Stochastic epidemics: the probability of extinction of an infectious disease at the end of a major outbreak

 The aim of this study is to derive an asymptotic expression for the probability that an infectious disease will disappear from a population at the end of a major outbreak (‘fade-out’). The study deals with a stochastic SIR-model. Local asymptotic expansions are constructed for the deterministic trajectories of the corresponding deterministic system, in particular for the deterministic trajectory starting in the saddle point. The analytical expression for the probability of extinction is derived by asymptotically solving a boundary value problem based on the Fokker-Planck equation for the stochastic system. The asymptotic results are compared with results obtained by random walk simulations. Received 20 July 1995; received in revised form 6 May 1996     

Morphological variability in response to palaeoenvironmental change – a case study on Cretaceous ammonites

Changes in ammonite morphology during the earliest Early Aptian (Cretaceous) of an epicontinental sea in northern Germany were investigated based on new and rich material of Deshayesites (Deshayesitidae). This is a globally distributed genus and one of the most important ammonites of the Cretaceous with respect to biostratigraphy and abundance. The purpose of our study was to describe changes in morphology over few hundred thousand years and to discuss their relationships to major palaeoenvironmental perturbations. Our material is derived from four different horizons, and the studied stratigraphical interval includes a sea‐level change, a period of warming and an oxygen depletion event. We observe variable patterns in the occurrence of different morphological groups through time, probably indicating that morphotypes were individually impacted by environmental change. These morphological groups, unequivocally attributed to Deshayesites, cannot be fit into the existing species classification system. The greatest morphological disparity in Deshayesites is multi‐causal. It may be the result of: (1) an invasion of two morphogroups new to the habitat, which thus immediately face considerably greater competition in an environment just recovered from low‐oxygen conditions in parts of the water column during an early interval within Oceanic Anoxic Event 1a (OAE 1a); or, (2) it may be due to a rise in sea level with a simultaneous invasion of competitors of Tethyan origin. This sheds new light on the current concept for high morphological variability within ammonite species and poses challenges to our current ideas about ammonite diversity and the use of well‐established index‐species in supraregional correlations.     

Extended Impact of Pin1 Catalytic Loop Phosphorylation Revealed by S71E Phosphomimetic

Pin1 is a two-domain human protein that catalyzes the cis–trans isomerization of phospho-Ser/Thr–Pro (pS/T–P) motifs in numerous cell-cycle regulatory proteins. These pS/T–P motifs bind to Pin1's peptidyl-prolyl isomerase (PPIase) domain in a catalytic pocket, between an extended catalytic loop and the PPIase domain core. Previous studies showed that post-translational phosphorylation of S71 in the catalytic loop decreases substrate binding affinity and isomerase activity. To define the origins for these effects, we investigated a phosphomimetic Pin1 mutant, S71E-Pin1, using solution NMR. We find that S71E perturbs not only its host loop but also the nearby PPIase core. The perturbations identify a local network of hydrogen bonds and salt bridges that is more extended than previously thought, and includes interactions between the catalytic loop and the α2/α3 turn in the PPIase core. Explicit-solvent molecular dynamics simulations and phylogenetic analysis suggest that these interactions act as conserved “latches” between the loop and PPIase core that enhance binding of phosphorylated substrates, as they are absent in PPIases lacking pS/T–P specificity. Our results suggest that S71 is a hub residue within an electrostatic network primed for phosphorylation, and may illustrate a common mechanism of phosphorylation-mediated allostery.     

The multiple-minima problem in the conformational analysis of polypeptides. III. An Electrostatically Driven Monte Carlo Method: Tests on enkephalin

The three-dimensional conformation of Met-enkephalin, corresponding to the lowest minimum of the empirical potential energy function ECEPP/2 (empirical conformational energy program for peptides), has been determined using a new algorithm, viz. the Electrostatically Driven Monte Carlo Method. This methodology assumes that a polypeptide or protein molecule is driven toward the native structure by the combined action of electrostatic interactions and stochastic conformational changes associated with thermal movements. These features are included in the algorithm that produces a Monte Carlo search in the conformational hyperspace of the polypeptide, using electrostatic predictions and a random sampling technique to locate low-energy conformations. In addition, we have incorporated an alternative mechanism that allows the structure to escape from some conformational regions representing metastable local energy minima and even from regions of the conformational space with great stability. In 33 test calculations on Met-enkephalin, starting from arbitrary or completely random conformations, the structure corresponding to the global energy minimum was found inall the cases analyzed, with a relatively small search of the conformational space. Some of these starting conformations wereright orleft-handed -helices, characterized by good electrostatic interactions involving their backbone peptide dipoles; nevertheless, the procedure was able to convert such locally stable structures to the global-minimum conformation.On leave from the National University of San Luis, Faculty of Sciences and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Instituto de Matemática Aplicada, San Luis, Ejército de los Andes 950, 5700 San Luis, Argentina.     

The effect of epidemics on genetic evolution

Mathematical models of a vector-borne infectious disease acting on a host population consisting of three genotypes which differ in susceptibility to, recovery from, and death due to the disease are presented and analyzed. Singular perturbation techniques are used to obtain a single differential equation describing the slow time evolution of gene frequencies.Karen Christine Beck died June 25, 1983 at home.Born February 8, 1952 in Madison, Wisconsin, She received a B.A. degree in 1974 from Luther College, Decorah, Iowa and a Ph.D. in mathematics in 1980 from the University of Iowa. Since that time she has been an instructor in the Mathematics Department at the University of Utah. She was to become an Assistant Professor at the University of Texas, Arlington, beginning Autumn, 1983. Dr. Beck's areas of specialization in mathematics were Mathematical Analysis and Mathematical Biology. She published numerous research articles that resolved various problems in these areas.     

Weak-field ELF magnetic interactions: Implications for biological change during paleomagnetic reversals

Contrary to the belief that paleomagnetic reversals are not biologically significant, we find good reason to think otherwise. Attention is drawn to polarity transitions, time intervals a few thousand years long that follow the collapse of the existing geomagnetic dipole moment and precede the establishment of the new, oppositely directed moment. The geomagnetic field during transitions is reduced to a maximal mean intensity about 10% of the stable field and can exhibit low-frequency perturbations comparable to numerous laboratory-based extremely low frequency (ELF) studies reporting biological interactions, making it very likely that similar interactions must occur over the course of a polarity transition. This conclusion is strengthened by reports of medical problems that significantly correlate with intense solar winds, events that also generate ELF perturbations similar to those that can occur during polarity transitions.     

The SEP domain of p47 acts as a reversible competitive inhibitor of cathepsin L

The solution structure of the human p47 SEP domain in a construct comprising residues G1-S2-p47(171-270) was determined by NMR spectroscopy. A structure-derived hypothesis about the domains' function was formulated and pursued in binding experiments with cysteine proteases. The SEP domain was found to be a reversible competitive inhibitor of cathepsin L with a K_i of 1.5 μM. The binding of G1-S2-p47(171-270) to cathepsin L was mapped by biochemical assays and the binding interface was investigated by NMR chemical shift perturbation experiments.     

Targeted reduction of complex models with time scale hierarchy--a case study

With the increasing flow of biological data there is a growing demand for mathematical tools whereby essential aspects of complex causal dynamic models can be captured and detected by simpler mathematical models without sacrificing too much of the realism provided by the original ones. Given the presence of a time scale hierarchy, singular perturbation techniques represent an elegant method for making such minimised mathematical representations. Any reduction of a complex model by singular perturbation methods is a targeted reduction by the fact that one has to pick certain mechanisms, processes or aspects thought to be essential in a given explanatory context. Here we illustrate how such a targeted reduction of a complex model of melanogenesis in mammals recently developed by the authors provides a way to improve the understanding of how the melanogenic system may behave in a switch-like manner between production of the two major types of melanins. The reduced model is shown by numerical means to be in good quantitative agreement with the original model. Furthermore, it is shown how the reduced model discloses hidden robustness features of the full model, and how the making of a reduced model represents an efficient analytical sensitivity analysis. In addition to yielding new insights concerning the melanogenic system, the paper provides an illustration of a protocol that could be followed to make validated simplifications of complex biological models possessing time scale hierarchies.     

Chaos and population control of insect outbreaks

We used small perturbations in adult numbers to control large fluctuations in the chaotic demographic dynamics of laboratory populations of the flour beetle Tribolium castaneum . A nonlinear mathematical model was used to identify a sensitive region of phase space where the addition of a few adult insects would result in a dampening of the life stage fluctuations. Three experimental treatments were applied: one in which perturbations were made whenever the populations were inside the sensitive region ("in-box treatment"), another where perturbations were made whenever the populations were outside the sensitive region ("out-box treatment"), and an unperturbed control. The in-box treatment caused a stabilization of insect densities at numbers well below the peak values exhibited by the out-box and control populations. This study demonstrates how small perturbations can be used to influence the chaotic dynamics of an ecological system.