Inference of the 3D pressure field exerted by a single cell from a thin membrane transverse deformation

TL;DR

A solution to the inverse problem of Protrusion Force Microscopy is proposed and its regime of applicability in the experimental context is explored.

Abstract

Numerous cell types relate to their immediate environment by exerting a three-dimensional pressure field on their environment, with components both longitudinal and transverse to the cell membrane. This pressure field can in principle be measured by traction force microscopy experiments. Compared to other approaches, the technique of Protrusion Force Microscopy gives access with high spatial resolution to the pressure field by measuring the deformation of a thin elastic membrane using atomic force microscopy (AFM). However, while the pressure field under interest is three-dimensional, the height profile measured by AFM is only one-dimensional. We propose a solution to this inverse problem and we explore its regime of applicability in the experimental context.

Figures (8)

• Figure 1: Top: Cartoon of the Protrusion Force Microscopy experimental setup (not at scale). The cell (in blue, its interface with the stimulatory substrate in dark blue) adheres on the lower face of the elastic membrane (in light gray), whereas the AFM measurement are performed on its upper face (AFM canterliver in dark gray). The Region of Interest (ROI) scanned by the AFM appears in white. Bottom: Point loading of the circular elastic membrane of radius $a$ at its center $O$. $\mathbf{F}_z$ and $\mathbf{F}_\parallel$ are respectively transverse and longitudinal point loadings.
• Figure 2: The height function $h$ (graph in green) results from the linear combination of the 2D longitudinal component $\mathbf{u}_\parallel$ and the transverse one $u_z$ (both in blue) of the total deformation field $\mathbf{u}$ at any point $\mathbf{r}$ of the membrane. This translates into Equation \ref{['ZE:eq']}.
• Figure 3: Ideal T-cell synapse. Pressure field $\mathbf{P}$ exerted by the cell at the synapse. Top-left: applied transverse pressure field $P_z(r)$ with $F_{z,{\rm tot}} =10$ nN. Top-right: reconstructed transverse pressure field. Bottom-left: applied longitudinal pressure field $P_\parallel(r) \mathbf{e}_r$ with $F_{\parallel,{\rm tot}} =10$ nN. Bottom-right: reconstructed longitudinal pressure field. In the bottom figures, the direction of the force is indicated by the arrow and its intensity by the color map. The square ROI side length is $L=15$$\mu$m. It is discretized in $N=63 \times 63$ pixels. The T-cell synapse radius is $R=5$$\mu$m. The coordinates on the axes are given in $\mu$m. The color maps give the local pressure in Pa.
• Figure 4: Left: Example of correlated AFM noise map. It is added to the height field $h$ before solving the inverse problem. Right: longitudinal pressure field reconstructed from the new height field. The applied pressure field is the same as in Figure \ref{['ideal:synapse']}. All distances are in $\mu$m.
• Figure 5: More realistic T-cell synapse with an irregular boundary. Left: applied longitudinal pressure field $P_\parallel(r) \mathbf{e}_r$. Right: reconstructed pressure field. Same parameter values as in Figure \ref{['ideal:synapse']}.