The mean size of exponentially dividing E. coli cells cultured in different nutrient conditions is known to depend on the mean growth rate only. However, the joint fluctuations relating cell size, doubling time and individual growth rate are only starting to be characterized. Recent studies in bacteria (i) revealed the near constancy of the size extension in a single cell cycle (adder mechanism), and (ii) reported a universal trend where the spread in both size and doubling times is a linear function of the population means of these variables. Here, we combine experiments and theory and use scaling concepts to elucidate the constraints posed by the second observation on the division control mechanism and on the joint fluctuations of sizes and doubling times. We found that scaling relations based on the means both collapse size and doubling-time distributions across different conditions, and explain how the shape of their joint fluctuations deviates from the means. Our data on these joint fluctuations highlight the importance of cell individuality: single cells do not follow the dependence observed for the means between size and either growth rate or inverse doubling time. Our calculations show that these results emerge from a broad class of division control mechanisms (including the adder mechanism as a particular case) requiring a certain scaling form of the so-called division hazard rate function, which defines the probability rate of dividing as a function of measurable parameters. This gives a rationale for the universal body-size distributions observed in microbial ecosystems across many microbial species, presumably dividing with multiple mechanisms. Additionally, our experiments show a crossover between fast and slow growth in the relation between individual-cell growth rate and division time, which can be understood in terms of different regimes of genome replication control.
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