• NP-Completeness and APX-Hardness • We prove that the problems MinFixMask OPT and MinFixMask BOUND are NP-complete. • We establish that FixMasks + is NP-complete and APX-hard. • We show NP-completeness of DFixMasks + . • Exact, FPT, and Approximation Algorithms for DFixMasks + • We design a fixed-parameter tractable (FPT) algorithm running in O *(2 M ) time, parameterized by the number of shield configurations M . • We present a polynomial-time 1 M -approximation algorithm. • We propose an exponential k M -approximation method (for a treatment plan length k ). • We provide a tight ( 1 − 1 e ) -approximation algorithm by exploiting monotonicity and submodularity of the objective function. • Quasi-Polynomial Algorithms and Logarithmic Approximations • We develop quasi-polynomial-time algorithms for MinFixMask OPT , MinFixMask OPT + , MinFixMask BOUND , and MinFixMask BOUND + , assuming that the maximum prescribed dose d ^ max is polynomially bounded. • Under the same assumption, we achieve polynomial-time approximation algorithms with a logarithmic factor log ( d ^ max ) for MinFixMask OPT and MinFixMask OPT + . • Polynomial-Time Solvable Special Cases • We identify polynomial-time solvable cases: FixMask and FixMasks + when M = 1 . • We show that the problems MinFixMask BOUND and MinFixMask BOUND + can be solved in polynomial time when T max 1, N ;. In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.

Understanding and predicting the pattern formation in groups of pluripotent stem cells has the potential to improve efficiency and efficacy of stem cell therapies. However, the underlying molecular mechanisms of pluripotent stem cell behaviors are highly complex and are currently still not fully understood. A key practical question is whether deep biological modelling of the cells is essential to predict their pattern formation, or whether there is sufficient predictive power in simply modelling their behaviors and interactions at a higher level. This study focuses on the social interactions and behaviors of pluripotent stem cells at a high-level to predict aggregate crowd behaviors within a level of uncertainty. Agent-based modelling was applied to study the pattern formation in pluripotent stem cells. Five models were established to test four biologically plausible rules of cell motility in terms of: a) velocity, b) directional persistence time, c) directional movements, and d) border effect. We found that it is possible that cells' directional movements based on local density play an important role of the pattern formation, and pattern formation in pluripotent stem cells is governed by a complex combination of rules in our agent-based model simulations, which account for much of the variability observed in experimental findings.

In this paper, we study circular words arising in the development of equipment using shields in brachytherapy. This equipment has physical constraints that have to be taken into consideration. From an algorithmic point of view, the problem can be formulated as follows: Given a circular word, find a constrained circular word of the same length such that the Manhattan distance between these two words is minimal. We show that we can solve this problem in pseudo polynomial time (polynomial time in practice) using dynamic programming.

MOTIVATION: Integrative network analysis methods provide robust interpretations of differential high-throughput molecular profile measurements. They are often used in a biomedical context-to generate novel hypotheses about the underlying cellular processes or to derive biomarkers for classification and subtyping. The underlying molecular profiles are frequently measured and validated on animal or cellular models. Therefore the results are not immediately transferable to human. In particular, this is also the case in a study of the recently discovered interleukin-17 producing helper T cells (Th17), which are fundamental for anti-microbial immunity but also known to contribute to autoimmune diseases. RESULTS: We propose a mathematical model for finding active subnetwork modules that are conserved between two species. These are sets of genes, one for each species, which (i) induce a connected subnetwork in a species-specific interaction network, (ii) show overall differential behavior and (iii) contain a large number of orthologous genes. We propose a flexible notion of conservation, which turns out to be crucial for the quality of the resulting modules in terms of biological interpretability. We propose an algorithm that finds provably optimal or near-optimal conserved active modules in our model. We apply our algorithm to understand the mechanisms underlying Th17 T cell differentiation in both mouse and human. As a main biological result, we find that the key regulation of Th17 differentiation is conserved between human and mouse. AVAILABILITY AND IMPLEMENTATION: xHeinz, an implementation of our algorithm, as well as all input data and results, are available at http://software.cwi.nl/xheinz and as a Galaxy service at http://services.cbib.u-bordeaux2.fr/galaxy in CBiB Tools. CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

International audience

We describe a theoretical unifying framework to express the comparison of RNA structures, which we call alignment hierarchy. This framework relies on the definition of common supersequences for arc-annotated sequences and encompasses the main existing models for RNA structure comparison based on trees and arc-annotated sequences with a variety of edit operations. It also gives rise to edit models that have not been studied yet. We provide a thorough analysis of the alignment hierarchy, including a new polynomial-time algorithm and an NP-completeness proof. The polynomial-time algorithm involves biologically relevant edit operations such as pairing or unpairing nucleotides. It has been implemented in a software, called gardenia, which is available at the Web server http://bioinfo.lifl.fr/RNA/gardenia.

Recent techniques increase rapidly the amount of our knowledge on interactions between proteins. The interpretation of these new information depends on our ability to retrieve known substructures in the data, the Protein-Protein Interactions (PPIs) networks. In an algorithmic point of view, it is an hard task since it often leads to NP-hard problems. To overcome this difficulty, many authors have provided tools for querying patterns with a restricted topology, i.e., paths or trees in PPI networks. Such restriction leads to the development of fixed parameter tractable (FPT) algorithms, which can be practicable for restricted sizes of queries. Unfortunately, Graph Homomorphism is a W[1]-hard problem, and hence, no FPT algorithm can be found when patterns are in the shape of general graphs. However, Dost et al. gave an algorithm (which is not implemented) to query graphs with a bounded treewidth in PPI networks (the treewidth of the query being involved in the time complexity). In this paper, we propose another algorithm for querying pattern in the shape of graphs, also based on dynamic programming and the color-coding technique. To transform graphs queries into trees without loss of informations, we use feedback vertex set coupled to a node duplication mechanism. Hence, our algorithm is FPT for querying graphs with a bounded size of their feedback vertex set. It gives an alternative to the treewidth parameter, which can be better or worst for a given query. We provide a python implementation which allows us to validate our implementation on real data. Especially, we retrieve some human queries in the shape of graphs into the fly PPI network.

In this paper, we are interested in the computational complexity of computing (dis)similarity measures between two genomes when they contain duplicated genes or genomic markers, a problem that happens frequently when comparing whole nuclear genomes. Recently, several methods ( [1], [2]) have been proposed that are based on two steps to compute a given (dis)similarity measure M between two genomes G_1 and G_2: first, one establishes a oneto- one correspondence between genes of G_1 and genes of G_2 ; second, once this correspondence is established, it defines explicitly a permutation and it is then possible to quantify their similarity using classical measures defined for permutations, like the number of breakpoints. Hence these methods rely on two elements: a way to establish a one-to-one correspondence between genes of a pair of genomes, and a (dis)similarity measure for permutations. The problem is then, given a (dis)similarity measure for permutations, to compute a correspondence that defines an optimal permutation for this measure. We are interested here in two models to compute a one-to-one correspondence: the exemplar model, where all but one copy are deleted in both genomes for each gene family, and the matching model, that computes a maximal correspondence for each gene family. We show that for these two models, and for three (dis)similarity measures on permutations, namely the number of common intervals, the maximum adjacency disruption (MAD) number and the summed adjacency disruption (SAD) number, the problem of computing an optimal correspondence is NP-complete, and even APXhard for the MAD number and SAD number.