Unit 4: Mechanics of skeletal muscle

Skeletal muscle is the biological actuator that converts chemical energy (ATP hydrolysis) into mechanical force and movement. In biomechanics it is treated as an active, force-generating material whose output depends on its internal structure, its neural activation, and its instantaneous length and velocity. This unit builds from anatomy up to lumped mechanical models.

• Function studied: how muscle produces tension, transmits it through tendon to bone, and moves body segments about joints.
• Governing hierarchy: muscle → fascicle → fiber → myofibril → sarcomere → myofilaments (actin, myosin).
• Contractile unit: the sarcomere, the repeating segment between two Z-lines (~2.0–2.5 μm at rest).
• Force convention: muscle is active only in tension (it pulls, never pushes); measured in newtons (N), stress in N/m².
• Key coupling: excitation (neural) → contraction (mechanical), mediated by Ca²⁺ release.

II. Structure of Skeletal Muscle

The gross-to-microscopic architecture that channels fiber force to bone.

Skeletal muscle is a hierarchically wrapped bundle of connective tissue and contractile cells, organized so that force generated at the molecular level sums along a mechanical path to the tendon.

A. Structure of skeletal muscle

The whole muscle is a layered composite of connective-tissue sheaths enclosing contractile cells.

• Epimysium: outer dense collagen sheath around the entire muscle belly; continuous with the tendon.
• Perimysium: sheath bundling fibers into fascicles (visible "grain" of meat).
• Endomysium: fine sheath around each individual fiber, carrying capillaries and nerve endings.
• Tendon: collagenous tissue transmitting muscle force to bone; acts as a series elastic element.
• Aponeurosis: broad flat tendon sheet from which fibers often originate internally.

B. Muscle fibers

The muscle fiber is a single multinucleate contractile cell, the structural unit of muscle.

• Dimensions: 10–100 μm diameter, up to several cm long; runs the fascicle length.
• Sarcolemma: cell membrane; propagates the action potential inward via T-tubules.
• Sarcoplasmic reticulum (SR): internal Ca²⁺ store; releases Ca²⁺ on excitation, reuptakes it for relaxation.
• Myofibrils: cylindrical contractile rods filling the fiber, made of serial sarcomeres.
• Sarcomere banding:
  • A-band: dark, thick (myosin) filaments; length constant during contraction.
  • I-band: light, thin (actin) only; shortens on contraction.
  • H-zone: central A-band region with no actin overlap; shortens on contraction.
  • Z-line: sarcomere boundary; anchors actin.

C. Motor units

A motor unit is one alpha motor neuron plus all the muscle fibers it innervates, the smallest controllable unit of contraction.

• All-or-none: activation of the neuron contracts every fiber in the unit fully.
• Innervation ratio: fibers per neuron; low (~10) in eye muscles for fine control, high (~1000) in gastrocnemius for gross force.
• Force gradation by two mechanisms:
  1. Recruitment: adding more motor units (Henneman's size principle — small, fatigue-resistant units recruited first).
  2. Rate coding: increasing firing frequency; summed twitches fuse into smooth tetanus.
• Twitch vs tetanus: a single stimulus gives a brief twitch; rapid repeated stimuli sum to a sustained plateau of higher force.

D. Structure of skeletal muscle — fiber types

Fibers differ in contractile speed and metabolic pathway, matching muscle to task.

1. Type I (slow-twitch, oxidative): slow myosin ATPase, many mitochondria, high myoglobin (red), fatigue-resistant; postural and endurance work.
2. Type II (fast-twitch): fast ATPase, larger diameter, higher peak force, faster fatigue; used for sprinting and jumping.
   • Type IIa: fast oxidative-glycolytic; intermediate fatigue resistance.
   • Type IIx (IIb): fast glycolytic; highest speed and force, quickest to fatigue.

• Mechanical consequence: fiber-type ratio shifts the muscle's force–velocity and fatigue curves; e.g., a soleus rich in Type I sustains tension, a gastrocnemius with more Type II delivers explosive power.

E. Fiber architecture

Fiber arrangement relative to the line of pull governs the trade-off between force and excursion.

• Physiological cross-sectional area (PCSA): area perpendicular to fibers; sets maximum force.

PCSA = (V · cos θ) / L_f
F_max = σ · PCSA

• Symbols: V = muscle volume (m³), θ = pennation angle, L_f = fiber length (m), σ = specific tension (~20–35 N/cm²), F_max = maximum force (N).

• Parallel (fusiform): fibers along the line of pull; long fibers give large shortening range and velocity, lower PCSA (e.g., biceps, sartorius).

• Pennate: fibers at a pennation angle θ to the tendon; packs more fibers → high PCSA and force, but effective force along tendon is reduced by cos θ and excursion is small.

  • Unipennate (fibers one side, e.g., flexor pollicis longus), bipennate (rectus femoris), multipennate (deltoid).

• Design trade-off: for fixed volume, force ∝ PCSA and shortening speed ∝ fiber length — architecture tunes a muscle toward force or velocity.

III. Sliding Element Theory of Skeletal Muscle

How sarcomere shortening arises from filament sliding, not filament shortening.

A. Sliding element theory of skeletal muscle

Proposed by A.F. Huxley and H.E. Huxley (1954), the theory states that muscle shortens because thin actin filaments slide past thick myosin filaments while both retain constant length.

• Constant filament length: A-band width unchanged; only I-band and H-zone narrow as Z-lines are drawn together.
• Cross-bridge cycle (the molecular engine):
  1. Attachment: myosin head binds actin (Ca²⁺ has exposed binding sites via troponin–tropomyosin shift).
  2. Power stroke: head pivots ~ from 90° to 45°, pulling actin ~5–10 nm toward the sarcomere center; ADP + Pi released.
  3. Detachment: fresh ATP binds myosin, releasing actin.
  4. Re-cocking: ATP hydrolysis re-primes the head for a new cycle.
• Excitation–contraction coupling: action potential → T-tubule → SR Ca²⁺ release → troponin C binds Ca²⁺ → tropomyosin uncovers actin sites → cycling begins.
• Force–length relationship: tension is maximal at optimal overlap (~2.0–2.2 μm); too short (filament collision) or too stretched (little overlap) reduces active force — direct evidence for sliding filaments.

IV. Skeletal Muscle Function

What muscles do mechanically once activated.

A. Skeletal muscle function

Muscle generates tension to produce, control, or resist joint motion, and to stabilize posture.

• Contraction types by length change:
  1. Isometric: force develops with no length change (holding a weight); joint angle fixed.
  2. Isotonic: force roughly constant while length changes:
     • Concentric: muscle shortens while producing force (lifting) — positive work.
     • Eccentric: muscle lengthens under tension (lowering) — negative work; produces the highest forces and stores elastic energy.
• Roles at a joint: agonist (prime mover), antagonist (opposes), synergist (assists), fixator (stabilizes origin).
• Force–velocity relationship: force falls hyperbolically as shortening velocity rises; peak force occurs at zero (isometric) velocity, and force exceeds isometric during lengthening.
• Elastic energy: series tendon stores and returns energy in stretch–shorten cycles, e.g., the Achilles tendon in running raises efficiency.

V. Contraction of Skeletal Muscle and Hill's Three-Element Model

The mechanical events of contraction and their lumped-parameter representation.

A. Contraction of skeletal muscle

Contraction is the coordinated, Ca²⁺-triggered cross-bridge cycling that develops tension along the fiber, expressed as an active plus passive mechanical response.

• Sequence: neural signal → depolarization → Ca²⁺ release → cross-bridge cycling → tension → Ca²⁺ reuptake → relaxation.
• Active force: generated by cycling cross-bridges; depends on activation, length (overlap), and velocity.
• Passive force: arises from stretched connective tissue and the protein titin; rises steeply beyond resting length even without activation.
• Total tension: sum of active and passive components at any given length.

B. Hill's three-element model

A.V. Hill's model (1938) represents muscle as three mechanical elements that reproduce its force–length–velocity behavior without modeling molecular detail.

• Elements:
  • Contractile element (CE): the active force generator; slack at rest, producing force only when activated; obeys the force–velocity relation.
  • Series elastic element (SEE): spring in series with CE (tendon + cross-bridge compliance); transmits CE force and stores elastic energy; explains the delay between activation and external force.
  • Parallel elastic element (PEE): spring parallel to CE (sarcolemma, connective tissue, titin); carries passive tension when the muscle is stretched.
• Arrangement: CE and SEE in series; this pair sits in parallel with PEE.

        ┌──[ CE ]──[ SEE ]──┐
Bone ───┤                   ├─── Bone
        └──────[ PEE ]──────┘

• Force sharing:
  • F_total = F_CE(via SEE) + F_PEE
  • Below rest length: PEE slack, force ≈ CE force transmitted through SEE.
  • Above rest length: PEE contributes rising passive tension.
• Hill's characteristic equation (force–velocity of the CE):

(F + a)(v + b) = (F_0 + a) · b

• Symbols: F = muscle force (N), v = shortening velocity (m/s), F_0 = maximum isometric force (N), a = force-dimension constant (N), b = velocity-dimension constant (m/s); a/F_0 ≈ 0.25 for many muscles.
• Utility: forms the basis of Hill-type muscle models in gait simulation, ergonomics, and prosthetic control, where activation dynamics multiply the CE force term.