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The study aims to increase the understanding regarding the role of regulatory T cells (Tregs) in lupus nephritis (LN) and ANCA-associated vasculitis (AAV) by comparing their localization in renal tissue and changes following immunosuppressive therapy. Kidney biopsies from 12 patients with LN and 7 patients with AAV were examined. Kidney biopsies had been performed both at active disease and following immunosuppressive treatment. Clinical data was collected at both biopsy occasions. Expression of Forkhead Box P 3 (Foxp3) in renal tissue was assessed by immunohistochemistry. An arbitrary scale was used to estimate the number of Foxp3+ cells. In LN, 8/12 (67%) had positive tissue staining for Foxp3 at baseline, most pronounced in inflammatory infiltrates, but also interstitially and in a peri-glomerular pattern. At second biopsies, after immunosuppressive treatment, 4/12 (33%) still had detectable Foxp3+ cells, found in persisting inflammatory infiltrates and some in the interstitium. Patients with a good clinical response to treatment had high grade of Foxp3+ cells in first biopsies. In AAV, only 2/7 (29%) had positive staining for Foxp3 at baseline, in inflammatory infiltrates and to a lesser extent in the interstitium, despite large areas of inflammatory infiltrates in all patients. At follow-up, 2/7 (29%) biopsies were positive for Foxp3. Our data show a higher presence of Foxp3+ cells in renal tissue from LN patients compared to AAV, suggesting that Tregs may be differently involved in the control of inflammatory mechanisms in these diseases. These findings could have further implication for therapeutic approaches aiming at restoring the immunological tolerance. Key Points • Foxp3+-cells are present in larger amount in renal tissue in lupus nephritis vs. ANCA-associated vasculitis. • Our data suggest that Foxp3+ regulatory T cells are involved in the control of inflammatory processes in lupus nephritis., (© 2023. International League of Associations for Rheumatology (ILAR).)

We use 57 recently found topological satellites of Broucke-Hadjidemetriou-Hénon's periodic orbits with values of the topological exponent k ranging from k=3 to k=58 to plot the angular momentum L as a function of the period T, with both L and T rescaled to energy E=-0.5. Upon plotting L(T/k) we find that all our solutions fall on a curve that is virtually indiscernible by the naked eye from the L(T) curve for nonsatellite solutions. The standard deviation of the satellite data from the sixth-order polynomial fit to the progenitor data is σ=0.13. This regularity supports Hénon's 1976 conjecture that the linearly stable Broucke-Hadjidemetriou-Hénon orbits are also perpetually, or Kol'mogorov-Arnol'd-Moser, stable.