Comparison of Tug-of-War Models Assuming Moran versus Branching Process Population Dynamics

Core Concepts

The core message of this article is that the Moran A model and the critical binomial branching process model exhibit similar behavior and statistics under various selection scenarios, making the computationally simpler Moran A model a viable alternative for modeling tumor evolution.

Abstract

The article examines the Tug-of-War model, which describes the joint effect of rare advantageous driver mutations and frequent but deleterious passenger mutations, under two common population dynamics frameworks: the Moran model and the branching process.
The key highlights and insights are:

The Moran A model and the critical binomial branching process model conditioned on the final population size being close to the initial count exhibit similar statistics, including the shape of the mutational Site Frequency Spectrum, under a range of selection scenarios.

The Moran B model, which differs from Moran A in the expected fitness change after a death-replacement event, exhibits a "drift barrier" that prevents deleterious passenger mutations from dominating the fitness trend, even when the mutation-selection balance condition favors them.

Relaxing the conditioning on the branching process to only non-extinction leads to higher variances in the statistics, but the means remain similar to Moran A and tightly conditioned branching process.

Changing the progeny cell count distribution in the branching process while retaining criticality can impact the allele and singleton counts, but not the averages of mutation and division/replacement event counts.

Fitting the mutational Site Frequency Spectrum from breast cancer samples shows that both the Moran A model and the critical binomial branching process can provide good fits, supporting the view that passenger mutations exert a deleterious effect during tumor progression.

The article concludes that the Moran A model can be a computationally efficient alternative to the branching process for modeling tumor evolution, as the two models exhibit comparable statistics in the Tug-of-War setting.

Stats

"The time until the next death - replacement event is exponentially distributed with parameter Σf =Σ i∈{1,…,N } fi."
"The time until the next mutation event is exponentially distributed with parameter Nμ, where μ is the mutation rate per cell."
"The fitness of a cell i of type (αi, βi) with αi drivers and βi passengers is defined by fi = (1 + s) αi (1 − d) βi."

Quotes

"The concept of a model involving the joint effect of rare advantageous and frequent neutral or slightly deleterious mutations can be applied to describe evolution of cancer genes."
"We explore similarities and differences between the two types of selection in cell populations. This contributes to the ongoing discussion of which models are most appropriate for proliferating cell populations under drift, mutation, and selection."
"Crucially, SFS fitting results for experimental breast cancer data (Section 3.2) are similar between the two models. This finding might hold mathematical importance, and requires further investigation."

Key Insights Distilled From

Comparison of Tug-of-War Models Assuming Moran versus Branching Process Population Dynamics

by Dinh,K. N., ... at www.biorxiv.org 10-23-2023

https://www.biorxiv.org/content/10.1101/2023.10.20.563302v2

Deeper Inquiries

How do the Moran A and critical binomial branching process models compare in their ability to capture other important tumor evolution dynamics beyond the Tug-of-War scenario, such as clonal competition and spatial heterogeneity

The Moran A and critical binomial branching process models have their strengths and limitations in capturing tumor evolution dynamics beyond the Tug-of-War scenario.
The Moran A model, with its discrete-time Markov chain approach, is well-suited for modeling scenarios where the population size remains constant and the focus is on the balance between driver and passenger mutations. It provides a straightforward framework for understanding the effects of selection, drift, and mutation on allele frequencies in a population. However, the Moran A model may not fully capture the stochastic fluctuations in population size that can occur in real tumor evolution scenarios.
On the other hand, the critical binomial branching process model allows for more flexibility in modeling population size dynamics, as the population can grow or decline over time. This is particularly important in capturing clonal competition dynamics, where the growth of one clone may suppress the growth of others. The branching process model also provides a more detailed representation of spatial heterogeneity within the tumor, as it allows for the consideration of spatially distinct subpopulations with varying growth rates.
In summary, while the Moran A model is useful for understanding the basic dynamics of selection and mutation in a constant population size setting, the critical binomial branching process model offers a more comprehensive approach for capturing complex tumor evolution dynamics, including clonal competition and spatial heterogeneity.

What are the theoretical conditions under which the Moran A model and critical binomial branching process model can be shown to be mathematically equivalent, and what are the limitations of this equivalence

The theoretical conditions under which the Moran A model and critical binomial branching process model can be shown to be mathematically equivalent are based on the assumption of a constant population size and specific selection scenarios.
For the Moran A model and critical binomial branching process model to be considered equivalent, the following conditions need to be met:

The population size remains constant throughout the simulation.
The selection coefficients for driver and passenger mutations are within a certain range that allows for a balance between the two types of mutations.
The mutation rate and probability of driver mutations are set at appropriate levels to maintain the equilibrium between mutation and selection.

However, there are limitations to this equivalence. The Moran A model simplifies the dynamics of tumor evolution by assuming a fixed population size and discrete time steps, while the branching process model allows for more flexibility in modeling population growth and spatial heterogeneity. In scenarios where the population size fluctuates significantly or spatial interactions play a crucial role in tumor evolution, the Moran A model may not fully capture the complexity of the dynamics observed in real tumors.
Therefore, while the Moran A model and critical binomial branching process model can be equivalent under certain conditions, their applicability and accuracy in capturing tumor evolution dynamics may vary depending on the specific characteristics of the tumor and the underlying biological processes.

Could the insights from this study be extended to develop hybrid models that combine the strengths of both the Moran and branching process frameworks to provide a more comprehensive description of tumor evolution

The insights from this study could be extended to develop hybrid models that combine the strengths of both the Moran and branching process frameworks to provide a more comprehensive description of tumor evolution.
One approach could be to integrate the discrete-time Markov chain framework of the Moran model with the continuous-time branching process model to create a hybrid model that captures both the discrete and continuous aspects of tumor evolution. This hybrid model could incorporate the ability of the branching process model to model population growth and spatial heterogeneity, while also leveraging the simplicity and interpretability of the Moran model for understanding selection and mutation dynamics.
Additionally, the hybrid model could incorporate additional features such as spatial constraints, varying mutation rates, and interactions between different subpopulations within the tumor. By combining the strengths of both models, the hybrid approach could provide a more realistic and detailed representation of tumor evolution dynamics, allowing for a better understanding of clonal competition, spatial heterogeneity, and the impact of selection pressures on tumor progression.