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Backward models for super-diffusion have been developed to identify pollutant source locations, but they are limited to a single observation and disregard field-measured concentrations. To overcome these limitations, this study derives the adjoint of the space-fractional advection–dispersion equation, incorporating measured concentrations from multiple observation data. Backward probabilities, such as the backward location probability density function (PDF), describe the likely source location(s) at a fixed time before sampling, offering a comprehensive modeling approach for source identification. By applying Bayes’ theorem, the individual PDFs from each observation and its corresponding concentration are combined into a joint PDF, enhancing both the information and reliability compared to the previous single PDF. Field applications show that the improved model enhances accuracy (with PDF peak locations closer to the actual source) and precision (with reduced variance) of backward PDFs for identifying point sources in a natural river and aquifer. The model’s performance is affected by observation count and measurement errors, with double peaks in the backward location PDF possible due to source mass uncertainty. Future refinements, such as incorporating backward travel time analysis and extending applications to reactive pollutants, could further enhance the utility of the conditioned backward fractional-derivative model developed in this study. 

Calculations provide the first comprehensive mechanistic study of carbene N–H insertion for His-ligated heme biocatalysts, as validatedvianew experiments. 

This article highlights recent computational research on heme-based carbene transfer reactions. Mechanistic insights reveal how cofactor components, coordination modes, substrates, and protein environments influence reactivity and selectivity. 

Nitric oxide (NO) is an important molecule that regulates many physiological processes in humans and plants and contributes to the formation of greenhouse gases. Bacterial NO reductases utilize a di-Fe heme/nonheme active site to couple two NOs to generate nitrous oxide (N2O) via a two-electron mechanism. Here, we report a previously unexplored Cr porphyrin NO complex with a Lewis acid (LA) BF3 for the NO reduction reaction. Density functional theory calculations were first employed to reveal its reaction mechanism with a reasonable barrier for experimental realization. Subsequent experimental synthesis work confirms this reactivity and reports the first nitrosyl Cr porphyrin X-ray crystal structure. Theoretical analysis uncovered a distinctive reaction feature for the Cr system compared to Fe and Co porphyrins: the electron transfer from the metal to the bound NO occurs before LA binding. A comparative study of the NO coupling mechanisms with the three representative metals suggests that the metal reduction potential should be finely tuned, as found in previous studies of NOR enzymatic systems. Overall, this study offers new theoretical and experimental insights to further facilitate the development of alternative NO reduction compounds with biological, environmental, and industrial applications. 

Abstract Some pathogens use heme‐containing nitric oxide reductases (NORs) to reduce NO to N₂O as their defense mechanism to detoxify NO and reduce nitrosative stress. This reduction is also significant in the global N cycle. Our previous experimental work showed that Fe and Co porphyrin NO complexes can couple with external NO to form N₂O when activated by the Lewis acid BF₃. A key difference from conventional two‐electron enzymatic reaction is that one electron is sufficient. However, a complete understanding of the entire reaction pathways and the more favorable reactivity for Fe remains unknown. Here, we present a quantum chemical study to provide such information. Our results confirmed Fe's higher experimental reactivity, showing advantages in all steps of the reaction pathway: easier metal oxidation for NO reduction and N−O cleavage as well as a larger size to expedite the N/O coordination mode transition. The Co system, with a similar product energy as the enzyme, shows potential for further development in catalytic NO coupling. This work also offers the first evidence that this new one‐electron NO reduction is both kinetically competitive and thermodynamically more favorable than the native pathway, supporting future initiatives in optimizing NO reduction agents in biology, environment, and industry.