FR19-C · Chemistry takes a step

ATP Molecular Motor

How does ATP drive a molecular motor against a load? ATP reactions maintain a chemical imbalance that favors forward cycles. An opposing load makes each forward step cost mechanical work and increases the relative chance of backward motion. At stall, forward and backward steps can continue even when their average motion cancels.

Pull back on a motor carrying a bead in an ideal force clamp. Watch its random steps slow, reverse near stall, or disappear into a narrow-band detector trace.

Motor displacement $80\,\mathrm{nm}$

Speed fitted to detector record $100\,\mathrm{nm\,s^{-1}}$

Model stall load $4.74\,\mathrm{pN}$

Forward / backward steps $12\,/\,1$

Pull against the motor

Opposing load

F\;(\mathrm{pN})

1.5

ATP concentration

c_{\rm ATP}\;(\mathrm{\mu M})

100

Detector bandwidth

f_c\;(\mathrm{Hz})

60

Record time

t\;(\mathrm{s})

1.2

Drag the gold ring left to pull harder. The track scrolls so the motor stays in view.

Load protocol and random record

Load protocolLoad after the steppiconewtonsRandom seed

Four seconds of simulated time play in sixteen seconds. Changing bandwidth preserves the molecular event sequence and the sampled bead trace.

Model, geometry and observation

The bead is linked to the motor by a linear spring. A prescribed force pulls left; its equilibrium lag is the load divided by linker stiffness. The gold ring is a schematic ideal force-clamp handle, not a simulated feedback-controlled laser focus. The bead, linker and track are magnified separately.

$d=8\,\mathrm{nm},\quad k_{\rm B}T=4.114\,\mathrm{pN\,nm}$

$k_\ell=0.06\,\mathrm{pN\,nm^{-1}},\quad \gamma=2.8\times10^{-5}\,\mathrm{pN\,s\,nm^{-1}}$

The bead follows an overdamped thermal process with these fixed parameters. Samples are taken every two milliseconds, then pass through a discrete one-pole detector. The chart separates the hidden motor, bead samples and detector output.

ATP changes the forward chemical rate by mass action. ADP and phosphate stay fixed; the illustrative equilibrium ATP concentration is 0.01 micromolar. The network is tightly coupled and has no futile cycles or detachment. It is not a fit to kinesin.

Use left and right arrow keys on the scene to adjust the opposing load.

Model assumptions

A reversible two-state teaching network couples one ATP reaction to an eight-nanometer period. Rates and the equilibrium ATP concentration are illustrative, with fixed ADP and phosphate reservoirs. An ideal external force clamp sets the motor load; a passive bead and sampled one-pole detector model observation. The motor rates use that imposed load rather than fluctuating linker tension. Detachment, futile cycles, head mechanics and feedback errors are omitted. All records are simulated.

Synthetic four-second record

Which steps reach the detector?

Hidden motor positionSampled thermal beadDetector output

Chemical state ABefore / after load step

v_{\rm ss}=100\,\mathrm{nm\,s^{-1}}

How the speed was inferred

The primary fitted speed is a straight-line least-squares fit to detector positions, excluding the first tenth of a second and, for a force step, the first tenth after that step. The two segment fits and both stationary network predictions are saved. A single short trajectory has stochastic uncertainty; agreement is judged across repeated records.

The cyan trajectory and step counts are simulation truth. They are not inferred from the detector. The saved record includes all reversible reaction events, raw bead samples, filtered samples, seed, rate constants, protocol and model version.

\mathcal A=6.29

Continue the single-molecule course

Start with the essentials

Focus question: How does ATP drive a molecular motor against a load?
One-sentence intuition: ATP biases a reversible cycle forward; an opposing load biases it backward.

Core mathematical model

The two reactions in one spatial period

A_n\mathop{\rightleftharpoons}^{u_0}_{w_0}B_n\mathop{\rightleftharpoons}^{u_1}_{w_1}A_{n+1}

The first reaction exchanges chemical free energy with ATP, ADP and phosphate reservoirs. The second moves the motor by eight nanometers. Both have reverse reactions. The drawn lobes mark the motor; they do not resolve a two-head gait.

How chemistry and force compete

\ln\frac{u_0u_1}{w_0w_1}=\ln\frac{c_{\rm ATP}}{c_{\rm eq}}-\frac{Fd}{k_{\rm B}T}

Load is positive when it opposes motion to the right. The chemical reservoirs set the ATP equilibrium concentration to the illustrative value 0.01 micromolar. A positive cycle affinity favors forward cycles; a zero affinity balances their long-time currents.

Stall in a tightly coupled teaching network

F_{\rm stall}=\frac{k_{\rm B}T}{d}\ln\frac{c_{\rm ATP}}{c_{\rm eq}}

At stall, net speed and output power vanish, but individual forward and backward steps continue. Real motors may keep hydrolyzing ATP at stall through additional reaction cycles; those cycles are absent here.

Common difficulties

A stalled motor still fluctuates

Typical misconceptionZero speed means that every chemical reaction and every molecular step has stopped.

Better mental modelZero speed is a statement about the difference between forward and backward step currents. The network can keep making steps in both directions. At equilibrium each edge balances separately; a finite record still fluctuates.

Run the experiment

01
Predict the effect of pulling harder
Start with Clear forward steps. Drag the gold force-clamp ring to the left, or increase Opposing load. Keep ATP concentration fixed.
What to observe: The model stall load stays fixed while forward motion slows. Near that load, a finite trace can drift either way because the two random step currents nearly cancel.
02
Let the detector hide the steps
Compare Clear forward steps and Blurred steps. They share the same seed, ATP concentration and load; only Detector bandwidth changes.
What to observe: The cyan motor trajectory is unchanged. The gold detector trace smooths short plateaus and lags behind sudden jumps. The fitted speed uses only the saved gold samples.
03
Test a new load protocol
Choose A sudden heavier load. Predict the direction of motion after two seconds, then compare both halves of the saved record with the two steady-load predictions.
What to observe: The reaction rates change at the load step. The bead also shifts backward as its elastic linker takes up the larger force, so the short mechanical transient should be excluded from a long-time drift fit.

ATP Molecular Motor

Which steps reach the detector?