Optimal Control of Cellular States

Recent experimental advancements have allowed for the precise spatial and temporal control of chemical potentials between proteins in the vicinity of one another through optogenetic techniques [3]. This new technique allows for the investigation of cell signalling. Cells communicate by sending chemical signals to induce changes in chemical potentials which then leads to a change of cellular state. In this thesis we apply non-equilibrium theory [6] to model the response of cells to changes in chemical potential, then, by assuming cells want to minimize wasted energy we derive the optimal protocol for cells to change their cellular state through changes in chemical potential. We provide the theoretical framework to derive this optimal protocol for three separate two state chemical reactions: a discrete open system attached to a bath of proteins, a discrete closed system where the total number of proteins is fixed and a continuous closed system where we consider both the spatial and temporal dependence. Although the theory developed is applicable to these reactions for any transition rates, we assume a specific form which closely resembles cell signalling. The resistance to changes in chemical potential is shown to increase exponentially with chemical potential for an open system, to increase exponentially then decay slowly with chemical potential for a closed system and decreases as 1/r where r is the distance from the change. From this we find the optimal protocol and compare the excess work required to change the cellular state using the optimal and naive (constant velocity) protocols. For an open system the optimal protocol is much better than the naive if the chemical potential is varied across a large distance. For a closed system we find similar behaviour for smaller chemical potentials but the improvement then peaks and decreases slowly for very large distances. The spatial dependence of the continuous system has the added effect of decreasing the improvement and smoothing out the peak. We show that our results are consistent with one another in the limiting cases. From this we conclude that cells which require changes in chemical potential within the peak region to change their cellular state will gain the largest benefit from the optimal protocols derived. The optimal protocol has a simple logarithmic form in time µ(t) = ln(ct + b), with c and b constants, for the open system, for the closed and continuous systems it has a more complex shape. Proof of concept of directly simulating the system for comparison is shown, and issues with simulation are discussed.