E. coli chemoreceptor: mesoscopic model

From Ilya Nemenman: Theoretical Biophysics @ Emory
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An E. coli chemoreceptor cluster consists of a mesoscopic number of homodimer receptors, each of which can bind a ligand. The entire receptor cluster then can be in an active or inactive state (in the simplest model). The transformation from the inactive to the active state may be a function of purely the number of bound receptors, or due to interactions among receptor homodimers. This is clearly an Ising model, where ligand concentration has the role of the magnetic field, and receptor interaction are the spin-spin coupling. In models of different complexity, the cluster of receptors may be heterogeneous or nonheterogeneous, the receptors may be considered on a regular or irregular lattice, and the interactions between the receptors may vary, from just nearest neighbor, to longer range, and from uniform to varying and even random.

This model is well described by various Ising model Hamiltonians of various complexity (see references below). However, dynamics of this model, especially in the case of time-varying ligand concentration requires more than just equilibrium statistical mechanics. Specifically, we have a mesoscopic number of receptors coupled to each other, and our methods can allow calculation of mean receptor activities and their variances even for time-dependent ligand concentrations. Different levels of modeling may be possible, including modeling ligand binding, activity state, and metilation state (where activity state depends not just on binding and neighbor activitie, but also on the metilation). For a simple proof of concept article, just modeling binding is probably good enough, but, if data becomes available, detailed modeling can be done (maybe numerically).

Relevant References

  1. B Mello and Y Tu. Quantitative modeling of sensitivity in bacterial chemotaxis: The role of coupling amonf different chemoreceptor specied. PNAS 100:8223-8228, 2003. PDF.
    We propose a general theoretical framework for modeling receptor sensitivity in bacterial chemotaxis, taking into account receptor interactions, including those among different receptor species. We show that our model can quantitatively explain the recent in vivo measurements of receptor sensitivity at different ligand concentrations for both mutant and wild-type strains. For mutant strains, our model can fit the experimental data exactly. For the wild-type cell, our model is capable of achieving high gain while having modest values of Hill coefficient for the response curves. Furthermore, the high sensitivity of the wild-type cell in our model is maintained for a wide range of ambient ligand concentrations, facilitated by near-perfect adaptation and dependence of ligand binding on receptor activity. Our study reveals the importance of coupling among different chemoreceptor species, in particular strong interactions between the aspartate (Tar) and serine (Tsr) receptors, which is crucial in explaining both the mutant and wild-type data. Predictions for the sensitivity of other mutant strains and possible improvements of our model for the wild-type cell are also discussed.
    A mean-field Ising coupling of receptors in a cluster is considered, and each of the receptors in a cluster may become active depending on its own ligand binding status, as well as on interactions with other receptors (many active receptors in a vicinity make a particular receptor active). Then comparison to Berg's data is used to derive various receptor parameters; importantly, the rate of ligand binding is found to be dependent on the receptor activity. For our purposes, this provides a model where each of receptors can be either active or inactive, and transition into an active state is a function of the ligand concentration plus the number of active receptors, while the transition out of the active state happens at a constant rate (there may be a problem, however, that it's not activity, but rather binding, which is ligand-dependent, and this incorrect dimensionality reduction may cause wrong statistics). A better model is to have a four-state receptor (active/inactive, bound/unbound), so that binding is proportional to ligand concentration, and activation is proportional to binding, or something similar. Then one can actually check if analysis of fluctuations allows to distinguish such Ising models from MWC models (below). This is worth thinking about.
  2. B Mello, L Shaw, and Y Tu. Effects of receptor interaction in bacterial chemotaxis. Biophys. J. 87:1578-1595, 2004. PDF
    Signaling in bacterial chemotaxis is mediated by several types of transmembrane chemoreceptors. The chemoreceptors form tight polar clusters whose functions are of great biological interest. Here, we study the general properties of a chemotaxis model that includes interaction between neighboring chemoreceptors within a receptor cluster and the appropriate receptor methylation and demethylation dynamics to maintain (near) perfect adaptation. We find that, depending on the receptor coupling strength, there are two steady-state phases in the model: a stationary phase and an oscillatory phase. The mechanism for the existence of the two phases is understood analytically. Two important phenomena in transient response, the overshoot in response to a pulse stimulus and the high gain in response to sustained changes in external ligand concentrations, can be explained in our model, and the mechanisms for these two seemingly different phenomena are found to be closely related. The model also naturally accounts for several key in vitro response experiments and the recent in vivo fluorescence resonance energy transfer experiments for various mutant strains. Quantitatively, our study reveals possible choices of parameters for fitting the existing experiments and suggests future experiments to test the model predictions.
    Almost the same model as in PNAS article above is considered, and references to simplified model (Shimizu et al.) are provided. Mostly numerics/mean-field paper with cute results, showing phase diagrams of the receptor cluster activity, including oscillatory regimes, etc. Interestingly, the paper pursues time scales separation -- fast binding/activation dynamics (which is treated here as equilibrated by stat mech tools) and slow methylation/demethylation processes. Here is one more suggestion for a possible analysis: study the system where the activation rates change slowly due to the mean activation level over long time; this will be similar to a Michaelis-Menten system, where forward or backward rates are functions of substrate/product balance, which itself is influenced by the rates over a long time. If we figure out how to do this, we would be able to put values on the mean and variance of adaptation processes. See also Lab Internal: Stochastic biochemistry.
  3. B Mello and Y Tu. An allosteric model for heterogeneous receptor complexes: Understanding bacterial chemotaxis responses to multiple stimuli. PNAS 102:17354-17359, 2005. PDF.
    The classical Monod-Wyman-Changeux model for homogeneous allosteric protein complex is generalized in this article to model the responses of heterogeneous receptor complexes to multiple types of ligand stimulus. We show that the recent in vivo experimental data of Escherichia coli chemotaxis responses for mutant strains with different expression levels of the chemo-receptors to different types of stimulus [Sourjik, V. & Berg, H. C. (2004) Nature 428, 437–441] all can be explained consistently within this generalized Monod-Wyman-Changeux model. Based on the model and the existing data, responses of all of the strains (studied in this article) to the presence of any combinations of ligand (Ser and MeAsp) concentrations are predicted quantitatively for future experimental verification. Through modeling the in vivo response data, our study reveals important information about the properties of different types of individual receptors, as well as the composition of the cluster. The energetic contribution of the nonligand binding, cytoplasmic parts of the cluster, such as CheA and CheW, is also discussed. The generalized allosteric model provides a consistent framework in understanding signal integration and differentiation in bacterial chemotaxis. It should also be useful for studying the functions of other heterogeneous receptor complexes.
    Compared to the standard Monod-Wyman-Changeus (MWC) model, this paper introduces heterogeneous receptors coupled to different ligands. The coupling between the receptors is still all-or-none. Comparison is made to the recent Berg's data on sensitivity to different ligands with different expressions of receptor genes. In the end of the paper, they discuss the difference between the Ising and the MWC model of a receptor complex.