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Isogenic E. coli cells growing in a constant environment display significant variability in growth rates, division sizes, and generation times. The guiding principle appears to be that each cell, during one generation, adds a size increment that is uncorrelated to its birth size. Here, we investigate the mechanisms underlying this "adder" behavior by mapping the chromosome replication cycle to the division cycle of individual cells using fluorescence microscopy. We have found that initiation of chromosome replication is triggered at a fixed volume per chromosome independent of a cell's birth volume and growth rate. Each initiation event is coupled to a division event after a growth-rate-dependent time. We formalize our findings in a model showing that cell-to-cell variation in division timing and cell size is mainly driven by variations in growth rate. The model also explains why fast-growing cells display adder behavior and correctly predict deviations from the adder behavior at slow growth., (Copyright © 2016 Elsevier Inc. All rights reserved.)