Home » Week 6: Integrative Muscle Structure and Function » Lecture 13: Integrative Muscle Structure and Function 3 — Organ and Limb Level

Lecture 13: Integrative Muscle Structure and Function 3 — Organ and Limb Level

37 slides

Slide 1

Title slide for "Integrative muscle structure & function" by Dr. Monica A. Daley, Professor, Ecology and Evolutionary Biology, University of California, Irvine. Background collage shows diverse animals and humans exercising: a cyclist, water polo player, swimmer, sprinter, an oxygen cascade schematic, and a row of comparative species (sea turtle, fish, snake, hummingbird, kangaroo, horse, seal, lizard, croc).

  • Opens Lecture 13, the third in the muscle structure-and-function sequence — moving from the cellular (Lecture 11) and tissue (Lecture 12) scales up to the organ (muscle–tendon unit) and limb (skeletal lever) scales.
  • Today’s lecture begins with a brief recap of how the intrinsic Hill-type properties of muscle vary with fiber type, activation, and operating velocity, and how those tissue-level properties shape whole-body performance.
  • It then introduces the next layer of structure: muscle–tendon architecture (PCSA, fascicle length, pennation, tendon design) and skeletal lever systems (moment arms, torque balance, effective mechanical advantage).

Slide 2

Slide titled "Intrinsic properties vary with fiber type (e.g., myosin isoforms)" with two side-by-side plots from Bottinelli et al. 1996 of human skeletal muscle. Left: velocity (L/s, 0–2.0) vs. P_o (kN/m², 0–60) for type 1 (filled circles), type 2A (open circles), and type 2B (filled circles) fibers — type 2B fibers reach the highest velocities at any given force. Right: power (W L⁻¹, 0–4) vs. velocity (L s⁻¹, 0.00–1.25) for the three fiber types — type IIB (fast glycolytic) has the highest peak power (~3.5 W L⁻¹) at velocity ~0.25 L s⁻¹; type IIA (fast oxidative) is intermediate; type I (slow oxidative) has the lowest peak power at the slowest optimum velocity. Magenta dots mark each curve's peak. Color labels at right name each fiber type.

Recap — Hill-Type Properties Vary With Fiber Type

  • The intrinsic Hill-type properties (force–length, force–velocity, and power–velocity) define the maximum performance envelopes of muscle — and these envelopes shift systematically with fiber type.
  • Fiber-type variation is driven primarily by which myosin heavy chain isoform the fiber expresses, with parallel variation in other contractile proteins (troponin, tropomyosin, SERCA isoforms).
  • In human skeletal muscle (Bottinelli et al. 1996):
    • Type IIB (fast glycolytic): highest Vmax and highest peak power, at the highest optimum velocity.
    • Type IIA (fast oxidative): intermediate.
    • Type I (slow oxidative): lowest Vmax, lowest peak power, at the slowest optimum velocity.
  • All three curves share the same fundamental hyperbolic shape — only their scale and curvature differ. This is why fiber-type composition of a muscle is so important for sport-specific performance.

Slide 3

Slide titled "Muscle efficiency as a function of velocity" with two panels from Barclay, Woledge, Curtin (2010) J Physiol. Left: relative rate of energy output (enthalpy and power) vs. shortening velocity (V/V_max, 0.0–1.0) at 20°C, 25°C, and 30°C — both enthalpy and power increase with velocity but plateau at high velocity. Right: mechanical efficiency (0.0–0.6) vs. shortening velocity (V/V_max, 0.0–1.0) at the same three temperatures, with a magenta arrow pointing to peak efficiency at low velocity (~0.1–0.2 V/V_max). Annotation: "Peak efficiency at low velocity."

Recap — Muscle Efficiency Peaks at Low Velocity

  • Mechanical efficiency = mechanical work output ÷ total energy expenditure (work + heat).
  • Across temperatures (20–30°C in mouse soleus), efficiency:
    • Peaks at low shortening velocity (~0.1–0.2 V/Vmax).
    • Falls off at higher velocities as more energy is lost as heat during rapid cross-bridge cycling.
  • The velocity that maximizes power (~0.2–0.3 V/Vmax) is higher than the velocity that maximizes efficiency (~0.1–0.2 V/Vmax).
  • Animals (and athletes) face a fundamental trade-off between going fast (peak power) and using fuel economically (peak efficiency). The optimal contraction velocity depends on which currency the task is optimizing.

Slide 4

Slide titled "Intrinsic properties influence whole-body performance — trade-offs between sprint speed & power vs fatigue resistance" with two scatterplots from Vanhooydonck et al. 2014 Proc Roy Soc B comparing 17 lizard species. Top: log-transformed mass-specific muscle power vs sprint speed — positive correlation (annotation: "Muscle power is positively correlated with sprint speed by enabling faster acceleration"). Bottom: log mass-specific muscle power vs fatigue resistance — negative correlation (annotation: "Muscle power output is negatively correlated with muscle fatigue resistance"). A photo of a green lizard appears at lower right.

Tissue Properties Predict Whole-Body Performance — Lizards

  • Tissue-level intrinsic properties can directly predict whole-body locomotor performance — and the same trade-offs visible at the cellular level reappear at the organismal level.
  • Across 17 lizard species (Vanhooydonck et al. 2014):
    • Mass-specific muscle power is positively correlated with sprint speed — high-power muscle enables faster acceleration.
    • Mass-specific muscle power is negatively correlated with fatigue resistance — species with high power have less endurance.
  • This is the whole-organism manifestation of the cellular zero-sum game from Lecture 11 and the F–V trade-off from Lecture 12: each species sits at a different point on the same trade-off surface, set by its ecological niche.

Slide 5

Slide titled "Intrinsic properties influence whole-body performance" with a photograph of a wildebeest with a tracking collar in tall savanna grass. Bullet points: wildebeest travel 20–40 km between drinking events; average daily temperatures > 38°C (100°F) in 9 of 12 months. Two efficiency-vs-stimulation phase plots (Curtin et al. 2018 Nature) compare wildebeest red muscle (peak efficiency ~0.6) and cow red muscle (peak efficiency ~0.4). Annotation: "If wildebeest completed the same muscle work with the cow efficiency, water loss would be 50% greater." Citation: Curtin et al., 2018. Remarkable muscles, remarkable locomotion in desert-dwelling wildebeest. Nature.

Tissue Properties Predict Whole-Body Performance — Wildebeest Efficiency

  • A second example linking tissue-level efficiency to whole-body performance: the migrating wildebeest.
  • Wildebeest travel 20–40 km between drinking events in temperatures above 38°C (100°F) for 9 of 12 months — a punishing combination of locomotor and thermal demand.
  • Curtin et al. (2018) Nature: wildebeest red muscle achieves peak efficiency ~0.6 (60%) — roughly 50% higher than typical mammalian muscle (cow: ~0.4).
  • Why the very high efficiency matters: low-efficiency muscle generates more heat for the same mechanical work, and dissipating that heat costs water (sweat, panting). The authors estimate that if wildebeest used cow-like muscle efficiency, water loss during migration would be ~50% greater.
  • This is a clean example of how selection on muscle’s intrinsic properties can be driven by the integrated thermal and water-balance demands of an animal’s ecology.

Slide 6

Slide titled "Measuring muscle dynamics non-invasively in humans" with two panels (Bohm et al. 2021). Panel A (left): an ultrasound image of muscle fascicles plus a schematic of a person seated on a dynamometer that measures ankle torque while ultrasound tracks the soleus fascicle; below, a force–fascicle-length plot shows force (N, 1000–5000) vs. fascicle length (mm, 20–80) with a parabolic active-force curve. Panel B (right): a similar ultrasound image with a different muscle and dynamometer-attachment configuration; below, a force–fascicle-length plot shows force (N, 0–8000) vs. fascicle length (mm, 60–100). Citation: Bohm et al. 2021.

Measuring Intrinsic Properties Non-Invasively in Humans

  • For decades, the F–L and F–V curves in Lectures 12–13 came from isolated-muscle bench-top experiments (mostly on animals). Modern techniques now allow these curves to be measured non-invasively in humans.
  • Two methods are combined (Bohm et al. 2021):
    • Dynamometer: a rigid rig that fixes joint angle and measures the resulting joint torque while the subject contracts maximally — the human equivalent of the muscle ergometer.
    • B-mode ultrasound: a probe taped over the muscle images fascicle length and pennation angle in real time during the contraction.
  • Combining the two yields a direct fascicle-level F–L curve for individual human muscles. Different muscles (panel A vs. panel B) show different optimum lengths and force ranges.
  • This opens the door to measuring subject-specific intrinsic properties — important for sport science, rehabilitation, and personalized musculoskeletal modeling.

Slide 7

Slide reproducing key panels from "Speed-specific optimal contractile conditions of the human soleus muscle from slow to maximum running speed" (Bohm, Mersmann, Schroll, Arampatzis, 2023). Four panels in a 2x2 grid show measurements at multiple running speeds (3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 8.4 m/s, indicated by shades of gray): (top-left, A) F_norm (F/F_max, 0–1.0) vs. L_norm (L/L_0, 0.4–1.6) on the active F-L curve, with data points clustering at L/L_0 ~0.8–1.0; (top-right, A) Efficiency (0–0.5) vs. V_norm (V/V_max, 0–1.0), with most data points near V/V_max ~0.2 (the peak-efficiency region); (bottom-left, B) F_norm vs. V_norm on the F-V curve, showing operating points at progressively higher velocities at faster running speeds; (bottom-right, B) P_norm (P/P_max, 0–1.2) vs. V_norm, with the operating points at fastest running speeds approaching the optimum velocity (~0.3) for peak power.

Operating Range Shifts From Economy to Power With Speed

  • Bohm et al. 2023 combined dynamometer and ultrasound to measure how the human soleus operates on its own F–L–V envelope during running at speeds from 3.0 to 8.4 m/s.
  • Each set of points shows the soleus’s operating point at one running speed, plotted on the muscle’s own intrinsic curves:
    • Top-left (F–L plane): at all speeds, the soleus operates near L/L0 ≈ 0.8–1.0 — close to the plateau for maximum force.
    • Top-right (Efficiency–V): at slower running speeds, the soleus operates near V/Vmax ≈ 0.2 — the peak efficiency region.
    • Bottom plots (F–V and P–V): as running speed increases, the operating point shifts rightward along the F–V curve toward higher velocities, and along the power curve toward the optimum velocity for peak power (~0.3 V/Vmax).
  • The soleus is not stuck at one operating condition. As running speed increases, it shifts from the economy regime (low velocity, near-isometric) toward the power regime (intermediate velocity, near-peak power) — a real-time demonstration of the trade-offs visualized on the F–V curve.

Slide 8

Slide titled "Intrinsic properties influence whole-body performance" with a single line plot from Hori et al. 2021 BMC Sports Sci Med Rehabil. Y-axis: peak knee-extensor torque per body mass (N·m/kg, 0–5). X-axis: knee angular velocity (°/s) at three settings: 0 (ISO), 60, 180. Red circles with red line: sprinters (n = 58). Blue squares with blue dashed line: body-mass-matched non-sprinters (n = 40). At 0°/s (ISO) the two groups are similar (~4.0 N·m/kg). At 60°/s, sprinters maintain ~2.9 vs. non-sprinters ~2.6 N·m/kg. At 180°/s, sprinters reach ~2.0 vs. non-sprinters ~1.7 N·m/kg. Caption: "Knee extensor peak torque vs. angular velocity. Mean ± SD for sprinters (red, n = 58) and body-mass–matched non-sprinters (blue, n = 40). Group difference is non-significant in isometric conditions and grows with contraction velocity, consistent with a velocity-specific training adaptation."

Sprinters vs. Non-Sprinters — Velocity-Specific F–V Adaptation

  • A direct human comparison of intrinsic properties between elite sprinters (n = 58) and body-mass-matched non-sprinters (n = 40) (Hori et al. 2021):
    • At isometric conditions (0°/s), the two groups produce similar peak knee-extensor torque — maximum static strength is not the differentiator.
    • As contraction velocity rises (60°/s, 180°/s), sprinters maintain higher torque than non-sprinters — a velocity-specific advantage that grows with speed.
  • Interpretation: the difference between sprinters and non-sprinters is not raw force but the shape of the F–V curve — sprinters resist the velocity-dependent decline in force more effectively, consistent with a velocity-specific training adaptation (and/or a fast-fiber-type genetic predisposition).
  • This neatly closes the loop from cellular fiber type (Slide 2) → tissue-level F–V curve (Lecture 12) → comparative animal performance (Slides 4–5) → individual human performance differences.

Slide 9

Slide titled "Perspective on musculoskeletal modelling and predictive simulations of human movement to assess the neuromechanics of gait" by Friedl De Groote and Antoine Falisse. Left: a series of skeletal figures walking, with red muscles overlaid on the right limb. Right: a flowchart showing optimal-control loop with central nervous system (control policy) → muscle excitations → muscle dynamics → musculoskeletal geometry (joint torques) → skeleton dynamics → movement, with feedback. A circled inset highlights a Hill-type muscle model (CE + PE element). Annotation at bottom right shows the equation F_tot = F_FV × F_FL × F_act and a note: "'Hill-type' models: force-length, force-velocity & activation; widely used in simulations for rehabilitation, design of mobility assistance devices."

Why Hill-Type Properties Matter — Musculoskeletal Modeling

  • The intrinsic F–L and F–V properties are the foundation of musculoskeletal models used in clinical and research biomechanics.
  • A predictive simulation of human walking can be built from a subject’s motion-capture video alone, run through a forward model with Hill-type muscle elements:
\[F\_{tot} = F\_{FV} \times F\_{FL} \times F\_{act}\]
  • These models are used for rehabilitation planning, prosthetic and orthotic design, surgical planning (e.g., tendon transfers), and the design of mobility-assistance devices (exoskeletons).
  • Hill-type models capture F–L, F–V, and activation effects but do not capture all of the complexities of contraction (e.g., shape changes with activation level, history-dependent effects, muscle-tendon interaction). The next slide highlights one of those limitations directly.

Slide 10

Recap slide titled "Intrinsic properties vary with activation" with four panels from Holt and Azizi 2016 Proc Roy Soc B. Top-left (a): F/F_0max vs L/L_0max for activation levels 1.0, 0.68, 0.40, 0.20 — F-L curves shrink and shift to longer L_0 at lower activation, with a magenta arrow showing the rightward/downward shift. Top-right (a): power (W kg⁻¹) vs velocity (L_0max s⁻¹) for the same activation levels — peak power and optimum velocity both decline with activation. Bottom-left (b): L_0/L_0max vs activation level — optimum length increases at lower activation. Bottom-right (b): V_opt vs activation level — optimum velocity for max power decreases at lower activation. Annotations: "Longer optimum length at lower activation level" and "Slower optimum velocity for max power at lower activation level." Citation: Holt and Azizi 2016 Proc Roy Soc B.

Recap — Intrinsic Properties Vary With Activation

  • Hill-type properties are not fixed — they depend on activation level (Holt and Azizi 2016).
  • At lower activation levels (more typical of natural submaximal movements):
    • F–L curve shifts to the right → optimum length L0 shifts to longer fiber lengths.
    • Power–V curve shifts left and down → optimum velocity for peak power Vopt shifts to slower velocities.
  • This is a known limitation of standard Hill-type models (Slide 9), which assume activation only multiplicatively scales the curve. Building better models that capture activation-dependent shape changes is an active area of research — and important for predicting muscle function in real, dynamic tasks.

Slide 11

Slide titled "Types of muscle action during contraction" with three labeled cartoon panels showing a person performing a biceps curl: (1) Concentric (Shortening) — muscle contracts with force greater than resistance and shortens; positive work; most energetically expensive (most ATP for a given force); F-V relation note: higher velocity → higher volume of active muscle. (2) Eccentric (Lengthening) — muscle contracts with force less than resistance and lengthens; negative work; most economic (for a given force demand: lower volume of active muscle, lower ATP for a given force); most likely to cause injury. (3) Isometric — muscle contracts but does not change length; no work (force only, to resist load); steady-state cost proportional to force; in cyclic contractions, frequency contributes to cost.

Types of Muscle Action — Concentric, Eccentric, Isometric

Action Description Work Energetic cost
Concentric (shortening) Fmuscle > load; muscle shortens Positive Most expensive ATP per unit force; via the F–V relation, higher velocity demands a higher volume of active muscle
Eccentric (lengthening) Fmuscle < load; muscle lengthens Negative Most economic — for a given force demand, lower volume of active muscle and lower ATP per unit force; highest injury risk
Isometric No length change; force only, to resist load None Steady-state cost proportional to force; in cyclic contractions, frequency itself contributes to cost
  • These distinctions are foundational for the rest of the lecture: a single muscle will perform very different work depending on whether it acts concentrically, eccentrically, or isometrically during a given task.
  • The relevance to architecture (next slides): a muscle’s fiber length, pennation, and tendon design strongly influence how much active fiber volume is required to do the same external task — and therefore how much ATP it costs.

Slide 12

Slide titled "Structural organization of muscle" with the now-familiar hierarchical anatomical diagram (whole muscle → fascicle → muscle fiber → myofibril → sarcomere → actin/myosin filaments). Text on the right reads: "Today: Relationships between muscle-tendon architecture and mechanical function in locomotion. Levers and gearing: Interactions between muscles and the skeletal system. Highlight trade-offs between force and displacement at the level of muscle-tendon organ and limb function."

Today — From Tissue to Organ to Limb

  • The recap is over — today’s new content builds upward in scale from the tissue-level properties:
    • Organ-level: muscle–tendon architecture (fascicle length, pennation angle, PCSA, tendon length and stiffness) and how it sets force, displacement, velocity, work, and power capacity at the whole muscle–tendon unit (MTU) level.
    • Limb-level: the lever and gearing systems through which muscles act on bones and joints.
  • Central theme: a trade-off between force and displacement appears at each level — independent of, but compounding with, the tissue-level trade-offs.

Slide 13

Slide titled "Integrative muscle structure & function" listing three learning objectives: (1) Relate muscle function to morphology — fascicle length, pennation angle, physiological cross-sectional area, and tendon length relative to fascicle length; (2) Use the lever system equation to relate muscle force demands to external loads; (3) Discuss changes in mechanical advantage with body size across diverse vertebrates.

Learning Objectives

  1. Relate muscle function to morphology: fascicle length, pennation angle, physiological cross-sectional area (PCSA), and tendon length relative to fascicle length.
  2. Use the lever-system equation to relate muscle-force demands to externally applied loads at a joint.
  3. Discuss how effective mechanical advantage (EMA) scales with body size across diverse vertebrates.

Slide 14

Slide titled "How does anatomy influence muscle function?" showing three labeled anatomical drawings of muscles with different fiber arrangements relative to the line of action (ML = muscle length; FL = fascicle length): Left — biceps brachii, with fibers running parallel to the long axis (longitudinal/parallel-fibered, ML = FL). Middle — vastus lateralis, with fibers acting at an angle to the tendon (pennate). Right — gluteus medius, with a fan-like multipennate arrangement.

Three Architectural Categories

  • Skeletal muscles fall on a continuum of architectural designs, illustrated by three textbook examples:
    • Longitudinal (parallel-fibered) — e.g., biceps brachii: fibers run along the line of action; muscle length ≈ fascicle length (ML = FL).
    • Unipennate / pennate — e.g., vastus lateralis: fibers act at an angle (the pennation angle) to the line of action; muscle length > fascicle length.
    • Multipennate — e.g., gluteus medius: multiple internal tendons with fibers fanning at multiple angles; cannot be characterized by a single pennation angle.
  • These categories influence the muscle’s force, displacement, velocity, work, and power capacities for a given volume of muscle.

Slide 15

Slide titled "How does anatomy influence muscle function?" with the same vastus lateralis pennate diagram and four bullet points: Force capacity of muscle per unit area (specific tension) ~ 18–30 N/cm²; highly conserved across vertebrates; varies with muscle fiber type; a typical value of ~25 N/cm² can be used to estimate force capacity (or, if known, the precise specific tension can be used).

Specific Tension — A Conserved Tissue Property

  • Specific tension (force per unit cross-sectional area of contractile tissue) is highly conserved across vertebrates:
\[\sigma\_{specific} \approx 18\text{–}30 \text{ N/cm}^2\]
  • The range reflects fiber-type variation:
    • Anaerobic fast-twitch fibers (high myofibril fraction) → higher specific tension.
    • Aerobic / slow-twitch fibers → lower specific tension.
  • This conservation lets researchers scale forces between species and estimate whole-muscle force from anatomical measurements.
  • A typical value of ~25 N/cm² is used as an estimate when the precise specific tension is not known.

Slide 16

Slide titled "How does anatomy influence muscle function?" listing five architectural relationships (with the same vastus lateralis pennate diagram): Maximum force ∝ number of sarcomeres in parallel ∝ cross-sectional area; Maximum displacement ∝ number of sarcomeres in series ∝ fiber length; Maximum velocity ∝ number of sarcomeres in series ∝ fiber length; Maximum work = force × displacement ∝ volume & mass; Maximum power = force × velocity ∝ volume & mass.

Architectural Relationships — Force, Displacement, Velocity, Work, Power

  • Maximum force ∝ number of sarcomeres in parallel ∝ cross-sectional area.
  • Maximum displacement ∝ number of sarcomeres in series ∝ fiber length.
  • Maximum velocity ∝ number of sarcomeres in series ∝ fiber length.
  • Maximum work = force × displacement ∝ volume ∝ mass.
  • Maximum power = force × velocity ∝ volume ∝ mass.
  • A clean way to remember the architecture rules: the muscle’s cross-sectional area sets force; its fiber length sets displacement and velocity; its volume (and therefore mass) sets work and power.
  • This is also why muscle mass is so often used as a proxy for power capacity in comparative studies — muscle has a highly conserved density (~1.06 g/cm³), so mass and volume are tightly coupled.

Slide 17

Slide titled "How does anatomy influence muscle function?" with two schematic diagrams illustrating the calculation of physiological cross-sectional area (PCSA). Top: a parallel-fibered muscle with a transverse cut (purple line) perpendicular to the fibers — annotation reads "Physiological cross-sectional area (PCSA): cross-section perpendicular to the fibers." Bottom: a unipennate muscle (feather-like arrangement of fibers off a central tendon). Right side shows the equation: PCSA = Volume / fiber length.

Physiological Cross-Sectional Area (PCSA)

  • For a parallel-fibered muscle, PCSA is just a transverse cut perpendicular to the fibers.
  • For a pennate or multipennate muscle, the fibers cross the muscle belly at angles — a single transverse cut does not give the correct cross-section.
  • The general formula avoids the geometry problem entirely:
\[\text{PCSA} = \frac{\text{Volume}}{\text{fiber length}}\]
  • Volume can be measured by:
    • Mass × density in dissection (muscle density is highly conserved at ~1.06 g/cm³).
    • Imaging (MRI, CT, ultrasound) in living human studies.
  • Fiber length is measured as the average length of dissected fascicles (or estimated from imaging).

Slide 18

Slide showing three schematic muscle architectures labeled A, B, and C with red lines representing fibers, blue lines showing transverse cross-section perpendicular to the muscle long axis, and green lines showing physiological cross-section perpendicular to the fiber axis. A: a parallel-fibered muscle (fibers along the long axis); B: a single bipennate muscle; C: a multi-compartment multipennate muscle with several internal tendons.

Anatomical vs. Physiological Cross-Sections

  • The blue lines show a transverse section through the muscle belly perpendicular to the long axis of the muscle — what an anatomist might cut in dissection.
  • The green lines show the actual physiological cross-section perpendicular to the fiber axis — the cross-section that the active sarcomeres present.
  • For complex muscles like the soleus (which internally resembles option C — many compartments with short fibers on internal tendons), there is no simple cut that yields the PCSA.
  • In these cases, only the Volume / fiber length formula is practical.

Slide 19

Slide titled "How does anatomy influence muscle function?" with a schematic of a long parallel-fibered muscle (green lines, top) and a short-fibered pennate muscle (cyan, bottom) on the left. A force–length plot on the right shows muscle force (N) vs. muscle length (cm, 5–25): the blue curve (short fibers, large PCSA) has a high, narrow peak (~100 N at length ~7 cm) that drops off rapidly; the green curve (long fibers, small PCSA) has a lower peak (~50 N) but spans a much wider range of lengths (5–20 cm).

Architectural Force–Length Trade-off

  • Two hypothetical muscles with the same volume but different architectures:
    • Short fibers, large PCSA (e.g., a bipennate calf-like muscle) → high peak force but a narrow operating range.
    • Long fibers, small PCSA (e.g., a parallel-fibered biceps-like muscle) → lower peak force but a wider operating range.
  • This is a whole muscle–tendon unit (MTU) level force–length trade-off that adds to the tissue-level force–length curve.
  • High-PCSA muscles are best for brief high-force tasks within a narrow range; long-fibered muscles are best for large excursions at moderate force.

Slide 20

Slide titled "How does anatomy influence muscle function?" with the same architectural diagrams on the left and a force–velocity plot on the right showing muscle force (N) vs. muscle velocity (cm/s, 5–25): the blue curve (short fibers, large PCSA) starts at ~100 N at low velocity and falls steeply, reaching zero at ~14 cm/s; the green curve (long fibers, small PCSA) starts at ~50 N and falls more gradually, reaching zero at ~20 cm/s. A highlighted box reads: "Trade-off between force and displacement in muscle architecture."

Architectural Force–Velocity Trade-off

  • The same architectural trade-off appears in the F–V relationship at the MTU level:
    • Short fibers, large PCSA → higher Fmax but lower Vmax (steeper drop with velocity).
    • Long fibers, small PCSA → lower Fmax but higher Vmax (more gradual drop).
  • This trade-off is independent of fiber type — it arises purely from muscle architecture and applies even when both muscles have identical tissue-level intrinsic properties.
  • The whole-muscle F–V curve combines fiber-type effects (Slide 2) with architectural effects (this slide) into a single envelope.

Slide 21

Slide titled "Design of tendon relative to muscle belly" with a schematic of a pennate muscle showing fibers, aponeurosis (internal tendon, on the muscle belly), and external tendon, with pennation angle α. Header reads "Tendon slack length (L_T): optimal muscle fiber length (L_o)." Two labeled blocks below: HIGH ratio — high tendon stretch, muscle shortens against tendon, good for economic force and elastic energy cycling, bad for position control and range of motion. LOW ratio — low tendon stretch, muscle shortening rotates the joint, good for position control and range of motion, bad for economic force and elastic energy cycling.

Tendon Slack Length to Optimal Fiber Length Ratio (LT / Lo)

  • A second architectural parameter: the ratio of tendon slack length (LT, including both aponeurosis and external tendon) to optimal muscle fiber length (Lo).
  • High LT/Lo ratio (long tendon, short fibers):
    • The tendon stretches a lot under load → muscle effectively shortens against the tendon.
    • Good for economic force and elastic energy cycling.
    • Bad for joint position control and range of motion.
  • Low LT/Lo ratio (short tendon, long fibers):
    • Muscle shortening directly rotates the joint.
    • Good for position control and range of motion.
    • Bad for elastic energy storage and economic force.

Slide 22

Slide titled "Design of tendon relative to muscle belly" with the same pennate muscle schematic and a comparative table of tendon-length to muscle-fiber-length ratios across species (human, cat, guinea fowl, wallaby, mallard, turkey). Soleus: 11 (human), 2 (cat). Gastrocnemius: 9 (human), 5 (cat), 11 (guinea fowl), 4 (mallard), 3 (turkey, unossified portion). Plantaris: 15 (wallaby). Quads (vasti): 3 (human), 3 (cat). Hamstrings (semitendinosus): 2 (human), 1 (cat). Hip uniarticular muscles: 0.2 (human), 1 (cat). A second figure on the right shows the muscle-tendon unit anatomy with overall length D, fiber length L, and free tendon length l_s.

Comparative LT/Lo Ratios — Distal vs. Proximal

Muscle Human Cat Guinea fowl Wallaby Mallard Turkey
Soleus 11 2 — — — —
Gastrocnemius 9 5 11 — 4 3
Plantaris — — — 15 — —
Quads (vasti) 3 3 — — — —
Hamstrings (semitendinosus) 2 1 — — — —
Hip uniarticular muscles 0.2 1 — — — —
  • A clear pattern: distal limb muscles (soleus, gastrocnemius, plantaris) have very high LT/Lo ratios (~5–15) — long tendons relative to short fibers, optimized for elastic energy cycling.
  • Proximal hip muscles have low ratios (~0.2–3) — short tendons, longer fibers, optimized for range of motion, work, and power.
  • Muscles typically only shorten by ~25% of their length in a single contraction. Combined with high LT/Lo, this means most of the MTU length change comes from tendon stretch, not fiber shortening.

Slide 23

Slide titled "Design of tendon relative to muscle belly" with the same muscle-tendon schematic. Header: "Tendon cross-sectional area (A_T): muscle physiological cross-sectional area (PCSA)." HIGH A_T/PCSA: 'stiffer' tendon, lower muscle shortening against tendon and lower elastic energy cycling, good for position control, muscle directly rotates joint, bad for economy. LOW A_T/PCSA: 'compliant' tendon, higher muscle shortening against tendon, higher elastic energy cycling, bad for position control, good for economy.

Tendon Cross-Sectional Area to PCSA Ratio (AT / PCSA)

  • A third architectural parameter: the ratio of tendon cross-sectional area to muscle PCSA. This sets how stiff the tendon is relative to the force the muscle can generate.
  • High AT/PCSA → stiff tendon:
    • Lower muscle shortening against the tendon, lower elastic energy cycling at a given force.
    • Good for position control — the muscle directly rotates the joint.
    • Bad for economy.
  • Low AT/PCSA → compliant tendon:
    • Higher muscle shortening against the tendon, higher elastic energy cycling at a given force.
    • Bad for position control.
    • Good for economy.
  • Combined with the LT/Lo ratio (Slide 21), AT/PCSA defines whether the MTU behaves as a spring or a direct-drive actuator.

Slide 24

Slide titled "Muscles act via bones and joints" with two photographs of vertebrate skeletons: a large elephant skeleton (left) with relatively straight, columnar limbs and an upright "graviportal" posture; a small shrew skeleton (right) with a crouched, flexed posture and visibly bent joints.

Why Skeletal Lever Systems Matter — Body Size

  • Muscles do not act in isolation: they pull on bones that rotate around joints. The skeleton is the lever system through which muscle force becomes movement.
  • Comparative observation: small animals (shrews, mice) have a crouched posture; large animals (elephants, rhinos) have a straight-legged, columnar (graviportal) posture.
  • Understanding lever systems is necessary to explain why this posture shift exists — and why it has implications for muscle force, work, and energetic cost.

Slide 25

Slide titled "Lever systems — trade-off between force and displacement" with a simple seesaw schematic: a horizontal bar balanced on a triangular fulcrum, with two equal-sized purple spheres labeled "Load (force)" sitting on the lever arms on either side. Both lever arms are labeled "lever arm" and the support point is labeled "fulcrum."

Lever Systems — The Basics

  • A lever is a rigid bar that rotates around a fulcrum (the joint).
  • Each load applies a force at some distance from the fulcrum — the lever arm (or moment arm).
  • A lever can be balanced when the torques on both sides are equal — and the trade-off between force and displacement is built directly into the geometry.

Slide 26

Slide titled "Lever systems — trade-off between force and displacement" with a tilted lever: a small purple ball sits at the end of a long left lever arm raised above the fulcrum; a much larger purple ball sits at the end of a shorter right lever arm pulled down to the ground. The lever is rotated so the larger mass has tipped its side down.

Imbalanced Loads — Lever Rotates Toward the Heavier Side

  • When the torques on the two sides of a lever are not equal, the lever rotates toward the heavier side.
  • A familiar example: on a playground seesaw, a heavier child sinks the seesaw toward themself.
  • To rebalance the lever without removing the heavier load, you have to change the moment arm rather than the force itself.

Slide 27

Slide titled "Lever systems — trade-off between force and displacement" showing a small ball at the end of a long lever arm (left) balanced against a much larger ball placed close to the fulcrum (right) — the lever is horizontal, indicating balance.

Balancing By Adjusting the Moment Arms

  • The larger force can be balanced if it sits closer to the fulcrum — i.e., with a shorter moment arm.
  • This is the basic intuition of a lever: a small force at a long lever arm can balance a large force at a short lever arm.
  • The trade-off is between force and displacement: a small force traveling a large distance does the same total work as a large force traveling a small distance. Levers convert one currency to the other.

Slide 28

Slide titled "Lever systems" with a 3D wooden-plank-on-fulcrum schematic showing a 100 kg ball on a short left arm and a 5 kg ball on a longer right arm. Equations on the left: T_1 = T_2; T_1 = F_1 D_1; T_2 = F_2 D_2; F_1 D_1 = F_2 D_2. On the right: F = mass × gravity, where gravity ≈ 10 m/s². Worked calculations: T_1 = F_1 D_1 = (100 kg × 10 m/s²) × 0.01 m = 10 Nm; T_2 = F_2 D_2 = (5 kg × 10 m/s²) × 0.20 m = 10 Nm.

Torque Balance — Worked Example

  • Torque ($T$) = the rotational effect of a force around a fulcrum:
\[T = F \times D\]
  • where $F$ is the force and $D$ is the perpendicular distance from the line of action of the force to the fulcrum.
  • For a balanced lever:
\[T_1 = T_2 \quad \Rightarrow \quad F_1 D_1 = F_2 D_2\]
  • Worked example with g ≈ 10 m/s² for simplicity:
    • 100 kg side: $T_1 = (100 \text{ kg} \times 10 \text{ m/s}^2) \times 0.01 \text{ m} = 10 \text{ N·m}$.
    • 5 kg side: $T_2 = (5 \text{ kg} \times 10 \text{ m/s}^2) \times 0.20 \text{ m} = 10 \text{ N·m}$.
  • Confirms balance: the 100 kg mass at 0.01 m balances the 5 kg mass at 0.20 m — short moment arm × large force = long moment arm × small force.
  • Lever systems are passive — work in equals work out. They convert force to displacement (or vice versa) without adding or subtracting energy.

Slide 29

Slide titled "Lever systems — Muscle moment arms (leverage)" with an anatomical drawing of a flexed elbow holding a 7 kg dumbbell at the wrist: the bicep insertion is at distance D_1 = 5 cm from the elbow joint center; the dumbbell is at D_2 = 25 cm. Equations: F_bicep × D_1 = F_weight × D_2 → F_bicep = F_weight × D_2/D_1. Worked example: F_weight = M × g = 7 kg × 10 m/s⁻² = 70 N; F_bicep = 70 N × 25/5 = 70 N × 5 = 350 N. Below: "Muscle forces required are often much larger than the external load! The ratio of the moment arms is the force multiplier."

Applied — The Biceps Curl

  • Biceps moment arm D1 ≈ 5 cm (insertion to elbow joint center).
  • External-load moment arm D2 ≈ 25 cm (dumbbell to elbow joint center).
  • For a 7 kg dumbbell:
\[F\_{weight} = 7 \text{ kg} \times 10 \text{ m/s}^2 = 70 \text{ N}\] \[F\_{bicep} = F\_{weight} \times \frac{D_2}{D_1} = 70 \text{ N} \times \frac{25}{5} = 350 \text{ N}\]
  • Muscle forces required to resist external loads are often many times larger than the external load itself — because muscles typically insert close to the joint while loads act far from the joint. The ratio of moment arms is the force multiplier.
  • This force amplification is paid for by a corresponding displacement amplification at the load — small muscle shortening produces large hand motion.

Slide 30

Slide reproducing the title page of "Scaling Body Support in Mammals: Limb Posture and Muscle Mechanics" by Andrew A. Biewener, Science (1989) Vol. 245, pp. 45–48. Below the citation are three skeletal hindlimb diagrams labeled T₁, T₂, T₃ showing different limb postures (crouched, upright, and intermediate) with the ground reaction force vector G drawn from the toe to the limb. Caption: "Relate ground forces to limb posture to determine the required muscle force and work to stand and walk."

Biewener (1989) — Scaling of Body Support

  • Classic comparative paper applying the lever-system concept across body sizes.
  • In standing or walking, the ground reaction force (GRF) must support body weight. This GRF acts at some perpendicular distance from each joint center — the external moment arm — which depends on limb posture.
  • The same GRF produces different muscle-force demands depending on whether the limb is crouched (long external moment arm) or upright (short external moment arm).

Slide 31

Slide reproducing the Biewener 1989 lever schematic: a hindlimb skeleton with the muscle (red, hatched) acting at moment arm r from the joint, and the ground reaction force F_g acting upward at perpendicular distance R from the joint. Annotations: "Calculate muscle force required to resist the ground reaction force (F_g); F_g must equate mass × gravity when averaged over a full stride cycle."

The Biewener Limb Lever System

  • Fm = muscle force, acting at moment arm r (the muscle’s insertion-to-joint distance — primarily set by skeletal morphology).
  • Fg = ground reaction force, acting at moment arm R (the perpendicular distance from the GRF vector to the joint center — primarily set by limb posture).
  • Fg averaged over a full stride must equal body weight (mass × gravity) for an animal moving at steady speed.

Slide 32

Slide reproducing the same Biewener schematic with three rearranged equations: F_muscle × r = F_g × R; F_muscle = F_g × R/r; F_g = F_muscle × r/R. A highlighted box defines: "Effective Mechanical Advantage (EMA): r/R." Annotation: "As R decreases, EMA increases, resulting in lower muscle force required to support F_g."

Effective Mechanical Advantage (EMA)

  • Torque balance at the joint:
\[F\_{muscle} \times r = F_g \times R\]
  • Solving for muscle force:
\[F\_{muscle} = F_g \times \frac{R}{r}\]
  • The ratio r / R is the effective mechanical advantage (EMA):
\[\text{EMA} = \frac{r}{R}\]
  • Higher EMA → lower muscle force required to support body weight.
  • A straight (upright) limb posture keeps the GRF vector closer to the joint center → smaller R → higher EMA.
  • This is why a straight-legged stance is energetically cheap — try standing with bent knees and you can feel the muscular effort rise immediately.

Slide 33

Slide reproducing the Biewener 1989 figure: a hindlimb schematic on the left showing F_m, F_g, r, and R, with EMA = r/R defined; a scatter plot on the right showing effective mechanical advantage (r/R, 0.0–2.0) vs. forward velocity (m/s, 0–8) for three species: horse (filled and open circles, EMA ~1.0), dog (filled and open triangles, EMA ~0.4–0.5), and ground squirrel (filled and open squares, EMA ~0.15–0.2). EMA is roughly flat with velocity within each species but offset by body size.

EMA Across Species — Larger Animals Have Higher EMA

  • Across species (horse, dog, ground squirrel):
    • EMA does not change much with running velocity within a species.
    • EMA increases dramatically with body size: horse ~1.0, dog ~0.4–0.5, ground squirrel ~0.15–0.2.
  • This means larger animals can produce body-weight-supporting forces with less mass-specific muscle force — they exploit a more upright posture to make their lever systems mechanically efficient.

Slide 34

Slide titled "Shift in leg posture with body size" with a striking photograph (top left) showing the foot of an elephant next to a small mouse. Below: an annotated F_m / F_g / r / R diagram, and a scatter plot on the right showing log–log effective mechanical advantage (EMA = r/R) vs. body mass (kg, 0.01–1000) for many species (mouse, chipmunk, squirrel, prairie dog, agouti, goat, deer, dog, horse — plus humans for walk/run). The plot shows a positive scaling: EMA increases from ~0.1 (mouse, chipmunk) up to ~1.0 (horse). Annotation across the top reads "Posture Shift" with red brackets at each end. Caption text below: "EMA increases with body size; allows muscle forces to scale similar to bone and muscle cross-sectional area."

Scaling of EMA With Body Size — Mammals

  • EMA scales positively with body mass across mammals:
    • Small animals (mouse, ~30 g): EMA ~0.1.
    • Large animals (horse, ~500 kg): EMA ~1.0.
  • The mechanism is a postural shift: larger animals adopt straighter limbs, which reduces R relative to r.
  • This scaling allows muscle and bone forces to scale similarly to muscle and bone cross-sectional area — preventing large animals from breaking under their own weight (or needing impossibly disproportionate muscle masses).
  • Humans are slightly off the mammalian regression: walking has a relatively high EMA (straight knee), but running EMA is lower (more flexed knee).

Slide 35

Slide titled "Shift in leg posture with body size" with two scaling plots (Daley and Birn-Jeffery 2018) for 23 bird species across multiple orders (Anseriformes, Cariamiformes, Charadriiformes, Ciconiiformes, Galliformes, Passeriformes, Ratites, Tinaniformes). Top: log hip height (m, 0.1–1) vs. log body mass (kg, 0.1–100), positive slope, ostriches at top right and small shorebirds and quail at lower left. Bottom: posture index (H / Σ L_seg) vs. log body mass — posture becomes more upright (higher index) at larger sizes, but with high diversity at intermediate (~1 kg) sizes. Annotations: "Log plots in scaling studies reveal 'high level' trends. High diversity in leg morphology and posture at a given size, particularly at intermediate body size."

The Same Pattern in Birds — With High Within-Size Diversity

  • Comparative study of 23 bird species (Daley and Birn-Jeffery 2018) shows the same trend of more upright posture with larger body size — the EMA scaling is not unique to mammals.
  • Caveats to log-log scaling plots:
    • They reveal high-level trends but average over substantial within-size diversity.
    • At intermediate body sizes (~1 kg), the same body mass can have very different limb postures depending on locomotor ecology (e.g., ground-foragers vs. take-off-flight specialists).
  • The scaling rules apply broadly, but selection for specific behaviors introduces species-specific deviations.

Slide 36

Slide titled "Variation in ankle function & human running economy" reproducing key panels from Scholz et al., "Running biomechanics: shorter heels, better economy." Top left: schematic of measuring foot anatomy — photographs of a foot from lateral and medial sides aligned with a reference block, with the moment arm calculated as the average of the perpendicular distances from the malleoli to the Achilles tendon. Bottom left: equation F_muscle = F_g × R/r, with annotations: "For smaller r, muscle-tendon force is higher; tendon energy storage increases with muscle force; as r decreases, tendon energy cycling increases. Improves running economy by minimizing muscle work." Right: scatter plot of V̇O₂ at 16 km/h (ml kg⁻¹ min⁻¹, 30–60) vs. ankle moment arm (cm, 4–5.5) showing a positive relationship — runners with shorter ankle moment arms had lower VO₂ (better economy).

Ankle Moment Arm and Human Running Economy

  • Within humans (Scholz et al.), individual variation in ankle moment arm (Achilles tendon to ankle joint center) significantly predicts running economy:
    • Shorter ankle moment arm → better running economy (lower V̇O2 at 16 km/h).
  • Mechanism — counterintuitive at first:
    • From the lever equation: smaller r → higher Fmuscle for a given Fg.
    • But higher muscle–tendon force increases tendon strain energy storage in the Achilles tendon.
    • The tendon’s elastic energy cycling can do work that the muscle would otherwise have to do — minimizing muscle work and improving economy.
  • This is a preview of the integrative theme of Lecture 14: muscle, tendon, lever, and limb posture all act together to determine in vivo muscle function.

Slide 37

Closing recap slide titled "Integrative muscle structure & function" repeating the three learning objectives from Slide 13: (1) Relate muscle function to morphology — fascicle length, pennation angle, physiological cross-sectional area, and tendon length relative to fascicle length; (2) Use the lever system equation to relate muscle force demands to external loads; (3) Discuss changes in mechanical advantage with body size across diverse vertebrates.

Learning Objectives — Recap

  1. Muscle morphology determines mechanical function: PCSA = volume / fiber length sets force capacity; fiber length sets displacement and velocity capacity; volume (mass) sets work and power. Tendon design (LT/Lo and AT/PCSA ratios) determines whether the MTU acts as a spring (economy, elastic cycling) or a direct-drive actuator (position control, range of motion).
  2. Lever-system equation: $F_{muscle} = F_g \times R/r$. Muscle forces required to support external loads are typically many times larger than the external loads themselves, because muscles insert close to the joint while loads act far from it.
  3. EMA scales positively with body mass across mammals and birds (mouse → horse). Larger animals adopt more upright postures — reducing R relative to r — so that mass-specific muscle and bone forces stay roughly constant despite increasing body size.

Key Equations

Equation Name Description
$\sigma_{specific} = F / \text{PCSA} \approx 18\text{–}30 \text{ N/cm}^2$ Specific tension Force per unit physiological cross-sectional area; highly conserved across vertebrates and varies modestly with fiber type. A typical value of ~25 N/cm² is used as an estimate.
$\text{PCSA} = \dfrac{\text{Volume}}{\text{fiber length}}$ Physiological cross-sectional area The cross-sectional area of muscle perpendicular to the fibers; used to compute maximum force capacity. Volume is obtained from mass × density (~1.06 g/cm³) or imaging.
$F_{max} \propto \text{PCSA}$ Force capacity Maximum force is proportional to the number of sarcomeres in parallel — i.e., the PCSA.
$\text{Displacement}, V_{max} \propto L_{fiber}$ Displacement and velocity capacity Maximum shortening distance and maximum shortening velocity are proportional to the number of sarcomeres in series — i.e., the fiber length.
$\text{Work}, P \propto \text{Volume} = \text{PCSA} \times L_{fiber}$ Work and power capacity Maximum work and power are proportional to the muscle’s volume (or, equivalently, mass via the conserved muscle density).
$T = F \times D$ Torque The rotational effect of a force around a fulcrum, where $D$ is the perpendicular distance from the force’s line of action to the fulcrum.
$F_1 D_1 = F_2 D_2$ Lever balance Torques on either side of a fulcrum balance at equilibrium.
$F_{muscle} = F_g \times \dfrac{R}{r}$ Limb lever equation Muscle force required to resist the ground reaction force, where r is the muscle’s moment arm (skeletal morphology) and R is the GRF moment arm (set by posture).
$\text{EMA} = \dfrac{r}{R}$ Effective mechanical advantage Ratio of muscle moment arm to GRF moment arm. Higher EMA → lower required muscle force per unit body weight; scales positively with body mass across vertebrates.

Glossary of Key Terms

Term Definition
Hill-type properties The intrinsic isometric force–length and isotonic force–velocity relationships of muscle, named after A.V. Hill, who developed the experimental methods to measure them.
Musculoskeletal model A computational model of the body that combines bones, joints, and Hill-type muscle elements to predict in vivo muscle force from observed motion (e.g., OpenSim).
B-mode ultrasound (in muscle) Non-invasive imaging method used in human studies to measure fascicle length and pennation angle in real time during contraction.
Dynamometer A rigid device that fixes joint angle and measures resulting joint torque during a maximal contraction; the human equivalent of a muscle ergometer.
Fiber arrangement The geometric organization of muscle fibers within a muscle belly: parallel (longitudinal), pennate (unipennate, bipennate), or multipennate.
Pennation angle The angle between the muscle fibers and the line of action of the muscle–tendon unit. Increasing pennation packs more fibers per unit volume.
Fascicle / fiber length (Lfiber) The length of a muscle fascicle (a bundle of fibers). Determines the displacement and velocity capacity of the muscle.
Physiological cross-sectional area (PCSA) The cross-section of muscle perpendicular to the fibers, computed as volume / fiber length. Determines the muscle’s maximum force capacity.
Specific tension (σ) Force per unit physiological cross-sectional area; ~18–30 N/cm² across vertebrate skeletal muscle, highly conserved with modest fiber-type variation.
Aponeurosis The flat, broad internal tendon that lies on (or within) the muscle belly and connects to the external tendon. Part of total tendon length LT.
External tendon The free portion of the tendon that connects the muscle to bone (e.g., the visible Achilles tendon).
Tendon slack length (LT) The length of the tendon (aponeurosis + external) at zero force; one of the key architectural parameters.
LT/Lo ratio Ratio of tendon slack length to optimal muscle fiber length. High (long tendon, short fibers) → economic force, elastic energy cycling. Low (short tendon, long fibers) → range of motion and position control.
AT/PCSA ratio Ratio of tendon cross-sectional area to muscle PCSA; sets the stiffness of the tendon relative to the force the muscle can apply. Low AT/PCSA → compliant tendon, high elastic energy cycling.
Compliant tendon A tendon that stretches substantially under physiological loads; stores and returns elastic strain energy. Low AT/PCSA.
Stiff tendon A tendon that stretches little under load; transmits force directly to rotate the joint. High AT/PCSA.
Concentric contraction Shortening contraction (Fmuscle > load); positive work; via the F–V relation, higher velocity demands a higher volume of active muscle.
Eccentric contraction Lengthening contraction (Fmuscle < load); negative work; lower volume of active muscle and lower ATP per unit force, but highest injury risk.
Isometric contraction No length change at the joint level; force only, to resist the load; cost is steady-state proportional to force, with frequency contributing in cyclic contractions.
Lever system A rigid bar (bone) that rotates around a fulcrum (joint), with forces applied at perpendicular distances called moment arms.
Fulcrum The pivot point of a lever — at the joint center of rotation in musculoskeletal systems.
Moment arm (lever arm) The perpendicular distance from the line of action of a force to the fulcrum; determines that force’s torque.
Torque (T) Rotational effect of a force, $T = F \times D$. Measured in newton-meters (N·m).
Muscle moment arm (r) The perpendicular distance from the muscle’s line of action to the joint center; determined primarily by skeletal morphology (e.g., calcaneal length for the Achilles tendon at the ankle).
External (GRF) moment arm (R) The perpendicular distance from the ground reaction force vector to the joint center; determined primarily by limb posture.
Effective mechanical advantage (EMA) The ratio r/R. Higher EMA means lower required muscle force to support a given GRF. Scales positively with body mass across mammals and birds.
Inverse dynamics A technique for inferring muscle and joint forces from external measurements of motion (motion capture or high-speed video) and ground reaction force. Requires lever-system analysis at each joint.
Graviportal posture The straight-legged, columnar limb posture seen in very large animals (elephants, rhinos), maximizing EMA to keep muscle forces tractable despite large body weight.
Crouched posture The flexed-limb posture seen in small animals (mice, shrews) and in humans during deep knee bends; lower EMA, higher muscle-force demand.