Highlights d Communicating cells can form spatial patterns without morphogen gradients d Disordered field of communicating cells forms dynamic patterns (e.g., spiral waves) d Simulations and theory found dynamic-pattern-forming cellcommunication methods d Dynamic patterns form via a three-stage (''order-fluctuatesettle'') process
12Cells form spatial patterns by coordinating their gene expressions. How a group of mesoscopic 13 numbers (hundreds-to-thousands) of cells, without pre-defined morphogens and spatial 14 organization, self-organizes spatial patterns remains incompletely understood. Of particular 15 importance are dynamic spatial patterns -such as spiral waves that perpetually move and 16 transmit information over macroscopic length-scales. We developed an open-source, 17 expandable software that can simulate a field of cells communicating with any number of cell-18 secreted molecules in any manner. With it and a theory developed here, we identified all 19 possible "cellular dialogues" -ways of communicating with two diffusing molecules -and core 20 architectures underlying them that enable diverse, self-organized dynamic spatial patterns that 21 we classified. The patterns form despite widely varying cellular response to the molecules, 22 gene-expression noise, and spatial arrangement and motility of cells. Three-stage, "order-23 fluctuate-settle" process forms dynamic spatial patterns: cells form long-lived whirlpools of 24 wavelets that, through chaos-like interactions, settle into a dynamic spatial pattern. These 25 results provide a blueprint to help identify missing regulatory links for observed dynamic-26 pattern formations and in building synthetic tissues. 60 emerge in a mesoscopic population of cells. 62We sought to resolve this shortcoming by developing an open-source software that analyzing these simulations. With the software and analysis algorithms, we sought to reveal 67 quantitative mechanisms by which mesoscopic numbers of cells can use their spatially 68 heterogeneous gene-expression levels as a seed to form spatial patterns. In particular, we 69 focused on dynamic patterns -patterns that constantly change over time without ever stopping 70 such as oscillations and spiral waves [Sgro et al., 2013] -instead of static patterns that remain 71 still after forming ( Figure 1B). Through an exhaustive computational search, we discovered all 72 the ways in which cells can communicate with just two diffusing molecules to form dynamic 73 patterns, including those that have been experimentally observed. We found that just a few 74 ways of communicating, which we refer to as "cellular dialogues", can generate a large palette 75 of complex, dynamic spatial patterns such as chaotic whirlpools of wavelets and travelling 76 waves of various shapes and orientations. Viewing these simulations as exact numerical 77 experiments, we devised an analytical (pen-and-paper) approach that recapitulates the 78 simulations and used it to understand why only certain cellular dialogues can sustain dynamic 79 spatial patterns. As we will show, we discovered that cells can form dynamic spatial patterns 80 through a three-stage, "order-fluctuate-settle" process. Starting from a configuration in which 81 there is no spatial correlation among cells' gene-expression levels, we observed that cells 82 rapidly become more spatially correlated over tim...
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