A Phenomenologic-mathematical Model of Growth Dynamics

Starting point of the modelling procedure are measured courses of the body length increase of man (inverse problem) reaching from the time of conception up to the end of adolescence. First assumption: The whole growth process can be subdivided into independent partial processes for succeeding time periods of the individual's development each of them producess a more or less marked growth spurt. 2. Superposition of these partial processes means addition of the portions of body length which are generated by the spurts yielding in this manner the measured course of body length increase. 3. There is no change in dynamics for producing the several growth spurts, and this dynamics will be described by the differential equation of the logistic law of growth. These steps will be interpreted in control-theoretical terms. In this sense growth is a follow-up control process which is governed by the genetically fixed “biological program of growth” in form of a step function of reference values.     

Phenomenologic-mathematical modeling of the human-body-length growth through the analysis of growth periods and their quantitative-analytical registration

The body height growth (of masculine beings) was modelled in a phenomenologic-mathematical manner by partitioning the time course of measured growth curve in parts every of which corresponds to a separated growth period. This partitioning was reached in a natural way so that a superposition of the single spurts yields the whole measured course. Every growth batch will be described in its time course by one term of inverse tangent function. The biological meaning and an explanation of the succession of the growth spurt as an effect of control circuits need further exploratory work. For detailed statements on acceleration phenomena concerning the body height growth this analysis gives possibilities for comparing the single growth spurts of the mean growth process of two populations in question. For measured values given by BROCK (1954) and SALZLER (1967) there are five growth periods in the time intervall reaching from time of conception until the end of the first year. Comparing the mathematical functions of the corresponding growth spurts for these two groups one can conclude that the second spurt (fetal spurt) is responsible for an increase of birth body height and the fourth for an increase of body height in the suckling age of the latter group against the former one.     

Quantitative mathematical evaluation of growth studies from the viewpoint of exogenously or genetically influenced variability

Firstly, the ideas are sketched which serve as the basis for the phenomenologic-mathematical kind of modeling of the body length growth process of man. For proving the biological relevance of the spurts analyzed by numerical procedures one has to consider the social and the biological circumstances in which the growth process takes place. An analysis of longitudinal data series of the body length of (monozygotic) twins will give further hints to the possible meaning of the growth spurts by way of separation of exogenous and of genetic determined endogenous agents on the growth process. The data available for our examinations cover the time interval of the praepubertal and the postpuberal development as well as the time interval in which the puberal growth spurt takes place. The way to proceed in evaluation these time series will be presented.     

A Continuum Mathematical Model of the Developing Murine Retinal Vasculature

Angiogenesis, the process of new vessel growth from pre-existing vasculature, is crucial in many biological situations such as wound healing and embryogenesis. Angiogenesis is also a key regulator of pathogenesis in many clinically important disease processes, for instance, solid tumour progression and ocular diseases. Over the past 10–20 years, tumour-induced angiogenesis has received a lot of attention in the mathematical modelling community and there have also been some attempts to model angiogenesis during wound healing. However, there has been little modelling work of vascular growth during normal development. In this paper, we describe an in silico representation of the developing retinal vasculature in the mouse, using continuum mathematical models consisting of systems of partial differential equations. The equations describe the migratory response of cells to growth factor gradients, the evolution of the capillary blood vessel density, and of the growth factor concentration. Our approach is closely coupled to an associated experimental programme to parameterise our model effectively and the simulations provide an excellent correlation with in vivo experimental data. Future work and development of this model will enable us to elucidate the impact of molecular cues upon vasculature development and the implications for eye diseases such as diabetic retinopathy and neonatal retinopathy of prematurity.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behaviour of a system, differential equations have proved to be powerful for modelling and unravelling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge–Kutta fourth-order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

(An algorithm for determining values of parameters of the Gompertz growth function)]

In the literature some attempts were made to analyse and to construct models for biological growth processes and to describe the quantitative aspects of a growth characteristic's changes in time using the Gompertz' function y=aexp(-exp(b--ct)). In this paper differential equations are derived having the Gompertz' function as solution. The goodness of fit after adjusting a chosen analytical expression to the courses of measured values is able to give hints at the reliability of that expression as a true model. This possibility of verification was hardly practiced in past because of lacking in proper numerical procedures for performing the nonlinear regression. An ALGOL program for iterative adjusting the parameters of the GOMPERTZ' function (with or without a constant term) to measured values is given in an appendix of the present paper. Starting values for the nonlinear parameters b and c will be evaluated by Internal Least Squares using one of the derived differential equations. For this algorithm an ALGOL program is given in the appendix too. The growth of human embryo serves as an example to demonstrate the numerical procedures and related programs for evaluating the starting values of the parameters and for their iterative improvement until reaching a minimum for the remainding variance between calculated and measured courses.     

Evolution of human growth spurts

This study investigates subadult growth spurts in a large sample of anthropoid primates, including humans. Analyses of body mass growth curves show that humans are not unique in the expression of female and male body mass growth spurts. Subadult growth spurts are observed in both New World and Old World anthropoid primates and are more common in males than in females. Allometric analyses of growth spurts indicate that many aspects of primate growth spurts are strongly correlated with species size. Small species tend not to exhibit growth spurts. Although male and female scaling patterns for velocity and size measures are comparable, scaling relations of variables that measure the timing of growth spurts differ by sex. These patterns can he related to sexual differences in life histories. Scaling analyses further show that humans do not depart substantially from patterns that describe other anthropoid primates. Thus, in relative terms, human growth spurts are not exceptional compared to this sample of primates. The long absolute delay in the initiation of the human growth spurt may be of substantial evolutionary importance and serves to distinguish humans from other primates. In essence, humans exhibit growth spurts that are comparable to other primates in many respects. However, human growth spurts are shifted to very late absolute ages. © 1996 Wiley-Liss, Inc.     

Mathematical models of the balance between apoptosis and proliferation

As our understanding of cellular behaviour grows, and we identify more and more genes involved in the control of such basic processes as cell division and programmed cell death, it becomes increasingly difficult to integrate such detailed knowledge into a meaningful whole. This is an area where mathematical modelling can complement experimental approaches, and even simple mathematical models can yield useful biological insights. This review presents examples of this in the context of understanding the combined effects of different levels of cell death and cell division in a number of biological systems including tumour growth, the homeostasis of immune memory and pre-implantation embryo development. The models we describe, although simplistic, yield insight into several phenomena that are difficult to understand using a purely experimental approach. This includes the different roles played by the apoptosis of stem cells and differentiated cells in determining whether or not a tumour can grow; the way in which a density dependent rate of apoptosis (for instance mediated by cell-cell contact or cytokine signalling) can lead to homeostasis; and the effect of stochastic fluctuations when the number of cells involved is small. We also highlight how models can maximize the amount of information that can be extracted from limited experimental data. The review concludes by summarizing the various mathematical frameworks that can be used to develop new models and the type of biological information that is required to do this.     

Analysis of differential growth of the right and the left leg

In spite of well documented standards for length and annual growth rates of the femur and tibia, there is little information on short term longitudinal bone growth. We investigated differential growth dynamics of the lower leg in 10 children, aged 6:3 to 14:2 years, by knemometry, a novel and non-invasive technique of accurate lower leg length measurement with a technical error of 0.09 to 0.16 mm. Mini growth spurts were detectable in 7 of the children and occurred synchroneously in both legs. Approximately half of the variance of the weekly lower leg length increments could be attributed to synchrony of leg length increments, but a significant amount of residual variance remained which exceeded the technical error of the measurements. Run-analysis of the individual series of right vs. left differences of the weekly lower leg length increments provided evidence for alternating periods of overgrowth of one leg compared to the contralateral side in 5 out of the 10 children. We concluded that there is suggestive evidence of partial independence of lower leg growth in the short term.     

Mathematical modelling of mycelia: a question of scale

Recent advances in systems biology have driven many aspects of biological research in a direction heavily weighted towards computational, quantitative and predictive analysis, based on, or assisted by mathematical modelling. In particular, mathematical modelling has played a significant role in the development of our understanding of the growth and function of the fungal mycelium. One of the main problems that faces modellers in this context is the choice of scale. In the study of fungal mycelia, the question of scale is expressed in an extreme manner: Their indeterminate growth habit ensures that the investigation of the growth and function of mycelial fungi has to consider scales ranging from the (sub) micron to the kilometer. An excellent and extensive review of the applications of mathematical modelling to fungal growth, conducted up to the mid-1990s, can be found in Prosser (1995). In this article, we will concentrate on work since that date, with the emphasis being on recent developments in understanding fungal mycelia at all scales.     

Comparison of plant and fungal gravitropic responses using imitational modelling

The mechanisms of gravity perception are still hypothetical, but there are sufficient data from experiments with plants to enable mathematical modelling to imitate the behaviour of gravitropic response systems. We have a much less complete picture of gravitropic kinetics in agaric mushrooms. However, some existing mathematical models which imitate plant responses are in principle universal because their conceptual components are not limited to any specific cellular entities. In this work we have used such models to compare plant and fungal gravitropism, using recently acquired kinetic data from the agarics Coprinus cinereius and Flammulina velutipes. The results show striking similarities between plants and fungi. First, it is evident that the basic assumptions of the plant models are logically applicable to fungi. Secondly, the mechanism of bending is the same (differential growth of opposite flanks of the growing organ). Thirdly, the distribution of growth seems very similar: in both plants and fungi growth of the organ is most intensive just behind the apex and is almost absent at the apex and at the base. Fourthly, in both fungi and plants the gravitropic response exhibits a substantial time delay suggesting that many time-consuming processes are involved in reception, transduction and realization of gravitropic stimuli. Important differences in plant and fungal gravitropism kinetics were: (i) the agaric stem apex always returned to the vertical, whereas some plant organs show stable plagiogravitropic growth; (ii) inflections were usually seen in C. cinereus stem gravitropism time courses suggesting that a curvature compensation process delayed bending for a time; (iii) C. cinercus stems very rarely overshot or oscillated around the vertical although many plant subjects oscillate and the (limited) data for F. velutipes showed a single, exaggerated overshoot and oscillation. In this latter case, experimental modelling with parameters characteristic of a low level of perception improved the fit to the F. velutipes data, indicating that the two fungi may differ in this factor. Application of the plant models focused future research attention on the urgent need for data bearing on angle-response and acceleration–response relationships in fungi, and their detection–level thresholds for gravitational acceleration. Since the modelling also highlighted some fundamental kinetic differences between the only two fungi for which sufficient data are available at the moment, it is also clear that detailed observations need to be made of gravitropism kinetics in a larger number and wider range of fungi.     

Comparative quantitative studies of mental and physical development of children up to age 3

Biomathematical-methodical aspects are exposed of quantitative recording of mean growth courses which will be taken as basis for comparisons between different sample curves. In the paper the body length growth process of children in the age period from 0 to 3 years is considered. There are samples of children growing up under different social conditions and data of corresponding samples collected about 20 years ago. After remarks on the correlation between physical and psychical development of infants several possibilities for representing mean body length growth curves are discussed. The advantages of empirical regression are explained as the best suited method for a modelfree data evaluation. The nonparametric location trend test of Cox and Stuart is used for statistical proving of global difference or parallelism in the course of 2 mean growth curves. Level differences in the global sense between 2 compared curves may be tested by constructing a Scheffé confidence region for a properly defined constrast. Local level differences can be proved by the t-test for those pairs of corresponding mean values for which the conditions of applying this test are fulfilled. The calculation of a curve-related normal belt as a succession of 95%-tolerance regions (without confidence probability) is demonstrated by an example.     

Mathematical Models of the Kinetics of the Cultivation of Microorganisms

Various equations of mathematical models for the kinetics of the development of various biological processes were obtained on the basis of the generalized differential equation of biomass growth. Aerobic periodic cultivation of the yeast Saccharomyces cerevisiae was carried out to provide a comparative evaluation of advantages and disadvantages of four types of mathematical models. It is shown that the exponential model is a particular solution to the generalized differential equation. The developed mathematical model can be used to predict the course of biological processes in time and can serve as a tool for a computational experiment in order to clarify the dependence of the rate of a biological process on changes in certain parameters that affect the development of cells.     

Quantifying the impact of human immunodeficiency virus-1 escape from cytotoxic T-lymphocytes

HIV-1 escape from the cytotoxic T-lymphocyte (CTL) response leads to a weakening of viral control and is likely to be detrimental to the patient. To date, the impact of escape on viral load and CD4(+) T cell count has not been quantified, primarily because of sparse longitudinal data and the difficulty of separating cause and effect in cross-sectional studies. We use two independent methods to quantify the impact of HIV-1 escape from CTLs in chronic infection: mathematical modelling of escape and statistical analysis of a cross-sectional cohort. Mathematical modelling revealed a modest increase in log viral load of 0.051 copies ml(-1) per escape event. Analysis of the cross-sectional cohort revealed a significant positive association between viral load and the number of "escape events", after correcting for length of infection and rate of replication. We estimate that a single CTL escape event leads to a viral load increase of 0.11 log copies ml(-1) (95% confidence interval: 0.040-0.18), consistent with the predictions from the mathematical modelling. Overall, the number of escape events could only account for approximately 6% of the viral load variation in the cohort. Our findings indicate that although the loss of the CTL response for a single epitope results in a highly statistically significant increase in viral load, the biological impact is modest. We suggest that this small increase in viral load is explained by the small growth advantage of the variant relative to the wildtype virus. Escape from CTLs had a measurable, but unexpectedly low, impact on viral load in chronic infection.     

From the biology of the individual to the dynamics of the population: bridging the gap in fish early life studies

The mean properties of larval fish populations do not necessarily reflect the properties of the mean individual. For example, the change in mean length in a population with time may not reflect the average individual growth rate, since individual growth rates and survival probability are linked so that slow growing individuals suffer higher mortality. Hence, mean growth rate indicated from population data could be biased upwards. Factors which influence the magnitude and variability of individual growth rates can exert nonlinear effects on population survival. Two categories of process must be considered: first, the variability in exposure of the average individual as a consequence of individual variability in dispersal through a patchy environment; and second, the intrinsic variability between individuals expressed even under equal exposure conditions. These two aspects have been addressed independently, the first by lagrangian modelling of individual fish larvae linked to spatially resolved hydrodynamic models, and the second by strategic biological modelling. In this paper, progress towards the goal of individually based larval fish ecosystem models is reviewed, highlighting the space and time scales which may be important in such systems, and identifying the gaps in current knowledge of larval biology.     

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.