# Predicting the Evolution of Syntenies--An Algorithmic Review
## 5 Accounting for Gene Trees
### Evolution through Segmental Duplications and Losses  
Another strategy for considering the co-evolution of adjacent genes in syntenies is to generalize the reconciliation model to account for segmental duplications and losses rather than single-gene events (see Figure 5). In [94; 95], the DL-reconciliation of a gene tree has been generalized to the DL-reconciliation of a \"synteny tree\" (defined bellow) accounting for segmental duplications and losses. In this study, a _synteny X_ is an ordered sequence of genes belonging to a genome \(s(X)\). The genes of a synteny all belong to different gene families and thus, in particular, tandem duplications are not allowed. We say that a set \(\mathcal{G}=\{\Gamma_{1},\Gamma_{2},\ldots,\Gamma_{t}\}\) of gene families are _organized into a set \(\mathcal{X}\) of syntenies_ iff there is a bijection between the genes of \(\mathcal{G}\) and the genes in \(\mathcal{X}\) (each gene of \(\mathcal{G}\) belongs to exactly one synteny of \(\mathcal{X}\)). A _synteny tree_ is a tree with leaves mapped to syntenies. In particular, a synteny tree \(T\) for \(\mathcal{X}\) is a tree with a one-to-one mapping between \(\mathcal{L}(T)\) and \(\mathcal{X}\). The syntenies of a _syntenty family_\(\mathcal{X}\) are considered to have evolved from a single ancestral synteny through speciation (defined as for single genes), segmental duplications and segmental losses, where:  
* a speciation \(Spe(X,[1,l])\) acting on a synteny \(X=g_{1}\cdots g_{l}\) belonging to a genome \(s(X)\) has the effect of reproducing \(X\) in the two genomes \(s_{l}\) and \(s_{r}\) children of \(s(X)\) in \(S\);
* a (segmental) duplication \(Dup(X,[i,j])\) acting on a synteny \(X\) belonging to a genome \(s(X)\) is an operation that copies a substring \(g_{i}\cdots g_{j}\) of \(X=g_{1}g_{2}\cdots g_{i}\cdots g_{j}\cdots g_{l}\)  
Figure 5: **(1)** Three gene families (red, blue and green) organized into four syntenies \(\mathcal{X}=\{A_{1},A_{2},B_{1},B_{2}\}\) located in two genomes \(A\) and \(B\), and the reconciled gene trees for each gene family (node labels are represented as in Figure 4). Gene orders, as well as gene trees are consistent; **(2)** The gene trees embedded in the species tree \(S=(A,B)\), illustrating an independent evolutionary history for each gene family; **(3)** A synteny tree for \(\mathcal{X}\) (left) and a Super-Reconciliation, i.e., an evolutionary scenario of segmental duplications and losses (right), embedded in the species tree \(S\). Each node of the Super-Reconciliation is labeled by the type of event (duplication represented by a rectangle, speciation by an oval and loss by a cross) and the segment affected by a duplication (inside the rectangle) or a loss (marked by the cross). The bold edge incident to a duplication node is the one leading to the duplicated synteny; **(4)** An unordered Super-Reconciliation for a new set of syntenies \(\{A_{3},A_{4},B_{3},B_{4}\}\) that are not order-consistent. The minimum number of duplications and losses explaining this synteny tree is 5, but rearrangements are still required on some branches as orders are not conserved. somewhere else into the genome \(s(X)\), creating a new _copied synteny_\(X^{\prime}=g^{\prime}_{i}\cdots g^{\prime}_{j}\), where each \(g^{\prime}_{k}\), for \(i\leq k\leq j\), belongs to the same gene family as \(g_{k}\);
* a (segmental) loss \(Loss(X,[i,j])\) acting on a \(X=g_{1}\cdots g_{l}\) is an operation that removes a substring \(g_{i}\cdots g_{j}\) of \(X\), leading to the _truncated synteny_\(X^{\prime}=g_{1}\cdots g_{i-1}g_{j+1}\cdots g_{l}\). A loss is called _full_ if \(X^{\prime}\) is the empty string (i.e., all genes of \(X\) are removed) and _partial_ otherwise. A partial loss event is denoted \(pLoss\). Thus, in contrast to a single-gene family, a tree representing the evolution of a set of syntenies is not only labeled by the type of event \(\vec{e}(x)\) corresponding to each internal node, but also by the segment of the synteny affected by the event.