🧬 biology

Optimization of sequential therapies to maximize extinction of resistant bacteria through collateral sensitivity

This paper develops a stochastic birth-death model to demonstrate that optimizing the switching periods of sequential antibiotic therapies based on collateral sensitivity can maximize bacterial extinction, revealing a nonmonotonic relationship between switching frequency and success while identifying a Pareto front that balances eradication efficacy against the risk of evolving resistance.

Original authors: Javier Molina-Hernández, José A. Cuesta, Beatriz Pascual-Escudero, Saúl Ares, Pablo Catalán

Published 2026-05-11

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CC BY 4.0

Original authors: Javier Molina-Hernández, José A. Cuesta, Beatriz Pascual-Escudero, Saúl Ares, Pablo Catalán

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are fighting a stubborn army of bacteria inside a patient. You have two different weapons (antibiotics), let's call them Weapon A and Weapon B.

The problem is that bacteria are clever. If you use Weapon A for too long, some bacteria learn to resist it. If you then switch to Weapon B, those same bacteria might be vulnerable again. But here's the tricky part: if you switch too fast, the bacteria don't have time to "learn" the weakness of the first weapon, and they just survive. If you wait too long, they might learn to resist both weapons.

This paper is like a math-based strategy guide for finding the perfect timing to switch between these two weapons to wipe out the entire bacterial army, even when the weapons aren't super strong (sub-inhibitory doses).

Here is the breakdown of their findings using simple analogies:

1. The "Collateral Sensitivity" Trick

The core idea is a phenomenon called Collateral Sensitivity. Think of it like a superhero's weakness.

When a bacteria evolves to become immune to Weapon A, it accidentally becomes super-sensitive to Weapon B.
Conversely, if it becomes immune to Weapon B, it becomes weak against Weapon A.
The researchers found that if you exploit this trade-off, you can trap the bacteria in a cycle where every time they try to survive one weapon, they become an easy target for the other.

2. The Goldilocks Timing (The Switching Period)

The researchers ran thousands of computer simulations to see how long you should wait before switching weapons. They found that timing is everything:

Switching too fast (Too early): It's like changing the lock on a door before the burglar has even tried to pick it. The bacteria haven't had time to evolve resistance to the first weapon yet, so when you switch, they are still vulnerable to the first one, but the second weapon doesn't kill them efficiently. The bacteria survive.
Switching too slow (Too late): You wait so long that the bacteria have fully mastered the first weapon. Worse, they might have evolved to resist both weapons.
The Sweet Spot: There is a specific "wait time" that is just right. You wait long enough for the bacteria to evolve resistance to the current weapon (making them vulnerable to the next one), but not so long that they become a "super-bug" resistant to everything.

3. The "Coin Flip" Game

The researchers discovered that extinction (killing all the bacteria) happens in a very specific way.

Imagine every time you switch antibiotics, you are flipping a coin.
If the bacteria are in the right state (mostly resistant to the old weapon and vulnerable to the new one), the coin lands on "Heads," and the whole army dies.
If the coin lands on "Tails," the bacteria survive and keep growing.
The paper shows that the longer you wait between switches (up to a point), the better the odds of that coin landing on "Heads." However, waiting too long increases the risk of the bacteria evolving a "double resistance" (resisting both weapons), which makes the coin almost impossible to win.

4. The "Pareto Front" (The Trade-off)

The paper highlights a difficult choice doctors might face, which they call a Pareto front.

Goal A: Maximize the chance of killing all the bacteria.
Goal B: Minimize the chance of creating a "super-bug" that resists both drugs.
The Conflict: To get the highest chance of total extinction, you often need to run the therapy for a long time with many switches. But running it longer also gives the bacteria more time to evolve that dreaded double-resistance.
The Solution: You can't have your cake and eat it too. You have to find a "Pareto optimal" point—a specific switching schedule that offers the best possible balance between killing the infection and avoiding the creation of an unstoppable super-bug.

5. The Role of Mutation Rates

The paper also looked at how fast bacteria mutate (change their DNA).

Too slow: They don't evolve resistance fast enough to trigger the "weakness" needed for the next drug to work.
Too fast: They evolve resistance so quickly that they accidentally create a "double-resistant" strain that can survive both drugs.
Just right: There is a "sweet spot" for mutation speed where the strategy works best. Interestingly, the paper suggests that slightly increasing mutation rates (up to a point) could actually help this strategy work better, but only if you are careful not to cross the line where double-resistance takes over.

Summary

The paper argues that sequential therapy (switching drugs one after another) is a powerful tool, but it requires precise math to work. It's not just about having two drugs; it's about knowing exactly when to switch them.

Fast switches fail because the bacteria don't evolve the necessary weakness.
Slow switches fail because the bacteria become too strong.
Optimal switches exploit the bacteria's evolutionary "blind spots," turning their survival tactics against them to wipe them out completely.

The authors provide a mathematical framework to calculate these perfect switching times, offering a blueprint for designing better antibiotic treatments that could potentially stop the rise of drug-resistant infections.

Problem Statement
Antimicrobial resistance (AMR) poses a critical threat to global health, with the development of new antibiotics failing to keep pace with the evolution of resistance. While combination therapies (simultaneous administration of multiple drugs) are a common strategy, they carry risks of toxicity and can sometimes accelerate resistance. Sequential therapies, which administer antibiotics one after another, offer a promising alternative based on the phenomenon of collateral sensitivity (CS). CS occurs when bacterial resistance to one antibiotic increases sensitivity to another. Despite the potential of CS-guided sequential regimens, there is a lack of quantitative design principles for optimizing the timing of drug switches to maximize bacterial extinction, particularly under subinhibitory antibiotic concentrations where resistance evolution is a significant concern.

Methodology
The authors develop a four-genotype stochastic birth–death model to simulate within-host evolutionary dynamics under two bacteriostatic antibiotics (A and B). The model tracks four bacterial populations:

$x_0$ : Wild type (susceptible to both).
$x_1$ : Resistant to A, susceptible to B.
$x_2$ : Resistant to B, susceptible to A.
$x_3$ : Double-resistant (resistant to both).

Key features of the model include:

Stochastic Dynamics: The system uses a tau-leaping algorithm to simulate birth, death, and mutation events ( $\mu_1$ for acquiring resistance, $\mu_2$ for losing it).
Sequential Switching: Simulations begin with antibiotic A. After a switching period $\tau$ , the antibiotic switches to B, then back to A, and so on, for a fixed total treatment duration $T$ .
Extinction Criteria: A population is considered extinct if it drops below a threshold of $0.05\gamma^{-1}$ (approx. 5 individuals), where $\gamma$ relates to the carrying capacity.
Parameters: The study focuses on subinhibitory concentrations ( $k=0.5$ , representing IC50) and strong reciprocal CS ( $k_{CS}=0.05$ ).

Key Contributions and Results

Nonlinear Dependence on Switching Period:
The probability of bacterial extinction depends nonlinearly on the switching period $\tau$ .
- Fast switching ( $\tau \to 0$ ): Results in no extinction because there is insufficient time for resistant mutants to evolve and dominate.
- Optimal switching: Extinction probability increases with $\tau$ and exhibits stepwise jumps aligned with discrete switch events (e.g., increasing from 1 to 2 switches).
- Mechanism: Extinction events occur shortly after a switch. For extinction to occur, the population must first evolve to be dominated by a single-resistant genotype (e.g., $x_1$ under drug A). When the drug switches to B, this dominant population (now sensitive) collapses. If the population contains significant numbers of other genotypes (wild type or double-resistant), extinction is less likely.
Geometric Distribution Framework:
The authors propose that extinction can be modeled as a series of Bernoulli trials (coin tosses) at each switch.
- They fit the cumulative extinction probability to a hierarchical geometric distribution, $P_{ext} = 1 - (1 - p)^n$ , where $p$ is the per-switch extinction probability and $n$ is the number of switches.
- The per-switch probability $p$ is not constant; it increases with the switching period $\tau$ and depends on the number of switches $n$ . A fit of the form $p = \alpha\tau + \beta n$ accurately predicts cumulative extinction probabilities.
Heuristic Approximation:
The authors derive an analytical heuristic for $p(\tau)$ based on three key time distributions:
- Time for single-resistant mutants to dominate the population.
- Time to extinction following a switch.
- Probability of non-dominance by single-resistant mutants.
  The resulting formula, $p(\tau) = p_d(\tau)[1 - p_r(\tau)]$ , where $p_d$ is the decay probability and $p_r$ is the probability of non-dominance, successfully matches simulation data.
Sensitivity to Parameters:
- Collateral Sensitivity (CS): Strong reciprocal CS is a necessary condition for extinction at subinhibitory doses. As CS weakens ( $k_{CS}$ increases), extinction probability drops to near zero.
- Antibiotic Dose: The relationship between dose and extinction is unimodal. At very high doses (near MIC), extinction occurs regardless of switching. At intermediate subinhibitory doses, there is an optimal dose that maximizes extinction. At very low doses, extinction probability decreases because the influx of resistant mutants slows down.
- Mutation Rates: Similar to dose, extinction probability shows a unimodal dependence on mutation rates. High mutation rates can lead to the emergence of double-resistant strains ( $x_3$ ), reducing extinction efficacy.
Pareto Front of Optimal Therapies:
There is a trade-off between maximizing extinction and minimizing the emergence of double-resistant mutants.
- Longer therapies (more switches) increase extinction probability but also increase the likelihood of $x_3$ appearing if the population survives.
- The authors identify a Pareto front of optimal switching periods. For the studied parameters, a switching period of $\tau \approx 42$ is most likely to lie on this front, allowing clinicians to choose between maximizing extinction or minimizing resistance by adjusting the number of switches $n$ .

Significance and Claims
The paper claims to provide quantitative design principles for in vitro and clinical sequential antibiotic therapies. By demonstrating that strong CS can lead to bacterial eradication even at subinhibitory concentrations (where monotherapies fail), the study underscores the potential of CS-guided regimens to suppress resistance evolution.

The authors emphasize that their stochastic framework captures extinction dynamics directly, unlike previous deterministic models that cannot predict eradication. They argue that the key to success is not just the existence of CS, but the timing of the switch: one must wait long enough for a single-resistant mutant to dominate the population before switching, but not so long that double-resistance emerges. The identification of a Pareto front offers a practical framework for balancing the dual goals of eradicating infection and preventing the evolution of pan-resistance.

The study concludes that while the model is minimal (two drugs, four genotypes), it isolates the core mechanisms of CS-guided switching and suggests that sequential therapies represent a rich opportunity to tackle the antibiotic crisis, with potential extensions to cancer treatment and epidemiological scales.