J Appl Physiol 116: 825–834, 2014.
First published January 9, 2014; doi:10.1152/japplphysiol.00962.2013.
A computational model of the response of adherent cells to stretch and
changes in substrate stiffness
Harikrishnan Parameswaran, Kenneth R. Lutchen, and Béla Suki
Department of Biomedical Engineering, Boston University, Boston, Massachusetts
Submitted 22 August 2013; accepted in final form 7 January 2014
fluidization; mechanotransduction; cortical actin network; active networks
NORMAL FUNCTIONING of living cells in the human body is
intricately tied to the ability of the cell to sense and respond to
its mechanical environment, which includes the composition
and stiffness of its surrounding tissue and also the stretch
patterns that it perceives in vivo. Mechanical stretch, in particular, has been shown to influence many basic cellular processes such as growth, remodeling, apoptosis, and gene expression (3, 22, 32, 40). However, the mechanisms by which
intracellular contractile elements of the cell sense and respond
to mechanical stretch are not fully understood.
Recent studies have shown that despite considerable diversity in function, the mechanical response of all adherent cells to
stretch is universal and independent of their cell type (39, 71).
For instance, when subjected to a transient stretch, cells
promptly lose their stiffness and traction force followed by a
slow recovery back to baseline. While the extent of decline in
stiffness and traction is an increasing function of the stretch
amplitude, the cell is able to modulate its recovery rate such
that the time taken to get back to its baseline mechanical state
is the same regardless of the stretch amplitude. Understanding
Address for reprint requests and other correspondence: H. Parameswaran,
Respiratory Physiology and Dynamics Laboratory, Dept. of Biomedical Engineering, 44 Cummington St., Boston, MA 02215 (e-mail: [email protected]).
http://www.jappl.org
the underlying mechanisms behind this universal response to
stretch is key to uncovering the cellular origins of many
clinically relevant problems such as the beneficial effect of
deep inspirations in healthy subjects (9, 53, 60) and the lack
thereof in asthmatics (8, 16, 33, 34) and the role of cyclic
stretch in maintaining mechanical homeostasis in multiple cell
types (14, 17).
A key element necessary for substrate sensing is the internal
indigenous tension or prestress in the actin cytoskeleton generated by myosin II motors that attach to and walk along actin
filaments drawing energy from ATP hydrolysis (26, 66, 73).
The rate of development of prestress by a living cell is a fast
process and reaches equilibrium in time scales of the order of
100 s (39, 50). Within this time frame, it has been experimentally shown that the prestress is borne by a network of randomly polarized actin filaments (2). The active motion of
myosin motors within such a network can also result in
rearrangement of the actin network (4), and the possibility of
self-organized structures emerging in such an active system has
been postulated by theoretical models (41, 76) and experimentally observed in actin gels (4, 64) and even in living cells
labeled for actin (15, 31, 67). Recently, Borau et al. (6) showed
that a network of actin, actin cross-linking proteins, and active
molecular motors can replicate response of living cells to
changes in substrate stiffness. Since dense networks of actin
with nonmuscle myosin II (NM2) motor proteins are present in
all adherent cell types, we hypothesized that such active
self-organization at the network level is a common pathway
that cells employ to sense and respond to both stretch and
alterations in the ECM stiffness.
Here, we introduce a simple network model of actin-myosin
interactions that links active self-organization of the actin
network to the stiffness of the network and the traction forces
generated by the network. We demonstrate that such a network
replicates not only the effect of changes in substrate stiffness
on cellular traction and stiffness and the dependence of rate of
force development by a cell on the stiffness of its substrate
(50), but also explains the physical response of adherent cells
to transient and cyclic stretch (14, 17, 39, 71). Our results
provide strong indication that network phenomena governed by
the active reorganization of the actin-myosin structure plays an
important role in cellular mechanosensing and response to both
changes in ECM stiffness and externally applied mechanical
stretch.
METHODS
Model setup. We first consider a randomly oriented 2D network of
springs in a square of size L as illustrated in Fig. 1, which represents
the subcortical actin network of a cell. Actin filaments are known to
carry high tensile forces but can easily buckle under very low
compressive loads (76). To model such elastic behavior, each actin
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Parameswaran H, Lutchen KR, Suki B. A computational model
of the response of adherent cells to stretch and changes in substrate
stiffness. J Appl Physiol 116: 825– 834, 2014. First published January
9, 2014; doi:10.1152/japplphysiol.00962.2013.—Cells in the body
exist in a dynamic mechanical environment where they are subject to
mechanical stretch as well as changes in composition and stiffness of
the underlying extracellular matrix (ECM). However, the underlying
mechanisms by which cells sense and adapt to their dynamic mechanical environment, in particular to stretch, are not well understood. In
this study, we hypothesized that emergent phenomena at the level of
the actin network arising from active structural rearrangements driven
by nonmuscle myosin II molecular motors play a major role in the
cellular response to both stretch and changes in ECM stiffness. To test
this hypothesis, we introduce a simple network model of actin-myosin
interactions that links active self-organization of the actin network to
the stiffness of the network and the traction forces generated by the
network. We demonstrate that such a network replicates not only the
effect of changes in substrate stiffness on cellular traction and stiffness and the dependence of rate of force development by a cell on the
stiffness of its substrate, but also explains the physical response of
adherent cells to transient and cyclic stretch. Our results provide
strong indication that network phenomena governed by the active
reorganization of the actin-myosin structure plays an important role in
cellular mechanosensing and response to both changes in ECM
stiffness and externally applied mechanical stretch.
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Cytoskeletal Structure and Mechanosensing
filament is discretized as a set of springs of identical length l0a and
spring constant ka. These springs can exert a force that depends on
their length l as f ⫽ ⌰(l ⫺ l0a)ka(l ⫺ l0a), where ⌰(x) ⫽ 0 for x ⬍ 0
and 1 for x ⱖ 0 is the Heaviside step function. The end points of each
spring that makes up an actin filament also serves as discrete binding
sites where myosin crosslinks can attach. Each actin filament is
modeled as a polarized filament with a positive and a negative end and
was created by picking a point at random within the square and
moving along a direction ␸ uniformly distributed in the interval [0,2␲)
in discrete steps of l0a until a maximum distance of L was reached or
if the boundary of the square was reached, in which case the filament
is terminated at the boundary. The start point of each actin filament is
assigned a negative polarity and the end point is assigned a positive
polarity. To this network of actin, we add Nm linear elastic springs
with spring constant km and initial length l0m representing myosin
crosslinks (shown in red in Fig. 1), which attach to binding sites along
two distinct actin filaments that are separated by a distance less than
a specified threshold. At present, there are no experimental measurements of the stiffness of the nonmuscle myosin. Here, we used
measurements from striated muscle myosin to specify that km was 1.6
times higher than ka (1, 54, 77). To find binding sites for myosins
along the actin filaments, randomly chosen pairs of binding sites along
two neighboring actin filaments are checked to see if the distance
between them is less than or equal to l0m. If this condition is satisfied,
a myosin crosslink is added to link these two binding sites. The
number of actin filaments (Na) was specified to be approximately
equal to the number of myosin crosslinks (Nm). All initial networks
Parameswaran H et al.
were checked for end-to-end connectivity from top to bottom and
from left to right.
Boundary conditions. At every point along the square boundary
where an actin filament terminates, a linear elastic spring with spring
constant KECM is added to the network (shown in green in Fig. 1).
These springs represent the combined effective stiffness of the focal
adhesion complexes and the substrate material, the elastic boundary
for the actin network in adherent cells. All substrate springs connect
to an actin filament at one end while the other end is held fixed.
Motor movement. Within a living cell, nonmuscle myosin II motor
proteins use the energy from ATP hydrolysis to repetitively bind to
actin and undergo conformational changes that result in the myosin II
motor walking along the actin fiber toward its plus (barbed) end. In the
model, the walk step for a myosin crosslink is implemented by
moving each end of the myosin crosslink (as shown in Fig. 1, inset)
to the next binding site on the filament to which it is attached in the
direction of the filament’s positive end. The rate of ADP release in
nonmuscle myosin II has been found to be load dependent and
decreases with increasing loads (38). Consequently, the forward
velocity of myosin motors decreases with increasing load and when
the load on the myosin crosslink approaches the stall force (fstall), the
myosin takes backward and forward steps around the same location
(56). For an individual myosin the stall force has been estimated to be
⬃2 pN (2, 51). Additionally, direct measurement of the characteristic
lifetime of an actin-myosin bond using the optical trap method shows
that the bond lifetime ␶ increases with increasing load up to ⬃6 pN,
a behavior known as a catch bond. Consistent with these experimental
observations, initially, when the network carries no force, the time
interval ␶ between two consecutive walk steps of a myosin crosslink
is specified to be the same for all myosin crosslinks. In the model, we
specify that once the myosin motor starts walking and the network
develops tension, ␶ increases in proportion to the force (f) that each
individual crosslink carries, ␶ ⬀ f for f ⱕ fstall. The exact functional
relationship between ␶ and f is not known for NM2 motors. A linear
form is assumed here for simplicity. The dynamic in-place cycling of
the myosin motor at stall is not considered here. Instead, we specify
that when the load on a myosin crosslink exceeds fstall, that myosin
motor stops (␶ ¡ ⬁).
Calculating the equilibrium configuration. After each walk step,
the entire network rearranges itself to balance local forces within the
network. The new equilibrium configuration is determined by minimizing the total elastic energy of the network given by the sum of the
elastic energy of the springs representing actin, myosin, and the
substrate. A typical network considered in this study consisted of 55
actin filaments and 50 crosslinks and the corresponding spring network had 3,347 nodes and 3,292 springs. The simulated annealing
algorithm (37, 49) was used to minimize the energy of the network.
This technique uses a control parameter usually referred to as temperature (T) which is set to a high value initially. The system is
perturbed by moving every node by a small amount and the resulting
configuration is accepted based on the probability p ⫽ e⫺(⌬U/T), where
⌬U is the change in free energy associated with the perturbation.
These steps are repeated until the system reaches thermal equilibrium,
at which point the temperature is reduced. The equilibrium criteria for
the minimization were set based on the condition that both the
magnitude of the maximum resultant force for every node in the
network and the magnitude of the mean resultant force for every node
in the network decreases below an arbitrarily small value (here 10⫺5).
After calculating the equilibrium configuration, the clock increment
for the entire network was set based on the lowest myosin cross force
in the network. An individual motor stops when the force it carries
exceeds a fstall or when both ends of the myosin crosslink have
reached the positive end of their respective actin filaments. This
process of walking and rearranging the network continues until the
traction forces in the network do not change for 150 iterations.
Substrate stiffness and rate of traction force development. Cells are
known to modulate their traction force and the rate of force develop-
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Fig. 1. A: to create a prestressed actin network, we start with a set of actin fibers
(blue) that are randomly oriented within a square space as shown. B, inset:
binding sites for myosin crosslinks are added at regular intervals along each
actin filament. Myosin crosslinkers can attach to binding sites on two actin
filaments if the binding sites are closer than a specified distance. At points
where the actin filaments intersect with the boundary, elastic elements representing the substrate material (green lines) are added to the network and
represent the boundary conditions of the actin network. One end of the
substrate is attached to the actin and the other end is held fixed.
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Cytoskeletal Structure and Mechanosensing
RESULTS
Actin-myosin network self-organizes to achieve equilibrium
in traction and stiffness. Starting from a network of random
filaments as shown in Fig. 1, the myosin motors are allowed to
Parameswaran H et al.
827
walk along the actin filaments. After each walk step the
equilibrium configuration of the network is obtained. Figure 2A
shows the resulting changes in structure of the actin network
over time. The corresponding traction force generated by the
network and the network stiffness are shown in Fig. 2, B and
C, respectively. At t ⬃ 100, the network reaches equilibrium in
both the traction and the stiffness plateaus, and the number of
myosin motors that walk in each time step also reduced to an
insignificant fraction of the total. The development of traction
forces in Fig. 2B is similar to that measured in living cells
plated on flexible substrates (50). Further, in Fig. 2D, we plot
the changes in network stiffness against the changes in traction
force developed by nine different networks with different
initial configuration at time t ⫽ 0. We find that changes in
stiffness are linearly related to changes in traction force during
the reorganization of the network by myosin (Fig. 2D). While
the model cannot be used to match the absolute stiffness of a
living cell as this is not solely a function of actin and myosin,
our network model can explain why changes in stiffness and
traction force are highly correlated as the cell redevelops
tension following a mechanical perturbation (39).
Influence of substrate stiffness. Cells are known to modulate
their prestress and traction force in response to changes in
substrate stiffness (78). For a given cell type, changes in
traction force can be divided into two regimes. In the first
regime, the cell develops a traction force in proportion to the
change in substrate stiffness for stiffness values below a
threshold. Beyond this threshold, however, the traction force
exerted by the cell remains largely insensitive to further
changes in substrate stiffness (50). While experiments with
blebbistatin have revealed that actin-myosin interactions are
responsible for the generation of traction (46, 50), why the cell
exhibits two regimes of behavior remains unexplained.
To investigate the response of our actin-myosin network
model to changes in substrate stiffness, we vary the stiffness of
the boundary springs (KECM) and allow the network to reach
equilibrium starting from a random configuration. When the
Fig. 2. A: starting from a random configuration of actin filaments at time t ⫽ 0, the myosin II motors walk along the actin filaments, actively rearranging the
network and developing traction forces on the boundary springs. B: the total traction force corresponding to the networks shown in A. C: change in stiffness of
the network as it develops traction. The ratcheting motion of the myosin crosslinks leads to decrease in length of actin filaments that carry tension; therefore the
network also gets stiffer as the network develop tension. D: the change in network stiffness is plotted against the change in traction force developed by nine
different networks with different initial configuration at time t ⫽ 0. When the network develops internal tension, the changes in stiffness and traction are highly
correlated.
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ment in response to changes in substrate stiffness (78). To simulate
this behavior using our network model, we vary the stiffness of the
substrate springs KECM over 6 orders of magnitude (10⫺4 to 102) and
measure the maximum traction force that can develop as well as the
rate of development of traction force. The rate of force development
by the network was quantified using an equivalent velocity calculated
as V ⫽ d␦/dt, where ␦ is the average displacement of the substrate
springs.
External mechanical stretch and bond rupture. When the load on
an actin-myosin crosslink exceeds its rupture force, the myosin
crosslink can detach from the actin filament. The average rupture
force of an actin-myosin bond has been found to be 4 – 6 times higher
than its stall force (⬃2 pN) and is estimated to be between 9 and 12
pN (12, 55). Preliminary simulations showed that in a prestressed
network that has reached equilibrium in traction, the maximum force
on a myosin crosslink never exceeded 1.2 fstall so that when the
network develops traction starting from a random network, the bond
rupture does not happen. However, when a prestressed network is
subjected to external mechanical stretch, actin-myosin bonds can rupture.
When an actin-myosin bond ruptures in the network, both ends of the
bond detach and the myosin can diffuse through the network and reform
at another location in the network where the binding sites on two actin
filaments are closer to each other than the specified threshold distance for
bond formation.
Measurements. At each time point, we keep track of the number of
myosin motors that walked in that time step, the traction force, the
stiffness of the network, and the configuration of the network. The
total traction force is measured as the sum of the magnitude of forces
carried by the substrate springs. To measure stiffness of the network,
we apply a small deformation ␦r equivalent to 0.01% strain on the
network simultaneously in the x and y directions and then measure
the resulting change in magnitude of the elastic recoil force of the
network, ␦f. The stiffness of the network is then calculated as the ratio
of change in magnitude of the force over change in displacement
(␦f/␦r).
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Cytoskeletal Structure and Mechanosensing
substrate stiffness is increased by a factor of 10, the equilibrium traction force increases too; however, the time to achieve
equilibrium does not change (Fig. 3A), which is consistent with
experimental data reported by Mitrossilis and colleagues (50).
In Fig. 3B, we plot the equilibrium traction force achieved by
the network for a range of substrate stiffness values spanning
six orders of magnitude. We find that, just as in living cells, the
response of this active network to changes in substrate stiffness
is also characterized by two regimes. For low values of sub-
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Parameswaran H et al.
Fig. 3. A: increase in traction force when the substrate stiffness (KECM) is
increased by 1 order of magnitude. The network develops a higher traction in
stiffer substrates, but the time taken to reach equilibrium is almost the same in
both cases. B: the maximum traction that a cell can develop scales linearly with
substrate stiffness up to a threshold, beyond which the cell cannot generate
additional traction. C shows the percentage of myosin crosslinks that carry stall
force when the network has reached equilibrium. The two regimes of macroscopic scale behavior of the network traction force in response to changes in
substrate stiffness result from different microscopic scale stopping conditions.
D: substrate stiffness also modulates the rate at which the network can generate
traction force. For low values of substrate stiffness, the rate of force generation
also increases in response to an increase in substrate stiffness. In inset, we plot
the rate at which the network is able to deform the substrate (velocity), as a
function of the substrate stiffness. The solid line is a fit to the nonlinear Hill
force-velocity curve empirically observed in muscle tissue strips, which relates
the velocity V to the load against with the muscle constricts (in this case, the
substrate stiffness, KECM) as (KECM ⫹ a)(V ⫹ b) ⫽ c, where a, b, and c are
constants. Note that the x-axis (KECM) is plotted on a logarithmic scale as
opposed to a linear scale as commonly plotted in experimental studies.
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strate stiffness, the traction force scales almost linearly with
substrate stiffness up to a threshold value beyond which further
increase in substrate stiffness has no effect on the traction force
developed by the network.
To understand the origin of this network response, we
determine the number of crosslinks with a force larger than the
stall force after the network has reached equilibrium for different values of substrate stiffness. We find that around the
substrate stiffness that separates the two regimes, there is a
sudden increase in the number of myosin crosslinks that have
reached stall force. Thus the two regimes correspond to two
different ways of the network reaching its equilibrium configuration. In the first regime, as the network develops tension, the
substrate, which is very compliant, deforms and releases tension from myosin motors. Since the average distance available
for a myosin motor to walk is limited by the size of the
network, in the first regime, the myosin crosslinks never
achieve stall force and stop only when they reach the positive
(barbed) end of the actin filaments. However, in the second
regime, the stiffer substrate does not deform as much and the
myosin motors are able to reach stall force within the distance
available for them to walk. Thus the mechanism by which cells
develop tension and sense substrate stiffness is a network
phenomenon that cannot be understood solely by considering
molecular interactions of actin and myosin in isolation.
As noted in Fig. 3A, not only does the maximum traction
force scale with substrate stiffness, but also the rate at which
the network can develop traction also depends on the substrate
stiffness. Indeed as shown in Fig. 3D, the rate of development
of traction force also increases with substrate stiffness to a
critical threshold beyond which it does not increase significantly in response to further changes in substrate stiffness.
Mitrossilis et al. (50) examined experimentally this behavior in
living cells and found that the rate of force development in
response to a change in matrix stiffness followed the empirical
Hill force-velocity relationship observed in muscle tissue strips
(25), where the velocity of constriction was measured as the
rate at which the cell deformed the substrate material. This was
found to be true not only for muscle cells, but also for
nonmuscle cell types, suggesting a common mechanism by
which cells develop traction. As described in METHODS, an
equivalent velocity can be calculated as V ⫽ d␦/dt, where ␦ is
Cytoskeletal Structure and Mechanosensing
Parameswaran H et al.
829
following a change in substrate stiffness can also explain the
amplitude-dependent decrease and recovery back to baseline
following a transient stretch. It should be noted that our present
model of force-dependent actomyosin interactions with an
elastic network is consistent with the continuous-binding crosslinker model proposed by Donovan et al. (13) and is complementary to other proposed mechanisms such as force-dependent translocation of zyxin from the focal adhesions (11),
which depends on the tension generated by the actin-myosin
network.
Effect of cyclic stretch. Previous studies have shown that
when subjected to cyclic stretch, both muscle and nonmuscle
cell types show a substantial decline in traction force (14, 17,
43). The mechanisms that underlie the loss in traction due to
cyclic stretch are currently not understood.
To understand the effect of repeated stretch patterns on our
model, we started with a prestressed network (Fig. 5A) and
subjected it to repeated equibiaxial strain pulse of the same
amplitude every 60 s apart. In between the stretch pulses, the
network was allowed to recover its force. The observed
changes in the network structure are shown in Fig. 5, A–F, and
the concurrent changes in network force are shown in Fig. 5G.
The average drop in traction force after 10 stretch cycles (time
point F in Fig. 5G) was 24 ⫾ 4%. Concomitant with the
changes in traction force, we find that over repeated stretch
cycles, stable “star”-like junctions develop that persist through
further stretch cycles. These star junctions are created by
geometrical configurations in the network where the barbed
(⫹ve) end of different actin filaments are brought together by
myosin motors. The formation of star junctions is associated
with a decline in network force because force generation
depends on the ratcheting motion of myosin crosslinks along
actin filaments, and when myosins reattach close to a star
junction, the average distance available for a myosin crosslink
to ratchet up an actin filament decreases, reducing the network
force.
In a prestressed network at equilibrium, each individual
motor can stop either because the force it carries has reached
fstall or because the myosin has walked all the way to the ⫹ve
end of the actin filament. In the latter case, the myosin carries
lower force and is also under lower strain than a myosin that
has reached fstall. Thus myosins that have reached the ⫹ve end
of the actin filaments are less likely to rupture when the
network is stretched and junctions persist over repeated
stretches. Further, when a myosin crosslink ruptures it will
more likely reform close to a junction simply because the star
junction has several actin filaments (with binding sites for
myosin) in close proximity to each other.
DISCUSSION
Fig. 4. Effect of a transient stretch applied at time t ⫽ 0 on the traction force
exerted by the network. Following the stretch, rupture of crosslinks causes the
traction to immediately decrease. The loss of traction is an increasing function
of the amplitude of the stretch pulse. The loss of prestress causes the myosin
motors to activate; the higher the amplitude of the stretch pulse, the faster the
myosin motors walk. The characteristic time of recovery is related to the
average distance that a typical myosin motor walks.
Dysfunction in the ability of the cell to sense and respond to
mechanical stretch as has been shown to be a common factor
in many pulmonary diseases (18, 28), cardiac and vascular
diseases (63), renal and bowel disorders (58), as well as cancer
cell growth (47), cancer cell invasion (48), and epithelial-tomesenchymal transitions (23). However, the mechanisms that
underlie the ability of the cell to sense and respond to stretch
are still unclear. Several previous studies have examined this
problem in the context of molecular signaling pathways triggered at the focal adhesions (59, 62). However, recent studies
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the average displacement of the substrate springs. The computed force-velocity curve (Fig. 3D, inset) is well described by
a fit of the Hill equation: (KECM ⫹ a)(V ⫹ b) ⫽ c, where KECM
is the stiffness of the substrate material and a, b, and c are
constants (25). Huxley (29) explained the Hill force-velocity
curve in the context of striated muscle. Given the new finding
of Mitrossilis that even nonmuscle cells obey the Hill curve, it
is not clear if the Huxley model can be applied to understand
the behavior of nonmuscle cells. We find that our model of the
cortical actin network is able to reproduce dynamics of shortening with no extra constraints or tuning. Thus mechanical
interactions in the cortical actin network may be the reason
why nonmuscle cells exhibit properties such as the Hill forcevelocity curve, which was previously believed to be a unique
property of muscle cells and tissues.
Effect of transient stretch. Since our active network accounts
for the dynamics of traction development and shortening velocity, we hypothesized that the model is also able reproduce
the response of cells to transient changes in external mechanical environment. Two recent studies uncovered a novel response of living cells, called fluidization: when subjected to
transient stretch using a single equibiaxial strain pulse lasting
⬃4 s, cells promptly lose their traction and stiffness, followed
by an ATP driven recovery phase, with larger stretches leading
to larger drops in traction and a faster recovery rate (39, 71).
Furthermore, the response to transient stretch was universal
and not specific to cell type.
We test this hypothesis by considering a network configuration that has reached equilibrium in traction force. An externally applied stretch causes crosslinks with forces above the
rupture threshold to detach (Fig. 4). The crosslinks that detach
are then allowed to reattach at locations where the binding sites
along two distinct actin filaments are closer than a threshold
distance. The sudden drop in internal prestress releases the
tension from the bound myosin crosslinks. These crosslinks, as
well as the ones that newly attach to actin, can now walk and
start rebuilding the traction force. Consistent with observed
data in living cells (39), in our network model, the extent of
decrease in traction force depends on the magnitude of the
applied stretch, and the recovery to the baseline traction takes
place in more or less the same time for different stretch
magnitudes. Thus the same pre-stress-dependent mechanisms
of network rearrangement that drive the recovery in traction
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Cytoskeletal Structure and Mechanosensing
have shown that mechanosensory apparatuses of the cell are
not simply localized to the focal adhesions; rather they involve
large-scale interactions originating in the actin cytoskeletal
network (50, 72).
Several previous studies have used computational modeling
to show that myosin crosslinks can actively rearrange actin
networks to form self-organized structures (41). Such networks
of actin and myosin have been shown to have active contractility and viscoelastic properties consistent with those observed
in living cells (35, 36, 76). Recent studies have also shown that
actin-myosin networks plays an important role in sensing
substrate stiffness (6). In this study, we investigated whether
such networks can also explain the physical response of cells to
Parameswaran H et al.
mechanical stretch. To this end, we developed a simple network model of actin-myosin interactions and examined the
effect of substrate stiffness and stretch on the ability of the
network to modulate its tension and stiffness and compared our
modeling results to experimental observations in living cells.
Understanding the mechanisms by which the cell responds
to stretch is important especially in organs such as the lung,
which is constantly subject to fluctuations in stretch due to tidal
breathing. For instance, it has been shown that tidal breathing
and especially tidal breathing at elevated minute ventilation
antagonizes experimentally induced bronchoconstriction in
normal animals and people (10). Disrupting the actin-myosin
connectivity and reorganization of the actin-myosin contractile
machinery has been proposed by Lavoie and colleagues (44) as
a potential novel approach for preventing prolonged bronchoconstriction in asthma. Of particular importance is the fluidization experiment (Fig. 4) which mimics the effect of deep
breaths on cell stiffness and traction (39, 57, 71). In this study,
we propose a physical mechanism for the response of cells to
a single transient stretch (39, 71). When subjected to a transient
pulse in stretch, local ruptures occur in the actin-myosin
network, thereby reducing the prestress in the network. When
the prestress reduces, the myosin II motors activate and start to
rearrange the configuration of the network, and this process
increases the traction and stiffness of the network. The greater
the loss in prestress, the faster the motors will walk. This leads
to a faster rate of recovery of stiffness for larger-amplitude
transient stretches. All recovery curves converge in the same
time, and this is related to the average distance that a motor
walks in the network, which is approximately the same for a
given network size. In other words, the motors walk approximately the same distance but at velocities that are linked to the
prestress in the network.
The active rearrangement of actin by myosin is not the only
possible explanation for the experimental observations considered here. An alternative explanation based on dynamic
changes in focal adhesion stiffness is feasible. However, while
the size of, shape of, and stress on focal adhesions are known
to be highly dynamic (68), there are no experimental measurements of the stiffness of focal adhesions (or whether it might
vary with stretch or ECM stiffness). For this reason, we have
only considered static boundary conditions. Further, given the
critical role of actin binding and internal force provided by
actin myosin activity in the formation of focal adhesion (45,
80), it is unlikely that these explanations are independent from
the mechanisms considered here. It should be also be noted that
our present model is consistent with the continuous-binding
crosslinker model proposed by Donovan et al. (13) and the
force-dependent translocation of zyxin from the focal adhesions (11), which depends on the tension generated by the actin
myosin network.
An important mechanism that is not accounted for by the
model is the dynamics of actin polymerization in the cell
cortex, where actin has been reported to turn over with a
half-life of 20 –25 s (52). However, it is difficult to translate the
observed actin turnover rates into rate of change of stiffness or
traction forces using a computational model as the turnover of
actin is influenced by a multitude of proteins, which cap, sever,
depolymerize, polymerize, or crosslink the cortical actin network(19). At present, there is not enough information on the
effect of stretch on all these processes to reliably account for
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Fig. 5. Starting from a prestressed network shown in A, the network is
subjected to repeated equibiaxial strain pulses of the same amplitude every 60
s apart. In between the stretch pulses, the network was allowed to recover its
force. A–F: over repeated stretch cycles, “star”-like junctions (circles in D)
develop that persist through further stretch cycles. These star junctions are
created by geometrical configurations in the network where the barbed (⫹ve)
end of different actin filaments are brought together by myosin motors. G: the
formation of star junctions is associated with a decline in network force. The
red arrows correspond to the network configurations shown in A–F.
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Parameswaran H et al.
831
network is able to reproduce the response of living cells to both
stretch and substrate stiffness? Within a prestressed elastic
network, the local force on a myosin crosslink is a function of
the total prestress in the network and the configuration of the
network. Therefore, the elastic network of actin allows interactions at two different length scales. At the microscale, the
myosin motors are able to elastically interact with each other as
they move along actin filaments and generate prestress in the
network, and the macroscopic scale prestress feeds back to the
individual myosin motors and regulates their movement. Thus
the strong link between internal prestress associated with
mechanical sensing and adaptation to stretch and changes in
substrate stiffness are therefore emergent network phenomena
that result from microscopic molecular processes attempting to
maintain local force balance within a large elastic network in
response to an external stimulus such as changes in ECM
stiffness or stretch.
Of the many filamentous proteins that form networks within
the cytoskeleton, we focused on actin networks because of the
experimental observation that the recovery in stiffness of cells
and the recovery in traction force are highly correlated (39).
Active remodeling of the cytoskeleton was also observed
concurrent with these changes (71). While the absolute stiffness is dependent on other filamentous proteins in the cytoskeleton such as microtubules and intermediate filaments as
well as a multitude of crosslinking proteins (7, 20, 74), the
correlation between the change in stiffness and change in
traction force during recovery following a mechanical perturbation indicates that actin is the key contributor to the recovery
process. Indeed, our model provides a simple explanation as to
why stiffness and traction force should be correlated while the
cell develops traction. The ratcheting motion of the myosin
crosslinks leads to shorter filaments that carry an increased
tension and hence the stiffness also increases as the network
generates traction.
It has previously been noted by Bond and Somlyo (5) that
polarity reversal of actin filaments occurs at the site of dense
bodies in the smooth muscle cell and that this polarity reversal
is an essential feature of the basic contractile unit in a vertebrate smooth muscle. Our network model does not present any
contradiction with the Bond study. Even in the network, the
basic principle is the same: force generation upon myosin
movement depends on the polarity of the actin filaments and
also how the actin is anchored. The key difference, however, is
that the contractile machinery referred to in the Bond study is
the parallel contractile array unique to the smooth muscle. The
parallel array’s structure (the parallel alignment of actin-myosin fibers as well as the polarity of actin referred to by Bond
and Somlyo) makes it a perfect arrangement for directed force
generation as seen in the smooth muscle. The structure is so set
up that all myosin movement translates to force generation.
The same is not true for the cortical actin because in this
network, the actin can be found oriented randomly (except at
the edges of an adherent cell where the actin is found in
dendritic network structures). So, the cortical actin network can
generate tension, but not as efficiently (not all myosin movement results in force generation) as the parallel contractile
arrays found in muscle cell types.
The substrate springs that form the boundary conditions on
our network represent the effective stiffness of the substrate
material and the focal adhesions. While the size of, shape of,
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this in our model. Tinevez et al. (70) showed that tension in the
cortex of an unstretched cell can be modulated by altering the
level of proteins that regulate actin depolymerization such as
gelsolin and actin depolymerizing factor (ADF)/cofilin. The
same study also reported myosin-inhibitory experiments that
showed a very significant, systematic decrease in tension with
myosin inhibitor dosage. Clearly, both these processes act in
conjunction to regulate tension in the resting cell. However, the
relative importance of myosin activity vs. actin polymerization/depolymerization on force and stiffness of the cell is not
clear.
Biochemical signaling also plays a critical role in the response of the cell to stretch and substrate stiffness. Integrins
have been shown to act as mechanosensors with the cytoplasmic domain of ␤-integrin, which interacts with focal adhesion
kinase, which in turn interacts with other signaling molecules
to trigger many downstream signaling cascades (21, 62). Mechanical forces can also directly cause the activation of ion
channels and be felt by the nucleus through the cytoskeleton.
Thus forces can directly alter gene expression within seconds
to minutes following mechanical perturbation (30). Forces
transferred via the cytoskeleton can also cause nuclear pores to
open and affect the posttranscriptional control of gene expression (75). Indeed, the mechanisms based on the actin-myosin
network presented here are only one part of a complex mechanosensing and response machinery in the cell and act in
conjunction with these already established molecular signaling
pathways that have important mechanosensory roles (21).
Our simple network model is able to replicate the following
important experimental observations reported in living cells. 1)
The network actively adjusts its tension in response to a change
in substrate stiffness KECM up to a critical value beyond which
the network tension becomes insensitive to further increases in
KECM (50, 79). 2) The rate of development of traction by the
network is dependent on KECM and is consistent with the Hill
force-velocity equation (50). 3) The development of traction
force and development of cell stiffness are highly correlated
processes (11, 27). 4) When subjected to a transient stretch,
cells promptly lose their traction force and stiffness in a
stretch-dependent manner followed by a slow recovery back to
their baseline mechanical state. While the loss in traction and
stiffness are stretch amplitude dependent, the time of recovery
is the same, irrespective of the stretch amplitude. 5) This active
network provides a physical basis for the experimental observation that cyclic stretching leads to a decline in traction force
and stiffness (14, 17). To verify that our results were not due
to the specific choice of parameters, we performed additional
simulations in which we separately varied the stiffness of
myosin relative to actin (KM:KA) by one order of magnitude
from 0.4 to 4. The simulations in Figs. 2 and 3 were repeated,
and the model showed a strong correlation between network
force and stiffness as well as a two-regime traction force
behavior identical to that in Fig. 3B. The simulations in Figs. 2
and 3 were also repeated after varying the concentration of
actin to myosin (NA:NM) from 1:1 to 6:1 as reported in smooth
muscle, and our results did not change. Therefore, structural
reorganization of the actin cytoskeletal network by active
crosslinkers could be a common pathway by which cells
respond to both stretch and changes in substrate stiffness.
How is it that a simple mechanical model based purely on
movement of myosin crosslinks rearranging an elastic actin
•
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Cytoskeletal Structure and Mechanosensing
Parameswaran H et al.
has been observed in active gels of actin and myosin crosslinks
(4) and also predicted in other actin-myosin network models
(76). The relative influence of stretch-induced actin polymerization (24, 34) on the formation of actin structures predicted in
Fig. 5 is presently unclear. Although there is some evidence of
such structures forming in the actin cytoskeleton of epithelial
cells subjected to equibiaxial stretch (11), it is not conclusive,
and the formation of these star junctions due to repeated
stretches remains a prediction of the model.
A commonly reported feature of cytoskeletal remodeling
with change in substrate stiffness is the formation of polarized
bundles of actin commonly referred to as stress fibers. The
absence of stress fibers in our model should not be a limitation
for the present study. Here, we focused on the recovery in
traction force and cell stiffness following the application of
transient stretch or change in substrate stiffness, both of which
achieve equilibrium in ⬃100 s. In a recent study, AratynSchaus et al. (2) showed that within this time frame, substantial
traction is exerted prior to stress fiber formation. Thus the
major pre-stress-bearing structure within the time frame we
consider here is a network and not stress fibers. Furthermore,
while stress fibers do not form on cells plated in soft substrates,
cells do develop traction forces within the time frame considered here. Conversely, as we do not consider them, our model
cannot explain cellular phenomena that depend on stress fibers
such as shape changes in cells with substrate stiffness (81).
Signaling in the cell has traditionally been viewed as solution chemistry. However, our results suggest that mechanosensing and the response of the cell to alterations in its
mechanical environment should be considered a network phenomenon. An important application of these ideas could be to
study the force-dependent binding kinetics of downstream
proteins in mechanotransduction pathways. By being able to
predict changes in cytoskeletal geometry in response to stretch,
it is possible to study the spatial variability of biochemical
reactions whose kinetics are dependent on the local forces on
cytoskeletal elements in the cell. Therefore, the principles of
actin network reorganization presented here may be able to
provide more insight into how mechanotransduction is coordinated spatially and temporally within the cell.
ACKNOWLEDGMENTS
We thank Dr. D. Stamenovic, Dr. M. Smith, and Dr. K. Morgan for
discussion and critical comments.
GRANTS
This work was supported by National Heart, Lung, and Blood Institute
Grants RO1 HL-098976 and HL-096797.
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
AUTHOR CONTRIBUTIONS
Author contributions: H.P. conception and design of research; H.P. designed and implemented the computational model; H.P., K.R.L., and B.S.
analyzed data; H.P. interpreted results of experiments; H.P. prepared figures;
H.P., K.R.L., and B.S. drafted manuscript; H.P., K.R.L., and B.S. edited and
revised manuscript; H.P., K.R.L., and B.S. approved final version of manuscript.
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