The switching dynamics of the bacterial flagellar motor

Many swimming bacteria are propelled by flagellar motors that stochastically switch between the clockwise and counterclockwise rotation direction. While the switching dynamics are one of the most important characteristics of flagellar motors, the mechanisms that control switching are poorly understood. We present a statistical-mechanical model of the flagellar rotary motor, which consists of a number of stator proteins that drive the rotation of a ring of rotor proteins, which in turn drives the rotation of a flagellar filament. At the heart of our model is the assumption that the rotor protein complex can exist in two conformational states corresponding to the two respective rotation directions, and that switching between these states depends on interactions with the stator proteins. This naturally couples the switching dynamics to the rotation dynamics, making the switch sensitive to torque and speed. Another key element of our model is that after a switching event, it takes time for the load to build up, due to polymorphic transitions of the filament. Our model predicts that this slow relaxation dynamics of the filament, in combination with the load dependence of the switching frequency, leads to a characteristic switching time, in agreement with recent observations.Comment: 7 pages, 6 figures, RevTeX

Differential Affinity and Catalytic Activity of CheZ in E. coli Chemotaxis

Push–pull networks, in which two antagonistic enzymes control the activity of a messenger protein, are ubiquitous in signal transduction pathways. A classical example is the chemotaxis system of the bacterium Escherichia coli, in which the kinase CheA and the phosphatase CheZ regulate the phosphorylation level of the messenger protein CheY. Recent experiments suggest that both the kinase and the phosphatase are localized at the receptor cluster, and Vaknin and Berg recently demonstrated that the spatial distribution of the phosphatase can markedly affect the dose–response curves. We argue, using mathematical modeling, that the canonical model of the chemotaxis network cannot explain the experimental observations of Vaknin and Berg. We present a new model, in which a small fraction of the phosphatase is localized at the receptor cluster, while the remainder freely diffuses in the cytoplasm; moreover, the phosphatase at the cluster has a higher binding affinity for the messenger protein and a higher catalytic activity than the phosphatase in the cytoplasm. This model is consistent with a large body of experimental data and can explain many of the experimental observations of Vaknin and Berg. More generally, the combination of differential affinity and catalytic activity provides a generic mechanism for amplifying signals that could be exploited in other two-component signaling systems. If this model is correct, then a number of recent modeling studies, which aim to explain the chemotactic gain in terms of the activity of the receptor cluster, should be reconsidered

Signaling Cascades Modulate the Speed of Signal Propagation through Space

Cells are not mixed bags of signaling molecules. As a consequence, signals must travel from their origin to distal locations. Much is understood about the purely diffusive propagation of signals through space. Many signals, however, propagate via signaling cascades. Here, we show that, depending on their kinetics, cascades speed up or slow down the propagation of signals through space, relative to pure diffusion.We modeled simple cascades operating under different limits of Michaelis-Menten kinetics using deterministic reaction-diffusion equations. Cascades operating far from enzyme saturation speed up signal propagation; the second mobile species moves more quickly than the first through space, on average. The enhanced speed is due to more efficient serial activation of a downstream signaling module (by the signaling molecule immediately upstream in the cascade) at points distal from the signaling origin, compared to locations closer to the source. Conversely, cascades operating under saturated kinetics, which exhibit zero-order ultrasensitivity, can slow down signals, ultimately localizing them to regions around the origin.Signal speed modulation may be a fundamental function of cascades, affecting the ability of signals to penetrate within a cell, to cross-react with other signals, and to activate distant targets. In particular, enhanced speeds provide a way to increase signal penetration into a cell without needing to flood the cell with large numbers of active signaling molecules; conversely, diminished speeds in zero-order ultrasensitive cascades facilitate strong, but localized, signaling

A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of E. coli MinE to E-ring formation

Many important cellular processes are regulated by reaction-diffusion (RD) of molecules that takes place both in the cytoplasm and on the membrane. To model and analyze such multicompartmental processes, we developed a lattice-based Monte Carlo method, Spatiocyte that supports RD in volume and surface compartments at single molecule resolution. Stochasticity in RD and the excluded volume effect brought by intracellular molecular crowding, both of which can significantly affect RD and thus, cellular processes, are also supported. We verified the method by comparing simulation results of diffusion, irreversible and reversible reactions with the predicted analytical and best available numerical solutions. Moreover, to directly compare the localization patterns of molecules in fluorescence microscopy images with simulation, we devised a visualization method that mimics the microphotography process by showing the trajectory of simulated molecules averaged according to the camera exposure time. In the rod-shaped bacterium _Escherichia coli_, the division site is suppressed at the cell poles by periodic pole-to-pole oscillations of the Min proteins (MinC, MinD and MinE) arising from carefully orchestrated RD in both cytoplasm and membrane compartments. Using Spatiocyte we could model and reproduce the _in vivo_ MinDE localization dynamics by accounting for the established properties of MinE. Our results suggest that the MinE ring, which is essential in preventing polar septation, is largely composed of MinE that is transiently attached to the membrane independently after recruited by MinD. Overall, Spatiocyte allows simulation and visualization of complex spatial and reaction-diffusion mediated cellular processes in volumes and surfaces. As we showed, it can potentially provide mechanistic insights otherwise difficult to obtain experimentally

Positional Information Generated by Spatially Distributed Signaling Cascades

The temporal and stationary behavior of protein modification cascades has been extensively studied, yet little is known about the spatial aspects of signal propagation. We have previously shown that the spatial separation of opposing enzymes, such as a kinase and a phosphatase, creates signaling activity gradients. Here we show under what conditions signals stall in the space or robustly propagate through spatially distributed signaling cascades. Robust signal propagation results in activity gradients with long plateaus, which abruptly decay at successive spatial locations. We derive an approximate analytical solution that relates the maximal amplitude and propagation length of each activation profile with the cascade level, protein diffusivity, and the ratio of the opposing enzyme activities. The control of the spatial signal propagation appears to be very different from the control of transient temporal responses for spatially homogenous cascades. For spatially distributed cascades where activating and deactivating enzymes operate far from saturation, the ratio of the opposing enzyme activities is shown to be a key parameter controlling signal propagation. The signaling gradients characteristic for robust signal propagation exemplify a pattern formation mechanism that generates precise spatial guidance for multiple cellular processes and conveys information about the cell size to the nucleus

Quantitative Modeling of Escherichia coli Chemotactic Motion in Environments Varying in Space and Time

Escherichia coli chemotactic motion in spatiotemporally varying environments is studied by using a computational model based on a coarse-grained description of the intracellular signaling pathway dynamics. We find that the cell's chemotaxis drift velocity vd is a constant in an exponential attractant concentration gradient [L]∝exp(Gx). vd depends linearly on the exponential gradient G before it saturates when G is larger than a critical value GC. We find that GC is determined by the intracellular adaptation rate kR with a simple scaling law: . The linear dependence of vd on G = d(ln[L])/dx directly demonstrates E. coli's ability in sensing the derivative of the logarithmic attractant concentration. The existence of the limiting gradient GC and its scaling with kR are explained by the underlying intracellular adaptation dynamics and the flagellar motor response characteristics. For individual cells, we find that the overall average run length in an exponential gradient is longer than that in a homogeneous environment, which is caused by the constant kinase activity shift (decrease). The forward runs (up the gradient) are longer than the backward runs, as expected; and depending on the exact gradient, the (shorter) backward runs can be comparable to runs in a spatially homogeneous environment, consistent with previous experiments. In (spatial) ligand gradients that also vary in time, the chemotaxis motion is damped as the frequency ω of the time-varying spatial gradient becomes faster than a critical value ωc, which is controlled by the cell's chemotaxis adaptation rate kR. Finally, our model, with no adjustable parameters, agrees quantitatively with the classical capillary assay experiments where the attractant concentration changes both in space and time. Our model can thus be used to study E. coli chemotaxis behavior in arbitrary spatiotemporally varying environments. Further experiments are suggested to test some of the model predictions

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system

<p>Abstract</p> <p>Background</p> <p>A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.</p> <p>Results</p> <p>Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations.</p> <p>Conclusions</p> <p>This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.</p

25th Annual Computational Neuroscience Meeting: CNS-2016

Abstracts of the 25th Annual Computational Neuroscience Meeting: CNS-2016 Seogwipo City, Jeju-do, South Korea. 2–7 July 201