Abstract
Cytoplasmic incompatibility (CI) relates to the manipulation by the parasite Wolbachia of its host reproduction. Despite its widespread occurrence, the molecular basis of CI remains unclear and theoretical models have been proposed to understand the phenomenon. We consider in this paper the quantitative Lock-Key model which currently represents a good hypothesis that is consistent with the data available. CI is in this case modelled as the problem of covering the edges of a bipartite graph with the minimum number of chain subgraphs. This problem is already known to be NP-hard, and we provide an exponential algorithm with a non trivial complexity. It is frequent that depending on the dataset, there may be many optimal solutions which can be biologically quite different among them. To rely on a single optimal solution may therefore be problematic. To this purpose, we address the problem of enumerating (listing) all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time. Interestingly, in order to solve the above problems, we considered also the problem of enumerating all the maximal chain subgraphs of a bipartite graph and improved on the current results in the literature for the latter. Finally, to demonstrate the usefulness of our methods we show an application on a real dataset.
Highlights
Wolbachia is an intracellular bacterium that infects numerous arthropod species
Calling m the number of edges in the graph, we provide an exact exponential algorithm which runs in time O∗((2 + ε)m) by combining our results on the enumeration of maximal chain subgraphs with the inclusion-exclusion technique [22]
Conclusions and open problems In this paper, we studied the problem of finding the minimum number of different Lock/Key molecules that explains the Cytoplasmic incompatibility (CI) for the observed data
Summary
Wolbachia is an intracellular bacterium that infects numerous arthropod species. It is transmitted vertically through the host’s eggs and is known for frequently influencing the reproductive development and behaviour of its host. In order to do this, we prove some upper bounds on the maximum number of maximal chain subgraphs of a bipartite graph G with n nodes and m edges. We provide a total quasi-polynomial time algorithm to enumerate all minimal covers by maximal chain subgraphs of a bipartite graph.
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