Nonequilibrium lattice dynamics: A new paradigm for modeling cooperative signaling in a large protein complex

How do bacteria navigate in complex environments?  This simple question has been studied for decades in the model bacteria Escherichia coli to reveal general principles of cellular information processing. Nevertheless, recent experiments and theory are beginning to show a new paradigm that highlights the importance of nonequilibrium cooperative interactions in signaling.

E. coli perform chemotaxis: by sensing their environment they can move up or down gradients in chemical concentrations to, for example, look for areas rich in sugars necessary to grow and reproduce. This behavior is achieved with a simple biochemical network: first, receptors on the cell surface bind to chemical attractants; this measurement is then propagated to activate messenger proteins within the cell, which in turn control the flagella, the motors that propel the bacteria forward or change its heading direction.

Time-reversal symmetry breaking: Recent experiments by Keegstra and colleagues [1] discovered a surprising behavior: even without any attractant in the environment, the signaling activity inside the cell can spontaneously switch between fully active and fully inactive states. Furthermore, the time it takes to switch depends on the direction of switching: backward switching takes longer. These switching statistics break time reversal symmetry, which hints towards an underlying nonequilibrium process. To elucidate this idea, consider the three-state system shown in the video below. For an equilibrium reversible system, the transitions between these states have no net clockwise or counterclockwise bias. Thus, playing the video forward or backward results in trajectories that are statistically identical. On the other hand, the irreversible system always transitions counterclockwise, so that the backward video (which always transitions clockwise) can be easily distinguished. This irreversibility, whether in the three-state model or in the switching statistics of the chemotaxis signaling system (Figure 1), demonstrates that energy is being dissipated by the underlying dynamics.

Video: Time-reversal symmetry breaking in a three-state system. Left: A reversible system has no clockwise or counterclockwise bias. When the movie is played backward (time reversed) its dynamics are statistically equivalent to the forward video. Right: The irreversible system always transitions counterclockwise, while the backward video always transitions clockwise, breaking time-reversal symmetry.

The nonequilibrium lattice model: Understanding the switching time asymmetry and its implications for signaling function requires two essential ingredients: cooperativity and dissipation. Cooperativity stems from coupling between many chemoreceptors, which form a spatially extended lattice on the cell membrane. Dissipation comes from continuous hydrolysis of ATP by a kinase protein that propagates the signal from the receptors to intracellular signal carriers via phosphorylation reactions. By introducing a nonequilibrium lattice model of the that, for the first time, incorporates these two mechanisms, our paper explains the switching asymmetry and elucidates how signaling response is enhanced by the interplay between dissipative chemical reaction cycles and the collective behavior of the receptor lattice.

Key insights on cooperative sensing: Our investigations reveal two key insights into the function of the chemosensory array.

1) Near-criticality: The system operates near a critical point, where receptors in the lattice transition from sensing independently to sensing collectively, leading to a higher sensitivity to external stimuli. Increasing receptor coupling strength promotes higher sensitivity, but this comes at the price of slow response speed because the strongly coupled lattice has a slow relaxation time.

2) Dissipation eases the speed-sensitivity trade-off: E. coli compensates for the sensitivity-speed trade-off problem by operating out of equilibrium. The energy dissipation (through ATP hydrolysis) in subunits of the receptor lattice breaks the fluctuation-dissipation theorem, which allows the sensory array to have high sensitivity and high response speed at the same time. Figure 2 shows examples of speed-sensitivity trade-off curves. If the cell must meet some speed and sensitivity operational thresholds (gray shaded region) to effectively sense its environment, then there is a minimum energy input (ΔG_min) that is required. Operating at this dissipation level requires fine-tuning of the array coupling to meet both speed and sensitivity thresholds (red point). Increasing dissipation further (ΔG_min>ΔG) enhances the robustness of the signaling, opening up a range of lattice parameters (red curve) where cells can achieve both sensitive and rapid response.

A structure-based modeling paradigm: Our work lays the groundwork for precise structure-based modeling of the chemosensory array and other large protein complexes. By combining details of the protein-protein interactions within these large protein complexes, unveiled by powerful experimental tools such as cryoEM, with the nonequilibrium kinetic reaction networks, we can develop quantitative models to understand and predict how protein complex structures give rise to biological functions. Our approach is highly generalizable; it will be exciting to see the range of biological complexes that can be studied within this new modeling paradigm.

[1] J. M. Keegstra, F. Avgidis, Y. Mullah, J. S. Parkinson, and T. S. Shimizu, Near-Critical Tuning of Cooperativity Revealed by Spontaneous Switching in a Protein Signalling Array, bioRxiv preprint https://doi.org/10.1101/2022.12.04.518992 (2022).

Poster image credit: Adapted from A. Burt, C.K. Cassidy, P.J. Stansfeld, I, Gutsche, Alternative Architecture of the E. coli Chemosensory Array. Biomolecules 11, 495 (2021). Reproduced under Creative Commons Attribution (CC BY) license, https://creativecommons.org/licenses/by/4.0/