Computational framework for the simulation of micro-swimming

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The aim of the study of swimming at the microscopic scale is to understand the propulsion strategies of biological micro-swimmers and the potential applications of these swimmers in bio-medicine, such as targeted drug delivery or cell manipulation. Natural micro-swimmers, like bacteria or sperm cells, move in groups in confined liquid environments. Thus, their trajectories are influenced by collisions and interactions with the solid walls of their environment. The development of our computational framework is based on mathematical models for fluid-structure interactions and a collision algorithm detecting contacts between arbitrarily-shaped strictly-convex bodies, and convex rigid boundaries. It allows the simulation of a wide variety of micro-swimmers, from rigid articulated bodies to deformable flagellated micro-swimmers.

Micro-swimming can be modeled as a system of differential equations. First, the dynamics of the Newtonian fluids in a moving domain are described by the Stokes equations. Then, the Newton and Euler ordinary differential equations model respectively the evolution of the linear and angular velocity of the swimmer. Finally, we add a model to express the swimmer's stroke. This model is specific to the micro-swimmer under examination, and can range from analytical expressions to solutions of reduced models, or even to elasticity equations. These three components of the system have to be connected according to some coupling conditions. Coupling conditions are always needed when addressing fluid-structure interaction problems. They impose the continuity between the fluid and the swimmer's velocities and stresses at the common interface, as well as a geometrical condition between the fluid-solid domains.

To give an example of a fluid-structure interaction, we show different snapshots of the simulation of a three-dimensional sperm cell swimming in bulk fluid. Its locomotion results from a sinusoidal planar wave of linearly increasing amplitude, propagating from the cell body to the tip of the flagellum (Fig 1). The stroke is modeled by prescribing the deformation velocity of the swimmer, which is given with respect to its reference configuration.

Fig 1. The figure shows, at different time instants, the position and shape of a sperm cell which propagates a sinusoidal wave along its flagellum. The effect of this swimming stroke is the straight propulsion in the opposite direction of the traveling wave.

Movie: Position and shape of a sperm cell which propagates a sinusoidal wave along its flagellum

To model the passive interactions between micro-swimmers and their environment, we add collision forces and torques to the fluid-structure system. Their magnitude is partly defined by the distance between the colliding interfaces. To evaluate explicitly these distances, we use a modified version of the Fast Marching Method algorithm: the distance functions are computed only in a close neighborhood of the interfaces. Using this narrow-band Fast Marching Method, the definition of the collision force is given by the following repulsive contact-avoidance model:

$$\begin{equation*} F^{int} = \left\{\begin{array}{rcl} 0 \;, \quad &\mbox{for }& d_{ij} > \rho , \\\ \frac{1}{\epsilon}(\arg\min_{x \in \partial \mathcal{S}_i} D_j(x) - \arg\min_{x \in \partial \mathcal{S}_j} D_i(x))(\rho - d_{ij})^2 \;, \quad &\mbox{for }& 0 \leq d_{ij} \leq \rho , \end{array}\right.\; \end{equation*}$$

where $Di\in \mathbb{R}^d$ and $Dj \in \mathbb{R}^d$ the distance functions from the interfaces of the micro-swimmers $\partial \mathcal{S}i$ and $\partial \mathcal{S}j$. This expression is a generalization of contact-avoidance models for circular and spherical bodies : the force is activated when the distance $d_{ij} \in \mathbb{R}$ between the two interfaces is smaller than the length of the collision zone $\rho > 0$; the direction of the repulsion force is computed by minimizing the distance functions, which provides the coordinates of the closest points to each interface, i.e. the contact points; and the intensity can be modulated by setting the parameter $\epsilon > 0$. If the contact produces a torque on the body, this latter is computed from $F^{int}$ and the lever arm vector connecting the center of mass of the body and the contact point.

We illustrate this collision model on the three-sphere swimmer, a micro-swimmer composed of three relatively-moving aligned spheres, heading towards a wall (Fig 2). The swimmer starts from a tilted position. Since its swimming stroke produces only a straight motion, it approaches the wall of the computational domain. Once it enters the collision zone, the combination of collision forces and torques with the constraint on the alignment of the spheres reorients the swimmer, which finally moves away from the wall.

Fig 2. This figure shows the trajectories of the three spheres as the swimmer approaches a horizontal wall. First, the right sphere enters the collision zone, where the collision torque causes a rotation of the swimmer. Similar torques are applied on the center and left spheres when getting close to the wall, resulting in the final orientation of the swimmer heading away from the wall. Only the centers of mass of the spheres have been represented to increase the readability of the picture.

Our computational framework allows simulating the swimming strategies of various micro-swimmers. We have the possibility to add different types of external forces, as collision or adhesion forces, enabling the simulation of realistic biological phenomena. We are currently working on further collision and swimmers' models for deformable elastic bodies, as well as on biological fluid models.

Movie 1: Objects flowing in the tail of a Zebra fish

Movie 2: Sedimentation in 2D of 100 particles in a cavity

Movie 3: 2 cells flowing through a stenosed artery