Randomly curved runs interrupted by tumbling: A model for bacterial motion

Small bacteria are strongly buffeted by Brownian forces that make completely straight runs impossible. A model for bacterial motion is formulated in which the effects of fluctuational forces and torques on the run phase are taken into account by using coupled Langevin equations. An integrated description of the motion, including runs and tumbles, is then obtained by the use of convolution and Laplace transforms. The properties of the velocity-velocity correlation function, of the mean displacement, and of the two relevant diffusion coefficients are examined in terms of the bacterial sizes and of the magnitude of the propelling forces. For bacteria smaller than E. coli, the integrated diffusion coefficient crosses over from a jump-dominated to a rotational-diffusion-dominated form.

Figure 1

Mean square speed as a function of time for three representative combinations of bacterial sizes and speeds. Starting from the top curve, these correspond to forces per unit mass equal to $6.1\times {10}^6\mathrm{cm}∕{\mathrm{s}}^2$, $4.9\times {10}^5\mathrm{cm}∕{\mathrm{s}}^2$, and $2.4\times {10}^4\mathrm{cm}∕{\mathrm{s}}^2$. Note the rapid evolution towards the steady-state speed given by Eq. (12).

Figure 2

Single run mean square displacements (divided by $4t$) as functions of time for the indicated values of the propelling force $G$ for a bacterium of radius (a) $R=1\mu \mathrm{m}$ and (b) $R=0.2\mu \mathrm{m}$. Steady state speeds are, starting from the bottom, $\overline{v}=8.2$, 16.4, and $24.6\mu \mathrm{m}∕\mathrm{s}$ in (a), and 130, 165, and $200\mu \mathrm{m}∕\mathrm{s}$ in (b). The diffusive regime becomes evident at long times.

Figure 3

Intrarun diffusion coefficient as a function of bacterial radius, for the indicated values of the propelling force. The dot on the extreme right curve corresponds to E. coli swimming at $16.4\mu \mathrm{m}∕\mathrm{s}$, while those on the left correspond to a $R=0.2\mu \mathrm{m}$ bacterium moving at 16.4 and $200\mu \mathrm{m}∕\mathrm{s}$, respectively.

Figure 4

Integrated mean square displacements (divided by $4t$) as functions of time for a bacterium of radius (a) $R=1\mu \mathrm{m}$ and (b) $R=0.2\mu \mathrm{m}$ as functions of time. Here ${\tau }_0=1\mathrm{s}$. Steady state speeds are, starting from the bottom, $\overline{v}=8.2$, 16.4, and $24.6\mu \mathrm{m}∕\mathrm{s}$ in (a), and 130, 165, and $200\mu \mathrm{m}∕\mathrm{s}$ in (b).

Figure 5

Integrated diffusion coefficient as a function of bacterial radius, for the indicated values of the propelling force. The approximate dependence of $D$ with $R$ is also indicated. The dot on the extreme right curve corresponds to E. coli swimming at $16.4\mu \mathrm{m}∕\mathrm{s}$, while those on the left correspond to the $R=0.2\mu \mathrm{m}$ bacterium moving at 16.4 and $200\mu \mathrm{m}∕\mathrm{s}$, respectively.

Figure 6

Integrated diffusion coefficient as a function of mean run length for typical bacterial values, as indicated. $R_m=0.2\mu \mathrm{m}$ and $R_{ec}=1\mu \mathrm{m}$ are the radii corresponding to a small marine bacterium and to E. coli, respectively, and the units of $G$ are $\mathrm{cm}∕{\mathrm{s}}^2$.