Consequences of chemosensory phenomena for leukocyte chemotactic orientation

The stochastic nature of cell surface receptor-ligand binding is known to limit the accuracy of detection of chemoattractant gradients by leukocytes (11, 12), thus limiting the orientation ability that is crucial to the chemotactic response in host defense. The probabilistic cell orientation model of Lauffenburger (11) is extended here to assess the consequences of recently discovered receptor phenomena: “down-regulation” of total surface receptor number, spatial asymmetry of surface receptors, and existence of a higher-affinity receptor subpopulation. In general, a reduction in orientation accuracy is predicted by inclusion of these phenomena. An orientation signal based on a simple model of chemosensory adaptation (i.e., a spatial difference inrelative receptor occupancy) is found to be functionally different from the signal suggested by an experimental correlation (i.e., a spatial difference inabsolute receptor occupancy). However, in the context of receptor “signal noise,” the signal based on adaptation yields predictions in better qualitative agreement with the experimental orientation data of Zigmond (10). From this cell orientation model we can estimate the effective timeaveraging period required for noise diminution to a level allowing orientation predictions to match observed levels. This time-averaging period presumably reflects the time constant for receptor signal transduction and locomotory response.     

Influence of external concentration fluctuations on leukocyte chemotactic orientation

The quantitative dependence of leukocyte chemotactic orientation on imprecision in the measurement of chemoattractant concentrations from thermal fluctuations is analyzed. First, a mathematical model relating orientation to differences in receptor occupancy across cell dimensions is developed. This is then coupled with an extension of Berg and Purcell's analysis (1) of the precision of attractant concentration measurements by means of receptor occupancy. Our results show that thermal fluctuations in external concentrations can limit the accuracy of orientation, unless the measurement noise is reduced by averaging the measurements over a period of time. Comparison of our model predictions to experimental orientation data suggests that leukocytes do overcome this limitation, and allows estimation of the time-averaging period necessary to do so. For the orientation observed in a visual bridge assay by Zigmond (2) using the attractant peptide FNLLP, we estimate that receptor occupancy measurements for spatial comparison across cell dimensions must be averaged for a few minutes. Otherwise, the fluctuations in the attractant concentration near the cell will be too great to allow the observed degree of orientation. Our analysis also suggests that the ratio of signal-to-signal noise does not adequately characterize orientation accuracy. Accurate orientation can, in some situations, occur when this ratio is substantially less than unity; in other situations, a ratio much greater than unity is required for accurate orientation.     

Biased random walk by stochastic fluctuations of chemoattractant-receptor interactions at the lower limit of detection

Binding of ligand to its receptor is a stochastic process that exhibits fluctuations in time and space. In chemotaxis, this leads to a noisy input signal. Therefore, in a gradient of chemoattractant, the cell may occasionally experience a "wrong" gradient of occupied receptors. We obtained a simple equation for P(pos), the probability that half of the cell closest to the source of chemoattractant has higher receptor occupancy than the opposite half of the cell. P(pos) depends on four factors, the gradient property delC/sq. root of C, the receptor characteristic R(t)/K(D), a time-averaging constant I, and nonreceptor noise sigma(B). We measured chemotaxis of Dictyostelium cells to known shallow gradients of cAMP and obtained direct estimates for these constants. Furthermore, we observed that in shallow gradients, the measured chemotaxis index is correlated with P(pos), which suggests that chemotaxis in shallow gradients is a pure biased random walk. From the observed chemotaxis and derived time-averaging constant, we deduce that the gradient transducing second messenger has a lifetime of 2-8 s and a diffusion rate constant of approximately 1 microm(2)/s. Potential candidates for such second messengers are discussed.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis. II. The effects of non-equilibrated ligand binding

Cells generally chemotax along a direction in which their receptor occupancy gradient--whether spatial or temporal--is maximum. Occupancy differentials are, however, often so small as to be masked by thermal noise; i.e., by fluctuations inherent in the stochastic nature of ligand binding. Such fluctuations therefore impose a fundamental limit on the sensitivity of a cell's ability to detect a chemoattractant gradient. In order to pursue the implications of this limit, fluctuation theories have been developed. The theories assume that the signal is some function of the receptor occupancy gradient, allow an estimate of the standard deviation about the mean signal, and permit an evaluation of, among other things, the extent to which a receptor defect can impair an effective response. Previous theories have assumed an equilibrated ligand-receptor interaction. In this paper we introduce a generalized definition of a signal caused by a receptor occupancy gradient that allows us to develop a non-equilibrium theory of thermal noise. We show that previous formulations are a special case of the current development. More specifically, we find the following. Swimming cells subject to Brownian tumbling must generally average their signals over a very long time period to achieve a signal-to-noise ratio less than or equal to 1. Spatial gradient detection is possible with ligand-receptor equilibrium constants less than 10(3)M-1, but since such ligands are rare, theory predicts that tumbling cells will generally not detect gradients by measuring spatial occupancy differentials. These conclusions hold irrespective of whether chemical equilibrium is achieved. For crawling cells not subject to Brownian tumbling, a range of affinities exists in which spatial or temporal gradient detection is possible. In general a spatial mechanism is more efficient for low affinity ligands (dissociation times less than 0.3 s), whereas a temporal mechanism is more efficient for higher K. In this case the detection of gradients in slowly dissociating ligand will be facilitated if signal processing begins prior to chemical equilibration. An important new parameter is indicated by the theory. The definitions of a temporal gradient signal is based on estimating and comparing average occupancy over two time intervals displaced by a time t1. The theory predicts an optimal t1, of order milliseconds, that leads to the shortest minimum averaging time. For t1 values at and longer than the optimum, and for all averaging times exceeding some minimum, the cell will detect a temporal signal.(ABSTRACT TRUNCATED AT 400 WORDS)     

A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations

Two central features of polymorphonuclear leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic "persistence time" between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an "orientation bias" characterizing the fraction of cells moving up the gradient. A coherent picture of cell movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper, we offer the possibility that "noise" in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we develop a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant/receptor binding. This model can simulate cell paths similar to those observed experimentally, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Furthermore, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. Thus, the concept of signal "noise" can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis

Cells generally chemotax along a direction in which their receptor occupancy gradient—whether spatial or temporal—is maximum. Occupancy differentials are, however, often so small as to be masked by thermal noise; i.e., by fluctuations inherent in the stochastic nature of ligand binding. Such fluctuations therefore impose a fundamental limit on the sensitivity of a cell's ability to detect a chemoattractant gradient. In order to pursue the implications of this limit, fluctuation theories have been developed. The theories assume that the signal is some function of the receptor occupancy gradient, allow an estimate of the standard deviation abouts the mean signal, and permit an evaluation of, among other things, the extent to which a receptor defect can impair an effective response. Previous theories have assumed an equilibrated ligand-receptor interaction. In this paper we introduce a generalized definition of a signal caused by a receptor occupancy gradient that allows us to develop a non-equilibrium theory of thermal noise. We show that previous formulations are a special case of the current development. More specifically, we find the following.

1. Swimming cells subject to Brownian tumbling must generally average their signals over a very long time period to achieve a signal-to-noise ratio≤1. Spatial gradient detection is possible with ligand-receptor equilibrium constants<10³ M ^?1, but since such ligands are rare, theory predicts that tumbling cells will generally not detect gradients by measuring spatial occupancy differentials.These conclusions hold irrespective of whether chemical equilibrium is achieved.
2. For crawling cells not subject to Brownian tumbling, a range of affinities exists in which spatial or temporal gradient detection is possible. In general a spatial mechanism is more efficient for low affinity ligands (dissociation times <0.3s), whereas a temporal mechanism is more efficients for higherK. In this case the detection of gradients in slowly dissociating ligand will be facilitated if signal processing begins prior to chemical equilibration.
3. An important new parameter is indicated by the theory. The definitions of a temporal gradient signal is based on estimating and comparing average occupancy over two time intervals displaced by a timet ₁. The theory predicts an optimalt ₁, of order milliseconds, that leads to the shortest minimum averaging time.
4. Fort ₁ values at and longer than the optimum, and for all averaging times exceeding some minimum, the cell will detect a temporal signal.
5. For values oft ₁ at and near the optimum, if the averaging time becomes too long, the cell enters a region of insensitivity in which it can no longer respond.
6. Finally, as the interval between estimates of average occupancy decreases below the optimum, a critical value oft ₁ is reached at which the minimum averaging time undergoes an abrupt transition from a relatively short value to a value five orders of magnitude longer.

The molecular process(es) controllingt ₁ are at present unknown, nor has any attempt been made to identify them since the parameter has not been previously recognized. We speculate that the search for its molecular basis might uncover a highly sensitive control mechanism, with defects in this mechanism predicted to have a far more pronounced effect on the cells behavior than defects in receptor number or affinity.     

Dynamic receptor team formation can explain the high signal transduction gain in Escherichia coli

Evolution has provided many organisms with sophisticated sensory systems that enable them to respond to signals in their environment. The response frequently involves alteration in the pattern of movement, either by directed movement, a process called taxis, or by altering the speed or frequency of turning, which is called kinesis. Chemokinesis has been most thoroughly studied in the peritrichous bacterium Escherichia coli, which has four helical flagella distributed over the cell surface, and swims by rotating them. When rotated counterclockwise the flagella coalesce into a propulsive bundle, producing a relatively straight "run," and when rotated clockwise they fly apart, resulting in a "tumble" which reorients the cell with little translocation. A stochastic process generates the runs and tumbles, and in a chemoeffector gradient, runs that carry the cell in a favorable direction are extended. The cell senses spatial gradients as temporal changes in receptor occupancy and changes the probability of counterclockwise rotation (the bias) on a fast timescale, but adaptation returns the bias to baseline on a slow timescale, enabling the cell to detect and respond to further concentration changes. The overall structure of the signal transduction pathways is well characterized in E. coli, but important details are still not understood. Only recently has a source of gain in the signal transduction network been identified experimentally, and here we present a mathematical model based on dynamic assembly of receptor teams that can explain this observation.     

Ligand-mediated internalization, recycling, and downregulation of the epidermal growth factor receptor in vivo [published erratum appears in J Cell Biol 1990 Jan;110(1):following 227]

EGF receptor internalization, recycling,a nd downregulation were evaluated in liver parenchyma as a function of increasing doses of injected EGF. The effect of ligand occupancy in vivo on the kinetics and extent of internalization was studied with changes in the receptor content of isolated plasmalemma and endosome fractions evaluated by direct binding, Scatchard analysis, and Western blotting. For all doses of injected EGF, receptor was lost from the plasmalemma and accumulated in endosomes in a time- and dose-dependent fashion. However, at doses of injected EGF equivalent to less than or equal to 50% surface receptor occupancy (i.e., less than or equal to 1 microgram/100 g body weight), receptor levels returned by 120 min to initial values. This return was resistant to cycloheximide and therefore did not represent newly synthesized receptor. Neither was the return due to replenishment by an intracellular pool of low-affinity receptors as such a pool could not be detected by Scatchard analysis or Western blotting. Therefore, receptor return was due to the recycling of previously internalized receptor. At doses of injected EGF greater than 50% receptor occupancy, net receptor loss-i.e., downregulation-was observed by evaluating the receptor content of total particulate fractions of liver homogenates. At the higher saturating doses of injected EGF (5 and 10 micrograms/100 g body weight), the majority of surface receptor content was lost by 15 min and remained low for at least an additional 105 min. As the kinetics of ligand clearance from the circulation and liver parenchyma were similar for all doses of EGF injected, then the ligand-mediated regulation of surface receptor content and downregulation were not a result of a prolonged temporal interaction of ligand with receptor. Rather, the phenomena must be a consequence of the absolute concentrations of EGF interacting with receptor at the cell surface and/or in endosomes.     

Environmental and dispersal controls of an annual plant’s distribution: how similar are patterns and apparent processes at two spatial scales?

At scales from microsites to entire ranges, species’ distributions reflect limited adaptation and/or limited dispersal. To what extent are specific distribution patterns and processes similar across scales? We investigated environmental effects—presumed because of adaptation—and independent spatial effects—presumed because of dispersal—on distribution at two scales (landscape patches of approximately 1,300 m², sampled along transects, and 4-m² cells, sampled in contiguous grids within populations) and on individual performance (water status, reproduction) in the California annual, Clarkia xantiana ssp. xantiana. Because water limitation helps set this species’ regional borders, we expected occupancy and performance at smaller scales to correlate with topographic and soil features affecting water relations. At the patch scale, environmental features associated with reduced water stress (i.e., steep slopes that face north; coarse, soft soils; igneous rather than metasedimentary parent rock) predicted occupancy. Spatial aggregation was not detected, but incomplete occupancy of apparently suitable patches indicated that dispersal limits occupancy. At the scale of small cells, relationships between environmental variables, occupancy, density, and performance varied among populations. Associations sometimes resembled those at the patch scale but sometimes opposed them. Spatial aggregation in cell occupancy and/or density occurred in all populations, implying limited dispersal, whereas spatial aggregation of water potential values in some populations might have arisen from spatially structured unmeasured environmental variables. Limited adaptation to drought and limited patch colonization appear to affect patch occupancy in C. xantiana ssp. xantiana, whereas smaller-scale patterns indicate consistent effects of limited dispersal and somewhat variable environmental effects.     

The effect of photoreceptor coupling and synapse nonlinearity on signal:noise ratio in early visual processing

Electrical coupling of vertebrate photoreceptors is well known to improve the signal: noise ratio in the photoreceptor layer for large-area stimuli. For example, if N photoreceptors are perfectly coupled to each other, the signal: noise ratio is improved for stimuli illuminating more than a number M = square root of N of the receptors but is made worse for small-area stimuli illuminating less than M of the N receptors. Using the model of Lamb & Simon (J. Physiol., Lond. 263, 257 (1976], which treats the photoreceptor layer as a square array of cells, each coupled through a resistive gap junction to the four cells around it, we show that the signal:noise ratio for small-area stimuli is much greater than would be expected from a model in which receptors are assumed to be perfectly coupled. Contrary to predictions made assuming perfect coupling, receptor coupling should not prevent rods from detecting single photons, but whether the single photon signal can be detected at the bipolar cell level depends on how signals are read out of the receptor layer. The signal:noise ratio in bipolar cells postsynaptic to the photo-receptor layer is determined partly by synaptic convergence and nonlinearity in synaptic transmission from receptors. If the synaptic gain decreases with light-induced receptor hyperpolarization, as is found experimentally, then receptor coupling can improve the postsynaptic signal:noise ratio for stimuli illuminating only one receptor, even though coupling decreases the presynaptic signal:noise ratio for such stimuli. Moreover, increasing the number of coupled receptors projecting to a bipolar cell can improve the signal:noise ratio for localized stimuli if the synapse is sufficiently nonlinear (although, for the degree of nonlinearity seen in lower vertebrates, synaptic convergence makes the ratio worse for the single photon event). The fact that receptor coupling and synaptic convergence can, under some circumstances, improve the signal:noise ratio in bipolar cells suggests a principle of retinal design that may compete with the requirements of high spatial resolution.     

Adaptation kinetics in bacterial chemotaxis.

Cells of Escherichia coli, tethered to glass by a single flagellum, were subjected to constant flow of a medium containing the attractant alpha-methyl-DL-aspartate. The concentration of this chemical was varied with a programmable mixing apparatus over a range spanning the dissociation constant of the chemoreceptor at rates comparable to those experienced by cells swimming in spatial gradients. When an exponentially increasing ramp was turned on (a ramp that increases the chemoreceptor occupancy linearly), the rotational bias of the cells (the fraction of time spent spinning counterclockwise) changed rapidly to a higher stable level, which persisted for the duration of the ramp. The change in bias increased with ramp rate, i.e., with the time rate of change of chemoreceptor occupancy. This behavior can be accounted for by a model for adaptation involving proportional control, in which the flagellar motors respond to an error signal proportional to the difference between the current occupancy and the occupancy averaged over the recent past. Distributions of clockwise and counterclockwise rotation intervals were found to be exponential. This result cannot be explained by a response regular model in which transitions between rotational states are generated by threshold crossings of a regular subject to statistical fluctuation; this mechanism generates distributions with far too many long events. However, the data can be fit by a model in which transitions between rotational states are governed by first-order rate constants. The error signal acts as a bias regulator, controlling the values of these constants.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis

In order that cells respond to environmental cues, they must be able to measure ambient ligand concentration. Concentrations fluctuate, however, because of thermal noise, and one can readily show that estimates based on concentration values at a particular moment will be subject to substantial error. Cells are therefore expected to average their estimates over some limited time period. In this paper we assume that a cell uses fractional receptor occupancy as a measure of ambient ligand concentration and develop general expressions for the error a cell makes because the length of the averaging period is necessarily limited. Our analysis is general, relieving many of the assumptions underlying the seminal work of Berg and Purcell. The most important formal difference is our inclusion of occupancy-dependent dissociation--a phenomenon that has been well-documented for many systems. In addition, our formulation permits signal averaging to begin before chemical equilibrium has been established and it allows binding kinetics to be nonlinear (i.e., biomolecular rather than pseudo-first-order). The results are applied to spatial and temporal concentration gradients. In particular we estimate the minimum averaging times required for cells to detect such gradients under typical in vitro conditions. These estimates involve assigning numerical values to receptor ligand rate constants. If the rate constants are at their maximum possible values (limited only by center of mass diffusion), then either temporal or spatial gradients can be detected in minutes or less. If, however, as suggested by experiments, the rate constants are several orders of magnitude below their diffusion-limited values, then under typical constant gradient conditions the time required to detect a spatial gradient is prohibitively long, whereas temporal gradients can still be detected in reasonable lengths of time. This result was obtained for large cells such as lymphocytes, as well as for the smaller, bacterial cells. The ratio of averaging times for the two mechanisms--amounting to several orders of magnitude--is well beyond what could be reconciled by limitations of the calculation, and strongly suggests heavy reliance on temporal sensing mechanisms under typical in vitro conditions with constant spatial gradients.     

Quantitative imaging of single live cells reveals spatiotemporal dynamics of multistep signaling events of chemoattractant gradient sensing in Dictyostelium

Activation of G-protein-coupled chemoattractant receptors triggers dissociation of Galpha and Gbetagamma subunits. These subunits induce intracellular responses that can be highly polarized when a cell experiences a gradient of chemoattractant. Exactly how a cell achieves this amplified signal polarization is still not well understood. Here, we quantitatively measure temporal and spatial changes of receptor occupancy, G-protein activation by FRET imaging, and PIP3 levels by monitoring the dynamics of PH(Crac)-GFP translocation in single living cells in response to different chemoattractant fields. Our results provided the first direct evidence that G-proteins are activated to different extents on the cell surface in response to asymmetrical stimulations. A stronger, uniformly applied stimulation triggers not only a stronger G-protein activation but also a faster adaptation of downstream responses. When naive cells (which have not experienced chemoattractant) were abruptly exposed to stable cAMP gradients, G-proteins were persistently activated throughout the entire cell surface, whereas the response of PH(Crac)-GFP translocation surprisingly consisted of two phases, an initial transient and asymmetrical translocation around the cell membrane, followed by a second phase producing a highly polarized distribution of PH(Crac)-GFP. We propose a revised model of gradient sensing, suggesting an important role for locally controlled components that inhibit PI3Kinase activity.     

Surface contact requirements for activation of cytotoxic T lymphocytes.

Cell activation resulting from binding of receptors on one cell to ligands on another is governed by receptor affinities and by ligand concentrations. Effective ligand concentration is determined by its density on the cell surface, but receptor occupancy level will also be influenced by the area of surface contact between the cells. The present study demonstrates the critical importance of a large, continuous surface contact area for effective CTL activation. Using class I alloantigen immobilized on latex microspheres, particle sizes of 4 to 5 microns were found to provide an optimum stimulus. Below 4 microns, responses decreased rapidly with decreasing particle size, and large numbers of small particles could not compensate for suboptimal size. Comparable size dependence was found for activation of degranulation by cloned CTL and for stimulation of in vitro generation of CTL responses by spleen cells from in vivo primed mice. In the presence of fluid-phase anti-TCR antibody, CD8-dependent binding to non-Ag class I (i.e., class I that is not recognized by the TCR) can provide a costimulatory signal to activate degranulation. This response is also critically dependent upon the class I being presented on a particle of 4 or 5 microns diameter. The results suggest that sufficient receptor occupancy (both TCR and CD8) over a contiguous region of the cell surface, as opposed to total interactions over the entire cell surface, is a critical determinant for activation. The ability of CTL to distinguish between Ag on cell-size vs subcellular fragments is probably necessary for their effective functioning, and may also explain the inability to significantly influence CTL activation in vivo with subcellular or soluble forms of Ag.     

Temporal adaptability and the inverse relationship to sensitivity: a parameter identification model

Following a prolonged period of visual adaptation to a temporally modulated sinusoidal luminance pattern, the threshold contrast of a similar visual pattern is elevated. The adaptive elevation in threshold contrast is selective for spatial frequency, may saturate at low adaptor contrast, and increases as a function of the spatio-temporal frequency of the adapting signal. A model for signal extraction that is capable of explaining these threshold contrast effects of adaptation is proposed. Contrast adaptation in the model is explained by the identification of the parameters of an environmental model: the autocorrelation function of the visualized signal. The proposed model predicts that the adaptability of threshold contrast is governed by unpredicted signal variations present in the visual signal, and thus represents an internal adjustment by the visual system that takes into account these unpredicted signal variations given the additional possibility for signal corruption by additive noise.     

A computational study of feedback effects on signal dynamics in a mitogen-activated protein kinase (MAPK) pathway model

Exploiting signaling pathways for the purpose of controlling cell function entails identifying and manipulating the information content of intracellular signals. As in the case of the ubiquitously expressed, eukaryotic mitogen-activated protein kinase (MAPK) signaling pathway, this information content partly resides in the signals' dynamical properties. Here, we utilize a mathematical model to examine mechanisms that govern MAPK pathway dynamics, particularly the role of putative negative feedback mechanisms in generating complete signal adaptation, a term referring to the reset of a signal to prestimulation levels. In addition to yielding adaptation of its direct target, feedback mechanisms implemented in our model also indirectly assist in the adaptation of signaling components downstream of the target under certain conditions. In fact, model predictions identify conditions yielding ultra-desensitization of signals in which complete adaptation of target and downstream signals culminates even while stimulus recognition (i.e., receptor-ligand binding) continues to increase. Moreover, the rate at which signal decays can follow first-order kinetics with respect to signal intensity, so that signal adaptation is achieved in the same amount of time regardless of signal intensity or ligand dose. All of these features are consistent with experimental findings recently obtained for the Chinese hamster ovary (CHO) cell lines (Asthagiri et al., J. Biol. Chem. 1999, 274, 27119-27127). Our model further predicts that although downstream effects are independent of whether an enzyme or adaptor protein is targeted by negative feedback, adaptor-targeted feedback can "back-propagate" effects upstream of the target, specifically resulting in increased steady-state upstream signal. Consequently, where these upstream components serve as nodes within a signaling network, feedback can transfer signaling through these nodes into alternate pathways, thereby promoting the sort of signaling cross-talk that is becoming more widely appreciated.     

Analysis of the signal transduction properties of a module of spatial sensing in eukaryotic chemotaxis

The movement of cells in response to a gradient in chemical concentration—known as chemotaxis—is crucial for the proper functioning of uni-and multicellular organisms. How a cell senses the chemical concentration gradient surrounding it, and what signal is transmitted to its motion apparatus is known as gradient sensing. The ability of a cell to sense gradients persists even when the cell is immobilized (i.e., its motion apparatus is deactivated). This suggests that important features of gradient sensing can be studied in isolation, decoupling this phenomenon from the movement of the cell. A mathematical model for gradient sensing in Dictyostelium cells and neutrophils was recently proposed. This consists of an adaptation/spatial sensing module. This spatial sensing module feeds into an amplification module, magnifying the effects of the former. In this paper, we analyze the spatial sensing module in detail and examine its signal transduction properties. We examine the response of this module to several inputs of experimental and biological relevance.     

Measurement of small mechanical vibrations of brain tissue exposed to extremely-low-frequency electric fields

Electromagnetic fields can interact with biological tissue both electrically and mechanically. This study investigated the mechanical interaction between brain tissue and an extremely-low-frequency (ELF) electric field by measuring the resultant vibrational amplitude. The exposure cell is a section of X-band waveguide that was modified by the addition of a center conductor to form a small TEM cell within the waveguide structure. The ELF signal is applied to the center conductor of the TEM cell. The applied ELF electric field generates an electrostrictive force on the surface of the brain tissue. This force causes the tissue to vibrate at a frequency equal to twice the frequency of the applied sinusoidal signal. An X-band signal is fed through the waveguide, scattered by the vibrating sample, and detected by a phase-sensitive receiver. Using a time-averaging spectrum analyzer, a vibration sensitivity of approximately 0.2 nmp-p can be achieved. The amplitude of the brain tissue vibrational response is constant for vibrational frequencies below 50 Hz; between 50 and 200 Hz resonant phenomena were observed; and above 200 Hz the amplitude fall-off is rapid.     

Ryanodine receptor adaptation and Ca2+(-)induced Ca2+ release-dependent Ca2+ oscillations.

A simplified mechanism that mimics "adaptation" of the ryanodine receptor (RyR) has been developed and its significance for Ca2+(-)induced Ca2+ release and Ca2+ oscillations investigated. For parameters that reproduce experimental data for the RyR from cardiac cells, adaptation of the RyR in combination with sarco/endoplasmic reticulum Ca2+ ATPase Ca2+ pumps in the internal stores can give rise to either low [Cai2+] steady states or Ca2+ oscillations coexisting with unphysiologically high [Cai2+] steady states. In this closed-cell-type model rapid, adaptation-dependent Ca2+ oscillations occur only in limited ranges of parameters. In the presence of Ca2+ influx and efflux from outside the cell (open-cell model) Ca2+ oscillations occur for a wide range of physiological parameter values and have a period that is determined by the rate of Ca2+ refilling of the stores. Although the rate of adaptation of the RyR has a role in determining the shape and the period of the Ca2+ spike, it is not essential for their existence. This is in marked contrast with what is observed for the inositol 1,4,5-trisphosphate receptor for which the biphasic activation and inhibition of its activity by Ca2+ are sufficient to produce oscillations. Results for this model are compared with those based on Ca2+(-)induced Ca2+ release alone in the bullfrog sympathetic neuron. This kinetic model should be suitable for analyzing phenomena associated with "Ca2+ sparks," including their merger into Ca2+ waves in cardiac myocytes.