Biochemical and Structural Basis of the Passive Mechanical Properties of Whole Skeletal Muscle

Abstract

Passive skeletal muscle mechanical properties of whole muscle are not as well understood as muscle’s active mechanical properties. Both the structural basis for passive mechanical properties and the properties themselves are challenging to determine because it is not clear which structures within skeletal muscle actually bear passive loads and there are not established standards by which to make mechanical measurements. Evidence suggests that titin bears the majority of the passive load within the single muscle cell. However, at larger scales, such as fascicles and muscles, there is emerging evidence that the extracellular matrix (ECM) bears the majority of the load. Complicating the ability to quantify and compare across size scales, muscles, and species, definitions of muscle passive properties such as stress, strain, modulus and stiffness can be made relative to many reference parameters. These uncertainties make a full understanding of whole muscle passive mechanical properties and modeling these properties very difficult. Future studies defining the specific load bearing structures and their composition and organization are required to fully understand passive mechanics of the whole muscle and develop therapies to treat disorders in which passive muscle properties are altered such as muscular dystrophy, traumatic laceration, and contracture due to upper motor neuron lesion as seen in spinal cord injury, stroke and cerebral palsy.

Graphical Abstract:

Schematic representation of passive load bearing at various skeletal muscle scales. In the sarcomere (upper left), load is borne by the giant elastic protein titin and, to a lesser extent, desmin. The myofilaments actin and myosin are responsible for active force production. Sarcomeres in series (myofibrils) connect to the fiber surface at specialized focal adhesions called costameres at which desmin and other cytoskeletal proteins converge (upper right). Muscle fibers are embedded in a connective tissue matrix (lower left) composed primarily of basal lamina collagen type IV. At larger scales, passive load is borne in the perimysium (lower right) by perimysial cables that seem to be composed of collagen types 1 and 3 and this is where the majority of passive load is borne.

Introduction

Skeletal muscle structural and functional studies have a long and rich history in physiology (Blix, 1895; Sandow, 1936; Hill, 1938; Eccles & O’Connor, 1939; Katz, 1939; Eccles, 1944; Huxley, 1953; Podolsky, 1960). Some of the best biological examples of structure-function relationships have been elucidated for skeletal muscle contractile properties at all levels ranging from the force-producing cross-bridge at the micro level to the dynamic isometric and isotonic properties at the macro-level. Similarly, exquisite details exist regarding the structural basis of excitation-contraction coupling ranging from the membranous invaginations, receptors and calcium regulating structures that allow precise activation and relaxation of skeletal muscle to the kinetics of calcium movement itself.

In contrast to our high resolution understanding of active properties, far less is known about how passive properties of skeletal muscle scale from the micro (single muscle cells) to the macro (whole muscle). This realization came to us based on our observations of surgeons performing tendon transfer surgery. We saw that muscle manipulation was based more on the “feel” of the muscle rather than the actual sarcomere length or functional muscle properties (Brand, 1974; Fridén, 2005). When we asked why this was the case, it was clearly assumed that passive mechanical properties were uniform across human muscles and that, at whole muscle “slack length,” the muscle was at optimal length for maximum force generation. Both of these are incorrect. Unlike active muscle mechanical properties, which were largely elucidated on frog skeletal muscle (Hill, 1938; Edman, 1966; Gordon et al., 1966), much of the recent work on passive muscle mechanics data was obtained from a variety of species; i.e., frog, rabbit, rat, mouse and human. It has become increasingly evident that variations in passive properties amongst species and differential scaling of these properties may have clouded the understanding of key issues in passive muscle mechanics.

While passive mechanical properties have been measured for over a hundred years, they were considered less important compared to active mechanical properties since most focus has been on sarcomere-based force generation. In muscle physiology, passive properties are often simply considered the “parallel elastic element” in lumped parameter models such as the classic models of A.V. Hill (Hill, 1953). These relationships appear as classic nonlinear curves, characteristic of most other connective tissues such as ligament, bone and tendon. Though, these connective tissue are composed of the same basic biological building blocks, their mechanical stiffness varies greatly based on their on their composition and structure, with calcified bone being the stiffest and muscle the most compliant (by about 1,000 fold, Fung, 1981).

Within skeletal muscle, there are two primary structures for force transmission and passive load bearing—the extensively studied muscle fiber, where a majority of the passive properties originate from the giant elastic protein (Granzier & Labeit, 2007), and the poorly understood connective tissue structures that surrounds these fibers, the muscle extracellular matrix (ECM). This review focuses primarily on the ECM which is classically divided into three levels (Fig. 1)—the endomysium, comprised primarily of type IV collagen that surrounds muscle fibers (Sanes, 2003), the perimysium, that bundles muscle fibers together into fascicles, and epimysium, which bundles the fascicles together to ensheathe a whole muscle (Gillies & Lieber, 2011). Apart from a small number of high resolution scanning electron microscope studies of endomysium (Trotter, 1990; Trotter & Purslow, 1992), skeletal muscle ECM structure is poorly understood. This is further complicated by the difficulty in fully characterizing its chemical composition in terms of collagen types and chemical modification that can affect mechanical properties. Therefore, the purpose of this review is to increase the level of awareness of skeletal muscle ECM passive functional and structural properties, summarize much of the current state of knowledge, and address areas in which we can advance our knowledge and understanding of passive muscle mechanics.

Figure 1:

Overview of skeletal muscle as a composite tissue. Structural hierarchy proceeds from the smallest level of the myofibril which is grouped inside a muscle fiber. The fiber is grouped into fascicles which is grouped into the whole muscle. In terms of connective tissue, the smallest level is the endomysium, surrounding the muscle fiber, which are then grouped into fascicles, surrounded by perimysium, which is grouped into the whole muscle, surrounded by epimysium. Perimysial cables are interspersed throughout the perimysium although their organization is not fully understood. In reality, these three levels of connective tissue are differentiated by anatomy more than function or composition. (Figure courtesy of Battista Illustration).

Functional Basis of Passive Muscle Mechanics

Significance of passive muscle properties:

The passive functional properties of muscle are measured in the absence of activation. In traditional muscle mechanics experiments, this value is often simply subtracted as a “baseline” offset value because those studies focus on active properties. While the differential design of muscles in terms of their architecture is easily appreciated and has been reviewed elsewhere (Lieber & Fridén, 2000; Lieber & Friden, 2004; Lieber & Ward, 2011), passive muscle “design” is less obvious. However, the stiffness (or, when normalized to area and zero strain, modulus) are rarely quantified but quite functionally important. Additionally there are many terms used to define the passive mechanical properties, all which convey slightly different information (Table 1 adapted from Meyer & Lieber, 2018). However, as we will describe below, there is significant variability among muscles with regard to passive mechanical properties that may have design implications and can change with altered use and disease. We focus on static properties, but skeletal muscles are slightly viscous (Sarver et al., 2003; Meyer et al., 2011) and thus, during movement, this viscosity may become functionally relevant. The viscous properties of skeletal muscle are beyond the scope of this review.

Table 1:

Passive Mechanics Property Definitions (adapted from Meyer & Lieber, 2018).

Passive Mechanical property	Definition
Passive tension	Tension (force) sustained at a given strain or sarcomere length
Passive stress	Passive tension divided by specimen cross-sectional area
Young’s (elastic) modulus	Slope of a linear elastic stress–strain or stress-sarcomere length curve
Regional elastic modulus	Slope of a non-linear stress–strain or stress–sarcomere length curve over a region that exhibits linearity
Tangent modulus	Slope of a non-linear stress–strain or stress–sarcomere length curve at a given strain or sarcomere length
Modulus coefficient(s)	Coefficient(s) obtained by regression of a stress–strain or stress–sarcomere length curve to a nonlinear model (typically polynomial, exponential or strain–energy density functions)

Active vs. passive muscle structure-function relationships:

To a first approximation, the active properties of a whole skeletal muscle can be considered a scaled version of its composite sarcomeres (Winters et al., 2011). Whole muscle isometric and isotonic properties generally reflect sarcomeres arrangement in series and in parallel (Gans & Bock, 1965; Lieber & Fridén, 2000). Unfortunately, no such scaling relationship exists for passive muscle properties. Numerous studies have demonstrated that passive load bearing occurs intracellularly at the sarcomere and single fiber levels and extracellularly at the larger fiber bundle and whole muscle levels as well. But where does the majority of load bearing occur in muscle? What are the structures responsible for bearing these loads? How do these structures change in situations where muscle use is altered or in disease states? These questions cannot currently be answered definitively and are not consistent across species, but a number of general themes and principles have emerged.

Examples of passive mechanical properties of human muscles:

High resolution studies of passive muscle properties are in their infancy with the passive mechanical scaling rules for skeletal muscle not yet established. In the few studies in which passive properties were measured at scales including whole muscles, it is clear that passive mechanical properties at the whole muscle level are also influenced by the gross connective tissue geometry observed at levels far above the fiber bundle. For example, we measured single muscle fiber properties of the human flexor carpi ulnaris (FCU) as well as fiber bundle properties (Fridén & Lieber, 2003; Lieber et al., 2003). Based on these measurements, we suggested that we could predict FCU muscle properties intraoperatively as a scaled version of the bundle properties. We then measured actual human muscle biomechanical properties intraoperatively with a dual-mode servo motor (see Fig. 2 of Lieber et al., 2005) and, to our surprise, stiffness of the whole muscle was at least 10-times greater than that of the fiber bundle which was about 4 times that of the fiber. A similar result was observed for relatively large rabbit muscle where, in three different muscles, the ratio of whole muscle:muscle bundle stiffness ranged from 10 to 50 for the three rabbit muscles studied (Ward et al., 2020). Again, a higher scale organization was present at the whole muscle level and this varied among muscles.

Figure 2:

Structure and function of perimysial cables. (A) Confocal microscopic view of a small group of stretched muscle fibers labelled with a type 1 collagen antibody. Note several collagen cables running alongside the fibers. White bars are the same as those in (B). (B) Phase contrast micrograph of fibers shown in A. Note the clearly delineated sarcomeres, but absence of perimysial cables that do not appear with this type of microscopy. White bars show groups of 10-sarcomeres used to calculate sarcomere length which, in this specimen averages ~4.3 µm. Collagen cable angles in (A) and sarcomere lengths in (B) are used for data plotted in (C). (C) Relationship between perimysial cable angle with respect to the long axis of the fiber and average sarcomere length. Note that cables become nearly parallel to fibers as sarcomere length increases. Data represent mean±SEM of 6 cables from two different muscles. (Experiments previously published in Gillies et al., 2017). (D) Sample scanning electron micrograph of one serial blockface section of muscle showing intercellular perimysial cable (PC). To define their structure, PCs are traced through serial sections (Fig. 3) to create three-dimensional reconstructions (Fig. 4).

Passive tension relates to sarcomere length operating range, not optimal length:

When we measured human passive mechanical properties, we were somewhat surprised to find that, at “slack length,” sarcomere length varied quite a bit across muscles (Fridén & Lieber, 1998). These data were most clear for the prime movers of the wrist for which we had the most data (Lieber et al., 1990; Loren & Lieber, 1995). When wrist muscles were measured throughout the range of motion, we found that the flexors tended to operate at shorter lengths that were biased toward the ascending limb of the length-tension curve while extensors tended to operate at longer lengths biased toward the descending limb of the length-tension curve (Loren et al., 1996). In this case, it appeared that the passive mechanical properties were tailored to the sarcomere length operating range of the muscle.

Consistent with the wrist flexors functional bias toward shorter sarcomere lengths; resting sarcomere length shifted shorter than optimal length. Whereas, within the extensor muscles whose functional bias is toward longer sarcomere lengths, their resting sarcomere length was closer to optimal length. Similar specialization was observed for spine muscles, specifically the lumbar extensor multifidus muscles that have very short resting sarcomere lengths, very stiff passive mechanical properties, and an operating range biased toward the ascending limb of the length-tension curve (Ward et al., 2009a; Ward et al., 2009b).

When we measured sarcomere length relationships while flexing and extending the human spine, these passive mechanical properties made sense—the multifidus appears to be suited for its function of lumbar spine extension since its passive properties tend to keep the lumbar spine extended or restore it to extension after flexion. Therefore, across many human muscles, passive mechanics are not uniquely related to sarcomere length but are more likely related to their operational range or functional use.

Skeletal muscle passive tension changes with injury and disease:

Perhaps the most dramatic change in muscle passive mechanical properties results from pathological insult to the muscle. Such changes have been extensively described for the mdx mouse model of Duchenne Muscular Dystrophy (DMD) in which loss of dystrophin results in cyclic muscle fiber degeneration/regeneration, increased collagen content and fibrosis observed histologically. We also reported that muscle contractures that occur secondary to cerebral palsy (CP) are fibrotic and may form some of the basis for the dysfunction that occurs in these muscle contractures (Lieber et al., 2003; Smith et al., 2019). While current surgical approaches are used to release these contractures in CP, there is no current therapy aimed at recovering from the fibrotic tissue. Similarly, in DMD, the therapeutic emphasis has been to restore or replace the function of dystrophin thus preventing the cyclic degeneration/regeneration cycles (Serrano & Munoz-Canoves, 2010; Smith & Barton, 2018). Other processes that result in muscle fibrosis include chronic inflammation (Meyer & Lieber, 2012), denervation (Lewis et al., 1978), neurotoxin injection (Thacker et al., 2012; Binder-Markey et al., 2019), and direct trauma (Omer, 1968; Lewis et al., 1978; Carraro et al., 1982).

Cellular basis of ECM production:

In all of these processes, it appears that multiple cell types located in the extracellular space lose their normal homeostatic relationship and overproduce collagen, creating fibrosis. While there is no generally agreed-upon “final common pathway” for muscle fibrosis, many of these processes include some type of inflammatory response, resulting in cytokine production and overproduction of collagen by fibroblasts, fibroadipogenic progenitors, and even satellite cells (Chapman et al., 2017). Collagen-producing cells can be studied by creating a type I collagen reporter mouse that expresses green fluorescent protein (GFP) under the control of the collagen α1(I) promoter (Yata et al., 2003). In a desmin knockout model (Li et al., 1997), in which mild chronic fibrosis results, the number of GFP+ cells was significantly increased compared to wild type (Chapman et al., 2017). In fact, we measured nearly a doubling of the number of cells per milligram of muscle that produced type 1 collagen in the knockout ECM (see Fig. 5B of Gillies et al., 2017). While we suspected that the predominant collagen producing cell type would be the fibroblast, we actually measured increased collagen production by all three of the major extracellular mononuclear cells—fibroblasts, satellite cells and fibroadipogenic progenitors.

Figure 5:

(A) Typical sarcomere length-stress relationship of single muscle fiber (circles) and muscle fiber bundle (squares). Note that both relationship are nonlinear and that the slack length of the muscle fiber bundle is shorter (2.21 µm) compared to that of the single fiber (2.46). Data obtained from the 5^th toe of the extensor digitorum longus (EDL) (Mus musculus, strain 129/Sv 7–9 weeks old)(Meyer & Lieber, 2018). (B) Same data as in A, but replotted as a stress-strain relationship, where 0% strain is defined as the slack length. Note that this way of plotting makes fiber and bundle properties appear to be more similar than in (A). (C) Comparison of the fiber:bundle stiffness ratio as a function of sarcomere length (squares, left vertical axis) or strain (circles, right vertical axis). Note that fiber stiffness appears to be closer to bundle stiffness for the stress-strain relationship plotted in B (reflected by a rightward shift in ratio for fiber data).

It is thus clear that the increase in ECM production represents a concerted effort across multiple cell types which may imply a cytokine-mediated endocrine effect or communication amongst cell types. Cellular regulation of muscle fibrosis represents and active area of research. Fortunately, fibrosis in other tissues also has significant clinical implications (lung, liver, cornea to name a few, (Wynn, 2008; Zeisberg & Kalluri, 2013), so muscle fibrosis is one area where muscle physiologist can join with other physiologists to attempt to understand a common problem. Because the ECM is constantly transducing the stress and strain imposed upon it and because a number of stem cell types exist in ECM (Pannerec et al., 2012) it is likely that ECM mechanical properties also affect stem cell maturation and differentiation. Certainly, in vitro models, in which stem cells are differentiated on substrates with different mechanical properties, show this to be the case (Engler et al., 2006).

Structural Basis of Whole Muscle Passive Muscle Mechanics

Titin bears the majority of passive load within muscle cells:

A resurgence of interest in muscle passive load bearing was ignited by the fascinating insight of Magid and Law who showed that, in frog skeletal muscle, the majority of load bearing occurred intracellularly within the fiber, rather than extracellularly, within the ECM (Magid & Law, 1985), as had been previously stated (Street, 1983). This was demonstrated by showing that the mechanical properties of myofibrils, muscle fibers and even whole frog muscles scaled uniformly with size. The search for this intracellular load bearer culminated in the discovery of the giant elastic protein, titin (Labeit & Kolmerer, 1995), also known as connectin (Maruyama et al., 1977) and unleashed a plethora of spectacular biophysical experiments that have revealed that titin is a giant molecular filament that bears significant intracellular load (Wang et al., 1991), serves as a template to the developing sarcomere (Tonino et al., 2017), provides a signaling function that can affect muscle development and hypertrophy (Lange et al., 2005; Brynnel et al., 2018), can influence active muscle contraction (Joumaa & Herzog, 2014; Eckels et al., 2018) and has the distinction of being the largest protein known that creates a single molecular filament (Neagoe et al., 2003). These fantastic studies have been reviewed elsewhere (Granzier & Labeit, 2007; Linke, 2018). While it is likely that titin bears the majority of the passive load in muscle cells (more than the intermediate filament network (Wang & Ramirez-Mitchell, 1983; Wang & Ingber, 1994), how is load borne at larger scales?

Passive force transmission from intra-to extra-cellular in skeletal muscle:

Active muscle force is generated by cyclic interaction between actin and myosin proteins (Huxley, 1957) within the muscle fiber and this force is transmitted laterally by the intermediate filament system that connects myofibrils laterally and, to a lesser extent, longitudinally (Lazarides, 1980). The intermediate filament network aligns adjacent myofibrils, serves as a mechanotransducer of fiber strain, and even mediates muscle injury due to eccentric contraction (Palmisano et al., 2015). Myofibrillar force is mechanically focused at punctate sites along the fiber surface known as costameres (Pardo et al., 1983) which are composed of a complex of force transducing proteins connecting the fiber interior to the extracellular matrix (Ervasti, 2003). As a testimony to the importance of lateral force transmission from the fiber interior to the exterior, it is noted that many muscle diseases result from mutations of costameric proteins (Gonatas et al., 1966; Dalakas et al., 2000; Neagoe et al., 2002; Nakada et al., 2003; Lange et al., 2009). The magnitude of lateral force transmission is functionally significant as well since, if a mechanical “yolk” is used to dissociate lateral and longitudinal force transmission, active force can change by up to 25% (Ramaswamy et al., 2011). This active area of “lateral force transmission” is fascinating but beyond the scope of this review for we are focusing largely on the extracellular force transmissions at larger scales.

Extracellular matrix bears increasing load at increasing size scales:

Quantitative biophysical studies of ECM are rare and most experiments that attempt to “quantify” ECM resort to simply describing bulk ECM with such course parameters as “area fraction” of ECM compared to whole muscle area or simply biochemically-measured total muscle collagen. Such studies have not provided deep insights into ECM function since these parameters correlate very poorly with mechanical properties of the whole muscle (Smith et al., 2011). Thus, unlike many of the high resolution single molecule or single sarcomere studies of titin, in which the molecular dimensions of the protein filament can predict its function (Labeit et al., 1997), or the quantitative relationship between sarcomere length and active muscle force (Gordon et al., 1966) there is no equivalent high resolution study of muscle ECM.

Passive mechanics at different scales:

We recently performed a systematic review of passive mechanical properties across the size scales from fiber to whole muscle and found that 106 studies tested at least 55 different muscles across 16 species (Binder-Markey & Lieber, 2020). Unfortunately, direct comparisons across studies are difficult because of large methodological and analytical variations among studies. Additionally, high resolution, quantitative scaling studies are very rare. In the few cases in which the same method is used within a single species across several scales, it appears that small muscle fiber bundles are quite a bit stiffer than single muscle cells and that larger fascicles and whole muscles are stiffer still. In one study, a bundle was defined as ~20 fibers and a fascicle as ~300 fibers so size increased dramatically among scales (see pictures in Fig. 1 of Ward et al., 2020). Identical methods were used on all three tissue samples in three different muscles, and the average ratio of bundle:fascicle:muscle modulus was nominally 1:3:95 (Ward et al., 2020). The fact that the ratio increased abruptly by size scale implies two important things: (1) ECM bears an increasing fraction of load at higher scales and (2) load bearing structures or geometry or composition must change as a function of scale. Unfortunately, neither the geometric nor the biochemical changes that occur across scale are fully understood.

Structural basis for change in stress with scale:

The two most likely candidates in ECM for which compositional changes would affect muscle stiffness are the collagen isoform ratio and the degree of collagen cross-linking. It is known in other connective tissues such as ligament and tendon, that the ratio of type I:type III collagen can affect compliance with an increasing ratio representing a stiffer specimen (Frank et al., 1999). However, there is no evidence in muscle that the type I:type III collagen ratio nor collagen cross-linking changes with size scale. Indeed, when collagen type specific antibodies are used to label tissue at the light microscopic level, uniform staining of type I and type III collagen is observed across the muscle cross-section. It has been shown that the type I:type III collagen ratio changes significantly in CP, but the magnitude of the effect is modest (Smith et al., 2019).

Connective tissue stiffness is also regulated by cross-linking across different collagen fibrils (van der Slot et al., 2004). Collagen crosslinks are formed both enzymatically and non-enzymatically. Enzymatic collagen crosslinks are formed when lysyl oxidase reacts with free lysyl or hydroxylysyl side chains within collagen fibrils, resulting in lysyl-pyridinoline (LP) and hydroxylysyl-pyridinoline (HP) crosslinks, respectively. Non-enzymatic crosslinks, such as pentosidine (PE), are created when glucose reacts with lysine and the resulting compound is oxidized. It is possible that there is a systematic increase in collagen cross-links as a function of muscle scale but, to our knowledge this has not been demonstrated. Indeed, it would be an extremely challenging experimental study to microdissect muscle at different scales and then perform these quantitative assays. Additionally, it is not clear how such systematic variability in enzymatic crosslinking would be accomplished by muscle tissue in which diffusion throughout the ECM appears to be relatively permissive (Minamoto et al., 2007).

Based on the observations above and the weak correlation between collagen content and mechanical properties observed, we posit that variations in stiffness across scales results from mesoscale structural variation across the muscle. One actual structure, observed in the perimysial space, is the so-called perimysial cable—a discrete structure that runs along the length of muscle fibers but is poorly appreciated in muscle cross-sections (see Figs. 1, 2 and 3A of Gillies et al., 2017). We have performed studies in which perimysial collagen cables are quantified using three methods. First, we simply immunolabeled type 1 collagen in isolated bundles of 2-5 fibers in order to visualize the cable (Fig. 2A) as a function of sarcomere length, measured by phase contrast microscopy (Fig. 2B). Note first, that the cable easily seen in Fig. 2A is “invisible” in Fig. 2B and thus it is not surprising that these cables have not received much attention. The angle between the cable and the long axis of the fibers can be quantified using this approach and it is seen that cables at long sarcomere lengths essentially align with the fibers (angle <10°) while they flex quite a bit at shorter sarcomere lengths (angle >40°) (Fig. 2C). If indeed these are load bearing cables, this geometric and intrinsic deformation pattern would give them very interesting nonlinear mechanical properties. Unfortunately, this confocal microscopy method requires disruption of the tissue itself. Thus, in a second method, we used very objective stereological methods used on perfusion-fixed tissue “frozen” in its natural state. In a study comparing the passive mechanical properties of mouse extensor digitorum longus (EDL) muscle from wildtype mice to desmin knockout mice (in which muscle increases its passive stiffness due to chronic inflammation, (Meyer et al., 2013), we measured the number and size of cables from high-magnification micrographs from stereological analysis and the cross-sectional area of each cable was calculated from classified high-magnification stereology grids (see Fig. 5A of Smith et al., 2019). We measured an increase in the number of collagen cables synthesized and assembled by the knockout muscle but not their size. Thus, the idea that number of perimysial cables could be altered and even regulated by the muscle represents a fertile area of future research. A third method we have used to quantify perimysial cables is serial blockface scanning electron microscopy, originally used to elucidate the microstructure of the cerebellum (Deerinck et al., 2010; Gillies et al., 2014). In this method, thousands of serial sections are imaged and a three-dimensional reconstruction of any structure of interest (e.g., perimysial cables, neuromuscular junctions, mitochondria, nuclei, satellite cells, capillaries, etc.) are made. Within a field, perimysial cables are easily identified (Fig. 2D) and can be traced through serial sections (Fig. 3). We have performed reconstructions of 12 of these cables (Fig. 4) and have seen many examples of undulating cables that run among fibers, associate with fibroblasts and other cell types (Fig. 3 of Gillies et al., 2014) and appear in some cases to even terminate on muscle fibers, reminiscent of the “perimysial plates” that have been reported for cardiac muscle (Passerieux et al., 2006). This work is extremely laborious and further quantitation and elucidation of the nature of these structures awaits future studies and development of more powerful experimental methods.

Figure 3:

(A-H) Successive slices through muscle in Figure 3 focusing on one perimysial cable (outlined in A). Images are shown approximately every 2.5 µm through the muscle (35 × 70 nm section thickness). Note that the cable “moves” away from one fiber toward another and displaces upward in the micrograph. This represents a two-dimensional view of cable undulation shown in Fig. 4.

Figure 4:

Three-dimensional reconstruction of perimysial cables (yellow), capillaries (magenta) and fibroblasts (blue) within a block of muscle sampled by serial blockface sectioning as shown in Figs. 2D and 3. Scale bar = 5 µm.

Bulk tissue explanation of passive mechanical properties:

A novel and qualitatively different structural basis for passive mechanical properties has been proposed that posits that muscle ECM constrains the muscle to a constant volume based on a sort of helical functional structure (Sleboda & Roberts, 2017), based partly on the incompressibility of water within muscle. Interestingly, there is experimental evidence supporting this model in that a physical representation of the structure recapitulates many known passive muscle mechanical properties (Sleboda & Roberts, 2017) and alterations in the structure (for example, by not allowing muscle to expand under external pressure (Sleboda et al., 2019)), decreases muscle active force generation (Sleboda & Roberts, 2020). This model is consistent with many observed experimental studies but validation of the structural basis of this model in actual muscles is in the early phases.

Barriers to Defining Passive Muscle Properties

The terms “resting length,” “in vivo length,” “slack length” and “in situ length” are often used interchangeably. They are often meant to refer to a muscle’s length at rest in the body but are usually poorly defined. Strictly speaking, the “slack length” of a muscle is its length under zero load. There are many muscles that are under load at rest with humans in the anatomical position. This is easily appreciated intraoperatively since, when a tendon is cut, the muscle retracts to leave a gap (Fridén & Lieber, 1994). The fact that there is no true “zero” length for skeletal muscle makes defining its passive mechanical properties, in terms of class stress-strain relationships, challenging. When one is reviewing muscle passive mechanical studies, the zero reference length must be known and critically evaluated.

The main difficulty in comparing passive muscle properties across scale and species is the wide variations in definitions of muscle stiffness (Table 1). While all investigators agree that there is a nonlinear load-deformation relationship measured experimental in muscles (see, for example, Fig. 2C of Smith et al., 2011), defining the stiffness or modulus from such experiments depends on many factors:

1. The length (or length range) at which stiffness is defined;

2. The scaling rules used to define stress and strain;

3. The definition of zero strain;

4. The curve-fitting method (if any) used to fit the data.

As a result, it is not yet clear whether passive mechanical differences among muscles are due to differences in experimental methods amongst experimentalists, true differences among muscles or some combination of the two.

Length at which stiffness is defined:

To illustrate this factor, consider the muscle fiber and fiber bundle comparison in Figure 5A. These data (one sample dataset from Meyer & Lieber, 2018) represent a single mouse muscle fiber and small muscle fiber bundle. There are numerous stiffness values that can be obtained from these curves. As one example, the tangent to the curve at a sarcomere length of 3 µm (if one posits that 3 µm is a physiologically important length) could be used as the length to define the stiffness and yields a value of 27 kPa/µm and 63 kPa/µm for the fiber and bundle respectively. However, note that if we had chosen the tangent at a sarcomere length of 4 µm the values would have been 78 kPa/µm and 228 kPa/µm for the fiber and bundle respectively. In the first case, the fiber:bundle modulus ratio is 43% and in the second only 34%. Thus the actual sarcomere length chosen affects our conclusions regarding relative stiffness. Which is correct? It is not clear.

The importance of choosing sarcomere length was dramatically emphasized in our recent comparison between muscle fibers and bundles from contractured muscle in children with CP (Smith et al., 2011). We obtained biopsies from these children and controls and found that CP fibers were the same as controls while CP bundles were about 25% stiffer (Figs. 3A and 3B of reference Smith et al., 2011) when subjected to traditional materials testing methods. However, the in situ sarcomere lengths of children with CP compared to control children was dramatically different (Fig. 6). Children with CP had very long sarcomere lengths (~3.6 µm) compared to controls (~3.0 µm) meaning that, functionally, the CP bundles were actually 200%-400% stiffer compared to control bundles at the in vivo length (dotted lines in Fig. 6). This has significant implications for surgical reconstruction and for understanding the etiology of contractures. This example illustrates the importance of placing mechanical measurements into a thoughtful physiological context.

Figure 6:

Relationship between specimen stress and sarcomere length measured in vitro for muscle fiber bundles obtained from typically developing (TD, squares) children and children with cerebral palsy (CP, circles). While the in vitro properties are fairly similar, the actual in situ properties (properties at the sarcomere length measured in the child’s leg, shown as open symbols for each curve) differ almost 10-fold. This illustrates the importance of placing passive mechanical properties into a physiological context. (Data replotted from Fig. 2C of reference Smith et al., 2011)

Scaling rules to define stress and strain.

For fibers and bundles in CP (Fig. 6), passive mechanical stress of the fiber and bundle were plotted against sarcomere length. Both relationships have been normalized to specimen cross-sectional area to calculate stress. Since the horizontal axis is sarcomere length, the stiffness or modulus is given in units of kPa/µm. But within the literature we find many definitions stress (e.g. N/cm², N/m², g/cm ²…) and the use of strain rather than sarcomere length to normalize these units. An additional factor affecting the measured stress is the assumed shape of the cross-sectional area; circular or oval, or at the whole muscle level the physiologic versus geometric cross-sectional area. These normalizations differences make it difficult to compare against other studies for the assumptions and units are not the same.

Definition of zero strain.

Note in Figure 5A, data obtained from the mouse EDL muscle (Meyer & Lieber, 2018), slack sarcomere length differed between the bundle and the fiber. For the fiber, slack length was 2.46 µm while for the bundle, slack length was only 2.21 µm. (We often note that bundle slack sarcomere length is shorter compared to fibers. This may be because the shortest fiber within the bundled determines its slack sarcomere length or perhaps the ECM has a resting strain shorter than most fibers.) Since the length at zero force is often defined as 0% strain, we replot these data as a stress strain curve, defining slack length at 0% strain (Fig. 5B). It is obvious from this simple example that the choice of “zero strain,” i.e., the length units used to compare samples, strongly affects the results. The fiber and bundle in Fig. 5B appear to have much more similar properties compared to the fiber and bundle in Fig. 5A even though they are precisely the same data. The fiber:bundle stiffness ratio (Fig. 5C) varies from about 0.34 to 0.53 while the stiffness ratio from the data in Fig. 5B ranges from 0.67 to 0.91, appearing more similar with ratios closer to one (rightward shifted in Fig. 5C).

Curve-fitting method used:

The last factor impacting the reporting of passive mechanics is the curve-fitting method used to define mechanics. Though an in-depth review of this is beyond the scope of this review, this is important to note because stiffness (or modulus) is the derivative of the stress-sarcomere length or stress-strain curve respectively. Hence, the equation used to fit that data will define the form of the stiffness relationship. For example, in the data of Fig. 5A, the points are equally well fit by an exponential function or a second order polynomial. However, the derivative of an exponential is an exponential while the derivative of a second order polynomial is a line. Thus, an investigator would reach two completely different conclusions about an experiment simply based on the form of the equation. The reader is referred to a recent review on this topic for a more detailed review of curve fitting for this application (Binder-Markey et al., 2021).

Implications for musculoskeletal modeling

The lack of a definitive passive muscle structure-function relationship presents tremendous challenges for those attempting to model human muscles. Most current modeling environments consist of generic active and passive length-tension relationships that are simply scaled to any particular muscle dimensions. While this may be acceptable for prediction of active muscle properties, as mentioned above, it is certainly not true for all muscles. “Fortunately,” there are so many unknowns in most of these models that passive tension is usually “adjusted” mathematically so that experimental and theoretical values of active and passive muscle force are as close as possible. In practice, it is also assumed that each muscle bears passive load in proportion to its physiological cross-sectional area and that “slack length” of the muscle defines the operating range of the muscle itself. While this is logical, there are almost no validation studies to confirm that such slack lengths actually correspond to any anatomical relationship possessed by the muscle. Stated another way, “slack length” in most models, optimizes the active force profile, but may have nothing to do with an actual length or zero tension level in a muscle-tendon unit. The problem is further confounded by the fact that passive mechanical properties of all muscles are assumed to scale identically, something for which we have already presented contradictory evidence, at least in rabbit muscle (Ward et al., 2020). We suggest that future studies are required to validate modeling assumptions and that new methods must be developed to study human muscle mechanics since extrapolating from muscle biopsy to whole muscle in terms of passive properties is no longer tenable. One reasonable possibility is that muscle models should at least increase passive tensile modulus as one proceeds from fiber to fascicle to muscle. The precise scaling factors are yet to be determined but a reasonable first approximation could be based on the data from rabbit study described above (Ward et al., 2020).

Conclusions

Passive structural and functional properties of whole muscles are not as well understood as active mechanical properties. This is based partly on the scaling problem (active stress scales with specimen size while passive stress does not). It is becoming increasingly clear that ECM bears a greater and greater fraction of the passive load at increasing muscle size scales within mammals. The structural basis for this observation is not yet known. Future studies might require the development of tools that probe muscle with much higher resolution at larger size scales than is currently available with ultrasound or even very high field strength magnetic resonance imaging (MRI). Also, the development of a standardized set of definitions for the reporting of passive mechanics will aid comparison across studies and advance the field. Improved understanding of the structural basis of passive muscle tension will impact the fields of muscle modeling, surgical reconstruction and may even apply to studies of other tissues that experience fibrosis.

Biographies

Rick Lieber is a physiologist who earned his Ph.D. in Biophysics from UC Davis developing a theory of light diffraction in single muscle cells. He joined the faculty at the UC San where he spent the first 30+ years of his career, achieving the rank of Professor and Vice-Chair of the Department of Orthopaedic Surgery. He received the M.B.A. in 2013 and is Chief Scientific Officer and Senior Vice President at the Shirley Ryan AbilityLab and Professor at Northwestern University.

Ben Binder-Markey is assistant professor in the Department of Physical Therapy and Rehabilitation Sciences and School of Biomedical Engineering, Science and Health Systems at Drexel University. He directs the Multiscale Neuromuscular Biomechanics Laboratory, integrating physical therapy with engineering principles to understand how muscle functions after injury or disease. He completed his DPT and PhD in biomedical engineering at Northwestern University, and bachelor’s in mechanical engineering at the University of Delaware.