A bidomain threshold model of propagating calcium waves

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca²⁺) in living cells. Modeling Ca²⁺ release as a threshold process allows the explicit construction of traveling wave solutions to probe the dependence of Ca²⁺ wave speed on physiologically important parameters such as the threshold for Ca²⁺ release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca²⁺ resequestration from the cytosol to the ER, and the total [Ca²⁺] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca²⁺ waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called ‘tango waves’ and the dependence of Ca²⁺ wave speed on the total [Ca²⁺].      

PDZ domains - common players in the cell signaling

PDZ domains are ubiquitous protein interaction modules that play a key role in cellular signaling. Their binding specificity involves recognition of the carboxyl-terminus of various proteins, often belonging to receptor and ion channel families. PDZ domains also mediate more complicated molecular networks through PDZ-PDZ interactions, recognition of internal protein sequences or phosphatidylinositol moieties. The domains often form a tandem of multiple copies, but equally often such tandems or single PDZ domain occur in combination with other signaling domains (for example SH3, DH/PH, GUK, LIM, CaMK). Common occurrence of PDZ domains in Metazoans strongly suggests that their evolutionary appearance results from the complication of signaling mechanisms in multicellular organisms. Here, we focus on their structure, specificity and role in signaling pathways.     

Analysis of dynamic and stationary pattern formation in the cell cortex

We study a sol-gel mechanochemical model for cellular cytoplasm. Using conservation equations and a force balance equation, we derive equations for the sol-gel dynamics. Regular perturbation analysis suggests the growth of patterns which may be either dynamic or stationary, depending on parameter values. Nonlinear analysis, which indicates that these patterns remain bounded, is confirmed by numerically solving the mechanochemical equations. We use these analytical and numerical results to model two different biological problems: the dynamic formation of filopodia in nerve growth cones, and the growth of microvilli in epithelial cells.     

A biomechanical model of the foot

The foot is modeled as a statically indeterminate structure supporting its load at the heads of the five metatarsals and the tuberosity of the calcaneous. The distribution of support is determined through an analysis of the deformations caused in the structure as a result of the forces at these locations. The analysis includes the effect of the plantar aponeurosis and takes into account the deformation of the metatarsals and bending of the joints. A parametric study is presented to illustrate the behavior of the solution under a broad range of conditions.     

Travelling waves and pattern formation in a model for fungal development

 Under a variety of conditions, the hyphal density within the expanding outer edge of growing fungal mycelia can be spatially heterogeneous or nearly uniform. We conduct an analysis of a system of reaction-diffusion equations used to model the growth of fungal mycelia and the subsequent development of macroscopic patterns produced by differing hyphal and hence biomass densities. Both local and global results are obtained using analytical and numerical techniques. The emphasis is on qualitative results, including the effects of changes in parameter values on the structure of the solution set. Received 22 November 1995; received in revised form 17 May 1996     

A dynamical model for plant cell wall architecture formation

We discuss a dynamical mathematical model to explain cell wall architecture in plant cells. The highly regular textures observed in cell walls reflect the spatial organisation of the cellulose microfibrils (CMFs), the most important structural component of cell walls. Based on a geometrical theory proposed earlier [A. M. C. Emons, Plant, Cell and Environment 17, 3–14 (1994)], the present model describes the space-time evolution of the density of the so-called rosettes, the CMF synthesizing complexes. The motion of these rosettes in the plasma membrane is assumed to be governed by an optimal packing constraint on the CMFs plus adherent matrix material, that couples the direction of motion, and hence the orientation of the CMF being deposited, to the local density of rosettes. The rosettes are created inside the cell in the endoplasmatic reticulum and reach the cell-membrane via vesicles derived from Golgi-bodies. After being inserted into the plasma membrane they are assumed to be operative for a fixed, finite lifetime. The plasma membrane domains within which rosettes are activated are themselves also supposed to be mobile. We propose a feedback mechanism that precludes the density of rosettes to rise beyond a maximum dictated by the geometry of the cell. The above ingredients lead to a quasi-linear first order PDE for the rosette-density. Using the method of characteristics this equation can be cast into a set of first order ODEs, one of which is retarded. We discuss the analytic solutions of the model that give rise to helicoidal, crossed polylamellate, helical, axial and random textures, since all cell walls are composed of (or combinations of) these textures. Received: 10 July 1999 / Revised version: 7 June 2000 / Published online: 16 February 2001     

A biomechanical model of the upper airways for simulating laryngoscopy

This paper describes a three-dimensional finite element model of the human upper airways during rigid laryngoscopy. In this procedure, an anaesthetist uses a rigid blade to displace and compress the tongue of the patient, and then inserts a tube into the larynx to allow controlled ventilation of the lungs during an operation. A realistic model of the main biomechanical aspects involved would help anaesthetists in training and in predicting difficult cases in advance. For this purpose, the finite element method was used to model structures such as the tongue, ligaments, larynx, vocal cords, bony landmarks, laryngoscope blade, and their inter-relationships, based on data extracted from X-ray, MRI, and photographic records. The model has been used to investigate how the tongue tissue behaves in response to the insertion of the laryngoscope blade, when it is subjected to a variety of loading conditions. In particular, the mechanical behaviour of the soft tissue of the tongue was simulated, from simple linear elastic material to complex non-linear viscoelastic material. The results show that, within a specific set of tongue material parameters, the simulated outcome can be successfully related to the view of the vocal cords achieved during real laryngoscopies on normal subjects, and on artificially induced difficult laryngoscopy, created by extending the upper incisors teeth experimentally.     

A logical model for growth and differentiation in multi-celled organisms

This paper is an attempt to provide a logical model for the process of growth and differentiation in a multi-cellular organism. More specifically it is intended to show how genetic information relating to macroscopic structure and coded in the form of a logical tree could be progressively embodied in the organism as it develops by repeated division from a single cell. The aim is to establish biological analogies rather than mathematical interest, and reproduction, adaption, and the coordinating action of hormones are discussed within the general logical framework.     

Travelling waves in a tissue interaction model for skin pattern formation

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed.     

Applications of a model for scale-invariant pattern formation in developing systems

A fundamental problem in developmental biology concerns the proportioning of the developing tissue of a morphallactic system into different cell types in a way that is independent of the overall size of the tissue. The two main models for positional information in pattern formation, the source-sink models and the Turing reaction-diffusion models, have shortcomings that limit their applicability. In a previous paper, we described a model that can produce perfectly scale-invariant spatial patterns and analyzed some of its mathematical properties. In the present paper, we demonstrate some of the shortcomings of the standard reaction-diffusion models and discuss the applicability of our model to developmental systems.     

Current shunting and formation of stationary shock waves during electric explosions of metal wires in air

Results of experiments on the generation of shock waves during electric explosions of fine copper and tungsten wires in air are analyzed. The generation mechanism of stationary shock wave by a plasma piston formed during the shunting breakdown of the electrode gap in the course of a wire explosion is investigated. The role of structural elements of such discharges, such as the core, corona, and wire environment, is analyzed.     

A 3D biomechanical model of the hand for power grip

A three-dimensional scalable biomechanical model of the four fingers of the hand to evaluate power grip is proposed. The model has been validated by means of reproducing an experiment in which the subjects exerted the maximal voluntary grasping force over cylinders of different diameters. The model is used to simulate the cylinder grip for two hand sizes and for five different handle diameters. The reduction of the muscle forces using different handle diameters has been studied. The model can be applied to the design and evaluation of handles for power grip and to the study of power grasp for normal and abnormal hands.     

Travelling lipid domains in a dynamic model for protein-induced pattern formation in biomembranes

Cell membranes are composed of a mixture of lipids. Many biological processes require the formation of spatial domains in the lipid distribution of the plasma membrane. We have developed a mathematical model that describes the dynamic spatial distribution of acidic lipids in response to the presence of GMC proteins and regulating enzymes. The model encompasses diffusion of lipids and GMC proteins, electrostatic attraction between acidic lipids and GMC proteins as well as the kinetics of membrane attachment/detachment of GMC proteins. If the lipid-protein interaction is strong enough, phase separation occurs in the membrane as a result of free energy minimization and protein/lipid domains are formed. The picture is changed if a constant activity of enzymes is included into the model. We chose the myristoyl-electrostatic switch as a regulatory module. It consists of a protein kinase C that phosphorylates and removes the GMC proteins from the membrane and a phosphatase that dephosphorylates the proteins and enables them to rebind to the membrane. For sufficiently high enzymatic activity, the phase separation is replaced by travelling domains of acidic lipids and proteins. The latter active process is typical for nonequilibrium systems. It allows for a faster restructuring and polarization of the membrane since it acts on a larger length scale than the passive phase separation. The travelling domains can be pinned by spatial gradients in the activity; thus the membrane is able to detect spatial clues and can adapt its polarity dynamically to changes in the environment.     

Regeneration and the need for simpler model organisms

The problem of regeneration is fundamentally a problem of tissue homeostasis involving the replacement of cells lost to normal 'wear and tear' (cell turnover), and/or injury. This attribute is of particular significance to organisms possessing relatively long lifespans, as maintenance of all body parts and their functional integration is essential for their survival. Because tissue replacement is broadly distributed among multicellular life-forms, and the molecules and mechanisms controlling cellular differentiation are considered ancient evolutionary inventions, it should be possible to gain key molecular insights about regenerative processes through the study of simpler animals. We have chosen to study and develop the freshwater planarian Schmidtea mediterranea as a model system because it is one of the simplest metazoans possessing tissue homeostasis and regeneration, and because it has become relatively easy to molecularly manipulate this organism. The developmental plasticity and longevity of S. mediterranea is in marked contrast to its better-characterized invertebrate cohorts: the fruitfly Drosophila melanogaster and the roundworm Caenorhabditis elegans, both of which have short lifespans and are poor at regenerating tissues. Therefore, planarians present us with new, experimentally accessible contexts in which to study the molecular actions guiding cell fate restriction, differentiation and patterning, each of which is crucial not only for regeneration to occur, but also for the survival and perpetuation of all multicellular organisms.     

A biomechanical model for the analysis of the cervical spine in static postures.

To gain a better understanding of the forces working on the cervical spine, a spatial biomechanical computer model was developed. The first part of our research was concerned with the development of a kinematic model to establish the axes of rotation and the mutual position of the head and vertebrae with regard to flexion, extension, lateroflexion and torsion. The next step was the introduction of lines of action of muscle forces and an external load, created by gravity and accelerations in different directions, working on the centre of gravity of the head and possibly a helmet. Although the results of our calculations should be interpreted cautiously in the present stage of our research, some conclusions can be drawn with respect to different head positions. During flexion muscle forces and joint reaction forces increase, except the force between the odontoid and the ligamentum transversum atlantis. This force shows a minimum during moderate flexion. The joint reaction forces on the levels C0-C1, C1-C2, and C7-T1 reach minimum values during extension, each in different stages of extension. Axial rotation less than 35 degrees does not need great muscle forces, axial rotation further than 35 degrees causes muscle forces and joint reaction forces to increase fast. While performing, lateral flexion muscle forces and joint reaction forces must increase rapidly to balance the head. We obtained some indications that the order of magnitude of the calculated forces is correct.     

A spatial model for the spread of invading organisms subject to competition

 A spatially explicit integrodifference equation model is studied for the spread of an invading organism against an established competitor. Provided the invader is initially confined to a bounded region, the invasion spreads asymptotically as a travelling wave whose speed depends on the strength of the competitive interaction and on the dispersal characteristics of the invader. Even an inferior, but established, competitor can significantly reduce the invasion speed. The invasion speed is also influenced by the exact shape of the dispersal kernel (especially the thickness of the tail) as well as the mean dispersal distance for each generation. Received 10 April 1996; received in revised form 21 August 1996