Many eukaryotic cell functions depend on dynamic instability, meaning the nucleotide-driven assembly and disassembly of microtubules. Assembly requires the constituent tubulin dimers to bind the nucleotide GTP, and its subsequent hydrolysis to GDP induces disassembly. The underlying structural mechanisms, however, are not well understood. Here, we determine the strength of contacts in the microtubule lattice by combining high precision measurements of the bending stiffness of analogues of GTP and GDP microtubules with a recent theoretical model. While previous structural studies have focussed on how the curvature of the tubulin dimer is affected by nucleotide binding, we present evidence of a dramatic regulation of the lateral interactions between the parallel protofilaments that dimers form in the microtubule. We conclude that the shear coupling between neighboring protofilaments is at least two orders of magnitude stronger in the GTP state than in the GDP state, and discuss the implications for the microtubule assembly.