Analysis of tracer experiments: VIII. Integro-differential equation treatment of partly accessible,partly injectable multicompartment systems

An integro-differential equation treatment of multi-compartment systems is developed which permits formal analysis of the incomplete data which is available from partly accessible, partly injectable systems. New transport functions are defined which can be obtained directly from the experimental data. These functions serve to characterize the communication and topology between different accessible compartments and also the reentrant contributions from inaccessible sites. The method gives solutions consistent with those of the differential equation approach when the system is uniformly contiguous and accessible, more complete solutions than those of the integral equation approach when all measured compartments are injectable, and in addition provides complete or partial solutions for certain otherwise analytically intractable systems. Detailed numerical illustrations of the method are given.     

Mathematical models for mixing in deep jet bioreactors: analysis

A mathematical model for single and multi step deep-jet bioreactors is presented. A stagewise approach based on macroscopic mechanistic model which divides the reactor into compartments with good quality of mixing and plug flow regions (macromixer), was used. For the mathematical representation of this model a system of differential equations, describing the concentration of tracer in structural elements based on mass balance, and the Runge-Kutta-Fehlberg numerical method of integration, was applied. The mixing time in a 300 dm³ tank was determined by conductivity method with NaCl as tracer.List of Symbols V _g dm³ total volume of liquid - V ₁; V ₆ dm³ volumes of ideally mixed compartments in the vessel - V ₂; V ₇ dm³ volumes of macromixer in the inner circulation flows - V ₃; V ₉ dm³ volumes of liquid phase in the pump - V ₄; V ₈ dm³ volumes of liquid phase in the pipe between the vessel and the pump - V ₅; V ₁₀ dm³ volumes of liquid phase in pipes between the pump and the air input system, including falling jet - F _E; F _E,1; F _E,2 dm³/s the inner volumetric circulation flow rates accross the macromixers - F _E,3; F _E,4 dm³/s exchanges volumetric flow rates between two ideally mixed compartments in the vessel - F _cir; F _1,cir; F _2,cir dm³/s external volumetric circulation flow rates (pumping capacity) - t _A s time interval of puls application - t _AA s time point of impuls application related to the free chosen point of simulation - t _end s end time of simulation - F _qu g²/dm⁶ sum of quadratic error - C _*,* kg/m³ concentration of the tracer in the indicated compartment - C ₀ kg/m³ concentration of the tracer before the injection - C _t kg/m³ concentration of the tracer at the indicated time - C kg/m³ theoretical concentration of full mixed tracer - i index of an arbitrary tank - C _sim kg/m³ calculated concentration of the tracer by numerical integration method     

Analysis of tracer experiments: VII. General multi-compartment systems imbedded in non-homogeneous inaccessible media

An integral equation analysis of generaln compartment steady state systems imbedded in static media of arbitrary complexity has been developed. A set of initial entry functions can be found which serve to determine a corresponding set of partitioned initial entry functions. The partitioned functions, in turn, can be used to predict the probabilities and time courses of various transport histories and to determine all steady state rates of flow between measured compartments. The method is quite general, being completely applicable, for example, to closed systems, to cyclic systems and to systems in which relatively rapid (but finite) exchange between compartments occurs.     

A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of <Emphasis Type="Italic">E. coli</Emphasis> MinE to E-ring formation

Many important cellular processes are regulated by reaction-diffusion (RD) of molecules that takes place both in the cytoplasm and on the membrane. To model and analyze such multicompartmental processes, we developed a lattice-based Monte Carlo method, Spatiocyte that supports RD in volume and surface compartments at single molecule resolution. Stochasticity in RD and the excluded volume effect brought by intracellular molecular crowding, both of which can significantly affect RD and thus, cellular processes, are also supported. We verified the method by comparing simulation results of diffusion, irreversible and reversible reactions with the predicted analytical and best available numerical solutions. Moreover, to directly compare the localization patterns of molecules in fluorescence microscopy images with simulation, we devised a visualization method that mimics the microphotography process by showing the trajectory of simulated molecules averaged according to the camera exposure time. In the rod-shaped bacterium Escherichia coli, the division site is suppressed at the cell poles by periodic pole-to-pole oscillations of the Min proteins (MinC, MinD and MinE) arising from carefully orchestrated RD in both cytoplasm and membrane compartments. Using Spatiocyte we could model and reproduce the in vivo MinDE localization dynamics by accounting for the previously reported properties of MinE. Our results suggest that the MinE ring, which is essential in preventing polar septation, is largely composed of MinE that is transiently attached to the membrane independently after recruited by MinD. Overall, Spatiocyte allows simulation and visualization of complex spatial and reaction-diffusion mediated cellular processes in volumes and surfaces. As we showed, it can potentially provide mechanistic insights otherwise difficult to obtain experimentally.     

Placental transport and utilization of amino acids and carbohydrates

Uteroplacental tissues have been shown to have a high rate of metabolism under in vivo steady-state conditions. Fully two-thirds of the glucose and one-half of the oxygen consumed by the uterus are utilized by these tissues rather than by the fetus. Its high metabolic rate must be borne in mind in any analysis of tracer kinetics, which prohibits the exclusion of these tissues and the use of a two-compartment model for analysis of carbohydrate and amino acid metabolism. Current techniques permit determination of utilization rates of nutrients in all three compartments (fetal, uteroplacental, and maternal) with considerable precision. Using tracer amino acids one can determine rates of protein synthesis and protein breakdown as well as rates of amino acid oxidation. These techniques should prove useful in investigating the role of various trophic factors in fetal life and in assessing the impact of changes in placental function or maternal nutritional state on fetal growth and metabolism.     

Ketone body kinetics in humans: a mathematical model

A model has been developed to account for ketone body kinetics in man based on data following bolus injections of [14C]acetoacetate (A) and [14C]beta-OH butyrate (B) into normal humans in the postabsorptive state. The model consists of separate compartments for blood A and B that are linked by a tissue compartment in which rapid interconversion of the ketone bodies occurs. The probability of movement from blood into this compartment was assumed to be the same for both ketone bodies. Two slowly equilibrating tissue compartments are required to account for the slow components in the tracer data, and thus a five-compartment model is proposed. By modeling the transient tracer data with the tracee in a steady state, ketone body kinetics were defined in terms of the rapid interconversions of A and B, and the slow exchanges of carbon within the tissues. The rates of release of new A and B into blood, (UA and UB) were calculated. These rates were less than the apparent production rates, PRA and PRB, as the PR's included carbon atoms first released as the other ketone body. The exchange constants between the compartments were determined in addition to the fractional catabolic rates (FCR) and metabolic clearance rates (MCR) of A and B. The initial space of distribution was 10 L and the mean values +/- SD (n = 11), normalized to this volume, were UA = 6.4 +/- 5.0, UB = 8.8 +/- 8.0 (mumol L-1 min-1), FCRA = 0.226 +/- 0.142, FCRB = 0.188 +/- 0.124 (min-1), MCRA = 2.26 +/- 1.42, MCRB = 1.87 +/- 1.23 (L min-1) and PRA = 11.1 +/- 7.6, PRB = 12.7 +/- 10.0 (mumol L-1 min-1).     

Solutions for the kinetic analysis of 45calcium uptake curves

The classic solutions based on specific activity curves for the kinetic analysis of ⁴⁵Ca movements in three compartment cellular systems cannot be used when the extracellular compartment is one to two orders of magnitude larger than the cellular or tissue compartments. However if the relative radioactivity curve (tracer uptake curve) is analyzed it is possible to calculate all the relevant kinetic parameters. This paper offers the solutions based on relative radioactivity measurements for the calculation of exchange rates, rate constants and compartment sizes of three compartment systems, for series and parallel cases, for closed and open systems.     

Matrix proof of flow,volume and mean transit time theorems for regional and compartmental systems

The relations (inflow) = (dose)/(area under indicator curve), and (volume of distribution) = (throughflow) × (mean transit time) are derived by a matrix method for a system of interconnected subsystems, within which spatial indicator activity gradients may exist, and for compartments, within which the indicator activity is spatially uniform. The inflow theorem, is different from the outflow theorem. Equivalent labeling of multi-input systems reduces them formally to single input systems. Foreign indicator flow-volume kinetics are more general than, and include as a special case, tracer flux-mass (metabolic) kinetics. Volume of distribution in the indicator steady state may be different from the equilibrium volume of distribution.     

The influence of noninstantaneous mixing on the measurement of fluxes of solute and water across a biological membrane

The usual method of tracer analysis for calculating the flow across a biological membrance is based on the assumption that the compartments on either side are well-stirred. Thus, the validity of the rate of flow determination is questionable for cases where the distribution of tracer is not homogeneous. In this study, a mathematical model is developed for the purpose of estimating the effect of slow mixing on the calculation of the flow rate. The model is applied to the measurement of the rate of flow of aqueous humor through the living eye by use of a fluorescent dye as a tracer. A transit time of several minutes for the passage of fluorescein through the posterior chamber and an extended period of nonuniform distribution of fluorescein in the anterior chamber was observed. The effect of slow mixing on the calculated flow rate is compared to rates derived from equations based on the assumption of rapid mixing. Aqueous flow rates determined by the two methods were found to agree to within ≈20%.     

Nitrate reduction, nitrous oxide formation, and anaerobic ammonia oxidation to nitrite in the gut of soil-feeding termites (Cubitermes and Ophiotermes spp.)

Soil-feeding termites play important roles in the dynamics of carbon and nitrogen in tropical soils. Through the mineralization of nitrogenous humus components, their intestinal tracts accumulate enormous amounts of ammonia, and nitrate and nitrite concentrations are several orders of magnitude above those in the ingested soil. Here, we studied the metabolism of nitrate in the different gut compartments of two Cubitermes and one Ophiotermes species using (15)N isotope tracer analysis. Living termites emitted N(2) at rates ranging from 3.8 to 6.8 nmol h(-1) (g fresh wt.)(-1). However, in homogenates of individual gut sections, denitrification was restricted to the posterior hindgut, whereas nitrate ammonification occurred in all gut compartments and was the prevailing process in the anterior gut. Potential rates of nitrate ammonification for the entire intestinal tract were tenfold higher than those of denitrification, implying that ammonification is the major sink for ingested nitrate in the intestinal tract of soil-feeding termites. Because nitrate is efficiently reduced already in the anterior gut, reductive processes in the posterior gut compartments must be fuelled by an endogenous source of oxidized nitrogen species. Quite unexpectedly, we observed an anaerobic oxidation of (15)N-labelled ammonia to nitrite, especially in the P4 section, which is presumably driven by ferric iron; nitrification and anammox activities were not detected. Two of the termite species also emitted substantial amounts of N(2) O, ranging from 0.4 to 3.9 nmol h(-1) (g fresh wt.)(-1), providing direct evidence that soil-feeding termites are a hitherto unrecognized source of this greenhouse gas in tropical soils.     

Measurement of the rate of aqueous humor flow

Techniques which estimate the rate of aqueous flow generally require the use of tracer substances. Determination of the distribution of the tracer in the relevant body compartments permits calculation of the rate of flow within the limits of accuracy of the method used. The underlying theory, as well as the advantages and limitations of methods employing systemic, topical, intracameral, and intravitreal administration of tracer substances are reviewed. Since these methods all assume that the rate of aqueous secretion is constant, yet the presence of a diurnal rhythm of flow has been demonstrated in both rabbits and humans, a compartmental model of a circadian system based upon the vitreous depot technique is presented. This model estimates the degree to which a continuously changing rate of aqueous flow limits the ability to determine aqueous flow rate accurately by this particular method and illustrates this limitation, which is common to all tracer methods.     

Spinal and cranial contributions to total cerebrospinal fluid transport

In this study, we quantified cerebrospinal fluid (CSF) transport from the cranial and spinal subarachnoid spaces separately in sheep and determined the relative proportion of total CSF drainage that occurred from both CSF compartments. Cranial and spinal CSF systems were separated by placement of an extradural ligature over the spinal cord between C(1) and C(2). In one approach, two different radiolabeled human serum albumins (HSA) were introduced into the appropriate CSF compartment by a perfusion system (method 1) or as a bolus injection (method 2). Plasma tracer recoveries in conjunction with a mass balance equation were used to estimate CSF transport. In method 3, catheters connected to reservoirs filled with artificial CSF were introduced into the cranial and spinal CSF compartments. Incremental CSF pressures were established in each CSF system, and the corresponding steady-state flow rates were measured. Total CSF drainage ranged from 0.51 to 0.75 ml. h(-1). cmH(2)O(-1). Expressed as a percentage of the total CSF transport, the ratios of cranial-to-spinal clearance estimated from methods 1, 2, and 3 were 75:25, 88:12, and 75:25, respectively. Primarily on the basis of the data derived from methods 1 and 3, we conclude that the spinal subarachnoid compartment has an important role in CSF clearance and is responsible for approximately one-fourth of total CSF transport.     

Small G proteins of two green algae are localized to exocytic compartments and to flagella

The Ypt/Rab proteins are small GTPases, which belong to the Ras superfamily and have been shown to be involved in endo-and exocytosis in mammalian cells and yeast. Using affinity-purified antibodies specific for four Ypt proteins, namely Ypt1p, Ypt4p, Ypt5p and Ypt6p, of the multicellular green alga Volvox carteri (YptVp) and its close unicellular relative Chlamydomonas reinhardtii (YptCp), we examined the abundance of the corresponding antigens during the asexual life cycle of Volvox, and their intracellular localization. The YptV proteins were found in all stages throughout the asexual life cycle and are tightly associated with intracellular membranes. Indirect immunofluorescence revealed that YptV4p, YptV5p and YptV6p are present in perinuclear regions of the cell, indicating an association with the Golgi region. Golgi localization of YptV4p and YptV6p in Volvox was confirmed by immunogold electron microscopy. In contrast, we found Ypt1p associated with the contractile vacuole in both V. carteri and C. reinhardtii. Furthermore, the YptV proteins were also detected along the entire length of the flagella of somatic Volvox cells. This flagellar location was substantiated by western blot analysis of extracts prepared from isolated flagella of both algae. While localization to exocytic compartments is in agreement with the established Ypt/Rab function in intracellular vesicle transport of eukaryotic cells, presence in the algal flagellum is the first hint of a possible role for small G proteins also in motility organelles.     

A comparative study of viral capsids and bacterial compartments reveals an enriched understanding of shell dynamics

In this work, we carry out a comparative study of the homo 360‐mer structures of viral capsids and bacterial compartments. Different from viral 360‐mers that all are arranged on a skewed right‐handed icosahedral lattice with a triangulation number T of 7, the new 360‐mer structure of AaLS‐13, an engineered bacterial compartment, offers a novel open conformation that has a unique unskewed lattice arrangement with a triangulation number T of 1 and large keyhole‐shaped pores in the shell. By comparing their differences, we are able to predict a closed conformation of AaLS‐13 that has the same lattice arrangement as existing viral capsid structures and in which all the keyhole‐shaped pores are shut. We find that there is a smooth transition pathway between the open and closed conformations. There exists another close conformation but with an opposite, left handedness, which, however, is not kinetically accessible from the open conformation. Our finding thus provides a clue why existing 360‐mer capsid structures all share the same right handedness. We further show that the conformation transition between the open and closed forms aligns extremely well with the intrinsic dynamics of the system as revealed from normal mode analysis, indicating that conformation transition can be fully driven by thermal fluctuations. The significance of this work is that it provides a better understanding of shell dynamics of both viral capsids and bacterial compartments, paving a way for future study of pore dynamics and the selective permeability of these systems.     

On transfer times in tracer experiments

A compartment is defined as a pool of material whose behavior can be described by a deterministic or by a stochastic equation; these two equations are used to define the transit time through the compartment, the total residence time, the time of entrance and the time of exit.If in a complex system one or more compartments are accessible, the transport of material through it can be studied using a tracer. Then the transfer time between any two compartments, or through the cycle around a compartment, can be analyzed under certain hypotheses, even if the transport along the route considered cannot be described by compartment equations.     

Stimulus-response method for flows and volumes in slightly perturbed constant parameter systems

The area and the first time moment of tracer kinetic data on steady state systems contains information on flows and volumes of distribution in the system. A simple and general method of obtaining such information is given. The method consists of equating the ratio of the time integrals of appropriate stimulus function and response function in an actually performed finite amount tracer experiment to the ratio of stimulus and response that would exist in the tracer steady state. The generality of the method is illustrated by deriving the volume of osmotically exchangeable water in an organ from measurement of its transient bulk water exchange during a small osmotic perturbation.     

ARF6 Targets Recycling Vesicles to the Plasma Membrane: Insights from an Ultrastructural Investigation

We have shown previously that the ADP- ribosylation factor (ARF)-6 GTPase localizes to the plasma membrane and intracellular endosomal compartments. Expression of ARF6 mutants perturbs endosomal trafficking and the morphology of the peripheral membrane system. However, another study on the distribution of ARF6 in subcellular fractions of Chinese hamster ovary (CHO) cells suggested that ARF6 did not localize to endosomes labeled after 10 min of horseradish peroxidase (HRP) uptake, but instead was uniquely localized to the plasma membrane, and that its reported endosomal localization may have been a result of overexpression. Here we demonstrate that at the lowest detectable levels of protein expression by cryoimmunogold electron microscopy, ARF6 localized predominantly to an intracellular compartment at the pericentriolar region of the cell. The ARF6-labeled vesicles were partially accessible to HRP only on prolonged exposure to the endocytic tracer but did not localize to early endocytic structures that labeled with HRP shortly after uptake. Furthermore, we have shown that the ARF6-containing intracellular compartment partially colocalized with transferrin receptors and cellubrevin and morphologically resembled the recycling endocytic compartment previously described in CHO cells. HRP labeling in cells expressing ARF6(Q67L), a GTP-bound mutant of ARF6, was restricted to small peripheral vesicles, whereas the mutant protein was enriched on plasma membrane invaginations. On the other hand, expression of ARF6(T27N), a mutant of ARF6 defective in GDP binding, resulted in an accumulation of perinuclear ARF6-positive vesicles that partially colocalized with HRP on prolonged exposure to the tracer. Taken together, our findings suggest that ARF activation is required for the targeted delivery of ARF6-positive, recycling endosomal vesicles to the plasma membrane.     

Parafusin and calcium-induced signal transduction in exocytosis

Summary The phosphoglycoprotein parafusin is a member of the phosphoglucomutase superfamily and has been shown, both via biochemical and localization studies, to be associated with the Ca²⁺-dependent regulated exocytosis process inParamecium tetraurelia. Stimulation of exocytosis in this cell leads to a Ca²⁺-dependent glucosylation of parafusin accompanied by its dissociation from the secretory vesicles and from cell membrane docking sites. These events are blocked in the presence of extracellular Mg²⁺ in wild-type cells and in either Ca²⁺ or Mg²⁺ in a temperature-sensitive mutant, nd9, stimulated at the nonpermissive temperature. Furthermore, laser scanning confocal localization studies with antibodies to parafusin whole protein versus antibody made to a specific peptide (insertion 2) show different localization patterns. While insertion-2 antibodies only label the organelles previously shown to have parafusin associated with them, i.e., cell membrane fusion (docking) sites and secretory vesicles, antibodies to whole protein outline in addition the alveolar sacs (subsurface cisterns) which are Ca²⁺ storage compartments in this cell. This may indicate tht other members of the phosphoglucomutase superfamily which interact specifically with this compartment are present inP. tetraurelia.     

The nonexistence of “hermaphoroditic” tracer systems

This communiction argues that so-called “hermaphroditic” tracer systems, which are neither open nor closed, do not exist physically. The argument is based on the assumption that any observable (possibly nonhomogeneous) macroscopic compartment can be approximated by a compartmentC with a finite number of entry points for the tracer, each associated with an abstract subcompartment ofC. It is shown that the “hermaphroditic” property requires that the mean waiting time be infinite in at least one of the subcompartments, or in a subcompartment elsewhere in the system. A subcompartment with infinite mean waiting time must have some sort of memory, of infinite duration, which knows how long a given particle has been retained, however long that is, and thereby determines its probability of departure. Assuming, as seems likely, that no physical basis exists for such an infinite memory, it follows that “hermaphroditic” systems do not exist. Supported in part by U.S. Public Health Service Research Grant GM 21269 from the National Institute of General Medical Sciences, and in part by Biomedical Research Support Grant S07 RR 05392 from the National Institutes of Health.