To function properly protein molecules require both flexibility and rigidity, therefore fast and accurate prediction of protein rigidity/flexibility is one of the important problems in protein science. In this work we used two theoretical approaches to determine flexible regions in four homologous pairs of proteins from thermophilic and mesophilic organisms. Protein pairs chosen in this study were selected to represent four typical folding classes. Our first approach, FoldUnfold, uses amino acid sequence and statistical information on the density of contacts of amino acids in tertiary structures of known globular proteins. The main advantages of such knowledge-based methodology are its computational speed and ability to make predictions in the absence of three-dimensional (3D) structure of a protein. The second approach uses a graph theory-based rigid cluster decomposition termed FIRST, applied together with Molecular Dynamics (MD) simulations of proteins with known structure. While MD simulations are time-consuming, they are the most direct way of studying physical properties of proteins, including their rigidity/flexibility. Flexible regions predicted by both methods in this work were in good agreement with each other. We also showed that high mobility of a site is not necessarily indicative of its high flexibility and vice versa. In our simulations thermophile proteins were less flexible than their mesophilic homologues. Longer flexible loops were found in mesophilic proteins of all classes.