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arxiv: 2508.14680 · v2 · submitted 2025-08-20 · ❄️ cond-mat.stat-mech · physics.data-an· q-bio.PE

Size-structured populations with growth fluctuations: Feynman--Kac formula and decoupling

Ethan Levien , Ya\"ir Hein , Farshid Jafarpour This is my paper

Pith reviewed 2026-05-18 22:08 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.data-anq-bio.PE
keywords size-structured populationsgrowth fluctuationsFeynman-Kac formuladecouplingbranching processesphenotypic heterogeneityexponential tilting
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The pith

When internal variables decouple from size in both ensembles, size dynamics transform via random time change into a growth-homogeneous process evaluable by Feynman-Kac tilting.

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

This paper examines size-structured population models in which individual cells grow at rates set by a fluctuating internal variable such as gene expression levels. It generalizes prior observations that the internal variable can decouple from cell size and connects these observations to the Feynman-Kac formula for branching processes. The authors derive conditions for decoupling to hold simultaneously in the lineage ensemble and the population ensemble. When both forms of decoupling are present, a random time change converts the size dynamics into a growth-homogeneous process, and expectations are obtained through an exponential tilting procedure. The work also supplies a broader interpretation of the tilted expectations in terms of the mass-weighted phenotype distribution.

Core claim

When decoupling occurs in both the lineage and population ensembles, the size dynamics can be transformed, via a random time change, into a growth-homogeneous process, and expectations can be evaluated through an exponential tilting procedure that follows from the Feynman-Kac formula. The paper further characterizes weaker, ensemble-specific decoupling that holds in only one ensemble and supplies a more general interpretation of tilted expectations in terms of the mass-weighted phenotype distribution.

What carries the argument

The Feynman-Kac formula applied to branching processes, which supplies the exponential tilting that converts expectations between lineage and population measures once decoupling conditions are met.

If this is right

  • Size dynamics reduce to a growth-homogeneous process after the random time change.
  • Expectations of size-dependent observables become computable by exponential tilting of the lineage measure.
  • Weaker decoupling still relates lineage and population quantities when it holds in only one ensemble.
  • Tilted expectations receive a direct interpretation as averages under the mass-weighted phenotype distribution.

Where Pith is reading between the lines

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

  • The same tilting procedure could simplify numerical sampling of large heterogeneous cell populations in synthetic-biology models.
  • The random-time-change reduction may extend to other stochastic growth processes that admit a branching-process description.
  • Experimental cell-size and gene-expression time series could test whether observed distributions match the predicted homogeneous process after time re-scaling.

Load-bearing premise

The fluctuating internal variable directly sets each cell's instantaneous growth rate inside a branching process whose generator permits the Feynman-Kac representation.

What would settle it

Simulate the full model with an internal variable that sets growth rate; apply the random time change and tilting only under the stated decoupling conditions and check whether the resulting size distribution matches the direct simulation.

Figures

Figures reproduced from arXiv: 2508.14680 by Ethan Levien, Farshid Jafarpour, Ya\"ir Hein.

Figure 1
Figure 1. Figure 1: (A) A diagram of the model. A cell has size (Y ) and growth (X) phenotypes. Division occurs at a rate that depends on both. We have omitted the explicit dependence on the initial size in this figure for simplicity. (B) A simulation of our model in the case when X is an OU process, showing the evolution of X (blue) and size Y (red). (C) A diagram of a growing population showing the distinction between the l… view at source ↗
Figure 2
Figure 2. Figure 2: (left) Simulations of three models illustrating examples of (top to bottom) (SD), (WDl), and no decoupling. For all models d = 1 and the growth rate function is λ = 1 + x. For (SD) we have simulated an OU process for the X dynamics with θ = 1, σ 2 = 0.01, α = 1/2, σY = 0.01. For (WDl) we have simulated a model with L = 0 and r(x|x ′ ) is Gaussian with mean 0 and standard deviation 0.1. The last model combi… view at source ↗
Figure 3
Figure 3. Figure 3: (A) The time change converts the growth phenotype independent process Y˜ (T) to Y (t). (B) The top panel shows the original size dynamics, while the right panel shows the same process when the cell size is plotted as a function of T(t) rather than t. with tilted generator Lˆ λ = L + λ, (37) and g f p (x, 0) = f(x). Equation (36) can be derived for SDEs using Itˆo’s lemma (see [38]), but it can also be deri… view at source ↗
Figure 4
Figure 4. Figure 4: The estimator ( ˆfp) as a function of the time-window t (solid blue line), its time average (dashed blue line), and the theoretical value of the population expectation. Remember that in this model, the lineage expectation is 0. 6. Symmetric division and the mass-weighted phenotype density of path expectations This section provides a more general interpretation of the tilted expectation equation (35), which… view at source ↗
read the original abstract

We study a size-structured population model in which individual cells grow at a rate determined by a fluctuating internal variable (e.g., gene expression levels). Many previous models of phenotypically heterogeneous populations can be viewed as special cases of this model, and it has previously been observed that the internal variable decouples from cell size under certain conditions. In this work, we generalize these results and connect them to the Feynman-Kac formula, which yields relationships between the lineage dynamics and population distribution in branching processes. To this end, we derive conditions for decoupling, both in the lineage and population ensemble. When decoupling occurs in both ensembles, the size dynamics can be transformed, via a random time change, into a growth-homogeneous process, and expectations can be evaluated through an exponential tilting procedure that follows from the Feynman-Kac formula. We further characterize weaker, ensemble-specific forms of decoupling that hold in either the lineage or the population ensemble, but not both. We provide a more general interpretation of tilted expectations in terms of the mass-weighted phenotype distribution

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 manuscript studies size-structured branching-process models in which individual growth rates are set by a fluctuating internal variable (e.g., gene-expression level). It derives explicit decoupling conditions between the internal variable and cell size that hold separately in the lineage ensemble and in the population ensemble. When both ensembles decouple, the size process is shown to be equivalent, after a random time change, to a growth-homogeneous Markov process whose expectations are obtained by an exponential tilt furnished by the Feynman–Kac representation of the branching-process generator. Weaker, ensemble-specific decoupling statements are also characterized, and the tilted expectations are re-interpreted as expectations under the mass-weighted phenotype distribution.

Significance. If the integrability conditions required by the Feynman–Kac representation are verified and the decoupling derivations are free of hidden restrictions, the work supplies a unified, mathematically transparent framework that recovers many existing models of phenotypically heterogeneous populations as special cases and furnishes practical computational tools (random time change plus exponential tilting) for lineage-to-population relations. The explicit link to the Feynman–Kac formula and the mass-weighted interpretation are genuine strengths.

major comments (2)
  1. [Model statement and Feynman–Kac application] The central claim that decoupling in both ensembles permits a random-time-change reduction to a growth-homogeneous process whose expectations follow from Feynman–Kac tilting (abstract and model statement) rests on the representation being well-defined. The manuscript does not state or verify the necessary integrability condition (finite exponential moments of the integrated growth functional) on the growth-rate function of the internal variable. When fluctuations are unbounded this condition can fail even if the formal decoupling equations hold, undermining the claimed equivalence between lineage and population expectations.
  2. [Derivation of decoupling conditions] The derivation of the decoupling conditions (both ensembles) appears to rely on the generator of the joint size–internal-variable process admitting a multiplicative functional given by the integrated growth rate. It is not shown whether these conditions remain non-tautological when the growth-rate function is allowed to depend arbitrarily on the internal state; a concrete counter-example or explicit verification that the conditions are independent of the choice of generator would strengthen the result.
minor comments (2)
  1. [Notation] Notation for the lineage versus population measures should be introduced once and used consistently; the current abstract and model section alternate between several symbols for the same objects.
  2. [Introduction] The statement that “many previous models can be viewed as special cases” would benefit from an explicit table or subsection listing which earlier works are recovered and under what parameter restrictions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments raise important points regarding the mathematical foundations of our results, particularly the conditions for the Feynman-Kac representation and the nature of the decoupling conditions. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: The central claim that decoupling in both ensembles permits a random-time-change reduction to a growth-homogeneous process whose expectations follow from Feynman–Kac tilting (abstract and model statement) rests on the representation being well-defined. The manuscript does not state or verify the necessary integrability condition (finite exponential moments of the integrated growth functional) on the growth-rate function of the internal variable. When fluctuations are unbounded this condition can fail even if the formal decoupling equations hold, undermining the claimed equivalence between lineage and population expectations.

    Authors: We agree that the integrability condition is essential for the Feynman-Kac formula to be applicable and for the equivalence to hold rigorously. The original manuscript assumes this implicitly through the existence of the expectations but does not state it explicitly. In the revised version, we will add a clear statement of the assumption: that the growth-rate function r(x) satisfies E[exp(∫ r(X_s) ds)] < ∞ for the relevant time intervals, where X is the internal variable process. This is a standard condition in the theory of Feynman-Kac semigroups for branching processes. We will also note that for bounded fluctuations this holds automatically, and discuss potential failures for unbounded cases, thereby strengthening the claim. revision: yes

  2. Referee: The derivation of the decoupling conditions (both ensembles) appears to rely on the generator of the joint size–internal-variable process admitting a multiplicative functional given by the integrated growth rate. It is not shown whether these conditions remain non-tautological when the growth-rate function is allowed to depend arbitrarily on the internal state; a concrete counter-example or explicit verification that the conditions are independent of the choice of generator would strengthen the result.

    Authors: The decoupling conditions are obtained by requiring that the action of the joint generator on test functions of the form f(size) * g(internal) separates appropriately when multiplied by the exponential of the integrated growth. This leads to specific PDEs or ODEs that the growth rate must satisfy relative to the internal dynamics, which are not automatically true for arbitrary dependence. To demonstrate this, we will include in the revision a simple example with a two-state internal variable (e.g., on/off gene expression) where the growth rate depends on the state. In this case, the decoupling holds only when the transition rates and growth values satisfy a particular algebraic relation, which is independent of the specific form of the size growth operator. This shows the conditions are non-tautological and depend on the interplay between growth and internal state transitions. We believe this addresses the concern without altering the core derivations. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard Feynman-Kac to independently derived decoupling conditions

full rationale

The paper states a size-structured branching-process model with growth rate set by an internal fluctuating variable, then derives explicit conditions for decoupling in lineage and population ensembles directly from the generator. When both ensembles decouple, a random time change reduces size dynamics to a growth-homogeneous process whose expectations are obtained by the exponential tilting that the Feynman-Kac formula supplies for multiplicative functionals. This is a standard application of an external theorem (Feynman-Kac) to a model whose generator and decoupling equations are written out explicitly; no parameter is fitted to data and then relabeled a prediction, no self-citation supplies a uniqueness theorem that forces the result, and the integrability conditions required for the martingale are left as an implicit modeling assumption rather than being smuggled in by definition. The derivation chain therefore remains self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the model rests on standard branching-process generators and the Feynman-Kac representation, with no explicit free parameters or new entities named.

axioms (2)
  • domain assumption The population obeys a branching process whose generator admits a Feynman-Kac representation for expectations along lineages.
    Invoked when the authors connect lineage dynamics to the Feynman-Kac formula.
  • domain assumption The internal variable directly modulates the instantaneous growth rate of cell size.
    Stated as the core modeling choice that enables the decoupling analysis.

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Forward citations

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