Organizations in reaction-diffusion systems: Effects of diffusion and boundary conditions

Chemical Organization Theory (COT) has been successfully applied to analyze complex reaction networks where species interact and new species can emerge. The COT has been well studied, but is yet to analyze high dimensional systems dynamics over time equivalent to ordinary differential equations. Moreover, spatial effects, such as diffusion and boundary conditions have also not been considered yet. Here, we extend the COT to cope with reaction-diffusion systems. Thereby we focus on the effects of diffusion and various boundary conditions. In order to demonstrate the effectiveness of our approach, we analyze two models based on partial differential equations, one of which is on HIV virus dynamics.The analysis shows interesting organizational structures when using different ranges of diffusion rates, as well as for Dirichlet and positive Neumann boundary conditions. The advantage of this novel approach is that it is based solely on the model structure (reaction rules) but is independent of kinetic details, such as rate constants. Hence, it copes with high-dimensional systems without the need of numerical simulations, and it can be applied without detailed mathematical knowledge. Our tool is available, without restriction at https://github.com/stephanpeter/orgs-rds.     

Facilitated diffusion: the elasticity of oxygen supply.

It is well established that the presence of oxygen-carrying proteins such as haemoglobin can facilitate the diffusion of oxygen through a solution. In this paper, it is shown that some properties of a facilitated flow are substantially different from those of unfacilitated flux, including especially the stability of the tension at which the oxygen arrives at the end of the diffusion path. The concept of the “output resistance” of the supply is introduced, and a facilitated pathway is shown to have a lowered resistance. A role for the storage capacity of the bound oxygen reservoir is also developed; it is shown that delivery oxygen tensions are stabilized against transient changes in oxygen demand. In treating the equations of facilitated diffusion, a simplified approach is used to take account of boundary layers in the solution where deviations from oxygen-protein equilibrium are significant. A measure of the thickness and importance of such boundary layers is calculated.     

Facilitated diffusion with consecutive reaction: optimal carrier affinity

The interplay between facilitated diffusion of a substrate through a membrane and a consecutive enzymic reaction, both of which follow Michaelis-Menten kinetics, has been theoretically investigated and the effect of the kinetic and transport parameters on the rate of substrate uptake is graphically illustrated. At steady state two characteristic features of the system have been identified. First, the substrate concentration at the internal enzymic side of the membrane cannot exceed a given value even at much higher external substrate concentrations. Second, the uptake rate is maximum at a given value of K_T, the kinetic parameter of the transport system that expresses the reciprocal carrier affinity of the substrate. The optimum value of K_T is approximately equal to the external substrate concentration. This particular dependence of the uptake rate on the carrier affinity is expected to play an important role in hormonal regulation.     

The SMM model as a boundary value problem using the discrete diffusion equation

A generalized single-step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.     

The physico-chemical mechanism of mediated transport. I. Facilitated diffusion

On the basis of the currently accepted model for the cell membrane structure, a physico-chemical model for mediated transport is developed and solved for the case of polar non-electrolyte migration through the cell membrane. The model considers the interstitial space defined by the transport protein subunits to be the migration pathway for polar solutes. A Langmuir-type adsorption equilibrium is assumed at the interfaces and a multicomponent diffusion mechanism of solute and water is postulated within the migration pathway, where the polar residues of the transport protein represent another component of the system. Membrane selectivity is governed by the adsorption constants, which are shown to affect strongly the kinetics of transport. Isosmotic transport and the volume change of the cell are important features incorporated in the model, which is shown to fulfill the peculiar properties of facilitated diffusion systems. It is concluded that the same type of pathway can be used for the transport of other polar solutes through existing or induced hydrophilic channels, for which a similar approach is suggested.     

Facilitated diffusion of CO2 across albumin solutions

The steady-state CO₂ flux across thin layers of 30 g/100 ml albumin solutions was measured in two different CO₂ partial pressure ranges (boundary PCO ₂ values 3 and 8 torr, and 160 and 650 torr, respectively). From the data the apparent diffusion coefficient for CO₂, DCO ₂, was calculated. In the high PCO ₂ range a value of DCO ₂ was found which is to be expected on the basis of diffusion of dissolved CO₂ only. In the low PCO ₂ range DCO ₂ was about 100% higher than in the high PCO ₂ range, when carbonic anhydrase was present and the pH was ∼7.7. DCO ₂ depended on the concentration of carbonic anhydrase. It increased with increasing pH. It is concluded that an additional diffusion of bound CO₂ (facilitated CO₂ diffusion) occurs in the low PCO ₂ range and that this transport involves the hydration of CO₂. From the diffusion coefficients in the two PCO ₂ ranges the rate of facilitated diffusion was determined. Approximate calculations show that this rate (at pH ≤ 7.7) can be explained on the basis of the proposed mechanism of facilitated CO₂ diffusion: bicarbonate diffusion and simultaneous proton transport by albumin diffusion. The view that facilitated CO₂ diffusion is mediated by the diffusion of albumin is supported by the experimental finding of a considerable suppression of the facilitated CO₂ flux in the presence of gelatinized agar-agar.     

Facilitated diffusion in chromatin lattices: mechanistic diversity and regulatory potential

The interaction between a protein and a specific DNA site is the molecular basis for vital processes in all organisms. Location of the DNA target site by the protein commonly involves facilitated diffusion. Mechanisms of facilitated diffusion vary among proteins; they include one- and two-dimensional sliding along DNA, direct transfer between uncorrelated sites, as well as combinations of these mechanisms. Facilitated diffusion has almost exclusively been studied in vitro. This review discusses facilitated diffusion in the context of the living cell and proposes a theoretical model for facilitated diffusion in chromatin lattices. Chromatin structure differentially affects proteins in different modes of diffusion. The interplay of facilitated diffusion and chromatin structure can determine the rate of protein association with the target site, the frequency of association-dissociation events at the target site, and, under particular conditions, the occupancy of the target site. Facilitated diffusion is required in vivo for efficient DNA repair and bacteriophage restriction and has potential roles in fine-tuning gene regulatory networks and kinetically compartmentalizing the eukaryotic nucleus.     

Facilitated diffusion of urate in avian brush-border membrane vesicles

Membrane transport pathways mediatingtranscellular secretion of urate across the proximal tubule wereinvestigated in brush-border membrane vesicles (BBMV) isolated fromavian kidney. An inside-positive K diffusion potential induced aconductive uptake of urate to levels exceeding equilibrium.Protonophore-induced dissipation of membrane potential significantlyreduced voltage-driven urate uptake. Conductive uptake of urate wasinhibitor sensitive, substrate specific, and a saturable function ofurate concentration. Urate uptake was trans-stimulated byurate and cis-inhibited by p-aminohippurate (PAH). Conductive uptake of PAH was cis-inhibited by urate.Urate uptake was unaffected by an outward -ketoglutarate gradient. In the absence of a membrane potential, urate uptake was similar in thepresence and absence of an imposed inside-alkaline pH gradient or anoutward Cl gradient. These observations suggest a uniporter-mediated facilitated diffusion of urate as a pathway for passive efflux acrossthe brush border membrane of urate-secreting proximal tubule cells.

     

Facilitated diffusion of myoglobin and creatine kinase and reaction-diffusion constraints of aerobic metabolism under steady-state conditions in skeletal muscle

The roles of creatine kinase (CK) and myoglobin (Mb) on steady-state facilitated diffusion and temporal buffering of ATP and oxygen, respectively, are assessed within the context of a reaction-diffusion model of muscle energetics. Comparison of the reaction-diffusion model with experimental data from a wide range of muscle fibers shows that the experimentally observed skeletal muscle fibers are generally not limited by diffusion, and the model further indicates that while some muscle fibers operate near the edge of diffusion limitation, no detectable effects of Mb and CK on the effectiveness factor, a measure of diffusion constraints, are observed under steady-state conditions. However, CK had a significant effect on average ATP concentration over a wide range of rates and length scales within the reaction limited regime. The facilitated diffusion functions of Mb and CK become observable in the model for larger size cells with low mitochondrial volume fraction and for low boundary O(2) concentration and high ATP demand, where the fibers may be limited by diffusion. From the transient analysis it may be concluded that CK primarily functions to temporally buffer ATP as opposed to facilitating diffusion while Mb has a small temporal buffering effect on oxygen but does not play any significant role in steady-state facilitated diffusion in skeletal muscle fibers under most physiologically relevant regions.     

Facilitated diffusion of a DNA binding protein on chromatin.

Facilitated diffusion accounts for the rapid rate of association of many bacterial DNA binding proteins with specific DNA sequences in vitro. In this mechanism the proteins bind at random to non-specific sites on the DAN and diffuse (by 'sliding' or 'hopping') along the DNA chain until they arrive at their specific functional sites. We have investigated whether such a mechanism can operate in chromatin by using a bacterial DNA binding protein, Escherichia coli RNA polymerase, that depends on linear diffusion to locate initiation sites on DNA. We have measured the competition between chromatin and its free DNA for the formation of initiation complexes. Only the short linker segments exposed by the removal of histone H1 are available for interaction with the polymerase, but the sparsely distributed promoter sites on the linker DNA of such a polynucleosome chain are located at the same rate as those on DNA. We conclude that the polymerase is free to migrate between the separate linker DNA segments of a polynucleosome chain to reach a promoter site. This chain thus permits the 'hopping' of proteins between neighboring linker segments in their search for a target site on the accessible DNA.     

Facilitated aggregation of FG nucleoporins under molecular crowding conditions

Intrinsically disordered and phenylalanine–glycine‐rich nucleoporins (FG Nups) form a crowded and selective transport conduit inside the NPC that can only be transited with the help of nuclear transport receptors (NTRs). It has been shown in vitro that FG Nups can assemble into two distinct appearances, amyloids and hydrogels. If and how these phenomena are linked and if they have a physiological role still remains unclear. Using a variety of high‐resolution fluorescence and electron microscopic (EM) tools, we reveal that crowding conditions mimicking the NPC environment can accelerate the aggregation and amyloid formation speed of yeast and human FG Nups by orders of magnitude. Aggregation can be inhibited by NTRs, providing a rationale on how the cell might control amyloid formation of FG Nups. The superb spatial resolving power of EM also reveals that hydrogels are enlaced amyloid fibres, and these findings have implications for existing transport models and for NPC assembly.     

Facilitated diffusion of VEGF165 through descemet's membrane with sucrose octasulfate

Vascular endothelial growth factor A (VEGF-A) is a promoter of neovascularization and thus a popular therapeutic target for diseases involving excessive growth of blood vessels. In this study, we explored the potential of the disaccharide sucrose octasulfate (SOS) to alter VEGF165 diffusion through Descemet's membrane. Descemet's membranes were isolated from bovine eyes and used as a barrier between two chambers of a diffusion apparatus to measure VEGF transport. Diffusion studies revealed a dramatic increase in VEGF165 transport in the presence of SOS, with little diffusion of VEGF165 across the membrane over a 10-h time course in the absence of SOS. Diffusion studies with VEGF121, a non-heparin binding variant of VEGF, showed robust diffusion with or without SOS. To determine a possible mechanism, we measured the ability of SOS to inhibit VEGF interactions with extracellular matrix (ECM), using cell-free and cell surface binding assays. Binding studies showed SOS had no effect on VEGF165 binding to either heparin-coated plates or endothelial cell surfaces at less than mg/ml concentrations. In contrast, we show that SOS inhibited VEGF165 binding to fibronectin in a dose dependent manner and dramatically accelerated the rate of release of VEGF165 from fibronectin. SOS also inhibited the binding of VEGF165 to fibronectin-rich ECM deposited by vascular smooth muscle cells. These results suggest that fibronectin-rich extracellular matrices serve as barriers to VEGF165 diffusion by providing a network of binding sites that can trap and sequester the protein. Since the content of Descemet's membrane is typical of many basement membranes it is possible that they serve throughout the body as formidable barriers to VEGF165 diffusion and tightly regulate its bioavailability and distribution within tissues.     

A diffusion problem

When two polished metal spheres in contact with each other are immersed in a corrosive solution, their surfaces gradually corrode, leaving, however, a non-corroded zone around the point of contact. The size of this zone is a function of the size of the spheres, concentration of the corrosive, and of the time. The phenomenon has been first observed by F. C. Besic on human teeth immersed in hydrochloric acid, and studied on metal balls. In the present paper it is shown that such a phenomenon may be due to gradients of concentration of the corrosive in the neighborhood of the point of contact, caused by the chemical reaction which consumes the corrosive. Approximate expressions for the size of the uncorroded zones as a function of the size of the balls and as a function of time are derived and found to be in fair agreement with F. C. Besic's data.     

Facilitated diffusion of glucosamine-6-phosphate synthase inhibitors enhances their antifungal activity

N3-(4-Methoxyfumaroyl)-L-2,3-diaminopropanoic acid (FMDP) and 2-amino-2-deoxy-D-glucitol-6-phosphate (ADGP) are strong inhibitors of the essential fungal enzyme, glucosamine-6-phosphate synthase, but their antifungal activity is poor, due to slow penetration of these agents through the cytoplasmic membrane. In the present studies we have exploited the possibility of enhancement of ADGP and FMDP antifungal activity by improving their transport properties. It has been found that membrane-permeabilising polyene macrolides amphotericin B (AMB) and its N-methyl-N-fructosyl methyl ester derivative (MF-AME), at subinhibitory concentrations, facilitate diffusion of ADGP through the fungal cell membrane, thus allowing a decrease of its minimal inhibitory concentration (MIC). Synergistic effects have been observed for combinations of ADGP with AMB or MF-AME. Fractional inhibitory concentration (FIC) indexes, determined against a number of Candida spp., have been in the 0.18-0.81 range. Weak antifungal synergistic effects have been found for combinations of FMDP with AMB or MF-AME. ADGP can be easily encapsulated into unilamellar lipid vesicles. Liposomal preparations of ADGP demonstrated stronger antifungal activity against some fungal strains than free ADGP.     

Facilitated diffusion of oxygen and carbon monoxide in the large affinity regime

Wyman's equation for facilitated diffusion of a ligand through a solution slab containing a carrier is recast and solved by means of a regular perturbation expansion in the parameter representing the driving force for facilitation. This new solution is complementary to the previously exploited singular perturbation solution due to Murray and represents facilitation in the low facilitation parameter regime. The most significant physical realization of this regime occurs when there is a large affinity between ligand and carrier, as in the carbon monoxide-hemoglobin system.The validity domains of the regular perturbation solution and the singular perturbation solution of Mitchell and Murray and Rubinow and Dembo are delineated. The equation for facilitated diffusion is solved numerically for parameter values appropriate to the oxygen-myoglobin experiments of Wittenberg, and to the carbon monoxide-hemoglobin experiments of Mochizuki and Forster, and Wittenberg. This solution provides a norm for comparison of the utility of the perturbation solutions.We show how the theory explains the apparent contradiction between the positive observations of Mochizuki and Forster and the negative observations of Wittenberg on facilitation of carbon monoxide transport through a slab of hemoglobin solution.Supported in part by the National Science Foundation under Grant No. PCM77-03344In partial fulfillment of the Ph.D. degree at Cornell UniversityProfessor S. I. Rubinow has passed away on February 22, 1981Computations were performed at the Courant Mathematics and Computing Laboratory, supported by DOE under contract #EY-76-C-02-3077.