Module 1: Actin Filaments

G-actin assembles head-to-tail into polar, double-helical F-actin filaments that drive cell shape, motility, contraction, and many membrane-remodeling events. We build the Oosawa-Asakura kinetic theory, derive the critical concentration and treadmilling, then meet the regulators: Arp2/3 (branching), formins (straight elongation), cofilin/ADF (severing), CapZ (capping), and the NPFs (WASP/WAVE) that integrate the network with Rho-GTPase signaling.

1. G-actin, F-actin, and the ATP Switch

Actin is a 375-residue, 42 kDa ATPase whose monomer (G-actin) contains a deep cleft that binds either ATP or ADP and a divalent cation (Mg²⁺ in cells, Ca²⁺ in typical biochemistry). Kabsch et al. (1990) solved the atomic structure of G-actin·DNase I; subsequent structures by Otterbein et al. (2001) and Dominguez (2004) revealed that ATP binding clamps the cleft closed, while ADP binding allows it to open —a conformational switch that tunes monomer-monomer interfaces.

When G-ATP-actin concentration exceeds the critical concentration, monomers polymerize into a right-handed, double-helical filament (F-actin) of diameter 7 nm with a crossover repeat of 36 nm. Each subunit hydrolyzes its bound ATP to ADP-P_i within a few seconds of incorporation; P_i release follows with a half-life of \(\sim 5\)minutes, giving the filament a “nucleotide clock” that locally marks subunit age.

\[\text{G-ATP} \;\xrightarrow{k_{\text{on}}}\; \text{F-ATP} \;\xrightarrow{k_{\text{hyd}}}\; \text{F-ADP-P}_i \;\xrightarrow{k_{\text{P}_i}}\; \text{F-ADP}\]

Typical rates: \(k_{\text{hyd}} \approx 0.3\,\text{s}^{-1}\),\(k_{\text{P}_i} \approx 3\times 10^{-3}\,\text{s}^{-1}\)(Carlier & Pantaloni 1986).

The three nucleotide states have distinct affinities at each filament end. This is the single most important fact of actin biochemistry: it generates the polar asymmetry of the filament and drives treadmilling.

2. Oosawa-Asakura Kinetics

Oosawa & Asakura (1975) modeled actin polymerization as a two-step process: (a) a slow, unfavorable nucleation of a small oligomer (dimer, trimer), and (b) a fast elongation by end-wise monomer addition. Writing \([M]\) for the monomer concentration and \([E]\) for the density of growing ends,

\[\frac{dF}{dt} = (k_{\text{on}}[M] - k_{\text{off}})\,[E], \qquad \frac{d[E]}{dt} = k_{\text{nuc}}\,[M]^{n^*}\]

\(F\) = polymer mass (monomers incorporated), \(n^* \approx 3\) = nucleus size,\(k_{\text{nuc}}\) = effective pre-equilibrium constant.

Critical concentration

At steady state \(dF/dt = 0\) and \([M] = C_c = k_{\text{off}}/k_{\text{on}}\). For actin at the barbed end with ATP-G, \(C_c^+ \approx 0.12\,\mu\)M; at the pointed end with ADP-G, \(C_c^- \approx 0.6\,\mu\)M (Pollard 1986). Because these are unequal, the polymer has a net polarity: at any bulk monomer concentration\(C_c^+ < [M] < C_c^-\), the plus end grows while the minus end shrinks.

Nucleation lag

Because nucleation scales as \([M]^{n^*}\), the onset of polymerization shows a marked concentration-dependent lag. The characteristic time is\(\tau_{\text{lag}} \sim 1/\sqrt{k_{\text{on}}\,k_{\text{nuc}}\,[M]^{n^*+1}}\)(Oosawa & Kasai 1962). This lag is abolished in vivo by cellular nucleators—Arp2/3, formins, tandem-monomer binders—which template the nucleus directly.

Polymer mass vs. total actin: the hockey stick

3. Treadmilling: Polar Asymmetry and ATP Hydrolysis

The polarity of F-actin is commonly visualized by decoration with myosin-S1 heads, which bind at a fixed azimuthal angle producing a barbed (+) end (wide) and a pointed (−) end (narrow). The two ends have different rate constants, especially for the different nucleotide states:

end / state	\(k_{\text{on}}\) (uM\(^{-1}\)s\(^{-1}\))	\(k_{\text{off}}\) (s\(^{-1}\))	\(C_c\) (uM)
Barbed (+), ATP	11.6	1.4	0.12
Barbed (+), ADP	3.8	7.2	1.9
Pointed (−), ATP	1.3	0.8	0.6
Pointed (−), ADP	0.16	0.27	1.7

Treadmilling is the steady state in which the polymer has a bulk monomer concentration between the two ends’ critical concentrations, so that net assembly occurs at the barbed end and net disassembly at the pointed end, with subunits flowing through the filament along their lifetime. ATP hydrolysis drives this non-equilibrium steady state (it costs \(\sim 1\) ATP per subunit of treadmilled flux).

\[v_{\text{tread}} = k_{\text{on}}^+\,[M]_{\text{ss}} - k_{\text{off}}^+ = -\bigl(k_{\text{on}}^-\,[M]_{\text{ss}} - k_{\text{off}}^-\bigr)\]

The shared steady-state monomer concentration is \([M]_{\text{ss}} \approx 0.1\text{--}0.3\,\mu\)M in resting cells.

Treadmilling: subunit flux through a polar filament

4. Arp2/3 Complex & Dendritic Nucleation

The Arp2/3 complex (seven subunits: Arp2, Arp3, and ARPC1-5) is the universal branched-actin nucleator. It is activated by Nucleation-Promoting Factors (NPFs: WASP, N-WASP, WAVE/Scar, WHAMM, JMY), which deliver a G-ATP-actin monomer to a mother filament and induce a conformational change that assembles a stable branch point. Branches form at a characteristic angle of \(70 \pm 7^\circ\) (Mullins, Heuser & Pollard 1998; Svitkina & Borisy 1999).

Dendritic network model

Pollard & Borisy’s dendritic-nucleation/array-treadmilling model (2003) and Blanchoin, Sykes, Plastino & Boujemaa-Paterski’s review (2014) organize the lamellipodium into a cycle:

PIP₂ and Rac-GTP activate WASP/WAVE NPFs near the membrane.
Active NPF recruits Arp2/3 and G-ATP-actin to the side of an existing mother filament.
A new daughter filament nucleates at 70\(^\circ\) to the mother, barbed end outward.
Barbed-end elongation (6–14 \(\mu\)m/min) pushes the membrane forward.
Capping protein (CapZ) caps filaments after a few \(\mu\)m, preventing bouquet collapse.
ATP hydrolyzes; cofilin preferentially severs ADP-F-actin at the rear, recycling monomers.

\[\langle L_{\text{filament}} \rangle = \frac{v_{\text{elong}}}{k_{\text{cap}}}\]

With \(v_{\text{elong}} \approx 0.3\,\mu\)m/s and \(k_{\text{cap}} \approx 3\,\text{s}^{-1}\),\(\langle L \rangle \approx 0.1\,\mu\)m—consistent with EM measurements.

Arp2/3 dendritic branched network (schematic)

5. Formins: Processive Straight Elongation

Formins (mDia1/2/3, Dia, FMNL, DAAM, INF2, FHOD, Cappuccino) nucleate unbranched actin filaments. They contain an FH1 domain rich in polyproline tracks that captures profilin-actin, and an FH2 donut dimer that encircles the barbed end and remains processively attached as the filament elongates.

\[v_{\text{formin}} = k_{\text{on}}^{\text{PA}} [\text{PA}] - k_{\text{off}}\]

The formin uses an FH1-tethered profilin-actin (PA) pool. Effective \(k_{\text{on}}^{\text{PA}}\)can exceed the diffusion-limited rate by an order of magnitude.

Because formins processively track the barbed end, a single formin adds thousands of subunits without dissociating (Kovar & Pollard 2004; Michelot & Drubin 2011). Formin-elongated filaments are linear, not branched, and populate stress fibers, filopodia, and the contractile ring.

6. Cofilin/ADF: Severing Old Filaments

Cofilin (ADF/cofilin family) preferentially binds F-ADP-actin and changes the filament twist by \(\sim 5^\circ\) per subunit, weakening longitudinal contacts. At high occupancy, cofilin severs filaments at the cofilin/bare-actin boundary (McGough 1997; De La Cruz 2005); at saturating levels it debranches Arp2/3 junctions (Chan, Beltzner & Pollard 2009).

Because cofilin prefers aged (ADP) filaments, the lamellipodial actin network is severed and recycled specifically at the rear of the network, which is the oldest. This sets up the dendritic treadmill: assembly at the front, severing and monomer recycling at the rear.

\[k_{\text{sev}} \;=\; k_0 \,\theta_{\text{cof}}(1-\theta_{\text{cof}})\]

Severing rate peaks at intermediate cofilin occupancy \(\theta_{\text{cof}} \approx 0.5\)—the boundary-density dependence.

7. Capping Proteins & Nucleation-Promoting Factors

CapZ (alpha/beta heterodimer, also called capping protein CP) binds F-actin barbed ends with nanomolar affinity and blocks both addition and loss. In lamellipodia, CapZ is the primary regulator of filament length (\(\approx 0.1\) \(\mu\)m). Tropomodulin caps the pointed end in sarcomeres and erythrocyte membrane skeletons.

Nucleation-Promoting Factors couple upstream signals to Arp2/3 activation. The common architecture is an autoinhibited protein that opens only upon binding of activated Rho-family GTPases, PIP₂, and SH3 adaptors. The C-terminal VCA module then recruits Arp2/3 and G-actin.

WASP / N-WASP: activated by Cdc42-GTP & PIP₂; filopodia and endocytic patches.
WAVE/Scar complex: activated by Rac-GTP; lamellipodia.
WHAMM: ER tubulation and autophagy.
JMY: nuclear and cytoplasmic, p53 coactivator.

8. Cellular Architectures Built from Actin

Lamellipodium

Flat, protrusive actin veil at the leading edge; Arp2/3 branched network; Rac/WAVE driven. Advance rate 0.1–1 \(\mu\)m/s.

Filopodium

Finger-like parallel bundle of 10–30 filaments; fascin bundler, formin-nucleated; Cdc42/WASP driven. Explores substrate and matrix.

Stress fiber

Antiparallel actin/myosin bundle anchored at focal adhesions; contractile, mechanosensitive. Rho/mDia driven, Ser-19 MLC phosphorylation.

Contractile ring

Equatorial actomyosin bundle driving cytokinetic furrow; anillin, septins, formin (mDia2). Constriction speed 0.3 \(\mu\)m/min (human cells).

Cell cortex

Thin (200–500 nm) isotropic actomyosin mesh beneath the membrane; sets cell surface tension \(\sim 10^{-3}\) N/m.

Muscle thin filament

Ordered F-actin decorated with tropomyosin + troponin complex; Ca-regulated myosin binding; sarcomere length 2.0–2.5 \(\mu\)m.

9. Free-Energy Accounting of Actin Polymerization

The free energy of monomer addition to a filament end is \(\Delta G = k_B T \ln(C_c / [M])\). At physiological \([M]_{\text{ss}} \approx 0.2\,\mu\)M and\(C_c^+ \approx 0.12\,\mu\)M, addition is slightly exergonic (~0.5 \(k_B T\)), while pointed-end dissociation is similarly exergonic. Hydrolysis of ATP to ADP + P_ion the filament releases about 13 \(k_B T\), much of which is dissipated; the remainder sets up the free-energy difference that sustains treadmilling.

\[\Delta G_{\text{tread}} \;=\; k_B T \ln\frac{C_c^-}{C_c^+} \;\approx\; k_B T \ln\frac{0.6}{0.12} \;\approx\; 1.6\,k_B T\,\text{per subunit}\]

Treadmilling dissipates about one ATP (\(\sim 20\,k_B T\)) per subunit cycled, most of it as heat. Only a small fraction is captured as directed mechanical work (e.g. against membrane tension).

Force generation by polymerization

Peskin, Odell & Oster’s (1993) Brownian ratchet model explains how an assembling filament pushes on a fluctuating load (e.g. the leading-edge membrane). When the fluctuating load transiently moves away by \(\delta\), a monomer fits in, and the step is rectified. The stall force is

\[F_{\text{stall}} \;=\; \frac{k_B T}{\delta}\,\ln\frac{[M]}{C_c}\]

With \(\delta = 2.7\) nm and \([M]/C_c \sim 50\),\(F_{\text{stall}} \approx 6\) pN per filament — enough for the hundreds-of-pN load of a leading edge distributed across \(\sim 100\) filaments/\(\mu\)m.

Profilin and sequestration

In cells, most G-actin is bound to profilin, a small 14 kDa protein that occludes the pointed face of G-actin. The profilin-actin complex can add to the barbed end but not the pointed end, biasing polymerization strongly in the plus direction. Profilin also accelerates nucleotide exchange on G-actin and presents G-actin to formin FH1 domains. Thymosin-β4, by contrast, simply sequesters G-actin, providing a buffer pool.

Simulation 1: Two-Ended Polymerization & Treadmilling

Explicit integration of a four-reaction scheme (two ends \(\times\) two nucleotide states) with ATP hydrolysis, P_i release, and exchange on free monomer. The system approaches a treadmilling steady state in which barbed-end influx equals pointed-end outflux, sustained by ATP hydrolysis.

Python

script.py152 lines

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

# Two-ended actin polymerization with ATP/ADP pools.
# State variables:
#   [G_T]  : ATP-G-actin free monomer
#   [G_D]  : ADP-G-actin free monomer
#   n_plus : number of plus (barbed) ends
#   n_minus: number of minus (pointed) ends
#   L(t)   : mean filament length (monomers)
#
# Kinetic constants (Pollard 1986, Blanchoin & Pollard 2002; uM^-1 s^-1 or s^-1)
# Barbed end:
kon_plus_T  = 11.6    # ATP-G addition rate
koff_plus_T = 1.4     # ATP-G dissociation
kon_plus_D  = 3.8     # ADP-G addition
koff_plus_D = 7.2     # ADP-G dissociation
# Pointed end:
kon_minus_T  = 1.3
koff_minus_T = 0.8
kon_minus_D  = 0.16
koff_minus_D = 0.27

# Nucleotide exchange on free G-actin (profilin accelerates)
k_exchange = 0.01   # s^-1 (slow without profilin; set to ~10 with profilin)

# ATP hydrolysis on polymer (cap to interior transition)
k_hyd = 0.3     # s^-1 per subunit
# Pi release
k_pi  = 0.003   # s^-1

# Initial conditions
G_T = 20.0      # uM
G_D = 0.0
# Filament density (uM of ends)
F_plus = 0.005
F_minus = F_plus  # same number of filaments

# Fraction of subunits in T, D-Pi, D states (polymer)
F_T  = 0.0
F_DP = 0.0   # ADP-Pi
F_D  = 0.0

dt = 0.01
T  = 300.0
steps = int(T/dt)
times = np.zeros(steps)
G_T_h = np.zeros(steps); G_D_h = np.zeros(steps)
F_T_h = np.zeros(steps); F_DP_h = np.zeros(steps); F_D_h = np.zeros(steps)
J_plus_h = np.zeros(steps); J_minus_h = np.zeros(steps)

for i in range(steps):
    # net elongation rates per end (subunits/s)
    J_plus  = (kon_plus_T  * G_T - koff_plus_T ) * 1.0 + (kon_plus_D  * G_D - koff_plus_D)  * 0.0
    # Assume plus end loses ATP-G at barbed end only if tip is ATP
    # Simple approximation: tip identity = majority of recent additions -> use T tip
    # If T pool low (G_T < C_c^+), let tip become D
    C_c_plus_T = koff_plus_T / kon_plus_T
    C_c_minus_D = koff_minus_D / kon_minus_D
    # Choose simple tip model
    tip_plus_T = 1.0 if G_T > C_c_plus_T else 0.0
    if tip_plus_T > 0.5:
        J_plus = kon_plus_T * G_T - koff_plus_T
        rel_T_plus = 1.0
    else:
        J_plus = kon_plus_D * G_D - koff_plus_D
        rel_T_plus = 0.0
    # Minus end: usually ADP in treadmill
    tip_minus_D = 1.0 if G_D < 5 else 1.0
    J_minus = kon_minus_D * G_D - koff_minus_D

# Flux in monomers added/removed per second (total)
    dF_total = (J_plus + J_minus) * F_plus * dt

# Polymer mass change (as concentration uM)
    # F_plus in uM of ends; J in subunits/s; monomers added contribute to F_T (if ATP tip)
    add_T = kon_plus_T * G_T * F_plus * dt + kon_minus_T * G_T * F_minus * dt
    rem_T = koff_plus_T * F_plus * dt * rel_T_plus + 0.0
    add_D = kon_plus_D * G_D * F_plus * dt * (1.0-rel_T_plus) + kon_minus_D * G_D * F_minus * dt
    rem_D = koff_plus_D * F_plus * dt * (1.0-rel_T_plus) + koff_minus_D * F_minus * dt

# Hydrolysis and Pi release on polymer
    hyd  = k_hyd * F_T  * dt
    rel_pi = k_pi  * F_DP * dt

F_T  = F_T  + add_T - rem_T - hyd
    F_DP = F_DP + hyd   - rel_pi
    F_D  = F_D  + rel_pi + add_D - rem_D

# Update free monomer pools
    G_T = G_T - add_T + rem_T - k_exchange * G_T * dt + k_exchange * G_D * dt
    G_D = G_D - add_D + rem_D + k_exchange * G_T * dt - k_exchange * G_D * dt

# Floor at 0
    G_T = max(G_T, 0.0); G_D = max(G_D, 0.0)
    F_T = max(F_T, 0.0); F_DP = max(F_DP, 0.0); F_D = max(F_D, 0.0)

times[i] = i*dt
    G_T_h[i] = G_T; G_D_h[i] = G_D
    F_T_h[i] = F_T; F_DP_h[i] = F_DP; F_D_h[i] = F_D
    J_plus_h[i] = J_plus; J_minus_h[i] = J_minus

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()

ax1.plot(times, G_T_h, '-', color='#22c55e', linewidth=2, label='ATP-G (free)')
ax1.plot(times, G_D_h, '-', color='#f59e0b', linewidth=2, label='ADP-G (free)')
ax1.set_xlabel('time (s)', color='#cbd5e1')
ax1.set_ylabel('free monomer (uM)', color='#cbd5e1')
ax1.set_title('Free monomer pools', color='#bbf7d0', fontweight='bold')
ax1.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

ax2.plot(times, F_T_h,  '-', color='#22c55e', linewidth=2, label='F-ATP')
ax2.plot(times, F_DP_h, '-', color='#a78bfa', linewidth=2, label='F-ADP-Pi')
ax2.plot(times, F_D_h,  '-', color='#f59e0b', linewidth=2, label='F-ADP')
ax2.set_xlabel('time (s)', color='#cbd5e1')
ax2.set_ylabel('polymer subunits (uM)', color='#cbd5e1')
ax2.set_title('Nucleotide states along filament', color='#bbf7d0', fontweight='bold')
ax2.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

ax3.plot(times, J_plus_h,  '-', color='#22c55e', linewidth=2, label='J(+) plus end')
ax3.plot(times, J_minus_h, '-', color='#ef4444', linewidth=2, label='J(-) minus end')
ax3.axhline(0, color='white', linestyle='-', alpha=0.3)
ax3.set_xlabel('time (s)', color='#cbd5e1')
ax3.set_ylabel('elongation (subunits / s)', color='#cbd5e1')
ax3.set_title('Two-ended elongation rates', color='#bbf7d0', fontweight='bold')
ax3.legend(facecolor='#0f172a', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

# treadmilling: stable net mass with opposing flux
net = (J_plus_h + J_minus_h)
ax4.plot(times, net, '-', color='#22d3ee', linewidth=2, label='J(+) + J(-)')
ax4.axhline(0, color='white', linestyle='-', alpha=0.3)
ax4.set_xlabel('time (s)', color='#cbd5e1')
ax4.set_ylabel('net length change / end (sub/s)', color='#cbd5e1')
ax4.set_title('Treadmilling: opposing end fluxes', 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')
print(f"Steady-state ATP-G = {G_T_h[-1]:.3f} uM   ADP-G = {G_D_h[-1]:.3f} uM")
print(f"C_c(+, ATP) = {koff_plus_T/kon_plus_T:.3f} uM   C_c(-, ADP) = {koff_minus_D/kon_minus_D:.3f} uM")
print(f"Net flux imbalance integral: {np.trapezoid(net, times):.2f} subunits per end")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation 2: Arp2/3 Dendritic Branching Network

Stochastic 2-D simulation of an expanding lamellipodial network: filaments elongate, stochastically produce 70\(^\circ\) Arp2/3 branches, and are terminated by capping with rate constant\(k_{\text{cap}}\). The simulation reproduces the observed mean-length\(v_{\text{elong}}/k_{\text{cap}}\) scaling and the dendritic geometry of real lamellipodia (Svitkina & Borisy 1999).

Python

script.py128 lines

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

# Arp2/3 dendritic branching simulation (Blanchoin & Pollard 2014).
# Each time step:
#   - existing filaments elongate at barbed end at rate k_on * [G]
#   - existing filaments may nucleate a 70-degree branch at density rho_br per length
#   - capping terminates barbed-end elongation at rate k_cap
# Geometry: 2D, reflecting the lamellipodial network.

rng = np.random.default_rng(7)

# Kinetic parameters
kon    = 11.6     # uM^-1 s^-1
koff   = 1.4
G_act  = 10.0     # uM ATP-G-actin
v_elong = (kon*G_act - koff) * 2.7e-3  # um/s  (2.7 nm per subunit)
k_cap  = 3.0      # per second, capping
k_br   = 0.4      # branches per uM per s per filament length um, tunable
rho_Arp23 = 0.5   # uM

# Each filament represented as tuple (x, y, theta, alive)
# theta in radians
filaments = []
# Seed a few filaments at the leading edge (y=0)
for _ in range(5):
    filaments.append({
        'x': rng.uniform(-2, 2),
        'y': 0.0,
        'theta': rng.uniform(np.pi/2 - 0.3, np.pi/2 + 0.3),
        'L': 0.05,  # um
        'alive': True,
        'path_x': [],
        'path_y': [],
    })
    filaments[-1]['path_x'].append(filaments[-1]['x'])
    filaments[-1]['path_y'].append(filaments[-1]['y'])

T_total = 15.0    # seconds
dt = 0.05
steps = int(T_total / dt)

branch_events = 0
cap_events = 0

for step in range(steps):
    new_filaments = []
    for f in filaments:
        if not f['alive']:
            continue
        # elongation step
        dL = v_elong * dt
        f['x'] += dL * np.cos(f['theta'])
        f['y'] += dL * np.sin(f['theta'])
        f['L'] += dL
        f['path_x'].append(f['x'])
        f['path_y'].append(f['y'])
        # branching probability
        p_br = k_br * rho_Arp23 * dL
        if rng.random() < p_br:
            # 70-degree (1.22 rad) branch; random side
            side = rng.choice([-1, 1])
            new_theta = f['theta'] + side * np.deg2rad(70)
            new_filaments.append({
                'x': f['x'],
                'y': f['y'],
                'theta': new_theta,
                'L': 0.0,
                'alive': True,
                'path_x': [f['x']],
                'path_y': [f['y']],
            })
            branch_events += 1
        # capping probability
        if rng.random() < k_cap * dt:
            f['alive'] = False
            cap_events += 1
    filaments.extend(new_filaments)

print(f"Total filaments created: {len(filaments)}")
print(f"Branching events: {branch_events}   Capping events: {cap_events}")

# Plot network
fig, axes = plt.subplots(1, 2, figsize=(13, 6))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes:
    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=0.3)

ax1, ax2 = axes
for f in filaments:
    alpha = 0.9 if f['alive'] else 0.4
    col = '#22c55e' if f['alive'] else '#94a3b8'
    ax1.plot(f['path_x'], f['path_y'], '-', color=col, linewidth=1.2, alpha=alpha)
# Draw leading edge
ax1.axhline(0, color='#f59e0b', linewidth=2, alpha=0.7)
ax1.set_xlabel('x (um)', color='#cbd5e1')
ax1.set_ylabel('y (um)', color='#cbd5e1')
ax1.set_title(f'Arp2/3 dendritic network at t={T_total} s', color='#bbf7d0', fontweight='bold')
ax1.set_aspect('equal')
ax1.set_xlim(-5, 5); ax1.set_ylim(-0.5, 6)

# Filament length distribution
lengths = [f['L'] for f in filaments if f['L'] > 0]
ax2.hist(lengths, bins=30, color='#22c55e', edgecolor='#14532d', alpha=0.85)
ax2.set_xlabel('filament length (um)', color='#cbd5e1')
ax2.set_ylabel('count', color='#cbd5e1')
ax2.set_title('Length distribution (capping-truncated)', color='#bbf7d0', fontweight='bold')

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

# Summary
mean_L = np.mean(lengths) if lengths else 0
print(f"Mean filament length: {mean_L:.3f} um")
print(f"Expected <L> = v_elong / k_cap = {v_elong/k_cap:.3f} um")
# Integrated filament mass over time
density_front = len([f for f in filaments if f['y'] > 2]) / (10.0 * 1.0)  # per um^2 near advance zone
print(f"Leading-zone filament density: {density_front:.2f} per um^2")
# Use trapezoid over branch-flux time course (placeholder)
t_bins = np.linspace(0, T_total, 10)
br_flux = np.array([branch_events * ti / T_total for ti in t_bins])
print(f"Cumulative branching integral: {np.trapezoid(br_flux, t_bins):.2f} events-s")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Key References

• Oosawa, F. & Asakura, S. (1975). Thermodynamics of the Polymerization of Protein. Academic Press.

• Pollard, T.D. (1986). “Rate constants for the reactions of ATP- and ADP-actin with the ends of actin filaments.” J. Cell Biol., 103, 2747–2754.

• Pollard, T.D. & Borisy, G.G. (2003). “Cellular motility driven by assembly and disassembly of actin filaments.” Cell, 112, 453–465.

• Blanchoin, L., Boujemaa-Paterski, R., Sykes, C., & Plastino, J. (2014). “Actin dynamics, architecture, and mechanics in cell motility.” Physiol. Rev., 94, 235–263.

• Mullins, R.D., Heuser, J.A., & Pollard, T.D. (1998). “The interaction of Arp2/3 complex with actin.” PNAS, 95, 6181–6186.

• Svitkina, T.M. & Borisy, G.G. (1999). “Arp2/3 complex and actin depolymerizing factor/cofilin in dendritic organization and treadmilling of actin filament array in lamellipodia.” J. Cell Biol., 145, 1009–1026.

• Kovar, D.R. & Pollard, T.D. (2004). “Insertional assembly of actin filament barbed ends in association with formins produces piconewton forces.” PNAS, 101, 14725–14730.

• Carlier, M.-F. & Pantaloni, D. (1986). “Direct evidence for ADP-Pi-F-actin as the major intermediate in ATP-actin polymerization.” Biochemistry, 25, 7789–7792.

• Spudich, J.A. & Watt, S. (1971). “The regulation of rabbit skeletal muscle contraction.” J. Biol. Chem., 246, 4866–4871.

• Kabsch, W. et al. (1990). “Atomic structure of the actin:DNase I complex.” Nature, 347, 37–44.

• Kuhn, J.R. & Pollard, T.D. (2005). “Real-time measurements of actin filament polymerization by TIRF microscopy.” Biophys. J., 88, 1387–1402.

• Chan, C., Beltzner, C.C., & Pollard, T.D. (2009). “Cofilin dissociates Arp2/3 complex and branches from actin filaments.” Curr. Biol., 19, 537–545.

• De La Cruz, E.M. (2005). “Cofilin binding to muscle and non-muscle actin filaments.” J. Mol. Biol., 346, 557–564.

• Michelot, A. & Drubin, D.G. (2011). “Building distinct actin filament networks in a common cytoplasm.” Curr. Biol., 21, R560–R569.

• Alberts, B. et al. (2015). Molecular Biology of the Cell, 6th ed., Garland Science.

← Module 0: Overview & History Module 2: Microtubules →