Optimal control of collective electrotaxis in epithelial monolayers

Abstract

Epithelial monolayers are some of the best-studied models for collective cell migration due to their abundance in multicellular systems and their tractability. Experimentally, the collective migration of epithelial monolayers can be robustly steered e.g. using electric fields, via a process termed electrotaxis. Theoretically, however, the question of how to design an electric field to achieve a desired spatiotemporal movement pattern is underexplored. In this work, we construct and calibrate an ordinary differential equation model to predict the average velocity of the centre of mass of a cellular monolayer in response to stimulation with an electric field. We use this model, in conjunction with optimal control theory, to derive physically realistic optimal electric field designs to achieve a variety of aims, including maximising the total distance travelled by the monolayer, maximising the monolayer velocity, and keeping the monolayer velocity constant during stimulation. Together, this work is the first to present a unified framework for optimal control of collective monolayer electrotaxis and provides a blueprint to optimally steer collective migration using other external cues.

Figures (8)

• Figure 1: Experimental data of MDCK monolayer electrotaxis with its electric field stimulation protocol. Left panel: phase field image of an MDCK epithelial monolayer indicating direction of the uniaxial electric field and bulk region where the mean velocity is measured. Right panel, top: electric field density during the experiment. Right panel, bottom: bulk velocity corresponding to the electric field trace in the top row.
• Figure 2: Experimental data of bulk velocity decay post-stimulation with an electric field pulse of 3 V/cm (black dots) together with least-squares fit of exponential decay in the form $C e^{-\gamma t}$ (solid blue line). The least-squares solution gives an excellent fit to the data and can be used to identify the decay rate, $\gamma$.
• Figure 3: Bayesian inference on a simple model of adaptation and excitation given by Equation \ref{['chapter3_eq:linearSystem']} using experimental data from an electrotaxis experiment. Left: bulk velocity using a constant field of 3V/cm. Posterior mean of the model together with a confidence interval between the 5% and 95% quantiles of the model posterior predictions in cyan, experimental data and their confidence interval in grey. Notice the small width of the posterior predictive intervals. Right panels, from top to bottom: posterior distributions for model parameters, $\alpha$, $\tau_e, \tau_a$, respectively.
• Figure 4: Stimulation patterns for optimal control of collective electrotaxis. In all plots, red corresponds to maximum tissue displacement, pink corresponds to maximum terminal velocity. Green curves represent 3V/cm electric field pulse stimulation pattern used experimentally. Left: the different stimulation patterns in normalized units. Middle: corresponding velocity curves using the different stimulation patterns. Right: data from 3V/cm pulse experiment (dashed line, green), compared to posterior distribution for optimal solution for distrance travelled (red) and terminal velocity (pink).
• Figure 5: Solution to optimal control problem for constant bulk speed during stimulation with an electric field. Left: bulk speed for constant velocity (solid line) plotted alongside experimental data with constant electric field stimulation protocol (dashed line). Middle: numerical solution of the optimal control problem for constant velocity (solid line) plotted against reference normalized electric field strength (dashed line). Right: effective signal, $s_{\text{eff}}$.