Run-and-tumble motion

Synechocystis cells can also undergo biased motility under directional illumination. Under directional light flux, Synehcocystis cells perform phototactic motility and head toward the light source (in positive phototaxis). Vourc’h et al. (2020) showed that this biased motility stems from the averaged displacements during run periods, which is no longer random (as it was in the uniform illumination).^[15] They showed the bias is the result of the number of runs, which is greater toward the light source, and not of longer runs in this direction.^[15] Brought together, these results suggest distinct pathways for the recognition of light intensity and light direction in this prokaryotic microorganism. This effect can be used in the active control of bacterial flows.^[8]

It has also been observed that very strong local illumination inactivates the motility apparatus.

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:[26]^[27]

• Runs are straight and performed at constant velocity v₀ (initial speed-up is instantaneous)
• Tumbling events 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

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

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

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

where

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

with ŝ parametrized as ŝ = (cos θ, sin θ).^[28]

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.^[29][30][31]

The mathematical abstraction of run-and-tumble motion also appears outside of biology—for example, in idealized models of radiative transfer and neutron transport.^[32][33][34]

See also

Notes