Ultrasensitivity and fluctuations in the Barkai-Leibler model of chemotaxis receptors in Escherichia coli

Abstract

A stochastic version of the Barkai-Leibler model of chemotaxis receptors in Escherichia coli is studied here with the goal of elucidating the effects of intrinsic network noise in their conformational dynamics. The model was originally proposed to explain the robust and near-perfect adaptation of E. coli observed across a wide range of spatially uniform attractant/repellent (ligand) concentrations. In the model, a receptor is either active or inactive and can stochastically switch between the two states. The enzyme CheR methylates inactive receptors while CheB demethylates active receptors and the probability for a receptor to be active depends on its level of methylation and ligand occupation. In a simple version of the model with two methylation sites per receptor (M = 2), we show rigorously, under a quasi-steady state approximation, that the mean active fraction of receptors is an ultrasensitive function of [CheR]/[CheB] in the limit of saturating receptor concentration. Hence the model shows zero-order ultrasensitivity (ZOU), similar to the classical two-state model of covalent modification studied by Goldbeter and Koshland (GK). We also find that in the limits of extremely small and extremely large ligand concentrations, the system reduces to two different two-state GK modules. A quantitative measure of the spontaneous fluctuations in activity is provided by the variance in the active fraction, which is estimated mathematically under linear noise approximation (LNA). It is found that peaks near the ZOU transition. The variance is a non-monotonic, but weak function of ligand concentration and a decreasing function of receptor concentration. Gillespie simulations are also performed in models with M = 2, 3 and 4. For M = 2, simulations show excellent agreement with analytical results obtained under LNA. Numerical results for M = 3 and M = 4 are qualitatively similar to our mathematical results in M = 2; while all the models show ZOU in mean activity, the variance is found to be smaller for larger M. The magnitude of receptor noise deduced from available experimental data is consistent with our predictions. A simple analysis of the downstream signaling pathway shows that this noise is large enough to affect the motility of the organism, and may have a beneficial effect on it. The response of mean receptor activity to small time-dependent changes in the external ligand concentration is computed within linear response theory, and found to have a bilobe form, in agreement with earlier experimental observations.

Steady-state running rate sets the speed and accuracy of accumulation of swimming bacteria

“…The underlying biochemistry is abstracted away but it is easy to generalize the model to incorporate more complex intracellular dynamics. The relevant length-scales in this framework are intermediate between those of PDE (partial differential equation) models ( 50 ), which capture average properties of populations but are insensitive to microscopic details, and mechanistic models that link behavior of individual bacteria to intracellular biochemistry ( 51 , 52 , 53 , 54 ) but are more difficult to scale to experimentally realistic large populations. Furthermore, the path-integral approach is well suited to studying chemotaxis in complex, time-varying environments and could easily be extended to incorporate cell-to-cell interactions in space and time.…”

Flagellar dynamics reveal fluctuations and kinetic limit in theEscherichia colichemotaxis network

“…Previous single-cell measurements of flagellar rotation by bead tracking 31,39 and of CheA activity using fluorescent reporters 40,41 have reported long-term noise in the chemotaxis network. This has largely been attributed to CheR-CheB methylation-demethylation dynamics 31,68 and, recently, to new sources such as receptor clustering and other dynamics 40,41,67 . A direct comparison between these results and ours is difficult because of the differences in timescales between the measurements (3-40 s vs. 10-1000 s).…”

A path-integral characterization of run and tumble motion and chemotaxis of bacteria

“…where ∆ is the time delay between the centres of the positive and negative lobes, κ is an empirical parameter which depends on the details of the underlying biochemical network and δ(t) is the Dirac delta function. The response function has also been computed explicitly [11,12] using variants of the Barkai-Leibler model [13] for the receptor methylation-demethylation processes, originally introduced to explain the perfect adaptation property of E. coli, and the robustness of the network output to cell-to-cell variations in enzyme concentrations. Using a combination of heuristic arguments and rigorous calculations, de Gennes [7] derived the following expression for the drift velocity of a bacterium in two dimensions, in a concentration gradient ∇c = α:…”