TY - JOUR AB - Geometry is crucial in our efforts to comprehend the structures and dynamics of biomolecules. For example, volume, surface area, and integrated mean and Gaussian curvature of the union of balls representing a molecule are used to quantify its interactions with the water surrounding it in the morphometric implicit solvent models. The Alpha Shape theory provides an accurate and reliable method for computing these geometric measures. In this paper, we derive homogeneous formulas for the expressions of these measures and their derivatives with respect to the atomic coordinates, and we provide algorithms that implement them into a new software package, AlphaMol. The only variables in these formulas are the interatomic distances, making them insensitive to translations and rotations. AlphaMol includes a sequential algorithm and a parallel algorithm. In the parallel version, we partition the atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented by a buffer zone to account for atoms whose ball representations may partially cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up compared to an independent serial implementation when using 32 processors. For instance, it takes 31 s to compute the geometric measures and derivatives of each atom in a viral capsid with more than 26 million atoms on 32 Intel processors running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant computations, which ultimately limit the impact of using multiple processors. AlphaMol is available as an OpenSource software. AU - Koehl, Patrice AU - Akopyan, Arseniy AU - Edelsbrunner, Herbert ID - 12544 IS - 3 JF - Journal of Chemical Information and Modeling SN - 1549-9596 TI - Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives VL - 63 ER - TY - JOUR AB - Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law. AU - Akopyan, Arseniy AU - Karasev, Roman ID - 7791 IS - 4 JF - European Journal of Mathematics SN - 2199-675X TI - When different norms lead to same billiard trajectories? VL - 8 ER - TY - JOUR AB - Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density. AU - Akopyan, Arseniy AU - Edelsbrunner, Herbert AU - Nikitenko, Anton ID - 10222 JF - Experimental Mathematics SN - 1058-6458 TI - The beauty of random polytopes inscribed in the 2-sphere ER - TY - JOUR AB - Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory. AU - Akopyan, Arseniy AU - Bobenko, Alexander I. AU - Schief, Wolfgang K. AU - Techter, Jan ID - 8338 JF - Discrete and Computational Geometry SN - 0179-5376 TI - On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs VL - 66 ER - TY - JOUR AB - We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods. AU - Akopyan, Arseniy AU - Schwartz, Richard AU - Tabachnikov, Serge ID - 8538 JF - European Journal of Mathematics SN - 2199-675X TI - Billiards in ellipses revisited ER - TY - CHAP AB - We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures. AU - Akopyan, Arseniy AU - Karasev, Roman ED - Klartag, Bo'az ED - Milman, Emanuel ID - 74 SN - 00758434 T2 - Geometric Aspects of Functional Analysis TI - Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures VL - 2256 ER - TY - JOUR AB - In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces. AU - Akopyan, Arseniy AU - Karasev, Roman ID - 10867 IS - 3 JF - International Mathematics Research Notices KW - General Mathematics SN - 1073-7928 TI - Waist of balls in hyperbolic and spherical spaces VL - 2020 ER - TY - JOUR AB - Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy. AU - Akopyan, Arseniy AU - Edelsbrunner, Herbert ID - 9157 IS - 1 JF - Computational and Mathematical Biophysics SN - 2544-7297 TI - The weighted mean curvature derivative of a space-filling diagram VL - 8 ER - TY - JOUR AB - The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy. AU - Akopyan, Arseniy AU - Edelsbrunner, Herbert ID - 9156 IS - 1 JF - Computational and Mathematical Biophysics SN - 2544-7297 TI - The weighted Gaussian curvature derivative of a space-filling diagram VL - 8 ER - TY - JOUR AB - We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible. AU - Akopyan, Arseniy AU - Fedorov, Roman ID - 6050 JF - Proceedings of the American Mathematical Society TI - Two circles and only a straightedge VL - 147 ER - TY - JOUR AB - Characterizing the fitness landscape, a representation of fitness for a large set of genotypes, is key to understanding how genetic information is interpreted to create functional organisms. Here we determined the evolutionarily-relevant segment of the fitness landscape of His3, a gene coding for an enzyme in the histidine synthesis pathway, focusing on combinations of amino acid states found at orthologous sites of extant species. Just 15% of amino acids found in yeast His3 orthologues were always neutral while the impact on fitness of the remaining 85% depended on the genetic background. Furthermore, at 67% of sites, amino acid replacements were under sign epistasis, having both strongly positive and negative effect in different genetic backgrounds. 46% of sites were under reciprocal sign epistasis. The fitness impact of amino acid replacements was influenced by only a few genetic backgrounds but involved interaction of multiple sites, shaping a rugged fitness landscape in which many of the shortest paths between highly fit genotypes are inaccessible. AU - Pokusaeva, Victoria AU - Usmanova, Dinara R. AU - Putintseva, Ekaterina V. AU - Espinar, Lorena AU - Sarkisyan, Karen AU - Mishin, Alexander S. AU - Bogatyreva, Natalya S. AU - Ivankov, Dmitry AU - Akopyan, Arseniy AU - Avvakumov, Sergey AU - Povolotskaya, Inna S. AU - Filion, Guillaume J. AU - Carey, Lucas B. AU - Kondrashov, Fyodor ID - 6419 IS - 4 JF - PLoS Genetics TI - An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape VL - 15 ER - TY - GEN AU - Pokusaeva, Victoria AU - Usmanova, Dinara R. AU - Putintseva, Ekaterina V. AU - Espinar, Lorena AU - Sarkisyan, Karen AU - Mishin, Alexander S. AU - Bogatyreva, Natalya S. AU - Ivankov, Dmitry AU - Akopyan, Arseniy AU - Avvakumov, Sergey AU - Povolotskaya, Inna S. AU - Filion, Guillaume J. AU - Carey, Lucas B. AU - Kondrashov, Fyodor ID - 9790 TI - A statistical summary of segment libraries and sequencing results ER - TY - GEN AU - Pokusaeva, Victoria AU - Usmanova, Dinara R. AU - Putintseva, Ekaterina V. AU - Espinar, Lorena AU - Sarkisyan, Karen AU - Mishin, Alexander S. AU - Bogatyreva, Natalya S. AU - Ivankov, Dmitry AU - Akopyan, Arseniy AU - Povolotskaya, Inna S. AU - Filion, Guillaume J. AU - Carey, Lucas B. AU - Kondrashov, Fyodor ID - 9797 TI - A statistical summary of segment libraries and sequencing results ER - TY - GEN AU - Pokusaeva, Victoria AU - Usmanova, Dinara R. AU - Putintseva, Ekaterina V. AU - Espinar, Lorena AU - Sarkisyan, Karen AU - Mishin, Alexander S. AU - Bogatyreva, Natalya S. AU - Ivankov, Dmitry AU - Akopyan, Arseniy AU - Avvakumov, Sergey AU - Povolotskaya, Inna S. AU - Filion, Guillaume J. AU - Carey, Lucas B. AU - Kondrashov, Fyodor ID - 9789 TI - Multiple alignment of His3 orthologues ER - TY - JOUR AB - In this paper we prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique, consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space; and establish waist-type results in terms of the Hausdorff measure. AU - Akopyan, Arseniy AU - Hubard, Alfredo AU - Karasev, Roman ID - 6634 IS - 2 JF - Topological Methods in Nonlinear Analysis TI - Lower and upper bounds for the waists of different spaces VL - 53 ER - TY - JOUR AB - The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here, we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry. AU - Akopyan, Arseniy AU - Izmestiev, Ivan ID - 6793 IS - 5 JF - Bulletin of the London Mathematical Society SN - 00246093 TI - The Regge symmetry, confocal conics, and the Schläfli formula VL - 51 ER - TY - JOUR AB - We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them. AU - Akopyan, Arseniy ID - 692 IS - 1 JF - Geometriae Dedicata TI - 3-Webs generated by confocal conics and circles VL - 194 ER - TY - JOUR AB - Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks. AU - Akopyan, Arseniy AU - Segal Halevi, Erel ID - 58 IS - 3 JF - SIAM Journal on Discrete Mathematics TI - Counting blanks in polygonal arrangements VL - 32 ER - TY - JOUR AB - We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the plane. These are congruences of straight lines in the plane with the combinatorics of the square grid, combinatorially colored as a checkerboard, such that all black coordinate quadrilaterals possess inscribed circles. We show how this larger class of IC-nets appears quite naturally in Laguerre geometry of oriented planes and spheres and leads to new remarkable incidence theorems. Most of our results are valid in hyperbolic and spherical geometries as well. We present also generalizations in spaces of higher dimension, called checkerboard IS-nets. The construction of these nets is based on a new 9 inspheres incidence theorem. AU - Akopyan, Arseniy AU - Bobenko, Alexander ID - 458 IS - 4 JF - Transactions of the American Mathematical Society TI - Incircular nets and confocal conics VL - 370 ER - TY - JOUR AB - The goal of this article is to introduce the reader to the theory of intrinsic geometry of convex surfaces. We illustrate the power of the tools by proving a theorem on convex surfaces containing an arbitrarily long closed simple geodesic. Let us remind ourselves that a curve in a surface is called geodesic if every sufficiently short arc of the curve is length minimizing; if, in addition, it has no self-intersections, we call it simple geodesic. A tetrahedron with equal opposite edges is called isosceles. The axiomatic method of Alexandrov geometry allows us to work with the metrics of convex surfaces directly, without approximating it first by a smooth or polyhedral metric. Such approximations destroy the closed geodesics on the surface; therefore it is difficult (if at all possible) to apply approximations in the proof of our theorem. On the other hand, a proof in the smooth or polyhedral case usually admits a translation into Alexandrov’s language; such translation makes the result more general. In fact, our proof resembles a translation of the proof given by Protasov. Note that the main theorem implies in particular that a smooth convex surface does not have arbitrarily long simple closed geodesics. However we do not know a proof of this corollary that is essentially simpler than the one presented below. AU - Akopyan, Arseniy AU - Petrunin, Anton ID - 106 IS - 3 JF - Mathematical Intelligencer TI - Long geodesics on convex surfaces VL - 40 ER -