To dynamically reshape the membrane, cells rely on a variety of intracellular mechanisms, ranging from forces exerted by the cytoskeleton to the spontaneous curvature induced by membrane–protein interactions. In this thesis, we present mathematical models in a continuum framework to understand the physics underlying membrane deformation by two different modes of curvature-generating mechanisms (1) protein-induced spontaneous curvature and (2) forces due to membrane-cytoskeleton interactions. In the first part of the thesis, we model the effects of curvature-generating proteins by extending the classical Helfrich-Canham bending energy and demonstrate how the local shape of the membrane can be used to infer the traction acting locally on the membrane. Particularly, we first propose a technique to extract effective line tension at the protein interface using the morphology and the composition of the membrane. We then analyze the beading morphology of membrane nanotubes due to heterogeneity in the membrane properties and protein distribution. We find that there exists a discontinuity in the energy that impedes two beads from fusing. Finally, we show the application of our continuum framework for studying curvature generation due to protein phase separation on membranes. In the second part of the thesis, we model the forces due to membrane-cytoskeleton interactions by adding an extra degree of freedom to the energy equation to account for heterogeneous forces representing the effects of actin polymerization and activity of molecular motors such as myosin on the plasma membrane. Using this framework, we show that a non-uniform force distribution coupled with membrane tension characterized the biconcave shape of Red Blood Cells (RBCs). We also explore the application of our mathematical framework to identify the possible force regimes that give rise to the classic shapes of dendritic spines which are bulbous protrusions along the dendrites of neurons and are sites of excitatory postsynaptic activity. We identify different mechanical pathways that are likely associated with different dendritic spine shapes, and find that some mechanisms may be energetically more favorable than others. We believe our models identify mechanisms of cell shape adaption by two modes of curvature generation, enabling future work to establish the contribution of cell membrane mechanics in many human diseases and designing better systems for drug and gene delivery.

The ability to adapt is a hallmark of the brain. Unlike most cells in the body, most neurons are not replaced as one ages.Therefore, neurons are highly plastic, modifying themselves biochemically and structurally across multiple lengthscales in a process called synaptic plasticity. The backbone of these modifications lies in a neuron's ability to modulate the strength of its synapses, the connections between neurons. Most excitatory synapses are housed in small subcompartments, ~ 0.1 femtoliter in volume, called dendritic spines. Dendritic spines have characteristic shapes and sizes that correlate to disease, learning, and aging. These spine shapes and sizes change during synaptic plasticity and more specifically structural plasticity. When a dendritic spine receives a signal, it triggers complex biochemical signaling networks that can modulate the synaptic strength of the dendritic spine, or its synapse, by modulating the number of receptors on its membrane, amongst other modifications. Therefore, there is a coupling and feedback between the biochemical signaling underlying synaptic plasticity and the morphology of dendritic spines.

In this work, we utilize computational and mathematical models to investigate this structure-function relationship in dendritic spines, as well as in larger lengthscale components such as dendrites and neurons. We start with an example of extreme structural plasticity, where a biochemical signal can rupture connections between cells. We show that biochemical signaling can trigger mechanical forces to modulate cellular morphology and neural connections. Next, we investigate how neuronal morphology influences biochemical signaling more closely in the context of synaptic plasticity in dendritic spines. We consider different timescales of signaling starting with voltage propagation during spine activation, and calcium influx into dendritic spines of different morphologies. We find that spine morphology and ultrastructure can impact signaling dynamics even at these early timescales. Finally, we use both a compartmental model and a spatial model to consider the biochemistry underlying synaptic plasticity, and how signaling and receptor trafficking couple during synaptic plasticity. We find that model architecture filters the influence of upstream signaling on synaptic plasticity readouts and that spine morphology, in particular spine size, can act to regulate synaptic plasticity to prevent excessive receptor change. This work strives to emphasize the coupling between spine morphology and biochemical signaling networks that govern synaptic plasticity, creating a foundation for future work to investigate structural plasticity in more detail.

Biological processes resulting in observable cell state changes, such as synaptic plasticity inneuronal dendritic spines and muscular hypertrophy in skeletal muscles, share common characteristics. Both are influenced by calcium signaling, metabolic networks, mitochondria, and protein kinase activation. This research presents computational models of these processes, offering insights into synaptic plasticity and the skeletal muscle response to exercise. The first model is a spatial representation of mitochondrial ATP generation during calcium signaling in neurons, emphasizing the role of mitochondrial ER contact sites in increasing calcium levels in the mitochondria, thus enhancing mitochondrial potential. This model underscores the importance of subcellular geometry, metabolic consumption rates, and transport processes in ATP production, in addition to enzymatic activity. The second model is a compartmental model exploring the role of signaling frequency in protein kinase activation in dendritic spines. This model analyzes the interaction between the canonical neuronal signaling cascade, the AMP-Activated Protein Kinase (AMPK) cascade, and the insulin signaling cascade in conditions of high neurotransmitter stimulation rates, which activate calcium signaling and consume significant cellular energy. The final model, a logic-based representation of skeletal muscle signaling in response to resistance, endurance, and sprint exercise, investigates potential mechanisms of exercise-induced changes in mitochondrial morphology. Together, these models deepen our understanding of the dynamic interplay between signaling and metabolism in synaptic plasticity and exercise physiology, laying the groundwork for further investigation and potential development of therapeutic strategies for neurological disorders and improvements in exercise efficiency.

Cell and tissue movement are essential to embryonic development, cancer metastasis, wound healing, cargo delivery etc. These movements span multiple length scales — collective cell behavior occurs at ~ 10^-2m, membrane trafficking occurs at ~ 10^-8m, and the growth of the actin cytoskeleton occurs at ~ 10^-10m. The forces needed to drive movement begins with actin polymerization and other molecular motors, enabling local deformations that can translate into movement across length scales. Experimental methods for quantification of such forces are often difficult to implement in a high-throughput context and can be disruptive. In this work, we present mathematical and computational models to understand the relationship between cell movements and forces at two different length scales. At the sub-cellular length scale, we use Helfrich-energy theory in an axisymmetric and continuum framework to probe traction stress distributions generated along membrane tubules and buds. After discussing the applicability of this model to predict traction stresses from 2D electron micrograph (EM) images of membrane bud shapes, we then use a 3D Finite Element Model (FEM) to analyze a spontaneous symmetry breaking instability of the membrane neck during the pinching step of membrane trafficking. We draw similarities with classical buckling in many thin elastic structures, and proceed to analyze the effect of helical loading to compare against polymers like Dynamin. We then pair a continuum membrane mechanics model with an agent based model of filament dynamics to show that actin filaments self-organize to promote axial force production towards the base of the endocytic pit. At the tissue length scale, we use a vertex model of colony morphogenesis to validate a data-driven force-inference toolkit applicable to time-series 2D images of cell monolayers. We show that including a regularization term in the opitimization formulation boosts model prediction across time. We also discuss the potential for high-throughput imaging to model pipelines through machine learning algorithms for segmentation, generation, and meshing of cellular structures. Our models identify mechanisms of cell movement at two different length scales, enabling future work to establish the contribution of endocytic pathways in directing cell topologies and tissue morphogenesis.

Ca2+ signaling within myofibers is fundamental for muscle contraction facilitating activities from everyday tasks to high-intensity exercise. Stimulation at the neuromuscular junction induces an action potential that propagates along and throughout the myofiber, leading to release of Ca2+ from the sarcoplasmic reticulum (SR). Ca2+ then binds to troponin, enabling muscle contraction. Recent research highlights the role of store-operated calcium entry (SOCE) in maintaining intracellular Ca2+ stores during muscle activation. Upon SR Ca2+ depletion, stromal interaction molecule 1 in the SR membrane forms assemblies with ORAI1 channels in the sarcolemma, facilitating Ca2+ influx into myoplasm that can then be transported back into the SR. In this thesis, we develop a multi-compartment model to capture the role of SOCE in myoplasmic Ca2+ dynamics and force production during exercise. This model builds on previous models in the literature, integrating features of Ca2+ signaling and APs by including a novel combination of relevant ion channels and pumps, as well as Ca2+ buffers, and SOCE. Model parameters were calibrated against experimental data to ensure physiological relevance and model robustness. The model was simulated at various frequencies and durations of stimuli relevant to endurance vs resistance exercises. The model predicted reductions in both the maximum and average myoplasmic Ca2+ in the absence of SOCE flux, implying a diminished ability to sustain high force. The strength of this effect varied according to the frequency and exercise type. This study offers insights into muscle performance, endurance, and recovery, relevant to athletic training, physical therapy, and medical research.