Myosin cross-bridge kinetics slow at longer muscle lengths during isometric contractions in intact soleus from mice

Small step analysis Striated muscle responds to a sudden length increase with a force response that has classically been described by four phases (1–4). In phase 1, force rises instantaneously as attached cross-bridges and elastic elements are strained (Fig. 2, σ0 to σ1). In phase 2, force decays as cross-bridges detach in a synchronized manner (Fig. 2, σ1 to σ2). Next, force rises in phase 3 due to cross-bridge recruitment, and finally the force response plateaus at an elevated level (phase 4) due to passive properties (i.e., of stretching tendon, connective tissue, titin and collagen) and changes in active (i.e. altered cross-bridge activity) biomechanical properties of the muscle at the new, longer muscle length. These phases are a time-domain representation of the same processes observed via smaller-amplitude length perturbation analysis (0.05-1.25% muscle length), in which frequency-dependent shifts in the elastic and viscous moduli (i.e. viscoelastic stress-strain stiffness response) describe enzymatic and mechanical properties of the muscle (5–7). Thus, the time-dependent modulus response, Y(t) (=σ(t)/ε(t), where σ(t) and ε(t) represent the muscle stress response and muscle strain as a function of time, respectively) a step-function change in muscle length can be characterized as described by Palmer et al. (7): Y(t) = At-k-Be-2πbt+Ce-2πct (1) The molecular, cellular, and tissue characteristics described by Eq. 1 are outlined below, with this model appropriately describing Y(t) for the measured stress-strain response between σ1 and σfinal, with the initial time-point occurring at the onset of the strain stimulus (i.e. time=0 at t0 for σ0)—as further described below and illustrated in Fig. 2. The A-process reflects the viscoelastic mechanical response of passive, structural elements in the muscle and holds no enzymatic dependence. Parameter A represents the combined mechanical stress of the muscle and tendon, while parameter k describes the viscoelasticity of these passive elements; a k value of 0 (minimum possible value for the fractional derivative) represents a purely elastic response and a k value of 1 represents a purely viscous response (8,9). The B- and C- processes characterize work-producing (cross-bridge recruitment) and work-absorbing (cross-bridge detachment) characteristics of the contracting muscle, respectively (7,10–12). Thus, these B- and C- processes are analogous to the work-producing and work-absorbing process described above during phase 3 for cross-bridge recruitment and phase 2 for cross-bridge detachment. The parameters B and C represent the mechanical stress from the cross-bridges, and the rate parameters b and c reflect cross-bridge kinetics (13). More specifically, molecular processes contributing to cross-bridge recruitment or force generation underlie the cross-bridge recruitment rate, 2πb, and processes contributing to cross-bridge detachment or force decay underlie the cross-bridge detachment rate, 2πc. To best emulate physiological length changes, both the proximal and distal tendons of the muscle preparations were left intact. However, the tendons add an additional in-series elastic element that should be considered when applying small step analysis to ensure that the muscle (and not tendon) is primarily experiencing the perturbation. Using methods described by Cui et al. 2008 (14), we calculated tendon area and length to provide an estimate of tendon stiffness with: (2) where E is the elastic modulus (reported to be approximately 62 N/mm2 for mouse soleus tendon (15)), AT is calculated tendon area (0.59 mm2), and lT is the calculated tendon length (5.67±0.14 mm) at Lo. Therefore, we can roughly estimate an in-series elastic element with a stiffness of 6.45 N/mm. Our average Lo was 15.22±0.37 mm, so a 1% length-step stretched the MTU on average 0.15 mm. During this step, we measured an average rise in force (represented as phase 1) around 0.16 N, for MTU stiffness (KMTU) of 1.05 N/mm. Therefore, we can solve for the stiffness of the muscle (KM) using: (3) which estimates Km to be on average 1.25 N/mm. These values suggest that the stiffness of the tendon is approximately five times that of the stiffness of the muscle, and thus we assume that for a given length increase, the muscle fibers experience the majority of the length change from the step-length perturbations imposed herein. Animal models All procedures were approved by the Institutional Animal Care and Use Committee at Washington State University and complied with the Guide for the Use and Care of Laboratory Animals published by the National Institutes of Health. Male C57BL/6 mice (7 weeks old) were sourced from Simonsen Laboratories. Ten mice were anesthetized by isoflurane inhalation (3% volume in 95% O2-5% CO2 flowing at 2 L/min), then a soleus muscle was removed, preserving as much of the proximal and distal tendons as possible. After removal, the whole MTU was immediately placed in the experimental chamber containing oxygenated Ringer’s solution. Tetanic step procedure Soleus muscles were suspended in a temperature controlled, oxygenated chamber with the distal tendon sutured onto a force transducer (Aurora Scientific 407A, Aurora, ON), and the proximal tendon sutured onto a length controller (Aurora Scientific 305C, Aurora, ON). Muscles equilibrated with the Ringer’s solution (154 mM NaCl, 5.6 mM KCl, 1 mM MgCl2, 2.2 mM CaCl2, 10 mM glucose, 20 mM HEPES, pH 7.4, (16)) for at least 5 minutes. Muscles were field stimulated (Aurora scientific 701C, Aurora, ON) via parallel electrode plates along the length of the chamber at 20 V and 100 Hz to measure maximum isometric tetanic force (Figure 1A). Optimal MTU length (Lo) was determined by tetanically stimulating the muscle at multiple lengths (minimum of 5 lengths, each followed by at least 3 minutes rest) to produce a force-length curve. Lo was then defined as the length that generated maximum active force along the MTU force-length curve. Once Lo was established, the MTU was set to one of three lengths: Lo, 90% of Lo, or 110% of Lo. Tetanic contraction was then induced in the muscle for 2 seconds, with a length step (1% increase in MTU length) applied at 1 second and held for the remainder of the activation period (Figure 2). This procedure was then repeated for the remaining two lengths and then again for all three lengths at the other test temperature. The order of MTU length and temperature of the chamber (17°C or 27°C) was randomized for each experiment. After mechanical experiments were completed, muscle and tendons were separated, measured, and weighed to estimate the physiological cross-sectional area (CSA, (17)): (4) where m is muscle mass, θ is pennation angle (=8.5° for soleus muscle (18)), Lf is fiber length calculated from fiber-to-muscle length ratio (=0.71 for soleus muscle (19)), and ρ is muscle density (=1.06 mg/mm3 (20)). Average soleus muscle CSA was 0.78±0.05 mm2. Force was then divided by the CSA to calculate stress. Statistical analysis All data are listed as mean ± SEM. Sequential quadratic programming methods in Matlab (v. 9.4.0, The Mathworks, Natick MA) was used for constrained nonlinear least-squares fitting of Eq. 1 to moduli data for each muscle. Statistical analysis of experimental data were performed in SPSS (IBM Statistics, Chicago, IL), implementing linear mixed models with muscle length and temperature as repeated measures where appropriate. This approach matches data from the same muscle to provide more statistical power than a one-way ANOVA. First-order autoregression was assumed for the covariance structure and post-hoc analyses were performed using least-significant difference corrections where appropriate. Statistical significance is reported at p<0.05. References 1. Huxley AF, Simmons RM. Proposed mechanism of force generation in striated muscle. Nature. 1971;233(5321):533–8. 2. Ford LE, Huxley AF, Simmons RM. Tension responses to sudden length change in stimulated frog muscle fibres near slack length. J Physiol. 1977;269(2):441–515. 3. Davis JS, Rodgers ME. Force generation and temperature-jump and length-jump tension transients in muscle fibers. Biophys J. 1995 May;68(5):2032–40. 4. Huxley AF. Muscular contraction. J Physiol. 1974;243(1):1–43. 5. Kawai M, Brandt PW. Sinusoidal analysis: a high resolution method for correlating biochemical reactions with physiological processes in activated skeletal muscles of rabbit, frog and crayfish. J Muscle Res Cell Motil. 1980 Sep;1(3):279–303. 6. Tanner BCW, Wang Y, Maughan DW, Palmer BM. Measuring myosin cross-bridge attachment time in activated muscle fibers using stochastic vs. sinusoidal length perturbation analysis. J Appl Physiol. 2011;110(4):1101–8. 7. Palmer BM, Suzuki T, Wang Y, Barnes WD, Miller MS, Maughan DW. Two-State Model of Acto-Myosin Attachment-Detachment Predicts C-Process of Sinusoidal Analysis. Biophys J. 2007;93(3):760–9. 8. Palmer BM, Tanner BCW, Toth MJ, Miller MS. An inverse power-law distribution of molecular bond lifetimes predicts fractional derivative viscoelasticity in biological tissue. Biophys J. 2013 Jun;104(11):2540–52. 9. Mulieri LA, Barnes WD, Leavett BJ, Ittleman FP, LeWinter MM, Alpert NR, et al. Alterations of myocardial dynamic stiffness implicating abnormal crossbridge function in human mitral regurgitation heart failure. Circ Res. 2002;90(1):66–72. 10. Kawai M, Wray JS, Zhao Y. The effect of lattice spacing change on cross-bridge kinetics in chemically skinned rabbit psoas muscle fibers. I: Proportionality between the lattice spacing and the fiber width. Biophys J. 1993;64(1):187–96. 11. Campbell KB, Chandra M, Kirkpatrick RD, Slinker BK, Hunter WC. Interpreting cardiac muscle force-length dynamics using a novel functional model. Am J Physiol Hear Circ Physiol. 2004;286:H1535-H1545. 12. Kawai M, Halvorson HR. Two step mechanism of phosphate release and the mechanism of force generation in chemically skinned fibers of rabbit psoas muscle. Biophys J. 1991 Feb;59(February):329–42. 13. Lymn RW, Taylor EW. Mechanism of adenosine triphosphate hydrolysis by actomyosin. Biochemistry. 1971;10(25):4617–24. 14. Cui L, Perreault EJ, Maas H, Sandercock TG. Modeling short-range stiffness of feline lower hindlimb muscles. J Biomech. 2008;41(9):1945–52. 15. Rigozzi S, Müller R, Snedeker JG. Local strain measurement reveals a varied regional dependence of tensile tendon mechanics on glycosaminoglycan content. J Biomech. 2009; 16. Barton ER, Lynch G. Measuring isometric force of isolated mouse muscles in vitro. 2008;(Id):1–14. 17. Sacks RD, Roy RR. Architecture of the hind limb muscles of cats: Functional significance. J Morphol. 1982 Aug 1;173(2):185–95. 18. Burkholder TJ, Fingado B, Baron S, Lieber RL. Relationship between muscle fiber types and sizes and muscle architectural properties in the mouse hindlimb. J Morphol. 1994;221(2):177–90. 19. Brooks S V, Faulkner JA. Contractile properties of skeletal muscles from young, adult and aged mice. 1988 Oct;404:71–82. 20. Mendez J, Keys A. Density and Composition of Mammalian Muscle. Metabolism. 1960;9(2):184–8.

小步长分析
横纹肌（striated muscle）对突发性长度增加的力响应经典上可分为四个阶段（1~4）。第1阶段中，随着结合的横桥（cross-bridge）与弹性元件被牵拉，力瞬时上升（图2，σ₀至σ₁）。第2阶段中，横桥以同步方式解离，力随之衰减（图2，σ₁至σ₂）。随后第3阶段中，因横桥募集使力再次上升；最终因被动特性（即肌腱（tendon）、结缔组织、肌联蛋白（titin）与胶原蛋白（collagen）的特性）及肌肉在新的更长长度下的主动（即横桥活性改变）生物力学特性变化，力响应在升高水平趋于平稳（第4阶段）。

上述阶段是通过小振幅长度扰动分析（0.05%~1.25%肌肉长度）观测到的相同过程的时域表征，该分析中弹性与黏性模量的频率依赖性偏移（即黏弹性应力-应变刚度响应）可反映肌肉的酶学与力学特性（5~7）。因此，肌肉长度阶跃变化下的时变模量响应Y(t)（=σ(t)/ε(t)，其中σ(t)与ε(t)分别代表随时间变化的肌肉应力响应与肌肉应变）可由Palmer等人（7）提出的式(1)表征：
Y(t) = At⁻ᵏ - Be⁻²πᵇᵗ + Ce⁻²πᶜᵗ (1)

下文将详述式(1)所描述的分子、细胞与组织特性，该模型可恰当地描述σ₁至σ_final区间内测得的应力-应变响应，初始时间点对应应变刺激 onset（即t₀时刻t=0，对应σ₀），详见下文及图2说明。

A过程反映肌肉中被动结构元件的黏弹性力学响应，无酶学依赖性。参数A代表肌肉与肌腱的复合机械应力，参数k表征这些被动元件的黏弹性：k=0（分数阶导数的最小可能值）代表纯弹性响应，k=1代表纯黏性响应（8,9）。B过程与C过程分别表征收缩肌肉的产功（横桥募集）与耗功（横桥解离）特性（7,10~12），因此二者分别对应前述第3阶段的横桥募集产功过程与第2阶段的横桥解离耗功过程。参数B与C代表横桥产生的机械应力，速率参数b与c反映横桥动力学（13）。更具体而言，横桥募集速率2πb对应参与横桥募集或力产生的分子过程，横桥解离速率2πc对应参与横桥解离或力衰减的分子过程。

为尽可能模拟生理性长度变化，肌肉标本的近端与远端肌腱均予以保留。但肌腱会引入额外的串联弹性元件，在应用小步长分析时需予以考量，以确保主要受扰动的是肌肉而非肌腱。采用Cui等人2008年（14）所述方法，我们计算了肌腱面积与长度以估算肌腱刚度，所用公式如式(2)：其中E为弹性模量（小鼠比目鱼肌腱约为62 N/mm²（15）），A_T为计算得到的肌腱面积（0.59 mm²），l_T为Lo处计算得到的肌腱长度（5.67±0.14 mm）。由此可粗略估算串联弹性元件的刚度为6.45 N/mm。本实验中平均Lo为15.22±0.37 mm，因此1%的步长拉伸平均使肌肌腱单元（muscle-tendon unit, MTU）延长0.15 mm。实验中测得平均力上升（对应第1阶段）约0.16 N，由此得到MTU刚度K_MTU为1.05 N/mm。通过式(3)可求解肌肉刚度K_M：估算得到平均K_M为1.25 N/mm。上述结果表明肌腱刚度约为肌肉刚度的5倍，因此我们假设在本次施加的步长扰动中，肌纤维承受了大部分长度变化。

## 动物模型
所有实验流程均经华盛顿州立大学机构动物护理与使用委员会批准，并符合美国国立卫生研究院（National Institutes of Health, NIH）发布的《实验动物使用与护理指南》。
雄性C57BL/6小鼠（7周龄）购自Simonsen实验室。10只小鼠经异氟烷（isoflurane）吸入麻醉（3%体积浓度，混合气体为95% O₂-5% CO₂，流速2 L/min），随后分离比目鱼肌，尽可能保留近端与远端肌腱。分离后，完整的MTU即刻被置于充氧的实验槽中，槽内为林格液（Ringer’s solution）。

## 强直刺激步长实验流程
比目鱼肌被悬挂于温控充氧槽中，远端肌腱缝合至力传感器（Aurora Scientific 407A，加拿大安大略省奥罗拉市），近端肌腱缝合至长度控制器（Aurora Scientific 305C，加拿大安大略省奥罗拉市）。肌肉在林格液（154 mM NaCl、5.6 mM KCl、1 mM MgCl₂、2.2 mM CaCl₂、10 mM葡萄糖、20 mM HEPES，pH 7.4（16））中平衡至少5分钟。通过沿槽长方向布置的平行电极板，以20 V、100 Hz的参数进行场刺激（Aurora Scientific 701C刺激器，加拿大安大略省奥罗拉市），以测量最大等长强直肌力（图1A）。通过在多个长度点（至少5个，每个长度点后至少休息3分钟）进行强直刺激，绘制力-长曲线，确定最优MTU长度Lo，即对应MTU力-长曲线最大主动力的长度。

确定Lo后，将MTU设置为以下三种长度之一：Lo、90%Lo或110%Lo。随后诱导肌肉强直收缩2秒，在收缩至1秒时施加1%MTU长度增加的步长扰动，并维持至激活期结束（图2）。对其余两种长度重复上述流程，随后在另一测试温度下再次重复所有三种长度的实验。每次实验的MTU长度与槽温（17°C或27°C）顺序均随机化。

机械实验完成后，分离肌肉与肌腱，进行测量与称重，以估算生理横截面积（physiological cross-sectional area, CSA），所用公式如式(4)：其中m为肌肉质量，θ为羽状角（比目鱼肌为8.5°（18）），L_f为通过肌纤维-肌肉长度比计算得到的肌纤维长度（比目鱼肌为0.71（19）），ρ为肌肉密度（1.06 mg/mm³（20））。实验测得平均比目鱼肌CSA为0.78±0.05 mm²。将肌力除以CSA即可得到应力。

## 统计分析
所有数据以均值±标准误（standard error of the mean, SEM）表示。使用Matlab（v.9.4.0，The Mathworks，马萨诸塞州内蒂克市）中的序列二次规划方法，对每个肌肉的模量数据进行式(1)的约束非线性最小二乘拟合。实验数据的统计分析采用SPSS（IBM Statistics，伊利诺伊州芝加哥市），在合适情况下采用以肌肉长度与温度为重复测量变量的线性混合模型，该方法较单因素方差分析（ANOVA）可提供更高的统计效力。协方差结构假设为一阶自回归，合适情况下采用最小显著差校正进行事后分析。统计显著性阈值设为p<0.05。

## 参考文献
1. Huxley AF, Simmons RM. Proposed mechanism of force generation in striated muscle. 《自然》. 1971;233(5321):533–8.
2. Ford LE, Huxley AF, Simmons RM. Tension responses to sudden length change in stimulated frog muscle fibres near slack length. 《生理学杂志》. 1977;269(2):441–515.
3. Davis JS, Rodgers ME. Force generation and temperature-jump and length-jump tension transients in muscle fibers. 《生物物理杂志》. 1995 May;68(5):2032–40.
4. Huxley AF. Muscular contraction. 《生理学杂志》. 1974;243(1):1–43.
5. Kawai M, Brandt PW. Sinusoidal analysis: a high resolution method for correlating biochemical reactions with physiological processes in activated skeletal muscles of rabbit, frog and crayfish. 《肌肉研究与细胞运动杂志》. 1980 Sep;1(3):279–303.
6. Tanner BCW, Wang Y, Maughan DW, Palmer BM. Measuring myosin cross-bridge attachment time in activated muscle fibers using stochastic vs. sinusoidal length perturbation analysis. 《应用生理学杂志》. 2011;110(4):1101–8.
7. Palmer BM, Suzuki T, Wang Y, Barnes WD, Miller MS, Maughan DW. Two-State Model of Acto-Myosin Attachment-Detachment Predicts C-Process of Sinusoidal Analysis. 《生物物理杂志》. 2007;93(3):760–9.
8. Palmer BM, Tanner BCW, Toth MJ, Miller MS. An inverse power-law distribution of molecular bond lifetimes predicts fractional derivative viscoelasticity in biological tissue. 《生物物理杂志》. 2013 Jun;104(11):2540–52.
9. Mulieri LA, Barnes WD, Leavett BJ, Ittleman FP, LeWinter MM, Alpert NR, et al. Alterations of myocardial dynamic stiffness implicating abnormal crossbridge function in human mitral regurgitation heart failure. 《循环研究》. 2002;90(1):66–72.
10. Kawai M, Wray JS, Zhao Y. The effect of lattice spacing change on cross-bridge kinetics in chemically skinned rabbit psoas muscle fibers. I: Proportionality between the lattice spacing and the fiber width. 《生物物理杂志》. 1993;64(1):187–96.
11. Campbell KB, Chandra M, Kirkpatrick RD, Slinker BK, Hunter WC. Interpreting cardiac muscle force-length dynamics using a novel functional model. 《美国生理学杂志：心脏与循环生理学》. 2004;286:H1535-H1545.
12. Kawai M, Halvorson HR. Two step mechanism of phosphate release and the mechanism of force generation in chemically skinned fibers of rabbit psoas muscle. 《生物物理杂志》. 1991 Feb;59(February):329–42.
13. Lymn RW, Taylor EW. Mechanism of adenosine triphosphate hydrolysis by actomyosin. 《生物化学》. 1971;10(25):4617–24.
14. Cui L, Perreault EJ, Maas H, Sandercock TG. Modeling short-range stiffness of feline lower hindlimb muscles. 《生物力学杂志》. 2008;41(9):1945–52.
15. Rigozzi S, Müller R, Snedeker JG. Local strain measurement reveals a varied regional dependence of tensile tendon mechanics on glycosaminoglycan content. 《生物力学杂志》. 2009;
16. Barton ER, Lynch G. Measuring isometric force of isolated mouse muscles in vitro. 2008;(Id):1–14.
17. Sacks RD, Roy RR. Architecture of the hind limb muscles of cats: Functional significance. 《形态学杂志》. 1982 Aug 1;173(2):185–95.
18. Burkholder TJ, Fingado B, Baron S, Lieber RL. Relationship between muscle fiber types and sizes and muscle architectural properties in the mouse hindlimb. 《形态学杂志》. 1994;221(2):177–90.
19. Brooks S V, Faulkner JA. Contractile properties of skeletal muscles from young, adult and aged mice. 1988 Oct;404:71–82.
20. Mendez J, Keys A. Density and Composition of Mammalian Muscle. 《代谢》. 1960;9(2):184–8.