Molecular systems within biological cells are frequently organised to respond to input signals that influence system output. These systems may be, for example, receptors on the cell surface that respond to external chemicals or molecular complexes on DNA that determine how a gene is expressed in response to regulatory inputs. An important property of these input-output responses is how sharply they can change their level of output when there is a small change in the input. We can imagine an input-output response plotted as a curve, as in the accompanying infographic. Sharpness is often characterised by comparison to a family of mathematical curves that were introduced in 1910 by the biophysicist Archibald Vivian Hill and are now named after him. Hill functions depend on a single parameter, called the Hill coefficient, which is used as a measure of sharpness. However, Hill functions were merely a convenient mathematical choice; although they have been widely used in biology, they have lacked any theoretical justification. Our paper provides that long-missing justification. This turns out, surprisingly, to depend on thermodynamics, specifically on whether or not the underlying system is using energy to convert its input to its output. It is possible to create sharp responses without energy and molecular systems sometimes use it and sometimes do not. In more detail, we quantify the sharpness of an input-output response with two measures - the maximum slope of the input-output curve, which we call "steepness", and the smallest input value for which the curve has that steepness, which we call "position". When we plot position and steepness for different underlying systems, assuming they are not using energy to generate their outputs, we find that these values occupy a two-dimensional region with a distinctive shape, shown in the infographic. It has tapering "wings" and a central "cusp", which falls exactly on the "Hill line", or the locus generated by the position-steepness values of the Hill functions as the Hill coefficient changes. In fact, the cusp falls just below the Hill coefficient given by the number of molecular sites in the system with which the input signal interacts. This position-steepness region is universal, as long as the systems being considered obey widely-accepted core assumptions, of which the most important is being "Markovian", or not retaining a memory. No matter what system of this kind is chosen, if it is not using energy and has N input sites, then it is impossible for its position and steepness to both be greater than that of the Hill function with coefficient N. However, if we consider any system with N input sites that does use energy, then we can bypass this barrier and find position and steepness values that both exceed that of the Hill function with coefficient N. The potential existence of thermodynamic barriers like this was first suggested in 1974 by another biophysicist, John Hopfield, and we have called them "Hopfield barriers" in his honour. Our results show that the Hill function with coefficient N is the universal Hopfield barrier for input-output systems with N input sites.

## Why is it important?

Our results show that Hill's choice of function, which was made for convenience - they have a simple algebraic form and provide excellent fits to experimental data - has a far deeper justification as a fundamental thermodynamic limit. Furthermore, the existence of this universal limit provides a new way to test whether systems are expending energy: they must be doing so if they bypass the Hopfield barrier. If we believe the underlying system obeys the core assumptions, then we can draw this strong conclusion no matter how complicated the underlying molecular system is - we do not even have to know all its details - and without having to fit experimental data to a model. Most theoretical findings in modern biology are based on specific models; it is striking to have a result that holds universally for a broad and widely-used class of models. Our results further suggest that other forms of cellular information processing may exhibit similar universal Hopfield barriers and encourage us to systematically identify and characterise them. This new direction of research may reveal some of the underlying physical principles that govern the inner workings of biological cells.