On the relationships between kinetic schemes and two-state single molecule trajectories

Trajectories of a signal that fluctuates between two states which originate from single molecule activities have become ubiquitous. Common examples are trajectories of ionic flux through individual membrane-channels, and of photon counts collected from diffusion, activity, and conformational changes of biopolymers. By analyzing the trajectory, one wishes to deduce the underlying mechanism, which is usually described by a multi-substate kinetic scheme. In previous works, we divided kinetic schemes that generate two-state trajectories into two types: reducible schemes and irreducible schemes. We showed that all the information in trajectories generated from reducible schemes is contained in the waiting time probability density functions (PDFs) of the two states. It follows that reducible schemes with the same waiting time PDFs are not distinguishable. In this work, we further characterize the topologies of kinetic schemes, now of irreducible schemes, and further study two-state trajectories from the two types of scheme. We suggest various methods for extracting information about the underlying kinetic scheme from the trajectory (e. g., calculate the binned successive waiting times PDF and analyze the ordered waiting times trajectory), and point out the advantages and disadvantages of each. We show that the binned successive waiting times PDF is not only more robust than other functions when analyzing finite trajectories, but contains, in most cases, more information about the underlying kinetic scheme than other functions in the limit of infinitely long trajectories. For some cases however, analyzing the ordered waiting times trajectory may supply unique information about the underlying kinetic scheme

Properties of best approximation with interpolatory and restricted range side conditions

AbstractAn alternation property of polynomials of best uniform approximation to a function | ϵ C[a, b] having restricted ranges of some of their derivatives is proven. For this purpose, the problem of best uniform approximation to continuous functions by polynomials having restricted ranges and satisfying interpolatory conditions on their derivatives is discussed. The method is an improved version of the one used in [3] and provides an easily computed lower bound for the number of alternations

Best uniform approximation with Hermite-Birkhoff interpolatory side conditions

AbstractBest approximation to continuous functions by polynomials satisfying Hermite-Birkhoff interpolation conditions is discussed. Characterization, sufficient conditions for uniqueness, and the alternation property of these polynomials are studied. The results obtained extend work on best approximation with interpolatory side conditions of Hermite type. By this extension the space of polynomials that plays a role in the approximation is no longer a Haar space, and the results depend strongly on the structure of the side conditions

A necessary condition for best approximation in monotone and sign-monotone norms

AbstractBest approximation to ƒ ϵ C[a, b] by elements of an n-dimensional Tchebycheff space in monotone norms (norms defined on C[a, b] for which ¦ƒ(x)¦ ⩽ ¦ g(x)¦, a ⩽ x ⩽ b, implies ∥ƒ∥ ⩽ ∥g∥) is studied. It is proved that the error function has at least n zeroes in [a, b], counting twice interior zeroes with no change of sign. This result is best possible for monotone norms in general, and improves the one in [5]. The proof follows from the observation that, for any monotone norm, sgn ƒ(x) = sgn g(x), a ⩽ x ⩽ b, implies ∥ƒ− λg ∥ < ∥ƒ∥ for λ > 0 small enough. This property is shown to characterize a class of norms wider than the class of monotone norms, namely “sign-monotone” norms defined by: ¦ƒ(x)¦ ⩽ ¦g(x)¦, ƒ(x) g(x) ⩾ 0, a ⩽ x ⩽ b, implies ∥ƒ∥ ⩽ ∥g∥. It is noted that various results concerning approximation in monotone norms, are actually valid for approximation in sign-monotone norms

Convergence properties of sequences of functions with application to restricted derivative approximation

AbstractConvergence properties of sequences of continuous functions, with kth order divided differences bounded from above or below, are studied. It is found that for such sequences, convergence in a “monotone norm” (e.g., Lp) on [a, b] to a continuous function implies uniform convergence of the sequence and its derivatives up to order k − 1 (whenever they exist), in any closed subinterval of [a, b]. Uniform convergence in the closed interval [a, b] follows from the boundedness from below and above of the kth order divided differences. These results are applied to the estimation of the degree of approximation in Monotone and Restricted Derivative approximation, via bounds for the same problems with only one restricted derivative

“Restricted derivatives” approximation to functions with derivatives outside the range

AbstractThe problem of best uniform approximation by polynomials with restricted ranges of some of their derivatives to functions not satisfying the same restrictions is treated. Results concerning the number of alternations of the best approximating polynomial are derived, and the impossibility of approximating arbitrarily closely functions with one derivative outside the range is proved