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

Quentin Thommen

arXiv:2608.08074v1cond-mat.stat-mechq-bio.SC

TL;DR

The study asks whether finite-size pausing leaves a distinct signature in stationary exit statistics beyond its effect on mean current. Using an open pausing TASEP, analytical approximation, and simulations, it finds that low-frequency output noise peaks at an order-one paused population while the required pausing rate scales inversely with system size.

  • Problem

    Whether the finite-size pausing regime identified from mean current has a distinct signature in stationary completed-exit statistics remains unresolved.

  • Method

    The paper combines exit-counting statistics, residence-time and cluster analyses, parameter scans, and a minimal analytical description of reversible pausing in an open exclusion process.

  • Results

    The low-frequency Fano factor is nonmonotonic, peaking near N_p ≃1.5–2 across studied lengths while the corresponding pausing rate scales as L^-1.

  • Takeaways & Limitations

    Maximum relative output noise occurs when productive and congested configurations are repeatedly sampled at an order-one paused population, not when congestion is persistent.

  • Takeaways & Limitations

    The numerical peak is not universal, shifting with boundary conditions, while the model omits several biological ingredients and structural measurements are limited to L = 100.

Abstract

from arXiv · show

Slow internal states reshape both the mean throughput and the temporal organization of a driven lattice gas. Exit-counting statistics reveal this effect in a finite open totally asymmetric simple exclusion process whose particles reversibly switch between active and paused states. Increasing pausing lowers the mean current smoothly, whereas the long-window Fano factor is strongly nonmonotonic. At the reference boundary rates, the maximum remains near a measured mean paused population \(N_p=Lρ_{\rm paused}\simeq1.5\)--\(2\) across lattice lengths L=50-500, while the corresponding pausing rate scales as \(k_p^{\rm max}\propto L^{-1}\). A minimal constant-birth, linear-death approximation translates an order-one collective crossover into this finite-size displacement and gives \(N_p^\star\simeq1.50\) in the independent-pause, strong-blocking limit. The simulations delimit this approximation: the pause number is overdispersed, and at fixed \(N_p\), slower unpausing increases both the correlation time and the noise amplitude. Residence-time and structural analyses further separate the relevant slow variables. The pause-free versus pause-containing residence-time scale tracks the fitted output-correlation time, whereas the noise amplitude follows fluctuations, rather than the mean size, of the largest particle cluster. Low-frequency output noise therefore identifies an intermittent finite-size regime shaped jointly by slow-defect kinetics and traffic-jam reorganization.

I. INTRODUCTION · II. MODEL, OBSERVABLES, AND ANALYTICAL FRAMEWORK · A. Open pausing TASEP

This work studies how reversible particle pausing reshapes temporal output fluctuations in a finite open TASEP. Exit-counting statistics reveal a low-frequency fluctuation regime organized by the realized paused population, while model structure links internal kinetics to interacting occupancy.

  • I. INTRODUCTION: TASEP represents unidirectional exclusion transport with boundary exchange, where local interruptions propagate upstream and reorganize traffic.Transcription and translation motivate slow active–paused states, whose persistent pauses form jams and reduce throughput beyond direct immobilization.
  • I. INTRODUCTION: Earlier pausing-TASEP work incorporated reversible active–paused switching and current reduction from paused particles plus blocked followers, while finite open-system theory identified boundary phases and a slow finite-size regime.That theory also connected pausing with TASEPs containing dynamical defects.
  • I. INTRODUCTION: The paper addresses the unresolved temporal signature of the stationary finite-size pausing regime by counting lattice exits over windows of duration T_w and analyzing the corresponding Fano factor.The stationary current alone cannot distinguish regular output from productive episodes separated by long congested intervals.
  • I. INTRODUCTION: At reference boundary rates, the low-frequency Fano factor peaks near N_p ≃1.5–2 across studied lengths, while the mean current decreases continuously with paused population.A minimal constant-birth, linear-death description maps this order-one optimum to k_p^max ∝ L^-1.
  • I. INTRODUCTION: The simulations show that N_p organizes crossover position but not amplitude: pause numbers are non-Poisson, and slower unpausing increases correlation time and low-frequency noise at comparable N_p.Residence-time and structural analyses associate output-correlation time with pause-related residence times, while noise amplitude follows fluctuations of the largest particle cluster.
  • A. Open pausing TASEP: The model is a one-dimensional lattice of L sites with exclusion, where active particles hop at rate ϵ and switch reversibly between active and paused states.Particles enter at rate α when site 1 is empty, active particles exit at rate β, and paused particles at the final site must resume before terminating.
  • A. Open pausing TASEP: Internal-state kinetics set stationary active and paused fractions within occupied particles, whereas boundaries and congestion determine total occupancy and therefore the absolute paused population.The measured N_p combines internal kinetics with interacting finite-system occupancy.

B. Output counting and structural observables · C. Minimal theory for the location of the noise maximum

Output-counting statistics distinguish short-time exit noise from slow current modulation, while structural observables connect fluctuations to paused particles and particle clustering. A minimal theory explains the noise maximum through an order-one paused population and maps it to an L^-1 shift in the microscopic pausing rate.

  • B. Output counting and structural observables: The output-counting framework defines cumulative exits N(t), stationary current, and window-dependent Fano factors for analyzing temporal fluctuations.These quantities provide the counting basis for distinguishing fast exit statistics from slow current organization.
  • B. Output counting and structural observables: The window-dependent Fano factor separates short-time exit statistics from slow modulation of the current, with a Poisson counting process serving as the F = 1 reference.Its exponential-correlation interpretation characterizes the integrated-count signature but does not assert an exact microscopic two-state process.
  • B. Output counting and structural observables: Structural analysis tracks paused-particle counts, the largest cluster Cmax and its position, downstream pauses, and pause-attached clusters, while ⟨Cmax⟩ and Var(Cmax) remain primary threshold-free observables.Residence-time diagnostics additionally use congestion states defined from Cmax/L.
  • C. Minimal theory for the location of the noise maximum: The maximum is organized by the measured paused population Np rather than the bare pausing rate kp, with stationary balance providing their explicit system-size-dependent conversion.The balance relates active and paused populations through the total occupancy, so longer lattices require smaller per-particle pausing rates to sustain the same order-one pause number.
  • C. Minimal theory for the location of the noise maximum: The finite-L correction is largest for the shortest lattice while preserving the leading L^-1 scaling of the predicted rate displacement.The approximation therefore attributes the system-size trend to the number of particles exposed to pausing, not to a change in the bare pausing kinetics.
  • C. Minimal theory for the location of the noise maximum: In the strong-blocking limit, the auxiliary constant-birth, linear-death model has a nonzero noise maximum at an order-one mean pause number.It replaces aggregate pausing by a stationary birth rate and independent unpausing, yielding a Poisson pause-number model whose slow modulation is strongest at an intermediate mean.
  • C. Minimal theory for the location of the noise maximum: An order-one mean number of paused particles maximizes slow output modulation, while stationary pause balance maps this crossover to an L^-1 displacement of the pausing rate.The construction is qualitative rather than a quantitative theory of peak amplitude because it neglects jam position, finite jam lifetime, pause correlations, boundary effects, and continuously varying blocked currents.

D. Physical calibration and polymerase numbers

A coarse-grained calibration maps simulation lengths to bacterial transcription units and rates to physical timescales. The simulated densities correspond to biologically plausible particle numbers and dense traffic on strongly transcribed units.

  • Physical mapping: Using roughly ten nucleotides per lattice site and 40 nt s−1 elongation, the model provides an order-of-magnitude bacterial-transcription calibration rather than an elementary-step interpretation.The ten-nucleotide mapping is described as a coarse graining of steric spacing.
  • Physical mapping: One simulation time unit represents approximately 0.25 s, with L = 50–500 corresponding to transcription units of roughly 0.5–5 kb.A 1 kb transcription unit corresponds to L ≃100.
  • Rate calibration: The calibrated rates α = 0.1, β = 1, and ku = 10−3 correspond respectively to 0.4 s−1, 4 s−1, and 4 × 10−3 s−1, with a mean pause duration of about 250 s.These values use hopping-rate units with ϵ = 1.
  • Polymerase numbers: Measured densities of 0.15–0.35 imply roughly 8–18 particles at L = 50 and 75–175 at L = 500, with mean spacings of a few tens of nucleotides.Intermediate sizes correspond to 15–35 particles for L = 100 and 30–70 for L = 200.

E. Simulation protocol

The study uses exact continuous-time event-driven kinetic Monte Carlo simulations, varying pausing rate and lattice length under defined reference rates. Main Fano-factor scans use ten independent, rate-relaxed trajectories per parameter point.

  • Simulation protocol: Exact continuous-time event-driven kinetic Monte Carlo simulations vary k_p and L under reference rates ϵ = 1, α = 0.1, β = 1, and k_u = 10−3.The pausing rate k_p and lattice length L are the varied parameters unless otherwise stated.
  • Simulation protocol: Ten independent trajectories sample each main Fano-factor scan parameter point after a rate-adapted relaxation period.A boundary-rate control uses L = 100 and α = 0.03 while retaining the other reference rates.
  • Simulation protocol: Exit-count statistics, structural observables, residence times, fitting procedures, and uncertainty estimates are documented in Appendix A.

III. RESULTS · A. Intermittent output and observation-time-dependent noise · B. Finite-size organization of the noise maximum

Pausing continuously reduces mean throughput, but low-frequency output noise is nonmonotonic because slow alternation between productive and inhibited configurations creates an intermittent finite-size regime. The noise maximum remains organized by an order-one paused population, although its location and amplitude depend on system size and boundary rates.

  • A. Intermittent output and observation-time-dependent noise: Near the noise maximum, productive intervals alternate with extended weak- or zero-output periods, whereas rare pauses yield regular output and large paused populations suppress current for most trajectories.For L = 100, the intermediate regime occurs at kp = 7.1 × 10^-5, between kp = 2×10^-5 and 3.6 × 10^-4.
  • A. Intermittent output and observation-time-dependent noise: Increasing the observation window raises F(Tw) above the short-time counting baseline until it reaches a long-window plateau after integrating over slow productive–inhibited alternation.The rise reflects extra covariance from slow switching, and saturation occurs when Tw exceeds the correlation time.
  • B. Finite-size organization of the noise maximum: Mean throughput decreases continuously with pausing, while relative low-frequency noise is nonmonotonic and peaks near Np ≃1.5–2.A large F∞ requires repeated sampling of configurations with substantially different output currents.
  • B. Finite-size organization of the noise maximum: Absolute fluctuations are also nonmonotonic, so the F∞ maximum is not produced solely by normalization by the decreasing mean current.The variance growth rate JF∞ reaches its maximum at a smaller paused population, typically Np = O(1).
  • B. Finite-size organization of the noise maximum: Across finite-size scans, the peak location shifts only weakly while kmax_p changes by more than one decade, and Lkmax_p remains approximately constant.The maxima satisfy kuτc = O(1), indicating an unpausing-related decorrelation scale; peak amplitude decreases for the largest systems.
  • B. Finite-size organization of the noise maximum: Np is a scaling variable for the crossover location but not a complete collapse variable for F∞, whose peak amplitude varies across system sizes.The order-one paused population identifies the crossover, while the largest systems show reduced peak amplitude.
  • B. Finite-size organization of the noise maximum: Reducing α from 0.1 to 0.03 preserves a pronounced intermittent regime, shifting the relative-noise maximum to kp = 1.6 × 10^-4 and Np = 3.26 with F∞ = 47.25.At neighboring Np values, F∞ = 40.41 at Np = 1.57 and 42.15 at Np = 4.51; JF∞ remains nonmonotonic and is largest in the sample at Np = 1.57.

C. Unpausing sets the time scale and changes the noise amplitude · D. Microscopic organization of the intermittent regime · 1. Fluctuations of the dominant jam

Slower unpausing lengthens output correlations and amplifies low-frequency noise even at nearly fixed paused population, showing that N_p sets crossover position but not noise amplitude. The noise maximum instead tracks fluctuations and reorganization of the dominant jam, not persistent mean congestion.

  • C. Unpausing sets the time scale and changes the noise amplitude: τ_c scales approximately as k_u^-1.1–1.2, identifying unpausing as the dominant microscopic clock for slow output modulation.The relation k_uτ_c remains of order unity except at the largest k_u, with residual deviations allowing additional traffic-relaxation times.
  • C. Unpausing sets the time scale and changes the noise amplitude: F∞ rises from about 17 to 242 as k_u decreases from 5 × 10^-3 to 5 × 10^-4 at N_p ≃2, despite N_p changing only from 1.85 to 1.95.Across target populations, the effective scaling is F∞∝k_u^-γF with γF ≃1.15.
  • C. Unpausing sets the time scale and changes the noise amplitude: At fixed k_u, N_p robustly sets the finite-size crossover position, but slower unpausing independently increases both correlation time and the contrast between productive and inhibited intervals.Pause-induced clusters persist and reorganize traffic over longer times when unpausing is slower.
  • 1. Fluctuations of the dominant jam: F∞ peaks near N_p ≃1.5–2 alongside Var(C_max), with an approximately 0.97 Pearson correlation across eight parameter sets.The cluster attached to the downstream-most pause shows a comparable association with output noise.
  • 1. Fluctuations of the dominant jam: The mean largest-cluster size and the probability of C_max > 0.2L remain large after F∞ declines, so persistent congestion suppresses current but does not determine nonmonotonic noise.Noise is largest when the dominant cluster explores a broad range of sizes through repeated growth, shrinkage, and reorganization.
  • 1. Fluctuations of the dominant jam: For N_p ≪1 output remains comparatively regular, N_p = O(1) maximizes largest-cluster variance, and N_p ≫1 produces persistent congestion with narrower configuration exploration.The intermittent regime alternates between weakly and strongly congested configurations at order-one paused population.
  • D. Microscopic organization of the intermittent regime: Structural analysis identifies fluctuations of the dominant cluster as the observable most closely associated with noise amplitude, while the paused-particle distribution remains necessary to explain the order-one crossover.The structural results distinguish the relevant noise carrier from the unresolved finite-size mechanism.

2. Pause-number statistics · 3. Residence times and effective telegraph variables

Pause-number statistics place the noise maximum in a finite-size crossover where pause-free, single-pause, and multipause states coexist, while deviations from Poisson behavior expose correlations beyond independent pausing. Residence-time tests show that paused-particle activity tracks output decorrelation better than a fixed largest-cluster threshold, whereas largest-cluster fluctuations better track noise amplitude.

  • 2. Pause-number statistics: Pause-number probabilities deviate systematically from Poisson predictions: pause-free states are overrepresented, while configurations with two or more pauses are underrepresented.The mean paused population locates sector coexistence but does not determine the full distribution because occupancy fluctuations, residence-time changes, and exclusion-induced congestion generate correlations.
  • 2. Pause-number statistics: At Np ≃1.92, P(M = 0) ≃0.295, P(M = 1) ≃0.239, and P(M ≥2) ≃0.467, so no pause-number sector dominates.The output-noise maximum therefore lies in a genuine crossover regime rather than a sharply defined single-pause state.
  • 2. Pause-number statistics: Across eight parameter sets, P(M = 1) correlates strongly with F∞, but Var(Cmax) is the closer structural correlate of output modulation.Pause-number statistics identify the order-unity crossover, whereas dominant-jam fluctuations better describe noise amplitude.
  • 3. Residence times and effective telegraph variables: The fitted F(Tw) form motivates an effective two-state output modulation, but the microscopic variable defining its states must be identified through residence-time comparisons.The analysis compares structural trajectories based on largest-cluster congestion and on the presence of at least one paused particle.
  • 3. Residence times and effective telegraph variables: The fixed largest-cluster threshold fails to track τc: τtel increases with Np while τc decreases, and this disagreement persists under moderate threshold changes.A congestion threshold therefore does not define the slow state variable underlying output decorrelation.
  • 3. Residence times and effective telegraph variables: The pause-presence residence-time scale remains close to τc across the scanned range, with replicate-wise τpause/τc approximately constant near 0.8.Appearance and disappearance of paused particles capture the dominant output-decorrelation time more closely than fixed congestion-threshold crossings.
  • 3. Residence times and effective telegraph variables: Near the noise maximum, weakly and strongly congested-state survival probabilities are approximately exponential over substantial residence-time ranges, supporting—but not proving—an effective telegraph parametrization.Residual deviations preclude an exact Markov reduction.
  • 3. Residence times and effective telegraph variables: Pauses initiate slow intermittent episodes, while ensuing largest-cluster fluctuations provide the closest correlate of noise amplitude.Thus pause-number dynamics, pause-presence residence times, and jam fluctuations describe complementary roles in the same intermittent output dynamics.

IV. DISCUSSION · V. CONCLUSION

Reversible pausing separates mean transport from temporal delivery: current decreases smoothly, but low-frequency output noise peaks when the paused population is order one and traffic alternates between productive and congested states. The peak reflects coupled pause kinetics and jam reorganization, with its location and amplitude constrained by system size, boundary density, and pause lifetime.

  • IV. DISCUSSION: The low-frequency Fano factor peaks at an intermediate paused population, because maximal noise requires repeated switching between productive and congested configurations rather than persistent congestion alone.For N_p ≪1, jams are rare; for N_p = O(1), pause-free, single-pause, and multiple-pause states coexist; for N_p ≫1, congestion persists.
  • IV. DISCUSSION: The maximum remains near N_p ≃1.5–2 across studied lengths and corresponds to an order-one finite-size crossover, extending earlier mean-current descriptions with a sharper exit-noise signature.The referenced studies established reversible pausing, dominant-cluster formation, and the importance of an order-one paused population; the present work identifies nonmonotonic low-frequency exit noise.
  • IV. DISCUSSION: A constant-birth, linear-death model places the independent-pause, strong-blocking maximum at N_p^⋆ ≃1.50 and maps the order-one crossover to k_p^max ∝ L^-1.The approximation captures the leading finite-size mechanism but not the full interacting dynamics; measured pause statistics are overdispersed and shift the numerical maximum near N_p ≃1.9.
  • IV. DISCUSSION: Pause-number dynamics sets output decorrelation, while fluctuations in the largest cluster—not its mean size—track the nonmonotonic noise amplitude.The pause-free versus pause-containing residence-time scale is approximately proportional to fitted τ_c, whereas Var(C_max) follows the noise more closely than mean cluster size or threshold exceedance.
  • IV. DISCUSSION: At comparable N_p, slower unpausing increases both τ_c and F_∞, indicating that pause lifetime changes the jam state reached during inhibited episodes as well as output persistence.A quantitative amplitude theory requires a dimensionless ratio comparing pause lifetime with jam formation and relaxation times; residual peak-size dependence likely reflects this competition.
  • V. CONCLUSION: The biological implication is temporal delivery: identical mean transcription rates can produce either regular completions or productive bursts separated by silent intervals, exposing downstream processes to different inputs.Reversible pausing separates mean transport from temporal delivery, while the noise maximum persists when initiation is reduced more than threefold, although its location shifts with boundary-driven density.
  • IV. DISCUSSION: The conclusions apply most directly to dense, strongly transcribed units exposed to rare, persistent interruptions and are not universal across boundary-driven phases or heterogeneous biological systems.The model uses point particles and homogeneous hopping and pausing, excluding promoter switching, sequence-dependent pauses, backtracking, termination, degradation, and template heterogeneity; the mapped ku = 10^-3 pause time is about 250 s.

Appendix A: Numerical methods and statistical analysis · 1. Continuous-time simulations · 2. Window counts and Fano-factor estimation

The appendix specifies exact continuous-time simulation and time-weighted measurement procedures, then defines window-count statistics and finite-size Fano-factor fitting protocols. It also details diagnostics and scan rules that constrain interpretation of sparse sampling and crossover maxima.

  • 1. Continuous-time simulations: Event-driven kinetic Monte Carlo exactly simulates all allowed entrance, exit, forward-hop, pausing, and unpausing events in continuous time.The total escape rate is R = R_in + R_out + R_hop + R_pause + R_unpause; events are selected proportionally to their rates.
  • 1. Continuous-time simulations: The baseline parameters are ϵ = 1, α = 0.1, β = 1, and k_u = 10^-3, while k_p and L vary across finite-size scans.These boundary rates define an initiation-limited, nonexit-limited pause-free reference and isolate slow-pause finite-size crossover effects.
  • 1. Continuous-time simulations: Simulations begin from an empty lattice, discard rate-adapted transients, sample stationary observables, and use ten independent replicates per Fano-factor and finite-size scan point.Relaxation and sampling are extended when pauses become rarer.
  • 1. Continuous-time simulations: Stationary observables use exact residence-time weighting, and exit-time differences provide inter-exit intervals from the interacting stationary process.This yields current and total, active, and paused-particle densities without inferring paused occupancy from isolated-particle behavior.
  • 2. Window counts and Fano-factor estimation: Each trajectory is divided into nonoverlapping observation windows, whose exit-count means and unbiased variances define the trajectory-level Fano factor.The analysis also records the fraction of windows with no exit to diagnose sparse sampling at long observation times or strongly inhibited parameters.
  • 2. Window counts and Fano-factor estimation: The Fano crossover is fitted by nonlinear ordinary least squares with positive free parameters F_0, F_∞, and τ_c, without uncertainty weighting.Finite-size parameters come from fits to replicate-averaged Fano curves, and the fit describes the crossover phenomenologically rather than imposing exact two-state microscopic dynamics.
  • 2. Window counts and Fano-factor estimation: Finite-size maxima are located from resolved local scans of F_∞ versus k_p or measured N_p, using only lengths with sampled points on both sides.The reported k_max is the simulated rate at the largest fitted plateau, with no interpolation between neighboring rates.

3. Unpausing-rate scan · 4. Initiation-rate control · 5. Event-resolved output trajectories

The initiation-rate control preserves a nonmonotonic fluctuation maximum, while congestion changes occupancy and prevents a simple rescaling of the pausing rate. Separate event-resolved trajectories use short-bin exit recordings to examine temporal organization without entering the quantitative analysis.

  • 3. Unpausing-rate scan: The unpausing-rate scan fixes L = 100, varies ku from 5×10−4 to 5×10−3, and adjusts kp to compare similar paused populations across three targets.
  • 3. Unpausing-rate scan: Each replicate is fitted separately by unweighted nonlinear least squares, with Fig. 5 power laws treated as effective interval fits rather than asymptotic exponents.
  • 4. Initiation-rate control: F∞ rises from 1.80 at Np = 0.038 to 47.25 at Np = 3.26, then falls to 4.69 at Np = 14.00, confirming a bracketed fluctuation maximum.Neighboring conditions give F∞ = 40.41 at Np = 1.57 and 42.15 at Np = 4.51, so interpolation is unnecessary.
  • 4. Initiation-rate control: The absolute fluctuation rate is also nonmonotonic, reaching JF∞ = 0.752 at Np = 1.57 and decreasing on either side.
  • 4. Initiation-rate control: Reducing α cannot be treated as uniformly rescaling kp because congestion raises residence times and exposes more particles to further pausing.Across the control, density increases from ρ = 0.038 to approximately 0.28.
  • 5. Event-resolved output trajectories: Event-resolved trajectories use five independent simulations for each of three representative pausing rates after a fixed relaxation interval, retaining exact exit times.
  • 5. Event-resolved output trajectories: The current is binned on intervals much shorter than ku^-1 while suppressing event discreteness, and a 50-bin moving average is visual only.

6. Structural sampling and cluster observables · 7. Pause-number statistics

Structural sampling quantified largest particle clusters and pause locations, while pause-number statistics used trajectory-based probabilities, Poisson references, and block-bootstrap uncertainties. Near the output-noise maximum, pause counts were reproducible and overdispersed across trajectories.

  • 6. Structural sampling and cluster observables: The analysis used L = 100, reference rates, ten independent replicates, and matched burn-in and sampling durations to the main Fano-factor scan.Complete structural trajectories were retained for replicate 1 at every parameter point, with additional trajectories stored for reproducibility controls.
  • 6. Structural sampling and cluster observables: The structural sampling interval limited resolution of residence episodes shorter than Δt_struct, despite being shorter than the characteristic correlation time near the noise maximum.The supplied passages identify the correlation-time scale as order k_u^-1 but do not provide a complete numerical value.
  • 6. Structural sampling and cluster observables: Structural analysis recorded the largest cluster size Cmax, its position, paused-particle locations, and the queue attached upstream of the downstream-most pause.A particle cluster was defined as a maximal sequence of consecutively occupied sites, regardless of internal particle state.
  • 6. Structural sampling and cluster observables: The binary indicator Cmax > Cth was restricted to state-occupancy and residence-time diagnostics rather than the main threshold-free structural comparison.Threshold-free quantities were used for the primary cluster analysis.
  • 6. Structural sampling and cluster observables: Pearson coefficients across eight sampled parameter sets were descriptive covariations, not independent causal tests.This limitation applies to the correlations quoted in the main text.
  • 7. Pause-number statistics: Pause-number probabilities were computed as sampled fractions in three classes, with Poisson references based on the measured mean N_p = ⟨M⟩ and a dispersion measure testing variance-to-mean departures.The supplied passages define the statistical construction but do not specify the three class labels or the dispersion value.
  • 7. Pause-number statistics: Because successive structural samples were temporally correlated, uncertainties used a circular moving-block bootstrap with 300 resamples and blocks twice the estimated integrated autocorrelation time.The block length was expressed in structural-sampling intervals, using random seed 20260729.
  • 7. Pause-number statistics: At the output-noise maximum, N_p = 1.946 ± 0.024, with P(M = 0) = 0.291 ± 0.004, P(M = 1) = 0.234 ± 0.004, and P(M ≥2) = 0.474 ± 0.006.These values are replicate-averaged across five trajectories; the dispersion index exceeded unity in all trajectories and increased across three control conditions.

8. Residence-time and telegraph analyses · 9. Uncertainty and reproducibility

Residence-time analyses compare pause-number and congestion telegraph descriptions of output dynamics, while independent replication tests the resulting timescale separation. The telegraph model is useful as an effective description but not an exact microscopic reduction.

  • 8. Residence-time and telegraph analyses: Residence episodes are formed from consecutive structural-trajectory samples, excluding episodes truncated at either trajectory boundary from means and survival curves.Durations equal the number of sampling intervals multiplied by ∆tstruct.
  • 8. Residence-time and telegraph analyses: The pause-number and congestion partitions yield τpause from τ0 and τP, and τtel from τU and τB, respectively.These estimates provide the two timescales compared with the fitted output-correlation time.
  • 9. Uncertainty and reproducibility: Residence-time uncertainties use 300 moving-block bootstrap resamples, with block lengths spanning approximately two integrated autocorrelation times for each condition.The resampling uses random seed 20260729 and converts correlation time to episode counts using the mean episode duration.
  • 8. Residence-time and telegraph analyses: Telegraph survival curves test an effective Markov description, but nonexponential deviations and state-definition sensitivity rule out an exact microscopic reduction.The exponential references use measured mean residence times.
  • 8. Residence-time and telegraph analyses: Pause-number dynamics tracks the output-correlation timescale across the crossover, with τpause/τc = 0.83 ± 0.27, 0.79 ± 0.19, and 0.80 ± 0.17 at Np ≃ 0.73, 1.95, and 3.12.These ratios are means across five independent trajectories, with uncertainties given as standard deviations.
  • 8. Residence-time and telegraph analyses: By contrast, the congestion-based ratio τtel/τc rises from 0.96 ± 0.31 to 1.83 ± 0.37 as Np increases across the same conditions.At the representative threshold Cth/L = 0.20, the intermediate ratio is 1.24 ± 0.31.
  • 9. Uncertainty and reproducibility: Independent simulation replicates are the primary numerical-replication unit for finite-size, initiation-rate, unpausing, and structural summary observables.This replication framework underlies the reported control across five independent trajectories.
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