General continuum analysis of transport through pores. II. Nonuniform pores.

A general continuum derivation of the nonelectrolyte (Js) and volume (Jv) flux through a pore whose cross section is a function of axial position (nonuniform) is given. In general, the flux equations cannot be reduced to the same form as for a uniform pore and it is not possible to characterize the pore kinetics by three constants as in the uniform pore case. However, it is shown that under certain conditions, the nonuniform pore equations can be approximated by the uniform pore form and can be characterized by three constants (omega, sigma, Lp). The only condition needed to reduce the Jv equation to the uniform form is that the solution be dilute. The deviation of the Js equation from the uniform form is characterized by an asymmetrical function of Jv whose maximum value is estimated. It is shown that the maximum posible fractional deviation of the Js equation from the uniform form is given by the parameter: 0:5sigmaJv/omegaRT. Since this parameter is less then 0.15 for most membrane studies, the nonuniform Js equation can usually be approximated by the uniform pore form. The general results are illustrated by explicit calculations on several models of nonuniform pores. It is shown, for example, that the "equivalent pore radius" defined in the usual way is a function of the experimental parameter that is measured and is not unique.     

Diagnosis of enzyme inhibition based on the degree of inhibition

In this work, a method for the diagnosis of kinetic inhibition, based on the dependence of the degree of inhibition (epsilon(i)) on the inhibitor concentration [I] and on the substrate concentration [S], is presented. Because the degree of inhibition is a ratio between rates, kinetic data are normalized by the introduction of an internal control-the rate of the uninhibited reaction. Therefore, the error associated with the kinetic measurements decreases and less experimental measurements are necessary to achieve the diagnosis. The process described, which uses graphical and/or non-linear fitting procedures, allows distinguishing between 20 different kinds of inhibition, including not only linear and hyperbolic, but also parabolic and rational 2,2 inhibitions. Rational 2,2 indicates a new type of inhibition corresponding to an incomplete parabolic inhibition, i.e. mechanistically it corresponds to an inhibitor that binds to two inhibition sites producing enzymatic complexes that are still active. In spite of its comprehensiveness, the diagnosis process is greatly facilitated by the division of the diagnosis of the inhibition in a step-by-step procedure, where only two rival models are evaluated in each step. In the non-linear fittings, the choice between rival models uses a test based on information statistics theory, the Akaike information criterion test, in order to penalize complex models that tend to be favoured in fittings. Finally, equations that allow the determination of inhibition kinetic constants were also deduced. The formalism presented was tested by examining inhibition of acid phosphatase by phosphate (a linear competitive inhibitor).     

The kinetics of hemolytic plaque formation. III. Inhibition of plaques by antigen.

Equations are presented which can be used to describe the inhibition of plaques by multifunctional antigen which binds γG antibody bivalently. The interaction is treated as a bimolecular reaction which is irreversible within the time of the experiment. It is shown that under these conditions the characteristics of the inhibition curve, and their relationship to kinetic and thermodynamic parameters are strongly dependent upon how antibody interacts with RBCs. When the epitope coating is dense, multivalent attachment of antibody is likely and the interaction is considered irreversible. When the epitope coating is sparse, only rapid, reversible univalent attachment is considered and local equilibrium is assumed to hold.The first case leads to an abrupt inhibition curve whose position is determined by the forward rate constant and RBC density. The second case leads to broad asymmetric curves. For this situation the relation between the extent of inhibition and the affinity of the population is generally complicated and reliable affinity information is difficult to obtain. This is contrasted to results obtained previously for unifunctional inhibitors from which reliable affinity information can, in principle, be obtained. The results emphasize the need for carefully designed experiments if affinity information is to be obtained from inhibition studies.     

The kinetics of hemolytic plaque formation. IV. IgM plaque inhibition.

An analysis of the inhibition of hemolytic plaques formed against IgM antibodies is presented. The starting point is the equations of DeLisi &; Bell (1974) which describe the kinetics of plaque growth, and DeLisi &; Goldstein (1975) which describe inhibition of IgG plaques. However, the physical chemical models which were used previously to describe IgG inhibition data are shown to be inadequate for describing the characteristics of IgM inhibition curves. Moreover, it is shown that the experimental results place severe restrictions on the possible choices of physical chemical models for IgM upon which to base the calculations. It is argued that in order to account even qualitatively for all the data, one must assume (1) a very restricted motion of IgMs about the Fab hinge region and (2) a very narrow secretion rate distribution of IgM by antibody secreting cells.     

Spatial patterns in coupled biochemical oscillators

Summary The effects of diffusion on the dynamics of biochemical oscillators are investigated for general kinetic mechanisms and for a simplified model of glycolysis. When diffusion is sufficiently rapid a population of oscillators relaxes to a globally-synchronized oscillation, but when diffusion of one or more species is slow enough, the synchronized oscillation can be unstable and a nonuniform steady state or an asynchronous oscillation can arise. The significance of these results vis-a-vis models of contact inhibition and zonation patterns is discussed.     

Role of iron in T cell activation: TH1 clones differ from TH2 clones in their sensitivity to inhibition of DNA synthesis caused by IgG Mabs against the transferrin receptor and the iron chelator deferoxamine.

TH1 and TH2 helper T cell clones have been studied with respect to their sensitivity to inhibition of DNA synthesis by an IgG anti-transferrin receptor antibody (ATRA), the iron chelator deferoxamine, and the combination of the two reagents. TH1 clones are very sensitive to ATRA-mediated inhibition of DNA synthesis while TH2 clones are very resistant, but both TH1 and TH2 clones show significant down-modulation of surface transferrin receptors after ATRA exposure. TH2 clones exhibit larger chelatable iron storage pools than TH1 clones, however, and even partial chelation of TH2 cell storage iron does not fully convert a TH2 clone to the ATRA sensitivity pattern of a TH1 clone. It is therefore proposed that the greater resistance of TH2 clones to ATRA mediated inhibition derives from the combined effects of larger and less labile iron storage pools. These studies provide novel evidence indicating that nonuniform iron metabolism can exist within the T cell compartment and thus raise questions as to why such differences exist and how they can be integrated into models of the T cell activation process. These studies also suggest that the cell-mediated immune response in vivo, which is known to be sensitive to iron deficiency, may be evoked by effector cells which resemble TH1 clones insofar as iron metabolism is concerned.     

A mathematical model of tumor growth. III. comparison with experiment

In this paper a model of nonuniform inhibitor production is used to discuss some experimental results obtained by Folkman and Hochberg for multicellular spheroids (of V-79 Chinese hamster lung tissue). Built into the mathematical model is a parameter b which is a measure of the degree of nonuniformity of the inhibitor production rate. The “inverse problem” is solved by finding the value of b necessary to account for results with which the uniform model is incompatible [6]. The resulting value of b enables a good estimate to be made for the observed width of peripheral mitotic zones in the V-79 spheroids, and the resulting stability parameters lie in appropriate ranges for the model to be a significant improvement over uniform models. Further improvements are discussed, including a heuristic model for estimating the destabilizing effect of vascularization on tissue growth.     

Psychoneural models of the auditory masking process.

Two mathematical models are presented which describe perception of the loudness of a tone masked by noise. The designation “psychoneural” indicates that the models are consistent with accepted findings in the fields of neuroanatomy, neurophysiology, and psychophysics. Each of the models consists of two channels, one which responds only to noise and another which responds to a weighted sum of tone and noise. Each channel is functionally equivalent to the model proposed by MacKay to explain intensity perception. In the first of the two masking models, it is assumed that the noise decreases the perceived loudness of the tone by lateral inhibition at a central location. In the second model, it is assumed that the noise decreases the loudness of the tone by efferent inhibition which acts at the periphery. The quantitative predictions of both models are identical and, with the appropriate adjustment of certain parameters, can be brought into close agreement with psychophysical masking data. The relative merits and handicaps of the models are discussed in the context of a more complex representation of the masking process in which a separate channel is assigned to each critical frequency band.     

Development of a computational technique to measure cartilage contact area

Computational measurement of joint contact distributions offers the benefit of non-invasive measurements of joint contact without the use of interpositional sensors or casting materials. This paper describes a technique for indirectly measuring joint contact based on overlapping of articular cartilage computer models derived from CT images and positioned using in vitro motion capture data. The accuracy of this technique when using the physiological nonuniform cartilage thickness distribution, or simplified uniform cartilage thickness distributions, is quantified through comparison with direct measurements of contact area made using a casting technique. The efficacy of using indirect contact measurement techniques for measuring the changes in contact area resulting from hemiarthroplasty at the elbow is also quantified. Using the physiological nonuniform cartilage thickness distribution reliably measured contact area (ICC=0.727), but not better than the assumed bone specific uniform cartilage thicknesses (ICC=0.673). When a contact pattern agreement score (s_agree) was used to assess the accuracy of cartilage contact measurements made using physiological nonuniform or simplified uniform cartilage thickness distributions in terms of size, shape and location, their accuracies were not significantly different (p>0.05). The results of this study demonstrate that cartilage contact can be measured indirectly based on the overlapping of cartilage contact models. However, the results also suggest that in some situations, inter-bone distance measurement and an assumed cartilage thickness may suffice for predicting joint contact patterns.     

Effects of nonuniform column packing in analytical gel chromatography. I. Monodisperse solutes

The effect of nonuniform column packing on solute profile shapes in analytical gel chromatography has been investigated for monodisperse solutes. This investigation considers the influence of a family of nonuniform partition cross sections on the concentration profiles for small-zone experiments. The nonuniformity causes the dispersive and translational transport coefficients to be functions of position. It is shown that for presently encountered amounts of nonuniformity, the principal effect is in the deviation of peak position from linear dependence on time. There is very little effect on peak shape, when concentrations are expressed in terms of bulk solution values.     

Information transfer in metabolic pathways. Effects of irreversible steps in computer models.

Various metabolic models have been studied by computer simulation in an effort to understand why allowing for the reversibility of the reaction catalysed by pyruvate kinase, normally considered as irreversible for all practical purposes, significantly altered the behaviour of the model of glycolysis in Trypanosoma brucei [Eisenthal, R. & Cornish-Bowden, A. (1998) J. Biol. Chem. 273, 5500-5505]. Studies of several much simpler models indicate that the enzymes catalysing early steps in a pathway must receive information about the concentrations of the metabolites at the end of the pathway if a model is to be able to reach a steady state; treating all internal steps as reversible is just one way of ensuring this. Feedback inhibition provides a much better way, and as long as feedback loops are present in a model it makes almost no difference to the behaviour whether the intermediate steps with large equilibrium constants are treated as irreversible. In the absence of feedback loops, ordinary product inhibition of all the enzymes in the chain can also transfer information; this is efficient for regulating fluxes but very inefficient for regulating intermediate concentrations. More complicated patterns of regulation, such as activation of a competing branch or forcing flux through a parallel route, can also serve to some degree as ways of passing information around an irreversible step. However, they normally do so less efficiently than inhibition, because the extent to which an enzyme or a pathway can be activated always has an upper limit (which may be below what is required), whereas most enzymes are inhibited completely at saturating concentrations of inhibitor.     

On the computation of the tertiary structure of globular proteins. III. Inter-residue distances and computed structures

An analysis of geometrical models for computing the tertiary structure of globular proteins from the primary structure is presented. The roles of initial configuration, input information on inter-residue distances and the errors in this information are delineated. It is shown that for local information like that on secondary structure, the calculated structure is very sensitive to errors and to the initial configuration. Thus, such information is far from adequate for predicting the tertiary structure. On the other hand, global information on all the inter-residue distances is quite insensitive to errors. A semi-empirical method is presented to estimate these distances and the calculated structures are given for two proteins—pancreatic trypsin inhibitor and parvalbumin. These structures have good resemblances to those determined by X-ray diffraction. A strategy for further refinement of the method is indicated.     

The kinetics of hemolytic plaque formation. II. Inhibition of plaques by hapten.

We present a mathematical theory of hapten inhibition of hemolytic plaque formation. The treatment is based upon the mathematical model for plaque growth presented by DeLisi &; Bell (1974). The lymphocyte under consideration is embedded in an infinite three-dimensional medium, and is secreting antibodies isotropically at a constant rate. As the antibodies diffuse from the source they can bind reversibly to hapten, and in the most general case reversibly to red blood cell (RBC) epitope. The model leads to a non-linear diffusion equation coupled to a set of first order differential equations. The system must, in general, be solved numerically. However, in many cases of experimental interest simplifications arise which permit closed form solutions to be obtained. In this paper we have developed solutions for three special cases.In the first example antibodies can bind only univalently to RBCs, as would be expected if the epitopes are sparsely distributed. In this case reaction between antibody site and RBC epitope is rapid ( ⪆ 1 sec) and reversible and local equilibrium is assumed. This leads to a “pure” diffusion equation in the free antibody concentration, but with a reduced diffusion coefficient.In another example univalent attachment of an antibody site to a RBC epitope is followed by a rapid irreversible intramolecular reaction. This might be expected for example if the epitope density is large. An exact solution to the resulting diffusion equation was also found in this case. In order to assess an intermediate situation, we also solved the equations for a model in which intramolecular reaction is slow and irreversible.The theory predicts that the type of information one can obtain from inhibition experiments depends critically upon the preparation of the RBC. If the cell is sufficiently haptenated so that rapid irreversible multivalent attachment is favorable, a differential plot of the inhibition curve will reflect the affinity distribution of antibody sites for free hapten. If only univalent attachment with RBCs is possible, so that antibody sites bind to RBC hapten in the same way they bind to free hapten, then a differential plot of the inhibition curve will reflect the secretion rate distribution.     

Dynamics of correlated genetic systems. II. Simulation studies of chromosomal segments under selection

The dynamics of chromosomal segments under selection are investigated by comparing experimental data to simulations of simple models of selection. The simulations assume 93 loci distributed evenly along an entire chromosome. The two issues addressed in this paper concern rates of decay of linkage disequilibria for chromosomes under selection and rates of gene frequency change after perturbation of gametic frequencies to states near the edge of the gametic frequency simplex. The findings are: (1) If reasonable values of inbreeding depression are assumed, linkage disequilibria decays to zero but at a rate nearly twice that expected from neutral theory. Experimental results also show accelerated decay rates. The acceleration of decay seems to be a simple consequence of the increased heterozygosity produced by selection. It is, therefore, argued that massive linkage disequilibria, of the kind found by Franklin and Lewontin (1970) in their simulations, are unlikely to characterize the genetic structure of natural populations of random mating organisms. (2) It is possible to distinguish between two time-honored models of multilocus selection, known as the symmetric overdominant and classical models, on the basis of gene frequency change near the edge of the gametic frequency simplex assuming linkage disequilibria is intense. (3) Examination of experimental data from perturbation experiments shows that neither of these elementary models adequately account for observed rates of gene frequency change, although the symmetric overdominant model does provide the better fit. Instead the experimental data suggest a markedly nonuniform distribution of selective effects along the chromosome. The data also suggest that these selective effects combine in markedly nonadditive ways in determining joint fitness.     

Biokinetic models for representing the complete inhibition of microbial activity at high substrate concentrations.

This paper reintroduces the Wayman and Tseng model for representing substrate inhibition effects on specific growth rate by further documenting its potential predictive capabilities. It also introduces a modification to this model in which an Andrews inhibition function is used in place of the Monod noninhibitory substrate function. This modification better represents the relationship between specific growth rate and substrate concentration for those substrates that show Andrews type inhibition at lower substrate concentrations, rather than the Monod type noninhibitory behavior described in the model of Wayman and Tseng. Results from nonlinear, least squares regression analysis are used to evaluate the ability of these models to empirically represent experimental data (both new and from the literature). The statistical goodness of fit is evaluated by comparing the regression results against those obtained using other empirical models. Finally, possible mechanisms of toxicity responsible for the observed inhibition trends are used to further justify use of these empirical models. The dominant mechanism considered to be relevant for conceptually explaining complete inhibition at high concentrations of solvents is the deterioration of cell membrane integrity. Literature citations are used to support this argument. This work should lead to improvements in the mathematical modeling of contaminant fate and transport in the environment and in the simulation of microbial growth and organic compound biodegradation in engineered systems.     

The Shannon information entropy of protein sequences.

A comprehensive data base is analyzed to determine the Shannon information content of a protein sequence. This information entropy is estimated by three methods: a k-tuplet analysis, a generalized Zipf analysis, and a "Chou-Fasman gambler." The k-tuplet analysis is a "letter" analysis, based on conditional sequence probabilities. The generalized Zipf analysis demonstrates the statistical linguistic qualities of protein sequences and uses the "word" frequency to determine the Shannon entropy. The Zipf analysis and k-tuplet analysis give Shannon entropies of approximately 2.5 bits/amino acid. This entropy is much smaller than the value of 4.18 bits/amino acid obtained from the nonuniform composition of amino acids in proteins. The "Chou-Fasman" gambler is an algorithm based on the Chou-Fasman rules for protein structure. It uses both sequence and secondary structure information to guess at the number of possible amino acids that could appropriately substitute into a sequence. As in the case for the English language, the gambler algorithm gives significantly lower entropies than the k-tuplet analysis. Using these entropies, the number of most probable protein sequences can be calculated. The number of most probable protein sequences is much less than the number of possible sequences but is still much larger than the number of sequences thought to have existed throughout evolution. Implications of these results for mutagenesis experiments are discussed.     

Effect of electric field gradients on lipid monolayer membranes.

Externally applied nonuniform electric fields can strongly affect thermodynamic phases in a lipid monolayer when applied under conditions of temperature, pressure, and composition that are near phase boundaries. Under such conditions nonuniform applied fields can produce or suppress phase separations. Field-induced phase-separated domains have sizes that are in good agreement with calculations. Field gradients can also produce large concentration gradients in binary mixtures just above their critical points. The present work elaborates our earlier studies of these field effects using thermodynamic models of the phase behavior of two-component liquid mixtures. The calculations are of interest in connection with biological membranes that, at the growth temperature, are in a liquid state close to a phase boundary.     

Functional multimodality of axonal tree in invertebrate neurons.

This review, based on invertebrate neuron examples, aims at highlighting the functional consequences of axonal tree organization. The axonal organization of invertebrate neurons is very complex both morphologically and physiologically. The first part shows how the transfer of information along sensory axons is modified by presynaptic inhibition mechanisms. In primary afferents, presynaptic inhibition is involved in: 1) increasing the dynamic range of the sensory response; 2) processing the sensory information such as increasing spatial and/or temporal selectivity; 3) discriminating environmental information from sensory activities generated by the animal's own movement; and 4) modulating the gain of negative feedback (resistance reflex) during active rhythmic movements such as locomotion. In a second part, the whole organization of other types of neurons is considered, and evidence is given that a neuron may not work as a unit, but rather as a mosaic of disconnected 'integrate-and-fire' units. Examples of invertebrate neurons are presented in which several spike initiating zones exist, such as in some stomatogastric neurons. The separation of a neuron into two functionally distinct entities may be almost total with distinct arborizations existing in different ganglia. However, this functional separation is not definitive and depends on the state of the neuron. In conclusion, the classical integrate-and-fire representation of the neuron, with its dendritic arborization, its spike initiating zone, its axon and axonal tree seems to be no more applicable to invertebrate neurons. A better knowledge of the function of vertebrate neurons would probably demonstrate that it is the case for a large number of them, as suggested by the complex architecture of some reticular interneurons in vertebrates.