intro1.qmd

# Introduction

## Getting started in cell division modeling

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 in Fig 1.

![Growth and cell division representation using automata notation.](hsys.png){fig-align="center" width="368"}

Cell size $s$ at a given experimental time $t$ can be written as the combination of sizes in individual cell cycles, that is, $$\begin{aligned}
    s(t)=\sum_{n=0}^{\infty} s_n(t) P_n(t),
\end{aligned}$$ where $$\begin{aligned}
    s_n(t)=&s_{0}\left[\prod_{i=1}^{n} \frac{e^{\alpha\left(t_{i}-t_{i-1}\right)}}{2}\right] e^{\alpha\left(t-t_{n}\right)}\\
    =&\frac{s_0}{2^n} e^{\alpha t}
\end{aligned}$$ is the cell size $s$ at experimental time $t$ given cell performed $n$ divisions. The sequence of experimental times at which division happens is $t_i$ with $t_i-t_{i-1}=\bar{\tau}$. The cell cycle time after $n$ divisions is given by $\tau=t-t_n$. $$P_n(t)=\begin{cases}
    1,  \text{ if }\quad n\,\bar{\tau} < t \leq (n+1)\bar{\tau}  \\
    0,  \text{ otherwise}
\end{cases}$$ dictates the number of cell cycles $n$ performed by the cell at time $t$. Cell size $s(t)$ is a periodic function in time, with a period $\bar{\tau}$ and a repeating dynamics represented by exponential growth $s_0e^{\alpha t}$ in the cell cycle interval $\tau \in [0,\bar{\tau}]$.

## Division mechanisms: the timer, the sizer, and the adder
We also can track key features of the cell cycle such as division time $\tau_d$, newborn $s_b$, division $s_d$, and added size $s_a$. The feature $s_c$ records the newborn cell size of the next division event. Let $$\mathbf{x}=[s_d,s_a,s_b,s_c,\tau_d, s, n, \tau]$$ be the vector of cell cycle features. At the beginning of the experiment $s_b(0)=s_c(0)=s_0$, $s_a(0)=\tau_d(0)=0$. At the end of the first cell cycle, the size at division resets to the current cell size $s_d \mapsto s$, then we compute the added cell size $s_a\mapsto s-s_b$, the newborn cell size becomes half the division size $s_b \mapsto s/2,$ the number of cell cycles performed resets to $n\mapsto n+1,$ and the cell cycle time resets to zero $\tau \mapsto 0.$ 

It is important to maintain the order in the resets performed. Preserving this order of updating features ensures that once a cell cycle ends, $s_c$ records the newborn size of the next cell cycle event. The updated vector of features is $$\phi(\mathbf{x})=[s,s-s_b,s_c,s/2,\tau,s/2,n+1,0].$$ The modified graph summarizing the cell division dynamics is shown in Fig 2.

![Growth and cell division representation using automata notation.](shsm.png){fig-align="center" width="368"}

The relationships between $s_b$, $s_d$, $s_a$, and $\tau_d$ describe the division control mechanism \cite{}. For any cell cycle, assuming that the cell cycle time $\tau_d[n]=\bar{\tau}$ is kept fixed over generations using the division rate $p(\tau)=\delta(\tau-\bar{\tau})$, we have $s_d=s_b e^{\alpha \bar{\tau}}$ and $s_a=s_d-s_b=s_b (e^{\alpha \bar{\tau}}-1)$. This is known as the **timer** strategy for cell division. For any cell cycle $n$ we see that the size of the newborn cell $s_b[n]$ is affected by the size of the newborn cell in the previous generation, i.e., $$s_b[n]=\frac{e^{\alpha \bar{\tau}}}{2}s_b[n-1]$$. This recursion can be rewritten as a function of the number of cell cycles $n$ $$s_b[n]= \frac{e^{n \alpha \bar{\tau}}}{2^{n}} s_0.$$ Division size at any cell cycle is given by $$s_d[n]= \frac{e^{(n+1) \alpha \bar{\tau}}}{2^{n}} s_0.$$ Added size series are $$s_a[n]= (e^{\alpha \bar{\tau}}-1)\frac{e^{n \alpha \bar{\tau}}}{2^{n}} s_0.$$ These series converges to a non-zero finite value only when $\tau_d=\bar{\tau}=\log{2}/\alpha.$ This reduces the size at division to $s_d=2 s_b$, and the added size in any cycle to $s_a=s_b$. This implies that for any timer strategy, we will find a slope of 2 when we plot the size at division $s_d$ vs. newborn size $s_b$. Plotting added vs. newborn size gives a slope of one. 

If we assume the cell divides by the division rate $p(s)=\delta(s-\bar{s})$, for any cell cycle $n>1$ then we have $$s_d[n]=\bar{s},\, s_b[n]=\bar{s}/2,\, \text{and } \, \tau_d[n]=\log{2}/\alpha.$$ Such division strategy is known as **sizer**.

An alternative to this division mechanism is considering that the cell aims to add a fixed amount of size at any cell cycle. Let the division rate be $p(s)=\delta(s-s_b-\Delta)$, then the size at birth at any cell cycle $n$ is $$\begin{aligned}
    s_b[n]=&\frac{s_b[n-1]+\Delta}{2}\\
    =&\frac{s_0+(2^n-1)\Delta}{2^n}
\end{aligned}$$
where 
$$\lim_{n\to \infty}s_b[n]=\Delta.$$ Such division strategy is know as **adder**. Size at division for the $n$ cell cycle is given by $$
s_d[n]=2^{-n}(s_0 -\Delta)+2 \Delta, \, \lim_{n\to \infty}s_d[n]=2 \Delta.
$$

Division time at any cell cycle is given by  $$\begin{aligned}
    \tau_d[n]=&\frac{1}{\alpha}\log \frac{s_d[n]}{s_b[n]}\\
    =&\frac{1}{\alpha}\log\left({2+\frac{\Delta-s_0}{s_0+(2^n-1)\Delta}}\right), 
\end{aligned}$$
where $$\lim_{n\to \infty}\tau_d[n]=\frac{\log 2}{\alpha}.$$
All division strategies can be summarize when assuming a division rate $\delta(s-b\,s_b-a)$. Size at birth in any cell cycle is given by 
$$
s_b[n]=\frac{b^n(a+(b-2)s_0)-2^n\,a}{2^n(b-2)}.
$$ Note the special cases $b=1$, $a=\Delta$ as adder and $b=0$ as sizer. The size at division is given by $$
s_d[n]=a+\frac{b^{n+1}(a+(b-2)s_0)-2^nba}{2^n(b-2)}.
$$ Cell cycle time is given by $$
\tau[n]=\frac{1}{\alpha}\log \left[b-\frac{2^n a(b-2)}{2^n a-b^n(a+(b-2) s_0)}\right].
$$