🧬 biology

Low-frequency output fluctuations in an open exclusion process with particle pausing

This paper demonstrates that in an open exclusion process with reversible particle pausing, low-frequency output noise exhibits a strong nonmonotonic dependence on pausing rates, peaking at a universal, order-one mean paused population of approximately 1.5–2 particles regardless of system size, a phenomenon driven by the interplay between slow-defect kinetics and traffic-jam reorganization.

Original authors: Quentin Thommen

Published 2026-08-11

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Quentin Thommen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a busy highway where cars are trying to get from point A to point B. In the world of physics, this is often modeled as a "traffic jam" of particles moving in one direction. Scientists call this the "Totally Asymmetric Simple Exclusion Process" (TASEP). Think of it like a single-lane road where cars can't overtake each other; if the car in front stops, the one behind it must stop too. Usually, we just care about the average speed: how many cars get through per hour? But in real life, traffic isn't just about average speed; it's about the pattern of the flow. Sometimes traffic is smooth, and sometimes it's a chaotic stop-and-go mess, even if the total number of cars passing by is the same.

This paper explores what happens when those "cars" (which could be tiny molecular machines like ribosomes building proteins) suddenly decide to take a break. They don't just stop because of a red light; they have an internal state where they pause, perhaps to fix a mistake or wait for a signal, before jumping back into the flow. The big question is: How do these random pauses change the rhythm of the traffic? Does a few pauses make the flow chaotic, or does it just slow everything down evenly? Understanding this helps us figure out how cells manage to produce proteins reliably, or why sometimes the production line suddenly goes haywire, even if the average speed looks fine.

The Story of the Stuttering Traffic Jam

In this study, the author, Quentin Thommen, sets up a virtual traffic simulation to see what happens when particles on a one-dimensional track can switch between "active" (zooming along) and "paused" (stuck in place) states. The goal was to count how many particles successfully exit the track over time and look at the noise in that count. Imagine you are counting cars leaving a parking garage. If they leave at a steady pace, your count goes up smoothly. If they leave in bursts followed by long silences, your count jumps up and down wildly. The paper asks: When does this wild jumping happen?

The simulations reveal a surprising twist. As the author increases the rate at which particles pause, the average number of cars getting through the exit drops smoothly, just as you'd expect. However, the "noise" or unpredictability of the exit count doesn't just get bigger and bigger. Instead, it behaves like a rollercoaster. It starts low, shoots up to a massive peak, and then crashes back down.

The most exciting discovery is where this peak happens. The chaos is at its absolute worst not when the road is completely gridlocked, and not when it's empty, but when there is, on average, about 1.5 to 2 paused particles on the track at any given time. The track length in the simulation varied from 50 to 500 sites, but this "sweet spot" for chaos always hovered around that tiny number of paused cars.

To understand why, think of the paused particles as "defects" or "roadblocks."

If there are almost no pauses (0 paused particles): The traffic flows smoothly. You get a steady stream of cars. No noise.
If there are too many pauses (say, 8 or 10): The road is permanently clogged. The cars are stuck in a massive, unchanging jam. The flow is slow, but it's consistently slow. The exit count is low, but it's predictable.
The "Sweet Spot" (1.5 to 2 paused particles): This is the chaotic middle ground. Here, the road is usually clear, but occasionally, a single car stops. Because cars can't overtake, that one stopper creates a temporary traffic jam behind it. Then, that car unpauses, the jam dissolves, and traffic flows again. But then another car stops, and a new jam forms. The system is constantly flipping between "free-flowing" and "jammed." This switching creates huge fluctuations in the exit count. The noise is highest because the system is spending equal time in two very different states.

The paper also digs into how long these pauses last. The author found that the "noise" isn't just about how many cars are paused, but how long they stay paused. If the cars unpausing is slow, the traffic jams last longer, and the noise gets even louder. The time it takes for a pause to appear and disappear acts like a clock that sets the rhythm of the chaos.

One of the key takeaways is that you can't just look at the average number of paused cars to predict the noise. The paper explicitly rules out the idea that the noise is simply a result of the average traffic density. Instead, the noise comes from the fluctuations in the size of the biggest traffic jam. When the paused population is just right (around 1.5 to 2), the size of the traffic jam swings wildly from "nothing" to "huge" and back again. This swinging is what creates the maximum noise.

The study uses a "minimal theory" to explain this, comparing the paused particles to a simple birth-and-death process (like people entering and leaving a room). This simple math predicts that the peak noise should happen when there are about 1.50 paused particles. The simulations confirmed this is very close to the truth (finding it around 1.5 to 2), though the real system is a bit more complex because the cars interact with each other.

In the end, this paper shows that in a system of interacting particles, the most chaotic moments happen when the system is balanced on a knife-edge between being free and being stuck. It's a regime where a single pause can trigger a massive reorganization of the whole line, creating a "stop-and-go" rhythm that is far more unpredictable than a simple slowdown. For biology, this suggests that cells might experience wild variations in protein production not just because of how many ribosomes are stuck, but because of the specific timing of when they get stuck and unstick, creating a fluctuating supply of finished products that could affect how genes are expressed.

Technical Summary: Low-frequency output fluctuations in an open exclusion process with particle pausing

Problem Statement
Driven transport in interacting particle systems, such as the Totally Asymmetric Simple Exclusion Process (TASEP), converts local stochastic events into collective fluctuations. While the standard TASEP describes ribosome and polymerase traffic, biological systems often involve slow internal states where particles reversibly switch between active and paused states (e.g., transcriptional pausing). Previous work established that reversible pausing reduces stationary current and creates a finite-size regime where the mean number of paused particles ( $N_p$ ) becomes the relevant control variable, particularly when $N_p$ is of order unity. However, the temporal statistics of the output (completed exits) in this regime remain unresolved. Specifically, it is unclear whether the finite-size pausing regime identified by mean current possesses a distinct signature in the stationary statistics of completed exits, and how slow internal states reshape the temporal organization of the output beyond simply lowering the mean throughput.

Methodology
The study employs a finite open TASEP on a lattice of length $L$ ( $50 \le L \le 500$ ). Particles hop unidirectionally with rate $\epsilon$ , enter at rate $\alpha$ , and exit at rate $\beta$ . Crucially, particles reversibly switch between an active state ( $A$ ) and a paused state ( $P$ ) with rates $k_p$ (pause) and $k_u$ (unpause). A paused particle at the exit site must resume activity before leaving.

The dynamics are simulated using an event-driven kinetic Monte Carlo algorithm. The primary observables are:

Output Counting Statistics: The cumulative number of exits $N(t)$ is analyzed over observation windows of duration $T_w$ . The study focuses on the window-dependent Fano factor, $F(T_w) = \text{Var}(N_w)/\langle N_w \rangle$ , to distinguish short-time counting noise from slow output modulation.
Structural Observables: The number of paused particles ( $N_p = L\rho_{\text{paused}}$ ), the size of the largest contiguous particle cluster ( $C_{\text{max}}$ ), and residence times of specific states (e.g., pause-free vs. pause-containing, or weakly vs. strongly congested).
Analytical Approximation: A minimal "constant-birth, linear-death" model is constructed to approximate the pause number dynamics as an immigration-death process, aiming to rationalize the location of the noise maximum.

Key Contributions and Results

Non-monotonic Low-Frequency Noise: While increasing the paused population ( $N_p$ ) smoothly lowers the mean stationary current ( $J$ ), the low-frequency Fano factor ( $F_\infty$ ) exhibits a strong non-monotonic behavior. $F_\infty$ reaches a pronounced maximum when the mean number of paused particles is of order unity ( $N_p \simeq 1.5\text{--}2$ ). This maximum persists across lattice lengths $L=50$ to $500$.
Finite-Size Scaling: The microscopic pausing rate ( $k_p$ ) required to reach this noise maximum scales as $k_p^{\text{max}} \propto L^{-1}$ . This scaling arises because a longer lattice contains more particles exposed to pausing; thus, a smaller per-particle rate is sufficient to maintain the critical order-one number of paused particles ( $N_p^\star$ ).
Mechanism of Fluctuation: The maximum noise does not occur in the limit of rare pauses (regular output) nor in the limit of persistent congestion (suppressed but steady output). Instead, it emerges in an intermittent regime where the system alternates between productive intervals and extended periods of weak or vanishing output. This alternation is driven by the nucleation and dissolution of large particle clusters (jams) initiated by pauses.
Role of Unpausing Rate ( $k_u$ ): While $N_p$ organizes the position of the crossover, it does not fully determine the amplitude of the noise. At a fixed $N_p$ , slower unpausing (smaller $k_u$ ) increases both the correlation time ( $\tau_c$ ) and the noise amplitude ( $F_\infty$ ). This indicates that the noise amplitude depends on the ratio of the pause lifetime to the timescales of jam formation and relaxation.
Structural Correlates:
- Correlation Time: The residence-time scale associated with the appearance and disappearance of paused particles ( $N_{\text{paused}}=0$ vs. $\ge 1$ ) tracks the fitted output correlation time ( $\tau_c$ ) closely. In contrast, a threshold-based definition of congestion (based on $C_{\text{max}}$ ) fails to track $\tau_c$ as $N_p$ increases.
- Noise Amplitude: The amplitude of the low-frequency noise correlates strongly with the variance of the largest cluster size ( $\text{Var}(C_{\text{max}})$ ), rather than the mean cluster size. The maximum noise occurs when the dominant cluster explores a broad range of sizes, reflecting repeated growth, shrinkage, and reorganization.
Limitations of Minimal Theory: A minimal analytical model assuming independent pauses (Poisson distribution) predicts a maximum at $N_p^\star \simeq 1.50$ . Simulations show the actual maximum occurs near $N_p \simeq 1.9$ and that the pause number is overdispersed (variance > mean). The deviations arise from spatial correlations, boundary effects, and exclusion-induced congestion, which the minimal theory neglects.

Significance
The paper demonstrates that reversible pausing in finite open exclusion processes creates a distinct fluctuation regime where the timing of molecular production is highly intermittent, even if the mean throughput is merely reduced. The study identifies that the most significant output noise arises not from the mean density of defects, but from the collective dynamics of slow defects interacting with traffic jams when the number of defects is small (order unity).

The work distinguishes between two levels of organization:

Pause-number dynamics sets the timescale for output decorrelation.
Jam reorganization dynamics (fluctuations in the largest cluster) determines the amplitude of the noise modulation.

This implies that for processes sensitive to transcript completion times (e.g., downstream gene expression), the temporal delivery of products can be highly variable due to elongation traffic alone, without requiring promoter switching or sequence-specific pausing sites. The findings extend the understanding of finite-size effects in driven lattice gases, showing that the "finite-size regime" is characterized not just by a reduction in current, but by a specific, non-monotonic enhancement of low-frequency output noise driven by the interplay of slow internal states and collective jamming.

Technical Summary: Low-frequency output fluctuations in an open exclusion process with particle pausing

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