In Takaji et al’s data the feedback gain for τ½= 1ms generated a half-maximal
displacement at 10ms. In the model, setting feedback gain for τ½= 1ms resulted in the
half-maximal displacement at 19ms, the feedback gain had to be almost doubled (τ½=
0.52ms in the feedback model) to achieve 10ms. The response surface model showed some indications as to why these discrepancies occurred.
The optical trap stiffness had a negligible influence on the calculation of feedback but did influence the mid-displacement time target in the response surface model.
Increasing the optical trap stiffness reduced the time as did reducing kmh, kb and bmax.
The loading applied via the optical trap is also opposed by the lever stiffnesses, kmh
and kb. Also working against the optical trap is the cofilament stiffness the behaviour
of which would follow the trend of kmh. By changing the balance of compliance
between the cofilament, lever arm and optical trap the half-maximal displacement time is modified. Levering stiffnesses kmh and kb are not present when setting the feedback
level.
Modifying optical trap, kmh, kb and bmax values within the response surface model
boundaries significantly compromised the ability to align with other targets while failing to bring the time down lower than ~ 16ms. At mid-displacement 12.8% of the crossbridge displacement was transferred to the cofilament and at initial maximum displacement 1.5%, result (2), Table 3.5.2. Dropping kmh, while lowering the time of τ½,
would confer greater displacement to the substrate, reducing the movement of actin and the apparent lever distance. The difference in response times, therefore, seems attributable to the setting of the feedback gain and the compliance used for it. This could be pursued further using the model, but was not deemed useful to the over all project.
Within the response surface model, the optical trap stiffness also influenced the magnitude of the forces at maximum displacement, a state where they are in equilibrium with the cofilament/substrate stiffness. The damping of the lever arm increases the force on the right trap as it slows the lever release so the right trap has to work against greater strain energy in the crossbridge with small losses in displacement results (2)-(5), Table 3.5.2. Increasing the feedback gain has a similar influence. The difference between the model and in vitro forces may be due to a variation in
performance between the motor and transducer in Takaji et al’s [63] experiment
compared to the symmetry of loading in the model or a potential variation in force profile at the end of the lever movement.
In the model, the crossbridge length is greater than the displacement of the actin bond site. To distinguish between these two lengths the crossbridge length at its maximum is the lever distance denoted by bmax and the actin bond displacement, the working
distance denoted z. The difference in length may be due to the deformation of the
for this as a known experimental error and it may contribute to the broad range of displacements reported for individual events: from 5.5nm and 15nm [32,53,64,65,92]. Studies that are more recent give values in the 6-10nm range, for example Kaya and
Higuchi [29] determined a working stroke mean of 7.6nm peaking at 8nm. The bmax =
7.9nm specified from the model fits comfortably within this more recent range of values (see Appendix B, Table 3).
Equation 3.5.1 with kb = 1.923 pN/nm and bmax = 7.9nm gave a peak force of 15.2pN
consistent with the peak forces observed by Takaji et al [63] of 15pN (although with a few outlying 17pN events). The model’s plateau force under isometric loading was 8.53pN which is comparable to the upper end of values measured elsewhere in related optical trap experiments: 0.8 to 7pN [30,31,64,65,71] where these author’s were inclined to consider their measurements lower estimates due to compliance in their experimental set-ups.
The speed of movement of the actin can be used to indicate if the crossbridge is releasing quickly enough, i.e. to determine whether cb is small enough while the kb
component of the time constant (Section 3.5.2) has been set by the peak force and bmax.
Typical muscle contraction speeds are 6 to 7 µm/s [10]. Actin filaments moving
across a substrate scattered with myosin fragments appear to move more quickly: 8 to 9 µm/s, [16,28,52,89,90,91]. An individual crossbridge’s displacement of actin, z (the working distance), divided by the time myosin remains bound to actin must achieve these velocities as no other means of increased speed presents itself. The duration of that attachment is dependent on the time the crossbridge spends releasing strain energy (levering actin), tlever, and the time the crossbridge is present before and after levering, tdwell. dwell lever t t z z + = & , (3.5.8),
From result (2),Table 3.5.2, z = 6.99nm, tlever = 0.3ms the crossbridge speed is
23.3µm/s. The crossbridge is levering rapidly enough with some time remaining for
attachment and release of the crossbridge tdwell.
The S1 head stiffness during levering was determined as kmh = 3.530 pN/nm and is of
cofilament stiffnesses, kms and kmh act in series in this version of the model. If kmh is
reduced, kms must be increased to maintain the same output. The substrate-cofilament
stiffness used here was taken from Section 3.2.6 and was for a cofilament attached to a substrate but in this instance a smaller fragment was bound to the substrate giving a smaller contact area leading to a lower stiffness. So it follows that kms, in this analysis,
was set high and therefore kmh is a lower estimate.
The crossbridge stiffness 1.3pN/nm during levering dropped due to the additional in
series compliance of kb, which was previously found to be 2.6 and 2.9pN/nm pre- and
post-lever (Section 3.2.4). This lower value aligns with crossbridge values from other experimenters, e.g. 0.13, 0.6, 0.48, 1.79pN/nm [62,64,65]. The damping component,
cb, applies 0.21% of the total force on the crossbridge so should have minimal
influence on the crossbridge stiffness.
bmax, kb, kmh and cb represent behaviours in the motor domain and S1 region of the
myosin II fragment. Therefore, they can all be considered open to modification when considering the isoforms of myosin II.
3.5.6
Summary of Parameter Values.
The model lever arm representation has been compared to a single crossbridge levering event under isometric loading. Comparable results have been generated indicating that the model is in good agreement with in vitro data. Values have been determined for the following parameters that are consistent with other published data.
bmax, Maximum lever displacement: 7.9nm,
kb, Elastic component of levering: 1.92 pN/nm,
kmh, S1 stiffness during levering, lower estimate: 3.53 pN/nm,
cb, Viscous damping component of levering: 0.04 pN/µm/s,
Sensitivity of the system to a five percent variation in the parameters kb, kmh,
cb, and bmax±0.1nm was ~2% for maximum displacement of actin and ~7% for the