Silvana Ilie

HiTEC: accurate error correction in high-throughput sequencing data

MOTIVATION High-throughput sequencing technologies produce very large amounts of data and sequencing errors constitute one of the major problems in analyzing such data. Current algorithms for correcting these errors are not very accurate and do not automatically adapt to the given data. RESULTS We present HiTEC, an algorithm that provides a highly accurate, robust and fully automated method to correct reads produced by high-throughput sequencing methods. Our approach provides significantly higher accuracy than previous methods. It is time and space efficient and works very well for all read lengths, genome sizes and coverage levels. AVAILABILITY The source code of HiTEC is freely available at www.csd.uwo.ca/~ilie/HiTEC/.

Multiple spaced seeds for homology search

MOTIVATION Homology search finds similar segments between two biological sequences, such as DNA or protein sequences. The introduction of optimal spaced seeds in PatternHunter has increased both the sensitivity and the speed of homology search, and it has been adopted by many alignment programs such as BLAST. With the further improvement provided by multiple spaced seeds in PatternHunterII, Smith-Waterman sensitivity is approached at BLASTn speed. However, computing optimal multiple spaced seeds was proved to be NP-hard and current heuristic algorithms are all very slow (exponential). RESULTS We give a simple algorithm which computes good multiple seeds in polynomial time. Due to a completely different approach, the difference with respect to the previous methods is dramatic. The multiple spaced seed of PatternHunterII, with 16 weight 11 seeds, was computed in 12 days. It takes us 17 s to find a better one. Our approach changes the way of looking at multiple spaced seeds.

SpEED: fast computation of sensitive spaced seeds

SUMMARY Multiple spaced seeds represent the current state-of-the-art for similarity search in bioinformatics, with applications in various areas such as sequence alignment, read mapping, oligonucleotide design, etc. We present SpEED, a software program that computes highly sensitive multiple spaced seeds. SpEED can be several orders of magnitude faster and computes better seeds than the existing leading software programs. AVAILABILITY The source code of SpEED is freely available at www.csd.uwo.ca/~ilie/SpEED/ CONTACT: ilie@csd.uwo.ca SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

Adaptivity and computational complexity in the numerical solution of ODEs

In this paper we analyze the problem of adaptivity for one-step numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating grids of minimal computational cost for which the local error is below a prescribed tolerance (optimal grids). The grids are generated by introducing an auxiliary independent variable τ and finding a grid deformation map, t=Θ(τ), that maps an equidistant grid {τj} to a non-equidistant grid in the original independent variable, {tj}. An optimal deformation map Θ is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a non-adaptive method.

Numerical solutions of index-1 differential algebraic equations can be computed in polynomial time

The cost of solving an initial value problem for index-1 differential algebraic equations to accuracy ɛ is polynomial in ln(1/ɛ). This cost is obtained for an algorithm based on the Taylor series method for solving differential algebraic equations developed by Pryce. This result extends a recent result by Corless for solutions of ordinary differential equations. The results of the standard theory of information-based complexity give exponential cost for solving ordinary differential equations, being based on a different model.

Fast computation of good multiple spaced seeds

Homology search finds similar segments between two biological sequences, such as DNA or protein sequences. A significant fraction of computing power in the world is dedicated to performing such tasks. The introduction of optimal spaced seeds by Ma et al. has increased both the sensitivity and the speed of homology search and it has been adopted by many alignment programs such as BLAST. With the further improvement provided by multiple spaced seeds in PatternHunterII, the sensitivity of dynamic programming is approached at BLASTn speed. Whereas computing optimal multiple spaced seeds was proved to be NP-hard, we show that, from practical point of view, computing good ones can be very efficient. We give a simple heuristic algorithm which computes good multiple seeds in polynomial time. Computing sensitivity is not required. When allowing the computation of the sensitivity for few seeds, we obtain better multiple seeds than previous ones in much shorter time.

An adaptive stepsize method for the chemical Langevin equation

Mathematical and computational modeling are key tools in analyzing important biological processes in cells and living organisms. In particular, stochastic models are essential to accurately describe the cellular dynamics, when the assumption of the thermodynamic limit can no longer be applied. However, stochastic models are computationally much more challenging than the traditional deterministic models. Moreover, many biochemical systems arising in applications have multiple time-scales, which lead to mathematical stiffness. In this paper we investigate the numerical solution of a stochastic continuous model of well-stirred biochemical systems, the chemical Langevin equation. The chemical Langevin equation is a stochastic differential equation with multiplicative, non-commutative noise. We propose an adaptive stepsize algorithm for approximating the solution of models of biochemical systems in the Langevin regime, with small noise, based on estimates of the local error. The underlying numerical method is the Milstein scheme. The proposed adaptive method is tested on several examples arising in applications and it is shown to have improved efficiency and accuracy compared to the existing fixed stepsize schemes.

Implicit Riquier Bases for PDAE and their semi-discretizations

Complicated nonlinear systems of pde with constraints (called pdae) arise frequently in applications. Missing constraints arising by prolongation (differentiation) of the pdae need to be determined to consistently initialize and stabilize their numerical solution. In this article we review a fast prolongation method, a development of (explicit) symbolic Riquier Bases, suitable for such numerical applications. Our symbolic-numeric method to determine Riquier Bases in implicit form, without the unstable eliminations of the exact approaches, applies to square systems which are dominated by pure derivatives in one of the independent variables. The method is successful provided the prolongations with respect to a single dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are nonsingular when evaluated at points on the zero sets defined by the functions of the pdae. For polynomially nonlinear pdae, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. Our method generalizes Pryces method for dae to pdae. Given a dominant independent time variable, for an initial value problem for a system of pdae we show that its semi-discretization is also naturally amenable to our symbolic-numeric approach. In particular, if our method can be successfully applied to such a system of pdae, yielding an implicit Riquier Basis, then under modest conditions, the semi-discretized system of dae is also an implicit Riquier Basis.

Adaptive time-stepping for the strong numerical solution of stochastic differential equations

Models based on stochastic differential equations are of high interest today due to their many important practical applications. Thus the need for efficient and accurate numerical methods to approximate their solution. In this paper, we propose several adaptive time-stepping strategies for the strong numerical solution of stochastic differential equations in Itô form, driven by multiple Wiener processes satisfying the commutativity condition. The adaptive schemes are based on I and PI control, and allow arbitrary values of the stepsize. The explicit Milstein method is applied to approximate the solution of the problem and the adaptive implementations are based on estimates of the local error obtained using Richardson extrapolation. Numerical tests on several models arising in applications show that our adaptive time-stepping schemes perform better than the fixed stepsize alternative and an adaptive Brownian tree time-stepping strategy.

Variable time-stepping in the pathwise numerical solution of the chemical Langevin equation

Stochastic modeling is essential for an accurate description of the biochemical network dynamics at the level of a single cell. Biochemically reacting systems often evolve on multiple time-scales, thus their stochastic mathematical models manifest stiffness. Stochastic models which, in addition, are stiff and computationally very challenging, therefore the need for developing effective and accurate numerical methods for approximating their solution. An important stochastic model of well-stirred biochemical systems is the chemical Langevin Equation. The chemical Langevin equation is a system of stochastic differential equation with multidimensional non-commutative noise. This model is valid in the regime of large molecular populations, far from the thermodynamic limit. In this paper, we propose a variable time-stepping strategy for the numerical solution of a general chemical Langevin equation, which applies for any level of randomness in the system. Our variable stepsize method allows arbitrary values of the time-step. Numerical results on several models arising in applications show significant improvement in accuracy and efficiency of the proposed adaptive scheme over the existing methods, the strategies based on halving/doubling of the stepsize and the fixed step-size ones.