For both datasets we extracted data from three experiments done on different days. We did not pool the data together to avoid statistical biases arising from day-to-day differences in experimental conditions. Each analysis was performed in parallel on the data corresponding to each experiment.

#### Numerical simulations and estimation procedures

All the estimation procedures and simulations were performed using MATLAB. Experimental age-size distributions, such as those shown in Figure

3A,B, were estimated from the size and age measurements of every cell at every time step using the MATLAB kde2D function, which estimates the bivariate kernel density. This estimation was performed on a regular grid composed of 2

^{7} equally spaced points on [ 0,

*A*
_{
m
a
x
}] and 2

^{7} equally spaced points on [ 0,

*X*
_{
m
a
x
}], where

*A*
_{
m
a
x
} is the maximal cell age in the data and

*X*
_{
m
a
x
} the maximal cell size (for instance

*A*
_{
m
a
x
}=60 min and

*X*
_{
m
a
x
}=10 µm for the experiment

*f*
_{1}, as shown in Figure

3A). To estimate the size-dependent division rate

*B*
_{s} for each experiment, the distribution of size at division was first estimated for the cell size grid [ 0,

*X*
_{
m
a
x
}] using the ksdensity function. This estimated distribution was then used to estimate

*B*
_{s} for the size grid using Equation (20) (for

*s*
_{
i
} data) or (22) (for

*f*
_{
i
} data) of Additional file

1. The age-size distributions corresponding to the Size Model (Figure

3E,F) were produced by running the Age & Size Model (Equation (

3) in the main text) using the estimated division rate

*B*
_{s} and an exponential growth function (

*v*(

*x*)=

*v*
*x*) with a rate

*v* directly estimated from the data as the average of single-cell growth rates in the population (e.g.

*v*=0.0274 min

^{−1} for the

*f*
_{1} experiment and

*v*=0.0317 min

^{−1} for

*s*
_{1}). For the Age & Size Model, we discretized the equation along the grid [ 0,

*A*
_{
m
a
x
}] and [ 0,

*X*
_{
m
a
x
}], using an upwind finite volume method described in detail in [

43]. We used a time step:

meeting the CFL: Courant-Friedrichs-Lewy stability criterion. We simulated *n*(*t*,*a*,*x*) iteratively until the age-size distribution reached stability (|(*n*(*t*+*d*
*t*,*a*,*x*)−*n*(*t*,*a*,*x*))|<10^{−8}). To eliminate the Malthusian parameter, the solution *n*(*t*,*a*,*x*) was renormalized at each time step (for details see [43]).

The age-dependent division rate *B*
_{a} for each experiment was estimated for the cell age grid [0,*A*
_{
m
a
x
}] using Equation (14) and (16) of Additional file 1. Using this estimated division rate, the age-size distributions corresponding to the Age Model (Figure 3C,D) were produced by running the Age & Size Model. As explained in the main text, we used various growth functions for small and large cells (i.e. for *x*<*x*
_{
m
i
n
} and *x*>*x*
_{
m
a
x
}; between *x*
_{
m
i
n
} and *x*
_{
m
a
x
} growth is exponential with the same rate as for the Size Model). For instance for the fit of the experiment *f*
_{1} shown in Figure 3C, for *x*<2.3 µm and *x*>5.3 µm, *v*(*x*)= max(*p*(*x*),0), with *p*(*x*)=−0.0033*x*
^{3}+0.036*x*
^{2}−0.094*x*+0.13. Likewise, for the fit of the experiment *s*
_{1} shown in Figure 3D, for *x*<3.5 µm and *x*>7.2 µm, *v*(*x*)= max(*p*(*x*),0), with *p*(*x*)=−0.0036*x*
^{3}+0.063*x*
^{2}−0.33*x*+0.67. For each dataset the polynomial *p*(*x*) was chosen as an interpolation of the function giving the length increase as a function of length (shown in Figure 2B for *f*
_{1} data).

Simulations of the extended size models with variability in growth rates or septum positioning (Equations (23) and (24) in Additional file 1) were performed as for the Age & Size Model, with an upwind finite volume scheme. To simulate Equation (23), we used a grid composed of 2^{7} equally spaced points on [ 0,*X*
_{
m
a
x
}] and 100 equally spaced points on [ 0.9*v*
_{
m
i
n
},1.1*v*
_{
m
a
x
}], where *v*
_{
m
i
n
} and *v*
_{
m
a
x
} are the minimal and maximal individual growth rates in the data.