Persistence of direction increases the drift velocity of run and tumble chemotaxis

Escherichia coli is a motile bacterium that moves up a chemoattractant gradient by performing a biased random walk composed of alternating runs and tumbles. Previous models of run and tumble chemotaxis neglect one or more features of the motion, namely (a) a cell cannot directly detect a chemoattractant gradient but rather makes temporal comparisons of chemoattractant concentration, (b) rather than being entirely random, tumbles exhibit persistence of direction, meaning that the new direction after a tumble is more likely to be in the forward hemisphere, and (c) rotational Brownian motion makes it impossible for an E. coli cell to swim in a straight line during a run. This paper presents an analytic calculation of the chemotactic drift velocity taking account of (a), (b) and (c), for weak chemotaxis. The analytic results are verified by Monte Carlo simulation. The results reveal a synergy between temporal comparisons and persistence that enhances the drift velocity, while rotational Brownian motion reduces the drift velocity. This work was supported by an Oliver Gatty Studentship from the University of Cambridge.     

Theoretical results for chemotactic response and drift of E. coli in a weak attractant gradient

The bacterium Escherichia coli (E. coli) moves in its natural environment in a series of straight runs, interrupted by tumbles which cause change of direction. It performs chemotaxis towards chemo-attractants by extending the duration of runs in the direction of the source. When there is a spatial gradient in the attractant concentration, this bias produces a drift velocity directed towards its source, whereas in a uniform concentration, E. coli adapts, almost perfectly in case of methyl aspartate. Recently, microfluidic experiments have measured the drift velocity of E. coli in precisely controlled attractant gradients, but no general theoretical expression for the same exists. With this motivation, we study an analytically soluble model here, based on the Barkai-Leibler model, originally introduced to explain the perfect adaptation. Rigorous mathematical expressions are obtained for the chemotactic response function and the drift velocity in the limit of weak gradients and under the assumption of completely random tumbles. The theoretical predictions compare favorably with experimental results, especially at high concentrations. We further show that the signal transduction network weakens the dependence of the drift on concentration, thus enhancing the range of sensitivity.     

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.     

Biased reorientation in the chemotaxis of peritrichous bacteria Salmonella enterica serovar Typhimurium

Many kinds of peritrichous bacteria that repeat runs and tumbles by using multiple flagella exhibit chemotaxis by sensing a difference in the concentration of the attractant or repellent between two adjacent time points. If a cell senses that the concentration of an attractant has increased, their flagellar motors decrease the switching frequency from counterclockwise to clockwise direction of rotation, which causes a longer run in swimming up the concentration gradient than swimming down. We investigated the turn angle in tumbles of peritrichous bacteria swimming across the concentration gradient of a chemoattractant because the change in the switching frequency in the rotational direction may affect the way tumbles. We tracked several hundreds of runs and tumbles of single cells of Salmonella enterica serovar Typhimurium in the concentration gradient of L-serine and found that the turn angle depends on the concentration gradient that the cell senses just before the tumble. The turn angle is biased toward a smaller value when the cells swim up the concentration gradient, whereas the distribution of the angle is almost uniform (random direction) when the cells swim down the gradient. The effect of the observed bias in the turn angle on the degree of chemotaxis was investigated by random walk simulation. In the concentration field where attractants diffuse concentrically from the point source, we found that this angular distribution clearly affects the reduction of the mean-square displacement of the cell that has started at the attractant source, that is, the bias in the turn angle distribution contributes to chemotaxis in peritrichous bacteria.     

Tumble Kinematics of Escherichia coli near a Solid Surface

Bacteria tumble periodically, following environmental cues. Whether they can tumble near a solid surface is a basic issue for the inception of infection or mineral biofouling. Observing freely swimming Escherichia coli near and parallel to a glass surface imaged at high magnification (×100) and high temporal resolution (500 Hz), we identified tumbles as events starting (or finishing, respectively) in abrupt deceleration (or reacceleration, respectively) of the body motion. Selected events show an equiprobable clockwise (CW) or counterclockwise change in direction that is superimposed on a surface CW path because of persistent propulsion. These tumbles follow a common long (about 300 ± 100 ms, N = 52) deceleration-reorientation-acceleration pattern. A wavelet transform multiscale analysis shows these tumbles cause in-plane diffusive reorientations with 1.5 rad²/s rotational diffusivity, a value that compares with that measured in bulk tumbles. In half of the cases, additional few-millisecond bursts of an almost equiprobable CW or counterclockwise change of direction (12 ± 90°, N = 89) occur within the reorientation stage. The highly dispersed absolute values of change of direction (70 ± 66°, N = 89) of only a few bursts destabilize the cell-swimming direction. These first observations of surface tumbles set a foundation for statistical models of run-and-tumble surface motion different from that in bulk and lend support for chemotaxis near solid surface.     

Growth of a root system described as diffusion. II. Numerical model and application

In simulation models for water movement and nutrient transport, uptake of water and nutrients by roots forms an essential part. As roots are spatially distributed, prediction of root growth and root distribution is crucial for modelling water and nutrient uptake. In a preceding paper, De Willigen et al. (2002; Plant and Soil 240, 225–234) presented an analytical solution for describing root length density distribution as a diffusion-type process. In the current paper, we present a numerical model that does the same, but which is more flexible with respect to where root input can occur. We show that the diffusion-type root growth model can describe well observed rooting patterns. We used rooting patterns for different types of crops: maize, gladiolus, eastern white cedar, and tomato. For maize, we used data for two different types of fertiliser application: broadcast and row application. In case of row application, roots extend more vertically than horizontally with respect to the broadcast application situation. This is reflected in a larger ratio of diffusion coefficients in vertical versus horizontal direction. For tomato, we considered tomatoes grown on an artificial rooting medium, i.e. rockwool. We have shown that, in principle, the model can be extended by including reduction functions on the diffusion coefficient in order to account for environmental conditions.     

Configurational and dynamic properties of different length superhelical DNAs measured by dynamic light scattering

The solution conformation and internal motions of five superhelical DNAs between 2100 and 10200 base-pairs in length have been characterized by dynamic light scattering (DLS). Variations in the diffusion coefficients and rotational relaxation times with molecular weight are both indicative of an anisotropic extended structure of these DNAs; we therefore conclude that under our conditions the interwound superhelical structure prevails. The internal dynamics can be described by a superposition of rotational diffusion and internal relaxation. The latter process is characterized by the internal diffusion of persistence length size segments within the DNA chain and faster bending motions within these segments.     

Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy

The rotational dynamics of short DNA fragments with or without intrinsic curvature were studied using time-resolved phase fluorimetry of intercalated ethidium with detection of the anisotropy. Parameters determined were the spinning diffusion coefficient of the DNA fragments about the long axis and the zero-time ethidium fluorescence anisotropy. We find a significant decrease in the spinning diffusion coefficient for all curved fragments compared to the straight controls. This decrease is likewise evident in rotational diffusion coefficients computed from DNA structures obtained by a curvature prediction program for these sequences. Using a hinged-cylinder model, we can identify the change in rotational diffusion coefficient with a permanent bend of 13-16 degrees per helix turn for the sequences studied. Moreover, for some of the curved fragments an increased flexibility has to be assumed in addition to the permanent bend in order to explain the data.     

Time-dependent absorption anisotropy and rotational diffusion of proteins in membranes.

The decay of flash-induced absorption anisotropy, r(t), of a chromophore in a membrane protein is closely correlated with rotational diffusion of the protein in the membrane. We develop a theory of time-dependent absorption anisotropy which is applicable to both linear chromophores and planar chromophores which have two different absorption moments at right angles to one another. The theory treats two types of rotational diffusion of membrane proteins: one is rotation of the whole protein about the normal to the plane of the membrane, and the other is restricted wobbling of the whole or part of the protein molecule. In the former case, r(t) is determined by a rotational diffusion coefficient and an angle between the absorption moment(s) and the normal to the plane of the membrane. Rotation of rigid transmembrane proteins can be described by this treatment. In the latter case, r(t) is characterized by a wobbling diffusion coefficient and the degree of orientational constraint. This treatment may be applicable to independent wobbling of the hydrophilic part of membrane proteins. We further show that, for linear and circularly degenerate chromophores, the effect of the excitation flash intensity on r(t) can be accounted for by a constant scaling factor.     

Fluctuations in rotation rate of the flagellar motor of Escherichia coli.

The purpose of this work was to study the changes in rotation rate of the bacterial motor and to try to discriminate between various sources of these changes with the aim of understanding the mechanism of force generation better. To this end Escherichia coli cells were tethered and videotaped with brief stroboscopic light flashes. The records were scanned by means of a computerized motion analysis system, yielding cell size, radius of rotation, and accumulated angle of rotation as functions of time for each cell selected. In conformity with previous studies, fluctuations in the rotation rate of the flagellar motor were invariably found. Employing an exclusively counterclockwise rotating mutant ("gutted" RP1091 strain) and using power spectral density, autocorrelation and residual mean square angle analysis, we found that a simple superposition of rotational diffusion on a steady rotary motion is insufficient to describe the observed rotation. We observed two additional rotational components, one fluctuating (0.04-0.6 s) and one oscillating (0.8-7 s). However, the effective rotational diffusion coefficient obtained after taking these two components into account generally exceeded that calculated from external friction by two orders of magnitude. This is consistent with a model incorporating association and dissociation of force-generating units.     

A theoretical study of rotational diffusion models for rod-shaped viruses. The influence of motion on 31P nuclear magnetic resonance lineshapes and transversal relaxation.

Information about the interaction between nucleic acids and coat proteins in intact virus particles may be obtained by studying the restricted backbone dynamics of the incapsulated nucleic acids using 31P nuclear magnetic resonance (NMR) spectroscopy. In this article, simulations are carried out to investigate how reorientation of a rod-shaped virus particle as a whole and isolated nucleic acid motions within the virion influence the 31P NMR lineshape and transversal relaxation dominated by the phosphorus chemical shift anisotropy. Two opposite cases are considered on a theoretical level. First, isotropic rotational diffusion is used as a model for mobile nucleic acids that are loosely or partially bound to the protein coat. The effect of this type of diffusion on lineshape and transversal relaxation is calculated by solving the stochastic Liouville equation by an expansion in spherical functions. Next, uniaxial rotational diffusion is assumed to represent the mobility of phosphorus in a virion that rotates as a rigid rod about its length axis. This type of diffusion is approximated by an exchange process among discrete sites. As turns out from these simulations, the amplitude and the frequency of the motion can only be unequivocally determined from experimental data by a combined analysis of the lineshape and the transversal relaxation. In the fast motional region both the isotropic and the uniaxial diffusion model predict the same transversal relaxation as the Redfield theory. For very slow motion, transversal relaxation resembles the nonexponential relaxation as observed for water molecules undergoing translational diffusion in a magnetic field gradient. In this frequency region T2e is inversely proportional to the cube root of the diffusion coefficient. In addition to the isotropic and uniaxial diffusion models, a third model is presented, in which fast restricted nucleic acid backbone motions dominating the lineshape are superimposed on a slow rotation of the virion about its length axis, dominating transversal relaxation. In an accompanying article the models are applied to the 31P NMR results obtained for bacteriophage M13 and tobacco mosaic virus.     

Backbone dynamics of the N-terminal domain of the HIV-1 capsid protein and comparison with the G94D mutant conferring cyclosporin resistance/dependence.

Nuclear magnetic resonance (NMR) (15)N relaxation methods have been used to characterize the backbone dynamics of the N-terminal core domain of the HIV-1 capsid protein (CA(151)). The domain, which has an unusually flat, triangular shape, tumbles in solution at 28 degrees C with an effective rotational correlation time of 11.5 ns. Relaxation data for backbone amides in the domain's seven alpha-helices are indicative of fully anisotropic rotational diffusion. The principal axes of the rotational diffusion tensor calculated from the NMR data are aligned to within 12-23 degrees of the principal axes of the inertial tensor, with the axis of fastest rotational diffusion coincident with that of minimal inertia, and vice versa. Large variations in the (15)N-(1)H nuclear Overhauser effects for individual amino acids correlate with the degree of convergence in the previously calculated NMR structure. In particular, the partially disordered residues Val86-Arg97 that contain the human cyclophilin A (CypA) packaging signal have (15)N heteronuclear NOEs and transversal relaxation rates consistent with a high degree of dynamic conformational averaging. The N-terminal domain of a CA mutant (G94D) that confers both resistance to and dependence on cyclosporin A analogues was also analyzed. Our results indicate that this mutation does not influence the conformation or dynamics of CA(151), and therefore probably affects the function of the protein by modifying essential intermolecular CA-CA interactions.     

Real-time imaging of fluorescent flagellar filaments of Rhizobium lupini H13-3: flagellar rotation and pH-induced polymorphic transitions

The soil bacterium Rhizobium lupini H13-3 has complex right-handed flagellar filaments with unusual ridged, grooved surfaces. Clockwise (CW) rotation propels the cells forward, and course changes (tumbling) result from changes in filament speed instead of the more common change in direction of rotation. In view of these novelties, fluorescence labeling was used to analyze the behavior of single flagellar filaments during swimming and tumbling, leading to a model for directional changes in R. lupini. Also, flagellar filaments were investigated for helical conformational changes, which have not been previously shown for complex filaments. During full-speed CW rotation, the flagellar filaments form a propulsive bundle that pushes the cell on a straight path. Tumbling is caused by asynchronous deceleration and stops of individual filaments, resulting in dissociation of the propulsive bundle. R. lupini tumbles were not accompanied by helical conformational changes as are tumbles in other organisms including enteric bacteria. However, when pH was experimentally changed, four different polymorphic forms were observed. At a physiological pH of 7, normal flagellar helices were characterized by a pitch angle of 30 degrees, a pitch of 1.36 micro m, and a helical diameter of 0.50 micro m. As pH increased from 9 to 11, the helices transformed from normal to semicoiled to straight. As pH decreased from 5 to 3, the helices transformed from normal to curly to straight. Transient conformational changes were also noted at high viscosity, suggesting that the R. lupini flagellar filament may adapt to high loads in viscous environments (soil) by assuming hydrodynamically favorable conformations.     

Intensity-fluctuation spectroscopy and tRNA conformation. II. Changes of size and shape of tRNA in the melting process

We have measured the translational (D^T) and rotational (D^R) diffusion coefficients of bulk tRNA from baker's yeast during the thermal unfolding process by means of photon-correlation spectroscopy. It should be noted that our estimate of the rotational diffusion coefficient represented, for the first time, measurements on a small macromolecule in solution by the photoelectron time-of-arrival technique with a delay-time resolution of 1 nsec. The melting curves expressed in terms of δD^T vs temperature were consistent with the literature data in revealing the melting steps and their dependence on NaCl concentration. Additionally, it was possible to prove the existence of an intermediate, more compact structure during the initial steps of the thermal unfolding process. We found that the temperature ranges over which this intermediate structure appears depend strongly on salt concentration. By utilizing both translational and rotational diffusion coefficients and Perrin's equations for ellipsoids of revolution, we have computed the values of the equivalent length and width of tRNA molecules in solution at four different temperatures for NaCl concentrations of 0.2, 0.5, and 1M. The approximate model of ellipsoids of revolution also permits us to obtain an estimate of the radius of gyration, which is in very good agreement with literature data measured by means of small-angle x-ray scattering. Furthermore, we have measured the shape and size changes of tRNA with varying NaCl concentrations at room temperatures (25°C). The molecule becomes smaller and more spherical when NaCl concentration increases. As a result of partial melting at 70°C, the macromolecule is surprisingly elongated with an approximate axial ratio of 8:1 and has dimensions of about 180/22Å. Such information on conformational changes by a simultaneous determination of rotational and translational diffusion coefficients illustrates the potential of this approach, not available by other methods.     

Two-dimensional determination of the cellular Ca2+ binding in bovine chromaffin cells.

The spatiotemporal profile of intracellular calcium signals is determined by the flux of calcium ions across different biological membranes as well as by the diffusional mobility of calcium and different calcium buffers in the cell. To arrive at a quantitative understanding of the determinants of these signals, one needs to dissociate the flux contribution from the redistribution and buffering of calcium. Since the cytosol can be heterogeneous with respect to its calcium buffering property, it is essential to assess this property in a spatially resolved manner. In this paper we report on two different methods to estimate the cellular calcium binding of bovine adrenal chromaffin cells. In the first method, we use voltage-dependent calcium channels as a source to generate calcium gradients in the cytosol. Using imaging techniques, we monitor the dissipation of these gradients to estimate local apparent calcium diffusion coefficients and, from these, local calcium binding ratios. This approach requires a very high signal-to-noise ratio of the calcium measurement and can be used when well-defined calcium gradients can be generated throughout the cell. In the second method, we overcome these problems by using calcium-loaded DM-nitrophen as a light-dependent calcium source to homogeneously and quantitatively release calcium in the cytosol. By measuring [Ca2+] directly before and after the photorelease process and knowing the total amount of calcium being released photolytically, we get an estimate of the fraction of calcium ions which does not appear as free calcium and hence must be bound to either the indicator dye or the endogenous calcium buffer. This finally results in a two-dimensional map of the distribution of the immobile endogenous calcium buffer. We did not observe significant variations of the cellular calcium binding at a spatial resolution of approximately 2 micron. Furthermore, the calcium binding is not reduced by increasing the resting [Ca2+] to levels as high as 1.1 microM. This is indicative of a low calcium affinity of the corresponding buffers and is in agreement with a recent report on the affinity of these buffers (Xu, T., M. Naraghi, H. Kang, and E. Neher. 1997. Biophys. J. 73:532-545). In contrast to the homogeneous distribution of the calcium buffers, the apparant calcium diffusion coefficient did show inhomogeneities, which can be attributed to restricted diffusion at the nuclear envelope and to rim effects at the cell membrane.     

Rod-like cholesterol micelles in aqueous solution studied using polarized and depolarized dynamic light scattering.

Micelles of cholesterol in aqueous solution have been investigated using polarized and depolarized dynamic light scattering. They are shown to be highly extended and characterized by a narrow size distribution. It is shown that a rod-like model is applicable with length, L = 580 nm. Determination of the rotational diffusion coefficient by analysis of the autocorrelation function gave a value of theta = 150 s-1, which is close to the calculated value for the rod with this dimension. Depolarized dynamic light scattering measurements as a function of angle gave a value of 110 s-1.     

The random walk of Azospirillum brasilense

The bacterium Azospirillum brasilense has been frequently studied in laboratory experiments. It performs movements in space where long forward and backward runs on a straight line occur simultaneously with slow changes of direction of the line. A model is presented in which a correlated random walk on a line is joined to diffusion on a sphere of directions. For this transport system, a hierarchy of moment approximations is derived, ranging from a hyperbolic system with four dependent variables to a scalar damped wave equation (telegraph equation) and then to a single diffusion equation for particle density. The original parameters are compounded in the diffusion quotient. The effects of these parameters, such as particle speed or turning rate, on the diffusion coefficient are discussed in detail.     

Detection of membrane packing defects by time-resolved fluorescence depolarization.

Packing defects in lipid bilayer play a significant role in the biological activities of cell membranes. Time-resolved fluorescence depolarization has been used to detect and characterize the onset of packing defects in binary mixtures of dilinoleoylphosphatidylethanolamine/1-palmitoyl-2- oleoylphosphatidylcholine (PE/PC). These PE/PC mixtures exhibit mesoscopic packing defect state (D), as well as one-dimensional lambellar liquid crystalline (L alpha) and two-dimensional inverted hexagonal (HII) ordered phases. Based on previous electron microscopic investigations, this D state is characterized by the presence of interlamellar attachments and precursors of HII phase between the lipid layers. Using a rotational diffusion model for rod-shaped fluorophore in a curved matrix, rotational dynamics parameters, second rank order parameter, localized wobbling diffusion, and curvature-dependent rotational diffusion constants of dipyenylhexatriene (DPH)-labeled PC (DPH-PC) in the host PE/PC matrix were recovered from the measured fluorescence depolarization decays of DPH fluorescence. At approximately 60% PE, abrupt increases in these rotational dynamics parameters were observed, reflecting the onset of packing defects in the host PE/PC matrix. We have demonstrated that rotational dynamics parameters are very sensitive in detecting the onset of curvature-associating packing defects in lipid membranes. In addition, the presence of the D state can be characterized by the enhanced wobbling diffusional motion and order packing of lipid molecules, and by the presence of localized curvatures in the lipid layers.     

Diffusion of methylene blue organic cation in the polymeric matrix of cell walls of the crustose lichen Cladonia rangiferina (L.) F. H. Wigg.

The diffusion within the polymeric matrix of the cell wall is a process that determines the possibility and the rate of ion penetration into cell. In this work, we quantitatively determined the diffusion of Methylene Blue cation within the cell walls isolated from the crustose lichen Cladonia rangiferina (L.) F. H. Wigg. and its possible contribution to the processes of absorption into the lichen thalli. The swelling coefficient of the cell wall matrix and the diffusion coefficient of the organic cation within the cell walls were determined. Our results show that lichen cell walls are characterized by a higher crosslinking degree and a smaller diffusion coefficient than plants.