---
title: "Hidden Entropy: How Much Dissipation Can You See? Interactive Lab | Sandbox Physics"
description: "How much of an enzyme’s dissipation can an observer prove from what it records?"
canonical_url: "https://sandboxphysics.com/labs/hidden-entropy-network/"
language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/labs/hidden-entropy-network/"
last_modified: "2026-10-29"
---

FR25 · Irreversibility

# Hidden Entropy: How Much Dissipation Can You See?

**How much of an enzyme’s dissipation can an observer prove from what it records?** An enzyme that burns ATP dissipates at a steady rate, but an observer who sees only part of it can prove only a lower bound. Counting completed cycles proves almost all of it near equilibrium and less under strong drive; snapshots prove less as the interval grows; a signal that only shows bound or empty proves nothing.

An ATP-driven enzyme cycles through three states. Compare the entropy production with what cycle counts, snapshots or a bound/empty signal can prove.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A single enzyme goes from empty (E) to ATP-bound (ET), releases phosphate to become ADP-bound (ED), then releases ADP and is empty again, consuming one ATP per cycle. Each step obeys local detailed balance on a tilted free-energy washboard; the attempt rates and barrier heights are schematic. 1024 independent enzymes are simulated exactly with the Gillespie algorithm; the bars and readouts are exact model values and the dots are estimates from the simulated records. No measured data are shown.

One enzyme, three records

What each record can prove

Slower snapshots prove less

Stronger fuel hides more from the counts

More numbers and how they are computed

- **Share proved by counting cycles**: $0$
- **Share proved by snapshots**: $0$
- **Entropy production from the simulated full records**: $0$
- **Counting bound from the simulated cycle counts**: $0$
- **Snapshot bound from the simulated snapshots**: $0$
- **Bound-or-empty estimate from the simulated signal**: $0$
- **Cycle rate**: $0$
- **Spread of the cycle count**: $0$
- **Uncertainty product and its minimum for three states**: $0$
- **Time spent in E, ET and ED**: $0$
- **Mean dwell time**: $0$
- **Records times duration**: $0$

Exact values come from the stationary distribution, the tilted generator and the matrix exponential of the three-state rate matrix. Simulated estimates use every jump of 1024 Gillespie records; the snapshot estimator adds one half to each pair count. The bound-or-empty record is exactly reversible because every exit from the bound states returns to the single empty state.

Physics tutorial

## Dissipation you cannot see

**Background** An enzyme that burns ATP goes round the same three states again and again. Each turn releases the free energy of one ATP into the water as heat.

**Why it matters** Experiments rarely see every state. A fluorescence signal may only tell whether something is bound; a camera may only take a frame every few milliseconds. How much of the dissipation does such a record still reveal?

Start with the essentials

- **Focus question**: How much of an enzyme’s dissipation can an observer prove from what it records?
- **One-sentence intuition**: Every partial record gives a lower bound on the entropy production. Counting cycles proves almost all of it near equilibrium but only a fraction under strong drive. Snapshots lose it as the interval grows. A bound-or-empty signal proves nothing, because its record looks the same played backwards.

### Core mathematical model

#### Entropy production of the cycle

$$
\sigma=\sum_{i\ne j}p_i w_{ij}\ln\frac{p_i w_{ij}}{p_j w_{ji}}=J\,\Delta\mu
$$

In the steady state the sum over all jumps reduces to the cycle rate $J$ times the fuel per cycle.

#### Bound from counting cycles

$$
\sigma\ge\frac{J^2}{D}
$$

Here $D$ measures how much the number of completed cycles spreads over time. The thermodynamic uncertainty relation turns a precise count into a minimum cost.

#### Bound from snapshots

$$
\sigma\ge\frac{1}{\Delta t}\sum_{i\ne j}P_{ij}\ln\frac{P_{ij}}{P_{ji}}
$$

$P_{ij}$ is the probability of seeing state $i$ and then state $j$ one interval later. A blurred record can only lose irreversibility, never gain it.

### Common difficulties

#### No current, no dissipation?

**Typical misconception** If the observed signal shows no net flow, the system is at equilibrium.

**Better mental model** The bound-or-empty signal has no net flow and is exactly time-symmetric, yet the enzyme keeps burning ATP. The dissipation is hidden inside the bound state.

### Run the experiment

1. 01

   #### Watch the walker

   Follow the large bead down the tilted staircase of wells.

   **What to observe:** Most hops go downhill and flash red as heat enters the water; occasional uphill hops flash blue. The small dots drift and spread.

2. 02

   #### Slow the camera

   Choose Slow camera misses the cycle.

   **What to observe:** The snapshot bound drops to a few percent, while counting cycles still proves almost half.

3. 03

   #### Reshape the landscape

   Drag the orange or pink wells up and down.

   **What to observe:** The entropy production changes with the cycle rate, but the bound-or-empty signal stays at zero.

Physical conclusion

Irreversibility is a property of the record as well as of the system: coarser records can only show less of it.

## Related experiments and articles

Stochastic thermodynamics · Reading path

### From a trapped bead to free energy from irreversible pulls

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How much entropy a fuel-driven enzyme produces each second**](https://sandboxphysics.com/stories/stochastic-thermodynamics/entropy-production-in-networks/)
2. [Article **How much dissipation an observer can see when part of the motion is hidden**](https://sandboxphysics.com/stories/stochastic-thermodynamics/hidden-dissipation/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)