An adaptive algorithm for simulation of stochastic reaction–diffusion processes

We propose an adaptive hybrid method suitable for stochastic simulation of diffusion dominated reaction–diffusion processes. For such systems, simulation of the diffusion requires the predominant part of the computing time. In order to reduce the computational work, the diffusion in parts of the domain is treated macroscopically, in other parts with the tau-leap method and in the remaining parts with Gillespie’s stochastic simulation algorithm (SSA) as implemented in the next subvolume method (NSM). The chemical reactions are handled by SSA everywhere in the computational domain. A trajectory of the process is advanced in time by an operator splitting technique and the timesteps are chosen adaptively. The spatial adaptation is based on estimates of the errors in the tau-leap method and the macroscopic diffusion. The accuracy and efficiency of the method are demonstrated in examples from molecular biology where the domain is discretized by unstructured meshes.     

QSAR with experimental and predictive distributions: an information theoretic approach for assessing model quality

We propose that quantitative structure–activity relationship (QSAR) predictions should be explicitly represented as predictive (probability) distributions. If both predictions and experimental measurements are treated as probability distributions, the quality of a set of predictive distributions output by a model can be assessed with Kullback–Leibler (KL) divergence: a widely used information theoretic measure of the distance between two probability distributions. We have assessed a range of different machine learning algorithms and error estimation methods for producing predictive distributions with an analysis against three of AstraZeneca’s global DMPK datasets. Using the KL-divergence framework, we have identified a few combinations of algorithms that produce accurate and valid compound-specific predictive distributions. These methods use reliability indices to assign predictive distributions to the predictions output by QSAR models so that reliable predictions have tight distributions and vice versa. Finally we show how valid predictive distributions can be used to estimate the probability that a test compound has properties that hit single- or multi- objective target profiles.     

Solubility and density measurements of palmitic acid in supercritical carbon dioxide + alcohol mixtures

Solubility data of palmitic acid (hexadecanoic acid) in supercritical carbon dioxide were measured in the pressure range from 10 to 25 MPa at temperatures of 313 and 318 K. Densities for this binary mixture in the homogeneous phase and at saturation conditions were measured in the same range of temperature. The influence of 3 and 6 mol% ethanol and 2-propanol as co-solvent on the solubility and density data for the CO₂ + palmitic acid mixture was also determined at 313 K. Measurements were carried out in a static-synthetic sapphire cell coupled to a vibrating-tube densitometer. The self-consistency of the data was tested according to the density-based models proposed by Mendez-Santiago and Teja.     

A jump operator on honest subrecursive degrees

It is well known that the structure of honest elementary degrees is a lattice with rather strong density properties. Let and denote respectively the join and the meet of the degrees and . This paper introduces a jump operator () on the honest elementary degrees and defines canonical degrees and low and high degrees analogous to the corresponding concepts for the Turing degrees. Among others, the following results about the structure of the honest elementary degrees are shown: There exist low degrees, and there exist degrees which are neither low nor high. Every degree above is the jump of some degree, moreover, for every degree above there exist degrees such that . We have and . The jump operator is of course monotonic, i.e. . We prove that every situation compatible with is realized in the structure, e.g. we have incomparable degrees such that and incomparable degrees such that etcetera. We are able to prove all these results without the traditional recursion theoretic constructions. Our proof method relies on the fact that the growth of the functions in a degree is bounded. This technique also yields a very simple proof of an old result, namely that the structure is a lattice. Received October 4, 1995     

Approximating the permanent: A simple approach

In this article, we present an efficient and very simple unbiased estimator for the Permanent of square (0–1) matrices. As the main result, we prove that our estimator, even though its worst case behavior is provably very bad, runs in time polynomial in the size of the input matrix for almost all matrices. We also generalize our technique and apply it to another enumeration problem, that of counting Hamilton cycles in directed graphs. The ease with which this was done suggests that the basic technique may have wide applicability. © 1994 John Wiley & Sons, Inc.