--- title: "How ATP gives random transitions a direction" description: "Construct a reversible two-state motor, impose local detailed balance, and derive its stationary speed and stall load." canonical_url: "https://sandboxphysics.com/stories/single-molecule/chemical-motor/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/single-molecule/chemical-motor/" last_modified: "2026-10-28" --- [Single-molecule physics from bead motion to motors](https://sandboxphysics.com/stories/single-molecule/) L3 · Chapter 6 # How ATP gives random transitions a direction Sandbox Physics · 2026-10-28 Thermal motion permits both forward and backward steps. A chemical-potential difference maintained by ATP and its products favors forward cycles; an opposing load spends that advantage. Keeping both directions in the network lets one model explain motion, reversal and stall. ## Before you begin Chapter four on transition rates and waiting times. A state network is a graph of effective molecular configurations and allowed transitions. One forward cycle here consumes one chemical free-energy increment and advances one step. [Review the preceding chapter](https://sandboxphysics.com/stories/single-molecule/loading-rate-inference/) - Explain the need for reverse transitions. - Relate cycle-rate ratios to chemical and mechanical driving. - Derive stationary current, velocity and stall. ## A visible step can contain hidden waiting At lattice position $n$, call the states $A_n$ and $B_n$. A chemical edge connects them with forward rate $u_0$ and reverse rate $w_0$. A mechanical edge connects $B_n$ to $A_{n+1}$ with rates $u_1$ and $w_1$. The mechanical displacement is $d$. These are coarse chemical–mechanical states, not assignments of every nucleotide-bound structure of a particular protein. A bead records displacement and can miss a chemical transition at fixed position. $A_n$ Chemical transition $B_n$ Mechanical step $A_{n+1}$ Original schematic of the teaching cycle. A chemical transition changes state at one position; a mechanical transition advances to the next position. Both edges allow reverse transitions. Reading: U. Seifert · K. Svoboda & S. M. Block ## The forward-to-backward cycle ratio is constrained Define opposing load $F$ as positive. Advancing one step delivers work $Fd$ against it. One forward cycle releases chemical free energy $\Delta\mu$. Multiplying forward-to-reverse rate ratios around the cycle cancels internal-state free energies. The remaining dimensionless quantity $\mathcal A$ is the cycle affinity. A positive value favors forward cycling; zero removes this preference. Local detailed balance constrains rate ratios. It leaves kinetic scales to be determined separately. $$ \mathcal A=\ln\frac{u_0u_1}{w_0w_1}=\frac{\Delta\mu-Fd}{k_BT} $$ Assume fixed temperature, maintained chemical reservoirs and a tightly coupled single cycle. The relation does not supply protein-specific rate constants. Reading: U. Seifert ## Where ATP enters the teaching network Let $c$ be ATP concentration divided by one micromolar. All rates below are in inverse seconds. The chemical forward rate grows with concentration; opposing force slows forward mechanical transitions and accelerates reverse ones. We choose $d=8\;\mathrm{nm}$ and $k_BT=4.114\;\mathrm{pN\,nm}$ as teaching scales. These choices give $\Delta\mu=k_BT\ln(c/0.01)$. The reference $0.01\;\mathrm{\mu M}$ absorbs fixed ADP, phosphate and standard chemical potentials. It is not a universal cellular equilibrium concentration. A real assay must specify the products and solution conditions. $$ u_0=3c,\quad w_0=3,\quad u_1=60e^{-Fd/(2k_BT)},\quad w_1=0.6e^{Fd/(2k_BT)} $$ These rates are not fitted kinesin parameters. Reversible chemical and mechanical channels allow equilibrium, stall and reversal within one model. Reading: U. Seifert ## From state occupancies to speed At stationarity, incoming and outgoing probability balance. The total rate leaving $A$ is $u_0+w_1$; the total leaving $B$ is $w_0+u_1$. Equating probability flows and normalizing gives the occupancies. Mechanical forward flow is $p_Bu_1$ and reverse flow is $p_Aw_1$. Their difference times step length gives mean speed. For $100\;\mathrm{\mu M}$ ATP and $1.50\;\mathrm{pN}$ load, the four rates are approximately $300$, $3.00$, $14.0$ and $2.58$ per second. Most time is spent in $B$. The net current is about $13.1$ cycles per second, giving $105\;\mathrm{nm\,s^{-1}}$. Individual four-second records fluctuate around this long-time mean. Increasing load reduces the current to zero while transitions continue. At the same ATP concentration the chemical affinity is $\ln10000$, giving a stall load of $4.74\;\mathrm{pN}$. Real kinesin data can involve reaction pathways omitted by this minimal graph. $$ S=u_0+w_0+u_1+w_1,\qquad p_A=\frac{w_0+u_1}{S},\qquad p_B=\frac{u_0+w_1}{S} $$ Probability balance determines the fraction of time spent in each state. $$ J=p_Bu_1-p_Aw_1=\frac{u_0u_1-w_0w_1}{S},\qquad v=dJ,\qquad F_{\rm stall}=\frac{\Delta\mu}{d} $$ These relations apply to the periodic two-state graph. Additional cycles can consume ATP without motion, requiring separate calculations of stall and consumption. Reading: U. Seifert · K. Svoboda & S. M. Block ## Efficiency and power behave differently near stall During forward operation, net chemical input per cycle is $\Delta\mu$ and mechanical output is $Fd$. Their ratio is the tight-coupling efficiency. It approaches one near stall, but current approaches zero and output power vanishes. A finite record still contains forward and backward fluctuations. Long-time or repeated-record averages reveal the small net speed. Testing equilibrium requires setting both zero load and the reference ATP concentration. $$ P_{\rm out}=JF d,\qquad P_{\rm in}=J\Delta\mu,\qquad \eta=\frac{Fd}{\Delta\mu} $$ The efficiency expression applies to steady operation with positive input and output. At stall both powers are zero; their direct ratio is undefined. Reading: U. Seifert ## One default trajectory can outpace its long-time mean The synthetic default uses $100\;\mathrm{\mu M}$ ATP, $1.50\;\mathrm{pN}$ load and a $60\;\mathrm{Hz}$ digital detector cutoff. Seed $4919$ generates $2001$ samples over four seconds. Position and step counts in the table are hidden model truth. Fitted speed uses only detector positions, excluding the first $0.10\;\mathrm{s}$. This particular trajectory advances faster than the long-time mean. Random molecular waiting changes the step count in a finite four-second record, so different seeds give different fitted speeds. Compare stationary theory with averages of independent records. Synthetic motor default: latent truth, detector inference and stationary theory. | Quantity | Result | | --- | --- | | Hidden motor displacement at four seconds | $520\;\mathrm{nm}$ | | Forward / reverse step counts | $65\,/\,0$ | | Fitted speed of this detector record | $127.855\;\mathrm{nm\,s^{-1}}$ | | Long-time stationary speed | $104.628\;\mathrm{nm\,s^{-1}}$ | Reading: Sandbox Physics ## Try the prediction ### [Balance chemical driving against load](https://sandboxphysics.com/labs/atp-molecular-motor/) 1. Choose Clear forward steps. Keep opposing load fixed and reduce ATP concentration alone. Predict which edge changes, then inspect the waits and displacement. 2. Increase opposing load, then choose Steps at stall. Compare forward and backward events and distinguish finite-record drift from stationary mean speed. 3. Set load to zero and ATP concentration to $0.01\;\mathrm{\mu M}$, removing chemical driving in this network. Export repeated records and compare forward and reverse flows on each edge. ## Exercises and explanations 1. Does doubling all four rates change stall force? Read the explanation No. The rate ratios and affinity stay fixed. Occupancies remain unchanged, current and speed double, and waiting times halve. Thermodynamic ratios do not fix kinetic time scales. 2. Do random forward and backward steps contradict zero current at stall? Read the explanation No. Zero current means cancellation on average. Individual trajectories fluctuate. This single-cycle model is at cycle equilibrium at stall; a real motor with dissipative branches may still consume ATP. [Previous: Inferring a barrier from many unfolding events](https://sandboxphysics.com/stories/single-molecule/loading-rate-inference/) [Next: What remains measurable when steps blur together?](https://sandboxphysics.com/stories/single-molecule/hidden-steps/) ## Sources and calculation scope 1. [Sandbox Physics, FR19 synthetic default records and SHA-256 manifest (2026-10-28)](https://sandboxphysics.com/data/fr19/manifest.json) Frozen default parameters, seeds and saved observation channels for the three teaching Labs. All records are synthetic; the manifest lists each JSON dataset and its checksum. 2. [K. Svoboda & S. M. Block, Force and Velocity Measured for Single Kinesin Molecules, Cell 77(5), 773–784 (1994)](https://doi.org/10.1016/0092-8674(94%2990060-4) Figures 1 and 4–5 describe the bead assay and linker compliance. The published experiment used a stationary trap. Our ideal-clamp two-state motor illustrates chemical affinity and load response with independently chosen rates. 3. [U. Seifert, Stochastic Thermodynamics, Fluctuation Theorems, and Molecular Machines, arXiv:1205.4176v1 (2012)](https://arxiv.org/html/1205.4176v1) Section 9.4.5, equations (240)–(244), gives local detailed balance. Section 10.5, equation (273), gives unicyclic input/output power. Our limited recomputation applies these relations to a chosen network. Figure 3 illustrates a general multicycle network; our two-state diagram is an original illustration.