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Circadian clocks in unicellular phototrophic organisms are known to display remarkable reliability. In contrast, not much is known about how circadian clocks perform in a multicellular setting. Are clocks in multicellular cyanobacteria coupled and synchronized with one another? Are clocks entrained only by external cues? What is the spatial extent of synchronization? What is the role of cell-cell variations in copy numbers of molecules comprising the core clock (demographic noise) in setting the temporal pattern and its robustness? To tackle quantitatively these and other questions, we studied the dynamics of a circadian clock-controlled gene in Anabaena sp. PCC 7120, a multicellular cyanobacterium in which cells are arranged one after the other and coupled by protein channels, in a one-dimensional structure. Our real-time, single-cell level measurements showed significant synchronization and spatial coherence along filaments, and clock coupling mediated by cell-cell communication. Furthermore, we found significant variability in expression between different cells along filaments. A stochastic one-dimensional toy model of coupled clocks and their phosphorylation states shows that demographic noise can seed stochastic oscillations outside the region where deterministic limit cycles with circadian periods occur. The model reproduces the observed spatio-temporal coherence along filaments and provides a robust description of coupled circadian clocks in a multicellular organism, despite significant stochasticity in biomolecular reactions. Lastly, we carried out experiments in which developmental processes were induced. Our experiments showed that gene expression in different vegetative intervals along a developed filament was discoordinated, and that differentiation took place preferentially within a limited interval of the circadian clock cycle. The transition to multicellularity demanded coordination between clocks via cell-cell communication, to optimize fitness in the presence of significant demographic noise.

A group is cubulated if it acts freely and cocompactly on a CAT(0) cube complex, a high-dimensional generalization of a tree. Cubulating hyperbolic groups has proven to be a fruitful endeavor over the past couple decades, as cubulation implies strong subgroup separability and linearity properties. Generalizing property FA, we consider the n-dimensional cubical fixed point property. A group has property FW_n if every action on an n-dimensional CAT(0) cube complex has a global fixed-point. We will show how to produce many examples of FW_n groups and prove that random groups have FW_n for every n.