Recent advances in quantifying phenotypic heterogeneity in bacteria have shown its broad prevalence in nature. However, evaluating the ultimate impact of URLanism heterogeneity at the population level remains an active area of research in microbiology. We show that the generalized Fibonacci recurrence provides a simple, mathematically rigorous and computationally tractable model for heterogeneous bacterial growth. We examine the impact of small-scale heterogeneity during bacterial division, of the sort observed due to the asymmetric partitioning of aged molecules. We find that small consistent differences in growth rates between daughter cells can have ecologically significant population growth rate benefits and that different models of the cost of heterogeneity may lead to an optimal intermediate amount of heterogeneity between daughter cells. The mathematical framework generalizes the previous combinatorial models developed for the division of yeast, inadequate for modeling bacteria due to the finer control required over the amount of heterogeneity encountered in the latter. Furthermore, when k and m are relatively prime, by juxtaposing the point of view of binary heterogeneous division to the original combinatorial perspective on (GF), in terms of diagonals in Pascal's triangle, we obtain new combinatorial insights. In particular, we obtain an explicit binomial identity for the sums of elements lying on the diagonals kx+my=n in Pascal's triangle \mathbb N_0\times \mathbb N_0\ni(x,y)\mapsto {x+y\choose x} . This provides a missing piece in the earlier work of Dickinson ~\mbox{%DIFAUXCMD Dickinson19500pt%DIFAUXCMD , Raab \mbox{%DIFAUXCMD Raab1963}0pt%DIFAUXCMD }  ]  , and Green \mbox{%DIFAUXCMD Green1968 .
 The generalized Fibonacci recurrence g_n=g_{n-k an explicit binomial identity for the sums of elements lying on the diagonals kx+my=n in Pascal's triangle \mathbb N_0\times \mathbb N_0\ni(x,y)\mapsto {x+y\choose x} 0pt%DIFAUXCMD , Raab \mbox{%DIFAUXCMD Raab1963}0pt%DIFAUXCMD } , previously sought by Dickinson Dic50], Raab Raa63 , and Green Gre68 .
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Recent advances in quantifying phenotypic heterogeneity in bacteria have shown its broad prevalence in nature. However, evaluating the ultimate impact of URLanism heterogeneity at the population level remains an active area of research in microbiology. We show that the generalized Fibonacci recurrence
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provides a simple, mathematically rigorous and computationally tractable model for heterogeneous bacterial growth. We examine the impact of small-scale heterogeneity during bacterial division, of the sort observed due to the asymmetric partitioning of aged molecules. We find that small consistent differences in growth rates between daughter cells can have ecologically significant population growth rate benefits and that different models of the cost of heterogeneity may lead to an optimal intermediate amount of heterogeneity between daughter cells. The mathematical framework generalizes the previous combinatorial models developed for the division of yeast, inadequate for modeling bacteria due to the finer control required over the amount of heterogeneity encountered in the latter. Furthermore, when k and m are relatively prime, by juxtaposing the point of view of binary heterogeneous division to the original combinatorial perspective on (GF), in terms of diagonals in Pascal's triangle, we obtain new combinatorial insights. In particular, we obtain
The generalized Fibonacci recurrence g_n=g_{n-k
D
. This provides a missing piece in the earlier work of Dickinson ~\mbox{%DIFAUXCMD Dickinson1950
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, previously sought by Dickinson
A
Dic50
A
, Raab
A
Raa63
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\mbox{%DIFAUXCMD Green1968
Gre68