Estimating surface growth rates from changes in curvature

The shape of a biological surface may be regarded as an observable. Here a method is given for deriving growth parameters from the change in shape of such a surface. Isotropy is assumed, and implies a conformal relationship between initial and final surfaces. One further assumption is necessary to specify the growth regime: in the case of radially symmetric surfaces, this is that the process is similarly symmetric; in the general case the assumption is that the Dirichlet integral of scale factors is miminized.     

Considerations on the role of surface tension and epidermal growth factor in the mechanism of integumental pattern formation

The formation of the posterior parietal hair whorl during normal human development is proposed as a morphogenetic problem in which the effects of surface tension and the appearance of singularities figure prominently. Surface tension is considered in the sense of "Langer's lines" which define local contours of tension in the skin of the organism. According to this view, an analysis of the organism's skin tension field is fundamental to problems of integumental pattern formation. A computational analysis of the skin tension field is proposed, potentially using methods from finite element theory applied to molecular and cellular mechanisms within the skin which resist deformation. To this end, surface tension is provisionally defined as the differential adsorption (adhesion) of molecular and supramolecular binding elements within the skin. In practical applications, it is suggested that epidermal growth factor (EGF) and similar molecules have the physicochemical features and the biological effects required to experimentally probe surface tension phenomena at supramolecular levels. In this regard, the concept of topological discontinuities is introduced as a potential theoretical bridge across levels of organization. Specific examples of these discontinuities are given and discussed in terms of the development of singularities in control surfaces. It is hoped that these considerations will be useful in the mechanistic analysis of hair whorl formation during human embryo-genesis and in other problems of integumental pattern formation and nonequilibrium surface behavior.     

PARATI – a dynamic model for radiological assessments in urban areas

The structure and mathematical model of PARATI, a detailed computer programme developed for the assessment of the radiological consequences of an accidental contamination of urban areas, is described with respect to the scenarios used for the estimation of exposure fields in a village or town, the models for the initial and secondary contamination with the radionuclide ¹³⁷Cs, the concepts for calculating the resulting radiation exposures and the changes with time of the contamination and radiation fields. Kerma rates at various locations in tropical urban areas are given, and the contribution of different contaminated surfaces to these rates after dry or wet deposition are discussed. Received: 12 April 1996 / Accepted in revised form: 30 August 1996     

A mathematical framework for modelling cambial surface evolution using a level set method

Background and Aims

During their lifetime, tree stems take a series of successive nested shapes. Individual tree growth models traditionally focus on apical growth and architecture. However, cambial growth, which is distributed over a surface layer wrapping the whole organism, equally contributes to plant form and function. This study aims at providing a framework to simulate how organism shape evolves as a result of a secondary growth process that occurs at the cellular scale.

Methods

The development of the vascular cambium is modelled as an expanding surface using the level set method. The surface consists of multiple compartments following distinct expansion rules. Growth behaviour can be formulated as a mathematical function of surface state variables and independent variables to describe biological processes.

Key Results

The model was coupled to an architectural model and to a forest stand model to simulate cambium dynamics and wood formation at the scale of the organism. The model is able to simulate competition between cambia, surface irregularities and local features. Predicting the shapes associated with arbitrarily complex growth functions does not add complexity to the numerical method itself.

Conclusions

Despite their slenderness, it is sometimes useful to conceive of trees as expanding surfaces. The proposed mathematical framework provides a way to integrate through time and space the biological and physical mechanisms underlying cambium activity. It can be used either to test growth hypotheses or to generate detailed maps of wood internal structure.     

Experimental measurement and quantification of frictional contact between biological surfaces experiencing large deformation and slip

Interactions between contacting biological surfaces may play significant roles in physiological and pathological processes. Theoretical models have described some special cases of contact, using one or more simplifying assumptions. Experimental quantification of contact could help to validate theoretical analyses. The objective of this study was to develop a general mathematical approach describing the dynamics of deformation and relative surface motion between contacting bodies and to implement this approach to describe the contact between two experimentally tracked tissue surfaces. A theoretical formulation (in 2-D and 3-D) of contact using the movement of discrete tissue markers is described. The method was validated using theoretically generated 3-D datasets, with <1% error for a wide range of parameters. The method was applied to the contact loading of opposing articular cartilage tissues, where displacements of cell nuclei were tracked optically and used to quantify the movements and deformations of the surfaces. Compared to tissues with matched material properties, tissues with mismatched material properties exhibited increased disparities in lateral expansion and relative motion (sliding) between the contacting surfaces.     

Comparison of fluorinated polymers against stainless steel, glass and polypropylene in microbial biofilm adherence and removal

Biofilm formation is a long-standing problem in ultrapure water and bioprocess fluid transport lines. The standard materials used in these applications (316L stainless steel, polypropylene and glass) have long been known to be good surfaces for the attachment of bacteria and other biological materials. To compare the relative tenacity of biofilms grown on materials used in manufacturing processes, a model system for biofilm attachment was constructed that approximates the conditions in industrial process systems. New fluorinated polymers were compared to the above materials by evaluating the surface area coverage of bacterial populations on materials before and after mild chemical treatment. In addition, contact angle studies compared the relative hydrophobicity of surfaces to suspensions of bacteria in growth media, and scanning electron microscopy and atomic force microscopy studies were used to characterize surface smoothness and surface defects. Biofilm adherence to polymer-based substrata was determined to be a function of both surface finish and surface chemistry. Specifically, materials that are less chemically reactive, as indicated by higher contact angle, can have rougher surface finishes and still be amenable to biofilm removal. Received 20 March 1997/ Accepted in revised form 15 July 1997     

Resonant waveguide grating biosensor for living cell sensing

This article presents theoretical analysis and experimental data for the use of resonant waveguide grating (RWG) biosensors to characterize stimulation-mediated cell responses including signaling. The biosensor is capable of detecting redistribution of cellular contents in both directions that are perpendicular and parallel to the sensor surface. This capability relies on online monitoring cell responses with multiple optical output parameters, including the changes in incident angle and the shape of the resonant peaks. Although the changes in peak shape are mainly contributed to stimulation-modulated inhomogeneous redistribution of cellular contents parallel to the sensor surface, the shift in incident angle primarily reflects the stimulation-triggered dynamic mass redistribution (DMR) perpendicular to the sensor surface. The optical signatures are obtained and used to characterize several cellular processes including cell adhesion and spreading, detachment and signaling by trypsinization, and signaling through either epidermal growth factor receptor or bradykinin B2 receptor. A mathematical model is developed to link the bradykinin-mediated DMR signals to the dynamic relocation of intracellular proteins and the receptor internalization during B2 receptor signaling cycle. This model takes the form of a set of nonlinear, ordinary differential equations that describe the changes in four different states of B2 receptors, diffusion of proteins and receptor-protein complexes, and the DMR responses. Classical analysis shows that the system converges to a unique optical signature, whose dynamics (amplitudes, transition time, and kinetics) is dependent on the bradykinin signal input, and consistent with those observed using the RWG biosensors. This study provides fundamentals for probing living cells with the RWG biosensors, in general, optical biosensors.     

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.     

Analysis of logistic growth models

A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most predictive models are shown to be based on variations of the classical Verhulst logistic growth equation. We review and compare several such models and analyse properties of interest for these. We also identify and detail several associated limitations and restrictions.A generalized form of the logistic growth curve is introduced which incorporates these models as special cases. Several properties of the generalized growth are also presented. We furthermore prove that the new growth form incorporates additional growth models which are markedly different from the logistic growth and its variants, at least in their mathematical representation. Finally, we give a brief outline of how the new curve could be used for curve-fitting.     

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.     

Modelling Biological Gel Contraction by Cells: Mechanocellular Formulation and Cell Traction Force Quantification

Traction forces developed by most cell types play a significant role in the spatial organisation of biological tissues. However, due to the complexity of cell-extracellular matrix interactions, these forces are quantitatively difficult to estimate without explicitly considering cell properties and extracellular mechanical matrix responses. Recent experimental devices elaborated for measuring cell traction on extracellular matrix use cell deposits on a piece of gel placed between one fixed and one moving holder. We formulate here a mathematical model describing the dynamic behaviour of the cell-gel medium in such devices. This model is based on a mechanical force balance quantification of the gel visco-elastic response to the traction forces exerted by the diffusing cells. Thus, we theoretically analyzed and simulated the displacement of the free moving boundary of the system under various conditions for cells and gel concentrations. This modelis then used as the theoretical basis of an experimental device where endothelial cells are seeded on a rectangular biogel of fibrin cast between two floating holders, one fixed and the other linked to a force sensor. From a comparison of displacement of the gel moving boundary simulated by the model and the experimental data recorded from the moving holder displacement, the magnitude of the traction forces exerted by the endothelial cell on the fibrin gel was estimated for different experimental situations. Different analytical expressions for the cell traction term are proposed and the corresponding force quantifications are compared to the traction force measurements reported for various kind of cells with the use of similar or different experimental devices. This revised version was published online in June 2006 with corrections to the Cover Date.     

A theoretical model of LDL-receptor trapping on a spherical cell.

The kinetics of the trapping of LDL-receptor complexes by coated pits on the surface of fibroblasts is examined in this paper. We have recently developed a mathematical formalism to extend Keizer's non-linear, non-equilibrium fluctuation-dissipation theory to the kinetics of chemical systems constrained to a spherical surface. Keizer's theory is ideally suited to the study of open biological systems. In the past it has been used to investigate endocytosis on fibroblasts. However, these applications have modeled the cell membrane with an infinite plane. As such, the finite size of the cellular membrane, as well as its precise symmetry, could not be incorporated into the previous studies. Thus in this paper we use our recently developed methodology to reexamine the trapping step in endocytosis on spherical cells. For cell surface processes, the theoretical consideration of a spherical symmetry or an infinite plane, in model calculations, will depend on the experimental or in vivo conditions of the processes of interest. For a spherical symmetry, we find that the finite size of the cell surface does not significantly affect the rate of the trapping step given the empirically determined values for the relevant parametes on fibroblasts. This result supports the approximation used in the previous investigation. However, this and other analyses indicate that the finitie size of the biological surface probably is an important parameter for processes which occur on smaller biological surfaces such as those found on organelles.     

Studies in the physicomathematical theory of organic form

In continuation of previous studies the theory is developed on the assumption that the form of any organism is determined by requirements to perform definite biological functions. A previously outlined theory of the form of plants is developed further, showing how the conditions of mechanical strength together with the specifications of the total mass and metabolism, may quantitatively determine not only the general form of the plant, but even the number, size, shape and shades of the leaves. Next the form of animals, as required by mechanical conditions and by the different types of possible locomotions is discussed. A mathematical theory of locomotion of snakes in relation to their shape is outlined. Next is discussed the form and locomotion of quadrupeds. A number of theoretical relations, which describe the shape of an animal, are derived and compared to available observations. After that the theory of flight of birds and insects is discussed, and again some form relations comparable with observations, are discussed. Finally a set of equations is outlined, which determines not only the external shape, but also the internal structure of animals. Different relations pertaining to some inner organs are derived and compared with available observations. The paper ends with a brief discussion on the shape of unicellular organisms.     

Cyclic coordinate descent: A robotics algorithm for protein loop closure

In protein structure prediction, it is often the case that a protein segment must be adjusted to connect two fixed segments. This occurs during loop structure prediction in homology modeling as well as in ab initio structure prediction. Several algorithms for this purpose are based on the inverse Jacobian of the distance constraints with respect to dihedral angle degrees of freedom. These algorithms are sometimes unstable and fail to converge. We present an algorithm developed originally for inverse kinematics applications in robotics. In robotics, an end effector in the form of a robot hand must reach for an object in space by altering adjustable joint angles and arm lengths. In loop prediction, dihedral angles must be adjusted to move the C-terminal residue of a segment to superimpose on a fixed anchor residue in the protein structure. The algorithm, referred to as cyclic coordinate descent or CCD, involves adjusting one dihedral angle at a time to minimize the sum of the squared distances between three backbone atoms of the moving C-terminal anchor and the corresponding atoms in the fixed C-terminal anchor. The result is an equation in one variable for the proposed change in each dihedral. The algorithm proceeds iteratively through all of the adjustable dihedral angles from the N-terminal to the C-terminal end of the loop. CCD is suitable as a component of loop prediction methods that generate large numbers of trial structures. It succeeds in closing loops in a large test set 99.79% of the time, and fails occasionally only for short, highly extended loops. It is very fast, closing loops of length 8 in 0.037 sec on average.     

Identification of nonlinear models of biological membranes by the method of fixed voltage

A mathematical substantiation of the method suggested by Hodkin and Huxley in 1952 for the identification of nonlinear systems is presented. A procedure for the application of this method was developed, which involves creating the structure of a mathematical model, carrying out a series of tests with specially chosen signals, and finding the unknown parameters. The basic requirements to admissible sets of entrance and target signals and the operator of the system were determined. It was shown that it should be quite continuous, the minimal number of unknown parameters and the minimal complexity of structure of the operator should provide the required quality of approximation. The merits and demerits of the mathematical models of Hodkin-Huxley and Noble, and the procedures used for their creation are discussed. The structure of the operator for the identification of mathematical models of excitable membranes when a large number of membrane currents is considered is offered. It was found that nonlinear electric properties of biological membranes can be identified using tests with other kinds of "fixed" parameters, for example, the method of "fixed" current, the fixed linearly increasing voltage, and others.     

Determination of electrostatic potentials at biological interfaces using electron-electron double resonance.

A new general method for the determination of electrostatic potentials at biological surfaces is presented. The approach is based on measurement of the collision frequency of a charged nitroxide in solution with a nitroxide fixed to the surface at the point of interest. The collision frequency is determined with 14N:15N double label electron-electron double resonance (ELDOR). As a test, the method is shown to give values for phospholipid bilayer surface potentials consistent with the Gouy-Chapman theory, a simple model shown by many independent tests to accurately describe charged, planar surfaces. In addition, the method is applied to determine the electrostatic potential near the surface of DNA. The results indicate that the potential is significantly smaller than that predicted from Poisson-Boltzmann analysis, but is in qualitative agreement with that predicted by Manning's theory of counter ion condensation. The method is readily extended to measurement of surface potentials of proteins.     

Reduced surface: An efficient way to compute molecular surfaces

Because of their wide use in molecular modeling, methods to compute molecular surfaces have received a lot of interest in recent years. However, most of the proposed algorithms compute the analytical representation of only the solvent-accessible surface. There are a few programs that compute the analytical representation of the solvent-excluded surface, but they often have problems handling singular cases of self-intersecting surfaces and tend to fail on large molecules (more than 10,000 atoms). We describe here a program called MSMS, which is shown to be fast and reliable in computing molecular surfaces. It relies on the use of the reduced surface that is briefly defined here and from which the solvent-accessible and solvent-excluded surfaces are computed. The four algorithms composing MSMS are described and their complexity is analyzed. Special attention is given to the handling of self-intersecting parts of the solvent-excluded surface called singularities. The program has been compared with Connolly's program PQMS [M. L. Connolly (1993) Journal of Molecular Graphics, Vol. 11, pp. 139–141] on a set of 709 molecules taken from the Brookhaven Data Base. MSMS was able to compute topologically correct surfaces for each molecule in the set. Moreover, the actual time spent to compute surfaces is in agreement with the theoretical complexity of the program, which is shown to be O[n log(n)] for n atoms. On a Hewlett-Packard 9000/735 workstation, MSMS takes 0.73 s to produce a triangulated solvent-excluded surface for crambin (1crn, 46 residues, 327 atoms, 4772 triangles), 4.6 s for thermolysin (3tln, 316 residues, 2437 atoms, 26462 triangles), and 104.53 s for glutamine synthetase (2gls, 5676 residues, 43632 atoms, 476665 triangles). © 1996 John Wiley & Sons, Inc.     

Virulence determinants of Escherichia coli: present knowledge and questions.

This paper describes the present state of research on the pathogenicity of Escherichia coli and points out the gaps in knowledge that should be filled in the future. First, the great versatility of E. coli in producing disease is noted, as well as the invaluable contributions that studies of it have made to the development of general knowledge on bacterial pathogenicity. Then, the biological requirements for pathogenicity: infection of mucous surfaces; penetration of those surfaces; multiplication in vivo; interference with host defence mechanisms; and damage to the host, are taken in turn, and an enquiry is made on how far studies have progressed toward identifying their molecular determinants and relating structure to biological action. Only for mucous surface adhesins and protein toxins are studies at the structure-function level. Some progress has been made on interference with host defence, but little is known about competition with commensals on mucous surfaces, invasion into the tissues, and growth in vivo.     

A growth-controlled model of the shape of a sunflower head

A mathematical model is presented which predicts the shape of a sunflower receptacle (or the compact receptacle of various other taxa) and the pattern of its floral parts (florets) from the time of their initiation to maturity. The model assumes that the expansion and curving of the receptacle surface is just sufficient to accommodate the development of the florets, thus minimizing the quantity of plant tissue involved. The model assumes a fixed angular separation (divergence) between successive florets, an S-shaped (sigmoidal) growth function followed by each florets, and a fixed time delay (period) between the initiation of successive florets. It is further assumes that the shape and relative position occupied by the florets on the receptacle surface are invariant in time. By this theory, the shape of the receptacle surface is fully determined once the mathematical form of the growth function is specified. Using the logistic growth function, the theory is tested against the measured shapes of plant receptacles from different taxa at various points in their development. The least-squares adjusted fits to the theory are, in most cases, very good indeed.