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).

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.

We consider an important class of nonsymmetric networks that lies between the class of general networks and the class of symmetric networks, where group theoretic methods still apply—namely, networks admitting “interior symmetries.” The main result of this paper is the full analogue of the equivariant Hopf theorem for networks with symmetries. We extend the result of Golubitsky, Pivato, and Stewart [Dyn. Syst., 19 (2004), pp. 389–407] to obtain states whose linearizations on certain subsets of cells, near bifurcation, are superpositions of synchronous states with states having spatio-temporal symmetries.

From the point of view of coupled systems developed by Stewart, Golubitsky and Pivato, lattice differential equations consist of choosing a phase space R^k for each point in a lattice, and a system of differential equations on each of these spaces R^k such that the whole system is translation invariant. The architecture of a lattice differential equation specifies the sites that are coupled to each other (nearest neighbour coupling (NN) is a standard example). A polydiagonal is a finite-dimensional subspace of phase space obtained by setting coordinates in different phase spaces as equal. There is a colouring of the network associated with each polydiagonal obtained by colouring any two cells that have equal coordinates with the same colour. A pattern of synchrony is a colouring associated with a polydiagonal that is flow-invariant for every lattice differential equation with a given architecture. We prove that every pattern of synchrony for a fixed architecture in planar lattice differential equations is spatially doubly-periodic, assuming that the couplings are sufficiently extensive. For example, nearest and next nearest neighbour couplings are needed for square and hexagonal couplings, but a third level of coupling is needed for the corresponding result to hold in rhombic and primitive cubic lattices. On planar lattices this result is known to fail if the network architecture consists only of NN. The techniques we develop to prove spatial periodicity and finiteness can be applied to other lattices as well.