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AbstractBreathing behaviour involves the generation of normal breaths (eupnoea) on a timescale of seconds and sigh breaths on the order of minutes. Both rhythms emerge in tandem from a single brainstem site, but whether and how a single cell population can generate two disparate rhythms remains unclear. We posit that recurrent synaptic excitation in concert with synaptic depression and cellular refractoriness gives rise to the eupnoea rhythm, whereas an intracellular calcium oscillation that is slower by orders of magnitude gives rise to the sigh rhythm. A mathematical model capturing these dynamics simultaneously generates eupnoea and sigh rhythms with disparate frequencies, which can be separately regulated by physiological parameters. We experimentally validated key model predictions regarding intracellular calcium signalling. All vertebrate brains feature a network oscillator that drives the breathing pump for regular respiration. However, in air‐breathing mammals with compliant lungs susceptible to collapse, the breathing rhythmogenic network may have refashioned ubiquitous intracellular signalling systems to produce a second slower rhythm (for sighs) that prevents atelectasis without impeding eupnoea.image Key pointsA simplified activity‐based model of the preBötC generates inspiratory and sigh rhythms from a single neuron population.Inspiration is attributable to a canonical excitatory network oscillator mechanism.Sigh emerges from intracellular calcium signalling.The model predicts that perturbations of calcium uptake and release across the endoplasmic reticulum counterintuitively accelerate and decelerate sigh rhythmicity, respectively, which was experimentally validated.Vertebrate evolution may have adapted existing intracellular signalling mechanisms to produce slow oscillations needed to optimize pulmonary function in mammals. 

Abstract The origin of phenotypic novelty is a perennial question of genetics and evolution. To date, few studies of biological pattern formation specifically address multi-generational aspects of inheritance and phenotypic novelty. For quantitative traits influenced by many segregating alleles, offspring phenotypes are often intermediate to parental values. In other cases, offspring phenotypes can be transgressive to parental values. For example, in the model organismMimulus(monkeyflower), the offspring of parents with solid-colored petals exhibit novel spotted petal phenotypes. These patterns are controlled by an activator-inhibitor gene regulatory network with a small number of loci. Here we develop and analyze a model of hybridization and pattern formation that accounts for the inheritance of a diploid gene regulatory network composed of either homozygous or heterozygous alleles. We find that the resulting model of multi-generational Turing-type pattern formation can reproduce transgressive petal phenotypes similar to those observed inMimulus. The model gives insight into how non-patterned parent phenotypes can yield phenotypically transgressive, patterned offspring, aiding in the development of empirically testable hypotheses. 

Abstract We show how equilibrium binding curves of receptor homodimers can be expressed as rational polynomial functions of the equilibrium binding curves of the constituent monomers, without approximation and without assuming independence of receptor monomers. Using a distinguished spanning tree construction for reduced graph powers, the method properly accounts for thermodynamic constraints and allosteric interactions between receptor monomers (i.e. conformational coupling). The method is completely general; it begins with an arbitrary undirected graph representing the topology of a monomer state-transition diagram and ends with an algebraic expression for the equilibrium binding curve of a receptor oligomer composed of two or more identical and indistinguishable monomers. Several specific examples are analysed, including guanine nucleotide-binding protein-coupled receptor dimers and tetramers composed of multiple ‘ternary complex’ monomers.