📄 evolutionary biology

Exploring the influence of mutation rates with stochastic perturbations on genome evolution

This paper presents a stochastic Ito calculus-based extension of the classical Luria-Delbruck model to demonstrate how randomly varying mutation rates significantly influence genome evolution, particularly near carrying capacity, offering new insights into viral molecular evolution and strategies for overcoming mutational load.

Original authors: Bohlin, J., Pettersson, J. H.- O.

Published 2026-10-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bohlin, J., Pettersson, J. H.- O.

Original paper licensed under CC BY 4.0 (https://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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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

Life is a constant battle between order and chaos, a struggle played out most vividly in the microscopic world of viruses and bacteria. To understand how these tiny organisms change over time, scientists have long relied on a classic idea called the Luria-Delbrück model. Think of this model as a way to count how many times a population of microbes accidentally changes its genetic code as it multiplies. For decades, this calculation assumed a predictable world: if you know how fast a population grows and how often it makes mistakes, you can predict exactly how many new variations will appear. It is a neat, straight-line logic that works well when things are steady. But the real world is rarely steady. Populations bump into limits, resources run dry, and random events—like a sudden shift in temperature or a burst of immune activity—can jolt a colony in unpredictable ways. The question that has lingered is whether these random jolts, which scientists call stochastic effects, are just minor background noise, or if they are powerful enough to rewrite the rules of evolution itself.

Two researchers, Jon Bohlin and John H.-O. Pettersson, decided to test this by building a new kind of mathematical map. Instead of drawing a single, smooth line to represent growth, they introduced a layer of randomness, using a sophisticated method of calculation known as Itô calculus. This approach allows them to simulate how a population grows when its speed is constantly being nudged by chance. They applied this to a scenario where a population cannot grow forever but must stop when it hits a limit, much like a crowd filling a room until there is no more space. In their simulations, they watched what happened when the population grew fast versus when it grew slowly, and how the random nudges affected the total number of genetic mutations that accumulated over time.

The results revealed a surprising twist in how evolution works near the end of a growth spurt. When a population is growing rapidly, the sheer speed of multiplication tends to drown out the random noise; the population is so busy expanding that the little jolts of chance don't matter much. However, as the population slows down and approaches its maximum size, the story changes completely. The researchers found that in this slowing phase, the random effects stop being background noise and begin to dominate the process. Instead of a steady accumulation of changes, the population enters a feedback loop where chance events dictate the direction of evolution. If the growth rate drops below a certain threshold, these random fluctuations can take over, driving the genetic makeup of the group in ways that a predictable model would never anticipate. The study suggests that as a population reaches its limit, the role of pure luck in shaping its genome becomes profound, potentially leading to a rapid increase in genetic diversity or a sudden shift in traits.

To see if this theory held water in a real-world context, the authors applied their model to the SARS-CoV-2 virus, the pathogen responsible for the recent global pandemic. They set up a simulation of the virus growing inside a human host, assuming a starting number of 1,000 viral particles and a maximum capacity of 10 million. Using a growth rate and mutation rate derived from existing data, they compared their new, chaotic model against the old, predictable one. The difference was striking. The traditional model suggested a certain number of mutations, but the new model, which accounted for the random ups and downs of viral growth, predicted a significantly higher number of genetic variations. This is particularly important because the virus often faces a "bottleneck" when it moves from one person to another, where only a few survivors make the trip. If the virus is accumulating more random variations than previously thought while it is still inside a host, it might be carrying a much richer, more diverse genetic toolkit into the next infection.

The researchers also explored how a population might survive this chaotic environment. They found that if a population can maintain a high growth rate relative to the size of the random jolts, it can remain stable. However, if the growth slows, the population becomes vulnerable to a "mutational load," where the accumulation of random errors could overwhelm it. The study suggests that one way for a population to survive is to evolve a form of "robustness," essentially learning to withstand the genetic drift that comes with a slowing growth rate. This is not a guaranteed fix, but a strategy that emerges naturally from the math: by increasing the carrying capacity of the environment or reducing the volatility of the growth, a population can better manage the chaos.

In the end, this work does not claim to have solved the mystery of viral evolution, but it offers a clearer lens through which to view it. By replacing the smooth, predictable lines of the past with the jagged, realistic paths of the present, the authors show that randomness is not just a minor detail to be ignored. It is a central character in the story of life, especially when things are slowing down. For viruses like SARS-CoV-2, this means that the diversity we see in the world today might be the result of a chaotic, chance-driven process happening inside our bodies long before the virus ever leaves us. The study concludes that incorporating these random effects into our models is not just a mathematical exercise, but a necessary step to understand how life evolves when it hits the wall.

Technical Summary: Exploring the influence of mutation rates with stochastic perturbations on genome evolution

Problem Statement
The classical Luria-Delbrück model, originally formulated in 1943 to understand bacterial mutation rates, relies on deterministic ordinary differential equations (ODEs) assuming fixed growth rates and mutation parameters. While useful, this approach overlooks the inherent randomness in biological systems, particularly regarding population fluctuations and mutation accumulation over time rather than just per generation. The authors argue that modeling population growth as a deterministic process is unrealistic, especially when considering constraints like carrying capacity ( $K$ ) and the random nature of environmental perturbations. Furthermore, existing models often fail to account for how stochastic effects might dominate genome evolution when growth rates decrease or approach specific thresholds, potentially leading to a "snowball effect" of genetic drift.

Methodology
The authors propose a stochastic extension of the Luria-Delbrück model using Itô calculus. The methodology involves the following steps:

Stochastic Growth Formulation: The deterministic growth rate $r$ is replaced by a stochastic process $r = \kappa + cW_t(\omega)$ , where $\kappa$ is the deterministic growth rate, $W_t$ is Gaussian white noise, and $c$ is a scaling coefficient. This transforms the population growth equation into a Stochastic Differential Equation (SDE).
Carrying Capacity Constraints: The model incorporates a carrying capacity $K$ (representing the stationary phase of growth) using a logistic growth framework. The deterministic version is given by $\frac{dN}{dt} = r(K-N)N$ .
Mathematical Derivation:
- The authors derive a general solution for a class of non-linear SDEs of the form $dX_t = f(t, X_t)dt + c(t)X_t dB_t$ .
- They utilize Itô's formula and the concept of an integrating factor ( $F_t$ ) to solve the specific SDE for logistic growth with stochastic perturbations.
- The resulting solution for the population size $X_t$ is expressed in a closed form involving exponential martingales and Brownian motion ( $B_t$ ).
Mutation Accumulation: The number of mutations is modeled as the integral of the mutation rate $\mu$ multiplied by the stochastic population size $X_t$ over time.
Numerical Simulation: The models are simulated using Julia (v1.6.1) with the DifferentialEquations library. The Tsit5 solver is used for ODEs, and the Euler-Maruyama (EM) solver is used for SDEs.
Case Study: The framework is applied to the SARS-CoV-2 virus, modeling within-host viral evolution with parameters reflecting viral load, growth rates, and mutation rates observed in literature.

Key Contributions

Stochastic Luria-Delbrück Model: The paper presents a novel Itô calculus-based version of the Luria-Delbrück model that explicitly incorporates random perturbations in population growth.
Analytical Solution for Non-Linear SDEs: The authors provide a general formula and explicit derivation for solving a class of non-linear SDEs with carrying capacity constraints, offering a closed-form solution for the stochastic population size $X_t$ .
Identification of Threshold Effects: The analysis reveals that stochastic effects are not merely additive noise but can fundamentally alter evolutionary trajectories. Specifically, when the effective growth rate approaches a threshold defined by the stochastic volatility ( $rK \to c^2/2$ ), stochastic effects dominate in a feedback-based manner.
Carrying Capacity Dynamics: The study demonstrates that unlike exponential growth models, stochastic effects in logistic growth models increase in influence as the population approaches its carrying capacity, a phenomenon not captured by deterministic counterparts.

Results

Dominance of Stochasticity: Simulations show that as the growth rate decreases or the population nears carrying capacity, random perturbations can dominate the evolution of the population. This leads to a "snowball effect" where genetic drift significantly influences genome evolution, even if the initial growth rate was high.
Robustness Strategies: The model suggests that populations can evolve "robustness" against mutational load and genetic drift by reducing volatility ( $c$ ), reducing the growth rate (provided it remains above the volatility threshold), or increasing the carrying capacity ( $K$ ).
SARS-CoV-2 Implications: When applied to SARS-CoV-2, the stochastic model predicts a higher number of within-host mutations compared to deterministic models and existing consensus-based studies. The results suggest that immune responses (which reduce viral growth rates) may inadvertently increase the influence of stochastic volatility, thereby enhancing viral diversity and potentially the emergence of variants. The model highlights that consensus sequences may underestimate the true diversity of within-host variants due to the "collapse" of diverse stochastic realizations.

Significance and Claims
The paper claims that Itô calculus provides a "very natural" framework for incorporating stochastic effects into mathematical models of biological systems. The authors assert that their findings challenge the assumption that deterministic models are sufficient for understanding genome evolution, particularly in scenarios where populations face carrying capacity limits or fluctuating growth rates.

The significance lies in the demonstration that stochastic effects are not negligible; rather, they can become the primary driver of evolution under specific conditions (low growth rates, high carrying capacity proximity). The paper concludes that ignoring these stochastic perturbations may lead to an underestimation of mutational load and genetic diversity, with specific relevance to understanding the molecular evolution of rapidly mutating pathogens like SARS-CoV-2. The authors emphasize that their model offers a strategy for understanding how populations might overcome increasing mutational loads through the interplay of growth rates, carrying capacity, and volatility.

Technical Summary: Exploring the influence of mutation rates with stochastic perturbations on genome evolution

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