Run-and-Tumble motion in active Brownian particles

Run-and-Tumble motion in active Brownian particles

modeling the motion of motile bacteria

Start

source: https://link.springer.com/content/pdf/10.1007/978-3-030-23370-9_7.pdf

Brownian motion

Passive Brownian motion

Langevin equation: dynamics of a particle in a fluid

The case where it is passive occure in a fluid : random collisions between molecules, described statistically and important for microscopic systems

Brownian motion

Overdamped Regime

At microscopic scales:

Inertia can be neglected because of the low mass and the relaxation time associated with inertia is then extremely short, leading to the overdamped Langevin equation

Brownian motion

Diffusion motion

In this regime, the position of the particle evolves according to a stochastic differential equation driven by noise.The diffusion coefficient is given by:

Position evolves as:

which connects thermal energy to viscous friction. This is known as the Einstein fluctuation-dissipation relation

Active particles

Active Brownian motion

Active particles (such like bacteria and artificial microswimmer) consume energy, move autonomously and are affected by noise. Therefore their motion is composed of two parts: self-propulsion and random fluctuations due to thermal noise.

Active particles

Active Brownian model

In the active Brownian particle model, the particle moves at a constant speed v along an orientation angle θ.The orientation itself changes randomly due to rotational diffusion.This produces trajectories that are persistent at short times but become random at longer times.

Run-and-Tumble motion

Biological motivation

Run-and-Tumble motion is observed in bacteria like Escherichia Coli (E. Coli). Many motile bacteria use that strategy of motion composed of two distinct movement.During a run, the bacterium swims approximately in a straight line.Then, a tumble event occurs, during which the bacterium stops and randomly changes its orientation before starting another run.

Mathematical model

Random tumble events

Tumbles occur randomly and are described by a Poisson process:

the probability that at least one tumble occurs in a time interval Δt is:

Mathematical model

Run-and-Tumble equations

the equations are discrete:

When the bacterium is running, it moves forward with velocity v. During tumbles, the orientation changes randomly. This process gives the same results as Brownian motion: this produces trajectories that are persistent at short times but become random at longer times.

Physical consequences

The trajectories are persistent at short times but become random at longer times. Leading to two regimes, characterizing the mean square displacement:

At short times:

It is a ballistic motion

At long times:

It is a diffusive motion

Conclusion

Brownian motion describes passive particlesActive particles self-propelRun-and-tumble explains bacterial motionMotion combines deterministic and stochastic dynamics