Invariant Distance Research Articles

Surface motifs by a computer vision technique: Searches, detection, and implications for protein–ligand recognition

We describe the application of a method geared toward structural and surface comparison of proteins. The method is based on the Geometric Hashing Paradigm adapted from Computer Vision. It allows for comparison of any two sets of 3-D coordinates, such as protein backbones, protein core or protein surface motifs, and small molecules such as drugs. Here we apply our method to 4 types of comparisons between pairs of molecules: (1) comparison of the backbones of two protein domains; (2) search for a predefined 3-D C alpha motif within the full backbone of a domain; and in particular, (3) comparison of the surfaces of two receptor proteins; and (4) comparison of the surface of a receptor to the surface of a ligand. These aspects complement each other and can contribute toward a better understanding of protein structure and biomolecular recognition. Searches for 3-D surface motifs can be carried out on either receptors or on ligands. The latter may result in the detection of pharmacophoric patterns. If the surfaces of the binding sites of either the receptors or of the ligands are relatively similar, surface superpositioning may aid significantly in the docking problem. Currently, only distance invariants are used in the matching, although additional geometric surface invariants are considered. The speed of our Geometric Hashing algorithm is encouraging, with a typical surface comparison taking only seconds or minutes of CPU time on a SUN 4 SPARC workstation. The direct application of this method to the docking problem is also discussed. We demonstrate the success of this method in its application to two members of the globin family and to two dehydrogenases.

Visual Orientation: Facts and Hypotheses

How pilots judge an airplane's position and movement relative to objects on the earth's surface is considered in relation to the size distance invariance hypothesis. Factors that cause systematic departures from size, distance, and angular position constancies are identified, including effects of viewing real and virtual imaging displays. Bias errors in spatial judgments and associated problems in flight are analyzed in terms of the independent and intervening variables, most notably the misaccommodation of the eyes. Two theoretical partial explanations for the anomalous effect the zoom-lens hypothesis and the neural-signals hypothesis—are discussed in relation to recent experimental findings.

Graviton fluctuations in de Sitter space

The two-point correlation function of metric fluctuations in de Sitter space is calculated. The results are expressed in a gauge-invariant and O(4,1)-invariant form in terms of elementary functions of z(x,x′) (a biscalar variable simply related to the invariant distance between x and x′). The Feynman functions for the transverse, trace-free, and scalar metric fluctuations grow without bounds, as ln z and z ln z, respectively, for large z. The consequences of this bad infrared behavior are discussed and it is shown that it leads to divergences in physical quantities at both the classical and quantum levels.

Second-order elliptic operators and heat kernels on Lie groups

Arendt, Batty, and Robinson proved that each second-order strongly elliptic operator C associated with left translations on the Lp-spaces of a Lie group G generates an interpolating family of semigroups T, whenever the coefficients of C are sufficiently smooth. We establish that T has an integral kernel K satisfying the bounds a It ee < Kt(g; h) < at e g te where d is the dimension of G, Igh 1 l is the right invariant distance from h to g, and a', b', co', etc. are positive constants. Both bounds are derived by generalization of Nash's arguments for pure second-order operators on Rd.

C369. Invariant distance in statistics: some miscellaneous comments

Click to increase image sizeClick to decrease image sizeKeywords: Cross-entrophyHellinger-Bhattacharyya distanceHomogeneous functionsInformation matrixInvariant distancesInvariant priorsTensors

Scissors-grip model for DNA recognition by a family of leucine zipper proteins.

C/EBP is a sequence-specific DNA binding protein that regulates gene expression in certain mammalian cells. The region of the C/EBP polypeptide required for specific recognition of DNA is related in amino acid sequence to other regulatory proteins, including the Fos and Jun transforming proteins. It has been proposed that these proteins bind DNA via a bipartite structural motif, consisting of a dimerization interface termed the "leucine zipper" and a DNA contact surface termed the "basic region." An evaluation of the properties of conserved amino acids within the basic region of 11 deduced protein sequences, coupled with the observation that they are located at an invariant distance from the leucine zipper, has led to the formulation of a "scissors-grip" model for DNA binding. The architectural features of this model are well suited for interaction with directly abutted, dyadsymmetric DNA sequences. Data supportive of the model were obtained with chemical probes of protein: DNA complexes.

Using constancy of distance to estimate position and displacement in space

The problem of computing structure and motion from the observation of points in two distinct images of a scene is considered. The proposed method explicitly utilizes the principle of conservation of distance during rigid-body motion. The formulation is such that it separates the problem of estimating object position from that of determining motion parameters. The equations of invariance of distance for a rigid body are solved for the points' position in space. When these coordinates in space are known, motion parameters are computed in a simple and straightforward manner. Examples are given to illustrate the efficiency of the algorithm. >

Invariant Distances Related to the Bergman Function

Let $D$ be a bounded domain in ${{\mathbf {C}}^n}$. The invariant distance in $D$ is given by \[ {\rho _D}(z,w) = {\left ( {1 - {{\left ( {\frac {{{K_D}(z,w){K_D}(w,z)}}{{{K_D}(z,z){K_D}(w,w)}}} \right )}^{1/2}}} \right )^{1/2}}.\] It is shown that one half of the length of a piecewise ${C^1}$ curve $\gamma :[a,b] \to D$ with respect to the Bergman metric is equal to the length of $\gamma$ measured by ${\rho _D}$, which implies that the associated inner distance $\rho _D^*$ coincides (up to the factor $\tfrac {1}{2}$) with the Bergman-distance. Also it was proved that ${\rho _D}$ is not an inner distance.

Invariant distances related to the Bergman function

Let $D$ be a bounded domain in ${{\mathbf {C}}^n}$. The invariant distance in $D$ is given by \[ {\rho _D}(z,w) = {\left ( {1 - {{\left ( {\frac {{{K_D}(z,w){K_D}(w,z)}}{{{K_D}(z,z){K_D}(w,w)}}} \right )}^{1/2}}} \right )^{1/2}}.\] It is shown that one half of the length of a piecewise ${C^1}$ curve $\gamma :[a,b] \to D$ with respect to the Bergman metric is equal to the length of $\gamma$ measured by ${\rho _D}$, which implies that the associated inner distance $\rho _D^*$ coincides (up to the factor $\tfrac {1}{2}$) with the Bergman-distance. Also it was proved that ${\rho _D}$ is not an inner distance.

Projectively invariant distances for affine and projective structures

Projectively invariant distances for affine and projective structures

Espaces de Banach superstables, distances stables et homeomorphismes uniformes

We introduce here the notion of superstable Banach space, as the superproperty associated with the stability property of J. L. Krivine and B. Maurey. IfE is superstable, so are theL p (E) for eachp∈[1, +∞[. If the Banach spaceX uniformly imbeds into a superstable Banach space, then there exists an equivalent invariant superstable distance onX; as a consequenceX contains subspaces isomorphic tol p spaces (for somep∈[1, ∞[). We give also a generalization of a result of P. Enflo: the unit ball ofc 0 does not uniformly imbed into any stable Banach space.

Invariant distances and invariant differential metrics in locally convex spaces

Invariant distances and invariant differential metrics in locally convex spaces

Moon illusion and spiral aftereffect: illusions due to the loom-zoom system?

The moon illusion and the spiral aftereffect are illusions in which apparent size and apparent distance vary inversely. Because this relationship is exactly opposite to that predicted by the static size--distance invariance hypothesis, the illusions have been called "paradoxical." The illusions may be understood as products of a loom-zoom system, a hypothetical visual subsystem that, in its normal operation, acts according to its structural constraint, the constancy axiom, to produce perceptions that satisfy the constraints of stimulation, the kinetic size--distance invariance hypothesis. When stimulated by its characteristic stimulus of symmetrical expansion or contraction, the loom-zoom system produces the perception of a rigid object moving in depth. If this system is stimulated by a rotating spiral, a negative motion-aftereffect is produced when rotation ceases. If fixation is then shifted to a fixed-sized disc, the aftereffect process alters perceived distance and the loom-zoom system alters perceived size such that the disc appears to expand and approach or to contract and recede, depending on the direction of rotation of the spiral. If the loom-zoom system is stimulated by a moon-terrain configuration, the equidistance tendency produces a foreshortened perceived distance for the moon as an inverse function of elevation and acts in conjunction with the loom-zoom system to produce the increased perceived size of the moon.

Dimensional analysis of interstitial phases with the AlCr2C structure

A dimensional analysis is reported of interstitial phases with the hexagonal AlCr2C, NM2C, structure, which is based on examining the variation of the unit-cell parameters a and c as functions of DM and DN, the diameters of the M and N components for CN 12. The observed behaviour is uniquely different to that of phases with all other structures so examined, there being a drastic change of the DN dependences of a and c in the region where a ≃ DN. Nevertheless, equations are derived which reproduce the a and c dependences on DM, DN and the valency of the M component to better than 0.5% and which indicate the atomic arrays that control the cell dimensions. It is found that the M-M distances that do not lie in (0001) planes and which form half of the sides of the M octahedra surrounding the metalloid atoms, are an invariant feature of the structure for phases with the same M component. This invariant distance couples a, c and z, the atomic parameters of the M atoms, and it causes e dependences on 6.5DN (a > DN) and -1.65DN (a < DN), which could not possibly arise through the intrinsic contacts of the N atoms. The electronic distribution from the M atoms to M-M and M-N contacts and also to bands derived from the metalloid 2p states and transition-metal d states, is further analysed to account for the apparent dependence of a and c on the M-atom valency. The cell dimensions of SnM2C phases with the AlCr2C structure do not follow the systematic variations discussed above; reasons are advanced for this.

Characterizing Motions by Unit Distance Invariance

It has been proved by P. Zvengrowski [1, appendix to chapter II] that a map of the euclidean plane, T: E2 E2, which preserves unit distance is an isometry. His proof uses an approximation procedure based on the theorem of Kronecker that the ring generated over Z by J3 is dense in R . A weaker version of this theorem, where the map is assumed to be one-to-one, is proved in [2]. Some of the unproved assertions in [2] are not so obvious as one might think, since the proofs we have, anyway, do not generalize to higher dimensions. However, the ideas in [2] could be used to considerably simplify Zvengrowski's proof, eliminating the reliance on algebraic number theory. Here we present elementary proofs for all dimensions of this theorem.

Characterizing Motions by Unit Distance Invariance

Characterizing Motions by Unit Distance Invariance

The Invariant Distance in the Theory of Pseudo-Conformal Transformations and the Lu Qikeng Conjecture

The Invariant Distance in the Theory of Pseudo-Conformal Transformations and the Lu Qikeng Conjecture

The invariant distance in the theory of pseudoconformal transformations and the Lu Qi-keng conjecture

Preliminary remarks. In the theory of pseudoconformal transformations the Bergman function plays an important role. It gives rise to numerous pseudoconformal invariants, for example, the Bergman metric form and the invariants J(n) (see [1], [2 ]). For a given domain DCCn, the Bergman function KD(Z, t) is uniquely determined as a holomorphic function of 2n complex variables, defined on the Cartesian product2 DX D. Therefore, a study of the relations between the geometrical properties of the domain D and the function-theoretical properties of its Bergman function is of considerable interest. To this class belong the investigations of the behavior of KD(Z, S) at the boundary (cf. [3], [7], [8], [11]). In the present paper we shall be concerned with another property of the Bergman function, namely, with the solvability of the equation KD (Z, ) = 0 in D X D. This property is invariant under pseudoconformal transformations. We will show also its relation with a certain property of the invariant distance in the domain D; the latter notion will be defined below.

Water, fog and the size-distance invariance hypothesis.

Objects generally appear too large under water. The usual explanation for this is that they are magnified by the change of refractive index between air and water. However, they are only magnified because they are optically nearer. If size constancy operates normally they should be seen the correct size but nearer, or enlarged in size but farther than the optical distance. The problem was investigated by requiring sixteen divers to make size and distance judgements of a range of white disks of different sizes set at various distances. Judgements were made for the same display both under water and on land. It was found that size judgements were greater under water, and that this corresponded to relative overestimation of distance. A weak form of size—distance invariance was found both under water and on land, but the relation between size and distance judgements was not the same in both conditions.Overestimation of size and distance was also found in a fog on land. It was suggested that the overestimation of distance in a fog and under water was due to the exaggerated reduction of brightness contrast with distance, which increases ‘aerial perspective’.

Invariant distances on complex manifolds and holomorphic mappings

Invariant distances on complex manifolds and holomorphic mappings