Harmonic force spectroscopy measures load-dependent kinetics of individual human β-cardiac myosin molecules

Molecular motors are responsible for numerous cellular processes from cargo transport to heart contraction. Their interactions with other cellular components are often transient and exhibit kinetics that depend on load. Here, we measure such interactions using ‘harmonic force spectroscopy'. In this method, harmonic oscillation of the sample stage of a laser trap immediately, automatically and randomly applies sinusoidally varying loads to a single motor molecule interacting with a single track along which it moves. The experimental protocol and the data analysis are simple, fast and efficient. The protocol accumulates statistics fast enough to deliver single-molecule results from single-molecule experiments. We demonstrate the method's performance by measuring the force-dependent kinetics of individual human β-cardiac myosin molecules interacting with an actin filament at physiological ATP concentration. We show that a molecule's ADP release rate depends exponentially on the applied load, in qualitative agreement with cardiac muscle, which contracts with a velocity inversely proportional to external load.


Supplementary Note 1:
Forces on a molecular motor attached to a harmonically oscillated compliant dumbbell A: Motion of an unattached, compliant dumbbell The next section presents a useful pedagogical toy, the rigid dumbbell. It does not represent our reality: Experimentally, we found that the dumbbell showed its compliance even in the unattached state. This observation is explained by the top panel in Supplementary Figure 1, which follows Fig. 5 in Ref. 1: The actin "handle" of the dumbbell is attached tangentially to the two dumbbell beads. Thus, the distance between the two dumbbell beads easily changes in response to changes in the force that stretches the dumbbell: It requires only one degree of change in the orientation of the actin filament at its point of attachment to a bead to allow the center of a 1 µm-diameter bead to displace itself 9 nm, which is seen as follows.
In a stretched dumbbell, the pull on the beads keeps them in close physical contact with the actin filament even if the link to the filament is flexible. The filament is tangential to both beads (Supplementary Figure 1), so 1 • of change of direction of the filament at a point of contact, makes the radius of the bead to the point of contact rotate by 1 • as well. Since the radius is 0.5 µm long, the center of the bead is displaced by 1 • /360 • × 2π × 500 nm = 9 nm. This displacement takes place in the direction parallel to the filament's direction at its point of contact, so the displacement's component along the x-axis is shorter than 9 nm.
In the dumbbell's unattached, oscillating state, the force stretching the dumbbell may oscillate as the dumbbell is dragged back and forth in the two traps holding it by the oscillatory flow of fluid surrounding it. If the two traps are Hookean springs with identical stiffnesses and the two beads have identical drag coefficients, the separation between the beads will not oscillate while the dumbbell's position in the traps does. Our traps were not sufficiently identical to ensure constant separation, so our dumbbells displayed compliance even in their unattached states.

B: Motion of an unattached, rigid dumbbell
Consider the motion of an unattached, rigid dumbbell in response to the harmonic force on it caused by harmonic stage-oscillations. This simplest possible case of dumbbell motion illustrates aspects of the motion of an attached, compliant dumbbell. So it is a good starting point.

B-i: Definitions
Let x 1 and x 2 denote the lab-coordinates of Bead 1 and Bead 2, respectively, measured along the direction of the dumbbell and stage oscillation. Let the origin on the x 1 -axis be the time-averaged position of Bead 1 in the unattached state of the dumbbell. With a similar definition of x 2 for Bead 2, the net trapping force on the dumbbell is zero when the beads are at these origins. In these considerations we leave out Brownian motion. It is easily added to Eq. (1) below, see Ref. 2, but by leaving it out, we arrive at trajectories that are already averaged with respect to Brownian motion, and hence ready for fitting to experimental trajectories.
Since the dumbbell is rigid, x 1 = x 2 , and a single x-coordinate, x db , will describe its motion in the direction of its axis and the stage motion. We use x db = x 1 = x 2 . The dumbbell is trapped with Hookean stiffness κ db ≈ κ 1 +κ 2 and experiences a Stokes drag with coefficient γ db ≈ γ 1 + γ 2 , where κ i and γ i are, respectively, the stiffness of Trap i and Stokes drag coefficient of Bead i, i = 1, 2. Actually we expect γ db to be larger than γ 1 + γ 2 because the actin "handle" of the dumbbell also contributes to the drag coefficient of the dumbbell.

B-ii: Equation of motion for unattached, rigid dumbbell
Newton's 2nd law with vanishing inertial term gives the equation of motion, It is linear in x db with an inhomogeneous source term proportional to the known stage-velocity, v drv , since this is also the velocity of the buffer fluid, which drives the motion of the dumbbell: v drv (t) = A drv ω drv cos(ω drv t) .
Here A drv is the amplitude of the harmonic oscillation of the position of the stage, and ω drv is its cyclic frequency: x drv (t) = A drv sin(ω drv t) . ( We have chosen the origin of our time axis to be a point in time when x drv = 0 and v drv > 0. Thus, phase-angles of other oscillations are measured relatively to those of the stage position; v drv , e.g., is phase-shifted π/2 ahead of x drv . B-iii: Solution x db (t) to equation of motion where we have introduced the period t drv of stage oscillations, the corner frequency f c,db for the trapped dumbbell, and its inverse, the characteristic relaxation time, τ db , for the dumbbell in the trap; B-iv: Typical parameter values; approximations to x db (t) Typical values are f drv = 200 Hz and f c,db ≈ 2 kHz. So (f c,db /f drv ) 2 ≈ 100. Consequently, we commit less than 1% error with the approximation which shows that the dumbbell oscillates with approximately 10% of the amplitude of the stage and trails the stage velocity by τ db .
At f c,db = 2 kHz, τ db = 80 µs, to be compared with the 5 ms period of the stage at 200 Hz. It shows that the dumbbell responds almost instantly to the drag force cause by the stage motion, with a delay of only 0.016 period, i.e., a phase angle of -0.1 radian or -6 • . Thus, within a 2% error we can ignore τ db and have i.e., the dumbbell coordinate is phase shifted π/2, one fourth of a stage period, ahead of the stage coordinate. This phase shift ahead should not confuse, if one remembers that it is the stage velocity that drives the position of the unattached dumbbell away from zero, and this happens essentially in synchrony, with the dumbbell position trailing the stage velocity only by τ db .

C: Motion of an attached, compliant dumbbell C-i: Definitions
We neglect the compliance of the attachment of the dumbbell to the stage, i.e., the compliance in the S1, its linkage to the platform bead, and in the platform bead. We trust it to be negligible compared to the compliance in the dumbbell itself (Dupuis et al., 1997). With this approximation, the point of attachment between dumbbell and S1 simply follows the stage. This approximation decouples the dynamics of the two parts of the dumbbell that connect at the attachment point. Consequently, we can treat those two parts independently. We start with the part including Bead 1. Let t 1 denote the point in time when the S1 attaches to the dumbbell handle. For t ≤ t 1 , x 1 (t) = x db (t), where x db (t) is given in Eq. (4) for the case of a rigid, unattached dumbbell. To keep things simple, we will proceed with this case, using x db (t 1 ) as initial condition for the dynamics of x 1 (t) for t > t 1 . It will disappear so fast from the description of the attached dumbbell that this choice does not matter. If we used the more correct initial condition, the coordinate x 1 (t 1 ) of a compliant, unattached dumbbell, it would disappear equally fast from the description, so we don't bother to find it by solving for the motion of a compliant, unattached dumbbell.
The S1 attaches at a point on the dumbbell handle that we don't see and hence don't know. Since it isn't the midpoint, typically, Bead 1's and Bead 2's connection to the attachment-point have different compliances. For moderate relative movement of beads with respect to their attachment points, it is reasonable to assume that each compliance is well modeled by a Hookean spring. Let κ The actual coordinate x 1 (t) of Bead 1 will differ from x (rg) 1 (t) due to compliance, and this difference then causes a restoring Hookean force The difference between x 1 (t) and x   Ignoring inertia, the total force on Bead 1 is zero at all times t according to Newton's 2nd law, This is the equation of motion for x 1 . The equation of motion for x 2 is obtained by substituting subscript 1 for 2 in Eqs. (8-10).

C-iii: Physical meaning of terms in equation of motion for x 1
Imagine x (rg) 1 (t) is positive and growing. This is the trajectory that Bead 1 would need to follow in order to follow the stage in unison. Imagine that it does not do that perfectly: Imagine x 1 (t) also is positive and growing, but trailing behind x (rg) 1 (t), because it is held back by the trap to some extent, and the compliance of its connection with the stage allows it to yield somewhat to the force from the trap (Supplementary Figure 1, Panel C). Thus, the force from the trap is negative, while the drag force from the buffer fluid is positive. The sum of these two forces is transmitted through the compliant dumbbell to the S1. It is not quite the trapping force −κ 1 x 1 that we can measure by monitoring Bead 1's displacement x 1 in Trap 1. The force transmitted to the S1 is reduced in magnitude by the drag force, and equals −κ 1 The question now is how much the drag-force matters in the case of finite compliance. If it is negligible, the load on the myosin from Bead 1 equals the force that we can measure as −κ 1 x 1 . To decide whether this simplification is reasonable, we solve the full problem below.
C-iv: How to recover the rigid dumbbell from the compliant dumbbell in the limit of no compliance Equation (10) simplifies in the limit of zero compliance. This is the limit of κ (cp) 1 → ∞. In that limit, x 1 differs infinitesimally from x (rg) 1 , just enough to make −κ (cp) 1 vanishes in that limit, because

C-v: Solution to equation of motion for x 1
The solution to Eq. (10) for t ≥ t 1 is where we have introduced the characteristic time τ 1 for the relaxation of Bead 1 in the combined Hookean force fields from the optical trap and from the compliant section of the dumbbell between Bead 1 and the dumbbell's point of attachment to the stage, The first term on the right-hand side of Eq. (11) is a transient term that dies out exponentially fast in time with characteristic time τ 1 . It represents the memory in x 1 (t) of its value at time t 1 . The same exponentially decreasing "memory weight-factor" is seen also under the integrand in Eq. (11). It weighs the importance of past values (i.e., values at times before t) of the rest of the integrand for the value of x 1 (t). In that rest of the integrand, x 1 . This is used in the second of the following approximations to Eq. (11).

C-vi: Approximate solution revealing physics at play
We could insert the known explicit expressions for x (rg) 1 (t) and v drv (t) on the right-hand side of Eq. (11) and obtain the exact analytical expression of this equation of motion. That approach yields a complicated result. The complications specify details on a time scale that we barely resolve, that of τ 1 . Instead, we use the following simple approximation. It is precise because it keeps term to order τ 1 /t drv in the description, and by doing only that it reveals the physics at play well: Consider the case of τ 1 t drv , i.e., the relaxation time τ 1 of the bead under the combined force of optical trap and dumbbell compliance is much shorter than the period of the stage oscillation. In this case, the functions v drv (t ) and x (rg) 1 (t ) in the integrand in Eq. (11) change negligibly in the brief range of t -values with t ≤ t where the exponential weight-factor exp(−(t−t )/τ 1 ) in Eq. (11) differs significantly from zero. We consequently can approximate v drv (t ) and x (rg) 1 (t ) with their values in t = t − τ 1 , the mean value of t with respect to the weight-factor exp(−(t − t )/τ 1 ). The same exponential factor makes the first term on the right-hand side of Eq. (11) negligible for t − t 1 τ 1 . Taken together, these approximations give where Eq. (15) involves yet an approximation. It uses that v drv is the velocity of x , and phase-shifted ahead of the stage position. It is ahead of the stage position, as far as phase is concerned, because it does not quite follow the stage as far as amplitude is concerned. This deficiency in amplitude causes a drag force on Bead 1 that pushes it in the direction the stage moves. It responds to this by shifting its position in the direction that the stage velocity points, which phase shifts the cycles of x 1 (t) towards the phase of v drv . This phase shift is positive. It is also always less than π/2, the phase of v drv relatively to x drv . This is physically obvious, and follows mathematically from Eq. (14) by inserting the explicit expressions for x (rg) 1 (t) and v drv (t) in it. Then it can be rewritten as where the cycle mean value x (0) 1 is irrelevant for the phase, the phase shift φ 1 ∈ [0, π/2] is defined by and the amplitude ∆x 1 is The phase shift of x 1 (t) in Eq. (16), φ 1 − ω drv τ 1 , is always positive for our parameter values, because its negative term is numerically small, −ω drv τ 1 ≈ −0.1, while φ 1 is the arctan of a number strictly larger than that, though it also is small, typically.
C-vii: Load on S1 from Bead 1 The load L 1 (t) on S1 from Bead 1 equals minus the force on Bead 1 from trap and drag force. Thus the load is where we have used Eq. (10). We find from Eq. (13) by using the very good approximation x (rg) This load has a cycle-average and oscillates harmonically with amplitude i.e., as if the only effect of compliance is to reduce the amplitude of oscillations by the factor κ C-viii: Load on S1 from Bead 1 plus Bead 2 The load on S1 from Bead 2 is modeled like that from Bead 1. It oscillates with the same frequency as that from Bead 1, but typically with different amplitude and phase shift. The two phase shifts, φ 1 − ω drv τ 1 and φ 2 − ω drv τ 2 , are both so small that even if they are not exactly the same, we can safely assume that the two loads, L 1 (t) and L 2 (t), are in phase. Consequently, the total load of the S1 from the dumbbell, F (t) = L 1 (t)+L 2 (t), has a mean that equals the sum of means, and its amplitude equals the sum of amplitudes, which we have used after measuring L 1 (t) and L 2 (t). Not that it matters: If one wants to account for a difference between the phase-shifts of the two beads, one simply calculates F 0 and ∆F directly from F (t) = L 1 (t) + L 2 (t) or includes the effect of the phase difference in Eqs. (23) and (24). We conclude that we can treat the experiment as if only the trapping forces load the S1 and hence do this in phase with each other. We can ignore the drag force on the attached dumbbell because it mainly phase-shifts the load, without changing its mean value and its amplitude of oscillations.