Mathematical description of bacterial traveling pulses

Abstract

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on Escherichia coli have shown the precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at the macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This can account for recent experimental observations with E. coli. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition, we can capture quantitatively the traveling speed of the pulse as well as its characteristic length. This work opens several experimental and theoretical perspectives since coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance, the particular response of a single cell to chemical cues turns out to have a strong effect on collective motion. Furthermore, the bottom-up scaling allows us to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion.

Figure 1. Experimental evidence for pulses of Escherichia coli traveling across a channel.

The propagation speed is constant and the shape of the pulse front is remarkably well conserved. Observe that the profile is clearly asymmetric, being stiffer at the back of the front (see also Fig. 2). Cell division may not play a crucial role regarding the short time scale.

Figure 2. Comparison between experimental data and numerical results obtained from the model.

Superposition of three time-snapshots of the experiments (dots, see also Fig. 1) and the numerical simulations of (2)–(4) (plain line). The time interval between snapshots is 2000s. The density profile is clearly asymmetric and preserved along the time course of the experiment. The number of bacteria in the pulse is approximately constant during the course (main contribution to growth takes place at the back of the pulse). The model reproduces faithfully the exponential tail at the back of the peak. The profile s do not coincide perfectly in the last snapshot due to uctuations in the experimental speed of propagation. Parameters chosen for the simulations are given in Table 1. The numerical speed is 1∶8µm.s⁻¹.

Figure 3. Propagation of a pulse wave.

(Top) Experimental results under abundant nutrient conditions: M9 minimal medium supplemented with 4% glucose and 1% casamino acids (both ten times more concentrated than in the case of Fig. 5). (Bottom) Numerical simulations of system (2)–(3) in the case of unlimited nutrient, and a stiff response function φ. We observe the propagation of a traveling pulse with constant speed and asymmetric profile. Specific parameters are: (δ = 10⁻¹ and N ₀ = 103 (arbitrary units).

Figure 4. Dispersion of the cell population (no pulse wave).

(Top) In this experiment, bacteria are cultivated at a concentration of 5.10⁸cells.ml⁻¹ in the same rich medium as in Fig. 3. After, they are resuspended in LB nutrient to an OD600 of 3.10⁸cells.ml⁻¹. We interpret the absence of pulse propagation as following. Bacteria are adapted to a rich environmnent before resuspension. Thus they are not able to sense small chemical uctuations necessary for clustering to occur when evolving in a relatively poor medium. (Bottom) Inuence of the internal processes stiffness. When the individual response function φ is not stiff, the effect of dispersion is too strong and no pulse wave propagates, as opposed to Fig. 3. Specific parameters are: δ = 10 and N ₀ = 10³. In mathematical models of bacterial chemotaxis, it is commonly accepted that adaptation of cells to large chemoattractant changes acts through the measurement of relative time variations: S ⁻¹ DS/Dt. In our context, this is to say that the stiffness parameter δ is proportional to the chemical level S. Hence after having dramatically changed the environment and before bacteria adapt themselves, we can consider that the response function φ is not stiff.

Figure 5. Coexistence of a stationary cluster and a traveling pulse.

At low level of nutrient the cell population splits into two subpopulations. A fraction remains trapped at the boundary (as a stationary profile) and a fraction travels accross the channel with constant speed. Specific parameters are: δ = 10⁻¹ and N ₀ = 10².

Figure 6. Relative tumbling frequencies (at the mesoscopic scale) obtained from the numerical experiment described in Fig. 3 : the tumbling probability is higher when moving to the left (upper red line) at the back of the pulse, whereas the tumbling probability when moving to the right is lower (lower green line), resulting in a net ux towards the right, as the pulse travels (see Fig. 3 ).

Notice that these two curves are not symmetric w.r.t. to the basal rate 1, but the symmetry defect is of lower order. The peak location is also shown for the sake of completeness (blue line).