Self-propulsion and interactions of catalytic particles in a chemically active medium

Enzymatic “machines,” such as catalytic rods or colloids, can self-propel and interact by generating gradients of their substrates. We theoretically investigate the behaviors of such machines in a chemically active environment where their catalytic substrates are continuously synthesized and destroyed, as occurs in living cells. We show how the kinetic properties of the medium modulate self-propulsion and pairwise interactions between machines, with the latter controlled by a tunable characteristic interaction range analogous to the Debye screening length in an electrolytic solution. Finally, we discuss the effective force arising between interacting machines and possible biological applications, such as partitioning of bacterial plasmids.

Figure 1

Scaling of $v$ for diffusion-limited cases. (a) $v$ (in units of $\mu c_0/r_0)$ grows linearly in $|g_1|$ until diverging. Solid and dashed lines show $v$ with $g_0=\pm \left|g_0\right|$, respectively. Inset: $v$ for $g_0>0$ with small $|g_1|$ (solid), $g_0<0$ with small $|g_1|$ (dashed), and $g_0<0$ with large $|g_1|$ (dotted). $v$ grows linearly with $|g_0|$ for small $|g_0|$ if $g_1$ is sufficiently small, but decays as $1/|g_0|$ for sufficiently negative $g_0$ (dashed) or diverges for large $|g_0|$ (solid). For large $|g_1|$, steady-state $v$ exists only for sufficiently negative $g_0$ (dotted). (b) $v$ is insensitive to $R$ at small $R$, but decays at large $R$ as $1/R$ if $g_0$ and $|g_1|$ are sufficiently small (solid) and diverges otherwise (dashed).

Figure 2

Force-distance relation and distance fluctuations from theory and simulation. (a) Without chemical activity, particles pushed together with force $F$ (units of $2k_{B}T/r_0)$ entropically repel (solid line). $\langle r\rangle$ as a function of $F$ is depressed for attractions in both the theory (dashed line) and simulation (solid circles). $\langle r\rangle$ is larger for repulsions in the theory (dash-dotted line) and simulation (open circles). Dotted line shows $\langle r\rangle =2r_0$ and $L=30$. (b) The corresponding distance fluctuations.

Figure 3

Medium activity controls mean distance and fluctuations. (a) $\langle r\rangle$ varies nonmonotonically with $k^{-}$ (solid line). $\langle r\rangle \approx L/2$ for strong repulsions and $\langle r\rangle \approx 2r_0$ for strong attractions. For weak interactions, i.e., $k^{-}\approx k_c^{-}{e}^{\beta \epsilon }$ (vertical line) or large $k^{-},\langle r\rangle \to L/4+2r_0$ (dotted). (b) $\sqrt{\langle \delta r^2\rangle }$ is small for $k^{-}<k_c^{-}{e}^{\beta \epsilon }$ and $k^{-}\gtrsim k_c^{-}{e}^{\beta \epsilon }$, but large for $k^{-}\approx k_c^{-}{e}^{\beta \epsilon }$ and large $k^{-}$. (c) As $k^{+}$ increases, $\langle r\rangle$ decreases/increases toward maximum compression/extension for attractions/repulsions (solid/dashed). (d) For large $k^{+},\sqrt{\langle \delta r^2\rangle }$ decreases as $1/k^{+}$ for attractions (solid) and $1/\sqrt{k^{+}}$ for repulsions (dashed). $L=50$ in all panels.