A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis

Cells generally chemotax along a direction in which their receptor occupancy gradient—whether spatial or temporal—is maximum. Occupancy differentials are, however, often so small as to be masked by thermal noise; i.e., by fluctuations inherent in the stochastic nature of ligand binding. Such fluctuations therefore impose a fundamental limit on the sensitivity of a cell's ability to detect a chemoattractant gradient. In order to pursue the implications of this limit, fluctuation theories have been developed. The theories assume that the signal is some function of the receptor occupancy gradient, allow an estimate of the standard deviation abouts the mean signal, and permit an evaluation of, among other things, the extent to which a receptor defect can impair an effective response. Previous theories have assumed an equilibrated ligand-receptor interaction. In this paper we introduce a generalized definition of a signal caused by a receptor occupancy gradient that allows us to develop a non-equilibrium theory of thermal noise. We show that previous formulations are a special case of the current development. More specifically, we find the following.

1. Swimming cells subject to Brownian tumbling must generally average their signals over a very long time period to achieve a signal-to-noise ratio≤1. Spatial gradient detection is possible with ligand-receptor equilibrium constants<10³ M ^?1, but since such ligands are rare, theory predicts that tumbling cells will generally not detect gradients by measuring spatial occupancy differentials.These conclusions hold irrespective of whether chemical equilibrium is achieved.
2. For crawling cells not subject to Brownian tumbling, a range of affinities exists in which spatial or temporal gradient detection is possible. In general a spatial mechanism is more efficient for low affinity ligands (dissociation times <0.3s), whereas a temporal mechanism is more efficients for higherK. In this case the detection of gradients in slowly dissociating ligand will be facilitated if signal processing begins prior to chemical equilibration.
3. An important new parameter is indicated by the theory. The definitions of a temporal gradient signal is based on estimating and comparing average occupancy over two time intervals displaced by a timet ₁. The theory predicts an optimalt ₁, of order milliseconds, that leads to the shortest minimum averaging time.
4. Fort ₁ values at and longer than the optimum, and for all averaging times exceeding some minimum, the cell will detect a temporal signal.
5. For values oft ₁ at and near the optimum, if the averaging time becomes too long, the cell enters a region of insensitivity in which it can no longer respond.
6. Finally, as the interval between estimates of average occupancy decreases below the optimum, a critical value oft ₁ is reached at which the minimum averaging time undergoes an abrupt transition from a relatively short value to a value five orders of magnitude longer.

The molecular process(es) controllingt ₁ are at present unknown, nor has any attempt been made to identify them since the parameter has not been previously recognized. We speculate that the search for its molecular basis might uncover a highly sensitive control mechanism, with defects in this mechanism predicted to have a far more pronounced effect on the cells behavior than defects in receptor number or affinity.