Topological Stability of Kinetic kk-Centers

We study the $k$-center problem in a kinetic setting: given a set of continuously moving points $P$ in the plane, determine a set of $k$ (moving) disks that cover $P$ at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of $k$-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution---the topological stability ratio---considering various metrics and various optimization criteria. For $k = 2$ we provide tight bounds, and for small $k > 2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant $k$

A Framework for Algorithm Stability

We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees

On the nonlinear dynamics of topological solitons in DNA

Dynamics of topological solitons describing open states in the DNA double helix are studied in the frameworks of the model which takes into account asymmetry of the helix. It is shown that three types of topological solitons can occur in the DNA double chain. Interaction between the solitons, their interactions with the chain inhomogeneities and stability of the solitons with respect to thermal oscillations are investigated.Comment: 16 pages, 16 figure

Macromolecular crowding modulates folding mechanism of alpha/beta protein apoflavodoxin

Protein dynamics in cells may be different from that in dilute solutions in vitro since the environment in cells is highly concentrated with other macromolecules. This volume exclusion due to macromolecular crowding is predicted to affect both equilibrium and kinetic processes involving protein conformational changes. To quantify macromolecular crowding effects on protein folding mechanisms, here we have investigated the folding energy landscape of an alpha/beta protein, apoflavodoxin, in the presence of inert macromolecular crowding agents using in silico and in vitro approaches. By coarse-grained molecular simulations and topology-based potential interactions, we probed the effects of increased volume fraction of crowding agents (phi_c) as well as of crowding agent geometry (sphere or spherocylinder) at high phi_c. Parallel kinetic folding experiments with purified Desulfovibro desulfuricans apoflavodoxin in vitro were performed in the presence of Ficoll (sphere) and Dextran (spherocylinder) synthetic crowding agents. In conclusion, we have identified in silico crowding conditions that best enhance protein stability and discovered that upon manipulation of the crowding conditions, folding routes experiencing topological frustrations can be either enhanced or relieved. The test-tube experiments confirmed that apoflavodoxin's time-resolved folding path is modulated by crowding agent geometry. We propose that macromolecular crowding effects may be a tool for manipulation of protein folding and function in living cells.Comment: to appear in Biophysical Journal (2009). to appear in Biophysical Journal (2009

Water and molecular chaperones act as weak links of protein folding networks: energy landscape and punctuated equilibrium changes point towards a game theory of proteins

Water molecules and molecular chaperones efficiently help the protein folding process. Here we describe their action in the context of the energy and topological networks of proteins. In energy terms water and chaperones were suggested to decrease the activation energy between various local energy minima smoothing the energy landscape, rescuing misfolded proteins from conformational traps and stabilizing their native structure. In kinetic terms water and chaperones may make the punctuated equilibrium of conformational changes less punctuated and help protein relaxation. Finally, water and chaperones may help the convergence of multiple energy landscapes during protein-macromolecule interactions. We also discuss the possibility of the introduction of protein games to narrow the multitude of the energy landscapes when a protein binds to another macromolecule. Both water and chaperones provide a diffuse set of rapidly fluctuating weak links (low affinity and low probability interactions), which allow the generalization of all these statements to a multitude of networks.Comment: 9 pages, 1 figur

Two Skyrmion Dynamics with Omega Mesons

We present our first results of numerical simulations of two skyrmion dynamics using an $\omega$-meson stabilized effective Lagrangian. We consider skyrmion-skyrmion scattering with a fixed initial velocity of $\beta=0.5$, for various impact parameters and groomings. The physical picture that emerges is surprisingly rich, while consistent with previous results and general conservation laws. We find meson radiation, skyrmion scattering out of the scattering plane, orbiting and capture to bound states.Comment: 19 pages, 22 figure

Topological phase separation in 2D hard-core Bose-Hubbard system away from half-filling

We suppose that the doping of the 2D hard-core boson system away from half-filling may result in the formation of multi-center topological defect such as charge order (CO) bubble domain(s) with Bose superfluid (BS) and extra bosons both localized in domain wall(s), or a {\it topological} CO+BS {\it phase separation}, rather than an uniform mixed CO+BS supersolid phase. Starting from the classical model we predict the properties of the respective quantum system. The long-wavelength behavior of the system is believed to remind that of granular superconductors, CDW materials, Wigner crystals, and multi-skyrmion system akin in a quantum Hall ferromagnetic state of a 2D electron gas.Comment: 6 pages, 1 figur