Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

A kinetic model for the activated sludge process which considers diffusion and reaction in the microbial floc

A mathematical model has been developed which describes substrate removal, oxygen utilization, and biomass production in an aggregated microbial suspension containing the substrate as a soluble biodegradable material and a uniform floc size. It is applicable to both steady-state and transient conditions. The model, consisting of three partial differential equations and two ordinary differential equations, takes into account the flow pattern in the reactor, intraparticle mass transport of oxygen and substrate, and biochemical reaction by individual cells embedded in the floc. Efficient numerical solution of the coupled nonlinear equations is obtained using an implicit finite difference approach for both the reactor and floc equations. A convergent solution is realized through block interation utilizing the tridiagonal algorithm. Results indicate that a unifying theory of activated sludge dynamics will have to consider coupling between floc chemical kinetics and changes in the bulk liquid characteristics. Floc size emerges as an important influence on system performance. It appears necessary to distinguish between a system response caused by diffuslonal resistances and nutrient limitations within the floc and a response caused by physiological adaption when analyzing the transient behavior of an activated sludge process. Future research should be devoted to rigorous laboratory determinations of model parameters along with extensions to include limitations of nutrients other than orgabnic carbon and oxygen.     

Complex dynamic behaviour of autonomous microbial food chains

 The dynamic behaviour of food chains under chemostat conditions is studied. The microbial food chain consists of substrate (non-growing resources), bacteria (prey), ciliates (predator) and carnivore (top predator). The governing equations are formulated at the population level. Yet these equations are derived from a dynamic energy budget model formulated at the individual level. The resulting model is an autonomous system of four first-order ordinary differential equations. These food chains resemble those occuring in ecosystems. Then the prey is generally assumed to grow logistically. Therefore the model of these systems is formed by three first-order ordinary differential equations. As with these ecosystems, there is chaotic behaviour of the autonomous microbial food chain under chemostat conditions with biologically relevant parameter values. It appears that the trajectories on the attractors consists of two superimposed oscillatory behaviours, a slow one for predator–top predator and a fast one for the prey–predator on one branch at which the top predator increases slowly. In some regions of the parameter space there are multiple attractors. Received 8 November 1995; received in revised form 7 January 1997     

Intrinsic cone adaptation modulates feedback efficiency from horizontal cells to cones.

Processing of visual stimuli by the retina changes strongly during light/dark adaptation. These changes are due to both local photoreceptor-based processes and to changes in the retinal network. The feedback pathway from horizontal cells to cones is known to be one of the pathways that is modulated strongly during adaptation. Although this phenomenon is well described, the mechanism for this change is poorly characterized. The aim of this paper is to describe the mechanism for the increase in efficiency of the feedback synapse from horizontal cells to cones. We show that a train of flashes can increase the feedback response from the horizontal cells, as measured in the cones, up to threefold. This process has a time constant of approximately 3 s and can be attributed to processes intrinsic to the cones. It does not require dopamine, is not the result of changes in the kinetics of the cone light response and is not due to changes in horizontal cells themselves. During a flash train, cones adapt to the mean light intensity, resulting in a slight (4 mV) depolarization of the cones. The time constant of this depolarization is approximately 3 s. We will show that at this depolarized membrane potential, a light-induced change of the cone membrane potential induces a larger change in the calcium current than in the unadapted condition. Furthermore, we will show that negative feedback from horizontal cells to cones can modulate the calcium current more efficiently at this depolarized cone membrane potential. The change in horizontal cell response properties during the train of flashes can be fully attributed to these changes in the synaptic efficiency. Since feedback has major consequences for the dynamic, spatial, and spectral processing, the described mechanism might be very important to optimize the retina for ambient light conditions.     

Modelling and simulation of vertebrate retina

In the present work we investigate the neuronal activities in a vertebrate retina by modelling and simulations using the results of (Oguztöreli, 1979). The basic retinal network considered here consists of interconnected five neurons: a receptor cell (rod or cone), a horizontal cell, a bipolar cell, an amacrine cell, and a retinal ganglion cell. The mathematical model for the basic network is a system of nonlinear ordinary integral differential difference equations. A number of simulations describing the dynamics of the neural activities in the basic network under different conditions are presented, actual and steady-state solutions are discussed. An algorithm is proposed for the determination of the system parameters experimentally.This work was supported by the Natural Sciences and Engineering Research Council Canada under Grant NSERCA-4345 through the University of Alberta     

Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused HIT-15 cells

Oscillatory secretion of insulin has been observed in many different experimental preparations ranging from pancreatic islets to the whole pancreas. Here we examine the mathematical features underlying a possible model for oscillatory secretion from the perifused, insulin-secreting cell line, HIT-15. The model includes the kinetics of uptake of glucose by GLUT transporters, the rate of glucose metabolism within the cell, and the effect of glucose on the rate of insulin secretion. Putative feedback by insulin on the rate of glucose transport into the cells is treated phenomenologically and leads to insulin oscillations similar to those observed experimentally in HIT cells. The resulting set of ordinary differential equations is simplified by time-scale analysis to a two-variable set of ordinary differential equations. Because of this simplification we can explore, in great detail, the characteristics of the oscillations and their sensitivity to parameter variation using phase plane analysis.     

The development of largangians for biological models

A modern theory of the calculus of variations is used to form necessary and sufficient conditions for the existence of a Lagrangian representation of a system of first-order ordinary differential equations. There exists a theorem to the effect that when a system of ordinary differential equations is variationally self-adjoint, the fulfillment of such conditions is guaranteed. In addition, self-adjointness, allows establishement of an algorithm by which a Lagrangian for the system may be explicitly constructed. Examples in mathematical biology are given to illustrate the use of the stated theorem.     

A mathematical study of a two-regional population growth model

The paper provides a mathematical study of a model of urban dynamics, adjusting to an ecological model proposed by Lotka and Volterra. The model is a system of two first-order non-linear ordinary differential equations. The study proposed here completes the original proof by using the main tools such as a Lyapunov function.     

A Model for Migratory B Cell Oscillations from Receptor Down-Regulation Induced by External Chemokine Fields

A long-standing paradigm in B cell immunology is that effective somatic hypermutation and affinity maturation require cycling between the dark zone and light zone of the germinal center. The cyclic re-entry hypothesis was first proposed based on considerations of the efficiency of affinity maturation using an ordinary differential equations model for B cell population dynamics. More recently, two-photon microscopy studies of B cell motility within lymph nodes in situ have revealed the complex migration patterns of B lymphocytes both in the preactivation follicle and post-activation germinal center. There is strong evidence that chemokines secreted by stromal cells and the regulation of cognate G-protein coupled receptors by these chemokines are necessary for the observed spatial cell distributions. For example, the distribution of B cells within the light and dark zones of the germinal center appears to be determined by the reciprocal interaction between the level of the CXCR4 and CXCR5 receptors and the spatial distribution of their respective chemokines CXCL12 and CXCL13. Computer simulations of individual-based models have been used to study the complex biophysical and mechanistic processes at the individual cell level, but such simulations can be challenging to parameterize and analyze. In contrast, ordinary differential equations are more tractable, but traditional compartment model formalizations ignore the spatial chemokine distribution that drives B cell redistribution. Motivated by the desire to understand the motility patterns observed in an individual-based simulation of B cell migration in the lymph node, we propose and analyze the dynamics of an ordinary differential equation model incorporating explicit chemokine spatial distributions. While there is experimental evidence that B cell migration patterns in the germinal center are driven by extrinsically regulated differentiation programs, the model shows, perhaps surprisingly, that feedback from receptor down-regulation induced by external chemokine fields can give rise to spontaneous interzonal and intrazonal oscillations in the absence of any extrinsic regulation. While the extent to which such simple feedback mechanisms contributes to B cell migration patterns in the germinal center is unknown, the model provides an alternative hypothesis for how complex B cell migration patterns might arise from very simple mechanisms.     

Analysis of the kinetic hairpin transfer model for parvoviral DNA replication

All linear DNA molecules face special problems in replicating their 5' ends, as DNA polymerases add nucleotides only to pre-existing strands with free 3'-OH groups. Parvoviruses, a group of small animal viruses with a linear single-stranded DNA genome, cope with this problem by having palindromic terminal sequences that can fold back on themselves to form hairpin structures essential in priming DNA replication. The 3' terminal sequence that initiates replication becomes reversed in orientation during the process, and if the palindrome is imperfect, two different, reverse-complementary terminal sequences are generated. The relative abundances of the terminal sequence orientations at each end of the DNA molecules can be measured and give information about the replication process. From such clues, we developed a "kinetic hairpin transfer model" based on differential rates of hairpin formation and inversion processes depending on the conformations of the 3' termini. Numerical studies showed that this simple idea can account for the diverse pattern of DNA distributions observed in the family Parvoviridae. In this paper, we simplify the model to a set of coupled linear first-order ordinary differential equations in order to delineate its essential properties by Perron-Frobenius theory. Secondly, we examine our assumption of linear kinetics by modeling enzyme catalysis of the component steps of the hairpin transfer process. We show that the rate-determining step of the process is the binding of initiation complex to the self-priming hairpin structures. Furthermore, we find that if the replication machinery is saturated by DNA substrate late in an infection, the differential equations become non-linear but the steady-state DNA distribution is still given by the solution of our original linear equations.     

Quantification of toluene dioxygenase induction and kinetic modeling of TCE cometabolism by Pseudomonas putida TVA8.

As measured by the toluene-induced bioluminescent response of Pseudomonas putida TVA8 in batch experiments, toluene dioxygenase (Tod) enzyme activities are dependent on toluene concentration between 0 and 30 mg/L. To provide a measure of the Tod activity for use in Michaelis-Menten competitive-inhibition kinetics, a correlation between toluene concentration and induced Tod activity as measured by an induced bioluminescent response of P. putida TVA8 is presented as a nondimensional Tod activity parameter. A packed-bed, radial-flow bioreactor (RFB) using the bioreporter P. putida TVA8A serves as the model system for studying the effect of the enzyme activity parameter on model predictions of vapor-phase toluene oxidation and trichloroethylene (TCE) cometabolism. Mass balances were performed on a differential section of the RFB to describe the radial transport of vapor-phase toluene and TCE through a bulk gas phase and the concomitant biological reaction in a stationary biofilm phase. The finite-element Galerkin weak-statement formulation with first-order basis functions was used to find the optimum solution to the highly nonlinear, coupled equations. For this RFB system with toluene concentrations less than 1 mg/L in the bulk gas phase, the Tod activity parameter enables accurate predictions of steady-state TCE degradation rate (0.27 microg TCE/min).     

Stochastic simulations on a model of circadian rhythm generation

Biological phenomena are often modeled by differential equations, where states of a model system are described by continuous real values. When we consider concentrations of molecules as dynamical variables for a set of biochemical reactions, we implicitly assume that numbers of the molecules are large enough so that their changes can be regarded as continuous and they are described deterministically. However, for a system with small numbers of molecules, changes in their numbers are apparently discrete and molecular noises become significant. In such cases, models with deterministic differential equations may be inappropriate, and the reactions must be described by stochastic equations. In this study, we focus a clock gene expression for a circadian rhythm generation, which is known as a system involving small numbers of molecules. Thus it is appropriate for the system to be modeled by stochastic equations and analyzed by methodologies of stochastic simulations. The interlocked feedback model proposed by Ueda et al. as a set of deterministic ordinary differential equations provides a basis of our analyses. We apply two stochastic simulation methods, namely Gillespie's direct method and the stochastic differential equation method also by Gillespie, to the interlocked feedback model. To this end, we first reformulated the original differential equations back to elementary chemical reactions. With those reactions, we simulate and analyze the dynamics of the model using two methods in order to compare them with the dynamics obtained from the original deterministic model and to characterize dynamics how they depend on the simulation methodologies.     

Steady-state analysis of a two-compartment barrier-limited capillary-tissue model with michaelis-menten saturation kinetics

The steady-state solution of the equations governing substrate exchange between vascular and extravascular compartments separated by a membrane with finite, symmetrical substrate permeability is presented. Substrate removal from the extravascular compartment by Michaelis-Menten saturation type kinetics with negligible diffusion in the axial and instantaneous diffusion in the transverse directions in both compartments are assumed. It is shown that the solution degenerates into known expressions for special linearized and asymptotic cases. The method of solution is also applied to an extension of the original model incorporating autoregulatory feedback effects upon the process responsible for substrate removal.     

Dynamics of chromatic adaptation in cones of freshwater turtle

The closer the wavelength of a steady background of monochromatic light is to the peak sensitivity of a cone that is being illuminated, the stronger is the desensitization of that cone; this is chromatic adaptation. A model of the freshwater turtle retina with the neural components of chromatic adaptation via negative feedback circuits is used to simulate and study various aspects of chromatic adaptation. An internal negative feedback circuit resides solely within the cone pedicle and thereby, its adaptive effects are relatively specific, so that univariance is maintained. The cone-L-horizontal cell circuit is an external negative feedback circuit and its adaptive effects are less specific since all 3 chromatic cone types are involved, so that univariance is violated. Chromatic adaptation is the result of the decrease in the cone gain due to the dependency of the gains of the negative feedback circuits on the mean illuminance level. The results of the model are consistent with von Kries law, but the changes in gains of the cones due to chromatic adaptation are dependent on wavelength, intensity of the adapting light and size.     

A complete set of control equations for the human musculo-skeletal system

A mathematical model of the total human musculo-skeletal system is presented. The model comprises a link-mechanical and a musculo-mechanical set of ordinary first-order differential equations which describe the dynamics of the segment model and muscle model respectively. The interdependence of the two sets of equations is demonstrated. The set of musculo-mechanical equations contains the two neuromuscular control parameters motor unit recruitment and stimulation rate, and the significance of such a representation for a control-theoretical treatment of musculo-skeletal systems is discussed. Finally, after a short discussion of the successful application of the present model in the prediction of an optimal human motion, further possibilities are indicated of the use of the model for investigations into the control behaviour of musculo-skeletal systems.     

A numerical model for the oxygenation of blood in lung capillaries--effect of nth order one-step kinetics of oxygen uptake by haemoglobin

A numerical model is described for the oxygenation of blood in lung capillaries by considering the transport mechanisms of molecular diffusion, convection and the facilitated diffusion due to the presence of haemoglobin. In order to represent the oxygen dissociation curve accurately in the model, the nth order one-step kinetics of oxygen uptake by haemoglobin has been used. The resulting system of coupled, non-linear partial differential equations is solved numerically. It is shown that the blood is required to traverse a larger distance in the capillary before becoming fully oxygenated with nth order one-step kinetics in comparison to first-order one-step kinetics.     

Unbuffered and buffered supply chains in human metabolism

The investigation of very complex dynamical systems like the human metabolism requires the comprehension of important subsystems. The present paper deals with energy supply chains as subsystems of the metabolism on the molecular, cellular, and individual levels. We form a mathematical model of ordinary differential equations and we show fundamental properties by Fourier techniques. The results are supported by a transition from a system of ordinary differential equations to a partial differential equation, namely, a transport equation. In particular, the behavior of supply chains with dominant pull components is discussed. A special focus lies on the role of buffer compartments.     

Model-based hypothesis testing of key mechanisms in initial phase of insulin signaling

Type 2 diabetes is characterized by insulin resistance of target organs, which is due to impaired insulin signal transduction. The skeleton of signaling mediators that provide for normal insulin action has been established. However, the detailed kinetics, and their mechanistic generation, remain incompletely understood. We measured time-courses in primary human adipocytes for the short-term phosphorylation dynamics of the insulin receptor (IR) and the IR substrate-1 in response to a step increase in insulin concentration. Both proteins exhibited a rapid transient overshoot in tyrosine phosphorylation, reaching maximum within 1 min, followed by an intermediate steady-state level after approximately 10 min. We used model-based hypothesis testing to evaluate three mechanistic explanations for this behavior: (A) phosphorylation and dephosphorylation of IR at the plasma membrane only; (B) the additional possibility for IR endocytosis; (C) the alternative additional possibility of feedback signals to IR from downstream intermediates. We concluded that (A) is not a satisfactory explanation; that (B) may serve as an explanation only if both internalization, dephosphorylation, and subsequent recycling are permitted; and that (C) is acceptable. These mechanistic insights cannot be obtained by mere inspection of the datasets, and they are rejections and thus stronger and more final conclusions than ordinary model predictions.