Synchronization of Metabolic Oscillations:Two Cells and Ensembles of Adsorbed Cells

We treat synchronization of metabolic oscillations in two cellsand in ensembles of cells adsorbed at the liquid-solid interface.(i) Synchronization of oscillations in two cells is assumedto occur via perturbation of the metabolite concentration nearone cell due to the metabolite diffusion flux from another cell.This direct channel of synchronization may be important ifthe distance between two cells is comparable with the cell diameter.The corresponding coupling coefficient is found to be proportionalto the metabolite diffusion coefficient and inversely proportionalto the cell radius and the distance between the cells.(ii) In the case of ensembles of adsorbed cells, synchronizationof oscillations is considered to be indirect, i.e., to occur viathe metabolite concentration formed outside the cells nearthe interface due to metabolite diffusion from the cells. We havederived a general integral equation relating the metaboliteconcentration near the interface with concentrations inside the cells.PACS: 82.37.-j, 82.40.Bj     

Fractal scaling mechanisms in biomembranes

A modified model of the Cohen-Turnbull free volume theory for lateral transport processes in biomembranes is presented. The model which is based on renormalization group theoretical concepts incorporates fractal rather than Markovian diffusion kinetics. It predicts harmonic oscillations in the lateral diffusion coefficient around a dominant power-law trend and clarifies, in addition, recently observed deviations from the Cohen-Turnbull exponential law.     

Spatial dynamics of invasion in sinusoidally varying environments

Range expansions of invading species in homogeneous environments have been extensively studied since the pioneer works by Fisher (Ann Eugen 7:255–369, 1937) and Skellam (Biometrika 38:196–218, 1951). However, environments for living organisms are often fragmented by natural or artificial habitat destruction. Here we address how such environmental heterogeneity affects the range expansion of invading species. We consider a single-species invasion in heterogeneous environments whose habitat parameters vary in a sinusoidal or quasi-sinusoidal manner. Accordingly, Fisher’s model is modified to make the intrinsic growth rate and diffusion coefficient spatially variable. By numerically solving the model, we examine the spatio-temporal pattern of propagating waves, and predict the speed as a function of the amplitude and the wave length of the diffusion coefficient and the intrinsic growth rate. Firstly, the results demonstrate that in the sinusoidally varying environment, if the intrinsic growth rate solely oscillates, the speed increases with increases in the amplitude of oscillation. Conversely, if the diffusion coefficient solely oscillates, the speed decreases with increases in the amplitude of oscillation. When both the intrinsic growth rate and diffusion coefficient oscillate, the speed is synergistically accelerated if the oscillations are in anti-phase, whereas it is decelerated if the oscillations are in same phase. Secondly, the increase in the wave length in either the intrinsic growth rate or the diffusion coefficient leads to decreases in the speed. Thirdly, in the irregularly varying environment, the irregularity in the amplitude of the intrinsic growth rate enhances the speed, while that of the diffusion coefficient attenuates the speed.     

Lateral diffusion in model membranes is independent of the size of the hydrophobic region of molecules.

We have systematically investigated the probe size and shape dependence of lateral diffusion in model dimyristoyl phosphatidylcholine membranes. Linear hydrophobic polymers, which differ in length by an order of magnitude, were used to explore the effect on the lateral diffusion coefficient of hydrodynamic restrictions in the bilayer interior. The polymers employed are isoprenoid alcohols--citronellol, solanesol, and dolichol. Tracer lateral diffusion coefficients were measured by fluorescence photobleaching recovery. Despite the large difference in lengths, the nitrobenzoxadiazole labelled alcohols all diffuse at the rate of lipid self-diffusion (5.0 x 10(-12) m2 s-1, 29 degrees C) in the liquid crystal phase. Companion measurements in isotropic polymer solution, in gel phase lipid membranes and with nonpolar fluorescent polyaromatic hydrocarbons, show a marked dependence of the lateral diffusion coefficient on the probe molecule size. Our results in the liquid crystal phase are in accord with free area theory which asserts that lateral diffusion in the membrane is restricted by the surface-free area. Probe molecules which are significantly longer than the host phospholipid, seven times longer in the case of dolichol, are still restricted in their lateral motion by the surface properties of the bilayer in the liquid crystal phase. Fluorescence quenching experiments indicate that the nitrobenzoxadiazole label does not reside at the aqueous interface, although it must reside in close proximity according to the diffusion measurements.     

Limited rotational diffusion of DPH in human erythrocyte membranes

The rotational diffusion of diphenylhexatriene (DPH) determines its fluorescence depolarization. Time-resolved polarization measurements were used to calculate the coefficient of diffusion of this probe in human crythrocyte ghost membranes on the basis of a diffusion theory of limited rotation. The diffusion coefficient is 5.9 × 10⁷ sec^?1 at 37°C; this was compared with the diffusion coefficient of DPH in liquid paraffin for an estimation of the microviscosity of the membrane bilayer.     

Diffusion coefficient of the cyclic GMP analog 8-(fluoresceinyl)thioguanosine 3'',5'' cyclic monophosphate in the salamander rod outer segment.

Cyclic GMP (cGMP) is the intracellular messenger mediating phototransduction in retinal rods, with its longitudinal diffusion in the rod outer segment (ROS) likely to be a factor in determining light sensitivity. From the kinetics of cGMP-activated currents in the truncated ROS of the salamander (Ambystoma tigrinum), the cGMP diffusion coefficient was previously estimated to be approximately 60 x 10(-8) cm2 s-1. On the other hand, fluorescence measurements in intact salamander ROS using 8-(fluoresceinyl)thioguanosine 3',5'-cyclic monophosphate (Fl-cGMP) led to a diffusion coefficient for this compound of 1 x 10(-8) cm2 s-1; after corrections for differences in size and in binding to cellular components between cGMP and Fl-cGMP, this gave an upper limit of 11 x 10(-8) cm2 s-1 for the cGMP diffusion coefficient. To properly compare the two sets of measurements, we have examined the diffusion of Fl-cGMP in the truncated ROS. From the kinetics of Fl-cGMP-activated currents, we have obtained a diffusion coefficient of 3 x 10(-8) cm2 s-1 for this analog; the cGMP diffusion coefficient measured from the same truncated ROSs was approximately 80 x 10(-8) cm2 s-1. Thus, a factor of 27 appears appropriate for correcting differences in size and intracellular binding between cGMP and Fl-cGMP. Application of this correction factor to the Fl-cGMP diffusion coefficient measurements by Olson and Pugh (1993) gives a cGMP diffusion coefficient of approximately 30 x 10(-8) cm2 s-1, in reasonable agreement with the value measured from the truncated ROS.     

Lateral diffusion in an archipelago. Distance dependence of the diffusion coefficient.

An understanding of the distance dependence of the lateral diffusion coefficient is useful in comparing the results of diffusion measurements made over different length scales, and in analyzing the kinetics of mobile redox carriers in organelles. A distance-dependent, concentration-dependent diffusion coefficient is defined, and it is evaluated by Monte Carlo calculations of a random walk by mobile point tracers in the presence of immobile obstacles on a triangular lattice, representing the diffusion of a lipid or a small protein in the presence of immobile membrane proteins. This work confirms and extends the milling crowd model of Eisinger, J., J. Flores, and W. P. Petersen (1986. Biophys J. 49:987-1001). Similar calculations for diffusion of mobile particles interacting by a hard-core repulsion yield the distance dependence of the self-diffusion coefficient. An expression for the range of short-range diffusion is obtained, and the distance scales for various diffusion measurements are summarized.     

Single-image diffusion coefficient measurements of proteins in free solution

Diffusion coefficient measurements are important for many biological and material investigations, such as studies of particle dynamics and kinetics, and size determinations. Among current measurement methods, single particle tracking (SPT) offers the unique ability to simultaneously obtain location and diffusion information about a molecule while using only femtomoles of sample. However, the temporal resolution of SPT is limited to seconds for single-color-labeled samples. By directly imaging three-dimensional diffusing fluorescent proteins and studying the widths of their intensity profiles, we were able to determine the proteins' diffusion coefficients using single protein images of submillisecond exposure times. This simple method improves the temporal resolution of diffusion coefficient measurements to submilliseconds, and can be readily applied to a range of particle sizes in SPT investigations and applications in which diffusion coefficient measurements are needed, such as reaction kinetics and particle size determinations.     

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.     

The steady-state transport of oxygen through hemoglobin solutions

The steady-state transport of oxygen through hemoglobin solutions was studied to identify the mechanism of the diffusion augmentation observed at low oxygen tensions. A novel technique employing a platinum-silver oxygen electrode was developed to measure the effective diffusion coefficient of oxygen in steady-state transport. The measurements were made over a wider range of hemoglobin and oxygen concentrations than previously reported. Values of the Brownian motion diffusion coefficient of oxygen in hemoglobin solution were obtained as well as measurements of facilitated transport at low oxygen tensions. Transport rates up to ten times greater than ordinary diffusion rates were found. Predictions of oxygen flux were made assuming that the oxyhemoglobin transport coefficient was equal to the Brownian motion diffusivity which was measured in a separate set of experiments. The close correlation between prediction and experiment indicates that the diffusion of oxyhemoglobin is the mechanism by which steady-state oxygen transport is facilitated.     

In situ effective diffusion coefficient profiles in live biofilms using pulsed‐field gradient nuclear magnetic resonance

Diffusive mass transfer in biofilms is characterized by the effective diffusion coefficient. It is well documented that the effective diffusion coefficient can vary by location in a biofilm. The current literature is dominated by effective diffusion coefficient measurements for distinct cell clusters and stratified biofilms showing this spatial variation. Regardless of whether distinct cell clusters or surface‐averaging methods are used, position‐dependent measurements of the effective diffusion coefficient are currently: (1) invasive to the biofilm, (2) performed under unnatural conditions, (3) lethal to cells, and/or (4) spatially restricted to only certain regions of the biofilm. Invasive measurements can lead to inaccurate results and prohibit further (time‐dependent) measurements which are important for the mathematical modeling of biofilms. In this study our goals were to: (1) measure the effective diffusion coefficient for water in live biofilms, (2) monitor how the effective diffusion coefficient changes over time under growth conditions, and (3) correlate the effective diffusion coefficient with depth in the biofilm. We measured in situ two‐dimensional effective diffusion coefficient maps within Shewanella oneidensis MR‐1 biofilms using pulsed‐field gradient nuclear magnetic resonance methods, and used them to calculate surface‐averaged relative effective diffusion coefficient (D_rs) profiles. We found that (1) D_rs decreased from the top of the biofilm to the bottom, (2) D_rs profiles differed for biofilms of different ages, (3) D_rs profiles changed over time and generally decreased with time, (4) all the biofilms showed very similar D_rs profiles near the top of the biofilm, and (5) the D_rs profile near the bottom of the biofilm was different for each biofilm. Practically, our results demonstrate that advanced biofilm models should use a variable effective diffusivity which changes with time and location in the biofilm. Biotechnol. Bioeng. 2010;106: 928–937. © 2010 Wiley Periodicals, Inc.     

Gas dispersion in volume-cycled tube flow. II. Tracer bolus experiments.

We present a new method for rapid measurement of local gas dispersion in volume-cycled tube flow. After a small bolus of tracer gas (argon) was injected into the oscillating flow, the time-averaged effective diffusion coefficient (mean value of Deff/D) for axial transport of a tracer gas is evaluated from local argon concentration measurements taken by a mass spectrometer. Two methods are presented for the evaluation of mean value of Deff/D from the concentration measurements: one uses all the sampled data, and the other uses only the local peaks of the concentration. Experiments were conducted in two tubes (radius = 0.85 or 1.0 cm) over a range of frequencies (0.42 less than or equal to f less than or equal to 8.5 Hz) and tidal volumes (7 less than or equal to VT less than or equal to 48 ml). The experimental results show very good agreement with the theoretical predictions of Elad et al. (J. Appl. Physiol. 72: 312-320, 1992). In the absence of oscillations (static fluid), the resulting mean value of Deff/D converges to that of molecular diffusion. We also show that concentration data may be acquired at any radial or axial position, not necessarily at the tracer gas injection point, and the resulting mean value of Deff/D is independent of the spatial position of the sampling catheter. This method is of similar accuracy and is substantially faster than previous methods for measuring gas dispersion in oscillatory flows. The rapidity of these measurements may permit this method to be used for the in vivo assessment of gas transport properties within the pulmonary system.     

Single-particle tracking: the distribution of diffusion coefficients.

In single-particle tracking experiments, the diffusion coefficient D may be measured from the trajectory of an individual particle in the cell membrane. The statistical distribution of single-trajectory diffusion coefficients is examined by Monte Carlo calculations. The width of this distribution may be useful as a measure of the heterogeneity of the membrane and as a test of models of hindered diffusion in the membrane. For some models, the distribution of the short-range diffusion coefficient is much narrower than the observed distribution for proteins diffusing in cell membranes. To aid in the analysis of single-particle tracking measurements, the distribution of D is examined for various definitions of D and for various trajectory lengths.     

Shear‐induced diffusion of red blood cells measured with dynamic light scattering‐optical coherence tomography

Quantitative measurements of intravascular microscopic dynamics, such as absolute blood flow velocity, shear stress and the diffusion coefficient of red blood cells (RBCs), are fundamental in understanding the blood flow behavior within the microcirculation, and for understanding why diffuse correlation spectroscopy (DCS) measurements of blood flow are dominantly sensitive to the diffusive motion of RBCs. Dynamic light scattering‐optical coherence tomography (DLS‐OCT) takes the advantages of using DLS to measure particle flow and diffusion within an OCT resolution‐constrained three‐dimensional volume, enabling the simultaneous measurements of absolute RBC velocity and diffusion coefficient with high spatial resolution. In this work, we applied DLS‐OCT to measure both RBC velocity and the shear‐induced diffusion coefficient within penetrating venules of the somatosensory cortex of anesthetized mice. Blood flow laminar profile measurements indicate a blunted laminar flow profile and the degree of blunting decreases with increasing vessel diameter. The measured shear‐induced diffusion coefficient was proportional to the flow shear rate with a magnitude of ~0.1 to 0.5 × 10^?6 mm². These results provide important experimental support for the recent theoretical explanation for why DCS is dominantly sensitive to RBC diffusive motion.      

How Does Intracellular Ca2+ Oscillate: By Chance or by the Clock?

Ca²⁺ oscillations have been considered to obey deterministic dynamics for almost two decades. We show for four cell types that Ca²⁺ oscillations are instead a sequence of random spikes. The standard deviation of the interspike intervals (ISIs) of individual spike trains is similar to the average ISI; it increases approximately linearly with the average ISI; and consecutive ISIs are uncorrelated. Decreasing the effective diffusion coefficient of free Ca²⁺ using Ca²⁺ buffers increases the average ISI and the standard deviation in agreement with the idea that individual spikes are caused by random wave nucleation. Array-enhanced coherence resonance leads to regular Ca²⁺ oscillations with small standard deviation of ISIs.     

Protein-ligand interactions measured by 15N-filtered diffusion experiments

NMR diffusion coefficient measurements have been shown to be sensitive to the conformational and oligomeric states of proteins. Recently, heteronuclear-filtered diffusion experiments have been proposed [Dingley et al. (1997) J. Biomol. NMR, 10, 1–8]. Several new heteronuclear-filtered diffusion pulse sequences are proposed which are shown to have superior sensitivity to those previously proposed. One of these new heteronuclear-filtered diffusion experiments has been used to study the binding of an SH3 domain to a peptide. Using this system, we show that it is possible to measure binding constants from diffusion coefficient measurements.     

Thermal diffusion of a stiff rod-like mutant Y21M fd-virus

We investigated the thermal diffusion phenomena of a rodlike mutant filamentous fd-Y21M virus in the isotropic phase by means of an improved infrared thermal-diffusion-forced Rayleigh scattering (IR-TDFRS) setup optimized for measurements of slowly diffusing systems. Because this is the first thermal diffusion study of a stiff anisotropic solute, we investigate the influence of the shape anisotropy on the thermal diffusion behavior. The influence of temperature, fd-Y21M concentration, and ionic strength in relation with the thermodiffusion properties is discussed. We characterize and eliminate the effect of these parameters on the absolute diffusion of the rods and show that diffusion determines the behavior of the Soret coefficient because the thermal diffusion coefficient is constant in the investigated regime. Our results indicate that for the thermal diffusion behavior structural changes of the surrounding water are more important than structural changes between the charged macroions. In the investigated temperature and concentration range, the fd-Y21M virus is thermophobic for the low salt content, whereas the solutions with the high salt content change from thermophobic to thermophilic behavior with decreasing temperature. A comparison with recent measurements of other charged soft and biological matter systems shows that the shape anisotropy of the fd-virus becomes not visible in the results.     

Permeability characteristics of the basement membrane (basal lamella) of the intestine of Ascaris suum

Permeability characteristics have been determined for isolated ribbons of the basement membrane of the intestine ofAscaris suum. The solute permeability coefficient (P_c) was measured for a series of hydrophobic, nonionic molecules of graded molecular size. The geometric pore area per unit path length (A_o/Δx) was estimated to be 24.0 cm from the diffusion rates for the various solute molecules. A filtration coefficient (L_p) of 18.1×10^?12 cm⁵/dyne-sec was determined by a method that employs osmotic pressure. The preceding values were used to calculate an average pore radius of 24.0 A for the membrane. The unstirred layer was estimated to be 30μm thick from measurements of the change in the rate of diffusion of water across the membrane with change in the rate of perfusion. The preceding values were used to calculate a reflection coefficien (σ), effective permeability coefficient (ω′), and a permeability coefficient (ω). The results support the view that this basement membrane functions as a filter and selective barrier to diffusion of constituents of the worm's body fluid.     

Translational diffusion of bovine prothrombin fragment 1 weakly bound to supported planar membranes: measurement by total internal reflection with fluorescence pattern photobleaching recovery.

Previous work has shown that bovine prothrombin fragment 1 binds to substrate-supported planar membranes composed of phosphatidylcholine (PC) and phosphatidylserine (PS) in a Ca(2+)-specific manner. The apparent equilibrium dissociation constant is 1-15 microM, and the average membrane residency time is approximately 0.25 s-1. In the present work, fluorescence pattern photobleaching recovery with evanescent interference patterns (TIR-FPPR) has been used to measure the translational diffusion coefficients of the weakly bound fragment 1. The results show that the translational diffusion coefficients on fluid-like PS/PC planar membranes are on the order of 10(-9) cm2/s and are reduced when the fragment 1 surface density is increased. Control measurements were carried out for fragment 1 on solid-like PS/PC planar membranes. The dissociation kinetics were similar to those on fluid-like membranes, but protein translational mobility was not detected. TIR-FPPR was also used to measure the diffusion coefficient of the fluorescent lipid NBD-PC in fluid-like PS/PC planar membranes. In these measurements, the diffusion coefficient was approximately 10(-8) cm2/s, which is consistent with that measured by conventional fluorescence pattern photobleaching recovery. This work represents the first measurement of a translational diffusion coefficient for a protein weakly bound to a membrane surface.     

Anisotropic diffusion of fluorescently labeled ATP in rat cardiomyocytes determined by raster image correlation spectroscopy

A series of experimental data points to the existence of profound diffusion restrictions of ADP/ATP in rat cardiomyocytes. This assumption is required to explain the measurements of kinetics of respiration, sarcoplasmic reticulum loading with calcium, and kinetics of ATP-sensitive potassium channels. To be able to analyze and estimate the role of intracellular diffusion restrictions on bioenergetics, the intracellular diffusion coefficients of metabolites have to be determined. The aim of this work was to develop a practical method for determining diffusion coefficients in anisotropic medium and to estimate the overall diffusion coefficients of fluorescently labeled ATP in rat cardiomyocytes. For that, we have extended raster image correlation spectroscopy (RICS) protocols to be able to discriminate the anisotropy in the diffusion coefficient tensor. Using this extended protocol, we estimated diffusion coefficients of ATP labeled with the fluorescent conjugate Alexa Fluor 647 (Alexa-ATP). In the analysis, we assumed that the diffusion tensor can be described by two values: diffusion coefficient along the myofibril and that across it. The average diffusion coefficients found for Alexa-ATP were as follows: 83 +/- 14 microm(2)/s in the longitudinal and 52 +/- 16 microm(2)/s in the transverse directions (n = 8, mean +/- SD). Those values are approximately 2 (longitudinal) and approximately 3.5 (transverse) times smaller than the diffusion coefficient value estimated for the surrounding solution. Such uneven reduction of average diffusion coefficient leads to anisotropic diffusion in rat cardiomyocytes. Although the source for such anisotropy is uncertain, we speculate that it may be induced by the ordered pattern of intracellular structures in rat cardiomyocytes.