SUPPLEMENTARY MATERIAL
Brownian dynamics using the Langevin equation
Displacement of each cylindrical segment is governed by the Langevin equation with
inertia neglected:
Fi   i
dri
 FiT  0
dt
(S1)
where ri is the position of either center point of ACPs or endpoint of segments constituting
actin filaments (F-actin) and motor backbones, ζi is an effective drag coefficient, t is time, and
Fi is a net deterministic force. FiT , a stochastic force, is determined based on the fluctuationdissipation theorem [1]:
2kBT  i ij
FiT  t  FjT  t  
t
δ
(S2)
where δij is the Kronecker delta, δ is a unit second-order tensor, and Δt = 2.3×10-6 s is time step.
For the cylindrical geometry of segments for actins, ACPs, and motors, we used the approximate
form of ζi [2]:
 i  3 rc,i
3  r0,i / rc,i
(S3)
5
where rc,i and r0,i are the diameter and length of a segment, respectively. Position of each
segment is updated using the Euler integration scheme:
ri (t  t )  ri (t ) 
dri
1
t  ri (t )   Fi  FiT  t
dt
i
(S4)
Deterministic forces
Harmonic potentials describe the extension and bending of actins, ACPs, and motors
with stiffness  s and  b , respectively (Fig. 1a):
1
1
1
2
U s   s (r  r0 ) 2 , U b   b    0 
2
2
(S5)
where r is a distance, θ is a bending angle, and the subscript 0 denotes the equilibrium value.
Extensional (  s,A ) and bending stiffnesses of F-actins (  b,A ) maintain an equilibrium length of
actin segments ( r0,A = 140 nm) and an equilibrium angle formed by adjacent actin segments (  0,A
= 0 rad), and we chose their values based on experimental measurements. Extensional (  s,ACP )
and bending stiffnesses of ACPs (  b,ACP ) keep an equilibrium length of ACP arms ( r0,ACP = 23.5
nm) and an equilibrium angle formed by two ACP arms (  0,ACP = 0 rad), and their values are set
to reasonable values. We assume that extensional stiffness of motor backbone (  s,M1 and  s,M2 )
that maintains an equilibrium length of the backbone ( rs,M1 = rs,M2 = 42 nm) is identical to  s,A ,
whereas its bending stiffness (  b,M ) keeping the backbone straight (  0,M = 0 rad) is much larger
than  b,A . Extension of each motor arm is governed by a two-spring model where a transverse
spring (  s,M3 ) maintains an equilibrium distance ( r0,M3 = 13.5 nm) between F-actin and the
endpoint of the motor backbone while a longitudinal spring (  s,M4 ) helps maintaining a right
angle formed by axis of the F-actin and a motor arm ( r0,M4 = 0 nm). Bending and extensional
forces acting on the binding spots due to interactions with ACPs or motors are distributed onto
the barbed and pointed endpoints as described in our previous work [3].
Repulsive forces due to volume-exclusion effects are computed following harmonic
potential depending on the minimum distance between a pair of actin segments, r12 [4]:
2
1
  r  r12  rc,A  if r12  rc,A
Ur   2
0
if r12  rc,A

where  r is strength of repulsive force.
2
(S6)
Dynamics of motors
We assumed that each of the arms attached to endpoints of the motor backbones
represents Nh myosin heads. Walking ( k w,M ) and unbinding rates ( ku,M ) of the arms are
determined by “parallel cluster model” (PCM) for given mechanochemical rates, step size, and
the number of heads [5,6]. Table S2 in Supplementary Material shows values for the major
parameters that we used for adopting PCM to our model. We assume that only forces acting on
the longitudinal spring ( Fs,M4  U s,M4 ) affect k w,M and ku,M . In our recent work [3], we
demonstrated that arms in a single motor are mechanically coupled; the motor with discrete Na
arms representing Nh myosin heads per each exhibits the force-velocity relationship and dwell
time corresponding to those of a myosin thick filament with NaNh heads predicted by PCM,
regardless of extent of coarse-graining (Nh).
Measurement of sustainability of stress
As explained in the main text, we calculated the sustainability of stress by dividing the
time average of σ(t) between a time point when σ is σmax and t = 100 s by σmax (i.e. S = <σ>/σmax).
We used data obtained only for initial 100 s for the calculation of S since actins and motors
formed severe aggregations under some conditions, causing quite slow computation. While σ
under most conditions reaches σmax at early time points, we observed some cases where σ reaches
σmax at time points close to 100 s, which might lead to overestimation of S. However, we found
that σ in such cases tends to be sustained for a very long time (data not shown) because the slow
stress buildup is attributed to high structural stability of networks (e.g. very high RACP).
Determination of ranges of RM and RACP explored in simulations
We probed the effects of a wide range of motor and ACP densities (RM = 0.0008-0.8 and
RACP = 0.001-0.2). The lower limit of RM corresponds to the density with a single motor thick
filament, and the lower limit of RACP is a critical point where percolation barely exists between
3
the two boundaries. By contrast, the upper limits of RM and RACP were determined somewhat
arbitrarily to avoid the saturation of binding sites located on actin filaments. Note that RM is
indicative of the density of myosin heads (NaNh), not motor arms (Na).
Changes in network morphology during stress generation and relaxation
When RM is varied at a fixed level of RACP = 0.1, mesh size of networks increases in
proportion to RM, and motors show minimal coalescence until σ reaches σmax because motors
temporarily act as cross-linkers in conjunction with force generation for a short period (Figs. 5c,
d). After σ ~ σmax, at RM > ~0.01 where S starts deviating from one (Fig. 5a), networks undergo
significant, additional increases in mesh size and motor aggregation. Since structural
reorganization of a significantly cross-linked network (RACP = 0.1) requires unbinding of
numerous ACPs, only high RM can increase mesh size and motor aggregation via the forceinduced ACP unbinding.
When RACP is changed with RM fixed at 0.014, an increase in mesh size is weakly
proportional to RACP at RACP < 0.1 until σ reaches σmax because ACPs help network deformation
by enhancing connectivity, but it is inversely proportional to RACP at RACP > 0.1 since too many
ACPs can make a network too rigid to deform (Fig. 6c). By contrast, motor aggregation does not
show any noticeable dependence on RACP (Fig. 6d). After σ ~ σmax, an increase in the mesh size is
maximal at intermediate levels of RACP. Since a given number of motors (RM = 0.014) can
activate the force dependence of ACP unbinding only below a certain level of RACP, the temporal
increase in mesh size shows inverse proportionality to RACP at RACP > ~0.01 from which S
abruptly rises (Fig. 6a). A discrepancy between RACP for the maximum increase in mesh size at σ
~ σmax and that at t = 100 s demonstrates that the force-induced destabilization of bonds between
ACPs and F-actins plays an important role largely for long-term structural reorientation of
networks. Motor aggregation is severe at RACP < 0.01 but becomes weaker with higher RACP at
RACP > 0.01, implying that motors can still aggregate without large changes in the mesh size at
low RACP (Fig. 6d).
An increase in network mesh size at σ ~ σmax is largely proportional to both RM and RACP
4
because network coarsening at this early stage occurs mostly via local deformation of F-actins
without large-scale network remodeling caused by numerous unbinding events of ACPs. The
local deformation is greater when a network is cross-linked sufficiently (high RACP) and subject
to large force (high RM). After σ reaches σmax, mesh size can drastically increase with large-scale
structural reorganization of networks if connectivity between F-actins is good enough, and if
force dependence of ACP unbinding is activated by high ratio of RM to RACP. Thus, the largescale network remodeling is maximized at intermediate RACP and relatively high RM. Motors do
not aggregate until σ reaches σmax since they walk and unbind less frequently due to generated
forces. Once the force-induced ACP unbinding becomes effective after σ ~ σmax, the motors lose
the generated forces and begin to aggregate severely.
Table S1 List of parameters employed in the model. Most of the parameter values are identical to
those used in our recent work [7]. Stiffness and dimension of motors are arbitrarily determined
0*
except κs,M4 [8]. Values for ku,ACP
and u,ACP are adopted from a single-molecule experiment [9]
Symbol
Definition
Value
r0,A
Length of an actin segment
1.4×10-7 [m]
rc,A
Diameter of an actin segment
7.0×10-9 [m]
θ0,A
Bending angle formed by adjacent actin segments
0 [rad]
κs,A
Extensional stiffness of F-actin
1.69×10-2 [N/m]
κb,A
Bending stiffness of F-actin
2.64×10-19 [N·m]
r0,ACP
Length of an ACP arm
2.35×10-8 [m]
rc,ACP
Diameter of an ACP arm
1.0×10-8 [m]
θ0,ACP
Bending angle formed by two ACP arms
0 [rad]
κs,ACP
Extensional stiffness of ACP
2.0×10-3 [N/m]
κb,ACP
Bending stiffness of ACP
1.04×10-19 [N·m]
r0,M1
Length of a bare zone of motor backbone
4.2×10-8 [m]
r0,M2
4.2×10-8 [m]
κs,M1
Length of a side segment of motor backbone
Bending angle formed by adjacent segments
constituting motor backbone
Extensional stiffness of a bare zone
κs,M2
Extensional stiffness of a side segment
1.69×10-2 [N/m]
θ0,M
5
0 [rad]
1.69×10-2 [N/m]
κb,M
Bending stiffness of motor backbone
5.07×10-18 [N·m]
r0,M3
Length of a motor arm
1.35×10-8 [m]
rc,M
Diameter of a motor arm
1.0×10-8 [m]
κs,M3
Extensional stiffness 1 of a motor arm
1.0×10-3 [N/m]
κs,M4
Extensional stiffness 2 of a motor arm
1.0×10-3 [N/m]
Nh
Number of heads represented by a single motor arm
8
Na
Number of arms per motor
8
κr
Strength of repulsive force
1.69×10-3 [N/m]
CA
Actin concentration
25 [μM]
RM
Ratio of motor concentration to CA
0.0008-0.8
RACP
Ratio of ACP concentration to CA
0.001-0.2
<Lf>
1.44 [μm]
Δt
Average length of F-actins
Time step
μ
Viscosity of medium
8.6×10-2 [kg/m·s]
0
ku,ACP
Zero-force unbinding rate coefficient of ACP
0*
0.115 [s-1] (= ku,ACP
)
u,ACP
Force sensitivity of ACP unbinding
1.04×10-10 [m]
Thermal energy
4.142×10-21 [J]
Young’s modulus of an elastic substrate
3.0×104 [Pa]
kBT
E
2.3×10-6 [s]
Table S2 List of parameter values employed to adopt “parallel cluster model” [5,6]
Symbol
Definition
Value
k01
A rate from unbound to weakly bound state
40 [s ]
k10
A rate from weakly bound to unbound state
2 [s-1]
k12
A rate from weakly bound to post-power-stroke state 1,000 [s-1]
k21
A rate from post-power-stroke to weakly bound state 1,000 [s-1]
k20
A rate from post-power-stroke to unbound state
20 [s-1]
F0
Constant for force dependence
5.04×10-12 [N]
Epp
Free energy bias toward the post-power-stroke state
-60×10-21 [J]
Eext
External energy contribution
0 [J]
d
Step size
km
Spring constant of the neck linkers
7×10-9 [m]
1×10-3 [N/m] (=  s,M4 )
6
-1
(a)
(b)
Fig. S1 Walking ( k w,M ) and unbinding rates ( ku,M ) of motor arms depending on force acting on
the arms. They behave as a catch bond, leading to lower k w,M and ku,M with higher applied
forces. Unloaded walking velocity is ~140 nm/s (= 7 nm× k w,M at zero force), and stall force
beyond which the arms stop walking is ~5.7 pN
7
(a)
(b)
(c)
(d)
Fig. S2 Viscoelastic properties of networks depending on frequency and time. (a-c) Phase delay,
tan-1(E”/E’), between applied strain and measured stress measured at three different time ranges
with three sets of RM and RACP. A legend in (a) is shared with (b) and (c). Cyan inverted triangles
show the phase delay of cases with the same sets of RM and RACP without ACP unbinding.
Perfectly elastic and viscous materials exhibit the phase delay of 0̊ and 90̊, respectively. (d) E’ at
1 Hz vs stress (σ) measured at t = 10-60 s with three sets of RM and RACP. Brighter symbols
represent early times while darker ones represent later times
8
(a)
(b)
(c)
(d)
Fig. S3 Influences of RM and RACP on (a, c) σmax and (b, d) S. In (a, b), RM is varied while RACP is
fixed at the reference value (red circles, RACP = 0.1) or other values (other symbols shown in the
legend). In (c, d), RACP is changed while RM is held at the reference value (red circles, RM = 0.014)
or other values (other symbols shown in the legend). Note that the legend in (a) is applied to (b),
and the legend in (c) is shared with (d)
9
(a)
(b)
(c)
(d)
0
0*
Fig. S4 Importance and effects of ku,ACP
/ ku,ACP
on (a, c) σmax and (b, d) S under the reference
condition (red circles) or with a change in a single parameter value (other symbols shown in the
legend). Note that the legend in (a) is applied to (b), and the legend in (c) is shared with (d)
10
(a)
(b)
Fig. S5 Influences of RM and RACP on (a) maximum, σmax, and (b) sustainability of the stress, S,
evaluated using a cubical actomyosin network (3×3×3 μm). Values of other parameters are
identical to those used for thin cortex-like networks
11
Movie S1 Morphology and force distribution during contraction of a cross-linked actomyosin
network with RM = 0.014 and RACP = 0.1 for 100 s against the elastic substrate (E = 30,000 Pa).
Cyan, red, and yellow represent F-actins, motors, and ACPs in the right column, respectively
while blue, white, and red represent low, intermediate, and high forces in the left column,
respectively. Thick white lines at edges indicate the location of boundaries
Movie S2 Morphology and force distribution during contraction of a network with RM = 0.014
and RACP = 0.01 for 100 s. Only a difference compared to Movie S1 is the value of RACP
Movie S3 Morphology and force distribution during contraction of a network with RM = 0.14 and
RACP = 0.1 for 100 s. Only a difference compared to Movie S1 is the value of RM
12
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