Spatial landscape and flux for exploring protein pattern formation in rod-shaped bacteria

Spatial patterns formed by biomacromolecules such as proteins are widely present in biological systems and are closely related to fundamental cellular processes. A classic example is the spatial patterning of Min proteins in bacteria, where pole-to-pole oscillations of these patterns guide symmetric cell division. To uncover the underlying mechanisms behind the formation and transition of spatial patterns in the Min protein system, we applied nonequilibrium landscape-flux theory combined with the mode expansion method. By quantifying and visualizing the potential landscape in mode space, we identified distinct stable spatial patterns as potential wells, providing a global perspective on the system's stability. Moreover, we revealed that nonequilibrium flux acts as the driving force for spatial pattern switching with increasing cell length or molecular detachment rates. Peaks in the average flux and entropy production rate near phase boundaries highlight significant changes in dynamical nature and thermodynamic cost during critical transitions, offering deeper insights into the physical mechanisms underlying spatial pattern transitions. These findings not only underscore how spatial landscape topography and flux dynamics collectively govern the formation, stability, and switching of protein patterns but also establish a powerful framework for linking nonequilibrium physical mechanisms to biological functions. Furthermore, this framework holds potential applications, such as the detection of early warning signals for cell division.