# Developmental scRNASeq Trajectories in Gene- and Cell-State Space--The Flatworm Example
## 3 Results
### Portraval of Developmental Paths  
In the next step, we aimed at characterizing gene-state expression dynamics during differentiation. For a simple first view, we ordered the expression portraits of the cells of epidermal differentiation along the linear, unbranched pseudotime path taken from [PERSON] and colleagues [29] (Figure 4A). Hereby the stemness module is highly expressed in the portraits of stem cells, of epidermal neoblast, as well as in epidermal progenitors. Its amplitude however progressively decays in the portraits of early and late epidermal progenitors but then markedly decays in the portraits of differentiated epidermal cells. On the other hand, the expression of the epidermal module progressively increases in an antagonistic fashion. Notably, this switching affects about 1000 genes included in these two modules. We replot the default \"logFC\"-scaled portrait into double-logarithmic \"loglogFC\" scale in order to better visualize subtle expression changes upon differentiation. The red overexpression region \"flows\" along the upper border of the map in direction from the stemness \"source\" spot at the right toward the epidermis \"sink\" module at the left. A gene-state trajectory of epidermal differentiation was computed using the maximum flow algorithm [10] as described in the methods section. Briefly, the trajectory is obtained as directed graph linking the source and sink module via the path of maximum cumulative expression between the adjacent modules along the path (Figure 4B). As expected, the trajectory follows the right-to-left shift of the red overexpression region seen in the portraits upon differentiation. The expression of stemness genes decays along the trajectory and it increases for genes involved in epidermis differentiation (Figure 4C). Gene-state trajectories of differentiation, pt profiles, and associated expression portraits of parenchymal, neuronal, muscular, and gut tissues are shown in Figure 4D. Hereby, differentiation is described by trajectories starting in the common stemness mode and ending in the respective tissue-specific expression modules. As an alternative method we applied a lowest cost path algorithm to the difference map between stemness and differentiated tissues (Figure 4E) [42]. The trajectories obtained closely resemble those obtained by means of the maximum flow algorithm. Interestingly, the gene states move \"downhill\" along the differentiation trajectories in the difference SOM, a picture, which resembles the epigenetic \"Waddington\" landscape [18; 45]. This landscape was introduced as a conceptual model to illustrate differentiation of tissues from stem cells. Note that the SOM-landscape is a \"real\" expression data landscape, which this way supports the abstract Waddington concept of developing cells in gene-state space. In summary, our approach identifies cascades of gene modules changing expression upon differentiation in a tissue-specific (pseudo-)temporal order. In SOM metagene space it appears as \"lava lamp\"-like flows of upregulated genes, which can be summarized into gene expression trajectories directed from stemness toward differentiated tissue expression programs. Figure 4: Gene expression trajectories of planarian tissue development. **(A)** Expression portraits (in log FC and loglog FC scales) of the epidermal lineage illustrate the change of module-patterns during  
differentiation. The \"loglogFC\" scale better resolves subtle changes between over- and under-expression in red and blue, respectively. Alterations of expression patterns become evident as \"lava lamp\"-like flow of the red spot from the right to the left. The expression of stemness and epidermal spots N and J change in an antagonistic fashion along the pseudotime. (**B**) The gene-state trajectory of epidermal development links the stemness spot with the \"epidermal\" spots in the expression landscape shown as cumulative loglogFC SOM). The expression profile along the trajectory assigns genes and functions along the path.