Object segmentation model: analytical results and biological implications

 A simple, biologically motivated neural network for segmentation of a moving object from a visual scene is presented. The model consists of two parts: an object selection model which employs a scaling approach for receptive field sizes, and a subsequent network implementing a spotlight by means of multiplicative synapses. The network selects one object out of several, segments the rough contour of the object, and encodes the winner object's position with high accuracy. Analytical equations for the performance level of the network, e.g., for the critical distance of two objects above which they are perceived as separate, are derived. The network preferentially chooses the object with the largest angular velocity and the largest angular width. An equation for the velocity and width preferences is presented. Additionally it is shown that for certain neurons of the model, flat receptive fields are more favourable than Gaussian ones. The network exhibits performances similar to those known from amphibians. Various electrophysiological and behavioral results – e.g., the distribution of the diameters of the receptive fields of tectal neurons, of the tongue-projecting salamander Hydromantes italicus and the range of optimal prey velocities for prey catching – can be understood on the basis of the model. Received: 7 December 2000 / Accepted: 13 February 2001     

A stochastic model of human reproduction: some preliminary results

The stochastic model of human reproduction presented in this paper removes some of the limitations of the Perrin—Sheps model. It considers the possibility of dependence of amenorrhea period on the preceding gestation period, and also for live births, on the breast-feeding practices of the mother. Two models, virtually identical in concepts, are considered depending on whether time is treated as a continuous or discrete variable. The basic distributions in the models are kept arbitrary, and, therefore, the results for all the existing homogeneous fertility models can be obtained directly from the general results as special cases. The results presented here include the distribution of the interval between two live births, and the probabilities of different states at a point in time. We also illustrate the applications of the models to special cases and generate many known results.     

An analytical model of the hydraulic aspects of stomatal dynamics

An analytical model of the hydraulic aspects of stomatal dynamics is formulated in this paper. The model consists of a coupled system of non-linear, ordinary differential equations, written in terms of water potentials, hydrostatic pressures, osmotic potentials, water vapor resistances and water fluxes. The model is validated by comparisons with the experimental literature. Numerical solutions of the model show qualitative agreement with most known stomatal responses.Stomatal opening in the model is dependent on the interaction of the guard and subsidiary cells in the following manner. Pore opening is initiated by a rise in the guard cell hydrostatic pressure. As the stomate opens, transpiration increases, causing the cell wall water potential to drop. The drop in cell wall water potential then causes the subsidiary cell pressure to drop, opening is accelerated, and the stomate literally “pops” open. Simulated opening proceeds in two distinct phases: a stress phase and a motor phase. During the stress phase, guard cell pressure rises but the pore remains closed. The motor phase commences when the guard cell pressure has risen sufficiently to initiate pore opening, beyond which point opening progresses rapidly.Hydropassive stomatal movements are found to be insufficient to regulate water loss at low leaf water potentials. Stable, hydraulically-based oscillations in stomatal aperture are shown in the model by the existence of a stable limit cycle. The period of these oscillations is strongly influenced by the cell membrane hydraulic conductivity. An increased conductivity results in a shorter period oscillation. Environmental conditions promoting oscillatory behavior are in qualitative agreement with the experimental literature.     

A numerical method for simulating the dynamics of human walking

This paper presents a general method for simulating the movement of the lower extremity during human walking. It is based upon two separate algorithms: one for single support (an open kinematic chain), and the other for the double support phase (a closed-loop linkage). Central to each of these is the recursive Newton-Euler inverse dynamics algorithm, applicable, as given, to any serial, spatial linkage.

For the unconstrained single support model, the Newton-Euler scheme is applied directly to numerically generate the equations of motion. In the case of double support, however, the kinematic constraint equations are used to first eliminate the redundant degrees of freedom, and then solve for the unknown ground reactions under the constrained limb.

The attractiveness of the method is that it offers a compact alternative to manually deriving the equations defining a mathematical model for human gait.     

A model for the human cornea: constitutive formulation and numerical analysis

The human cornea (the external lens of the eye) has the macroscopic structure of a thin shell, originated by the organization of collagen lamellae parallel to the middle surface of the shell. The lamellae, composed of bundles of collagen fibrils, are responsible for the experimentally observed anisotropy of the cornea. Anomalies in the fibril structure may explain the changes in the mechanical behavior of the tissue observed in pathologies such as keratoconus. We employ a fiber-matrix constitutive model and propose a numerical model for the human cornea that is able to account for its mechanical behavior in healthy conditions or in the presence of keratoconus under increasing values of the intraocular pressure. The ability of our model to reproduce the behavior of the human cornea opens a promising perspective for the numerical simulation of refractive surgery.     

Predator-prey interactions in Lake Michigan: model predictions and recent dynamics

Synopsis Several years ago, we used a bioenergetics model to evaluate the impact of increasing salmonid stocking on the highly variable alewife forage base in Lake Michigan. At that time, we forecast an alewife population decline and the following system-wide effects: increased abundances of large zooplankton, decreased salmonid growth rates, increased diet breadth of salmonids, niche shifts among competitors of the alewife, increased alewife growth rates and increased densities of fishes suppressed by alewife. Alewives have continued to decline steadily since 1981 and are now reduced to a density similar to early outbreak levels in the early 1960s. Recent reports on fish growth rates, zooplankton size and fish community structure support our projections regarding system-wide responses to the alewife decline.     

Using numerical approximation as an intermediate step in analytical derivations: some observations from biomechanics

We present four examples to illustrate the use of a type of numerical approximation as an intermediate step in analytical derivation of seemingly complicated biomechanical equations. The method involves examination of curve shapes to elucidate useful underlying trends, which may otherwise be overlooked through consideration of only the equations themselves. Two examples of the method's use are drawn from recently published results in the area of experimental methods in biomechanics of very soft tissues, and two others are taken from our current work on cartilage tissue mechanics. We think that such observations provide a useful means of circumventing complexity issues when deriving models for biomechanical analysis, and further that the method, while simple in concept, could be effective in a range of biomechanics applications.     

Structure and dynamics of a DNA-based model system for the study of electron spin-spin interactions

We report on the structure and dynamics of a model system for measuring long-range distances in biological macromolecules by saturation-recovery EPR. Four DNA duplexes that incorporate a paramagnetic dysprosium ion (Dy(III)) and a nitroxide spin-label were examined by electron paramagnetic resonance (EPR), circular dichroism (CD), and ultra-violet absorbance (UV) spectroscopy. Dy(III) is chelated by the modified base deoxythymidine-EDTA, (dT-EDTA). Electron spin-spin interactions between the Dy(III) ion and the nitroxide radical are observed at distances as great as ∼5.3 nm. A slight change in the conformation of those nucleotides lying between the EDTA(Dy(III)) complex and the nitroxide spin-label results in a “stiffening” of the DNA helix on the EPR time scale. Changes in conformation and helix dynamics are due to the binding of the EDTA(Dy(III)) complex to the phosphodiester backbone of the complementary strand. Molecular mechanics calculations indicate that binding occurs in the 5′ direction on the complementary strand, at a position 3 or 4 phosphates distant from the dT-EDTA(Dy(III)) * dA base pair.     

A model for the dynamics of bipolar disorders

Bipolar disorders are characterized by recurrent, alternating episodes of mania and depression. To examine the dynamical bases of this cyclical illness we consider a minimal model for bipolar disorders based on the observation that the two poles of the disease are mutually exclusive. We assume that the propensities to mania and depression, which are correlated with the activity of two putative neural circuits that promote, respectively, the manic or the depressive state, inhibit each other. When mutual inhibition is sufficiently strong, the model predicts bistability: the bipolar system is then either in a depressive or in a manic state and can display abrupt switches between these stable states. We consider two simple mechanisms which, when added to mutual inhibition, allow the model to pass from bistability to oscillations. Self-sustained oscillations provide a mechanism for the spontaneous, recurrent switching between mania and depression. The model can generate oscillations with a variety of waveforms, including simple periodic oscillations with comparable or unequal durations of the manic and depressive episodes, or small-amplitude oscillations around one of the two states preceding large-amplitude periodic changes in the propensities to mania or depression. The model provides a theoretical framework that covers the bipolar spectrum, i.e., cycling between the two poles of the disease, or evolution to either mania or depression or to an intermediate state without alternating between the two poles of the disease. The model accounts for the clinical observation that antidepressants can trigger the transition to mania or increase the frequency of bipolar cycling.     

Macromolecule--small molecule interactions: analytical and graphical reexamination

An analysis of multiple binding of small molecules by a macromolecule has been made in terms of probabilistic considerations instead of equilibrium ones. This leads to algebraic expressions which suggest alternative nonlinear graphical representations of binding data. One of these, a double-logarithmic presentation of moles bound versus free ligand has been analyzed in detail to show how the number of binding sites and the intrinsic affinity constant can be obtained from the graph.     

A numerical simulation for the dynamics of the sexual phase of monogonont rotifera

A numerical simulation for the dynamics of a model that describes the sexual phase of Monogonont Rotifera reproduction is presented. The simulation is carried out by means of a numerical method based on the integration along the characteristic curves. The numerical experiments cover two basic situations: the existence of an asymptotic stable equilibrium state and the existence of an stable periodic solution. Our results are in agreement with the theoretical analysis made by Calsina and Ripoll (J. Math. Biol. 45 (2002) 22).     

A mathematical model for photoreceptor interactions

The interactions between rods and cones in the retina have been the focus of innumerable experimental and theoretical biological studies in previous decades yet the understanding of these interactions is still incomplete primarily due to the lack of a unified concept of cone photoreceptor organization and its role in retinal diseases. The low abundance of cones in many of the non-primate mammalian models that have been studied make conclusions about the human retina difficult. A more complete knowledge of the human retina is crucial for counteracting the events that lead to certain degenerative diseases, in particular those associated with photoreceptor cell death (e.g., retinitis pigmentosa). In an attempt to gain important insight into the role and interactions of the rods and the cones we develop and analyze a set of mathematical equations that model a system of photoreceptors and incorporate a direct rod-cone interaction. Our results show that the system can exhibit stable oscillations, which correspond to the rhythmic renewal and shedding of the photoreceptors. In addition, our results show the mathematical necessity of this rod-cone direct interaction for survival of both and gives insight into this mechanism.     

Active contraction of cardiac cells: a reduced model for sarcomere dynamics with cooperative interactions

We propose a reduced ODE model for the mechanical activation of cardiac myofilaments, which is based on explicit spatial representation of nearest-neighbour interactions. Our model is derived from the cooperative Markov Chain model of Washio et al. (Cell Mol Bioeng 5(1):113–126, 2012), under the assumption of conditional independence of specific sets of events. This physically motivated assumption allows to drastically reduce the number of degrees of freedom, thus resulting in a significantly large computational saving. Indeed, the original Markov Chain model involves a huge number of degrees of freedom (order of \(10^{21}\)) and is solved by means of the Monte Carlo method, which notoriously reaches statistical convergence in a slow fashion. With our reduced model, instead, numerical simulations can be carried out by solving a system of ODEs, reducing the computational time by more than 10, 000 times. Moreover, the reduced model is accurate with respect to the original Markov Chain model. We show that the reduced model is capable of reproducing physiological steady-state force–calcium and force–length relationships with the observed asymmetry in apparent cooperativity near the calcium level producing half activation. Finally, we also report good qualitative and quantitative agreement with experimental measurements under dynamic conditions.     

A cell cycle model for somitogenesis: mathematical formulation and numerical simulation

After many years of research, the mechanisms that generate a periodic pattern of repeated elements (somites) along the length of the embryonic body axis is still one of the major unresolved problems in developmental biology. Here we present a mathematical formulation of the cell cycle model for somitogenesis proposed in Development105 (1989), 119-130. Somite precursor cells in the node are asynchronous, and therefore, as a population, generate continuously pre-somite cells which enter the segmental plate. The model makes the hypothesis that there exists a time window within the cell cycle, making up one-seventh of the cycle, which gates the pre-somite cells so that they make somites discretely, seven per cycle. We show that the model can indeed account for the spatiotemporal patterning of somite formation during normal development as well as the periodic abnormalities produced by heat shock treatment. We also relate the model to recent molecular data on the process of somite formation.     

A lock-and-key model for protein-protein interactions

MOTIVATION: Protein-protein interaction networks are one of the major post-genomic data sources available to molecular biologists. They provide a comprehensive view of the global interaction structure of an organism's proteome, as well as detailed information on specific interactions. Here we suggest a physical model of protein interactions that can be used to extract additional information at an intermediate level: It enables us to identify proteins which share biological interaction motifs, and also to identify potentially missing or spurious interactions. RESULTS: Our new graph model explains observed interactions between proteins by an underlying interaction of complementary binding domains (lock-and-key model). This leads to a novel graph-theoretical algorithm to identify bipartite subgraphs within protein-protein interaction networks where the underlying data are taken from yeast two-hybrid experimental results. By testing on synthetic data, we demonstrate that under certain modelling assumptions, the algorithm will return correct domain information about each protein in the network. Tests on data from various model organisms show that the local and global patterns predicted by the model are indeed found in experimental data. Using functional and protein structure annotations, we show that bipartite subnetworks can be identified that correspond to biologically relevant interaction motifs. Some of these are novel and we discuss an example involving SH3 domains from the Saccharomyces cerevisiae interactome. AVAILABILITY: The algorithm (in Matlab format) is available (see http://www.maths.strath.ac.uk/~aas96106/lock_key.html).     

A multigroup model for predator-prey interactions

The predator-prey interactions between the protozoan Tetrahymena pyriformis and the bacterium Aerobacter aerogenes have been studied experimentally and mathematically. A mathematical model for the ciliates defines the mass distribution of cells within the population. The resulting model equations are solved by the use of multigroup theory. Experimental data from batch and continuous flow reactors are compared with the results of the numerical integration.     

A model for protein-DNA interaction dynamics

We map a simplified version of the protein-DNA interaction problem into an Ising-model in a random magnetic field. The model includes a "head" which moves along the chain while interacting with the underlying spins. The head moves by using the statistical fluctuations of base openings. A Monte Carlo (MC) simulation of this model reveals the possibility of biased diffusion in one direction, followed by sequence identification and binding. The model provides some insight into the mechanisms used by some repressor proteins to diffuse and bind to specific DNA-binding sites.     

A Simple model of the leaf daily water potential dynamics of some forest tree species

A simple model was developed to characterize the daily water potential dynamics (Ψ_x of sun and shade leaves of three forest tree species (Quercus cerris, Acer campestre andCarpinus betulus) under anticyclonic weather types. Input data used for this-model were the vapour pressure deficit (d) and the soil moisture content (w.). The model is usable for the calculation of the actual Ψ_x-values with a probable error 0.18 –0.28 MPa and limits the maximum and minimum Ψ_x-values which may occur with the particular tree species. The model makes it possible to establish for each species the regime, determined byd andw, at which the water potential of the leaves reacts most sensitively to the changes of the environmental parameters.     

A two-current model for the dynamics of cardiac membrane

In this paper we introduce and study a model for electrical activity of cardiac membrane which incorporates only an inward and an outward current. This model is useful for three reasons: (1) Its simplicity, comparable to the FitzHugh-Nagumo model, makes it useful in numerical simulations, especially in two or three spatial dimensions where numerical efficiency is so important. (2) It can be understood analytically without recourse to numerical simulations. This allows us to determine rather completely how the parameters in the model affect its behavior which in turn provides insight into the effects of the many parameters in more realistic models. (3) It naturally gives rise to a one-dimensional map which specifies the action potential duration as a function of the previous diastolic interval. For certain parameter values, this map exhibits a new phenomenon—subcritical alternans—that does not occur for the commonly used exponential map.