arXiv:2306.10949v1 [physics.bio-ph] 19 Jun 2023
Cooperation of myosin II in muscle contraction through nonlinear elasticity
Beibei Shen and Yunxin Zhang
∗
Shanghai Key Laboratory for Contemporary Applied Mathematics,
School of Mathematical Sciences, Fudan University, Shanghai 200433, China.
(Dated: June 21, 2023)
Myosin II plays a pivotal role in muscle contraction by generating force through the cooperative
action of multiple motors on actin filaments. In this study, we integrate the nonlinear elasticity of the
neck linker in individual myosin II and comprehensively investigate the evolution of cooperativity and
dynamics at microstate and mesostate levels using a combined model of single and multiple motors.
We find that a substantial proportion of actin-b ou nd motors reside in the mid- and post-power stroke
states, and our nonlinear model reveals their increased capacity for load sharing. Additionally, we
systematically explore t he impact of mechanical load and ATP concentration on myosin II motors.
Notably, we observe that the average net distance of actin undergoes a transition from a weak load-
sensitive regime at low ATP concentrations to a load-sensitive regime at higher ATP concentrations.
Furt hermore, increasing the load or raising the ATP concentration to saturation can enhance the
efficiency and output power of myosin filament. Moreover, the efficiency of the myosin filament
increases with the power stroke strength, reaching a maximum at a specific range, and subsequently
declining b eyond that threshold. Finally, we explore the mean run time/length and mean existence
probability of myosin filament, shedding light on its overall behavior.
I. INTRODUCTION
In cells, molecular motors such as myosin, kinesin, and
dynein play essential roles in powering cellular activities, and
their coordinated efforts are critical for the proper function-
ing of various biologic al systems, including muscle contrac-
tion, vesicle transport, and spindle formation [
1–4].
The myosin II family, encompassing skeletal, cardiac,
smooth, and non-muscle myosins, can interact with a c tin fil-
aments, and ultimately re sult in actin filaments sliding and
macroscopic force production [
5, 6]. The mechanical pro p-
erties of myosin II have been studied at the sing le -molecule
level extensively, with a predominant focus on the optimiza-
tion as an individual motor [
7–12]. However, the generation
of mus cle contraction requires the coordinated efforts of mul-
tiple myosin II motors to pull a c tin fila ments [
13–15]. In this
context, myosin II interacts with actin filaments not as in-
depe ndent force generators, but rather as cooperative force
generators [
16–18].
Recently, the collective behaviors of multiple myosin II
in vitro have been studied deeply to understand how indi-
vidual motor functions are integrated into collective motor
systems and contribute to the ove rall performance of coop-
erative motor systems [
19–24]. But so far the coop e ration
mechanism among multiple myosin II in muscle contraction
remains elusive. Meanwhile, the system of myosin II might
have been evolved with distinct mechano-kinetic properties,
which maximize output power or energy efficiency. These
properties remain incompletely characterized and warrant
further investigation.
To fully understand the collective behaviors of motor
proteins, it is crucial to consider both the single-molecule
properties and the intricate mechanochemical and/or chem-
ical interactions among motor proteins [
25, 2 6]. In this
study, we will try to discover the coop eration mechanism
among myosin II motors by taking both the microstate a nd
mesostate of the myosin II system into account. In view of
this, we will employ a model that allows for the detection
∗
Email:
and analysis of the dynamic behavior of individual myosin
II, as well as their ens emble effects. Different with previ-
ous study, our model incorporates the nonlinear elasticity
of the neck linker in myosin II, which closely approximates
the mechanical behavior of myosin during muscle contraction
[
9, 13]. Furthermore, we investigate the intricate effects of
mechanical load and ATP concentration on the functionality
of myosin II system.
II. THEORETICAL DESCRIPTION OF MUSCLE
CONTRACTION
A. The mechanochemical cycle of single myosin II
According to previous models of actomyosin [
5, 13, 14, 27],
we describe the work cycle of myosin II by a mechanochem-
ical model with five states, see Fig.
1. The cycle begins
with the unbound, recovered state, where the motor head is
loaded with ADP and phosphate (Pi), and the lever arm is
in a primed conformatio n. Notably, the head of myosin II
possesses a lever a rm that amplifies even the s lightest con-
formational changes resulting from the binding or dissoci-
ation of var ious ligands at the nucleotide binding site [
28].
Subsequently, the head of myo sin II binds to actin in a pre-
power stroke state, which is a weakly bound state and does
not generate force [
11]. Once the head attaches to a ctin,
the rapid release of Pi triggers a substantial rotation, known
as the “power stroke”, of the lever arm [
9, 28]. As the le ver
arm swings to its stretched conformation, the motor executes
power stroke, thereby increasing the strain on neck linker and
enhancing the force generated by the motor. Throughout ex-
ecution of power stroke, the lever arm continues to rotate,
pulling actin filament forward a gainst the external load F .
During power stroke, the lever arm swings forward by a
distance of d = 8 nm [
28, 29]. The power stro ke co mprises
two distinct stages, marked by transitions towards mid-power
stroke and post-power stroke states. During the first stage,
the motor strain elo ngates by d
′
nm, while in the second
stage, the strain elongates by d − d
′
nm. Generally, the
elongation of the strain in the first stage is slightly greater,
ranging from 4 to 8 nm [
14].
2
FIG. 1: The mechanochemical cycle of single active myosin II.
The myosin head incorporates a lever arm and is connected to
the backbone of myosin filament via a neck linker. For simplicity,
we draw the lever arm and myosin head together as an oval shape.
Actin filaments are represented by a double helix structure. The
cycle of interaction between the myosin and actin starts from
the recovered state. During ATP hydrolysis, the myosin binds to
actin, pulls actin against the external load, and then detaches.
The lever arm and neck linker play crucial roles in this process,
allowing the myosin to undergo conformational changes and per-
form the power stroke, which drives the movement of actin fila-
ments. ω is the transition rate of motor between two states.
After power stroke, ADP dissoc iates from myosin II with
a slow rate [
18, 30], which is immediately followed by the
binding of a ATP molecule. The binding of ATP destabi-
lizes the acto myosin interaction, causing the myosin head to
dissociate from actin filament and complete the cycle [
28].
Finally, along with ATP hydrolysis, myosin II returns to the
beginning of the cycle, that is, to the recovered state again.
As illustrated in Fig.
1, the transition rates betwee n these
five states are influenced by both ATP concentration [ATP]
and external load F . Specifically, the transition rates ω
on
,
ω
′
pre
and ω
′
mid
are solely dependent on the concentration of
ATP, w hile the rate ω
′
off
is exclusively influenced by load
F . The remaining transition rates ω
pre
, ω
mid
, ω
off
and ω
′
on
are affected by both ATP concentration and load F . The
cycle of a single myosin II motor is actually mirrored by the
cycle of ATP hydrolysis. The specific expressions for each
transition ra te can be found in Sec tion E of the Supporting
Material.
We should also focus on the change in basic free energy
across the five transitio ns. From the recovered state to the
pre-power stroke sta te, the basic free energy of a myosin II
motor undergoes a change of g
on
. When moving clockwise
around the cycle fr om the pre-power stroke state, the basic
free energy undergoes changes across the other transitions,
which are labeled as g
ps
/2, g
ps
/2, g
off
, g
recovery
. Due to the
changes in conformational free energy sum to zero across
a full cycle, the sum of these components is e qual to the
free energy gained from ATP hydrolysis: g
on
+ g
ps
+ g
off
+
g
recovery
= −∆µ
ATP
, where ∆µ
ATP
≥ 0, as it represents the
chemical potential energy of the system. The value of ∆µ
ATP
is dependent on the concentration of ATP. The formula is
∆µ
ATP
= 25 + ln
[ATP]
2000
. (1)
As ATP concentration increases, ∆µ
ATP
also increases. At a
standard concentration of [ATP] = 2000µM, the free energy
change of ATP hydrolysis is 25 k
B
T , where k
B
T is used as
the unit of energy, k
B
is Boltzmann’s constant and T is the
absolute room temperature.
B. The cooperation mechanism among multiple
myosin II motors
The fundamental building block of muscle tissue is half-
sarcomere, which is composed of overlapping sets of myosin
and actin filaments [
19]. In musc le contraction, multiple
myosin motors bind and output mechanical force on actin
filaments cooperatively to produce relative sliding b etween
myosin and actin filaments [
14, 19, 20].
FIG. 2: Schematic of N myosin II motors interacting with actin
filaments. The external load F pulling to the right is balanced
by forces in neck linkers of myosin II motors. The spatial interval
between each two motors on the backbone of myosin filament is
fixed at 14.5 nm [
19]. In muscle contraction, myosin II motors
must act cooperatively. To increase the possibility of coordinated
power stroke, motors need t o be in pre-power stroke state or mid-
power stroke state transiently. In our model, we assume that
among t he N bound myosin II motors, there are n
pre
myosin
II m otors in pre-power stroke state, n
mid
motors in mid-power
stroke state, and the other n
post
ones in post-power stroke state.
For more details of the coordinated microstate vector cycle, see
Fig. S2 in the Supplemental Material.
Multiple myosin motors are arranged into well-organized
sup erstructures, see Fig.
2. In the structures, a vast as-
sembly of myosin functions collectively as a single functional
unit, operating in a coordinated manner to facilitate effi-
cient relative sliding of actin and myosin fila ment [
19]. The
backbone of myosin filament displays a pattern of myosin
distribution, with clusters of myosin emerging at fixed 14.5
nm intervals [
19]. A myosin half-filament is found to have a
total of N
0
= 294 myosin motors, each of which can be either
active or inactive. The number of active myosin motors N
T
depe nds on load F , see Eq. (S3) [
20].
The mechanochemical cycle of each active myosin motor
consists of five possible states. Myosin motor in pre- , mid-,
or post-power stroke state binds to actin filament, while the
one in detached or recovered state is separated from actin fil-
ament, as shown in Fig.
1. We consider both the microstate
and the mesostate of this myosin system. The microstate is
described by a vector n = (n
pre
, n
mid
, n
post
, n
det
, n
rec
), where
the subscript indicates the mechanochemical s tate and n is
the number of myosin motors in that state. The mesostate
is described by the number N of myosin motors bound to
actin filament (N = n
pre
+ n
mid
+ n
post
). So the num-
ber of active myosin motors detached from actin filament
is N
T
− N = n
det
+ n
rec
.
3
Molecular pro perties of myosin II are specifically tuned
to perform cooperative force g eneration for efficient muscle
contractions [
13, 14, 20]. The bound myosin motors in power
stroke states transit in a specific order. To describe the mech-
anism of multiple myosin motors which act cooperatively, we
define the fo llowing sequence of vectors in microstate cycle,
n = ( n
pre
, n
mid
, n
post
, n
det
, n
rec
) ,
n
′
= (n
pre
, n
mid
, n
post
− 1, n
det
+ 1, n
rec
) ,
n
′′
= (n
pre
, n
mid
, n
post
− 1, n
det
, n
rec
+ 1) ,
n
′′′
= (n
pre
+ 1, n
mid
, n
post
− 1, n
det
, n
rec
) ,
n
′′′′
= (n
pre
, n
mid
+ 1, n
post
− 1, n
det
, n
rec
) ,
(2)
with the microstate vec tor cycle given as (see Fig. S2)
n
′′
→ n
′′′
→ n
′′′′
→ n → n
′
→ n
′′
.
During each cycle, o ne ATP molecule is hydrolyzed, and the
actin filament is pulled forward with a certain distance. In
the following, we w ill describ e the microstate cycle in detail.
We assume the microstate cycle starts from n
′′
, and the
actin filament is bound with N − 1 myosin motors. We give
each bound myosin an index from left to right, indicating
their o rder of duration bound with actin filament, where
i = 1 represents the one most recently bound, and therefore
in the pre-power stroke state, while i = N − 1 represents the
first bound one, and most probably in the post-power stroke
state, see Fig. S2.
Subsequently, a n additional motor binds to actin in the
pre-power stroke state, thereby increasing the number of
bound motors to N. This transition in the cy cle leads to
microstate n
′′′
.
To increase the likeliho od of coordinated power strokes,
multiple myosin motors must br iefly “stay” in either the
pre-power stroke or mid-power stroke state [
14]. Com-
pelling evidence from recent e xperimental results suggests
that the stepwise displacements of actin are prima rily gener-
ated through the coordination of power strokes among two
myosin motors [
14]. According to myosin motors transition
in a first- in, first-out manner [13], the transition from n
′′′
to
n
′′′′
corres ponds to motor i = n
pre
completing the first half
of the power stroke and transitioning to the mid-post power
stroke state.
During the firs t stage of the power stroke, the strain of
motor i = n
pre
elongates by d
′
nm, simultaneously exerting
an increasing force. This disrupts the initial force bala nc e
relationship, resulting in the actin filament being pulled to
the forward by a distance ∆x
1
.
After that, the transition from n
′′′′
to n corresponds to
motor i = (n
pre
+ n
mid
) completing the second stage of the
power stro ke and entering the post-power stroke state. Dur-
ing this phase, the rotation of the lever arm causes the strain
of the n
pre
+ n
mid
-th motor to elongate by d − d
′
nm. Sim-
ilarly, motors pull the actin filament to the forward by a
distance ∆x
2
.
Then, the release trans ition from n to n
′
corres ponds to
the unbinding of the i = N motor. When a post-power stroke
state motor detaches from the actin filament, the force ex -
erted by the bound motors suddenly decreases, breaking the
previous force equilibrium. As a result, the actin filament
slips backward by a distance of ∆x
slip
, causing the strain of
the other motors to be stretched until this remaining N − 1
motors are equilibrated with the load F a gain. This process
is similar to when someone quits a tug-of-war, the rope im-
mediately slides in the direction of the opposing team. See
Fig. S2 for illustration.
Finally, a motor in the detached s tate transitions to the
recovery state, which corresponds to the cycle transitioning
from n
′
to n
′′
, ultimately bringing the cycle back to n
′′
.
After such a cycle, the size of the net distance moved by
the actin filament is ∆x
net
= ∆x
1
+ ∆x
2
− ∆x
slip
. Here we
only provide a simplified expression for ∆x
net
for the sake
of clarity, as shown in Fig. S2. The specific formula with
detailed elements can be found in the Method Eq. (
10) and
Section G of the Supplemental Material.
C. Nonlinear elasticity of myosin II
In contrast to other models [
13, 21, 28, 31, 32], we pro-
pose that the mechanism of genera ting force during posi-
tively strained and negatively str ained neck linker is differ-
ent, and that the neck linker is not simply approximated
as a linear spr ing model. Actually, experiments measuring
the elastic properties of myosin motors for a wide range of
positive and negative strains demonstrate tha t myosin mo-
tors exhibit nonlinear elasticity [
9]. Taking account of the
nonlinear nature of myosin motor elasticity is essential to
relate myosin’s internal structural cha nges to physiological
force generation and filament sliding.
If myosin motors are under positive strain, they are con-
sidered to be “stretched”. In such cases, the neck linker can
be approximated as a worm-like chain (WLC) model [
33–
35]. On the other hand, if motors are under neg ative strain,
they are referred to as “drag” motors, exhibiting sig nificantly
lower stiffness compared to positively strained myosin mo -
tors. Low stiffness minimizes drag of negatively strained
motors during muscle contraction at loaded conditions [
14].
The nonlinear elasticity implies that active myosin motors
with high stiffness is primar ily responsible for the forward
step generation, w hereas the drag myosin moto rs with low
stiffness does not contribute to forward [
9].
The force-e xtension relationship of a bound motor i de-
pends on the strain, y
i
, as follows:
f(y
i
) =
k
B
T
L
p
1
4(1 − (y
i
/L
C
))
2
−
1
4
+
y
i
L
C
, y
i
> 0,
k
m
γ
[exp(γ · y
i
) − 1], y
i
≤ 0.
(3)
where y
i
represents the strain of motor i, and f(y
i
) is a con-
tinuous function with continuous derivatives. The constant
k
m
is defined a s 3k
B
T/ (2L
p
L
C
). The dimensionless constant
γ is a parameter that characterizes the nonlinearity of the
motor’s response. The quantities L
C
and L
p
represent the
contour length and persistence length of the motor chain,
respectively, and they are both p ositive. The ratio y
i
/L
C
corres ponds to the extension of the motor, and it takes val-
ues in the range (0, 1).
Fig. S1(a) shows the no nlinear elastic force of the b ound
myosin motor i as a function o f the extending ratio y
i
/L
c
.
When y
i
< 0, the slope of the curve is k
m
exp(γ · y
i
) > 0,
which decreases and becomes flatter as γ increases. As y
i
becomes more negative, the r e sulting force f(y
i
) rapidly ap-
proaches a negative constant −k
m
/γ, and the magnitude of
this constant | − k
m
/γ| decreases with inc reasing γ, indicat-
4
ing that the force f (y
i
) decreases with increasing γ. When y
i
is pos itive and approaches zero, the behavior o f the motor’s
neck linker is similar to that of a linear spring. However,
as y
i
increases, the no nlinearity of the neck linker becomes
more prominent. In contrast to the linear model, our non-
linear model reveals that motors in the mid- and post-power
stroke states effective ly distribute a larger load.
When N motors attach to actin filament, the force e xerted
by these motors should be equal the exter nal load F ,
N
X
i=1
f(y
i
) = F. (4)
Eq. (
4) illustrates the relationship between the external lo ad
F and the forces generated by all motors. This force s en-
sitivity enable s any myosin motor to coordinate its actions
with other bound myos in motors. As the external load F
increases, a greater number o f motors are needed to counter
balance it. This phenomenon is demonstr ated by the exper-
imental results shown in Fig. 3(b).
We calculate the change of motors’ strain through mathe-
matical derivation based on the first-in, first-out transitions
of myosin motors. Meanwhile, we observed a self-consistent
strain distribution at the end of microstate vector cyc le,
which was identical to that at the beginning of microstate
cycle except for an increase in the indices of all bound mo-
tors by one and a movement of actin filament by a certain
distance. The detailed derivation of which is presented in
Section C of Supplemental Material.
Finally, we find tha t each microstate n
′′
has a unique set of
strain values, which enables us to capture the distribution of
strain acr oss the bound motors. Specifically, we can describe
the strain of the bound motor i as follows:
y
i
(n
′′
, F ) = i ∆y(n
′′
, F ) + d
i
(n
′′
), (5)
where i = 1, · · · , N − 1, and d
i
(n
′′
) = 0, d
′
and d for a motor
in the pre-power stroke, mid-power stroke, and post-power
stroke states , respectively. We assume that the strain on a
pair of consecutive motors differs by a constant ∆y(n
′′
, F ).
III. RESULTS
A. Biophysics of myosin motion along actin filament
We parameterize our model based on data from Piazzesi
et. al. [
19], which provides information on the velo c ity V ,
the average number of bound motors hNi, and the sliding
distance L as functions of the external load F .
Using parameter values listed in Tab.
I, our model suc-
cessfully reproduces the relevant experimental r e sults shown
in Fig.
3. Detailed methods of theoretical prediction are
presented in Method section and the Section G of Supple-
mental Material. In the subseque nt sections, we will con-
sistently refer to the pa rameter values from Tab.
I to in-
vestigate the mechanochemical properties of myosin II. In
Fig.
3, the myosin motors are assumed be in biological en-
vironment with ATP concentration of 2000 µm. In Fig. S8,
we provide predicted curves of myosin II proper ties versus
load F at different ATP concentrations. Next, we will ana-
lyze the effect of load F on myosin II motors by combining
the information presented in both Fig. 3 and Fig. S8.
FIG. 3: Theoretical predictions (solid lines) and experimental
data (markers) of various biophysical properties of myosin II from
muscles of frogs. The data in ( a-f) are from the study by Piazzesi
et. al. [
19]. The data in (a-c) are the original data used for model
fitting, while the data in th e (d-f) are obtained from the rela-
tionship between hNi, F/ hNi, V , L and F . V is the velocity
of the actin filament. In (b), the left axis is for average number
of myosin II motors bound to actin hNi, while t he right axis is
for force per attach ed motor F/ hNi. L is the sliding distance
of actin filament. Theoretical results are obtained from formu-
lations given in Eqs. (
8), (9) and (6), with model parameters
listed Tab.
I. The depicted data in this figure corresponds to a
physiologically ATP concentration of 2000 µM.
Fig.
3(a) demonstrates a monotonic decrease of velocity
with increasing load F , consistent with the downward con-
vex velocity-force rela tionship at constant [ATP] described
by the Hill relation [
36]. On the other hand, the decrease in
ATP concentration from the mechanosensitive regime, oc-
curring at near-vanishing load, leads to a rapid reduction in
velocity, as shown in Fig. S8(a). However, as the load in-
creases, the rate at which velocity decrea ses gradually slows
down. This phenomenon has been thoroughly investigated
in skeletal muscle and previously ex amined in motility assays
[
37].
With the increases of load F , the number of bound mo-
tors hN i exhibits an increasing s ensitivity to variations in
ATP conc entration, see Fig. S8(b). The load F is shared
by all myosin motors bound to actin filament. Fig.
3(b)
shows that both the average number hNi of myosin motors
bound to actin filament and the average force shared by each
myosin, defined as F/ hN i, increase with load F . We may
need to point out that, the load shared by each myosin motor
are actually different, and can be obtained by Eq. (4).
Next, we introduce the sliding dista nce L of actin filament.
5
During each microstate cycle , the actin filament is displaced
by a net distance ∆x
net
, which can be referred to as the net
step size of the actin, and we denote h∆x
net
i as average net
distance. So the output work of the total myosin ensemble is
F ∆x
net
. As in [
19], the sliding distance L of actin filament
is calculated as work divided by the force p er motor F/N.
This motivates the relation
L = hN∆x
net
i . (6)
The distance that the actin filament slides from one motor
attachment to detachment can be interpreted as the sliding
distance. In other words, L represents the distance that the
actin filament slides while a motor remains a tta ched to it
[
13, 19].
The sliding distance L gradually decreases with incr e as-
ing load F , as shown in Fig.
3(c). However, the load F
has a limited effect on the sliding distance L, which remains
around 6 nm at large loads and 8 nm at small loa ds. On the
other hand, when the ATP concentration exceeds the phys -
iologica l level of [ATP]=2000 µm, h∆x
net
i becomes almost
insensitive to ATP concentration. When the load F is small,
h∆x
net
i decreases sharply, demonstrating a notable sensitiv-
ity to small loads, as depicted in Fig. S8(d). As the load F
increases, its effect o n h∆x
net
i diminishes.
According to the relationship betwee n V, hNi , F/ hNi , L
and F , we can naturally get Figs.
3(d-f). Specifically, a s the
load F increases, Figs. 3(d,e) demonstrate that both the
numbe r of bound motors hNi and the force per motor F/ hNi
exhibit opposite trends to velocity. In contrast, Fig.
3(f)
demonstrates a consistent trend between the sliding distance
and the velocity.
Since these biophysical quantities are related not only to
the magnitude of external load F but also to the ATP con-
centration, we provide the effect on each biophysical quantity
as the AT P concentration varies, as shown in Fig. S7(a).
The velocity increases with increasing ATP concentration
because the motor cycle is accelerated, Fig. S7(a). At very
low ATP concentrations, the velocity is primarily determined
by the slow detaching from the actin. When the concentra-
tion of ATP exceeds 10
4
, the rate of velocity ascent exhibits
TABLE I: Model parameter values obtained by fitting to experi-
mental data of myosin II purified from muscles of frog measured
in [
19], see Fig. 3. The g
recovery
is limited by the presence of other
parameters, resulting in the equation: g
on
+g
ps
+g
off
+g
recovery
=
−∆µ
ATP
.
Parameter Range tested Fitting value
g
on
−10 to 0 k
B
T [13] −5.637 k
B
T
g
ps
−20 to −15 k
B
T [
13] −17.438 k
B
T
g
off
−10 to 0 k
B
T [
13] −3.499 k
B
T
g
recovery
− 1.574 k
B
T
ω
on
0.01 − 10 s
−1
[
13] 0.178 s
−1
ω
off
0.1 − 100000 s
−1
[
13] 194.009 s
−1
d
′
4 − 8 nm [
14] 4.187 nm
L
p
0.1 − 1 n m [
33–35, 38] 0.162 nm
L
c
9 − 40 nm [
33–35, 38] 24.561 nm
k
pre
0 − 10000s
−1
[
14] 5.03E−06 s
−1
k
mid
0 − 10000s
−1
[
14] 0.714 s
−1
γ 0.1 − 1 [
9, 14] 0.192
a tendency towards attenuation and velocity tends to flatten
out, although this is not shown in the Fig. S7(a).
As ATP concentration increases, the average number of
bound motors hN i decreases and becomes weakly dependent
on load at high ATP concentrations. This means that the
effect of load F on hNi gradually decreases with increa s-
ing [ATP]. When the concentration of ATP decre ases to
1 µm, the value of hNi tends to approach N
T
. However,
as ATP concentration approaches saturation, the downward
tendency of hNi slows and e ssentially stabilizes, as shown in
Fig. S7(b). This is mainly because the inc rease in [ATP]
directly leads to an increa se in the rate ω
off
, which further
promotes the detachment of the motor from the actin fila-
ment, resulting in a decrease in hN i.
The increase in force per motor F/ hNi is a result o f the
decrease in hNi with increasing ATP concentration, while
the e xternal load F is fixed. Moreover, the rate of increase
in F/ hN i is initially sluggish and then accelerates rapidly,
as depicted in Fig. S7(c).
The average net distance h∆x
net
i increases with ATP
concentrations, but passes through a ma ximum and then
decreases with further increasing [ATP], as shown in
Fig. S7(e). h∆x
net
i reaches its peak after the ATP con-
centration reaches 2000 µm. However, the degree of this
increase in h∆x
net
i is reduced as F increases, as shown in
Fig. S7(e). At low ATP concentrations, the effect of the
external loa d F on h∆x
net
i is not significant, and the values
of h∆x
net
i are small. As the ATP concentration increases
and reaches saturation, the influence of F on h∆x
net
i be-
comes increasingly ev ide nt. Sp ecifically, as F increases from
65 pN to 400 pN, the magnitude of h∆x
net
i increment be-
comes progressively less pronounced. At F = 65 pN, the
greatest increase in h∆x
net
i is observed, reaching 0.49 nm.
However, at F = 400 pN, the increase is minimal, with only
0.08 nm, as illustrated in Fig. S7(e).
The trend of the sliding distance plateauing and subse-
quently decreasing with increasing ATP concentration, as
shown in Fig. S7(d). One possible reason for this is that
the continuously increase in ATP concentration leads to a
decrease in the number of bound hNi.
B. The energy efficiency and power output of myosin
half-filament
In this section, we predict the thermodynamic efficiency
η and power output P of muscle contraction over different
external load F and [ATP]. The thermodynamic efficiency
of the myosin half-filament is
η =
F h∆x
net
i
∆µ
ATP
, (7)
where F h∆x
net
i represents the average output work W ex-
erted on the actin filament during one complete microstate
cycle [
13].
Furthermore, the power o utput is determined by the prod-
uct of load F and sliding velo city, given by P = F · V .
In Fig.
4(a), the efficiency incr eases as the external load
F increas es and eventually approaches a limiting value of
around 31%. At low ATP concentrations, the effect of ex-
ternal load F on efficiency is more pronounce d, res ulting in
larger changes in efficiency, as shown in Figs. 4(a,b). Af-
ter the ATP concentration reached saturation, the effect of
6
FIG. 4: The efficiency (η) and power output (P ) of muscle con-
traction over different external load F and [ATP].
F change on efficienc y is relatively less, and the rise in ef-
ficiency became more moderate. For instance, at an AT P
concentration of 500 µM, increasing load F caused efficiency
to rise from 20.05% to 31.57%, corresponding to a change of
11.52%. By contrast, at an ATP concentration of 2000 µM,
the incre ase in efficienc y with increasing F is only 4.01%,
with efficiency rising from 27.88% to 31.89%, as shown in
Fig.
4(a). Furthermore, the efficiency curves are very close
to each other and almost overlap at [ATP]=2000 µM and
[ATP]=3000 µM.
In Fig.
4(b), the efficiency of myosin half-filament under-
goes a transition from the load-sensitive regime (observed at
low AT P concentrations) to the weak load-sensitive regime
(occurring at higher ATP concentrations). Additionally, ef-
ficiency increases slowly at low ATP concentrations ranging
from 1 to 10 µM. As ATP concentration increases from 10µM
to 10 00 µM, efficiency increases more rapidly until it reaches
a maximum at ATP saturation, after which it begins to de-
crease. The position of the peak efficiency shifts towards
lower ATP concentrations as F increases. At F = 65 pN,
120 pN, 240 pN, and 400 pN, the peak efficiencies are 30.48%,
31.69%, 32.71%, and 33.41%, respectively, occurring at ATP
concentrations of 3162 µM, 3162 µM, 1995 µM, and 1000
µM, respe ctively.
The power exhibits a positive correlation with the incre-
ment in external load F , and its growth rate gradually slows
down, as illustrated in Fig.
4(c). Fig. 4(d) clearly depicts
the exponential growth of power with increasing ATP con-
centration. Additionally, power is affected by the ve locity,
which increases with rising ATP conce ntration when F is
fixed. The rate of power increase tends to level off when the
ATP concentration is above 10
4
.
C. The steady-state distribution of myosin II motors
Generally, when [ATP]=2000 µm and the external load
F is fixed, the steady-state mesostate probability p(N|F )
increases initially and then decrease s with the number of
bound motors N, forming a peak, as shown in Fig.
5(a).
The distribution of p(N|F ) is mainly concentrated around
the average number of bound motors hNi. However, as
the external load F increases, the maximum value of this
0 20 40 60 80 100
0
0.1
0.2
0.3
(a)
(b)
17 18 19 20 21
0
0.1
0.2
0.3
0.4
(c)
17 18 19 20
0
0.1
0.2
0.3
(d)
0 1 2
0
0.1
0.2
0.3
FIG. 5: The steady-state mesostate probability p(N|F ) and
the microstate conditional probability distribution p(n|N, F ) of
myosin II motors. Since [ATP] is maintained constant at 2000 µm,
we omit [ATP] here. (a) p(N|F ) versus the number of bound mo-
tors N at different load F . (b) p(n|N, F ) of N = 28 at load F =120
pN versus the number of motors in post-power stroke state n
post
.
(c) p(n|N, F ) of N=48 at load F =240 pN versus the number of
motors in mid-power stroke state n
mid
. (d) p(n|N, F ) of N=77 at
load F =480 pN versus the number of motors in pre-power stroke
state n
pre
.
peak gradually decreases, and the shape of the distribution
changes from tall and thin to short and wide.
When the external load F is given, we identify the
value of N corresponding to the maximum steady-state
mesostate probability p(N|F ), which corresponds to the
peak of p(N|F ). For this particula r N, there can be multi-
ple sets of n
pre
, n
mid
, and n
post
that satisfy the constraint
n
pre
+ n
mid
+ n
post
= N. Therefo re, this value of N can
corres pond to multiple microstate configura tions n. To fur-
ther explore the distribution of microstate configurations n
within this given N, we plot the conditional probability dis-
tribution p(n|N, F ) in two and three dimensions, as shown
in Figs.
5(b-d) and Figs. S5(b-d).
When F =120 pN, the steady-state mesostate probabil-
ity p(N|F ) reaches its maximum value at N=28. The
corres ponding conditional distribution p(n|N =28, F =120
pN) is primarily conc e ntrated in the range 17≤n
post
≤21,
n
pre
=0, and 7≤n
mid
≤11. This distribution exhibits an in-
creasing and then decreasing trend in response to changes
in the variable n
post
. The max imum probability value of
p(n|N=28, F =120 pN) is observed a t the configuration
(n
pre
=0, n
mid
=9, n
post
=19), with a value of 0.39, as illus-
trated in Fig.
5(b) and Fig. S5(b).
Similarly, at external loads of F =240 pN and 4 80 pN,
the steady-state probability p(N |F ) achieves its maximum
values a t N=48 and N=77, respectively. The condi-
tional distribution p(n|N=48, F =240 pN) pr imarily ex-
hibits concentration in the range 28≤n
post
≤31, n
pre
=0, and
17≤n
mid
≤20. Furthermore, p(n|N=77, F =480 pN) is pre-
dominantly concentrated in the range 40≤n
post
≤44, n
pre
=0,
and 33≤n
mid
≤37. The conditional distribution p(n|N, F )
increases and then decreases with changes in both n
mid
and
7
n
pre
. The maximum value of p(n|N =48, F =240 pN) is 0.28
and occurs at (n
pre
=0, n
mid
=18, n
post
=30). For p(n|N=77,
F =480 pN), the maximum value is 0.24 and it occurs at
(n
pre
=0, n
mid
=35, n
post
=42). These results are shown in
Figs.
5(c-d).
From Figs. 5(b-d), it is apparent that a large majority
of the motors are in the mid-power stroke a nd post-power
stroke states when N motors are bound. Since motors in
the mid-power stroke and post-power stroke states generate
a relatively large force, having a larger number of motors in
the mid-power stroke a nd post-power stroke states allows for
more efficient sharing of the load.
In addition, with the incre ase in externa l loa d F , there is
a shift in the predominant location (n
pre
, n
mid
, n
post
) where
the conditio nal distribution p(n|N, F ) is concentrated, ac-
companied by an increase in the number of bound mo-
tors N. Meanwhile, the peak of the conditional distribu-
tion p(n|N, F ) decreases, as depicted in Figs.
5(b-d) and
Fig. S5(b-d).
D. Influence of power stroke on muscle performance
FIG. 6: The effects of variations in free energy bias g
ps
on muscle
performance at different values of external load F . (a) Velocity
V . (b) Average number of bound motors hNi. (c) Average net
distance h∆x
net
i. (d) Sliding distance L. (e) Efficiency η . (f)
Output power P . ∆g
ps
= g
ps
− g
ps0
and the g
ps0
is fitting value
given in Tab.
I. In order to maintain the balance of basic free
energy terms and ensure their sum remains eq ual to −∆µ
ATP
,
we evenly distribute the variation of g
ps
by ∆g
ps
among the re-
maining basic free energy terms. g
on
, g
off
, and g
recovery
undergo
a variation of −∆g
ps
/3.
Muscle contraction is initiated by the power stroke of
myosin II motors, which generates force and motion. In ad-
dition, the strength of the power stroke is determined by the
free energy bias between the pre-power stroke and post-power
stroke states, g
ps
[
13]. In simpler terms, ATP hydrolysis sup-
plies the energy required for the power-stroke process, and
the magnitude of this energy is represented as |g
ps
|. Con-
sequently, a more negative value of g
ps
corres ponds to a
stronger power stroke, resulting in a greater proportion of
motors being in the mid-power stroke and post-power stroke
state.
To investigate the impact of g
ps
on muscle performance, we
set a range of values for g
ps
from −20 k
B
T to −5 k
B
T , and
we observe the trends in biophysical quantities with respect
to different g
ps
values, as shown in Fig.
6.
A more negative value of g
ps
leads to an increase in the
rates of ω
pre
and ω
mid
, which accelerates the motor’s cycle
and ultimately results in an incre ase in velocity, as shown
in Fig.
6(a). When [ATP] is 2000 µM, the total energy
released by ATP hydrolysis remains co nstant, given by g
on
+
g
ps
+ g
off
+ g
recovery
= −25. Consequently, a decrease in |g
ps
|
results in an increase in |g
on
| and |g
off
|, thereby pr oviding
more energ y for the mo tor to attach to and deta ch from the
actin filament. This change influences the increase in ω
on
and
the decrease in ω
′
on
, but the rate ω
′
on
is significantly smaller
than ω
on
. Therefore, the ultimate effect is an incr ease in
the attachment r ate of the motor. This leads to an increase
in the average number of bound motors hNi, as shown in
Fig.
6(b).
Fig. 6(c) illustrates that as the absolute value of g
ps
de-
creases, the average net distance h∆x
net
i of the motor also
decreases. One possible explanation for this reduction is due
to the increase in attached motors. The more attached mo-
tors, the shorter the average net distance h∆x
net
i, which is
consistent with the conclusion of [
13].
The peculiar trend observed in the sliding distance, as
depicted in Fig.
6(d), arises from the combined influence of
the number of bo und motor N and the net distance ∆x
net
.
In Fig. 6(e), the efficiency exhibits an increasing trend
with the a bsolute value of g
ps
and gradually re aches a max-
imum as g
ps
approaches the range of −17 k
B
T to −1 9 k
B
T .
However, beyond this rang e , the efficiency starts to decline
as |g
ps
| further increases. Importantly, our fitted value of g
ps
at −17.438 k
B
T precisely falls within this range.
The power output exhibits a n increase as the absolute
value of g
ps
rises, as shown in Fig.
6(f). This trend of
power can be attributed to the influence of velocity when
force F remains constant.
E. The mean run time/length, mean existence
probability and half-life period of myosin filament
In this section, we investigate the mean run time/length,
mean existence probability and half-life period of myosin
half-filament, shedding light on actomyosin overall behavior.
Figs.
7(c,d) demonstrate a rapid increase in the mea n
run time hti and mean run length hli of myosin filament as
the load F varies, with [ATP]=2000 µm. This implies that
the mo tor remains almost constantly attached to actin, even
at low loads (2 0 pN≤F ≤ 60 pN), where the mean run time
hti is already considerably long.
Figs.
7(e,f) illustrate that the mean run time hti and
mean r un length hl i of myosin filament detachment from
actin decreas e monotonically with increasing [ATP]. At low
ATP concentration, myosin II motors will s tay on actin for
more time, since the period of single cycle becomes long due
8
to the lack of ATP molecule, namely ω
off
is small. Further-
more, at low ATP concentrations, both the hti and h li are
load-dependent, wherea s they gradually become less depen-
dent on force as the [ATP] approaches saturatio n.
In addition, we can evaluate the mean existence proba-
bility o f myosin filament detaching from the actin filament
after time t, denoted as h˜ρ(t)i, as shown in Eq. (
13). We
can observe from the Fig. 7(a) that the decay of the mean
existence probability approximates an exponential decrea se.
However, the ln h˜ρ(t)i curve is not a strictly linear one, be-
cause the s econd derivative of ln h˜ρ(t)i decays rapidly and
approaches zero. Half-life T
1/2
refers to the time required
for the mean existence probability h˜ρ(t)i to decay to 1/2, as
shown in Fig.
7(b).
The curves in Figs.
7(b-d) are not smooth, but rather
exhibit fluctuations. One possible expla nation for this is
the discontinuity of N
T
caused by our use of integer values.
These fluctuations are inherent in the N
T
model and cannot
be avoided. Nevertheless, the overall trend of the curves is
correct, indicating that the hti, hl i and T
1/2
are generally
positively correlated with the load F . To ensure the accu-
racy of our findings, we have acknowledged the issue and
conducted multiple parameter sets to validate our results.
0 0.2 0.4 0.6 0.8 1
10
-6
10
-3
10
0
(a)
0 5 10 15 20
10
0
(b)
0 20 40 60
10
0
10
10
10
20
10
30
(c)
0 20 40 60
10
0
10
10
10
20
10
30
10
40
(d)
10
0
10
1
10
2
10
3
10
4
[ATP] ( M)
10
0
10
20
10
40
10
60
10
80
(e)
10
0
10
1
10
2
10
3
10
4
[ATP] ( M)
10
0
10
20
10
40
10
60
10
80
(f)
FIG. 7: (a) Mean existence probability h˜ρ(t)i. (b) Half-life T
1/2
.
(c) Mean run time hti as load F varies at [ATP] = 2000 µM. (d)
Mean run length hli as load F varies at [ATP] = 2000 µM. (e)
Mean run time hti as [ATP] changes at different loads. (f) Mean
run length hli as [ATP] changes at different loads.
IV. DISCUSSION
By integrating the mechanochemical model of a single
myosin II motor with a cooperative model encompassing
multiple myosin II motors, we comprehensively investigate
both the microstate and mesostate dynamics of this myosin
system. Our analy sis re veals intriguing insights into the dis-
tribution of p(N|F ) in the steady state, which primarily c on-
centrates around the average number of bound motors hNi.
As the load F increases, the maximum value of the peak
gradually diminishes, inducing a notable shift in the distri-
bution shape from being tall and thin to becoming short and
wide.
Significantly, a substantial fraction of motors bound to
actin is found to reside in the mid-power stroke and post-
power stroke states. This observation bears crucial impli-
cations as motors in the mid-power stroke and post-power
stroke state possess the capability to generate relatively
larger forces. Moreover, our nonlinear model uncovers their
enhanced load-sharing capabilities. Consequently, this pop-
ulation of mo tors facilitates more efficient load sharing by
augmenting the number of motors engaged in this particular
state.
Additionally, we conducted a systematic analysis of the
impact of loa d F and [ATP] on the biophysical quantities of
myosin II motors. By systematically ex ploring these two key
factors, we aimed to gain dee per insights into the underlying
mechanisms governing the behavior of myosin II in a com-
plex and dynamic environment. This contributes to a more
comprehensive understanding of the regulatory mechanisms
governing the functionality of this vital molecular motor sys-
tem.
Increasing the load F results in a monotonic decrease in ve-
locity. Meanwhile, the average number of motors hN i bound
to actin increase s, and hNi exhibits an increasing sensitiv-
ity to variations in [ATP]. Additionally, the sliding distanc e
gradually decreases with increasing load F .
At near-vanishing load, the decrease in [ATP] leads to a
rapid reduction in velocity, accompanied by a sharp decrease
in the h∆x
net
i. However, as the load F increases, the rate
at which velocity decreases gradually slows down, and the
impact of load F on h∆x
net
i diminishes.
With increasing [ATP], the velocity of the actin inc reases
due to the acceleration of the motor cycle. At the same time,
the hNi rapidly decreases and becomes weakly dependent on
load F at high ATP concentrations.
h∆x
net
i initially increases with low ATP concentrations
but reaches a maximum and then decreases w ith further in-
creases in ATP concentration. At low ATP concentrations,
the effect of the load F on h∆x
net
i is not significant. How-
ever, as ATP concentration increases and reaches sa turation,
the influence of load F on h∆x
net
i becomes increasingly ev-
ident. Nevertheless, the degree of this h∆x
net
i increase di-
minishes as the load F intensifies. When the ATP conce n-
tration exceeds the physiological level of [ATP] = 2000 µm,
h∆x
net
i becomes almost insensitive to ATP concentration
while remaining highly dependent on the external load F .
Furthermore, we conducted an analysis to investigate the
influence of F and [ATP] on the efficiency a nd output power
of myosin filament. The efficiency shows a dependence on
the load F , with an increas e in load F leading to an increase
in efficie ncy. This trend continues until it reaches a limiting
value of approximately 31%, which is consistent with the
model described by Wagoner et. al. [
13].
In addition, the efficiency undergoes a transition from a
load-sensitive regime at low ATP concentrations to a weak
load-sensitive regime at higher ATP concentrations. Finally,
our findings reveal that increasing the lo ad F or elevating
the ATP concentration to saturation levels results in a sub-
stantial enhancement in both efficiency and output power.
Interestingly, as the |g
ps
| is enhanced, it has two significant
effects on the myosin system. Firstly, it results in an increase
9
in velocity, reflecting the more fo rceful contraction of half-
sarcomere. Secondly, it leads to a decrease in the average
numbe r of bound motor hN i, suggesting that stronger power
strokes make it more difficult for motors to attach to actin.
We observe an intriguing relationship be tween efficiency
and the g
ps
. The efficiency shows an increasing trend with
the |g
ps
|, indicating that a stronger power stroke generally
improves the efficiency of the myosin filament. This trend
continues until g
ps
reaches the range of approximately −17
k
B
T to −19 k
B
T , where the efficiency reaches a maximum
value. However, beyond this ra nge, further increa ses in |g
ps
|
lead to a decline in efficiency. This finding highlights that en-
hancing the power stroke strength initially boosts efficie ncy,
but there is a threshold beyond which the excessive power
stroke strength becomes counterproductive and reduces effi-
ciency.
Finally, the results of our study sugge st tha t under physi-
ologica l concentrations, the mean run time hti and mea n run
length hli of myosin filament ex hibit a rapid increase as the
load F varies. This implies that as the load F increases, the
myosin II motors spend significantly longer periods attached
to the actin filament.
On the other hand, the hti and hli s how a monotonous
decrease with increasing ATP concentration. Specifically,
at low ATP concentrations, myosin filament tends to remain
attached to the ac tin filament for longer durations, and both
the hti and hli are load-dependent. However, as the ATP
concentration approaches satura tion, the dependence on load
gradually decreases. These observations may have significant
implications for shedding light on the overall be havior of the
myosin system and enhancing our understanding of it.
V. METHOD
When the system enters the steady state, the biophysical
quantities of myosin II can be obtained theoretically accord-
ing to our model, including velocity V , the ave rage number
of bound motors hNi, and sliding distance L.
The number of bound motors N can be regarded as a
linear Markov chain with mesostate rates, and the steady-
state probability p(N|F, [ATP]) can be obtained by using the
master equation, we describe in Supplemental Material,
section F. We can calculate the average number of bound
motors as
hNi =
N
T
X
N=1
Np(N |F, [ATP]). (8)
For the same N, there are many cases of n
pre
, n
mid
and
n
post
, so one N corresponds to multiple n. The local equi-
librium approximation solves p(n|N, F, [ATP]) for the condi-
tional distribution of all microstates within a given mesostate
N [
13]. The explicit expression for p(n|N, F, [ATP]) is given
in Supplemental Material, section E. Then, we can obtain
p (n|F, [ATP]) = p(n|N, F, [ATP])p(N |F, [ATP]).
The velocity-load relationship repr e sents a steady state
in which the motors repeatedly attach to, stroke and then
detach from the actin. In our model, the expression of veloc-
ity that sums the transition ra tes multiplied by the distance
moved by the actin filament across all possible transitions of
the cycle. Thus, the velocity of the actin fila ment is
V =
X
n
h
p(n
′′′
)k
n
′′′′
,n
′′′
− p(n
′′′′
)k
n
′′′
,n
′′′′
∆x
1
(n
′′′
)
i
+
X
n
h
p(n
′′′′
)k
n,n
′′′′
− p(n)k
n
′′′′
,n
∆x
2
(n
′′′′
)
i
−
X
n
h
p(n)
N
X
i=n
pre
+n
mid
+1
ω
off
(i, n)∆x
slip
(i, n)p(i)
i
+
X
n
h
p(n
′
)
N
X
k=n
pre
+n
mid
+1
ω
′
on
(k, n
′
)∆x
slip
(k, n)p
′
(k)
i
,
(9)
where p (n
′′′
) = p (n
′′′
|F, [ATP]) and similar for the other
probabilities. p(i) is the probability of motor i releasing from
post-power storke state and p
′
(k) signifies the probability of
a motor in the detached state spontaneously extending to a
sufficient extent to bind at position k in the post-power stroke
state. The distances ∆x
1
(n
′′′
), ∆x
2
(n
′′′′
), ∆x
slip
(i, n), and
∆x
slip
(k, n) are dependent on the load F , while ω
off
(i, n)
and ω
′
on
(k, n
′
) are influenced by both the load F and [ATP],
which we abbreviate here. Next, we obtain the expression
for average net distance of the actin filament during one full
microstate cycle. The ave rage net distance is
h∆x
net
i =
X
n
h
p (n
′′′
)
P
n
p (n
′′′
)
∆x
1
(n
′′′
)
i
+
X
n
h
p (n
′′′′
)
P
n
p (n
′′′′
)
∆x
2
(n
′′′′
)
i
(10)
−
X
n
h
p (n)
P
n
p (n)
N
X
i=n
pre
+n
mid
+1
∆x
slip
(i, n)p(i)
i
.
Denoting by F
N
(t) the pro bability density that myo sin
filament separates from the actin (i.e., reaches 0 motor
binding) for the first time at time t, starting from the
state where there are N motors binding at time t = 0
where N ∈ [1 , 2, · · · , N
T
]. It c an be shown that F(t) =
[F
1
(t), · · · , F
N
(t), · · · , F
N
T
(t)]
T
satisfy the following back-
ward master equations,
dF(t)
dt
= AF(t) + [r(1)F
0
(t), 0, · · · , 0]
T
, (11)
where F
0
(t) = δ(t) in the first equation means that if myosin
filament detaches from actin, the first-passage process is ac-
complished immediately, r(1) is the detachment rate of only
one motor from actin, and ma trix A is given in Supplemen-
tal Material, section H.
The run time of myosin filament initiated with N motors
bound is T
N
=
R
+∞
0
tF
N
(t)dt. So the mean run time of
myosin filament along actin is
hT i =
N
T
X
N=1
P (N|F ) T
N
. (12)
And the mean r un length of the myosin filament along actin
is hli = hT i V.
The existence probability can be repr esented as ˜ρ(t) =
e − ρ(t) =
R
∞
t
F(t)dt = [˜ρ
1
(t), · · · , ˜ρ
N
T
(t)]
T
, see Supple-
mental Material, section H. We can obtain the expression
10
for existence probability ˜ρ(t) = e
At
e, with ˜ρ(0) = e. The
mean existence probability is then obtained by
h˜ρ(t)i =
N
T
X
N=1
P (N|F ) ˜ρ
N
(t). (13)
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