## Section: New Results

### Modelling and analysing a network of individuals, or a network of individuals' networks

**Robustness of the parsimonious reconciliation method in cophylogeny**

The currently most used method in cophylogenetic studies is the so-called
*phylogenetic tree reconciliation*. In this model, we are given the
phylogenetic tree of the hosts $H$, the one of the symbionts $S$, and a mapping
$\phi $ from the leaves of $S$ to the leaves of $H$ indicating the known symbiotic
relationships among present-day organisms. The common evolutionary
history of the hosts and of their symbionts is then explained through a number of macroevolutionary
events (four in general). A reconciliation is then a function
$\lambda $ which is an extension of the mapping $\phi $
between leaves to a mapping that includes all internal nodes and that can be
constructed using the different types of events considered. An optimal reconciliation is
usually defined in a parsimonious way: a cost is associated to each event and
a solution of minimum total cost is searched for.

An important issue in this model is that it makes strong assumptions on the input data which may not be verified in practice. We examine two cases where this situation happens. The first is related to a limitation in the currently available methods for tree reconciliation where the association $\phi $ of the leaves is for now required to be a function. This is not realistic as a single symbiont species can infect more than one host. For each present-day symbiont involved in a multiple association, one is currently forced to choose a single one. The second case addresses a different type of problem related to the phylogenetic trees of hosts and symbionts. These indeed are assumed to be correct, which may not be the case. In this work, we addressed the problem of correctly rooting a phylogenetic tree.

We thus explored the robustness of the parsimonious tree reconciliation method under "editing" (multiple associations) or "small perturbations" of the input (rooting problem) [29].

An extended version of this paper has been submitted to *IEEE/ACM Transactions on Computational Biology and Bioinformatics*.

**Insights on the virulence of swine respiratory tract mycoplasmas through genome-scale metabolic modelling**

The respiratory tract of swines is colonised by several bacteria among which are three Mycoplasma species: *Mycoplasma flocculare*, *Mycoplasma hyopneumoniae* and *Mycoplasma hyorhinis*. While colonisation by *M. flocculare* was shown to be virtually asymptomatic, *M. hyopneumoniae* is known to be the causative agent of enzootic pneumonia and *M. hyorhinis* to be present in cases of pneumonia, polyserositis and arthritis. Nonetheless, the elevated genomic resemblance among these three mycoplasmas combined with their different levels of pathogenicity is an indication that they have unknown mechanisms of virulence and differential expression. In 2015, we performed whole-genome metabolic network reconstructions for these three mycoplasmas. The results obtained were then submitted for publication to *BMC Genomics*. The paper has since been published [13].

**Maximal chain subgraphs and covers of bipartite graphs motivated by analysis of cytoplasmic incompatibility**

In a previous work of the team (Nor *et al.* *American Naturalist*, 182(1):15-24, 2013; Nor*et al.*
*Information and Computation*, 213:23-32, 2012), we showed that a minimum chain subgraph cover of a given bipartite graph provides a good model for identifying the minimum genetic architecture enabling to explain one type of manipulation, called *cytoplasmic incompatibility*, by some parasite bacteria
on their hosts.
This phenomenon results in the death of embryos produced in crosses
between males carrying the infection and uninfected females. The
observed cytoplasmic compatibility relationships, can then be
represented by a bipartite graph with males and females in different
classes. Moreover, as different minimum (resp. minimal) covers may correspond to
solutions that differ in terms of their biological interpretation, the
capacity to enumerate all such minimal chain covers becomes crucial.

We recently addressed three related problems that bear some interest for the above problem besides raising interesting theoretical questions [35]. One is the enumeration of all the maximal *edge induced* chain subgraphs of a bipartite graph, for which we provided a polynomial delay algorithm. We gave bounds on the number of maximal chain subgraphs for a bipartite graph and used them to establish the input-sensitive complexity of the enumeration problem.
The second problem we treated was the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this, we provided an exact exponential algorithm with a non trivial complexity. Finally, we approached the problem of enumerating all minimal chain subgraph covers of a bipartite graph and showed that it can be solved in quasi-polynomial time.

An extended version of the conference paper has been submitted to a journal in December 2016.