
\end{equation}
This gives a cycle time (doubling time) of
\begin{equation}
\tau= \frac{\ln 2}{\mu}=19.4 \, \mbox{min}.
\label{eqn:tau}
\end{equation}

\begin{figure}
\hskip 2.8cm
\includegraphics[width=0.7\linewidth]{67250Fig2.eps}
\caption{
(Color online)
Upper left (lower left) picture:
snapshot of the cell filament at time
$T=125$ min ($T=245$ min).
The upper right (lower right) graph shows
the log-log plots of $N_T(\varepsilon)$ versus $\varepsilon$.
We find linear regions in the log-log plots,
and the fractal dimensions of the filament configurations
are evaluated as $D(125)=1.16$ and $D(245)=1.96$, respectively.
}
\label{fig:fractal1}
\end{figure}

\begin{figure}
\hskip 4.5cm
\includegraphics[width=0.4\linewidth]{67250Fig3.eps}
\caption{
Time dependence of fractal dimension $D(T)$
of the filamentous cell configuration on the agar plate. 
The data are well described by
the sigmoid function given by Eq. (\ref{eqn:sigmoid}). 
}
\label{fig:fractal2}
\end{figure}

The snapshots of filament configurations at
$T=125$ and 245 min are shown on the left-hand side in
Fig. \ref{fig:fractal1} with a resolution of $2040 \times1536$ pixels. 
They are analyzed by the box-counting method to evaluate the fractal 
dimensions of the filament configurations of bacterial cells on the agar plate. 
The procedure is as follows. 
At each time $T$, the snapshot picture is divided by
squares (two-dimensional boxes) of linear size $\varepsilon$, 
and then is counted the number $N_T(\varepsilon)$ of squares containing the pixels 
occupied by the filament of bacterial cell.
 If the configuration has a fractal structure, $N_T(\varepsilon)$ is scaled as
\begin{equation}
   N_T(\varepsilon) \sim \varepsilon^{-D(T)}
\label{eqn:fractal1}
\end{equation}
with the fractal dimension $D(T)$. 
We changed the value of $\varepsilon$ from 
1 to 1536 pixels (from $2.1 \times 10^{-1}$
to $3.2 \times 10^2$ $\mu$m in the real scale).
As shown in the log-log plots given on the right-hand side in Fig. \ref{fig:fractal1}, 
the data for $T=125$ min show a power law 
in the range of 3.5 $\mbox{$\mu$m} < \varepsilon < 5.6 \times 10^1$ \mbox{$\mu$m}
in the real scale,  and 
those for $T=245$ min show a power law
in the range of 2.5 $\mbox{$\mu$m} < \varepsilon < 2.1 \times 10^1$ \mbox{$\mu$m}
in the real scale.
The slopes $-1.16$ and $-1.96$ in these log-log plots in
Fig. \ref{fig:fractal1} provide the fractal dimensions 
$D(125)=1.16$ and $D(245)=1.96$, respectively. 
The result implies that the filament configurations have fractal structures. 
We evaluated the fractal dimensions of the filament 
at $T=45, 85, 165, 205, 285, 325$, and 365 min
and the results are plotted in Fig. \ref{fig:fractal2}.
We found that the data can be fitted to the following
sigmoid function:
\begin{equation}
D(T)=c_1 \tanh\bigl[\sigma(T-T_0)\bigr]+c_2
\label{eqn:sigmoid}
\end{equation}
with $T_0=1.82 \times$ $10^{2}$ min,
$\sigma=1.61 \times 10^{-2}$ \mbox{min$^{-1}$},
$c_1=5.1 \times 10^{-1}$, and $c_2=1.54$.
The time evolution of the multiple fission of bacterial cells
from the exponential phase to the stationary phase
is well described by the time dependence
of the fractal dimension of 
the filament configuration on the agar plate.

\SSC{Folding Processes and Experimental Measurements}
\label{sec:measurement}

In this section, we denote the observation time by $t$ [min] instead of $T$ [min],
since we will set $t=0$ at the time when the first folding of a filament
occurs as explained below.

Let $L(t)$ [$\mu$m] be the total length of a filament of bacterial cells at time $t$.
Here, we write the specific growth rate as $\alpha$, 
and then $L(t)$ should obey the following differential equation:
\begin{equation}
\frac{d}{dt} L(t)=\alpha L(t).
\label{eqn:Lt2}
\end{equation}
On the semisolid agar plate,
the linear growth is unstable, and a filament of cells
starts folding after a short time.
It repeats the folding processes and the configuration of
the filament on the plate becomes complicated very rapidly.
Over time, the multifold structure becomes
dense  and its region spreads over the plates.
In this way, the configuration of a filament of cells
shows a crossover from the one-dimensional structure
to the two-dimensional structure, as shown in the previous section.

\begin{figure}
\hskip 5cm
\includegraphics[width=0.4\linewidth]{67250Fig4.eps}
\caption{
Illustrations of elementary processes found in the
self-elongating process of a filament of bacterial cells
in the early stage.
[1] Elongation of simple segment.
[2] Creation of a twofold segment by folding a simple segment.
Twice the length $r_2$ is added to the length $l_2$.
[3] Creation of a fourfold segment by elongation of a twofold segment.
Four times the length $r_4$ is added to the length $l_4$.
[4] Creation of a threefold segment by folding of a twofold segment
on a simple segment.
Three times the length $r_3$ is added to the length $l_3$.
}
\label{fig:processes}
\end{figure}

In an interval of a filament, if bacterial cells are linked in tandem
and form a curved line without folding, 
the interval is called a simple segment.
At each time $t$, we consider the union of all simple segments
in a filament and call it the simple part.
We represent its length by $l_1(t)$ [$\mu$m].
As illustrated by [1] in Fig. \ref{fig:processes}, we see the following process:
$$
\mbox{Elementary Process [1] : elongation of a simple segment}, 
$$
and $l_1(t)$ rapidly increases with time $t$.

We set $t=0$ at the time when the first folding
of a filament is observed.
In this experiment, $t=T-60$ [min].
For $t >0$, the following process occurs:
\begin{eqnarray}
\mbox{Elementary Process [2]} &:& \mbox{creation of a twofold segment by folding of}
\nonumber\\
&& \mbox{a simple segment,}
\nonumber
\end{eqnarray}
as illustrated by [2] in Fig. \ref{fig:processes}.
The twofold part is defined by the union of all twofold segments
in the filament, whose length is denoted by $l_2(t)$ [$\mu$m].
By definition, $l_1(t)=L(t), l_2(t)=0$ for $t \leq 0$,
while $l_1(t)=L(t)-l_2(t), l_2(t)>0$
for $t > 0$.
Sooner or later, we will see threefold segments, fourfold segments,
and so forth.
We call the union of all $k$-fold segments the $k$-fold part, 
and write the length of the $k$-fold part
as $l_k(t)$ [$\mu$m], $k =2, 3, 4, \dots$. See the elementary processes 
[3] and [4] in Fig. \ref{fig:processes},
which create a fourfold segment and a threefold segment, respectively.
We will attempt to characterize