Christopher K. R. T. Jones

Proteolytic Activity of Human Osteoclast Cathepsin K EXPRESSION, PURIFICATION, ACTIVATION, AND SUBSTRATE IDENTIFICATION

Human cathepsin K is a recently identified protein with high primary sequence homology to members of the papain cysteine protease superfamily including cathepsins S, L, and B and is selectively expressed in osteoclasts (Drake, F. H., Dodds, R., James, I., Connor, J., Debouck, C., Richardson, S., Lee, E., Rieman, D., Barthlow, R., Hastings, G., and Gowen, M.(1996) J. Biol. Chem. 271, 12511-12516). To characterize its catalytic properties, cathepsin K has been expressed in baculovirus-infected SF21 cells and the soluble recombinant protein isolated from growth media was purified. Purified protein includes an inhibitory pro-leader sequence common to this family of protease. Conditions for enzyme activation upon removal of the pro-sequence have been identified. Fluorogenic peptides have been identified as substrates for mature cathepsin K. In addition, two protein components of bone matrix, collagen and osteonectin, have been shown to be substrates of the activated protease. Cathepsin K is inhibited by E-64 and leupeptin, but not by pepstatin, EDTA, phenylmethylsulfonyl fluoride, or phenanthroline, consistent with its classification within the cysteine protease class. Leupeptin has been characterized as a slow binding inhibitor of cathepsin K (k/[I] = 273,000 M•s). Cathepsin K may represent the elusive protease implicated in degradation of protein matrix during bone resorption and represents a novel molecular target in treatment of disease states associated with excessive bone loss such as osteoporosis.

A topological invariant arising in the stability analysis of travelling waves

Travelling waves are special Solutions of partial differential equations in one space variable. They are characterized by their time invariant profile; indeed, s Solutions they evolve by translating at constant speed in the one spatial dimension. These Solutions are often the centerpiece of a physical System s they represent the transport of Information in a single direction. It is of fundamental importance for a given travelling wave to determine its stability relative to perturbations in the initial conditions for Solutions of the f ll partial differential equations. Stable Solutions are the most physically realistic since the external world provides enough perturbations that we can only expect to see waves which will dampen out these perturbations.

Invariant Manifolds for Semilinear Partial Differential Equations

When studying the behaviour of a dynamical system in the neighbourhood of an equilibrium point the first step is to construct the stable, unstable and centre manifolds. These are manifolds that are invariant under the flow relative to a neighbourhood of the equilibrium point and carry the solutions that decay or grow (or neither) at certain rates. These ideas have a long history, see for instance Poincare [32] and Hadamard [11]. Sophisticated recent results can be found in Fenichel [7], Hirsch, Pugh and Shub [17] and Kelley [22].

A Method for Assimilation of Lagrangian Data

Abstract Difficulties in the assimilation of Lagrangian data arise because the state of the prognostic model is generally described in terms of Eulerian variables computed on a fixed grid in space, as a result there is no direct connection between the model variables and Lagrangian observations that carry time-integrated information. A method is presented for assimilating Lagrangian tracer positions, observed at discrete times, directly into the model. The idea is to augment the model with tracer advection equations and to track the correlations between the flow and the tracers via the extended Kalman filter. The augmented model state vector includes tracer coordinates and is updated through the correlations to the observed tracers. The technique is tested for point vortex flows: an NF point vortex system with a Gaussian noise term is modeled by its deterministic counterpart. Positions of ND tracer particles are observed at regular time intervals and assimilated into the model. Numerical experiments demon...

Ultra-short pulses in linear and nonlinear media

We consider the evolution of ultra-short optical pulses in linear and nonlinear media. For the linear case, we first show that the initial-boundary value problem for Maxwells equations in which a pulse is injected into a quiescent medium at the left endpoint can be approximated by a linear wave equation which can then be further reduced to the linear short-pulse equation (SPE). A rigorous proof is given that the solution of the SPE stays close to the solutions of the original wave equation over the time scales expected from the multiple scales derivation of the SPE. For the nonlinear case we compare the predictions of the traditional nonlinear Schrodinger equation (NLSE) approximation with those of the SPE. We show that both equations can be derived from Maxwells equations using the renormalization group method, thus bringing out the contrasting scales. The numerical comparison of both equations with Maxwells equations shows clearly that as the pulse length shortens, the NLSE approximation becomes steadily less accurate, while the SPE provides a better and better approximation.

Quantifying transport in numerically generated velocity fields

Abstract Geometric methods from dynamical systems are used to study Lagrangian transport in numerically generated, time-dependent, two-dimensional (2D) vector fields. The flows analyzed here are numerical solutions to the barotropic, β-plane, potential vorticity equation with viscosity, where the partial differential equation (PDE) parameters have been chosen so that the solution evolves to a meandering jet. Numerical methods for approximating invariant manifolds of hyperbolic fixed points for maps are successfully applied to the aperiodic vector field where regions of strong hyperbolicity persist for long times relative to the dominant time period in the flow. Cross sections of these 2D “stable” and “unstable” manifolds show the characteristic transverse intersections identified with chaotic transport in 2D maps, with the lobe geometry approximately recurring on a time scale equal to the dominant time period in the vector field. The resulting lobe structures provide time-dependent estimates for the transport between different flow regimes. Additional numerical experiments show that the computation of such lobe geometries are very robust relative to variations in interpolation, integration and differentiation schemes.

Tracking invariant manifolds up to exponentially small errors

This work establishes a new tool for proving the existence of multiple-pulse homoclinic orbits in perturbed Hamiltonian systems and general multidimensional singular-perturbation problems. The center-stable and center-unstable manifolds of slow manifolds in these problems intersect transversely at angles that are of the same order as the asymptotically small parameter in the problem, which can be either an amplitude or a frequency. To deal with the difficulties associated with small angles of intersection, we develop the exchange lemma with exponentially small error (ELESE), which is the main technical result of this work. This lemma enables highly accurate tracking of invariant manifolds while orbits on them spend long intervals of time near slow manifolds.

On the infinitely many solutions of a semilinear elliptic equation

A dynamical systems approach is developed for studying the spherically symmetric solutions of

Lagrangian Motion and Fluid Exchange in a Barotropic Meandering Jet

\Delta u + f(u) = 0

Instability of standing waves in nonlinear optical waveguides

, where