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Single-cell size dynamics
Imagine a single cell growing at a given rate and then, suddenly out of blue, it divides its size in two. Is this random? is there an internal clock, a size sensing mechanisms or any control that fires such abrupt change?
Let the cell size \(s\) grow exponentially according to the equation
\[\frac{ds}{dt}=\alpha s, \, s(0)=s_0,\]where \(\alpha\) is the growth rate, \(s_0\) is the initial cell size, and \(t\) is the experiment time. Also, let the cell cycle time \(\tau\) be the time it takes for a cell to grow and divide. The cell cycle progress is represented by the equation
\[\frac{d\tau}{dt}=1, \, \tau(0)=0.\]A division event occurs when a cell splits into two daughter cells. This event resets both the cell size to half and the cell cycle time to zero, that is,
\[s \mapsto s / 2, \quad \tau \mapsto 0,\]marking the end of one cycle and the start of a new one. Let \(P(\tau)\) define when division happens as per
\[P(\tau)=U(\tau-\bar{\tau})=\begin{cases} 1, \text{ if } \tau > \bar{\tau} \\ 0, \text{ otherwise,} \end{cases}\]where \(U(\tau-\bar{\tau})\) is the unit step function and \(\bar{\tau}\) is the time to division since the start of the cell cycle. Cell performs a division event if \(P(\tau)=1\).
The division rate can be defined as
\[\begin{aligned} p(\tau)&=\frac{d P}{d\tau}\\ &=\delta(\tau-\bar{\tau}), \end{aligned}\]where \(\delta(\tau)\) is the Delta Dirac function. The above description is summarized using graph or automata notation
