The interpretation of voltage-concentration relations

The dependence of a cell's membrane potential upon ion concentrations is often described via a Goldman constant field equation (with coefficients called permeabilities) or what may be called the Hodgkin-Horowicz equation. The latter is derived by assuming that different ion species traverse separate pathways of at least formally constant conductance. If one assumes truly constant “permeabilities” or “conductances”, then these two equations have essentially different forms. So the literature has been culled for voltage-concentration data which unambiguously fit an equation of one of these two forms. Eight or nine are found to fit a Hodgkin-Horowicz relation; none, a Goldman one. This adds to the evidence that different ion species traverse relatively separate pathways through cell membranes (and that these pathways are long compared to a Debye length). It also indicates the desirability of generally describing voltage-concentration data with a Hodgkin-Horowicz relation rather than a Goldman one.     

Effect of temperature on plant elongation and cell wall extensibility

Lockhart equation was derived for explaining plant cell expansion where both cell wall extension and water uptake must occur concomitantly. Its fundamental contribution was to express turgor pressure explicitly in terms of osmosis and wall mechanics. Here we present a new equation in which pressure is determined by temperature. It also accounts for the role of osmosis and consequently the role of water uptake in growing cell. By adopting literature data, we also attempt to report theoretically the close relation between plant elongation and cell wall extensibility. This is accomplished by the modified equation of growth solved for various temperatures in case of two different species. The results enable to interpret empirical data in terms of our model and fully confirm its applicability to the investigation of the problem of plant cell extensibility in function of environmental temperature. Moreover, by separating elastic effects from growth process we specified the characteristic temperature common for both processes which corresponds to the resonance energy of biochemical reactions as well as to the rapid softening of the elastic modes toward the high temperature end where we encountered viscoelastic and/or plastic behavior as dominating. By introducing analytical formulae connected with growth and elastic properties of the cell wall, we conclude with the statement how these both processes contribute quantitatively to the resonance-like shape of the elongation curve. In addition, the tension versus temperature "phase diagram" for a living plant cell is presented.     

Cell cycles and growth laws: the CCC model

In the cell-cycle-with-control model (CCC model), cells have to satisfy a condition before they are allowed to pass a control point during G1. Different cycle durations within a cell population are explained by individual time spans needed to satisfy the passing condition. If the distribution of cycle durations is time invariant, the population will grow exponentially. However, if the average cycle duration becomes longer, while the population grows, non-exponential population growth results. Simple functions for the lengthening of the average cycle duration, like linear or exponential ones, yield the well-known growth laws found in the biological literature. The same functions can be represented by an "S-system" differential equation that was derived earlier as an approximation for biochemical systems with many fast reactions (metabolism) and one slow process (e.g. ageing).     

Theory of interaction of polymer and small molecules which can aggregate in solution

The theory of interaction of small molecules with polymers is extended to the case where the small molecules can aggregate in solution to form dimers, trimers, etc., and where the aggregates can also interact with the polymer. The secular equation is found to be identical in form with that obtained for the binding of nonaggregating monomer to the polymer. Thus, all formulas derived for the latter case apply to the present one also, except that, now the concentration of the free monomer is the relevant factor. It is shown how the experimental data can be analyzed by using information on the molal activity coefficients of the small molecules. As an illustration these formulas are applied to data on the interaction of purines and nucleosides with poly(uridylic acid) in concentrated solutions.     

Application of the osmotic virial equation in cryobiology

The multisolute osmotic virial equation is the only multisolute thermodynamic solution theory that has been derived from first principles and can make predictions of multisolute solution behaviour in the absence of multisolute solution data. Other solution theories either (i) include simplifying assumptions that do not take into account the interactions between different types of solute molecules or (ii) require fitting to multisolute data to obtain empirical parameters. The osmotic virial coefficients, which are obtained from single-solute data, can be used to make predictions of multisolute solution osmolality. The osmotic virial coefficients for a range of solutes of interest in cryobiology are provided in this paper, for use with concentration units of both molality and mole fraction, along with an explanation of the background and theory necessary to implement the multisolute osmotic virial equation.     

Transport-limited growth in the chemostat and its competitive inhibition; A theoretical treatment

Summary An equation expressing the specific growth rate of heterotrophic cell populations in terms of yield factor and transport rate is proposed. From this equation expressions are derived for the specific growth rate when the transport of the energy source is growth0limiting. These expressions are applied to cell population growth in the chemostat limited by the transport of the energy source or of other substrates and simple mathematical tools are provided for obtaining estimates of the transport parameters. An equation is derived which predicts that at constant dilution rate in the chemostat the concentration of any substrate (whether or not the source of energy) the transport of which is growth limiting, is a linear function of the concentration of a competitive inhibitor of its transport. With this equation estimates of the Michaelis constants of competitive transport inhibitors can be obtained. The growth rate equation of Monod (1942) is discussed.     

Linear constraint relations in biochemical reaction systems: I. Classification of the calculability and the balanceability of conversion rates

Measurements provide the basis for process monitoring and control as well as for model development and validation. Systematic approaches to increase the accuracy and credibility of the empirical data set are therefore of great value. In (bio)chemical conversions, linear conservation relations such as the balance equations for charge, enthalpy, and/or chemical elements, can be employed to relate conversion rates. In a pactical situation, some of these rates will be measured (in effect, be calculated directly from primary measurements of, e.g., concentrations and flow rates), as others can or cannot be calculated from the measured ones. When certain measured rates can also be calculated from other measured rates, the set of equations, the accuracy and credibility of the measured rates can indeed be improved by, respectively, balancing and gross error diagnosis. The balanced conversion rates are more accurate, and form a consistent set of data, which is more suitable for further application (e.g., to calculate nonmeasured rates) than the raw measurements. Such an approach has drawn attention in previous studies. The current study deals mainly with the problem of mathematically classifying the conversion rates into balanceable and calculable rates, given the subset of measured rates. The significance of this problem is illustrated with some examples. It is shown that a simple matrix equation can be derived that contains the vector of measured conversion rates and the redundancy matrix R. Matrix R plays a predominant role in the classification problem. In supplementary articles, significance of the redundancy matrix R for an improved gross error diagnosis approach will be shown. In addition, efficient equations have been derived to calculate the balanceable and/or calculable rates. The method is completely based on matrix algebra (principally different from the graph-theoretical approach), and it is easily implemented into a computer program. (c) 1994 John Wiley & Sons, Inc.     

Hormone gradients and cartilage cell kinetics

We present a model of growth control in mammalian cartilage growth plates by hormones. The model is based on the distribution of insulin-like growth factors I and II (IGF-I and IGF-II) and their receptors, and assumes that a hormone-receptor complex of IGF controls cells proliferation. A system of differential equations is derived and solved with simplifications in extreme cases, for the one-dimensional time independent case. Even if opposite extremes, such as proliferation control by factors extrinsic to the cell versus intrinsic to the cell, are assumed, similar distributions of hormones and proliferating cells are produced. Hence, choice between alternative models of growth control must be based on empirical observations. On the positive side, similarities between our model for cartilage growth and other models for differentiation and proliferation are evident and might be exploited for unifying these systems on an abstract level.     

Growth of the African elephant (Loxodonta africana)

The three-stage desk calculation of the von Bertalanffy equation to describe growth in height and weight with age in the elephant is compared with a new approach to calculating the three coefficients in the function by a computer. The two methods give different results with respect to the weight/age calculations. Theoretical von Bertalanffy equations calculated by both methods to describe growth in height and weight with age in the African elephant in Zambia are compared with previously published equations for the elephant in East Africa. Details are given of growth in height in two known-age African elephants, a female ‘Diksie’ and a male ‘Kartoum’. Theoretical growth in height curves for the female African and Asiatic elephant are compared. The coefficients K and / for growth in height are not transferable to the growth in weight equations. Inherent inaccuracies in the calculation of the coefficients in the von Bertalanffy equation are discussed, and it is concluded that in animals which have a long life-span such as the elephant, the equation serves as a purely empirical representation of weight-at-age data and that there is little biological significance in the parameters it contains. The computer-calculated curves give the best fit to the data. The regression of log age on log shoulder height from 2–20 years of age provides a more realistic approach to comparative growth studies. The increase in adrenal weight with age is linear. Tusk growth in relation to age and sex in Zambia is compared with East Africa. It is concluded that the tusks in Zambia are smaller and are more difficult to sex correctly than their East African counterparts, possibly a consequence of the Zambian elephant having a greater degree of tusk wear. Allometric growth is described with emphasis on the estimation of body weight from shoulder height. The most reliable estimates are obtained from a purely empirical representation of the data, a semilog plot of log body weight on shoulder height.     

A theoretical derivation of the Contois equation for kinetic modeling of the microbial degradation of insoluble substrates

Although developed as an empirical model to describe microbial growth on soluble substrates, the Contois equation has been widely accepted for kinetic modeling of insoluble substrate degradation. Yet, the mechanistic basis underlining these successful applications remains unanswered. Unlike soluble substrates that mainly cultivate suspended cultures, microbes cultivated on insoluble substrates have the populations that attach to the substrate surface or remain suspended in the bulk solution, while those attached usually grow faster than those suspended due to their proximity to food resources. This imbalanced growth provides a plausible explanation to the inverse relationship between microbial concentration and their specific growth rate as conveyed in the Contois equation. Based on a theoretical derivation, this study revealed that the Contois equation holds true only when attached microbes substantially obstruct the access of food to their suspended counterparts. On the other hand, when plentiful insoluble substrate surfaces are exposed for cell attachment, the Contois equation will be reduced back to the classic Monod equation.     

A macroscopic formulation of electromagnetophoresis

A macroscopic conservation equation is derived for electromagnetophoresis of a dilute suspension. The governing equation and auxiliary conditions are formulated for transport in a rectangular cell, these being reduced to standard form by dimensional methods.     

Molecular Dynamic Simulation of Transmembrane Pore Growth

A molecular dynamic approach was applied for simulation of dynamics of pore formation and growth in a phospholipid bilayer in the presence of an external electric field. Processing the simulation results permitted recovery of the kinetic coefficients used in the Einstein–Smoluchowski equation describing the dynamics of pore evolution. Two different models of the bilayer membrane were considered: membrane consisting of POPC and POPE lipids. The simulations permitted us to find nonempirical values of the pore energy parameters, which are compared with empirical values. It was found that the parameters are sensitive to membrane type.     

On the dynamics of vegetation: Patterns of environmental and vegetational change

Summary Ecological succession theory deals with temporal change in biological communities. It consists largely of generalizations based on temporal sequences inferred from spatial ones. The predictive content of the theory is low, since predictions are derived from unconditional trends rather than conditional laws. There exist several conflicting theories purporting to explain successional change, but their empirical vacuousness prevents an assessment on empirical terms. It is argued here that one can nevertheless advocate a theory which accounts for the ubiquity of successional change and explains the most conspicuous characteristics of the successional process, even though it cannot predict the detailed dynamics. Such a theory is derived here from an analysis of adaptive strategies.It is also pointed out that a persistant confusion exists in the ecological literature between what are considered to be the driving forces of successional change, competition and reaction. The former is taken to be an instantaneous type of interaction, whereas the latter has historical (cumulative) aspects. It is not at all obvious whether interactions of the historical type play an important role in driving vegetational change, although it is usually suggested that they do.     

The kinetics of monazomycin-induced voltage-dependent conductance. II. Theory and a demonstration of a form of memory

The empirical differential equation that describes the kinetics of monazomycin-induced voltage-dependent conductance is derived using a standard chemical kinetic formulation and the assumption that monazomycin entry into and exit from the membrane is autocatalytic. The predicted form of gating currents is shown and numerical calculations for this process are made using a range of values for two unmeasured variables. A form of "memory" is then demonstrated, along with the ability of the theoretical equation to explain the nature of the memory.     

Analysis of cell growth in a polymer scaffold using a moving boundary approach

Two mathematical models of chondrocyte generation and nutrient consumption are developed to analyze the behavior of cell growth in a biodegradable polymer matrix. Substrate reaction and diffusion are analyzed in two regions: one consisting of cells and nutrients and the other consisting of only nutrients. A pseudo-steady state approximation for the transport of nutrients in these two regions is utilized. The rate of growth is determined by a moving boundary equation that equates the rate at which the interfacial region between the cells and the void space moves to a substrate dependent growth reaction. The change in the location of this interfacial region with time therefore depicts the rate at which the cells propagate. The two limiting cases discussed in this article represent extremes in how the cells will grow in the polymer matrix; one case assumes that cells grow inward from the external boundary, and the other case assumes that cells grow parallel to the external boundary. The results of both models are compared to experimental data found in the literature. It is found through these comparisons that the model parameters, including the unit cell spacing parameter L, the metabolic rate constant k, the growth rate constant k(G), and external mass transfer coefficient, K, may vary as the thickness of the polymer matrix is changed, however, unrealistic and large changes in the diffusion coefficients were required to account for the full range of experimental data. Furthermore, these results suggest modification of the functional form of the growth kinetics to include substrate or product inhibition, or death terms. Based upon diffusion/reaction concepts, these models for cell growth in a biodegradable polymer give bounds for the upper and lower limits of the cellular growth rate and nutrient consumption in a polymer matrix and will aid in the development of more extensive models. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 422-432, 1997.     

Modeling substrate inhibition of microbial growth

This article presents a general equation for substrate inhibition of microbial growth using a statistical thermodynamic approach. Existing empirical models adapted from enzyme kinetics, for example, the Haldane-Andrews equation, often criticized for not being physically based for microbial growth, are shown to derive from the general equation in this article, and their empirical parameters are shown to be well defined physically. Three sets of experimental data from the literature are used to test the modeling abilities of the general equation to represent experimental data. The results are compared with those obtained by fitting the same data set to a widely used empirical model existing in the literature. The general equation is found to represent all three experimental data sets better than the alternative model tested. In addition, a graphical method existing in enzyme kinetics is successfully adapted and further developed to determine the number of inhibition sites of a basic functional unit of a bacterial cell. (c) 1996 John Wiley & Sons, Inc.     

Apple Fruit Bud Development and Growth; Analysis and an Empirical Model

Analysis of the information available on apple bud developmentand growth after dormancy leads to an empirical model of growthto full bloom. The analysis and model are set in the frameworkof the physiological mechanisms considered to be responsiblefor dormancy and subsequent bud growth. It is necessary to introducean arbitrary ‘growth unit’ scale to describe theseprocesses quantitatively, which is done by the equation G = A/(I+be^–k(I).P) where G and A are in growth units, the value of k is controlledby a dormancy index I and P is a temperature summation. Themodel fulfils the main requirements laid down for it and thevalues of P at full bloom, derived from controlled environmentwork and field observations, are very similar.