Modelling the dynamics of cellular motililty, from adhesion dynamics to cellular migration

Cellular migration is a complex phenomenon where cells displace themselves by integrating many intracellular processes that autonomously respond to motility cues in their environments.As it usually allows for the cells to carry out their biological functions in a specific location, it is considered essential to a number of physiological processes (e.g., embryonic development, wound healing, and immune responses).Defects in cellular migration are implicated in pathological conditions (e.g., immune disorders and cancer metastasis).Understanding the interdependencies of the various cellular processes and how they integrate external cues to produce motility is a challenging task, due to both the intrinsic complexity of the processes as well as the large variability in migration modes.Here we have focused on mathematical modeling of processes related to integrin-based adhesions, which are macromolecular complexes that serve as force-transmission points and signaling hubs during mesenchymal migration.We began by reviewing the literature of mathematical models of integrin-ligand bonds, and largerscale models of adhesions that incorporate the molecular properties of integrins.We then developed a novel model of nascent adhesion formation that accounted for experimentally observed variations in integrin density, and predicted the mechanical conditions required for self-assembly of adhesions.Subsequently, we focused on cellular-scale models of motility and studied how the spatio-temporal dynamics of motility are influenced by two different adhesion-associated proteins, paxillin and VASP.First we studied how GTPase signaling can be used by a cell to produce a pattern formation system that defines its front and rear directions, and how adhesion-dependent signaling through paxillin influences this process.Pattern formation properties were studied using theoretical analysis and spatiotemporal simulations.The latter simulations were then coupled to a phenomenological model of cell motility, demonstrating that paxillin phosphorylation can directly increase motility of cells.Finally, in order to gain insight into a detailed spatio-temporal model of VASP-dependent actin-based protrusion dynamics, we derived a simplified model and characterized its full set of dynamics.Using this characterization, we then combined spatio-temporal simulations and a theoretical analysis of population-level statistics to identify how the model can be used to explain the poorly understood rough motility phenotype.i Statement of ContributionsThis thesis is comprised of four main chapters, each of which is based on an article that I co-authored.Chapters 1, 2, and 4 have been published in peer reviewed journals: [MK20], [MK19b], and [MLK20], respectively.Chapter 3 is currently in preparation to be submitted, and will likely constitute at least 2 publications.Chapter 1 is a literature review of mathematical models of integrin-based adhesions, and as such does not contain significant contributions to original knowledge.Below we have briefly summarized the contributions to original knowledge by chapter. Chapter 2• We developed a novel data analysis methodology, termed Conditional Expectation Analysis, that allows us to infer a constitutive relationship between two physical variables from histograms of their measurements.Such a relationship was then used to derive a model of nascent adhesion formation dynamics.• This combination of modelling and data analysis contributed to the understanding of the molecular level organization of nascent adhesions, their macromolecular assembly dynamics in response to applied force, and provides a more physiologically-relevant description of their disassembly than previous models in the field. Chapter 3• We presented a deterministic macroscopic framework for understanding the spatio-temporal dynamics of chemical reactions, and use results from geometric singular perturbation analysis to show how one can systematically impose the conservation of matter when employing quasisteady state assumptions.-This framework should be more approachable for non-mathematicians compared to what can be found in the literature, as it is based on the familiar concept of chemical reactions (rather than abstract mathematical formulation, formal definitions, proofs, and lemmas), it should be much more approachable for non-mathematicians.iii• We extended the non-linear stability analysis termed local perturbation analysis (LPA) to better reconcile its results with observed pattern formation phenomenon.• Using LPA, we demonstrated that a pattern-formation model published in [TBBK18] and fit to data from Chinese Hamster Ovary (CHO-K1) cell data lies inside a Turing unstable regime.-This allows simulated CHO-K1 cells to self-polarize (a finding that is consistent with experimental observations).-This regime is poorly studied in the field, but our findings suggest it could be highly relevant to the migration of some cell-types.Chapter 4• Using a model of cellular protrusion from [BAL `17], we were able to propose the dynamical origins underlying a poorly understood keratocyte motility phenotype, known as rough motility.• We also demonstrated that the previously described waving regime is produced by canard explosion with orbits passing through a folded-saddle.• We incorporated some mechanosensitive properties of nascent adhesions (discussed in Chapter 1) and identified novel parameter regimes that appear to be relevant for fibroblast-like cells.I was responsible for writing all of the text in Chapters 1-4, and obtained significant feedback from my supervisor, Anmar Khadra.I was solely responsible for the development of the code in Chapters 2 and 4. Furthermore, the code in Chapter 3 can be found at https://github.com/laurentmackay/ CPM_RD_matlab, and is the cumulative effort of Zhang Tan, Ashesh Momi, Ellie Zhang with myself contributing through guidance and extensive re-writes to speed up the computations.Etienne Lehman contributed significantly to the initial model analysis in Chapter 4. Chapter 1Literature Review: The bioenergetics of integrinbased adhesion, from single molecule dynamics to stability of macromolecular complexes.

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