Part IV: Electrophysiology & Signaling | Chapter 4

Mechanotransduction

Mechanosensitive channels, cell adhesion mechanics, the molecular clutch model for durotaxis, and actomyosin contractility

Why Mechanotransduction?

Cells are not passive recipients of chemical signals — they actively sense and respond to mechanical forces. Mechanotransduction, the conversion of mechanical stimuli into biochemical signals, governs processes from hearing (hair cells detect sub-nanometre deflections) to embryonic development (tissue morphogenesis requires coordinated force generation). Aberrant mechanosensing underlies diseases from atherosclerosis to cancer metastasis.

This chapter derives the biophysics of mechanosensitive channels, force-dependent bond kinetics (Bell model and catch bonds), the molecular clutch model explaining cell migration on substrates of different stiffness (durotaxis), and the Hill model of actomyosin contractility in stress fibres.

1. Mechanosensitive Channels

Mechanosensitive (MS) channels are ion channels that open in response to membrane tension. Found in all kingdoms of life, they serve as safety valves (bacterial MscL/MscS), touch receptors (Piezo1/2 in mammals), and osmosensors. The gating mechanism involves a direct coupling between membrane mechanics and protein conformational change.

Derivation: Tension-Gated Channel Mechanics

The channel exists in closed (C) and open (O) states with different cross-sectional areas in the membrane. The key thermodynamic quantity is the work done by membrane tension$\gamma$ (force per unit length, units N/m) when the channel changes its area by $\Delta A = A_O - A_C$ upon opening:

$$W = -\gamma \Delta A$$

The negative sign means that tension favours opening (since $\Delta A > 0$, work is done on the system, lowering the free energy of the open state). The total free energy difference between open and closed states is:

$$\Delta G = \Delta G_0 - \gamma \Delta A$$

where $\Delta G_0 > 0$ is the intrinsic energy cost of opening (without tension). Applying the Boltzmann distribution:

$$\frac{P_O}{P_C} = \exp\left(-\frac{\Delta G}{k_B T}\right) = \exp\left(-\frac{\Delta G_0 - \gamma \Delta A}{k_B T}\right)$$

Since $P_O + P_C = 1$:

$$\boxed{P_O = \frac{1}{1 + \exp\left(\frac{\Delta G_0 - \gamma \Delta A}{k_B T}\right)}}$$

This is a Boltzmann sigmoid. The channel is half-open ($P_O = 0.5$) at the critical tension:

$$\gamma_{1/2} = \frac{\Delta G_0}{\Delta A}$$

The slope of the $P_O$ vs $\gamma$ curve at $\gamma_{1/2}$ is:

$$\left.\frac{dP_O}{d\gamma}\right|_{\gamma_{1/2}} = \frac{\Delta A}{4 k_B T}$$

Example: MscL (large conductance mechanosensitive channel).For the bacterial MscL channel: $\Delta G_0 \approx 50$ $k_B T$,$\Delta A \approx 20$ nm$^2$, giving$\gamma_{1/2} \approx 10$ mN/m. This is close to the lytic tension of a lipid bilayer ($\sim 12$ mN/m), consistent with MscL's role as a last-resort safety valve.

Piezo channels: Mammalian Piezo1 has a much smaller $\Delta G_0$ and opens at lower tensions ($\gamma_{1/2} \approx 1$–$3$ mN/m), enabling it to sense gentle touch and blood flow shear stress. The large propeller-like structure of Piezo1 ($\Delta A \approx 10$–$20$ nm$^2$) amplifies the tension sensitivity.

Membrane Tension and Laplace's Law

Membrane tension can be related to measurable quantities using Laplace's law. For a spherical membrane with radius $R$ under pressure difference $\Delta P$:

$$\gamma = \frac{\Delta P \cdot R}{2}$$

For micropipette aspiration experiments, where a cell is partially aspirated into a pipette of radius $R_p$ under suction pressure $\Delta P$:

$$\gamma = \frac{\Delta P}{2}\left(\frac{1}{R_p} - \frac{1}{R_c}\right)^{-1}$$

where $R_c$ is the radius of the cell body outside the pipette. This allows precise control and measurement of membrane tension in patch clamp experiments on mechanosensitive channels.

2. Cell Adhesion — Force-Dependent Bond Kinetics

Cells adhere to extracellular matrix and other cells through specific receptor-ligand bonds (integrins, cadherins, selectins). These bonds experience mechanical forces that profoundly affect their lifetime, creating a direct link between mechanics and biochemistry.

Derivation: Bell Model for Slip Bonds

George Bell (1978) proposed that force tilts the energy landscape of a receptor-ligand bond, lowering the dissociation barrier. Starting from Kramers theory, an applied force$F$ reduces the barrier by $F x_\beta$, where $x_\beta$is the distance from the bound state to the transition state along the reaction coordinate:

$$\boxed{k_{\text{off}}(F) = k_0 \exp\left(\frac{F x_\beta}{k_B T}\right)}$$

where $k_0 = k_{\text{off}}(0)$ is the zero-force off-rate. The bond lifetime is:

$$\boxed{\tau(F) = \frac{1}{k_{\text{off}}(F)} = \tau_0 \exp\left(-\frac{F x_\beta}{k_B T}\right)}$$

where $\tau_0 = 1/k_0$ is the zero-force lifetime. This is a slip bond: the lifetime decreases monotonically with force (the bond slips off faster under tension).

Characteristic parameters: For selectin-ligand bonds (important in leukocyte rolling): $k_0 \approx 1$ s$^{-1}$,$x_\beta \approx 0.5$ nm. Thus a force of $k_BT/x_\beta \approx 8$ pN accelerates unbinding by a factor of $e \approx 2.7$. At 50 pN:$\tau \approx \tau_0 e^{-50 \times 0.5/(4.1)} \approx \tau_0/440$.

Survival probability: Under constant force, the probability that the bond survives until time $t$ is:

$$S(t) = \exp(-k_{\text{off}}(F) \cdot t) = \exp\left(-k_0 e^{Fx_\beta/(k_BT)} t\right)$$

Under a linearly increasing force ($F = rt$, where $r$ is the loading rate), the most probable rupture force is:$$F^* = \frac{k_BT}{x_\beta} \ln\frac{r x_\beta}{k_0 k_BT}$$This Bell-Evans formula predicts that $F^*$ increases logarithmically with loading rate — a key prediction confirmed by AFM force spectroscopy experiments.

Catch Bonds: Non-Monotonic Lifetime

Counterintuitively, some bonds are strengthened by moderate forces — these are catch bonds. The lifetime first increaseswith force (catch regime) before eventually decreasing (slip regime). This behaviour was predicted theoretically by Dembo et al. (1988) and confirmed experimentally for P-selectin/PSGL-1 bonds by Marshall et al. (2003).

The two-pathway model explains catch bonds: the bond can dissociate through either a slip pathway (rate $k_s$) or a catch pathway (rate $k_c$). Force accelerates the slip pathway but decelerates the catch pathway:

$$k_s(F) = k_s^0 \exp\left(\frac{F x_s}{k_BT}\right), \quad k_c(F) = k_c^0 \exp\left(-\frac{F x_c}{k_BT}\right)$$

The total off-rate is the sum:

$$k_{\text{off}}(F) = k_s(F) + k_c(F) = k_s^0 e^{Fx_s/(k_BT)} + k_c^0 e^{-Fx_c/(k_BT)}$$

The lifetime $\tau(F) = 1/k_{\text{off}}(F)$ has a maximum at the optimal force:

$$\frac{dk_{\text{off}}}{dF} = 0 \implies F_{\text{opt}} = \frac{k_BT}{x_s + x_c} \ln\frac{k_c^0 x_c}{k_s^0 x_s}$$

Catch bonds are physiologically important: they allow leukocytes to roll along blood vessel walls at optimal flow rates. Too little force means rapid detachment (slip), while moderate shear flow strengthens the bonds (catch), enabling stable rolling. This "shear threshold" behaviour of leukocyte rolling was a puzzle until catch bonds were discovered.

3. Cellular Mechanosensing — Durotaxis

Cells can sense the stiffness of their substrate and migrate preferentially toward stiffer regions — a phenomenon called durotaxis. This was first demonstrated by Lo et al. (2000) for fibroblasts on polyacrylamide gels with stiffness gradients. The molecular clutch model provides a quantitative framework for understanding this behaviour.

Derivation: The Molecular Clutch Model

The molecular clutch model describes force transmission between the actin cytoskeleton (flowing rearward due to myosin contraction) and the substrate (through integrin-mediated focal adhesions). The key components are:

• $v_{\text{retro}}$: retrograde actin flow velocity (driven by myosin)
• $n_{\text{clutch}}$: number of engaged molecular clutches (integrins connected to actin)
• $k_{\text{clutch}}$: stiffness of each clutch (integrin-talin-vinculin complex)
• $k_{\text{sub}}$: substrate stiffness (per adhesion site)

The clutches and substrate are in series mechanically. When $n$ clutches are engaged, each clutch extends at rate $v_{\text{retro}}$ and transmits force to the substrate. The force balance gives:

$$F_{\text{total}} = n_{\text{clutch}} \cdot k_{\text{clutch}} \cdot \Delta x_{\text{clutch}}$$

where $\Delta x_{\text{clutch}}$ is the clutch extension. The same force is transmitted to the substrate:

$$F_{\text{total}} = k_{\text{sub}} \cdot \Delta x_{\text{sub}}$$

The total retrograde displacement is shared between clutch extension and substrate deformation: $v_{\text{retro}} \cdot t = \Delta x_{\text{clutch}} + \Delta x_{\text{sub}}$. At steady state, the traction force transmitted to the substrate is:

$$\boxed{F = \frac{n_{\text{clutch}} \cdot k_{\text{clutch}} \cdot k_{\text{sub}}}{n_{\text{clutch}} \cdot k_{\text{clutch}} + k_{\text{sub}}} \cdot v_{\text{retro}} \cdot t}$$

The effective stiffness of the system is springs in series:

$$k_{\text{eff}} = \frac{n_{\text{clutch}} k_{\text{clutch}} k_{\text{sub}}}{n_{\text{clutch}} k_{\text{clutch}} + k_{\text{sub}}}$$

Stiffness sensing mechanism: On soft substrates ($k_{\text{sub}} \ll n k_{\text{clutch}}$):$k_{\text{eff}} \approx k_{\text{sub}}$ — forces are low, clutches do not reach the force threshold for reinforcement, and adhesions remain small. On stiff substrates ($k_{\text{sub}} \gg n k_{\text{clutch}}$):$k_{\text{eff}} \approx n k_{\text{clutch}}$ — forces build rapidly on clutches, triggering catch-bond reinforcement of integrins and growth of focal adhesions.

The force per clutch is:

$$F_{\text{clutch}} = \frac{k_{\text{clutch}} \cdot k_{\text{sub}}}{n_{\text{clutch}} k_{\text{clutch}} + k_{\text{sub}}} \cdot v_{\text{retro}} \cdot t$$

When $F_{\text{clutch}}$ exceeds a threshold $F^*$ (the catch-to-slip transition force), clutches begin breaking. On stiff substrates, many clutches engage before the threshold is reached (because $k_{\text{eff}}$ is high, force builds quickly, and catch-bond strengthening occurs). On soft substrates, force builds slowly and clutches slip before reinforcement. This asymmetry between the cell's front (stiffer side) and rear (softer side) generates net traction toward the stiffer region — durotaxis.

4. Stress Fibres and Actomyosin Contractility

Stress fibres are contractile actin-myosin bundles that generate intracellular tension. They connect focal adhesions, transmitting force across the cell. The force-velocity relationship of actomyosin contraction follows the Hill model, originally derived for whole muscle.

Derivation: Hill's Force-Velocity Relationship

A.V. Hill (1938, Nobel Prize 1922) derived the hyperbolic force-velocity relation from thermodynamic considerations. The power output of a muscle is the product of force and shortening velocity. Hill proposed that the total rate of energy liberation (heat + work) above the resting rate is proportional to the force deficit from isometric tension:

$$\text{(Heat rate)} + F \cdot v = (F_0 - F) \cdot b$$

where $F_0$ is the isometric (zero-velocity) force and $b$ is a constant with dimensions of velocity. Hill also showed that the extra heat rate is proportional to velocity:

$$\text{Heat rate} = a \cdot v$$

where $a$ has dimensions of force. Substituting:

$$a \cdot v + F \cdot v = (F_0 - F) \cdot b$$

$$(F + a) \cdot v = (F_0 - F) \cdot b$$

$$\boxed{(F + a)(v + b) = (F_0 + a) \cdot b = \text{const}}$$

This is a rectangular hyperbola. Solving for $v$:

$$v = \frac{b(F_0 - F)}{F + a}$$

Key limits:

• Isometric ($v = 0$): $F = F_0$ (maximum force)
• Unloaded ($F = 0$): $v_{\max} = b F_0 / a$ (maximum shortening velocity)
• Maximum power: occurs at $F^* = \sqrt{a(F_0+a)} - a$, $v^* = \sqrt{b(v_{\max}+b)} - b$

The power output is:

$$P = F \cdot v = F \cdot \frac{b(F_0 - F)}{F + a}$$

Typical values for stress fibres: $F_0 \sim 1$–$10$ nN per fibre,$v_{\max} \sim 0.1$–$1$ $\mu$m/s,$a/F_0 \approx 0.25$, $b/v_{\max} \approx 0.25$.

Force balance in a cell: The total contractile force from stress fibres must balance the traction forces at focal adhesions. For a cell with$N_{\text{SF}}$ stress fibres each generating force $F_{\text{SF}}$:$$\sum_i \vec{F}_{\text{FA},i} + \sum_j \vec{F}_{\text{SF},j} = 0$$This force balance, combined with the Hill model, determines the steady-state cell shape, stress fibre tension, and focal adhesion size on substrates of different stiffness.

Focal Adhesion Dynamics

Focal adhesions (FAs) are mechanosensitive: they grow under force and disassemble when force is removed. The simplest model couples FA size $A$ to applied force$F$:

$$\frac{dA}{dt} = k_{\text{on}} \cdot f(F/A) - k_{\text{off}} \cdot A$$

where $f(F/A)$ is an increasing function of stress (force per area). The stress-dependent growth captures the experimental observation that FAs grow in the direction of applied force. At steady state:

$$A^* \propto F \quad (\text{linear force-size scaling})$$

This linear relationship between force and FA area has been confirmed experimentally (Balaban et al. 2001), with a stress of approximately 5.5 nN/$\mu$m$^2$at FAs, remarkably consistent across different cell types.

5. Applications

Wound Healing

Wound healing requires coordinated cell migration driven by mechanosensing. Cells at the wound edge experience an asymmetric mechanical environment: free space ahead (low stiffness) and intact tissue behind (high stiffness). Leader cells extend lamellipodia into the wound and generate traction forces that pull follower cells. The molecular clutch model explains why cells at the wound edge show enhanced focal adhesion assembly and increased traction forces compared to cells in the bulk tissue.

Cancer Metastasis

Cancer cells exploit mechanotransduction during metastasis. Tumour stroma is typically stiffer than normal tissue (due to increased collagen cross-linking), creating stiffness gradients that promote invasion through durotaxis. Cancer cells also show altered mechanosensitive channel expression: Piezo1 overexpression correlates with enhanced invasion in breast cancer. The mechanical properties of cancer cells (typically softer than healthy cells, with $E \sim 0.5$ kPa vs $\sim 2$ kPa) allow them to squeeze through narrow spaces during extravasation.

Tissue Engineering

Stem cell differentiation is guided by substrate stiffness (Engler et al. 2006): mesenchymal stem cells on soft gels ($E \sim 0.1$–$1$ kPa) differentiate into neurons; on medium-stiffness gels ($E \sim 10$ kPa), into muscle; on stiff gels ($E \sim 30$–$40$ kPa), into bone. This "mechanobiology of stem cells" guides rational design of biomaterial scaffolds for regenerative medicine. The molecular basis involves integrin-dependent mechanotransduction through the YAP/TAZ signaling pathway, which translocates to the nucleus on stiff substrates.

Hearing: Hair Cell Mechanotransduction

The inner ear hair cell is perhaps the most exquisitely sensitive mechanotransducer in biology. Stereocilia deflection of just $\sim 0.3$ nm (smaller than the diameter of a hydrogen atom) opens mechanotransduction channels at the tips. The gating spring model describes the channel:

$$P_O = \frac{1}{1 + \exp(-z(x - x_0)/(k_BT))}$$

where $x$ is stereocilia deflection, $z$ is the single-channel gating force ($\sim 0.7$ pN), and $x_0$ is the set point. The operating point is maintained near $P_O \approx 0.05$ by adaptation motors (myosin-1c) that adjust the resting tension in the tip links. This allows the hair cell to detect sounds spanning six orders of magnitude in pressure (0 to 120 dB SPL).

6. Python Simulations

Mechanosensitive Channel Gating vs Tension

Python

script.py120 lines

import numpy as np

# =========================================================
# MECHANOSENSITIVE CHANNEL GATING vs TENSION
# =========================================================

print("=" * 60)
print("MECHANOSENSITIVE CHANNEL: TENSION-DEPENDENT GATING")
print("=" * 60)

kBT = 4.114  # pN*nm at 25C

# Channel parameters for different MS channels
channels = {
    "MscL (E. coli)": {"dG0": 51.0, "dA": 20.0},    # kBT, nm^2
    "MscS (E. coli)": {"dG0": 22.0, "dA": 12.0},
    "Piezo1 (mammal)": {"dG0": 10.0, "dA": 15.0},
    "TREK-1 (mammal)": {"dG0": 8.0, "dA": 6.0},
}

print(f"\nkBT = {kBT:.3f} pN*nm at 25C")
print(f"\nChannel parameters:")
print(f"  {'Channel':>20} {'dG0 (kBT)':>10} {'dA (nm^2)':>10} {'gamma_1/2 (mN/m)':>18}")
print("-" * 62)

for name, params in channels.items():
    dG0 = params["dG0"]
    dA = params["dA"]
    # gamma_1/2 = dG0 * kBT / dA (in pN/nm = nN/nm^2)
    # 1 mN/m = 1 pN/nm
    gamma_half = dG0 * kBT / dA  # pN/nm = mN/m
    print(f"  {name:>20} {dG0:>10.1f} {dA:>10.1f} {gamma_half:>18.2f}")

# Compute P_O vs tension for each channel
print(f"\nOpen probability vs membrane tension:")
tensions = np.linspace(0, 15, 30)  # mN/m

print(f"\n  {'gamma':>8}", end="")
for name in channels:
    short = name.split("(")[0].strip()
    print(f"  {short:>12}", end="")
print()
print("-" * 60)

for gamma in tensions:
    if gamma % 1.0 < 0.3:
        print(f"  {gamma:>8.1f}", end="")
        for name, params in channels.items():
            dG0 = params["dG0"]
            dA = params["dA"]
            exponent = (dG0 * kBT - gamma * dA) / kBT
            exponent = min(exponent, 500)  # prevent overflow
            Po = 1.0 / (1.0 + np.exp(exponent))
            print(f"  {Po:>12.4f}", end="")
        print()

# Sensitivity (slope at gamma_1/2)
print(f"\nSensitivity at half-activation (dPo/d_gamma):")
for name, params in channels.items():
    dA = params["dA"]
    sensitivity = dA / (4 * kBT)  # per mN/m
    print(f"  {name}: {sensitivity:.3f} (mN/m)^-1, or {1/sensitivity:.1f} mN/m per e-fold")

# --- Single-channel gating simulation ---
print("\n" + "=" * 60)
print("STOCHASTIC GATING SIMULATION: MscL")
print("=" * 60)

np.random.seed(42)

dG0_mscl = 51.0  # kBT
dA_mscl = 20.0   # nm^2
gamma_half_mscl = dG0_mscl * kBT / dA_mscl

# Simulate at different tensions
dt = 0.0001  # seconds
T = 1.0      # seconds
N = int(T / dt)

# Rate model: k_open = k0 * exp(gamma*dA/(2kBT)), k_close = k0 * exp(-gamma*dA/(2kBT))
# such that Po = k_open/(k_open+k_close) matches Boltzmann
k0 = 100.0  # attempt rate (s^-1)

print(f"\n{'Tension (mN/m)':>16} {'P_O theory':>12} {'P_O sim':>10} {'Mean open (ms)':>16} {'Mean closed (ms)':>18}")
print("-" * 75)

for gamma in [5.0, 8.0, 10.0, 10.5, 11.0, 12.0, 15.0]:
    dG = (dG0_mscl * kBT - gamma * dA_mscl) / kBT
    Po_theory = 1.0 / (1.0 + np.exp(dG))

k_open = k0 * np.exp(-dG / 2)
    k_close = k0 * np.exp(dG / 2)

# Simulate
    state = 0
    n_open = 0
    open_times = []
    closed_times = []
    dwell = 0

for step in range(N):
        dwell += dt
        if state == 0:
            if np.random.rand() < k_open * dt:
                closed_times.append(dwell)
                dwell = 0
                state = 1
        else:
            if np.random.rand() < k_close * dt:
                open_times.append(dwell)
                dwell = 0
                state = 0
            n_open += 1

Po_sim = n_open / N
    mean_open = np.mean(open_times) * 1000 if open_times else 0
    mean_closed = np.mean(closed_times) * 1000 if closed_times else 0

print(f"  {gamma:>14.1f} {Po_theory:>12.4f} {Po_sim:>10.4f} {mean_open:>16.2f} {mean_closed:>18.2f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Bell Model, Catch Bonds & Durotaxis Simulation

Python

script.py227 lines

import numpy as np

# =========================================================
# BELL MODEL: BOND LIFETIME vs FORCE
# =========================================================

print("=" * 60)
print("BELL MODEL: SLIP AND CATCH BOND KINETICS")
print("=" * 60)

kBT = 4.114  # pN*nm

# --- Slip Bond (Bell model) ---
print("\n--- SLIP BONDS ---")

bonds = [
    ("P-selectin/PSGL-1", 1.0, 0.5),     # k0 (s^-1), x_beta (nm)
    ("Integrin/fibronectin", 0.1, 0.3),
    ("E-cadherin/E-cadherin", 0.5, 0.12),
    ("Biotin/streptavidin", 0.001, 0.05),
]

print(f"\n{'Bond':>25} {'k0 (s^-1)':>10} {'x_beta (nm)':>12} {'tau_0 (s)':>10} {'F* at 8pN':>12}")
print("-" * 72)

for name, k0, xb in bonds:
    tau0 = 1.0 / k0
    F_char = kBT / xb  # characteristic force
    tau_8 = tau0 * np.exp(-8.0 * xb / kBT)
    print(f"  {name:>23} {k0:>10.3f} {xb:>12.2f} {tau0:>10.3f} {tau_8:>12.4f}")

# Lifetime vs force for P-selectin
print(f"\nP-selectin/PSGL-1: Lifetime vs Force")
k0_sel = 1.0
xb_sel = 0.5
tau0_sel = 1.0 / k0_sel

forces = np.linspace(0, 80, 20)
print(f"  {'F (pN)':>8} {'k_off (s^-1)':>14} {'tau (ms)':>12} {'Survival at 1s':>16}")
print("-" * 55)

for F in forces:
    k_off = k0_sel * np.exp(F * xb_sel / kBT)
    tau = 1.0 / k_off
    surv = np.exp(-k_off * 1.0)
    if F % 8 < 2:
        print(f"  {F:>8.0f} {k_off:>14.2f} {tau*1000:>12.3f} {surv:>16.6f}")

# --- Bell-Evans: Most probable rupture force vs loading rate ---
print(f"\nBell-Evans: Rupture force vs loading rate")
loading_rates = np.logspace(0, 5, 20)  # pN/s

print(f"  {'Rate (pN/s)':>12} {'F* (pN)':>10}")
print("-" * 25)

for r in loading_rates:
    F_star = kBT / xb_sel * np.log(r * xb_sel / (k0_sel * kBT))
    if F_star > 0 and np.log10(r) % 1 < 0.3:
        print(f"  {r:>12.1f} {F_star:>10.2f}")

# --- Catch Bonds (two-pathway model) ---
print("\n" + "=" * 60)
print("CATCH BONDS: TWO-PATHWAY MODEL")
print("=" * 60)

# Parameters (inspired by P-selectin catch bond data)
k_s0 = 5.0    # slip pathway zero-force rate (s^-1)
x_s = 0.2     # slip pathway distance (nm)
k_c0 = 50.0   # catch pathway zero-force rate (s^-1)
x_c = 0.8     # catch pathway distance (nm)

print(f"\nParameters:")
print(f"  Slip pathway: k_s0 = {k_s0} s^-1, x_s = {x_s} nm")
print(f"  Catch pathway: k_c0 = {k_c0} s^-1, x_c = {x_c} nm")

# Optimal force
F_opt = kBT / (x_s + x_c) * np.log(k_c0 * x_c / (k_s0 * x_s))
print(f"  Optimal force: F_opt = {F_opt:.2f} pN")

forces_cb = np.linspace(0, 60, 50)
lifetimes_catch = []
lifetimes_slip_only = []

print(f"\n{'F (pN)':>8} {'k_s':>10} {'k_c':>10} {'k_total':>10} {'tau (ms)':>12} {'tau_slip (ms)':>14}")
print("-" * 70)

for F in forces_cb:
    k_s = k_s0 * np.exp(F * x_s / kBT)
    k_c = k_c0 * np.exp(-F * x_c / kBT)
    k_total = k_s + k_c
    tau_catch = 1000.0 / k_total
    tau_slip = 1000.0 / (k_s0 * np.exp(F * x_s / kBT))

lifetimes_catch.append(tau_catch)
    lifetimes_slip_only.append(tau_slip)

if F % 5 < 1.5:
        print(f"  {F:>6.0f} {k_s:>10.2f} {k_c:>10.2f} {k_total:>10.2f} {tau_catch:>12.3f} {tau_slip:>14.3f}")

max_lifetime = max(lifetimes_catch)
max_idx = lifetimes_catch.index(max_lifetime)
F_at_max = forces_cb[max_idx]

print(f"\nMaximum lifetime: {max_lifetime:.3f} ms at F = {F_at_max:.1f} pN")
print(f"Zero-force lifetime: {1000/(k_s0+k_c0):.3f} ms")
print(f"Catch bond enhancement: {max_lifetime / (1000/(k_s0+k_c0)):.1f}x")

# --- Molecular Clutch & Durotaxis ---
print("\n" + "=" * 60)
print("MOLECULAR CLUTCH MODEL: DUROTAXIS")
print("=" * 60)

n_clutch = 50       # number of clutches
k_clutch = 0.8      # pN/nm per clutch
v_retro = 50.0      # nm/s retrograde flow
F_break = 40.0      # pN, force to break a clutch (catch-slip transition)

# Stiffness range (substrate)
k_sub_range = np.logspace(-1, 2, 40)  # pN/nm

print(f"\nParameters:")
print(f"  n_clutch = {n_clutch}")
print(f"  k_clutch = {k_clutch} pN/nm")
print(f"  v_retro = {v_retro} nm/s")
print(f"  F_break = {F_break} pN")

# For each substrate stiffness, compute steady-state force and clutch fate
print(f"\n{'k_sub (pN/nm)':>14} {'k_eff':>10} {'F_clutch/F_break':>16} {'Traction (pN)':>14} {'Regime':>12}")
print("-" * 70)

for k_sub in k_sub_range:
    # Effective spring constant
    k_eff = n_clutch * k_clutch * k_sub / (n_clutch * k_clutch + k_sub)

# Time to reach breaking force on a single clutch
    k_single = k_clutch * k_sub / (n_clutch * k_clutch + k_sub)
    t_break = F_break / (k_single * v_retro) if k_single * v_retro > 0 else 1e10

# Average force before breaking (assuming linear loading)
    F_avg_clutch = k_single * v_retro * t_break / 2
    F_total = n_clutch * F_avg_clutch

# Force per clutch at steady state (simplified)
    F_per_clutch = k_single * v_retro * min(t_break, 10)  # cap at 10s

ratio = F_per_clutch / F_break

if ratio > 1:
        regime = "breaking"
    elif ratio > 0.5:
        regime = "loaded"
    else:
        regime = "slipping"

if np.log10(k_sub) % 0.5 < 0.15:
        print(f"  {k_sub:>12.2f} {k_eff:>10.2f} {ratio:>16.4f} {F_total:>14.1f} {regime:>12}")

# Traction force vs substrate stiffness (biphasic)
print(f"\nTraction force vs substrate stiffness:")
print(f"  {'k_sub (pN/nm)':>14} {'Traction (nN)':>14}")
print("-" * 32)

for k_sub in np.logspace(-1, 2, 20):
    k_eff = n_clutch * k_clutch * k_sub / (n_clutch * k_clutch + k_sub)
    # Time to clutch failure
    k_per = k_clutch * k_sub / (n_clutch * k_clutch + k_sub)
    t_fail = F_break / (k_per * v_retro) if k_per * v_retro > 0 else 100

# On stiff substrates: reinforcement (catch bonds)
    # On soft: clutches don't load, no reinforcement
    if t_fail < 2.0:  # fast loading -> stiff
        F_trac = n_clutch * F_break * 0.7  # near isometric
    else:
        F_trac = k_eff * v_retro * min(t_fail, 5) * 0.5

print(f"  {k_sub:>12.2f} {F_trac/1000:>14.3f}")

# --- Hill Model for stress fibre ---
print("\n" + "=" * 60)
print("HILL MODEL: ACTOMYOSIN FORCE-VELOCITY")
print("=" * 60)

F0 = 5.0    # nN (isometric force per stress fibre)
a_hill = 1.25  # nN (Hill constant a = 0.25 * F0)
b_hill = 0.05  # um/s (Hill constant b = 0.25 * vmax * F0/(F0) )
v_max = b_hill * F0 / a_hill

print(f"\nHill parameters:")
print(f"  F0 = {F0} nN (isometric force)")
print(f"  a = {a_hill} nN, b = {b_hill} um/s")
print(f"  v_max = bF0/a = {v_max:.4f} um/s")
print(f"  a/F0 = {a_hill/F0:.2f}, b/v_max = {b_hill/v_max:.2f}")

# Maximum power
F_pmax = np.sqrt(a_hill * (F0 + a_hill)) - a_hill
v_pmax = b_hill * (F0 - F_pmax) / (F_pmax + a_hill)
P_max = F_pmax * v_pmax

print(f"\n  Force at max power: {F_pmax:.3f} nN ({F_pmax/F0*100:.1f}% of F0)")
print(f"  Velocity at max power: {v_pmax:.4f} um/s ({v_pmax/v_max*100:.1f}% of v_max)")
print(f"  Maximum power: {P_max:.6f} nN*um/s = {P_max*1e-3:.6f} pW")

print(f"\n{'F/F0':>8} {'v/v_max':>10} {'Power (fW)':>12} {'Efficiency':>12}")
print("-" * 45)

for F_frac in np.linspace(0, 1.0, 15):
    F = F_frac * F0
    if F < F0:
        v = b_hill * (F0 - F) / (F + a_hill)
    else:
        v = 0
    P = F * v
    eff = P / P_max if P_max > 0 else 0
    print(f"  {F_frac:>6.2f} {v/v_max:>10.4f} {P*1e3:>12.4f} {eff:>12.4f}")

print("\n" + "=" * 60)
print("SUMMARY")
print("=" * 60)
print("1. MS channels: P_O = 1/(1+exp((dG0-gamma*dA)/kBT))")
print("   MscL opens near lysis tension (~10 mN/m)")
print("2. Bell model: tau(F) = tau_0 * exp(-F*x_beta/kBT)")
print("   Catch bonds: lifetime peaks at optimal force")
print("3. Molecular clutch: k_eff = n*k_c*k_s/(n*k_c+k_s)")
print("   Durotaxis: cells migrate toward stiffer substrates")
print("4. Hill model: (F+a)(v+b) = (F0+a)*b")
print("   Power peaks at ~30% of F0 and ~30% of v_max")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Chapter Summary

• Mechanosensitive channels: Gating follows $P_O = 1/(1+\exp((\Delta G_0 - \gamma \Delta A)/k_BT))$. MscL opens near lysis tension; Piezo channels sense gentle touch.
• Bell model (slip bonds): $k_{\text{off}}(F) = k_0 \exp(Fx_\beta/k_BT)$. Bond lifetime decreases exponentially with force. Bell-Evans predicts $F^* \propto \ln(r)$.
• Catch bonds: Two-pathway model produces non-monotonic lifetime vs force. Optimal force $F_{\text{opt}} = (k_BT/(x_s+x_c))\ln(k_c^0 x_c / k_s^0 x_s)$. Explains leukocyte rolling shear threshold.
• Molecular clutch: Durotaxis arises from stiffness-dependent force loading on integrin clutches. Stiffer substrates trigger catch-bond reinforcement and focal adhesion growth.
• Hill model: Actomyosin force-velocity: $(F+a)(v+b) = (F_0+a)b$. Maximum power at intermediate force and velocity ($\sim 30\%$ of maximum).

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