Natural odors typically consist of many molecules at different concentrations. It is unclear how the numerous odorant molecules and their possible mixtures are discriminated by relatively few olfactory receptors. Using an information theoretic model, we show that a receptor array is optimal for this task if it achieves two possibly conflicting goals: (i) Each receptor should respond to half of all odors and (ii) the response of different receptors should be uncorrelated when averaged over odors presented with natural statistics. We use these design principles to predict statistics of the affinities between receptors and odorant molecules for a broad class of odor statistics. We also show that optimal receptor arrays can be tuned to either resolve concentrations well or distinguish mixtures reliably. Finally, we use our results to predict properties of experimentally measured receptor arrays. Our work can thus be used to better understand natural olfaction, and it also suggests ways to improve artificial sensor arrays.iscrimination of olfactory signals occurs in a high-dimensional space of odor stimuli in which a large number of distinct molecules and their mixtures can be distinguished by a much smaller number of receptors (1-3). For example, humans have about 300 distinct olfactory receptors (4), which can sense at least 2,100 odorant molecules (5), and the real number might be much larger (1). Moreover, humans can differentiate between mixtures of up to 30 odorants (6). Such remarkable molecular discrimination is thought to use a combinatorial code (7,8), where typical odorant molecules bind to receptors of multiple types (1, 3). Each receptor type is expressed in many cells (9), and the information from all receptors of the same type is accumulated in corresponding glomeruli in the olfactory bulb (10, 11) (see Fig. 1A). The activity of a single glomerulus is thus the total signal of the associated receptor type, so the information about the odor is encoded in the activity pattern of the glomeruli (11,12). This activity pattern is interpreted by the brain to learn about the composition and the concentration of the inhaled odor. We here study how receptor arrays can maximize the transmitted information.It is known (13, 14) that the input−output characteristics of sensory apparatuses of many organisms are tailored to the statistics of the organism's natural environment to maximize information transmission. For example, in the visual circuit of the fly, the input−output relationship of neurons is matched to the cumulative distribution of the input distribution (13). Similar observations have since been made in many sensory systems (14, 15) and even in transcriptional regulation (16). In all these cases, the distinguishable outputs of the sensory system must be dedicated to equal parts of the input distribution, which is known as Laughlin's principle (13) or histogram equalization (17). Intuitively, more of the response range is dedicated to common stimuli, at the expense of less frequent stimuli (13).Similarly, t...