On the kinetic depth effect

The problem of the kinetic depth effect is revisited. We study how many points in how many views are necessary and sufficient to recover structure. The constraints in the cases where the velocities of the image points are known, and the positions of the image points are known with the correspondence between them established, are different and they have to be studied separately. In the case of two projections of any number of points there are infinitely many solutions, but if we regularize the problem we get a unique solution under some assumptions. Finally, an algorithm is discussed for learning this particular kind of regularization.     

A computational aspect of kinetic depth effect

A method is proposed to determine the rigid structure as well as the three dimensional motion of an object from a sequence of orthographically projected images. It is assumed that the velocities as well as the positions of the points attached to the object are observable in the images. The instantaneous rigidity condition wich states that the relative position vector between any two points is orthogonal to the corresponding relative velocity vector is derived from the condition of rigidity. Assuming further that the rotational velocity component is constant throughout the period of observation, each of the projected relative velocity vectors is shown to move along one of the elliptical trajectories, all of which are similar to one another. The orientations of the longer axes are common to every trajectory. Then one can determine the equations of the ellipses by observing only two points in three views or three points in two views. The solution is obtained from a set of linear equations. The length of the longer axis enables one to determine the relative velocity. The instantaneous rigidity condition is then used to obtain the relative position. The special cases where the rotational axis is either perpendicular to or parallel with the image plane are discussed. The above results are discussed in relation to the relevant psychophysical observations as well as theoretical studies.     

Tangle analysis of Xer recombination reveals only three solutions, all consistent with a single three-dimensional topological pathway

The product of Xer recombination at directly repeated psi sites on a circular unknotted DNA molecule is a right-hand four-noded catenane. Here, we use tangle equations to analyze the topological changes associated with Xer recombination at psi. This mathematical method allows computation of all possible topological pathways consistent with the experimental data. We give a rigorous mathematical proof that, under reasonable biological assumptions, there are only three solutions to the tangle equations. One of the solutions corresponds to a synaptic complex with antiparallel alignment of recombination core sites, the other two correspond to parallel alignment of cores. We show that all three solutions can be unified into a single three-dimensional model for Xer recombination. Thus the three distinct mathematical solutions do not necessarily represent distinct three-dimensional pathways, and in this case the three distinct tangle solutions are different planar projections of the same three-dimensional configuration.     

Studies of macromolecular heterogeneous associations involving cross-linking: a re-examination of the ovalbumin-lysozyme system

A system is considered in which a multivalent acceptor interacts with a bivalent ligand in solution to form an array of complexes via multiple binding and cross-linking reactions. With the use of reacted site probability functions expressions are derived in terms of a site binding constant which are of potential use in the interpretation of sedimentation equilibrium and binding results obtained with such systems. Their potential use is explored in relation to results obtained on the interacting ovalbumin-lysozyme system at pH 6.80, ionic strength 0.02. A comparison is made of this interpretation with that based on an interaction pattern involving only multiple binding of ligand in the absence of cross-linking effects. While both interpretations quantitatively describe certain results, it is shown, by invoking further experimental observations on apparent weight-average molecular weight and precipitation behavior, that the more favored interpretation is that involving the operation of a spectrum of forces leading to a large array of ovalbumin-lysozyme complexes, including those of the cross-linked type. It is stressed that the particular ovalbumin-lysozyme system is but one example of interaction between oppositely charged macromolecules and therefore that the derived equations may find wider application to such systems and those known to involve more specific cross-linking interactions.     

Predictions of co-contraction depend critically on degrees-of-freedom in the musculoskeletal model

In biomechanics, the calculation of individual muscle forces during movements is based on a model of the musculoskeletal system and a method for extracting a unique set of muscle forces. To obtain a unique set of muscle forces, non-linear, static optimisation is commonly used. However, the optimal solution is dependent on the musculoskeletal geometry, and single joints may be represented using one, two or three degrees-of-freedom. Frequently, a system with multiple degrees-of-freedom is replaced with a system that contains a subset of all the possible degrees-of-freedom. For example, the cat ankle joint is typically modelled as a planar joint with its primary degree-of-freedom (plantar-dorsiflexion), whereas, the actual joint has three rotational degrees-of-freedom. Typically, such simplifications are justified by the idea that the reduced case is contained as a specific solution of the more general case. However, here we demonstrate that the force-sharing solution space of a general, three degrees-of-freedom musculoskeletal system does not necessarily contain the solutions from the corresponding one or two degrees-of-freedom systems. Therefore, solutions of a reduced system, in general, are not sub-set solutions of the actual three degrees-of-freedom system, but are independent solutions that are often incompatible with solutions of the actual system. This result shows that representing a three degrees-of-freedom system as a one or two degrees-of-freedom system gives force-sharing solutions that cannot be extrapolated to the actual system, and vice-versa. These results imply that general solutions cannot be extracted from models with fewer degrees-of-freedom than the actual system. They further emphasise the need for precise geometric representation of the musculoskeletal system, if general force-sharing rules are to be derived.     

Gene Conversion at the Gray Locus of SORDARIA FIMICOLA: Fit of the Experimental Data to a Hybrid DNA Model of Recombination

A hybrid DNA (hDNA) model of recombination has been algebraically formulated, which allows the prediction of frequencies of postmeiotic segregation and conversion of a given allele and their probability of being associated with a crossing over. The model considered is essentially the "Aviemore model." In contrast to some other interpretations of recombination, it states that gene conversion can only result from the repair of heteroduplex hDNA, with postmeiotic segregation resulting from unrepaired heteroduplexes. The model also postulates that crossing over always occurs distally to the initiation site of the hDNA. Eleven types of conversion and postmeiotic segregation with or without associated crossover were considered. Their theoretical frequencies are given by 11 linear equations with ten variables, four describing heteroduplex repair, four giving the probability of hDNA formation and its topological properties and two giving the probability that crossing over occurs at the left or right of the converting allele. Using the experimental data of Kitani and coworkers on conversion at the six best studied gray alleles of Sordaria fimicola, we found that the model considered fit the data at a P level above or very close (allele h4) to the 5% level of sampling error provided that the hDNA is partly asymmetric. The best fitting solutions are such that the hDNA has an equal probability of being formed on either chromatid or, alternatively, that both DNA strands have the same probability of acting as the invading strand during hDNA formation. The two mismatches corresponding to a given allele are repaired with different efficiencies. Optimal solutions are found if one allows for repair to be more efficient on the asymmetric hDNA than on the symmetric one. In the case of allele g1, our data imply that the direction of repair is nonrandom with respect to the strand on which it occurs.     

The minimal model of the hypothalamic–pituitary–adrenal axis

This paper concerns ODE modeling of the hypothalamic–pituitary– adrenal axis (HPA axis) using an analytical and numerical approach, combined with biological knowledge regarding physiological mechanisms and parameters. The three hormones, CRH, ACTH, and cortisol, which interact in the HPA axis are modeled as a system of three coupled, nonlinear differential equations. Experimental data shows the circadian as well as the ultradian rhythm. This paper focuses on the ultradian rhythm. The ultradian rhythm can mathematically be explained by oscillating solutions. Oscillating solutions to an ODE emerges from an unstable fixed point with complex eigenvalues with a positive real parts and a non-zero imaginary parts. The first part of the paper describes the general considerations to be obeyed for a mathematical model of the HPA axis. In this paper we only include the most widely accepted mechanisms that influence the dynamics of the HPA axis, i.e. a negative feedback from cortisol on CRH and ACTH. Therefore we term our model the minimal model. The minimal model, encompasses a wide class of different realizations, obeying only a few physiologically reasonable demands. The results include the existence of a trapping region guaranteeing that concentrations do not become negative or tend to infinity. Furthermore, this treatment guarantees the existence of a unique fixed point. A change in local stability of the fixed point, from stable to unstable, implies a Hopf bifurcation; thereby, oscillating solutions may emerge from the model. Sufficient criteria for local stability of the fixed point, and an easily applicable sufficient criteria guaranteeing global stability of the fixed point, is formulated. If the latter is fulfilled, ultradian rhythm is an impossible outcome of the minimal model and all realizations thereof. The second part of the paper concerns a specific realization of the minimal model in which feedback functions are built explicitly using receptor dynamics. Using physiologically reasonable parameter values, along with the results of the general case, it is demonstrated that un-physiological values of the parameters are needed in order to achieve local instability of the fixed point. Small changes in physiologically relevant parameters cause the system to be globally stable using the analytical criteria. All simulations show a globally stable fixed point, ruling out periodic solutions even when an investigation of the ‘worst case parameters’ is performed.     

Simplified models for packed-bed biofilm reactors

For a packed-bed biofilm reactor two reactor models are proposed. One model is for the limiting case of a biofilm with a constant biofilm thickness in which diffusion within the biofilm is shown to be negligible. The second model assumes that the thickness of the biofilm is limited by the concentration of substrate within the biofilm. The analytical solutions for these reactor models are shown to agree very well with the numerical solutions to the exact differential equations.     

The interpretations of fox possession: Illness as metaphor

Although fox possession is a rare phenomenon in contemporary Japan, it is a matter of concern in psychiatry and in folk healing. The case study presented here deals with the social process of role transformation, from client to shaman, as a way of self healing. The discussion then assumes that fox possession renders multiple interpretations and argues that it is, in fact, a metaphorical representation in much the same way that biomedical interpretation is. The argument also proposes that an etiological (causal) understanding of fox possession is less productive than a metaphorical one.     

On the stability of separable solutions of a sexual age-structured population dynamics model

The Sharpe-Lotka-McKendrick-von Foerster equations for non-dispersing age-sex-structured populations with a harmonic mean type mating law are considered and their separable solutions are analysed. For certain forms of the demographic rates the underlying evolution equations are reduced to systems of ODEs, the long time behavior of their solutions is studied, and the stability of separable solutions is discussed. It is found that for the constant death rates and constant sex ratio of newborns with stationary birth rates this model admits one one-parameter class of separable solutions, two such classes (repeated or different) or no such ones. In the case of special forms of age-dependent birth rates, solutions of one of these two different classes corresponding to the greater root of the characteristic equation are locally stable, solutions of the other one corresponding to the smaller root are unstable, and the population dies out if the model does not admit separable solutions or if initial densities of newborns are small enough in the case of the existence of separable solutions. In the case of constant vital rates, the model has no separable solutions or admits only one class of such ones that are globally stable.     

Molecular clock fork phylogenies: closed form analytic maximum likelihood solutions

Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary trees, but finding the global optimum is a hard computational task. Because no general analytic solution is known, numeric techniques such as hill climbing or expectation maximization (EM) are used in order to find optimal parameters for a given tree. So far, analytic solutions were derived only for the simplest model-three-taxa, two-state characters, under a molecular clock. Quoting Ziheng Yang, who initiated the analytic approach,"this seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogenetic estimation."In this work, we give general analytic solutions for a family of trees with four-taxa, two-state characters, under a molecular clock. The change from three to four taxa incurs a major increase in the complexity of the underlying algebraic system, and requires novel techniques and approaches. We start by presenting the general maximum likelihood problem on phylogenetic trees as a constrained optimization problem, and the resulting system of polynomial equations. In full generality, it is infeasible to solve this system, therefore specialized tools for the molecular clock case are developed. Four-taxa rooted trees have two topologies-the fork (two subtrees with two leaves each) and the comb (one subtree with three leaves, the other with a single leaf). We combine the ultrametric properties of molecular clock fork trees with the Hadamard conjugation to derive a number of topology dependent identities. Employing these identities, we substantially simplify the system of polynomial equations for the fork. We finally employ symbolic algebra software to obtain closed formanalytic solutions (expressed parametrically in the input data). In general, four-taxa trees can have multiple ML points. In contrast, we can now prove that each fork topology has a unique(local and global) ML point.     

The visual ambiguity of a moving plane

It is shown that the optic flow field arising from motion relative to a visually textured plane may be characterized by eight parameters that depend on the observer's linear and angular velocity and the coordinate vector of the plane. These three vectors are not, however, uniquely determined by the values of the eight parameters. First, the optic flow field does not supply independent values for the observer's speed and distance from the plane; it only gives the ratio of these two quantities. But more unexpectedly, the equations relating the observer's linear velocity and the plane's coordinate vector to the eight parameters are still satisfied if the two vectors are interchanged or reversed in direction, or both. So in addition to the veridical interpretation of the optic flow field there exist three spurious interpretations to be considered and if possible excluded. This purpose is served by the condition that an interpretation can be seriously entertained only if it attributes every image element to a light source in the observer's field of view. This condition immediately eliminates one of the spurious interpretations, and exhibits the other two as mutually inconsistent: one of them is tenable only if all the visible sources lie on the forward half of the plane (relative to the observer's linear velocity); the other only if they all lie on the backward half-plane. If the sources are distributed over both halves of the plane, only the veridical interpretation survives. Its computation involves solving a 3 X 3 eigenvalue problem derived from the flow field. If the upper two eigenvalues coincide, the observer must be moving directly towards the plane; if the lower two eigenvalues coincide, his motion must be directly away from it; in both cases the spurious interpretation merges with the veridical one. If all three eigenvalues are equal, it may be inferred that either the observer's linear velocity vanishes or the plane is infinitely distant.     

A suggestion for another statistical interpretation of the fundamental equations of the mathematical biophysics of the central nervous system

In a preceding paper an interpretation of the ε andj factors has been given in terms of an average effect of a large number of interneurons. In the present paper, a different interpretation is given in terms of the probability of a sufficient number of afferents to fire within the period of latent addition of the efferent. From this interpretation it follows that the old equations for ε andj are only first linear approximations to more complicated equations, the nature of which is suggested by this interpretation.     

NULL MODELS FOR THE NUMBER OF EVOLUTIONARY STEPS IN A CHARACTER ON A PHYLOGENETIC TREE

Random trees and random characters can be used in null models for testing phylogenetic hypothesis. We consider three interpretations of random trees: first, that trees are selected from the set of all possible trees with equal probability; second, that trees are formed by random speciation or coalescence (equivalent); and third, that trees are formed by a series of random partitions of the taxa. We consider two interpretations of random characters: first, that the number of taxa with each state is held constant, but the states are randomly reshuffled among the taxa; and second, that the probability each taxon is assigned a particular state is constant from one taxon to the next. Under null models representing various combinations of randomizations of trees and characters, exact recursion equations are given to calculate the probability distribution of the number of character state changes required by a phylogenetic tree. Possible applications of these probability distributions are discussed. They can be used, for example, to test for a panmictic population structure within a species or to test phylogenetic inertia in a character's evolution. Whether and how a null model incorporates tree randomness makes little difference to the probability distribution in many but not all circumstances. The null model's sense of character randomness appears more critical. The difficult issue of choosing a null model is discussed.     

Rate theory models for ion transport through rigid pores. I. Time-dependent analysis in the case of vanishing interactions

In this first of a series of papers concerning the theoretical analysis of rate theory models for ion transport through rigid pores, the case of vanishing interactions is investigated. "Rigidity" means that ions crossing membranes through pores see a fixed structure of the pores, not changing in time. A single pore is considered to be a sequence of (n + 1) activation barriers separated by n energy minima. The explicit analytical treatment is restricted to pores with regular internal barrier structure, including the nonequilibrium situation of an applied electric field. In this case the connection with continuum diffusion models is demonstrated by performing in the limit n leads to infinity (n = number of binding sites within the pores) the transition to continuum. Thus, from diffusion equations describing a discrete number of jumps, the corresponding diffusion-like partial differential equations and boundary conditions are generated. For regular pores, from the time dependent solutions of the discrete equations, the corresponding solutions of the continuum equations are explicitly generated. The time-dependent relaxation behaviour of the discrete model is in good agreement with the continuum model if one assumes more than two binding sites in the pores.     

The computation of structure from fixed-axis motion: Nonrigid structures

We show that an assumption of rigidity or quasi-rigidity is not necessary, in principle, for the computation of three-dimensional structure and motion from changing retinal images. In particular, we show that the three-dimensional structure of certain nonrigid objects, namely objects whose texture elements rotate about a common axis but at varying angular velocities, can in principle be computed from three successive retinal images of four texture elements, or from four successive images of two texture elements. We then show that in both cases the computed structure matches the actual structure of the object with probability one.     

Steady state enzyme kinetics for systems with three enzyme-binding species

The steady state enzyme kinetics of those systems are discussed, which involve three species binding to enzymes. Two specific systems are considered. In one system, all three species bind only once to the enzyme. In the other system, two species bind once and one binds twice to the enzyme. The species are labeled S, A and B. The general case is considered, in which all possible complexes involving enzyme E and species S generate product P. Species A and B may become co-substrates, activators or inhibitors. The steady state enzyme kinetic equations for the general case for both systems are presented. These equations are further discussed for a number of special cases, which may be of interest to enzymologists and others using enzymes.     

Spatial reconstruction of the human motion based on images of a single camera

The inverse dynamic analysis procedures used in the study of the human gait require that the kinematics of the supporting biomechanical model is known beforehand. The first step to obtain the kinematic data is the reconstruction of human spatial motion, i.e., the evaluation of the anatomic points positions that enables to uniquely define the position of all anatomical segments. In photogrammetry, the projection of each anatomical point is described by two linear equations relating its three spatial coordinates with the two coordinates of the projected point. The need for the image of two cameras arises from the fact that three equations are necessary to find the original spatial position of the anatomical point. It is shown here that the kinematic constraint equations associated with a biomechanical model can be used as the extra set of equations required for the reconstruction process, instead of the equations associated with the second camera. With this methodology, the system of equations arising from the point projections and biomechanical model kinematic constraints are solved simultaneously. Since the system of equations has multiple solutions for each image, a strategy based on the minimization of the cost function associated to the smoothness of the reconstructed motion is devised, leading to an automated computer procedure enabling a unique reconstruction.     

Fiber stress profiles in the left ventricle of the heart during diastole: Effects of distension and hypertrophy

Pressure-volume and volume-dimensions relationships, obtained from excised dog left ventricles were used for calculating the stresses acting along the longitudinal axis of the individual myocardial fibers. The calculations were based on a set of empirical and theoretical equations. The pressure-volume relationship as well as the volume-dimensions relationships for the excised left ventricle were expressed in the form of empirical equations; the fiber orientation was written as a function of the fiber location within the left ventricular wall; finally, the fiber stress was determined by means of theoretically derived formulas. Simultaneous solutions for the fibers of a meridian cut through the left ventricular myocardial shell were obtained by means of a digital computer and presented in the form of diagrams. The results showed that at low degrees of distension of the left ventricle there are two zones of higher stresses at the equatorial area, one near the epicardium and one near the endocardium. As the distension proceeds under the effect of progressively increasing intraventricular pressure, these two zones become less well defined, whereas a new zone of higher stresses appears near the apex. At high degrees of distension, the ventricle assumes a more spherical shape and the equatorial zones of higher stresses are replaced by zones of lower stresses. Increase in the myocardial mass results in appearance of the equatorial lower stress zones at lower degrees of distension.