Cells interact with both tethered and motile ligands in their extra-cellular environment, which mediates, initiates and regulates a series of cellular functions, such as cell adhesion, migration, morphology, proliferation, apoptosis, bi-directional signal transduction, tissue homeostasis, wound healing among others. A fundamental understanding of the thermodynamics of receptor-mediated cell interaction is necessary not only from the aspect of physiology, but also for bioengineering applications, e.g. drug discovery, tissue engineering and biomaterial fabrication. Our models on free energy calculations of receptor mediated cell-matrix interactions supplement computational endeavors based on continuum mechanics. By incorporating conformational, entropic, solvation, steric effect, implicit and explicit interactions of receptors and extra-cellular ligand molecules, we can predict free energy, chemical equilibrium constant of binding, spatial and conformational distributions of biopolymers, adhesion force as functions of a set of key variables, e.g. surface coverage of receptor, interaction distance between cell and substrate, specific binding energy, implicit interaction strength, constraint in ligand's conformation, size of motile nano-ligand, aggregation of receptors, sliding velocity relative to fluid. Our work has improved understanding of phenomena in cell-matrix interactions at both cellular and the molecular scales.