Mathematical models of cell colonization of uniformly growing domains

During the development of vertebrate embryos, cell migrations occur on an underlying tissue domain in response to some factor, such as nutrient. Over the time scale of days in which this cell migration occurs, the underlying tissue is itself growing. Consequently cell migration and colonization is strongly affected by the tissue domain growth. Numerical solutions for a mathematical model of chemotactic migration with no domain growth can lead to travelling waves of cells with constant velocity; the addition of domain growth can lead to travelling waves with nonconstant velocity. These observations suggest a mathematical approximation to the full system equations, allowing the method of characteristics to be applied to a simplified chemotactic migration model. The evolution of the leading front of the migrating cell wave is analysed. Linear, exponential and logistic uniform domain growths are considered. Successful colonization of a growing domain depends on the competition between cell migration velocity and the velocity and form of the domain growth, as well as the initial penetration distance of the cells. In some instances the cells will never successfully colonize the growing domain. These models provide an insight into cell migration during embryonic growth, and its dependence upon the form and timing of the domain growth.     

Global analysis of dynamical decision-making models through local computation around the hidden saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity.     

Principles of quantitation of viral loads using nucleic acid sequence-based amplification in combination with homogeneous detection using molecular beacons

For quantitative NASBA-based viral load assays using homogeneous detection with molecular beacons, such as the NucliSens EasyQ HIV-1 assay, a quantitation algorithm is required. During the amplification process there is a constant growth in the concentration of amplicons to which the beacon can bind while generating a fluorescence signal. The overall fluorescence curve contains kinetic information on both amplicon formation and beacon binding, but only the former is relevant for quantitation. In the current paper, mathematical modeling of the relevant processes is used to develop an equation describing the fluorescence curve as a function of the amplification time and the relevant kinetic parameters. This equation allows reconstruction of RNA formation, which is characterized by an exponential increase in concentrations as long as the primer concentrations are not rate limiting and by linear growth over time after the primer pool is depleted. During the linear growth phase, the actual quantitation is based on assessing the amplicon formation rate from the viral RNA relative to that from a fixed amount of calibrator RNA. The quantitation procedure has been successfully applied in the NucliSens EasyQ HIV-1 assay.     

Modelling of root growth and bending in two dimensions

A special co-ordinate system is developed for modelling the gravitropic bending of plant roots. It is based on the Local Theory of Curves in differential geometry and describes, in one dimension, growth events that may actually occur in two, or even three, dimensions. With knowledge of the spatial distributions of relative elemental growth rates (RELELs) for the upper and lower flanks of a gravistimulated root, and also their temporal dependencies, it is possible to compute the development of curvature along the root and hence describe the time-course of gravitropic bending. In addition, the RELEL distributions give information about the velocity field and the basipetal displacement of points along the root's surface. According to the Fundamental Theorem of Local Curve Theory, the x and y co-ordinates of the root in its bending plane are then determined from the associated values of local curvature and local velocity. With the aid of this model, possible mathematical growth functions that correspond to biological mechanisms involved in differential growth can be tested. Hence, the model can help not only to distinguish the role of various physiological or biophysical parameters in the bending process, but also to validate hypotheses that make assumptions concerning their relative importance. However, since the model is constructed at the level of the organ and treats the root as a fluid continuum, none of the parameters relate to cellular behaviour; the parameters must instead necessarily apply to properties that impinge on the behaviour of the external boundary of the root.     

Kinematics of surface growth

 In this paper a general mathematical framework is developed to describe cases of fixed and moving growth surfaces. This formulation has the mathematical structure suggested by Skalak (1981), but is extended herein to include discussion of possible singularities, incompatibilities, residual stresses and moving growth surfaces. Further, the general theoretical equations necessary for the computation of the final form of a structure from the distribution of growth velocities on a growth surface are presented and applied in a number of examples. It is shown that although assuming growth is always in a direction normal to the current growth surface is generally sufficient, growth at an angle to the growth surface may represent the biological reality more fully in some respects. From a theoretical viewpoint, growth at an angle to a growth surface is necessary in some situations to avoid postulating singularities in the growth velocity field. Examples of growth on fixed and moving surfaces are developed to simulate the generation of horns, seashells, antlers, teeth and similar biological structures. Received 20 February 1996; received in revised form 15 October 1996     

ARTIFICIAL SELECTION ON HORN LENGTH-BODY SIZE ALLOMETRY IN THE HORNED BEETLE ONTHOPHAGUS ACUMINATUS (COLEOPTERA: SCARABAEIDAE)

Males of the horned beetle Onthophagus acuminatus Har. (Coleoptera: Scarabaeidae) exhibit horn length dimorphism due to a sigmoidal allometric relationship between horn length and body size: the steep slope of the allometry around the inflection of the sigmoid curve separates males into two groups; those larger than this inflection possess long horns, and those smaller than this inflection have short horns or lack horns. I examined the genetic basis of the allometric relationship between horn length and body size by selecting males that produced unusually long horns, and males that produced unusually short horns, for their respective body sizes. After seven generations of selection, lines selected for relatively long horns had significantly longer horn lengths for a given body size than lines selected for relatively short horns, indicating a heritable component to variation in the allometry. The sigmoidal shape of the allometry was not affected by this selection regime. Rather, selected lines differed in the position of the allometry along the body size axis. One consequence of lateral shifts in this allometric relationship was that the body size separating horned from hornless males (the point of inflection of the sigmoid curve) differed between selection lines: lines in which males were selected for relatively long horns began horn production at smaller body sizes than lines selected for relatively short horns. These results suggest that populations can evolve in response to selection on male horn length through modification of the growth relationship between horn length and body size.     

Interplay between Graph Topology and Correlations of Third Order in Spiking Neuronal Networks

The study of processes evolving on networks has recently become a very popular research field, not only because of the rich mathematical theory that underpins it, but also because of its many possible applications, a number of them in the field of biology. Indeed, molecular signaling pathways, gene regulation, predator-prey interactions and the communication between neurons in the brain can be seen as examples of networks with complex dynamics. The properties of such dynamics depend largely on the topology of the underlying network graph. In this work, we want to answer the following question: Knowing network connectivity, what can be said about the level of third-order correlations that will characterize the network dynamics? We consider a linear point process as a model for pulse-coded, or spiking activity in a neuronal network. Using recent results from theory of such processes, we study third-order correlations between spike trains in such a system and explain which features of the network graph (i.e. which topological motifs) are responsible for their emergence. Comparing two different models of network topology—random networks of Erdős-Rényi type and networks with highly interconnected hubs—we find that, in random networks, the average measure of third-order correlations does not depend on the local connectivity properties, but rather on global parameters, such as the connection probability. This, however, ceases to be the case in networks with a geometric out-degree distribution, where topological specificities have a strong impact on average correlations.     

Repulsive expansion dynamics in colony growth and gene expression

Spatial expansion of a population of cells can arise from growth of microorganisms, plant cells, and mammalian cells. It underlies normal or dysfunctional tissue development, and it can be exploited as the foundation for programming spatial patterns. This expansion is often driven by continuous growth and division of cells within a colony, which in turn pushes the peripheral cells outward. This process generates a repulsion velocity field at each location within the colony. Here we show that this process can be approximated as coarse-grained repulsive-expansion kinetics. This framework enables accurate and efficient simulation of growth and gene expression dynamics in radially symmetric colonies with homogenous z-directional distribution. It is robust even if cells are not spherical and vary in size. The simplicity of the resulting mathematical framework also greatly facilitates generation of mechanistic insights.     

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.     

A Novel Application of Musculoskeletal Ultrasound Imaging

Ultrasound is an attractive modality for imaging muscle and tendon motion during dynamic tasks and can provide a complementary methodological approach for biomechanical studies in a clinical or laboratory setting. Towards this goal, methods for quantification of muscle kinematics from ultrasound imagery are being developed based on image processing. The temporal resolution of these methods is typically not sufficient for highly dynamic tasks, such as drop-landing. We propose a new approach that utilizes a Doppler method for quantifying muscle kinematics. We have developed a novel vector tissue Doppler imaging (vTDI) technique that can be used to measure musculoskeletal contraction velocity, strain and strain rate with sub-millisecond temporal resolution during dynamic activities using ultrasound. The goal of this preliminary study was to investigate the repeatability and potential applicability of the vTDI technique in measuring musculoskeletal velocities during a drop-landing task, in healthy subjects. The vTDI measurements can be performed concurrently with other biomechanical techniques, such as 3D motion capture for joint kinematics and kinetics, electromyography for timing of muscle activation and force plates for ground reaction force. Integration of these complementary techniques could lead to a better understanding of dynamic muscle function and dysfunction underlying the pathogenesis and pathophysiology of musculoskeletal disorders.     

Human hand modelling: kinematics, dynamics, applications

An overview of mathematical modelling of the human hand is given. We consider hand models from a specific background: rather than studying hands for surgical or similar goals, we target at providing a set of tools with which human grasping and manipulation capabilities can be studied, and hand functionality can be described. We do this by investigating the human hand at various levels: (1) at the level of kinematics, focussing on the movement of the bones of the hand, not taking corresponding forces into account; (2) at the musculotendon structure, i.e. by looking at the part of the hand generating the forces and thus inducing the motion; and (3) at the combination of the two, resulting in hand dynamics as well as the underlying neurocontrol. Our purpose is to not only provide the reader with an overview of current human hand modelling approaches but also to fill the gaps with recent results and data, thus allowing for an encompassing picture.     

Global dynamics of microbial communities emerge from local interaction rules

Most microbes live in spatially structured communities (e.g., biofilms) in which they interact with their neighbors through the local exchange of diffusible molecules. To understand the functioning of these communities, it is essential to uncover how these local interactions shape community-level properties, such as the community composition, spatial arrangement, and growth rate. Here, we present a mathematical framework to derive community-level properties from the molecular mechanisms underlying the cell-cell interactions for systems consisting of two cell types. Our framework consists of two parts: a biophysical model to derive the local interaction rules (i.e. interaction range and strength) from the molecular parameters underlying the cell-cell interactions and a graph based model to derive the equilibrium properties of the community (i.e. composition, spatial arrangement, and growth rate) from these local interaction rules. Our framework shows that key molecular parameters underlying the cell-cell interactions (e.g., the uptake and leakage rates of molecules) determine community-level properties. We apply our model to mutualistic cross-feeding communities and show that spatial structure can be detrimental for these communities. Moreover, our model can qualitatively recapitulate the properties of an experimental microbial community. Our framework can be extended to a variety of systems of two interacting cell types, within and beyond the microbial world, and contributes to our understanding of how community-level properties emerge from microscopic interactions between cells.     

Validation of a mathematical approach to estimate dynamic scapular orientation

The goal of this study was to develop and validate a non-invasive approach to estimate scapular kinematics in individual patients. We hypothesized that individualized mathematical algorithms can be developed using motion capture data to accurately estimate dynamic scapula orientation based on measured humeral orientations and acromion process positions. The accuracy of the mathematical algorithms was evaluated against a gold standard of biplane fluoroscopy using a 2D to 3D fluoroscopy/model matching process. Individualized linear models were developed for nine healthy adult shoulders. These models were used to predict scapulothoracic kinematics, and the predicted kinematics were compared to kinematics obtained using biplane fluoroscopy to determine the accuracy of the algorithms. Results showed strong correlations between mathematically predicted kinematics and validation kinematics. Estimated kinematics were within 8° of validation kinematics. We concluded that individualized linear models show promise for providing accurate, non-invasive measurements of scapulothoracic kinematics in a clinical environment.     

生长模型的误差函数及其数学特征

生长曲线是估计动物年龄的重要方法之一,在野生动物生态学中,动物的体重往往被用作估计动物年龄的主要指标。然而,在动物体重测定过程中经常会出现一些偏差。例如,动物的日常活动（ 进食、饮水、排泄等）通常会引起动物体重的变化,这样在不同时间测定动物的体重就会产生偏差;此外,在测定动物体重的过程中,我们往往称量到一定的精确度。这些偏差将直接导致对动物年龄的估计误差。本文分析了4种常见生长模型（Logistic、Gompertz、Bertalanffy、Richards)的误差函数的数学特征。结果表明,由动物日常活动导致的年龄估计误差在动物的幼龄阶段为量小,而由称量精确度导致的年龄估算误差在生长曲线的拐点处为最小。     

RNA Unwinding by NS3 Helicase: A Statistical Approach

The study of double-stranded RNA unwinding by helicases is a problem of basic scientific interest. One such example is provided by studies on the hepatitis C virus (HCV) NS3 helicase using single molecule mechanical experiments. HCV currently infects nearly 3% of the world population and NS3 is a protein essential for viral genome replication. The objective of this study is to model the RNA unwinding mechanism based on previously published data and study its characteristics and their dependence on force, ATP and NS3 protein concentration. In this work, RNA unwinding by NS3 helicase is hypothesized to occur in a series of discrete steps and the steps themselves occurring in accordance with an underlying point process. A point process driven change point model is employed to model the RNA unwinding mechanism. The results are in large agreement with findings in previous studies. A gamma distribution based renewal process was found to model well the point process that drives the unwinding mechanism. The analysis suggests that the periods of constant extension observed during NS3 activity can indeed be classified into pauses and subpauses and that each depend on the ATP concentration. The step size is independent of external factors and seems to have a median value of 11.37 base pairs. The steps themselves are composed of a number of substeps with an average of about 4 substeps per step and an average substep size of about 3.7 base pairs. An interesting finding pertains to the stepping velocity. Our analysis indicates that stepping velocity may be of two kinds- a low and a high velocity.     

Effect of Water Stress on Cortical Cell Division Rates within the Apical Meristem of Primary Roots of Maize

We characterized the effect of water stress on cell division rates within the meristem of the primary root of maize (Zea mays L.) seedlings. As usual in growth kinematics, cell number density is found by counting the number of cells per small unit length of the root; growth velocity is the rate of displacement of a cellular particle found at a given distance from the apex; and the cell flux, representing the rate at which cells are moving past a spatial point, is defined as the product of velocity and cell number density. The local cell division rate is estimated by summing the derivative of cell density with respect to time, and the derivative of the cell flux with respect to distance. Relatively long (2-h) intervals were required for time-lapse photography to resolve growth velocity within the meristem. Water stress caused meristematic cells to be longer and reduced the rates of cell division, per unit length of tissue and per cell, throughout most of the meristem. Peak cell division rate was 8.2 cells mm-1 h-1 (0.10 cells cell-1 h-1) at 0.8 mm from the apex for cells under water stress, compared with 13 cells mm-1 h-1 (0.14 cells cell-1 h-1) at 1.0 mm for controls.     

Functional mapping of quantitative trait loci underlying the character process: a theoretical framework

Unlike a character measured at a finite set of landmark points, function-valued traits are those that change as a function of some independent and continuous variable. These traits, also called infinite-dimensional characters, can be described as the character process and include a number of biologically, economically, or biomedically important features, such as growth trajectories, allometric scalings, and norms of reaction. Here we present a new statistical infrastructure for mapping quantitative trait loci (QTL) underlying the character process. This strategy, termed functional mapping, integrates mathematical relationships of different traits or variables within the genetic mapping framework. Logistic mapping proposed in this article can be viewed as an example of functional mapping. Logistic mapping is based on a universal biological law that for each and every living organism growth over time follows an exponential growth curve (e.g., logistic or S-shaped). A maximum-likelihood approach based on a logistic-mixture model, implemented with the EM algorithm, is developed to provide the estimates of QTL positions, QTL effects, and other model parameters responsible for growth trajectories. Logistic mapping displays a tremendous potential to increase the power of QTL detection, the precision of parameter estimation, and the resolution of QTL localization due to the small number of parameters to be estimated, the pleiotropic effect of a QTL on growth, and/or residual correlations of growth at different ages. More importantly, logistic mapping allows for testing numerous biologically important hypotheses concerning the genetic basis of quantitative variation, thus gaining an insight into the critical role of development in shaping plant and animal evolution and domestication. The power of logistic mapping is demonstrated by an example of a forest tree, in which one QTL affecting stem growth processes is detected on a linkage group using our method, whereas it cannot be detected using current methods. The advantages of functional mapping are also discussed.     

Gas dispersion in volume-cycled tube flow. I. Theory.

A mathematical theory is derived for the dispersion of a contaminant bolus introduced into a fully developed volume-cycled oscillatory pipe flow. The convection-diffusion equation is solved for a tracer gas bolus by expressing the local concentration field as a series expansion of derivatives of the area-averaged concentration. The local, as well as the area-averaged, concentration is determined for a uniform initial slug or Gaussian bolus. The effect of various flow parameters such as Womersley parameter, Schmidt number, and tidal volume is investigated. The overall dispersion is characterized by a time-averaged effective diffusion coefficient, which for long duration coincides with previous dispersion theories based on a constant linear axial concentration profile. The effective diffusion coefficient can be determined from the local time history of concentration, independent of the spatial location or the initial tracer bolus. Furthermore the local peaks of the concentration-time curve follow a decaying curve dictated by the time-averaged effective diffusion coefficient. Thus the theory is directly applicable for dispersion measurements in oscillatory tube flows, a basis for the pulmonary airways application, as shown by Gaver et al. (J. Appl. Physiol. 72: 321-331, 1992).     

Fitting rhythmic biological data by means of (combinations of) Voltera Integro‐Differential equations

Abstract

A method of analysing rhythmic biological data is presented utilising a curve fitting technique based on solutions of the Volterra Integro‐Differential Equation (VIDE). The technique involves the use of difference equations and the method of moments as the fitting criterion. It is conjectured that combinations of the VIDE can provide a more meaningful fit for such data than does harmonic analysis. The (generating, inhibiting and heredity) parameters of the VIDE may be related to the true biological mechanisms and thus may give the biologist opportunities to understand the underlying processes. A practical example is presented to illustrate the merits and limitations of the method.