A quantitative method for the analysis of mammalian cell proliferation in culture in terms of dividing and non-dividing cells

The application of the exponential growth equation is the standard method employed in the quantitative analyses of mammalian cell proliferation in culture. This method is based on the implicit assumption that, within a cell population under study, all division events give rise to daughter cells that always divide. When a cell population does not adhere to this assumption, use of the exponential growth equation leads to errors in the determination of both population doubling time and cell generation time. We have derived a more general growth equation that defines cell growth in terms of the dividing fraction of daughter cells. This equation can account for population growth kinetics that derive from the generation of both dividing and non-dividing cells. As such, it provides a sensitive method for detecting non-exponential division dynamics. In addition, this equation can be used to determine when it is appropriate to use the standard exponential growth equation for the estimation of doubling time and generation time.     

用麦夸方法最优拟合逻辑斯谛曲线

一般对非线性逻辑斯谛生长曲线的拟合是采用首先对原方程线性化以后用线性最小二乘的方法。此方法不是最优的。本文提出用麦夸方法对曲线进行拟合,并且比较了Gause、Andrewartha、May、Pearl、Krebs、万昌秀等人提出的方法与计算结果。麦夸方法对生物实验及生态学中诸多非线性曲线的参数估计具有普遍的意义。     

One equation to rule them all: a philosophical analysis of the Price equation

This paper provides a philosophical analysis of the Price equation and its role in evolutionary theory. Traditional models in population genetics postulate simplifying assumptions in order to make the models mathematically tractable. On the contrary, the Price equation implies a very specific way of theorizing, starting with assumptions that we think are true and then deriving from them the mathematical rules of the system. I argue that the Price equation is a generalization-sketch, whose main purpose is to provide a unifying framework for researchers, helping them to develop specific models. The Price equation plays this role because, like other scientific principles, shows features as abstractness, unification and invariance. By underwriting this special role for the Price equation some recent disputes about it could be diverted.     

On the kinetics of enzyme reactions

The Abel type differential equation governing the kinetics of the enzyme reactions is derived. Approximate solutions of this equation corresponding to the transient phase of the reaction, before a steady state is reached, are considered. It is shown that in several cases it is possible to obtain explicit, approximate solutions to the transient phase.     

Evaluation of a simplified generic bi-substrate rate equation for computational systems biology

The evaluation of a generic simplified bi-substrate enzyme kinetic equation, whose derivation is based on the assumption of equilibrium binding of substrates and products in random order, is described. This equation is much simpler than the mechanistic (ordered and ping-pong) models, in that it contains fewer parameters (that is, no K(i) values for the substrates and products). The generic equation fits data from both the ordered and the ping-pong models well over a wide range of substrate and product concentrations. In the cases where the fit is not perfect, an improved fit can be obtained by considering the rate equation for only a single set of product concentrations. Due to its relative simplicity in comparison to the mechanistic models, this equation will be useful for modelling bi-substrate reactions in computational systems biology.     

Parallel lines: nothing has changed?

The exponential increase in mortality rate with age is a universal feature of aging and is described mathematically by the Gompertz equation. When this equation is transformed semilogarithmically, it conforms to a straight line, the slope of which is generally used to reflect the rate of senescence. Historical and contemporary data of human and nonhuman populations show that adverse environmental conditions do not always change the slope of the log mortality rate over age. From these latter observations it is sometimes mistakenly inferred that the rate of senescence is unaffected by environmental conditions. Current biological inference emphasizes that gene action is dependent on the environment in which it is expressed. Here, we propose using the tangent line of the Gompertz equation to assess whether the rate of senescence has altered. Such an approach unmasks different rates of senescence when parameter G has remained constant, an observation that is in line with the notion that a plastic life history trait such as the rate of senescence results from the interplay of both genes and environment.     

One-dimensional food chain equations and their generalization

Two equations describing one-dimensional food chains are known to possess soliton solutions. It is demonstrated that both equations are embraced within another equation, which arises in the theory of chains of enzymic reactions. We find an elliptic function solution to this equation. We obtain a one-soliton solution from it and re-derive the elliptic function solutions of the two ecological equations.     

A study of the effect of non-linearities in the equation of bone remodeling

In this paper, we introduced a high-order non-linear equation of bone remodeling to combine with FEM by introducing two non-linearities, i.e. the remodeling coefficient B(t) and the order of non-linear remodeling equation. The influence of each non-linearity was tested based on its mechanical and physiological implications discussed. We use two finite element models to investigate the influences of non-linearities in this equation: a plate subjected to a ramp load, and a 2D model of the cross-section of a vertebra. By importing the idea of topology optimization in engineering, their external shapes and internal density distributions were simulated from unfixed configurations. To a certain extent, the high-order non-linear equation of bone remodeling we suggested here can control the remodeling processes of bones in different stages of growth or at different anatomic sites more effectively, and make it more consistent with physiological reality, i.e. express the remodeling characteristic that bone's best morphology is adapted to its mechanical environment. Furthermore, it is likely to describe the process of bone growth and evolution.     

An equation for flow in the renal proximal tubule

Approximate equations for epithelial solute and water transport have been combined with the relations of mass conservation to yield a single differential equation representing volume flow along the proximal tubule. This flow equation is first order, quasilinear and may be integrated directly. For the steady state, the result is an implicit relation between volume flow and distance along the tubule. For two time-dependent problems (step change of tubule inlet velocity or osmolality) the trajectories (distance as a function of transit time) of a fluid element starting at the inlet are obtained. Differentiation of the steady-state relation with respect to the inlet velocity yields a first-order differential equation relating inlet and outlet velocity. This equation is considered in detail, particularly with regard to the influence of solute-linked water reabsorption. Model calculations with parameters representing rat proximal tubule indicate that it will be difficult to discern coupled water flux in this epithelium from only outlet and inlet flows. Calculations using lower transport rates and lower permeabilities suggest that this equation may be useful in quantifying coupled water flow in proximal tubules from other species.     

The Validity of the Ussing Flux Ratio Equation in a Three-Dimensionally Inhomogeneous Membrane

Derivations of the Ussing flux ratio equation have, until now, required the membrane to be both bounded by parallel planes and homogeneous, except in the transmembrane direction. These constraints have been necessary for the theoretical demonstration that the equation is independent of membrane parameters in the absence of carriers, coupling, solvent drag, or “single-file” diffusion. In a new derivation, the flux ratio equation is shown to be valid in this kind of diffusion regime without regard to the three-dimensional structure of the membrane. Thus the constraints on both membrane homogeneity and membrane geometry are shown to be unnecessary. The general use of this equation to differentiate between simple, uncoupled diffusion and other membrane transport phenomena is thus placed on a firmer base. However, as in earlier derivations, it is necessary that isopotential, isobaric, constant concentration surfaces exist sufficiently close to the membrane on both of its sides.     

Global asymptotic stability of a periodic solution to an epidemic model

In this paper a periodic delay differential equation with spatial spread is investigated. This equation can be used to model the growth of malaria which is transmitted by a mosquito. Using monotone techniques, it is shown that the following bifurcation holds: either the disease dies out or the density of infectious people tends to a spatially homogeneous, time periodic and positive solution.Research partially supported by NSF Grant MCS 810-4837     

A generic rate equation for catalysed,template-directed polymerisation

Biosynthetic networks link to growth and reproduction processes through template-directed synthesis of macromolecules such as polynucleotides and polypeptides. No rate equation exists that captures this link in a way that it can effectively be incorporated into a single computational model of the overall process. This paper describes the derivation of such a generic steady-state rate equation for catalysed, template-directed polymerisation reactions with varying monomer stoichiometry and varying chain length. The derivation is based on a classical Michaelis–Menten mechanism with template binding and an arbitrary number of chain elongation steps that produce a polymer composed of an arbitrary number of monomer types. The rate equation only requires the identity of the first dimer in the polymer sequence; for the remainder only the monomer composition needs be known. Further simplification of a term in the denominator yielded an equation requiring no positional information at all, only the monomer composition of the polymer; this equation still gave an excellent estimate of the reaction rate provided that either the monomer concentrations are at least half-saturating, or the polymer is very long.     

Vibrational dynamics of bio- and nano-filaments in viscous solution subjected to ultrasound: implications for microtubules

In this paper, using a new analytical method, we solved the beam equation for a uniform bio- and nano-filament in a viscous solution. The filament is assumed to be attached at its two ends and driven by ultrasound plane waves. To obtain analytical solutions, we converted the beam equation to an equation that allows us the use of the method of separation of variables. We then reconstructed the solution of the original beam equation from the solution of the converted equation. Subsequently, we have used the parametric equations derived in this paper to investigate the resonance condition for a microtubule (MT) in an aqueous solution. We show that by using ultrasound plane waves, one cannot satisfy a resonance condition for MTs treated as rigid rods. In order to achieve resonance, a single mode of the MT vibration must be excited with a harmonic number larger than a threshold value found here. Single mode excitation not only helps to transfer a minimum amount of energy to the surrounding medium compared with multi-mode excitation, but it also allows for a simultaneous high amplitude and high mode quality that is impossible using plane waves. In order to overcome this difficulty, we propose to use an ultrasound generation device as a potential technical solution characterized by both frequency control and optimized energy transfer to the MT. Finally, the minimum required intensity of the ultrasound at the location of the MT in order to break it is shown to be on the order of 10⁵ W/m², which corresponds to 170 dB.     

A mathematical derivation of size spectra in fish populations

Taking into account a predator/prey size ratio in a size-structured population model leads to a partial derivative equation of which we study the properties. By expliciting the structure of the attractor of this equation, it is shown that a simple mechanism, size-based opportunistic predation, can explain the stability in the shape of size spectra observed in various marine ecosystems.     

On true and apparent Michaelis constants in enzymology. II. Is the equation k(m)app = k(s) + k)cat)/k(1) true for enzyme-catalysed reactions with activator participation?

The article is dedicated to analysis of equation which expresses apparent Michaelis constant K(m)app) of enzyme-catalysed reactions with activator participation by means of the substrate constant K(s) and rate constant of enzyme-substrate complex decomposition k(cat). It has been shown that although it is possible to record the mechanisms of such reactions as a scheme similar to Michaelis-Menten model and to derive equation of apparent Michaelis constant as K(m(app) = K(s) + k(cat)/k(1), but this approach cannot be used for investigation of all reactions with activator participation. The equation mentioned above is not obeyed in the general case, it may be true for some mechanisms only or under certain ratio of kinetic parameters of enzyme-catalysed reactions.     

A futher theoretical treatment of the turnover of protein bound phospate in the presence of both protein kinase and phosphatase activities.

Many subcellular fractions contain both protein kinase and phosphatase enzymes which can act on endogenous proteins in the fractions.When studying the phosphorylation of the endogenous protein it is necessary to take into account the presence of both enzymes.In a previous paper (Weller, 1974) an equation was derived which described the time-course of phosphorylation of a protein in the presence of both kinase and phosphatase activities. To derive this equation the assumption was made that the activity of the kinase was very much greater than that of the phosphatase. This assumption may not be valid in all cases and the present paper describes the derivation of a similar equation without making any assumptions about the relative rates of kinase and phosphatase activities.The time-course of protein phosphorylation predicted by the new equation is compared to that predicted by the previous equation with varying rates of protein phosphatase activity. It is found that the new equation should be used to describe the time-course of protein phosphorylation in the presence of protein kinase and phosphatase activities if the activity of the phosphatase is more than about 7% of that of the kinase.     

红松单木高生长模型的研究

１引言生长模型是定量研究树木生长过程的有效手段。它既可对林木生长作出现实的评价,也可用来预估将来各测树因子的变化;既是编制修订各种数表的基础,也是森林经营中各种措施实施的依据。在林学上,生长模型主要包括单木生长模型和林分生长模型,其中单木生长模型是林...     

Metabolic cost of walking: equation and model

Twenty years of published experience with the Workman-Armstrong equation for predicting walking VO2 is reviewed. The equation is reexpressed in currently accepted terminology, and it is shown that the equation serves well as a basic model of normal walking. Employing this model to analyze VO2/step leads to the elaboration of a three-compartment model of the metabolic cost of walking. This three-compartment model provides a rational estimate of the fraction of walking's metabolic cost that powers the actual walking movement. Doubt is expressed that "comfortable speed of walking" is definable in energy terms. It is suggested that the requirements of maintaining balance while walking may determine both the comfortable speed of walking and the curvilinearity of the relationship between ground-speed and freely chosen step frequency of walking.     

A comparative investigation of certain difference equations and related differential equations: Implications for model-building

Many mathematical models for physical and biological problems have been and will be built in the form of differential equations or systems of such equations. With the advent of digital computers one has been able to find (approximate) solutions for equations that used to be intractable. Many of the mathematical techniques used in this area amount to replacing the given differential equations by appropriate difference equations, so that extensive research has been done into how to choose appropriate difference equations whose solutions are “good” approximations to the solutions of the given differential equations. The present paper investigates a different, although related problem. For many physical and biological phenomena the “continuum” type of thinking, that is at the basis of any differential equation, is not natural to the phenomenon, but rather constitutes an approximation to a basically discrete situation: in much work of this type the “infinitesimal step lengths” handled in the reasoning which leads up to the differential equation, are not really thought of as infinitesimally small, but as finite; yet, in the last stage of such reasoning, where the differential equation rises from the differentials, these “infinitesimal” step lengths are allowed to go to zero: that is where the above-mentioned approximation comes in. Under this kind of circumstances, it seems more natural tobuild themodel as adiscrete difference equation (recurrence relation) from the start, without going through the painful, doubly approximative process of first, during the modeling stage, finding a differential equation to approximate a basically discrete situation, and then, for numerical computing purposes, approximating that differential equation by a difference scheme. The paper pursues this idea for some simple examples, where the old differential equation, though approximative in principle, had been at least qualitatively successful in describing certain phenomena, and shows that this idea, though plausible and sound in itself, does encounter some difficulties. The reason is that each differential equation, as it is set up in the way familiar to theoretical physicists and biologists, does correspond to a plethora of discrete difference equations, all of which in the limit (as step length→0) yield the same differential equation, but whose solutions, for not too small step length, are often widely different, some of them being quite irregular. The disturbing thing is that all these difference equations seem to adequately represent the same (physical or biological) reasoning as the differential equation in question. So, in order to choose the “right” difference equation, one may need to draw upon more detailed (physical or) biological considerations. All this does not say that one should not prefer discrete models for phenomena that seem to call for them; but only that their pursuit may require additional (physical or) biological refinement and insight. The paper also investigates some mathematical problems related to the fact of many difference equations being associated with one differential equation.