to confirm the rationale of our selection, we employed
circular dichroism (CD) spectroscopy to investigate the structure
of the BSA protein in the presence and absence of disulphide bonds
both in solution and in the hydrogel.Figures c and d
show the spectra of BSA hydrogels in the absence and presence of DTT,
respectively, in the pre-gelation solution, in the hydrogel immediately
post-gelation and one hour after gelation. From these spectra, a reduction
in the protein’s CD signal at 222 nm is observed pre- to post-gelation,
which can be interpreted as a decrease in the amount of folded protein
in the hydrogel network (α-helix secondary structure in this
case) and from which a proportion of folded protein can be extracted.
The CD solution spectra (Figure S1a) and
the solution structure from small-angle X-ray scattering (SAXS) profiles
(Figure S1b) of BSA in the presence and
absence of DTT show no significant differences. This demonstrates
that the increase in BSA unfolding only occurs in the presence of
DTT and internal stress due to gelation. A time course of the CD signal
at 222 nm measured in situ for hydrogel maturation
over 6 h (Figure e)
shows that the proportion of folded protein in each hydrogel decays
over time, approaching end point values after 6 h (Materials and Methods). For BSA, there is a striking difference
between the proportion of folded protein in the absence and presence
of DTT. While in both chemical conditions there is a decrease in the
amount of folded protein after gelation, the extent and rate of decrease
is far greater in the presence of DTT (exponential fits to the curves
in Figure e extract
end point values of 9% and 28% in the absence and presence of DTT,
respectively). This result implies that removal of these structural
staples increases the extent of the gelation-induced unfolding, consistent
with the view that structural disulphide bonds provide molecular reinforcement
in the BSA folded structure. In accordance with this hypothesis, another
folded globular protein that lacks any disulphide bonds[46] and so is also labile to force (MBP[47,48]) showed behavior similar to disulphide-reduced, unstapled BSA (Figure e,f). These results
show that the intramolecular disulphide bonds act as molecular staples,
reinforcing the BSA building block against force-induced unfolding
due to gelation, which leads to the low degree of unfolding observed in situ. In contrast, the unstapled BSA in the presence
of DTT shows a higher degree of unfolding in situ due to their force labile structures. Comparison of these three
systems demonstrates that force induced unfolding occurs as a consequence
of gelation.To investigate whether gelation-induced unfolding
affected the
structure of the cross-linked BSA network, we used small-angle scattering
of neutrons (SANS) and X-rays (SAXS). These techniques provide structural
information over the length scales of tens to hundreds of Ångstroms.The SANS and SAXS curves of the BSA hydrogels in the absence and
presence of DTT are shown in Figures a and b, respectively. A qualitative assessment of
the scattering curves suggests there are significant structural differences
between BSA hydrogels in the absence and presence of DTT, as shown
by the reduced intensity at low q values and the shallower slope in
the mid q range. Interestingly, the BSA hydrogels
with DTT show markedly similar profiles to MBP hydrogels (Figure S2). Previous SAS characterization of
folded globular protein hydrogels of MBP has shown the presence of
discrete fractal-like clusters of cross-linked folded protein in the
network structure.[12] With this model in
mind we use eq to extract
quantitative information from the curves in Figures a and b:where ϕ is the volume
fraction of protein, Vblock is the volume
of the protein building
block, Δρ is the contrast difference between the building
block and the solvent, F(q) is the
ellipsoidal form factor of the building block, p is the proportion of protein in fractal-like
clusters within the gel network, and S(q) is a fractal structure factor.[12] Two
fitting parameters are of core interest in this work, the fractal
dimension, Df, and the correlation length,
ξ (see Materials and Methods), values
which are displayed in Figures c and d. Df can be defined as
the space-filling capacity of a fractal object which can (and often
does) differ from the dimension of the topological space in which
the object is embedded. Df can also be
interpreted as the measure of how the structural detail in an object
changes with the scale at which the object is measured.[49] Sometimes called the mass fractal dimension,
it gives a measure of how the “mass” (an intrinsic property)
of an object scales with its size (an extrinsic property). For example,
if the size of the object increased by a factor of 2, then the “mass”
of the object would increase by a factor 2. For our system Df can
be thought of intuitively as related to the density of the clusters
of cross-linked folded protein. ξ, then, is an imposed parameter
representing the upper limit length scale over which Df is a valid measure of hierarchical structure. We interpret
this as indicative of the size of the fractal clusters within the
network, with the associated lower limit of fractal behavior being
the size of an individual protein building block.
Figure 2
(a) SANS curves and (b)
SAXS curves of folded BSA hydrogels (final
concentrations: 100 mg/mL BSA, 50 mM NaPS, 100 μM Ru(BiPy)3) in the absence (light red) and presence (dark red) of DTT.
(c) Fractal dimension and (d) correlation length of clusters present
in BSA hydrogels (in the absence and presence of DTT) and MBP hydrogels.
(e) Number of protein monomers in a cluster as a function of distance
from the center of the cluster. (f) Volume fraction of a cluster (solid
color) and intercluster region (red and white striped) for each hydrogel
system. Line added at 7.4% to denote the initial volume fraction of
the system pre-gelation, ϕinitial. (g, h) Schematic
representation of the predicted structures of the BSA hydrogel networks
in the absence (light red) and presence (dark red) of DTT. Networks
consist of cross-linked fractal-like clusters with a volume fraction
of ϕC (represented by solid circles and highlighted
by light blue rings) connected by an inter-cluster region of protein
with a volume fraction of φIC (represented by white
striped circles). Solid dark red lines represent unfolded BSA protein
strands in the intercluster region in the presence of DTT. Error bars
show the standard errors, where number of repeats, N = 3.
(a) SANS curves and (b)
SAXS curves of folded BSA hydrogels (final
concentrations: 100 mg/mL BSA, 50 mM NaPS, 100 μM Ru(BiPy)3) in the absence (light red) and presence (dark red) of DTT.
(c) Fractal dimension and (d) correlation length of clusters present
in BSA hydrogels (in the absence and presence of DTT) and MBP hydrogels.
(e) Number of protein monomers in a cluster as a function of distance
from the center of the cluster. (f) Volume fraction of a cluster (solid
color) and intercluster region (red and white striped) for each hydrogel
system. Line added at 7.4% to denote the initial volume fraction of
the system pre-gelation, ϕinitial. (g, h) Schematic
representation of the predicted structures of the BSA hydrogel networks
in the absence (light red) and presence (dark red) of DTT. Networks
consist of cross-linked fractal-like clusters with a volume fraction
of ϕC (represented by solid circles and highlighted
by light blue rings) connected by an inter-cluster region of protein
with a volume fraction of φIC (represented by white
striped circles). Solid dark red lines represent unfolded BSA protein
strands in the intercluster region in the presence of DTT. Error bars
show the standard errors, where number of repeats, N = 3.The results in Figure c show that the measured fractal
dimension of a cross-linked
cluster is significantly larger in BSA hydrogels formed in the presence
of DTT (Df = 2.66 ± 0.01 and 2.17
± 0.01 in the presence and absence of DTT, respectively). While
the correlation length also increases in the presence of DTT (ξ
= 130 ± 1 Å and 123 ± 1 Å in the presence and
absence of DTT, respectively), it is not a significant increase (Figure d). This suggests
that hydrogels made from force-labile reduced BSA form “denser”
fractal-like clusters of a slightly larger size compared to the relatively
“sparser” clusters present in hydrogels in the absence
of DTT. These results suggest that while the cluster size is unchanged
with and without molecular reinforcement, the dimensionality or “density”
of the clusters is increased in BSA hydrogels in the presence of DTT.
Previously characterized hydrogels constructed from MBP (which has
no disulphide bonds) are shown for reference[12] and are again similar to the BSA hydrogels in the presence of DTT,
exhibiting clusters of protein with similar dimensionality (Df = 2.6 ± 0.01) but larger cluster size.
MBP hydrogels were the same volume fraction (ϕ = 7.4%) as the
BSA hydrogels (ϕ = 7.4%) presented in this work, but due to
the difference in protein size (rMBP ∼
24 Å vs rBSA ∼ 33 Å)
a greater number of MBP monomers are required to make up the same
volume fraction. This difference in number of monomers results in
larger clusters containing more protein in MBP hydrogels compared
to BSA:DTT hydrogels, though the network topology remains the same.To gain more insight into the hydrogel cluster size and morphology,
we consider the radial distribution function, g(r), determined by Teixeira[50,51] to derive
the fractal structure factor (eq ):where ρ is the maximum packing
density of the system, i.e. for randomly assembled
spheres 0.637, and r0 is minimum cutoff
distance of the fractal cluster, i.e. the effective
radius of the building block. The exponential
term is introduced with the parameter ξ to act as a cutoff distance,
imposing a maximum size on the fractal cluster, as discussed earlier.[50] Multiplying the radial distribution function
by the volume fraction of the system and integrating over r gives an expression for the number of individual building
blocks in a sphere of radius, r, from the center
of the cluster (eq ):where γ(D, R/ξ)
is the lower incomplete
gamma function. Figure e shows how the number of protein building blocks varies as a function
of distance from the center of a fractal-like cluster in units of
the building block radius. In all cases the curves increase at a rate
related to the fractal dimension of the cluster and plateau at large
distances from the cluster center. This is expected given the exponential
term in eq , giving
a measure of the maximum number of building blocks in a cluster. We
find that BSA hydrogels have approximately 7 times more protein in
each cluster in the presence of DTT, suggesting an extremely important
role of protein unfolding in determining the cross-linked cluster
density. From the calculated curves in Figure e we can extract an estimate of the radius
of the fractal-like clusters (Figure S3). The increase in the cluster size in the presence of DTT is consistent
with the same increase observed in the correlation length, which is
to be expected as the distance at which N(r) plateaus (Figure e) is completely dependent upon the correlation length. Using
an estimate of the cluster size and proportion of protein in the fractal-like
clusters, we can calculate the volume fraction of a fractal-like cluster
in isolation (φC) and the associated intercluster
region (φIC) in the hydrogel (Figure f). In both the absence and presence of DTT,
φC is larger than φIC, suggesting
a heterogeneous hydrogel network dominated by clusters of proteins.
The volume fractions of clusters and intercluster regions change upon
addition of DTT, with denser clusters and a sparser intercluster region,
suggesting a more heterogeneous network. Interestingly, the clusters
formed in the absence of DTT have a volume fraction very close to
the initial volume fraction pregel solution (7.89 ± 0.08% versus
7.4%). This result is consistent with what is expected from diffusion-limited
cluster aggregation theory,[52,53] in which individual
particles undergoing Brownian motion aggregate together to form clusters
of such particles. A consequence of this theory is that clusters will
continue to grow in size, until their volume fraction is equal to
the initial volume fraction of the solution. This result implies that
the predominate mechanism in the formation of BSA hydrogels in the
absence of DTT is diffusion-limited cluster aggregation, as opposed
to reaction-limited aggregation.[54,55] In the presence
of DTT, however, the volume fraction of BSA clusters (13.6 ±
0.1%) differs significantly from the initial pregel volume fraction,
suggesting an additional mechanism involved in the formation of these
hydrogels. For reference, this analysis was also performed on previously
published SAS data on MBP, included in Figures c–f, showing large levels of heterogeneity
in the network comparable to that of BSA hydrogels in the presence
of DTT. All the data therefore suggests that both BSA in the presence
of DTT and native MBP will yield and unfold to applied force, whereas
native stapled BSA does not yield to applied force.Combining
the force lability of the protein building block and
CD and SAS structural analysis, we propose a model of the network
structure of folded globular protein hydrogels, shown in Figure g and h. Folded proteins
with covalent intramolecular disulphide bonds are unyielding to force
and form hydrogel networks with fractal-like clusters made up of proteins
connected by intermolecular dityrosine cross-links, with clusters
linked together by multiple folded proteins (Figure g). Breakage of the intramolecular disulphide
bonds yields a force labile protein, and denser fractal-like clusters
are formed, with clusters connected by unfolded protein (Figure h). The force lability
of the protein is crucial in modulating the structure of the hydrogel
networks.In order to gain insight into the evolution of the
structure from
monodispersed solution to a self-supported network, we employ a previously
used dynamic computational model: BioNet.[56,57] BioNet can model individual folded protein monomers by representing
them as freely diffusing and rotating, pseudo-deformable (soft-core
potential) spheres with explicit cross-linking sites defined at the
sphere surface. When within 3 Å of one another, a rigid bond
will form between these sites to represent the cross-linking mechanism.
To approximately model the BSA subunit, each sphere was given a radius
of 33 Å with 14 evenly spaced cross-linking sites defined (representing
the tyrosine residues in BSA) and another 4 randomly placed in the
remaining space. These spheres then undergo a Brownian dynamics protocol
with a local drag on each sphere. We are also able to model unfolded
BSA as a chain of interacting binding sites connected by Hookean springs.
These Hookean springs represent the end-to-end fluctuations expected
of the worm-like chain polymer model, specifically where the length
of the overall polymer is significantly greater than its persistence
length as is the case for fully unfolded protein. To ensure the relative
diffusion time scales of the unfolded protein components were appropriate,
we assigned a local drag to each point-like binding site to approximately
match the drag on each segment of the amino-acid chain between binding
sites.We consider two key cases: one where the simulation is
initialized
with 91% of the monomers in a “folded” state and one
initialized where 72% of the monomers are in the “folded”
state. These were chosen as close approximations of BSA hydrogel in
the absence and presence of DTT, respectively (Figure f). The simulations were robustly initialized
with periodic boundary conditions applied and continued until the
networks were sufficiently percolated (Materials
and Methods). We emphasize that in these simulations, the explicit
unfolding of protein monomers during the “gelation”
process is not modeled, as protein unfolding is highly non-trivial
due to the complexity of the pulling direction[58] and also the effects of crowding.[59]Once the simulations are complete, a box counting method was
employed
to extract an explicit value for the fractal dimension from each of
the simulated cross-linked clusters (Materials and
Methods, see Figure S4). The fractal
dimensions extracted from both the simulations and experimental data
are shown in Figure . The fractal dimension extracted from the simulation with 91% folded
monomers (2.28 ± 0.01) is in reasonably good agreement with experimentally
measured BSA hydrogels in the absence of DTT (2.16 ± 0.01). However,
the fractal dimension extracted from simulations containing only 72%
folded monomers (2.13 ± 0.01) is significantly different to our
experimental results of BSA hydrogel in the presence of DTT (2.66
± 0.01). However, previous BioNet simulations of monodisperse
systems of 100% folded proteins (i.e. spheres only) showed that a
lower fractal dimension is to be expected for systems at lower volume
fractions56, showing that the homogeneous presence of mechanically
weak polymeric chains effectively acts as almost empty space with
respect to the fractal dimension. Hence, comparison of the simulation
and experimental results (Figure ) indicates that a general, homogeneous presence of
unfolded protein throughout the system during gelation is not sufficient
to cause the large structural changes in the hydrogel architecture
observed in our experimental SAS data (Figure ). The combination of experimental and computational
results therefore suggests that it is the act of unfolding of specific
force labile protein building blocks during gelation itself that is
crucial in defining the hydrogel architecture. However, it is possible
that the change in hydrogel network structure observed experimentally
involves aggregation of the unfolded protein chains, which is not
captured by our simulation. To confirm our hypothesis that it is the
act of unfolding that defines hydrogel architecture, further investigation
beyond the scope of this work would be required, including rapid frame
acquisition SAXS and computational modeling that accurately models
dynamic unfolding during gelation and the behavior of unfolded protein
chains.
Figure 3
(a, b) Schematics representing the explicit structures calculated
using BioNet simulations, where the blue spheres represent folded
protein and the lines represent connections made by unfolded protein.
(c) Fractal dimensions of cross-linked clusters extracted from both
BioNet simulations of a “single cluster” (calculated
using a box counting method) and experimental SAS data (Figure ). The proportion of unfolded
protein for the experimental results is the post-relaxation end point
values taken from Figure e. In contrast, for the simulations the proportions of folded
protein represent the fixed amount of unfolded present in the simulation
box over the course of the simulation. Error bars show the standard
errors, where number of repeats, N = 3.
(a, b) Schematics representing the explicit structures calculated
using BioNet simulations, where the blue spheres represent folded
protein and the lines represent connections made by unfolded protein.
(c) Fractal dimensions of cross-linked clusters extracted from both
BioNet simulations of a “single cluster” (calculated
using a box counting method) and experimental SAS data (Figure ). The proportion of unfolded
protein for the experimental results is the post-relaxation end point
values taken from Figure e. In contrast, for the simulations the proportions of folded
protein represent the fixed amount of unfolded present in the simulation
box over the course of the simulation. Error bars show the standard
errors, where number of repeats, N = 3.The results above show that removal of disulphide cross-links
within
BSA monomers affects the resulting hydrogel structure at the molecular
and network level, due to the force-induced unfolding of the BSA building
block that occurs in the absence of disulphide “staples”.
To investigate the effects of these structural changes on the macroscopic
mechanics, we performed rheology experiments on the BSA hydrogels
in the absence and presence of DTT.Figures a and b
show how the components of the complex shear modulus of BSA hydrogels, G′ and G′′ (storage
and loss moduli, respectively), and the loss ratio, tan(δ) (defined
as G′′/G′),
vary with applied oscillatory frequency in the absence and presence
of DTT. The storage modulus, which is a measure of the hydrogel elasticity,
is approximately 3-fold higher in the presence of DTT, while the loss
modulus, which is a measure of the hydrogel viscosity, is approximately
5-fold larger. Fitting a linear function to the storage modulus allows
for the extraction of the power law exponent giving an insight into
the dynamics of the system (Materials and Methods). The extracted exponent, known as the relaxation exponent, n, is a measure of the frequency dependent behavior of the
system and for gel-like systems gives information on the relative
dominance of elastic or viscous behavior. A gel with n approaching 1 is a purely viscous gel whereas a gel with n approaching 0 is a purely elastic gel.[60−62] The n values are 0.027 ± 0.002 and 0.061 ± 0.001 in
the absence and presence of DTT, respectively. From these values it
can be seen that both gels exhibit elastically dominated behavior;
however, the increase in exponent in the presence of DTT suggests
an increased level of viscosity in the BSA:DTT hydrogel. This increased
level of viscosity is likely due to the increased proportion of unfolded
protein present in the hydrogel (Figure f). This is also consistent with the results
in Figure b in which
tan(δ) is higher in the presence of DTT, showing a higher level
of viscosity in BSA hydrogels in the presence of DTT.
Figure 4
(a) Frequency sweeps
showing the (filled) storage, G′, and (open)
loss moduli, G′′,
of chemically cross-linked BSA hydrogels (final concentrations: 100
mg/mL BSA, 50 mM NaPS, 100 μM Ru(BiPy)3) in the absence
and presence of 3 mM DTT. An oscillatory strain of 0.5% was applied
to each sample. (b) tan(δ) of BSA hydrogels as a function of
applied frequency in the absence and presence of DTT. An oscillatory
strain of 0.5% was applied to each sample. (c) Gelation curves, showing
storage (closed symbols) and loss moduli (open symbols) vs time of
BSA hydrogels in the absence (light red) and presence (dark red) of
DTT. Illuminated at t = 60 s till t = 360 s. (d) Time scales of relaxation modes in BSA hydrogels with
MBP hydrogels added for reference. Error bars and ribbons show the
standard errors, where number of repeats, N = 3.
(a) Frequency sweeps
showing the (filled) storage, G′, and (open)
loss moduli, G′′,
of chemically cross-linked BSA hydrogels (final concentrations: 100
mg/mL BSA, 50 mM NaPS, 100 μM Ru(BiPy)3) in the absence
and presence of 3 mM DTT. An oscillatory strain of 0.5% was applied
to each sample. (b) tan(δ) of BSA hydrogels as a function of
applied frequency in the absence and presence of DTT. An oscillatory
strain of 0.5% was applied to each sample. (c) Gelation curves, showing
storage (closed symbols) and loss moduli (open symbols) vs time of
BSA hydrogels in the absence (light red) and presence (dark red) of
DTT. Illuminated at t = 60 s till t = 360 s. (d) Time scales of relaxation modes in BSA hydrogels with
MBP hydrogels added for reference. Error bars and ribbons show the
standard errors, where number of repeats, N = 3.In addition to the enhancement of the storage modulus
in the presence
of DTT, there is also an increase in the viscous behavior of the hydrogel
as denoted by an increase in the loss ratio, which is consistent with
the increased amount of unfolded protein in the system in the presence
of DTT as confirmed by CD (Figure c–f). This increase in the storage modulus of
BSA hydrogels gelated in the presence of DTT may be due to additional
cross-links in the intercluster region, either physical or chemical,
between the force unfolded protein chains. To explore this further,
a BSA hydrogel formed in the absence of DTT (i.e., fully folded) was soaked in situ on the rheometer
in a 3 mM DTT solution post-gelation (Figure S5a), which resulted in a decrease in the storage modulus (Figure S5b). This decrease in storage modulus
can be attributed to the force induced unfolding of load bearing BSA
building blocks as the DTT diffuses into the gel. Upon soaking of
the BSA hydrogel in DTT, an increase in the loss ratio (Figure S5c) is noted, demonstrating an increase
in the viscous behavior, which is consistent with an increase in the
amount of unfolded protein in the gel. This decrease in the storage
modulus (and simultaneous increase in the viscous behavior) upon soaking
suggests that no additional physical cross-links are being formed
between the unfolded protein chains, as we would expect this to increase
the value of G′. The enhancement to the storage
modulus of BSA hydrogels in the presence of DTT (Figure a,c) is therefore not due to
additional physical cross-links but a result of additional chemical
cross-links formed by the force unfolded protein chains during the
photochemical gelation process.To further investigate the molecular
reinforcement of the protein
and its impact on hydrogel formation, we analyzed the changes in storage
(G′) and loss (G′′)
moduli of BSA hydrogels in the absence and presence of DTT during
gelation, as a function of time, again using rheology (Figure c). In both cases the curves
initially show a dramatic increase in G′ during
the photochemical cross-linking process, which is then followed by
a large relaxation to a final value of G′.
Fitting these curves with a previously used empirical function[12] (eq , Materials and Methods) allows us to extract
information on the relaxation behavior of the system. Figure d shows the extracted time
constants of relaxation of the system. The difference between the
relaxation behavior of BSA hydrogels in the absence and presence of
DTT is striking, with the former having one mode of relaxation (τ1 = (1290 ± 30) s) while the later has two distinct relaxation
modes (τ1DTT = (480 ± 10) s, τ2DTT = (3800 ± 200) s). In a previous study
of a folded protein hydrogel using MBP,[12] we measured two modes of relaxation. A fast relaxation was attributed
to the formation of a percolated hydrogel network, and a second, slower
relaxation was attributed to the unfolding of the protein building
block. Interestingly, BSA in the absence of DTT displays one relaxation
mode, while in the presence of DTT we see two (Examplar fits shown
in Figure S6). The τ2DTT values extracted from the gelation curves in the presence
of DTT are similar to the time scale of unfolding observed in CD and
in our previous work.[12] In combination
with CD data (Figure c–f) this suggests that the emergence of two-relaxation modes
is inherently linked to force lability of the protein during gelation.
Note, this same behavior is observed for BSA hydrogels which are then
soaked in DTT solution (Figure S5a and Figure S5d). Therefore, we have demonstrated
that the relaxation behavior of folded protein-based hydrogels is
intimately linked to the force lability of the protein building block.In addition to the analysis of the linear mechanics of BSA hydrogels,
the behavior of the system under load was explored. Figure a shows shear stress–strain
loading curves of BSA hydrogels from applied rotational rheology in
the absence and presence of DTT. In either condition, the stress–strain
curves show linear elasticity at shear strains of less than 25% (Figure a). Fitting this
linear region yields storage moduli (G′) of
2.6 ± 0.3 kPa and 6.3 ± 0.2 kPa (in the absence and presence
of DTT, respectively) in good agreement with the values in Figure a. The stress strain
curves also display hysteresis behavior upon unloading of the sample
and are particularly prominent in BSA hydrogels in the presence of
DTT. The hysteresis area enclosed by the stress–strain curves
is a quantitative measure of the energy dissipated to the internal
energy of the system. Integrating the stress–strain curves
allows for the extraction and calculation of the energy dissipated
and the efficiency of the hydrogels (shown in Figure b). In the presence of DTT there is an over
4-fold increase in the energy dissipated, while there is only a 3%
reduction in the efficiency of the gels from 97% in the absence and
94% in the presence of DTT. This increase in energy dissipation is
likely due to larger amounts of unfolded protein in samples in the
presence of DTT, as upon unloading energy is lost to the rearrangement
of the unfolded peptide chains. This interpretation is consistent
with the CD results (Figure c–f), our structural model (Figure g,h), and previous literature[13,63] (which observed greater hysteresis behavior in samples with more
unfolded protein).
Figure 5
(a) Stress–strain curves of chemically cross-linked
BSA
hydrogels (final concentrations: 100 mg/mL BSA, 50 mM NaPS, 100 μM
Ru(BiPy)3) in the absence and presence of 3 mM DTT. (b)
Energy dissipation (open) and efficiency (striped) during the load–unload
cycle of BSA hydrogels in the absence (light red) and presence (dark
red) of DTT. Data on MBP hydrogels (gray) added for reference. (c)
Storage and loss moduli and (d) tan(δ) of BSA hydrogel in the
absence and presence of DTT as a function of applied oscillation strain
at 1 Hz. (inset) Enlargement of the strain-stiffening regime, plotted
without error bars for clarity. Error bars and ribbons show the standard
errors, where number of repeats, N = 3.
(a) Stress–strain curves of chemically cross-linked
BSA
hydrogels (final concentrations: 100 mg/mL BSA, 50 mM NaPS, 100 μM
Ru(BiPy)3) in the absence and presence of 3 mM DTT. (b)
Energy dissipation (open) and efficiency (striped) during the load–unload
cycle of BSA hydrogels in the absence (light red) and presence (dark
red) of DTT. Data on MBP hydrogels (gray) added for reference. (c)
Storage and loss moduli and (d) tan(δ) of BSA hydrogel in the
absence and presence of DTT as a function of applied oscillation strain
at 1 Hz. (inset) Enlargement of the strain-stiffening regime, plotted
without error bars for clarity. Error bars and ribbons show the standard
errors, where number of repeats, N = 3.Finally, the nonlinear mechanical behavior of the BSA hydrogels
was investigated; the shear moduli and loss ratio of the hydrogels
under increasing strain are shown in Figures c and d, respectively. The graphs show a
linear trend up to strains of approximately 40%, after which there
is a stiffening region until rupture at strains of approximately 300%.
Interestingly a far larger degree of strain stiffening is noted in
BSA hydrogels in the absence of DTT than those in the presence of
DTT. This difference is likely due to the difference in structures
(Figure g,h), in which
the native BSA gels have many folded domains that will act as load
bearing molecules as the strain is increased, whereas gels in the
presence of DTT have a large proportion of unfolded proteins which
uncoil toward their full contour length under strain meaning very
little change would be seen in the shear moduli.
Conclusions
We
have demonstrated that control of protein force lability has
an important role in defining the architecture and mechanics of cross-linked
protein hydrogels. We show that a network made from an internally
stapled protein building block retains 91% of its protein in the folded
state compared to 72% folded protein in a network made from unstapled
protein (Figure f).
This result implies protein reinforcement reduces the probability
of force induced protein unfolding during gelation. The network structure
formed from a disulphide-protected folded protein, that is unyielding
to force, consists of fractal-like clusters made up of cross-linked
protein, with the linking intercluster region populated by folded
proteins (Figure g).
Without molecular reinforcement, the protein building block is force
labile and denser fractal-like clusters are formed, with the connecting
intercluster region populated by unfolded protein (Figure h). By complementing our experimental
work with computational modeling (Figure ), we infer that it is the act of unfolding
of specific force labile protein building blocks during gelation,
rather then just the presence of unfolded protein, that is crucial
in defining the hydrogel architecture. To confirm this hypothesis,
further investigation beyond the scope of this work would be needed
including rapid frame acquisition SAXS and computational modeling
that accurately models unfolding during gelation.Controlling
the force lability of the protein building block also
has a significant impact on the mechanics as well as the architecture
of the protein hydrogel network. Networks formed from force labile
protein exhibit a higher elasticity (storage modulus approximately
3-fold higher), enhanced viscous behavior, and energy dissipation,
relative to the networks formed from internally stapled protein (Figures and 5). We suggest that this increase in viscous behavior is due
to a higher prevalence of unfolded protein in the intercluster region
of the hydrogel network constructed from the force labile protein
building block. The increase in the elasticity of the network is attributed
to additional chemical cross-links in the intercluster region of the
network, formed between the strands of force-induced unfolded protein
in this region. These results suggest that controlling the building
block unfolding and cross-link density in the intercluster region
is key in regulating and defining the mechanics of the network. Interestingly,
the dominance of the intercluster region on the mechanical response
of a network has been observed by other groups in colloidal systems.
Del Gado et al. and Frust et al. have found that the connections between
clusters in the intercluster region, termed the “rigidity percolation
network”, are key in regulating the mechanics of colloidal
networks both theoretically[64,65] and experimentally.[66] These studies similarly found that heterogeneity
in the network structure was crucial in governing the mechanical response
of the network.Conversely, studies on peptide-based hydrogels
have shown that
as the network becomes more homogeneous[33,34] the mechanical
strength is enhanced. We speculate that this difference is due to
the contrasting structures between the two systems, with interconnected
clusters and web-like fibrous structures exhibited by folded protein
and peptide gels, respectively.Restriction of unfolding of
the protein building block also has
a significant impact on the relaxation behavior of the hydrogels.
While a single mode of relaxation describes the networks formed from
stapled protein, a dual relaxation mode is necessary for networks
formed from the unstapled protein. We propose the additional mode
of relaxation in protein networks constructed from unstapled protein
corresponds to unfolding of the force labile protein (Figure ). Additionally, the relaxation
mode corresponding to unfolding is observed in hydrogels constructed
from a stapled protein soaked in DTT post-gelation (Figure S5) accompanied by an approximate factor of 2 reduction
in the storage modulus, consistent with previously published literature[41] comparing rigid and flexible building blocks.The modulation of the force lability of the building block plays
a fundamental role in defining the network architecture and mechanics.
The transition of the building block from a rigid folded state to
a flexible unfolded state emerges as a powerful method for controlling
the intercluster region of the network structure and the subsequent
mechanical response. This knowledge provides a powerful route to be
exploited in protein engineering, whereby the force lability of the
protein can be manipulated in subtle and specific ways. The single
molecule community has provided rich information on protein mechanics
and their rational design through approaches including protein engineering,
ligand binding, and external stimuli (including pH and temperature)
and routes to manipulating the local and global mechanical properties
of protein and unfolding.[36,67−71] TThese studies provide inspiration for the continued development
of a tool-box of force labile protein building blocks and their incorporation
into protein hydrogels. Furthermore, the results of single molecule
force experiments and computational models may be able to act as a
direct parallel to the internal behavior of protein hydrogels. This
study has demonstrated the necessity of combined structural and mechanical
characterization to understand the translation of complex molecular
properties across length scales. By understanding the crucial role
of building block unfolding on hierarchical networks, we demonstrate
the importance of in situ unfolding in defining the
structural and mechanical behavior of the network and reveal building
block unfolding as a method for the design of biomimetic and bioinspired
materials.
Materials and Methods
Materials
Bovine
serum albumin (heat shock fraction,
protease free, fatty acid free, and essentially immunoglobulin free),
tris(2,2′-bipyridyl)dichlororuthenium(II) hexahydrate (Ru(BiPy)3), sodium persulfate (NaPS), 1,4-dithiothreitol (DTT), d-(+)maltose monohydrate, sodium phosphate dibasic, and sodium
phosphate monobasic were obtained from Sigma-Aldrich and used without
further treatment. N-Terminal hexa-histidine tagged
MBP was expressed and purified as described below.
Protein Preparation
For completeness the preparation
method of MBP has been included. MBP was prepared using a mutated
pMal-c5x vector, with a stop codon inserted at position 378 by Q5
mutagenesis. The mutated vector was transformed into the expression
host Escherichia coli BL21 (DE3) pLysS competent
cells. Selected colonies were grown overnight in Lysogeny Broth (LB)
at 37 °C, 200 rpm to form starter cultures. 2 mL of these starter
cultures were used to inoculate 0.5 L of autoinduction media[72] in 2.5 L conical flasks. Cultures were incubated
for 48 h at 37 °C, 200 rpm before cells were harvested at 8000
rpm for 45 min. The harvested pellets were resuspended in lysis buffer
(0.1% Triton X-100, 1 mM PMSF, 20 mM benzamidine, 20 mM Tris, 300
mM NaCl, 10 mM imidazole, pH 8), homogenized and incubated for 1 h
in the presence of DNAase. Cell solutions were passed through a cell
disruptor (30 Kpsi, 25 °C), to ensure complete lysis, before
centrifuging at 25,000 rpm for 25 min to pellet the cell debris and
collect the lysate.To purify the MBP from the lysate, it was
loaded onto a Ni-NTA resin column overnight at 2 mL/min to ensure
maximum binding of the hexa-histidine-tagged MBP. 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 purified protein was dialyzed into water and freeze-dried for
storage at −20 °C. Average MBP yields of 300 mg per liter.
Sample Preparation
As previously published, hydrogel
samples are prepared by mixing in a 1:1 ratio a 200 mg/mL stock of
either BSA or MBP protein and 2× cross-link reagent stock for
final protein and reagent concentrations of 100 mg/mL BSA (MBP), 50
mM (30 mM) NaPS, and 100 μM Ru(BiPy)3.
Circular Dichroism
(CD)
Far-UV circular dichroism spectra
of MBP hydrogels were acquired on a Chirascan plus circular dichroism
spectrometer (Applied PhotoPhysics) with a bandwidth of 2 nm, a step
size of 1 nm, and a commercially available cuvette (Hellma) with a
path length of 10 μm. The BSA samples contained either 0 mM
or 3 mM DTT. Time-course CD measurements were acquired at 23 °C
over a long-time scale (approximately 10 h) to allow for correction
to the data due to dehydration. Dehydration was corrected for by fitting
the natural log of the data at large t (>6 h)
to
determine the rate of dehydration. This rate was used to fit the whole
data set with a double exponential decay function (in which one of
the rates was fixed to the rate of dehydration), and the exponential
decay term corresponding to the dehydration was removed.
Small Angle
Scattering (SAS)
SAS curves were fitted
using SasView (http://www.sasview.org) in accordance with eq .where F(Q) is an ellipsoidal form factor[73] and S(Q) is
a fractal structure factor to model
the geometry of the clustering of objects of the form F(Q).[50]D, ξ, and R0 are defined
as the mass fractal dimension, correlation length, and minimum cutoff
length scale defined by the ellipsoid form factor, respectively.
Small Angle Neutron Scattering (SANS)
SANS measurements
were conducted on the time-of-flight instrument Sans2d at the ISIS
Neutron and Muon Source (STFC Rutherford Appleton Laboratory, Didcot,
UK). Sans2d front and rear detectors were set up at 5 and 12 m, respectively,
from the sample, defining the accessible q-range
as 0.002–0.5 Å–1. Temperatures were
controlled by an external circulating thermal bath. Samples were loaded
and gelled in 1 mm path length quartz cuvettes. The raw SANS data
were processed using the Mantid framework[74] following the standard procedures for the instrument (detector efficiencies,
measured sample transmissions, absolute scale using the scattering
from a standard polymer, etc.).[75]
Small Angle X-ray Scattering (SAXS)
SAXS measurements
were conducted in the Materials Characterization Laboratory of the
ISIS Neutron and Muon Source, on the Nano-inXider instrument (Xenocs,
Sassenage, France) using a microfocus sealed-tube Cu 30 W/30 μm
X-ray source (Cu Kα, λ = 1.54 Å). Samples were loaded
and gelled in 1 mm path length glass capillary tubes. The q-range investigated was 0.0045–0.37 Å–1, and measurements were made at room temperature.
Rheometry
Mechanical characterization experiments of
BSA hydrogel samples were performed on a Anton Parr MCR 502 stress
controlled rheometer (Anton Parr GmbH, Austria) in parallel plate
configuration (with a plate diameter of 8 mm). Photochemical cross-linking
was initiated and controlled via illumination by
blue LED (peak emission at 452 nm) at a current of 0.48 A. To prevent
evaporation, during this process low viscosity silicone oil (approximately
5 ct) was placed around the geometry. The silicone oil should present
no schematic error on rheometric data as this is below the rheometer’s
torque range. Time sweep gelation measurements were conducted at a
frequency and shear strain of 1 Hz and 0.5%, respectively. Gelation
curves were fitted with an empirical function,where C and t0 are the rate and midpoint of increase of G′ due to photochemical cross-linking, respectively, B1 and B2 are the
coefficients of the first and second relaxation mode, respectively, G∞′ is the plateau value of the storage modulus post-gelation
and post-relaxation, and finally G0′ is the storage modulus
of the sample pre-gelation. All other terms are defined within the
main text. This function has previously been used to fit gelation
curves of folded protein hydrogels assembled using this photochemical
cross-linking method.[12] Frequency sweeps
were fit with a linear function to extract the relaxation exponent, n:where f is the fundamental
oscillation frequency of measurement and A is a prefactor.
Computational Modeling—BioNet
Our computational
modeling utilized BioNet, a dynamic simulation platform designed to
simulate network formation such as that observed within protein hydrogels.[56] The specific implementations used in this work
consisted of Brownian dynamics simulations of soft-core spheres (globular
proteins) and chains of point-like particles connected by Hookean
springs (unfolded proteins). The soft-core spheres and chains have
specifically defined point-like sites representing cross-linking tyrosine
residues, which interact with one another when within 3 Å by
forming a permanent Hookean bond of length 1.5 Å, representing
a carbon–carbon covalent cross-link. Further details not specific
to this work can be found in a recent work by Hanson et al.[56]We designed a system consisting of 5000
spheres, representing 100% folded protein, placed in a simulation
box with periodic boundary conditions with a size defined to give
a volume fraction of 0.074. Using this box, we created two systems,
one with 91% (4550) spheres and another with 72% (3600) spheres. The
remaining 9% and 28% were expanded into amino-acid chains: point-like
particles connected by Hookean springs with properties defined to
give the appropriate worm-like chain polymeric behavior of a disordered
polymeric chain. We performed the simulation of each system three
times using a statistically independent initial state each time. Statistical
independence was achieved between initial states by first randomizing
the initial positions of all particles, then allowing the system to
relax under a steric potential in isolation, and finally performing
a small amount of full Brownian dynamics (thermal noise, steric interaction, etc.) in the absence of any kinetic cross-linking. Simulations
were run until sufficiently percolated; that is, 99% of all objects
were connected to the biggest cluster in the box. All results quoted
in the main work (fractal dimensions, elastic moduli, etc) were averaged over these three repeats to generate errors. These
errors also include any fitting uncertainties carried forward (for
example, from our box counting method).Box counting for the
simulations was equivalent to that performed
previously, and the now present amino-acid chains were not accounted
for.
Authors: Olga V Stepanenko; Olga I Povarova; Anna I Sulatskaya; Luisa A Ferreira; Boris Y Zaslavsky; Irina M Kuznetsova; Konstantin K Turoverov; Vladimir N Uversky Journal: J Biomol Struct Dyn Date: 2016-01-06
Authors: Thomas Suren; Daniel Rutz; Patrick Mößmer; Ulrich Merkel; Johannes Buchner; Matthias Rief Journal: Proc Natl Acad Sci U S A Date: 2018-10-26 Impact factor: 11.205
Authors: Kathryn A Whitaker; Zsigmond Varga; Lilian C Hsiao; Michael J Solomon; James W Swan; Eric M Furst Journal: Nat Commun Date: 2019-05-20 Impact factor: 14.919
Authors: Jing Chen; Bo Wang; Julia S Caserto; Kaavian Shariati; Peng Cao; Yang Pan; Qixuan Xu; Minglin Ma Journal: Adv Healthc Mater Date: 2021-11-16 Impact factor: 11.092