John Hopfield pointed out, in his seminal paper on kinetic proofreading, that if a biochemical system operates at thermodynamic equilibrium there is a barrier to how well it can achieve high-fidelity in transcription and translation. Hopfield showed that the only way to bypass this barrier is to dissipate energy and maintain the system away from equilibrium. Eukaryotic gene regulation uses dissipative mechanisms, such as nucleosome remodelling, DNA methylation and post-translational modification of histones, which are known to play a critical regulatory role but have been largely ignored in quantitative treatments. I will describe joint work with my colleague Angela DePace in which we use the recently-developed, graph-theoretic linear framework to show that the sharpness with which a gene is turned on or off in response to an upstream transcription factor is limited if the regulatory system operates at equilibrium, even with arbitrary degrees of higher-order cooperativity. In contrast, if the regulatory system is maintained away from equilibrium, substantially higher degrees of sharpness can be achieved. We suggest that achieving sharpness in gene regulation exhibits a Hopfield Barrier, and uncover, along the way, a new interpretation for the ubiquitously used, but poorly justified, Hill function.