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One of the most persistent and intriguing discrepancies between our computer simulations of chemotaxis and the actual performance of living bacteria is in the matter of chemotactic gain. For example, most computer simulations of the chemotaxis pathway based on experimentally determined rates and concentrations predict a minimum detectable concentration of the attractant aspartate of around 200 nM. However, classic experiments by Segall et al. in 1986, in which E. coli cells tethered to a coverslip were exposed to small quantities of chemoattractant delivered iontophoretically, indicated that a receptor occupancy of as little as 1/600 could produce an detectable change in swimming behaviour (Segall et al., 1986). With a Kd of 1 µM, this corresponds to a minimum detectable concentration of about 2 nM aspartate (that is, less than 1% of the calculated value). Note that this large difference follows as an inevitable consequence of a mechanism in which attractants are detected by the inhibition of kinase molecules associated with the receptors (Jasuja et al., 1999). It cannot easily be rectified by making qualitative changes to the kinetics of the pathway — such as increasing the specific activity of the kinase, or increasing the "Hill coefficient" for the interaction between CheYp and the motor switch complex.
In 1998, we raised the possibility that the remarkable sensitivity and range of response of bacterial chemotaxis might depend on the clustering of chemotactic receptors on the surface of the bacterium (Bray et al., 1998) (see Receptor clusters). Specifically, we suggested that when a ligand bound to a receptor, the change in activity might propagate to neighbouring receptors in a cluster. We calculated that if the size of this "infective" spread was changed by adaptation, then the system could readily reproduce the chemotactic response of actual bacteria.
Snapshot of an array of receptors showing a spread of activity. The receptors are portrayed as hexagons in an instantaneous snapshot showing their conformation as being either inactive (white) or active (orange). Binding of a ligand to one receptor (cross) favours the inactive state. A mechanism that allows this inhibited state to spread to neighbouring receptors — shown on the right — will generate a much larger change for downstream signalling (from Bray et al., 1998).
How might the activity of receptors spread across a cluster? Discussions with Tom Duke and Yu Shi, two physicists at the Cavendish Laboratory, led to the suggestion that the energetic exchanges between proteins in a two-dimensional lattice could be likened to the interactions of magnetic dipoles in a spin glass and analysed by means of an Ising model (Shi & Duke, 1998). A more explicit statistical mechanical analysis of a simplified array of receptors was then performed, in which the receptors were portrayed as allosteric enzymes. Each receptor could exist in one of two conformations, one of which was favoured by the binding of a ligand. Receptor conformations were also affected by the conformations of the four nearest neighbours in the lattice — due to a so-called "coupling energy". Choice of a suitable value of this energy could integrate the activities of receptors over an extended lattice (Duke & Bray, 1999).
Basis of the Monte Carlo simulation. A square lattice of receptors, each of which is in either an active ("1") or inactive ("0") conformation, is sampled at regular time intervals. At each time step, a randomly selected receptor flips its conformation if R > exp(-E), where R is a random number between 0 and 1, and E is an activation energy. The value of E depends on (i) whether or not the receptor has a bound ligand (favours inactive conformation) (ii) whether or not it has adapted (favours active conformation) and (iii) the current activity of the four nearest neighbours (tend to couple the central receptor to their activity state). The system changes on a millisecond time scale because of the diffusive binding and dissociation of ligand molecules, and on a second time scale because of the chemical modification causing adaptation.
Representative results from the Monte Carlo simulation. A 50 x 50 array of receptors is shown that are in either active or inactive conformation (displayed with toroidal coordinates, so that they "wrap around" from right to left and top to bottom). One selected receptor has been "pinned" by having a ligand molecule permanently bound (circle plus "X"). A second receptor is permanently adapted (empty circle). The image on the left shows an instantaneous snapshot of a continually changing field. That on the right shows the same array with the activities of the receptors averaged over 10,000 time intervals. We see that the ligand bound receptor has on average "nucleated" a patch of inactive receptors and the adapted receptor has nucleated a patch of active receptors. This capacity to affect neighbouring receptors forms the basis of the high sensitivity of the system.
A series of simulations were run using this Duke-Bray model in which an array of 50 x 50 receptors was exposed to different concentrations of ligand. The signal output of the array was measured as the number of active receptors at a particular instant of time. With suitable selection of parameters, and especially with a critical value of the interaction energy between neighbouring receptors, the system showed a high sensitivity and wide dynamic range. It's performance was at least an order of magnitude better than the same array in which coupling between neighbouring receptors did not occur.
Sample results from the Duke-Bray model. Plots of number of active receptors (vertical axis) against time. Each time course represents a 50 x 50 array that was allowed to adapt to the concentration of ligand indicated on the left. At 10 s, this concentration was abruptly doubled. The system can detect changes from less than 0.01 µM to over 100 µM.
We recently applied the postulated mechanism of conformational spread to the one-dimensional, unbounded case of a closed ring of proteins (Duke et al., 2001). The simple geometry of this new situation allows us to define rigorously the conditions under which a ring will show cooperative switching and to relate the physics of conformational spread to classical models of allostery (the canonical MWC and KNF models emerge naturally as limiting cases of conformational spread). This study also revealed the interesting fact that the time taken for a large multiprotein assembly to change its state may be much greater than that of individual allosteric transitions — a phenomenon we suspect may be of general significance in the living cell.
Bray, D., Levin, M. D., & Morton-Firth, C. J. (1998). Receptor clustering as a cellular mechanism to control sensitivity. Nature 393, 85-88.
Duke, T. A. J., & Bray, D. (1999). Heightened sensitivity of a lattice of membrane receptors. Proc. Natl. Acad. Sci. USA 96, 10104-10108.
Duke, T. A. J., Le Novère, N., & Bray, D. (2001). Conformational spread in a ring of proteins: a stochastic view of allostery. J. Mol. Biol. 308, 541-553.
Jasuja, R., Yu-Lin, Trentham, D. R., & Khan, S. (1999). Response tuning in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA 96, 11346-11351.
Segall, J. E., Block, S. M., & Berg, H. C. (1986). Temporal comparisons in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA 83, 8987-8991.
Shi, Y., & Duke, T. (1998). Cooperative model of bacterial sensing. Phys. Rev. E 58, 6399-6406.
| Site designed by Matthew Levin. This page updated 3 May 2006. |