Budding Yeast, Branching Processes, and Generalized Fibonacci Numbers

Olofsson, Peter; Daileda, Ryan C.

doi:10.4169/math.mag.84.3.163

“…We subsequently use the model to examine the cost in generating offspring asymmetry and potential differences between stressed and unstressed states. Analytical models for heterogeneous growth have been developed and utilized in many microbial contexts, from understanding the noisy growth of bacterial colonies [BH48,Ken52] to predicting growth rates of the inherently heterogeneous budding yeast [OD11,SBJ98]. Such models focus on situations with large discrepancies in growth rates and cannot be applied to bacteria such as Escherichia coli.…”

Section: Introductionmentioning

confidence: 99%

Aging a little: On the optimality of limited senescence in Escherichia coli

Blitvić

¹

,

Fernández

²

2020

Journal of Theoretical Biology

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Recent studies have shown that even in the absence of extrinsic stress, the morphologically symmetrically dividing model bacteria Escherichia coli do not generate offspring with equal reproductive fitness. Instead, daughter cells exhibit asymmetric division times that converge to two distinct growth states. This represents a limited senescence / rejuvenation process derived from asymmetric division that is stable for hundreds of generations. It remains unclear why the bacteria do not continue the senescence beyond this asymptote. Although there are inherent fitness benefits for heterogeneity in population growth rates, the two growth equilibria are surprisingly similar, differing by a few percent. In this work we derive an explicit model for the growth of a bacterial population with two growth equilibria, based on a generalized Fibonacci recurrence, in order to quantify the fitness benefit of a limited senescence process and examine costs associated with asymmetry that could generate the observed behavior. We find that with simple saturating effects of asymmetric partitioning of subcellular components, two distinct but similar growth states may be optimal while providing evolutionarily significant fitness advantages.

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“…Starting with this division and for all divisions thereafter, each bacterium in the system splits into a "faster-growing" child that will undergo division at time k and "slower-growing" one, which will undergo division at time m > k. (See Figure 1 (a).) This model accurately describes the bimodal morphologically stable asymmetric cell division (see [BF] for some examples) and is a generalization of analogous models for the division of yeast [OD11] (case k = 1). Beyond biology, the model is more generally applicable to any situation that exhibits binary branching with persistent asymmetry.…”

Section: Generalized Fibonacci Numbers As a Model For Asymmetric Bran...mentioning

confidence: 94%

“…the references for the entry A005686 in [Slo19] for the case k = 2, m = 5). The case k = 1 has previously attracted attention in computer science [PN77] and biology, especially in the context of asymmetric cell division in yeast [SBJ98,OD11].…”

Section: Introductionmentioning

confidence: 99%

On a Generalized Fibonacci Recurrence

Blitvić¹,

Fernández²

2019

Preprint

0

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The generalized Fibonacci recurrence g n = g n−k +g n−m was recently used to demonstrate the theoretically optimal nature of limited senescence in morphologically symmetrically dividing bacteria. Here, we study this recurrence from a more abstract viewpoint, as a general model for asymmetric branching, and interpret solutions for different initial conditions in terms of branching-related quantities. We provide a compact diagrammatic representation for the evolution of this process which leads to an explicit binomial identity for the sums of elements lying on the diagonals kx + my = n in Pascal's triangle N 0 × N 0 (x, y) → x+y x , previously sought by Dickinson [Dic50], Raab [Raa63], and Green [Gre68].

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“…Both of these sequences have been extensively studied and satisfy a wealth of mathematical properties [Kno19,Vaj08,Kos17], often unexpectedly featuring in various areas of mathematics and natural sciences. Models of asymmetric yeast division have also been based on a similar recurrence with k = 1 [OD11], but the generalized recurrence with integer 1 ≤ k ≤ m is required to capture the much smaller growth asymmetries generated by bacteria with morphologically symmetric division. By expanding the recurrence relation (2) in terms of the sequence of divisions d…”

Section: Population Growth Ratementioning

confidence: 99%

Aging a little: The optimality of limited senescence in Escherichia coli

Blitvić¹,

Fernández²

2019

Preprint

0

View full text Add to dashboard Cite

Recent studies have shown that even in the absence of extrinsic stress, the morphologically symmetrically dividing model bacteria Escherichia coli do not generate offspring with equal reproductive fitness. Instead, daughter cells exhibit asymmetric division times that converge to two distinct growth states. This represents a limited senescence / rejuvenation process derived from asymmetric division that is stable for hundreds of generations. It remains unclear why the bacteria do not continue the senescence beyond this asymptote. Although there are inherent fitness benefits for heterogeneity in population growth rates, the two growth equilibria are surprisingly similar, differing by a few percent. In this work we derive an explicit model for the growth of a bacterial population with two growth equilibria, based on a generalized Fibonacci recurrence, in order to quantify the fitness benefit of a limited senescence process and examine costs associated with asymmetry that could generate the observed behavior. We find that with simple saturating effects of asymmetric partitioning of subcellular components, two distinct but similar growth states may be optimal while providing evolutionarily significant fitness advantages.

show abstract

Budding Yeast, Branching Processes, and Generalized Fibonacci Numbers

Cited by 6 publications

References 4 publications

Aging a little: On the optimality of limited senescence in Escherichia coli

Aging a little: On the optimality of limited senescence in Escherichia coli

On a Generalized Fibonacci Recurrence

Aging a little: The optimality of limited senescence in Escherichia coli

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