Appendix 1 - Mathematical Analysis for the Time-Independent Case

Problem Definition

We consider a population of cells where each cell can either divide at a rate $\gamma(\tau)$ or die at a rate $\delta(\tau)$, both dependent only on the cell's age $\tau$. In this scenario, the McKendrick–Von Foerster equation governing the population dynamics is:

$$\begin{aligned} \left(\partial_t + \partial_\tau\right) n(\tau, t) &= -\left(\gamma(\tau) + \delta(\tau)\right) n(\tau, t), \\ n(0, t) &= 2 \int_0^\infty \gamma(\tau) n(\tau, t) \, d\tau. \end{aligned}$$

Key Components:

1. Cell Division ($\gamma(\tau)$): A cell can divide at an age-dependent rate.
2. Cell Death ($\delta(\tau)$): A cell can die before dividing, also at an age-dependent rate.
3. Birth Condition: Division creates two new cells of age zero, forming the boundary condition at $n(0, t)$.

Our goal is to determine the population dynamics $n(\tau, t)$, particularly the asymptotic growth rate $\lambda$, under the assumption of time-independent division and death rates.

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Separation of Variables

To solve the PDE, we assume the solution can be expressed as: $$n(\tau, t) = \sum_\lambda N_\lambda(t) \pi_\lambda(\tau),$$ where: - $N_\lambda(t)$: Time-dependent population size associated with eigenvalue $\lambda$. - $\pi_\lambda(\tau)$: Age distribution for the eigenvalue $\lambda$.

Substituting into the PDE and separating variables, we obtain: $$\frac{\pi_\lambda'(\tau)}{\pi_\lambda(\tau)} + \left(\gamma(\tau) + \delta(\tau)\right) = -\frac{N_\lambda'(t)}{N_\lambda(t)} = -\lambda.$$

From this, we solve the time-dependent and age-dependent parts separately: $$\begin{aligned} N_\lambda(t) &= N_\lambda(0) e^{\lambda t}, \\ \pi_\lambda(\tau) &= \pi_\lambda(0) e^{-\lambda \tau} e^{-\int_0^\tau \left(\gamma(s) + \delta(s)\right) \, ds}. \end{aligned}$$

The eigenvalue $\lambda$ determines the growth or decay rate of the population. It is obtained from the boundary condition: $$n(0, t) = 2 \int_0^\infty \gamma(\tau) n(\tau, t) \, d\tau.$$

Substituting the expression for $n(\tau, t)$, we arrive at: $$\pi_\lambda(0) = 2 \int_0^\infty \gamma(\tau) \pi_\lambda(\tau) \, d\tau.$$

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Deriving the Euler-Lotka Equation

To further simplify, we define the division probability density function $\phi(\tau)$ and the death survival function $\varphi(\tau)$ as follows: 1. Division rate as a hazard function: $$\gamma(\tau) = \frac{\phi(\tau)}{1 - \int_0^\tau \phi(s) \, ds}, \quad \phi(\tau) = \gamma(\tau) e^{-\int_0^\tau \gamma(s) \, ds}.$$ 2. Death survival function: $$\varphi(\tau) = 2 e^{-\int_0^\tau \delta(s) \, ds}.$$

Combining these, the boundary condition reduces to the Euler-Lotka equation: $$1 = \int_0^\infty e^{-\lambda \tau} \phi(\tau) \varphi(\tau) \, d\tau,$$ where: - $e^{-\lambda \tau}$: Discount factor due to exponential population growth. - $\phi(\tau)$: Probability density of division at age $\tau$. - $\varphi(\tau)$: Probability of survival from death up to age $\tau$.

The equation can be solved numerically or analytically for $\lambda$, depending on the specific forms of $\gamma(\tau)$ and $\delta(\tau)$.

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Special Cases

1. Immortal Cells ($\delta(\tau) = 0$)

If cells do not die, the division rate is constant, $\gamma(\tau) = R$. The probability density function of division is: $$\phi(\tau) = R e^{-R \tau}, \quad \varphi(\tau) = 2.$$

The Euler-Lotka equation becomes: $$1 = 2R \int_0^\infty e^{-(\lambda + R) \tau} \, d\tau.$$ Evaluating the integral: $$1 = \frac{2R}{\lambda + R}.$$ Solving for $\lambda$: $$\lambda = R.$$ This result shows that the population growth rate equals the division rate when no cells die.

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2. Normal Cells ($\delta(\tau) = D$)

If cells die at a constant rate $D$, the death survival function becomes: $$\varphi(\tau) = 2 e^{-D \tau}.$$

The Euler-Lotka equation becomes: $$1 = 2R \int_0^\infty e^{-(\lambda + R + D) \tau} \, d\tau.$$ Evaluating the integral: $$1 = \frac{2R}{\lambda + R + D}.$$ Solving for $\lambda$: $$\lambda = R - D.$$

Interpretation

1. When $R > D$, the population grows at a rate $\lambda = R - D$.
2. When $R = D$, the population size remains constant ($\lambda = 0$).
3. When $R < D$, the population declines ($\lambda < 0$).