The excluded volume effect induced by poly(ethylene glycol) modulates the motility of actin filaments interacting with myosin

Authors

  • Shinsuke Munakata,

    1. Department of Bio-Systems Engineering, Graduate School of Science and Engineering, Yamagata University, Yonezawa, Japan
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  • Kuniyuki Hatori

    Corresponding author
    1. Department of Bio-Systems Engineering, Graduate School of Science and Engineering, Yamagata University, Yonezawa, Japan
    • Correspondence

      K. Hatori, Department of Bio-Systems Engineering, Graduate School of Science and Engineering, Yamagata University, Yonezawa 992-8510, Japan

      Fax +81 238 26 3727

      Tel: +81 238 26 3727

      E-mail: khatori@yz.yamagata-u.ac.jp

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Abstract

To examine the motility of actomyosin complexes in the presence of high concentrations of polymers, we investigated the effect of poly(ethylene glycol) on the sliding velocities of actin filaments and regulated thin filaments on myosin molecules in the presence of ATP. Increased concentrations and relative molecular masses of poly(ethylene glycol) decreased the sliding velocities of actin and regulated thin filaments. The decreased ratio of velocity in regulated thin filaments at − log[Ca2+] of 4 was higher than that of actin filaments. Furthermore, in the absence of Ca2+, regulated thin filaments were moderately motile in the presence of poly(ethylene glycol). The excluded volume change (∆V), defined as the change in water volume surrounding actomyosin during the interactions, was estimated by determining the relationship between osmotic pressure exerted by poly(ethylene glycol) and the decreased ratio of the velocities in the presence and absence of poly(ethylene glycol). The ∆V increased up to 3.7 × 105 Å3 as the Mr range of poly(ethylene glycol) was increased up to 20 000. Moreover, the ∆V for regulated thin filaments was approximately two-fold higher than that of actin filaments. This finding suggests that differences in the conformation of filaments according to whether troponin–tropomyosin complexes lie on actin filaments alter the ∆V during interactions of actomyosin complexes and influence motility.

Abbreviations
V

excluded volume change

HMM

heavy meromyosin

pCa

− log[Ca2+]

S1

myosin subfragment 1

Tn-Tm

troponin–tropomyosin

TRITC

tetramethylrhodamine isothiocyanate

Introduction

In living cells, proteins must interact and carry out their functions in a highly complex and crowded environment. Poly(ethylene glycol) is widely employed to mimic the intracellular milieu, and its effects on protein–protein interactions, enzyme–substrate binding, protein folding and DNA structure have been extensively investigated [1-11]. Crowded environments can induce an excluded volume around proteins that is inaccessible to crowding agents by the distance of their gyration radius [4]. Consequently, proteins tend to associate so that the excluded volume is minimized to increase the area possible for the translational motion of crowding agents [12, 13]. Moreover, the presence of poly(ethylene glycol) in solution affects the behavior of water molecules that surround proteins and can influence protein activity.

Rand et al. [1] detected a change in hydration when hexokinase binds glucose in the presence of poly(ethylene glycol) to induce osmotic stress. They concluded that the change in the conformation of hexokinase upon binding glucose is accompanied by dehydration of the binding pocket and the substrate. Similarly, Highsmith et al. [2] showed that the interactions between actin filaments and myosin subfragment 1 (S1) during ATP hydrolysis occur with the dehydration of actomyosin, and that higher Mr poly(ethylene glycol) induces more dehydration. In muscle fibers, the presence of poly(ethylene glycol) increases isometric tension and decreases shortening velocity and ATPase activity [3]. These studies also suggest that the presence of poly(ethylene glycol) mainly strengthens the binding of myosin heads to actin filaments, but does not affect the conformation and dehydration of myosin alone [2, 3]. Because poly(ethylene glycol) is an osmolyte with the potential to produce the excluded volume around proteins, further verification of the effect of poly(ethylene glycol) on the motility of actomyosin complexes is required to understand the action of the motors surrounded by water molecules and crowding agents.

In the present study, we demonstrate the effect of ethylene glycol, di(ethylene glycol) and poly(ethylene glycol) molecules ranging in Mr up to 20 000 on the motility of actin filaments interacting with myosin molecules. The sliding velocity of actin filaments decreased with the increase in concentration and Mr of poly(ethylene glycol). Because Amitani et al. proposed that sliding velocity is proportional to the square root of the Km of actin-activated ATPase activity [14], we applied this relationship to the equation describing the correspondence of free energy change (based on Km) with the osmotic work [2]. The logarithm of the decreased ratio of velocity was proportional to the osmotic pressure exerted by poly(ethylene glycol), and this proportional constant was postulated to account for the excluded volume change (∆V) around actomyosin during their interactions. The ∆V increased as the Mr of poly(ethylene glycol) was increased, and regulated thin filaments raised the ∆V. These results suggest that the conformation of the surface of filaments decorated with troponin–tropomyosin (Tn-Tm) affects the excluded volume for actomyosin interactions.

Results

Sliding velocities of actin filaments and regulated thin filaments in the presence of poly(ethylene glycol)

Figure 1A shows the decrease in the sliding velocity of actin filaments in the presence of ethylene glycol, di(ethylene glycol), and poly(ethylene glycol). Ethylene glycol and di(ethylene glycol) at 20% (w/w) decreased the velocity by approximately one-half. Higher Mr poly(ethylene glycol) preparations further decreased the velocity, and poly(ethylene glycol) 20 000 completely suppressed sliding at 8% (w/w). The velocity of regulated thin filaments at − log[Ca2+] (pCa) 4 also decreased with increases in both the concentration and Mr of poly(ethylene glycol) (Fig. 1B). The decrease in the ratio of sliding velocities caused by poly(ethylene glycol) was greater in regulated thin filaments than in actin filaments. Interestingly, in the absence of Ca2+, regulated thin filaments were motile in the presence of poly(ethylene glycol) molecules ranging in Mr between 1000 and 20 000 (Fig. 1C). The velocity of induced motility was decreased with further increases in poly(ethylene glycol) concentration. The poly(ethylene glycol) concentration range in which complete suppression occurred in the absence of Ca2+ was almost consistent with that at pCa 4. The presence of poly(ethylene glycol) induced changes in the lengths of actin and regulated thin filaments at pCa 4. Figure 2 shows typical images of regulated thin filaments in the motility assay in the presence of poly(ethylene glycol) 3000. In the absence of poly(ethylene glycol), the mean velocity and mean length of regulated thin filaments were 6.4 μm s−1 and 1.7 μm, respectively. In contrast, 5% (w/w) poly(ethylene glycol) 3000 induced a velocity of 1.1 μm s−1 and length of 0.5 μm (nearly the limit of optical resolution). At 10% (w/w) poly(ethylene glycol) 3000, which completely suppressed motility, the length was 2.3 μm. The presence of poly(ethylene glycol) induced the fragmentation of actin filaments during movement, whereas an excess of poly(ethylene glycol) that completely inhibited motility did not alter filament length.

Figure 1.

Dependence of relative velocity on the concentration and Mr of poly(ethylene glycol). (A) Actin filaments without Tn-Tm. (B) Regulated actin filaments with Tn-Tm at pCa 4. (C) Regulated actin filaments in the absence of Ca2+. Relative velocity is defined as the ratio Vpeg/V0; here, Vpeg and V0 are velocities in the presence and absence of poly(ethylene glycol), respectively.

Figure 2.

Micrographs of fluorescently labeled regulated thin filaments on an HMM-fixed surface in the presence of ATP and poly(ethylene glycol) 3000 at pCa 4. (A) In the absence of poly(ethylene glycol), the mean velocity and length of actin filaments were 6.4 μm s−1 and 1.7 μm, respectively. (B) At 5% poly(ethylene glycol), the mean velocity and filament length were 1.1 μm s−1 and 0.5 μm, respectively. (C) At 10% poly(ethylene glycol), the mean velocity and filament length were 0.0 μm s−1 and 2.3 μm, respectively. Scale bar: 10 μm. All three images are at the same magnification.

Relationship between osmotic pressure and sliding velocity of actomyosins

Poly(ethylene glycol) exerts osmotic pressure on proteins in solution. It is likely that osmotic pressure affects the interaction between proteins by changing the extent of hydration on the surface of proteins or the volume of peripheral water molecules [1]. We attempted to estimate the change in preferential water volume during actomyosin interactions by defining the decrease in the ratio of sliding velocity as a function of the osmotic pressure exerted by poly(ethylene glycol). Figure 3 shows the relationship between osmotic pressure and the logarithm of the decrease in the ratio of velocity (V0/Vpeg) without and with poly(ethylene glycol), based on the data shown in Fig. 1 (see also Experimental procedures). The results for ethylene glycol and di(ethylene glycol) are shown in Fig. S1. The values of 2 log(V0/Vpeg) were proportional to the osmotic pressure at all conditions tested. The magnitude of the slope of the line increased with increases in the Mr of poly(ethylene glycol). The slopes reflect the ∆V of the actomyosin interface during their interactions (Eqn (1) in Experimental prodedures). Figure 4 shows the ∆V for actin filaments and regulated thin filaments at pCa 4, and without Ca2+, as a function of the Mr of poly(ethylene glycol). The ∆V for regulated thin filaments was, on average, approximately two-fold greater than that for actin filaments from ethylene glycol to poly(ethylene glycol) 20 000. Furthermore, regulated thin filaments showed almost identical ∆V values in the presence and absence of Ca2+. The relationship between the ∆V and Mr of poly(ethylene glycol) could be described by power functions of 0.94 and 0.74 for actin filaments and regulated thin filaments, respectively.

Figure 3.

Plot of the logarithm of inverse relative velocity [2 log(V0/Vpeg)] versus osmotic pressure exerted by poly(ethylene glycol) according to Eqn (1). A straight line through the origin was fitted to the data, and the slope of the line was defined as the change in osmolyte-excluded volume (∆V). (A) Actin filaments. (B) Regulated thin filaments at pCa 4. (C) Regulated thin filaments in the absence of Ca2+. The value of osmotic pressure was obtained from the relationship between the concentration (%, w/w) of poly(ethylene glycol) and osmotic pressure, as shown by Rand. Symbols: ●, poly(ethylene glycol) 1000; ○, poly(ethylene glycol) 2000; ■, poly(ethylene glycol) 3000; □, poly(ethylene glycol) 10 000; ▼, poly(ethylene glycol) 20 000.

Figure 4.

Logarithmic plot of the excluded volume change (∆V) during actomyosin interactions versus the Mr of poly(ethylene glycol) [insert: linear plot of ∆V versus Mr of poly(ethylene glycol)]. The data were fitted by use of a power function (ΔV ∞ Mrα; here, Mr is the relative molecular mass of poly(ethylene glycol), and α is a variable). Filled circles, filled triangles and open triangles represent actin filaments (α = 0.94), regulated thin filaments at pCa 4 (α = 0.74) and regulated thin filaments without Ca2+ (α = 0.76), respectively.

Discussion

Effect of poly(ethylene glycol) on the motility of actin filaments and regulated thin filaments

Although some studies have shown that crowded environments have only a small effect on protein–protein interactions [8, 9], poly(ethylene glycol) can decrease the dissociation constant of myosin for actin [2, 3]. Accordingly, we were interested in how the increase in the binding affinity of actomyosin induced by osmotic pressure is related to the decrease in velocity. Moreover, Chase et al. have already reported that the velocity of actin filaments in an in vitro motility assay decreases with increasing concentrations of poly(ethylene glycol) of Mr ≤ 3000 [15]. They proposed that poly(ethylene glycol) induces a further decrease in velocity beyond the effect of viscosity, and strengthens the electrostatic interactions of actomyosin. We attempted to further determine the effect of poly(ethylene glycol) over three orders of magnitude of Mr on the sliding velocity of both actin and regulated thin filaments. If viscosity mainly affected cross-bridge motions, we would expect the effect of poly(ethylene glycol) to be similar between actin filaments and regulated thin filaments decorated with Tn-Tm. We demonstrated that poly(ethylene glycol) molecules with Mr values ranging up to 20 000 suppressed velocity. Furthermore, we found that poly(ethylene glycol) exerted a greater influence on the motility of regulated thin filaments than on the motility of actin filaments. However, the effects of viscosity on the excluded volume effect induced by poly(ethylene glycol) cannot be individually determined with our present system.

Some aggregation of regulated thin filaments was observed during the motility assay in the presence of higher Mr preparations of poly(ethylene glycol). This occurred after addition of the filaments to a heavy meromyosin (HMM)-fixed surface in the presence of a solution containing ATP and poly(ethylene glycol), and aggregation increased gradually thereafter. This may have been caused by the bundling of actin filaments induced by poly(ethylene glycol) [16]. To avoid the effects of aggregation on the sliding velocity of filaments, we measured filament velocity early during the assay (< 5 min), before aggregates appeared.

To verify whether phalloidin affects poly(ethylene glycol)-induced suppression of motility, Cy3-bound actin filaments modified by Cy3-maleimide introduced into Cys374 of actin were tested instead of tetramethylrhodamine isothiocyanate (TRITC)–phalloidin-bound filaments. In the presence of 5% poly(ethylene glycol) 20 000, the decreased ratio of velocity of Cy3-bound actin filaments was consistent with that of TRITC–phalloidin-bound actin filaments, indicating that phalloidin is not a major factor in poly(ethylene glycol)-induced suppression.

Actin filaments and regulated thin filaments fragmented while their velocities decreased in the presence of poly(ethylene glycol). Moreover, it is known that, during actin-activated ATP hydrolysis, the Km for actin decreases with an increase in poly(ethylene glycol) concentration [2]. Furthermore, Chinn et al. [3] proposed that poly(ethylene glycol) induces an increase in the affinity of the otherwise weak binding state of actomyosin before the release of inorganic phosphate. It is likely that the increase in binding affinity strengthens friction between the filaments and myosin heads against a driving force, because Tawada and Sekimoto [17] theoretically proposed that the weak binding interaction induces an increase in friction. Once friction induces constriction within the filaments, the filaments distort and eventually fragment. Although single actin filaments break at forces of 200–400 pN in an in vitro stretching test [18], it is possible that actin filaments are readily broken at 3–4 pN in myosin-mediated motility conditions [19]. Furthermore, this mechanism can be explained by poly(ethylene glycol) induction of an increase in isometric tension and a decrease in the shortening velocity of muscle fibers [3]. Interestingly, we did not detect fragmentation of filaments when motility was completely suppressed by an excess of poly(ethylene glycol) [e.g. 10% of poly(ethylene glycol) 3000]. We determined the values of Km and Vmax for actin-activated ATPase activity in the presence of poly(ethylene glycol) 20 000 (Figs S2 and S3). As reported previously [2], Km decreased with the increase in poly(ethylene glycol) concentrations in the range of 0–5%, whereas Vmax was almost unchanged. On the other hand, higher concentrations of poly(ethylene glycol) (> 5%) decreased Vmax. An interpretation of this finding is that higher concentrations of poly(ethylene glycol) inhibited the structural change in actomyosin required for catalysis, and therefore impaired the generation of the force required for motility. Therefore, we cannot easily refer to our Eqn (1) for data at higher concentrations of poly(ethylene glycol), in which the velocity was below 20% of the original velocity. Although velocity was affected by the decrease in Vmax with higher concentrations of poly(ethylene glycol), data obtained at lower concentrations appeared to obey a linear relationship between osmotic pressure and 2 log(V0/Vpeg).

There is a consensus that, in striated muscles, Tn-Tm bound to actin filaments inhibits the binding of myosin heads to the filaments in the absence of Ca2+ [20]. When the concentration of Ca2+ is raised above 1 μm (pCa 6), the structural change of Ca2+-bound troponin C is transmitted to tropomyosin along actin filaments, resulting in the transition from a blocked to a closed state in thin filaments [21]. Subsequently, the binding of myosin heads to thin filaments induces the open state in thin filaments that is required to generate force. Our study showed that poly(ethylene glycol) at certain concentrations and relative molecular masses allowed the regulated thin filaments to release the inhibition of motility of actomyosin, even in the absence of Ca2+. It is expected that the interaction between actin filaments and myosin heads for driving force will occur if their binding affinity overcomes the barrier of blocking the binding of myosin by Tn-Tm in the absence of Ca2+. In fact, ADP-induced strong binding of myosin heads to regulated thin filaments can decrease the sliding velocity, and increases the Ca2+ sensitivity [22]. However, the possibility that poly(ethylene glycol) affects the Tn-Tm system on actin filaments is not precluded.

Excluded volume effect of poly(ethylene glycol) on actomyosin interactions

A significant finding of the present study is that the decrease in velocity was also dependent on the Mr of poly(ethylene glycol). This indicates that the gyration radius of poly(ethylene glycol) molecules affects the excluded volume near the actomyosin interface. There is evidence that poly(ethylene glycol) induces the preferential hydration of proteins caused by steric exclusion of poly(ethylene glycol) molecules that are inaccessible to the protein surface [4]. The steric exclusion of solutes around protein surfaces generates the depletion force between proteins, which minimizes the excluded volume [12]. The present study shows that the ∆V is a function of Mr to the power 0.94 (Mr0.94) for actin filaments and to the power of 0.74 for regulated thin filaments (Fig. 4). Because the ∆V value predicted geometrically for the binding between two spheres was larger than the experimental data and was a function of Mr1.38 (Fig. S4), the shape of the actomyosin interface cannot simply be treated as a sphere. Because the ∆V in ethylene glycol and di(ethylene glycol) deviated from a power regression line in Fig. 4, the effect of such lower weight osmolytes might be distinct from that of poly(ethylene glycol). In contrast, the gyration radius of poly(ethylene glycol) 20 000 is large (64 Å), in the same order of magnitude as the size of a myosin head; nonetheless, such large poly(ethylene glycol) molecules could also affect motility.

The ∆V values for poly(ethylene glycol) 1000 and poly(ethylene glycol) 3000 were estimated as 1.6 × 104 and 7.4 × 104 Å3, respectively, from the changes in velocity; these are consistent with the values for the Km of actin-activated ATPase activity measured by Highsmith et al. [2]. Furthermore, our finding that the ∆V was nearly two-fold greater in regulated thin filaments than in actin filaments indicates that the binding of Tn-Tm molecules to actin filaments increases the contact area in thin filaments for binding of myosin heads. Figure 5 illustrates the binding of a myosin head to actin units with an excluded volume change. Contact areas in the atomic model of a tropomyosin–actin–S1 complex (Protein Data Bank ID: 4A7L) were estimated as 890 Å2 for the binding of S1 to actin and as 1056 Å2 for the binding to tropomyosin–actin. This calculation indicates that S1 contacts tropomyosin–actin over a larger area than it contacts actin.

Figure 5.

Illustration of the excluded volume change (∆V  ) during the binding of a myosin head to an actin filament (cross-section). Poly(ethylene glycol) can induce the excluded volume change around actin filaments and myosin molecules to a value equal to its gyration radius (Rg). When a myosin head binds to an actin filament, the excluded volume decreases (hatched areas). Our results suggest that the excluded volume change is larger in regulated thin filaments than in actin filaments, owing to the increase in the contact region caused by addition of Tm-Tn. An atomic model of S1 bound to actin filament with tropomyosin was obtained from Protein Data Bank ID 4A7L [32], and was drawn with pymol (Schrödinger LLC). With the ccp4 program [33], contact areas for the binding of S1 to actin units and for binding to tropomyosin–actin units were estimated as 890 Å2 and 1069 Å2, respectively. Tm, tropomyosin.

We approximated the osmotic work needed to stall actomyosin by calculating the ∆V multiplied by ∏stall, at which the sliding velocity became zero. For example, values of ∏stall at 16% poly(ethylene glycol) 1000 and at 8% poly(ethylene glycol) 20 000 (Fig. 1A) were 8.4 × 106 dyne cm−2 and 6.7 × 105 dyne cm−2, respectively. The average osmotic work for stalling of actin filaments in the range of poly(ethylene glycol) 1000 to poly(ethylene glycol) 20 000 was estimated as 4.4 ± 1.4 kBT (± indicates upper and lower limits) per myosin head. This value corresponds to 22% of the magnitude of free energy (− 20 kBT) generated by the hydrolysis of a single ATP molecule, which is somewhat lower than the maximum efficiency (36%) of the contraction of muscle fibers [23]. Furthermore, higher energy (5.8 ± 1.7 kBT) was obtained for regulated thin filaments. In the absence of Ca2+, the onset of motility of regulated thin filaments occurred at 7.6 × 105 dyne cm−2 for 2% poly(ethylene glycol) 1000 and at 2.1 × 104 dyne cm−2 for 1% poly(ethylene glycol) 20 000. This osmotic work that induced the onset of motility was estimated as 0.9 ± 0.5 kBT. Therefore, osmotic work on the order of thermal energy allows regulated thin filaments to interact with myosin heads for motility without calcium ions. Evidence to support this interaction is that the equilibrium constant between the blocked and closed conformation of thin filaments is 0.3 in the absence of Ca2+ [24].

We found that the geometrically intricate shape of actin filaments, induced by its decoration with Tn-Tm, produced large changes in the volume of water surrounding the interface of actomyosin during interactions, resulting in a significant effect on their motility. Because poly(ethylene glycol) suppresses the motility of actomyosin in vitro, this raises the question of whether crowded environments in situ adversely affect actomyosin functions. One explanation for this discrepancy is that ionic strength is higher in muscle cells (~ 0.14 m) [25] than in our experimental conditions (~ 0.05 m). Electrostatic interactions between actin filaments and myosin heads tend to weaken as ionic strength increases. Moreover, the suppression of motility of actomyosin by poly(ethylene glycol) could be recovered by increasing the concentration of KCl [15]. Interactions of actomyosin might be governed by the balance between the excluded volume and hydrophobic effects, as well as the electrostatic force [26].

Experimental procedures

Reagents

Ethylene glycol, di(ethylene glycol), poly(ethylene glycol) 1000 and poly(ethylene glycol) 3000 were purchased from Wako Pure Chemical Industries (Osaka, Japan). Poly(ethylene glycol) 2000 and poly(ethylene glycol) 20 000 were purchased from Nacalai Tesque (Kyoto, Japan). Poly(ethylene glycol) 10 000 was purchased from Sigma-Aldrich Co. LLC (St Louis, MO, USA). Each poly(ethylene glycol) number denotes the average Mr. We used these poly(ethylene glycol) molecules without further purification.

Preparation of actin filaments, HMM, and regulated thin filaments

Actin and myosin were prepared from rabbit skeletal muscle with a standard protocol [27]. HMM was prepared by chymotryptic digestion of myosin [27]. To minimize the influence of phalloidin on motility [28], actin monomers at 1.6 μm were polymerized into Mg-bound actin filaments in a standard solution (25 mm KCl, 25 mm imidazole-HCl, pH 7.4, 4 mm MgCl2, 0.5% 2-mercaptoethanol) containing TRITC-conjugated phalloidin (Sigma-Aldrich Co. LLC) at equimolar ratios. After incubation for 3 days at 4 °C, we used these fluorescently labeled actin filaments in all experiments. Tn-Tm complexes were obtained from rabbit skeletal muscle by the method of Ebashi and Kodama [29], and purified by fractionation with 25–37.5 g dL−1 ammonium sulfate. Regulated thin filaments were prepared according to the method of Ishiwata and Kondo [30], in which a mixture of 0.05 mg mL−1 Tn-Tm and 1 μg mL−1 fluorescently labeled actin filaments in a standard solution (25 mm KCl, 25 mm imidazole-HCl, pH 7.4,, 4 mm MgCl2, 0.5% 2-mercaptoethanol) was heated at 45 °C for 30 s, and cooled immediately on ice to ensure binding of Tn-Tm to actin filaments.

Motility assay

Motility assays were performed according to a standard method [27]. HMM molecules were fixed onto a collodion-coated glass slide (No. 1, 24 × 50 mm; Matsunami Glass Industries, Osaka, Japan) by perfusing 0.05 mg mL−1 HMM in standard solution between the slide and the cover glass (No. 1, 18 × 18 mm; Matsunami Glass Industries) separated by 0.1 mm. Sixty seconds after perfusion, the solution was replaced with 3 mg mL−1 BSA in standard solution to remove unbound HMM molecules. The slide was then perfused with the actin solution (1 μg mL−1 TRITC–phalloidin-bound actin filaments in the standard solution), to which the ATP solution [2 mm ATP, 3 mg mL−1 glucose, 0.1 mg mL−1 glucose oxidase, 0.02 mg mL−1 catalase, and various poly(ethylene glycol) concentrations in the standard solution] was immediately added. The movement of the actin filaments was observed with a fluorescence microscope (objective TIRF × 100 H, Ti-U; Nikon, Tokyo, Japan) equipped with a fluorescence illuminator (TI-SFL; Nikon) and optical interference filters for rhodamine. For analysis of regulated thin filaments, the specimens were supplemented with 4 mm EGTA and 3.95 mm CaCl2 (pCa 4).

Images were acquired by use of an EM-CCD camera (DE-500; Hitachi Kokusai Electric, Tokyo, Japan), and recorded on a personal computer (Vistro 220; Dell, Austin, Texas, USA) with a video grabber board (PIXCI-SV5; Epix, Buffalo Grove, IL, USA) and streampix software (NorPix, Montreal, Quebec, Canada). Spacing between the nearest-neighbor pixels was 0.10 μm. The velocity of the actin filaments was determined by measuring the displacement at intervals of 0.5 s for 5 s with imagej software (Rasband, W.S., ImageJ; National Institutes of Health, Bethesda, MD, USA). The mean velocity was calculated from the data for 100 filaments under each condition. The velocities were normalized to the original velocity in the absence of poly(ethylene glycol). Velocities of regulated thin filaments in the absence of Ca2+ were normalized to those at pCa 4 in the absence of poly(ethylene glycol).

Estimation of apparent dehydration under osmotic pressure

At the osmotic pressure exerted by poly(ethylene glycol), the dissociation constant for actin and the myosin head is decreased [2], when the Michaelis constant (Km) for actin-activated ATPase activity is defined as the dissociation constant. At equilibrium, the free energy change derived from the change in the Km value caused by poly(ethylene glycol) is equal to the work produced by osmotic pressure, as follows:

display math(1)

where math formula and math formula are Michaelis constants for actin in the absence and presence of poly(ethylene glycol), respectively. ∏0 and ∏peg are osmotic pressures in the absence and presence of poly(ethylene glycol), respectively. R and T denote the gas constant and absolute temperature, respectively. We defined ∏ ∆V as osmotic work. The original paper proposed that ∆V represents the change in hydration of proteins associated with water activity [1]. At the same time, we defined ∆V as the osmolyte-excluded volume change resulting from the decrease in preferential water volume around actomyosin accompanying the binding of myosin and actin [2].

Amitani et al. derived an equation to describe the relationship between motility and ATPase activity as follows: sliding velocity is proportional to the square root of Km multiplied by the Vmax of actin-activated ATPase activity [15]. Because poly(ethylene glycol) is chemically inert with respect to myosin, and the Vmax of actomyosin is independent of poly(ethylene glycol) concentration [2], Eqn (1) is transformed into Eqn (1) for velocity as follows:

display math(2)

where V0 and Vpeg are the sliding velocities of actin filaments on HMM molecules in the absence and presence of poly(ethylene glycol), respectively. We refer to V0/Vpeg as inverse relative velocity. The data for the osmotic pressure exerted by each poly(ethylene glycol) as a function of its concentration were obtained from the website of Rand (http://www.brocku.ca/researchers/peter_rand/osmotic/osfile.html), and were interpolated by use of a quadratic function according to the method of Cohen and Highsmith [31]. Osmotic pressures exerted by ethylene glycol and di(ethylene glycol) were calculated by use of the Morse equation (∏ = MRT, where M is molarity).

Acknowledgements

We thank Editage for providing editorial assistance.

Author contributions

S. Munakata: performed experiments and analyzed data. K. Hatori: designed research and wrote the article.

Disclosure

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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