Mean residence times in linear compartmental systems. Symbolic formulae for their direct evaluation

A complete analysis has been performed of the mean residence times in linear compartmental systems, closed or open, with or without traps and with zero input. This analysis allows the derivation of explicit and simple general symbolic formulae to obtain the mean residence time in any compartment of any linear compartmental system, closed or open, with or without traps, as well as formulae to evaluate the mean residence time in the entire system like the above situations. The formulae are given as functions of the fractional transfer coefficients between the compartments and, in the case of open systems, they also include the excretion coefficients to the environment from the different compartments. The relationship between the formulae derived and the particular connection properties of the compartments is discussed. Finally, some examples have been solved.     

Modified veranda‐trap hut for improved evaluation of vector control interventions

Experimental huts with veranda traps have been used in Tanzania since 1963 for the study of residual insecticides for use with insecticide‐treated nets and indoor residual spraying. Mosquitoes are allowed unrestricted entry through the eaves to facilitate the collection of an estimable proportion of mosquitoes that attempt to exit through the eave gaps, which are left open on two sides of the hut. This study was designed to validate the use of eave baffles to funnel entry and to prevent mosquito escape, and to determine biting times of Anopheles arabiensis (Patton) (Diptera: Culicidae). Anopheles arabiensis and Culex quinquefasciatus (Say) (Diptera: Culicidae) were released into the room at 20.30 hours and collected the following morning from veranda traps, window traps and the room. Centers for Disease Control light traps hung overnight next to volunteers were emptied every 2 h to determine peak biting times. A total of 55% of An. arabiensis were trapped before 22.30 hours and the highest peak in ‘biting’ was recorded during 18.30–20.30 hours. Of the released An. arabiensis that exited into veranda traps, 7% were captured in veranda traps entered through baffles and 93% were captured in traps entered through unmodified eaves. When veranda screens were left open to allow for escape outdoors, recapture rates were 68% for huts with eave baffles and 39% for huts with unmodified eaves. The comparison of open eaves with baffled eaves validated the assumption that in huts of the traditional non‐baffled design, 50% of mosquitoes escape through open eaves. Eave baffles succeeded in reducing the potential for mosquito exit and produced more precise estimates of effect.     

An efficient method for calculation of dynamic logarithmic gains in biochemical systems theory

Biochemical systems theory (BST) characterizes a given biochemical system based on the logarithmic gains, rate-constant sensitivities and kinetic-order sensitivities defined at a steady state. This paper describes an efficient method for calculation of the time courses of logarithmic gains, i.e. dynamic logarithmic gains L(Xi, Xj; t), which expresses the percentage change in the value of a dependent variable Xi at a time t in response to an infinitesimal percentage change in the value of an independent variable Xj at t=0. In this method, one first recasts the ordinary differential equations for the dependent variables into an exact canonical nonlinear representation (GMA system) through appropriate transformations of variables. Owing to the structured mathematical form of this representation, the recast system can be fully described by a set of numeric parameters, and the differential equations for the dynamic logarithmic gains can be set up automatically without resource to computer algebra. A simple general-purpose computer program can thus be written that requires only the relevant numeric parameters as input to calculate the time courses of the variables and of the dynamic logarithmic gains for both concentrations and fluxes. Unlike other methods, the proposed method does not require to derive any expression for the partial differentiation of flux expressions with respect to each independent variable. The proposed method has been applied to two kinds of reaction models to elucidate its usefulness.     

The computerized derivation of rate equations for enzyme reactions on the basis of the pseudo-steady-state assumption and the rapid-equilibrium assumption.

A computer program is developed for the derivation of the rate equation for enzyme reactions on the basis of the pseudo-steady-state assumption and the combination of the pseudo-steady-state and the rapid-equilibrium assumptions. The program not only has an easy input method, but also can obtain a complete rate equation in itself on only one run. The usefulness of the program is demonstrated by deriving the rate equations for some typical enzyme reactions. Details of the program have been deposited as Supplementary Publication SUP 50141 (42 pages) at the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7QB, U.K., from whom copies may be obtained as indicated in Biochem. J. (1988), 249, 5.     

Development of individual insulin infusion profiles for open loop infusion systems

A computer controlled syringe-type insulin infusion pump storing up to 254 different infusion rates, eight different meal programs and two different basal rates automatically changeable during 24 h in EPROM was used for insulin infusion applying a wavy step profile. This profile approaching the physiological postprandial insulin secretion of the B-cell was calculated by an algorithm following the biphasic insulin secretion model proposed by E. Cerasi . The computer program for the open loop infusion device simulated the feed-back structure of a closed loop insulin secretion control by an algorithm based upon a theoretical postprandial blood sugar profile. Fifteen unstable juvenile onset insulin requiring diabetics could be well controlled after two to three days of an intravenous open loop insulin infusion program. The programs consisted of two constant basal rates and superimposed wavy step profile programs activated at the beginning of each meal. The preabsorptive bolus or cephalic phase was an additional tool both for improved postprandial blood sugar control and further reduction of insulin consumption. The programmable insulin infusion device proved as a valuable tool for the study of a sophisticated insulin infusion profile suitable as well for open loop as for closed loop insulin infusion systems.     

High-performance liquid chromatography study of the pharmacokinetics of various spin traps for application to in vivo spin trapping.

In vivo spin trapping is potentially a very useful tool to investigate the role of free radicals in physiologic processes and disease development. Unfortunately, knowledge on the stability and distribution of spin traps in living systems is limited. Therefore, in our study, we selected 11 acyclic and cyclic nitrone spin traps with diverse properties to determine their pharmacokinetics in mice. At varying times after intraperitoneal administration, we measured the concentration of the spin traps in the liver, heart, and blood. Our results showed that most spin traps were rapidly absorbed and were approximately evenly distributed throughout the mouse body. It was also found that most of the traps were relatively stable in vivo with more than half of the injected amount still available for spin trapping free radicals after an hour. Two of the 11 tested spin traps, however, decomposed after injection. These results indicate that for a successful in vivo spin trapping experiment, the stability of the spin trap is not of major concern, but the time course of distribution may be important.     

Modeling auditory system nonlinearities through Volterra series

A computational method is reviewed, in which the solution of systems of nonlinear differential equations is written in terms of a Volterra functional series. Results of implementing the aforementioned technique in a computer program (exploiting new software for symbolic manipulation) and of applying it to a nonlinear oscillator model are presented. The relevance of this approach to Auditory System modeling is discussed. Suggestions are given, regarding possible applications to Speech Recognition problems.     

Pre-clinical validation of a closed surface system (Cryotop SC) for the vitrification of oocytes and embryos in the mouse model

Vitrification is currently a well-established technique for the cryopreservation of oocytes and embryos. It can be achieved either by direct (open systems) or indirect (closed systems) contact with liquid nitrogen. While there is not a direct evidence of disease transmission by transferred cryopreserved embryos, it was experimentally demonstrated that cross-contamination between liquid nitrogen and embryos may occur, and thus, the use of closed devices has been recommended to avoid the risk of contamination. Unfortunately, closed systems may result in lower cooling rates compared to open systems, due to the thermal insulation of the samples, which may cause ice crystal formation resulting in impaired results. In our study, we aimed to validate a newly developed vitrification device (Cryotop SC) that has been specifically designed for being used as a closed system. The cooling and warming rates calculated for the closed system were 5.254 °C/min and 43.522 °C/min, respectively. Results obtained with the closed system were equivalent to those with the classic Cryotop (open system), with survival rates in oocytes close to 100%. Similarly, the potential of the survived oocytes to develop up to good quality blastocysts after parthenogenetic activation between both groups was statistically equivalent. Assessment of the meiotic spindle and chromosome distribution by fluorescence microscopy in vitrified oocytes showed alike morphologies between the open and closed system. No differences were found either between the both systems in terms of survival rates of one-cell stage embryos or blastocysts, as well as, in the potential of the vitrified/warmed blastocysts to develop to full-term after transferred to surrogate females.     

On the stochastic theory of compartments: The leaving process of the two-compartment systems

In this paper three stochastic models are developed for a class of two-compartment systems to analyse the randomness of the leaving process of the particles in the system. Results in closed form for the distribution of the leaving process of the particles in the system are given both for general and exponential sojourn time distributions and also in association with forward recurrence time distributions with and without Poisson input.     

Inferring Mathematical Equations Using Crowdsourcing

Crowdsourcing, understood as outsourcing work to a large network of people in the form of an open call, has been utilized successfully many times, including a very interesting concept involving the implementation of computer games with the objective of solving a scientific problem by employing users to play a game—so-called crowdsourced serious games. Our main objective was to verify whether such an approach could be successfully applied to the discovery of mathematical equations that explain experimental data gathered during the observation of a given dynamic system. Moreover, we wanted to compare it with an approach based on artificial intelligence that uses symbolic regression to find such formulae automatically. To achieve this, we designed and implemented an Internet game in which players attempt to design a spaceship representing an equation that models the observed system. The game was designed while considering that it should be easy to use for people without strong mathematical backgrounds. Moreover, we tried to make use of the collective intelligence observed in crowdsourced systems by enabling many players to collaborate on a single solution. The idea was tested on several hundred players playing almost 10,000 games and conducting a user opinion survey. The results prove that the proposed solution has very high potential. The function generated during weeklong tests was almost as precise as the analytical solution of the model of the system and, up to a certain complexity level of the formulae, it explained data better than the solution generated automatically by Eureqa, the leading software application for the implementation of symbolic regression. Moreover, we observed benefits of using crowdsourcing; the chain of consecutive solutions that led to the best solution was obtained by the continuous collaboration of several players.     

A general pre-steady-state solution to complex kinetic mechanisms

We have developed a general method for solving transient kinetic equations using Laplace transforms. Laplace transforms can be used to transform systems of differential equations that describe pre-steady-state kinetics to systems of linear algebraic equations. The general form of the pre-steady-state solution is (formula; see text) where I(t) is the time dependence of the physically observed property of the system, n is the number of intermediates, lambda i are the observed rate constants (reciprocals of the relaxation times), t is time, and Ii are the amplitude coefficients associated with each observed rate constant. We have written a program in compiled BASIC to run on a personal computer to evaluate Ii and lambda i. The program will evaluate the rate constants and coefficients of a mechanism with eight intermediates and seven relaxation times in 4 s on an 8-MHz PC-XT equipped with a math coprocessor. The most complex mechanism that we have solved, a mechanism containing 20 intermediates and 19 relaxation times, required approximately 5 min. We believe that this method will be useful to evaluate the differences in transient properties of complex biochemical mechanisms.     

Kinetic analysis of channel gating. Application to the cholinergic receptor channel and the chloride channel from Torpedo californica.

Identification of the minimum number of ways in which open and closed states communicate is a crucial step in defining the gating kinetics of multistate channels. We used certain correlation functions to extract information about the pathways connecting the open and closed states of the cation channel of the purified nicotinic acetylcholine receptor and of the chloride channel of Torpedo californica electroplax membranes. Single channel currents were recorded from planar lipid bilayers containing the membrane channel proteins under investigation. The correlation functions are conveniently computed from single channel current records and yield information on E, the minimum number of entry/exit states into the open or closed aggregates. E gives a lower limit on the numbers of transition pathways between open and closed states. For the acetylcholine receptor, the autocorrelation analysis shows that there are at least two entry/exit states through which the open and closed aggregates communicate. The chloride channel fluctuates between three conductance substates, here indentified as C, M, and H for closed, intermediate, and high conductance, respectively. Correlation analysis shows that E is greater than or equal to 2 for the M aggregate, indicating that there are at least two distinct entry/exit states in the M aggregate. In contrast, there is no evidence for the existence of more than one entry/exit state in the C or H aggregates. Thus, these correlation functions provide a simple and general strategy to extract information on channel gating kinetics.     

Equations for nonlinear MHD convection in shearless magnetic systems

A closed set of reduced dynamic equations is derived that describe nonlinear low-frequency flute MHD convection and resulting nondiffusive transport processes in weakly dissipative plasmas with closed or open magnetic field lines. The equations obtained make it possible to self-consistently simulate transport processes and the establishment of the self-consistent plasma temperature and density profiles for a large class of axisymmetric nonparaxial shearless magnetic devices: levitated dipole configurations, mirror systems, compact tori, etc. Reduced equations that are suitable for modeling the long-term evolution of the plasma on time scales comparable to the plasma lifetime are derived by the method of the adiabatic separation of fast and slow motions.     

Some Formal Approaches to the Analysis of Kinetic Data in Terms of Linear Compartmental Systems

A formal approach to the routine analysis of kinetic data in terms of linear compartmental systems is presented. The methods of analysis are general in that they include much of the theory in common use, such as direct solution of differential equations, integral equations, transfer functions, fitting of data to sums of exponentials, matrix solutions, etc. The key to the formalism presented lies in the fact that a basic operational unit—called “compartment”—has been defined, in terms of which physical and mathematical models as well as input and output functions can be expressed. Additional features for calculating linear combinations of functions and for setting linear dependence relations between parameters add to the versatility of this method. The actual computations for the values of model parameters to yield a least squares fit of the data are performed on a digital computer. A general computer program was developed that permits the routine fitting of data and the evolution of models.     

Extension of the theory of the multiple-indicator dilution technique to metabolic systems with an arbitrary number of rate constants

The multiple indicator dilution technique consists in the instantaneous injection of a mixture of tracers into the arterial perfusate flow of a catheterized or isolated perfused organ, followed by the analysis of the effluent perfusate. The theory of this technique, which has hitherto been developed for cases where metabolism of a tracer is confined to sequestration described by a single rate constant, is extended in this paper to include an arbitrary number of metabolic rate constants. Partial differential equations with constant coefficients describing the events in a single capillary are derived by applying conventional compartmental analysis to infinitesimally small sections of the capillary. Methods for solving such systems in the time as well as in the frequency domain are developed. From the solutions, the impulse response of the whole organ is evaluated assuming variable capillary and uniform large-vessel transit times. In addition, an efficient method using much less computer time was developed, based on the approximation of the distribution of the capillary transit times by a sum of exponentials. Evaluation of moments (recoveries and mean transit times) is also treated. The results are applied to an example from hepatic lactate metabolism.     

The optical cross section and absolute size of a photosynthetic unit

The concepts of a photosynthetic unit (PSU) and of an optical cross section are defined. The various estimates of sizes of photosynthetic units are described, and it is shown how an unambiguous measurement of the size of a unit can be obtained by measurement of its optical cross section via the saturation response to a single turnover light flash. The Emerson-Arnold unit must be divided by the quantum requirement for oxygen to obtain the true size of the unit. The size so obtained is the average number of chlorophylls per trap or reaction center. The effects of escape from open and closed traps are considered and it is shown that when these escape probabilities are equal, their effect on the saturation curve vanishes, leaving the simple cumulative one hit Poisson distribution.     

Chloride channels in toad skin

A study of the voltage and time dependence of a transepithelial Cl- current in toad skin (Bufo bufo) by the voltage-clamp method leads to the conclusion that potential has a dual role for Cl- transport. One is to control the permeability of an apical membrane Cl-pathway, the other is to drive Cl- ions through this pathway. Experimental analysis of the gating kinetics is rendered difficult owing to a contamination of the gated currents by cellular ion redistribution currents. To obtain insight into the effects of accumulation-depletion currents on voltage clamp currents of epithelial membranes, a mathematical model of the epithelium has been developed for computer analysis. By assuming that the apical membrane Cl- permeability is governed by a single gating variable (Hodgkin-Huxley kinetics), the model predicts fairly well steady-state current-voltage curves, the time course of current activations from a closed state, and the dependence of unidirectional fluxes on potential. Other predictions of the model do not agree with experimental findings, and it is suggested that the gating kinetics are governed by rate coefficients that also depend on the holding potential. Evidence is presented that Cl- transport through open channels does not obey the constant-field equation.     

Analytical and Simulation Results for Stochastic Fitzhugh-Nagumo Neurons and Neural Networks

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represented by systems of nonlinear ordinary stochastic differential equations—the Fitzhugh-Nagumo system with Gaussian white noise current. For a single neuron, five equations hold for the first- and second-order central moments of the voltage and recovery variables. From this system we obtain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use these quantities to estimate the probability that a neuron is emitting an action potential at any given time. The differential equations are solved by numerical methods. We also perform simulations on the stochastic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent deterministic current inputs withsuperimposed noise. For intermittent currents, which mimic synaptic input, the agreement between the analytical and simulation results for the moments is excellent. For sustained input, the analytical approximations perform well for small noise as there is excellent agreement for the moments. In addition, the probability that a neuron is spiking as obtained from the empirical distribution of the potential in the simulations gives a result almost identical to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accurate for short time intervals. Using the simulation method, we study the distribution of the interspike interval directly from simulated sample paths. We confirm that noise extends the range of input currents over which (nonperiodic) spike trains may exist and investigate the dependence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations for the means, variances, and covariances of the voltage and recovery variables and show how solving them leads to an expression for the probability that a given neuron, or given set of neurons, is firing at time t. Using such expressions one may implement dynamical rules for changing synaptic strengths directly without sampling. The present analytical method applies equally well to temporally nonhomogeneous input currents and is expected to be useful for computational studies of information processing in various nervous system centers.     

MAFFT: a novel method for rapid multiple sequence alignment based on fast Fourier transform

A multiple sequence alignment program, MAFFT, has been developed. The CPU time is drastically reduced as compared with existing methods. MAFFT includes two novel techniques. (i) Homo logous regions are rapidly identified by the fast Fourier transform (FFT), in which an amino acid sequence is converted to a sequence composed of volume and polarity values of each amino acid residue. (ii) We propose a simplified scoring system that performs well for reducing CPU time and increasing the accuracy of alignments even for sequences having large insertions or extensions as well as distantly related sequences of similar length. Two different heuristics, the progressive method (FFT-NS-2) and the iterative refinement method (FFT-NS-i), are implemented in MAFFT. The performances of FFT-NS-2 and FFT-NS-i were compared with other methods by computer simulations and benchmark tests; the CPU time of FFT-NS-2 is drastically reduced as compared with CLUSTALW with comparable accuracy. FFT-NS-i is over 100 times faster than T-COFFEE, when the number of input sequences exceeds 60, without sacrificing the accuracy.     

A program for simulation of nerve equations with branching geometries

A computer program has been developed for simulation of electrical activity in neurons with complex branching morphology, multiple channel types, and inhomogeneous channel distribution. The program is based around an interpreter and screen editor for flexible specification of nerve properties and analysis of simulation results. Efficient simulation of the nerve specification is accomplished with procedure calls to fast, compiled routines for integration of the nerve equations.