Communicated by E.M. Friedlander MSC: 35B32 37G40 a b s t r a c tIn this paper we present results for the systematic study of reversible-equivariant vector fields -namely, in the simultaneous presence of symmetries and reversing symmetries -by employing algebraic techniques from invariant theory for compact Lie groups. The Hilbert-Poincaré series and their associated Molien formulae are introduced, and we prove the character formulae for the computation of dimensions of spaces of homogeneous anti-invariant polynomial functions and reversible-equivariant polynomial mappings. A symbolic algorithm is obtained for the computation of generators for the module of reversible-equivariant polynomial mappings over the ring of invariant polynomials. We show that this computation can be obtained directly from a well-known situation, namely from the generators of the ring of invariants and the module of the equivariants.

Homeostasis refers to a phenomenon whereby the output $$x_o$$ x o of a system is approximately constant on variation of an input $${{\mathcal {I}}}$$ I . Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs $${{\mathcal {G}}}$$ G with a distinguished input node $$\iota $$ ι , a different distinguished output node o, and a number of regulatory nodes $$\rho _1,\ldots ,\rho _n$$ ρ 1 , … , ρ n . In these models the input–output map $$x_o({{\mathcal {I}}})$$ x o ( I ) is defined by a stable equilibrium $$X_0$$ X 0 at $${{\mathcal {I}}}_0$$ I 0 . Stability implies that there is a stable equilibrium $$X({{\mathcal {I}}})$$ X ( I ) for each $${{\mathcal {I}}}$$ I near $${{\mathcal {I}}}_0$$ I 0 and infinitesimal homeostasis occurs at $${{\mathcal {I}}}_0$$ I 0 when $$(dx_o/d{{\mathcal {I}}})({{\mathcal {I}}}_0) = 0$$ ( d x o / d I ) ( I 0 ) = 0 . We show that there is an $$(n+1)\times (n+1)$$ ( n + 1 ) × ( n + 1 ) homeostasis matrix$$H({{\mathcal {I}}})$$ H ( I ) for which $$dx_o/d{{\mathcal {I}}}= 0$$ d x o / d I = 0 if and only if $$\det (H) = 0$$ det ( H ) = 0 . We note that the entries in H are linearized couplings and $$\det (H)$$ det ( H ) is a homogeneous polynomial of degree $$n+1$$ n + 1 in these entries. We use combinatorial matrix theory to factor the polynomial $$\det (H)$$ det ( H ) and thereby determine a menu of different types of possible homeostasis associated with each digraph $${{\mathcal {G}}}$$ G . Specifically, we prove that each factor corresponds to a subnetwork of $${{\mathcal {G}}}$$ G . The factors divide into two combinatorially defined classes: structural and appendage. Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of $$\det (H)$$ det ( H ) without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of $$\det (H)$$ det ( H ) . There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).

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The causative agent of COVID-19 pandemic, SARS-CoV-2, has a 29,903 bases positive-sense single-stranded RNA genome. RNAs exhibit about 150 modified bases that are essential for proper function. Among internal modified bases, the N6-methyladenosine, or m6A, is the most frequent, and is implicated in SARS-CoV-2 immune response evasion. Although the SARS-CoV-2 genome is RNA, almost all genomes sequenced thus far are, in fact, reverse transcribed complementary DNAs. This process reduces the true complexity of these viral genomes because the incorporation of dNTPs hides RNA base modifications. Here, we present an initial exploration of Nanopore direct RNA sequencing to assess the m6A residues in the SARS-CoV-2 sequences of ORF3a, E, M, ORF6, ORF7a, ORF7b, ORF8, N, ORF10 and the 3′-untranslated region. We identified fifteen m6A methylated positions, of which, six are in ORF N. Additionally, because m6A is associated with the DRACH motif, we compared its distribution in major SARS-CoV-2 variants. Although DRACH is highly conserved among variants, we show that variants Beta and Eta have a fourth position C > U change in DRACH at 28,884b that could affect methylation. This is the first report of direct RNA sequencing of a Brazilian SARS-CoV-2 sample coupled with the identification of modified bases.