Facilitated diffusion and membrane permeation of fatty acid in albumin solutions

Mathematical Models Suggest Facilitated Fatty Acids Crossing of the Luminal Membrane in the Cardiac Muscle

Long-chain fatty acids cross a few membranes on their way from the capillary blood to the cardiomyocyte cytosol, where they are utilized as an essential source of energy. Details of the transport mechanism across those membranes remained elusive despite decades of laboratory and theoretical work. Here we inspect several optional scenarios for the crossing of the luminal membrane of the endothelial cell, the first barrier that should be crossed: a passive diffusion, facilitation by receptors for albumin and facilitation by fatty acids transporters. Related measured rate constants are incorporated in a theoretical simulation that is based on reaction-diffusion equations. Asymptotic analytical solutions for the resulting stiff boundary value problems are formulated based on singular perturbations theory. We conclude that a passive diffusion has to be supplemented with facilitation mechanisms in order to meet energy requirements. Binding sites for albumin, scattered on the membrane face, might enhance the flux provided that they internalize the captured fatty acids and speed up the dissociation of the albumin-fatty acids complex. As such enhancement is moderate, another mechanism seems to be essential for an adequate supply of fatty acids. Lack of experimental data prohibits us from computing the quantitative effect of membrane fatty acids transporters but their involvement in the membrane crossing is inferred.

Modeling Fatty Acid Transfer from Artery to Cardiomyocyte

Despite the importance of oxidation of blood-borne long-chain fatty acids (Fa) in the cardiomyocytes for contractile energy of the heart, the mechanisms underlying the transfer of Fa from the coronary plasma to the cardiomyocyte is still incompletely understood. To obtain detailed insight into this transfer process, we designed a novel model of Fa transfer dynamics from coronary plasma through the endothelial cells and interstitium to the cardiomyocyte, applying standard physicochemical principles on diffusion and on the chemical equilibrium of Fa binding to carrier proteins Cp, like albumin in plasma and interstitium and Fatty Acid-Binding Proteins within endothelium and cardiomyocytes. Applying these principles, the present model strongly suggests that in the heart, binding and release of Fa to and from Cp in the aqueous border zones on both sides of the cell membranes form the major hindrance to Fa transfer. Although often considered, the membrane itself appears not to be a significant hindrance to diffusion of Fa. Proteins, residing in the cellular membrane, may facilitate transfer of Fa between Cp and membrane. The model is suited to simulate multiple tracer dilution experiments performed on isolated rabbit hearts administrating albumin and Fa as tracer substances into the coronary arterial perfusion line. Using parameter values on myocardial ultrastructure and physicochemical properties of Fa and Cp as reported in literature, simulated washout curves appear to be similar to the experimentally determined ones. We conclude therefore that the model is realistic and, hence, can be considered as a useful tool to better understand Fa transfer by evaluation of experimentally determined tracer washout curves.

The energy required for pump work of the heart is generated primarily by oxidation of long-chain fatty acids (Fa), transferred to the heart by plasma albumin. In coronary capillaries, Fa detach from albumin, cross endothelial cells, pericapillary interstitium and cardiomyocyte membrane prior to oxidation. The exact mechanism underlying the transfer process, however, is unknown. We designed a computer model of this Fa transfer using parameter values in the physiological range. We postulate that known physical principles of diffusion, Fa-protein complex binding, and Fa solubility in water and lipid membranes can describe intra-cardiac transfer. Model simulations were compared with multiple indicator dilution experiments, administrating a bolus of labeled Fa and albumin into the coronary artery. The resulting dilution time courses of label concentrations in the coronary veins compared favorably with the model simulations. We conclude that the model appears to be realistic, providing a useful tool to study in detail the mechanisms of intra-cardiac Fa transfer.

Transport of Free Fatty Acids from Plasma to the Endothelium of Cardiac Muscle: A Theoretical Study

Fatty acids are transported in a multistep process from the plasma to the mitochondria, where they are oxidized in order to meet energy requirements of the myocardium. Some of those steps, mainly the crossing of the involved cells' membranes are far from being understood. Here, by means of mathematical modeling we address the problem of the fatty acid transport from the microvascular compartment to the endothelium. Values of parameters that are incorporated in the model are deduced from relevant experimental work. Concentration profiles are established as solutions of diffusion-reaction equations both numerically and using an analytical asymptotic approximation. The analytical solution accurately determines the fatty acid flux for any set of parameter values in contrast to off-the-shelf numerical solvers that fail under quite a few circumstances due to the stiffness of the differential equation system. Sensitivity analysis indicates that in spite of few uncertain parameter values, most of our conclusions are expected to be valid throughout the physiological range of operation. We find that in order to have an adequate fatty acid uptake rate it is essential for the luminal endothelial membrane to have very fast fatty acid transporters and/or specific sites that interact with the albumin-fatty acids complex.

JSim, an open-source modeling system for data analysis

JSim is a simulation system for developing models, designing experiments, and evaluating hypotheses on physiological and pharmacological systems through the testing of model solutions against data. It is designed for interactive, iterative manipulation of the model code, handling of multiple data sets and parameter sets, and for making comparisons among different models running simultaneously or separately. Interactive use is supported by a large collection of graphical user interfaces for model writing and compilation diagnostics, defining input functions, model runs, selection of algorithms solving ordinary and partial differential equations, run-time multidimensional graphics, parameter optimization (8 methods), sensitivity analysis, and Monte Carlo simulation for defining confidence ranges. JSim uses Mathematical Modeling Language (MML) a declarative syntax specifying algebraic and differential equations. Imperative constructs written in other languages (MATLAB, FORTRAN, C++, etc.) are accessed through procedure calls. MML syntax is simple, basically defining the parameters and variables, then writing the equations in a straightforward, easily read and understood mathematical form. This makes JSim good for teaching modeling as well as for model analysis for research.   For high throughput applications, JSim can be run as a batch job.  JSim can automatically translate models from the repositories for Systems Biology Markup Language (SBML) and CellML models. Stochastic modeling is supported. MML supports assigning physical units to constants and variables and automates checking dimensional balance as the first step in verification testing. Automatic unit scaling follows, e.g. seconds to minutes, if needed. The JSim Project File sets a standard for reproducible modeling analysis: it includes in one file everything for analyzing a set of experiments: the data, the models, the data fitting, and evaluation of parameter confidence ranges. JSim is open source; it and about 400 human readable open source physiological/biophysical models are available at http://www.physiome.org/jsim/.

How fatty acids of different chain length enter and leave cells by free diffusion

Opposing views exist as to how unesterified fatty acids (FA) enter and leave cells. It is commonly believed that for short- and medium-chain FA free diffusion suffices whereas it is questioned whether proteins are required to facilitate transport of long-chain fatty acid (LCFA). Furthermore, it is unclear whether these proteins facilitate binding to the plasma membrane, trans-membrane movement, dissociation into the cytosol and/or transport in the cytosol. In this mini-review we approach the controversy from a different point of view by focusing on the membrane permeability constant (P) of FA with different chain length. We compare experimentally derived values of the P of short and medium-chain FA with values of apparent permeability coefficients for LCFA calculated from their dissociation rate constant (k(off)), flip-flop rate constant (k(flip)) and partition coefficient (Kp) in phospholipid bilayers. It was found that Overton's rule is valid as long as k(flip)<<k(off). With increasing chain length, the permeability increases according to increasing Kp and reaches a maximum for LCFA with chain length of 18 carbons or longer. For fast flip-flop (e.g. k(flip)=15s(-1)), the apparent permeability constant for palmitic acid is very high (P(app)=1.61 cm/s). Even for a slow flip-flop rate constant (e.g. k(flip)=0.3s(-1)), the permeability constant of LCFA is still several orders of magnitude larger than the P of water and other small non-electrolytes. Since polyunsaturated FA have basically the same physico-chemical properties as LCFA, they have similar membrane permeabilities. The implications for theories involving proteins to facilitate uptake of FA are discussed.

Simultaneous blood-tissue exchange of oxygen, carbon dioxide, bicarbonate, and hydrogen ion

A detailed nonlinear four-region (red blood cell, plasma, interstitial fluid, and parenchymal cell) axially distributed convection-diffusion-permeation-reaction-binding computational model is developed to study the simultaneous transport and exchange of oxygen (O2) and carbon dioxide (CO2) in the blood-tissue exchange system of the heart. Since the pH variation in blood and tissue influences the transport and exchange of O2 and CO2 (Bohr and Haldane effects), and since most CO2 is transported as HCO3(-) (bicarbonate) via the CO2 hydration (buffering) reaction, the transport and exchange of HCO3(-) and H+ are also simulated along with that of O2 and CO2. Furthermore, the model accounts for the competitive nonlinear binding of O2 and CO2 with the hemoglobin inside the red blood cells (nonlinear O2-CO2 interactions, Bohr and Haldane effects), and myoglobin-facilitated transport of O2 inside the parenchymal cells. The consumption of O2 through cytochrome-c oxidase reaction inside the parenchymal cells is based on Michaelis-Menten kinetics. The corresponding production of CO2 is determined by respiratory quotient (RQ), depending on the relative consumption of carbohydrate, protein, and fat. The model gives a physiologically realistic description of O2 transport and metabolism in the microcirculation of the heart. Furthermore, because model solutions for tracer transients and steady states can be computed highly efficiently, this model may be the preferred vehicle for routine data analysis where repetitive solutions and parameter optimization are required, as is the case in PET imaging for estimating myocardial O2 consumption.

Computational evidence for protein-mediated fatty acid transport across the sarcolemma

Long-chain fatty acids (FAs) are important substrates used by the heart to fulfil its energy requirements. Prior to mitochondrial oxidation, blood-borne FAs must pass through the cell membrane of the cardiac myocyte (sarcolemma). The mechanism underlying the sarcolemmal transport of FAs is incompletely understood. The aim of the present study was to estimate the trans-sarcolemmal FA uptake rate using a comprehensive computer model, in which the most relevant mechanisms proposed for cardiac FA uptake were incorporated. Our in silico findings show that diffusion of FA, present in its unbound form (uFA) in close proximity to the outer leaflet of the sarcolemma and serving as sole FA source, is insufficient to account for the physiological FA uptake rate. The inclusion of a hypothetical membrane-associated FA-TFPC (FA-transport-facilitating protein complex) in the model calculations substantially increased the FA uptake rate across the sarcolemma. The model requires that the biological properties of the FA-TFPC allow for increasing the rate of absorption of FA into the outer leaflet and the 'flip-flop' rate of FA from the outer to the inner leaflet of the sarcolemma. Experimental studies have identified various sarcolemma-associated proteins promoting cardiac FA uptake. It remains to be established whether these proteins possess the properties predicted by our model. Our findings also indicate that albumin receptors located on the outer leaflet of the sarcolemma facilitate the transfer of FA across the membrane to a significant extent. The outcomes of the computer simulations were verified with physiologically relevant FA uptake rates as assessed in the intact, beating heart in experimental studies.

Multiscale modeling of cardiac cellular energetics

Multiscale modeling is essential to integrating knowledge of human physiology starting from genomics, molecular biology, and the environment through the levels of cells, tissues, and organs all the way to integrated systems behavior. The lowest levels concern biophysical and biochemical events. The higher levels of organization in tissues, organs, and organism are complex, representing the dynamically varying behavior of billions of cells interacting together. Models integrating cellular events into tissue and organ behavior are forced to resort to simplifications to minimize computational complexity, thus reducing the model's ability to respond correctly to dynamic changes in external conditions. Adjustments at protein and gene regulatory levels shortchange the simplified higher-level representations. Our cell primitive is composed of a set of subcellular modules, each defining an intracellular function (action potential, tricarboxylic acid cycle, oxidative phosphorylation, glycolysis, calcium cycling, contraction, etc.), composing what we call the "eternal cell," which assumes that there is neither proteolysis nor protein synthesis. Within the modules are elements describing each particular component (i.e., enzymatic reactions of assorted types, transporters, ionic channels, binding sites, etc.). Cell subregions are stirred tanks, linked by diffusional or transporter-mediated exchange. The modeling uses ordinary differential equations rather than stochastic or partial differential equations. This basic model is regarded as a primitive upon which to build models encompassing gene regulation, signaling, and long-term adaptations in structure and function. During simulation, simpler forms of the model are used, when possible, to reduce computation. However, when this results in error, the more complex and detailed modules and elements need to be employed to improve model realism. The processes of error recognition and of mapping between different levels of model form complexity are challenging but are essential for successful modeling of large-scale systems in reasonable time. Currently there is to this end no established methodology from computational sciences.

Distributed diffusion-clearance model for transient drug distribution within the skin

Quantitative predictions of molecular transport rates through the skin are key to the development of topically applied and transdermally delivered drugs, as well as risk assessment associated with dermal exposure. Most research to date has focused on correlations for the permeability of the stratum corneum, and transient diffusion models that oversimplify vascular clearance processes in terms of a perfect-removal boundary condition at an artificially introduced lower boundary. Considerations of the spatially distributed nature and action of blood vessels have usually been limited to the steady-state case. This article describes a more comprehensive transient model of percutaneous absorption formulated in terms of volumetric dispersion and clearance coefficients reflecting the spatial distribution of vascular processes. The model was implemented through an analysis of published experimental results on in vivo permeation of salicylic acid (SA) in de-epidermized rat skin. With regard to the characterization of SA in rat dermis ("de") in vivo, it was found that: (i) SA is likely to have a dermal effective partition coefficient (relative to pH 7.4 aqueous buffer "pH7.4") around unity (K(de/pH7.4) = 0.9 +/- 0.3); (ii) vascular processes seem not to increase drug dispersion significantly beyond molecular diffusion [D(de) approximately (D(de))(mol) = (8 +/- 3) . 10(-7) cm(2) s(-1)]; and (iii) vascular clearance is characterized by a rate coefficient k(de) = (7 +/- 2) . 10(-4) s(-1). Application of a whole-skin variant of the model (including the stratum corneum and viable epidermis) allowed realistic predictions to be made of transient subsurface concentration levels after application from a finite dose.

Rapid flip-flop of oleic acid across the plasma membrane of adipocytes

Nonesterified long-chain fatty acids may enter cells by free diffusion or by membrane protein transporters. A requirement for proteins to transport fatty acids across the plasma membrane would imply low partitioning of fatty acids into the membrane lipids, and/or a slower rate of diffusion (flip-flop) through the lipid domains compared to the rates of intracellular metabolism of fatty acids. We used both vesicles of the plasma membrane of adipocytes and intact adipocytes to study transmembrane fluxes of externally added oleic acid at concentrations below its solubility limit at pH 7.4. Binding of oleic acid to the plasma membrane was determined by measuring the fluorescent fatty acid-binding protein ADIFAB added to the external medium. Changes in internal pH caused by flip-flop and metabolism were measured by trapping a fluorescent pH indicator in the cells. The metabolic end products of oleic acid were evaluated over the time interval required for the return of intracellular pH to its initial value. The primary findings were that (i) oleic acid rapidly binds with high avidity in the lipid domains of the plasma membrane with an apparent partition coefficient similar to that of protein-free phospholipid bilayers; (ii) oleic acid rapidly crosses the plasma membrane by the flip-flop mechanism (both events occur within 5 s); and (iii) the kinetics of esterification of oleic acid closely follow the time dependence of the recovery of intracellular pH. Any postulated transport mechanism for facilitating translocation of fatty acid across the plasma membrane of adipocytes, including a protein transporter, would have to compete with the highly effective flip-flop mechanism.