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The binding in protein-protein interactions exhibits a kind of biochemical stability in cells. The mathematical notion of fixed points also describes stability. A point is a fixed point if it remains unchanged after a transformation by a function. Many points may not be a fixed point, but they may approach a stable status after multiple steps of transformation. In this paper, we define a point as a protein motif pair consisting of two traditional protein motifs. We propose a function and propose a method to discover stable motif pairs of this function from a large protein interaction, sequence data set. There are many interesting properties for this function (for example, the convergence). Some of them are useful for gaining much efficiency in the discovery of those stable motif pairs; some are useful for explaining why our proposed fixed point theorems are a good way to model the binding of protein interactions. Our results are also compared to biological results to elaborate the effectiveness, of our method.