Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

An $r \times s$ partial Latin rectangle $(l_{ij})$ is an $r \times s$ matrix containing elements of $\{1,2,\ldots,n\} \cup \{\cdot\}$ such that each row and each column contain at most one copy of any symbol in $\{1,2,\ldots,n\}$. An entry is a triple $(i,j,l_{ij})$ with $l_{ij} \neq \cdot$. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of $m$-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) $r=s=n$, i.e., partial Latin squares, (b) $r=2$ and $s=n$, and (c) $r=2$ and $s \neq n$

Quantifying donor-to-donor variation in macrophage responses to the human fungal pathogen Cryptococcus neoformans

Cryptococcosis remains the leading cause of fungal meningitis worldwide, caused primarily by the pathogen Cryptococcus neoformans. Symptomatic cryptococcal infections typically affect immunocompromised patients. However, environmental exposure to cryptococcal spores is ubiquitous and most healthy individuals are thought to harbor infections from early childhood onwards that are either resolved, or become latent. Since macrophages are a key host cell for cryptococcal infection, we sought to quantify the extent of individual variation in this early phagocyte response within a small cohort of healthy volunteers with no reported immunocompromising conditions. We show that rates of both intracellular fungal proliferation and non-lytic expulsion (vomocytosis) are remarkably variable between individuals. However, we demonstrate that neither gender, in vitro host inflammatory cytokine profiles, nor polymorphisms in several key immune genes are responsible for this variation. Thus the data we present serve to quantify the natural variation in macrophage responses to this important human pathogen and will hopefully provide a useful "benchmark" for the research community