Marcel Jackson

Cells do not act in isolation; they communicate with each other through a complex network of signals. In multicellular organisms, cell signaling buffers against fluctuations in gene expression. Its effect, however, on multistability—the ability of genetically identical cells to take on and maintain diverse stable states or phenotypes—is unclear. We develop spatially explicit stochastic models that couple fine-grained gene regulatory dynamics with intercellular signaling to study cell fate control in multicellular tissues. We show that intercellular signaling acts as a switch-like controller of cell fate, driving transitions from transient to stable states across tissues. Even weak signaling stabilizes short-lived expression states, and we derive mathematical expressions for the threshold signaling strength required to trigger tissue-wide stabilization. We also derive a universal spatial limit: stable region size grows only sublinearly with signaling strength, implying that large stable regions are prohibitively costly to maintain. The resulting trade-off between robustness and signaling in cell fate control highlights the delicate balance required during developmental processes to maintain spatial precision and provides insight into how organisms regulate tissue structure. Overview: Cell fate decisions in developing tissues are governed by gene regulatory networks, but how communication between neighboring cells influences these decisions is poorly understood. We develop a mathematical framework showing that intercellular signaling acts as a controller of cell fate, suppressing spontaneous switching, and inducing stable, coordinated behaviors across tissues. Even weak signaling induces a sharp transition from unstable to stable population-wide dynamics. We also derive fundamental scaling laws showing that the size of stable regions increases sublinearly with signaling strength, revealing a trade-off between developmental robustness and resource cost. These results offer insight into how organisms achieve reliable spatial organization during development, despite molecular noise, and provide design principles for engineering synthetic tissues that balance stability, scalability, and resource efficiency.

We use high girth, high chromatic number hypergraphs to show that there are finite models of the equational theory of the semiring of non‐negative integers whose equational theory has no finite axiomatisation, and show this also holds if factorial, fixed base exponentiation and operations for binomial coefficients are adjoined. We also derive the decidability of the equational logical entailment operator for antecedents true on by way of a form of the finite model property. Two appendices contain additional basic development of combinatorial operations. Amongst the observations are an eventual dominance well‐ordering of combinatorial functions and consequent representation of the ordinal in terms of factorial functions; the equivalence of the equational logic of combinatorial algebra over the natural numbers and over the positive reals; and a candidate list of elementary axioms.

In this paper, we examine some flexible notions of constraint satisfaction, observing some relationships between model theoretic notions of universal Horn class membership and robust satisfiability. We show the NP-completeness of 2-robust positive 1-in-3SAT in order to give very small examples of finite algebras with NP-hard variety membership problem. In particular, we give a 3-element algebra with this property, and solve a widely stated problem by showing that the 6-element Brandt monoid has NP-hard variety membership problem. These are the smallest possible sizes for a general algebra and a semigroup to exhibit NP-hardness for the membership problem of finite algebras in finitely generated varieties.