Cell migration is key to many biological processes including embryonic development, wound healing, and disease progression, and the mechanical stiffness of a cell's environment exerts a strong, but variable, influence on this migration. Many cells display a stiffness optimum at which migration is maximal, however, these stiffness optima span several orders of magnitude, from ~1-1000 kPa, suggesting that different cell types possess distinct operating parameters. Firstly, we describe how a motor-clutch model of cell traction, which exhibits a maximum in traction force with respect to substrate stiffness, may provide a mechanistic basis for understanding how cells are "tuned" to sense the stiffness of specific microenvironments. We found that the optimal stiffness is generally more sensitive to clutch parameters than to motor parameters, but that single parameter changes are generally only effective over a small range of values. By contrast, dual parameter changes, such as coordinately increasing the numbers of both motors and clutches, offer a larger dynamic range for tuning the optimum. The model exhibits distinct regimes with "frictional slippage" at both low and high substrate stiffness where clutches are inefficiently utilized. Between the two extremes, we find the maximum traction force where clutches are most efficiently utilized, which occurs when the substrate load-and-fail cycle time equals the expected time for all clutches to bind. Secondly, we also present a master equation-based ordinary differential equation (ODE) description of the motor-clutch model, from which we derive an analytical expression for a cell's optimum stiffness. This analytical expression provides insight into the requirements for stiffness sensing by establishing fundamental relationships between the key controlling cell-specific parameters. Both the ODE solution and the analytical expression show good agreement with Monte Carlo motor-clutch output, and reduce computation time by several orders of magnitude, which potentially enables long time scale behaviors (hours-days) to be studied computationally in an efficient manner. The ODE solution and the analytical expression may be incorporated into larger scale models of cellular behavior to bridge the gap from molecular time scales to cellular and tissue time scales. Thirdly, to create a unified theoretical framework for cell migration, we have developed and experimentally tested a whole cell migration simulator based on the motor-clutch model by imposing coupled force balances and mass balances on molecular motors, adhesion molecules ("clutches"), and actin subunits in a compliant microenvironment. The model predicts a stiffness optimum that can be shifted by altering the number of active molecular motors and clutches. This prediction was verified experimentally by comparing cell traction and F-actin retrograde flow for two cell types with differing amounts of active motors and clutches: embryonic chick forebrain neurons (ECFNs; optimum ~1 kPa) and U251 glioma cells (optimum ~1000 kPa). In addition, the model predicted, and experiments confirmed, that the stiffness optimum of U251 glioma cell migration, projected area, aspect ratio, F-actin flow rate, and traction strain energy can be shifted to lower stiffness by simultaneous drug inhibition of myosin II motors and integrin-mediated adhesions. Overall, the motor-clutch cell migration simulator provides a unified theoretical framework with which to predict cell adhesion and migration in defined mechanochemical environments.