Biophysical Journal

Volume 80

January 2001

379 –397

379

Link between the Enzymatic Kinetics and Mechanical Behavior in an
Actomyosin Motor
Ichiro Amitani, Takeshi Sakamoto, and Toshio Ando
Department of Physics, Faculty of Science, Kanazawa University, Kanazawa 920-1192, Japan

ABSTRACT We have attempted to link the solution actomyosin ATPase with the mechanical properties of in vitro actin
filament sliding over heavy meromyosin. To accomplish this we perturbed the system by altering the substrate with various
NTPs and divalent cations, and by altering ionic strength. A wide variety of enzymatic and mechanical measurements weremade under very similar solution conditions. Excellent correlations between the mechanical and enzymatic quantities wererevealed. Analysis of these correlations based on a force-balance model led us to two fundamental equations, which can beAA
described approximately as follows: the maximum sliding velocity is proportional to 公VmaxKm, where Km
is the actin
concentration at which the substrate turnover rate is half of its maximum (Vmax). The active force generated by a cross-bridgeA
under no external load or under a small external load is proportional to 公Vmax/Km. The equations successfully accounted forthe correlations observed in the present study and observations in other laboratories.

INTRODUCTION
Cyclic interactions of myosin heads with actin coupled to
ATP hydrolysis are the molecular basis of muscle contraction, cytokinesis, certain types of vesicle transport, and soforth. Several types of muscles contain myosin heavy andlight chains characteristic of each type. They differ in the
speed of unloaded shortening and in the magnitude ofisometric tension. Even in a specific type of muscle thesemechanical outputs vary depending on conditions. Nonmuscle myosins and actins also exhibit distinct rates of
movement. How does the actomyosin ATPase kinetics determine the mechanical performance of this molecular motor? One approach to this issue is to isolate myosins fromvarious sources, and then compare the kinetics of actinactivated ATPase reaction of these myosins and their mechanical performance. This approach was pioneered byBárány (1967) with myosins isolated from various muscles
having known maximum speeds of shortening. He foundthat the ATPase activity was proportional to the speed ofshortening of the parent muscles. Qualitatively similar correlations have been observed also by other groups withdifferent mammalian smooth muscles (Malmqvist andArner, 1991; Helper et al., 1988), frog skeletal muscle fibers
with different myosin heavy chain composition (Edman etal., 1988), crab muscle fibers of different types (Galler andRathmayer, 1992), and rabbit aorta and small arterial muscles containing myosin heavy chains differing at the NH2-

Received for publication 28 February 2000 and in final form 16 October
2000.
Address reprint requests to Dr. Toshio Ando, Dept. of Physics, Faculty of
Science, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan. Tel.: 81-76-264-5663; Fax: 81-76-264-5739; E-mail: tando@kenroku.
kanazawa-u.ac.jp.
Abbreviations used: NTP, nucleotide triphosphate; BSA, bovine serum
albumin; HMM, heavy meromyosin; NEM, N-ethyl maleimide; S1, myosin
subfragment 1.
© 2001 by the Biophysical Society
0006-3495/01/01/379/19 $2.00

termini (DiSanto et al., 1997). Another approach is to substitute various ATP analogs (NTPs) for ATP, using a given
type of myosin. This approach was initiated by Blum
(1955). Myosin can use a wide variety of NTPs as theenergy source for the motile activity. Hasselbach (1956)
reported an excellent correlation between the rate of NTP
hydrolysis by actomyosin and the isometric tension of glycerinated muscle fibers. More recently, this study has beenrefined by several groups. It has been recognized that various NTPs have different affinities for actomyosin, andmyosins complexed with these NTPs or their products havedifferent affinities for actin (Pate et al., 1993; White et al.,1993). This was not carefully considered in the previous
study by Hasselbach. Moreover, accumulation of the NTP
hydrolysis products in fibers without an NTP-regeneration
system must have reduced the tension. Therefore, the excellent correlation observed in 1956 has to be reexamined.
Recent studies have shown that the isometric tension andunloaded shortening speed of skinned fibers in various
NTPs have only moderate correlation with the acto-HMM
or acto-S1 NTPase activities (Pate et al., 1993; Regnier etal., 1998). We note, however, that measurements of themechanical properties and the chemical kinetics have notbeen made under similar conditions. For technical reasons,
it is often so in studies with muscle fibers, where themechanical measurements are made at higher ionic strength,
while the measurements of solution chemical kinetics are
made at low ionic strength. Moreover, the solution and fiber
enzymatic activities have been measured in limited concentrations of NTPs.
Sheetz and Spudich (1983) developed a motility assay
wherein myosin-coated particles move along parallel tracksof Nitella actin bundles. After the success of fluorescencemicroscopy in visualizing fluorescently labeled single actin
filaments (Yanagida et al., 1984), a more versatile andstable assay was developed wherein actin filaments moveover surfaces coated with myosin filaments (Kron and Spudich, 1986), or proteolytic fragments of myosin (Toyoshima

380

et al., 1987). These developments allow us to investigate
actomyosin motility under wide varieties of solution conditions with myosins and actins from diverse species and cell
types. It has now become possible to investigate the fundamental question regarding the mechanochemical couplingof the actomyosin motor mentioned above in greater detail.
First, it has been shown that the velocity of actin filaments
sliding over myosin-coated surfaces and the velocity ofmyosin-coated beads along actin cables is analogous (withslight differences, depending on conditions) to the speed of
unloaded shortening of muscle fibers (Sheetz et al., 1984;
Homsher et al., 1992). Despite this analogy, many previous
studies have reported no direct correlation between thesteady-state actin-activated MgATPase activity of myosin
and the rate of movement in the in vitro motility assays.
However, we have to be careful in interpreting these results
because identical, or very similar, solution conditions havenot always been used in the motility and enzymatic assays.
Having this caution in mind, let us briefly review several
types of previous studies where rates of movement measured in the in vitro motility assays have been comparedwith the corresponding actin-activated substrate turnoverrates in solution. 1) Dependence on myosin species: beads
coated with myosin from skeletal muscle and Dictyostelium
move on actin cables from Nitella at distinct velocities, andthese velocities are proportional to the respective actinactivated ATPase activities (Sheetz et al., 1984). Phosphorylated platelet myosin and phosphorylated turkey gizzard
myosin have similar actin-activated ATPase activities, yetshow very dissimilar rates of movement in the Nitella-based
assay (Umemoto et al., 1989). Skeletal muscle myosin lightchain isoforms have the same maximum actin-activatedATPase activity, yet again they translocate actin filaments atdistinct velocities (Lowey et al., 1993). Chimera myosins,
constructed from Dictyostelium myosin by substituting the50K/20K junction region with those from other species ofmyosins, exhibit actin-activated ATPase activities characteristic of the activity of the myosins from which the junction region was donated. The velocities of actin sliding
propelled by these chimeras, on the contrary, have no correlation with those driven by the donor myosins (Uyeda etal., 1994). Similarly, no correlation has been found withmyosins from other sources (Vale et al., 1984; HigashiFujime, 1991). However, we have to be cautious about
interpreting these observations, because in most cases actinactivated ATPase activity has been measured at one concentration of actin (Note that Bárány also used one concentration of actin in his 1967 study), and also because theNitella-based assay tends to show slower motility than thesliding actin filament assay, depending on the myosin typeused (Wolenski et al., 1993). Chimeric heavy meromyosins,
which are produced by substituting the 50K/20K loop ofsmooth muscle heavy meromyosin with that from skeletalor ␤-cardiac myosin, lose regulation by regulatory lightchain phosphorylation. The chimeras and the wild type,
Biophysical Journal 80(1) 379 –397

Amitani et al.

however, show moderate correlation of the motile activity
observed in the sliding actin filament assay with the maximum acto-HMM ATPase activity and with the Km for actin(Rovner et al., 1995). Tryptic cleavage of the 25K/50K loopof skeletal myosin inhibits its motor function without any
significant changes in the enzymatic properties of actomyosin, while cleavage of the 50K/20K loop increases the Km
for actin without significant effect on the motor function(Bobkov et al., 1996). The rate constant for dissociation of
ADP from various muscle myosins complexed with actin
has been compared with the maximum shortening speed ofthe parent fibers (Siemankowski et al., 1985). These quantities show a strong correlation, although it is still open toquestion whether the state formed by adding ADP to actomyosin is on the ATP hydrolysis pathway (Sleep and Hutton, 1980). Contrary to this observation, a chimeric myosincontaining the Dictyostelium myosin heavy chain with the
25K/50K loop from skeletal myosin propels actin filaments
at a velocity slightly slower than the wild type, while therate of mant-ADP release from the actin-chimera myosin S1
is more than two times larger than that from the actin-wildtype myosin S1 (Murphy and Spudich, 1998) (the affinities
of mant-ADP and ADP binding to actin-S1 are similar). 2)
Dependence on ATP analogs: the velocity of actin filamenttranslocation has been examined by Shimizu et al. (1991)
using 15 ATP analogs. The relative velocities do not correlate well with the actin-activated substrate turnover rates.
The actin-activated substrate turnover rate varies in a relatively small range compared to the variation in the velocityof actin translocation. A similar study has been made using
naturally occurring nucleotides by another group (HigashiFujime and Hozumi, 1996). Again, no correlation is found.
In these studies, however, one concentration of actin and
one concentration of substrate have been used in the measurements of the enzymatic activities. However, in similar
studies a moderate correlation is found between the motile
activity and the maximum substrate turnover rate (Pate etal., 1993; Regnier et al., 1998). In these studies the actin
concentration is varied, but one concentration of substrate isused in the measurements of the substrate turnover rate. 3)
Dependence on actin species: the maximum MgATPase
activity of skeletal muscle myosin activated with yeast actin
is significantly lower than with skeletal muscle actin,
whereas the apparent Km for yeast actin of myosin heads isslightly larger than for skeletal actin (Cook et al., 1993).
Nevertheless, the sliding velocities of both actins are quite
similar, while force production of skeletal muscle heavy
meromyosin with yeast actin is lower than with skeletal
muscle actin (Cook et al., 1993; Kim et al., 1996; Miller etal., 1996). Subtilisin cleavage of skeletal muscle actin at
Met-47 and Gly-48 markedly increases the Km for actin of
myosin heads without changing the maximum ATP turnover rate. The sliding velocity of the cleaved actin filaments
is slower than that of intact actin filaments (Schwyter et al.,1990). In contrast to the effect of subtilisin cleavage of actin

Link of Chemistry and Mechanics in Actomyosinon the actin-activated ATPase of myosin, replacement by
mutagenesis of aspartic acid residues of Dictyostelium actinwith histidine residues reduces the maximum ATP turnoverrate with moderate reduction in the Km for actin. Thisreplacement also reduces the sliding velocity of actin filaments (Sutoh et al., 1991). Actins modified at Cys-374 bydifferent fluorophores have parallel effects on the velocityof actin sliding over HMM, the maximum actin-activatedS1 ATPase activity, and the affinity of actin for S1 in ATP(Crosbie et al., 1994). 4) Dependence on other factors:
binding of tropomyosin to actin has a parallel enhancement
effect on the actomyosin ATPase activity and the rate of
movement (Umemoto et al., 1989; Umemoto and Sellers,
1990; Okagaki et al., 1991; Wang et al., 1993). This effectis ascribed to an increase in the maximum ATPase activity(Umemoto et al., 1989); the optimum pH is around pH 7.0
for the actomyosin ATPase activity (Stone and Prevost,
1973). For the motile activity, the optimum pH has beenreported to be around pH 7.0 (Sheetz et al., 1984; Warshaw
et al., 1990; Sugiura et al., 1992), or pH 8.5 (Homsher et al.,1992). In the former three studies the rate of movement at
pH 8.5 is very much less than at pH 7.0. The cause of thisdiscrepancy is unknown. We have to note, however, thatparallel measurements of the pH effects on the ATPase andthe motile activities have never been carried out under theidentical or very similar solution conditions. Within a rangeof ionic strength where smooth movement of either actin
filaments or myosin-coated beads is observed, the rate of
movement increases with increasing ionic strength(Homsher et al., 1992; Umemoto and Sellers, 1990; Warshaw et al., 1990; Harada et al., 1987; Takiguchi et al.,
1990; Saito et al., 1994; Vale and Oosawa, 1990). The speedof isotonic and unloaded shortening of skinned muscle
fibers also increases with increasing ionic strength when theionic strength is lower than 100 mM (Gulati and Podolsky,
1981; Arheden et al., 1988). Although an increase in ionicstrength is known to reduce the affinity (particularly in thepresence of ATP), of myosin heads for actin, a systematiccomparison of the effect of ionic strength on the motile
activity with that on the actomyosin ATPase activity has
never been made.
As seen in recent many reports, the relationship between
the rate of movement and the actomyosin ATPase kinetics
does not seem as simple as first suggested by Bárány in1967. Our present situation regarding this central issue ofthe actomyosin motor seems confused. In part, this confusion may be because the identical or very similar solution
conditions have not been used for measurements of the
motile and enzymatic activities. Discrepancy is often seen in
the previous studies wherein the ATPase assay is performed
in a solution whose ionic strength is a few times lower thanthe motility assay solution. If the affinities for myosin of thetwo actin species have a different dependence on ionicstrength, the order of the affinities can be reversed bychanging ionic strength. Thus, we would misconstrue the381

relationship between the motile activity and the actin affinity for myosin. In the present study, care was taken in thisrespect. To examine the links between the motor function
and the enzymatic reaction of actin-heavy meromyosin thesystem was perturbed by altering the substrate with various
NTPs and divalent cations, and by altering ionic strength. It
was also taken into account that various NTPs have different affinities for actomyosin, and that myosins complexedwith these NTPs or their products have different affinities
for actin. The experiments made with these precautions
revealed an excellent correlation between the mechanicalproperties and the kinetics of substrate hydrolysis. Moreover, to account for these correlations we constructed amodel based on an idea that the balance of a positive force
and a velocity-dependent negative force determines the
maximum velocity of movement. This idea was originallyproposed by Huxley (1957). Equations derived from thismodel linked the chemical kinetics to the actin translocation
velocity and the force generated without external load orunder a small external load. The equations coincided withthe experimentally revealed correlations in the present
study, and also with observations from other studies.
MATERIALS AND METHODS
Preparation of proteins
Myosin and actin were prepared from rabbit skeletal muscle according to
the methods of Tonomura et al. (1966) and Spudich and Watt (1971),
respectively. HMM was obtained by chymotryptic digestion of freshly
prepared (not glycerinated) myosin according to Weeds and Pope (1977).
After centrifugal removal of the nondigested myosin and light meromyosin
in a solution of low ionic strength, HMM was quickly frozen in liquid
nitrogen and stored in liquid nitrogen. The molar concentration of HMM
was estimated on the basis of E1%
280 ⫽ 7.0 and a molecular weight of 3.5 ⫻105, with correction for the turbidity (1.93 times the 330 nm absorbancewas subtracted from the 280 nm absorbance). The molar concentration of
F-actin was estimated on the basis of E1%
290 ⫽ 6.5 and a molecular weightof 4.2 ⫻ 104 (1.68 times the 330 nm absorbance was subtracted from the
290 nm absorbance to correct for turbidity artifacts).

Actin-activated HMM NTPase assays
In the measurements below the concentration of HMM was adjusted,depending on the actin concentrations, NTPs and divalent cations, so thathydrolyzed NTP at the last time point amounts to ⬍15% of the initialamount of NTP. In the first set of experiments we used various NTPs toalter the enzymatic kinetics of acto-HMM. NTPs we used are ATP, CTP,
TTP, UTP, ITP, and GTP. Actin was polymerized in a solution containing
100 mM KCl, 1 mM MgCl2, 0.2 mM CaCl2, 5 mM Tris-HCl (pH 8.0), and
0.2 mM ATP. To remove ATP from the actin sample, the polymerizedactin was centrifuged at 150,000 ⫻ g for 1 h. The resulting pellet wasdispersed in Buffer A (25 mM KCl, 25 mM imidazole-HCl (pH 7.6), 2 mM
MgCl2, 0.2 mM CaCl2) and dialyzed against the same buffer. HMM wasmixed with various concentrations of F-actin in Buffer A. The NTPasereaction was initiated by the addition of NTP. The concentration of NTP
was varied. In the second set of experiments we used various divalentcations as the complexing agents with ATP. Divalent cations we used are
Mg2⫹, Mn2⫹, Ni2⫹, and Sr2⫹, all in the chloride form. Except for the caseof Mg2⫹, actin pellet was dispersed in Buffer B (25 mM KCl, 25 mM
Biophysical Journal 80(1) 379 –397

382

Amitani et al.

imidazole-HCl (pH 7.6), 0.2 mM CaCl2) and dialyzed against Buffer B.
The ATPase reaction was initiated by adding 2 mM ATP and 2 mM
divalent cation together to a solution containing HMM and various concentrations of F-actin in Buffer B. In the third set of experiments actinactivated HMM MgATPase activity was measured in solutions of various
ionic strengths. The solvents (Buffer C) contained various concentrationsof KCl, 25 mM imidazole-HCl (pH 7.6), 2 mM MgCl2, and 0.2 mM CaCl2.
The concentration of MgATP was 2 mM. When the acto-HMM MgATPase
activities were measured at [ATP] ⬍ 0.2 mM, an ATP-regeneration system
consisting of phosphoenol pyruvate, pyruvate kinase, NADH, and lactate
dehydrogenase was used. The reaction was monitored by measuring thetime course of the change in absorption of NADH at 340 nm. In all theother cases, amounts of phosphate liberated at 25°C were quantified by themethod of Fiske and Subbarow (1925). The activities of HMM alone ateach condition of substrate or ionic strength were subtracted from eachdetermined activity of acto-HMM under the identical condition. The NTPase activities were determined by analyzing the amounts of phosphateliberated at six time points by the least-squares criterion. The absence ofthe perturbation effect of Mn2⫹, Ni2⫹, and Sr2⫹ upon the phosphate assay
was confirmed.

Determination of kinetic parameters
At a given concentration of NTP the acto-HMM NTPase activity, VNTP,
was measured as above at seven different actin concentrations. The maxA
imum activity (Vm) at infinite concentration of actin and the actin concentration, KA
0.5, that gave the half maximum activity were determined, asimple hyperbolic dependence of VNTP on actin concentration being asA
sumed. The values of Vm
and KA
0.5 were obtained at six different concentrations of NTP. The maximum activity (Vmax; see Table 1 for parameters)A
at infinite concentrations of actin and NTP, and the actin concentration, Km
A(K0.5 at infinite concentration of NTP), were determined, simple hyperbolicA
dependence of Vm
and KA
0.5 on MgNTP concentration being assumed. The
N
A
MgNTP concentration, Km, at which Vm
is half of Vmax, was determinedAfrom the relationship of Vm
versus [MgNTP].

TABLE 1

List of parametersd
具fs典

Length of power stroke
Magnitude of sliding force (active force) averaged over thesubstrate turnover time
具fr典 Time-averaged magnitude of resistive force
Fu
Force required to unbind a rigor head⌫
Elastic constant of cross-bridgek⫺w First-order dissociation rate constant of weakly bound cross-bridgek⫹w Second-order binding rate constant in the weak-binding statesk⫹1 Second-order rate constant of substrate binding to A 䡠 M
k⫹6 Forward rate constant in the rate-limiting step (isomerization ratefrom A 䡠 M 䡠 ADP 䡠 Pi to A 䡠 M* 䡠 ADP 䡠 Pi)
AKm
Actin concentration for half-maximum NTPase rate
N
Km
Substrate concentration for half-maximum NTPase rate
KsN Substrate concentration for half-maximum sliding velocity
␭w Fraction of weakly bound cross-bridge
␭rig Fraction of rigor cross-bridgepMagnitude of power stroke (force spike)␳
Molar ratio of NEM-HMM over intact HMM[S] Substrate concentration
Tc
Substrate turnover time
Tp
Duration of power stroke
Tw Lifetime of weakly bound cross-bridge
Vmax Maximum substrate turnover rate
Vs
Sliding velocity of actin filament
Vsmax Maximum velocity of actin sliding
Biophysical Journal 80(1) 379 –397

In vitro motility assay
Sliding filament in vitro motility assays were carried out as described in(Toyoshima et al., 1987) with some modifications. The temperature wasmaintained at 25°C for all assays. To pre-remove ATP-insensitive HMM,
5 ␮M HMM was mixed with 10 ␮M F-actin in a solution containing 0.1
M KCl, 3 mM MgCl2, 0.2 mM CaCl2, and 5 mM Tris-HCl (pH 8.0) at
25°C. The mixture was incubated for 20 min and then cooled to 0°C. Afteradding 2 mM ATP and 1 mM potassium pyrophosphate, the mixture wasimmediately centrifuged at 150,000 ⫻ g for 1 h. HMM in the supernatant
was diluted to 0.3 ␮M with Buffer A, and then applied to a flow cell (⬃50
␮l in volume) made of two coverslips (24 ⫻ 36 mm2 and 22 ⫻ 22 mm2),
where the larger coverslip had been coated with nitrocellulose. The cell
was incubated for 2 min. Unattached HMM was washed out by applying 50
␮l of Buffer A from one side of the cell and by sucking, with a piece offilter paper, from the other side of the cell. This wash was repeated fivetimes. BSA (1 mg/ml, 100 ␮l) dissolved in Buffer A was then applied tothe cell, incubated for 2 min, and unattached BSA was washed either withBuffer A (when NTPs were the alterant), Buffer B (when divalent cations
were the alterant), or Buffer C (when [KCl] was the alterant). Tetramethylrhodamine-phalloidine-labeled F-actin (2 nM, 100 ␮l) in the same buffer
solution was applied to the cell and incubated for 30 s. The motion of actin
filaments was initiated by applying 100 ␮l of a test solution that additionally contained oxygen-scavenging reagents (1% 2-mercaptoethanol, 4.5
mg/ml glucose, 0.216 mg/ml glucose-oxidase, 0.36 mg/ml catalase). When
Ni2⫹ was used as a complexing agent with ATP, 2-mercaptoethanol wasomitted from the oxygen-scavenging reagents. The test solution conditions
were the same as those used for the actin-activated HMM NTPase assays.
Both open sides of the flow cell were sealed with white Vaseline. The
sliding motion of individual actin filaments was observed under an epiluminescence fluorescence microscope (Olympus IX70, Tokyo, Japan;
equipped with an oil-immersion objective, 100⫻, NA 1.35), the images
being taken with a SIT video camera (C2400-08, Hamamatsu Photonics,
Shizuoka, Japan) and being recorded with a Hi8 video cassette recorder(EVO-9650, Sony, Tokyo, Japan). The recorded images were digitizedwith an image processor (Excel, Nippon Avionics, Osaka, Japan) and thetwo-dimensional coordinates of the rear end of each actin filament werechased frame by frame. Length of a track connecting these coordinates
during a given period of time was calculated and averaged over ⬎50 actin
filaments. The sliding velocity was obtained by dividing the average valueof the lengths of tracks by the given period of time.

In vitro motility assays withnoncycling cross-bridges
Chemically damaged and therefore noncycling HMM was prepared byextensive treatment of HMM with NEM. HMM (30 ␮M) dialyzed against
30 mM KCl, 25 mM Tris-HCl (pH 8.0), and 0.1 mM PMSF was mixed
with 9 mM NEM and incubated for 1 h at 25°C. The sample was cooled in
an ice-water bath, mixed with 180 mM dithiothreitol, and dialyzed against
a large volume of 25 mM KCl, 2 mM MgCl2, 0.2 mM CaCl2, 25 mM
TES-KOH (pH 7.0), 1 mM 2-mercaptoethanol, and 0.1 mM PMSF. The
modified sample was frozen in liquid nitrogen and stored in liquid nitrogen.
Intact HMM pretreated for removing ATP-insensitive HMM heads was
mixed with NEM-treated HMM at given ratios in either Buffer A, Buffer
B, or Buffer C. The total concentration of intact and noncycling HMM wasadjusted to 0.3 ␮M. The succeeding procedures were the same as thosementioned in the preceding subsection. Immediately after initiating the
movement of actin filaments, the fluorescent images of actin filaments
were recorded at five different surface areas. The recording time per one
area was adjusted depending on the velocity of sliding actin filaments.
When the velocity was relatively high, it was minimized to be 20 s. Whenthe velocity was very low, it was ⬃2 min. After recording, ⬃40 actin
filaments on one observation area were arbitrarily picked up for analysis of
the average sliding velocity as a function of the molar ratio (␳) of NEM-

Link of Chemistry and Mechanics in Actomyosin

383

HMM to intact HMM. The data were obtained from the five observation
areas and then averaged.

Determination of the relative magnitude ofsliding force
When actin filaments are sliding freely on HMM, the time-averaged active
force generated by a cross-bridge must be balanced with the time-averaged
resistive force generated by a cross-bridge that is attached to actin, but not
generating active force. Let’s call the active force in this situation “sliding
force.” We estimated the relative magnitude of sliding force as follows. We
measure the average sliding velocity, Vs, as a function of ␳. Vs would
decrease with increasing ␳. The intercept of the initial tangent of the Vs
versus ␳ relationship to the abscissa gives a value of ␳ ⫽ ␳s. The ␳s value
is taken as the relative magnitude of sliding force. The theoretical basis of
this method is given in the Discussion.

RESULTS
Sliding Velocity as a Function of
NTP Concentration
Sliding velocity, Vs, of actin filaments was measured in an
in vitro motility assay using various NTPs. Various nucleotides have different affinities for HMM and acto-HMM(Pate et al., 1993; White et al., 1993). Rigor complexes ofacto-HMM, even when in small fractions, may produce
resistive drag force against actin filaments sliding, resulting
in a reduction of the velocity. To estimate the full ability ofeach NTP to support acto-HMM motility without rigor
complexes and to estimate the apparent Michaelis-Menten
constant for each NTP (KsN: NTP concentration at which Vs
is half of its maximum value), Vs was measured at various
NTP concentrations. Fig. 1, a and b give plots of Vs versus[NTP]. At first glance, Vs seemed to show Michaeliansaturation as a function of [NTP], i.e., Vs ⫽ Vsmax/(1 ⫹KsN/[NTP]). Similar experiments have previously been madewith ATP (Sheetz et al., 1984; Homsher et al., 1992;
Umemoto and Sellers, 1990; Warshaw et al., 1990; Haradaet al., 1987) and with the other nucleotides (Cooke andBialek, 1979; Pate et al., 1993; Regnier et al., 1998). Someof these studies, where data of Vs versus [MgNTP] havebeen analyzed quantitatively, have assumed Michaelian saturation behavior. Careful inspection of these and our own
data showed that they did not exactly obey the Michaelian
relationship. At [NTP] lower than KsN, Vs tends to deviate
downward from the Michaelian fitting curve. However, at[NTP] moderately higher than KsN, Vs tends to deviate upward (see, e.g., Fig. 5 in Homsher et al., 1992; Fig. 3 in
Tawada and Sekimoto, 1991). A theoretical consideration
regarding this issue under a certain condition led to a
mixture of a modified Michaelian equation, i.e., Vs ⫽ Vsmax/(1 ⫹ (KsN)2/[NTP]2) and the original Michaelian equation(see Discussion). So, we assumed a modified saturation
behavior, Vs ⫽ Vsmax/(1 ⫹ (KsN)n/[NTP]n), where the parameter, n, is one of the parameters to be determined from aleast-squares fitting of the data to this modified Michaelian

FIGURE 1 Substrate concentration dependence of velocity of actin filament translocation over HMM for (a) ATP (E), CTP (‚), TTP (䊐); and(b) UTP (F), GTP (ƒ), and ITP ({). Lines are fits to the modified
Michaelian equation with n as a variable to be determined, i.e., Vs ⫽Vsmax/(1 ⫹ (KsN)n/[NTP]n).

equation. As listed in Table 2, the value of n varied from 1.4
to 2.2 depending on the data, the average value being 1.66.
The results are summarized in Table 2, together with theresults obtained from the Michaelian data fitting. The standard deviations and ␹2 tests were always smaller when themodified Michaelian saturation with n as a variable wasassumed than when the original Michaelian saturation was
assumed (Table 2). Although we hereafter describe only
values for Vsmax and KsN obtained with the modified Michaelian data fitting, there were no major differences in thecorresponding values obtained with the two types of data
fitting (except for the case with ITP).
The value of Vsmax for CTP was similar to that obtainedwith ATP. This similarity was observed previously in in
vitro motility assays (Regnier et al., 1998), although inmuscle fibers CTP produces shortening velocity 30 –50%
Biophysical Journal 80(1) 379 –397

384

Amitani et al.

TABLE 2 Kinetic parameters of NTP hydrolysis in solution and mechanical properties in the actin sliding assays obtained using
various NTPs as substraten*␹2 tests*
Vsmax (␮m/s)*
KsN (mM)*
Vsmax (␮m/s)†
KsN (mM)†
␹2 tests†
Vmax (s⫺1)‡
A
Km
(␮M)
N
Km
(mM)
VHMM (s⫺1)§
␳s具fs典 (pN)k⫹w/␭w (M⫺1 s⫺1)k⫺w/␭w (s⫺1)

ATP

CTP

TTP

UTP

GTP

ITP

1.42
0.064
4.3 ⫾ 0.18
0.059 ⫾ 0.006
4.6 ⫾ 0.21
0.07 ⫾ 0.02
0.096
23.6
29.1
0.009
0.05
0.144
0.648
6.8 ⫻ 107
1992

1.58
0.058
3.9 ⫾ 0.27
0.76 ⫾ 0.12
4.5 ⫾ 0.32
0.97 ⫾ 0.25
0.079
22.0
28.6
0.11
0.038
0.118
0.531
7.7 ⫻ 107
2202

1.41
0.018
2.2 ⫾ 0.16
0.83 ⫾ 0.14
2.5 ⫾ 0.17
1.12 ⫾ 0.23
0.023
18.5
12.0
0.25
0.075
0.159
0.716
7.7 ⫻ 107922

1.74
0.00671.38 ⫾ 0.07
0.82 ⫾ 0.09
1.68 ⫾ 0.14
1.19 ⫾ 0.29
0.014
13.5
6.87
0.20
0.18
0.207
0.932
6.5 ⫻ 107444

1.57
0.000450.47 ⫾ 0.05
1.76 ⫾ 0.35
0.66 ⫾ 0.09
3.34 ⫾ 1.0
0.00074
1.8
0.27
0.50
0.35
0.169
0.76
6.9 ⫻ 108185

2.23
0.000030.16 ⫾ 0.01
2.10 ⫾ 0.18
0.37 ⫾ 0.12
8.7 ⫾ 4.0
0.00012
4.71
8.0
1.00
0.50
0.177
0.800
7.5 ⫻ 10660

*Obtained by fitting data in Fig. 1, a and b to the modified Michaelian equation with n as a variable.
†
Obtained by fitting data in Fig. 1, a and b to the original Michaelian equation (i.e., n ⫽ 1).
‡§
The unit is Pi/s per head.

less than those obtained with ATP (Pate et al., 1993; Wahr
and Metzger, 1998). The KsN for ATP agreed with publishedvalues (Cooke and Bialek, 1979; Ferenczi et al., 1984;
Homsher et al., 1992; Regnier et al., 1998). The values of
KsN for CTP, TTP, and UTP were quite similar to each other(⬃0.8 mM, 14 times greater than for ATP), while the valuesof Vsmax for these NTPs varied over a relatively wide range.
This indicates that the apparent affinity of NTP for actoHMM does not correlate with Vsmax. The values of KsN for
GTP and ITP were ⬎29 times larger than for ATP, andthese nucleotides were poor substrates for producing thesliding motion of actin filaments, as was previously observed with muscle fibers (Pate et al., 1993; Regnier et al.,1998) and in the in vitro motility assays (Regnier et al.,
1998; Higashi-Fujime and Hozumi, 1996).
Kinetic parameters of acto-HMM NTPase
At a given concentration of NTP the acto-HMM NTPase
activity, VNTP, was measured at various actin concentrations. The NTPs used here were the same as used above.
With all the NTPs used VNTP displayed simple hyperbolic
saturation behavior as a function of [actin] (data not shown).
When the relationship of VNTP versus [actin] was analyzed
with the modified Michaelian equation, with n as a variable tobe determined, we obtained n ⫽ 1.0 on the average. The values
Aof VA
m (VNTP at infinite actin concentration) and K0.5 ([actin] atAwhich VNTP is half of Vm) for all six nucleotides were obtainedby analyzing the hyperbolic saturation curves by the leastAsquares criterion to fit the equation VNTP ⫽ VA
m/(1 ⫹ K0.5/
AA[actin]). Although these parameters, Vm and K0.5, must be lesssensitive to the presence of a small amount of rigor complexes
than Vs, they (except for those with ATP) may not be saturated
in the presence of a millimolar concentration of NTP. We,
Biophysical Journal 80(1) 379 –397

therefore examined these parameters as a function of NTP
Aconcentration. VAm and K0.5 with CTP were nearly saturated at
1 mM CTP. However, these parameters with TTP, UTP, GTP,
and ITP were not saturated at 1 mM [substrates]. VA
m displayed
Michaelian saturation behavior as a function of [MgNTP]. K0.5
versus [MgNTP] also showed Michaelian saturation behavior.
From these saturation curves Vmax (VNTP at infinite [NTP] andA
infinite [actin]) and KA
m (K0.5 at infinite [NTP]) were estimated.
These values are listed in Table 2. Both Vmax and KA
m decreasedin the same order, ATP ⬎ CTP ⬎ TTP ⬎ UTP ⬎ ITP ⬎ GTP,
with KAm varying more widely than Vmax. The NTP concentraA
tions, KNm, at which Vm is half of Vmax, are also listed in TableN
2. Km with ATP was 9 ␮M, 6.6 times less than the value of KsN.
KN
m with CTP could not be accurately determined because thisvalue seemed so small and the hydrolysis rate is so high thatVA
m could not be measured at low concentrations of [MgCTP](effective CTP-regeneration systems are not available). Thisparameter, however, seemed to be ⬃0.1 mM, 6.9 times lessthan the value of KsN. TTP, UTP, and GTP gave values of KN
mthat were in the submillimolar range, 3– 4 times less than therespective values of KsN. The value of KN
m for ITP was 1 mM,
2.1 times less than the value of KsN. Thus the ratio, KsN/KNm, ismarkedly dependent on NTP. The values obtained for the
NTPase activity of HMM alone, VHMM, at 6 mM [NTP] arealso listed in Table 2. VHMM increased in the order CTP ⬍ATP ⬍ TTP ⬍ UTP ⬍ GTP ⬍ ITP, approximately the inverse
order to those with Vmax and KAm.
Comparison of motile activity and NTPasekinetics with various NTPs
Here, we examine the relationships between the sliding
velocity and the kinetic parameters obtained above. Twokinds of plots, (a) Vsmax versus Vmax and (b) Vsmax versus KAm,Link of Chemistry and Mechanics in Actomyosin

are given with open circles and with circles with a cross in
Figs. 2 and 3, respectively. Although the curve of Vsmax
versus Vmax relationship was concave downward, it suggests
a strong correlation between the rate of hydrolysis and thesliding velocity. Previous studies with muscle fibers havereported only moderate correlations between the maximum
rate of actin-activated hydrolysis of NTPs by acto-S1 or
acto-HMM and the unloaded shortening velocity of muscle
fibers (White et al., 1993; Pate et al., 1993; Regnier et al.,1998). In these studies, however, the solution conditions
were different in the mechanical and enzymatic measurements, and the concentrations of NTP used for measurements of the hydrolysis rates were fixed at 1 mM, irrespective of the NTP used. Moderate or very poor correlations
were also reported in studies with the in vitro actin filamentsliding assays (Higashi-Fujime and Hozumi, 1996; Regnier
et al., 1998). In the study reporting very poor correlation(Higashi-Fujime and Hozumi, 1996), the hydrolysis rate
was measured at a fixed concentration of actin (3.6 ␮M). A
correlation as good as is observed here between the sliding
velocity and apparent Km for actin (KAm) (Fig. 3) has neverbeen reported before. Similar examinations have been done
previously using various NTPs in muscle fibers and in in
vitro motility assays (White et al., 1993; Pate et al., 1993;
Regnier et al., 1998). However, results similar to ours were
not observed. This may be in part because the previous
studies used different solution conditions for measuring thesliding velocity and the enzymatic kinetics.
We have to be careful in concluding that the two excellent correlations shown with open circles and circles with across in Figs. 2 and 3 are significant. The two kinetic

385

FIGURE 3 The relationship between the maximum sliding velocity
A(Vsmax) and the Km for actin (i.e., Km). The substrate was altered with
MgATP (Q), MgCTP, MgTTP, MgUTP, MgGTP, MgITP (E), or with
MnATP, NiATP, SrATP (F). The ionic strength was altered with KCl (䊐,AQ). In this case MgATP was used as substrate. The values of Km
are thoseAat infinite [NTP]. The inset is the relationship between Vsmax vs. 公Km
obtained when ionic strength was varied.

parameters, Vmax and KA
m are, in principle, independent ofeach other. Yet, they varied in a similar way when thesubstrate was altered with various MgNTPs. It is, therefore,possible that only one of the two types of correlations is
significant, and that another is just accidental. To clarify this
aspect we have to perturb the acto-HMM system by othermethods and examine the two relationships again.
Motility and kinetics with substrates, ATPcomplexed with various divalent cations

FIGURE 2 The relationship between the maximum sliding velocity(Vsmax) and the maximum acto-HMM NTPase activity (Vmax, Pi/s/head).
The substrate was altered with MgATP (Q), MgCTP, MgTTP, MgUTP,
MgGTP, MgITP (E), or with MnATP, NiATP, SrATP (F). The ionicstrength was altered with KCl (䊐, Q). In this case MgATP was used as
substrate. The maximum acto-HMM NTPase activities are those at infinite[NTP] and infinite [actin].

As second perturbants of the acto-HMM system we used
substrates, ATP complexed with various divalent cations(Me2⫹). We chose Mg2⫹, Mn2⫹, Ni2⫹, and Sr2⫹ for the
Me2⫹. All of these ions, with ATP, supported movement of
actin filaments over HMM. First, we measured the dependence of sliding velocity on the substrate concentration. The
data for the four substrates were analyzed by least-squares
fitting to the modified Michaelian equation with n as avariable to be determined (i.e., Vs ⫽ Vsmax/(1 ⫹ (KsN)n/[S]n))and to the original Michaelian equation. The results are
summarized in Table 3. The standard deviations and ␹2 tests
were again often smaller with the former analysis than thelatter. Because of this we choose the results obtained fromthe modified Michaelian data fitting. The data for Vsmax
yielded the relationship Mg ⬎ Mn ⬎ Ni ⬎ Sr. MnATP wasan effective substrate, its Vsmax being 77% of that with
MgATP. SrATP was a very poor substrate, its Vsmax being⬍2% of that with MgATP. These substrates showed relatively high affinities for acto-HMM, and their values of KsN
Biophysical Journal 80(1) 379 –397

386

Amitani et al.

TABLE 3 Kinetic parameters of Me2ⴙ ATP hydrolysis in solution and mechanical properties in the actin sliding assays obtainedusing various divalent cations as agents complexing with ATPn*␹2 tests*
Vsmax (␮m/s)*
KsN (mM)*
Vsmax (␮m/s)†
KsN (mM)†
␹2 tests†
Vmax (s⫺1)‡
A
Km
(␮M)
VHMM (s⫺1)§
␳s具fs典 (pN)k⫹w/␭w (M⫺1 s⫺1)k⫺w/␭w (s⫺1)

MgATP

MnATP

NiATP

SrATP

1.42
0.064
4.3 ⫾ 0.18
0.059 ⫾ 0.006
4.6 ⫾ 0.21
0.07 ⫾ 0.02
0.096
23.6
29.1
0.05
0.144
0.648
6.8 ⫻ 107
1192

1.25
0.029
3.3 ⫾ 0.17
0.18 ⫾ 0.03
3.5 ⫾ 0.15
0.20 ⫾ 0.04
0.031
41.0
22.0
0.67
0.12
0.54
8.3 ⫻ 107
1826

2.32
0.00820.65 ⫾ 0.03
0.47 ⫾ 0.09
0.73 ⫾ 0.1
0.36 ⫾ 0.23
0.015
15.0
17.0
0.5
0.14
0.63
1.82 ⫻ 107
309

1.52
0.00004
0.065 ⫾ 0.007
0.67 ⫾ 0.17
0.073 ⫾ 0.007
0.83 ⫾ 0.28
0.00004
0.29
9.6
0.023
N.D.
N.D.
N.D.
N.D.

*Obtained by fitting data (Vs vs. [substrate], not shown except for MgATP) to the modified Michaelian equation with n as a variable.
†
Obtained by fitting data (Vs vs. [substrate], not shown except for MgATP) to the original Michaelian equation (i.e., n ⫽ 1).
‡§
The unit is Pi/s per head.

were in the submillimolar range. Because KN
m must be a few
times less than KsN, several measurements with these substrates were thereafter made at a fixed substrate concentration, 2 mM, which is sufficient to provide almost complete
saturation of the nucleotide binding sites. The acto-HMM
Me2⫹ ATPase activity exhibited again a simple hyperbolic
saturation behavior as a function of actin concentration. The
values Vmax and KA
m are listed in Table 3. MnATP gave thehighest Vmax, 1.7 times greater than that for MgATP, while
its KAm value was smaller than that for MgATP. NiATP wasan effective substrate with Vmax, which was 64% of thatobtained with MgATP, and its KA
m value was about half ofthat for MgATP. SrATP was a very poor substrate. However, HMM complexed with SrATP or its product had thehighest affinity for actin. Plots Vsmax versus Vmax, shownwith closed circles and a circle with a cross in Fig. 2,
indicate only a moderate correlation between the sliding
velocity and the maximum hydrolysis rate of Me2⫹ ATP byacto-HMM. As mentioned above, the MnATPase activity
was 1.7 times greater than the MgATPase activity. Nevertheless, the sliding velocity with MnATP was 77% of thatwith MgATP. The relationship Vsmax versus KAm, shown withclosed circles and a circle with a cross in Fig. 3, however,indicated that the sliding velocity was highly correlated withKAm, although the fitted line did not intercept the origin. The
hydrolysis rates, VHMM, of these Me2⫹ ATP by HMM alone
are also listed in Table 3. The 1/VHMM increased in the order
SrATP ⬍ MgATP ⬍ NiATP ⬍ MnATP. This order wasquite different from that of Vsmax, indicating no correlation
between the two quantities.
Comparison of motile activity and MgATPasekinetics at various ionic strengths
As a third perturbant of the acto-HMM system, we chosediffering ionic strengths. When ionic strength in the in vitro
Biophysical Journal 80(1) 379 –397

motility assays is higher than ⬃80 mM, actin filaments or
myosin-coated beads tend to dissociate from the respective
partners. At lower ionic strengths the rate of movement
increases with increasing ionic strength (Harada et al., 1987;
Umemoto and Sellers, 1990; Warshaw et al., 1990; Takiguchi et al., 1990; Vale and Oosawa, 1990; Homsher et al.,
1992; Saito et al., 1994). With the help of methylcellulose,
which reduces the lateral diffusion of actin filaments fromthe myosin-coated surface (Uyeda et al., 1990), the sliding
velocity of actin filaments can increase as ionic strength iselevated even higher than 80 mM (Homsher et al., 1992).
Despite general acceptance of this acceleration effect ofionic strength, quantitative analysis of the effect has notbeen performed, and therefore the underlying mechanism
has not been well understood. Here, we reexamined theionic strength effect on the rate of actin filament translocation. The ionic strength of solution was varied with KCl.
MgATP was used as substrate and its concentration was
fixed at 2 mM. As shown in Table 4, Vsmax increasedsmoothly with increasing [KCl], with a slight upward deviation from a linear relationship. The maximum rate (Vmax)
of MgATP hydrolysis by acto-HMM was nearly constantover [KCl] from 5 mM to 50 mM (Table 4). KA
m was,
however, increased with increasing [KCl], with upward
deviation at higher [KCl] from a linear relationship (Table4). These distinct behaviors of Vmax and KA
m as a function of[KCl] made the relationships Vsmax versus Vmax and Vsmax
versus KA
m very different (squares and circles with a cross in
Figs. 2 and 3, respectively). Vsmax varied widely, withoutalmost no changes in Vmax. In contrast to this, Vsmax increased with increasing KA
m. The MgATPase activity of
HMM alone, VHMM, increased, in a linear manner, withincreasing [KCl] (Table 4). This enhancement effect of[KCl] gave a relationship between Vsmax versus VHMM thatwas completely opposite to that observed when the substrate
was altered with various MgNTPs (Table 2).

Link of Chemistry and Mechanics in Actomyosin

387

TABLE 4 Kinetic parameters of MgATP hydrolysis in solution and mechanical properties in the actin sliding assays obtained
under various ionic strengths
[KCl]
(mM)

Vsmax
(␮m/s)

Vmax*
(s⫺1)

A
Km
(␮M)

VHMM†
(s⫺1)

5
10
25
30
40
43
46
50

1.45
1.81
4.00
4.38
5.25
5.30
5.26
7.35

21.4
23.0
23.6
23.1
19.6
18.3
19.5
18.9

5.7
9.1
29.1
37.0
46.2
53.9
56.2
70.6

0.043
0.042
0.060
0.065
0.065
0.065
0.070
0.065

␳s

具fs典
(pN)

⑀Vmax/Vsmax‡(pN)

A
公Vmax/Km⫺1/2(s 䡠 ␮M)

0.20
0.17
0.144
0.115
0.108
0.123
0.126
0.117

0.9
0.77
0.65
0.52
0.49
0.55
0.56
0.52

1.36
1.16
0.54
0.49
0.34
0.31
0.34
0.24

1.94
1.59
0.90
0.79
0.65
0.58
0.59
0.52

*†The unit is Pi/s per head.
⑀ is the work done by one shot of power stroke. The value used is 9.2 ⫻ 10⫺20 J (see Discussion).

‡

Sliding force
It has been recognized that actin filaments would not slide
smoothly over HMM when the HMM sample is partiallydamaged. Based on this phenomenon, Haeberle developed amethod for estimating relative magnitude of active force
exerted on actin filaments in the in vitro motility assays(Haeberle, 1994). Chemically damaged and therefore noncycling HMM that is mixed with intact HMM imposes anexternal load on sliding actin filaments and thereby slows
the sliding motion. When the molar ratio (␳) of noncycling
HMM to intact HMM is increased with keeping the total
HMM amount constant, the movement of actin filaments iseventually stalled. We prepared noncycling HMM by extensively treating HMM with NEM. This NEM-HMM had
neither Ca2⫹ NTPase nor actin-activated NTPase activities.
Actin filaments attached to the surface that had been coatedwith this damaged HMM never detached from the surface inthe presence of any Mg2⫹-NTPs. NEM-HMM, therefore,
seemed unable to associate with NTP.
We estimated the relative magnitude of active force thatis generated by a cross-bridge when cross-bridges are propelling actin filaments to slide without external load or under asmall external load. We call this active force “sliding force.”
The method for this estimation and the basis of this method aredescribed in Materials and Methods and Discussion, respectively. In the first set of experiments we studied the dependence of the relative magnitude of sliding force on NTPs. The
concentration of NTP was first fixed at 2 mM. The sliding
velocity decreased linearly with increasing ␳, as demonstrated
in the inset of Fig. 4. The intercept to the abscissa (i.e., ␳s) of
the initial tangent of the Vs versus ␳, which is supposed to be
proportional to the magnitude of sliding force, was roughly
constant over the NTPs used (Fig. 4 and Table 2). Since 2 mM
of NTPs (except for ATP) do not saturate the nucleotide
binding site of HMM, these measurements were repeated atvarious concentrations of NTP (from 1 to 6 mM). We could
not, however, find NTP-concentration dependence of the ␳svalues, even with GTP and ITP, whose affinities for actoHMM were low compared with the other NTPs. Also in thesecond set of experiments, where divalent cations that complex
with ATP were varied, the ␳s values were quite similar to eachother (Table 3). In this experiment Sr2⫹ ATP was omitted
because of the very small rate of movement. In the third set ofexperiments, where MgATP was used as substrate and theionic strength was varied with various [KCl], ␳s showed atendency to decline with increasing [KCl] (Table 4). In thisexperiment the resistive force by NEM-HMM may vary depending on [KCl]. It is likely that the higher [KCl] may resultin less resistive force. The tendency for sliding force to declinewith increasing [KCl] may, therefore, be more significant thanthat of ␳s.
DISCUSSION
A kinetic scheme of the actomyosin ATPase
Before we discuss the experimental results obtained in thepresent study, we briefly describe the kinetics of the acto-

FIGURE 4 Relative magnitude of sliding forces in various MgNTPs andits dependency on [MgNTP]. The ionic strength was kept constant by
adjusting [KCl]. The inset shows dependence of sliding velocity in MgATP
upon the molar ratio (␳) of NEM-HMM to intact HMM. The linear line
intercepts the abscissa at ␳ ⫽ ␳s.
Biophysical Journal 80(1) 379 –397

388

Amitani et al.

myosin ATPase reaction. The kinetic scheme shown hereaccounts for the results of various kinetic experiments.
Myosin heads are distinguished according to their actinbinding kinetics (Stein et al., 1979), i.e., “weak-binding
states” and “strong-binding states.” ATP binds to a rigor
cross-bridge (step 1) to form a weakly bound state (A 䡠 M 䡠ATP), followed rapidly by dissociation of actin (A) frommyosin (M) (step 2), or by hydrolysis of the ␥-phosphate(step 3). After hydrolysis of the ␥-phosphate (step 3 and step5), a weakly bound state (A 䡠 M 䡠 ADP 䡠 Pi) forms, which isin a rapid equilibrium with the M 䡠 ADP 䡠 Pi state (step 4).
The weakly bound cross-bridge then isomerizes to a strongbinding state (step 6). The bound phosphate is released (step7) to form A 䡠 M* 䡠 ADP, which is followed by isomerization (step 8). Finally, ADP is released (step 9) to form arigor cross-bridge. Rate limitation of the ATPase cycle insolution has been variously thought to occur at the isomerization step (step 6) (Stein et al., 1979, 1984), at the Pi
release step (step 7) (Webb and Trentham, 1981; Hibberd
and Trentham, 1986; Barman et al., 1998) (the two states,A 䡠 M 䡠 ADP 䡠 Pi and A 䡠 M* 䡠 ADP 䡠 Pi, are not distinguished), or at the cleavage step (step 3) (Rosenfeld andTaylor, 1984; White et al., 1997) (deduced from observations that a high concentration of actin suppresses the ATPase activity at very low ionic strength). Using a fluorescent
Pi-probe and caged-ATP in single muscle fibers, He et al.
(1997, 1998) have observed that the onset of tension development is slightly earlier than that of Pi release, indicatingthat A 䡠 M* 䡠 ADP 䡠 Pi contributes to the force-generatingstate. Other studies have also presented evidence that bothA 䡠 M* 䡠 ADP 䡠 Pi and A 䡠 M* 䡠 ADP contribute to theforce-generating state (Lund et al., 1987; Dantzig et al.,
1992; Cooke, 1995). Although a consensus about the ratelimiting step in the ATPase cycle has not been reached, it
has been generally accepted that 1) myosin heads reside
predominantly in the weak-binding state(s) in which they
are in a rapid equilibrium between the actin-attached anddetached states, and 2) an intermediate just after the powerstroke is A 䡠 M 䡠 ADP. Although in a part of the subsequentdiscussion we assume the rate-limiting step to be step 6, we
do not intend to validate this assumption. The results given
below are not strongly associated with this assumption. The
ATPase kinetic scheme cited here probably hold for almost
all the other nucleotides, as suggested by Pate et al. (1993);

White et al. (1993); and Regnier et al. (1998); although the
GTPase probably has a different kinetic mechanism (Eccleston and Trentham, 1979; White et al., 1997).
Resistive drag forces
Actin-attached cross-bridges that are not executing thepower stroke produce a resistive force opposing sliding
movement, because these cross-bridges are pulled and deformed passively by moving actin filaments. In a highconcentration of substrate, such resistive cross-bridges arein the post-power stroke state (from which NDP is released)
and in the weak-binding states. To the authors, it seems
controversial whether weakly bound cross-bridges produce
resistive force to such an extent that it affects or determines
the sliding velocity (Warshaw et al., 1990; Homsher et al.,
1992; Cuda et al., 1997; Brenner, 1990). First, we examine
this issue quantitatively, since this issue seems to have thekey to understanding the link between the chemistry and themechanics in skeletal actomyosin. If for a time T a resistive
cross-bridge remains attached to a sliding actin filament(velocity, Vs), it is pulled for a distance of VsT. When
averaged over the cycle time (Tc ⫽ 1/Vmax) of the NTPasereaction, the resistive force, fr, becomes

具fr典 ⫽ 12 ␭⌫VsT

where ⌫ is an elastic constant of a resistive cross-bridge and␭ is the fraction of this resistive state over all states. Now,
we roughly estimate the magnitude of 具fr典 in ATP, using
values of ␭, ⌫, and T. Although ⌫ may vary depending on
the state concerned, the reported values for different states
are similar to each other: 0.65 pN/nm, measured directly
using a single-molecule technique in the midst of an ATPase cycle (Mehta et al., 1997); 0.58 pN/nm, measured
directly using a single-molecule technique in the rigor state(Nishizaka et al., 1995). So, we assume ⌫ ⫽ 0.6 pN/nm for
all the resistive states. For weakly bound cross-bridges, thetime, T, equals 1/k⫺2 or 1/k⫺4. Hereafter, we assume k⫺2 ⫽k⫺4 (⬅k⫺w) and k⫹2 ⫽ k⫹4 (⬅k⫹w). The rate constant ofdissociation of myosin heads with ATP from actin is likely
to be ⬃2000 s⫺1, inferring from the values obtained previously under various solution conditions. So, we assume T ⫽0.5 ms. Skeletal actomyosin has a low duty ratio, the frac-

Scheme 1
Biophysical Journal 80(1) 379 –397

(1)

Link of Chemistry and Mechanics in Actomyosintion of time that a cross-bridge spends generating active
force. The ratio is suggested to be ⬃0.1 (Cooke, 1997), andmyosin heads reside predominantly in the weak-binding
state(s). Taking these features into account, the total fractionof weakly bound cross-bridges is supposed to be ⬃0.8.
These values for ⌫, T, ␭, and Vs ⫽ 4.3 ␮m/s result in 具fr典 ⫽0.51 pN per a weakly bound cross-bridge. As will be shown
later, this magnitude is comparable to the time-averaged
active force per a cross-bridge, but significantly lower than
the force required to unbind a rigor head from actin. The
lifetime of the post-power stroke state is determined by therate constant of ADP dissociation from the actin-boundcross-bridge (here we distinguish the resistive forces produced by A 䡠 M 䡠 ADP and A 䡠 M 䡠 ATP. The latter iscounted among those produced by weakly bound crossbridges.) Although the rate constant of ADP dissociation on
the ATP hydrolysis pathway has not been determined, we
assume that it is similar to the rate constant (k⫺AD) of ADP
dissociation from the state formed by externally adding
ADP to actomyosin. For rabbit skeletal actomyosin, k⫺AD is⬃1000 s⫺1 (700 –1000 s⫺1 at 25°C, Siemankowski et al.,
1985; 1400 s⫺1 at 20°C, Borejdo et al., 1985). So, we
assume T ⫽ 1 ms. The fraction of the post-power stroke
state is supposed to be very small (Barman et al., 1998).
Tentatively, we assume ␭ ⫽ 0.023, because ␭ must equal
T/Tc ⫽ 1/42 (Tc ⫽ 1/Vmax ⫽ 42.4 ms). These values lead toa rough estimate, 具fr典 ⫽ 0.029 pN, 17.6 times less than that
produced by a weakly bound cross-bridge. We have to notehere that a cross-bridge in the post-power stroke state produces a resistive force just once during an ATPase cycle,
while in the weak-binding states a cross-bridge produces it
many times. Here, for simplicity, we neglected the reductive
effect of elastic deformation on the lifetime (T) of weakly
bound cross-bridges. The rate of ADP dissociation may beaffected by the elastic deformation. Here, we do not consider it. When the deformation energy, Ed ⫽ 0.5 ⫻ ⌫(VsT)2,
is comparable to or larger than the thermal energy, E0 ⬃4 ⫻ 10⫺21 J, T should be reduced significantly, as has been
demonstrated in the acto-HMM rigor bond (Nishizaka et al.,1995). The deformation accelerates the rate of dissociationfrom actin by a factor of eEd/E0 (Bell, 1978). So, the probability of finding a cross-bridge that bound to actin at time
zero and keeps associating with actin till time t is proportional to P(t) ⬅ exp(⫺兰t0 k⫺weEd/E0 dt). Therefore, the life⬁
time, T, can be determined by T ⫽ 兰⬁
0 t P(t)dt/兰0 P(t)dt.
Performing the integrations numerically, we find that the
lifetime of a weakly bound cross-bridge is slightly reduced
to 0.3 ms (the resistive force, 0.31 pN). A slight increase in
the dissociation rate hardly affects the fraction of weakly
bound cross-bridges because a high concentration of actin
shifts the rapid equilibrium to the actin-attached side. Although the difference in the resistive forces produced in the
weak-binding states and in the post-power stroke state became smaller (10.7 times), it is still large. Even when wechoose a lower value, 700 s⫺1, for the ADP dissociation rate389

(␭ becomes 0.034), the difference in the resistive forces isstill 5 times. After all, we reached an estimate that resistive
force is produced predominantly by weakly bound crossbridges. Hereafter, we neglect resistive forces produced bycross-bridges in the post-power stroke state. We also neglect
the deformation effect on the lifetime of weakly boundcross-bridges, because it is not so large. As will be seen
later, deformation energy stored in weakly bound crossbridges is rather independent of nucleotides (a longer lifetime and a slower rate of movement cancel out each other togive a similar deformation energy).

Dependence of sliding velocity on
substrate concentration
Sliding velocity (Vs) of actin filaments decreases with decreasing substrate concentration, [S] (Fig. 1, a and b). Thisis certainly due to the resistive forces produced by the
nucleotide-free rigor cross-bridges (i.e., AM). The lifetime,
Trig, of an AM rigor cross-bridge is determined by

Trig ⫽ 1/共k⫹1关S兴兲,(2)

where k⫹1 is the second-order rate constant of nucleotide
binding. When averaged over Trig ⫹ Tc, the resistive force
per an AM rigor cross-bridge is given by␭rig ⫻ 12 ⌫VsTrig ,(3)

where ␭rig ⫽ Trig/(Trig ⫹ Tc), and ⌫VsTrig cannot exceed theforce (Fu) required to rupture an AM rigor cross-bridge. If
it exceeds Fu, the average resistive force should be replacedwith 0.5 ⫻ ␭rigFu. For a while we consider the case where⌫VsTrig ⬍ Fu holds. Including the resistive force produced
by a weakly bound cross-bridge, a balance of the timeaveraged forces regarding one cross-bridge is expressed asfollows.

Tc
Tc
1
具fs典 ⫽⫻ ␭w⌫VsTw
Trig ⫹ Tc
Trig ⫹ Tc 2
⫹

Trig
1
⫻ ⌫V T ,Trig ⫹ Tc 2 s rig(4)

where 具fs典 is sliding force (active force) averaged over Tc,
␭w is the fraction of weakly bound cross-bridges in Tc, andTw(⬅1/k⫺w) is the lifetime of weakly bound cross-bridges.
This equation is independent of the concentration of HMM
on the coverslip (above certain threshold level). Solving thisequation for Vs, we obtain

Vs ⫽

2具fs典
␭w⌫Tw

冒冉

1⫹

2
Trig
␭ wT w T c

冊(5)

By substituting 1/k⫺w, 1/(k⫹1[S]), and 1/Vmax into Tw, Trig,
Biophysical Journal 80(1) 379 –397

390

Amitani et al.

For a range [S] ⬃ KsN, Eq. 9 can roughly be approximatedto the original Michaelian equation (n ⫽ 1) as

solution as used in the present study with [KCl] ⫽ 25 mM).
The second order rate constant (k⫹1) of MgATP binding toactomyosin is around 2 ⫻ 106 M⫺1 s⫺1 (White and Taylor,
1976; Ando, 1984). So, F0(⬅⌫Vsmax/(2k⫹1KsN) becomes 11
pN, slightly larger than the unbinding force, Fu ⫽ 9 pN.
This means that at [MgATP] between ([S]c ⫺ ⌬) and([S]c ⫹ ⌬), the relationship Vs versus [MgATP] should obey
the original Michaelian equation (or Eq. 9), and in the otherranges of [MgATP], it should obey the modified Michaelian
equation with n ⫽ 2. In practice, the borders ([S]c ⫺ ⌬) and([S]c ⫹ ⌬) are blurred by stochastic variation of the unbinding force around its average (Fu). The shifts between the
two types of Michaelian relationships, therefore, do notoccur in discrete steps. This seems to be the reason why theleast-squares analysis of our data of Vs versus [MgATP]
gave n ⫽ 1.42 (Table 2). Since analysis of our data of Vs
versus [S] (Fig. 1, a and b) gave n ⫽ 1.4 –1.74 for thevarious nucleotides (except for ITP (n ⫽ 2.23)), similarbalances between Fu and F0 must hold for these nucleotides.
Now we are going to find the basis for the large difference in the [NTP] dependence of the velocity of actin
filaments, and of acto-HMM NTPase activity (Table 2). The
NTP concentration, KsN, that gives half of the maximum
sliding velocity (Vsmax), is larger than the NTP concentration, KN
m, at which the acto-HMM NTPase is half of its
maximum (Vmax). Suppose that isomerization step (step 6)
is the rate-limiting step and k⫹6 ⬍⬍ k⫹7, k⫹8, k⫹9, we obtainapproximations: Vmax ⯝ ␭wk⫹6 and KNm ⯝ ␭wk⫹6/k⫹1 ⫽Vmax/k⫹1 (when k⫺1 ⬍⬍ k⫹3). The condition k⫺1 ⬍⬍ k⫹3 hasbeen proven to hold for MgATP (Brune et al., 1994).
Although these approximations may not hold for the poor
substrates, GTP and ITP, let’s continue to explicate further.
By putting these approximations into Eqs. 8 and 10, KsN canbe related to KN
m as

Vs ⯝ Vsmax/共1 ⫹ KsN/关S兴兲

KsN ⫽ KmN 冑 k⫺w/k⫹6/␭w ,and Tc, respectively, we obtainVs ⫽冒冉1 ⫹ ␭k k V关S兴 冊

2k⫺w具fs典␭w ⌫⫽ Vsmax

冒冉⫺w max
2
2
w ⫹1

1⫹

冊

共KsN兲2,关S兴2(6)

where Vsmax and KsN are respectively defined by

Vsmax ⫽
KsN ⫽

1
k⫹1

冑

2k⫺w具fs典 2k⫹wKmA具fs典
⫽
␭w ⌫
␭ w⌫

k⫺wVmax
1
⫽
␭w
k⫹1

冑

k⫹wKmA Vmax
␭w

(7)

(8)

Here, Vsmax is the maximum sliding velocity, KsN is thesubstrate concentration at which Vs is half of Vsmax, and KAmis the actin concentration at which the enzymatic activity ishalf of Vmax and approximately equals k⫺w/k⫹w. Here, we
assumed that 具fs典 is constant, although it must vary depending on Vs. Equation 6 is the same as the modified Michaelian equation, with n ⫽ 2. Now, we consider the case where⌫VsTrig ⬎ Fu holds. In this case, substitution of Fu for⌫VsTrig in Eq. 4 leads to冉

Vs ⫽ Vsmax 1 ⫺冒 冊 共V ⬎ 0兲,

KsN
关S兴
2

s

(9)

where Vsmax is the same as that in Eq. 7, but KsN is newly
given byKsN ⫽ VmaxFu/共k⫹1具fs典兲

(10)

(11)

Equation 6 holds for [S] that satisfies the condition, Fu ⬎⌫VsTrig. Putting this condition and Trig ⫽ 1/(k⫹1[S]) into Eq.
6, we obtain the following quadratic inequality:

关S兴2 ⫺⌫V关S兴 ⫹ 共KsN兲2 ⬎ 0
k⫹1Fu
max
s

(12)

Solving this inequality for [S], we find that in the case Fu ⬎F0(⬅⌫Vsmax/(2k⫹1KsN)), any [S] satisfies Eq. 12, while in thecase Fu ⬍ F0, Eq. 12 can be satisfied by [S] that are higher
than ([S]c ⫹ ⌬), or lower than ([S]c ⫺ ⌬), where [S]c and ⌬are (F0/Fu)KsN and 公([S]c)2 ⫺ (KsN)2, respectively. Here, we
examine quantitatively whether Fu is smaller or larger thanF0. The magnitude of force required for unbinding a rigor
complex of a single head of HMM and actin has beenmeasured by a laser optical trap (Nishizaka et al., 1995), or
by atomic force microscopy (Nakajima et al., 1997). In thecase where an actin filament is pulled along its length, theunbinding force is about 9 pN (measured in the same ionic
Biophysical Journal 80(1) 379 –397

(13)

or

KsN ⫽ KmN

Fu
具fs典

(14)

From the relationships, k⫺w ⬎⬎ k⫹6 or Fu ⬎⬎ 具fs典, we canunderstand the observed relationship, KsN ⬎⬎ KN
m.
Using KN
⯝
V
/k
and
␭
⫽
T
/(T
m
max ⫹1rig
rig
rig ⫹ Tc) ⫽1/(1 ⫹ k⫹1[S]/Vmax), the modified Michaelian equation
gives the relationship of Vs versus ␭rig.

Vs ⫽ V

冒冉 冉

max
s

冊冉 冊

␭rig n KsN n
1⫹
共1 ⬍ n ⬍ 2兲
1 ⫺ ␭rig KmN(15)

The initial slope of Vs/Vsmax as a function of ␭rig is zero,coinciding with the observation (open circles in Fig. 5). The
solid line in Fig. 5 was obtained by fitting the data (closed
circles in Fig. 1 b to Eq. 15. The fitting was not good athigher ␭rig, wherein Vs/Vsmax ⬍ 0.3. This arises mainly from

Link of Chemistry and Mechanics in Actomyosin

391

FIGURE 5 Resistive effects of rigor complexes on the velocity of actin
filament translocation in assay 1 and assay 2. The substrate was MgUTP.
In assay 1, the fraction of rigor complexes (␭rig) was altered by changing
the concentration of UTP (E); while in assay 2, the fraction of rigor
complexes (␰) was altered by mixing various amounts of NEM-HMM with
intact HMM while keeping the total amount of NEM-HMM plus intact
HMM constant (F). In assay 1, the fraction of rigor complexes wasNN
calculated using ␭rig ⫽ 1/(1 ⫹ k⫹1[S]/Vmax) ⯝ (Km/[UTP])/(1 ⫹ Km/
[UTP]). A line is fit to Eq. 15 with n ⫽ 1.74. The inset shows therelationship between Vs/Vsmax vs. ␳(⬅␰/(1 ⫺ ␰)) obtained in assay 2.

that 具fs典 is not constant in this range of ␭rig. We observed theresistive effect of nucleotide-free rigor complexes on Vs also
in a different assay system (assay 2), where noncycling
HMM (NEM-HMM) is present together with intact HMM
in the presence of [S] that is sufficient to saturate thenucleotide binding sites. As shown in Fig. 5, this resistive
effect (closed circles) was stronger than that in the previous
assay system (assay 1, open circles). In assay 2 the force
balance equation is given by

共1 ⫺ ␰兲具fs典 ⫽ 共1 ⫺ ␰兲21 ␭w⌫VsTw ⫹ ␰ 21 Fu ,(16)

where ␰ is the fraction of NEM-HMM. Solving this equation
for Vs, we obtain冉

Vs ⫽ Vsmax 1 ⫺冊␰Fu
1 ⫺ ␰ 2具fs典(17)

The initial slope of Vs/Vsmax as a function of ␰ is ⫺0.5Fu/具fs典,
markedly different from that in assay 1. Equation 17 predicts that plots of Vs versus ␳ ⬅ ␰/(1 ⫺ ␰) should give astraight line. As shown in the insets of Figs. 4 and 5, this istrue as far as Vs/Vsmax is larger than 0.3. At Vs/Vsmax ⬍ 0.3,
the noncycling HMM is less efficient in reducing the rate of
movement, indicating that 具fs典 increases under a large external load.
Sliding force
From Eq. 17 the sliding force, 具fs典, equals 0.5␳sFu, where ␳sis the intercept of the abscissa of the initial tangent in therelationship of Vs versus ␳. The ␳s values with various
MgNTPs varied from 0.12 to 0.2 (Fig. 4, Table 2). Whendivalent cations that complex with ATP were varied, the ␳svalues were 0.144, 0.12, and 0.14 for MgATP, MnATP, and
NiATP, respectively (Table 3). When [KCl] was varied
using MgATP as substrate, ␳s varied from 0.2 to 0.11, witha tendency to decrease with increasing ionic strength (Table4). The time-averaged magnitude of sliding force with
MgATP at 25 mM KCl is estimated to be 0.65 pN (⫽0.5 ⫻0.144 ⫻ 9). Force generation by a single myosin head has
been measured directly using optical traps (Mehta et al.,1997) or micro-needles (Ishijima et al., 1991, 1996). The
magnitude of the power stroke (force spike) has been foundto be 2–10 pN, depending on the methods. The higher
values seem reliable, considering a compliance problem
involved in the single-molecule techniques. So, the maximum power stroke must be 5–10 pN. Since the duty ratio is⬃0.1 (Cooke, 1997), the active force per cross-bridge is
0.5–1.0 pN on a time-average, similar to 0.65 pN, the value
estimated above as the sliding force. The values of sliding
force for the other substrates are listed in Tables 2 and 3.
The values vary in a small range around 0.7 pN. A possible
mechanism that accounts for the approximate constancy of
具fs典 over various substrates will be discussed later.
Dissociation and association rate constants
Equation 7 can be re-written as follows.

⌫Vsmaxk⫹w⫽ A
␭w 2Km具 fs典(18)

The right-hand side of this equation contains kinetic andmechanical parameters whose values, except for ⌫, wereobtained in the present study. As mentioned already, thevalue of ⌫ has been estimated to be ⬃0.6 pN/nm in otherlaboratories. Using these values (Table 2), we calculated thevalues of k⫹w/␭w and k⫺w/␭w using Eq. 18 and k⫺w ⫽k⫹wKAm, respectively. The results are listed in Tables 2 and3. The values of k⫹w/␭w for MgATP, MgCTP, MgTTP, and
MgUTP were quite similar to each other. Since k⫹w and ␭w
are independent parameters, this similarity suggests that therespective values of k⫹w and ␭w are similar for these substrates. However, the value k⫹w/␭w for MgGTP was oneorder of magnitude larger than these substrates, while thevalue for MgITP was one order of magnitude smaller. As
mentioned already, this probably arises from different hydrolysis mechanisms of the poor nucleotides. The dissociation rate constant divided by ␭w (i.e., k⫺w/␭w) decreased in
the order CTP ⬇ ATP ⬎ TTP ⬎ UTP ⬎ GTP ⬎ ITP, quite
similar to the order with the maximum sliding velocity,
Vsmax. Because the fraction of weakly bound cross-bridges in
ATP is very likely to be 0.8 – 0.9, the value of k⫺w for ATP
is 1600 –1800 s⫺1 (1992 ⫻ 0.8 – 0.9), similar to those reported previously for the maximum rate of acto-HMM
Biophysical Journal 80(1) 379 –397

392

dissociation (Regnier et al., 1998) or acto-S1 dissociation(White et al., 1993) by ATP, although the previous measurements were made at relatively high ionic strength.

Relationships of Vsmax versus Vmax, and Vsmax
A
versus Km
Analysis of changes in Vsmax (the maximum sliding velocity)
and Vmax (the maximum substrate turnover rate) dependingon MgNTPs (open circles and circle with a cross in Fig. 2)
revealed that these quantities correlated well with eachother. When various MeATPs were used as substrates(closed circles and circle with a cross in Fig. 2), the correlation was poor. When MgATP was used as substrate andionic strength was varied, Vmax was nearly constant, while
Vsmax increased with increasing ionic strength (squares andcircle with a cross in Fig. 2). How can we interpret thisresult? Before getting into this issue, let’s look at the observed relationships between Vsmax and KA
m. The relationshipwas approximately linear when the substrate was alteredwith MgNTPs (open circles and circle with a cross in Fig.
3). When the system with MgATP was perturbed by altering
the ionic strength, Vsmax increased with increasing KA
m, withthe curve being concave upward (squares and circle with across in Fig. 3). These similar behaviors guarantee that thisexcellent correlation is not accidental, but intrinsic to theprotein motor system. This supports our theoretical result,Eq. 7, that indicates Vsmax is proportional to KAm (the otherparameters contained in Eq. 7, i.e., k⫹w/␭w and 具fs典, are notstrongly dependent on nucleotide substrate except for GTP
and ITP (Table 2)). A previous study (Regnier et al., 1998)
also seems to support this relationship. The rate constant,k⫺2, estimated from the rate of acto-HMM dissociation after
adding NTP, increased in the order CTP ⬍ UTP ⬍ ATP ⬍dATP, while the shortening speed of fibers increased in theorder UTP ⬍ ATP ⬇ CTP ⬍ dATP. Neglecting the datawith CTP, the shortening speed linearly increased withincreasing k⫺2 (Tables 3 and 4 in Regnier et al., 1998).
The maximum turnover rate, Vmax, is not visibly contained in Eq. 7. To link Vmax to the maximum rate of
movement (Vsmax) we need another perspective. We consider
a situation where an actin filament is sliding at maximum
velocity on HMM without external load. In this situation,
the actin filament receives power stroke(s) from crossbridges at any moment. Each power stroke (the duration, Tp)
displaces the actin filament by a distance, d. However,power strokes by several cross-bridges that happen to overlap at one moment produce the same step displacement, d.
According to this model, the power stroke speed (d/Tp)
should equal the maximum sliding velocity (Vsmax) (Uyeda
et al., 1991). The average sliding force per a cross-bridge,
具fs典, equals pTp/Tc, where p is the magnitude of power stroke
Biophysical Journal 80(1) 379 –397

Amitani et al.

(force spike). From these relationships we can obtain具fs典 ⫽ pd

Vmax
Vsmax

(19)

From Eqs. 7 and 19, two mechanical variables, Vsmax and具fs典, can be expressed separately.

冑 冑冉冑 冑 冊冑 冑冉 冑 冑 冊Vsmax ⫽⫽

具fs典 ⫽

2pdk⫹w
KmA Vmax
␭w ⌫

2pd
⌫

k⫺wk⫹6

pd␭w⌫
2k⫹w

Vmax
KmA

pd⌫
2

k⫹6
k⫺w

⫽␭w

(20)

(21)

These two equations link the mechanical behaviors of actomyosin to the enzymatic properties. They tell us thatimportant parameters of the enzymatic kinetics in the determination of the mechanical behaviors are the dissociation
rate constant (k⫺w) in the weak-binding states and the rate
constant of the rate-limiting step (here, we assumed it to be
step 6). Vmax (or k⫹6) is involved in Vsmax and 具fs典 in the samemaxway, while KAandm (or k⫺w) is inversely involved in Vs
具fs典. These two equations successfully account for many of
our observations. When the acto-HMM system was perturbed by altering the substrate with various MgNTPs, bothKAm and Vmax changed roughly in parallel. So, Eq. 20 predicts approximate linear relationships between Vsmax and KAm,and between Vsmax and Vmax, under an assumption thatpdk⫹w/␭w⌫ is approximately constant. This coincides withwhat we observed (open circles and circles with a cross in
Figs. 2 and 3). When the acto-HMM system in MgATP was
perturbed by changing [KCl], Vmax was nearly constant,while KAm increased with increasing [KCl] (Table 4). In thiscase, Eq. 20 predicts that Vsmax increases in proportion to
公KA
m. This again approximately coincides with what we
observed (the inset of Fig. 3). Equation 21 predicts that 具fs典
Ais proportional to 公Vmax/KA
m. As mentioned above, both Km
and Vmax changed roughly in parallel when various
MgNTPs were used. Therefore, 具fs典 should be nearly constant under the assumption that pdk⫹w/␭w⌫ is approximately unchanged. This is what we observed (Fig. 4, Table2). When [KCl] was altered for acto-HMM in MgATP, 具fs典
declined with increasing [KCl]. As can be seen in Table 4,
具fs典 was roughly proportional to 公Vmax/KA
m. At [KCl] ⬍ 25
mM, however, 具fs典 was smaller than expected from thecorresponding 公Vmax/KAm values, while at [KCl] ⬎ 25 mM
具fs典 was larger than expected from the corresponding
公Vmax/KAm values. We have to note that the 具fs典 values havebeen estimated on the assumption that Fu (measured at 25mM KCl) is constant irrespective of [KCl]. Corrections ofLink of Chemistry and Mechanics in Actomyosin

the 具fs典 values by a likely dependence of Fu on [KCl] maylead to better agreement.
Equations 20 and 21 predict that plots of Vsmax versus
公2k⫹wKAmVmax/(␭w⌫) (the right-hand side of Eq. 20 divided
by 公pd) and plots of 具fs典 versus 公␭w⌫Vmax/(2k⫹wKAm) (theright-hand side of Eq. 21 divided by 公pd) give straightlines with a slope of 公pd. As shown in Fig. 6, a and b, thiswas the case, although the data with the poor substrates
were deviated from the straight line (Fig. 6 b). The leastsquares analysis of these plots gave pd ⫽ 9.2 ⫻ 10⫺20 J(from Fig. 6 a) and pd ⫽ 9.3 ⫻ 10⫺20 J (from Fig. 6 b),
surprisingly close to the free energy change associated with
ATP hydrolysis in living cells, 8.3 ⫻ 10⫺20 J (Stryer, 1981).
These linear relationships also indicate that these MgNTPs(except for the poor substrates) are similar to each other inthe extent of energy liberation by hydrolysis and in theefficiency of energy transduction (nearly 100%). Here, we

393

have to say that this efficient energy transduction does notmean that all the chemical energy is used to perform external mechanical work. Under no external load, a large part ofthe energy is used to deform actin-bound cross-bridges, andtherefore eventually dissipates as heat.
Now we discuss the experimental results with various
MeATPs. The sliding forces with MnATP and NiATP weresimilar to those with MgATP. This observation, as well as
Eq. 19, predicts that the maximum sliding velocity should
be proportional to the maximum hydrolysis rates of thesesubstrates. This was not the case. The maximum sliding
velocities with MnATP and NiATP were smaller than expected from their Vmax values (closed circles in Fig. 2). Thissuggests that the energy liberation by hydrolysis of thesesubstrates or the efficiency of energy transduction by actoHMM with these substrates is smaller than those with
MgATP. The former possibility is unlikely. The type of
cation that is bound to ATP should not strongly affect theextent of energy liberation, because there exist several enzymes that utilize CaATP, Na-ATP, or K-ATP as energy
sources. Equation 7 predicts that the maximum sliding velocity, Vsmax, should be proportional to k⫹wKAm/␭w. The valueof Vsmax with NiATP was smaller than expected from the KAmvalue (Fig. 3). This is understandable because k⫹w/␭w with
NiATP is smaller than those with MgATP and MnATP (seeTable 3). Since the values of Vmax/KAm for MgATP and
NiATP (0.8 ⫻ 106 M⫺1 s⫺1 and 0.9 ⫻ 106 M⫺1 s⫺1, respectively) are quite similar, the smaller pd and k⫹w/␭w
cancel out to afford NiATP 具fs典 similar to that with MgATP(see Eq. 21).
Relevance to other studies

A
FIGURE 6 The relationships (a) between Vsmax vs. 公2k⫹wKm
Vmax/
A
公
(␭w⌫) (E) and (b) between 具fs典 vs. ␭w⌫Vmax/(2k⫹wKm) (F).

It has been studied how noncycling unphosphorylatedsmooth muscle or nonmuscle myosin affects the actin sliding produced by cycling myosins (Warshaw et al., 1990;
Cuda et al., 1997). These noncycling myosins weakly bindto actin. The sliding velocity generated by phosphorylatedsmooth muscle myosin was only slightly reduced by unphosphorylated smooth muscle myosin. For instance, whenthe mixing ratio (r) of unphosphorylated myosin over phosphorylated myosin was 1, the reduction rate was ⬃20%. A
similar small reduction was also observed in the mixture ofphosphorylated platelet and unphosphorylated platelet myosin. These small reductions apparently imply that weakly
bound cross-bridges generate little resistive force. According to our force-balance model, a main determinant for the
reduction in this mixing assay is the ratio of the dissociation
rate constant (k⫺w) of cycling myosin in the weak-bindingustates over that of unphosphorylated species (k⫺w), i.e.,maxuuVs ⫽ Vs /(1 ⫹ rk⫺w/(␭wk⫺w)). The larger ratio (k⫺w/k⫺w)
will give a greater reduction in the actin filament velocity.
Under an assumption, ␭w ⫽ 0.8, analysis of the data (Figs.
8 and 9 of Cuda et al., 1997) based on this model predictsuthat k⫺w
is ⬃5 times larger than k⫺w of corresponding
Biophysical Journal 80(1) 379 –397

394

phosphorylated myosin in the weak-binding states. Thisprediction may come true, because in ATP unphosphorylated smooth muscle myosin has 4 –10-times lower actin
affinity than its phosphorylated form (Ikebe et al., 1981;
Sellers et al., 1982). When unphosphorylated platelet myosin was mixed with skeletal muscle myosin, the reduction
rate was approximately proportional to the mixing ratio(Cuda et al., 1997). This suggests, according to our forcebalance model, that the dissociation rate of unphosphorylated platelet myosin is similar to that of skeletal muscleu⯝ k⫺w). Pairmyosin in the weak-binding states (i.e., k⫺wwise mixings of different types of actively cycling myosins
revealed that only small proportions of a more slowly translating myosin type significantly inhibited the sliding velocity generated by a more rapidly translocating myosin type(Warshaw et al., 1990; Cuda et al., 1997). The dissociation
rate of weakly bound cross-bridges probably differs greatly
between the two myosin types, so a large inhibition of actin
sliding is expected to be caused by the slower myosin type.
In addition to this, the cross-bridges that are slowly executing the power strokes in the strong-binding states (the) also produce resistive forces
power-stroke time, Tslowpagainst the actin filaments that are sliding at Vs larger thanthe power-stroke speed, d/Tslow.
p
Myosin heads with ATP␥S weakly attach to actin (thehydrolysis rate is negligibly small). The actin sliding velocity in the presence of 1 mM ATP ⫹ 5 mM ATP␥S is ⬃80%of that in 1 mM ATP without ATP␥S (Homsher et al.,1992), apparently implying that weakly attached crossbridges do not play a predominant role in limiting the rate of
movement. However, in this report there are not presented
experimental data as to the fraction of actin-attached myosin
heads that are occupied with ATP␥S, so we measured theacto-HMM ATPase activity in the presence of 1 mM ATP
and various concentrations (0 –5 mM) of ATP␥S (data notshown). The ATPase activity was little inhibited (5% atmost) even by 5 mM ATP␥S, indicating a very low affinityof ATP␥S for acto-HMM compared with ATP, coinciding
with Goody and Hofmann (1980).
In addition to k⫺w and the rate constant of the ratelimiting step, the elasticity of actin-attached cross-bridges
seems an important factor in determining the mechanicalbehavior of actomyosin. The more stiff cross-bridges resultin slower sliding movement and higher force generation, as
indicated by Eqs. 20 and 21. This effect may account, inpart, for variation of the rate of movement among different
myosin species, and also possibly among different actin
species that have different structures at the myosin bindingsites. A series of mutant Dictyostelium myosins that havedifferent neck lengths propel movement of actin filaments atdistinct velocities. The velocity increases with increasing
neck length (Uyeda et al., 1996). Although this behavior has
been interpreted by the “swinging lever arm model,” it canbe accounted for by differences in the elasticity of the neckregion that depends on length.
Biophysical Journal 80(1) 379 –397

Amitani et al.

We surveyed previous studies that have compared therates of actin translocation with the ATPase activities using
various myosin species, actin species, chemically modified
actins, tropomyosin, or different solution conditions. A
number of papers have reported complete sets of data formax
Vmax, KA
. Almost all the works have, however,m, and Vs
maxmeasured Vsand (Vmax, KAm) under different solution
conditions. Nevertheless, we did analyze, using Eq. 20, thedata that appeared in seven reports (Cook et al., 1993;
Umemoto and Sellers, 1990; Sutoh et al., 1991; Johara et al.,
1993; Crosbie et al., 1994; Rovner et al., 1995; Hozumi etal., 1996). The results are shown in Fig. 7, where the valuesof Vsmax and 公VmaxKA
m are normalized to the corresponding
control values. Although scattered in a somewhat wide
range, the plots are distributed around the diagonal straightline connecting the coordinates (0,0) and (1.8,1.8). Thisstrongly suggests that Vsmax is approximately proportional to
公VmaxKA
m. Johara et al. (1993) produced a mutant of Dictyostelium actin, and measured Vmax, KA
m, and active force
generated by a single actin filament and rabbit skeletal
HMM. The force and 公Vmax/KAm with the mutant actin are
0.84 and 0.7, respectively, when normalized to those withthe wild actin. This supports the proportional relationshipbetween active force and 公Vmax/KAm.

A
FIGURE 7 The relationship between Vsmax vs. 公VmaxKm. The values ofmaxA
Vs , Vmax, and Km were quoted from seven published sources that havecompared the rate of actin translocation with the ATPase activities using
various myosin species, actin species, chemically modified actins, orAtropomyosin (F). The values of Vsmax and 公VmaxKm
were normalized tothe corresponding control values. The numbers attached to each data point
represent the literature quoted here. (1) Dictyostelium actin mutants (Sutoh
et al., 1991). (2) Chemically modified actins (Crosbie et al., 1994). (3)50K/20K loop-substituted smooth muscle HMMs (Rovner et al., 1995). (4)Yeast myosin mutant (Cook et al., 1993). (5) Mono and diphosphorylatedsmooth muscle HMMs and effect of tropomyosin (Umemoto et al., 1989).
(6) Chemically modified actin (Hozumi et al., 1996). (7) DictyosteliumAactin mutant (Johara et al., 1993). The values of Vsmax, Vmax, and Km
arethose obtained in the present study with various MgNTPs (䊐). The values
Aof Vsmax and 公VmaxKm
were normalized to the corresponding values for
MgATP.

Link of Chemistry and Mechanics in Actomyosin

In this report we showed that the resistive force produced
by weakly bound cross-bridges plays an important role indetermining the sliding velocity. We considered only asituation where actin filaments are sliding on HMM without
external load or under small external load. Within thislimitation, Eqs. 20 and 21 can be fundamental equations thatlink the mechanical behaviors of skeletal actomyosin to the
enzymatic kinetics. However, for different types of myosins
whose kinetic mechanisms are largely different from that ofskeletal myosin, these equations have to be modified.
We thank Dr. S. Ishiwata (Waseda University) for informing us of several
articles relevant to the present study, Dr. H. Sugi (Teikyo University) for
comments on the measurements of active forces, and Dr. M. F. Morales(University of the Pacific) for reviewing this manuscript and for valuablecomments.
This work was supported in part by the Ministry of Education, Science,Sports and Culture of Japan (grant-in-aid for Scientific Research on Priority Areas (A) to T.A.).

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