The motile behavior of the bacterium Escherichia coli depends on the direction of rotation of its flagellar motors. Binding of the phosphorylated signaling molecule CheY to a motor component FliM is known to enhance clockwise rotation. It is difficult to study this interaction in vivo, because the dynamics of phosphorylation of CheY by its kinase CheA and the hydrolysis of CheY (accelerated by CheZ) are not under direct experimental control. Here, we examine instead the interaction with the flagellar motor of a double mutant CheY13DK106YW that is active without phosphorylation. The behavioral assays were carried out on tethered cells lacking CheA and CheZ. The effects of variation in intracellular concentration of the mutant protein were highly nonlinear. However, they can be explained by a thermal isomerization model in which the free energies of clockwise and counterclockwise states depend linearly on the amount of CheY bound.