Robustness and optimal control in Turing patterns

The regulation of self-organized systems has long been of interest to the biophysical community, especially in the context of developmental biology. In this talk, we demonstrate a novel approach to investigating biological pattern formation via techniques from optimal control theory. We consider the behavior of a toy model consisting of a simple substrate-inhibitor system in a supercritical Turing regime, subject to feedforward modulation by upstream morphogen gradients. In this context, a biologically motivated question is posed: what control is needed to drive the system to a desired steady state of chosen wavelength? An optimal control formulation is proposed, for which optimality conditions are derived and solved numerically, yielding control solutions that rapidly achieve pattern selection. Further, our approach allows us to investigate the dependence of control strategies on natural biological constraints by manipulating cost function terms associated with control cost or parametric sensitivity. These results enable us to draw conclusions about the intrinsic trade-offs and synergies between robustness and controllability in Turing patterns and related systems.