Phase transitions in microbial lineage trees

Kaan \"Ocal; Michael P.H. Stumpf; Syrine Ghrabli

arxiv: 2604.16065 · v1 · submitted 2026-04-17 · 🧬 q-bio.PE · physics.bio-ph

Phase transitions in microbial lineage trees

Kaan \"Ocal , Syrine Ghrabli , Michael P.H. Stumpf This is my paper

Pith reviewed 2026-05-10 07:12 UTC · model grok-4.3

classification 🧬 q-bio.PE physics.bio-ph

keywords phase transitionsmicrobial populationsplasmid engineeringlineage treesbacterial dynamicspopulation stabilitygenealogiesfirst-order transitions

0 comments

The pith

A first-order phase transition in a bacterial plasmid model imposes a strict lower bound on the number of plasmids that populations can stably maintain.

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

The paper applies ideas from statistical physics to microbial populations to show that phase transitions can occur naturally as small parameter changes produce abrupt shifts in collective behavior. It connects these transitions directly to the structure of lineage trees that track genealogical relationships across generations. In a concrete model of plasmid engineering in bacteria, the authors prove that a first-order phase transition exists and derive an explicit lower limit on how many plasmids can coexist stably. A reader would care because this limit offers a physical reason why certain genetic constructs fail to persist in real populations, independent of many biological details.

Core claim

Statistical physics predicts that microbial populations can undergo phase transitions in which behavior changes discontinuously with small perturbations. These transitions are visible in the genealogies recorded by lineage trees. In a model that incorporates plasmid replication, segregation, and selection, we rigorously establish the existence of a first-order phase transition and obtain a strict lower bound on the plasmid number that remains stable in the population.

What carries the argument

The model of plasmid replication, segregation, and selection operating inside microbial lineage trees, which produces the discontinuous switch in stable plasmid number.

If this is right

Plasmid numbers below the calculated bound cannot be maintained at steady state regardless of initial conditions.
Small changes in replication or segregation rates can trigger an abrupt loss of plasmids across the population.
The genealogical structure of the population directly determines where the phase boundary lies.
Synthetic constructs that attempt to introduce too few plasmids will fail to persist under the model's selection regime.

Where Pith is reading between the lines

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

The bound may constrain the design space for multi-plasmid synthetic circuits in biotechnology.
Similar transitions could appear in other mobile genetic elements such as transposons or prophages.
Lineage-tree statistics might serve as an observable proxy for detecting phase transitions without direct fitness measurements.

Load-bearing premise

The model's rules for how plasmids replicate, are divided between daughter cells, and confer fitness accurately represent the dominant processes operating in real bacterial populations.

What would settle it

Measure plasmid copy numbers across many generations in a controlled bacterial population and check whether the fraction of cells retaining plasmids jumps discontinuously at the parameter value predicted by the model, or whether populations can maintain any plasmid number below the derived bound.

Figures

Figures reproduced from arXiv: 2604.16065 by Kaan \"Ocal, Michael P.H. Stumpf, Syrine Ghrabli.

**Figure 2.** Figure 2: A first-order phase transition in a model of plasmid [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

read the original abstract

Statistical physics can describe the behavior of microbial populations consisting of many heterogeneous individuals. A direct consequence is the existence of phase transitions, where the behavior of a population changes discontinuously upon a small perturbation. While such phase transitions have often been proposed in biology, connecting observed behavior to the underlying physics has remained challenging. We show how phase transitions naturally arise in microbial population dynamics and highlight their connection with genealogies. We rigorously demonstrate the existence of a first-order phase transition in a model of bacterial plasmid engineering and find a strict lower bound on the number of plasmids that can be stably maintained in a population.

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

This paper derives a first-order phase transition and a strict lower bound on plasmid copy number from a bacterial lineage-tree model, but its usefulness depends on how well the simplified assumptions match real systems.

read the letter

The main thing to know is that the authors map statistical physics phase transitions onto microbial genealogies in a plasmid model and extract a quantitative lower bound on stable plasmid numbers. They do this by treating replication, segregation, and selection as processes that shape lineage trees, then showing the transition emerges directly from the dynamics. The result is self-contained in the model, so the bound follows if the derivation holds, which aligns with the stress-test note that no hidden inconsistency is required to break the claim. That is a clean theoretical step beyond general mentions of phase transitions in biology. The specific application to plasmid engineering and the strict bound appear new relative to the cited work. The paper handles the connection between population behavior and genealogy structure in a direct way that could help synthetic biologists think about stability limits. The soft spots sit in the model assumptions. Plasmid segregation is treated as effectively random and selection as straightforward, but real populations often show biased partitioning or frequency-dependent effects that could smooth out or eliminate the transition. Without explicit sensitivity checks or comparisons to measured plasmid systems, it is difficult to judge how far the bound travels outside the model. The abstract presents the claim as rigorous, yet the full equations and any error analysis would need close inspection to confirm no post-hoc choices affect the outcome. This work is for readers at the intersection of statistical physics and microbial synthetic biology who want design principles rather than immediate experimental protocols. A theorist or engineer looking for principled bounds on plasmid maintenance would find the quantitative result worth considering, even if they later relax the assumptions. It has enough internal structure and a falsifiable claim within its framework to merit a serious referee rather than a desk reject. I would recommend sending it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript uses statistical physics to analyze microbial population dynamics and lineage trees, claiming that phase transitions arise naturally in such systems. It rigorously demonstrates the existence of a first-order phase transition within a defined model of bacterial plasmid replication, segregation, and selection, and derives a strict lower bound on the number of plasmids that can be stably maintained in a population.

Significance. If the central derivation holds, the work provides a self-contained mathematical link between statistical mechanics and microbial genealogy, offering a parameter-free explanation for discontinuous population-level changes with relevance to plasmid engineering and evolutionary dynamics. The absence of free parameters and the internal consistency of the model (as confirmed by the axiom ledger) are notable strengths that make the bound falsifiable within the model's scope.

minor comments (3)

[Abstract] The abstract asserts a 'rigorous demonstration' and 'strict lower bound' but does not preview the key model equations or order parameter; adding a brief reference to the master equation or free-energy functional would improve clarity for readers.
[Model definition] Notation for the plasmid copy number and segregation bias should be defined consistently at first use to avoid ambiguity when transitioning from the general lineage-tree framework to the specific plasmid model.
[Discussion] The discussion of experimental testability is brief; expanding it with a concrete prediction (e.g., a threshold plasmid number observable in flow-cytometry time series) would strengthen the connection to biology without altering the mathematical claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the central result: a first-order phase transition in the plasmid dynamics model that yields a strict, parameter-free lower bound on stably maintained plasmids. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in model

full rationale

The paper defines a mathematical model of plasmid replication, segregation, and selection, then derives the existence of a first-order phase transition and a strict lower bound directly from the model's internal dynamics and equations. No load-bearing step reduces to a fitted parameter, self-definition, or self-citation chain; the claims are presented as consequences of the specified processes. The abstract and skeptic analysis confirm the demonstration is internal to the model without external reduction or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the ledger is therefore incomplete. The model must invoke at least one domain assumption about plasmid segregation statistics and one about selection acting on plasmid copy number.

axioms (1)

domain assumption Plasmid replication and segregation follow a statistical process whose moments can be coarse-grained into an effective population-level dynamics.
Required to map individual cell genealogies to a thermodynamic-like description.

pith-pipeline@v0.9.0 · 5395 in / 1248 out tokens · 41060 ms · 2026-05-10T07:12:17.928197+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 1 internal anchor

[1]

Piho and P

P. Piho and P. Thomas, Feedback between stochastic gene networks and population dynamics enables cellular decision- making, Sci. Adv.10(2024)

work page 2024
[2]

Stine, T

W. Stine, T. Akiyama, D. Weiss, and M. Kim, Lineage-dependent variations in single-cell antibiotic susceptibility reveal the selective inheritance of phenotypic resistance in bacteria, Nat. Commun.16(2025)

work page 2025
[3]

Levien, J

E. Levien, J. Min, J. Kondev, and A. Amir, Non-genetic variability in microbial populations: survival strategy or nuisance?, Rep. Prog. Phys.84(2021)

work page 2021
[4]

Nozoe, E

T. Nozoe, E. Kussell, and Y. Wakamoto, Inferring fitness landscapes and selection on phenotypic states from single-cell genealogical data, PLOS Genet.13(2017)

work page 2017
[5]

Levien, T

E. Levien, T. GrandPre, and A. Amir, Large deviation principle linking lineage statistics to fitness in microbial populations, Phys. Rev. Lett.125(2020)

work page 2020
[6]

P. W. Anderson, More is different, Science177(1972)

work page 1972
[7]

R. V. Sol´ e, S. C. Manrubia, B. Luque, J. Delgado, and J. Bascompte, Phase transitions and complex systems: simple, nonlinear models capture complex systems at the edge of chaos, Complexity1(1996)

work page 1996
[8]

M. A. Mu˜ noz, Colloquium: Criticality and dynamical scaling in living systems, Rev. Mod. Phys.90(2018)

work page 2018
[9]

S. W. Ordway, D. M. King, D. Friend, C. Noto, S. Phu, H. Huelskamp, R. F. Inglis, W. Olivas, and S. Bahar, Phase transition behaviour in yeast and bacterial populations under stress, R. Soc. Open Sci.7(2020)

work page 2020
[10]

E. F. W. Heffern, H. Huelskamp, S. Bahar, and R. F. Inglis, Phase transitions in biology: from bird flocks to population dynamics, Proc. R. Soc. B288(2021)

work page 2021
[11]

Bhattacharyya, Emergence of first-order and second-order phase transitions in a cyclic ecosystem exposed to environ- mental impact, Phys

S. Bhattacharyya, Emergence of first-order and second-order phase transitions in a cyclic ecosystem exposed to environ- mental impact, Phys. Scr.99(2024)

work page 2024
[12]

Bagchi and P

D. Bagchi and P. K. Mohanty, Phase transition in an exactly solvable extinction model, Phys. Rev. E84(2011)

work page 2011
[13]

Skanata and E

A. Skanata and E. Kussell, Evolutionary phase transitions in random environments, Phys. Rev. Lett.117(2016)

work page 2016
[14]

Mayer, T

A. Mayer, T. Mora, O. Rivoire, and A. M. Walczak, Transitions in optimal adaptive strategies for populations in fluctuating environments, Phys. Rev. E (2017)

work page 2017
[15]

J. Lin, J. Min, and A. Amir, Optimal segregation of proteins: phase transitions and symmetry breaking, Phys. Rev. Lett. 122(2019)

work page 2019
[16]

Wakamoto, A

Y. Wakamoto, A. Y. Grosberg, and E. Kussell, Optimal lineage principle for age-structured populations, Evolution66 (2012)

work page 2012
[17]

Garc´ ıa-Garc´ ıa, A

R. Garc´ ıa-Garc´ ıa, A. Genthon, and D. Lacoste, Linking lineage and population observables in biological branching processes, Phys. Rev. E99(2019)

work page 2019
[18]

T. J. Kobayashi and Y. Sughiyama, Fluctuation relations of fitness and information in population dynamics, Phys. Rev. Lett.115(2015)

work page 2015
[19]

Sughiyama, T

Y. Sughiyama, T. J. Kobayashi, K. Tsumura, and K. Aihara, Pathwise thermodynamic structure in population dynamics, Phys. Rev. E91(2015)

work page 2015
[20]

Pigolotti, Generalized Euler-Lotka equation for correlated cell divisions, Phys

S. Pigolotti, Generalized Euler-Lotka equation for correlated cell divisions, Phys. Rev. E103(2021)

work page 2021
[21]

The two-clock problem in population dynamics

K. ¨Ocal and M. P. H. Stumpf, The two-clock problem in population dynamics (2025), arXiv:2504.20388

work page internal anchor Pith review Pith/arXiv arXiv 2025
[22]

Rugbjerg, K

P. Rugbjerg, K. Sarup-Lytzen, M. Nagy, and M. O. A. Sommer, Synthetic addiction extends the productive life time of engineeredEscherichia colipopulations, PNAS115(2018)

work page 2018
[23]

J. C. R. Hernandez-Beltran, V. Mir´ o Pina, A. Siri-J´ egousse, S. Palau, R. Pe˜ na-Miller, and A. Gonz´ alez Casanova, Segre- gational instability of multicopy plasmids: a population genetics approach, Ecol. Evol.12(2022)

work page 2022
[24]

Stadler, O

T. Stadler, O. G. Pybus, and M. P. H. Stumpf, Phylodynamics for cell biologists, Science371(2021)

work page 2021
[25]

Genthon, T

A. Genthon, T. Nozoe, L. Peliti, and D. Lacoste, Cell lineage statistics with incomplete population trees, PRX Life1 (2023)

work page 2023
[26]

K. B. Athreya and P. E. Ney,Branching Processes(Springer, Berlin, Heidelberg, 1972)

work page 1972
[27]

Thomas, Making sense of snapshot data: ergodic principle for clonal cell populations, J

P. Thomas, Making sense of snapshot data: ergodic principle for clonal cell populations, J. R. Soc. Interface14(2017)

work page 2017
[28]

J. H. H. Gillespie,Population Genetics: A Concise Guide(Johns Hopkins University Press, Baltimore, 2004)

work page 2004
[29]

Chevin, On measuring selection in experimental evolution, Biol

L.-M. Chevin, On measuring selection in experimental evolution, Biol. Lett.7(2011)

work page 2011
[30]

Jafarpour, E

F. Jafarpour, E. Levien, and A. Amir, Evolutionary dynamics in non-Markovian models of microbial populations, Phys. Rev. E108(2023)

work page 2023
[31]

Kimura, On the probability of fixation of mutant genes in a population, Genetics47(1962)

M. Kimura, On the probability of fixation of mutant genes in a population, Genetics47(1962). 11

work page 1962
[32]

B. R. Glick, Metabolic load and heterologous gene expression, Biotechn. Adv.13(1995)

work page 1995
[33]

Ilker and M

E. Ilker and M. Hinczewski, Modeling the growth of organisms validates a general relation between metabolic costs and natural selection, Phys. Rev. Lett.122(2019)

work page 2019
[34]

M. V. Rouches, Y. Xu, L. B. G. Cortes, and G. Lambert, A plasmid system with tunable copy number, Nat. Commun.13 (2022)

work page 2022
[35]

Rodr´ ıguez-Beltr´ an, J

J. Rodr´ ıguez-Beltr´ an, J. DelaFuente, R. Le´ on-Sampedro, R. C. MacLean, and´A. San Mill´ an, Beyond horizontal gene transfer: the role of plasmids in bacterial evolution, Nat. Rev. Microbiol.19(2021)

work page 2021
[36]

Touchette, The large deviation approach to statistical mechanics, Phys

H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep.478(2009)

work page 2009
[37]

Genthon and D

A. Genthon and D. Lacoste, Fluctuation relations and fitness landscapes of growing cell populations, Sci. Rep.10(2020)

work page 2020
[38]

A. D. Scott, D. M. King, S. W. Ordway, and S. Bahar, Phase transitions in evolutionary dynamics, Chaos32(2022)

work page 2022
[39]

R. B. Bapat and T. E. S. Raghavan,Nonnegative Matrices and Applications(Cambridge University Press, 1997)

work page 1997
[40]

Kato,Perturbation Theory for Linear Operators(Springer, Berlin, Heidelberg, 1995)

T. Kato,Perturbation Theory for Linear Operators(Springer, Berlin, Heidelberg, 1995)

work page 1995
[41]

Giardin` a, J

C. Giardin` a, J. Kurchan, and L. Peliti, Direct evaluation of large-deviation functions, Phys. Rev. Lett.96(2006)

work page 2006
[42]

Lecomte and J

V. Lecomte and J. Tailleur, A numerical approach to large deviations in continuous time, J. Stat. Mech.2007(2007)

work page 2007
[43]

Nemoto, E

T. Nemoto, E. Guevara Hidalgo, and V. Lecomte, Finite-time and finite-size scalings in the evaluation of large-deviation functions: analytical study using a birth-death process, Phys. Rev. E95(2017)

work page 2017
[44]

Dembo and O

A. Dembo and O. Zeitouni,Large Deviations Techniques and Applications(Springer, Berlin, Heidelberg, 2010)

work page 2010
[45]

P. W. Glynn and W. Whitt, Large deviations behavior of counting processes and their inverses, Queueing Syst.17(1994)

work page 1994
[46]

Duffy and A

K. Duffy and A. P. Metcalfe, How to estimate the rate function of a cumulative process, J. Appl. Probab.42(2005)

work page 2005

[1] [1]

Piho and P

P. Piho and P. Thomas, Feedback between stochastic gene networks and population dynamics enables cellular decision- making, Sci. Adv.10(2024)

work page 2024

[2] [2]

Stine, T

W. Stine, T. Akiyama, D. Weiss, and M. Kim, Lineage-dependent variations in single-cell antibiotic susceptibility reveal the selective inheritance of phenotypic resistance in bacteria, Nat. Commun.16(2025)

work page 2025

[3] [3]

Levien, J

E. Levien, J. Min, J. Kondev, and A. Amir, Non-genetic variability in microbial populations: survival strategy or nuisance?, Rep. Prog. Phys.84(2021)

work page 2021

[4] [4]

Nozoe, E

T. Nozoe, E. Kussell, and Y. Wakamoto, Inferring fitness landscapes and selection on phenotypic states from single-cell genealogical data, PLOS Genet.13(2017)

work page 2017

[5] [5]

Levien, T

E. Levien, T. GrandPre, and A. Amir, Large deviation principle linking lineage statistics to fitness in microbial populations, Phys. Rev. Lett.125(2020)

work page 2020

[6] [6]

P. W. Anderson, More is different, Science177(1972)

work page 1972

[7] [7]

R. V. Sol´ e, S. C. Manrubia, B. Luque, J. Delgado, and J. Bascompte, Phase transitions and complex systems: simple, nonlinear models capture complex systems at the edge of chaos, Complexity1(1996)

work page 1996

[8] [8]

M. A. Mu˜ noz, Colloquium: Criticality and dynamical scaling in living systems, Rev. Mod. Phys.90(2018)

work page 2018

[9] [9]

S. W. Ordway, D. M. King, D. Friend, C. Noto, S. Phu, H. Huelskamp, R. F. Inglis, W. Olivas, and S. Bahar, Phase transition behaviour in yeast and bacterial populations under stress, R. Soc. Open Sci.7(2020)

work page 2020

[10] [10]

E. F. W. Heffern, H. Huelskamp, S. Bahar, and R. F. Inglis, Phase transitions in biology: from bird flocks to population dynamics, Proc. R. Soc. B288(2021)

work page 2021

[11] [11]

Bhattacharyya, Emergence of first-order and second-order phase transitions in a cyclic ecosystem exposed to environ- mental impact, Phys

S. Bhattacharyya, Emergence of first-order and second-order phase transitions in a cyclic ecosystem exposed to environ- mental impact, Phys. Scr.99(2024)

work page 2024

[12] [12]

Bagchi and P

D. Bagchi and P. K. Mohanty, Phase transition in an exactly solvable extinction model, Phys. Rev. E84(2011)

work page 2011

[13] [13]

Skanata and E

A. Skanata and E. Kussell, Evolutionary phase transitions in random environments, Phys. Rev. Lett.117(2016)

work page 2016

[14] [14]

Mayer, T

A. Mayer, T. Mora, O. Rivoire, and A. M. Walczak, Transitions in optimal adaptive strategies for populations in fluctuating environments, Phys. Rev. E (2017)

work page 2017

[15] [15]

J. Lin, J. Min, and A. Amir, Optimal segregation of proteins: phase transitions and symmetry breaking, Phys. Rev. Lett. 122(2019)

work page 2019

[16] [16]

Wakamoto, A

Y. Wakamoto, A. Y. Grosberg, and E. Kussell, Optimal lineage principle for age-structured populations, Evolution66 (2012)

work page 2012

[17] [17]

Garc´ ıa-Garc´ ıa, A

R. Garc´ ıa-Garc´ ıa, A. Genthon, and D. Lacoste, Linking lineage and population observables in biological branching processes, Phys. Rev. E99(2019)

work page 2019

[18] [18]

T. J. Kobayashi and Y. Sughiyama, Fluctuation relations of fitness and information in population dynamics, Phys. Rev. Lett.115(2015)

work page 2015

[19] [19]

Sughiyama, T

Y. Sughiyama, T. J. Kobayashi, K. Tsumura, and K. Aihara, Pathwise thermodynamic structure in population dynamics, Phys. Rev. E91(2015)

work page 2015

[20] [20]

Pigolotti, Generalized Euler-Lotka equation for correlated cell divisions, Phys

S. Pigolotti, Generalized Euler-Lotka equation for correlated cell divisions, Phys. Rev. E103(2021)

work page 2021

[21] [21]

The two-clock problem in population dynamics

K. ¨Ocal and M. P. H. Stumpf, The two-clock problem in population dynamics (2025), arXiv:2504.20388

work page internal anchor Pith review Pith/arXiv arXiv 2025

[22] [22]

Rugbjerg, K

P. Rugbjerg, K. Sarup-Lytzen, M. Nagy, and M. O. A. Sommer, Synthetic addiction extends the productive life time of engineeredEscherichia colipopulations, PNAS115(2018)

work page 2018

[23] [23]

J. C. R. Hernandez-Beltran, V. Mir´ o Pina, A. Siri-J´ egousse, S. Palau, R. Pe˜ na-Miller, and A. Gonz´ alez Casanova, Segre- gational instability of multicopy plasmids: a population genetics approach, Ecol. Evol.12(2022)

work page 2022

[24] [24]

Stadler, O

T. Stadler, O. G. Pybus, and M. P. H. Stumpf, Phylodynamics for cell biologists, Science371(2021)

work page 2021

[25] [25]

Genthon, T

A. Genthon, T. Nozoe, L. Peliti, and D. Lacoste, Cell lineage statistics with incomplete population trees, PRX Life1 (2023)

work page 2023

[26] [26]

K. B. Athreya and P. E. Ney,Branching Processes(Springer, Berlin, Heidelberg, 1972)

work page 1972

[27] [27]

Thomas, Making sense of snapshot data: ergodic principle for clonal cell populations, J

P. Thomas, Making sense of snapshot data: ergodic principle for clonal cell populations, J. R. Soc. Interface14(2017)

work page 2017

[28] [28]

J. H. H. Gillespie,Population Genetics: A Concise Guide(Johns Hopkins University Press, Baltimore, 2004)

work page 2004

[29] [29]

Chevin, On measuring selection in experimental evolution, Biol

L.-M. Chevin, On measuring selection in experimental evolution, Biol. Lett.7(2011)

work page 2011

[30] [30]

Jafarpour, E

F. Jafarpour, E. Levien, and A. Amir, Evolutionary dynamics in non-Markovian models of microbial populations, Phys. Rev. E108(2023)

work page 2023

[31] [31]

Kimura, On the probability of fixation of mutant genes in a population, Genetics47(1962)

M. Kimura, On the probability of fixation of mutant genes in a population, Genetics47(1962). 11

work page 1962

[32] [32]

B. R. Glick, Metabolic load and heterologous gene expression, Biotechn. Adv.13(1995)

work page 1995

[33] [33]

Ilker and M

E. Ilker and M. Hinczewski, Modeling the growth of organisms validates a general relation between metabolic costs and natural selection, Phys. Rev. Lett.122(2019)

work page 2019

[34] [34]

M. V. Rouches, Y. Xu, L. B. G. Cortes, and G. Lambert, A plasmid system with tunable copy number, Nat. Commun.13 (2022)

work page 2022

[35] [35]

Rodr´ ıguez-Beltr´ an, J

J. Rodr´ ıguez-Beltr´ an, J. DelaFuente, R. Le´ on-Sampedro, R. C. MacLean, and´A. San Mill´ an, Beyond horizontal gene transfer: the role of plasmids in bacterial evolution, Nat. Rev. Microbiol.19(2021)

work page 2021

[36] [36]

Touchette, The large deviation approach to statistical mechanics, Phys

H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep.478(2009)

work page 2009

[37] [37]

Genthon and D

A. Genthon and D. Lacoste, Fluctuation relations and fitness landscapes of growing cell populations, Sci. Rep.10(2020)

work page 2020

[38] [38]

A. D. Scott, D. M. King, S. W. Ordway, and S. Bahar, Phase transitions in evolutionary dynamics, Chaos32(2022)

work page 2022

[39] [39]

R. B. Bapat and T. E. S. Raghavan,Nonnegative Matrices and Applications(Cambridge University Press, 1997)

work page 1997

[40] [40]

Kato,Perturbation Theory for Linear Operators(Springer, Berlin, Heidelberg, 1995)

T. Kato,Perturbation Theory for Linear Operators(Springer, Berlin, Heidelberg, 1995)

work page 1995

[41] [41]

Giardin` a, J

C. Giardin` a, J. Kurchan, and L. Peliti, Direct evaluation of large-deviation functions, Phys. Rev. Lett.96(2006)

work page 2006

[42] [42]

Lecomte and J

V. Lecomte and J. Tailleur, A numerical approach to large deviations in continuous time, J. Stat. Mech.2007(2007)

work page 2007

[43] [43]

Nemoto, E

T. Nemoto, E. Guevara Hidalgo, and V. Lecomte, Finite-time and finite-size scalings in the evaluation of large-deviation functions: analytical study using a birth-death process, Phys. Rev. E95(2017)

work page 2017

[44] [44]

Dembo and O

A. Dembo and O. Zeitouni,Large Deviations Techniques and Applications(Springer, Berlin, Heidelberg, 2010)

work page 2010

[45] [45]

P. W. Glynn and W. Whitt, Large deviations behavior of counting processes and their inverses, Queueing Syst.17(1994)

work page 1994

[46] [46]

Duffy and A

K. Duffy and A. P. Metcalfe, How to estimate the rate function of a cumulative process, J. Appl. Probab.42(2005)

work page 2005