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ABSTRACT Inspired by Waddington's illustration of an epigenetic landscape, cell-fate transitions have been envisioned as bifurcating dynamical systems, wherein exogenous signaling dynamics couple to the enormously complex signaling and transcriptional machinery of a cell to elicit qualitative transitions in its collective state. Single-cell RNA sequencing (scRNA-seq), which measures the distributions of possible transcriptional states in large populations of differentiating cells, provides an alternate view, in which development is marked by the variations of a myriad of genes. Here, we present a mathematical formalism for rigorously evaluating, from a dynamical systems perspective, whether scRNA-seq trajectories display statistical signatures consistent with bifurcations and, as a case study, pinpoint regions of multistability along the neutrophil branch of hematopoeitic differentiation. Additionally, we leverage the geometric features of linear instability to identify the low-dimensional phase plane in gene expression space within which the multistability unfolds, highlighting novel genetic players that are crucial for neutrophil differentiation. Broadly, we show that a dynamical systems treatment of scRNA-seq data provides mechanistic insights into the high-dimensional processes of cellular differentiation, taking a step toward systematic construction of mathematical models for transcriptomic dynamics. 

Some bacteria and archaea possess an immune system, based on the CRISPR-Cas mechanism, that confers adaptive immunity against viruses. In such species, individual prokaryotes maintain cassettes of viral DNA elements called spacers as a memory of past infections. Typically, the cassettes contain several dozen expressed spacers. Given that bacteria can have very large genomes and since having more spacers should confer a better memory, it is puzzling that so little genetic space would be devoted by prokaryotes to their adaptive immune systems. Here, assuming that CRISPR functions as a long-term memory-based defense against a diverse landscape of viral species, we identify a fundamental tradeoff between the amount of immune memory and effectiveness of response to a given threat. This tradeoff implies an optimal size for the prokaryotic immune repertoire in the observational range.