TY - JOUR AB - Selection on polygenic characters is generally analyzed by statistical methods that assume a Gaussian (normal) distribution of breeding values. We present an alternative analysis based on multilocus population genetics. We use a general representation of selection, recombination, and drift to analyze an idealized polygenic system in which all genetic effects are additive (i.e., both dominance and epistasis are absent), but no assumptions are made about the distribution of breeding values or the numbers of loci or alleles. Our analysis produces three results. First, our equations reproduce the standard recursions for the mean and additive variance if breeding values are Gaussian; but they also reveal how non-Gaussian distributions of breeding values will alter these dynamics. Second, an approximation valid for weak selection shows that even if genetic variance is attributable to an effectively infinite number of loci with only additive effects, selection will generally drive the distribution of breeding values away from a Gaussian distribution by creating multilocus linkage disequilibria. Long-term dynamics of means can depart substantially from the predictions of the standard selection recursions, but the discrepancy may often be negligible for short-term selection. Third, by including mutation, we show that, for realistic parameter values, linkage disequilibrium has little effect on the amount of additive variance maintained at an equilibrium between stabilizing selection and mutation. Each of these analytical results is supported by numerical calculations. AU - Turelli, Michael AU - Barton, Nicholas H ID - 3649 IS - 1 JF - Theoretical Population Biology SN - 0040-5809 TI - Dynamics of polygenic characters under selection VL - 38 ER - TY - JOUR AB - It is widely held that each gene typically affects many characters, and that each character is affected by many genes. Moreover, strong stabilizing selection cannot act on an indefinitely large number of independent traits. This makes it likely that heritable variation in any one trait is maintained as a side effect of polymorphisms which have nothing to do with selection on that trait. This paper examines the idea that variation is maintained as the pleiotropic side effect of either deleterious mutation, or balancing selection. If mutation is responsible, it must produce alleles which are only mildly deleterious (s & 10(-3)), but nevertheless have significant effects on the trait. Balancing selection can readily maintain high heritabilities; however, selection must be spread over many weakly selected polymorphisms if large responses to artificial selection are to be possible. In both classes of pleiotropic model, extreme phenotypes are less fit, giving the appearance of stabilizing selection on the trait. However, it is shown that this effect is weak (of the same order as the selection on each gene): the strong stabilizing selection which is often observed is likely to be caused by correlations with a limited number of directly selected traits. Possible experiments for distinguishing the alternatives are discussed. AU - Barton, Nicholas H ID - 3651 IS - 3 JF - Genetics SN - 0016-6731 TI - Pleiotropic models of quantitative variation VL - 124 ER - TY - CONF AB - This paper proves an O(m 2/3 n 2/3+m+n) upper bound on the number of incidences between m points and n hyperplanes in four dimensions, assuming all points lie on one side of each hyperplane and the points and hyperplanes satisfy certain natural general position conditions. This result has application to various three-dimensional combinatorial distance problems. For example, it implies the same upper bound for the number of bichromatic minimum distance pairs in a set of m blue and n red points in three-dimensional space. This improves the best previous bound for this problem. AU - Edelsbrunner, Herbert AU - Sharir, Micha ID - 4067 SN - 978-3-540-52921-7 T2 - Proceedings of the International Symposium on Algorithms TI - A hyperplane Incidence problem with applications to counting distances VL - 450 ER - TY - JOUR AB - We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m 2/3 n logn +n 2); we can calculatem such cells specified by a point in each, in worst-case timeO(m 2/3 n log3 n+n 2 logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m 2/3 n logn+n 2), but this number is onlyO(m 3/5– n 4/5+2 +m+n logm), for any>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m 3/4– n 3/4+3 +m) log2 n+n logn logm) for any>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m 3/4– n 3/4+3 +m] log2 n+n logn logm) for any>0. (v) The maximum number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m 2/3 n d/3 logn+n d–1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight. AU - Edelsbrunner, Herbert AU - Guibas, Leonidas AU - Sharir, Micha ID - 4066 IS - 1 JF - Discrete & Computational Geometry SN - 0179-5376 TI - The complexity of many cells in arrangements of planes and related problems VL - 5 ER - TY - JOUR AB - We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m 2/3– n 2/3+2 +n) for any>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of thesem faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity isO(m 2/3– n 2/3+2 logn+n logn logm). If instead of lines we have an arrangement ofn line segments, then the maximum number of edges ofm faces isO(m 2/3– n 2/3+2 +n (n) logm) for any>0, where(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m 2/3– n 2/3+2 log+n(n) log2 n logm). AU - Edelsbrunner, Herbert AU - Guibas, Leonidas AU - Sharir, Micha ID - 4072 IS - 1 JF - Discrete & Computational Geometry SN - 0179-5376 TI - The complexity and construction of many faces in arrangements of lines and of segments VL - 5 ER - TY - CONF AB - A number of rendering algorithms in computer graphics sort three-dimensional objects by depth and assume that there is no cycle that makes the sorting impossible. One way to resolve the problem caused by cycles is to cut the objects into smaller pieces. The problem of estimating how many such cuts are always sufficient is addressed. A few related algorithmic and combinatorial geometry problems are considered. AU - Chazelle, Bernard AU - Edelsbrunner, Herbert AU - Guibas, Leonidas AU - Pollack, Richard AU - Seidel, Raimund AU - Sharir, Micha AU - Snoeyink, Jack ID - 4073 SN - 0-8186-2082-X T2 - 31st Annual Symposium on Foundations of Computer Science TI - Counting and cutting cycles of lines and rods in space ER - TY - JOUR AB - Let S be a set of n closed intervals on the x-axis. A ranking assigns to each interval, s, a distinct rank, p(s)∊ [1, 2,…,n]. We say that s can see t if p(s)