Robust filtering: Correlated noise and multidimensional observation

In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff] pointed out that it would be natural for $\pi_t$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y=\{Y_s,s\in[0,t]\}$. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function $f$, there exists a continuous map $\theta^f_t$, defined on the space of continuous paths $C([0,t],\mathbb{R}^d)$ endowed with the uniform convergence topology such that $\pi_t(f)=\theta^f_t(Y)$, almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43-56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125-139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160-167], Kushner [Stochastics 3 (1979) 75-83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260-278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505-528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403-424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process $Y$ is "lifted" to the process $\mathbf{Y}$ that consists of $Y$ and its corresponding L\'{e}vy area process, and we show that there exists a continuous map $\theta_t^f$, defined on a suitably chosen space of H\"{o}lder continuous paths such that $\pi_t(f)=\theta_t^f(\mathbf{Y})$, almost surely.Comment: Published in at http://dx.doi.org/10.1214/12-AAP896 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stretching Single Domain Proteins: Phase Diagram and Kinetics of Force-Induced Unfolding

Single molecule force spectroscopy reveals unfolding of domains in titin upon stretching. We provide a theoretical framework for these experiments by computing the phase diagrams for force-induced unfolding of single domain proteins using lattice models. The results show that two-state folders (at zero force) unravel cooperatively whereas stretching of non-two-state folders occurs through intermediates. The stretching rates of individual molecules show great variations reflecting the heterogeneity of force-induced unfolding pathways. The approach to the stretched state occurs in a step-wise "quantized" manner. Unfolding dynamics depends sensitively on topology. The unfolding rates increase exponentially with force f till an optimum value which is determined by the barrier to unfolding when f=0. A mapping of these results to proteins shows qualitative agreement with force-induced unfolding of Ig-like domains in titin. We show that single molecule force spectroscopy can be used to map the folding free energy landscape of proteins in the absence of denaturants.Comment: 12 pages, Latex, 6 ps figure

Mechanical response of random heteropolymers

We present an analytical theory for heteropolymer deformation, as exemplified experimentally by stretching of single protein molecules. Using a mean-field replica theory, we determine phase diagrams for stress-induced unfolding of typical random sequences. This transition is sharp in the limit of infinitely long chain molecules. But for chain lengths relevant to biological macromolecules, partially unfolded conformations prevail over an intermediate range of stress. These necklace-like structures, comprised of alternating compact and extended subunits, are stabilized by quenched variations in the composition of finite chain segments. The most stable arrangements of these subunits are largely determined by preferential extension of segments rich in solvophilic monomers. This predicted significance of necklace structures explains recent observations in protein stretching experiments. We examine the statistical features of select sequences that give rise to mechanical strength and may thus have guided the evolution of proteins that carry out mechanical functions in living cells.Comment: 10 pages, 6 figure

Reversible stretching of homopolymers and random heteropolymers

We have analyzed the equilibrium response of chain molecules to stretching. For a homogeneous sequence of monomers, the induced transition from compact globule to extended coil below the $\theta$-temperature is predicted to be sharp. For random sequences, however, the transition may be smoothed by a prevalence of necklace-like structures, in which globular regions and coil regions coexist in a single chain. As we show in the context of a random copolymer, preferential solvation of one monomer type lends stability to such structures. The range of stretching forces over which necklaces are stable is sensitive to chain length as well as sequence statistics.Comment: 14 pages, 4 figure

Projet « cabinets de groupe » : collaboration entre médecine de premier recours et psychiatrie [« Group medical practices » project : collaboration between primary care medicine and institutional public psychiatry]

Collaboration between primary care medicine and psychiatry is a well-known challenge. In order to improve access to psychological care for patients undergoing primary care, the « group medical practices » project proposes a collaborative care model in which a psychiatrist employed by a public psychiatric institution integrates group medical practices in order to provide assistance to primary care physicians. It is thus able to evaluate patients directly in the practices and to offer supervision and consilium spaces to primary care physicians

Mechanical Strength of 17 134 Model Proteins and Cysteine Slipknots

A new theoretical survey of proteins' resistance to constant speed stretching is performed for a set of 17 134 proteins as described by a structure-based model. The proteins selected have no gaps in their structure determination and consist of no more than 250 amino acids. Our previous studies have dealt with 7510 proteins of no more than 150 amino acids. The proteins are ranked according to the strength of the resistance. Most of the predicted top-strength proteins have not yet been studied experimentally. Architectures and folds which are likely to yield large forces are identified. New types of potent force clamps are discovered. They involve disulphide bridges and, in particular, cysteine slipknots. An effective energy parameter of the model is estimated by comparing the theoretical data on characteristic forces to the corresponding experimental values combined with an extrapolation of the theoretical data to the experimental pulling speeds. These studies provide guidance for future experiments on single molecule manipulation and should lead to selection of proteins for applications. A new class of proteins, involving cystein slipknots, is identified as one that is expected to lead to the strongest force clamps known. This class is characterized through molecular dynamics simulations.Comment: 40 pages, 13 PostScript figure

Spatio-temporal dynamics of quantum-well excitons

We investigate the lateral transport of excitons in ZnSe quantum wells by using time-resolved micro-photoluminescence enhanced by the introduction of a solid immersion lens. The spatial and temporal resolutions are 200 nm and 5 ps, respectively. Strong deviation from classical diffusion is observed up to 400 ps. This feature is attributed to the hot-exciton effects, consistent with previous experiments under cw excitation. The coupled transport-relaxation process of hot excitons is modelled by Monte Carlo simulation. We prove that two basic assumptions typically accepted in photoluminescence investigations on excitonic transport, namely (i) the classical diffusion model as well as (ii) the equivalence between the temporal and spatial evolution of the exciton population and of the measured photoluminescence, are not valid for low-temperature experiments.Comment: 8 pages, 6 figure

Single Molecule Statistics and the Polynucleotide Unzipping Transition

We present an extensive theoretical investigation of the mechanical unzipping of double-stranded DNA under the influence of an applied force. In the limit of long polymers, there is a thermodynamic unzipping transition at a critical force value of order 10 pN, with different critical behavior for homopolymers and for random heteropolymers. We extend results on the disorder-averaged behavior of DNA's with random sequences to the more experimentally accessible problem of unzipping a single DNA molecule. As the applied force approaches the critical value, the double-stranded DNA unravels in a series of discrete, sequence-dependent steps that allow it to reach successively deeper energy minima. Plots of extension versus force thus take the striking form of a series of plateaus separated by sharp jumps. Similar qualitative features should reappear in micromanipulation experiments on proteins and on folded RNA molecules. Despite their unusual form, the extension versus force curves for single molecules still reveal remnants of the disorder-averaged critical behavior. Above the transition, the dynamics of the unzipping fork is related to that of a particle diffusing in a random force field; anomalous, disorder-dominated behavior is expected until the applied force exceeds the critical value for unzipping by roughly 5 pN.Comment: 40 pages, 18 figure