Theoretical results for chemotactic response and drift of E. coli in a weak attractant gradient

Abstract

The bacterium Escherichia coli (E. coli) moves in its natural environment in a series of straight runs, interrupted by tumbles which cause change of direction. It performs chemotaxis towards chemo-attractants by extending the duration of runs in the direction of the source. When there is a spatial gradient in the attractant concentration, this bias produces a drift velocity directed towards its source, whereas in a uniform concentration, E. coli adapts, almost perfectly in case of methyl aspartate. Recently, microfluidic experiments have measured the drift velocity of E. coli in precisely controlled attractant gradients, but no general theoretical expression for the same exists. With this motivation, we study an analytically soluble model here, based on the Barkai-Leibler model, originally introduced to explain the perfect adaptation. Rigorous mathematical expressions are obtained for the chemotactic response function and the drift velocity in the limit of weak gradients and under the assumption of completely random tumbles. The theoretical predictions compare favorably with experimental results, especially at high concentrations. We further show that the signal transduction network weakens the dependence of the drift on concentration, thus enhancing the range of sensitivity.

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

“…After using Eqs (37) and (38) in Eq (58), and performing the inversion, we arrive at the following multi-exponential form for χ b ( t ): The rate constants A and B , as well as the coefficients X m , Y m and Z m ( m = 0, 2) are expressed in terms of β mn and γ m as follows: In Fig 10, we show plots of the mathematical expression for χ b ( t ) obtained in Eq (59), (a) by varying B 0 and (b) varying ℓ . The curves closely resemble the experimentally measured responses to short-lived stimuli [9, 30], for somewhat large values of B 0 , in agreement with earlier results in a mean-field BL model [15]. In (a), with increase in B 0 , there is an overall reduction in time scales and an increase in the depth of the negative lobe.…”

“…It follows that fractional changes in the CW bias P CW and Y are related as where, in the second part, we have defined the response function χ b ( t ) for the bias. Using Eqs (55) and (54) in Eq (57), it follows that, in steady state, χ b ( t ) is related to χ a ( t ) through From Eq (58), it follows that the Laplace transforms of χ b ( t ) and χ a ( t ) are related linearly: , hence , i.e., the response curve for the bias too encloses zero area [9, 15]. After using Eqs (37) and (38) in Eq (58), and performing the inversion, we arrive at the following multi-exponential form for χ b ( t ): The rate constants A and B , as well as the coefficients X m , Y m and Z m ( m = 0, 2) are expressed in terms of β mn and γ m as follows: In Fig 10, we show plots of the mathematical expression for χ b ( t ) obtained in Eq (59), (a) by varying B 0 and (b) varying ℓ .…”

“…We assume that the lowest state ( m = 0) is always inactive, while m = M is always active; the intermediate state(s) can be either active or inactive, depending on whether it is attractant-bound. A simple mathematical form satisfying these conditions is a m ( L ) = a m (0) K m /( L + K m ) where K m is the dissociation constant for attractant binding to a receptor in state m [14, 15, 20]. To satisfy the earlier assumption, we choose K 0 = 0 and K M = ∞ (these are ideal values).…”

“…The specific numerical values for various parameters correspond to the methylation–demethylation reactions of chemotaxis receptors in the bacterium E. coli , previously used by various authors [2, 15]. Initially, we choose all the receptors to be in inactive, unbound configuration at the lowest methylation/inactive level ( m = 0).…”

“…As a consequence of these assumptions, the mean receptor activity in the model in steady state turns out to be independent of the attractant concentration. Further, mathematical modeling has shown that, with suitable extensions, the model also successfully predicts the response of the network to a short-lived spike in attractant concentration [15]. …”