The mathematics of gnomonic seashells

Parametric Cartesian vector-valued functions are constructed for the purpose of systematically describing various features of spiral shell geometry. The underlying geometrical hypothesis is that molluscan shell shapes can usually, to at least a good first approximation, be developed by rotating a generating curve about a fixed axis whilst simultaneously diminishing it by an “equiangular spiral” scale factor. A first-order symmetry equation is derived; then variational calculus is used to construct energy functionals which establish that Hooke's law is inherent in the formalism and that naturally occurring shell geometries are analogous to those of elastic spiral “clock springs”. The biological requirement that shelly structures must exist in a three-dimensional space is shown to be a sufficiently powerful mathematical constraint to ensure the existence of geometrical artifacts which can, perhaps, be likened to the conservation laws, pseudoforces, and fields of classical physics.     

The transducer model for contrast detection and discrimination: formal relations, implications, and an empirical test

The transducer function mu for contrast perception describes the nonlinear mapping of stimulus contrast onto an internal response. Under a signal detection theory approach, the transducer model of contrast perception states that the internal response elicited by a stimulus of contrast c is a random variable with mean mu(c). Using this approach, we derive the formal relations between the transducer function, the threshold-versus-contrast (TvC) function, and the psychometric functions for contrast detection and discrimination in 2AFC tasks. We show that the mathematical form of the TvC function is determined only by mu, and that the psychometric functions for detection and discrimination have a common mathematical form with common parameters emanating from, and only from, the transducer function mu and the form of the distribution of the internal responses. We discuss the theoretical and practical implications of these relations, which have bearings on the tenability of certain mathematical forms for the psychometric function and on the suitability of empirical approaches to model validation. We also present the results of a comprehensive test of these relations using two alternative forms of the transducer model: a three-parameter version that renders logistic psychometric functions and a five-parameter version using Foley's variant of the Naka-Rushton equation as transducer function. Our results support the validity of the formal relations implied by the general transducer model, and the two versions that were contrasted account for our data equally well.     

Some comments on coordinate-free and scale-invariant methods in morphometrics

The unusual strategy for comparing biological shapes is to use some kind of superimposition of the two forms under study and then look at the "residuals" as the shape change. In this paper, I take a careful look at this general strategy and point out some subtle but inherent and important pitfalls. Additionally an alternative approach based on Euclidean Distance Matrix representation is presented. It is applicable to two- as well as three-dimensional objects.     

On the diffusion of ions in membranes

A simple model of a membrane is used to obtain relations among five measurable quantities: the inward and outward ionic fluxes, the internal and external ionic concentrations, and the difference of electrical potential across the membrane. The Goldman equation is generalized to arbitrary geometrical shapes. Predoctoral Fellow of the National Science Foundation.     

A two-dimensional morphospace for cyanobacteria and microalgae: Morphological diversity,evolutionary relatedness,and size constraints

1. Body metrics are considered as master traits that regulate physiological, behavioural and life history features of planktic cyanobacteria and microalgae. Although the distribution of their morphological traits reflects the various trade-offs and strategies needed for survival in pelagic habitats, previous methods for quantifying phytoplankton body shape do not adequately represent the intricate details of surface variation that are so important for their nutrient- and light-harvesting capabilities. Therefore, here we provide a new framework to quantify and illustrate the morphological diversity of cyanobacteria and microalgae.
2. Essential components of formulae used for surface area (A = Cs × d²) and volume (V = Cv × d³) calculations are provided by the shape-specific surface area and volume constants (Cs, Cv). Cs, the surface shape factor, characterises the coarseness of the object surface, and Cv, the volumetric shape factor, characterises the shape deviation from a sphere. Using these morphologically and biologically relevant variables, we defined a two-dimensional morphological space, in which all three-dimensional objects have well-defined positions.
3. By analysing morphologies of taxa representing various forms in major cyanobacterial and microalgal groups, we demonstrated that these groups show considerable differences in the area occupied within the morphospace and these differences are not affected by evolutionary relatedness. We showed that the ratio of surface and volume constants correlated with organism size, suggesting that the development of basic morphologies is constrained by their linear dimensions.
4. Using surface and volumetric shape factors, we created a two-dimensional Euclidean morphospace in which all three-dimensional objects have a specific position. Positioning uni- and multicellular cyanobacteria and microalgae of various shapes into this morphospace allows their geometrical and ecological limitations to be understood. Because of the close linkage between phytoplankton morphology and ecology, the proposed morphospace may serve as a proxy for an ecospace. Thus, in future the proposed morphospace can be used to visualise current ecological processes such as eutrophication or seasonal succession of phytoplankton.

     

Advances in Geometric Morphometrics

Geometric morphometrics is the statistical analysis of form based on Cartesian landmark coordinates. After separating shape from overall size, position, and orientation of the landmark configurations, the resulting Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, the mathematical space induced by the shape coordinates, is a metric space that can be approximated locally by a Euclidean tangent space. Thus, notions of distance (similarity) between shapes or of the length and direction of developmental and evolutionary trajectories can be meaningfully assessed in this space. Results of statistical techniques that preserve these convenient properties—such as principal component analysis, multivariate regression, or partial least squares analysis—can be visualized as actual shapes or shape deformations. The Procrustes distance between a shape and its relabeled reflection is a measure of bilateral asymmetry. Shape space can be extended to form space by augmenting the shape coordinates with the natural logarithm of Centroid Size, a measure of size in geometric morphometrics that is uncorrelated with shape for small isotropic landmark variation. The thin-plate spline interpolation function is the standard tool to compute deformation grids and 3D visualizations. It is also central to the estimation of missing landmarks and to the semilandmark algorithm, which permits to include outlines and surfaces in geometric morphometric analysis. The powerful visualization tools of geometric morphometrics and the typically large amount of shape variables give rise to a specific exploratory style of analysis, allowing the identification and quantification of previously unknown shape features.     

The tooth of perfection: functional and spatial constraints on mammalian tooth shape

This paper addresses the question of how close mammalian teeth are to ideal functional forms. An 'ideal' form is a morphology predicted to be the best functional shape according to information of the relationships between shape and function. Deviations from an ideal form are likely to indicate the presence of developmental or genetic constraints on form. Model tools were constructed to conform to functional principles from engineering and dental studies. The final model shapes are very similar to several mammalian tooth forms (carnassial teeth and tribosphenic-like cusps), suggesting that these tooth forms very closely approach ideal functional forms. Further evidence that these tooth forms are close to ideal comes from the conservation over 140 million years, the independent derivation and/or the occurrence over a size range of several orders of magnitude of these basic tooth forms. One of the main functional shapes derived here is the 'protoconoid', a fundamental design for double-bladed tools that fits a large number of functional parameters. This shape occurs in tooth forms such as tribosphenic, dilambdodont and zalambdodont. This study extends our understanding of constraints on tooth shape in terms of geometry (how space influences tooth shape) and function (how teeth divide food).  © 2003 The Linnean Society of London . Biological Journal of the Linnean Society , 2003, 78 , 173–191.     

Fractal morphometry of cell complexity.

Irregularity and self-similarity under scale changes are the main attributes of the morphological complexity of both normal and abnormal cells and tissues. In other words, the shape of a self-similar object does not change when the scale of measurement changes, because each part of it looks similar to the original object. However, the size and geometrical parameters of an irregular object do differ when it is examined at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how irregular shapes and structures in the physical and biological sciences can be analysed. Dominant influences have been the discovery of a new practical geometry of Nature, now known as fractal geometry, and the continuous improvements in computation capabilities. Unlike conventional Euclidean geometry, which was developed to describe regular and ideal geometrical shapes which are practically unknown in nature, fractal geometry can be used to measure the fractal dimension, contour length, surface area and other dimension parameters of almost all irregular and complex biological tissues. We have used selected examples to illustrate the application of the fractal principle to measuring irregular and complex membrane ultrastructures of cells at specific functional and pathological stage.     

Variation in lifetime male fitness in Ipomopsis aggregata: tests of sex allocation theory

Sex allocation theory assumes that a shift in allocation of resources to male function both increases male fitness and decreases female fitness. Moreover, the shapes of these fitness gain functions determine whether hermaphroditism or another breeding system is evolutionarily stable. In this article, I first outline information needed to measure these functions in flowering plants. I then use paternity analysis to describe the shapes of the fitness gain functions in natural populations of the hermaphroditic herb Ipomopsis aggregata. I also explore the relationships of male fitness (number of seeds sired) and female fitness (number of seeds produced) to the number of flowers produced by a plant. Plants with greater investment of biomass in the androecium, compared to the gynoecium and seeds, showed increased success at siring seeds, assumed by the models. That sex allocation trait, however, explained only 9% of the variance in estimates of male fitness. The shapes of the fitness gain functions were consistent with theoretical expectations for a hermaphroditic plant, but the model predicted a more female-biased evolutionarily stable strategy (ESS) allocation than was observed. These results lend only partial support the classical sex allocation model.     

Euclidean distance matrix analysis: a coordinate-free approach for comparing biological shapes using landmark data.

For problems of classification and comparison in biological research, the primary focus is on the similarity of forms. A biological form can be conveniently defined as consisting of size and shape. Several approaches for comparing biological shapes using landmark data are available. Lele (1991a) critically discusses these approaches and proposes a new method based on the Euclidean distance matrix representation of the form of an object. The purpose of this paper is to extend this new methodology to the comparison of groups of objects. We develop the statistical versions of various concepts introduced by Lele (1991a) and use them for developing statistical procedures for testing the hypothesis of shape difference between biological forms. We illustrate the use of this method by studying morphological differences between normal children and those affected with Crouzon and Apert syndromes and craniofacial sexual dimorphism in Cebus apella.     

Mathematical Models of Naturally “Morphed” Human Erythrocytes: Stomatocytes and Echinocytes

We present two mathematical models that describe human red blood cells (RBCs) with morphologies that are attained naturally under certain patho-physiological conditions, namely stomatocytes and echinocytes. Muñoz San Martín et al. (Bioelectromagnetics 27:521–527, 2006) recently presented models of these shapes based on our previous set of parametric equations (Kuchel and Fackerell, Bull. Math. Biol. 61:209–220, 1999) that involve Jacobi elliptic functions and integrals. Thus, both discocytes and stomatocytes are described. Here, we derived the Cartesian forms of these new equations; and, in addition, present a realistic model of a Type III echinocyte, using prolate spheroids ‘decorating’ a central sphere at the vertices of an internal dodecahedron. The RBC models based on Cartesian equations have been used for representing the shape changes (morphological transformations or “morphing”) that occur in RBCs under various experimental conditions; specifically, when the shape changes have been monitored by nuclear magnetic resonance (NMR) micro-imaging.     

Footprints of carbon and nitrogen: Revisiting the paradigm and exploring their nexus for decision making

Paradigms used for developing footprint metrics define the efficacy of these metrics for guiding decisions in the correct direction. Several metrics in use are developed from a narrow viewpoint of only looking at the output side impact of human activities on natural systems. This underestimates or overlooks other impacts on natural systems that occur due to input side interaction of human systems with natural systems. We revisit the paradigms used for development of footprint metrics and analyze the carbon and nitrogen footprints from these perspectives. Based on our analysis, utilizing both input and output side footprint measures is suggested such that heterogeneity of the nature of impact on both sides is not ignored. Utilizing these viewpoints, updated carbon and nitrogen footprints are proposed. Eco-LCA data for 2002 US economy is used to study the effect on decisions if these updated metrics are used. We also highlight the importance of understanding the multidimensional nature of environmental impacts that does not get captured by a single footprint, thus necessitating attention to the nexus of various footprints in decision making. Thus, the nexus between the updated carbon and nitrogen metrics is studied to show the shift in decisions resulting from looking at multidimensional impacts. Results for the US economy show that revisiting the paradigm for development of footprint metrics is a crucial step to avoid pushing decisions in incorrect directions due to use of narrowly focused footprint measures.     

Heat transfer from spheres and other animal forms.

A general predictive relation for the convection heat transfer from animal forms is developed. This relation is based on the convection equation for a sphere, and employs a simple, unique characteristic dimension to represent the animal which is the cube root of the animal volume. The accuracy of this relation is established through comparison with available convection results from animal shapes ranging in size and shape from spiders to cows. This relation allows an extrapolation to animal shapes for which data are not available. Results are also presented for the enhancement of convection heat transfer due to natural turbulence. A procedure is outlined for estimating the convecture heat loss from an animal in the natural outdoor environment.     

The morphogenesis of liposomes viewed from the aspect of bending energy

It is known that liposomes transform their shapes sequentially through one of several transformation pathways. Using the mechanical principle of the least bending energy of membranes, we investigate the stability and shape transformation of liposomes with geometrical symmetry. We have done this by computer simulations and theoretical analyses, in which three-dimensional liposome shapes have been generated by the modified Cassini equation. We show first that there are energetically stable liposome shapes having intrinsic geometrical symmetry. We find that by reducing the volume, the stable shape can change from a circular biconcave shape as in red blood cells, to elliptical, triangular, square, and other polygonal shapes. It is also found that the preceding two results hold true irrespective of the overall surface area of liposome.     

Simulation tools for biochemical networks: evaluation of performance and usability

MOTIVATION: Simulation of dynamic biochemical systems is receiving considerable attention due to increasing availability of experimental data of complex cellular functions. Numerous simulation tools have been developed for numerical simulation of the behavior of a system described in mathematical form. However, there exist only a few evaluation studies of these tools. Knowledge of the properties and capabilities of the simulation tools would help bioscientists in building models based on experimental data. RESULTS: We examine selected simulation tools that are intended for the simulation of biochemical systems. We choose four of them for more detailed study and perform time series simulations using a specific pathway describing the concentration of the active form of protein kinase C. We conclude that the simulation results are convergent between the chosen simulation tools. However, the tools differ in their usability, support for data transfer to other programs and support for automatic parameter estimation. From the experimentalists' point of view, all these are properties that need to be emphasized in the future.     

Prediction of shape and internal structure of the proximal femur using a modified level set method for structural topology optimisation

A computational framework was developed to simulate the bone remodelling process as a structural topology optimisation problem. The mathematical formulation of the Level Set technique was extended and then implemented into a coronal plane model of the proximal femur to simulate the remodelling of internal structure and external geometry of bone into the optimal state. Results indicated that the proposed approach could reasonably mimic the major geometrical and material features of the natural bone. Simulation of the internal bone remodelling on the typical gross shape of the proximal femur, resulted in a density distribution pattern with good consistency with that of the natural bone. When both external and internal bone remodelling were simulated simultaneously, the initial rectangular design domain with a regularly distributed mass reduced gradually to an optimal state with external shape and internal structure similar to those of the natural proximal femur.     

Prediction of shape and internal structure of the proximal femur using a modified level set method for structural topology optimisation

A computational framework was developed to simulate the bone remodelling process as a structural topology optimisation problem. The mathematical formulation of the Level Set technique was extended and then implemented into a coronal plane model of the proximal femur to simulate the remodelling of internal structure and external geometry of bone into the optimal state. Results indicated that the proposed approach could reasonably mimic the major geometrical and material features of the natural bone. Simulation of the internal bone remodelling on the typical gross shape of the proximal femur, resulted in a density distribution pattern with good consistency with that of the natural bone. When both external and internal bone remodelling were simulated simultaneously, the initial rectangular design domain with a regularly distributed mass reduced gradually to an optimal state with external shape and internal structure similar to those of the natural proximal femur.     

A quantum chemistry evaluation of the stereochemical activity of the lone pair in Pb<Superscript>II</Superscript> complexes with sequestering ligands

The stereochemical activity of the lone pair on Pb^II complexes is assessed using several theoretical methods, including structural analyses, computations of Fukui functions, natural bond orbitals, electron localization function, investigation of the electron density and of its laplacian. The attention is focused on four octadentate N-carbamoylmethyl-substituted tetraazamacrocycles of various ring sizes ranging from 8 to 14 atoms associated with the Pb^II cation. The theoretical study illustrates the geometrical constraints imposed by the ring structure which limits the spatial development of the lone pair but without fully preventing it. For a given coordination number, the lone pair activity is strongly correlated to the geometry of the ligand and in particular to the size of the cage that the ligand forms around the Pb^II cation. Some limitations of the theoretical tools used are also evidenced, among them the necessity to sample around a critical point instead of just analyzing its nature. In the case of the laplacian of the electron density, a visualization method is introduced to moderate the results based only on the nature of a critical point. These limitations should also be related to the difficulty to extend the lone pair concept for the heaviest atoms of the classification.