INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Over thirty years ago, Barany (1967) showed that a muscle's maximum shortening velocity is correlated with its actomyosin ATPase rate, suggesting that myosin's hydrolytic and mechanical processes are coupled. Given that the hydrolysis of ATP by actomyosin is a multi-step biochemical process (Lymn and Taylor, 1971), the challenge over the past several decades has been to characterize the mechanical properties of the individual biochemical states and to determine at what point in its ATPase cycle myosin generates actin movement (i.e., myosin's mechanical step). In an attempt to achieve these goals, various experimental approaches have been used (for reviews see Goldman, 1987; Cooke, 1997; Sellers, 1999), including transient kinetic techniques in solution and mechanical studies in skinned fibers performed simultaneously with spectroscopic (Irving et al., 1995; Baker et al., 1998) or fiber diffraction measurements (Linari et al., 2000). These studies, along with structural data from electron micrographic (Reedy et al., 1965, Taylor et al., 1999) and crystallographic studies (Rayment et al., 1993; Dominguez et al., 1998; Houdusse et al., 2000) suggest the minimal mechanochemical scheme in Fig. 1.

View larger version (13K): [in this window] [in a new window]   FIGURE 1   A model of the mechanochemistry of the actomyosin ATPase reaction. A myosin head (ovals) has a weak affinity for an actin filament (helix) when ATP or ADP and inorganic phosphate, Pi, are bound to myosin. Upon Pi release, myosin's affinity for actin increases by several orders of magnitude and, upon strong binding to actin, undergoes a conformational change (a light chain domain rotation). Myosin remains strongly bound to actin even after ADP release.

When ATP (T) or the products of hydrolysis, i.e., ADP (D) and inorganic phosphate (P_i), are bound to myosin (M), myosin has a weak affinity for actin (A) and is in, what are referred to as, weak binding states (M·T, M·D·P_i). Upon release of P_i, myosin's affinity for actin increases by several orders of magnitude (White and Taylor, 1976; Eisenberg and Hill, 1985), resulting in strong actin binding (A·M·D) that is maintained even after the release of ADP (A·M). Myosin returns to a weak binding state when an ATP molecule binds to myosin's active site, allowing myosin to detach from actin and to begin its cycle once again.

With the development of the single molecule laser trap assay, direct access to the mechanochemistry of the actomyosin ATPase reaction is now possible (Finer et al., 1994). In this assay, the distance, d, that myosin moves an actin filament can be measured as well as the period of time, t_on, that myosin maintains this displacement (Guilford et al., 1997). For smooth and cardiac muscle myosin, changes in the average t_on, or lifetime (_on), with ATP concentration have been used to determine a second-order ATP-induced detachment rate, k_T, and an effective ADP release rate, k_D (Lauzon et al., 1998; Palmiter et al., 1999). In the present study, we use a laser trap to estimate values for k_T, k_D, k_D (the second-order ADP binding rate), and k_Pi (the second-order P_i-induced actin dissociation rate) for skeletal muscle myosin at the level of a single molecule.

Myosin's mechanical step is believed to result from a discrete rotation of the myosin light chain domain, or neck (Rayment et al., 1993; Dominguez et al., 1998; Baker et al., 1998; Houdusse et al., 2000), with the neck presumably acting as a lever that amplifies small structural changes in the motor domain upon strong actin binding (Uyeda et al., 1996; Anson et al., 1996; Warshaw et al., 2000; Ruff et al., 2001). However, the timing of the mechanical step relative to P_i release remains unclear. Does the mechanical step occur prior to (Dantzig et al., 1992), concomitant with (Eisenberg and Hill, 1985), or after P_i release? Here, we address this question by determining how _on is affected by [P_i] and by a smooth muscle heavy meromyosin (HMM) mutation that dramatically reduces the actin-activated P_i release rate (Joel et al., 2001).

Finally, in the past, our laboratory (Harris and Warshaw, 1993) and others (Homsher et al., 1993) have used actin filament velocities, V, measured in an in vitro motility assay as a means of estimating actomyosin detachment kinetics by assuming the relationship V ~ /_on (Huxley, 1990). This relationship predicts that changes in ligand concentrations that alter single molecule kinetics (i.e., change _on) should similarly affect V in a motility assay. Implicit in this relationship is the assumption that actin-attached myosin heads in a motility assay impose a load that fully limits actin movement without affecting the detachment kinetics measured in a minimally loaded laser trap assay. In the present study, we test this hypothesis by performing motility assays that parallel our laser trap studies. We observe that, under certain conditions, V measured in a motility assay equals /_on determined at the single molecule level, but that at physiological ATP concentrations, V is considerably faster than /_on. To better understand these data, we develop a simple model in which myosin's mechanical step is partitioned between moving an actin filament and stretching internal compliant linkages in the actomyosin system. This model serves as a framework within which single molecule and ensemble data can be formally compared.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Proteins

Fast skeletal muscle myosin was prepared from chicken pectoralis as previously described (Warshaw et al., 1990). To investigate how reducing the P_i release rate affects single myosin molecule behavior in the laser trap, a mutant smooth muscle HMM was expressed in which two highly conserved lysines in loop 2 were replaced with alanines (Joel et al., 2001). The kinetics and motile properties of this mutant have been characterized extensively and suggest that the predominant effect of the mutation is to dramatically reduce the rate of actin-activated P_i release (Joel et al., 2001). All myosins were stored in glycerol at 20°C (Warshaw et al., 1990).

Immediately before use in the motility and laser trap assays, myosin was further purified to eliminate "dead heads" by centrifugation with equimolar actin and 1 mM ATP in myosin buffer (see Buffers below). N-ethylmaleimide-modified skeletal myosin (NEM-myosin) was prepared as previously described (Warshaw et al., 1990) and was used to bind actin filaments to polystyrene beads (1.0 µm dia. polystyrene; Polysciences Inc., Warrington PA; Guilford et al., 1997). Actin was isolated from chicken pectoralis (Pardee & Spudich, 1982) and incubated overnight with tetramethylrhodamine isothiocyanate (TRITC)-labeled phalloidin as previously described (Warshaw et al., 1990).

Buffers

Myosin buffer (MB) contained 0.3 M KCl, 25 mM imidazole, 1 mM EGTA, 4 mM MgCl₂, and 10 mM DTT, adjusted to pH 7.4. Actin buffer (AB) contained 25 mM KCl, 25 mM imidazole, 1 mM EGTA, 4 mM MgCl₂, 10 mM DTT, and oxygen scavengers (0.1 mg ml¹ glucose oxidase, 0.018 mg ml¹ catalase, 2.3 mg ml¹ glucose) adjusted to pH 7.4. For experiments in which a range of ligands were tested (i.e., 0.1 µM to 1 mM ATP, 0-5 mM MgADP, and 0-40 mM inorganic phosphate, P_i), the desired ligand concentration was added to AB. To maintain a constant ionic strength and a 3-mM free Mg⁺² concentration, the KCl and MgCl₂ concentrations were adjusted according to an algorithm based on Fabiato and Fabiato (1979).

Laser trap

Detailed protocols for the laser trap assay have been previously described (Dupuis et al., 1997; Guilford et al., 1997; Palmiter et al., 2000). Contractile proteins were added to the experimental flow cell chamber with the following series of solution incubations: 1) 20 µl of 1 µg ml¹ skeletal myosin or mutant HMM for 2 min.; 2) 20 µl of 0.5 mg ml¹ bovine serum albumin in MB for 1 min.; 3) 3 × 20 µl AB; and 4) 3 × 20 µl of AB with desired ligands, TRITC-actin, and NEM-coated beads. For the mutant HMM, two additional incubations were required to attach HMM to the flow cell surface (Trybus and Henry, 1989): an initial incubation with 20 µl of 0.1 M monoclonal antibody S2.1 for 2 min. followed by 20 µl of 0.5 mg ml¹ bovine serum albumin in MB for 1 min. Experiments were performed at 25°C.

By manipulating the microscope stage, a bead was captured in each of the two laser traps and the ends of a single actin filament were then attached to the beads. After pre-tensioning the filament (~4 pN), the bead-actin-bead was brought near a silica pedestal sparsely coated with myosin. The bright-field image of one of the beads was projected onto a quadrant photodiode detector and separate signals were acquired for bead movement in directions parallel and perpendicular to the actin filament's long axis. Both signals were recorded for at least ~120 s (a data record) before moving the bead-actin-bead to another pedestal within the flow cell. Data records were rejected if displacements were detected in the direction perpendicular to the filament's long axis. The remaining data records were digitized at 4 kHz after initial filtering at 2 kHz.

In vitro motility

Fluorescent images of actin filament movement over a myosin-coated surface were recorded as previously described (Warshaw et al., 1990). The solutions used in these experiments were nearly identical to those used in our laser trap experiments with the exceptions that the myosin concentration was 100 µg/ml and the final AB contained methylcellulose (Palmiter et al., 2000). Experiments were performed at 25°C. Actin filament velocities, V, were determined by analyzing video recordings of filament motility using an ExpertVision motion analysis system (Motion Analysis, Santa Rosa, CA) as previously described (Homsher et al., 1993).

Laser trap data analysis

When myosin strongly binds to an actin filament in a laser trap, it displaces the filament and causes a reduction in the Brownian motion of the bead-actin-bead system (see Fig. 2 a) by adding its stiffness to the bead-actin-bead system (Molloy et al., 1995; Guilford et al., 1997). Both phenomena are used to determine when in a data trace myosin is strongly bound to actin and to calculate the duration, t_on, of these strong binding events.

View larger version (55K): [in this window] [in a new window]   FIGURE 2   Effects of [ATP] on ton. (a) Four characteristic data records acquired at different ATP concentrations illustrate an increase in event durations, ton, with decreasing [ATP]. (b) Characteristic plots of MV densities, on(tw), versus window widths, tw, obtained from individual data records acquired at 100 () and 20 µM () ATP fitted to Eq. 2 (lines). Densities at 15 ms (corresponding to N = 107 and 45 events for 20 and 100 µM ATP, respectively) are normalized to one. (c) Characteristic plots of MV densities, on(tw), versus window width, tw, obtained from individual data records acquired at 1 () and 2 µM () ATP are fitted to Eq. 4 (lines). Densities at 50 ms (corresponding to N = 32 and 36 events for 1 and 2 µM ATP, respectively) are normalized to one. (d) Characteristic plots of non(t) obtained from data records acquired at 0.5 (, t = 0.1 s) and 0.1 (, t = 0.2 s) µM ATP are fitted to Eq. 3. Numbers of events at 200 ms (14 and 53 for 0.1 and 0.5 µM ATP, respectively) are normalized to one. The symbols indicate the center of a bin of width t.

Depending on the number of events observed in a given data trace, one of two methods was used for determining kinetic rate constants from our t_on data. For experimental conditions that resulted in data records containing relatively few events (i.e., <40 events in a 2-min trace), t_on for each event was directly measured, and for a set of data records the number of events, n_on, having t_on values between t and t + t was plotted in a histogram. This distribution, n_on(t), was then used to estimate kinetic rate constants as described below. For experimental conditions that resulted in data records containing a relatively large number of events, we used a mean-variance (MV) analysis (Patlak, 1993; Guilford et al., 1997). Briefly, this approach involves moving a time window of width t_w, point-by-point through a displacement trace and then plotting the mean and variance of each window in a two-dimensional MV histogram (see Guilford et al., 1997). These histograms are typically characterized by two regions within the MV space that contain a large number (high density) of points. One region of high variance and zero mean displacement corresponds to baseline data, whereas another region of low variance and net mean displacement corresponds to myosin displacement events (Guilford et al., 1997). Because only events with durations t_w appear in the event region of an MV histogram, the event density, _on, varies with window width, t_w, reflecting the stochastic nature of event durations (i.e., the detachment kinetics). Thus, kinetic rate constants can be determined from an analysis (described below) of _on(t_w), without tallying individual events (Guilford et al., 1997).

Rate constant determination from n_on(t) and _on(t_w) data

Two advantages of determining kinetic rate constants from single-myosin-molecule event-duration data rather than from solution kinetic or in vitro motility data are that, for each recorded event, myosin is synchronized at t = 0 to be in the biochemical state occupied at the onset of an event, and that we can directly determine the coupling between myosin's mechanics and kinetics without making assumptions about possible cooperative mechanical effects that may or may not exist in an ensemble preparation.

Given the scheme in Fig. 1, we can determine kinetic rate constants from an analysis of n_on(t) and _on(t_w) distributions acquired at various ligand concentrations. According to this scheme, detachment is a two-step biochemical process and, in the absence of P_i, three rates contribute to a t_on distribution: the effective ADP release rate, k_D, the ADP binding rate, k_D, and the second-order ATP-induced dissociation rate, k_T. Lu et al. (1998) showed that, for a two-step process, the number of events, n_on(t), having t_on values between t and t is

(1)

where

and A is the product of t and the number of events in the data set, N. Values for k_D, k_D, and k_T can thus be determined by fitting a distribution, n_on(t), of individually measured t_on values to Eq. 1. These rate constants can also be determined by fitting MV window densities, _on(t_w), to the equation

(2)

where B is the product of N and the sampling frequency (Patlak, 1993), and p and q are defined as above for Eq. 1.

When detachment is limited by a single biochemical step, Eq. 1 reduces to

(3)

where k_rls is the rate of the limiting step. In the absence of ADP and at sufficiently low ATP concentrations ([ATP]k_T k_D), detachment is limited by ATP binding, and k_rls equals k_T[ATP]. At sufficiently high ATP concentrations ([ATP]k_T k_D), detachment is limited by ADP release, and k_rls equals k_D. The analogous equation for data gathered through MV analysis is

(4)

Event lifetime determination

The duration of a given event is t_on, and the average duration of many events is the event lifetime, _on, which can be calculated using one of two methods. One approach is to simply average individually measured t_on values. However, this approach is valid only if an event population does not contain a significant number of events with durations shorter than the laser trap dead time (<2 ms). The second approach is to calculate _on from the rate constants determined above (i.e. k_D, k_D, and k_T). Solving
substituting n_on(t) from Eq. 1, we obtain an expression for _on as a function of k_D, k_D, and k_T

(5)

where p and q are defined as above for Eq. 1.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX REFERENCES

The effects of [ATP] on single molecule and ensemble kinetics

The lifetime of the strongly bound state in the absence of ADP and P_i is related to k_D and k_T (Palmiter et al., 1999) as

(6)

where K_M(on) = k_D/k_T is the ATP concentration at which _on is twice its minimum value (i.e., 1/k_D at saturating ATP). This relationship is obtained by setting [ADP] = 0 in Eq. 5. Eq. 6 predicts that _on should increase when [ATP] is decreased, reaching an ATP-limited value of _on ~ 1/k_T[ATP] ~ 1/k_rls at sufficiently low [ATP]. As predicted, Fig. 2 qualitatively shows that _on for skeletal muscle myosin increased when we decreased [ATP] from 100 to 0.1 µM, and at 2 µM ATP _on appeared to be at least an order of magnitude longer than _on at 100 µM ATP (i.e., 1/k_T[ATP] 1/k_D thus _on is limited by k_T[ATP]). Therefore, for each experiment performed at [ATP]  2 µM, we obtained a value for k_rls either by fitting MV densities, _on(t_w), to Eq. 4 (1 and 2 µM ATP; Fig. 2 c) or by fitting distributions of individually measured t_on values, n_on(t), to the corresponding relationship in Eq. 3 (0.1 and 0.5 µM ATP; Fig. 2 d). In Fig. 3, we plotted the average k_rls estimated at each [ATP] with the slope of the regression giving a value for k_T of 7.6 × 10⁶ M¹ s¹ (see Fig. 3, legend).

View larger version (11K): [in this window] [in a new window]   FIGURE 3   krls versus [ATP]. Values for krls determined from on(tw) and non(t) data acquired at [ATP]  2 µM are plotted for different ATP concentrations. Each point is the average value obtained from n data records (each containing >40 events) with error bars representing the SEM. The slope of a regression weighted by 1/SEM (line) gives a value for kT of 7.6 (± 1.3 SEM) × 106 M1 s1. A 95% confidence interval gives values for kT ranging from 5 to 12.5 × 106 M1 s1. It is possible that, at 2 µM ATP, the ton distribution might not be fully limited by kT[ATP], which might explain why this point appears not to be described by the linear regression. A regression that excludes this point gave a value for kT of 5 × 106 M1 s1.

At 10 µM ATP, _on was not an order of magnitude longer than _on at 100 µM, and thus was not limited by k_T[ATP]. Therefore, to account for the effects of both ADP release and ATP binding, we fitted MV densities, _on(t_w), acquired at [ATP]  10 µM (10, 20, and 100 µM) to Eq. 2 (for examples, see Fig. 2 b). Setting k_T equal to the value determined above, the fits gave an average value for k_D of 100 s¹ (Table 1). Based on our estimates for k_D and k_T, we calculate a value for K_M(on) = k_D/k_T of 13 ± 3 µM.

To compare the detachment kinetics (k_D and k_T) of a single skeletal muscle myosin molecule with the motion-generating biochemistry of an ensemble of myosin molecules, we measured actin filament velocities, V, in a motility assay over [ATP] ranging from 0.01 to 1.0 mM. The relationship between V and [ATP] is often assumed to follow a Michaelis-Menten equation,

(7)

where K_M(vel) is the ATP concentration at which the velocity is half its maximum value, V_max. We fitted our motility data to Eq. 7, with the fit giving a value for K_M(vel) of 76 µM (Fig. 4). This value is comparable to previously reported values for monomeric skeletal muscle myosin in a motility assay (Homsher et al., 1993), but it is approximately five-fold greater than the value for K_M(on) estimated from our single molecule data above. This five-fold difference is surprising because actin velocities are often assumed to be limited by _on, in which case K_M(vel) should be equal to K_M(on) (see Discussion and Appendix).

View larger version (11K): [in this window] [in a new window]   FIGURE 4   V/Vmax versus [ATP]. Actin velocities measured in a motility assay are plotted for different ATP concentrations and are fitted to Eq. 7 (line), giving a value for KM of 76 ± 17 µM (SEM). Vmax is 2.6 µm s1. Each data point represents the average velocity from four different experiments, and the error bars represent the SEM. All four experiments showed a similar [ATP] dependence.

The effects of [ADP] on single molecule and ensemble kinetics

We determined the effective ADP binding rate, k_D, from MV distributions, _on(t_w), acquired at 100 µM ATP and various ADP concentrations (0-5 mM; Fig. 5). We chose 100 µM ATP rather than 1 mM ATP because event durations at 1 mM ATP are only marginally within our temporal resolution (_on was estimated by Finer et al. (1994) to be <5 ms). We observed that _on increased when we added ADP. We fitted MV distributions, _on(t_w), acquired at different [ADP] to Eq. 2 (Fig. 5, solid lines), setting k_T = 7.6 × 10⁶ M¹ s¹ and k_D = 100 s¹ as determined above. The fits gave an average value for k_D of 2.7 × 10⁵ M¹ s¹ (Table 1).

View larger version (17K): [in this window] [in a new window]   FIGURE 5   Effects of [ADP] on ton. Characteristic plots of MV densities, on(tw), versus window width, tw, obtained from data records acquired at 100 µM ATP and 0 (), 1 (), 2.5 (), and 5 () mM ADP are fit to Eq. 2 (lines). Plots are normalized to the MV density at 20 ms.

To determine the effects of [ADP] on actin filament velocities, V, we performed motility experiments under conditions (100 µM ATP and 0-5 mM ADP) similar to those used above in our single-molecule experiments. The velocities acquired at different ADP concentrations are plotted in Fig. 6, and show that V slowed when we added ADP. We fitted these data to a variation of Eq. 7 (see Fig. 6, legend), and the fit gave a value for an ADP inhibition constant, K_I(vel), of 222 µM.

View larger version (10K): [in this window] [in a new window]   FIGURE 6   V/Vmax versus [ADP]. Actin velocities measured in a motility assay are plotted for different ADP concentrations and fitted to V/Vmax = [ATP]/{[ATP] + KM(vel)(1 + [ADP]/KI(vel))}. This relationship is obtained by substituting KM(vel) for KM(vel)(1 + [ADP]/KI(vel)) in Eq. 7, where KI(vel) is an inhibition constant. The fit gave a value for KI(vel) of 222 µM, when KM(vel) was set to the value of 76 µM obtained from our ATP-dependent motility data. Vmax is 2.6 µm s1. Each data point represents the average velocity from three different experiments, and the error bars represent the SEM.

The effects of P_i on single molecule and ensemble kinetics

According to the scheme in Fig. 1, myosin can detach from actin through one of two pathways: ATP binding to the A·M state or P_i binding to the A·M·D state. To test this hypothesis, we acquired single-molecule displacement data from skeletal muscle myosin at 0.1 µM ATP both in the presence (Fig. 7 a) and absence of 40 mM P_i. We chose these conditions in an attempt to clearly separate the lifetimes of the two detachment processes. For P_i to induce myosin detachment, P_i must bind to myosin prior to ADP release. Thus, if we are to observe a significant population of P_i-induced detachments, the concentration of P_i must be sufficiently high so that the probability of P_i binding is comparable to the probability of ADP release. To distinguish a P_i-dependent t_on population from an ATP-dependent t_on population, we chose a low ATP concentration of 0.1 µM ATP. We estimate the ATP-dependent lifetime of the A·M state to be ~1300 ms at 0.1 µM ATP, which is significantly longer than the ~10 ms lifetime of the A·M·D state determined above.

View larger version (50K): [in this window] [in a new window]   FIGURE 7   Effects of Pi on ton. (a) A characteristic data trace acquired at 0.1 µM ATP and 40 mM Pi contains some long events (lower trace) and an inordinately large number of short events (expanded scale, upper trace). (b) Event densities, non(t), acquired at 0.1 µM ATP and 40 mM Pi indicate a short event population that is contained within the first t = 200-ms bin (point), with the remainder of the points belonging to a significantly longer event population. (Lower inset) The long event population is replotted and fit to Eq. 3 (dashed line), giving a value for kT of 7.8 × 106 M1 s1 that is similar to the value of 11.3 × 106 M1 s1 obtained from a fit (solid line) to the data (solid circles) acquired in the absence of added Pi. Plots are normalized to the number of events at 400 ms. Upper inset: The first 200-ms bin (containing the short event population) is expanded into smaller 10-ms bins (open circles) and is fit to Eq. 8, setting kD = 100 s1 and kT = 7.8 × 106 M1 s1. The fit gives a value for kPi of 1.1 × 103 M1 s1. The curve (solid line) in Fig. 7 b (large graph) is a plot of Eq. 8, with kD = 100 s1, kPi = 1.1 × 103 M1 s1, and kT = 7.8 × 106 M1 s1.

In Fig. 7 a, a portion of a data trace acquired at 0.1 µM ATP and 40 mM P_i contains some long events along with an inordinate number of short events. In Fig. 7 b, we plotted the n_on(t) distribution from these experiments, and the data clearly indicate the presence of two event populations. One population has long event durations characteristic of those expected at 0.1 µM ATP and the other event population, only observed in the presence of P_i, has short event durations. Thus it appears that these long- and short-event populations correspond to the ATP- and P_i-induced detachment processes, respectively. These two independent processes should result in two independent n_on(t) distributions. The short-event population appears to be completely contained within the first 200-ms bin (Fig. 7 b), and thus the remainder of the data should be described by the n_on(t) distribution predicted for the long (ATP-induced) population alone, which, at low [ATP], is Ak_T[ATP]exp(k_T[ATP]t) (Eq. 3). In the lower inset of Fig. 7 b, we fitted the long-event data (open circles) to this equation, setting k_D = 100 s¹. The fit (dashed line) gave a value for k_T of 7.8 (± 0.7) × 10⁶ M¹ s¹, which is comparable to the value for k_T of 11.3 (± 0.6) × 10⁶ M¹ s¹ obtained when we fitted (solid line) 0.1 µM ATP data (closed circles) acquired in the absence of P_i to the same equation. Analogous to n_on(t) for the ATP-induced population, the n_on(t) distribution for the P_i-induced population is Ak_Pi[P_i]exp(k_Pi[P_i]t). Because the fraction of the total number of events in the n_on(t) distribution attributable to the ATP-dependent pathway is k_D/(k_D + k_Pi[P_i]) and the fraction attributable to the P_i-dependent pathway is k_Pi/(k_D + k_Pi[P_i]), the overall n_on(t) distribution predicted for these experiments is

(8)

In the upper inset of Fig. 7 b, we expanded the first 200-ms bin (which contains the short-event population) into smaller 10-ms bins, and we fitted these data to Eq. 8 after setting k_D = 100 s¹ and k_T = 7.8 × 10⁶ M¹ s¹. The fit gave a value for k_Pi of 1.1 (± 0.4) × 10³ M¹ s¹. In the discussion, we use these results to assess the timing of myosin's mechanical step relative to P_i release.

The effects of a myosin mutation that inhibits P_i release

To investigate further the timing of the mechanical step relative to P_i release, we used an expressed smooth muscle HMM mutation that has been shown to limit actin's ability to accelerate P_i release (k_Pi in Fig. 1 is reduced to less than 0.2 s¹) without significantly altering the kinetics of other steps in the biochemical cycle (Joel et al., 2001). A similar mutation in skeletal muscle myosin would have provided a more direct comparison with the above data but was unavailable. Nevertheless, because the same mechanochemical scheme (Fig. 1) is thought to apply to all muscle myosin types, the relationship between myosin's mechanical step and P_i release measured in smooth muscle HMM should be the same as that for skeletal muscle myosin. If myosin's mechanical step precedes P_i release, then we would expect this mutation would significantly increase the event lifetime, whereas, if the mechanical step occurs with or after P_i release, then we would expect this mutation would reduce the event frequency without affecting the event lifetime.

We acquired displacement data from the mutant at 10 µM ATP so that we could compare these data with our previous displacement data acquired from wild-type smooth muscle HMM (Lauzon et al., 1998). We observed a displacement event frequency of <0.1 s¹, which is considerably less than the frequency of ~2 s¹ observed for the wild type. This may explain why the mutant does not support actin filament movement in an in vitro motility assay (Joel et al., 2001). From the events that we did observe (Fig. 8), we measured an average step size of ~8 nm, comparable to that previously reported for the wild-type control (Lauzon et al., 1998). Moreover, we calculated an event lifetime of _on = 198 ± 76 ms that is also comparable to the lifetime of _on = 158 ms previously reported for the wild-type myosin at 10 µM ATP (Lauzon et al., 1998). These results, showing that the mutation decreases event frequency without significantly altering event lifetimes, indicate that the mechanical step occurs with or after P_i release.

View larger version (33K): [in this window] [in a new window]   FIGURE 8   Effects of a Pi release rate myosin mutation on on. (a) Characteristic data records are shown for both wild-type smooth muscle myosin (upper trace; data from Lauzon et al., 1998) and mutant smooth muscle myosin (lower trace; see text for mutant details). (b) The non(t) distribution obtained from mutant data records.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSIONS APPENDIX REFERENCES

The kinetics of the actomyosin ATPase reaction have been characterized extensively both in solution and in muscle fiber preparations (for reviews see Goldman 1987, and Sellers 1999), and a minimal scheme is presented in Fig. 1. In the present study, we used the laser trap assay to characterize the effects of substrate (ATP) and product (ADP, P_i) concentrations on event durations, t_on, of single myosin molecules, and the observed effects were accounted for by this scheme. Using a kinetic analysis based on this scheme, we determined rate constants (summarized in Table 1) from t_on distributions acquired under nearly unloaded conditions, low ionic strength, a temperature of 25°C, and a pH of 7.4. In parallel experiments, we used an in vitro motility assay to characterize how changes in substrate and product concentrations affected the actin velocity generated by an ensemble of myosin molecules. Here we compare our single molecule data (acquired under nearly unloaded conditions) with previous solution kinetic studies (acquired under fully unloaded conditions), and discuss apparent discrepancies, focusing on possible differences between the two assays. Moreover, we compare our single-molecule data with our ensemble velocity data (presumably limited by the load of actin-attached myosin), using a model in which myosin's mechanical step is partitioned between moving an actin filament and generating force.

Kinetics of the actomyosin ATPase cycle

By varying [ATP] from 0.1 to 100 µM, we determined values for the second-order ATP-induced dissociation rate (k_T = 7.6 × 10⁶ M¹ s¹) and the effective ADP release rate (k_D = 100 s¹). The value we obtained for k_T is similar to the value of 5.5 × 10⁶ M¹ s¹, whereas the value for k_D is lower than the value of >300 s¹ estimated from solution studies of fast chicken skeletal muscle subfragment-1 (Marston and Taylor, 1980). One possible explanation for this three-fold difference between k_D values is that the slight positive strain that exists in the laser trap (~0.4 pN) may decrease the ADP release rate below that of unloaded myosin in solution. This explanation is supported by evidence suggesting that k_D is highly strain sensitive. For example, in skinned psoas fiber studies k_D was estimated to be 13.5 s¹ for positively strained myosin but increased to as high as 400 s¹ when the myosin was negatively strained (Dantzig et al., 1991). Moreover if, as assumed in the original Huxley (1957) two-state crossbridge model, the rate of crossbridge detachment (g) is strain dependent, then, considering that detachment is limited by ADP release under physiological conditions (Siemankowski et al., 1985; Weiss et al., 2001), ADP release would be the strain-dependent step in the detachment process.

Another possible explanation for the three-fold difference between the value for k_D estimated in the present study and that measured in solution studies is that k_D in our analysis may not be the ADP release rate measured in solution studies. In our analysis, k_D is the effective rate at which myosin releases ADP, starting from the state occupied at the onset of strong binding, whereas, in solution studies, k_D is the rate at which myosin releases ADP starting from the state occupied following the addition of ADP to the A·M state. Thus a relatively slow myosin isomerization that follows strong actin binding and precedes ADP release (A·M*·D  A·M·D; Sleep and Hutton, 1980) would account for the relatively low value for k_D estimated from the laser trap assay.

From the laser trap experiments in which we varied [ADP], we determined a second-order ADP-binding rate, k_D, of 2.7 × 10⁵ M¹ s¹. This value is similar to the value of 1.5 to 3.8 × 10⁵ M¹ s¹ reported for skinned psoas fibers (Dantzig et al., 1991) but is an order of magnitude less than the ADP binding rate obtained from solution studies (Geeves, 1989). One possible explanation for why values for both k_D and k_D determined in the laser trap are significantly less than the ADP binding and release rates measured in solution studies is that structural constraints in the laser trap assay may restrict the thermal fluctuations of the actomyosin system, effectively increasing the activation energy barrier for both ADP binding and release rates. Another possible explanation is that, as argued above for k_D, a significantly populated A·M*·D state exists, which is not included in our kinetic analysis, resulting in an underestimate of k_D.

Kinetics of the mechanical step

The largest free energy drop within the actomyosin ATPase cycle is associated with P_i release (White and Taylor, 1976). Therefore, the motion-generating step of the cycle has been linked to this biochemical transition. However, the timing of the mechanical step relative to the release of P_i has yet to be determined. To address this question, we took advantage of a myosin mutation that significantly reduces the actin-activated P_i release rate. We showed that the mutation had little effect on the event lifetime, implying that the mechanical step could not occur prior to the biochemical transition affected by the mutation (i.e., P_i release) and thus must occur after or concomitant with P_i release.

To further test the hypothesis that the mechanical step is closely associated with P_i release, we analyzed t_on distributions for skeletal muscle myosin acquired at 0.1 µM ATP both in the presence and absence of added P_i. Our data imply that P_i can induce myosin detachment from actin via a pathway distinct from the ATP-induced actin dissociation pathway (Fig. 1). Our data do not rule out the existence of a strong-binding ternary complex (A·M·D·P_i), but they do place limits on its lifetime and temporal relationship to the mechanical step. We consider the following scheme:
If the A·M·D·P_i state in Scheme 1 is a strong binding state that precedes a mechanical step, the lifetime of this state must be shorter than the time resolution of our laser trap assay (<2 ms). This follows from the fact that we do not detect a change in variance (the signature of strong binding) that precedes a mechanical step. A similar conclusion was reached based on single-myosin molecule fluorescence polarization measurements (Warshaw et al., 1998) and rapid freezing EM images (Walker et al., 1999).

If the A·M·D·P_i state in Scheme 1 is a strong binding state that follows a mechanical step, as concluded from several different muscle fiber mechanics studies (Kawai and Halverson, 1991; Dantzig et al., 1992; Walker et al., 1992), the lifetime of this state must also be extremely short. This follows from the fact that we do not observe an event population with a short lifetime in the absence of P_i, and so, either the lifetime of this state must be shorter than the time resolution of our assay (i.e., in Scheme 1, 1/k_A < 2 ms), or the probability of actin dissociation from this state is low relative to P_i release (i.e., in Scheme 1, k_Pi k_A). For the latter case, our data show that k_A is at least 44 s¹ (k_A > 1.1 × 10³ M¹ sec¹ × 40 mM), and so k_Pi in Scheme 1 must have a value greater than 400 s¹ to explain the absence (<10%) of a short t_on population at low [ATP] and no added P_i. Thus, if a strongly bound A·M·D·P_i state follows the mechanical step, it must have a relatively short lifetime (<3 ms). Of those muscle mechanics studies that imply the existence of this state, most imply that it is short-lived (Fortune et al., 1991; Ranatunga, 1999). In general, our single-molecule data suggest that strong-binding A·M·D·P_i states, whether they precede or follow the mechanical step, are not metastable states; i.e., they are extremely short lived (<3 ms). Within the current time resolution of our detection system, it appears that myosin's mechanical step is closely associated with both P_i release and strong actin binding (Fig. 1).

Comparison of single molecule and ensemble data

We have shown that a laser trap assay can be used to directly correlate the mechanics and kinetics of the actomyosin ATPase cycle. In the past, the in vitro motility assay was thought to be capable of serving a similar purpose. Two equations have typically been used for extrapolating kinetic constants from the ATP-dependence of actin velocities, V, measured in a motility assay; they are V = /_on and the Michaelis-Menten equation, V = V_max[ATP]/([ATP] + K_M(vel)). However, neither expression accurately describes our velocity data acquired over all ATP concentrations. First, we observed that K_M(vel) differed from K_M(on). Second, the relationship between actin velocities and ATP concentration was not accurately described by a Michaelis-Menten equation (see least squares fit in Fig. 4). Implicit in the two equations above is the assumption that the load of actin-attached myosin molecules in a motility assay fully limits actin movement without affecting actin-myosin detachment kinetics measured in a single molecule laser trap assay. It appears that a better understanding of the relationship between actin-myosin ATPase kinetics, myosin's mechanical step, load, and actin velocity is needed.

In the Appendix, we develop a simple model of myosin-based actin movement. Briefly, we assume that, when a mechanical step (defined here to include the mechanism by which actin is moved in an unloaded laser trap) occurs against a resistive load, it is partitioned between moving an actin filament and stretching internal compliant elements in the actomyosin system. In one extreme limit of the model (Appendix, Case 2), actin-attached myosin heads in an ensemble impose a high-resistive load against which a mechanical step can only generate force (i.e., it stretches compliant elements). This is the classic detachment-limited model of actin motility, and indeed for the case in which a single biochemical step limits detachment, we derived the widely used relationship V = /_on. The general expression we derived for the ATP-dependence of V (Eq. A2) is non-Michaelian, but its deviation from a Michaelis-Menten (hyperbolic) relationship is relatively subtle and still does not account for the more dramatic departure from a hyperbolic relationship exhibited by our data at high [ATP] (see Fig. 9 a). Nevertheless, we use this model as a starting point for a quantitative comparison of our single molecule and ensemble data as follows.

View larger version (15K): [in this window] [in a new window]   FIGURE 9   Effects of [ATP] and [ADP] on on and V. (a) From single molecule data acquired at [ATP]  10 µM and no added ADP or Pi, on values were calculated using Eq. 6, with kT = 7.6 × 106 M1 s1 and kD set to the values obtained at each [ATP]. These on values (black circles), along with values for /V (red circles,  = 8 nm), are plotted versus 1/[ATP]. The black line is Eq. 6, with kT = 7.6 × 106 M1 s1 and kD = 100 s1. The velocity data acquired at [ATP] < 100 µM is fit to Eq. 10 (red line), with the fit giving values for kD, kT, and KM = kD/kT of 128 s1, 7.0 × 106 M1 s1, and 18 µM respectively. (b) For single-molecule data acquired at 100 µM ATP and various [ADP], on values were calculated from Eq. 5, setting kD = 100 s1 and kT = 7.6 × 106 M1 s1. These values (black circles), along with values for /V (red circles,  = 8 nm), are plotted versus [ADP]. The velocity data are fit to Eq. 9, setting kD = 128 s1 and kT = 7 × 106 M1 s1. The fit gave a value for kD of 5.1 × 105 M1 s1.

In Fig. 9 a, we replotted our velocity data as /V (which, according to a detachment-limited model, should roughly equal _on) versus 1/[ATP], and, on the same graph, we plotted our single molecule _on data versus 1/[ATP]. Figure 9 a shows that, at [ATP] < 100 µM, our motility data roughly follow our _on data (i.e., V = /_on) and exhibit Michaelis-Menten behavior (i.e., a linear relationship in a double reciprocal plot) as predicted by a detachment-limited model. Even the apparent slight downward curvature in the low [ATP] velocity data is predicted by our detachment-limited model (Eq. A2) as a transition from velocities limited by ATP binding (at low [ATP]) to velocities limited by ADP release (at high [ATP]). A least squares fit of these low [ATP] motility data to Eq. A2 (Fig. 9 a, red line) gave values for k_D, k_T, and K_M = k_D/k_T of 128 ± 9 s¹, 7.0 (± 0.8) × 10⁶ M¹ s¹, and 18 ± 3 µM respectively. These values are in excellent agreement with those obtained from our single-molecule data for k_D, k_T, and K_M of 100 s¹, 7.6 × 10⁶ M¹ s¹, and 13 µM, respectively.

We also used this model to compare the ADP-dependence of our single-molecule and ensemble data by plotting both /V for motility data and our single-molecule _on data versus [ADP] (Fig. 9 b). Once again, our motility data appear to follow our _on data as V ~ /_on. A fit of our motility data to Eq. A1 provides estimates for k_D of 5.1 × 10⁵ M¹ s¹ (see Fig. 9 b, legend), which is comparable to the value of k_D = 2.7 × 10⁵ M¹ s¹ estimated from our single-molecule experiments. Thus it appears that the detachment-limited model of actin velocities (Appendix, Case 2) accurately describes our velocity data acquired at [ATP]  100 µM both in the absence and presence of ADP.

However at [ATP] > 100 µM (Fig. 9 a, arrow) our motility data deviate both from our _on data (i.e., V  /_on) and from a Michaelis-Menten [ATP] dependence. When we fitted our velocity data acquired at [ATP] > 100 µM to Eq. 7, we obtained a value for K_M(vel) of 174 µM. This value is similar to values for K_M of 150-200 µM obtained from in vitro motility (Homsher et al., 1993) and muscle fiber studies (Cooke and Bialek, 1979) but is roughly an order of magnitude greater than the values for K_M(on) and K_M(vel) obtained from velocity data acquired at [ATP] < 100 µM. Similarly, from the velocity data acquired at [ATP] > 100 µM, we estimated a value for k_D (= V_max/) of 313 s¹, which is significantly greater than the value of 100 s¹ obtained from our single-molecule _on data. These apparent discrepancies between kinetic constants determined from V and _on data are not limited to chicken skeletal myosin. For many different myosin types, the values for k_D that we estimate from actin velocities (by assuming V = /_on) are consistently more than two-fold greater than the values for k_D that we estimate from _on data (Tyska and Warshaw, 2002). What we have shown in this paper is that this apparent discrepancy has an [ATP] dependence. Our V data depart both from the relationship V = /_on and from a Michaelis-Menten relationship only when [ATP] is increased above 100 µM. Because the mean step size, , does not depend on [ATP] (Lauzon et al., 1998