A notion of sectional regularity for a homogeneous ideal I, which measures the regularity of its generic sections with respect to linear spaces of various dimensions, is introduced. It is related to axial constants defined as the intercepts on the coordinate axes of the set of exponents of monomials in the reverse lexicographic generic initial ideal of I. The equivalence of these notions and several other homological and ideal-theoretic invariants is shown. It is also established that these equivalent invariants grow linearly for the family of powers of a given ideal.

Objectives The goal of this project was to understand how compounds naturally present in foods interact with the bacteria in the human gut microbiome. Methods Information on the compounds in different foods was obtained from FooDB. Microbiome pathway information was obtained from BioCyc. This information was synthesized into a web interface built with Flask. The interface allows a user to select a combination of food items. Upon selection, the user is presented with all microbiome reactions for which the constituent compounds of the foods are substrates, and the respective metabolic pathways of which they are part. The computations were done using Python. The interface was applied to investigate the ketogenic diet. Two ketogenic one-day meal plans were generated via an online tool (eatthismuch.com). The meal plans were entered into a macronutrient tracker “myfitnesspal,” and modified to ensure similar macronutrient distributions. The average difference between the macronutrient content of the diets was 11.6%. The interface was used to determine the number of microbiome pathways in which a food compound served as a substrate. Results The following is a list of the foods in each of the meal plans and the number of interacting microbiome pathways. The first ketogenic meal plan consisted of: breakfast - 367 (bacon, eggs, butter); lunch - 547 (tuna, tomato, spinach); dinner - 362 (pork chop, cauliflower) and a snack - 327 (macadamia) yielding 581 unique bacterial pathways. The second meal plan consisted of: breakfast - 367 (bacon, eggs, butter); lunch - 382 (chicken, avocado); dinner - 339 (meatballs, cheese) and a snack - 357 (yogurt, cashews) yielding 428 unique bacterial pathways. This reflects a 30.3% difference between the two meal plans in interacting pathways. Excluding snacks, the highest number of interacting pathways was with a vegetable/fish meal (547) while the lowest was with a meat/cheese meal (339). Conclusions An assumption in many studies regarding diet is that two diets of similar macronutrient compositions will have similar physiological effects on the microbiome. However, the differences in the total number of interacting bacterial pathways between the two macronutrient similar ketogenic diets suggests that the effects on the microbiome are more complex, suggesting a new approach to the study of diet. Funding Sources N/A.

We introduce a notion of sectional regularity for a homogeneous ideal I I , which measures the regularity of its general sections with respect to linear spaces of various dimensions. It is related to axial constants defined as the intercepts on the coordinate axes of the set of exponents of monomials in the reverse lexicographic generic initial ideal of I I . We show the equivalence of these notions and several other homological and ideal-theoretic invariants. We also establish that these equivalent invariants grow linearly for the family of powers of a given ideal.