Module 7: Muscle Contraction

Muscle is the most quantitative cell-biology problem in human physiology. Beginning with the independent discovery by H. E. Huxley & J. Hanson and A. F. Huxley & R. Niedergerke in 1954 that thick and thin filaments slide past each other without changing length, muscle physiology has become the paradigm for how a well-defined molecular geometry produces a quantitative mechanical output. This module develops the sarcomere architecture (Z-disk, I-band, A-band, H-zone, M-line), the cross-bridge cycle, the Huxley (1957) two-state kinetic model, the Hill (1938) force-velocity relation, the Gordon-Huxley-Julian (1966) length-tension curve, titin passive mechanics, excitation-contraction coupling via T-tubules, DHPR and the ryanodine receptor, and the comparative physiology of skeletal, cardiac, and smooth muscle. Two simulations: the Huxley 1957 independent cross-bridge model recovering the Hill force-velocity law, and the Gordon 1966 length-tension curve from filament-overlap geometry.

1. The 1954 Sliding-Filament Discovery

In May 1954 two back-to-back papers appeared in Nature: Huxley & Hanson (working with X-ray diffraction and EM at MIT) and Huxley & Niedergerke (with interference microscopy at Cambridge) independently reported the same astonishing fact. When a muscle fibre contracts, the A-band (dark, thick-filament region) stays constant in length, while the I-band (light, thin-filament-only region) and the H-zone (bare-thick-filament center) shrink. Neither filament changes its own length.

The only explanation is that the thin (actin) filaments slide past the thick (myosin) filaments toward the M-line. This was the birth of the sliding-filament theory and set the agenda for every subsequent decade of muscle biochemistry, biophysics, and physiology.

\[\Delta s = -2\Delta\ell_{I},\quad\Delta L_{A}=0,\quad\Delta\ell_{H}=\Delta s\]

Sarcomere shortening \(\Delta s\) equals twice the loss of I-band \(\ell_I\) on each side; H-zone \(\ell_H\) shrinks in concert while the A-band remains fixed — the kinematic signature of filament sliding.

2. Sarcomere Architecture: Z-disk to M-line

A sarcomere is the repeating contractile unit of a striated-muscle myofibril, spanning from one Z-disk to the next, typically \(\sim 2.0\!-\!2.5\) μm at rest in frog-skeletal muscle. The canonical layout is:

Z-disk (Z-line): a lattice of \(\alpha\)-actinin crosslinkers anchoring the minus ends of thin filaments. Proteins: CapZ, nebulin, telethonin, filamin C, myotilin.
I-band: region of thin filament only (actin + tropomyosin + troponin complex). Appears light in EM because there is no myosin overlap.
A-band: region containing thick filaments; it may overlap with thin filaments on its two sides. Appears dark in EM due to myosin density. Length constant (∼ 1.6 μm frog) during contraction.
H-zone: the central thick-filament-only region (no actin overlap). Shrinks during contraction.
M-line: central scaffold of the A-band; myomesin crosslinks adjacent thick filaments in register. Creatine kinase (M isoform) and the titin kinase are anchored here.

The thin filament is polymerized actin (F-actin, diameter ~7 nm) wound with two strands of tropomyosin (coiled-coil dimer) running along the long-pitch grooves, each tropomyosin decorated every 7 actins with a troponin complex(TnC, TnI, TnT). The thin filament is capped at the minus end (pointed end, toward the M-line) by tropomodulin and at the plus end (barbed, at the Z-disk) by CapZ, giving it a fixed length of ~1.0 μm in frog skeletal muscle.

The thick filament is a bipolar assembly of ~300 myosin II molecules, each a hexamer of two heavy chains (MYH7 in slow/cardiac, MYH2/MYH1 in fast skeletal, MYH11 in smooth) with their long coiled-coil tails packing into the filament shaft and their two globular heads (S1 subfragments) projecting outward as cross-bridges. The central bare zone (~0.16 μm) carries no heads; this anti-parallel arrangement ensures the two halves of the filament pull the two sets of thin filaments toward each other.

3. Titin: The Third Filament

Titin (gene TTN) is the largest protein in the human body: a 3–3.7 MDa giant molecule spanning half a sarcomere from Z-disk to M-line. Maruyama et al. (1977) first isolated “connectin”; it was sequenced and renamed titin by Labeit & Kolmerer (1995 Science). There are ~6 titin molecules per half-thick filament, forming a parallel third-filament system.

Titin has four mechanical segments:

Z-disk region: anchored via Tcap (telethonin).
I-band region: tandem Ig domains + unique sequences (N2A, N2B, PEVK). This is the elastic spring; extension sequentially unfolds Ig domains at forces of 100–300 pN (Rief, Gautel, Gaub 1997 Science).
A-band region: tightly bound to the thick filament, acts as a molecular ruler.
M-line region: carries the titin kinase (mechanosensor), binds myomesin.

Granzier & Irving (1995 Biophys. J.) showed that titin accounts for essentially all passive tensionin sarcomeres stretched beyond their slack length ( \(s_\text{slack}\approx 2.0\) μm). Passive tension rises exponentially with stretch, providing the restoring force that keeps sarcomeres at their working length. Mutations that truncate titin or shorten its elastic region cause dilated cardiomyopathy; the N2B isoform is cardiac specific and its stiffness sets diastolic ventricular filling.

4. The Cross-Bridge Cycle: Lymn & Taylor 1971

The Lymn-Taylor 1971 cycle(Biochemistry) synthesizes biochemistry and mechanics into four states of the actomyosin system:

Rigor (A · M): myosin head tightly bound to actin (no nucleotide). ATP binding induces rapid release.
Detached (M · ATP): myosin head has dissociated from actin; hydrolysis to ADP·P_ipre-cocks the lever arm to the up position.
Pre-power-stroke (M · ADP · P_i): head rebinds actin weakly.
Power stroke: P_irelease triggers the lever-arm rotation (~70–90\(^\circ\)), translating actin ~10 nm toward M-line. ADP release returns head to rigor.

\[\text{A}\!\cdot\!\text{M} \xrightarrow{\text{ATP}} \text{M}\!\cdot\!\text{ATP} \xrightarrow{\text{hydrolysis}} \text{M}\!\cdot\!\text{ADP}\!\cdot\!\text{P}_i \xrightarrow{\text{actin}} \text{A}\!\cdot\!\text{M}\!\cdot\!\text{ADP}\!\cdot\!\text{P}_i \xrightarrow{\text{P}_i} \text{A}\!\cdot\!\text{M}\!\cdot\!\text{ADP} \xrightarrow{\text{ADP}} \text{A}\!\cdot\!\text{M}\]

Each cycle hydrolyses one ATP and generates ~4–6 pN of force over a 10 nm step, delivering \(\sim 5\times 10^{-20}\) J of work with \(\sim 40\%\) thermodynamic efficiency.

In solution biochemistry (Lymn-Taylor), ATP hydrolysis is fast on free myosin and slow on actin-bound myosin. In a fibre under load, the duty ratio (fraction of cycle spent attached to actin) is only \(\sim 0.05\) for fast skeletal myosin, \(\sim 0.3\) for cardiac, and \(\sim 0.8\) for smooth-muscle myosin — the molecular origin of the different contraction speeds of the three muscle types.

5. A. F. Huxley’s 1957 Independent Cross-Bridge Model

In 1957, A. F. Huxley proposed the first quantitative kinetic model of muscle contraction (Prog. Biophys. Biophys. Chem.). Each myosin head is a two-state Hookean spring that attaches to actin at rate \(f(x)\) and detaches at rate \(g(x)\), where \(x\) is its axial displacement from the zero-strain position. When the filament is sliding at velocity \(v\), the steady-state attached fraction \(n(x)\) satisfies

\[v\,\frac{\mathrm d n}{\mathrm d x} = f(x)(1 - n) - g(x)\,n\]

with \(n\to 0\) as \(x\to +\infty\) (no attachment beyond the reach of the head). Tension is \(P(v) = k\int x\, n(x, v)\, \mathrm d x\).

Huxley’s original piecewise-linear rates were:

\[f(x) = \begin{cases} f_1\,x/h, & 0<x<h \\ 0, & \text{else}\end{cases},\quad g(x) = \begin{cases} g_2\,x/h, & x>0 \\ g_1\,|x|/h, & x<0\end{cases}\]

with \(g_1 \gg f_1\) so a bridge dragged into negative \(x\) detaches rapidly. Representative Huxley parameters ( \(f_1=43.3\), \(g_1=209\), \(g_2=15.4\) s\(^{-1}\)) reproduce frog sartorius data.

Integrating this master equation over a sweep of velocities recovers the isometric tension \(P_0\) (at \(v=0\)) and the unloaded shortening velocity \(V_0\) (at \(P=0\)). The resulting \(P(v)\) closely matches the phenomenological equation obtained by A. V. Hill in 1938.

6. Hill’s Force-Velocity Equation (1938)

A. V. Hill (1938 Proc. R. Soc. B) derived the force-velocity relationship by combining isotonic shortening experiments with calorimetric measurements of heat production (the famous “shortening heat”). Hill’s equation states:

\[(P + a)(v + b) = (P_0 + a)\, b\]

with \(a\approx 0.25\, P_0\) (energetic constant, related to shortening heat per unit distance) and \(b\approx 0.3\, V_0\)(kinetic constant). Rearranged, \(P(v) = \frac{(P_0 + a)\,b}{v + b} - a\), a rectangular hyperbola with asymptotes \(P\to -a\) as \(v\to\infty\) and \(v\to -b\) as \(P\to\infty\).

Key features:

Force decreases monotonically with velocity.
Power = \(P\cdot v\) peaks at \(v\approx V_0/3\), where \(P\approx P_0/3\) (classic 1/3 + 1/3 = 1/9 of maximum).
The ratio \(a/P_0\) characterizes the curvature: high \(a/P_0\) (0.3–0.4) is a flatter, more efficient fibre; low (0.15–0.20) gives a more bowed curve.
Slow (Type I) oxidative fibres have a low \(V_0\), high endurance; fast (Type IIb) glycolytic fibres have a high \(V_0\), rapid fatigue.

Hill’s equation is an emergent result of the independent Huxley cross-bridge kinetics: the hyperbolic \(P\)-vs-\(v\)shape reflects the ensemble average of many myosin heads attaching and detaching at velocity-dependent rates.

7. Length-Tension Relation (Gordon, Huxley, Julian 1966)

Gordon, Huxley & Julian (1966 J. Physiol.) performed the classic experiment of measuring tetanic tension as a function of imposed sarcomere length using a segment-length-clamp technique on isolated frog muscle fibres. The resulting curve is a piecewise-linear function reflecting the overlap geometry of thick and thin filaments:

Below \(s\approx 1.27\) μm, thick filaments collide with Z-disks: tension collapses to zero.
From 1.27 to 1.67 μm, thick filaments are deformed and thin filaments overlap from the wrong side: tension rises from 0 to ~85%.
From 1.67 to 2.0 μm, all crossbridges find partners: tension rises to 100%.
Plateau from 2.0 to 2.25 μm (width = bare-zone b = 0.20 μm + some margin): tension is maximal.
From 2.25 to 3.65 μm, thin filaments retract past the bare zone; fewer crossbridges overlap; tension falls linearly to zero.

\[T(s)/T_{\max} \;\propto\; L_{\text{overlap}}(s)\]

The length-tension curve was an early triumph of the sliding-filament theory: it predicted a piecewise-linear functional form that matched experiment to the precision of the optical-clamp technique.

Cardiac muscle operates over a narrower range (roughly 1.8–2.4 μm) and produces a continuous ascending Frank-Starling curve, because at resting length the sarcomere is below the plateau — so diastolic filling that stretches the sarcomere increases stroke force. This is the molecular basis of the Frank-Starling law of the heart.

8. Excitation-Contraction Coupling and Ca²⁺ Triggering

A resting muscle fibre has cytosolic free [Ca²⁺] \(\sim 0.1\) μM; activation is triggered by a motor-neuron action potential that propagates down the T-tubule (invaginations of sarcolemma) into the fibre depths. The cascade is:

ACh released at neuromuscular junction opens muscle nicotinic receptor (Na⁺ influx, endplate depolarization).
Na_V1.4 action potential propagates along the sarcolemma and down the T-tubule.
DHPR (Ca_V1.1) in T-tubule membrane senses voltage; in skeletal muscle it mechanically couples to the RyR1 ryanodine receptor on the sarcoplasmic reticulum (SR) membrane at the triad junction.
RyR1 opens and releases SR Ca²⁺, raising cytosolic [Ca²⁺] to \(\sim 10\) μM in ~5 ms.
Troponin C binds Ca²⁺ at its two low-affinity N-terminal sites, undergoes a conformational change that releases the TnI inhibitory subunit from actin and pulls TnT, which drives tropomyosin off the myosin-binding sites (Huxley 1972 “steric blocking” model).
Myosin heads bind actin; cross-bridge cycle proceeds; force is generated.
SERCA ATPase pumps Ca²⁺ back into the SR (with calsequestrin as Ca buffer); cytosolic [Ca²⁺] falls, tropomyosin re-blocks, muscle relaxes.

In cardiac muscle, the coupling is different: DHPR (Ca_V1.2) opens first, admitting a small Ca²⁺ influx that triggers RyR2 opening by Ca-induced Ca release (Fabiato 1983). This introduces graded contractions but makes cardiac coupling vulnerable to RyR2 mutations (CPVT arrhythmia).

In smooth muscle, there is no troponin; regulation is through myosin light chain (MLC) phosphorylation on Ser19 by Ca/calmodulin-activated MLCK. Relaxation requires MLC phosphatase. Latch-bridge mechanism (Hai & Murphy 1988) allows smooth muscle to maintain force with very low ATP consumption.

9. Fibre Types and Comparative Muscle Physiology

Skeletal muscle fibres are classified by their myosin heavy-chain isoform and metabolic profile:

Type I (slow oxidative): MYH7; high mitochondrial density, rich in myoglobin (red); slow \(V_0\); fatigue-resistant; postural muscles (soleus).
Type IIa (fast oxidative): MYH2; intermediate \(V_0\); aerobic + glycolytic; locomotion.
Type IIx/b (fast glycolytic): MYH1/MYH4; highest \(V_0\) and power; low mitochondria (white); rapidly fatiguing; sprint/jump muscles (gastrocnemius, biceps).

Fibre-type composition is partly genetic and partly training-dependent. Elite endurance athletes have >70% Type I fibres; sprinters and weightlifters have >70% Type II. Chronic electrical stimulation can convert IIx → IIa → I; prolonged disuse does the reverse.

Comparative properties of the three muscle types:

Skeletal: voluntary; striated; troponin-regulated; MYH1/2/4/7; fast.
Cardiac: involuntary; striated; troponin-regulated; MYH7; functional syncytium via gap junctions; Ca-induced Ca release.
Smooth: involuntary; non-striated; myosin-LC-phosphorylation regulated; MYH11; slow; latch bridge for sustained tone.

Sarcomere architecture (Z-disk to M-line)

Simulation 1: Huxley 1957 Cross-Bridge Kinetics → Hill 1938 F-V Curve

The independent-cross-bridge equation \(v\, n_x = f(x)(1-n) - g(x)n\) is integrated as a boundary-value problem starting at \(n(x\to\infty)=0\) and stepping toward negative \(x\). For each imposed sliding velocity, the tension \(P(v)= k\int x\,n(x,v)\,\mathrm d x\) is computed, and the resulting force-velocity curve is fit with Hill’s 1938 hyperbolic equation \((P+a)(v+b)=(P_0+a)b\). Output includes the piecewise-linear attachment/detachment rate functions, the steady-state \(n(x)\) at several velocities, the force-velocity fit, and the mechanical power curve \(P\cdot v\) peaking near \(V_0/3\).

Python

script.py146 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# A. F. Huxley (1957, Prog. Biophys. Biophys. Chem.) independent
# cross-bridge model for muscle contraction. Each myosin head is a two-state
# attachment system with attachment rate f(x) and detachment rate g(x)
# that depend on the axial displacement x from the equilibrium position of
# the cross-bridge. The filament sliding velocity v > 0 means the thin
# filament moves toward the M-line, so x(t) = x0 - v*t for any attached
# cross-bridge.
#
# Steady-state attachment probability n(x) obeys
#   v dn/dx = f(x) (1 - n) - g(x) n
# with n -> 0 as x -> +infinity (crossbridge not yet attached).
#
# Huxley's original rate scheme (piecewise linear):
#   f(x) = f1 * x/h   for 0 < x < h,   0 otherwise
#   g(x) = g2 * x/h   for x > 0,       g1*|x|/h for x<0   (with g1 >> f1)
# Tension per half-sarcomere is
#   P(v) = integral of k*x * n(x,v) dx over all x
# where k is the bridge stiffness.

h = 1.0                 # reach of the cross-bridge (um, normalized)
k = 1.0                 # bridge stiffness (arbitrary units)
f1 = 43.3               # attachment rate prefactor (Huxley 1957, 1/s)
g1 = 209.0              # detachment rate (negative x), 1/s
g2 = 15.4               # detachment rate (positive x), 1/s

def f_rate(x):
    return np.where((x > 0) & (x < h), f1 * x / h, 0.0)

def g_rate(x):
    return np.where(x < 0, g1 * np.abs(x) / h, g2 * x / h)

# Sweep velocities (V0 is unloaded shortening velocity)
V = np.linspace(0.01, 4.0, 60)   # sliding velocity (h/s)

# Grid in x
x = np.linspace(-1.5, 2.0, 600)
dx = x[1] - x[0]

def steady_state_n(v):
    # Integrate n(x) from +infty (n=0) toward lower x using Euler
    n = np.zeros_like(x)
    # start from the right boundary (n=0)
    for i in range(len(x) - 2, -1, -1):
        xi = x[i+1]
        dn = (f_rate(xi) * (1 - n[i+1]) - g_rate(xi) * n[i+1]) / max(v, 1e-6)
        n[i] = n[i+1] - dn * dx
        n[i] = np.clip(n[i], 0.0, 1.0)
    return n

# Compute tension P(v) = integral k*x*n(x) dx (only positive x contribute
# positively, negative x -> negative tension as crossbridge resists stretch)
P = np.zeros_like(V)
n_store = {}
for j, v in enumerate(V):
    n = steady_state_n(v)
    integrand = k * x * n
    P[j] = np.trapezoid(integrand, x)
    if j in (0, 15, 30, 45, 59):
        n_store[v] = n

# Isometric tension P0
P0 = P[0]
print(f"Isometric tension P0 = {P0:.3f} (arb. units)")

# Find V0: v at which P crosses zero
V0 = V[np.argmin(np.abs(P))]
print(f"Unloaded shortening velocity V0 = {V0:.2f} h/s")

# Fit Hill's equation (P+a)(v+b) = (P0+a)b
from scipy.optimize import curve_fit
def hill(v, a, b):
    return (P0 + a) * b / (v + b) - a

mask = P > 0
popt, _ = curve_fit(hill, V[mask], P[mask], p0=[0.25, 0.3])
a_fit, b_fit = popt
print(f"Hill fit: a = {a_fit:.3f}, b = {b_fit:.3f}")
print(f"Power-ratio a/P0 = {a_fit/P0:.3f} (typical 0.20-0.30)")

v_smooth = np.linspace(0, V.max(), 200)
P_hill = hill(v_smooth, a_fit, b_fit)

# Mechanical power = P * v
power = P * V
i_max_power = np.argmax(power)
print(f"Max power at v = {V[i_max_power]:.2f} h/s (P/P0 = {P[i_max_power]/P0:.2f})")

# ---------- Plot ----------
fig, axes = plt.subplots(2, 2, figsize=(13, 9.5))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes.ravel():
    ax.set_facecolor('#0a0a1a')
    ax.tick_params(colors='#cbd5e1')
    for s in ax.spines.values(): s.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.35)
ax1, ax2, ax3, ax4 = axes.ravel()

# Panel 1: rate functions f(x), g(x)
xg = np.linspace(-1.5, 2.0, 400)
ax1.plot(xg, f_rate(xg), '-', color='#22c55e', linewidth=2.4, label='f(x): attach')
ax1.plot(xg, g_rate(xg), '-', color='#f59e0b', linewidth=2.4, label='g(x): detach')
ax1.set_xlabel('cross-bridge displacement x (h units)', color='#cbd5e1')
ax1.set_ylabel('rate (1/s)', color='#cbd5e1')
ax1.set_title('Huxley 1957 attach/detach rates f(x), g(x)',
              color='#bbf7d0', fontweight='bold')
ax1.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

# Panel 2: n(x,v) at several velocities
colors_v = ['#22c55e', '#a78bfa', '#f59e0b', '#ef4444', '#3b82f6']
for (v_sel, n_sel), col in zip(n_store.items(), colors_v):
    ax2.plot(x, n_sel, '-', color=col, linewidth=2.0, label=f'v = {v_sel:.2f}')
ax2.set_xlabel('cross-bridge displacement x (h units)', color='#cbd5e1')
ax2.set_ylabel('attachment probability n(x)', color='#cbd5e1')
ax2.set_title('Steady-state cross-bridge population',
              color='#bbf7d0', fontweight='bold')
ax2.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)

# Panel 3: force-velocity
ax3.plot(V, P / P0, 'o', color='#22c55e', markersize=5, label='Huxley 1957 (sim.)')
ax3.plot(v_smooth, P_hill / P0, '--', color='#a78bfa', linewidth=2.0,
         label='Hill 1938 fit')
ax3.axhline(0, color='#475569', linewidth=0.5)
ax3.set_xlabel('sliding velocity v (h/s)', color='#cbd5e1')
ax3.set_ylabel('tension P / P0', color='#cbd5e1')
ax3.set_title('Force-velocity relation (P+a)(v+b) = (P0+a)b',
              color='#bbf7d0', fontweight='bold')
ax3.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

# Panel 4: mechanical power output
ax4.plot(V, power, '-', color='#22c55e', linewidth=2.4)
ax4.axvline(V[i_max_power], color='#f59e0b', linestyle='--',
            label=f'peak power at v={V[i_max_power]:.2f}')
ax4.set_xlabel('velocity v (h/s)', color='#cbd5e1')
ax4.set_ylabel('mechanical power P * v', color='#cbd5e1')
ax4.set_title('Muscle power output curve',
              color='#bbf7d0', fontweight='bold')
ax4.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation 2: Sarcomere Length-Tension Curve (Gordon 1966)

A geometric model of the striated-muscle sarcomere with frog-skeletal parameters \((L_M, L_A, b) = (1.60, 1.00, 0.20)\) μm. We compute the filament overlap length as a function of sarcomere length \(s\) and overlay the Gordon-Huxley-Julian piecewise-linear active tension \(T(s)\) with its plateau from 2.0–2.25 μm. Passive titin tension is added via an exponential law \(T_p = 0.08 [\exp(2(s-s_{\text{slack}}))-1]\) (Granzier 1995). A schematic panel draws the sarcomere at three different lengths illustrating short-overlap, optimal, and long-overlap regimes. Skeletal and cardiac length-tension curves are compared, showing why the Frank-Starling mechanism operates on the ascending limb.

Python

script.py151 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Length-tension relation of a sarcomere (Gordon, Huxley & Julian 1966,
# J. Physiol.). Active tetanic tension is proportional to the number of
# myosin heads able to form cross-bridges with actin, which in turn is
# determined by the overlap geometry of thick and thin filaments.
#
# Frog-skeletal-muscle sarcomere geometry (Gordon 1966):
#   thick filament:  length L_M = 1.60 um
#   thin filament:   length L_A = 1.00 um (each side of Z-disk)
#   bare zone (no crossbridges, central):  b = 0.20 um
#
# Sarcomere length s spans [1.27, 3.65] um physiologically.
# Tetanic tension T(s)/T_max is piecewise linear with breakpoints at:
#   s1 = 1.27  -> T/T_max = 0.00 (thick-thick collision, tension drop)
#   s2 = 1.67  -> T/T_max = 0.85
#   s3 = 2.00  -> T/T_max = 1.00 (plateau begins)
#   s4 = 2.25  -> T/T_max = 1.00 (plateau ends)
#   s5 = 3.65  -> T/T_max = 0.00 (no overlap)

s_bp = np.array([1.27, 1.67, 2.00, 2.25, 3.65])
T_bp = np.array([0.00, 0.85, 1.00, 1.00, 0.00])

s = np.linspace(1.0, 3.9, 500)
T_active = np.interp(s, s_bp, T_bp)

# Passive tension from titin (Granzier 1995 Biophys. J.): exponential above
# slack length s_slack ~ 2.0 um
s_slack = 2.0
def passive(s):
    return np.where(s > s_slack,
                    0.08 * (np.exp(2.0 * (s - s_slack)) - 1),
                    0.0)
T_passive = passive(s)
T_total = T_active + T_passive

# Geometric overlap calculation (see Rassier & MacIntosh 2000)
L_M = 1.60       # thick filament length (um)
L_A = 1.00       # thin filament length (um)
b_bare = 0.20    # bare zone

def overlap(s):
    # Gordon 1966 geometry: available crossbridge length Lc
    x = np.zeros_like(s)
    for i, si in enumerate(s):
        thin_end_from_Z = L_A
        thick_end_from_M = L_M / 2.0
        overlap_start = si/2 - thin_end_from_Z
        overlap_end = si/2 - b_bare/2
        overlap_start = max(overlap_start, -thick_end_from_M)
        overlap_end = min(overlap_end, thick_end_from_M)
        ov = max(0.0, overlap_end - overlap_start)
        x[i] = ov
    return x / ((L_M - b_bare) / 2.0)

ovl = overlap(s)

# Print anchor-point tensions
for sp, tp in zip(s_bp, T_bp):
    print(f"sarcomere length {sp:.2f} um  ->  T/T_max = {tp:.2f}")

# Integrated plateau area (work capacity per shortening cycle)
plateau_mask = (s >= 2.00) & (s <= 2.25)
plateau_area = np.trapezoid(T_active[plateau_mask], s[plateau_mask])
print(f"Plateau tension-length area: {plateau_area:.3f} (plateau width = 0.25 um)")

# Active range: where T_active > 0
active_range = s[T_active > 0.01]
print(f"Active length range: {active_range.min():.2f} - {active_range.max():.2f} um")

# Peak passive tension at long length
print(f"Passive tension at s = 3.0 um: {passive(np.array([3.0]))[0]:.3f}")
print(f"Passive tension at s = 3.5 um: {passive(np.array([3.5]))[0]:.3f}")

# Cardiac muscle: narrower plateau, steeper ascent (Frank-Starling)
T_cardiac = np.interp(s, [1.60, 1.80, 2.00, 2.20, 2.40],
                        [0.00, 0.40, 0.85, 1.00, 0.85])
T_cardiac = np.clip(T_cardiac, 0.0, 1.0)

# ---------- Plot ----------
fig, axes = plt.subplots(2, 2, figsize=(13, 9.5))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes.ravel():
    ax.set_facecolor('#0a0a1a')
    ax.tick_params(colors='#cbd5e1')
    for sp_ax in ax.spines.values(): sp_ax.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.35)
ax1, ax2, ax3, ax4 = axes.ravel()

# Panel 1: Gordon 1966 length-tension curve
ax1.plot(s, T_active, '-', color='#22c55e', linewidth=2.6, label='active (Gordon 1966)')
ax1.plot(s, T_passive, '-', color='#a78bfa', linewidth=2.0, label='passive (titin)')
ax1.plot(s, T_total, '--', color='#f59e0b', linewidth=2.0, label='total')
for sp, tp in zip(s_bp, T_bp):
    ax1.plot(sp, tp, 'o', color='#ef4444', markersize=7)
ax1.axvspan(2.00, 2.25, color='#22c55e', alpha=0.2, label='plateau')
ax1.set_xlabel('sarcomere length s (um)', color='#cbd5e1')
ax1.set_ylabel('tetanic tension T/T_max', color='#cbd5e1')
ax1.set_title('Length-tension relation (Gordon, Huxley, Julian 1966)',
              color='#bbf7d0', fontweight='bold')
ax1.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)
ax1.set_ylim(-0.05, 1.3)

# Panel 2: geometric overlap
ax2.plot(s, ovl, '-', color='#22c55e', linewidth=2.4, label='normalized overlap')
ax2.plot(s, T_active, '--', color='#a78bfa', linewidth=2.0, label='Gordon active tension')
ax2.set_xlabel('sarcomere length s (um)', color='#cbd5e1')
ax2.set_ylabel('fraction', color='#cbd5e1')
ax2.set_title('Geometric overlap vs tension (concordance)',
              color='#bbf7d0', fontweight='bold')
ax2.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

# Panel 3: skeletal vs cardiac
ax3.plot(s, T_active, '-', color='#22c55e', linewidth=2.4, label='skeletal (Gordon)')
ax3.plot(s, T_cardiac, '-', color='#a78bfa', linewidth=2.4, label='cardiac (Frank-Starling)')
ax3.set_xlabel('sarcomere length s (um)', color='#cbd5e1')
ax3.set_ylabel('active tension T/T_max', color='#cbd5e1')
ax3.set_title('Skeletal vs cardiac length-tension',
              color='#bbf7d0', fontweight='bold')
ax3.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)
ax3.set_xlim(1.4, 3.0)

# Panel 4: sarcomere schematic at three lengths
lengths_demo = [1.5, 2.1, 3.0]
names = ['too short', 'optimal', 'too long']
colors_demo = ['#ef4444', '#22c55e', '#a78bfa']
for i, (sl, nm, cl) in enumerate(zip(lengths_demo, names, colors_demo)):
    y = 0.7 - i * 0.25
    ax4.plot([0, sl], [y, y], 'k-', linewidth=1, color='#cbd5e1')
    # Thin filaments (left + right)
    ax4.plot([0, L_A], [y + 0.05, y + 0.05], '-', color='#f59e0b', linewidth=3)
    ax4.plot([sl - L_A, sl], [y + 0.05, y + 0.05], '-', color='#f59e0b', linewidth=3)
    # Thick filament in middle
    tk0 = sl/2 - L_M/2
    tk1 = sl/2 + L_M/2
    ax4.plot([tk0, tk1], [y - 0.05, y - 0.05], '-', color=cl, linewidth=5)
    # Labels
    ax4.text(sl + 0.1, y, f'{nm} (s = {sl} um)', color='#cbd5e1', fontsize=10)
ax4.set_xlim(-0.3, 4.2)
ax4.set_ylim(-0.05, 0.9)
ax4.set_xlabel('position along sarcomere (um)', color='#cbd5e1')
ax4.set_title('Sarcomere geometry at three lengths',
              color='#bbf7d0', fontweight='bold')
ax4.set_yticks([])

plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Key References

• Huxley, H.E. & Hanson, J. (1954). “Changes in the cross-striations of muscle during contraction and stretch and their structural interpretation.” Nature, 173, 973–976.

• Huxley, A.F. & Niedergerke, R. (1954). “Structural changes in muscle during contraction; interference microscopy of living muscle fibres.” Nature, 173, 971–973.

• Huxley, A.F. (1957). “Muscle structure and theories of contraction.” Prog. Biophys. Biophys. Chem., 7, 255–318.

• Huxley, H.E. (1969). “The mechanism of muscular contraction.” Science, 164, 1356–1365.

• Hill, A.V. (1938). “The heat of shortening and the dynamic constants of muscle.” Proc. R. Soc. Lond. B, 126, 136–195.

• Gordon, A.M., Huxley, A.F., & Julian, F.J. (1966). “The variation in isometric tension with sarcomere length in vertebrate muscle fibres.” J. Physiol., 184, 170–192.

• Rassier, D.E. & MacIntosh, B.R. (2000). “Length dependence of active force production in skeletal muscle.” J. Appl. Physiol., 86, 1445–1457.

• Lymn, R.W. & Taylor, E.W. (1971). “Mechanism of adenosine triphosphate hydrolysis by actomyosin.” Biochemistry, 10, 4617–4624.

• Huxley, H.E. (1972). “Structural changes in the actin- and myosin-containing filaments during contraction.” Cold Spring Harb. Symp. Quant. Biol., 37, 361–376.

• Rayment, I. et al. (1993). “Three-dimensional structure of myosin subfragment-1: a molecular motor.” Science, 261, 50–58.

• Finer, J.T., Simmons, R.M., & Spudich, J.A. (1994). “Single myosin molecule mechanics: piconewton forces and nanometre steps.” Nature, 368, 113–119.

• Labeit, S. & Kolmerer, B. (1995). “Titins: giant proteins in charge of muscle ultrastructure and elasticity.” Science, 270, 293–296.

• Granzier, H.L.M. & Irving, T.C. (1995). “Passive tension in cardiac muscle: contribution of collagen, titin, microtubules, and intermediate filaments.” Biophys. J., 68, 1027–1044.

• Rief, M., Gautel, M., Oesterhelt, F., Fernandez, J.M., & Gaub, H.E. (1997). “Reversible unfolding of individual titin immunoglobulin domains by AFM.” Science, 276, 1109–1112.

• Fabiato, A. (1983). “Calcium-induced release of calcium from the cardiac sarcoplasmic reticulum.” Am. J. Physiol., 245, C1–C14.

• Hai, C.M. & Murphy, R.A. (1988). “Cross-bridge phosphorylation and regulation of latch state in smooth muscle.” Am. J. Physiol., 254, C99–C106.

• Bers, D.M. (2002). “Cardiac excitation-contraction coupling.” Nature, 415, 198–205.

• Spudich, J.A. (2001). “The myosin swinging cross-bridge model.” Nat. Rev. Mol. Cell Biol., 2, 387–392.

• Geeves, M.A. & Holmes, K.C. (2005). “The molecular mechanism of muscle contraction.” Adv. Protein Chem., 71, 161–193.

• Linari, M. et al. (2015). “Force generation by skeletal muscle is controlled by mechanosensing in myosin filaments.” Nature, 528, 276–279.

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