Calculation of the conductance and selectivity of an ion-selective potassium channel (IRK1) from simulation of atomic scale models

We present the equations and methodology for the theoretical prediction of the conductance, permeability and selectivity of a K⁺ channel on the basis of atomic scale models for it. The methodology involves the use of Langevin dynamics and activated trajectories in order to obtain translocation free energies, rate constants and transmission coefficients for an ion going through the channel. The models are for the Inward Rectifier K⁺ channel (IRK1) which is a member of a family of ion-selective K⁺ channels. The IRK1 channel is biologically important because of its role in cardiac pacemaker function. The models we use for the IRK1 channel are developed from a model of the Shaker voltage-gated K⁺ channel. We find that the theoretically predicted conductance is too low by three orders of magnitude. We attribute this underestimate to a specific structural defect in the model used. Perhaps our most significant result is that the computed conductance is tremendously sensitive to the structural details of the so-called ‘P-loop’ that lines the outer half of the permeation pathway of the channel. This sensitivity may be useful in future studies on ion channel proteins for which the structure is not known from X-ray crystallography. In addition, this sensitivity may help determine whether X-ray structures of these proteins correspond to open or closed conformations.     

The Economics of Social Stratification in Premodern Societies

We present a microeconomic model of social stratification, which includes an endogenous fertility component. In the model, egalitarian and stratified societies coexist. The latter are divided into 2 hereditary classes: a warrior elite and a productive class. The model entails that the extra cost warriors must incur to train and equip their children for war determines the relative sizes of both classes and the degree of economic inequality. Higher costs of warrior children imply a greater economic advantage for warriors and a smaller ratio of warriors to producers. These results are consistent with the historical evidence. Finally, we explore conditions under which the social contributions of the warrior elite could discourage a revolution.     

Singularities and the Rayleigh Hypothesis for Solutions to the Helmholtz Equation

Possible and actual singularities in the analytic continuationof the solution to the Dirichlet problem for the two-dimensionalHelmholtz equation are studied in order to investigate the Rayleighhypothesis of scattering theory. The procedure uses a representationfor the solution to the analytic Cauchy problem and relatesone type of singularity in the Schwarz function associated withthe boundary curve to singularities in the solution. The resultsprovide confiration of criteria for the validity of the Rayleighhypothesis that have been given by van den Berg and Fokkema.     

ROTIFERS AND HUBBELL'S UNIFIED NEUTRAL THEORY OF BIODIVERSITY AND BIOGEOGRAPHY

ABSTRACT. This paper reviews several neutral models as possible mechanisms for simulating and explaining so‐called “hollow curve distributions.” The models include simple random assignment, MacArthur's broken stick, Preston's lognor‐mal (but as simulated by Sugihara), Fisher's logseries, and Hubbell's unified neutral theory of biodiversity and biogeog‐raphy. Each model is fitted to data describing the distribution of genera over families in the phylum Rotifera.     

On the Differential Equation for the Generation of Frequency Modulation

Representations are presented for the general solution of thelinear differential equation associated with the electronicgeneration of frequency modulated signals. The frequency modulationcomponent is determined explicitly by an orthogonal fundamentalsolution matrix, while the amplitude modulation component, whicharises as an impurity in the generation process, is discussedwith the aid of elementary aspects of the theory of Lie algebras.The Lie algebra thus defined is three dimensional and algebraicallysimple. Asymptotic representations of the amplitude modulationcomponent to the first-order in the absolute frequency deviationare shown to be completely prescribed by the basis vectors chosenfor the Lie algebra.     

Modified Damped Least Squares: An Algorithm for Non-linear Estimation

A new method for non-linear regression is described and comparedto three other well-known methods for the non-linear least squaresproblem. Convergence theorems and computational results aregiven which indicate that the new method is more efficient andreliable than previously published methods.