The Role of Adaptation in Bacterial Speed Races

Evolution of biological sensory systems is driven by the need for efficient responses to environmental stimuli. A paradigm among prokaryotes is the chemotaxis system, which allows bacteria to navigate gradients of chemoattractants by biasing their run-and-tumble motion. A notable feature of chemotaxis is adaptation: after the application of a step stimulus, the bacterial running time relaxes to its pre-stimulus level. The response to the amino acid aspartate is precisely adapted whilst the response to serine is not, in spite of the same pathway processing the signals preferentially sensed by the two receptors Tar and Tsr, respectively. While the chemotaxis pathway in E. coli is well characterized, the role of adaptation, its functional significance and the ecological conditions where chemotaxis is selected, are largely unknown. Here, we investigate the role of adaptation in the climbing of gradients by E. coli. We first present theoretical arguments that highlight the mechanisms that control the efficiency of the chemotactic up-gradient motion. We discuss then the limitations of linear response theory, which motivate our subsequent experimental investigation of E. coli speed races in gradients of aspartate, serine and combinations thereof. By using microfluidic techniques, we engineer controlled gradients and demonstrate that bacterial fronts progress faster in equal-magnitude gradients of serine than aspartate. The effect is observed over an extended range of concentrations and is not due to differences in swimming velocities. We then show that adding a constant background of serine to gradients of aspartate breaks the adaptation to aspartate, which results in a sped-up progression of the fronts and directly illustrate the role of adaptation in chemotactic gradient-climbing.


Extracting bacterial densities in race channels
The lateral channels of the microfluidic device described in the main text feature the 24 speed races performed by bacteria. The density of bacteria in the channels was obtained 25 from fluorescence images and the progression of the fronts as follows. 26 For each time step and each position in the channel, 20 fluorescence images were 27 acquired with an exposure time of 0.2 s. After subtraction of the background and 28 smoothing, the variation in the fluorescence intensity over noise (∆F/F ) corresponding 29 to the bacteria was systematically over 1, as shown in Fig. S1. The bacteria could then 30 be assimilated to moving objects and identified by keeping only the local maxima with 31 ∆F/F > const. The constant was fixed by experimentation to the value 5 and the 32 filtering was implemented by a standard process in the Fiji plugin [2]. Note that images 33 are taken only in the lateral channels, i.e. bacteria that reach the reservoirs are not  In order to verify that the progression within the channels reported in the main text is 54 germane to chemotaxis, we submitted the bacteria to a constant level (100µM) of serine 55 or aspartate. These controls (see Figure S2) show that the progression due to diffusion is 56 indeed much slower (roughly one order of magnitude) than the progressions obtained in 57 the presence of gradients. The square-root diffusive behavior is obviously not negligible 58 with respect to the linear-in-time drift at very short times yet this is limited to the very 59 first few minutes. We verified that including or leaving aside those initial few minutes 60 plays a negligible role in the estimation of the drift velocity. It is also worth noting that 61 constant backgrounds of aspartate or serine have comparable rates of progression. The goal of this Section is to provide estimates for the consumption of chemoattractants 65 by the bacteria in our experimental conditions. The analysis shows that the linear 66 attractant profiles established in the lateral channels before the injection of bacteria, are 67 not significantly altered during the experimental speed races.

68
The uptake rate of aspartate is ≈ 10 nmol/min/(mg of cell protein) [3]. Using that 69 E. coli has a cell protein content of ≈ 0.2pg per cell (see page 33 in [4]), it follows that 70 the uptake rate is α ≈ 20, 000 aspartate molecules per second per cell. The cell number 71 density in the injection channel of our microfluidic setup is n ≈ 4 · 10 7 cells/ml and the 72 number of bacteria in the lateral channels (where speed races actually take place) is at 73 least three orders of magnitude smaller. The latter estimate is obtained noting that few 74 hundred bacteria are present in the entire lateral channels even at the latest times of the uptake rate α above and the value of n for the injection channel, we obtain that the 79 rate of consumption αn ≈ 1 nM/s. Dividing the rate of consumption by the velocity of 80 drift v 1µm/s that we measured in our set-up (see main text), we obtain for the 81 gradient created by consumption ≈ 1 nM/µm. This value is already more than two 82 orders of magnitude smaller than the gradient 0.25 mM/mm that we maintain in the 83 lateral channels. Moreover, the consumption rate in the lateral channels is at least three 84 orders of magnitude smaller, as discussed above. We conclude that the aspartate linear 85 profiles in the lateral channels where speed races take place, are unaffected by bacterial 86 consumption.

87
Consumption rates of serine have the same order of magnitude as those for aspartate 88 (see Fig. 2 in [5]) ; the same conclusions drawn above for aspartate apply then to serine. saturation process and the mechanism intuitively explained in the main text. 96 We take images in the lateral channels only, as discussed in the main text and above. 97 It follows that those bacteria which penetrate into the reservoirs (where a constant level 98 of chemoattractant is maintained) stop being processed. This is equivalent to having an 99 absorbing boundary condition at the end of the channels. As long as all bacteria are for the dynamics of bacteria in aspartate (left) and serine (right) channels of length L = 4mm. The choice of the parameters of the model is discussed and motivated in the text. The progression function is defined as the cumulative distribution of the number of bacteria summed from a given location (shown on the abscissae in µm) to the end of the channel, where an absorbing boundary condition is imposed. The absorbing boundary reflects the fact that we experimentally take images only in the lateral channels and the large size of the reservoirs. Note that the slope of the curves decreases with x, i.e. bacterial density is higher in the back than at the edge of the fronts. 110 We performed numerical simulations of the following partial differential equation profile c(x) (see [6] for details) : Here, D 0 is the effective diffusion coefficient, which accounts for the random 115 component of the run-and-tumble motion. Its order of magnitude is v 2 t r , where v is the 116 bacterial speed and t r is the run time. The parameters χ and γ in Eq. (1) account for 117 the response of the chemotactic pathway to chemical stimuli. In particular, χ is the 118 chemotactic coefficient mentioned also in the main text. The parameter γ relates to the 119 level of adaptation : the response is perfectly adapted for γ = 0 while increasing values 120 of γ correspond to loss of precise adaptation, namely a positive lobe more pronounced 121 than the negative one in the impulse response function [1,7].

122
To determine the parameters in Eq. (1), we combined known results on the 123 chemotactic pathway with our measurements of the bacterial run time and speed. The 124 diffusion constant was set to D 0 = 10 3 µm 2 /s both for aspartate and serine channels. 125 We used γ asp = 0 for bacteria in the aspartate channel whilst for serine we chose a value 126 to account for the experimentally measured increase in the run time. Specifically, since 127 the run time doubles as the serine concentration changes from 10µM to 10 3 µM (cf.  Fig. 4 in the main text). The order ρ ser > ρ asp is consistent with the fact that 142 bacteria entering the lateral channels already sense the chemoattractant gradients and 143 with the higher affinity for serine than aspartate that was reported in the main text.  Fig. S3. Time and length scales of the saturation are consistent with the experimental 150 findings reported in the main text. We conclude that the leveling off of the progression 151 curves is well captured by the simple physical ingredients introduced into Eq. (1). 152 We also checked that our results hold when additional effects are included, e.g. 153 receptors saturate at high concentrations and Weber's law is then violated. To have a 154 sense of the importance of those effects, it is convenient to use the phenomenological 155 approach proposed in [8]. Specifically, the concentration c is replaced by a function φ(c), 156 which leads to replacing Eq. (1) by 157 ∂ t n = −∇ (ξn∇φ(c)) + D 0 ∇ 2 (1 + γφ(c)) n .
(2) 158 The general function φ is chosen to ensure that Weber's law obtains at 159 concentrations below the saturation of receptors and a faster decay (∝ ∇c/c 2 in 160 standard allosteric models, see e.g. [9]) is obtained beyond saturation. A convenient 161 functional form that respects those desiderata is : where the modulation factor Q(t) is controlled by the chemotactic response function and τ 0 is the mean run time in the absence of attractant (c = 0). The response function 180 that we specifically consider is 181 which has a unique positive lobe and is therefore manifestly non-adapted. While the 182 arguments in [6] were developed for the general case, we assume here for simplicity that 183 tumbles have zero duration. The chemoattractant concentration profile is assumed 184 linear: c(x) = c 0 + gx and bacteria all start at the origin at the initial time.

185
Setting D rot = 0 (again for simplicity only), one has α = σ (see eqs (1) and (2) where τ r ≈ τ 0 /(1 − K 0 c 0 /λ). 188 We simulated the stochastic dynamics described above and report in Figure S6 the 189 average position for 10 7 bacteria of the x component along the direction of the gradient. 190 Results agree with the prediction (7) above (i.e. eq (1) of the main text and Ref. [6]).

191
Conversely, for the current choice of parameters and more generally when λτ 0 = cos ϕ , 192 Ref. [11] predicts a zero chemotactic velocity -see eqs.