Forces generated during stretch in the heart of the lobster Homarus americanus are anisotropic and are altered by neuromodulators

Crustacean hearts can provide a model for understanding the neuromechanics of heart function because the heart in the crustacean open circulatory system exhibits the same primary phases of the cardiac cycle as does the heart in the mammalian closed circulatory systems: isovolumetric contraction, ventricular emptying, isovolumetric relaxation and ventricular filling (e.g. in shrimp hearts, Guadagnoli et al., 2007). Furthermore, in terms of passive mechanical properties, both vertebrate hearts (Demer and Yin, 1983) and crustacean hearts (Meyhöfer, 1993) have viscoelastic material properties that are important to the function of the heart.

The biophysical characteristics of both invertebrate and vertebrate cardiac muscle have been described by length–tension curves (Hibberd and Jewell, 1982; Demer and Yin, 1983; Meyhöfer, 1993; Linke et al., 1994). Individual cardiac fibers in the crab Cancer magister exhibit a non-linear increase in passive force across the physiological range of sarcomere length (Meyhöfer, 1993), as do vertebrate cardiac myocytes. In vertebrates, passive stiffness is lower at a given sarcomere length in skeletal myocytes than it is in cardiac myocytes (Granzier and Irving, 1995). This altered passive stiffness in cardiac passive tension resists the filling of the heart and may help maintain the physiological sarcomere length in the ascending limb of the length–tension curve.

Passive tension is important in the Frank–Starling effect, in which the contraction force of the heart muscle increases with stretch throughout its operating range. Resting length is at or near the start of the ascending limb. This contrasts with length–tension curves of skeletal muscle, in which the largest contraction force typically occurs in the middle of the operating range of muscle lengths (Allen and Kentish, 1985; Burkholder and Lieber, 2001). Vertebrate cardiac length–tension curves have several potential mechanisms by which the ascending limb is effectively extended, for instance by length-dependent changes in calcium sensitivity (de Tombe et al., 2010). These mechanisms generate a length–tension curve that effectively has only increasing tension, or only modest decreases in force at longer sarcomere lengths. A match of length–tension curves and stroke length ensures the Frank–Starling imperative of a continuously increasing contractile force with increasing stretch in some vertebrate hearts. For example, fish and amphibians, which tend to vary stroke volume more than mammals, have correspondingly longer sarcomere lengths at which peak active forces occur (Shiels and White, 2008).

The extent to which the Frank–Starling law applies to the lobster cardiac system is presently unknown. In fact, the myocardial length–tension curve of the American lobster, Homarus americanus, has not been characterized. The only reported length–tension curve is from the ostial muscle, where the passive force increased linearly with stretch and the active force was much less than the passive force (Yazawa et al., 1999). Given the different functions of ostial and other cardiac muscles and the anatomical similarities between crab hearts (Meyhöfer, 1993) and lobster hearts (Alexandrowicz, 1932), the passive length–tension curve for the lobster heart could closely resemble that of crab cardiac muscle: a nonlinear increase in passive force at greater extensions.

The range of orientations of the muscle fibers of the heart complicates the understanding of muscle length–tension curves in the context of whole hearts. Because hearts are naturally loaded three-dimensionally via pressure, the forces generated and resisted by the heart arise from the passive and active components of force acting across many axes. Thus, as the heart contracts and stretches, heart muscle fibers are unlikely to generate or resist force equally in all directions. For example, in mammals, cardiac tissue has been shown to be anisotropic, suggesting that the direction of extension may have functional implications (Demer and Yin, 1983).

In contrast to vertebrate hearts, crustacean hearts are neurogenic, and are controlled directly by the cardiac ganglion (CG), a central pattern generator (CPG) that is embedded in the myocardium (Fig. 1) (Alexandrowicz, 1932; Cooke, 2002). Rhythmic behaviors, such as the neurogenic heartbeat, can be hormonally and locally modulated by neurochemicals, thus generating flexibility in the hard-wired CPG motor output (reviewed in Marder, 1991; Simmers et al., 1995; Pearson, 2000; Hooper and DiCaprio, 2004; Goulding, 2009; Guertin and Steuer, 2009; Brezina, 2010; Christie et al., 2010a; Selverston, 2010).

Although the effects of neuromodulators on crustacean CPG-effector systems are well documented, these studies have only examined the physiological effects at single locations on the length–tension curve (Mercier and Russenes, 1992; Fort et al., 2004;Wilkens et al., 2005; Fort et al., 2007a,b; Stevens et al., 2009; Christie et al., 2010b; Dickinson et al., 2015). However, it is important to consider the functional space in which the neurotransmitters are working. The lobster heart does not work at a single location on its length–tension curve; rather, it functions over a range of lengths. Therefore, it is important to understand how neuromodulators might affect the heart over a range of lengths, and to determine the effects of neuromodulators on active as well as passive length–tension curves. Moreover, neuromodulators might interact with the neuromuscular system differently along various axes. Therefore, a multi-directional analysis of the cardiac length–tension curve can provide a more complete understanding of how the lobster heart generates force, and how that force is altered by neuromodulators.

Thus, this study characterizes the force–length curves for whole lobster hearts subjected to slow lengthening and shortening stretches at constant speeds, both uniaxially and biaxially. Additionally, this study characterizes the same curves in the presence of the neuropeptides SGRN and GYS, in intact (with the CG) and stimulated (without the CG) preparations, to describe how mechanical and physiological properties interact in the cardiac neuromuscular system.

Homarus americanus H. Milne-Edwards 1837, weighing 250–450 g, were obtained from local seafood retailers (Brunswick, ME, USA). Lobsters were maintained for up to 3 weeks in recirculating natural seawater aquaria at 10–12°C with a 12 h:12 h light:dark cycle. Lobsters were fed chopped squid weekly. Before dissection, lobsters were packed on ice for 30–45 min.

All experiments were carried out in physiological saline (composition in mmol l⁻¹: 479.12 NaCl, 12.74 KCl, 13.67 CaCl₂, 20.00 MgSO₄, 3.91 Na₂SO₄, 11.45 Trizma base and 4.82 maleic acid; pH 7.45) maintained at 10–12°C by an in-line Peltier temperature regulator (CL-100 bipolar temperature controller and TCM-1 solution heater/cooler; Warner Instruments, Hamden, CT, USA).

GenScript Corporation (Scotch Plains, NJ, USA) synthesized SGRNFLRFamide (SGRN) and GYSDRNYLRFamide (GYS). The peptides were dissolved in deionized water to make 10⁻³ mol l⁻¹ stock solutions that were stored at −20°C until they were further diluted in physiological saline to the desired concentrations 15–20 min before use. The concentrations of SGRN (10⁻⁹ mol l⁻¹) and GYS (5×10⁻¹⁰ mol l⁻¹) were selected as the lowest concentrations that consistently produced a significant increase in contraction amplitude and frequency (Dickinson et al., 2015).

The heart, attached to the overlying dorsal carapace, was removed from the lobster. The carapace was pinned in a Sylgard 184 (KR Anderson, Santa Clara, CA, USA)-lined Petri dish filled with chilled physiological saline such that the ventral side of the heart faced upwards. The suspensory ligaments connecting the heart to the carapace were left intact to help preserve the natural stretch required for contraction.

The posterior small artery was cannulated with polyethylene tubing that continuously perfused chilled physiological saline at 2.5 ml min⁻¹ and was secured with 6/0 suture silk. A second tube delivered saline to the posterior end of the dish while two outflow tubes were located at the anterior end to facilitate a constant flow of saline across and through the heart. Two Aurora force transducers (Dual-Mode Lever Systems, Model 300C, Aurora Scientific, Aurora, ON, Canada) simultaneously recorded force and imposed stretches on the heart; the sampling rate for force and length was 200 Hz. The signals from the force transducers were amplified by a Model 44 Brownlee Precision Instrumentation amplifier and recorded with a Cambridge Electronic Design (CED) Micro 1401 digitizer paired with Spike2 version 7.03 software (CED, Cambridge, UK). The output of the force transducer is reported in grams rather than Newtons for consistency with other studies that have studied single locations on the length–tension curve (Fort et al., 2004, 2007a,b; Stevens et al., 2009; Christie et al., 2010b; Dickinson et al., 2015).

The levers of the longitudinal and transverse force transducers were attached to the heart via suture silk. The silk was attached at an angle of 90 deg to each lever arm. The longitudinal attachment to the heart was via the five anterior arteries, which were tied together with the suture silk; the transverse attachment to the heart was via a glass hook that was hooked anteriorly through the left ventral ostium. Both force transducers were oriented such that the silk pulled upward at an angle of 30 deg from horizontal; the longitudinal pull on the heart was along the line of the sagittal plane (Fig. 2A). The force–extension relationship for the silk thread is f=88x^1.7, where f is the force in grams and x is the extension in millimeters (R²=0.999). Typical maximum forces exerted by the string on the heart range from 2 to 15 g (0.02 to 0.15 N), which correspond to extensions of the silk of 0.11 to 0.35 mm.

Both force transducers were mounted on micromanipulators to allow for fine adjustments in the baseline force. Baseline longitudinal passive force was set at either 1 or 2 g in different experiments. Baseline transverse passive force was set between 0.15 and 0.33 g in each lobster. This lower transverse baseline was used because a transverse baseline greater than 0.5 g resulted in tearing around the ostium in more than 50% of preparations; the lower baseline never caused tears. Thus, all following references to a 1 or 2 g baseline force refer specifically to the longitudinal, baseline passive force. The hearts were stabilized for 1 h at the baseline extensions. Force decreased slightly during the stabilization period; at the end of the stabilization period extensions were manually increased so as to return force to the pre-determined baseline.

An Aurora force transducer was used to generate length–tension curves for the lobster heart by imposing a displacement of 2.50 mm at an extension rate of 0.05 mm s⁻¹ (50 s extension and 50 s return). To characterize the length–tension curve for the lobster heart, it is essential that the stretch interval be long enough that the interval of a heartbeat is a relatively small fraction of the total stretch, thus necessitating the slow extension rate used in the present study. It is unknown how this extension rate corresponds to in vivo strains associated with potential increases in heart filling during periods of higher activity.

Because the Aurora force lever was at a 30 deg angle to the plane of the heart, this displacement of the lever arm resulted in 2.17 mm of displacement of the heart, ΔL_max, at a constant extension rate, ⁠, of 0.0434 mm s⁻¹ along the two major axes of the heart, longitudinal and transverse, while recording the changes in passive, active and total force (Fig. 2). Actual extensions varied slightly because of the compliance of the thread; for example, the average displacement of the connection of the thread to the heart, as measured from the three videos used for strain analysis (see below), was 2.10 mm (±0.13 mm s.d.). To determine the effects of uniaxial extension, stretches were imposed on the heart across one axis, longitudinal or transverse, while extension on the perpendicular axis remained at baseline force; to determine the effects of biaxial extension, stretches were imposed across both axes simultaneously.

Because the heart is a structure composed of viscoelastic materials, and thus experiences stress relaxation, it was conditioned to generate reproducible hysteresis loops within the extension range used in the experiments. Three longitudinal conditioning stretch pyramids were imposed on each heart after an hour-long stabilization at the baseline force. Upon completion of each pyramid, the baseline force, which had experienced slight stress relaxation, was manually returned to either 1 or 2 g of passive force. The heart then recovered for 5 min before the subsequent conditioning pyramid. Using the same protocol, three transverse conditioning stretch pyramids were subsequently imposed on each heart.

To characterize the control mechanical properties of the heart in saline, three control stretch pyramids were imposed on the heart following the conditioning cycle. Each of the three pyramids (longitudinal, transverse and biaxial), was separated from the previous stretch by a 5-min recovery period. Baseline passive force was readjusted if necessary.

To assess the effects of SGRN (1 and 2 g baseline) and GYS (1 g baseline only) on the force–extension curves, each peptide was applied separately to the same lobster heart.

Following the primary control pyramids, one of the peptides was perfused for 30 min through the heart to allow the effects of the neurotransmitter to stabilize. After the peptide stabilization period, three stretch pyramids (uniaxial longitudinal, uniaxial transverse and biaxial), separated by 5-min recovery periods, were imposed. Immediately after the final stretch pyramid, the peptide was washed from the heart for 1 h with physiological saline; previous work has shown that the effects of GYS and SGRN generally wash out after approximately 20 min (see fig. 1A in Dickinson et al., 2015). Baseline passive force was readjusted if necessary.

The heart was then subjected to a final set of control pyramids (longitudinal, transverse and biaxial) to assess the levels of muscle fatigue or tissue strain. This provided a post-peptide control for the first peptide and, simultaneously, the pre-peptide control for the second peptide. After the control pyramids, the second peptide was applied to the heart using the same procedure as outlined above, including the peptide wash and subsequent set of control stretch pyramids.

Neuropeptides, including SGRN and GYS, can exert modulatory effects at several locations within the cardiac neuromuscular system, including the CG and the cardiac muscle or neuromuscular junction (Dickinson et al., 2015). To eliminate effects of the peptides on ganglionic output and record the responses of the periphery (muscle/neuromuscular junction) to peptides, the CG was removed from the whole heart, while the motor nerves remained intact.

The whole heart was prepared as described above until just after the first control stretches in saline. After the first control stretch, a small incision was made on the ventral surface of the heart. The portion of the CG containing the nine neurons was removed. The remaining end of the left anterolateral nerve was gently pulled into a suction electrode. Using stimulation patterns generated in Spike2, the nerve was stimulated in bursts (235 ms, consisting of 15 pulses, each 0.5 ms duration, at 60 Hz) every 1.5 s over 80 s. After the first 15 bursts, a longitudinal uniaxial stretch was imposed on the heart. The group of 15 bursts before the initiation of stretch allowed sufficient time for facilitation (Anderson and Cooke, 1971) and burst-to-burst increases in contraction amplitude during stimulation to stabilize (Stevens et al., 2009). To prevent over-stimulation and nerve damage, the heart was only stimulated during the extension phase of stretch; following completion of the stretch, the heart rested for 5 min without stimulation before a transverse uniaxial stretch was initiated. After the control uniaxial stretches during stimulation, each heart was perfused with either SGRN or GYS for 30 min before longitudinal and transverse uniaxial stretches with stimulation were imposed on the heart. To assess any time-dependent effects, after 1 h of rinsing the peptide from the heart with saline, each heart was stretched uniaxially, in longitudinal and transverse directions, during stimulation.

An iSight camera (Apple, Cupertino, CA, USA, 1920×1080 pixel resolution, 30 frames s⁻¹), mounted 10 cm above the preparation, recorded the movement of each heart during each type of test stretch pyramid (longitudinal, transverse and biaxial). Carmine particles (Fisher Scientific, Fair Lawn, NJ, USA) sprinkled over the surface of the heart created fixed visual points on the heart. Heart movement was tracked using Tracker Video Analysis and Tracking Tool software (Open Source Physics, www.opensourcephysics.org).

From these videos, we measured longitudinal and transverse L_o and ΔL_max and calculated ε_max and ⁠. Along both axes, length was measured when the heart was most contracted. Measuring length at a consistent phase of heart contraction was necessary because the compliance in the suture silk attaching the heart to the force transducer allowed the heart to shorten during nearly isometric beats. The nearly isometric beats allowed the heart to shorten by approximately 0.3 mm during each heartbeat as compared with a total lengthening of approximately 2.1 mm during a stretch pyramid. Thus, although the Aurora lever continuously pulled on the end of the silk attached at the lever, the compliance of the suture silk allowed the end of the heart held by the suture silk to shorten with each heartbeat. This effect tends to decrease or increase muscle-shortening rate during the lengthening and shortening phases of the extension cycle, respectively.

Data were analyzed using in-house scripts written for Spike2 that measured the contraction parameters of the heart: amplitude (active force), baseline passive force and peak force (total force) (Fig. 2); data were collected from the onset of the stretch pyramid and for every heartbeat for the subsequent 100 s.

Data were further analyzed using programs written by O. Ellers in Mathematica 9.0.1.0 (Wolfram Research, Champaign, IL, USA). Length–tension curves were generated for the active, passive and total force in each of the lobsters during extension (uniaxial longitudinal, uniaxial transverse and biaxial) in the three treatments (saline, 10⁻⁹ mol l⁻¹ SGRN and 5×10⁻¹⁰ mol l⁻¹ GYS). Length–tension curves for each lobster heart were generated from interpolated (first order) values of the forces (active, passive and total) during the imposed stretch pyramid from 0.1 to 2.0 mm; using an interpolation avoided assumptions that would have been required in a fitted model. The endpoints (0.0–0.1 mm; 2.0–2.1 mm) were eliminated to reduce extraneous noise caused by contraction occurring during the change in the extension direction. Averaging the interpolated values at 0.1 mm increments generated grouped data curves; the error bars were calculated as the standard error of the interpolated values.

There is significant variation among lobsters; for this reason, all statistical comparisons are paired and thus there are no statistical comparisons between 1 and 2 g longitudinal baseline experiments. All statistical tests are two-tailed.

The variable used for comparison of stretch-dependent changes in passive, active and total forces was the difference between a given force at the highest and lowest extensions, ΔF. The variable called passive force hysteresis was calculated as the area between the extension and return curves of the force versus length graph, divided by the total area under the extension curve calculated relative to the lowest force in the whole curve. Passive force hysteresis represents the passive viscous energy losses during each cycle. For the active force, this calculation represents the difference between active forces generated against a background of lengthening and active forces generated against a background of shortening; for the purposes of parallel language, we call this the active force hysteresis.

For tests conducted without peptides, repeated-measures ANOVA (SPSS version 22.0, IBM, Armonk, NY, USA) were used to compare mean ΔF and hysteresis between test types (uniaxial and biaxial) and measurement axes (longitudinal and transverse), excluding data from the axis perpendicular to the test axis in uniaxial tests.

Paired t-tests were used to compare control and neuropeptide results for pre-stretch passive force, pre-stretch active force, ΔF_passive, ΔF_active, passive force hysteresis and active force hysteresis. For a given peptide treatment, only experiments that included all relevant comparison pairs were included in the analyses.

For the three videorecorded hearts, the longitudinal L_o and transverse L_o were, respectively, 2.05 and 1.44 cm, 2.16 and 1.50 cm, and 2.29 and 1.17 cm. Peak strains and strain rates imposed by the stretch pyramid were similar between loading treatments and measurement axes (Fig. 3). The range of strain is similar to the normal physiological strains (0.1) in sarcomeres of the hearts of the crab Cancer magister (Meyhöfer, 1993). In videos, the hearts shortened during a heartbeat by an average of 0.28 mm (±0.074 mm s.d., N=36; measured at the start, apex and end of all four stretch pyramids for all three lobster hearts).

Stretch always elicited an increase in passive, active and total forces along the axis of stretch (Figs 4–6). Passive forces tended to increase in slope with increasing stretch, whereas active forces began to level off above 1.5 mm extension along the longitudinal axis, but increased more steadily along the transverse axis.

Biaxial stretch increased transverse ΔF_passive relative to transverse uniaxial stretch, but biaxial stretch did not affect longitudinal ΔF_passive relative to longitudinal uniaxial stretch. For both biaxial and uniaxial stretches, the increase in ΔF_passive was an order of magnitude greater along the longitudinal than the transverse measurement axis (Fig. 4; see figure legend for statistics).

Biaxial stretch increased ΔF_active and ΔF_total relative to uniaxial stretch along both measurement axes. When comparing within a given pull type (uniaxial or biaxial), the transverse ΔF_active was greater than the longitudinal ΔF_active for tests starting from a 1 g baseline, but did not differ between measurement axes for tests starting from a 2 g baseline. Mirroring trends seen in ΔF_passive, longitudinal ΔF_total was always greater than transverse ΔF_total (Figs 5, 6; see legends for statistics).

Passive force hysteresis was greater along the transverse axis whereas active force hysteresis was greater along the longitudinal axis (Figs 5–7; see legend to Fig. 7 for statistics).

Although neither peptide altered longitudinal pre-stretch passive force, both peptides significantly increased transverse pre-stretch passive force. Similarly, longitudinal ΔF_passive did not change, or decreased slightly, while transverse ΔF_passive increased in peptide treatments relative to controls, except during biaxial tests in GYS (Fig. 8; see legend for statistics).

Both peptides also significantly increased pre-stretch active force and ΔF_active in peptide treatments relative to controls (Fig. 9; see legend for statistics).

Passive force hysteresis decreased with peptide treatment, although the effect of GYS reached significance in only one comparison. Longitudinal active force hysteresis also tended to decrease with peptide treatment; however, transverse active force hysteresis was unaffected by peptides except for one comparison (Figs 8–10; see legend to Fig. 10 for statistics).

In most cases, neither peptide altered either pre-stretch passive forces or ΔF_passive in stimulated preparations in which the CG was removed. However, similar to intact preparations, SGRN did elicit an increase in the transverse pre-stretch passive force (Fig. 11, see legend for statistics).

Both peptides caused increases in the pre-stretch active force as well as increases in ΔF_active along one measurement axis. Interestingly, longitudinal ΔF_active increased in SGRN whereas transverse ΔF_active increased in GYS (Fig. 12, see legend for statistics).

Anisotropy in mechanical properties of biological tissues can contribute to tissue-specific functions (Koehl, 1977; Shadwick and Gosline, 1985). For instance, changing the orientation of transverse myofibers in a finite element model of the human left ventricle altered the circumferential–radial shear strains, such that the model matched in vivo strains at some orientations (Ubbink et al., 2006). Anisotropy has been measured in canine myocardium, where tissue stiffness was greater under biaxial than uniaxial loads and depended on the direction of extension, with the greatest stiffness aligned with the direction of the predominant fiber (Demer and Yin, 1983). Although anisotropic behavior in myocardium is well characterized in vertebrates (e.g. Yin et al., 1987; Smaill and Hunter, 1991), the present study is the first to characterize whole-heart lobster cardiac length–force curves and to show that the lobster heart is mechanically anisotropic.

One potential contributor to anisotropy is that the heart is approximately 1.4 to 2 times longer than it is wide. Because we applied the same absolute extension to the heart in the longitudinal and transverse directions, this has the potential to exert different strain and strain rates in the longitudinal and transverse directions. Strain rate increases passive muscle stiffness in cardiac myocytes in both vertebrates (Demer and Yin, 1983) and crabs (Meyhöfer, 1993). But our observed strains and strain rates were similar along both axes of stretch (Fig. 3). This occurred because the heart was attached by suture silk on both ends in the longitudinal direction (at the perfusion and at the five anterior arteries) but attached on only one side in the transverse direction (at the hook). Thus, some of the transverse pull resulted in lateral displacement rather than stretch. The observed similarity of strain and strain rates along both axes suggests that passive mechanical differences measured along these axes represent intrinsic differences in material properties and/or differences in effective material cross-section in the different directions rather than extrinsic differences in strain or strain rate.

Two important contributors to passive force in mammalian myocardium are titin and collagen (Linke et al., 1994, 2002; Granzier and Irving, 1995). Titin contributes more than collagen to passive forces close to slack length because at short sarcomere lengths, collagen is wavy and the passive force necessary to unkink the collagen is low (Hanley et al., 1999; Wu et al., 2000). As sarcomere length increases, collagen straightens and stiffens, such that at the maximal working sarcomere length, collagen contributes as much or more than titin to the passive material properties of cardiac cells (Granzier and Irving, 1995; Hanley et al., 1999; Wu et al., 2000). Because titin tends to have a straighter passive force–extension curve than does collagen, the J-shape of the passive, whole-heart, length–tension curves we observed in lobster heart (Fig. 4) suggests that we might be seeing the effects of collagen at the larger extensions during stretch pyramids. Collagen stiffness may determine end-diastolic volume (Wu et al., 2000); thus the visibility of the J-shape suggests that the lobster's working range may coincide with the range over which we tested.

The low cross-transmission of passive force between the longitudinal and transverse axes of the heart during uniaxial extension (bottom lines in Fig. 4A–D) suggests that muscle or other fibers being pulled are oriented primarily along those axes; greater cross-transmission of force would be expected if obliquely oriented fibers were being loaded. Furthermore, we observed a relatively larger increase in longitudinal passive forces (ΔF_passive; Fig. 4, compare E and F); this pattern suggests that the cross-sectional area or stiffness of longitudinal fibers is higher along the longitudinal axis. Locally different mechanical properties within a heart have been observed in bovine atrium versus bovine ventricle due in part to differential expression of titin isoforms of different stiffness (Wu et al., 2000). We also observed higher fractional passive energy losses (hysteresis) along the transverse axis (Fig. 7). Hysteresis is inherently independent of cross-sectional area and high hysteresis is more characteristic of a matrix than a fiber, further suggesting that there is a smaller cross-sectional area of fibers along the transverse axis. However, transverse ΔF_passive was greater during biaxial tests than during uniaxial tests (Fig. 4F) but longitudinal ΔF_passive was not (Fig. 4E), indicating that there is some small cross-transmission of passive force to the transverse axis, but not to the longitudinal axis.

In contrast to passive forces, both longitudinal and transverse ΔF_active were significantly higher during biaxial than during uniaxial tests. Higher ΔF_active during biaxial tests could be explained if muscles were operating at higher sarcomere lengths. For transverse active forces, this explanation is plausible because transverse ΔF_passive values were also higher during biaxial tests, suggesting that the parallel elastic component represented by the passive forces was more stretched. All other things being equal, higher passive forces on the ascending limb correspond to higher sarcomere lengths. But as there were no increases in longitudinal passive forces, the higher active forces observed during biaxial tests invites other explanations. One possibility is that there is stretch feedback such that transverse stretch causes an inotropic increase in active forces in the longitudinal direction.

An increase in active force with extension (Fig. 5A–D) has been observed on the ascending limb of length–tension curves in invertebrate and vertebrate skeletal and cardiac muscle (Brady, 1967; Muhl, 1982; Josephson and Stokes, 1989). Operation on the ascending limb is typically associated with cyclical loading (e.g. hawkmoth wing depressor muscle: Tu and Daniel, 2004; squid mantle muscle: Thompson et al., 2014) and enables larger active forces to be generated at higher extensions. In hearts, the active force increases over the entire working length range, whereas in skeletal muscle, the working range of the active length–tension curve also often includes the descending limb (Burkholder and Lieber, 2001). The continuous increase in active force seen in vertebrate cardiac muscle may be due to: (1) operating only at shorter sarcomere lengths, i.e. on the ascending limb of the length–tension curve; (2) length-dependent sensitivity to calcium; or (3) closer spacing of actin and myosin filaments as length increases, among other possibilities (de Tombe et al., 2010). The length–tension curve of single lobster cardiomyocytes is unknown, but a parabolic-shaped active length–tension curve of a single lobster ostial muscle shows a maximum force at 1.2 times slack length (Yazawa et al., 1999). In comparison, we observed whole-heart length–tension curves that represent 1.1 times baseline length (0.1 strain), where the baseline length is slightly longer than slack length. Thus, assuming similar length–tension curve shapes of ostial and cardio myocytes, the observed whole-heart length–tension curves were likely in the ascending limb. Indeed, although the active length–tension curves are nearly linear, they show a hint of a leveling off at the longer extensions, suggesting that we are reaching the maximum force point on the ascending limb at the highest extensions in our experiment.

The lobster ostial muscle had a linear passive force–extension curve with a maximum passive force almost 10 times larger than the maximum active force (Yazawa et al., 1999). In the whole lobster heart, we similarly recorded force–extension curves in which passive forces were greater than active forces, but only in the longitudinal direction.

Active force was less influenced by direction of stretch than was passive force. Because all of the heart muscles contract when the heart is stimulated, differences in force generation must be due either to pre-load differences in fiber or sarcomere length or to differences in inotropy. At the level of the whole heart, differences in active force measured along each axis could also be due to differences in the angle of muscle fibers relative to the axis of measurement. However, because there was little effect of stretch along one axis on active force along the perpendicular axis (flat lines in Fig. 5A–D), we concluded that the fiber angles are unlikely to be important contributors to active force anisotropy. In any case, the active force anisotropy was much less than the passive force anisotropy.

Total force did not plateau (Fig. 6) even though the heart approached the length range at which longitudinal active forces did start to plateau. In lobsters, longitudinal and transverse total forces are reaction forces to the forces generated by pressure that is exerted across the heart wall by hemolymph and due to ligaments attaching the heart to external tissues. The total force comprises two components: the passive force, which maintains the diastolic wall pressure, and the active force, which produces the heart contraction. Thus, in a beating heart at the longitudinal extensions at which the active force starts to plateau, the longitudinal passive forces increase sharply, resulting in longitudinal total forces that could continue to increase in response to increasing hemolymph pressure within the heart. This combination of forces should enable the heart to pump effectively even at extensions greater than that at which peak active forces are reached.

There are some functional parallels between our mechanical findings for lobster heart and those for octopus aorta. Shadwick and Gosline (1985) found that the octopus aorta exhibits greater passive stiffness along the longitudinal axis than in the circumferential direction. This is similar to the lobster heart, which also has greater longitudinal passive forces. Furthermore, in the octopus aorta, the active forces increase in both longitudinal and transverse directions with increasing extension and are approximately twice as high in the transverse direction as in the longitudinal direction; the same is observed in the lobster heart at the 1 g baseline. Similar to what was observed for the octopus aorta, these anisotropies mean that at a given working pressure, the lobster heart will expand more circumferentially than longitudinally.

Mechanical anisotropy may reflect anisotropy both in how pressure translates into circumferential and longitudinal tension in the walls of the heart and in how circumferential versus longitudinal contractions result in volume changes in the heart. For example, if the heart were shaped as a simple cylinder, then for walls of uniform thickness, pressure would induce twice the hoop stresses relative to the longitudinal stresses in the walls of the heart (Wainwright et al., 1976; Koehl, 1977). However, consider another thought experiment: if the heart is ovoid, with a greater radius of curvature along the longitudinal direction than along the transverse direction, then by LaPlace's law, pressure-induced tension in the wall of the heart would be greater along the longitudinal axis; the greater passive resistance along the longitudinal axis might act to compensate for that added tension. For a cylindrical heart, consider also the consequences of longitudinal or transverse contraction to volume of blood pumped out of the heart during a contraction. For an equivalent strain in the longitudinal and circumferential muscles, a changing radius doubles the volume change relative to a changing length (see Appendix for derivation). Thus, circumferential (and the proportional radial) transverse length changes will be the most significant contributor to the volume of blood pumped.

Active forces can be modulated not only via stretch feedback from the muscles to the CG, but also by the action of neuromodulators acting directly on the CG and/or the periphery. Previous experiments have shown that the effects of the neuropeptides SGRN and GYS differ in experiments on unstretched, intact lobster hearts versus isolated CG (Dickinson et al., 2015). In the present study, we found that both neuropeptides increased pre-stretch active forces not only in intact hearts (Fig. 9C,D), but also in stimulated hearts (Fig. 12C,D), indicating that SGRN and GYS also exert at least some of their effects directly on the heart muscle or neuromuscular junction.

Previous work on lobster hearts has not examined the effect of these neuropeptides on passive forces. We found that neither SGRN nor GYS altered longitudinal pre-stretch passive force (Figs 8C and 11C). Interestingly, both peptides elicited increases (SGRN: 22%; GYS: 27%) in transverse pre-stretch passive forces in intact hearts (Fig. 8D). However, when the CG was removed, GYS no longer caused such an increase, and the rise in passive pre-stretch forces in SGRN was decreased to only 5% (Fig. 11D). These data suggest that much of the increase in passive force is mediated by the CG, perhaps as a result of repeated contractions at higher frequencies (Dickinson et al., 2015), with the consequence that the baseline force does not return to completely relaxed levels, i.e. that some active level of force persists between contractions. However, at least some of the increase seen in the stimulated heart with SGRN is likely a direct effect of the neuropeptide on the muscle because there is some increase in pre-stretch transverse passive force even in the absence of the CG. To our knowledge, this is the first report of a significant effect of neuroactive compounds on passive muscle properties in crustaceans; however, serotonin has been shown to decrease passive muscle tension in leeches (Gerry and Ellerby, 2011; Gerry et al., 2012). Because passive forces at low extensions are dominated by titin, perhaps neuropeptides affect titin function in lobster cardiomyocytes. Alternatively, some baseline, non-rhythmic tension could be generated by actin–myosin interactions.

Both passive and active forces are lower during the descending ramp than during the ascending ramp of the stretch pyramid. The simplest explanation for this difference for passive forces is that viscoelastic materials always show such hysteresis. But the active force hysteresis requires a different explanation. One possibility is that muscles generate different forces depending on the rate of contraction and the direction of length change. For instance, an eccentrically contracting muscle develops twice as much force as does the same isometrically contracting muscle. And the same concentrically contracting (shortening) muscle develops less force the faster it contracts. Because the pyramids lengthen and shorten the muscle, they will tend to change the eccentric/concentric status of the muscle contraction as well as the velocity of contraction. This is possible because the imposed lengthening speed is much slower than the contraction speed of the heart muscle, and the compliance of the suture silk is such that during the up-ramp, the imposed slow lengthening rate will tend to slightly decrease the shortening velocity of the heartbeat, thus increasing the force slightly relative to the down-ramp. This hypothesized effect is consistent with the observed active force hysteresis (Fig. 7). An alternative explanation is that lengthening of the heart muscle causes different feedback than does shortening and that this feedback alters the force produced. Indeed, neurotransmitters, which might alter feedback, did cause slight decreases in active force hysteresis (Fig. 10C), at least in the longitudinal direction. Additionally, neurotransmitters cause small changes in the passive force hysteresis (Fig. 10A,B), although it is unclear whether this effect on the passive forces is direct or whether it is a result of persistent active forces between heartbeats.

The present study found that these neuromodulators also enhanced the effects of stretch on active forces in both intact and stimulated hearts (Figs 9E,F and 12E,F). One way in which SGRN or GYS might interact with the active length–tension curve is via the combined effects of the initial peptide-induced increase in active force and the stretch-dependent increase in active force, i.e. the neuropeptides shift the curve upwards on the graph. Alternatively, the peptide-induced increase in the slope of active forces as a function of stretch could be the result of stretch feedback from the periphery (Fig. 1). Alexandrowicz (1932) postulated that dendrites with ramifications on the peripheral muscles within the heart might be stretch-sensitive and relay sensory information back to the CG. Thus, the additive effect of stretch and the increased initial contraction force due to the peptides could sufficiently increase the contraction amplitude to trigger the mechanosensitive dendrites, thereby positively regulating the active forces at greater extensions.

One explanation for the apparent multiplicative effects of GYS and SGRN is that both GYS and SGRN increase the duration and frequency, as well as the duty cycle, of cardiac motor neuron bursts (Dickinson et al., 2015), which would result in increased Ca²⁺ availability in cardiac muscles. Given the non-linear nature of the neuromuscular transform (Williams et al., 2013), it is likely that Ca²⁺ also increases non-linearly with stretch. Such a non-linear increase in Ca²⁺ availability, when combined with greater sarcomere lengths, could in turn result in an increase in contraction force that is multiplicative rather than simply additive. Furthermore, the neuropeptides increased ΔF_active equally in the longitudinal and transverse directions, supporting the hypothesis that the multiplicative effect was due to the increased availability of actin–myosin interactions at each sarcomere length.

The effect of neuropeptides on ΔF_active was maintained in the stimulated preparations (Fig. 12), albeit with some axis-specific differences, further supporting the hypothesis that some of the multiplicative effects of peptides and stretch occur at the neuromuscular junction and/or muscle. Ultimately, the diversity of targets for the neuropeptides enables flexibility in cardiac activity.

The effects of both neuropeptides on stimulated preparations were anisotropic: SGRN only significantly increased longitudinal ΔF_active whereas GYS only significantly increased transverse ΔF_active. These results suggest that these neuropeptides differ in the location and/or mechanism of action in the muscle or at the neuromuscular junction. Thus, the mechanical anisotropy may be paralleled by a neurophysiological anisotropy. Together, both mechanical and neurophysiological flexibility and anisotropy mediate and control the three-dimensional response of the lobster heart to its environment.

For a cylindrical heart, a fractional circumferential contraction causes twice the volume change as does the same fractional length contraction.

where h is the initial length of the cylinder, v is the initial volume of the cylinder, Δv_l is the change in cylinder volume due to length change and r is the radius of the cylinder.

where Δv_r is the change in cylinder volume due to radial change, and r_i and r_f are the initial and final radii after contraction, respectively.

We thank Hadley Horch, Julie Hewitt and Andrew Calkins for feedback on these experiments, and Marko Melendy for assistance with animal care.