Extended Data Fig. 2: Concentrations versus yields near domain boundaries and the need to sample domain interiors.

a, Taking as an example a basic bimolecular reaction A + B → C (with kinetic rate k[A][B], reaction time t = 1 and kinetic constant k = 1), the plot has the yield and concentration of C for various combinations of A and B substrates. b, The plot shows the yield surface explicitly (black curve is the boundary at which the limiting reactant switches from A to B). As seen, yield—defined with respect to the concentration of the limiting reagent—tends to maximize at the boundaries of the region (orange curve in a) but the concentration of the product tends to maximize in its interior (blue curve in a). There are two implications of this ‘inverse’ dependence: (1) yields at very low concentrations may be artificially high, whereas—in reality—they involve very high (wasteful) excesses of some reagents; (2) detecting and quantifying the product reliably may be compromised in such a low-concentration regime. Accordingly, in the current work—especially for the multicomponent mixtures—we have worked over concentration ranges that do not extend to very small values, thereby assuring reliable detection while avoiding highly wasteful excess ratios. Such a ‘practical’ range is shown schematically in a by the dashed vertical lines. Another corollary is that discovery of unexpected products may be severely compromised when using a strategy of investigating only the edges and boundaries of an n-dimensional region while leaving the interior unexplored—a strategy that may otherwise seem intuitive to a chemist and has some merit in maximization of the yield of known products. c,d, Comparison of final concentration map (c) to yield map (d) for product 19e in Hantzsch reaction. Note the difference between the conditions that maximize concentration versus ones that maximize yield: yield is maximized at the vertices of the cube, whereas concentration is maximized in the middle of the front face and top edge, with higher concentrations protruding deeper into the interior of the cube (see blue curve in a). The isosurfaces correspond to 5 mM, 10 mM, 15 mM, 20 mM, 25 mM and 30 mM concentrations of 19e in c, 20%, 30%, 40%, 50% and 60% yields of 19e in d and are calculated as explained in the caption of Fig. 2c,d.