MOTIVATION: The evolution of complexity is one of the most fascinating and challenging problems in modern biology, and tracing the evolution of complex traits is an open problem. In bacteria, operons and gene blocks provide a model of tractable evolutionary complexity at the genomic level. Gene blocks are structures of co-located genes with related functions, and operons are gene blocks whose genes are co-transcribed on a single mRNA molecule. The genes in operons and gene blocks typically work together in the same system or molecular complex. Previously, we proposed a method that explains the evolution of orthologous gene blocks (orthoblocks) as a combination of a small set of events that take place in vertical evolution from common ancestors. A heuristic method was proposed to solve this problem. However, no study was done to identify the complexity of the problem. RESULTS: Here, we establish that finding the homologous gene block problem is NP-hard and APX-hard. We have developed a greedy algorithm that runs in polynomial time and guarantees an O(ln⁡n) approximation. In addition, we formalize our problem as an integer linear program problem and solve it using the PuLP package and the standard CPLEX algorithm. Our exploration of several candidate operons reveals that our new method provides more optimal results than the results from the heuristic approach, and is significantly faster. AVAILABILITY AND IMPLEMENTATION: The software and data accompanying this paper are available under the GPLv3 and CC0 license respectively on: https://github.com/nguyenngochuy91/Relevant-Operon.

MOTIVATION: Complexity is a fundamental attribute of life. Complex systems are made of parts that together perform functions that a single component, or subsets of components, cannot. Examples of complex molecular systems include protein structures such as the F1Fo-ATPase, the ribosome, or the flagellar motor: each one of these structures requires most or all of its components to function properly. Given the ubiquity of complex systems in the biosphere, understanding the evolution of complexity is central to biology. At the molecular level, operons are classic examples of a complex system. An operon's genes are co-transcribed under the control of a single promoter to a polycistronic mRNA molecule, and the operon's gene products often form molecular complexes or metabolic pathways. With the large number of complete bacterial genomes available, we now have the opportunity to explore the evolution of these complex entities, by identifying possible intermediate states of operons. RESULTS: In this work, we developed a maximum parsimony algorithm to reconstruct ancestral operon states, and show a simple vertical evolution model of how operons may evolve from the individual component genes. We describe several ancestral states that are plausible functional intermediate forms leading to the full operon. We also offer Reconstruction of Ancestral Gene blocks Using Events or ROAGUE as a software tool for those interested in exploring gene block and operon evolution. AVAILABILITY AND IMPLEMENTATION: The software accompanying this paper is available under GPLv3 license on: https://github.com/nguyenngochuy91/Ancestral-Blocks-Reconstruction. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

We introduce a new tool for approximation and testing algorithms called partitioning oracles. We develop methods for constructing them for any class of bounded-degree graphs with an excluded minor, and in general, for any hyperfinite class of bounded-degree graphs. These oracles utilize only local computation to consistently answer queries about a global partition that breaks the graph into small connected components by removing only a small fraction of the edges. We illustrate the power of this technique by using it to extend and simplify a number of previous approximation and testing results for sparse graphs, as well as to provide new results that were unachievable with existing techniques. For instance:1. We give constant-time approximation algorithms for the size of the minimum vertex cover, the minimum dominating set, and the maximum independent set for any class of graphs with an excluded minor.2. We show a simple proof that any minor-closed graph property is testable in constant time in the bounded degree model.3. We prove that it is possible to approximate the distance to almost any hereditary property in any bounded degree hereditary families of graphs. Hereditary properties of interest include bipartiteness, k-colorability, and perfectness.

We present a technique for transforming classical approximation algorithms into constant-time algorithms that approximate the size of the optimal solution. Our technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. The technique is based on greedily considering local improvements in random order.The problems amenable to our technique include Vertex Cover, Maximum Matching, Maximum Weight Matching, Set Cover, and Minimum Dominating Set. For example, for Maximum Matching, we give the first constant-time algorithm that for the class of graphs of degree bounded by $d$, computes the maximum matching size to within $\eps n$, for any $\eps ≫ 0$, where $n$ is the number of nodes in the graph. The running time of the algorithm is independent of $n$, and only depends on $d$ and $\eps$.