Oscillatory Flow Accelerates Autocrine Signaling due to Nonlinear Effect of Convection on Receptor-Related Actions

We study effects of oscillatory convective flow in extracellular space on the velocity of chemical signal propagation having a form of a front wave above a cellular layer. We found that the time-averaged propagation velocity under oscillatory flow for a particular Péclet number amplitude is slower than the velocity under steady laminar flow regime for the same value of the Péclet number, but significantly faster than under no-flow conditions. We derive asymptotic values of the propagation velocity and asymptotic characteristics of the corresponding concentration fronts in high- and low-frequency regimes and show that the reason for the observed velocity increase under the oscillatory flow stems from a nonlinear dependence of the propagation velocity on the Péclet number, particularly from the convex character of the dependence. Our findings suggest that the specific responses of cellular cultures to different flow conditions in the extracellular space (for example, expression of atherosclerosis protective genes under steady laminar flow but not under oscillatory flow) is a consequence of a nonlinear coupling between the extracellular transport and complex intracellular reaction cascades forming a positive feedback loop of the autocrine signaling. This mechanism can operate independently of, or in conjunction with, a direct stress-sensing due to mechanotransduction.     

Oscillatory Flow Accelerates Autocrine Signaling due to Nonlinear Effect of Convection on Receptor-Related Actions

We study effects of oscillatory convective flow in extracellular space on the velocity of chemical signal propagation having a form of a front wave above a cellular layer. We found that the time-averaged propagation velocity under oscillatory flow for a particular Péclet number amplitude is slower than the velocity under steady laminar flow regime for the same value of the Péclet number, but significantly faster than under no-flow conditions. We derive asymptotic values of the propagation velocity and asymptotic characteristics of the corresponding concentration fronts in high- and low-frequency regimes and show that the reason for the observed velocity increase under the oscillatory flow stems from a nonlinear dependence of the propagation velocity on the Péclet number, particularly from the convex character of the dependence. Our findings suggest that the specific responses of cellular cultures to different flow conditions in the extracellular space (for example, expression of atherosclerosis protective genes under steady laminar flow but not under oscillatory flow) is a consequence of a nonlinear coupling between the extracellular transport and complex intracellular reaction cascades forming a positive feedback loop of the autocrine signaling. This mechanism can operate independently of, or in conjunction with, a direct stress-sensing due to mechanotransduction.     

Retinal spreading depression and the extracellular milieu

We used isolated chick retina in vitro to study the participation of the extracellular milieu in the occurrence and propagation of spreading depression. The propagation was followed by visual observation or microphotometry and the ionic changes in the extracellular compartment were recorded with double-barreled ion-selective microelectrodes. The front of the spreading wave is accompanied by increased light scattering in the tissue and by decrease of Cl-, Na+, and Ca2+, increase of K+, and an alkaline-acid shift in the extracellular space, concomitant with the slow voltage changes characteristic of the wave. As the spread is related to the chemical steady-state of the extracellular milieu, the velocity of propagation is influenced by a balanced interplay of the chemical constituents of the superfusing solution, e.g., K+, HCO-3, and glucose facilitate, while Cl- and Mg2+ hinder the wave. Steady-state alterations induced by physical factors (temperature) or related to experimental conditions (speed and direction of superfusate flow) change markedly the velocity of propagation. Generally the procedures that cause increase of velocity augment the susceptibility of the preparation to the reaction and eventually may trigger it. Propagated spreading depression is considered as a chemical diffusion reaction pervading more intensively the inner plexiform layer of the retina.     

Kinetics of ligand binding to receptor immobilized in a polymer matrix, as detected with an evanescent wave biosensor. I. A computer simulation of the influence of mass transport.

The influence of mass transport on ligand binding to receptor immobilized in a polymer matrix, as detected with an evanescent wave biosensor, was investigated. A one-dimensional computer model for the mass transport of ligand between the bulk solution and the polymer gel and within the gel was employed, and the influence of the diffusion coefficient, the partition coefficient, the thickness of the matrix, and the distribution of immobilized receptor were studied for a variety of conditions. Under conditions that may apply to many published experimental studies, diffusion within the matrix was found to decrease the overall ligand transport significantly. For relatively slow reactions, small spatial gradients of free and bound ligand in the gel are found, whereas for relatively rapid reactions strong inhomogeneities of ligand within the gel occur before establishment of equilibrium. Several types of deviations from ideal pseudo-first-order binding progress curves are described that resemble those of published experimental data. Extremely transport limited reactions can in some cases be fitted with apparently ideal binding progress curves, although with apparent reaction rates that are much lower than the true reaction rates. Nevertheless, the ratio of the apparent rate constants can be semiquantitatively consistent with the true equilibrium constant. Apparently "cooperative" binding can result from high chemical on rates at high receptor saturation. Dissociation in the presence of transport limitation was found to be well described empirically by a single or a double exponential, with both apparent rate constants considerably lower than the intrinsic chemical rate constant. Transport limitations in the gel can introduce many generally unknown factors into the binding progress curve. The simulations suggest that unexpected deviations from ideal binding progress curves may be due to highly transport influenced binding kinetics. The use of a thinner polymer matrix could significantly increase the range of detectable rate constants.     

Concentration wave of a solute in an artery: the influence of curvature

Mass transport and diffusion phenomena in the arterial lumen are studied through a mathematical model. Blood flow is described by the unsteady Navier-Stokes equation and solute dynamics by an advection-diffusion equation, the convective field being provided by the fluid velocity. A linearization procedure over the steady state solution is carried out and an asymptotic analysis is used to study the effect of a small curvature with respect to the straight tube. Analytical and numerical solutions are found: the results show the characteristics of the long wave propagation and the role played by the geometry on the solute distribution and demonstrate the strong influence of curvature induced by the fluid dynamics.     

Wave propagation in discrete media

 We have considered infinite systems of nonlinear ODEs on the one-dimensional integer lattice which describes the activity in an excitatorily coupled network of excitable cells. For an ideal nonlinearity, we calculated the speed of propagation of an activity and derived the condition for its existence. We also studied the existence and stability of the traveling wave solution and gave, in the simplest case, its explicit expression. We established that some unstable traveling waves lead to propagation with an enlarging profile defined by a front velocity and a wake velocity. We generalized some results to inhomogeneous medium and network with long range connections. Received: 3 July 2000 / Revised version: 17 April 2001 / Published online: 7 December 2001     

Analytical solution to the initial-value problem for traveling bands of chemotactic bacteria

The governing parabolic partial differential equations for the diffusion and chemotactic transport of a distribution of bacteria and for the diffusion and bacterial degradation of a distribution of chemotactic agent are supplemented with boundary and initial conditions that model the recent capillary tube experiments on the formation and propagation of traveling bands of chemotactic bacteria. An iteration procedure that takes the exact solution to the “diffusionless” problem as a first approximation is applied to solve the equations of the complete theoretical model. It is shown that satisfactory agreement with experiment obtains for the analytical results of the first approximation which relate the velocity of propagation and total number of bacteria cells per unit cross-sectional area in a traveling band to the constant parameters in the governing equations and supplementary conditions. The second approximation is shown to yield approximate analytical expressions for the solution functions which are in close correspondence with previously derived traveling band solutions for values of time after the initial period of formation.     

Mathematical modelling and computational study of two-dimensional and three-dimensional dynamics of receptor–ligand interactions in signalling response mechanisms

Cell signalling processes involve receptor trafficking through highly connected networks of interacting components. The binding of surface receptors to their specific ligands is a key factor for the control and triggering of signalling pathways. But the binding process still presents many enigmas and, by analogy with surface catalytic reactions, two different mechanisms can be conceived: the first mechanism is related to the Eley–Rideal (ER) mechanism, i.e. the bulk-dissolved ligand interacts directly by pure three-dimensional (3D) diffusion with the specific surface receptor; the second mechanism is similar to the Langmuir–Hinshelwood (LH) process, i.e. 3D diffusion of the ligand to the cell surface followed by reversible ligand adsorption and subsequent two-dimensional (2D) surface diffusion to the receptor. A situation where both mechanisms simultaneously contribute to the signalling process could also occur. The aim of this paper is to perform a computational study of the behavior of the signalling response when these different mechanisms for ligand-receptor interactions are integrated into a model for signal transduction and ligand transport. To this end, partial differential equations have been used to develop spatio-temporal models that show trafficking dynamics of ligands, cell surface components, and intracellular signalling molecules through the different domains of the system. The mathematical modeling developed for these mechanisms has been applied to the study of two situations frequently found in cell systems: (a) dependence of the signal response on cell density; and (b) enhancement of the signalling response in a synaptic environment.     

Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species

Presented is a reaction-diffusion model for the interaction of pioneer and climax species. For certain parameters the system exhibits bistability and traveling wave solutions. Specifically, we show that when the climax species diffuses at a slow rate there are traveling wave solutions which correspond to extinction waves of either the pioneer or climax species. A leading order analysis is used in the one-dimensional spatial case to estimate the wave speed sign that determines which species becomes extinct. Results of these analyses are then compared to numerical simulations of wave front propagation for the model on one and two-dimensional spatial domains. A simple mechanism for harvesting is also introduced.     

The mechanism of propagation of variation potentials in wheat leaves

Here we examined the mechanism of propagation of variation potential (VP) induced by burning in wheat leaves. Participation of hydraulic and chemical mechanisms in VP transmission was analyzed by optical coherent tomography and a radioactive tracer method, respectively. The speed of the hydraulic signal considerably exceeded the VP velocity. Investigation of a chemical substance spreading from the zone of local wounding was based on experimental data for radioactive marker transmission derived with a one-dimensional diffusion equation. The speed of the marker transmission was in accordance with VP velocity. The elimination of the potential transmission of a chemical signal by a timed severing of the leaf between the burn site and the recorded site blocked VP propagation. We suggest that a VP is formed by the transmission of a wound substance, the velocity of which is likely increased by hydraulic wave propagation.     

Ligand-receptor interaction rates in the presence of convective mass transport.

The rate of binding of a ligand to receptors on the cell surface can be diffusion limited. We analyze the kinetics of binding, diffusion-limited in a stationary liquid, in the presence of convective mass transport. We derive a formula that expresses the reaction kinetics in terms of the mass transfer coefficient. A moderately transport-limited kinetics is not readily recognizable from the shape of the binding curve and may lead to erroneous estimates of the rate coefficients. We apply our results to practically important cases: a cell suspension in a stirred volume of liquid and a confluent cell colony under a laminar stream. Using typical numbers characterizing the ligand-receptor interactions, we show that stirring and perfusion can be important factors determining the reaction rates. With the confluent colony, the early reaction kinetics requires a different treatment, and we provide it for the case of low receptor occupancy. We show that, even with a fast perfusion, a cell monolayer can transiently generate a zone of depletion of the ligand, and that would affect the early stages of the reaction. Our results are expressed in a simple analytical form and can be used for the design and interpretation of experimental data.     

Autocrine ligand binding to cell receptors. Mathematical analysis of competition by solution "decoys".

Autocrine ligands have been demonstrated to regulate cell proliferation, cell adhesion, and cell migration in a number of different systems and are believed to be one of the underlying causes of malignant cell transformation. Binding of these ligands to their cellular receptors can be compromised by diffusive transport of ligand away from the secreting cell. Exogenous addition of antibodies or solution receptors capable of competing with cellular receptors for these autocrine ligands has been proposed as a means of inhibiting autocrine-stimulated cell behavioral responses. Such "decoys" complicate cellular binding by offering alternative binding targets, which may also be capable of aiding or abating transport of the ligand away from the cell surface. We present a mathematical model incorporating autocrine ligand production and the presence of competing cellular and solution receptors. We elucidate effects of key system parameters including ligand diffusion rate, binding rate constants, cell density, and secretion rate on the ability of solution receptors to inhibit cellular receptor binding. Both plated and suspension cell systems are considered. An approximate analytical expression relating the key parameters to the critical concentration of solution "decoys" required for inhibition is derived and compared to the numerical calculations. We find that in order to achieve essentially complete inhibition of surface receptor binding, the concentration of decoys may need to be as much as four to eight orders of magnitude greater than the equilibrium disociation constant for ligand binding to surface receptors.     

Nonlinear propagation of spherical calcium waves in rat cardiac myocytes.

Spontaneous calcium waves in enzymatically isolated rat cardiac myocytes were investigated by confocal laser scanning microscopy (CLSM) using the fluorescent Ca2+-indicator fluo-3 AM. As recently shown, a spreading wave of enhanced cytosolic calcium appears, most probably during Ca2+ overload, and is initiated by an elementary event called a "calcium spark." When measured by conventional fluorescence microscopy the propagation velocity of spontaneous calcium waves determined at several points along the cardiac myocyte was previously found to be constant. More precise measurements with a CLSM showed a nonlinear propagation. The wave velocity was low, close to the focus, and increased with increasing time and propagation length, approaching a maximum of 113 microns/s. This result was surprising, inasmuch as for geometrical reasons a decrease of the propagation velocity might be expected if the confocal plane is not identical with that plane where the focus of the wave was localized. It is suggested that the propagation velocity is essentially dependent on the curvature of the spreading wave. From the linear relationship of velocity versus curvature, a critical radius of 2.7 +/- 1.4 microns (mean +/- SD) was worked out, below which an outward propagation of the wave will not take place. Once released from a sufficiently extended cluster of sarcoplasmic reticulum release channels, calcium diffuses and will activate its neighbors. While traveling away, the volume into which calcium diffuses becomes effectively smaller than at low radii. This effect is the consequence of the summation of elementary events (Ca2+ sparks) and leads to a steeper increase of the cytosolic calcium concentration after a certain diffusion path length. Thus the time taken to reach a critical threshold of [Ca2+]i at the neighboring calcium release sites decreases with decreasing curvature and the wave will propagate faster.     

Receptor dissociation constants and the information entropy of membranes coding ligand concentration

The binding of ligands to receptor proteins embedded in cell membranes drives cellular responses that involve either second messenger cascades or directly gated ion channels. It is known that a single class of receptor proteins expresses approximately 98% of its graded response to ligand concentrations over four orders of magnitude, where the response is measured by the equilibrium proportion of bound ligand-receptor complexes. This four-decadic concentration range is centered on a logarithmic scale around logK, where K is the dissociation constant defined by the ratio of ligand-receptor unbinding (k-) to binding (k+) rates. Remarkably, this four-decadic concentration range is intrinsic to all homogeneous ligand-receptor (or, equivalently, enzyme-substrate) systems. Thus, adapting the sensitivity of cell membranes to narrower or wider ranges of ligand concentrations, respectively, requires multivalent receptors or heterogeneous populations of receptors. Here we use a normalized Shannon-Weaver measure of information entropy to represent the efficiency of coding over given concentrations for membranes containing a population of univalent receptors with a specified distribution of dissociation constants, or a homogeneous population of strongly cooperative multivalent receptors. Assuming a specified level of resolution in the response of cellular or neural systems downstream from the membrane that 'read' the ligand concentration 'code', we calculate the range of concentrations over which the coding efficiency of the membrane itself is maximized. Our results can be used to hypothesize the number of receptor types associated with the membranes of particular cells. For example, from data in the literature, we conclude that the response of most general olfactory sensory neurons can be explained in terms of a homogeneous population of receptor proteins, while the response of pheromone sensory neurons is satisfactorily explained by the presence of two types of membrane receptor protein with pheromone-binding dissociation constants that have values at least one to two orders of magnitude apart.     

Spatial distribution and dose-response relationship for different operation modes in a reaction-diffusion model of the MAPK cascade

The mitogen-activated protein kinase (MAPK) cascade plays a critical role in the control of cell growth. Deregulation of this pathway contributes to the development of many cancers. To better understand its signal transduction, we constructed a reaction-diffusion model for the MAPK pathway. We modeled the three layers of phosphorylation-dephosphorylation reactions and diffusion processes from the cell membrane to the nucleus. Based on different types of feedback in the MAPK cascade, four operation modes are introduced. For each of the four modes, spatial distributions and dose-response curves of active kinases (i.e. ppMAPK) are explored by numerical simulation. The effects of propagation length, diffusion coefficient and feedback strength on the pathway dynamics are investigated. We found that intrinsic bistability in the MAPK cascade can generate a traveling wave of ppMAPK with constant amplitude when the propagation length is short. ppMAPK in this mode of intrinsic bistability decays more slowly than it does in all other modes as the propagation length increases. Moreover, we examined the global and local responses to Ras-GTP of these four modes, and demonstrated how the shapes of these dose-response curves change as the propagation length increases. Also, we found that larger diffusion constant gives a higher response level on the zero-order regime and makes the ppMAPK profiles flatter under strong Ras-GTP stimulus. Furthermore, we observed that spatial responses of ppMAPK are more sensitive to negative feedback than to positive feedback in the broader signal range. Finally, we showed how oscillatory signals pass through the kinase cascade, and found that high frequency signals are damped faster than low frequency ones.     

New capabilities in determining the binding parameters for ligand-receptor interaction

A new method for determining the binding parameters of ligand-receptor interaction is suggested. The method is based on the application of the so-called coordinate of dilution, suggested by us earlier. We demonstrated that it is possible to determine the binding characteristics of ligand-receptor interaction using either the measurement of the concentration of the ligand-receptor complex at a state of equilibrium or the concentration of free receptors at different dilutions of the studying ligand-receptor mixture. The method also allows the determination of the concentration of the ligand in a pre-existing ligand-receptor mixture without preliminary separation of the interacting counterparts. For this reason the suggested method could be especially useful when the studying very labile receptors for which purification from the corresponding ligand is very difficult or impossible.     

Velocity-curvature relationship of colliding spherical calcium waves in rat cardiac myocytes.

Colliding spherical calcium waves in enzymatically isolated rat cardiac myocytes develop new wavefronts propagating perpendicular to the original direction. When investigated by confocal laser scanning microscopy (CLSM), using the fluorescent Ca2+ indicator fluo-3 AM, "cusp"-like structures become visible that are favorably approximated by double parabolae. The time-dependent position of the vertices is used to determine propagation velocity and negative curvature of the wavefront in the region of collision. It is evident that negatively curved waves propagate faster than positively curved, single waves. Considering two perfectly equal expanding circular waves, we demonstrated that the collision of calcium waves is due to an autocatalytic process (calcium-induced calcium release), and not to a simple phenomenon of interference. Following the spatiotemporal organization in simpler chemical systems maintained under conditions far from the thermodynamic equilibrium (Belousov-Zhabotinskii reaction), the dependence of the normal velocity on the curvature of the spreading wavefront is given by a linear relation. The so-called velocity-curvature relationship makes clear that the velocity is enhanced by curvature toward the direction of forward propagation and decreased by curvature away from the direction of forward propagation (with an influence of the diffusion coefficient). Experimentally obtained velocity data of both negatively and positively curved calcium waves were approximated by orthogonal weighted regression. The negative slope of the straight line resulted in an effective diffusion coefficient of 1.2 x 10(-4) mm2/s. From the so-called critical radius, which must be exceeded to initiate a traveling calcium wave, a critical volume (with enhanced [Ca2+]i) of approximately 12 microm3 was calculated. This is almost identical to the volume that is occupied by a single calcium spark.     

Free boundary morphogenesis in living matter

Morphogenetic theories investigate the creation and the emergence of form in living organisms. A novel approach for studying free boundary problems during morphogenesis is proposed in this work. The presence of mass fluxes inside a biological system is coupled with the local gradient of diffusing morphogens. The contour stability of a growing material is studied using a two-dimensional system model with a rectilinear free border inside a Hele-Shaw cell. Modeling mass transport during morphogenesis allows fixing the velocity at the traveling wave solution as a function of one-dimensionless parameter. Performing a perturbation of the free boundary, the dispersion relation is derived in an implicit form. Although both the velocity of the moving front and the surface tension act as stabilizing effects at small wavelengths, the dispersion diagrams show that the rectilinear border is always unstable at large wavelengths. Further applications of this model can help give insights into a number of free boundary problems in biological systems.     

Long-Range Signal Transmission in Autocrine Relays

Intracellular signaling induced by peptide growth factors can stimulate secretion of these molecules into the extracellular medium. In autocrine and paracrine networks, this can establish a positive feedback loop between ligand binding and ligand release. When coupled to intercellular communication by autocrine ligands, this positive feedback can generate constant-speed traveling waves. To demonstrate that, we propose a mechanistic model of autocrine relay systems. The model is relevant to the physiology of epithelial layers and to a number of in vitro experimental formats. Using asymptotic and numerical tools, we find that traveling waves in autocrine relays exist and have a number of unusual properties, such as an optimal ligand binding strength necessary for the maximal speed of propagation. We compare our results to recent observations of autocrine and paracrine systems and discuss the steps toward experimental tests of our predictions.     

A theoretical approach to the analysis of axonal transport.

A theoretical model of intra-axonal transport is proposed that presupposes a carrier system moving down the axon in a distal direction. Protein and particle transport is achieved by their reversible association with the distally moving carriers. Mathematical equations representing the concentrations of moving carriers and proteins and/or particles within the axon at any position and time are proposed. Analysis of the equations demonstrates that a traveling wave solution for the particle concentration (an experimental fact) is possible provided the chemical interaction between particles and carriers exhibits positive cooperativity. The phase velocity of the wave solution is interpreted as the observed velocity of the intra-axonal transport, known to be independent of position of observation. In addition, the theory predicts a spectrum of transport velocities for different proteins, in agreement with observations. The velocity of a given protein is dependent on its affinity to the carrier.