Using non-rigid-foldable origami patterns to design mechanical metamaterials could 14 potentially offer more versatile behaviors than the rigid-foldable ones, but their applications are 15 limited by the lack of analytical framework for predicting their behavior. Here, we propose a 16 theoretical model to characterize a non-rigid-foldable square-twist origami pattern by its rigid origami 17 counterpart. Based on the experimentally observed deformation mode the square-twist, a virtual 18 crease was added in the central square to turn the non-rigid-foldable pattern to a rigid-foldable one. 19 Two possible deformation paths of the non-rigid-foldable pattern were calculated through kinematic 20 analysis of its rigid origami counterpart, and the associated energy and force were derived 21 analytically. Using the theoretical model, we for the first time discovered that the non-rigid-foldable 22 structure bifurcated to follow a low-energy deformation path, which was validated through 23 experiments. Furthermore, the mechanical properties of the structure could be programmed by the 24 geometrical parameters of the pattern and material stiffness of the creases and facets. This work thus 25 paves the way for development of non-rigid-foldable origami-based metamaterials serving for 26 mechanical, thermal, and other engineering applications.