Physical biology offers powerful tools for quantitatively dissecting the various aspects of cellular life that one cannot attribute to inanimate matter. Signature examples of living matter include adaptation, self-organization, and division. In this thesis, we explore different interconnected facets of these processes using statistical mechanics, nonequilibrium thermodynamics, and biophysical modeling. One of the key mechanisms underlying physiological and evolutionary adaptation is allosteric

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... egulation. It allows cells to dynamically respond to changes in the state of the environment often expressed through altered levels of different environmental cues. The first thread of our work is dedicated to exploring the combinatorial diversity of responses available to allosteric proteins that are subject to multi-ligand regulation. We demonstrate that proteins characterized through the Monod-Wyman-Changeux model of allostery and operating at thermodynamic equilibrium are capable of eliciting a wide range of response behaviors which include the kinds known from the field of digital circuits (e.g., NAND logic response), as well as more sophisticated computations such as ratiometric sensing. Despite the fact that biomolecules at thermodynamic equilibrium are able to orchestrate a variety of fascinating behaviors, the cell is ultimately 'alive' because it constantly metabolizes nutrients and generates energy to drive functions that cannot be sustained in the absence of energy consumption. One prominent example of such a function is nonequilibrium error correction present in high-fidelity processes such as protein synthesis, DNA replication, or pathogen recognition. We begin the second thread of our work by providing a conceptual understanding of the prevailing mechanism used in explaining this high-fidelity behavior, namely that of kinetic proofreading. Specifically, we develop an allostery-based mechanochemical model of a kinetic proofreader where chemical driving is replaced with a mechanical engine with tunable knob [...]