Mathematical Description of Bacterial Traveling Pulses

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.

Supporting Information "Text S1" Computation of the speed of the pulse We provide below analytical details yielding Formulae (8) and (9) in Section "Results".
Assuming that the sign of the chemical gradient switches at z = 0, the cell density of the traveling pulse is given by The chemotactic field is given by S(z) = (K * βρ)(z), where the fundamental solution of the diffusion equation for S(z) is To match the transition in monotonicity condition, the chemical signal should satisfy S (0) = 0, that is (K * βρ)(0) = 0, which leads to This leads to the following equation that we shall invert to obtain the front speed:

Dispersion relation for the cluster formation issue
We perform below the linear stability analysis refered to in Section "Results". Consider the following simple model for cluster formation, in the absence of an external nutrient N .
We rewrite α/D S = l −2 ,where l denotes the range of action of the chemical signal. We investigate the linear stability of the stationary state (ρ, S) = (ρ 0 , βα −1 ρ 0 ) where ρ 0 denotes the (constant) reference density over the domain [0, L].
We introduce the deviation to the stationary state:ρ = ρ − ρ,S = S − S. Then the linearized system writes close to (ρ, S): We have introduced our stiffness parameter δ = −φ (0) −1 . The associated eigenvalue problem reduces to the following dispersion relation for ξ = 2πk/L, Due to the conservation of mass, we shall only consider k ≥ 1. The eigenvalue remains negative for any frequency if the following inequality is fulfilled: (3)

Derivation of the macroscopic model from the kinetic equation
We perform below the drift-diffusion limit which yields to equations (15)-(16) in Materials and Methods. We start from the nondimensional kinetic equation which reads as follows, Therefore the dominant contribution in the tumbling operator is a relaxation towards a uniform distribution in velocity at each position: Notice that more involved velocity profiles can be handled [1,2], but this is irrelevant in our setting as the tumbling frequency does not depend on the posterior velocity v. The space density ρ(t, x) remains to be determined. For this purpose we first integrate with respect to velocity v and we obtain the equation of motion for the local density ρ(t, x) = v∈V f (t, x, v)dv: To determine the bacterial flow j we integrate (4) against v: We obtain formally, as → 0: Finally, the drift-diffusion limit equation reads in one dimension of space: To sum up, we have derived a macroscopic drift-diffusion equation, where the bacterial diffusion coefficient and the chemotactic flux are given by: In the limiting case where the internal response function φ is bivaluated: φ(Y ) = φ 0 1 {Y <0} − φ 0 1 {Y >0} , the flux rewrites simply as For simplicity we set = 0 in the main text. This does not change the qualitative and quantitative results for the degree of precision we are looking for.