Mathematical model predicts best way to build muscle


This article is a repost which originally appeared on

University of Cambridge – AUGUST 23, 2021

Edited for content and readability
Images sourced from Pexels

DOI: 10.1016/j.bpj.2021.07.023

Mathematical model predicts best way to build muscle

Figure 1. The “textbook” hierarchy in the anatomy of skeletal muscle. The overall muscle is characterized by its cross-sectional area (CSA), which contains a certain number (Nc) of muscle fibers (the muscle cells with multiple nuclei or multinucleate myocytes). A given muscle has a nearly fixed number of myocytes: between Nc ≈ 1000 for the tensor tympani and Nc > 1,000,000 for large muscles (gastrocnemius, temporalis, etc. Credit: DOI: 10.1016/j.bpj.2021.07.023

Researchers have developed a mathematical model that can predict the optimum exercise regimen for building muscle.

The researchers, from the University of Cambridge, used methods of theoretical biophysics to construct the model, which can tell how much a specific amount of exertion will cause a muscle to grow and how long it will take. This model, could allow users to optimize their exercise regimens by entering a few details of their individual physiology.

The model is based on earlier work by the same team, which found that a component of muscle called titin is responsible for generating the chemical signals that affect muscle growth.

The Results

In the Biophysical Journal, the results suggest that there is an optimal weight at which to do resistance training for each person and each muscle growth target. Muscles can only be near their maximal load for a very short time. It is that max load, integrated over time which activates the cell signaling pathway that leads to the combining of new muscle proteins. However, below a certain value, the load is too insufficient to cause much signaling, and exercise time would have to increase exponentially to compensate. The value of this critical load is likely to depend on the particular physiology of the individual.

We all know that exercise builds muscle. Or do we? “Surprisingly, not very much is known about why or how exercise builds muscles: there’s a lot of anecdotal knowledge and acquired wisdom, but very little in the way of hard or proven data,” said Professor Eugene Terentjev from Cambridge’s Cavendish Laboratory.

Exercise as we know it

When exercising, the higher the load, the more repetitions, or the greater the frequency, then the greater the increase in muscle size. However, even when looking at the whole muscle, why or how much this happens isn’t known. The answers to both questions get even trickier as the focus goes down to a single muscle or its individual fibers.

Muscles are made up of individual filaments, which are only 2 micrometers long and less than a micrometer across, smaller than the size of the muscle cell. “Because of this, part of the explanation for muscle growth must be at the molecular scale,” said co-author Neil Ibata. “The interactions between the main structural molecules in muscle were only pieced together around 50 years ago. How the smaller, accessory proteins fit into the picture is still not fully clear.”


This is because the data is very difficult to obtain:

People differ greatly in their physiology and behavior, making it almost impossible to conduct a controlled experiment on muscle size changes in a real person. “You can extract muscle cells and look at those individually, but that then ignores other problems like oxygen and glucose levels during exercise,” said Terentjev. “It’s very hard to look at it all together.”

History of the Research

Terentjev and his colleagues started looking at the ability of cells to sense mechanical cues in their environment several years ago. This research was noticed by the English Institute of Sport, who was interested in whether it might relate to their observations in muscle rehabilitation. Together, they found that muscle hyper/atrophy was directly linked to the Cambridge work.

In 2018, the Cambridge researchers started a project on how the proteins in muscle filaments change under force. They found that main muscle constituents, actin and myosin, lack binding sites for signaling molecules. This meant that it had to be the third-most abundant muscle component—titin—that was responsible for signaling the changes in applied force.

Molecule Level

Whenever part of a molecule is under tension for a sufficiently long time, it toggles into a different state, exposing a previously hidden region. If this region can then bind to a small molecule involved in cell signaling, it can activate that molecule, and generate a chemical signal chain. Titin is a giant protein, a large part of which is extended when a muscle is stretched. Also, a small part of the molecule is also under tension during muscle contraction. This part of titin contains the so-called titin kinase domain, which is the one that generates the chemical signal that affects muscle growth.

The molecule will be more likely to open if it is under more force, or when kept under the same force for longer. Both conditions will increase the number of activated signaling molecules. These molecules then induce the synthesis of more messenger RNA, leading to production of new muscle proteins, and the cross-section of the muscle cell increases.

This realization led to the current work, started by Ibata, himself a keen athlete. “I was excited to gain a better understanding of both the why and how of muscle growth,” he said. “So much time and resources could be saved in avoiding low-productivity exercise regimens, and maximizing athletes’ potential with regular higher value sessions, given a specific volume that the athlete is capable of achieving.”

Woman in White and Red Dress Figurine

The Mathematical Model

Terentjev and Ibata set out to create a mathematical model that could give quantitative predictions on muscle growth. They started with a simple model that kept track of titin molecules opening under force and starting the signaling cascade. They used microscopy data to determine the force-dependent probability that a titin kinase unit would open or close under force and activate a signaling molecule.

After that, they made the model more complex by including additional information, such as metabolic energy exchange, repetition length, and recovery. The model was validated using past long-term studies on muscle hypertrophy.

“Our model offers a physiological basis for the idea that muscle growth mainly occurs at 70% of the maximum load. Which is the idea behind resistance training,” said Terentjev. “Below that, the opening rate of titin kinase drops steeply and prevents mechanosensitive signaling from taking place. Above that, rapid exhaustion prevents a good outcome, which our model has quantitatively predicted.”

Eventually, the researchers hope to produce a user-friendly software-based application that could give individualized exercise regimens for specific goals. The researchers also hope to improve their model by extending their analysis with detailed data for both men and women. This is because many exercise studies are heavily biased towards male athletes.