On three conjectures by K. E. Shuler

Some fifteen years ago, Shuler formulated three conjectures relating to the large-time asymptotic properties of a nearest-neighbor random walk on ² that is allowed to make horizontal steps everywhere but vertical steps only on a random fraction of the columns. We give a proof of his conjectures for the situation where the column distribution is stationary and satisfies a certain mixing codition. We also prove a strong form of scaling to anisotropic Brownian motion as well as a local limit theorem. The main ingredient of the proofs is a large-deviation estimate for the number of visits to a random set made by a simple random walk on . We briefly discuss extensions to higher dimension and to other types of random walk.Dedicated to Prof. K. E. Shuler on the occasion of his 70th birthday, celebrated at a Symposium in his honor on July 13, 1992, at the University of California at San Diego, La Jolla, California.     

Structural aspects of antibody-antigen interaction revealed through small random peptide libraries

Summary Two small random peptide libraries, one composed of 4550 dodecapeptides and one of 8000 tripeptides, were synthesized in newly developed credit-card format miniPEPSCAN cards (miniPEPSCAN libraries). Each peptide was synthesized in a discrete well (455 peptides/card). The two miniPEPSCAN libraries were screened with three different monoclonal antibodies (Mabs). Two other random peptide libraries, expressed on the wall of bacteria (recombinant libraries) and composed of 10⁷ hexa- and octapeptides, were screened with the same three Mabs. The aim of this study was to compare the amino acid sequence of peptides selected from small and large pools of random peptides and, in this way, investigate the potential of small random peptide libraries. The screening of the two miniPEPSCAN libraries resulted in the identification of a surprisingly large number of antibody-binding peptides, while the screening of the large recombinant libraries, using the same Mabs, resulted in the identification of only a small number of peptides. The large number of peptides derived from the small random peptide libraries allowed the determination of consensus sequences. These consensus sequences could be related to small linear and nonlinear parts of the respective epitopes. The small number of peptides derived from the large random peptide libraries could only be related to linear epitopes that were previously mapped using small libraries of overlapping peptides covering the antigenic protein. Thus, with respect to the cost and speed of identifying peptides that resemble linear and nonlinear parts of epitopes, small diversity libraries based on synthetic peptides appear to be superior to large diversity libraries based on expression systems.Abbreviations ABTS 2,2-azino-di-3-ethylbenzthiazoline sulfonate - EGF epidermal growth factor - Mab monoclonal antibody - OD_ccd optical density obtained using CCD camera - RAMPO rabbit-antimouse peroxidase - SDS sodium dodecylsulfate - TGEV transmissible gastroenteritis virus     

A “Multiplicative Coboundary” theorem for some sequences of random matrices

Schmidt(¹⁰) showed that if the family of distributions of the partial sums of a strictly stationary random sequence is tight, then that random sequence is a coboundary. Here an analog of that result is proved for some sequences of random matrices, with partial sums replaced by matrix products.     

Critical behavior of self-avoiding walks on fractals

Using a new graph counting technique suitable for self-similar fractals, exact 18th-order series expansions for SAWs on some Sierpinski carpets are generated. From them, the critical fugacityx _c and critical exponents _SAW and _SAW are obtained. The results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied. We find for some carpets with low lacunarity that _SAW<0.75, thus violating the relation _SAW(fractal) > _SAW (d) for SAWs on the fractals which are embedded in ad-dimensional Euclidean space.     

Automated molecular design: A new fragment-joining algorithm

Summary A popular first step in the problem of structure-based, de novo molecule design is to identify regions where specific functional groups or chemical entities would be expected to interact strongly. When the three-dimensional structure of the receptor is not available, it may be possible to derive a pharmacophore giving the three-dimensional relationships between such chemical groups. The task then is to design synthetically feasible molecules which not only contain the required groups, but which can also position them in the desired relative orientation. One way to do this is to first link the groups using an acyclic chain. We have investigated the application of the tweak algorithm [Shenkin, P.S. et al., Biopolymers, 26 (1987) 2053] for generating families of acyclic linkers. These linking structures can subsequently be braced using a ring-joining algorithm [Leach, A.R. and Lewis, R.A., J. Comput. Chem., 15 (1994) 233], giving rise to an even wider variety of molecular skeletons for further studies.     

Thermal degradation of polystyrene in the presence of hydrogen by catalyst in solution

For the first time, low temperature degradation (170-240 °C) of polystyrene in benzene is carried out in the presence of hydrogen using iron(III) oxide catalyst. The effect of temperature, catalyst loading and polymer loading on degradation are studied in hydrogen atmosphere. Degradation is also carried out at different initial hydrogen partial pressure. The time dependent molecular weight is calculated using viscosity average method. It is found that the degradation is enhanced considerably in the presence of hydrogen and followed random degradation chain scission. A random degradation kinetic model of Kelen [Kelen T. Polymer degradation. New York: Van Nostrand Reinhold Company; 1983.] is used to estimate the degradation rate constants. Empirical correlations are proposed to account for the effect of catalyst loading and initial hydrogen partial pressure on degradation. The true thermal degradation rate constants are calculated using these proposed correlations at given catalyst loading and initial hydrogen partial pressure with varying temperature. The frequency factor and activation energy are also determined using Arrhenius equation considering the true thermal degradation rate constants.     

Computation of suboptimal randomized strategies for steering the random motion of a point under partial observation

Consider the random motion in the plane of a pointM, whose velocityv=(v ₁,v ₂) is perturbed by an ²-valued Gaussian white noise. Only noisy nonlinear observations taken on the point location (state) are available toM. The velocityv is of the formv(y)= _u (u ₁,u ₂) _y (du), wherey denotes the value of the observed signal,U is the range of the velocity, and, for eachy, _y is a probability measure on (U). Using the available observations, the pointM wishes to steer itself into a given target set by choosing a randomized strategy ={ _y :y ²}. Sufficient conditions on weak optimal randomized strategies are derived. An algorithm for computing weak suboptimal randomized strategies is suggested, and the strategies are computed for a variety of cases.This work was partially supported by a grant from Control Data.     

Trapping of random walks on the line

Several features of the trapping of random walks on a one-dimensional lattice are analyzed. The results of this investigation are as follows: (1) The correction term to the known asymptotic form for the survival probability ton steps is O(( ₂n)^–1/3), where =–ln(1–c), andc is the trap concentration. (2) The short time form for the survival probability is found to be exp[–a(c)n ^1/2], wherea(c) is given in Eq. (21). (3) The mean-square displacement of a surviving random walker is found to go liken ^2/3for largen. (4) When the distribution of trap-free regions is changed so that very large regions are much rarer than for ideally random trap placement the asymptotic survival probability changes its dependence onn. One such model is studied.     

On the statistical independence of various column contributions to band broadening. Part 1: Second moment contributions from statistically independent,longitudinal diffusion in both phases

The total length-based second moment contribution from longitudinal sample diffusion in both phases on a column, σD , is derived by adding individual partial differential contributions to a partial differential equation accounting for the longitudinal diffusion processes only. Although each diffusion-dispersed sample part is equilibrated between two phases, the resulting σ,D (= 2D _mt _m + 2D _st _s) can be interpreted as the sum of two independent contributions in accordance with the variance addition rule. (D _m and D _s are the mean diffusion coefficients and t _mand t _s the mean residence times of the sample in the mobile and stationary phases, respectively.) The same σD expression is derived from the random walk model of Giddings by treating the diffusional process in each phase as statistically independent of the other processes. Under these conditions the broadening contribution from longitudinal diffusion in the mobile phase is shown to be independent of the velocity profile.     

Random walks on inhomogeneous lattices

For lattices with two kinds of points (black and white), distributed according to a translation-invariant joint probability distribution, we study statistical properties of the sequence of consecutive colors encountered by a random walker moving through the lattice. The probability distribution for the single steps of the walk is considered to be independent of the colors of the points. Several exact results are presented which are valid in any number of dimensions and for arbitrary probability distributions for the coloring of the points and the steps of the walk. They are used to derive a few general properties of random walks on lattices containing traps.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.