Three-dimensional chiral active Ornstein-Uhlenbeck model for helical motion of microorganisms

TL;DR

This work theoretically analyze the case of finite internal correlation time for microorganisms with helical trajectories as chiral active particles with an Ornstein-Uhlenbeck process for torque and shows that, for this type of internal noise, chirality and rotation increase the persistence of motion and results in helical trajectories that have a larger long-time mean squared displacement than straight trajectories at the same propulsion speed.

Abstract

Active movement is essential for the survival of microorganisms like bacteria, algae and unicellular parasites. In three dimensions, both swimming and gliding microorganisms often exhibit helical trajectories. One such case are malaria parasites gliding through 3D hydrogels, for which we find that the internal correlation time for the stochastic process generating propulsion is similar to the time taken for one helical turn. Motivated by this experimental finding, here we theoretically analyze the case of finite internal correlation time for microorganisms with helical trajectories as chiral active particles with an Ornstein-Uhlenbeck process for torque. We present an analytical solution which is in very good agreement with computer simulations. We then show that for this type of internal noise, chirality and rotation increase the persistence of motion and results in helical trajectories that have a larger long-time mean squared displacement than straight trajectories at the same propulsion speed. Finally we provide experimental evidence for this prediction for the case of the malaria parasites.

Figures (7)

• Figure 1: a: Reconstructed trajectories of malaria parasites gliding through synthetic hydrogels. Because this environment is nearly isotropic, the right-handed helical trajectories persist for long times. The typical (rescaled) turning time $T$ is 22 s as indicated. b: The direction of the angular velocity $\hat{\mathbf{\mathbf{\Omega}}}$ displays a decay of autocorrelation with a fast ($\tau=$ 20 s) and a slow ($\tau=$ 100 s) regime (see Fig. \ref{['fig:A2']} for a version with more details).
• Figure 2: a: Time course of different moments obtained from numerical simulation (Eq. \ref{['eq:dOm']}-\ref{['eq:dr']}) in comparison with numerical solution of the truncated system (Eq. \ref{['eq:hier1full']}-\ref{['eq:hier4trunc']}, gray dashed lines) and analytical approximation predicting exponential decay with eigenvalue $\lambda$ (Eq. \ref{['eq:eigval']}, yellow dotted lines). Parameter values: potential strength $k=$ 0.2, noise amplitude $h=$ 0.3, angular speed $\Omega_0=$ 1, angle $\alpha=\pi/6$. b: Same as a, but now for $k=$ 2, $h=$ 0.1 and $\Omega_0=$ 2, i.e. much reduced noise and faster turning. Here, the agreement between simulations and theory is even better. c: Simulated trajectories at parameters from a. d: Simulated trajectories at parameters from b. The reduced noise leads to more regular trajectories. See also Supp. Movies 1+2.
• Figure 3: a: Mean distance traveled in $z$-direction (the initial orientation of the helical axis) for different $\Omega_0$ at $k=$ 2, $h=$ 0.1. Full lines show particles moving straight while turning ($\alpha$=0), dashed lines particles on helical trajectories ($\alpha$=$\pi/4$), which can be seen overtaking slower turning straight particles. Colored and black lines are theoretical and numerical, respectively, and in very good agreement. b: Mean squared displacement for the same parameters as shown in a, theoretical results from Eq. \ref{['eq:MSD']} in color. c: Effective long-time diffusion constant $D_\infty$, cf. Eq. \ref{['eq:Diffconst']}, as a function of noise amplitude $h$ and angular speed $\Omega_0$. Black lines mark contours of constant $D_\infty$.
• Figure 4: a: Mean squared displacement for $\alpha$ close to $\pi/2$, such that the particles are close to describing circles, with $k=$ 1, $h=$ 0.5, $\Omega_0=$ 2. Black dotted lines are averages from numerical simulations. b: Theoretical expectation value of trajectories (Eq. \ref{['eq:expvalPos']}) for the two lower values of $\pi/2-\alpha$.
• Figure 5: a: Log-log plot of the mean squared displacement extracted from observed malaria parasite trajectories shown in Fig. \ref{['fig:Intro']} (purple, dotted), with five percent percentiles (purple, shaded) and the fitted model (orange). The gray dashed line is a fitted power law. b: Deviation from fitted power law. The vertical dashed line marks one period of rotation as extracted from the fitted model. c: Trajectories simulated with parameters obtained from MSD fit resembling Fig. \ref{['fig:Intro']} (cf. Supp. Movie 3).