Run-and-tumble motion

Run-and-tumble motion is a movement pattern exhibited by certain bacteria and other microscopic agents. It consists of an alternating sequence of "runs" and "tumbles": during a run, the agent propels itself in a fixed (or slowly varying) direction, and during a tumble, it remains stationary while it reorients itself in preparation for the next run.

The tumbling is erratic or "random" in the sense of a stochastic process—that is, the new direction is sampled from a probability density function, which may depend on the organism's local environment (e.g., chemical gradients). The duration of a run is usually random in the same sense. An example is wild-type E. Coli in a dilute aqueous medium, for which the run duration is exponentially distributed with a mean of about 1 second.

Run-and-tumble motion forms the basis of certain mathematical models of self-propelled particles, in which case the particles themselves may be called run-and-tumble particles.

Description

In bacteria such as E. Coli, trajectories appear as a sequence of nearly straight segments interspersed by erratic reorientation events, during which the bacterium remains stationary. The straight segments are called "runs", and the reorientation events are called "tumbles". Because they exist at low Reynolds number, bacteria starting at rest quickly reach a fixed terminal velocity, so the runs can be approximated as constant velocity motion. The deviation of real-world runs from straight lines is usually attributed to rotational diffusion, which causes small fluctuations in the orientation over the course of a run.

In contrast to the more gradual effect of rotational diffusion, the change in orientation (turn angle) during a tumble is large; for an isolated E. Coli in a uniform aqueous medium, the mean turn angle is about 70 degrees, with a relatively broad distribution. In more complex environments, the tumbling distribution depends on the agent's local environment and allows for goal-oriented navigation (taxis). For example, a tumbling distribution that depends on a chemical gradient can guide bacteria toward a food source or away from a repellant, a behavior referred to as chemotaxis. Tumbles are typically faster than runs: tumbling events of E. Coli last about 0.1 seconds, compared to ~ 1 second for a run.

Mathematical modeling

Theoretically and computationally, run-and-tumble motion can be modeled as a stochastic process. One of the simplest models is based on the following assumptions:

• Runs are straight and performed at constant velocity v₀ (initial speed-up is instantaneous)
• Tumbling events are uncorrelated and occur at average rate α, i.e., the number of tumbling events in a given time interval has a Poisson distribution. This implies that the run durations are exponentially distributed with mean α^-1.
• Tumble duration is negligible
• Interactions with other agents are negligible (dilute limit)

With a few other simplifying assumptions, an integro-differential equation can be derived for the probability density function f (r, ŝ, t), where r is the particle position and ŝ is the unit vector in the direction of its orientation. In d-dimensions, this equation is

${\frac {\partial f(\mathbf {r} ,{\hat {\mathbf {s} }},t)}{\partial t}}+v_{0}\,{\hat {\mathbf {s} }}\cdot \nabla f(\mathbf {r} ,{\hat {\mathbf {s} }},t)=\xi ^{-1}\nabla \cdot (\nabla V(\mathbf {r} )f(\mathbf {r} ,{\hat {\mathbf {s} }},t))-\alpha f(\mathbf {r} ,{\hat {\mathbf {s} }},t)+{\frac {\alpha }{\Omega _{d}}}\int g({\hat {\mathbf {s} }}-{\hat {\mathbf {s} }}')f(\mathbf {r} ,{\hat {\mathbf {s} }}',t)d{\hat {\mathbf {s} }}'$

where Ω_d = 2π^d/2/Γ(d/2) is the d-dimensional solid angle, V(r) is an external potential, ξ is the friction, and the function g (ŝ - ŝ') is a scattering cross section describing transitions from orientation ŝ' to ŝ. For complete reorientation, g = 1. The integral is taken over all possible unit vectors, i.e., the d-dimensional unit sphere.

In free space (far from boundaries), the mean squared displacement ⟨r(t)²⟩ generically scales as ⟨r(t)²⟩ ~ t² for small t and ⟨r(t)²⟩ ~ t for large t. In two dimensions, the mean squared displacement corresponding to initial condition f (r, ŝ, 0) = δ(r)/(2π) is

$\langle \mathbf {r} ^{2}\rangle ={\frac {2v_{0}^{2}}{\alpha ^{2}(1-\sigma _{1})}}\left[\alpha \,t+{\frac {e^{-\alpha (1-\sigma _{1})t}-1}{1-\sigma _{1}}}\right]$

where

$\sigma _{1}=\int _{0}^{2\pi }g({\hat {\mathbf {s} }})(\cos \theta )d\theta$

with ŝ parametrized as ŝ = (cos θ, sin θ).

In real-world systems, more complex models may be required. In such cases, specialized analysis methods have been developed to infer model parameters from experimental trajectory data.

Examples

Run-and-tumble motion is found in many peritrichous bacteria, including E. coli, Salmonella typhimurium, and Bacillus subtilis. It has also been observed in the alga Chlamydomonas reinhardtii.

See also

• Animal navigation
• Bacterial motility
• Aquatic locomotion
• Quorum sensing
• Microswimmer
• Active matter
• Squirmer

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