Theory of large-bead AFM compression of pressurized rod-shaped bacteria

Motivated by our atomic force microscopy (AFM) experiments measuring turgor pressure in living bacteria, we develop a theoretical description of pressurized spherocylindrical cells compressed by spherical probes whose radius greatly exceeds the cell radius. In this global-loading regime, the mechanical response is governed primarily by cell geometry and turgor pressure, while the elastic properties of the cell envelope enter as higher-order corrections. The strong confinement imposed by the probe drives water efflux from the cytoplasm, thereby increasing the turgor pressure. We present a theoretical framework for determining bacterial turgor pressure from AFM force--displacement curves, accounting for cell geometry and probe dimensions. Combining numerical simulations with analytical theory, we show that the mechanical response can be approximated in terms of cylindrical and hemispherical contributions. The leading-order force--displacement dependency is captured by an analytical description derived in the large-aspect-ratio limit, where the mechanics reduce to a tension-dominated cross-sectional problem. Finite cell aspect ratio and probe curvature are incorporated through geometric renormalizations. The resulting framework provides a quantitative basis for extracting bacterial turgor pressure from AFM measurements while simultaneously predicting cell deformation geometry.