Using tabu search and genetic algorithms in mathematics research

This paper discusses an ongoing project which uses computational heuristic search techniques such as tabu search and genetic algorithms as a tool for mathematics research. We discuss three ways in which such search techniques can be useful for mathematicians: in nding counterexamples to conjectures, in enumerating examples, and in nding sequences of transformations between two objects which are conjectured to be related. These problem-types are discussed using examples from topology

Single DNA conformations and biological function

From a nanoscience perspective, cellular processes and their reduced in vitro imitations provide extraordinary examples for highly robust few or single molecule reaction pathways. A prime example are biochemical reactions involving DNA molecules, and the coupling of these reactions to the physical conformations of DNA. In this review, we summarise recent results on the following phenomena: We investigate the biophysical properties of DNA-looping and the equilibrium configurations of DNA-knots, whose relevance to biological processes are increasingly appreciated. We discuss how random DNA-looping may be related to the efficiency of the target search process of proteins for their specific binding site on the DNA molecule. And we dwell on the spontaneous formation of intermittent DNA nanobubbles and their importance for biological processes, such as transcription initiation. The physical properties of DNA may indeed turn out to be particularly suitable for the use of DNA in nanosensing applications.Comment: 53 pages, 45 figures. Slightly revised version of a review article, that is going to appear in the J. Comput. Theoret. Nanoscience; some typos correcte

Folding Kinetics of Protein Like Heteropolymers

Using a simple three-dimensional lattice copolymer model and Monte Carlo dynamics, we study the collapse and folding of protein-like heteropolymers. The polymers are 27 monomers long and consist of two monomer types. Although these chains are too long for exhaustive enumeration of all conformations, it is possible to enumerate all the maximally compact conformations, which are 3x3x3 cubes. This allows us to select sequences that have a unique global minimum. We then explore the kinetics of collapse and folding and examine what features determine the various rates. The folding time has a plateau over a broad range of temperatures and diverges at both high and low temperatures. The folding time depends on sequence and is related to the amount of energetic frustration in the native state. The collapse times of the chains are sequence independent and are a few orders of magnitude faster than the folding times, indicating a two-phase folding process. Below a certain temperature the chains exhibit glass-like behavior, characterized by a slowing down of time scales and loss of self-averaging behavior. We explicitly define the glass transition temperature (Tg), and by comparing it to the folding temperature (Tf), we find two classes of sequences: good folders with Tf > Tg and non-folders with Tf < Tg.Comment: 23 pages (plus 10 figures included in a seperate file) LaTeX, no local report nu

Experimental approximation of the Jones polynomial with DQC1

We present experimental results approximating the Jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor. This is the first experimental implementation of a complete problem for the deterministic quantum computation with one quantum bit model of quantum computation, which uses a single qubit accompanied by a register of completely random states. The Jones polynomial is a knot invariant that is important not only to knot theory, but also to statistical mechanics and quantum field theory. The implemented algorithm is a modification of the algorithm developed by Shor and Jordan suitable for implementation in NMR. These experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91% of the time.Comment: 5 figures. Version 2 changes: published version, minor errors corrected, slight changes to improve readabilit

Multiple Testing and Variable Selection along Least Angle Regression's path

In this article, we investigate multiple testing and variable selection using Least Angle Regression (LARS) algorithm in high dimensions under the Gaussian noise assumption. LARS is known to produce a piecewise affine solutions path with change points referred to as knots of the LARS path. The cornerstone of the present work is the expression in closed form of the exact joint law of K-uplets of knots conditional on the variables selected by LARS, namely the so-called post-selection joint law of the LARS knots. Numerical experiments demonstrate the perfect fit of our finding. Our main contributions are three fold. First, we build testing procedures on variables entering the model along the LARS path in the general design case when the noise level can be unknown. This testing procedures are referred to as the Generalized t-Spacing tests (GtSt) and we prove that they have exact non-asymptotic level (i.e., Type I error is exactly controlled). In that way, we extend a work from (Taylor et al., 2014) where the Spacing test works for consecutive knots and known variance. Second, we introduce a new exact multiple false negatives test after model selection in the general design case when the noise level can be unknown. We prove that this testing procedure has exact non-asymptotic level for general design and unknown noise level. Last, we give an exact control of the false discovery rate (FDR) under orthogonal design assumption. Monte-Carlo simulations and a real data experiment are provided to illustrate our results in this case. Of independent interest, we introduce an equivalent formulation of LARS algorithm based on a recursive function.Comment: 62 pages; new: FDR control and power comparison between Knockoff, FCD, Slope and our proposed method; new: the introduction has been revised and now present a synthetic presentation of the main results. We believe that this introduction brings new insists compared to previous version

Flexible RNA design under structure and sequence constraints using formal languages

The problem of RNA secondary structure design (also called inverse folding) is the following: given a target secondary structure, one aims to create a sequence that folds into, or is compatible with, a given structure. In several practical applications in biology, additional constraints must be taken into account, such as the presence/absence of regulatory motifs, either at a specific location or anywhere in the sequence. In this study, we investigate the design of RNA sequences from their targeted secondary structure, given these additional sequence constraints. To this purpose, we develop a general framework based on concepts of language theory, namely context-free grammars and finite automata. We efficiently combine a comprehensive set of constraints into a unifying context-free grammar of moderate size. From there, we use generic generic algorithms to perform a (weighted) random generation, or an exhaustive enumeration, of candidate sequences. The resulting method, whose complexity scales linearly with the length of the RNA, was implemented as a standalone program. The resulting software was embedded into a publicly available dedicated web server. The applicability demonstrated of the method on a concrete case study dedicated to Exon Splicing Enhancers, in which our approach was successfully used in the design of \emph{in vitro} experiments.Comment: ACM BCB 2013 - ACM Conference on Bioinformatics, Computational Biology and Biomedical Informatics (2013

McGenus: A Monte Carlo algorithm to predict RNA secondary structures with pseudoknots

We present McGenus, an algorithm to predict RNA secondary structures with pseudoknots. The method is based on a classification of RNA structures according to their topological genus. McGenus can treat sequences of up to 1000 bases and performs an advanced stochastic search of their minimum free energy structure allowing for non trivial pseudoknot topologies. Specifically, McGenus employs a multiple Markov chain scheme for minimizing a general scoring function which includes not only free energy contributions for pair stacking, loop penalties, etc. but also a phenomenological penalty for the genus of the pairing graph. The good performance of the stochastic search strategy was successfully validated against TT2NE which uses the same free energy parametrization and performs exhaustive or partially exhaustive structure search, albeit for much shorter sequences (up to 200 bases). Next, the method was applied to other RNA sets, including an extensive tmRNA database, yielding results that are competitive with existing algorithms. Finally, it is shown that McGenus highlights possible limitations in the free energy scoring function. The algorithm is available as a web-server at http://ipht.cea.fr/rna/mcgenus.php .Comment: 6 pages, 1 figur

RNA secondary structure design

We consider the inverse-folding problem for RNA secondary structures: for a given (pseudo-knot-free) secondary structure find a sequence that has that structure as its ground state. If such a sequence exists, the structure is called designable. We implemented a branch-and-bound algorithm that is able to do an exhaustive search within the sequence space, i.e., gives an exact answer whether such a sequence exists. The bound required by the branch-and-bound algorithm are calculated by a dynamic programming algorithm. We consider different alphabet sizes and an ensemble of random structures, which we want to design. We find that for two letters almost none of these structures are designable. The designability improves for the three-letter case, but still a significant fraction of structures is undesignable. This changes when we look at the natural four-letter case with two pairs of complementary bases: undesignable structures are the exception, although they still exist. Finally, we also study the relation between designability and the algorithmic complexity of the branch-and-bound algorithm. Within the ensemble of structures, a high average degree of undesignability is correlated to a long time to prove that a given structure is (un-)designable. In the four-letter case, where the designability is high everywhere, the algorithmic complexity is highest in the region of naturally occurring RNA.Comment: 11 pages, 10 figure