††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Noise-induced survival resonances during fractional killing of cell
populations

Francesco Puccioni  Johannes Pausch  Paul Piho  Philipp Thomas  Imperial
College of London, South Kensington, London SW7 2AZ, United Kingdom

###### Abstract

Fractional killing in response to drugs is a hallmark of non-genetic cellular
heterogeneity. Yet how individual lineages evade drug treatment, as observed
in bacteria and cancer cells, is not quantitatively understood. We analyse a
stochastic population model with age-dependent division and death rates and
characterise the emergence of fractional killing as a stochastic phenomenon
under constant and periodic drug environments. In constant environments,
increasing cell cycle noise induces a phase transition from complete to
fractional killing, while increasing death noise can induce the reverse
transition. In periodic drug environments, we discover survival resonance
phenomena that give rise to peaks in the survival probabilities at division or
death times that are multiples of the environment duration not seen in
unstructured populations.

Under adverse conditions such as repeated drug treatments, most cells in a
population die while few cells persist – a phenomenon called fractional
killing [1, 2, 3]. The short timescale of drug exposure often excludes the
evolution of drug resistance but requires non-genetic mechanisms underlying
fractional killing, which are still not fully understood [4, 5, 6]. A well-
accepted view is that heterogeneous survival arises from fluctuations in
intracellular pathways influencing cell division, growth and apoptosis in
coordination with the cell cycle [7, 8, 9]. Recent advances in single-cell
imaging allow tracking cellular heterogeneity in individual lineages that can
drive insights into persistence against antimicrobial or anti-cancer
treatments aided by quantitative stochastic models.

Cellular heterogeneity in division times, as observed in experiments, can be
described through age-structured branching processes [10, 11, 12, 13, 14, 15].
Most of these models operate on a mean-field level as they often implicitly
assume cell survival or large populations, and therefore cannot explain the
fractional killing. Here, we provide a quantitative model of fractional
killing for populations established from a single ancestor by analysing an
age-structured branching process of cell division and death in time-dependent
environments [16, 17, 18] (Fig. 1). As fractional killing is the survival of
cells despite long or repeated exposure to adverse conditions, we focus on
long-term, i.e. infinite-time, survival. We will refer to it simply as
survival unless stated otherwise.

In this letter, we present several results towards understanding fractional
killing as a stochastic phenomenon that emerges due to the interplay of
heterogeneity and time-dependent treatments. To achieve this, we first
consider the survival probability in constant environments and confirm
universal behaviour that unifies all possible age-dependence. Next, we
generalize the age-dependent branching model of Bellman and Harris [10],
allowing arbitrary age-dependence of division and death rates in periodic
environments. Here, we discover that periodic treatment settings can lead to
resonance phenomena in the survival probability. Our results reveal an
intricate dependence of resonances on division or death times that could be
observed for cell-cycle dependent drugs such as many chemotherapeutic drugs or
antibiotics targeting replication and division machinery [19, 20, 21].

Figure 1: Illustration of age-dependent population dynamics. Top left panel:
the distribution of division and death ages. Bottom left: a cell either
divides into two newborns or dies. Top right: constant environment. Bottom
right: periodic on- and off-switching of a treatment. Some cells survive
treatment and persist, fractional killing occurs.

Model: We consider a population of cells that age deterministically and are
subject to spontaneous division or death events in changing environments. At
time $t$, a cell $s(x)$ with age $x$ divides into two newborns with rate
$\gamma(x,t)$ or dies with rate $d(x,t)$:

$s(x)\xrightarrow{\gamma(x,t)}2s(0),\ \ s(x)\xrightarrow{d(x,t)}\emptyset$ (1)

Here, the age $x$ labels the time since the last division. The age-dependence
of the rates means that the division and death time distributions of reactions
(1) are non-exponential and cell cycle-dependent as observed in experiments
[17, 22]. Together with the time-dependence of these rates, they determine not
only the fate of an individual cell but ultimately the survival probability
$p_{\text{surv}}$ of the entire population. We follow the evolution of the
cell population with a function $n$ where $n(x)$ is the density of cells with
age $x$. Starting at time $t=0$ with a single ancestor of age $x_{0}$, the
probability of observing a specific density is denoted by the functional
$\mathcal{P}[n,t|\delta_{x_{0}}]$ such that
$p_{\text{surv}}(x_{0})=1-\lim\limits_{t\rightarrow\infty}\mathcal{P}[0,t|\delta_{x_{0}}]$.
$\mathcal{P}$ evolves according to a functional Master equation with age-
drift:

$\displaystyle\frac{\partial\mathcal{P}[n,t|\delta_{x_{0}}]}{\partial
t}=\int_{0}^{\infty}\\!\\!{\rm{d}}x\biggl{(}\gamma(x,t)(\varepsilon^{+1}_{x}\varepsilon^{-2}_{0}-1)n(x)+$
$\displaystyle\quad
d(x,t)(\varepsilon^{+1}_{x}-1)n(x)+\partial_{x}n(x)\frac{\delta}{\delta
n(x)}\biggr{)}\mathcal{P}[n,t|\delta_{x_{0}}],$ (2)

whose mean-field is the McKendrick-von Foerster model [23, 24]. Here, the step
operator $\varepsilon^{\pm m}_{x}$ shifts the argument of the functional
$\mathcal{F}$: $\varepsilon^{\pm m}_{x}\mathcal{F}[n]=\mathcal{F}[n\pm
m\delta_{x}]$, where $\delta_{x}$ denotes the Dirac-$\delta$ function.

Figure 2: Survival probability $p_{\text{surv}}(0)$ in a constant environment.
(a): samples of survival probability (markers) collapse on
$p_{\text{surv}}(0)$ as a function of the effective death probability $\nu$
(black line, Eq. 3). (b) Noise-induced transition in $p_{\text{surv}}(0)$ as a
function of the coefficients of variation of division and death times,
$CV^{2}_{\gamma}$ and $CV^{2}_{d}$. We assume $\Gamma$-distributed division
and death times with $\mu_{d}=1$, $\mu_{\gamma}=1$, $CV^{2}_{d}=0.5$ and
$CV^{2}_{\gamma}=0.5$ unless stated otherwise.

Constant environment:  The survival probability in a constant environment can
be obtained from simple arguments. Cells can either divide or die, which
occurs with a constant death probability $\nu$ since cells are born with zero
age. The generating function of offspring (over one generation) thus equals
$z(h)=\nu+(1-\nu)h^{2}$ and its fixed point
$1-p_{\text{surv}}(0)=z(1-p_{\text{surv}}(0))$ yields:

$p_{\text{surv}}(0)=\frac{1-2\nu}{1-\nu}\Theta\left(\frac{1-2\nu}{1-\nu}\right).$
(3)

In constant environments, the rates are time-independent
($\gamma(x,t)=\gamma(x)$ and $d(x,t)=d(x)$) and, therefore, the effective
death probability $\nu$ is the integral over the death rate multiplied by the
probability that the cell reaches a certain age:
$\nu=\int_{0}^{\infty}{\textrm{d}}ud(u)e^{-\int_{0}^{u}{\textrm{d}}y(\gamma(y)+d(y))}$.

Eq. (3) demonstrates that the system exhibits a second-order phase transition
between a subcritical phase ($\nu>1/2$) with almost sure extinction and a
supercritical phase ($\nu<1/2$) with positive survival probability. The known
second-order phase transition for age-independent processes [25, 26] thus
extends to a broad class of age-structured processes.

We simulated the process assuming rates corresponding to $\Gamma$-distributed
division and death times, i.e.,
$\varphi_{y}(x)=y(x)e^{-\int_{0}^{x}{\rm{d}}uy(u)}$ with $y=\gamma,d$. We
sampled the survival probability while varying the means and the coefficients
of variation of division ($\mu_{\gamma}$, $CV^{2}_{\gamma}$) and death times
($\mu_{d}$, $CV^{2}_{d}$). As shown in Fig. 2 (a), all simulations collapse
onto the phase diagram given by Eq. (3). As expected, the survival probability
increases (decreases) with $\mu_{d}$ ($\mu_{\gamma}$), Fig. 2 (a).
Interestingly, the survival probability also increases (decreases) with noise
in division times $CV^{2}_{\gamma}$ (death times $CV^{2}_{d}$). This
dependence can cause a noise-induced transition from complete to fractional
killing as division noise increases (Fig. 2 (b)). The reverse transition is
observed as death timing noise increases.

Figure 3: Survival resonances under periodic on-off treatments of length $T$.
(a): Survival probability ($p_{\text{surv}}(0)$) shows a crossover from
deterministic (red) to noise-induced resonances (yellow and orange) as
division noise ($CV^{2}_{\gamma}$) increases. Theoretical locations of
deterministic (red, Eq. (7)) and noise-induced resonances (black ticks, Eq.
(9)) are shown as top ticks. (c): Phase diagram displaying the location of
peaks of the surviving probability in rescaled $T$-$\mu_{d}/\mu_{\gamma}$
space. Division and death ages are $\Gamma$ distributed with parameters
$\mu_{\gamma}=2$, $\mu_{d}=1.8$, $CV^{2}_{d}=10^{-2}$ in (a) and
$\mu_{\gamma}=1.8$, $\mu_{d}=2$, $CV^{2}_{\gamma}=1$ in (b).

Time-dependent environments: Motivated by periodic treatments, as used in
microbial and cancer cell populations, we consider the repeated on-off-
switching of the death process (Fig. 1). Specifically, we let $d(x,t)=d(x)$ in
an on-environment of length $T$ and $d(x,t)=0$ in an off-environment of the
same length while cells keep dividing regardless $\gamma(x,t)=\gamma(x)$.
Stochastic simulations display a complex landscape of survival probabilities
with several peaks, Fig. 3 (a-b).

We consider the offspring distribution of the underlying branching process to
calculate the survival probability. Here, the generating functionals of the
offspring distributions originating from a cell with age $x_{0}$ after an on-
or off-environment starting at time $0$ and ending at $T$ are
$\mathcal{Z}_{\text{on}}[h,T|\delta_{x_{0}}]:=\int\mathcal{D}[n]\exp(\int_{0}^{\infty}{\rm{d}}x\ln(h(x))n(x))\mathcal{P}[n,T|\delta_{x_{0}}]$
and similarly $\mathcal{Z}_{\text{off}}[h,T|\delta_{x_{0}}]$. Note that the
auxiliary variable is a general age-dependent function $h(x)$, and the
integral is over non-negative measures $n$ with positive support. The
generating functional of offspring after each period is then
$\mathcal{Z}_{\text{on}}[\mathcal{Z}_{\text{off}}[h,T|\delta_{\bullet}],T|\delta_{x_{0}}]$,
where $\bullet$ is a placeholder for the function argument. Since the
environments are periodic, the offspring distributions after one period are
invariant across environments. This means that the long-term survival
probability $p_{\text{surv}}(x_{0})$, starting from a single cell with age
$x_{0}$, is obtained from the survival probability of the embedded homogeneous
Galton-Watson chain that satisfies

$1-p_{\text{surv}}(x_{0})=\mathcal{Z}_{\text{on}}[\mathcal{Z}_{\text{off}}[1-p_{\text{surv}},T|\delta_{\bullet}],T|\delta_{x_{0}}]$
(4)

as a functional generalisation of the multi-type branching process [10, Ch.
II.7].

We now derive a method to calculate the generating functional of the offspring
distributions from the Master equation (Noise-induced survival resonances
during fractional killing of cell populations). The standard solution method
[27] involves transforming the Master equation into a partial differential
equation (PDE) for the probability generating function and solving it using
the method of characteristics. The same method can be adapted to the age-
structured case using the generating functionals of offspring after an on- or
off-environment of length $T$. To this end, we transform the functional Master
equation (Noise-induced survival resonances during fractional killing of cell
populations) into a functional differential equation for
$\mathcal{Z}_{\text{on}}$:

$\displaystyle\frac{\partial\mathcal{Z}_{\text{on}}[h,t|\delta_{x_{0}}]}{\partial
t}=\int_{0}^{\infty}\\!\\!{\rm{d}}x\Big{[}\partial_{x}h(x)-(\gamma(x)+d(x))h(x)$
$\displaystyle\qquad\qquad+\gamma(x)h^{2}(0)+d(x)\Big{]}\frac{\delta\mathcal{Z}_{\text{on}}[h,t|\delta_{x_{0}}]}{\delta
h(x)}.$ (5)

The characteristics of Eq. (Noise-induced survival resonances during
fractional killing of cell populations) are described by PDEs, whose solution
satisfies a non-linear integral equation (SM A):

$\displaystyle\mathcal{Z}_{\text{on}}[h,t|\delta_{x_{0}}]\Pi(x_{0})=\Pi(t+x_{0})h(t+x_{0})+\int_{0}^{t}\\!\\!{\rm{d}}u\,\Pi_{d}(u+x_{0})$
$\displaystyle\qquad\qquad+\int_{0}^{t}{\rm{d}}u\,\Pi_{\gamma}(u+x_{0})\mathcal{Z}_{\text{on}}^{2}[h,t-u|\delta_{0}],$
(6)

and we simply set $d(x)=0$ in an off-environment to obtain
$\mathcal{Z}_{\text{off}}$. The function
$\Pi(x)=e^{-\int_{0}^{x}{\rm{d}u}(\gamma(u)+d(u))}$ is the probability for a
newborn cell to survive until age $x$ and $\Pi_{\gamma}(x)=\gamma(x)\Pi(x)$
and $\Pi_{d}(x)=d(x)\Pi(x)$ are the probabilities for a newborn cell to divide
or die at age $x$, respectively. Eq. (Noise-induced survival resonances during
fractional killing of cell populations) generalises the integral equation
derived by Bellman and Harris [10] similar to [28, 29] where cell death occurs
independently from divisions, but here we account explicitly for
division-/death rates and for initial age $x_{0}$. Using Eq. (Noise-induced
survival resonances during fractional killing of cell populations) in (4)
gives a coupled system of integral equations that can be solved using an
iterative numerical procedure (SM B).

In agreement with stochastic simulations, the survival probability displays
narrow peaks of survival probabilities that emerge when the environment length
$T$ is tuned precisely to the mean division or death times, $\mu_{\gamma}$ and
$\mu_{d}$ respectively. These peaks are absent in the unstructured models with
constant division and death rates (SM C). Reminiscent of overtones, the peaks
repeat for higher integer multiples of $T$, and we therefore call this
phenomenon survival resonances.

We find two types of resonances. Firstly, deterministic resonances occur when
division and death times fluctuate little around their means, i.e.
$CV^{2}_{d},CV^{2}_{\gamma}\ll 1$, Fig. 3 (a). In this limit, division and
death occur at fixed ages $\mu_{\gamma}$ and $\mu_{d}$. If
$\mu_{\gamma}<\mu_{d}$, death never occurs because no cell ever reaches the
age of death $\mu_{d}$, and $p_{\text{surv}}(0)=1$, see Fig. 3 (c). If
$\mu_{\gamma}>\mu_{d}$, cells die if they reach the age of death $\mu_{d}$ in
any of the on-environments $A_{m}:=[2mT,(2m+1)T)$. Starting with a newborn
ancestor at time $t=0$, the age of its descendants follows a sawtooth wave
$t-\mu_{\gamma}\lfloor t/\mu_{\gamma}\rfloor$ because divisions are
synchronous. Thus, the descendants synchronously exceed the deadly age
$\mu_{d}$ in time intervals
$B_{n}=(n\mu_{\gamma}+\mu_{d},(n+1)\mu_{\gamma}),n\in\mathbb{N}_{0}$.
Moreover, all of them die simultaneously if this occurs in any on-environment
$A_{m}$, leading to the survival condition that $\forall m,n:A_{m}\cap
B_{n}=\emptyset$ implies $p_{\text{surv}}(0)=1$. Consequently, only singular,
fine-tuned combinations of $T$ and $\mu_{\gamma}$ allow escaping certain death
and deterministic resonance only appears for the following environment
lengths:

$\displaystyle\\{T:\mu_{\gamma}=2nT,n\in\mathbb{N},0<\mu_{\gamma}-\mu_{d}<T\\}.$
(7)

Physically, Eq. (7) is a renewal condition: all the generations are
approximately born at the beginning of an on-environment.

We make the following observations on the effect of noise seen in Fig. 3 (a).
Firstly, higher overtones of survival probabilities are progressively
attenuated because they have a higher absolute variance of division ages and
therefore are less tuned to the environment length. Secondly, the asymmetry
and shift of peaks are due to noise since division times longer than $2nT$
lead to death, while cells with shorter division times avoid it.

As more and more noise in division timing is introduced (Fig. 3 (a)),
deterministic resonances disappear and give way to the second type of
resonance, which we call noise-induced resonances. Here, peaks in the survival
probability appear for specific combinations of $\mu_{d}$ and $T$ with only
small fluctuations in death times. These resonances appear also for
$\mu_{\gamma}<\mu_{d}$ and their emergence can be explained qualitatively by
considering the death probability of the first cell. Assuming that death is
deterministic with death time $\tau_{d}$ and that the cell was born at time
$t=0$ at the start of the first on-environment, we can identify two regimes.
If $\tau_{d}$ falls in an on-environment, the survival probability of the
first generation is given by the probability that the cell has divided before
age $\tau_{d}$:
$\phi_{\gamma}(\tau_{d})=e^{-\int_{0}^{\tau_{d}}\rm{d}u\gamma(u)}$. If
$\tau_{d}$ falls in an off-environment, the cell evades death if it divides
before the next on-environment. This consideration leads to the effective
death probability of the first cell:

$\nu_{1}=\mathbb{E}\left(\mathbbm{1}_{\text{on}}(\tau_{d})\phi_{\gamma}(\tau_{d})+\mathbbm{1}_{\text{off}}(\tau_{d})\phi_{\gamma}\left(\left\lfloor\frac{\tau_{d}}{T}+1\right\rfloor
T\right)\right),$ (8)

where $\mathbbm{1}_{\text{off/on}}(\tau_{d})$ is $1$ if $\tau_{d}$ falls into
an off/on environment and zero otherwise and the average is over the
distribution of death times $\tau_{d}$. As a rough approximation, we assume
that the cells of later generations have the same death probability as the
first cell and then use Eq. (8) in Eq. (3). The approximation, strictly valid
only for homogeneous environments, predicts resonances at the following
environment lengths:

$\displaystyle\\{T:\mu_{d}=(2n+1)T,n\in\mathbb{N}\\},$ (9)

which reproduces all the noise-induced resonances seen in the simulations and
numerics (dashed lines, Fig. 3 (a, b)). The approximation also produces a
ramp-like shape followed by sudden drops and thus provides qualitative
agreement with the exact results. We also calculated the transient survival
probabilities, which show that the first generation determines the locations
of all noise-induced resonances in the following generations (SM D).

From Eqs. (7) and (9), we observe that, for a fixed $\mu_{d}$ and
$\mu_{\gamma}$, there are infinitely many noise-induced resonances, but there
exists only a finite band with deterministic resonances for
$\mu_{d}<\mu_{\gamma}$. The latter appears in parameter regions where noise-
free models would not predict any survival. We summarise our findings in a
phase diagram showing the locations of deterministic (red) and noise-induced
survival resonances (black lines, Fig. 3 (c)). Discussion: We examined a
stochastic population model with cell cycle-dependent division and death rates
to understand how fractional killing may emerge from the interplay of cellular
heterogeneity with drug treatments.

We first considered cells under continuous treatments and predicted their
survival chances increase with noise in division timing but decrease with
noise in death timing (Fig. 2(a)). Such a dependence can induce a transition
between complete to fractional killing phases. This suggests that division
heterogeneity combined with controlled cell death could represent a strategic
or selective advantage for cells to persist in continuous drug treatment.

Then, we analysed survival probabilities under periodic treatments and
discovered deterministic and noise-induced resonances observed under a tuning
of division and death rates to the environment duration. In populations with
pre-existing genetic variability, such as in genetic screens or cancer, these
resonances could be difficult to avoid as division and death rates are
heterogeneous and can be selected by adjusting the environmental duration. The
survivors of this selection process will be located predominantly along peaks
of the survival probability, shown as lines in the phase diagram, Fig 3(c). In
this hypothetical situation, deterministic resonances would select highly
predictable phenotypes within a relatively narrow band of division times that
are even integer multiples of the environmental duration, Eq. (7).

Noise-induced resonances select on cells with death times being odd multiples
of the environmental duration. Interestingly, for $\mu_{d}>\mu_{\gamma}$, the
signal-to-noise ratio of these resonances increases with division time noise
and, in some cases, reaches a maximum at a finite noise value (SM D), which is
a hallmark of stochastic resonance [30]. The intuition behind this effect is
that cells are killed when they stochastically exceed an age-dependent death
threshold and these events are resonantly filtered by periodic forcing.
Stochastic threshold fluctuations attenuate resonances but enhance the
survival probability away from them (Fig. 3 (b)) in contrast to constant
environments. Since thresholds of cell death are implemented by complex
intracellular pathways [7, 31, 16], our findings link fluctuations in these
biochemical networks with evolvable survival strategies in time-dependent
environments that could be exploited for cellular decision making or synthetic
biology.

A limitation of our approach is that we ignored generational inheritance [14,
32] and transient drug responses that can be exposure-time dependent [33]. The
latter effect can be incorporated by generalising Eq. (Noise-induced survival
resonances during fractional killing of cell populations) to the time-
dependent case as done in SM A. It would be interesting to explore extensions
to random environments with fluctuating drug duration. Such random
environments likely select several resonances along a vertical section of the
phase diagram (Fig. 3(c)) but these effects are beyond the scope of our study.

In summary, we have identified a possible non-genetic mechanism for fractional
killing emerging in response to periodic drug treatments. These stochastic
mechanisms could be one of many contributing factors to drug persistence,
among other active mechanisms such as phenotypic switching [34]. Since
periodically forced age-structured branching processes are ubiquitous, such as
in epidemics and ecology, we anticipate survival resonances to be widespread
in stochastic population dynamics.

Acknowledgments: This work was supported through a UKRI Future Leaders
Fellowship (MR/T018429/1).

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Supplemental materials

##  Supplemental Material A Offspring generating functional in time-dependent
environments

Here, we provide the general integral equations characterising the offspring
distributions in a time-dependent environment, which contains (Noise-induced
survival resonances during fractional killing of cell populations) as a
special case. The probability $\mathcal{P}[n,t|\delta_{x_{0}},t_{0}]$ of
observing a population with number density $n(x)$ at time $t$ arising from a
single ancestor of age $x_{0}$ at time $t_{0}$. The probability generating
functional associated with this offspring distribution is given by
$\mathcal{Z}[h,t|\delta_{x_{0}},t_{0}]:=\int\mathcal{D}[n]\exp(\int_{0}^{\infty}{\rm{d}}x\ln(h(x))n(x))\mathcal{P}[n,t|\delta_{x_{0}},t_{0}]$
where the auxiliary variable is a general age-dependent function $h(x)$ and
the integral is over all non-negative measures $n$ with positive support.
Using the definition of the generating functional in the Master equation
(Noise-induced survival resonances during fractional killing of cell
populations) we find a functional derivative equation:

$\begin{split}\frac{\partial\mathcal{Z}[h,t|\delta_{x_{0}},t_{0}]}{\partial
t}=&\int_{0}^{\infty}\big{[}\partial_{x}h(x)-(\gamma(x,t)+d(x,t))h(x)+\gamma(x,t)h^{2}(0)+d(x,t)\big{]}\frac{\delta\mathcal{Z}[h,t|\delta_{x_{0}},t_{0}]}{\delta
h(x)}{\rm{d}}x,\\\ \end{split}$ (10)

which can be manipulated via the method of characteristics. Doing so, the
characteristic curves are parameterised in terms of a continuous variable $s$
and are described by two ODEs:

$\frac{dt(s)}{ds}=1,\ \ \frac{d\mathcal{Z}(s)}{ds}=0$ (11)

and a PDE for $h(x,s)$:

$\begin{split}&(\partial_{s}+\partial_{x})h(x,s)=(d(x,s)+\gamma(x,s))h(x,s)-\gamma(x,s)h(0,s)^{2}-d(x,s).\\\
\end{split}$ (12)

We define
$\Pi(s,x)=\exp\Bigl{(}-\int_{0}^{s}\gamma(x+u,u)+d(x+u,u){\textrm{d}}u\Bigr{)}$
and integrate Eq. (12) to obtain:

$h(x,0)=\Pi(s,x)h(x+s,s)+\int_{0}^{s}\Pi(y,x)\big{(}\gamma(y+x,y)h^{2}(0,y)+d(y+x,y)\big{)}dy.$
(13)

We fix $t_{0}=0$ so that $t(s=0)=0$ and
$\mathcal{Z}[h(\bullet,0),t(0)|\delta_{x_{0}},0]=h(x_{0},0)$. Under these
initial conditions, we obtain $t(s)=s$ and
$\mathcal{Z}[h(\bullet,s),t(s)|\delta_{x_{0}},0]=h(x_{0},0)$ from Eqs. (11).
Therefore, Eq. (13) can be expressed as:

$\mathcal{Z}[h(\bullet,s),s|\delta_{x_{0}},0]=\Pi(s,x)h(x+s,s)+\int_{0}^{s}\Pi(y,x)\big{(}\gamma(y+x,y)h^{2}(0,y)+d(y+x,y)\big{)}dy.$
(14)

The final step is replacing $h^{2}(0,y)$ with a known function in Eq. (14). It
follows from Eq. (12) that $\mathcal{Z}$ is constant along any characteristic
starting at $s=y$ and hence:

$h(x_{0},y)=\mathcal{Z}[h(\bullet,s),t(s)|\delta_{x_{0}},y].$ (15)

Plugging Eq. (15) in Eq. (14), we obtain an integral equation for the
offspring initial generating function in time-dependent environments:

$\mathcal{Z}[h,t|\delta_{x_{0}},0]=\Pi(t,x_{0})h(x_{0}+t)+\int_{0}^{t}\Pi(y,x_{0})\bigg{(}\gamma(x_{0}+y,y)\mathcal{Z}^{2}[h,t|\delta_{0},y]+d(y+x_{0},y)\bigg{)}dy.$
(16)

Finally, we restrict our analysis to a homogeneous environment (so that
$\mathcal{Z}[h,t+T|m,T]=\mathcal{Z}[h,t|m,0]$ for any range of time $T$) and
adopt the notation proposed in Eq. (Noise-induced survival resonances during
fractional killing of cell populations):

$\begin{split}\mathcal{Z}[h,t|\delta_{x_{0}},0]\Pi(x_{0})=\Pi(t+x_{0})h(t+x_{0})+\int_{0}^{t}\Pi_{d}(u+x_{0})du+\int_{0}^{t}\Pi_{\gamma}(u+x_{0})\mathcal{Z}^{2}[h,t-u|\delta_{0},0]du\\\
\end{split}$ (17)

The long-term extinction probability follows from
$\lim_{t\rightarrow\infty}\mathcal{Z}[h=0,t|\delta_{x_{0}},0]=p^{*}(x_{0})=1-p_{\text{surv}}(x_{0})$
in Eq. (17), which gives

$p^{*}(x_{0})\Pi(x_{0})=\int_{x_{0}}^{\infty}\Pi_{d}(u)du+(p^{*}(0))^{2}\int_{x_{0}}^{\infty}\Pi_{\gamma}(u)du.\\\
$ (18)

We recover Eq. (3) of the main text by setting $x_{0}=0$ and identifying
$\nu=\int_{0}^{\infty}{\rm{d}}x\Pi_{d}(x)$ as the effective death probability.
Together with Eq. (18), this provides an exact solution for the extinction
probability of a population starting from a single ancestor with initial age
$x_{0}$. It can be observed that $p^{*}(0)=1$ implies $p^{*}(x_{0})=1$ for all
$x_{0}$ and this includes the phase transition from extinction to fractional
killing at $\nu=\frac{1}{2}$.

##  Supplemental Material B Long-term survival probability in periodic on-off
environments

Here, we present a detailed derivation the equations governing the survival
probability during periodic on-off treatments. such framework which, in
substance, is an age-dependent generalization of the multi-type Galton-Watson
processes [10, Ch. II.7]. We start observing that, for any branching
processes, the evolution of each branch is independent. This means that the
generating functional for a number of branches with density $q(x_{0})$ can be
expressed in terms of evolution of each individual branch starting with age
$x_{0}$:

$\mathcal{Z}[h,t|q,t^{\prime}]=\exp\int_{0}^{\infty}q(x_{0})\ln(\mathcal{Z}[h,t|\delta_{x_{0}},t^{\prime}])dx_{0},$
(19)

for $t^{\prime}<t$. Using the above in the Chapman-Kolmogorov integral
equation,
$\mathcal{Z}[h,t|m,0]=\int\mathcal{Z}[h,t|q,t^{\prime}]\mathcal{P}[q,t^{\prime}|m,0]\mathcal{D}q$,
gives:

$\displaystyle\mathcal{Z}[h,t|m,0]$ $\displaystyle=\int
e^{-\int_{0}^{\infty}q(x)\ln(\mathcal{Z}[h,t|\delta_{x},t^{\prime}]dx)}\mathcal{P}[q,t^{\prime}|m,0]\mathcal{D}q.$
(20)

Eq. (20) can be expressed via nesting generating functionals:

$\displaystyle\mathcal{Z}[h,t|m,0]$
$\displaystyle=\mathcal{Z}[\mathcal{Z}[h,t|\delta_{\tiny{\bullet}},t^{\prime}],t^{\prime}|m,0],$
(21)

where $\bullet$ is a placeholder for the function argument, i.e.,
$\mathcal{Z}[h,t|\delta_{\bullet},t^{\prime}](x_{0})=\mathcal{Z}[h,t|\delta_{x_{0}},t^{\prime}]$.
Furthermore, we can model periodic on-off treatment by nesting the generating
functionals of homogeneous on-environment
$\mathcal{Z}_{\text{on}}[h,t|\delta_{x_{0}},0]$ and homogeneous off-
environments $\mathcal{Z}_{\text{off}}[h,t|\delta_{x_{0}},0]$. Since the on-
off environment sequence is repeated with period $2T$, we define the offspring
generating functional at time $t=2T$ as:

$\mathcal{W}[h,x_{0}]=\mathcal{W}[h,\bullet]({x_{0}})=\mathcal{Z}_{\text{on}}[\mathcal{Z}_{\text{off}}[h,T|\delta_{\bullet},0],T|\delta_{{x_{0}}},0].$
(22)

Therefore, the offspring generating functional evaluated at time $t=2kT$ (with
$k\in\mathbb{N}$) can be expressed by nesting Eq. (22) $k$ times:

$\mathcal{Z}[h,t=2kT|\delta_{x_{0}},0]=\mathcal{W}^{k}[h,x_{0}],$ (23)

where we defined the $k^{th}$ composition of functions as
$\mathcal{W}^{k}[h;x_{0}]=(\mathcal{W}\circ\mathcal{W}^{k-1})[h,x_{0}]=\mathcal{W}[\mathcal{W}^{k-1}[h,\tiny{\bullet}];x_{0}]$.
Following a similar argument to Harris [10], we set $h=0$ to obtain the
extinction probability, for which we assume the existence of an asymptotic
steady state for $t>2kT$:

$\mathcal{W}[\mathcal{W}^{k}[0,\bullet],x_{0}]=\mathcal{W}[0,x_{0}].$ (24)

Finally, we obtain an expression (equal to Eq. (4)) for the asymptotic
extinction probability $p^{*}(x_{0})=1-p_{\text{surv}}(x_{0})$ of a population
starting from a single individual with age $x_{0}$ :

$\begin{split}p^{*}(x_{0})=\mathcal{W}[p^{*},T](x_{0})=\mathcal{Z}_{\text{on}}[\mathcal{Z}_{\text{off}}[p^{*},T|\delta_{\bullet},0],T|\delta_{x_{0}},0].\end{split}$
(25)

Manipulating Eq. (25) via Eq. (Noise-induced survival resonances during
fractional killing of cell populations), we obtain a system of integral
equations:

$\begin{split}\mathcal{G}[x_{0},t;p^{*}]\Pi^{on}(x_{0})&=\Pi^{on}(x_{0}+t)\mathcal{H}[x_{0}+t,t;p^{*}]+\int_{0}^{t}\Pi^{on}_{\gamma}(u+x_{0})\big{(}\mathcal{G}[0,t-u;p^{*}]\big{)}^{2}du+\int_{0}^{t}\Pi^{on}_{d}(u+x_{0})du,\end{split}$
(26)

$\begin{split}\mathcal{H}[x_{0},t;p^{*}]\Pi^{off}(x_{0})&=\Pi^{off}(x_{0}+t)\mathcal{G}[x_{0}+t,T;p^{*}]+\int_{0}^{t}\Pi^{off}_{\gamma}(x_{0}+u)(\mathcal{H}[0,t-u;p^{*}])^{2}du,\end{split}$
(27)

where we defined
$\mathcal{G}[x,t;p^{*}]=\mathcal{Z}_{\text{on}}[\mathcal{Z}_{\text{off}}[p^{*},t|\delta_{\bullet},0],T|\delta_{x},0]$
and $\mathcal{H}[x,t;p^{*}]=\mathcal{Z}_{\text{off}}[p^{*},t|\delta_{x_{0}}]$.
Setting $t=T$ in Eqs. (26) and (27), we obtain a system of coupled integral
equation for $p^{*}(x_{0})=\mathcal{G}[x_{0},t;p^{*}]$. Moreover, the
probabilities with upper index on and off are the defined as in Eq. (Noise-
induced survival resonances during fractional killing of cell populations)
with rate functions respectively $\\{\gamma(x),d(x)\\}$ and
$\\{\gamma(x),0\\}$. Eqs. (26) and (27) can be solved numerically by 1)
discretising functions and integrals and 2) iteratively inserting them into
each other until the fixed point is reached.

##  Supplemental Material C Constant rates in time-dependent environments

We discuss the age-independent branching processes in a periodic on-off
environment.

In a homogeneous environment, an exact solution of generating function can be
obtained [25]:

$\mathcal{Z}(H,t|1,0)=\frac{d+d(H-1)e^{(\gamma-d)T}-\gamma
H}{d+\gamma(H-1)e^{(\gamma-d)T}-\gamma H},$ (28)

where $d$ and $\gamma$ are the age-independent death and division rates
(respectively equal to the inverse of the average death time $\mu_{d}$ and
average division time $\mu_{\gamma}$). From comparison with Eq. (3), we find
the survival probability from Eq. (28) :
$p_{surv}=1-\lim_{t\rightarrow\infty}\mathcal{Z}(H=0,t|1,0)=\frac{\gamma-d}{\gamma}\Theta(\frac{\gamma-d}{\gamma})$.

We now move to time-dependent environment characterised by a fixed division
rate $\gamma$ and a periodically switching death rate $d(t)$. The asymptotic
extinction probability function $p_{surv}$ solves the following fixed point
equation:
$1-p_{surv}=\mathcal{Z}_{\text{on}}(\mathcal{Z}_{\text{off}}(1-p_{surv},T|1,0),T|0,1)$,
where $\mathcal{Z}_{\text{on}}$ and $\mathcal{Z}_{\text{off}}$ labels the
generating function for homogeneous on and off environments. This gives:

$p_{surv}=1-\frac{d\big{(}e^{2\gamma
T}-e^{(d+\gamma)T}\big{)}}{de^{dT}-de^{(d+\gamma)T}-\gamma\big{(}e^{dT}+e^{2\gamma
T}\big{)}}.$ (29)

As shown in Fig. S1, we observe a phase transition between extinction (blue,
$p_{surv}=0$) and fractional killing (red shading, $p_{surv}>0$). In contrast
to age-dependent results, the survival probability is monotonic in
$\frac{\mu_{d}}{\mu_{\gamma}}$ and $\frac{T}{\mu_{\gamma}}$ and there are no
survival resonances.

Figure S1: Asymptotic, age-independent survival probability in a periodic
time-dependent environments treatments.

##  Supplemental Material D Alternative approximation through transient
behaviour of the survival probability

This section provides an alternative approximation that captures the key
features of noise-induced resonances. As explained in the main text, we
consider death times $\tau_{d}$ are deterministic and roughly equal to their
mean $\mu_{d}$, while no conditions has to hold for division times
$\tau_{\gamma}$ which follow a cumulative distribution
$\phi(\tau_{\gamma}<y)=1-e^{-\int_{0}^{y}{\rm{d}}u\gamma(u)}$. Following the
arguments used in the main text, if $\mu_{d}$ falls in an on-environment
$(\mu_{d}\in(2nT,(2n+1)T$), the survival probability of the first generation
is given by $\phi(\mu_{d})$. On the other hand, if a death time falls in an
off-environment ($\mu_{d}\in((2n+1)T,(2n+2)T$), the extinction probability of
the first generation is $\phi((mod(\frac{\mu_{d}}{T})+1)T)$.

A different point of view on these discontinuities is proposed in Fig. S2 (a)
where the survival probability of the initial cell ($p_{\text{surv}}^{I}$)
exhibits a discontinuous jump when $\mu_{d}=(2n+1)T$. Another way to display
the effects of noise-induced resonances can be found by considering a fixed
value of $\mu_{d}$ and varying the environment length $T$, see Fig. S2 (b).
These jumps of $p_{\text{surv}}^{I}$ are evident any time that
$\mu_{d}=(2n+1)T$ and the height of each jump is given by:

$\Delta_{n}(T,\mu_{d})=\phi(2nT)-\phi(\mu_{d})=e^{-\int_{0}^{\mu_{d}}\gamma(u)du}\big{(}1-e^{-\int_{\mu_{d}}^{2nT}\gamma(u)du}\big{)}.$
(30)

Such a discontinuity propagates through the following generations, shaping the
asymptotic behaviour of the survival probability $p_{\text{surv}}$. The jumps
are a consequence of noise-induced resonances occurring when:

$\displaystyle\\{T:\mu_{d}=(2n+1)T,n\in\mathbb{N}\\}.$ (31)

Figure S2: Noise-induced jumps of the survival probability as function of
$\mu_{d}$ (panel (a)) or $T$ (panel (b)). The survival probabilities of the
first and second generations (Eqs. (32) and (33)) approximate the asymptotic
behaviour displayed in Fig. 3 (a) and (b) (respectively panel (c) and panel
(d)

We provide analytic solutions for the survival probability in the first
($p^{I}_{surv}$) and second generation ($p_{\text{surv}}^{II}$). The survival
probability for the first generation (i.e. an initial newborn cell) is given
by:

$p_{\text{surv}}^{I}(\mu_{d},T)=\mathbbm{1}_{\text{on}}(\mu_{d})\phi(\mu_{d})+\mathbbm{1}_{\text{off}}(\mu_{d})\phi\left(\left(\left\lfloor\frac{\mu_{d}}{T}\right\rfloor+1\right)T\right),$
(32)

where we defined the indicator function $\mathbbm{1}_{\text{off/on}}(\mu_{d})$
which returns $1$ if $\mu_{d}$ falls in an off/on environment or $0$
otherwise. Moving on, we observed that ages are synchronous in the second
generation and thus he survival probability $p_{\text{surv}}^{II}$ can be
written as:

$\begin{split}p_{\text{surv}}^{II}&=\int_{0}^{\mu_{d}}P(\tau_{\gamma}=x|\tau_{\gamma}<\mu_{d})(p_{\text{surv}}^{I}(\mu_{d}+x,T))^{2}dx=\\\
&=\int_{0}^{\mu_{d}}\frac{\partial_{x}\phi(x)}{\phi(\mu_{d})}(p_{\text{surv}}^{I}(\mu_{d}+x,T))^{2}dx\\\
\end{split}$ (33)

We note that Eq. (33) can also be recalled from Eq. (Noise-induced survival
resonances during fractional killing of cell populations). In Fig. S2 (c, d),
we compare the survival probability of the first and second generations and
how they approach long-term. The result highlights the imprint of resonances
in the first generation on the following ones.

### Stochastic resonance behaviour of noise-induced resonances

We conclude by discussing the role of noise-induced survival resonances in
relation to the phenomenon of stochastic resonances. To do so, we study the
relationship between noise-induced jumps ($p_{\text{surv}}$) and the noise
affecting division times. Using Eq. (30), we label the jump at the main
resonance of the survival probability ($p_{\text{surv}}^{I}$) as
$\Delta_{1}(T,\mu_{d})=\Delta$ as:

$\Delta=\phi(2T)-\phi(\mu_{d})=e^{-\int_{0}^{\mu_{d}}\gamma(u)du}\big{(}1-e^{-\int_{\mu_{d}}^{2T}\gamma(u)du}\big{)}.$
(34)

From the latter equation we obtain the output-performance ($\Delta$) versus
the input noise magnitude ($CV^{2}_{\gamma}$) in the case of
$\Gamma$-distributed division times, see Fig. S3. We observe that, for
specific values of $\mu_{d}$, the jump height displays a non-monotonic
dependence on $CV^{2}_{\gamma}$. This peaked behaviour shown in Fig. S3
matches an increase in the signal-to-noise ratio with the noise intensity
associated with stochastic resonance.

Figure S3: Dominant jump (of the survival probability in the first generation)
versus the noise affecting division times at $\frac{\mu_{d}}{T}=1$. For some
values of $\mu_{\gamma}$, the survival jumps display a non-monotonic behaviour
following Eq. (34).