Effects of diffusion on the electrophoretic behavior of associating systems: The Gilbert-Jenkins theory revisited

The Gilbert-Jenkins theory predicts the asymptotic shape of moving-boundary sedimentation and electrophoretic patterns and broad zone molecular sieve chromatographic elution profiles for the class of interacting systems, A + B ⇄ C, in which two dissimilar macromolecules react reversibly to form a complex. A particularly provocative case is the one in which the complex has a greater migration velocity than that of either reactant, each of which has a different velocity. Depending upon conditions, this case predicts, for example, that in the asymptotic limit an ascending electrophoretic pattern or a frontal gel chromatographic elution profile can show two hypersharp reaction boundaries separated by a plateau. This prediction is now confirmed by numerical solution of transport equations which retain the second-order diffusional term and extrapolation of the computed patterns to zero diffusion coefficient. For finite diffusion coefficient, however, the two hypersharp reaction boundaries are separated by a weak negative gradient. These calculations are extended to an examination of the transitions between the three types of patterns admitted by the case under consideration in order to gain physical understanding and to define criteria for recognizing the transitions. Studies of this kind not only establish confidence in the Gilbert-Jenkins theory, but, in addition, they provide new insights which make for more effective application of the theory to real systems.     

Defects in associating systems: example of actin.

A statistical mechanical model is given for linear associating systems that contain defects, using the double-stranded actin polymer as an example. We treat the system as a one-dimensional lattice that can desorb monomers (giving defects) using grand partition function techniques. The main difference from a standard adsorption problem is that the monomer units are also responsible for the structural integrity of the lattice (polymer) and if too many desorb the polymer will be broken. We use literature data to estimate the density of defects in the actin polymer.     

Global asymptotic behavior in single-species discrete diffusion systems

We consider a single-species dynamical system which is composed of several patches connected by discrete diffusion. Based on recently developed cooperative system theory and the property of a cooperative matrix, we obtain sufficient and necessary conditions for the system with linear diffusion to be extinct and for one with nonlinear diffusion to be globally stable. We also obtain a critical patch number in the system with linear diffusion for the species to go extinct. These results extend some recent known ones for discrete diffusion systems.Research partly supported by the Ministry of Education, Science and Culture, Japan, under Grant 01540177     

Broad-zone active-enzyme chromatography: Keto-acid dehydrogenases as associating systems

We have extended the method of active-enzyme chromatography to include the use of broad zones of enzyme. This allows examination of interacting systems in a way formally analogous to sedimentation velocity so that simulation of the observed activity profiles is possible. The method has been applied using pyridine nucleotide-linked active enzyme assays. At the concentrations presently accessible by this technique, hexokinase and glucose-6-phosphate dehydrogenase, both associating systems, show single symmetrical boundaries, as does isolated diaphorase, while pyruvate and α-ketoglutarate dehydrogenases show more complex patterns, with the position of the reaction boundary for diaphorase activity being dependent on enzyme concentration.     

The theory of functional systems and purposeful behavior

The role of probability forecasting in the purposive behavior under conditions of subjective uncertainty is considered in terms of the theory of functional systems. Participation of the probability forecasting in the afferent synthesis, goal formation, formation of the acceptor of action result and action program, and, finally, in the action program actualization is substantiated. The model of behavior under conditions of subjective uncertainty is advanced. It includes all the classical elements of the model of behavioral act developed by P.K. Anokhin. In order to take into account the probability aspects of behavior, the role of probability forecasting is emphasized at every stage of the system functioning. In addition to the classical elements, two novel components are introduced. These are the "memory buffer" (results of searching reactions) and the apparatus of probability decisions about changes in the action program. By the memory buffer an apparatus is meant, which gathers and stores the information about the results of many behavioral acts performed during the actualization of the action program. This information is used in the process of making a probability decision as whether to alter or not the action program after each specific behavioral act. Such an approach integrates the probability forecasting and the theory of functional systems. The theory becomes universal, i.e., applicable not only to deterministic but also to probabilistic environments.     

Effects of diffusion on the stability of the equilibrium in multi-species ecological systems

Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.     

Anomalous permeation of a reversibly associating substance: Hydraulic conductivity and tracer water diffusion

Summary The permeability coefficient determined with isotopically labeled solvent (water) and the permeability coefficient determined from the volume flow under the influence of an osmotic gradient are different, if the solvent is considered to be reversibly associated. This is shown by application of the equations of irreversible thermodynamics to systems with an associating substance. An equation is derived which relates the permeability ratio to the average cluster size of the solvent.     

The theory of functional systems and probabilistic prognosis of behavior

The article is dedicated to possible probability prediction of behavior in system organization of behavioral acts. System mechanisms of anticipation of required results of behavioral activity by living organisms adapted to stable and changing conditions of life are discussed. The author proposes that in all forms of behavior, an organism strictly predicts parameters for satisfaction of dominating needs of results that constitute the goal of a given form of behavior. In author's opinion, the probabilistic prognosis concerns only methods, acts and attendant emotional states, as well as possible ways of attaining the results (i.e., the means rather than action parameters).     

Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems

Theoretical models of populations on a system of two connected patches previously have shown that when the two patches differ in maximum growth rate and carrying capacity, and in the limit of high diffusion, conditions exist for which the total population size at equilibrium exceeds that of the ideal free distribution, which predicts that the total population would equal the total carrying capacity of the two patches. However, this result has only been shown for the Pearl-Verhulst growth function on two patches and for a single-parameter growth function in continuous space. Here, we provide a general criterion for total population size to exceed total carrying capacity for three commonly used population growth rates for both heterogeneous continuous and multi-patch heterogeneous landscapes with high population diffusion. We show that a sufficient condition for this situation is that there is a convex positive relationship between the maximum growth rate and the parameter that, by itself or together with the maximum growth rate, determines the carrying capacity, as both vary across a spatial region. This relationship occurs in some biological populations, though not in others, so the result has ecological implications.     

A theory of associating food types with their postingestive consequences

Animals often face complex and changing food environments. While such environments are challenging, an animal should make an association between a food type and its properties (such as the presence of a nutrient or toxin). We use information theory concepts, such as mutual information, to establish a theory for the development of these associations. In this theory, associations are assumed to maximize the mutual information between foods and their consequences. We show that associations are invariably imperfect. An association's accuracy increases with the length of a feeding session and the relative frequency of a food type but decreases as time delay between consumption and postingestive consequence increases. Surprisingly, the accuracy of an association is independent of the number of additional food types in the environment. The rate of information transfer between novel foods and a forager depends on the forager's diet. In light of this theory, an animal's diet may have two competing goals: first, the provision of an appropriate balance of nutrients, and second, the ability to quickly and accurately learn the properties of novel foods. We discuss the ecological and behavioral implications of making associational errors and contrast the timescale and mechanisms of our theory with those of existing theory.     

Towards a definition of living systems: A theory of ecological support for behavior

It is proposed that the Darwinian theoretical approach and account of living systems has not yet been clearly given. A first approximation to this is attempted, focussing on behavior in evolving environments. A theoretical terminology is defined emphasizing the mutuality of organism and environment and the existence of biologically theoretical entities.     

Comment on Gallistel: behavior theory and information theory: some parallels

In this article, Gallistel proposes information theory as an approach to some enduring problems in the study of operant and classical conditioning.     

On the diffusion theory of phyllotaxis

An inhibitor diffusion theory of phyllotaxis is examined in the steady-state approximation for cylindrical shoot apex models. The model calculations give rise naturally to common patterns of spiral phyllotaxis, as well as to higher whorled patterns. The model also predicts commonly observed subpatterns of axillary organs superimposed on primary phyllotaxis patterns. Application of the model to phyllotaxis patterns in other organisms and in flowers is proposed.     

Note on the theory of mass behavior

A differential equation has been derived by A. Rapoport,Bull. Math. Biophysics,14, 159 (1952), giving the time course of the fraction of the population who have performed a given act. The general solution of this equation is obtained, some properties of the solution are deduced, and a special case presented in detail.     

Sedimentation velocity analysis of heterogeneous protein-protein interactions: sedimentation coefficient distributions c(s) and asymptotic boundary profiles from Gilbert-Jenkins theory

Interacting proteins in rapid association equilibrium exhibit coupled migration under the influence of an external force. In sedimentation, two-component systems can exhibit bimodal boundaries, consisting of the undisturbed sedimentation of a fraction of the population of one component, and the coupled sedimentation of a mixture of both free and complex species in the reaction boundary. For the theoretical limit of diffusion-free sedimentation after infinite time, the shapes of the reaction boundaries and the sedimentation velocity gradients have been predicted by Gilbert and Jenkins. We compare these asymptotic gradients with sedimentation coefficient distributions, c(s), extracted from experimental sedimentation profiles by direct modeling with superpositions of Lamm equation solutions. The overall shapes are qualitatively consistent and the amplitudes and weight-average s-values of the different boundary components are quantitatively in good agreement. We propose that the concentration dependence of the area and weight-average s-value of the c(s) peaks can be modeled by isotherms based on Gilbert-Jenkins theory, providing a robust approach to exploit the bimodal structure of the reaction boundary for the analysis of experimental data. This can significantly improve the estimates for the determination of binding constants and hydrodynamic parameters of the complexes.     

On the theory of diffusion of electrolytes

In the theory of diffusion of electrolytes the following assumptions are frequently made: (i) the electrolytic solution is electrically neutral everywhere, (ii) the ionic concentrations and the electric potential all depend on a single Cartesian coordinate as the only space variable. Often the electric potential of the solution is determined on the basis of the Poisson equation alone, disregarding any other relation between this potential and the ionic concentrations. Since the Poisson equation only represents a condition which the potential fulfills, the use of this equation alone may lead to error unless the explicit relation for the potential involving a space integration of ionic concentrations is also taken into account. But if this relation is used the Poisson equation becomes redundant and, more important, assumptions (i) and (ii) appear unacceptable, the former because it leads to a zero electric potential everywhere, the latter because it is mathematically incorrect. The present paper is based on general equations of diffusion of ions, excluding the Poisson equation. These equations form a system of nonlinear integrodifferential quations whose number equals the number of ionic species present in the solution. It appears that when all ions are distributed symmetrically around a point all functions related to the above system of equations can be made dependent on a single space coordinate: the distance from the center of symmetry. Two methods of successive approximations are given for the solution of the equations in the case of spherical symmetry with limitation to the steady state. These methods are then applied to the study of the distribution of ionic concentrations and electrical potentials inside a cell of spherical shape in equilibrium with its surroundings. These methods are rapidly convergent; exact theoretical values of the electric potential are calculable on the boundary of the cell. It appears that the potential at the center of the cell is not more than ∼50% higher than at its boundary and that variation of concentration inside the cell is not very large. For instance, with 100 mV on the boundary the ionic concentration there is about four times higher than at the center. Calculations show that extremely small amounts of electricity are sufficient to account for the electric potentials currently observed. In a cell of 100 micra diameter an average concentration of only 10⁻¹⁴ mole/cm³ of a monovalent ion would be sufficient to give 1 millivolt on the boundary. This concentration is directly proportional to the voltage and inversely proportional to the square of the cell diameter. Most of the numerical results given above are obtained by considering only those ions whose electrical charge is not compensated for by ions of an opposite sign. The total concentrations may be much higher than those quoted. The theory does not take into account possible effects of structural heterogeneities which may exist in the cell, particularly of various phase boundaries. An incidental result shows that the Boltzmann distribution function in the form employed in modern theory of electrolytes is fundamentally a consequence of the mathematical theory of diffusion alone. It is pointed out, however, that Boltzmann distribution is not always compatible with the definition of the electric potential.     

Single-file diffusion and neurotransmitter transporters: Hodgkin and Keynes model revisited

Norepinephrine transporters (NETs) use the Na gradient to remove norepinephrine (NE) from the synaptic cleft of adrenergic neurons following NE release from the presynaptic terminal. By coupling NE to the inwardly directed Na gradient, it is possible to concentrate NE inside cells. This mechanism, which is referred to as co-transport or secondary transport (L?uger, 1991, Electrogenic Ion Pumps, Sinauer Associates) is apparently universal: Na coupled transport applies to serotonin transporters (SERTs), dopamine transporters (DATs), glutamate transporters, and many others, including transporters for osmolites, metabolites and substrates such as sugar. Recently we have shown that NETs and SERTs transport norepinephrine or serotonin as if Na and the transmitter permeated through an ion channel together 'Galli et al., 1998, PNAS 95, 13260-13265; Petersen and DeFelice, 1999, Nature Neurosci. 2, 605-610'. These data are paradoxical because it has been difficult to envisage how NE, for example, would couple to Na if these ions move passively through an open pore. An 'alternating access' model is usually evoked to explain coupling: in such models NE and Na bind to NET, which then undergoes a conformational change to release NE and Na on the inside. The empty transporter then turns outward to complete the cycle. Alternating-access models never afford access to an open channel. Rather, substrates and co-transported ions are occluded in the transporter and carried across the membrane. The coupling mechanism we propose is fundamentally different than the coupling mechanism evoked in the alternating access model. To explain coupling in co-transporters, we use a mechanism first evoked by 'Hodgkin and Keynes (1955) J. Physiol. 128, 61-88' to explain ion interactions in K-selective channels. In the Hodgkin and Keynes model, K ions move single-file through a long narrow pore. Their model accounted for the inward/outward flux ratio if they assumed that two K ions queue within the pore. We evoke a similar model for the co-transport of transmitter and Na. In our case, however, coupling occurs not only between like ions but also between unlike ions (i.e. the transmitter and Na ). We made a replica of the Hodgkin and Keynes mechanical model to test our ideas, and we extended the model with computer simulations using Monte Carlo methods. We also developed an analytic formula for Na coupled co-transport that is analogous to the single-file Ussing equation for channels. The model shows that stochastic diffusion through a long narrow pore can explain coupled transport. The length of the pore amplifies the Na gradient that drives co-transport.