1. Topic 16: Muscle Mechanics Elliot Howard Cardiac Mechanics Research Group Ph.D. Candidate ehoward@ucsd.edu

2. Physiology Review (Key points from previous lecture) Structural Hierarchy of Muscle (skeletal) The Muscle Cell (aka fiber) Ionic Species and Calcium Activation The Sarcomere Cross-bridge Cycling

3. Structural Hierarchy Whole Muscle (organ/tissue)  Fascicle (tissue)  Fiber (cell)  Myofibril (sub-cellular)  Sarcomere (contractile unit)  Myofilaments (molecular) Tendon s Biceps Whole Muscle Sarcomere Fascicle Fiber Myofibril Myofilaments

4. The Muscle Fiber (cell) Key Components: Multinucleated Longitudinally arranged, circular cross section Sarcolemma (plasma membrane) Sarcoplasm (cytosol) Sarcoplasmic Reticulum (ER) Mitochrondria (ATP powerhouse) Transverse (T) Tubule Terminal Cisternae Network of Triads (T-tuble + 2 Terminal Cisternaue)

5. Ionic Species Key Features Negative resting membrane potential -High Intracellular [K + ] -High Extracellular [Na 2+ ] Action Potential travels along sarcolemma and down the T-tubule activates DHP. What is the DHP receptor? a.L-type (long-lasting) calcium voltage gated channel b.Mechanically connected to the Ryandine receptors (RyRs). c.How is cystosolic [Ca 2+ ] increased? What is the primary difference in Ca 2+ release between skeletal and cardiac muscle? - Calcium Induced Calcium Release (CICR)

6. Calcium Activation Key Features Force-development is highly dependent on [Ca 2+ ]. How is this regulated? Ca 2+ binds to troponin-C sites, which are separated by neighboring troponin-C sites every 7 nm (Cooperative Activity). What is cooperative activity? How does [Ca 2+ ] regulate force- relaxation? How is this achieved?

7. The Sarcomere The “Contractile Unit” Thick Myosin Filaments Thin Actin Filaments Z-line to Z-line A and I bands Sliding Filament Theory Titin a.Molecular Spring b.Connects Z-line to M-line, allows for force transmission. c.Limits range of motion of sacromere in tension, primary contributor to passive stiffness.

8. The Crossbridge

9. Crossbridge Cycling

10. Step 1: Cross-bridge (high-energy state) is formed Step 2: Release of stored chemical energy is transferred to mechanical motion, myosin head “power strokes” to it’s rigor position (45 o ). Actin slides towards the M-line. Step 3: In rigor conformation, ATP molecule binds to myosin head, releasing myosin (rigor state) from actin filament. Step 4: ATP hydrolysis (ATP  ADP + PI) supplies potential energy to the myosin head. Myosin head is in it’s upright position (90 o ) and awaits for next open binding site of actin to form crossbridge.

11. Muscle Mechanics Twitch Tetanus Isometric contraction Isotonic contraction Concentric contraction Eccentric contraction Transient loading

12. Muscle Contraction We know that muscle shortening corresponds to the sliding of thin (actin) filaments past thick (myosin) filaments Most widely accepted mechanism for generation of force is the formation of connections between these filaments by crossbridges

13. Tests of Contractile Mechanics Isometric - constant length Isotonic - constant force Transient

14. Isometric (constant length) Preparations Force (N) Time (s) 0 Frog Semitendinosis Muscle 0.4 Passive Active Total

15. The Twitch Definition: Response of a muscle (force developed) during a brief period of threshold stimulation. What defines the threshold? 3 Phases of the Twitch: (1)Latent Period (2)Contraction Period (3)Relaxation Period

16. Determining Factors of the Twitch Graded Response Muscle developed-force depends on: 1.Threshold value 2.Maximal stimulus The “all or none” response 1.A muscle exposed to a threshold stimulus will undergo a complete twitch. Summation: 1.If multiple stimuli are delivered rapidly, the developing twitch will be larger in magnitude that the preceding one.

17. Fast vs. Slow Twitch Muscle Duration of twitch corresponds to functional requirements of muscle Fast twitch muscle White muscle lower concentrations of RBCs and myoglobin, e.g. ocular muscle Slow twitch muscle Red muscle e.g. soleus muscle Soleus (calf) Gastrocnemius (calf) Ocular (eye) 10100 time (ms) T Isometric Twitch Tension T

18. Stimulation Frequency Increasing stimulus frequency will result into wave summation of multiple twitches. As frequency occurs, tetanus can occur (unfused/fused) Fused tetanus occurs via recruitment of multiple muscle fibers at high stimulation frequencies, resulting in a maximum force development (>> twitch) that is sustained over time.

19. When a series of stimuli is given, isometric muscle force rises to a plateau (unfused tetanus) which ripples at the stimulus frequency. As stimulus frequency is increased, the plateau rises and becomes a smooth fused tetanus Twitch and Tetanus

20. Preload and Afterload Preload: Load on muscle prior to stimulation Can also be related to length of the muscle. Afterload: Load at which muscle must do work against during stimulation. If developed tension > afterload, then muscle will shorten in length.

21. Passive Properties Non-Stimulated Skel. Muscle: Soft mechanical tissue Nonlinear stress/strain Viscoelastic properties: (1)Hysteresis (2)Stress Relaxation (3)Creep Major Contributors: ECM proteins -Collagen -Elastin Titin Vinculin Fiber arrangement (anisotropy)

22. Isometric Length-Tension Relationships Tension-length curves for frog sartorius muscle at 0ºC Fixed length Electrical Stimulation Measure Force Generation Increasing Preload

23. Ascending limb is dependent on Ca 2+ concentration Isometric Tension: Sliding Filament Theory Developed tension versus length for a single fiber of frog semitendinosus muscle Results in Isometric Length-Tension Relationship of muscle L max

24. Isotonic (constant force) Preparations Time (ms) F L Muscle shortens

25. Reality: Isometric + Isotonic The majority of skeletal muscles under go both: Isometric Contraction (constant length): T muscle < afterload Isotonic Contraction (constant force): T muscle > afterload IsometricIsotonic Force Time

26. Eccentric and Concentric Contraction L0 L < L0 L > L0 Concentric Contraction: Muscle shortens as it generates tension. T muscle > afterload Eccentric Contraction: Muscle lengths as it generates. T muscle < afterload Serves to decelerate muscle velocity as it lengthens. Ex: placing

27. Isotonic Shortening: Force Velocity Relations (a)Quick-release isotonic apparatus (b)Isometric tetanic stimulation followed by quick release and isotonic shortening by  x 1 and  x 2 “Instantaneous” Events

28. The shortening part (V>0) of the curve was computed from Hill’s equation with c = 4 The asymptotes for Hill’s hyperbola (broken lines) are parallel to the T/T 0 and V/V max axes Mechanical power output is the product of T (force) and V (velocity) Hill’s Force-Velocity Curve

29. Hill's Equation V max T0T0 Original Form: (T+a)(V+b)=b(T 0 +a) Dimensionless forms: a,b = asymptotes T 0 =Isometric force V max = velocity at T = 0 c = T 0 /a (ranges from 1.2-4.0) “Hyperbolic” Behavior Isometric: (V  0) bT+ab = bT 0 +ab; T = T 0 Vmax: (T  0) aV max +ab = bT 0 + ab V max = b/a T 0

30. Hill's Two Element Model Fundamental Assumptions: model consists of linear elastance in series with contractile element. Active contractile element (c.e.) depends on both muscle length and shortening velocity. Velocity can be determined from Hill’s equation. T T c.e. T1T1 T2T2 L1L1 L2L2 L L = L 1 + L 2 T = T 1 = T 2 L’ = L 1 ’ + L 2 ’ T1 = α*ΔL 1 L’ = L 1 ’ + V c.e.

31. Hill’s 2-element model cont. L’ = L 1 ’ + V c.e; ; where V c.e. = L 2 ’ (shortening velocity of contractile element) From Hill’s Equation: V c.e. = - b (P-P 0 )/(P+a) T = α*ΔL 1  T’ = α L 1 ’ = α (L’ – V c.e. ) Typical Parameters (Hill 1927, Holmes 2005): a = 380 * 0.098 [mN/mm 2 ] b = 0.325 [mm/s] P 0 = a/0.257 [mN/mm 2 ] V max = b/a * P 0 [mm/s] L s.e. 0 = 0.3; assume L s.e. 0 is 30% of length T = α (L’ + b(P-P 0 )/(P+a)

32. Solving in MATLAB Ref: Holmes 2005 (AJP) Input Time and Length Profile Define Parameters Initialize Variables (Element Lengths, Tension) Numerical Solver (generic form)

33. MATLAB Output (Isometric) S.E.  linear elastic

34. MATLAB Output (Quick Release)

35. Hill's Three Element Model Fundamental Assumptions: Resting length-tension relation is governed by an elastic element in parallel with a contractile element. In other words, active and passive tensions add. The parallel elastic element is the passive properties. Active contractile element is determined by active length- tension and velocity-tension relationships only. Series elastic element becomes evident in quick-release experiments. Fig 9.8:1 from text. Hill’s functional model of muscle

36. Hill's Three-Element Model (basic equations) L   TT

37. Example analysis of Hill's model -

38. Special Cases of Hill's Model Need third experiment to identify Series Elastic Element. Use Isometric-Isotonic cross-over experiment

39. Cross-Over Experiment Length Tension time Isometric Isotonic

40. Limitations of Hill Model Division of forces between parallel and series elements and division of displacements between contractile and elastic elements is arbitrary (i.e. division is not unique). Structural elements cannot be identified for each component. Hill model is only valid for steady-shortening of tetanized muscle. 1)For a twitch we must include the time-course of activation and hence define "active state" 2)Transient responses observed not reproduced Series elasticity is not observed. Properties of tendon and crossbridge itself

41. Small Length Step Response Tetanized single frog muscle fiber at 0ºC during a 1% shortening step lasting 1 ms

42. Instantaneous and Plateau Tension Solid lines: sarcomere length = 2.2  m (near maximal myofilament overlap). Broken lines: sarcomere length = 3.1  m (39% myofilament overlap). Thus instantaneous tension T 1 reflects crossbridge stiffness and number of attached crossbridges which varies with myofilament overlap.

43. Huxley and Simmons Model (1971) Two stable attached states of S-1 head. Thin filament displacement y stretches S-2 spring generating force. Calculated curves of T 1 and T 2 versus length step y showing predictions of Huxley and Simmons model

44. Both Elastic Elements are Inside the Contractile Element Z-line Myosin Actin Titin T T

45. Muscle Mechanics: Summary of Key Points Skeletal muscle contractions can be twitches or tetani, isometric or isotonic, eccentric or concentriccontractions Twitch duration varies 10-fold with muscle fiber typemuscle fiber type Tetanic contraction is achieved by twitch summation Tetanic contraction The isometric length-tension curve is explained by the sliding filament theoryisometric length-tension curve sliding filament theory Isotonic shortening velocity is inversely related to force in Hill’s force-velocity equation Isotonic shortening velocity Hill’s force-velocity equation Hill’s three-element model assume passive and active stresses combine additively Hill’s three-element model The series elastance is Hill’s model is probably experimental artifact, but crossbridges themselves are elasticcrossbridges