Modelling of mechanosensitive morphogenesis and maintenance of cell structure: Application to musculoskeletal cells and tissues

Abstract

My dissertation presents some of the first key steps towards an analytical understanding of mechanosensitive morphogenesis and structure maintenance in biological systems. In order to make this problem tractable and to make a significant advance during the PhD, I focussed specifically on musculoskeletal applications of mechanosensing. My approach to this problem was to break it into two pieces: nucleation of structure, and how external and internal loads maintain and adapt existing structures. From a physical perspective, the first of these involves symmetry-breaking and can be studied with the tools of phase transitions, whereas the second involves out-of-equilibrium dynamics and requires a numerical modelling approach of time-dependent non-linear differential equations. Nucleation of clusters of adhesive molecules. In order to investigate the nucleation of structure under load, I considered the initial nucleation of the adhesive molecules integrin and cadherin. These are of paramount importance in holding animal tissues together, because they allow cells to bind either to the extracellular matrix or to other cells. Both of these molecules form similar chains of molecules that strengthen under load and bind to backbone of the cell, the cytoskeleton. By doing so, they help to develop many of the structures in the body: integrin-rich fibroblasts lay out collagen fibres in tendon and bone, whilst cadherins hold epithelial soft tissues as well as cardiac muscle together. The nucleation of clusters of adhesive molecules is the first step towards specialising the cells that contain them in order to develop and strengthen many of the structures in the body. It has recently been shown that they can aggregate if they can display some binding sites when extended under load. The density of integrins can change if a cell is placed on a stiff surface, allowing it to spread, while cadherin density fluctuates significantly during cytokinesis. I considered how changes in the density of stretched integrins or cadherins could pattern an initially random distribution of adhesive molecules. I used statistical field theory methods to find the Ginzburg-Landau free energy of a distribution of adhesive molecules modelled as a lattice gas, and demonstrated that the distributions could undergo two separate phase transitions at low or high density. By investigating the growth of instabilities at the phase transition, I was able to predict the density of clusters as well as the number of molecules per cluster, and found a good match with experimental results. Because clusters of adhesive molecules are essential for cell structure and function, this work was a first quantitative explanation for the initial development of mechanically-induced patterns at the sub-cellular level. Titin kinase controls muscle growth under load. Buoyed by this initial success, I shifted my focus during the Covid-19 lockdown to focus in a second stage on the maintenance and adaptation of cell structure to load. Specifically, I was taken aback by the lack of modelling for muscle hypertrophy or atrophy, despite the clear medical and commercial interest in the area. I found it striking that the vast majority of the literature to date focussed on the metabolic contributions to muscle size, even though muscle constituents can locally signal that they are under load. Using the existing literature, as well as structural arguments (e.g. both concentric and eccentric muscle contractions cause hypertrophy), I explained how the part of the titin molecule known as the titin kinase domain (TK) was perfectly placed for muscle mechanosensing. By bringing together the force-length kinetics of TK, the reaction kinetics of the TK signalling pathway, and energy metabolism interactions, I successfully modelled how lengthening TK under load could lead to increased signalling and synthesis of new muscle proteins. I found that mechanosensitive signalling would sharply increase around 70% of the maximum one-repetition force, and that adaptations to non-adaptive resistance exercise plateau after a few months - both of which are observed in the literature. My pioneering work strongly suggests that mechanosensitive signalling controls muscle size, and a medical tool could be developed from this work. This will require training data to be gathered for individuals with different physiological parameters. These ideas can also be applied to the modelling of population dynamics in evolutionary biology which take into account predator-prey interactions that are affected by muscle mechanics, and I look forward to working on this in the future. Adhesive strength, adaptation, and breaking of myotendinous junctions. Finally, in a last section, I brought both questions of mechanically-induced structure formation and structure maintenance in biology together. Integrin-mediated adhesion and muscle adaptation to load intersect at the interface between muscle and tendon, the myotendinous junction (MTJ). This region showcases many of the strategies that vertebrates deploy to adapt to load during and after development. Collagen fibres in tendons are very strong but, although they can widen up to an extent, their number does not change after birth. On the other hand, muscle fibres expand very significantly and can fluctuate in size during life depending on exercise and diet. This means that when the force in the muscle increases in tandem with the muscle cross-sectional area, the force per unit area through the tendon and MTJ drastically increases. Although the integrin bonds that hold MTJs together strengthen under load, they do fail past a critical value. Only so many integrins can be packed into an area of membrane, so there is a maximum adhesive force per unit area for a muscle cell. Therefore, the only strategy for the organism to avoid injury is to increase the area of the MTJ. Macroscopic changes to the contact area between muscle and tendon have been reported after exercise, but MTJs are fundamentally constrained by the size of the muscle: e.g., the length and width of the contact area cannot exceed the length and width of the muscle. I used measurements of MTJ area to compare the adhesive force estimated macroscopically from real MTJs with the highest forces experienced in the hamstrings during sprinting. These comparisons showed that many muscles must have a much higher surface area than can be estimated macroscopically. MTJs have solved this surface contact area limitation by developing micrometer scale interdigitations of muscle and collagen at the MTJ. I modelled the position of the muscle-tendon interface in elements of muscle that are connected to collagen fibres and elements that are not, and found that the difference in position can account for the size of the protrusions. These can then lengthen progressively under load as the muscle force and tendon stiffness change during development. This mechanism can explain how these finger-like protrusions are seen to lengthen in exercised rats (and likely in humans too). Fascinatingly, this model predicts limitations of the MTJ area that explain why large animals could be limited in their locomotion (e.g. elephants cannot jump) because of injury potential at the MTJ. I am excited to publish this work, because it shows that integrin and molecules involved in its force chain are a clear target for the prevention of the most common muscle-tendon injuries. Summary. My academic style has been to concentrate on writing comprehensive articles on the subjects I have studied in the hope that these will become seminal references in the field, while also writing some shorter papers to present timely side results. I expect my thesis work to motivate better and more numerous experimental studies of adhesive molecular structures, muscle hypertrophy and atrophy, and injuries related to myotendinous junction structure. I hope that it will invigorate a discussion of mechanosensitive structure formation and structure maintenance in medicine and biology, and allow me to capitalise on timely applications of my pioneering models to other areas of biology.