Coordinated reversal of flagellar motors on a single Escherichia coli cell

An Escherichia coli cell transduces extracellular stimuli sensed by chemoreceptors to the state of an intracellular signal molecule, which regulates the switching of the rotational direction of the flagellar motors from counterclockwise (CCW) to clockwise (CW) and from CW back to CCW. Here, we performed high-speed imaging of flagellar motor rotation and show that the switching of two different motors on a cell is controlled coordinatedly by an intracellular signal protein, phosphorylated CheY (CheY-P). The switching is highly coordinated with a subsecond delay between motors in clear correlation with the distance of each motor from the chemoreceptor patch localized at a cell pole, which would be explained by the diffusive motion of CheY-P molecules in the cell. The coordinated switching becomes disordered by the expression of a constitutively active CheY mutant that mimics the CW-rotation stimulating function. The coordinated switching requires CheZ, which is the phosphatase for CheY-P. Our results suggest that a transient increase and decrease in the concentration of CheY-P caused by a spontaneous burst of its production by the chemoreceptor patch followed by its dephosphorylation by CheZ, which is probably a wavelike propagation in a subsecond timescale, triggers and regulates the coordinated switching of flagellar motors.     

Modelling of microscale patch encounter by chemotactic protozoa

A model of protozoan chemotaxis, based on the rate of change of chemoreceptor occupancy, was used to analyse the efficiency of chemotaxis in a variety of situations. Simulated swimming behaviour replicated patterns observed experimentally. These were classified into three forms of chemosensory behaviour; run-tumble, steered turning, and helical klinotaxis. All three could be simulated from a basic model of chemotaxis by modifying memory times and rotational velocities. In order to steer during helical klinotaxis, the cell must have a short term memory for responding to a signal within a fraction of the time period of the helix. Steered turning was identified as a form where cells react to negative changes in concentration by steering around the turn to swim back up the gradient. All 3 forms were quite effective for encountering targets within the response radius. A response to negative changes in concentration, experienced when the cell is moving away from a target, was found to be important in the absence of periodic changes in swimming direction. The frequency of patch encounter at a fixed density was calculated to be roughly proportional to swimming speed. On the basis of the model, cells are only able to sense point sources within a radius of a few mm. However, even a response radius of 1 mm is enough to increase encounter probability of otherwise minute targets by 2 orders of magnitude. The mean time for patch encounter was calculated to be an exponential function of the mean distance between patches. This results in a very sharp threshold at approximately 6 cm, above which they are not encountered by protozoa within time periods of several days.     

Determination of dependence of binding parameters on receptor occupancy

Measurements of the binding of ligand to receptors that are macromolecules, either free or components of biomembranes, often show deviation from what is expected of a simple reaction described by an association and a dissociation rate constant. A more versatile model and more discriminating experiments are required for a satisfactory explanation. This paper is based on a general model of the binding reaction in which the rate constants and equilibrium constant are dependent upon occupancy of receptors. The analysis of the model leads to three kinds of experiments: (1) equilibrium measurements which permit quantitative determination of a dissociation equilibrium parameter as a function of receptor occupancy; (2) measurements prior to equilibrium which yield the same information; and (3) measurements prior to equilibrium which reveal quantitatively the dependence of both association and dissociation rate parameters separately, on occupancy.     

Ionic channels with conformational substates.

Recent studies of protein dynamics suggest that ionic channels can assume many conformational substates. Long-lived substates have been directly observed in single-channel current records. In many cases, however, the lifetimes of conformational states will be far below the theoretical limit of time resolution of single-channel experiments. The existence of such hidden substates may strongly influence the observable (time-averaged) properties of a channel, such as the concentration dependence of conductance. A channel exhibiting fast, voltage-dependent transitions between different conductance states may behave as an intrinsic rectifier. In the presence of more than one permeable ion species, coupling between ionic fluxes may occur, even when the channel has only a single ion-binding site. In special situations the rate of ion translocation becomes limited by the rate of conformational transitions, meaning that the channel approaches the kinetic behavior of a carrier. As a result of the strong coulombic interaction between an ion in a binding site and polar groups of the protein, rate constants of conformational transitions may depend on the occupancy of the binding site. Under this condition a nonequilibrium distribution of conformational states is created when ions are driven through the channel by an external force. This may lead to an apparent violation of microscopic reversibility, i.e., to a situation in which the frequency of transitions from state A to state B is no longer equal to the transition frequency from state B to state A.     

Temperature dependence of switching of the bacterial flagellar motor by the protein CheY(13DK106YW).

The behavior of the bacterium Escherichia coli is controlled by switching of the flagellar rotary motor between the two rotational states, clockwise (CW) and counterclockwise (CCW). The molecular mechanism for switching remains unknown, but binding of the response regulator CheY-P to the motor component FliM enhances CW rotation. This effect is mimicked by the unphosphorylated double mutant CheY13DK106YW (CheY**). To learn more about switching, we measured the fraction of time that a motor spends in the CW state (the CW bias) at different concentrations of CheY** and at different temperatures. From the CW bias, we computed the standard free energy change of switching. In the absence of CheY, this free energy change is a linear function of temperature (. Biophys. J. 71:2227-2233). In the presence of CheY**, it is nonlinear. However, the data can be fit by models in which binding of each molecule of CheY** shifts the difference in free energy between CW and CCW states by a fixed amount. The shift increases linearly from approximately 0.3kT per molecule at 5 degrees C to approximately 0.9kT at 25 degrees C, where k is Boltzmann's constant and T is 289 Kelvin (= 16 degrees C). The entropy and enthalpy contributions to this shift are about -0. 031kT/ degrees C and 0.10kT, respectively.     

CheX in the three-phosphatase system of bacterial chemotaxis

Bacterial chemotaxis involves the regulation of motility by a modified two-component signal transduction system. In Escherichia coli, CheZ is the phosphatase of the response regulator CheY but many other bacteria, including Bacillus subtilis, use members of the CheC-FliY-CheX family for this purpose. While Bacillus subtilis has only CheC and FliY, many systems also have CheX. The effect of this three-phosphatase system on chemotaxis has not been studied previously. CheX was shown to be a stronger CheY-P phosphatase than either CheC or FliY. In Bacillus subtilis, a cheC mutant strain was nearly complemented by heterologous cheX expression. CheX was shown to overcome the DeltacheC adaptational defect but also generally lowered the counterclockwise flagellar rotational bias. The effect on rotational bias suggests that CheX reduced the overall levels of CheY-P in the cell and did not truly replicate the adaptational effects of CheC. Thus, CheX is not functionally redundant to CheC and, as outlined in the discussion, may be more analogous to CheZ.     

Finding motion parameters from spherical motion fields (or the advantages of having eyes in the back of your head)

A theory is developed for determining the motion of an observer given the motion field over a full 360 degree image sphere. The method is based on the fact that for an observer translating without rotation, the projected circular motion field about any equator can be divided into disjoint semicircles of clockwise and counterclockwise flow, and on the observation that the effects of rotation decouple around the three equators defining the three principal axes of rotation. Since the effect of rotation is geometrical, the three rotational parameters can be determined independently by searching, in each case, for a rotational value for which the derotated equatorial motion field can be partitioned into 180 degree arcs of clockwise and counterclockwise flow. The direction of translation is also obtained from this analysis. This search is two dimensional in the motion parameters, and can be performed relatively efficiently. Because information is correlated over large distances, the method can be considered a pattern recognition rather than a numerical algorithm. The algorithm is shown to be robust and relatively insensitive to noise and to missing data. Both theoretical and empirical studies of the error sensitivity are presented. The theoretical analysis shows that for white noise of bounded magnitude M, the expected errors is at worst linearly proportional to M. Empirical tests demonstrate negligible error for perturbations of up to 20% in the input, and errors of less than 20% for perturbations of up to 200%.     

Multiple kinetic states for the flagellar motor switch.

By means of a computerized video processing system, the flagellar motors of Escherichia coli were shown to have multiple kinetic states for each rotational direction. High-resolution analysis of flagellar motors revealed new kinetic states both in wild-type cells and in a strain deleted of other signal-transducing genes to which CheY had been introduced. This strain, RP1091, retained residual kinase activity that could phosphorylate CheY, complicating the biochemical identification of certain kinetic states. The behavioral effect of CheY on single flagellar motors was ultrasensitive, with an apparent Hill coefficient of 5.5 +/- 1.9 (SD) and a half-maximal effect at 10.1 +/- 0.5 (SD) microM CheY. Based on the CheY concentration dependence, a two-state model is clearly excluded, even for the simpler system of CheY-induced rotational reversals in the deletion strain. The data are best described by a four-state model, with two clockwise and two counterclockwise states.     

Free energy and the kinetics of biochemical diagrams, including active transport.

In earlier papers on muscle contraction it was found very useful to relate the actual (not standard) free energy levels of the different states in the biochemical diagram of the myosin cross-bridge to the first-order rate constants governing transitions between these states and to the details of the conversion of ATP free energy into mechanical work. This same approach is applied here to other macromolecular biochemical systems, for example, carriers in active transport, and simple enzyme reactions. With the definition of free energy changes between states of diagram used here (and in the muscle papers), the rate constants of the diagram are firat order, the macromolecular transitions are effectively isomeric, the equilibrium constants are dimensionless, the free energy changes are directly related to first-order rate constant ratios, and the ratio of products of forward and backward rate constants around any cycle of the diagram is related to operational free energy changes (e.g. the in vivo free energy of ADP HYDROLYSIS). These general points are illustrated by means of particular arbitrary models, especially transport models. In contrast to the muscle case, the free energy conversion question in other biochemical systems can be handled at the less detailed, complete-cycle level rather than at the elementary transition level. There is a corresponding complete-cycle kinetics, with composite first-order rate constants for the different possible cycles (in both directions). An introductory stochastic treatment of cycle kinetics is included.     

Interactions between chemotaxis genes and flagellar genes in Escherichia coli

Escherichia coli mutants defective in cheY and cheZ function are motile but generally nonchemotactic; cheY mutants have an extreme counterclockwise bias in flagellar rotation, whereas cheZ mutants have a clockwise rotational bias. Chemotactic pseudorevertants of cheY and cheZ mutants were isolated on semisolid agar and examined for second-site suppressors in other chemotaxis-related loci. Approximately 15% of the cheZ revertants and over 95% of the cheY revertants contained compensatory mutations in the flaA or flaB locus. When transferred to an otherwise wild-type background, most of these suppressor mutations resulted in a generally nonchemotactic phenotype: suppressors of cheY caused a clockwise rotational bias; suppressors of cheZ produced a counterclockwise rotational bias. Chemotactic double mutants containing a che and a fla mutation invariably exhibited flagellar rotation patterns in between the opposing extremes characteristic of the component mutations. This additive effect on flagellar rotation resulted in essentially wild-type swimming behavior and is probably the major basis of suppressor action. However, suppression effects were also allele specific, suggesting that the cheY and cheZ gene products interact directly with the flaA and flaB products. These interactions may be instrumental in establishing the unstimulated swimming pattern of E. coli.     

Reconstitution of signaling in bacterial chemotaxis.

Strains missing several genes required for chemotaxis toward amino acids, peptides, and certain sugars were tethered and their rotational behavior was analyzed. Null strains (called gutted) were deleted for genes that code for the transducers Tsr, Tar, Tap, and Trg and for the cytoplasmic proteins CheA, CheW, CheR, CheB, CheY, and CheZ. Motor switch components were wild type, flaAII(cheC), or flaBII(cheV). Gutted cells with wild-type motors spun exclusively counterclockwise, while those with mutant motors changed their directions of rotation. CheY reduced the bias (the fraction of time that cells spun counterclockwise) in either case. CheZ offset the effect of CheY to an extent that varied with switch allele but did not change the bias when tested alone. Transducers also increased the bias in the presence of CheY but not when tested alone. However, cells containing transducers and CheY failed to respond to attractants or repellents normally detected in the periplasm. This sensitivity was restored by addition of CheA and CheW. Thus, CheY both enhances clockwise rotation and couples the transducers to the flagella. CheZ acts, at the level of the motor, as a CheY antagonist. CheA or CheW or both are required to complete the signal pathway. A model is presented that explains these results and is consistent with other data found in the literature.     

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.     

Sodium channel inactivation from closed states: evidence for an intrinsic voltage dependency.

The time course of Na channel inactivation from closed states was determined on inside-out excised patches from neuroblastoma N1E 115. Closed-state inactivation develops as a single exponential with mean time constants of 66.4 ms at -80 mV, 29.6 ms at -70 mV, 20.1 ms at -60 mV, and 15.1 ms at -50 mV. Corresponding mean steady-state values of the fitted exponentials were 0.321, 0.098, 0.035, and 0. Closed-state inactivation, in general, should develop either with a delay or as more than one exponential, depending on which closed state(s) directly inactivate. The absence of additional components cannot be attributed to a rate of exchange between closed states too rapid to detect. The time course is simply accounted for if all closed states directly inactivate and do so with the same rate constant for each closed state, suggesting that those conformational changes constituting the transitions between closed states have little effect on the structural components involved in inactivation. Closed to inactivated rate constants ranged from a mean of 0.0108 ms-1 at -80 mV to 0.0690 ms-1 at -50 mV. This voltage dependency is entirely intrinsic to closed-state inactivation with closed to inactivated rate constants similar for all closed states. Over the potential range studied nearly all the inactivation is from closed states.     

Methyltransferase CheR binds to its chemoreceptor substrates independent of their signaling conformation yet modifies them differentially

Methylation of specific chemoreceptor glutamyl residues by methyltransferase CheR mediates sensory adaptation and gradient sensing in bacterial chemotaxis. Enzyme action is a function of chemoreceptor signaling conformation: kinase‐off receptors are more readily methylated than kinase‐on, a feature central to adaptational and gradient‐sensing mechanisms. Differential enzyme action could reflect differential binding, catalysis or both. We investigated by measuring CheR binding to kinase‐off and kinase‐on forms of Escherichia coli aspartate receptor Tar deleted of its CheR‐tethering, carboxyl terminus pentapeptide. This allowed characterization of the low‐affinity binding of enzyme to the substrate receptor body, otherwise masked by high‐affinity interaction with pentapeptide. We quantified the low‐affinity protein–protein interactions by determining kinetic rate constants of association and dissociation using bio‐layer interferometry and from those values calculating equilibrium constants. Whether Tar signaling conformations were shifted by ligand occupancy or adaptational modification, there was little or no difference between the two signaling conformations in kinetic or equilibrium parameters of enzyme‐receptor binding. Thus, differential methyltransferase action does not reflect differential binding. Instead, the predominant determinants of binding must be common to different signaling conformations. Characterization of the dependence of association rate constants on Deybe length, a measure of the influence of electrostatics, implicated electrostatic interactions as a common binding determinant. Taken together, our observations indicate that differential action of methyltransferase on kinase‐off and kinase‐on chemoreceptors is not the result of differential binding and suggest it reflects differential catalytic propensity. Differential catalysis rather than binding could well be central to other enzymes distinguishing alternative conformations of protein substrates.     

A model for bacterial flagellar motor: free energy transduction and self-organization of rotational motion

A bacterial flagellar motor is an energy transducing molecular machine which shows some attractive characteristics. First, this motor is driven by a protonmotive force (PMF) across the membrane, two components of which, electric potential delta psi and chemical potential -(2.3RT/F)delta pH, are equivalently transduced to the mechanical work of the motor rotation. Second, a PMF threshold for rotation is observed. Third, this motor can rotate reversibly either counterclockwise (CCW) or clockwise (CW) at almost the same speed. To clarify the osmomechanical coupling of this motor, these characteristics must be explained consistently at the molecular level. In this paper, in order to allow quantitative analyses of the above characteristics, a theoretical model of a bacterial flagellar motor is constructed assuming that the torque generating sites are electrodes which can be charged by protons and that the electrostatic interaction between the electrodes generates the rotation torque. Electrode reaction reasonably derives the equivalence of delta psi and -(2.3RT/F)delta pH. In this model, rates of charging and discharging of protons are influenced by the motor rotation rate, so that the torque generating sites co-operatively work through the motor rotation. We named this kind of co-operativity among them "dynamic co-operativity" in torque generation. This co-operativity causes autocatalytic generation of motor torque and the existence of the rotation threshold. In this model, the appearance of the stable rotational states can be described by phase transition caused by the dynamic co-operativity among torque generating sites. According to this model, the flagellar motor has two stable rotational states corresponding to CCW and CW, which show the same torques. The motor selects one direction from them to rotate, and that is self-organization of rotational motion. Interpretation of the transition between the two stable rotational states as the chemotactic reversals of the flagellar motor is also discussed.     

Stochastic model of chemoattractant receptor dynamics in leukocyte chemosensory movement

Mammalian white blood cells are known to bias the direction of their movement along concentration gradients of specific chemical stimuli, a phenomenon called chemotaxis. Chemotaxis of leukocyte cells is central to the acute inflammatory response in living organisms and other critical physiological functions. On a molecular level, these cells sense the stimuli termed chemotactic factor (CF) through specific cell surface receptors that bind CF molecules. This triggers a complex signal transduction process involving intracellular biochemical pathways and biophysical events, eventually leading to the observable chemotactic response. Several investigators have shown theoretically that statistical fluctuations in receptor binding lead to “noisy” intracellular signals, which may explain the observed imperfect chemotactic response to a CF gradient. The most recent dynamic model (Tranquillo and Lauffenburger,J. Math. Biol. 25, 229–262. 1987) couples a scheme for intracellular signal transduction and cell motility response with fluctuations in receptor binding. However, this model employs several assumptions regarding receptor dynamics that are now known to be oversimplifications. We extend the earlier model by accounting for several known and speculated chemotactic receptor dynamics, namely, transient G-protein signaling, cytoskeletal association, and receptor internalization and recycling, including statistical fluctuations in the numbers of receptors among the various states. Published studies are used to estimate associated constants and ensure the predicted receptor distribution is accurate. Model analysis indicates that directional persistence in uniform CF concentrations is enhanced by increasing rate constants for receptor cytoskeletal inactivation, ternary complex dissociation, and binary complex dissociation, and by decreasing rate constants for receptor internalization and recycling. For most rate constants, we have detected an optimal range that maximizes orientation bias in CF gradients. We have also examined different desensitization and receptor recycling mechanisms that yield experimentally documented orientation behavior. These yield novel insights into the relationship between receptor dynamics and leukocyte chemosensory movement behavior.     

Rate constants for binding, dissociation, and internalization of EGF: effect of receptor occupancy and ligand concentration

We measured the kinetic parameters for interaction of epidermal growth factor (EGF) with fetal rat lung (FRL) cells under two sets of experimental conditions and applied sensitivity analysis to see which parameters were well-defined. In the first set of experiments (method 1), the kinetics of internalization and dissociation of radiolabeled EGF were measured with a temperature-shift protocol in medium initially devoid of free ligand. The initial concentration of radiolabeled EGF bound to the cell surface corresponded to levels of receptor occupancy ranging from approximately 200 receptors per cell to approximately 18,000 receptors per cell, a level at which EGF binding approaches saturation. In the second set of experiments (method 2), carried out at a constant temperature, we began with no surface-bound or internalized ligand. The initial free ligand concentration was varied from 0.2 to 50 ng/mL. In both sets of experiments, we measured surface-bound, internalized, and free 125I-EGF as functions of time and evaluated the parameters of a mathematical model of endocytosis. Sensitivity analysis showed that three rate constants were well-defined in this combination of two experimental approaches: ke, the endocytic rate constant; ka, the association rate constant; and kd, the dissociation rate constant. The endocytic parameter ke was found to be independent of initial surface receptor occupancy (method 1); there was some indication that it increased with initial free ligand concentration in method 2. Neither kd nor ka was found to change with extent of initial surface receptor occupancy or initial free ligand concentration, respectively, a finding of significance, since diffusion theory predicts these parameters will vary with surface receptor occupancy.(ABSTRACT TRUNCATED AT 250 WORDS)     

Analysis of the influences of the E5 transforming protein on kinetic parameters of epidermal growth factor binding and metabolism.

The E5 protein of the bovine papillomavirus induces cellular transformation when transfected into NIH 3T3 cells, and the extent of focal transformation is enhanced by cotransfection with the epidermal growth factor (EGF) receptor (Martin et al., Cell 59:21-32, 1989). To determine whether E5 affects EGF:receptor interactions we analyzed the kinetics of 125I-EGF processing using a mathematical model that enabled us to evaluate rate constants for ligand association (ka), dissociation (kd), internalization (ke), recycling (kr), and degradation (kh). These rate constants were measured in NIH 3T3 cells transfected with the human EGF receptor (ER cells) and in cells transfected with both the EGF receptor and E5 (E5/ER cells). We found that the rate constant for 125I-EGF association ka was significantly decreased in E5/ER cells, but was apparently occupancy-independent in both cell lines. The 125I-EGF dissociation rate constant kd was significantly lower in E5 transformed cells, and increased with occupancy in both cell lines. This suggests that E5 alters the receptor before or during EGF binding so that ligand association is slower; however, once complexes are formed, EGF is bound more tightly to the receptor. Rate constants for internalization ke were also found to be occupancy-dependent, although at a given level of occupancy ke was similar for both cell lines. Also, there was no apparent effect of E5 on the recycling rate constant kr. The 125I-EGF degradation rate constant kh was 30% lower in E5 transformed cells, and was occupancy-independent. The overall effect of E5 is to stabilize intact EGF:receptor complexes which may alter mitogenic signaling of the receptor.     

Gating kinetics of batrachotoxin-modified Na+ channels in the squid giant axon. Voltage and temperature effects.

The gating kinetics of batrachotoxin-modified Na+ channels were studied in outside-out patches of axolemma from the squid giant axon by means of the cut-open axon technique. Single channel kinetics were characterized at different membrane voltages and temperatures. The probability of channel opening (Po) as a function of voltage was well described by a Boltzmann distribution with an equivalent number of gating particles of 3.58. The voltage at which the channel was open 50% of the time was a function of [Na+] and temperature. A decrease in the internal [Na+] induced a shift to the right of the Po vs. V curve, suggesting the presence of an integral negative fixed charge near the activation gate. An increase in temperature decreased Po, indicating a stabilization of the closed configuration of the channel and also a decrease in entropy upon channel opening. Probability density analysis of dwell times in the closed and open states of the channel at 0 degrees C revealed the presence of three closed and three open states. The slowest open kinetic component constituted only a small fraction of the total number of transitions and became negligible at voltages greater than -65 mV. Adjacent interval analysis showed that there is no correlation in the duration of successive open and closed events. Consistent with this analysis, maximum likelihood estimation of the rate constants for nine different single-channel models produced a preferred model (model 1) having a linear sequence of closed states and two open states emerging from the last closed state. The effect of temperature on the rate constants of model 1 was studied. An increase in temperature increased all rate constants; the shift in Po would be the result of an increase in the closing rates predominant over the change in the opening rates. The temperature study also provided the basis for building an energy diagram for the transitions between channel states.