Fluorescence Correlation Spectroscopy Measures Molecular Transport in Cells

Fluorescence correlation spectroscopy (FCS) can measure dynamics of fluorescent molecules in cells. FCS measures the fluctuations in the number of fluorescent molecules in a small volume illuminated by a thin beam of excitation light. These fluctuations are processed statistically to yield an autocorrelation function from which rates of diffusion, convection, chemical reaction, and other processes can be extracted. The advantages of this approach include the ability to measure the mobility of a very small number of molecules, even down to the single molecule level, over a wide range of rates in very small regions of a cell. In addition to rates of diffusion and convection, FCS also provides unique information about the local concentration, states of aggregation and molecular interaction using fluctuation amplitude and cross-correlation methods. Recent advances in technology have rendered these once difficult measurements accessible to routine use in cell biology and biochemistry. This review provides a summary of the FCS method and describes current areas in which the FCS approach is being extended beyond its original scope.     

Mass transport in a microchannel enzyme reactor with a porous wall: Hydrodynamic modeling and applications

A two-dimensional flow model, incorporating mass transport, has been developed to simulate a microchannel enzyme reactor with a porous wall. A two-domain approach based on the finite volume method was implemented. Two parameters are defined to characterize the mass transports in the fluid and porous regions: the porous Damkohler number and the fluid Damkohler number. For reactions close to first-order type (enzyme reactor), the concentration results are found to be well correlated by the use of a reaction–convection distance parameter which incorporates the effects of axial distance, substrate consumption and convection. The reactor efficiency reduces with reaction–convection distance parameter because of reduced reaction (or flux) due to the lower concentration. Increased fluid convection improves the efficiency but it is limited by the diffusion in the fluid region. The correlated results can find applications for the design of enzyme reactors with a porous wall.     

Fluorescence correlation spectroscopy. I. Conceptual basis and theory

We describe a method for determining chemical kinetic constants and diffusion coefficients by measuring the rates of decay of spontaneous concentration fluctuations. The equilibrium of the system is not disturbed during the measurement. We measure the number of molecules of a specified type in a defined open volume as a function of time and compute the time course of the deviations from the thermodynamic mean concentration. The method is based on the principle that the rates of decay of spontaneous microscopic fluctuations are determined by the same phenomenological rate coefficients as those of macroscopic departures from equilibrium which result from external perturbations. Hence, an analysis of fluctuations yields the same chemical rate constants and diffusion coefficients as are measured by conventional procedures. In practice the number of the specified molecules is measured by a property such as absorbance or fluorescence which is specific and sensitive to chemical change. The sample volume is defined by a light beam which traverses the cell. As the molecules appear in or disappear from the light beam, either due to diffusion or chemical reaction, their concentration fluctuations give rise to corresponding fluctuations of the intensity of absorbed or emitted light. This paper presents the theory needed to derive chemical rate constants and diffusion coefficients from these fluctuations in light intensity. The theory is applied to three examples of general interest: pure diffusion in the absence of chemical reaction; the binding of a small rapidly diffusing ligand to a larger slowly diffusing macromolecule; and a unimolecular isomerization. The method should be especially useful in studying highly cooperative systems, relatively noncooperative systems with intermediate states closely spaced in free energy, small systems, and systems not readily subject to perturbations of state.     

Optimization and mathematical modeling of the transtubular bioreactor for the production of monoclonal antibodies from a hybridoma cell line

This report describes the use of a transtubular bioreactor to study the relative effects of diffusion versus perfusion of medium on antibody production by a hybridoma cell line. The study was performed with a high-density cell culture maintained in a serum-free, low-protein medium for 77 days. It was determined that the reactor possessed a macro-mixing pattern residence time distribution similar to a continuous stirred tank reactor (CSTR). However, due to the arrangement of the medium lines in the reactor, the flow patterns for nutrient distribution consist of largely independent medium path lengths ranging from short to long. When operated with cyclic, reversing, transtubular medium flow, some regions of the reactor (with short residence times) are more accessible to medium than others (with long residence times). From this standpoint, the reactor can be divided into three regions: a captive volume, which consists of medium primarily delivered via diffusion; a lapped volume, which provides nutrients through unilateral convection; and a swept volume, which operates through bilateral convection. The relative sizes of these three volumes were modified experimentally by changing the period over which the direction of medium flow was reversed from 15 min (larger captive volume) to 9 h (larger swept volume). The results suggest that antibody concentration increases as the size of the diffusion-limited (captive) volume is increased to a maximum at around 30 min with a sharp decrease thereafter. As reflected by changes in measured consumption of glucose and production of lactate, no significant difference in cellular metabolism occurred as the reactor was moved between these different states. These results indicate that the mode of operation of the transtubular bioreactor may influence antibody productivity under serum-free, low-protein conditions with minimal effects on cellular metabolism.     

Extrasynaptic transmission and the diffusion parameters of the extracellular space

Extrasynaptic volume transmission, mediated by the diffusion of neuroactive substances in the extracellular space (ECS), plays an important role in short- and long-distance communication between nerve cells. The ability of a substance to reach extrasynaptic high-affinity receptors via diffusion depends on the ECS diffusion parameters, ECS volume fraction alpha (alpha=ECS volume/total tissue volume) and tortuosity lambda (lambda2=free/apparent diffusion coefficient), which reflects the presence of diffusion barriers represented by, e.g., fine astrocytic processes or extracellular matrix molecules. These barriers channel the migration of molecules in the ECS, so that diffusion may be facilitated in a certain direction, i.e. anisotropic. The diffusion parameters alpha and lambda differ in various brain regions, and diffusion in the CNS is therefore inhomogeneous. Changes in diffusion parameters have been found in many physiological and pathological states, such as development and aging, neuronal activity, lactation, ischemia, brain injury, degenerative diseases, tumor growth and others, in which cell swelling, glial remodeling and extracellular matrix changes are key factors influencing diffusion. Changes in ECS volume, tortuosity and anisotropy significantly affect the accumulation and diffusion of neuroactive substances and thus extrasynaptic transmission, neuron-glia communication, mediator "spillover" and synaptic crosstalk as well as, cell migration. The various changes occurring during pathological states can be important for diagnosis, drug delivery and treatment.     

Effectiveness factor in biological external convection: study in high viscosity systems

A study regarding the influence of mixing on the efficiency of alginate and xanthan synthesis was carried out. The effectiveness of these two systems in terms of a dimensionless reaction convection number (N(RC)), which is a function of the power input per unit volume, was analysed and compared with low viscosity systems. The results revealed that a decrease in the power input caused a reduction in the growth rate as well as in production rate. It was observed that the effectiveness factor (eta) both for alginate production and xanthan synthesis resulted weakly dependent on the biomass content. A good correlation between effectiveness and N(RC) was obtained for the two tested models. N(RC) number could be appropriately employed in correlating the effectiveness factor for processes with viscosities from 0.001 to more than 1 kg m(-1) s(-1). Due to the parallelism with the conventional internal diffusion approach (Thiele modulus), the advantages for applying N(RC), in particular to correlate the efficiency in systems limited both for external convection and internal diffusional resistance, are shown.     

Astrocytes and extracellular matrix in extrasynaptic volume transmission

Volume transmission is a form of intercellular communication that does not require synapses; it is based on the diffusion of neuroactive substances across the brain extracellular space (ECS) and their binding to extrasynaptic high-affinity receptors on neurons or glia. Extracellular diffusion is restricted by the limited volume of the ECS, which is described by the ECS volume fraction α, and the presence of diffusion barriers, reflected by tortuosity λ, that are created, for example, by fine astrocytic processes or extracellular matrix (ECM) molecules. Organized astrocytic processes, ECM scaffolds or myelin sheets channel the extracellular diffusion so that it is facilitated in a certain direction, i.e. anisotropic. The diffusion properties of the ECS are profoundly influenced by various processes such as the swelling and morphological rebuilding of astrocytes during either transient or persisting physiological or pathological states, or the remodelling of the ECM in tumorous or epileptogenic tissue, during Alzheimer''s disease, after enzymatic treatment or in transgenic animals. The changing diffusion properties of the ECM influence neuron–glia interaction, learning abilities, the extent of neuronal damage and even cell migration. From a clinical point of view, diffusion parameter changes occurring during pathological states could be important for diagnosis, drug delivery and treatment.     

Mass transfer effects on DNA hybridization in a flow-through microarray

The goal of this study is to assess the influence of mass transfer phenomena on DNA hybridization kinetics in a flow-through, porous microarray for fast molecular testing. We present a scaled mathematical model of coupled convection, diffusion and reaction in porous media, which was used to simulate hybridization kinetics and to analyze the influence of convective transport on the reaction rate. In addition to computer simulations, we also present experimental data of hybridization collected on our microarray system for different flow rates. The results reported in this paper provide for a better understanding of the interaction between reaction and mass transfer processes during flow-through hybridization and suggest criteria for system design and optimization.     

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.     

Convective and diffusive gas transport in canine intrapulmonary airways.

The significance of convective and diffusive gas transport in the respiratory system was assessed from the response of combined inert gas and particle boluses inhaled into the conducting airways. Particles, considered as "nondiffusing gas," served as tracers for convection and two inert gases with widely different diffusive characteristics (He and SF6) as tracers for convection and diffusion. Six-milliliter boluses labeled with monodisperse di-2-ethylhexyl sebacate droplets of 0.86-microns aerodynamic diameter, 2% He, and 2% SF6 were inspired by three anesthetized mechanically ventilated beagle dogs to volumetric lung depths up to 170 ml. Mixing between inspired and residual air caused dispersion of the inspired bolus, which was quantified in terms of the bolus half-width. Dispersion of particles increased with increasing lung depth to which the boluses were inhaled. The increase followed a power law with exponents less than 0.5 (mean 0.39), indicating that the effect of convective mixing per unit volume was reduced with depth. Within the pulmonary dead space, the behavior of the inert gases He and SF6 was similar to that of the particles, suggesting that gas transport was almost solely due to convection. Beyond the dead space, dispersion of He and SF6 increased more rapidly than dispersion of particles, indicating that diffusion became significant. The gas and particle bolus technique offers a suitable approach to differential analysis of gas transport in intrapulmonary airways of lungs.     

Transport of O2 along a model pathway through the respiratory region of the lung

A pathway through the system of branching in the respiratory region of the lung is modelled by a circular cylinder, closed at one end, with partitions which define the component respiratory units. In this model the transport of O₂ during inspiration, generated by diffusion is compared with that produced by diffusion together with convection and the importance of convection in the respiratory region in promoting oxygen uptake at the alveolar wall is discussed. For this discussion it is only necessary to consider inspiration. The equations are solved numerically for flow rates of 10, 85 and 200 liters/min. O₂ uptake at the wall and curves of constant O₂ concentration are shown to illustrate the influence of convection. It is found that after a 2 sec inspiration from an O₂ tension of 98 mm Hg and a lung volume of 2300 ml, convection is about 12 per cent as important as diffusion at a flow rate of 85 liters/min, whereas at 10 liters/min convection is only about 0.4 per cent as important as diffusion.     

Role of nutrient supply on cell growth in bioreactor design for tissue engineering of hematopoietic cells

In the present study, a dynamic mathematical model for the growth of granulocyte progenitor cells in the hematopoietic process is developed based on the principles of diffusion and chemical reaction. This model simulates granulocyte progenitor cell growth and oxygen consumption in a three-dimensional (3-D) perfusion bioreactor. Material balances on cells are coupled to the nutrient balances in 3-D matrices to determine the effects of transport limitations on cell growth. The method of volume averaging is used to formulate the material balances for the cells and the nutrients in the porous matrix containing the cells. All model parameters are obtained from the literature. The maximum cell volume fraction reached when oxygen is depleted in the cell layer at 15 days and is nearly 0.63, corresponding to a cell density of 2.25 x 10(8) cells/mL. The substrate inhibition kinetics for cell growth lead to complex effects with respect to the roles of oxygen concentration and supply by convection and diffusion on cell growth. Variation in the height of the liquid layer above the cell matrix where nutrient supply is introduced affected the relative and absolute amounts of oxygen supply by hydrodynamic flow and by diffusion across a gas permeable FEP membrane. Mass transfer restrictions of the FEP membrane are considerable, and the supply of oxygen by convection is essential to achieve higher levels of cell growth. A maximum growth rate occurs at a specific flow rate. For flow rates higher than this optimal, the high oxygen concentration led to growth inhibition and for lower flow rates growth limitations occur due to insufficient oxygen supply. Because of the nonlinear effects of the autocatalytic substrate inhibition growth kinetics coupled to the convective transport, the rate of growth at this optimal flow rate is higher than that in a corresponding well-mixed reactor where oxygen concentration is set at the maximum indicated by the inhibitory kinetics.     

Extracellular space diffusion and extrasynaptic transmission

The diffusion of neuroactive substances in the extracellular space (ECS) plays an important role in short- and long-distance communication between nerve cells and is the underlying mechanism of extrasynaptic (volume) transmission. The diffusion properties of the ECS are described by three parameters: 1. ECS volume fraction alpha (alpha=ECS volume/total tissue volume), 2. tortuosity lambda (lambda2=free/apparent diffusion coefficient), reflecting the presence of diffusion barriers represented by, e.g., fine neuronal and glial processes or extracellular matrix molecules and 3. nonspecific uptake k'. These diffusion parameters differ in various brain regions, and diffusion in the CNS is therefore inhomogeneous. Moreover, diffusion barriers may channel the migration of molecules in the ECS, so that diffusion is facilitated in a certain direction, i.e. diffusion in certain brain regions is anisotropic. Changes in the diffusion parameters have been found in many physiological and pathological states in which cell swelling, glial remodeling and extracellular matrix changes are key factors influencing diffusion. Changes in ECS volume, tortuosity and anisotropy significantly affect the accumulation and diffusion of neuroactive substances in the CNS and thus extrasynaptic transmission, neuron-glia communication, transmitter "spillover" and synaptic cross-talk as well as cell migration, drug delivery and treatment.     

Signaling in small subcellular volumes. II. Stochastic and diffusion effects on synaptic network properties

The synaptic signaling network is capable of sophisticated cellular computations. These include the ability to respond selectively to different patterns of input, and to sustain changes in response over long periods. The small volume of the synapse complicates the analysis of signaling because the chemical environment is strongly affected by diffusion and stochasticity. This study is based on an updated version of a previously proposed synaptic signaling circuit (Bhalla and Iyengar, 1999) and analyzes three network computation properties in small volumes: bistability, thresholding, and pattern selectivity. Simulations show that although there are diffusive regimes in which bistability may persist, chemical noise at small volumes overwhelms bistability. In the deterministic situation, the network exhibits a sharp threshold for transition between lower and upper stable states. This transition is broadened and individual runs partition between lower and upper states, when stochasticity is considered. The third network property, pattern selectivity, is severely degraded at synaptic volumes. However, there are regimes in which a process similar to stochastic resonance operates and amplifies pattern selectivity. These results imply that simple scaling of signaling conditions to femtoliter volumes is unlikely, and microenvironments, such as reaction complex formation, may be essential for reliable small-volume signaling.     

Turing instabilities in a mathematical model for signaling networks

GTPase molecules are important regulators in cells that continuously run through an activation/deactivation and membrane-attachment/membrane-detachment cycle. Activated GTPase is able to localize in parts of the membranes and to induce cell polarity. As feedback loops contribute to the GTPase cycle and as the coupling between membrane-bound and cytoplasmic processes introduces different diffusion coefficients a Turing mechanism is a natural candidate for this symmetry breaking. We formulate a mathematical model that couples a reaction–diffusion system in the inner volume to a reaction–diffusion system on the membrane via a flux condition and an attachment/detachment law at the membrane. We present a reduction to a simpler non-local reaction–diffusion model and perform a stability analysis and numerical simulations for this reduction. Our model in principle does support Turing instabilities but only if the lateral diffusion of inactivated GTPase is much faster than the diffusion of activated GTPase.     

Fluorescence correlation spectroscopy. II. An experimental realization

This paper describes the first experimental application of fluorescence correlation spectroscopy, a new method for determining chemical kinetic constants and diffusion coefficients. These quantities are measured by observing the time behaviour of the tiny concentration fluctuations which occur spontaneously in the reaction system even when it is in equilibrium. The equilibrium of the system is not disturbed during the experiment. The diffusion coefficients and chemical rate constants which determine the average time behaviour of these spontaneous fluctuations are the same as those sought by more conventional methods including temperature-jump or other perturbation techniques. The experiment consists essentially in measuring the variation with time of the number of molecules of specified reactants in a defined open volume of solution. The concentration of a reactant is measured by its fluorescence; the sample volume is defined by a focused laser beam which excites the fluorescence. The fluorescent emission fluctuates in proportion with the changes in the number of fluorescent molecules as they diffuse into and out of the sample volume and as they are created or eliminated by the chemical reactions. The number of these reactant molecules must be small to permit detection of the concentration fluctuations. Hence the sample volume is small (10^?8 ml) and the concentration of the solutes is low (～ 10^?9 M). We have applied this technique to the study of two prototype systems: the simple example of pure diffusion of a single fluorescent species, rhodamine 6G, and the more interesting but more challenging example of the reaction of macromolecular DNA with the drug ethidium bromide to form a fluorescent complex. The increase of the fluorescence of the ethidium bromide upon formation of the complex permits the observation of the decay of concentration fluctuations via the chemical reaction and consequently the determination of chemical rate constants.     

Effect of breathing pattern on gas mixing in a model with asymmetrical alveolar ducts

A model of the pulmonary airways was used to study three single-breath indices of gas mixing, dead space (VD), slope of the alveolar plateau, and alveolar mixing inefficiency (AMI). In the model, discrete elements of airway volume were represented by nodes. Using a finite difference technique the differential equation for simultaneous convection and diffusion was solved for the nodal network. Conducting airways and respiratory bronchioles were modeled symmetrically, but alveolar ducts asymmetrically, permitting interaction between convection and diffusion. VD, alveolar slope, and AMI increased with increasing flow. Similar trends were seen with inspired volume, although slope decreased at high inspired volumes with constant flow. VD was affected most by inspiratory flow and AMI and alveolar slope by expiratory time. VD fell approximately exponentially with time of breath holding. Eight different breathing patterns were compared. They had a small effect on alveolar slope and AMI and a greater effect on VD. The model shows how series and parallel inhomogeneity occur together and interact in asymmetrical systems: the old argument as to which is the more important should be abandoned.     

Heat transfer from spheres and other animal forms.

A general predictive relation for the convection heat transfer from animal forms is developed. This relation is based on the convection equation for a sphere, and employs a simple, unique characteristic dimension to represent the animal which is the cube root of the animal volume. The accuracy of this relation is established through comparison with available convection results from animal shapes ranging in size and shape from spiders to cows. This relation allows an extrapolation to animal shapes for which data are not available. Results are also presented for the enhancement of convection heat transfer due to natural turbulence. A procedure is outlined for estimating the convecture heat loss from an animal in the natural outdoor environment.     

Flow-distributed spikes for Schnakenberg kinetics

We study a system of reaction–diffusion–convection equations which combine a reaction–diffusion system with Schnakenberg kinetics and the convective flow equations. It serves as a simple model for flow-distributed pattern formation. We show how the choice of boundary conditions and the size of the flow influence the positions of the emerging spiky patterns and give conditions when they are shifted to the right or to the left. Further, we analyze the shape and prove the stability of the spikes. This paper is the first providing a rigorous analysis of spiky patterns for reaction-diffusion systems coupled with convective flow. The importance of these results for biological applications, in particular the formation of left–right asymmetry in the mouse, is indicated.