Mechanisms of Frank-Starling law of the heart and stretch activation in striated muscles may have a common molecular origin

Abstract

Vertebrate cardiac muscle generates progressively larger systolic force when the end diastolic chamber volume is increased, a property called the “Frank-Starling Law”, or “length dependent activation (LDA)”. In this mechanism a larger force develops when the sarcomere length (SL) increased, and the overlap between thick and thin filament decreases, indicating increased production of force per unit length of the overlap. To account for this phenomenon at the molecular level, we examined several hypotheses: As the muscle length is increased, (1) lattice spacing decreases, (2) Ca²⁺ sensitivity increases, (3) titin mediated rearrangement of myosin heads to facilitate actomyosin interaction, (4) increased SL activates cross-bridges (CBs) in the super relaxed state, (5) increased series stiffness at longer SL promotes larger elementary force/CB to account for LDA, and (6) stretch activation (SA) observed in insect muscles and LDA in vertebrate muscles may have similar mechanisms. SA is also known as delayed tension or oscillatory work, and universally observed among insect flight muscles, as well as in vertebrate skeletal and cardiac. The sarcomere stiffness observed in relaxed muscles may significantly contributes to the mechanisms of LDA. In vertebrate striated muscles, the sarcomere stiffness is generated by titin, a single filamentary protein spanning from Z-line to M-line and tightly associated with the myosin thick filament. In insect flight muscles, kettin connects Z-line and the thick filament to stabilize the sarcomere structure. In vertebrate cardiac muscles, titin plays a similar role, and may account for LDA and may constitute a molecular mechanism of Frank-Starling response.

1. Frank-Starling law of the heart

Frank-Starling relationship is the observation that ventricular stroke volume increases as preload (end-diastolic pressure) increases, a beat-to-beat adjustment to maintain cardiac output. It is a positive relationship between end-diastolic volume and ejection volume (de Tombe et al., 2010; Farman et al., 2011). Since preload is positively related to the end diastolic length of cardiomyocytes, it is widely believed that length dependent activation (LDA) of myocardium underlies the Frank-Starling law (Kobirumaki-Shimozawa et al., 2014). As ventricular end-diastolic pressure increases, the end-diastolic sarcomere length (SL) increases, which reduces the overlap between thick and thin filaments. This may decrease the number of available myosin cross-bridges (CB). While Frank-Starling law appears as a length dependent activation (LDA) mechanism, the increase in active force as SL is stretched (Bers, 2002; Muir & Hamlin, 2020) may involve activating an increased number of available CBs. To understand this intriguing phenomenon at the molecular level, researchers have proposed several hypotheses.

1. Lattice spacing effect: While the volume of muscle cells maintains constant when stretched, the distance between the thick and thin filaments (lattice spacing) becomes smaller. Therefore, the distance between myosin heads and F-actin is shorter to facilitate CB attachment and activation. This mechanism was reviewed by (Millman, 1998). This hypothesis was later abandoned because isovolumetric behavior also takes place in skeletal muscles without causing LDA.

2. Increase in Ca²⁺ sensitivity: It has been known that Ca²⁺ sensitivity increases when cardiac sarcomeres are stretched, which may cause increased number of CBs (Fabiato & Fabiato, 1978; Shiels & White, 2008; Farman et al., 2011). However, this does not happen in insect muscles.

3. Involvement of titin in LDA: Titin (also named connectin in early literatures (Maruyama et al., 1976; Maruyama et al., 1981)) mediated arrangement of the myosin head for interaction with actin as reported (Fukuda et al., 2008; Farman et al., 2011; Ait-Mou et al., 2016). When a sarcomere is stretched, titin is stretched concomitantly, which may rearrange myosin heads’ orientation towards actin to facilitate actomyosin interaction.

4. LDA of the thick filament, in which increased SL promotes more CBs to participate in contraction (Irving, 2017). In this mechanism, length increase releases CBs from super relaxed state (SRX) to facilitate activation.

5. Increased series stiffness to CBs at longer sarcomere length accounts for the increased force/CB, and resulting in LDA.

6. Here we propose a hypothesis that stretch activation (SA) observed in insect muscles and LDA in vertebrate muscles involve similar mechanisms: Increased stiffness and resting tension at longer SL may account for both stretch activation and LDA, which may also involve stretch enhanced force production.

These hypotheses are based on experimental observations; hence they are primarily phenomenological descriptions. It is important to note that the above hypotheses are not mutually exclusive but have significant overlaps. Although a recent review summarized that LDA in the heart and SA in IFM of insect may involve different molecular mechanisms (Bullard & Pastore, 2019), our hypothesis presented in this perspective focuses on exploring controversial observations for new mechanistic insights, especially commonalities between LDA and SA.

2. Sarcomere length (SL) dependence of force production

2.1. Overlap length between thick and thin filaments in cardiac muscles

It was established more than 50 years ago that, in skeletal muscles, force development results from the interaction of actin and myosin, in which the overlap length between thick and thin filaments is proportionate to the amount of force generation (Gordon et al., 1966). This implicitly assumes that force generated by each CB does not vary when SL is changed, hence force development is proportionate to the overlap length of the thick and thin filaments. Consequently, it is essential to know the lengths of the thick and thin filaments when SL is changed. While the thin filament length (TFL) has been measured extensively in skeletal muscles (Page & Huxley, 1963; Ohtsuki, 1979; Littlefield & Fowler, 2002), TFL of cardiac muscles has not been established until lately. TFL of mouse atrial cardiac muscles was measured using antibodies against tropomodulin that caps the thin filament at the pointed (minus) end, and α-actinin at the Z-line (Littlefield & Fowler, 2002). TFL of atrial cardiac muscle was somewhat increased with an increase in SL, because TF is compliant. TFL follows empirically the regression Eq. 1 for left atrial muscles (Kolb et al., 2016).

$$\mathrm{T}\mathrm{F}\mathrm{L}=0.30\mathrm{S}\mathrm{L}+0.20\mathrm{\mu }\mathrm{m}$$ (1)

In this regression, half of the Z-line width is included in TFL. The overlap length (OVL) of thick and thin filaments of a sarcomere is shown in Eq. 2, where A is the width of the A-band.

$$\mathrm{O}\mathrm{V}\mathrm{L}=A+2\mathrm{x}\mathrm{T}\mathrm{F}\mathrm{L}-\mathrm{S}\mathrm{L}=A-0.40\mathrm{S}\mathrm{L}+0.40\mathrm{\mu }\mathrm{m}$$ (2)

The length of the thick filament (width of A-band) was measured by using structured illumination microscopy (SIM) in mouse ventricular muscle (Tonino et al., 2017). With SIM, the shrinkage artefact common to electron micrographs can be avoided. A was about 1.60 μm, although it increased slightly in mouse ventricular muscles. From Eq. 2,

$$\mathrm{O}\mathrm{V}\mathrm{L}=2.00\mathrm{\mu }\mathrm{m}-0.40\mathrm{S}\mathrm{L}$$ (3)

This situation is depicted in Fig. 1 as the function of SL. In particular, if SL=1.8 μm, OVL=1.28 μm (Fig. 1A); if SL=2.3 μm, OVL=1.08 μm (Fig. 1B). Therefore, OVL is less by 0.20 μm at the longer SL. Similar calculation follows for right atrial muscles. There are slight differences in the regression parameters when phalloidin was used instead of tropomodulin, but the conclusions are just about the same. These observations are summarized in Table 1 and Fig. 1.

Fig. 1. Model sarcomere and overlap between thick and thin filaments in cardiac muscle.

Based on Eqs. 1–3. The thin filament elongates with SL (Eq. 1). The thick filament stays approximately at the same length. The overlap decreases as SL is increased from 1.6 μm and extrapolates to 0 at SL=5 μm. Broken lines represent titin. Z=Z-line, M=M-line.

Table 1.

Overlap length between thick and thin filaments in cardiac muscles, where A=A-band width. Adopted from (Kolb et al., 2016).

		Overlap length (cardiac muscles)
		Measured with tropomodulin	Measured with phalloidin
Left atrium	Equation	A-0.40 SL + 0.40 μm	A-0.36 SL + 0.30μm
	SL=1.9 μm	1.24 μm	1.22 μm
	SL=2.4 μm	1.04 μm	1.04 μm
	Difference	0.20 μm	0.18 μm
Right atrium	Equation	A-0.24 SL + 0.15 μm	A-0.28 SL + 0.84 μm
	SL=1.9 μm	1.29 μm	1.91 μm
	SL=2.4 μm	1.17 μm	1.77μm
	Difference	0.12 μm	0.14μm

Consequently, it is important to note that OVL decreases as SL is increased in the range between 1.6 μm and 5 μm (Fig. 1 and Eqs. 2 and 3), which is essential to understand the length-tension relationship and LDA in cardiac muscle. Because active tension is larger at longer SL, these results predict larger force production per unit length of the overlap. These seemingly paradoxical observations can be explained if the number of actively cycling CBs per unit length of overlap increases at larger SL (mechanisms (2) and (4) above), or the force/CB increases with SL (mechanisms (1), (3), (5) and (6)), thereby explaining LDA.

In rabbit psoas muscle, active tension increases when sarcomeres are stretched from 2.0 to 2.4 μm, because of the removal of steric problems (eg, thick filament abutting the Z-line, and/or thin filament hitting the M-line) as it happens in frog semitendinosus muscle fibers at SL=1.4–2.0 μm (Gordon et al., 1966). Thus, the molecular mechanisms of the ascending limb of the length-tension diagram is not the same between skeletal and cardiac muscles.

2.2. Lattice spacing hypothesis (Mechanism 1)

This was the first mechanism proposed and reviewed by (Millman, 1998). Because myocytes keep their volumes constant as the cell shape is changed, such as when muscle length is increased, lattice spacing (the distance between thick and thin filaments) becomes less at longer SL to result in a shorter distance between the myosin head and actin, facilitating actomyosin interaction to produce more force. The major problem of this mechanism is that the isovolumetric behavior also takes place in skeletal muscles, but force per unit overlap length does not increase with SL as reported by (Gordon et al., 1966). When the lattice spacing was osmotically compressed with dextran, the equilibrium constant of the force generation step remained the same (Zhao & Kawai, 1993), which negates this hypothesis. Thus, this mechanism is largely rejected.

2.3. Increase in Ca²⁺ sensitivity with SL (Mechanism 2)

Increasing the diastolic length of sarcomeres due to increased preload is anticipated to impose a strain in the thin filament. An interesting question is how this strain alters myofilament conformation, and in turn to increase the Ca²⁺ sensitivity (Fabiato & Fabiato, 1978; Shiels & White, 2008; Farman et al., 2011). This mechanism may be related to the structure and function of the thin filament in the sarcomere (Mijailovich et al., 2019). To understand the Ca²⁺ sensitivity as a function of passive tension may also be important for understanding the stretch activation in striated muscles.

Ca²⁺ activation of striated muscle contraction is regulated via the troponin complex in the thin filaments (Ebashi & Endo, 1968). Troponin has three protein subunits: The Ca²⁺-binding subunit troponin C (TnC), the inhibitory subunit troponin I (TnI) which also binds to actin and tropomyosin (Tpm) (Zhang et al., 2011), and the Tpm-binding subunit troponin T (TnT) (Ebashi & Endo, 1968; Gordon et al., 2000; Jin et al., 2008; Sheng & Jin, 2016; Wei & Jin, 2016). Contraction results from a rise of cytosolic [Ca²⁺] that binds to TnC to induce allosteric changes in TnC and subsequently in TnI, TnT and Tpm, which allows myosin heads to form strongly bound CBs with actin in the thin filament and activate myosin ATPase, generating power strokes to shorten sarcomeres. Increasing the diastolic length of sarcomeres due to increased preload imposes a stain on myofilaments to alter their conformation, and in turn to increase the Ca²⁺ sensitivity of troponin.

Inhibition of strong CB attachment with blebbistatin had no effect on the length-dependent modulation of Ca²⁺ sensitivity or cooperativity of activation, suggesting that LDA originates upstream of CB attachment. This supports a notion that LDA arises primarily from the influence of length on the modulation of Ca²⁺cooperativity of the thin filament. Partial replacement of TnC with non-Ca²⁺ binding mutant caused larger decreases in Ca²⁺ sensitivity and cooperativity at short SL than at long SL (Farman et al., 2010). Truncation of N-terminal 43 aa in myosin essential light chain (which may reach to the thin filament) in mice resulted in no length dependence of Ca²⁺ sensitivity, and lowered the maximum tension (Michael et al., 2013).

It has been shown that the function of cardiac TnI contributes to the length-tension relationship in cardiac muscles (Konhilas et al., 2003). The length dependent increase in Ca²⁺ sensitivity is cardiac TnI phosphorylation dependent. Pseudo-phosphorylation of Thr143 of cTnI (PKC site) in humans (Thr144 in rats and mice) increased Ca²⁺ sensitivity (Wijnker et al., 2014). The LDA requires phosphorylation of both Ser23 and Ser24 (PKA sites). At SL=1.8 μm, the increase was not significant, but at SL=2.2 μm, the increase was about 0.11 pCa unit (29%) and statistically significant (Wijnker et al., 2014). Furthermore, the maximum tension at saturating [Ca²⁺] was not changed when Ca²⁺ sensitivity was increased. Consequently, the phosphorylation effect alone may not be large enough to account for the Frank-Starling response. Such change in Ca²⁺ sensitivity does not happen in asynchronous insect flight muscles, hence this mechanism is not common for LDA of cardiac muscles and SA of insect muscles.

2.4. Role of titin in Frank-Starling relationship (Mechanism 3)

Titin (also called connectin in early literatures (Maruyama et al., 1976; Maruyama et al., 1981)), which is mechanically in parallel to CBs, plays a major role in determining the passive tension of muscle while its role in active force generation may also be of significance via modulating CB functions. It has been proposed that when SL is increased, the concomitant strain in titin may rearrange myosin heads’ orientation towards actin to facilitate actomyosin interaction (Fukuda et al., 2008; Farman et al., 2011; Ait-Mou et al., 2016), an interesting insight into the mechanism of LDA. Additionally, passive tension increased with Ca²⁺ binding to titin (Fujita et al., 2004) and decreased with phosphorylation (Fukuda et al., 2005) in a splice-form specific manner, and these changes in titin may modulate LDA. In fast twitch EDL and slow twitch soleus muscles of mice, (Nocella et al., 2012) reported that Ca²⁺ binding to titin caused a small increase in stiffness (<4%), which may contribute to LDA as discussed next.

2.5. LDA of the thick filament (Mechanism 4)

In this mechanism, increased SL promotes more CBs to participate in contraction (reviewed in (Irving, 2017)). This is caused by the presence of “super relaxed state (SRX)”, in which CBs are tightly bound to the thick filament backbone, hence these CBs are not available for the actomyosin interaction even when the thin filament is fully activated in the presence of Ca²⁺. The factor that controls the activation of these CBs is an increased load, as studied by small angle X-ray diffraction (Irving, 2017; Reconditi et al., 2017; Piazzesi et al., 2018). Historically, many people use the terms stress and strain interchangeably, although they are different parameters. Stress (or tension) is force per cross-sectional area, whereas strain is dimensionless, as it is length change relative to initial length. Force has the unit N, whereas stress has the unit kPa.

3. Role of series stiffness in length dependent force generation (mechanism (5))

To look into the molecular mechanism of elementary force generation step, the amount of elementary force developed ($\Delta F$) is proportional to the stroke distance ($\Delta x$, step size) of the lever arm (LA) and the stiffness ($\sigma$) of in-series structures, and it follows the Hooke’s law (Eq. 4) (Wang & Kawai, 2013):

$$\Delta F=\sigma \Delta x$$ (4)

The stroke distance $\Delta x$ is determined by the physical dimension of the myosin motor, hence it is fixed and does not change with SL. Any biological materials including titin (Li et al., 2002) is a nonlinear spring, e.g., $\sigma$ changes with the length, but proportionality (Eq. 4) holds for a small length change ($\Delta x$), such as the CB stroke distance, which is estimated to be 5.3–12 nm. Because stiffness $\sigma$ increases with SL (Li et al., 2002), it may account for the LDA at least partially. Muscle stiffness ($\sigma$) originates from serially interrelated structures including thick filament, thin filament, CBs, titin, nebulin (skeletal muscles), Z-line, and M-line, of which the stiffness of one (or more) elements could increase with an increase in SL. Titin is the main contributor, which spans from Z-line to M-line and interacts tightly with the thick filament. Stiffness of titin is known to increase substantially with SL, as reflected by an exponential increase in the resting tension, whereas stiffness of other serially arranged molecules may not change as much. If $\sigma$ increases with SL, then $\Delta F$ increases according to Eq. 4, and explains LDA. Cardiac muscles have relatively rigid sarcomere compared to skeletal muscles, which results in larger σ that may result in larger LDA. Consequently, Frank-Starling law may be explained by the length dependent increase in titin stiffness. The observation that a cleavage of titin with mild trypsin treatment in skinned rat myocardium diminished active force generation (Fukuda et al., 2001) is in accord with this mechanism. Another observation that a compliant titin mutant decreased LDA (Ait-Mou et al., 2016) also supports this hypothesis.

3.1. Does Frank-Starling law and stretch activation involve a common molecular mechanism? (mechanism 6)

The forgoing discussion stated the significance of the sarcomere stiffness in force generation. Insect flight muscles have more rigid (stiffer) sarcomere structure than skeletal and cardiac muscles, and they have large stretch activation known for over 60 years as reviewed by (Pringle, 1967). Stretch activation is also known as delayed tension (step analysis) and oscillatory work (sinusoidal analysis) depending on how it was observed, and these terms have been used synonymously and interchangeably.

Stretch activation is a process that is typically found in insect asynchronous flight muscles (Iwamoto, 2011) such as that of Drosophila (Wang et al., 2014) and Lethocerus (giant water bug) (Pringle, 1967). It is also exhibited in the synchronous flight muscles of insects (Josephson et al., 2000) such as locust and dragonfly. Studies of the stretch activation of locust flight muscles have provided valuable insights for understanding similar properties of the synchronous vertebrate striated muscles.

While SL is changed during the contraction and relaxation of vertebrate cardiac muscles, it does not change much in insect muscles due to a rigid structure owing to kettin (a titin-like filamentous protein) (Bullard et al., 2000; Bullard et al., 2006) that connects the end of the thick filament to the Z-line. Then, the question is whether stretch activation in insect muscles and LDA in cardiac muscles involve the same molecular mechanisms? To address this intriguing question, we need to look into the known mechanisms of the stretch activation.

3.2. Stretch activation

If an active muscle fiber is stretched by a small amount ($\Delta L<0.3\mathrm{\%}$), there is a delayed rise in tension ($\Delta F$) which is substantial. Conversely, if the muscle is released, there is a delayed decay in tension. Hence, these phenomena are called “Delayed tension” (Pringle, 1967), which can be approximated by an exponential function with the rate constant $\beta$.

$$\Delta F\left(t\right)=B[1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\beta t\right)]\Delta L\left(t\right)$$ (5)

where $\Delta F\left(t\right)$ is force time ($t$) course (plotted in Fig. 2A), $B$ is the amplitude/magnitude of the exponential process, and indicates the strength of this process. $\Delta L\left(t\right)$ is a step function with amplitude $L_0$, where $\Delta L\left(t\right)=0$ for $t<0$, $\Delta L\left(t\right)=L_0$ for $t\ge 0$; $L_0>0$ indicates stretch, and $L_0<0$ release.

Fig. 2. Delayed tension and oscillatory work.

(A) Delayed rise of tension (Eq. 5). Stretch occurs at ↑. β is the rate constant. (B) Oscillatory work (Eq. 6) plotted in the complex plane, where abscissa is the real axis (elastic modulus), and ordinate is the imaginary axis (viscous modulus). This plot is called Nyquist plot in muscle mechanics literature.

In sinusoidal analysis, the same exponential process is studied as a function of frequency $\left(f\right)$ instead of time, and represented by Eq. 6 in place of Eq. 5.

$$Y\left(\omega \right)\equiv \frac{\Delta F\left(\omega \right)}{\Delta L\left(\mathrm{\omega }\right)}=B\left(1-\frac{\omega i}{\beta +\omega i}\right)=\frac{B\beta }{\beta +\omega i}=\frac{B\beta \left(\beta -\omega i\right)}{{\beta }^2+{\omega }^2}$$ (6)

where $\omega$ is the angular frequency, which is $2\mathrm{\pi }$ times frequency $(\omega =2\mathrm{\pi }f)$, and $i=\sqrt{-1}$; “angular” is often abbreviated for simplicity. Here, the muscle performs work on oscillating length driver and its maximum output is at frequency b, or its angular frequency $\beta =2\mathrm{\pi }b$. In Eq. 6, $\Delta L\left(\omega \right)$ is the length change in the frequency domain, $\Delta F\left(\omega \right)$ is the force change, and $Y\left(\omega \right)$ is a frequency response function. When strain and stress (tension) is used for $\Delta L$ and $\Delta F$, respectively, $Y\left(\omega \right)$ has the dimension of Young’s elastic modulus, and because it can be expressed conveniently with complex numbers, $Y\left(f\right)$ is called “complex modulus”. Its real part is elastic modulus, and imaginary part viscous modulus.

Note that the imaginary part of Eq. 6 (last term) is negative, indicating the negative viscous modulus, which is called oscillatory work (Pringle, 1967). Also note that the real part of Eq. 6 is positive. $Y\left(\omega \right)$ is plotted in Fig. 2B on the complex plane. As seen here, the plot falls on the 4^th quadrant. Furthermore, as the frequency is increased (as $0\to \beta \to \mathrm{\infty }$), the plot falls on a semi-circle with the diameter B and its center on the abscissa, and the negative viscous modulus becomes maximum at frequency $\beta$. The plot in Fig. 2B is called the Nyquist plot in muscle mechanics literature, which is convenient because one exponential process is represented by one semicircle on the complex plain, hence its quality can be easily judged by the eye. In contrast, the quality of exponential time course (such as that in Fig. 2A) cannot be judged by the eye. Eq. 6 can be derived from Eq. 5 by Laplace transformation and by setting $s=\omega i$, where $s$ is the Laplace parameter (Kawai & Brandt, 1980; Kawai, 2018). Actually, $\Delta F\left(s\right)$ is the Laplace transform of $\Delta F\left(t\right)$, and these are two different (but related) functions. Consequently, the delayed tension or stretch activation in step analysis, and oscillatory work in sinusoidal analysis are two views of the same phenomenon. The oscillatory work is also seen in vertebrate fast twitch skeletal muscles (Kawai et al., 1977) with relative magnitude/amplitude smaller than that of insect muscles. In following discussions, we will simply mention the case of stretch because the tension response to release is approximately the mirror image of the stretch.

In step analysis when an active muscle is stretched in a stepwise fashion, there is a simultaneous increase in tension, called “phase 1” (Huxley & Simmons, 1971) (Fig. 3) or “phase 0” (Abbott & Steiger, 1977). The phase 1 originates from elastic structures of the muscular tissue. Here the fast length change does not allow time for CBs to adjust to the length change, hence the tension increase reflects the frozen structure of the muscle. The phase 1 is followed by a quick recovery of tension, called “phase 2” (Huxley & Simmons, 1971) (Fig. 3) or “phases 1 & 2” (Abbott & Steiger, 1977), which can be fitted by 1–2 exponential functions with the rate constant $\gamma$. The phase 2 is followed by “phase 3” (Heinl et al., 1974; Huxley, 1974; Abbott & Steiger, 1977), which is either a pause in tension or a delayed rise of tension with the rate constant $\beta$ (Fig. 3) that can be approximated by Eq. 5. This phase depends on the condition of the activation, notably on the phosphate and ATP concentrations (Kawai, 1978, 1986). The phase 3 is followed by phase 4 (Heinl et al., 1974; Huxley, 1974; Abbott & Steiger, 1977), which is a further recovery of tension with the rate constant α (Fig. 3). The entire time course is mathematically represented in Eq. 7.

$$\mathrm{\Delta }F\left(t\right)=\left[H+A\mathrm{e}\mathrm{x}\mathrm{p}\left(-\alpha t\right)-B\mathrm{e}\mathrm{x}\mathrm{p}\left(-\beta t\right)+C\mathrm{e}\mathrm{x}\mathrm{p}\left(-\gamma t\right)\right]\mathrm{\Delta }L$$ (7)

where three rate constants are in the order of $\alpha <\beta <\gamma$. A, B, and C (all ≥0) are magnitudes (amplitudes, strength) of respective exponential functions, called exponential processes.

Fig. 3. The time course of stretch activation.

A record of the time course of force development occurring in response to a step-length increase (at ↑, 1.5 nm/half sarcomere, which is about 0.12%) in rabbit psoas muscle fibers at 5°C during Ca²⁺ activation. Numbers 1–4 indicate the four phases of tension transients. Modified from Fig. 3 of (Davis et al., 2002), and reproduced with permission from the Biophysical Society.

This time course is plotted in Fig. 4A with log time axis. With sinusoidal analysis, phases 2 and 4 are realized as exponential advances (leads), in which the phase of tension advances the phase of length change. Here, the muscle absorbs work from oscillating length driver, and the absorbed energy is converted to the heat to result in the positive viscous modulus. Because of this property, the sinusoidal analysis is equated to the mechanical spectroscopy. Phases 2 and 4 are called exponential processes C and A, respectively, where viscous modulus is positive. In contrast, phase 3 is realized as exponential delay, in which the phase of tension delays the phase of length change. Here muscle performs oscillatory work on oscillating length driver, viscous modulus is negative, and associated energy comes out from ATP hydrolysis (Ruegg & Tregear, 1966). This is called exponential process B and its optimal frequency is at $\beta$. The complex modulus of active muscle fibers is analytically shown in Eq. 8 (Kawai & Brandt, 1980).

Fig. 4. Correlation between Step analysis (A) and sinusoidal analysis (B).

These are related by a linear transformation LT—LT’. (A) is the tension time course of step analysis: step increase in length by L₁ takes place at t=1 ms, and concomitant tension time course is plotted in log time axis. “1”, “2”, “3” and “4” indicates respective phases of step analysis (cf. Fig. 3). Plot of Eq. 7. (B) is a Nyquist plot of rabbit psoas (fast twitch) fibers with the elastic modulus plotted in the abscissa, and viscous modulus in the ordinate. Plot of Eq. 8. (C) represents mechanical equivalence of the fast twitch fibers. α, β, γ are apparent rate constants, and A, B, C are their respective magnitudes; H is a small constant. Ca²⁺ activation closes the switch. Y_∞=H+A−B+C, and represents stiffness extraporated to infinite frequency (∞). Modified and redrawn from Kawai and Brandt (1980).

$$Y\left(\omega \right)=H+\frac{A\omega i}{\alpha +\omega i}-\frac{B\omega i}{\beta +\omega i}+\frac{C\omega i}{\gamma +\omega i}$$ (8)

Eq. 8 is derived from Eq. 7 by Laplace transformation and plotted in Fig. 4B in the Nyquist plot. The relationship between step analysis and sinusoidal analysis is graphically illustrated in Fig. 4 (Kawai & Brandt, 1980). Nyquist plot of rabbit psoas fibers (fast twitch type IID muscle) is shown in Figs. 4A and 5B. It consists of three semicircles (each is shown in Fig. 2B) with the diameters A, B, and C, and called exponential processes A, B, and C, respectively. These are compared to step analysis results of Figs. 3 and 4B. As shown in this figure, $Y_{\mathrm{\infty }}(=H+A-B+C)$ corresponds to phase 1, process C to phase 2, process B to phase 3, and process A to phase 4. The viscous modulus becomes either maximum (for processes A and C at frequency α and γ, respectively) or minimum (for process B at frequency $\beta$). Among fast twitch skeletal muscle fibers (type II), there are subtypes (A, D, B), but their Nyquist plots are very similar to that shown in Fig. 5A; only their rate constants differ by the order IIA<IID<IIB (Galler et al., 2005).

Fig. 5. Nyquist plots of Y(ω) of active fibers.

(A) rabbit psoas fast-twitch type IID fibers (Kawai & Brandt, 1980), (B) rabbit soleus slow-twitch fibers (Wang & Kawai, 1997), and (C) bovine ventricular cardiac muscle fibers (Lu et al., 2005); similar data in (de Tombe et al., 2010). Decade frequencies (0.13, 1, 11, 100Hz) are marked by ●. hf=high frequency end.

Fig. 4C represents physical demonstration of Eqs. 7 and 8. Resting stiffness is primarily supported by titin. When muscle is activated (switch closes); here A represents a serial combination of an elastic element and a dash pot with the rate constant α. Similarly, C represents a serial combination of an elastic element and a dash pot with the rate constant γ. B represents a work generator (driven by ATP hydrolysis) with the rate constant β, and written with intertwined two rings. H is a weak elastic element with the corresponding rate constant close to 0.

In slow twitch skeletal muscles (type I fibers; Fig. 5B), there are also 3 basic exponential processes as that seen in fast twitch muscles (Fig. 5A), except that magnitudes (amplitudes) of processes A and B (phases 4 and 3, respectively) are significantly less than those of fast twitch fibers (Wang & Kawai, 1996; Kawai et al., 2018). In cardiac muscles (Fig. 5C), the basic feature is the same as that in the slow twitch skeletal muscles, which is not surprising considering that many of their constituent proteins are encoded by homologous genes with conserved sequences. The slow process A is not visible in Fig. 5C that was obtained at 25°C, but it becomes evident at temperatures ≥ 30°C (Lu et al., 2006).

3.3. The molecular mechanisms of stretch activation based on length sensitivity of the rate constants

The two-state model proposed by (Abbott, 1972) has an attached (force generating) state Z and a detached state Y without force (Fig. 6A). Abbott assumed that the attachment rate (β) increases (or the detachment rate β’ decreases) with a stretch, which can explain the delayed tension of Eq. 5 (Fig. 2A). However, this model was not well received at that time because it does not follow the Le Chatelier-Braun principle, which states: “When any system at equilibrium is subjected to a change in concentration, temperature, volume, or pressure, the system changes to a new equilibrium, and this change partly counteracts the applied change” (https://en.wikipedia.org/wiki/Le_Chatelier%27s_principle; see also (Le Chatelier & Boudouard, 1898; Evans et al., 2001)). When applied to a muscle system, it states that as the muscle is stretched the equilibrium shifts to less force (less energy) state to compensate for the increased force (more energy) due to the stretch as used by (Huxley & Simmons, 1971) for their proposed model. This principle is applicable to a system where no new energy is placed into or out of the system. To make Abbott’s model to be consistent to the Le Chatelier-Braun principle, (Kawai & Halvorson, 2007) proposed a three-state model with two attached states (X and Z) and a detached state (Y) (Fig. 6B), where Z is the force generating state, and X is the rigor-like state that bears force. If X→Y transition (γ) is fast and accelerated by stretch (or γ’ is decelerated by the stretch), then Y accumulates with fast time course. If Y→Z transition (β) is slower, then the accumulation in the Y state will cause delayed rise in tension. If Z→X transition is slowest of all three transitions, this mechanism can explain the delayed tension without violating the Le Chatelier-Braun principle. This mechanism is in agreement with experimental results of the ATP and Pi effects on two apparent rate constants β and γ (Kawai & Halvorson, 1989; Kawai & Halvorson, 1991; Kawai & Halvorson, 2007) although the calculated delayed tension is not as large as experimentally observed, such as shown in Fig. 3.

Fig. 6. Simple CB models historically used.

(A) Two state model proposed by Abbott (1972). (B) Three state model proposed by Kawai & Halvorson (2007).

One important point associated with this mechanism is that the Le Chatelier-Braun principle is based on a passive phenomenon without consideration of external energy (increase or decrease). Muscle is an active system in which ATP is hydrolyzed, therefore, the effect of the free energy liberated from ATP must be added. That is, how to place the hydrolysis energy to the Le Chatelier-Braun principle is an important task that needs to be answered. One solution is the “ratchet” mechanism as suggested by some investigators (Vale & Oosawa, 1990; Smith & Geeves, 1995; Smith, 1998), in which preferred detachment of unstrained and/or negatively strained CBs takes place on the X→Y transition (CB detachment), which is propelled by ATP. This mechanism is not in accordance with the Le Chatelier-Braun principle (Kawai & Halvorson, 2007), and the reason is because this reaction is propelled by ATP binding to the myosin head, which subsequently detaches myosin head from actin. If the ratchet mechanism is placed on the X→Y transition, then accumulation of Y is much more significant than otherwise, which subsequently causes very large delayed tension, such as observed in Fig. 3.

3.4. Stretch activation (oscillatory work) can be enhanced by making sarcomeres rigid

There are other approaches to explain the molecular mechanisms of oscillatory work. It was reported earlier that by covalently cross-linking 18% of myosin heads to the thin filament in rabbit psoas fibers, a large oscillatory work (stretch activation) can be produced (Fig. 7: A, B → C, D), similar to that of insect muscles (Tawada & Kawai, 1990). The large increase of oscillatory work was apparently caused by elimination of slow process A (phase 4 in Fig. 3) without changing process B (β, r₃ or B). Because process A absorbs work and compromises the power production by process B, the elimination of process A results in an apparent increase in oscillatory work production. Fig. 7C also demonstrates that a stretch enhances the oscillatory work. The numbers in the plot indicate the amount of strain applied to the cross-linked fibers prior to the activation. As the strain is increased, both oscillatory work and tension are increased, as in the case of insect flight muscles. These results indicate that extra stiffness parallel to CBs apparently enhances oscillatory work and stretch activation significantly.

Fig. 7. Nyqust plots of cross-linked fibers.

(A, C) Nyquist plots of rabbit psoas muscle fibers activated in solution of physiological ionic strength (200 mM), which contained (mM:) 6 Na₂Ca-EGTA, 5.3 Na₂Mg-ATP, 4.7 Na₄-ATP, 8 Pi, 15 phosphocreatine, 32 K propionate, 26 Na-propionate, 10 MOPS, 160 U/mL creatine kinase, pCa 4.86, and pH 7.00. Frequencies used are (clockwise) 0.25, 0.5, 1, 2, 3.1, 5, 7.5, 11, 17, 25, 35, 50, 70, 100, 135, 186, 250, 350 Hz. Decade frequencies (1, 11, 100 Hz) are shown in filled symbols. (A) is from a native fiber, (C) is from EDC cross-linked fiber. The cross-linked fibers were stretched before activation, and the stress on the fiber is shown in kPa in the corresponding plot. (B, D) Best fit of the data in (A) and (C) to Eq. 8. From (Tawada & Kawai, 1990).

This mechanism is consistent with the sarcomere structure of insect flight muscles, which is rigid and difficult to stretch, primarily owing to the link of the end of the thick filament to the Z-line by kettin (and projectin). Kettin has an analogous structure to titin in vertebrate striated muscles. Kettin/projectin (Bullard et al., 2002; Bullard et al., 2006) connects the thick filament to the Z-line and stabilizes the position of the thick filament in the sarcomere of insect muscles. In an insect sarcomere, the kettin connection can be considered in parallel with CBs, because CBs also connect the thick filament to the Z-line through the thin filament. Additional feature in insect muscles is the presence of Tn bridge (Wu et al., 2010; Perz-Edwards et al., 2011), which connects thin filament to thick filament. Both of these linkages are in parallel with CBs, and the situation is similar to titin in vertebrate cardiac muscles, where titin connects the thick filament to the Z-line with less compliance than that in vertebrate skeletal muscles. The kettin connection in insect sarcomeres is more rigid than the titin connection in mammalian cardiac muscles, hence the I-band width hardly changes in insect muscles in contrast to that in vertebrate cardiac muscles. It appears that this sarcomere stability may contribute to the large oscillatory work (delayed tension, stretch activation) of insect muscles (Bullard et al., 2002). The data presented in Fig. 7 show that a stabilization of sarcomeres by crosslinking apparently enhances the oscillatory work in rabbit psoas muscles, demonstrating the importance of sarcomere stability for the stretch activation.

Vertebrate cardiac muscles also show stretch activation (Fig. 5C, indicated by b). However, its magnitude is smaller than that of skeletal muscles. This is primarily due to a large magnitude of process C (work absorption), which eclipses the presence of process B (work production). In cardiac muscles, additional parallel structures to CBs present are cardiac myosin binding protein C (cMyBP-C) and essential light chain (ELC) of myosin. cMyBP-C crosslinks the thick and thin filament (Whitten et al., 2008), hence it may contribute to the stretch activation and LDA. The N-terminus of ELC may interact with actin (Trayer et al., 1987; Lowey et al., 2007; Kazmierczak et al., 2009) or actin in the thin filament (Muthu et al., 2011). These potential structural links form parallel crosslinks to CBs, which in turn stabilize the sarcomere structure, decreases the magnitude of process A, and results in an increased oscillatory work. Consequently, the increased stretch activation may be the root cause of LDA.

Cardiac muscle differs from skeletal muscle in sarcomere structure by the lack of nebulin in the thin filament, which may reduce thin filament rigidity. A more compliant thin filament may allow passive tension and strain to produce larger degree of conformational changes in the thin filament and regulatory proteins. Vertebrate cardiac muscle sarcomere is more responsive to passive tension and stretch than that of skeletal muscles, hence providing a structural basis for the cardiac specific phenomenon of Frank-Starling response. It is known that while Ca²⁺ activation of cardiac myofibrils is less cooperative than that of skeletal muscles, in particular fast twitch muscles, cardiac thin filament cooperative activation is length dependent and play a role in Frank-Starling response (Terui et al., 2010). The stiffness of the thin filament is not a unique source of the difference, because it has been known that the Yong’s elastic modulus of insect muscles and vertebrate skeletal muscles are similar (Perz-Edwards et al., 2011). The underlying mechanisms may involve thin filament regulatory protein, because a partial replacement of cardiac TnT with a fast skeletal muscle isoform significantly increased the cooperativity in mouse cardiac muscle strips (Huang et al., 1999). This hypothesis requires continuing investigation.

4. Conclusions and future investigations

Frank-Starling relationship is a fundamental property of vertebrate cardiac muscle and a longstanding topic of active research. The molecular mechanism underlying Frank-Starling response is based on the observation that ventricular preload increases the resting length of cardiac muscle and sarcomeres to increase the stroke volume. This process confers an increase in passive tension in the myofilaments mimicking a stretch effect. Therefore, Frank-Starling relationship may be viewed as a positive relationship between the end diastolic passive tension and the subsequent active force generation.

Although less predominant than that of insect asynchronous flight muscle, stretch activation is an intrinsic property of vertebrate cardiac muscles, consistent with a sensibility of the myofilaments to passive tension. Stretch of myocardium during isometric contraction results in a spike of force increase before a rapid decrease to a minimum followed by a re-development of force that involves stretch activation to reach levels higher than the pre-stretch force. Experimental studies have implicated that stretch activation may contribute to the mechanisms underlying Frank-Starling response of the heart. It has been shown that at both maximal and submaximal activations, increased sarcomere length significantly reduced the initial rate of force decay following stretch. At submaximal activations (but not at maximal) the rate of force redevelopment was accelerated, showing stretch activation’s contribution to the steepness of the Frank-Starling relationship in cardiac muscle (Stelzer & Moss, 2006).

From the data and hypotheses discussed above, we conclude that stretch activation (delayed tension, oscillatory work) in insect and vertebrate skeletal muscles, length dependent activation (LDA), and the Frank-Starling response of vertebrate cardiac muscle may involve the same molecular mechanisms, and emerged as a result of the balance of sarcomere rigidity vs compliance. It is mediated by molecular structures such as titin, titin-like and other connecting myofilament proteins which are in parallel to CBs in the sarcomere. It is important to understand the SL dependence of force/CB and CB kinetics, and to confirm why force/CB is larger at longer SL in myocardium. It is likely that a simple Hooke’s law can explain LDA of force/CB as shown in Eq. 4. Since force generation requires Ca²⁺ activated strong CBs, the effects of rigid sarcomeres on the conformation of myosin heads which affects the ATP hydrolysis rate, and the compliance of the thin filament on the conformation and Ca²⁺ sensitivity of troponin lay at the center of this mechanism. On the basis of numerous steady state force-pCa studies, it would be especially informative to investigate the contribution of thin filament and troponin regulation in a dynamic system to further understand the molecular mechanism of Frank-Starling law of the heart and stretch activation of striated muscles.

Availability of data and material

The data are available by the authors on reasonable request.