Hidden pools,hidden modes,and visible repeated eigenvalues in compartmental models

In compartmental analysis, when time-series data are initially fitted by sums of exponential functions, it is usually assumed that the eigenvalues are real and distinct and the number of pools (compartments) equals the number of exponentials. However, repeated real or complex eigenvalues, although more difficult to detect, may be inherent in the data, and the number of pools may be larger than the number of exponentials. In order to describe compartmental models with such properties, the visible multiplicity of eigenvalues and concepts of hidden pools and visible and hidden modes are defined. It is then shown that if a model is state observable, each mode is visible (not hidden) in the zero-input response for some choice of initial state, but not conversely, and also that the visible multiplicity of eigenvalues is determined by the submodel of input-output connectable compartments. Compartmental models are analyzed, using their decomposition into what we define as strongly connected components. An upper bound is given for the visible multiplicity of eigenvalues in terms of the model's strongly connected components. For models with one input and one output, this bound is shown to be attained for what we call generalized trees.     

Oscillating reactions in open systems

The probable existence of oscillating chemical reactions has been attracting some interest in recent years for their possible role in explaining certain biological phenomena. Perhaps the simplest model of oscillating reactions is that of Lotka (1910), which consists of a chain of autocatalytic reactions. Two “reactor systems” in which such a chain of reactions could take place are considered in this work and are called homogeneous and compartmental models, respectively. The differential equations governing the temporal behavior of the reacting species are solved on an analog computer, and the conditions under which sustained oscillations occur are obtained and discussed. Comparisons of the solution obtained in the two models are discussed.     

Evolution of avian eggshell structure

Data are presented suggesting that birds have evolved eggs with shells containing different structures (numbers of mammillae per unit of inner eggshell surface area, i.e., mammillary densities) to cope up with different calcium requirements imposed by different growth rates and modes of development. Precocial bird species grow slowly, but have high mammillary density, while altricial bird species grow rapidly, but have low mammillary density. These results suggest an adaptation associated with growth rate and mode of development and show, moreover, that the mammillary layer is indicative of the breeding biology of the bird. J. Morphol. 2012. © 2011 Wiley Periodicals, Inc.     

Ionic and osmotic disruptions of the lily pollen tube oscillator: testing proposed models

Two mechanisms have been proposed as the primary control of oscillating tip growth in Lilium longiflorum Thunb. pollen tubes: changes in cell wall strength (Holdaway-Clarke et al. 1997) or alternatively, changes in turgor pressure (Messerli et al. 2000). Here we have modified the ionic and osmotic concentrations of the growth medium to test predictions derived from both models. Raising the [Ca2+]o tenfold above normal reduced the amplitude of the [Ca2+]i oscillations and growth oscillations while it raised the basal [Ca2+]i and growth rate such that the average growth rate did not change. Raising the [H+] of the growth medium tenfold reversibly decreased and sometimes eliminated the [Ca2+]i and growth oscillations without changing the average growth rate. Lowering the [H+] tenfold led to irregular frequency and amplitude [Ca2+]i oscillations, reduced the average growth rate of tubes and led to cell bursting in 33% of tubes. Addition of 50 mM H+ buffer, MES, to prevent pH changes in the cell wall increased the period, amplitude and duration of both [Ca2+]i and growth oscillations. Changing the [K+]o did not markedly effect [Ca2+]i oscillations. Reducing the osmolarity of the medium led to transient large-amplitude [Ca2+]i and growth oscillations while reducing large-amplitude oscillations over long periods. In many different conditions under which growth still occurs, lily pollen tubes maintain growth oscillations, albeit with modified frequency, amplitude and duration. We conclude that modifications to both proposed models are necessary to explain oscillating growth in this system.     

Steady-state dosage regimen calculations in linear pharmacokinetics

This paper describes a general approach to compute steady-state dosage regimens. The method deals with individual linear compartmental mammillary pharmacokinetic models and requires estimated values of model parameters. It is based on the linearity principle and computes dosage regimens to achieve and maintain specified steay-state plasma concentrations, e.g. mean value or minimum effective and maximum safe levels. An interactive computer program performing the calculations has been developed.     

On the probability of reaching a threshold in a stochastic mammillary system

The multivariate distribution over time of a particular stochastic mammillary compartmental model is obtained for any point in time. The maximum expectation of the peripheral compartments is then derived and used to determine lower bounds on the probability that the maximum of the peripheral compartments reaches any arbitrary threshold level. A bound on the probability is illustrated by an example and some of its implications are explored.     

On the oscillatory phenomenon in an oil/water interface

A simple theoretical model is presented for simulating the self-sustained oscillations of electric potential and pH at an oil/water interface appearing in a two-phase system composed of 2-nitropropane solution containing picrate acid and an aqueous solution of cetyltrimethylammonium bromide. In the present model, a well-known condition necessary for the occurrence of self-sustained oscillations, i.e., the presence of a positive feedback process far from equilibrium, is taken into account in a set of kinetic equations to describe simplified characters of the following two processes: (i) a cooperative formation of ion pair complexes at the interface, and (ii) supply of picrate anions and cetyltrimethylammonium cations to the interface accompanied by release of ion pair complexes to the organic phase. The numerical solutions of the present equations are shown to reproduce fairly well the characteristic properties of the oscillation of electric potential and pH such as wave forms and frequencies.     

Oscillating actin polymerization/depolymerization responses in human polymorphonuclear leukocytes

Leukotriene B4 and platelet-activating factor induced a rapidly oscillating actin polymerization/depolymerization response in human polymorphonuclear leukocytes. N-Formylpeptides were deficient in the ability to induce these oscillations. Flow cytometric analysis of filamentous actin verified that all cells were synchronously responding in this cyclic manner. The hypothesis was tested that these oscillations were analogous to chemical oscillations, i.e. oscillations of intermediate species in chemical systems that are far from equilibrium (Epstein, I. R., Kustin, K., DeKepper, P., and Orban, M. (1983) Sci. Am. 248, 112). Actin polymerization/depolymerization cycles were terminated by adding receptor antagonist a few seconds after initiation of the response by agonists. Thus the oscillations did not represent chemical oscillations that hypothetically could result from a rapid jump of the intracellular milieu to a state far from equilibrium. Rather, continued occupancy of receptors and/or occupancy of new receptors was required to sustain the oscillations. This suggested that the oscillations resulted from regulated polymerization and depolymerization pathways. In simultaneous measurements of actin-associated right angle light scatter and intracellular calcium, no calcium oscillations were detected. Thus, cycles of actin polymerization/depolymerization were not regulated by calcium oscillations.     

Frequency Response and Gap Tuning for Nonlinear Electrical Oscillator Networks

We study nonlinear electrical oscillator networks, the smallest example of which consists of a voltage-dependent capacitor, an inductor, and a resistor driven by a pure tone source. By allowing the network topology to be that of any connected graph, such circuits generalize spatially discrete nonlinear transmission lines/lattices that have proven useful in high-frequency analog devices. For such networks, we develop two algorithms to compute the steady-state response when a subset of nodes are driven at the same fixed frequency. The algorithms we devise are orders of magnitude more accurate and efficient than stepping towards the steady-state using a standard numerical integrator. We seek to enhance a given network''s nonlinear behavior by altering the eigenvalues of the graph Laplacian, i.e., the resonances of the linearized system. We develop a Newton-type method that solves for the network inductances such that the graph Laplacian achieves a desired set of eigenvalues; this method enables one to move the eigenvalues while keeping the network topology fixed. Running numerical experiments using three different random graph models, we show that shrinking the gap between the graph Laplacian''s first two eigenvalues dramatically improves a network''s ability to (i) transfer energy to higher harmonics, and (ii) generate large-amplitude signals. Our results shed light on the relationship between a network''s structure, encoded by the graph Laplacian, and its function, defined in this case by the presence of strongly nonlinear effects in the frequency response.     

Mean residence time in multicompartmental models with time delays

Calculation of the mean residence time (MRT) of a drug in a stationary compartmental model is classically carried out from several expressions. Nevertheless, one or more time delays between compartments modify the mean residence times. It is the aim of this paper to propose a general method for MRT calculations, in any n-compartmental models which may include time delays. As examples, catenary and mammillary models are considered.     

Phase resetting and bifurcation in the ventricular myocardium.

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.     

Four open mammillary and catenary compartment models for pharmacokinetics studies

A mathematical method is proposed to solve a fourth degree equation which gives the eigenvalues of the matrix connected with a four open compartment model used in pharmacokinetic studies. As examples, the method is applied to mammillary and catenary models.     

Effect of local anesthetics on the electrical characteristics of an excitable model membrane composed of dioleyl phosphate II. Dynamic response

The effects of local anesthetics (tetracaine, procaine and lidocaine) on self-sustained electrical oscillations were studied for a lipid membrane comprising dioleyl phosphate (DOPH). This model membrane exhibits oscillation of the membrane potential in a manner similar to that of nerve membranes, i.e., repetitive firing, in the presence of an ion-concentration gradient, on the application of d.c. electric current. Relatively weak anesthetics such as procaine and lidocaine increased the frequency of self-sustained oscillation, and finally induced aperiodic, rapid oscillation. The strong anesthetic tetracaine inhibited oscillation.     

Complexity of compartmental models

A family of complexity indices for compartmental models is introduced. These are analogous to the diversity indices used to study ecological communities. Each index is shown to be related to the eigenvalues of the model, and in one case (the Shannon index) is equal to the negative of the sum of all eigenvalues. Relations between these indices and equivalent models as well as intrinsic complexity are investigated. In each case, large values of the complexity index lead to greater stability, but not necessarily conversely.     

Eggshell structure, mode of development and growth rate in birds

The shell of an egg contributes to successful embryogenesis in many ways, such as through protection, respiration and water exchange. The shell is also the major source of calcium for the development of high-calcium consuming organs, e.g. the skeleton, muscles and brain. Some studies show, moreover, that growth rate may play a fundamental role in the pattern of skeletal development in birds: the faster the growth the less ossified the skeleton is at hatching. We predicted, therefore, that slow (precocial) and fast (altricial) growing bird species should lay eggs encased in shells with different structures adapted to support different rates of calcium removal by developing embryos. We tested this prediction by comparing the fine structure of the inner eggshell surface (mammillary layer) from 36 bird species belonging to 18 orders ranging from Struthioniformes to Passeriformes. Using scanning electron microscopy, we compared the mammillary layer of both non-incubated eggs and eggs at the time of hatching, i.e., before and after embryonic development and the accompanying calcium removal. The results were consistent with the prediction, i.e., the number of mammillary tips per unit of surface area was associated with mode of development and growth rate. The number was higher, and calcium removal was also more extensive, in shells from precocial bird species than in shells from altricial bird species.     

Qualitative theory of compartmental systems with lags

Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems. Often, these systems include time lags; in this context, continuous probability density functions (pdfs) of lags are far more important than discrete lags. There is a relatively complete theory of compartmental systems without lags, both linear and non-linear [SIAM Rev. 35 (1993) 43]. The authors extend their previous work on compartmental systems without lags to show that, for discrete lags and for a very large class of pdfs of continuous lags, compartmental systems with lags are equivalent to larger compartmental systems without lags. Consequently, the properties of compartmental systems with lags are the same as those of compartmental systems without lags. For a very large class of compartmental systems with time lags, one can show that the time lags themselves can be generated by compartmental systems without lags. Thus, such systems can be partitioned into a main system, which is the original system without the lags, plus compartmental subsystems without lags that generate the lags. The latter may be linear or non-linear and may be inserted into main systems that are linear or non-linear. The state variables of the compartmental lag subsystems are hidden variables in the formulation with explicit lags.     

Transcranial electrical currents to probe EEG brain rhythms and memory consolidation during sleep in humans

Previously the application of a weak electric anodal current oscillating with a frequency of the sleep slow oscillation (～0.75 Hz) during non-rapid eye movement sleep (NonREM) sleep boosted endogenous slow oscillation activity and enhanced sleep-associated memory consolidation. The slow oscillations occurring during NonREM sleep and theta oscillations present during REM sleep have been considered of critical relevance for memory formation. Here transcranial direct current stimulation (tDCS) oscillating at 5 Hz, i.e., within the theta frequency range (theta-tDCS) is applied during NonREM and REM sleep. Theta-tDCS during NonREM sleep produced a global decrease in slow oscillatory activity conjoint with a local reduction of frontal slow EEG spindle power (8-12 Hz) and a decrement in consolidation of declarative memory, underlining the relevance of these cortical oscillations for sleep-dependent memory consolidation. In contrast, during REM sleep theta-tDCS appears to increase global gamma (25-45 Hz) activity, indicating a clear brain state-dependency of theta-tDCS. More generally, results demonstrate the suitability of oscillating-tDCS as a tool to analyze functions of endogenous EEG rhythms and underlying endogenous electric fields as well as the interactions between EEG rhythms of different frequencies.     

RhoA Regulates Calcium-Independent Periodic Contractions of the Cell Cortex

When microtubules are depolymerized in spreading cells, they experience morphological oscillations characterized by a period of about a minute, indicating that normal interactions between the microfilament and microtubule systems have been significantly altered. This experimental system provides a test bed for the development of both fine- and coarse-grained models of complex motile processes, but such models need to be adequately informed by experiment. Using criteria based on Fourier transform analysis, we detect spontaneous oscillations in spreading cells. However, their amplitude and tendency to operate at a single frequency are greatly enhanced by microtubule depolymerization. Knockdown of RhoA and addition of various inhibitors of the downstream effector of RhoA, Rho kinase, block oscillatory behavior. Inhibiting calcium fluxes from endoplasmic reticulum stores and from the extracellular medium does not significantly affect the ability of cells to oscillate, indicating that calcium plays a subordinate regulatory role compared to Rho. We characterized the dynamic structure of the oscillating cell by light, fluorescence, and electron microscopy, showing how oscillating cells are dynamically polarized in terms of their overall morphology, f-actin and phosphorylated myosin light chain distribution, and nuclear position and shape. Not only will these studies guide future experiments, they will also provide a framework for the development of refined mathematical models of the oscillatory process.     

Genetic Variability of Flight Metabolism in DROSOPHILA MELANOGASTER . II. Relationship between Power Output and Enzyme Activity Levels

The major goal of the studies reported here was to determine the extent to which genetic variation in the activities of the enzymes participating in flight metabolism contributes to variation in the mechanical power output of the flight muscles in Drosophila melanogaster. Isogenic chromosome substitution lines were used to partition the variance of both types of quantitative trait into genetic and environmental components. The mechanical power output was estimated from the wingbeat frequency, wing amplitude and wing morphology of tethered flies by applying the aerodynamic models of Weis-Fogh and Ellington. There were three major results. (1) Chromosomes sampled from natural populations provide a large and repeatable genetic component to the variation in the activities of most of the 15 flight metabolism enzymes investigated and to the variation in the mechanical power output of the flight muscles. (2) The mechanical power output is a sensitive indicator of the rate of flight metabolism (i.e., rate of oxygen consumption during tethered flight). (3) In spite of (1) and (2), no convincing cases of individual enzyme effects on power output were detected, although the number and sign of the significant enzyme-power correlations suggests that such effects are not totally lacking.     

Equality of average and steady-state levels in some nonlinear models of biological oscillations

Nonlinear oscillatory systems, playing a major role in biology, do not exhibit harmonic oscillations. Therefore, one might assume that the average value of any of their oscillating variables is unequal to the steady-state value. For a number of mathematical models of calcium oscillations (e.g. the Somogyi–Stucki model and several models developed by Goldbeter and co-workers), the average value of the cytosolic calcium concentration (not, however, of the concentration in the intracellular store) does equal its value at the corresponding unstable steady state at the same parameter values. The average value for parameter values in the unstable region is even equal to the level at the stable steady state for other parameter values, which allow stability. This holds for all parameters except those involved in the net flux across the cell membrane. We compare these properties with a similar property of the Higgins–Selkov model of glycolytic oscillations and two-dimensional Lotka–Volterra equations. Here, we show that this equality property is critically dependent on the following conditions: There must exist a net flux across the model boundaries that is linearly dependent on the concentration variable for which the equality property holds plus an additive constant, while being independent of all others. A number of models satisfy these conditions or can be transformed such that they do so. We discuss our results in view of the question which advantages oscillations may have in biology. For example, the implications of the findings for the decoding of calcium oscillations are outlined. Moreover, we elucidate interrelations with metabolic control analysis. This paper is dedicated to the memory of the late Reinhart Heinrich, who was the academic teacher of S.S. and, to a great extent, also of M.M.