We probe the hypothesis of corresponding states, according to which homologues from mesophilic and thermophilic organisms are in corresponding states of similar rigidity and flexibility at their respective optimal temperatures. For this, the local distribution of flexible and rigid regions in 19 pairs of homologous proteins from meso- and thermophilic organisms is analyzed and related to activity characteristics of the enzymes by constraint network analysis (CNA). Two pairs of enzymes are considered in more detail: 3-isopropylmalate dehydrogenase and thermolysin-like protease. By comparing microscopic stability features of homologues with the help of stability maps, introduced for the first time, we show that adaptive mutations in enzymes from thermophilic organisms maintain the balance between overall rigidity, important for thermostability, and local flexibility, important for activity, at the appropriate working temperature. Thermophilic adaptation in general leads to an increase of structural rigidity but conserves the distribution of functionally important flexible regions between homologues. This finding provides direct evidence for the hypothesis of corresponding states. CNA thereby implicitly captures and unifies many different mechanisms that contribute to increased thermostability and to activity at high temperatures. This allows to qualitatively relate changes in the flexibility of active site regions, induced either by a temperature change or by the introduction of mutations, to experimentally observed losses of the enzyme function. As for applications, the results demonstrate that exploiting the principle of corresponding states not only allows for successful thermostability optimization but also for guiding experiments in order to improve enzyme activity in protein engineering. Proteins 2011. © 2010 Wiley-Liss, Inc.