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arxiv: 2605.16693 · v1 · pith:FW5UCLMCnew · submitted 2026-05-15 · 🧬 q-bio.PE · cond-mat.stat-mech· math.PR· physics.bio-ph

Branching under First-Passage Resetting

Aanjaneya Kumar , James Holehouse This is my paper

Pith reviewed 2026-05-19 20:49 UTC · model grok-4.3

classification 🧬 q-bio.PE cond-mat.stat-mechmath.PRphysics.bio-ph
keywords first-passage processesbranching processespopulation growth raterenewal equationstochastic timingyield-delay trade-offbacteriophage lysisreplication optimization
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The pith

Stochastic timing fluctuations in first-passage triggered replication enhance population growth for fixed offspring number and mean time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Branching under First-Passage Resetting, a framework in which replication events arise when an internal stochastic variable reaches a threshold rather than from fixed lifetime clocks. It derives an exact renewal equation that directly connects the first-passage statistics of individual trajectories to the long-term population growth rate. For fixed offspring number and fixed mean replication time, this equation shows that variability in timing necessarily increases growth relative to a deterministic schedule. When offspring yield varies with the first-passage time, the same mapping exposes a yield-delay trade-off that can be solved analytically for optimal strategies. The framework is applied to bacteriophage lysis, producing an optimal lysis time and growth rate that align with empirical observations.

Core claim

The population dynamics under first-passage resetting obey an exact renewal equation linking single-trajectory first-passage statistics to the population growth rate. This mapping shows that, for fixed offspring number and fixed mean replication time, stochastic timing fluctuations necessarily enhance growth relative to a deterministic clock. When offspring yield depends on the first-passage time, fluctuations have non-trivial effects and expose a fundamental yield-delay trade-off: waiting longer can increase the number of descendants, but delays all future lineages. The framework solves this optimization problem analytically and, when applied to bacteriophage lysis, yields an optimal lysis,

What carries the argument

The exact renewal equation that links single-trajectory first-passage statistics to the population growth rate.

If this is right

  • For fixed offspring number and fixed mean replication time, stochastic timing fluctuations enhance growth relative to a deterministic clock.
  • When offspring yield depends on first-passage time, a yield-delay trade-off appears that can be solved analytically for the optimal replication strategy.
  • Application to bacteriophage lysis produces an optimal lysis time and corresponding growth rate consistent with empirical data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The renewal mapping could be used to compare growth rates across different threshold distributions in other threshold-triggered biological systems such as cell division.
  • The yield-delay trade-off identified here suggests that selection on replication timing may balance immediate yield against the compounding effect of earlier subsequent generations.
  • Because the equation is exact, it offers a route to test whether observed timing variability in natural populations is consistent with growth-rate maximization.

Load-bearing premise

The population dynamics obey an exact renewal equation linking single-trajectory first-passage statistics to the population growth rate, which assumes that first-passage processes across independent lineages are statistically identical and that branching occurs precisely at the first-passage event.

What would settle it

Measure population growth rates in a controlled biological system under stochastic first-passage triggering versus an otherwise identical deterministic timing schedule with the same mean and offspring number, and check whether the stochastic case produces the higher growth rate predicted by the renewal equation.

Figures

Figures reproduced from arXiv: 2605.16693 by Aanjaneya Kumar, James Holehouse.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The density [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Many biological processes, from cell division to viral lysis, are triggered when an internal stochastic variable reaches a threshold. Here we introduce Branching under First-Passage Resetting, a general framework in which replication events arise endogenously from first-passage dynamics rather than from externally imposed lifetime clocks. We show that the resulting population dynamics obey an exact renewal equation linking single-trajectory first-passage statistics to the population growth rate. This mapping shows that, for fixed offspring number and fixed mean replication time, stochastic timing fluctuations necessarily enhance growth relative to a deterministic clock. When offspring yield depends on the first-passage time, however, fluctuations have non-trivial effects and expose a fundamental yield-delay trade-off: waiting longer can increase the number of descendants, but delays all future lineages. Our framework allows us to address this optimization problem analytically, and upon application to bacteriophage lysis, gives an optimal lysis time and growth rate consistent with empirical data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Branching under First-Passage Resetting, a framework in which replication events are endogenously triggered when an internal stochastic variable reaches a threshold, using a resetting mechanism. It claims that the resulting population dynamics obey an exact renewal equation that directly links single-trajectory first-passage time statistics to the population growth rate. For fixed offspring number and fixed mean replication time, stochastic timing fluctuations are shown to enhance growth relative to a deterministic clock. When offspring yield depends on first-passage time, the framework identifies a yield-delay trade-off and solves the associated optimization problem analytically; application to bacteriophage lysis yields an optimal lysis time and growth rate stated to be consistent with empirical data.

Significance. If the renewal mapping is rigorously derived and the assumptions hold, the work provides a valuable analytical bridge between microscopic first-passage processes and macroscopic branching population dynamics, with direct relevance to cell division and viral replication. The explicit identification of the yield-delay trade-off and its analytical optimization constitute a clear strength, as does the demonstration that fluctuations can be beneficial under fixed-mean constraints. The bacteriophage application, if shown to be a genuine a priori prediction rather than a fit, would add biological credibility. The paper is credited for attempting an exact (rather than approximate) renewal link from single-trajectory statistics to growth rate.

major comments (2)
  1. [Abstract, paragraph 2 and §2] Abstract, paragraph 2 and §2 (model definition): The central claim rests on an 'exact renewal equation' mapping first-passage statistics to population growth rate. The text must supply the full derivation, explicitly stating and justifying the assumptions that (i) every lineage is an independent, statistically identical copy of the same first-passage process and (ii) branching occurs precisely at the instant the internal variable hits the threshold with no additional state-dependent delay. Without this, the independence premise highlighted in the skeptic note remains unverified.
  2. [§4] §4 (bacteriophage application): The statement that the analytically obtained optimal lysis time and growth rate are 'consistent with empirical data' is presented as validation. The manuscript must clarify whether the functional form relating offspring yield to lysis time was taken from independent measurements or adjusted to produce consistency. If the latter, the result becomes a post-hoc fit and does not constitute an independent test of the yield-delay trade-off.
minor comments (2)
  1. [§2] The notation for the first-passage time distribution and the resetting kernel should be introduced with a dedicated equation block early in §2 to avoid ambiguity when the renewal equation is stated.
  2. [Figure 1] Figure 1 (schematic): Add explicit labels for the threshold value, the reset event, and the offspring initial condition to make the mapping from single trajectory to branching process visually immediate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and have revised the manuscript to strengthen the presentation of the renewal derivation and the bacteriophage application.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2 and §2] The central claim rests on an 'exact renewal equation' mapping first-passage statistics to population growth rate. The text must supply the full derivation, explicitly stating and justifying the assumptions that (i) every lineage is an independent, statistically identical copy of the same first-passage process and (ii) branching occurs precisely at the instant the internal variable hits the threshold with no additional state-dependent delay.

    Authors: We agree that the derivation and assumptions require explicit treatment. In the revised manuscript we have inserted a new subsection in §2 that derives the renewal equation from first principles. The derivation begins from the probability density of the first-passage time for a single trajectory and uses the fact that each daughter initiates an independent copy of the identical process. We explicitly state assumption (i): after division, each offspring inherits the same internal stochastic dynamics and resets to the initial condition, rendering lineages i.i.d. We state assumption (ii): the model defines replication as occurring exactly when the internal variable reaches threshold, with no additional state-dependent delay; this is justified by the biological interpretation that lysis or division is triggered at threshold crossing. These clarifications directly address the independence premise. revision: yes

  2. Referee: [§4] The statement that the analytically obtained optimal lysis time and growth rate are 'consistent with empirical data' is presented as validation. The manuscript must clarify whether the functional form relating offspring yield to lysis time was taken from independent measurements or adjusted to produce consistency.

    Authors: We appreciate the need for this clarification. The functional form relating yield to lysis time is taken directly from independent experimental measurements reported in the bacteriophage literature (specifically, burst-size versus lysis-time data from prior studies). The optimization is performed with this fixed, externally determined relation; the resulting optimal lysis time is then compared with observed values. We have revised §4 to state the provenance of the yield function explicitly and to note that the agreement tests the framework’s predictive capacity rather than constituting a post-hoc fit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces Branching under First-Passage Resetting and derives an exact renewal equation from the branching process definition, linking first-passage statistics to growth rate as a direct mathematical consequence rather than a redefinition or fit. The result that fluctuations enhance growth for fixed offspring number and mean replication time follows analytically from this equation without reducing to inputs by construction. The yield-delay trade-off and bacteriophage optimization are obtained by analyzing the same mapping under time-dependent yield, then compared to external empirical data for consistency; no parameter fitting is described that would force the reported optimum to match data tautologically. No self-citation chains, ansatz smuggling, or uniqueness theorems imported from prior author work appear load-bearing. The core chain remains independent of the target claims.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on one primary domain assumption about endogenous first-passage triggering and introduces at most one fitted quantity (optimal lysis time) in the bacteriophage application; no new physical entities are postulated.

free parameters (1)
  • optimal lysis time
    Value obtained by maximizing the growth rate under the yield-delay trade-off for the bacteriophage case and reported as consistent with data.
axioms (1)
  • domain assumption Replication events arise endogenously from first-passage dynamics rather than from externally imposed lifetime clocks.
    Stated in the first sentence of the abstract as the modeling premise that enables the renewal mapping.

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Reference graph

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