Erratum to: Clustering of $$ \overline{B}\to {D}^{\left(\ast \right)}{\tau}^{-}{\overline{v}}_{\tau } $$ kinematic distributions with ClusterKinG

However, this changes the interpretation of our results but slightly, because the exact probability distribution of the metric is only important for the choice and interpretation of the stopping criterium of the clustering.

Blinking statistics and molecular counting in direct stochastic reconstruction microscopy (dSTORM)

Motivation Many recent advancements in single-molecule localization microscopy exploit the stochastic photoswitching of fluorophores to reveal complex cellular structures beyond the classical diffraction limit. However, this same stochasticity makes counting the number of molecules to high precision extremely challenging, preventing key insight into the cellular structures and processes under observation. Results Modelling the photoswitching behaviour of a fluorophore as an unobserved continuous time Markov process transitioning between a single fluorescent and multiple dark states, and fully mitigating for missed blinks and false positives, we present a method for computing the exact probability distribution for the number of observed localizations from a single photoswitching fluorophore. This is then extended to provide the probability distribution for the number of localizations in a direct stochastic optical reconstruction microscopy experiment involving an arbitrary number of molecules. We demonstrate that when training data are available to estimate photoswitching rates, the unknown number of molecules can be accurately recovered from the posterior mode of the number of molecules given the number of localizations. Finally, we demonstrate the method on experimental data by quantifying the number of adapter protein linker for activation of T cells on the cell surface of the T-cell immunological synapse. Availability and implementation Software and data available at https://github.com/lp1611/mol_count_dstorm. Supplementary information Supplementary data are available at Bioinformatics online.

Trees grown under young-age preferential attachment

We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.

Exact Distribution of Divergence Times from Fossil Ages and Tree Topologies

Being given a phylogenetic tree of both extant and extinct taxa in which the fossil ages are the only temporal information (namely, in which divergence times are considered unknown), we provide a method to compute the exact probability distribution of any divergence time of the tree with regard to any speciation (cladogenesis), extinction, and fossilization rates under the Fossilized Birth–Death model. We use this new method to obtain a probability distribution for the age of Amniota (the synapsid/sauropsid or bird/mammal divergence), one of the most-frequently used dating constraints. Our results suggest an older age (between about 322 and 340 Ma) than has been assumed by most studies that have used this constraint (which typically assumed a best estimate around 310–315 Ma) and provide, for the first time, a method to compute the shape of the probability density for this divergence time. [Divergence times; fossil ages; fossilized birth–death model; probability distribution.]

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height h(x,t)h(x,t) on the positive half line with boundary condition \partial_x h(x,t)|_{x=0}=A∂xh(x,t)|x=0=A. It is equivalent to a continuum directed polymer (DP) in a random potential in half-space with a wall at x=0x=0 either repulsive A>0A>0, or attractive A<0A<0. We provide an exact solution, using replica Bethe ansatz methods, to two problems which were recently proved to be equivalent [Parekh, arXiv:1901.09449]: the droplet initial condition for arbitrary A \geqslant -1/2A≥−1/2, and the Brownian initial condition with a drift for A=+\inftyA=+∞ (infinite hard wall). We study the height at x=0x=0 and obtain (i) at all time the Laplace transform of the distribution of its exponential (ii) at infinite time, its exact probability distribution function (PDF). These are expressed in two equivalent forms, either as a Fredholm Pfaffian with a matrix valued kernel, or as a Fredholm determinant with a scalar kernel. For droplet initial conditions and A> - \frac{1}{2}A>−12 the large time PDF is the GSE Tracy-Widom distribution. For A= \frac{1}{2}A=12, the critical point at which the DP binds to the wall, we obtain the GOE Tracy-Widom distribution. In the critical region, A+\frac{1}{2} = \epsilon t^{-1/3} \to 0A+12=ϵt−1/3→0 with fixed \epsilon = \mathcal{O}(1)ϵ=𝒪(1), we obtain a transition kernel continuously depending on \epsilonϵ. Our work extends the results obtained previously for A=+\inftyA=+∞, A=0A=0 and A=- \frac{1}{2}A=−12.

On testing for seed sample heterogeneity with the exact probability distribution of the germination count range

Seed lot heterogeneity is often evaluated through the range between germination percentages of four seed samples, considering normal and binomial approximations for calculating the tolerated range (S). In this paper, an exact test for the germination count range (R) is derived based on the hypergeometric and the binomial probability model for germination count. Through Monte Carlo simulations, the empirical distribution of R is built to evaluate the quantiles of the exact distributions. Moreover, a power analysis is performed by simulation. Sample size and germination rate effects are evaluated. In lots with a high germination rate, the proposed test based on the hypergeometric model is about 20% more powerful than the test based on the S-value. A table containing the critical values is presented and recommended to be used in off-range heterogeneity testing.

Blinking Statistics and Molecular Counting in direct Stochastic Reconstruction Microscopy (dSTORM)

Many recent advancements in single molecule localization microscopy exploit the stochastic photo-switching of fluorophores to reveal complex cellular structures beyond the classical diffraction limit. However, this same stochasticity makes counting the number of molecules to high precision extremely challenging. Modeling the photo-switching behavior of a fluorophore as a continuous time Markov process transitioning between a single fluorescent and multiple dark states, and fully mitigating for missed blinks and false positives, we present a method for computing the exact probability distribution for the number of observed localizations from a single photo-switching fluorophore. This is then extended to provide the probability distribution for the number of localizations in a dSTORM experiment involving an arbitrary number of molecules. We demonstrate that when training data is available to estimate photo-switching rates, the unknown number of molecules can be accurately recovered from the posterior mode of the number of molecules given the number of localizations.

Distributionally Robust Stochastic Model Predictive Control for Collision Avoidance

This paper focuses on distributionally robust controller design for avoiding dynamic and stochastic obstacles whose exact probability distribution is unknown. The true probability distribution of the disturbance associated with an obstacle, although unknown, is considered to belong to an ambiguity set that includes all the probability distributions that share the same first two moment. The controller thus focuses on ensuring the satisfaction of the probabilistic collision avoidance constraints for all probability distributions in the ambiguity set, hence making the solution robust to the true probability distribution of the stochastic obstacles. Techniques from robust optimization methods are used to model the distributionally robust probabilistic or chance constraints as a semi-definite programming (SDP) problem with linear matrix inequality (LMI) constraints that can be solved in a computationally tractable fashion. Simulation results for a robot obstacle avoidance problem shows the efficacy of our method.

Exact distribution of divergence times from fossil ages and tree topologies

Being given a phylogenetic tree of both extant and extinct taxa in which the fossil ages are the only temporal information (namely, in which divergence times are considered unknown), we provide a method to compute the exact probability distribution of any divergence time of the tree with regard to any speciation (cladogenesis), extinction and fossilization rates under the Fossilized-Birth-Death model.We use this new method to obtain a probability distribution for the age of Amniota (the synapsid/sauropsid or bird/mammal divergence), one of the most-frequently used dating constraints. Our results suggest an older age (between about 322 and 340 Ma) than has been assumed by most studies that have used this constraint (which typically assumed a best estimate around 310-315 Ma) and provide, for the first time, a method to compute the shape of the probability density for this divergence time.