Hierarchies, multiple energy barriers, and robustness
govern the fracture mechanics of ?-helical and
?-sheet protein domains
Theodor Ackbarow*, Xuefeng Chen*†, Sinan Keten*, and Markus J. Buehler*‡
*Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental Engineering, and†Department of Mechanical Engineering,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139
Edited by Shu Chien, University of California at San Diego, La Jolla, CA, and approved August 13, 2007 (received for review June 20, 2007)
The fundamental fracture mechanisms of biological protein mate-
rials remain largely unknown, in part, because of a lack of under-
standing of how individual protein building blocks respond to
mechanical load. For instance, it remains controversial whether the
free energy landscape of the unfolding behavior of proteins
consists of multiple, discrete transition states or the location of the
transition state changes continuously with the pulling velocity.
This lack in understanding has thus far prevented us from devel-
oping predictive strength models of protein materials. Here, we
report direct atomistic simulation that over four orders of magni-
tude in time scales of the unfolding behavior of ?-helical (AH) and
?-sheet (BS) domains, the key building blocks of hair, hoof, and
wool as well as spider silk, amyloids, and titin. We find that two
discrete transition states corresponding to two fracture mecha-
nisms exist. Whereas the unfolding mechanism at fast pulling rates
is sequential rupture of individual hydrogen bonds (HBs), unfold-
ing at slow pulling rates proceeds by simultaneous rupture of
several HBs. We derive the hierarchical Bell model, a theory that
explicitly considers the hierarchical architecture of proteins, pro-
viding a rigorous structure–property relationship. We exemplify
our model in a study of AHs, and show that 3–4 parallel HBs per
turn are favorable in light of the protein’s mechanical and ther-
modynamical stability, in agreement with experimental findings
use of building materials.
?-helix ? deformation ? intermediate filaments ? rupture ? structure
displaying highly specific hierarchical structures, from nano to
macro. Some of these features are commonly found and highly
conserved universal building blocks of protein materials. Examples
include ?-helices (AHs) (1, 2) and ?-sheets (BSs) (1). Both the AH
and BS domains are typically only one of the many domains within
a larger protein structure.
The AH motif is commonly found in structural protein networks
and plays an important role in biophysical processes that involve
mechanical signals, including mechanosensation and mechano-
transduction, and provide mechanical stability to cells (1–4). For
instance, AH-rich intermediate filament networks forward signals
from the cellular environment to the DNA (3, 4), aspects that are
critical for cell mitosis or apoptosis. The BS motif is an integral
properties of proteins and the link to associated atomistic-scale
chemical reactions are not only of vital importance in biology but
protein structure, inducing unfolding of the protein. Typically, a
variety of unfolding processes exist for a given protein structure,
each of which has a specific reaction pathway and an associated
roteins constitute critical building blocks of life, forming bio-
interplay between processes with different activation barriers Eb
operating at different activation distances xb.
A variety of AH- and BS-based structures have been studied in
experiment and molecular dynamics (MD) simulation (10–19).
However, earlier MD simulations were carried out at rather large
pulling rates, and therefore, no direct link between simulation and
experiment has been reported. Transitions of unfolding mecha-
nisms have been suggested (20, 21) but have thus far not been
observed directly in either experiment or simulation. It remains
controversial whether the free energy landscape of the unfolding
behavior of proteins consists of multiple, discrete transition states
velocity (20, 21).
Further, structure–property relationships for the force–
extension behavior and associated strength models have not been
reported. No links exist between the details of the molecular
architecture, the resulting free energy landscape, and the mechan-
strength models of protein materials.
Here, we present studies of three model protein domains (for
molecular geometries, see Fig. 1). We consider two AH models.
filament dimer (22–24), and AH2 is a domain from bacteriophage
T4 fibritin (25). The BS model is a protein structure proposed for
Alzheimer’s amyloid ?-fibril (5).
Theoretical Model for Protein Unfolding Mechanics. Several theories
describe competing processes due to mechanically induced insta-
a phenomenological theory originally postulated by Bell (26), or
Kramer’s diffusion model (27). Here, we extend Bell’s approach so
that simulations at various pulling speeds can be used to gain
information about the free energy landscape of a protein.
frequency ?0 ? 1 ? 1013s?1(26), and the quasi-equilibrium
The energy barrier is reduced by mechanical energy f?xb?cos(?) due
to the externally applied force f, where xbis the distance between
Author contributions: T.A. and M.J.B. designed research; T.A., X.C., and S.K. performed
research; T.A., X.C., S.K., and M.J.B. analyzed data; and T.A. and M.J.B. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Abbreviations: AH, ?-helical; AP, angular point; BS, ?-sheet; HB, hydrogen bond; FDM,
fast-deformation mode; MD, molecular dynamics; SDM, slow-deformation mode; SMD,
steered molecular dynamics.
Mechanics, Department of Civil and Environmental Engineering, Massachusetts Institute
of Technology, 77 Massachusetts Avenue, Room 1–272, Cambridge, MA 02139. E-mail:
This article contains supporting information online at www.pnas.org/cgi/content/full/
© 2007 by The National Academy of Sciences of the USA
October 16, 2007 ?
vol. 104 ?
the equilibrated state and the transition state, and ? is the angle
between the direction of the reaction pathway of bond breaking (x
direction) and the direction of applied load [f direction; see
supporting information (SI) Fig. 6]. The angle can be determined
by analyzing the protein geometry. The off-rate is given by
? ? ?0?exp??Eb? f?xb?cos???
and describes how often a bond is broken per unit time (the
reciprocal of the bond lifetime).
However, Eq. 1 does not describe the dependence of the pulling
speed v (the controlled parameter in experiment and MD simula-
tion) at which a bond breaks because of the pulling force f. We thus
modify Eq. 1 based on the following idea. The speed v at which a
bond is broken is equal to the distance that needs to be overcome
to break the bond (xb), divided by the time for the bond breaking.
Consequently, v is the product of ??xb, thus v ? ??xb ? ?x/?t.
Macroscopically, the pulling speed is equal to the displacement ?x
be rewritten, leading to
v ? v0?exp?
with v0 as the natural bond-breaking speed (speed of bond
dissociation when no load is applied), defined as
This modified framework enables one to calculate the force at
which a bond breaks, at a certain pulling rate:
xb?cos(?)?ln v ?
xb?cos????ln v0? a1?ln v ? a2,
where a1? kB?T/(xb?cos(?)) and a2? ?kB?T/(xb?cos(?))?ln v0. Eq. 4
predicts that the bond-breaking force depends logarithmically on
the pulling speed in a nonequilibrated system. The parameters a1
temperature and angle. Note that if the free energy landscape is
dominated by several transition states, each of the states is char-
acterized by a combination of Eband xb. This results in segments of
multiple straight lines in the f–ln(v) plane. The model reduces to a
phenomenological model when the cos(?) term is removed; the
phenomenological model contains only the energy barrier Eband
xband no structural information. Note that an expression similar to
Eq. 4 was reported in ref. 28.
Results of MD Simulations. We carry out a series of classical MD
simulations (for details, see Materials and Methods). The goal is a
at varying pulling rates.
For the vimentin AH protein domain (AH1), two characteristic
force–strain curves are shown in Fig. 2 for two pulling speeds. The
simulations reveal existence of three distinct deformation regimes.
The first regime shows a linear increase in strain until the angular
point (AP) is reached. The second regime is a plateau of approx-
occurs. The last regime displays a significant strain hardening due
to pulling of the protein’s backbone [only partly visible in the
fast-deformation mode (FDM) plot]. A similar behavior is ob-
served for the AH2 structure. The change from the first to the
second regime is referred to as the AP, denoting the protein-
unfolding force. Unfolding of the protein is characterized by
rupture of hydrogen bonds (HBs) that destroys the protein struc-
ture as the displacement is increased. In the remainder of this
article, we focus on the force at the AP as a function of the pulling
We carry out computational experiments by systematically vary-
0.05 to 100 m/s. The unfolding force is plotted as a function of the
pulling speed in Fig. 3A for AH1 and AH2. Fig. 3B shows the
a maximum peak at which the structure fractures.
Notably, in all three cases we observe two distinct regimes, each
of which follows a logarithmic dependence of the unfolding force
indicates two different energy barriers and thus two different
unfolding mechanisms over the simulated pulling velocity regime.
The results clearly suggest a free energy landscape that consists of
two transition states. In the following text we refer to these two
regimes as the slow-deformation mode (SDM) and the FDM. The
change in mechanism from the FDM to the SDM occurs at v ? 0.4
m/s (AH1) and v ? 4 m/s (AH2), and at a force of ?350 pN (AH1)
and ?400 pN (AH2). For the BS structure, the transition occurs at
v ? 10 m/s at a force of ?4,800 pN.
To the best of our knowledge, up to now, neither any unfolding
behavior in the SDM nor the change from the FDM to the SDM
has been observed in direct MD simulation or in experiment. We
emphasize that the change in mechanism has thus far only been
suggested or inferred (21, 29). For example, a comparison between
MD simulation and experimental results revealed that force–
(AH1, AH2, and BS). Surrounding water molecules are not shown for reasons
loading for AH1 and AH2 and shear loading for BS). The BS structure consists
of two stacks of ?-sheets in the out-of-plane direction.
Atomistic geometries of the three protein domains studied here
mode (FDM) is represented by a curve taken at a pulling speed of 10 m/s. The
slow deformation mode (SDM) is represented by a pulling experiment at 0.1
in strain until the AP is reached (indicated by arrows) when the first HBs
constant force, during which unfolding of the entire protein occurs; and (III)
strain hardening (only partly shown for the FDM).
Examples for force–extension curves of AH1. The fast deformation
Ackbarow et al.
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pulling speed dependence must lie on two different curves in the
f–ln(v) plane (21, 29), suggesting a change in unfolding mechanism.
By fitting the extended Bell theory to the MD results of the AH1
structure, we obtain for the FDM Eb? 4.7 kcal/mol and xb? 0.20
Å. In the SDM (with ? ? 16°), Eb? 11.1 kcal/mol and xb? 1.2 Å.
Eb are slightly lower (see Table 1). Considering that the bond-
breaking energy Ebof a HB in water ranges typically from 3 to 6
kcal/mol (30), the results indicate that in the FDM, individual HBs
rupture sequentially. In contrast, in the SDM approximately three
HBs rupture at once. Studies of both AH structures clearly support
For the BS structure, we obtain for the FDM Eb? 2.2 kcal/mol
and xb? 0.024 Å. In the SDM, Eb? 11.1 kcal/mol and xb? 0.138
Å (the angular term is not considered here). Notably, the force
levels in the BS domain are much higher than in the AH structure,
indicating that this protein domain may be mechanically sturdy and
approaches rupture forces of 1 nN at experimental and physiolog-
ical pulling rates. These strength values agree quantitatively with
recent experimental studies of amyloid structures, possibly explain-
ing how the shear loading of arrays of HBs can lead to extremely
strong resistance against rupture, reaching the strength of covalent
bonds (31, 32).
The details of the atomistic rupture mechanisms are summa-
rized in Table 1. An analysis of the atomistic structure during the
rupture event is shown in Fig. 4A for the SDM in the vimentin
AH1 domain. In the SDM, three HBs rupture simultaneously,
within ?20-ps time scale. It was reported that the time for HB
breaking is ?20–40 ps (30), clearly supporting the notion that
these HBs rupture at once.
Further evidence for the change in mechanism is obtained by an
analysis of the HB rupture dynamics. In Fig. 4B we plot the HB
rupture as a function of the molecular strain for the vimentin AH1
domain. This provides a strategy to normalize the different time
scales by the pulling velocity (here, 0.1 and 10 m/s). In agreement
with the results shown in Fig. 2, the unfolding of the protein in the
SDM starts at ?10% strain, in contrast to 20% strain in the FDM
regime. This difference is indicated in Fig. 4B by the rupture of the
first HB. The data shown in Fig. 4 clearly show that, in the FDM,
HBs rupture sequentially as the lateral load is increased from 20 to
40% tensile strain. In contrast, in the SDM, several HBs rupture
and a BS amyloid domain (B), as a function of varying pulling speed over four
orders of magnitude, ranging from 0.05 to 100 m/s. The results clearly reveal
a change in protein-unfolding mechanism from the FDM to the SDM. The
arrows in A indicate the representative pulling speeds used for the analysis
reported in Figs. 2 and 4.
Table 1. Summary of the differences between the SDM and FDM, for AH1, AH2, and BS
AH1 (AH2) domainBS domain
Pulling speed, m/s
Unfolding force, pN
v ? 0.4 (4)
F ? 350 (400)
v ? 0.4 (4)
F ? 350 (400)
v ? 10
F ? 4,800
v ? 10
F ? 4,800
The values in parentheses in the AH columns represent the results for AH2.
Fig. 1 for v ? 0.1 m/s). (A) Atomistic representation of the rupture dynamics. The
time interval between these snapshots is 20 ps (between I and II) and 40 ps
the first four HBs as a function of the applied strain [residue number represents
whereas in the SDM, several HBs rupture almost simultaneously, within 20 ps. In
fixed residue, whereas in the SDM, the unfolding ‘‘wave’’ runs in the opposite
direction, nucleating at a random residue within the protein sequence.
www.pnas.org?cgi?doi?10.1073?pnas.0705759104 Ackbarow et al.
almost simultaneously, within ?20 ps, at a tensile strain of ?10%.
in the SDM, the HBs in the SDM rupture significantly faster.
Similar observations are made in the BS structure. An analysis of
in the SDM.
Notably, force spectroscopy results of individual AH or BS
domains are rare, even though the first studies with atomic force
microscopy were reported ?10 years ago. This is partly due to
experimental difficulties caused by the small size of the protein
probes. For instance, it is difficult in an experiment to stretch
individual proteins or protein domains rather than bundles (13,
33, 34). The unfolding forces measured in experiments are
typically between 100 and 200 pN for pulling velocities in the
SDM (13, 33, 34). This finding is close to the values obtained in
our simulations, which predict forces below 350 pN in the SDM.
Pulling experiments of coiled-coils, which consist of two AHs
arranged in a helical geometry, can be carried out in a more
controlled fashion. In these systems, the experimentally mea-
sured unfolding force ranges between 25 and 110 pN, also for
deformation speeds in the SDM (10, 35). These force values are
also in proximity to the unfolding forces predicted by our
simulations and the theoretical model.
Hierarchical Strength Model for the AH Structure. The remainder of
this article is focused on the AH structure. Even though the
phenomenological model (Eq. 1) explicitly considers chemical
protein architectures that include several bonds. For instance,
whether a single HB ruptures or several HBs rupture simulta-
in mechanism is not explicitly noted in the theory.
To estimate the strength and the energy landscape of a protein
without performing any simulations or experiments, we extend
the theory to explicitly consider the structural hierarchies of the
protein structure with the only input parameters being the
energy of a HB and the rupture distance. The AH represents a
hierarchical structure, reaching from individual HBs at the
lowest, atomistic level to a collection of HBs at the next higher,
molecular protein scale.
and xb, and the higher hierarchy consists of parallel HBs. Here we
assume that b bonds in a structure are in parallel and d bonds out
of these b bonds break simultaneously. Thus,bCd(the binomial
coefficient is defined asbCd) possible combinations for this rupture
mechanism exist. The probability that one of these combinations
constitutes a particular rupture event is one divided bybCd. Also,
if d bonds break simultaneously, the total energy barrier increases
by a factor d, to d?Eb
0. This leads to the following expression for the
We rewrite Eq. 5 so that the binomial coefficient appears in the
exponential, which enables us to compare Eq. 6 with Eq. 1,
?H? ?0?exp??? d?Eb
d? ? f?xb?cos????
The parameter Ebin Eq. 1 can thus be split up as
0is the energy of a single bond and the term
is the contribution to the energy barrier due to the hierarchical
structure. The unfolding force is
f (v, b, d; Eb
0, xb,?) ?
Note that f ? fv ? fh1 ? fh0, where the fv, fh1, and fh0 are the
(strength of bonds, Eband xb). This expression quantifies how the
text we refer to this model as the hierarchical Bell model.
This approach can easily be extended to three hierarchies,
which enables one to predict the rupture force of a tertiary
structure consisting of 2, 3, . . . , n AHs, of which k unfold
simultaneously (see SI Text, Extension of the Hierarchial Bell
in principle, valid for any protein structure that consists of several
parallel bonds, for example, ?-sheets or ?-helices.
Here, we apply this theory to predict unfolding force of an AH
domain. The AH is a two-hierarchy system, where the lowest
hierarchical scale is represented by an individual HB. A collection
of b bonds form the next higher hierarchical scale. Thus, Eq. 8
enables us to estimate the unfolding force at any pulling speed.
AHs Maximize the Robustness at Minimal Use of Building Resources.
Protein folding and thus the generation of hierarchical structures
are essential for biological function. First, folding allows distant
parts of the amino acid chain to come physically closer together,
creating local sites with specific chemical properties that derive
from the collection of particular residues. Second, folding permits
collective, localized motion of different regions (36). The AH
(2), forming a spring-like protein structure with high elasticity and
large deformation capacity.
But why does an AH fold in such a way that 3.6 parallel HBs,
instead of 2, 5, or 6, appear in parallel, per turn? Notably, all AHs
universally show this particular molecular architecture. To the best
of our knowledge, there has been no explanation for this particular
molecular feature, despite the fact that the AH is such an abundant
considering the robustness of the AH structure against mechanical
and thermodynamical unfolding.
We calculate robustness based on the definition of robustness as
parameter insensitivity, postulated by Kitano (37). This definition
of the protein strength in regard to missing HBs. Starting with the
hierarchical Bell model (Eq. 8), we calculate robustness as the ratio
in the failed system all except one HB contribute to the strength.
the pulling speed part of this equation is not taken into account
because we compare systems at identical pulling speeds. The
robustness is defined as
r?b? ?fh1?d ? b ? 1? ? fh0?d ? b ? 1?
fh1?d ? b? ? fh0?d ? b?
? 1 ?kB?T?ln(b) ? Eb
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The robustness converges toward fault tolerance when b 3 ?.
Fig. 5 depicts the robustness of an AH as a function of parallel
to four HBs per turn, 80% robustness is achieved (0% robustness
means that the system is highly fragile, and 100% represents
complete fault tolerance). This level of robustness in a biological
structure enables it to minimize waste of resources (that is, amino
acids), weight, and volume, and thus makes the structure overall
efficient and able to sustain extreme mechanical conditions (such
as high loading rates and deformations).
This finding is significant because the only input parameters in
this model are the dissociation energy of an HB, Eb
fundamental, ‘‘first-principles’’ property of protein structures. This
parameter can be determined reliably from either experiment or
atomistic simulation (both approaches lead to similar values). The
remainder of the parameters required to predict the robustness
properties can be derived from the geometry of the protein
Synthetic materials typically do not have such high levels of
robustness. Lack of robustness makes it necessary to introduce
safety factors that guarantee a structure’s functionality even under
extreme conditions. For instance, an engineering structure such as
a tall building must be able to withstand loads that are 10 times
higher than the usual load, even if this load will never appear
globally. This safety factor is necessary because these structures are
very fragile because of their extremely high sensitivity to material
instabilities such as cracks, which might lead to such high local
of the material is wasted. This calculation shows the potential of
engineering bio-inspired robust and efficient structures. The key
may be to include multiple hierarchies and an optimal degree of
redundancies, as illustrated here for the AH structure.
much stronger bond is also energetically favorable, in particular, in
light of the moderate assembly temperatures in vivo. However, this
finding only makes sense if three HBs rupture simultaneously so
that they can provide considerable mechanical and thermodynam-
ical resistance, which has indeed been shown to be the case at
physiological strain rates in Fig. 2 (33). The intimate connection of
structural properties, assembly, and functional processes is an
overarching trait of protein materials.
0, which is a
Using an integrated approach of theory and simulation, we have
unfolding behavior during stretching of AH and BS protein do-
mains. Our results prove that the unfolding mechanism at fast
pulling rates is rupture of a single HB, whereas the unfolding
mechanisms at slow pulling rates proceed by simultaneous rupture
of several parallel HBs (Figs. 2–4 and Table 1). This phenomenon
has been consistently observed for the three protein structures
studied here, including AH and BS protein domains.
At present, MD simulations are the only means to directly
observe these mechanisms, because experiments still lack appro-
priate spatial and temporal resolution. Advances in computing
power have enabled us to carry out direct atomistic simulation of
unfolding phenomena, including explicit solvent, at time scales
approaching a significant fraction of a microsecond.
In previous atomistic simulations, unfolding forces were signifi-
cantly larger than those measured in experiments, likely because
they were carried out in the FDM so that forces increase to several
nanonewtons for individual AHs. This finding is clearly an artifact
of large pulling speeds (20, 21). Our analysis shows that, in addition
also be significantly different if the pulling speed is too high. The
estimate for v?provides a ‘‘maximum’’ pulling rate that could be
used in MD studies to still allow a reasonable interpretation of MD
results in light of biological relevance. The quantitative values
The SDM is most relevant for biological function, but the FDM
could be important during tissue injuries that may be incurred
under large deformation rates (e.g., shock impact, bullets, and
The fact that this behavior is observed for three protein struc-
domains and shear loading for the larger BS domain) suggests that
the discrete change in mechanism from single HB rupture to
In particular, the results obtained from the BS structure illustrate
believe that the results reported here are applicable to the mechan-
ical behavior of many other protein domains and possibly larger
Note that the interface of different proteins or even the super-
molecular structure is significant and may be most relevant for
many biological functions [for instance, the unfolding of globular
two AHs under strain (21)]. However, to predict the deformation
mechanisms of more complex protein structures, studies like the
one reported here are critical because they enable one to compare
the strength of different competing deformation modes.
We have developed a hierarchical Bell theory that explicitly
considers the hierarchical arrangement of HBs in the AH protein,
structure, here exemplified for the vimentin AH motif. This theory
and HB energy parameter Eb
the characteristic parameter in capturing the time scale of protein-
folding events, the theory enables one to link the geometry of the
protein structure, distinct time scales, and consequently, experi-
extrapolate MD simulation results to experimental and in vivo
We have discovered that three to four parallel HBs are the most
favorable bond arrangement in light of mechanical and thermody-
namical stability, leading to a robustness of ?80%. This result
indicates that AHs are efficient according to Pareto’s principle (38,
39), which is also known as the 80/20 rule. This rule is an empirical
political, and natural phenomena. Our results indicate that this
concept may also be applicable to explaining the nanoscopic
architecture of the AH protein motif. In this study, we have found
substantial evidence for the applicability of the Pareto principle to
0and xb. Because the pulling speed is
predicted by the hierarchical Bell model. Robustness is defined as the ratio of
strength of a failed system and an intact system. In the intact system, all HBs
contribute to strength, whereas in the failed system, all except one HB
contribute to the strength. The shaded bar indicates the number of parallel
HBs per turn (3.6 HBs) as observed in nature. This particular molecular geom-
etry corresponds to a robustness value of ?80%, indicating that the AH is
efficient in Pareto’s sense (38, 39).
Robustness of an AH as a function of parallel HBs per turn, b,
www.pnas.org?cgi?doi?10.1073?pnas.0705759104Ackbarow et al.
explain the molecular structure of proteins. In Pareto’s sense, the
more robust the structure becomes with each additional HB, the
higher the barrier is to implementing an additional HB, because
each HB introduces an additional ‘‘cost’’ due to increased material
use, that is, the additional weight and volume. In light of these
considerations, it is not surprising that a robustness value of 80% is
found in AHs, which equals the optimal state due to these com-
peting mechanisms. We note that other reasons, such as steric
of the AH protein motif.
The theoretical progress in understanding protein materials at
the atomistic scale will enable us to understand, and eventually to
exploit, the extended physical space that is realized by utilization of
hierarchical features. These traits may be vital to enable biological
systems to overcome intrinsic limitations due to particular building
blocks: chemical bonds and chemical elements. By using a bot-
tom-up structural design and synthesis approach, the hierarchical,
extended design space could serve as a means to realize new
physical realities that are not accessible at a single scale [e.g.,
material synthesis at moderate temperatures, fault-tolerant hierar-
chical assembly mechanisms, and robust and strong materials (40)].
contribute to the understanding of which driving forces in nature
are most important for the evolutionary development of biological
materials, and what role the abundant nanoscopic features play in
determining their properties at different scales.
Materials and Methods
Atomistic-Protein Structures. The AH1 structure is taken from the
2B segment of the vimentin intermediate filament (IF) (22–24)
(PDB ID code 1ox3) [length, 60 Å (25)]. The BS domain is taken
from an Alzheimer’s amyloid-?(1–42) fibril (5) (PDB ID code
Atomistic-Simulation Methods. We use a classical MD approach,
(42). All simulations were performed at a temperature of 300 K
[NVT ensemble, Berendsen thermostat (43)], with a 1-fs time step.
Careful energy minimization and finite temperature equilibration
of all structures are simulated before the protein is loaded. The
in a TIP3 water skin. In all cases, the entire protein is embedded in
water, before and during deformation of the protein. The water is
essential to capture the correct HB rupture dynamics.
to the molecule to induce deformation, we use SMD (44), with the
SMD spring constant kSMD ? 10 kcal?mol?1?Å?2. We obtain
force-versus-displacement data by monitoring the time-averaged
applied force (f) and the position of the atom that is pulled at (x)
over the simulation time.
Force Application Boundary Conditions. To apply load, C?atoms at
one end are fixed and the force is applied on the C?atom at the
other end in the AH structure, with a pulling speed v (Fig. 1
Bottom). The tensile boundary conditions chosen for the AH
domain are closest to the physiological conditions. Several other
boundary conditions have been used (changing fixed and pulled
been observed, suggesting that the results reported here are robust
with respect to changes in the boundary conditions. In the BS
structure, we pull on the middle chain of the assembly (third chain
from top or bottom) at the midpoint of the turn that connects the
two ?-strands. We fix all C?atoms on the top and bottom chains
during pulling (Fig. 1 Bottom). These boundary conditions are
similar to those reported in recent atomic force microscopy exper-
iments of a comparable amyloid structure (32).
Analysis of HB Rupture Dynamics. We use Visual Molecular Dynam-
ics (VMD) for visualization of protein structures (45), as well as for
the analysis of the length of HBs. The rupture length of a HB is
defined as 5 Å [the equilibrium length of HBs is ?3 Å (46)]. The
the dynamics of bond breaking and rebinding caused by thermal
This research was supported by Army Research Office Grant W911NF-
06–1-0291 (program officer Dr. Bruce LaMattina), the Solomon Buchs-
baum AT&T Research Fund, and National Science Foundation Career
Award CMMI-0642545 (program officer Dr. Jimmy Hsia). T.A. was
supported by the German National Academic Foundation and the
Dr.-Juergen-Ulderup Foundation. X.C. acknowledges Massachusetts
Institute of Technology’s Undergraduate Research Opportunities Pro-
gram. S.K. was supported by the Presidential Graduate Fellowship
Program and Massachusetts Institute of Technology’s Department of
Civil and Environmental Engineering.
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