Fig 1.
Cilia orientation map in human bronchial cultures obtained using over 500 images, from Khelloufi et al. [38]. Scale bar represents m.
Fig 2.
Illustration of a 3D beating cilium simplified as a rotating rod with varying length.
(a) The rod’s motion over one full revolution, with red circles marking the rod tips. (b) Projection of the rod’s profile onto the (x,z)-plane, (c) Projection of the rod’s profile onto the (x,y)-plane.
Fig 3.
Illustration of the tissue-scale cilia cluster model.
(a) The top view of an array of three cilia clusters, each containing nine equally spaced cilia. The spacing between individual cilia within a cluster is m, while the inter-cluster spacing is Dc (variable). (b) A large patch containing multiple clusters, covering an area of
.
is the variable ciliary density. Blue dots indicate the position of individual cilia.
Table 1.
Parameters used in this study are based on experimental measurements.
Fig 4.
Typical snapshot of the swirl pattern in viscous flow driven by cilia at a density of .
Blue arrows indicate the velocity field. Black lines represent cilia, beating synchronously with an initial phage .
Fig 5.
Swirl motion of passive particles as a function of ciliary density: (a-d) (e-h)
, (i-l)
.
The top row (a, e, i) shows the locations of cilia clusters. The remaining rows show the top view of the distribution of passive particles at t = 1.38s (b, f, j), t = 2.78s (c,g,k), (d, h, l). These time points correspond to 50, 100, and 400 cilia beating revolutions, respectively, with all cilia beating synchronously. Particles are initially distributed in four color-coded groups (black, blue, green, and red) within the ciliated region
m and
m. The simulation domain is
m. Each time sequence shows a representative simulation out of three independent, random realizations.
Fig 6.
Net particle transport driven by mucociliary flow at ciliary densities (red),
(blue), and
(green).
(a) Center of mass (CoM) displacement in the x-direction, (b) CoM displacement in the y-direction, (c) CoM displacement in the z-direction. For each density , three independent simulation were performed. The solid lines represent the mean displacement and the shaded regions are the standard deviation.
Fig 7.
Ripley’s K function for particles at three ciliary densities (red),
(blue), and
(green) at
.
(a) Particles initially located at a height of m. (b) All particles within the vertical range of
m. The gray dashed reference line indicates complete spatial randomness. For each density
, three independent simulation were performed. The solid lines represent the mean displacement and the shaded regions are the standard deviation.
Fig 8.
Vertical mixing at tissue level with ciliary density .
Particles are initially partitioned into two groups: the bottom half (black) and the top half (cyan), to visualize vertical stratification and subsequent mixing. (a–e) Snapshots of particle distributions at t = 0, 50, 100, 200, and 400 cilia beating cycles, shown for a phase lag of . All cilia beat synchronously.
Fig 9.
Vertical mixing at tissue level with ciliary densities .
Particles are initially partitioned into two groups: the bottom half (black) and the top half (cyan). (a) Snapshots of particle distributions after 400 cilia beating cycles (, for
, 0.4. (b) Normalized mixing number m/m0 vs. time on a log-log scale. For each density
, three independent simulation were performed. Solid lines represent the mean values and shaded regions represent the standard deviation. All cilia beat synchronously (
).
Fig 10.
Particle trajectories, velocity field, and vorticity distributions at tissue level: (a-c) Trajectories for selected particle over 400 ciliary beating cycles for three densities .
Initial locations indicated by empty circles. (d-f) Time-averaged (over one cilia beating cycle) velocity vector fields in the xy-plane near the cilia tips (). (g-i) Corresponding vorticity maps, on the x-y plane, showing the spatial distribution of rotational flow. All cilia beat synchronously with phase lag
.
Fig 11.
Velocity field generated by a single cilia cluster over one full beating cycle.
(a-d) Top-down views at m at four phases of the cilia beating cycle (
). (e-h) Corresponding side views at
m. The black box marks the boundary of the cilia cluster, with the cluster center located at (0,0) (red circle). All cilia beat synchronously with an initial phase angle of
.
Fig 12.
Particle swirl motion generated by a single cilia cluster: Distribution of passive particles visualized in top view (left column: (a-d)) and side view (right column: (e-h)).
Time progresses from top to bottom, with snapshots at t = 0s in (a) and (e); t = 1.38s in (b) and (f); t = 2.78s in (c) and (g); and in (d) and (h). These time points correspond to 0, 50, 100, and 400 cilia revolutions, respectively. The black box in (a) outlines the boundary of the cilia cluster boundary, and the blue dots indicate the positions of the 9 individual cilia. In (e), particles are color-coded by their height (z): blue, red, and black represent different elevation range. All cilia beat synchronously with initial
.
Fig 13.
Ripley’s K function for particles at various heights (, 3, 5, 7, 10)
m at
s for a single cilia cluster.
The gray dashed reference line indicates complete spatial randomness.
Fig 14.
Uniform horizontal motion in one large cluster form by placing three cilia clusters next to each other, .
The 3-dimensional velocity field and the orientation of all cilia at (a), and
(b). The red dots indicate the cluster centers. All cilia beat synchronously.
Fig 15.
Transport and mixing of particles in a cluster of three adjacent cilia clusters, separated by .
The left column (a-d) shows a top view, and the right column (e-h) shows a side view. From top to bottom, sequential snapshots represent time points: t = 0s (a,e), t = 1.38s (b,f), t = 2.78s (c,g), and (d,h), corresponding to 0, 50, 100, and 400 cilia revolutions, respectively. All cilia movesynchronously. Each cluster is marked by black boxes. Particle colors indicate their initial positions along the x-axis, illustrating the extend of horizontal transport and mixing.
Fig 16.
Particle swirls from three widely separated cilia clusters, spaced at .
The left column shows the top view (a-d) and the right column shows the side view (e)-(h). From top to bottom, sequential snapshots represent time points: t = 0s (a,e), t = 1.38s (b,f), t = 2.78s (c,g), and (d,h), corresponding to 0, 50, 100, and 400 cilia revolutions, respectively. All cilia beat synchronously. Cluster centers are marked with red dots. Particle colors represent their initial positions along the x-axis, illustrating the dynamics of mixing and transport.
Fig 17.
Maximal transport at intermediate cluster spacing.
The trajectories of the center of mass (CoM) for all particles for three cilia clusters, in the (a) x-direction, (b) y-direction, and (c)z-direction. All cilia beat synchronously.
Fig 18.
Top-view vorticity fields at varying cilia cluster spacings .
Each cilia cluster generates a localized zone of high voticity, corresponding to rotational flow. At the small spacing (Dc = 3D), the clusters connect to a single elongated cluster and their vortex zones merge into a single extended zone confined primarily along the x-axis. As spacing increases to 5D and 7D, the vorticity zones begin to separate but remain partially connected, forming hydrodynamically coupled flow zones. At the largest spacings (9D and 12D), the high-vorticity regions are fully isolated, reducing inter-cluster interactions and coordinated transport.
Fig 19.
Ripley’s K function analysis of particle distributions in three cilia clusters with varying spacing from to 12D.
This analysis focused on particle positions at a height of m. The gray dashed reference line indicates complete spatial randomness.
Fig 20.
Metachronal wave reduces directional transport in three connected cilia clusters with spacing .
Snapshot of massless particles at are shown for varying phase shifts applied along the x direction. (a)
, (b)
, (c)
, (d)
. At
, the three columns exhibit a metachronal wave with a wavelength of
. Particles are color-coded based on their initial positions in the x direction at t = 0, same as in Fig 15a.
Fig 21.
Center-of-mass displacement for massless particles in a system of three cilia cluster with metachronal coordination, shown for phase lags , under three inter-cluster spacings: (a)
(
), (b)
(
only), and (c)
(
only).
The black curve corresponds to synchronous beating ().
Fig 22.
Vertical mixing of passive particles over time in a three-cluster cilia array.
Particles are initially divided into two vertical groups: the bottom half (black) and the top half (cyan) to visualize vertical stratification and subsequent mixing. (a-d) Snapshots of particles after 0,50,100,200, and 400 cycles of cilia beating, shown for a phase lag of . (e) Temporal evolution of the normalized mixing number m/m0 on a log-log scale for varying phase lag
. Lower values of m/m0 indicate greater vertical mixing. The black curve represents synchronous beating. Colored curves correspond to metachronal waves with varying phase lags.
Fig 23.
Time-averaged velocity field near the cilia tip () at the tissue scale under metachronal coordination.
Each panel corresponds to a different ciliary density , and 0.4, with an associated phase lag (
, and
). The velocity fields are averaged over one full cilia beating cycle. Red dots mark the centers of cilia clusters.