2D Axonal Transport: Kinesin vs Dynein Tug-of-War, Gillespie Stochastic Motors, Stall Force and Cargo Run Length Lab

Presets

Live readouts

Graphs

Position of every cargo against time

Velocity and bound motors of the highlighted cargo

Force-velocity of the two motor teams: the cargo moves where they cross

Net displacement per second, all cargos (two peaks = switching)

What is happening now

Starting.

Numerical checks

running...

How to use

Pick a preset or set the number of kinesin and dynein motors on the cargo. The picture has two parts: the strip at the top is the whole 40 um axon with every cargo (yellow = the highlighted one), and the large view is a molecular close-up of the highlighted cargo: motors that are bound to the microtubule are drawn with their feet on the tubulin lattice, free motors dangle. Arrows show the pulls of the two teams. Graph 1 is position against time; graph 2 the velocity and the bound motor counts; graph 3 the force-velocity curves (the cargo moves where they cross); graph 4 collects the displacement of all cargos each second - two peaks mean back-and-forth switching. Things to try: (1) one kinesin, and read the mean run length in the checks; (2) the Balanced preset (2 kinesin vs 11 dynein) and watch the cargo reverse; (2b) 4 vs 4, to see that equal numbers are not equal strength; (3) lower the ATP to 30 uM; (4) coat the track with tau and see dynein win; (5) find the number of dynein that balances 2 kinesin (12 pN / 1.1 pN = 11). Space pauses, R restarts, the mouse wheel zooms the close-up.

The science

Neurons make their proteins in the cell body, but the synapse can be a metre away. Vesicles, mitochondria and lysosomes are hauled along microtubules by two motor families: kinesin-1 walks toward the plus end (the synaptic terminal) and cytoplasmic dynein toward the minus end (the cell body). Most cargos carry both, so the cargo is a rope in a tug-of-war. This lab is the standard Muller-Klumpp-Lipowsky model (PNAS 105, 4609, 2008) solved with an exact stochastic simulation.

Force-velocity of one motor. A bound motor pulling against a load F steps more slowly. Below stall the relation is close to linear; kinesin resists backward sliding strongly, dynein is dragged backward faster:

kinesin: F_K(v) = F_s(1 − v/v_f) for v ≥ 0, F_s(1 − v/v_b) for v < 0 (F_s = 6 pN, v_f = 1000 nm/s, v_b = 6 nm/s) dynein: F_D(v) = F_s(1 − |v|/v_f) for v ≤ 0, F_s(1 + v/v_b) for v > 0 (F_s = 1.1 pN, v_f = 650 nm/s, v_b = 72 nm/s)

Force balance. All motors that are bound move together at one cargo speed v. With n_k kinesin and n_d dynein bound, the net force is zero (drag is about 10⁻³ pN, negligible):

n_k F_K(v) = n_d F_D(v), load per motor = n_kF_K/n_k or n_dF_D/n_d

The root is found by bisection (graph 3 shows the crossing). For one kinesin against one dynein it has a closed form, v = (F_sK − F_sD)/(F_sK/v_fK + F_sD/v_bD) = 230 nm/s, and for equal team stall forces (n_kF_sK = n_dF_sD) the cargo stalls.

Load-dependent detachment. A motor under load lets go faster, e(F) = e₀ exp(F/F_d) (e₀ = 1.0 /s and F_d = 3 pN for kinesin; 0.27 /s and 0.75 pN for dynein). Free motors on the cargo re-attach at constant rates π₀ (5 /s kinesin, 1.6 /s dynein). When a team is stretched in a tug-of-war, the weaker side detaches under the stronger side's pull, which is why the cargo moves in bursts.

Stochastic simulation (Gillespie). The state is (n_k, n_d). Four event types have the rates (N_k − n_k)π_K, (N_d − n_d)π_D, n_ke_K, n_de_D. The next event time is −ln(1−u)/R_total; the cargo moves at the force-balance speed between events, so the position is integrated exactly, with no time-step error. Waiting times are memoryless, so the playback can be cut into frames without changing the statistics.

ATP and the track. Free speeds follow Michaelis-Menten, v_f([ATP]) = v_max[ATP]/(K_M + [ATP]) with K_M = 70 µM for kinesin and 250 µM for dynein (typical in-vitro values; the table speeds are those at 1 mM). Stall forces are left unchanged. The tau slider multiplies kinesin binding by (1 − 0.7 tau) and detachment by (1 + 1.5 tau); the dynein-inhibition slider reduces dynein speed and binding. Both are illustrative scalings of real, qualitative effects, not measured curves.

Numerical checks run in the page. (1) The mean run length of a lone kinesin, from 4000 simulated runs, against v₀/e₀ = 1000 nm; (2) the force-balance solver for 1 kinesin versus 1 dynein against its closed form; (3) 11 kinesin against 60 dynein (equal stall force 66 pN) has zero speed; (4) the time-averaged bound fraction of an unopposed 6-kinesin team against the binomial value π/(π + e₀) = 5/6; (5) the Michaelis-Menten speed at [ATP] = K_M equals half of v_max.

Simplifications. The model is one-dimensional: no sideways steps, no switching between protofilaments or microtubules, and no cargo rotation. Forces are shared equally among the bound motors of a species; coupling stiffness (the elasticity of the stalks) and the dynein catch-bond or super-stall behaviour are not included. Parameters are the published in-vitro means of Muller et al.; in cells, adaptors, other motors (kinesin-3, myosin V), MAP crowding and organelle drag change them. The 40 um axon is a stylised straight segment; the cargo that reaches an end is re-launched from the middle. The motors in the pictures are drawn schematically at the correct size (kinesin 80 nm, vesicle as set).