We generalize the Susceptible-Infected-Removed (SIR) model for epidemics to take into account generic effects of heterogeneity in the degree of susceptibility to infection in the population. We introduce a single new parameter corresponding to a power-law exponent of the susceptibility distribution at small susceptibilities. We find that for this class of distributions the gamma distribution is the attractor of the dynamics. This allows us to identify generic effects of population heterogeneity in a model as simple as the original SIR model which is contained as a limiting case. Because of this simplicity, numerical solutions can be generated easily and key properties of the epidemic wave can still be obtained exactly. In particular, we present exact expressions for the herd immunity level, the final size of the epidemic, as well as for the shape of the wave and for observables that can be quantified during an epidemic. In strongly heterogeneous populations, the herd immunity level can be much lower than in models with homogeneous populations as commonly used for example to discuss effects of mitigation. Using our model to analyze data for the SARS-CoV-2 epidemic in Germany shows that the reported time course is consistent with several scenarios characterized by different levels of immunity. These scenarios differ in population heterogeneity and in the time course of the infection rate, for example due to mitigation efforts or seasonality. Our analysis reveals that quantifying the effects of mitigation requires knowledge on the degree of heterogeneity in the population. Our work shows that key effects of population heterogeneity can be captured without increasing the complexity of the model. We show that information about population heterogeneity will be key to understand how far an epidemic has progressed and what can be expected for its future course.
About three-quarters of eukaryotic DNA is wrapped into nucleosomes; DNA spools with a protein core. The affinity of a given DNA stretch to be incorporated into a nucleosome is known to depend on the base-pair sequence-dependent geometry and elasticity of the DNA double helix. This causes the rotational and translational positioning of nucleosomes. In this study we ask the question whether the latter can be predicted by a simple coarse-grained DNA model with sequence-dependent elasticity, the rigid base-pair model. Whereas this model is known to be rather robust in predicting rotational nucleosome positioning, we show that the translational positioning is a rather subtle effect that is dominated by the guanine-cytosine content dependence of entropy rather than energy. A correct qualitative prediction within the rigid base-pair framework can only be achieved by assuming that DNA elasticity effectively changes on complexation into the nucleosome complex. With that extra assumption we arrive at a model which gives an excellent quantitative agreement to experimental in vitro nucleosome maps, under the additional assumption that nucleosomes equilibrate their positions only locally.
Cells tend to divide along the direction in which they are longest, as famously stated by Oscar Hertwig in 1884 in his long axis rule. The orientation of the mitotic spindle determines the cell division axis, and the long axis rule is usually ensured by forces stemming from microtubules. Pulling on the spindle from the cell cortex can give rise to unstable behaviors, and we here set out to understand how the long axis rule is realized in early embryonic divisions where cortical pulling forces are prevalent. We focus on early C. elegans development, where we compressed embryos to reveal that cortical pulling forces favor an alignment of the spindle with the short axis of the cell. Strikingly, we find that this misalignment is corrected by an actomyosin-based mechanism that rotates the entire cell, including the mitotic spindle. We uncover that myosin-driven contractility in the cytokinetic ring generates inward forces that align it with the short axis, and thereby the spindle with the long axis. A theoretical model together with experiments using slightly compressed mouse zygotes suggest that a constricting cytokinetic ring can ensure the long axis rule in cells that are free to rotate inside a confining structure, thereby generalizing the underlying principle.
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