Kinetic theories of state- and generation-dependent cell populations

Mingtao Xia; Tom Chou

arxiv: 2403.05691 · v1 · submitted 2024-03-08 · 🧬 q-bio.PE · cond-mat.stat-mech· q-bio.QM

Kinetic theories of state- and generation-dependent cell populations

Mingtao Xia , Tom Chou This is my paper

Pith reviewed 2026-05-24 03:26 UTC · model grok-4.3

classification 🧬 q-bio.PE cond-mat.stat-mechq-bio.QM

keywords kinetic theorycell population dynamicsgeneration trackingstochastic processesmarginalizationinternal cell statedemographic stochasticitygene expression

0 comments

The pith

A kinetic theory describes internal cell states and generation numbers in stochastically dividing populations.

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

The paper develops a general high-dimensional kinetic framework for cells whose internal attributes evolve with noise while the population grows through random division events. It incorporates both the randomness inside each cell and the demographic uncertainty in when divisions occur, and it indexes subpopulations by generation. Marginalization of the kinetic equations produces reduced descriptions for population densities structured by state and generation, for moments of those states, and for total population size. If the marginalizations remain closed and faithful, the approach connects detailed cellular processes to observable population statistics in contexts such as tissue development.

Core claim

We formulate a general, high-dimensional kinetic theory describing the internal state of cells in a stochastically evolving population. The resolution of our kinetic theory also allows one to track subpopulations associated with each generation. Both intrinsic noise of the cell's internal attribute and randomness in a cell's division times are fundamental to the development of our model. Based on this general framework, we are able to marginalize the high-dimensional kinetic PDEs in a number of different ways to derive equations that describe the dynamics of marginalized or macroscopic quantities such as structured population densities, moments of generation-dependent cellular states, andmom

What carries the argument

High-dimensional kinetic partial differential equations that couple stochastic evolution of cell internal states with birth-death processes indexed by generation.

If this is right

Structured population densities can be obtained by marginalizing the high-dimensional kinetic PDEs.
Moments of generation-dependent cellular states follow from the same marginalization procedure.
Moments of the total population size are likewise derivable.
Nonlinear interaction terms appear in lower-dimensional integrodifferential equations when cell rates depend on variables of other cells.
The framework resolves the coevolution of cell populations and cell states for applications such as gene expression in developing tissues.

Where Pith is reading between the lines

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

The same marginalization steps could be applied to models that add spatial position as an additional cell attribute.
Lineage-specific accumulation of mutations might be tracked by extending the generation index to include state-dependent mutation rates.
Comparison against individual-based simulations would test whether the reduced equations preserve fluctuation statistics for small populations.
Feedback from total population density back to individual rates could be inserted at the high-dimensional level before marginalization.

Load-bearing premise

The high-dimensional kinetic PDEs can be marginalized in multiple ways to obtain closed or useful equations for structured population densities, moments, and total population without introducing uncontrolled approximations or losing essential dynamical features.

What would settle it

Direct stochastic simulation of a cell population with state-dependent division rates where the derived marginalized moment equations deviate measurably from the full-model statistics would show that the marginalization step loses essential features.

Figures

Figures reproduced from arXiv: 2403.05691 by Mingtao Xia, Tom Chou.

read the original abstract

We formulate a general, high-dimensional kinetic theory describing the internal state (such as gene expression or protein levels) of cells in a stochastically evolving population. The resolution of our kinetic theory also allows one to track subpopulations associated with each generation. Both intrinsic noise of the cell's internal attribute and randomness in a cell's division times (demographic stochasticity) are fundamental to the development of our model. Based on this general framework, we are able to marginalize the high-dimensional kinetic PDEs in a number of different ways to derive equations that describe the dynamics of marginalized or "macroscopic" quantities such as structured population densities, moments of generation-dependent cellular states, and moments of the total population. We also show how nonlinear "interaction" terms in lower-dimensional integrodifferential equations can arise from high-dimensional linear kinetic models that contain rate parameters of a cell (birth and death rates) that depend on variables associated with other cells, generating couplings in the dynamics. Our analysis provides a general, more complete mathematical framework that resolves the coevolution of cell populations and cell states. The approach may be tailored for studying, e.g., gene expression in developing tissues, or other more general particle systems which exhibit Brownian noise in individual attributes and population-level demographic noise.

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.

Desk Editor's Note private letter to a colleague

The paper gives a high-dimensional kinetic PDE setup that tracks cell states plus generations under both intrinsic and demographic noise, then marginalizes to lower-dimensional equations for densities and moments.

read the letter

The core contribution is a unified kinetic framework that keeps track of both a cell's internal state (gene expression or protein levels) and its generation number while including stochastic division times and state-dependent rates. From that high-dimensional linear PDE they derive several marginalizations that produce equations for structured population densities, generation-dependent moments, and total population size. They also show how nonlinear interaction terms can appear in the reduced equations even though the starting model is linear, simply because birth and death rates depend on other cells' variables. That part is cleanly stated and gives a concrete route from microscopic assumptions to macroscopic observables that standard structured population models often handle separately. The setup looks general enough to apply to developing tissues or similar systems with both Brownian noise in attributes and demographic fluctuations. The main uncertainty is whether the marginalizations stay exact or require closures that lose cross-generation correlations or introduce uncontrolled errors; the abstract asserts multiple exact routes but does not display the steps or error checks. If the full derivations hold without hidden approximations, the framework is a useful extension. This is aimed at mathematical biologists who already work with kinetic or structured population models and want a single formalism that includes generation tracking. A reader who needs to derive moment equations or check consistency between single-cell noise and population dynamics would find the marginalization strategy worth examining. It is coherent enough on its own terms to merit a serious referee who can verify the algebra in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript formulates a high-dimensional linear kinetic PDE framework for stochastically evolving cell populations that tracks both internal cell states (e.g., gene expression levels) and generation-specific subpopulations. It incorporates intrinsic Brownian noise in cell attributes and demographic stochasticity in division times, then derives marginalized equations for structured population densities, generation-dependent moments, and total population size. The work also shows how nonlinear interaction terms can emerge in lower-dimensional integrodifferential equations from the underlying linear high-dimensional model when birth/death rates depend on variables associated with other cells.

Significance. If the marginalizations are rigorously shown to be exact (or controlled) without loss of essential cross-generation correlations or introduction of unclosed hierarchies, the framework would supply a general, parameter-free route from microscopic stochastic assumptions to macroscopic structured-population equations. This is potentially useful for modeling coevolution of cell states and demographics in tissues or other particle systems with both intrinsic and demographic noise; the explicit separation of linear high-dimensional dynamics from emergent nonlinearities is a conceptual strength.

major comments (2)

[Abstract and marginalization sections] The central claim that multiple marginalizations over internal states and generations produce closed or useful macroscopic equations without uncontrolled approximations is load-bearing but not yet verified in the supplied abstract; the skeptic concern that marginalization over state-dependent division times typically yields integro-differential or unclosed moment equations must be addressed by explicit derivation steps (e.g., in the section presenting the marginalization procedure).
[Section deriving nonlinear terms] It is unclear whether the emergence of nonlinear interaction terms preserves all cross-generation correlations or requires additional closure assumptions when rates depend on other cells' states; this needs to be shown with a concrete example (e.g., a two-cell or low-dimensional case) that compares the marginalized equation to the full high-dimensional solution.

minor comments (2)

[Model formulation] Notation for the high-dimensional state variable and generation index should be introduced with a clear table or list of symbols to aid readability.
[Abstract] The abstract states that the approach 'resolves the coevolution' but does not specify the precise sense in which the marginal equations are closed or exact; a short clarifying sentence would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which help clarify the presentation of our kinetic framework. We address each major comment below and will revise the manuscript to strengthen the exposition of the marginalization procedures and the emergence of nonlinear terms.

read point-by-point responses

Referee: [Abstract and marginalization sections] The central claim that multiple marginalizations over internal states and generations produce closed or useful macroscopic equations without uncontrolled approximations is load-bearing but not yet verified in the supplied abstract; the skeptic concern that marginalization over state-dependent division times typically yields integro-differential or unclosed moment equations must be addressed by explicit derivation steps (e.g., in the section presenting the marginalization procedure).

Authors: The full manuscript contains explicit step-by-step derivations of the marginalizations (in the sections developing the high-dimensional kinetic PDE and the subsequent marginalization procedures) that demonstrate closure for the structured densities, generation-dependent moments, and total population size. These derivations rely on the linearity of the underlying model and the specific stochastic assumptions, which permit exact integration even when division times depend on internal states, avoiding unclosed hierarchies. The abstract summarizes the outcomes rather than the derivations. We will revise the abstract to note the rigorous, approximation-free nature of the marginalizations and expand the marginalization section with additional explicit derivation steps to address the skeptic concern directly. revision: yes
Referee: [Section deriving nonlinear terms] It is unclear whether the emergence of nonlinear interaction terms preserves all cross-generation correlations or requires additional closure assumptions when rates depend on other cells' states; this needs to be shown with a concrete example (e.g., a two-cell or low-dimensional case) that compares the marginalized equation to the full high-dimensional solution.

Authors: The nonlinear interaction terms arise directly from embedding state-dependent rates (coupled across cells) into the otherwise linear high-dimensional kinetic model; the marginalization remains exact and preserves cross-generation correlations because no additional closures are imposed beyond the original linear stochastic dynamics. The lower-dimensional equations inherit the correlations from the full system. To make this transparent, we will add a concrete low-dimensional example (e.g., a two-generation case with explicit state-dependent birth rates) to the section on nonlinear terms, explicitly comparing the marginalized integrodifferential equation against the solution of the corresponding high-dimensional linear system. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is forward from stochastic assumptions

full rationale

The paper starts from explicit stochastic assumptions on intrinsic cell-state noise and demographic division-time randomness, constructs a high-dimensional linear kinetic PDE, and then performs marginalizations over states or generations to obtain structured densities, moments, and total-population equations. No quoted step reduces a derived macroscopic equation to a fitted parameter or to a self-citation whose content is itself the target result; the abstract and provided text present the marginalizations as exact consequences of the linear high-dimensional model under the stated assumptions. Self-citation load-bearing, ansatz smuggling, or renaming of known results are absent from the given material. The central claim therefore remains a self-contained forward derivation rather than a tautological re-expression of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard stochastic-process assumptions for cell division and internal-state evolution; no explicit free parameters, new entities, or ad-hoc axioms are named in the abstract.

axioms (2)

domain assumption Cell internal states evolve according to stochastic processes and division times are random (demographic stochasticity).
Stated as fundamental to the model development in the abstract.
domain assumption High-dimensional kinetic PDEs admit useful marginalizations to lower-dimensional equations for densities and moments.
Central to the claim that macroscopic equations can be derived.

pith-pipeline@v0.9.0 · 5757 in / 1291 out tokens · 26929 ms · 2026-05-24T03:26:48.908879+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We formulate a general, high-dimensional kinetic theory describing the internal state ... marginalize the high-dimensional kinetic PDEs in a number of different ways to derive equations that describe ... structured population densities, moments ... nonlinear 'interaction' terms ... from high-dimensional linear kinetic models
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Both intrinsic noise of the cell's internal attribute and randomness in a cell's division times (demographic stochasticity) are fundamental ... SDE-jump-process hybrid model

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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(A9) By induction, Eq

It is straightforward to verify that F m(0; X(0)n(0), 0) = 1; therefore, we have F m(t; X(0)n(0), 0) ≡ 1, ∀t ≥ 0, which indicates that ∑ n ∫ pm n (Xn, t |X(0)n(0), 0)dXn ≤ 1. (A9) By induction, Eq. (A7) holds true for all m ∈ N+. Finally, it is easy to show that pm n (X(0)n, t |X(0)n(0), 0) ≥ 0, so by the monotone convergence theorem, ∑ n ∫ p∗ n(Xn, t |X(...

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We shall apply Theorem 6.2 in [27]

Proof of Proposition 1 Here, we prove Proposition 1 and provide the needed technical ass umptions. We shall apply Theorem 6.2 in [27]. If n ̸= n(0), then by deﬁnition ˆ pn = 0, which solves Eq. (A1). If n(s) ≡ n(0), s ∈ [0, t ], for any smooth function φ ∈ C∞ (R|n|1), |n|1 := ∑k i=1 ni, we deﬁne the measure γ m(φ, t ) := ∫ C|n|1 φ(Xn(t; ω ))S ( t; X(t, ω ...

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timer” models where a new daughter cell acquires age 0 at its birth, while the other cell is assumed to be th e “mother

Proof of Proposition 2 We prove Proposition 1 by induction on m. Clearly, when m = 0, 1, p0 and p1 solve Eq. (A6) by using Proposition 1. If the conclusion holds for m ≥ 1, then when n ̸= n(0), we have ∂pm+1 n ∂t =E [ S ( t; X(t)n(t) ) J m(t, t; Xn, n(0)) ⏐ ⏐ ⏐Xn(0)(0), 0; n(0 < s < t ) = n(0) ] + ∫ t 0 E [ S( τ ; Xn ) ∂tJ m( t, τ; Xn, n(0)) ⏐ ⏐ ⏐X(0)n(0)...

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[1] [1]

landscapes

or protein [14, 15] levels from experimental data, mathematica l models of how expression levels or cell states evolve is often couched in terms of transport along Waddington or ﬁtness landscapes [16, 17]. The value of the landscape may represent an “energy” function that is shaped by diﬀerent ge nes, or a proliferation rate that is a function diﬀerent ge...

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[2] [2]

Some remarks on changing population s

Heinz von Foerster. Some remarks on changing population s. In Jr. F. Stohlman, editor, The Kinetics of Cellular Prolifer- ation, pages 382–407. Grune and Stratton, New York, 1959

work page 1959

[3] [3]

Modelling the i mpact of birth control policies on China’s population and age: eﬀects of delayed births and minimum birth age constrai nts

Yue Wang, Renaud Dessalles, and Tom Chou. Modelling the i mpact of birth control policies on China’s population and age: eﬀects of delayed births and minimum birth age constrai nts. Royal Society Open Science , 9:211619, 2022

work page 2022

[4] [4]

S Taheri-Araghi, S Bradde, J. T. Sauls, N. S. Hill, P. A. Le vin, J Paulsson, M Vergassola, and S Jun. Cell-size control a nd homeostasis in bacteria. Current Biology, 25(3):385–391, 2015

work page 2015

[5] [5]

Effective Potential for Cellular Size Control

Stanislav Burov and David Kessler. Eﬀective potential f or cellular size control. arXIv:1701.01725, 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017

[6] [6]

Division in Escherichia coli is triggered by a size-sensing rather than a timing mechanis m

Lydia Robert, Marc Hoﬀmann, Nathalie Krell, St´ ephane A ymerich, J´ erˆ ome Robert, and Marie Doumic. Division in Escherichia coli is triggered by a size-sensing rather than a timing mechanis m. BMC Biology , 12:17, 2014

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[8] [8]

J. A. J. Metz and O. Diekmann. The Dynamics of Physiologically Structured Populations . Springer Berlin, Heidelberg, 1986

work page 1986

[9] [9]

Greenman and Tom Chou

Chris D. Greenman and Tom Chou. Kinetic theory of age-str uctured stochastic birth-death processes. Physical Review E , 93(1):012112, 2016

work page 2016

[10] [10]

A hierarchical kinetic the ory of birth, death and ﬁssion in age-structured interactin g populations

Tom Chou and Chris D Greenman. A hierarchical kinetic the ory of birth, death and ﬁssion in age-structured interactin g populations. Journal of Statistical Physics , 164(1):49–76, 2016

work page 2016

[11] [11]

Kinetic theory for structured populations: application to stochastic sizer-timer model s of cell proliferation

Mingtao Xia and Tom Chou. Kinetic theory for structured populations: application to stochastic sizer-timer model s of cell proliferation. Journal of Physics A: Mathematical and Theoretical , 54(38):385601, 2021

work page 2021

[12] [12]

Kynaston, Chris Guiver, and Christian A

Joshua C. Kynaston, Chris Guiver, and Christian A. Yate s. Equivalence framework for an age-structured multistage representation of the cell cycle. Physical Review E , 105:064411, 2022

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(A9) By induction, Eq

It is straightforward to verify that F m(0; X(0)n(0), 0) = 1; therefore, we have F m(t; X(0)n(0), 0) ≡ 1, ∀t ≥ 0, which indicates that ∑ n ∫ pm n (Xn, t |X(0)n(0), 0)dXn ≤ 1. (A9) By induction, Eq. (A7) holds true for all m ∈ N+. Finally, it is easy to show that pm n (X(0)n, t |X(0)n(0), 0) ≥ 0, so by the monotone convergence theorem, ∑ n ∫ p∗ n(Xn, t |X(...

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We shall apply Theorem 6.2 in [27]

Proof of Proposition 1 Here, we prove Proposition 1 and provide the needed technical ass umptions. We shall apply Theorem 6.2 in [27]. If n ̸= n(0), then by deﬁnition ˆ pn = 0, which solves Eq. (A1). If n(s) ≡ n(0), s ∈ [0, t ], for any smooth function φ ∈ C∞ (R|n|1), |n|1 := ∑k i=1 ni, we deﬁne the measure γ m(φ, t ) := ∫ C|n|1 φ(Xn(t; ω ))S ( t; X(t, ω ...

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timer” models where a new daughter cell acquires age 0 at its birth, while the other cell is assumed to be th e “mother

Proof of Proposition 2 We prove Proposition 1 by induction on m. Clearly, when m = 0, 1, p0 and p1 solve Eq. (A6) by using Proposition 1. If the conclusion holds for m ≥ 1, then when n ̸= n(0), we have ∂pm+1 n ∂t =E [ S ( t; X(t)n(t) ) J m(t, t; Xn, n(0)) ⏐ ⏐ ⏐Xn(0)(0), 0; n(0 < s < t ) = n(0) ] + ∫ t 0 E [ S( τ ; Xn ) ∂tJ m( t, τ; Xn, n(0)) ⏐ ⏐ ⏐X(0)n(0)...

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