Mathematics, Department of

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Prediction of 492 Human Protein Kinase Substrate Specificities

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2011
Abstract: 

Background: Complex intracellular signaling networks monitor diverse environmental inputs to evoke appropriateand coordinated effector responses. Defective signal transduction underlies many pathologies, including cancer,diabetes, autoimmunity and about 400 other human diseases. Therefore, there is high impetus to define thecomposition and architecture of cellular communications networks in humans. The major components ofintracellular signaling networks are protein kinases and protein phosphatases, which catalyze the reversiblephosphorylation of proteins. Here, we have focused on identification of kinase-substrate interactions throughprediction of the phosphorylation site specificity from knowledge of the primary amino acid sequence of thecatalytic domain of each kinase.Results: The presented method predicts 488 different kinase catalytic domain substrate specificity matrices in 478typical and 4 atypical human kinases that rely on both positive and negative determinants for scoring individualphosphosites for their suitability as kinase substrates. This represents a marked advancement over existing methodssuch as those used in NetPhorest (179 kinases in 76 groups) and NetworKIN (123 kinases), which consider onlypositive determinants for kinase substrate prediction. Comparison of our predicted matrices with experimentallyderivedmatrices from about 9,000 known kinase-phosphosite substrate pairs revealed a high degree ofconcordance with the established preferences of about 150 well studied protein kinases. Furthermore for many ofthe better known kinases, the predicted optimal phosphosite sequences were more accurate than the consensusphosphosite sequences inferred by simple alignment of the phosphosites of known kinase substrates.Conclusions: Application of this improved kinase substrate prediction algorithm to the primary structures of over23, 000 proteins encoded by the human genome has permitted the identification of about 650, 000 putativephosphosites, which are posted on the open source PhosphoNET website (http://www.phosphonet.ca).

Document type: 
Article

A Novel Approach to Modelling Water Transport and Drug Diffusion Through the Stratum Corneum

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2010
Abstract: 

Background: The potential of using skin as an alternative path for systemicallyadministering active drugs has attracted considerable interest, since the creation ofnovel drugs capable of diffusing through the skin would provide a great steptowards easily applicable -and more humane- therapeutic solutions. However, fordrugs to be able to diffuse, they necessarily have to cross a permeability barrier: thestratum corneum (SC), the uppermost set of skin layers. The precise mechanism bywhich drugs penetrate the skin is generally thought to be diffusion of moleculesthrough this set of layers following a “tortuous pathway” around corneocytes, i.e.impermeable dead cells.Results: In this work, we simulate water transport and drug diffusion using a threedimensionalporous media model. Our numerical simulations show that diffusiontakes place through the SC regardless of the direction and magnitude of the fluidpressure gradient, while the magnitude of the concentrations calculated areconsistent with experimental studies.Conclusions: Our results support the possibility for designing arbitrary drugs capableof diffusing through the skin, the time-delivery of which is solely restricted by theirdiffusion and solubility properties.

Document type: 
Article

Wavelength Isolation Sequence Design: Supplementary Data

Peer reviewed: 
No, item is not peer reviewed.
Date created: 
2013-01-04
Abstract: 

This technical report accompanies the paper J. Jedwab and M. Strange. Wavelength isolation sequence design. Submitted to IEEE Trans. Inform. Theory, 2012. It gives a complete listing of the inequivalent nontrivial wavelength isolation sequence triples(WISTs) of length at most 26 and their corresponding aperiodic cross correlations.

Document type: 
Technical Report
File(s): 

The Cellular Dynamics of Bone Remodeling: a Mathematical Model

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2010
Abstract: 

The mechanical properties of vertebrate bone are largely determined by a processwhich involves the complex interplay of three different cell types. This process is called bone remodelingand occurs asynchronously at multiple sites in the mature skeleton. The cells involvedare bone resorbing osteoclasts, bone matrix producing osteoblasts, and mechanosensing osteocytes.These cells communicate with each other by means of autocrine and paracrine signaling factors andoperate in complex entities, the so-called bone multicellular units (BMUs). To investigate the BMUdynamics in silico, we develop a novel mathematical model resulting in a system of nonlinear partialdifferential equations (PDEs) with time delays. The model describes the osteoblast and osteoclastpopulations together with the dynamics of the key messenger molecule RANKL and its decoy receptorOPG. Scaling theory is used to address parameter sensitivity and predict the emergence ofpathological remodeling regimes. The model is studied numerically in one and two space dimensionsusing finite difference schemes in space and explicit delay equation solvers in time. The computationalresults are in agreement with in vivo observations and provide new insights into the role ofthe RANKL/OPG pathway in the spatial regulation of bone remodeling.

Document type: 
Article

An integral representation of the Green’s function for a linear array of acoustic point sources

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2010
Abstract: 

We present a new algorithm for the evaluation of the periodized Green’s function fora linear array of acoustic point sources such as those arising in the analysis of linearray loudspeakers. A variety of classical algorithms (based on spatial and spectralrepresentations, Ewald transformation, etc.) have been implemented in the past toevaluate these acoustic fields. However as we show, these methods become unstableand/or impractically expensive as the frequency of use of the sources increases. Herewe introduce a new numerical scheme that overcomes some of these limitations allowingfor simulations at unprecedentally large frequencies. The method is based ona new integral representation derived from the classic spatial form, and on suitablefurther manipulations of the relevant integrands to render the integrals amenable toefficient and accurate approximations through standard quadrature formulas. Weinclude a variety of numerical results that demonstrate that our algorithm comparesfavorably with several classical method both for points close to the line where thepoles are located and at high-frequencies while remaining competitive with them inevery other instance.

Document type: 
Article

On the Well-Posedness of the Stochastic Allen-Cahn Equation in Two Dimensions

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2011-12-31
Abstract: 

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d \geq 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen-Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that a series of published numerical studies are problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit.

Document type: 
Article

Numerical Integration for High Order Pyramidal Elements

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2011
Abstract: 

We examine the effect of numerical integration on the accuracy of high order conformingpyramidal finite element methods. Non-smooth shape functions are indispensable to the construction ofpyramidal elements, and this means the conventional treatment of numerical integration, which requiresthat the finite element approximation space is piecewise polynomial, cannot be applied. We developan analysis that allows the finite element approximation space to include non-smooth functions andshow that, despite this complication, conventional rules of thumb can still be used to select appropriatequadrature methods on pyramids. Along the way, we present a new family of high order pyramidalfinite elements for each of the spaces of the de Rham complex.

Document type: 
Article

Error Analysis of the DtN-FEM for the Scattering Problem in Acoustics via Fourier Analysis

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2011
Abstract: 

In this paper, we are concerned with the error analysis for the nite elementsolution of the two-dimensional exterior Neumann boundary value problem inacoustics. In particular, we establish an explicit priori error estimates in H1and L2- norms including both the e ect of the truncation of the DtN mappingand that of the numerical discretization. To apply the nite element method(FEM) to the exterior problem, the original boundary value problem is reducedto an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann(DtN) mapping represented in terms of the Fourier expansion series. We discussessential features of the corresponding variational equation and its modi cationdue to the truncation of the DtN mapping in appropriate function spaces. Nu-merical tests are presented to validate our theoretical results.

Document type: 
Article

Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2008
Abstract: 

For exterior scattering problems one of the chief difficulties arises from the unboundednature of the problem domain. Inhomogeneous obstacles may require a volumetricdiscretization, such as the Finite Element Method (FEM), and for this approach to be feasiblethe exterior domain must be truncated and an appropriate condition enforced at thefar, artificial, boundary. An exact, non-reflecting boundary condition can be stated usingthe classical DtN-FE method if the Artificial Boundary’s shape is quite specific: circularor elliptical. Recently, this approach has been generalized to permit quite general ArtificialBoundaries which are shaped as perturbations of a circle resulting in the “EnhancedDtN-FE” method. In this paper we extend this method to a two-dimensional FEM featuringhigh-order polynomials in order to realize a high rate of convergence. This is more involvedthan simply specifying high-order test and trial functions as now the scatterer shape andArtificial Boundary must be faithfully represented. This entails boundary elements whichconform (to high order) to the true boundary shapes. As we show, this can be accomplishedand we realize an arbitrary order FEM without spurious reflections.

Document type: 
Article
File(s): 

High-order Finite Elements on Pyramids: Approximation Spaces, Unisolvency and Exactness

Peer reviewed: 
Yes, item is peer reviewed.
Date created: 
2010-10-21
Abstract: 

We present a family of high-order finite element approximation spaces on a pyramid, and associated unisolvent degrees of freedom. These spaces consist of rational basis functions. We establish conforming, exactness and polynomial approximation properties.

Document type: 
Article
File(s):