# Differential equation modeling of cell population dynamics in skeletal muscle regeneration from single-cell transcriptomic data

## TL;DR

This paper introduces a new differential equation model to study how different cell populations interact during skeletal muscle regeneration using single-cell RNA sequencing data.

## Contribution

A novel non-linear ordinary differential equation model is developed to capture cell population dynamics during muscle regeneration, incorporating a new regulatory interaction between M2 macrophages and satellite cells.

## Key findings

- The model successfully captures key features of muscle regeneration dynamics using calibrated time-series scRNA-seq data.
- Incorporating a novel regulatory interaction between M2 macrophages and satellite cells improves model accuracy.
- The model provides a foundation for future computational studies of muscle regeneration and therapeutic testing.

## Abstract

Skeletal muscle regeneration is a complex process orchestrated by diverse cell populations within a dynamic niche. In response to muscle damage and intercellular signaling, these cells undergo cell fate and migration decisions including quiescence, activation, proliferation, differentiation, infiltration, apoptosis, and exfiltration. The emergence of single-cell RNA sequencing (scRNA-seq) studies of muscle regeneration offers a significant opportunity to refine models of regeneration and enhance our understanding of cellular interactions. To better understand how crosstalk between cell types governs cell fate decisions and cell population dynamics, we developed a novel non-linear ordinary differential equation model guided by scRNA-seq data. Our model consists of 9 variables and 17 parameters, capturing the dynamics of key myogenic lineage and immune cell types. We calibrated time-series scRNA-seq data to units of cells per cubic millimeter of tissue and fit our model’s parameters to capture the observed dynamics, validating on an independent time series. The model successfully captures key features of regeneration dynamics, particularly after incorporating a novel regulatory interaction between M2 macrophages and satellite cells that has been hypothesized in the literature. Our model lays a foundation for future computational explorations of muscle regeneration, modeling of disease conditions, and in silico testing of therapeutic strategies.

Skeletal muscles in humans and animals have the ability to regenerate—an ability that enables recovery from injury but also underlies muscle strengthening in response to exercise. Conversely, failures of muscle regeneration are implicated in muscular dystrophies and age-related muscle loss. Muscle regeneration depends on stem cells, called satellite cells, within the muscle, but they cannot do the job alone. Various other types of cells are necessary, including immune cells, which infiltrate the muscle after injury and clean up damaged tissue. Cross-talk between these cell types is necessary to coordinate their activity and ensure successful regeneration. Recent advances in single-cell RNA-sequencing allow us to measure the states and activities of cells within regenerating tissue. Here, we propose a differential equation model of cell population dynamics during muscle regeneration, which describes the numbers and activities of different cell types over time. We show that the single-cell data can be used to tune the parameters of the model. Unlike many existing approaches to studying dynamics from single-cell data, such as pseudotime and RNA velocity methods, the differential equation model summarizes the dynamics of the data compactly, and utilizes and tests our extensive prior biological knowledge of muscle regeneration, rather than discarding that knowledge.