The dispersion of bacteria moving through porous environments follows an astonishingly simple and universal law.
The Earth is populated by swimming microorganisms, such as bacteria, which are constantly on the move. Their swimming strategies vary from one species to another, but because they involve random reorientations in chosen directions, they all constitute "random walks."
Over long periods, this mode of movement is diffusive: bacteria initially released at the same location gradually disperse, much like tea steeping in still water. This spreading of the population, also called dispersion, is quantitatively characterized by a parameter known as the "diffusion coefficient."
About one-third of the 10
30 bacteria living on Earth today thrive in porous environments—structures made up of interconnected cavities found in sediments, soils, rocks, food, and even inside the human body.
Predicting bacterial dispersion in these environments is therefore a relevant problem in many contexts. Whether it's an infection in the human body or the contamination of food or aquifers, knowing the diffusion coefficients associated with dissemination in these porous media—as well as their dependence on the characteristics of each system—is crucial for developing effective control strategies.
The difficulty, however, lies in the enormous diversity of encountered situations. Not only do bacteria exhibit many variations in their swimming strategies, but there is also a myriad of porous environments, differing in structure, morphology, and characteristic sizes. Given the vast number of parameters that can vary from one system to another, how can dispersion be predicted? Is there a unifying principle that can simplify this problem?
The answer, surprisingly simple, has just been provided in a study involving the iLM laboratory in Lyon and ETH Zürich. Researchers have shown that bacterial dispersion in porous media has a universal character: regardless of pore structure or swimming strategy, dispersion follows a general law that condenses the diversity of situations into a single mathematical relationship.
To reach this conclusion, the first step was to numerically simulate bacteria moving through porous media. By rotating their flagella, a bacterium moves in a straight line for about a second before abruptly reorienting in another direction.
This swimming strategy is called "run-and-tumble," and there are multiple variations among species, particularly in how reorientation occurs. Additionally, when a bacterium encounters the wall of the porous medium, it remains stuck there until a reorientation allows it to move back into the liquid.
(a) Bacterium moving through porous media. Its movement strategy is "run-and-tumble," with sudden and random reorientations.
(b) Simulations of bacterial trajectories in three porous structures.
(c) Universal curve of dispersion as a function of mean "run" time: the dots represent 37 different situations, and the black curve is the theoretical prediction. The scientists considered very different porous structures in their simulations: ordered or disordered, with circular or rectangular obstacles, low or high porosity, etc. Each time, the diffusion coefficient D was measured as a function of the mean "run" time τ—the average time between two (random) reorientation events of the bacterium. In all cases, the D(τ) curve shows a peak.
The surprising observation is that, with a simple scaling around the peak, all the curves—covering dozens of different situations—can be superimposed onto a single master curve, defining a universal law of dispersion.
What is the origin of this generic behavior? The answer was found through an elementary model, in which the crucial quantity is the average time between two successive contacts with the wall. Counterintuitively, this average time does not depend on the movement strategy but only on the average chord length of the structure. This geometric property, called Cauchy invariance, is what makes the dispersion law universal.
Although inspired by bacterial swimming, the dispersion law is actually relevant for a broad class of microorganisms, with potential applications in ecology. The next challenge is to understand dispersion when microorganism movement is biased by an external factor, such as fluid flow or a chemical gradient. These results have just been published in
Physical Review Letters.