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Topological Excitations govern Ordering Kinetics in Endothelial Cell Layers
Iris Ruider,1, 2, 3, 4, ∗Kristian Thijssen,5, ∗Daphné Raphaëlle Vannier,1, 2, 4 Valentina
Paloschi,6, 7 Alfredo Sciortino,1, 2 Amin Doostmohammadi,5, †and Andreas R. Bausch1, 2, 3, 4, ‡
1Lehrstuhl für Zellbiophysik E27, Technical University of Munich, 85748 Garching, Germany
2Center for Functional Protein Assemblies (CPA),
Technical University of Munich, 85748 Garching, Germany
3Matter to Life Program, Max Planck School, München, Germany
4Center for Organoid Systems and Tissue Engineering (COS),
Technical University of Munich, 85748 Garching, Germany
5The Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100 Copenhagen O, Denmark
6Department of Vascular and Endovascular Surgery, Klinikum rechts der Isar,
Technical University of Munich, 80333 Munich, Germany
7German Center for Cardiovascular Research DZHK,
Partner Site Munich Heart Alliance, 80336 Berlin, Germany
Many physiological processes, such as the shear flow alignment of endothelial cells in
the vasculature, depend on the transition of cell layers between disordered and ordered
phases. Here, we demonstrate that such a transition is driven by the non-monotonic
evolution of nematic topological defects and the emergence of topological strings that
bind the defects together, unveiling an intermediate phase of ordering kinetics in bi-
ological matter. We used time-resolved large-scale imaging and physical modeling to
resolve the nature of the non-monotonic decrease in the number of defect pairs. The
interaction of the intrinsic cell layer activity and the alignment field determines the
occurrence of defect domains, which defines the nature of the transition. Defect pair
annihilation is mediated by topological strings spanning multicellular scales within the
cell layer. We propose that these long-range interactions in the intermediate order-
ing phase have significant implications for a wide range of biological phenomena in
morphogenesis, tissue remodeling, and disease progression.
Introduction: Since 1858, the relationship between a homogeneous endothelium and blood flow patterns has been
recognized, with endothelial cells displaying high levels of alignment in the aorta, in contrast to their misalignment at
bifurcation points [1, 2]. Subsequent in vitro studies revealed that endothelial cell layers have an intrinsic ability to
realign and elongate in response to shear stress [3–5]. Since then, the extent of this tissue-wide global order has been
considered a defining characteristic of various cell layers, influencing processes such as vascularization [6], tissue regen-
eration [7], development [8], and morphogenesis [9–11]. However, it is increasingly evident that disrupting the order
in localized areas, known as topological defects, serves as a hotspot for biological activity such as the regulation of cell
apoptosis [12], cell layer homeostasis, and stem cell accumulation [13–15]. In non-living systems, it is well-established
that local defects are crucial for transitions between disordered and globally ordered phases [16]. Understanding those
transitions has provided the basis for applications in material science, ranging from superfluidity and melting [17–19]
to superconductors [20, 21]. Yet, transitions between local and global order in living, cellular systems remain largely
unexplored, partly due to the large length and time scales involved [22]. Instead, our understanding remains limited to
characterizing the steady-state properties of disordered or ordered cellular assemblies. This lack of knowledge about
intermediary states between disordered and ordered states of cell organization leads to heterogeneous and sometimes
contradictory observations, as seen in endothelial cell alignment [23–27]. To unravel the kinetics of shear flow-driven
ordering in primary endothelial cells, we employed time-resolved large-scale imaging paired with physical modeling.
We show that the flow-induced order transition in endothelial cell layers is governed by a three-stage process in which
the ordering kinetic is disrupted by an intermediate stage where cells temporarily misalign, and multicellular defect
strings emerge. We further demonstrate that an intricate interplay between the external shear flow and the activity
of the living endothelial cell layer governs the emergence of defect strings and the concomitant intermediate ordering
phase.
Ordering kinetics characterized by three distinct phases: To gather sufficient statistical data to fully describe
the phase ordering kinetics between disordered and ordered cellular organization, we examined the effect of a constant
∗These authors contributed equally
†doostmohammadi@nbi.ku.dk
‡abausch@mytum.de
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FIG. 1. Shear flow induced alignment and defect dynamics in endothelial cell monolayers. (a) The cells experience
a shear flow along the x-z plane. (b) Cell alignment is captured by the nematic director field, where the orientation α= 0
is defined with respect to the direction of the shear flow. (c) The nematic director is color-coded to visually represent cell
alignment. (d) Without a flow-aligning field, inter-cellular forces are randomly oriented and cancel each other out. (e) The
shear in the x-z plane acts as an affine alignment field, inducing elongation and ordering of the cells along that direction.
As local ordering is established, inter-cellular forces stop canceling out, resulting in mesoscopic active forces Fact (collective
active stresses), approximated as extensile forces (ζ > 0) [28]. (f) Field of view of the endothelial cell monolayer during a
flow experiment. The zoom-ins show a phase contrast image of a cellular monolayer and simultaneously illustrate analysis
steps with increasing details containing the nematic director field, the orientation field, and nematic topological defects. (g) A
representative +1/2defect indicated by a red dot. (h) A representative −1/2defect indicated by a blue dot. (i) The number
of defects and the global nematic ordering of the endothelial cell monolayer during the alignment process in the experiments.
The light green area indicates the standard deviation from the mean. (j) The number of defects (black dots) and the global
nematic ordering of the endothelial cell monolayer during the alignment process in the simulations.
superjacent shear flow (≈20 dyn/cm2) on a confluent layer of human aortic endothelial cells (HAOEC) [5, 29, 30] with
an automated microscope setup, which allowed stitching of high-resolution images on a scale of several millimeters
during periods of up to 64 hours (Fig. 1a). During HAOEC alignment and elongation (Ext. Data Fig. 1), the
emergence of nematic order was observed in phase contrast images (Fig. 1b,c and Supplementary Movie 1). To
quantify the order, we extracted the local nematic director field of the endothelial cell monolayer in the full field
of view (Fig. 1b,f). From this, we obtained the orientation angle α, color-mapped throughout the field of view to
spatially resolve and visualize the alignment. We identified the presence of nematic +1/2and −1/2defects within the
cellular monolayer (Fig. 1c,f,g,h). Upon application of shear flow, an initially disordered layer of cells evolved into
a well-aligned ordered layer, and the density of topological defects dropped significantly from over 400 defect pairs
in the initial state to around 0 at the final state. (Fig. 1i). While in passive materials, the ordering kinetics would
be characterized by a monotonous decrease of the topological defect density [31], here we observed three distinct
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phases driven by the non-equilibrium nature of the cell layer: In the initial ordering phase (I) exposure to shear stress
induced the cells’ elongation and alignment along the flow direction. This was accompanied by a decrease in defect
density in the first nine hours after flow was applied. In the subsequent intermediate disordering phase (II), from
t= 10 and t= 25 hours, a transient reduction in cellular alignment and an increase in defect density was observed.
In the final ordering phase, which we call the steady-state ordering phase (III), the system realigned again, almost all
nematic defects disappeared, and a long-range orientational order was established (Fig. 1i and Supplementary Movie
1).
Interplay between local activity and global shear: The appearance of the intermediate disordering phase
suggests that while shear flow drives the ordering of the cells, the active motion and cell-cell interactions within
the endothelial layer oppose the induced ordering. To test this, we simulated the transient behavior using active
nematic theory (’Simulations’ in Materials and Methods). Here, we used a coarse-grained model to solve the local
velocity v and the nematic orientation tensor Q. The superjacent shear flow was modeled as an external aligning field
that energetically penalizes orientations perpendicular to the flow direction, similar to an anisotropic quenching field
[32]. This induced a global nematic orientation of the cell layer. We modeled the activity of the endothelial cells by
out-of-equilibrium force contributions, often called active force, dominated by dipolar forces along the resulting local
elongation axis (Fig. 1d,e). The activity and anisotropic field competed at different timescales, which resulted in a
variety of behaviors (Fig. 2), including the non-monotonic coarsening behavior (Fig. 1j). As the cells elongated under
the shear flow and local orientational order was established, the active stress from fluctuations in the orientation field
was sufficient to nucleate topological defects and transiently destroyed the local orientational order in the system.
Eventually, the orientation of the cells became almost perfectly aligned along the preferred axis set by the external field.
In our experimental setup, rising cell density entailed an increase in the activity of the monolayer due to enhanced
cell migration and cell-cell interactions, which collectively augmented the intrinsic active stress [33], if cell density
was low enough that no jamming occured. Experiments carried out at different cell densities revealed that at low
cell densities and thus lower active stress, endothelial cells aligned monotonously in the flow direction as the external
field dominated the system (Fig. 2 a,b). However, with increasing cell density and thus increasing activity within the
cellular monolayer, we observed the emergence of an intermediate phase during cell alignment (Fig. 2 a,c). While the
experimental setup offered only a limited range to explore this transition, simulations provided a more comprehensive
understanding of how the activity of the monolayer and the strength of the external flow field influence the nature of
the cell ordering process. The phase diagram illustrates how the interplay between activity and the external field’s
strength affects the kinetics of transition between disordered and ordered states. In congruence with experiments,
and as expected for passive materials, in the absence of or at low values of active stresses, when the aligning field
dominates on all time scales, the ordering kinetic follows a monotonic rarefaction of topological defects (Fig. 2d,e).
In contrast, larger active stresses result in a defect-loaded steady-state, called active turbulence (Fig. 2d,h). Only
when the active stress and the external field compete on similar timescales can we recover the experimental behavior,
where the global nematic order parameter exhibits three distinct transient phases (Fig. 2d,f). We point out that for a
weak anisotropic aligning field, the system remains in an isotropic state with no emergence of global order (Fig. 2g).
The existence of a threshold shear stress, below which endothelial cells no longer respond to flow, has already been
reported in the literature [26, 34]. Therefore, only in an intermediate regime does the intricate interplay between
activity-driven defect nucleation and flow alignment forces result in an intermediate increase in defect numbers.
Anisotropic correlation lenghts: Because of the preferred orientation axis set by the direction of the superjacent
flow, the coarsening of the local nematic order was not isotropic. Rather, the system can be split into domains of
negative and positive rejection to the preferred axis δn, which divides these domains (Fig. 3 a,e). The multicellular
orientationally ordered domains slowly coarsened in an anisotropic manner as global order was established (Supple-
mentary Movie 2 and 3). This is best quantified by measuring the correlation functions Cδn∥(∆r∥)and Cδn⊥(∆r⊥)
that resolve the orientation correlation along the perpendicular ∆r⊥and parallel ∆r∥directions to the preferred
axis (‘Correlation Functions’ in Materials and Methods). The correlation lengths found from the correlation function
along the parallel ξ∥and perpendicular axes ξ⊥to the preferred orientation showed asymmetric growth of similarly
aligned regions in both the model and experiments, where coarsening of the correlation length occurred primarily
along the preferred axis (Ext. Data Fig. 2a,e). Interestingly, the domain growth along the axis parallel the flow direc-
tion was not uniform in time and was closely linked to the three previously observed ordering phases, as we discuss next.
Three-stage kinetic ordering: In the initial ordering phase (I), orientation correlation functions along the pre-
ferred axis r∥decayed to zero, and the characteristic correlation lengths ξ∥increased exponentially. From the simple
dimensional analysis of diffusive isotropic-nematic transitions [35], the growth of ξ∥is expected to follow t1/2which
is consistent with our simulation results and close to the experimental data (Fig. 3b,f ). This suggests a negligible
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FIG. 2. Interplay between cellular activity and external aligning field governs alignment dynamics. (a) The phase
diagram for the experimental data obtained from several experiments at different cell densities shows a transition from the
regime with monotonous alignment dynamics in red towards the regime with a non-monotonous alignment behavior in blue. (b)
At lower cell densities, the endothelial cells align monotonously along the flow direction, lacking the three-stage dynamics. The
light green area indicates the standard deviation from the mean. (c) At higher cell densities, the endothelial monolayer displays
non-trivial alignment dynamics characterized by a non-monotonous alignment. The light green area indicates the standard
deviation from the mean. (d) The phase diagram obtained from simulations shows that the non-trivial regime, highlighted in
blue, only exists within a very defined region. The trivial region is highlighted in red. (e) The system displays a monotonous
alignment behavior in the trivial region for high anisotropic field strength and low activity within the nematic director field.
(f) In the non-trivial regime, the interplay between the anisotropic field strength and the intrinsic activity gives rise to a three-
stage, non-monotonous alignment behavior. (g) When activity and flow anisotropic field are both low, the cells don’t display
localized order, resulting in an isotropic state. The number of defects saturates as orientational order becomes ill-defined. (h)
When the activity is the dominating parameter, the system fails to align and evolves towards a state with only local nematic
alignment.
impact from defect annihilations on the ordering kinetic in this initial phase. As the correlation functions did not
collapse with the distance normalized by the correlation length ∆r∥/ξ∥, the growth of the ordering did not exhibit
dynamic scaling behavior, that would be expected for kinetic ordering in passive materials (Fig. 3c,g) [36].
In the subsequent intermediate disordering phase (II), the asymptotic value of the correlation function at long distances
did not drop to zero. It reached a finite value that increased with time, indicating the appearance of a true long-range
order (Ext. Data Fig. 2e,h). The characteristic correlation length ξ∥ceased to increase and remained stable while
the number of defects rose, which is attributed to the increase of cellular forces as cells begin to align. As the active
forces reached a critical value, an active instability set in which caused defects to nucleate [37].
In the steady-state ordering phase (III), the correlation functions converged to a finite value and entirely collapsed
with ∆r∥/ξ∥, indicating a dynamical scaling of length-scales as the system became more ordered (Fig. 3d,h). The
characteristic correlation length ξ∥increased again, and for final times, this length scale is expected to follow 1/2
power law with a logarithmic correction due to the interaction between topological defects [31, 36]. While, due
to the limited experimental accessible timespan, this precise form of temporal scaling could not be unambiguously
determined in the experimental data set, it was possible to use prolonged simulations to confirm the existence of a
logarithmic correction to the 1/2power law growth in the final ordering phase (Fig. 3b,f). This indicated that the
coarsening in the final stage was dominated by long-range logarithmic interaction between defects [36].
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FIG. 3. Three distinct scaling behaviors of the domain growth. (a) Illustration of the domains with positive or
negative projection δn⊥values corresponding to the experimental data extending anisotropically along the flow axis. (b) The
correlation length ξ∥exhibits distinct growth rates. (c) The correlation functions normalized by the length scale ξ∥during the
initial ordering phase (I) for the experimental data. (d) The correlation functions normalized by the length scale ξ∥for late
times during the steady-state ordering phase (III) for the experimental data. (e-h) Corresponding simulation figures to (a-d).
In (b) and (f) the dashed and dotted lines denote 1/2-power law scaling and 1/2-power law scaling with logarithmic correction,
respectively.
Heterogenous distribution of bound defect pairs: The emergence of three-stage ordering kinetics, linked to the
non-monotonic evolution of topological defects during the establishment of long-range orientational order (Fig. 1i,j,
and Fig. 3b,f), prompted us to investigate the spatiotemporal defect dynamics leveraging the extensive field of view.
A detailed examination of defect dynamics revealed that the emergence of new defect pairs was highly localized at
the onset of the defect nucleation phase. During this phase, multiple defect pairs nucleated in close proximity to one
another, while a significant proportion of the field of view remained defect-free. We characterized this heterogeneity
by analyzing the distance between the kth nearest neighbors of oppositely charged defects (Fig. 4a). We observed
both in experiment and simulation that after defect pairs nucleated, they did not move further apart as expected in
traditional active turbulence, nor did they immediately annihilate, as seen previously in flow-tumbling active systems
with external fields [32]. Instead, the defects remained relatively bound to each other (Fig. 4b,c; k0). Hence, in
contrast to a homogeneous increase in defect spacing, we identified regions densely populated with defects interspersed
with void areas where no defects were present (Fig. 4d). This heterogeneity in the spatial defect distribution impacted
the later stages of ordering kinetics. Consequently, during the final coarsening, the distance between nearest neighbor
defect pairs k0did not increase. Instead, the distance of the higher-order pairs ki>0increased as the distance between
defect-rich regions increased (Fig. 4b,c; ki>0). Eventually, at late times, the logarithmic interactions between the
separate defect pair configurations became dominant, and all topological defect pairs annihilated. Even as the number
of defects decreased and the domains grew larger (Ext. Data Fig. 3), we did not observe a significant change in the
nearest neighbor distance between oppositely charged defects. This is because defect pairs tend to cluster in groups,
and only these groups become increasingly sparse. This unique behavior did not occur in the absence of activity or
the external flow field, indicating an intricate interplay between the cell activity, aligning field, domain interfaces,
and the topological defects residing on them.
String excitations: In the absence of an external field, it is well-established that activity unbinds pairs of oppositely
charged nematic topological defects [37, 38]. Interestingly, in both experiments and simulations, we measured a fixed,
small distance (≈123 ±11 µm, equivalent to up to 11 cell sizes) between nearest neighbor k0oppositely charged
±1/2defects throughout the ordering process. This indicates that, despite the intrinsic activity of the cells, the
oppositely-charged defect pairs remained bound during coarsening, even after the nucleation phase. We hypothesized
that the emergence of such stable defect configurations is governed by topological strings that bind defect pairs
together [39], which are driven by the interplay of active stresses within the cell layer and the external aligning field.
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FIG. 4. Nematic defects interact through strings. (a) The distance to the kth nearest neighbour between two defects
of opposite charge. (b) The time evolution of the distance between the kth nearest neighbors for the experimental data. The
brighter areas indicate the standard deviation from the mean for the corresponding color. (c) The time evolution of the distance
between the kth nearest neighbors for the simulation data. The brighter areas indicate the standard deviation from the mean
for the corresponding color. (d) In the experimental data, the nematic defects sit on the domain boundaries. The nearest
neighbor pairs remain at a close distance from each other, while the distance to the second nearest neighbor increases during
coarsening. (e) The nematic defect sitting on the domain walls interacts through strings with parallel orientation (green) and
perpendicular orientation (red) to the preferred orientation axis. (f) The transition from one domain to the other can be energy
intensive, indicated by the red arrow, or energy efficient, indicated by the green arrow. (g) The average length of the strings
with perpendicular orientation (red) decreases to zero as the system coarsens. The length of the green string initially increases
and eventually plateaus during coarsening for experimental data. The brighter areas illustrate the interquartile range for data
of the corresponding color. (h) Force field of the nematic director field. The forces (blue arrows) are highest along the domain
boundaries.
To test our hypothesis, we identified regions that separated orientation quadrant 1 (domain 1, with positive δn⊥)
from quadrant 4 (domain 2, with negative δn⊥) (Fig. 4e,f ). Along these domain boundaries, the director angle α
displayed maximum or minimum gradients in reference to the preferred axis. Remarkably, in both experiments and
simulations, we found distinct strings between oppositely charged topological defect pairs, binding them together.
Due to the symmetry-breaking field, we could define strings as director isolines that connected two oppositely charged
defects along the preferred direction. In general, in the absence of an external flow, an infinite number of isolines
exist along all possible directions, all of them equivalent. However, with the presence of a preferred direction, the
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isoline along the preferred direction gains physical significance as there is a different energy cost associated with the
director transitioning from domain 1 to domain 4 through the preferred axis (green lines) compared to perpendicular
transitions (red lines) (Fig. 4f). The evidence for this physical significance is manifested by the localization of the
topological defect trajectories on these isolines in the experiments (Fig. 4e). As a result, string excitations stabilized
the oppositely-charged defect pair configurations until the final ordering phase. Consequently, since defect motility
was highly localized to movement along the strings, the strings provided a path for the annihilation of defects (Sup-
plementary Movie 4 and 5), and the strings that lined up perpendicular to the preferred axis (Fig. 4g; red ) eventually
decayed as all defects were annihilated at later times. Meanwhile, the strings extending along the preferred axis (Fig.
4g; green) increased during the course of ordering, but the average string length appeared to remain constant, as
small strings continuously nucleated. Examining snapshots of the ordering kinetics, however, revealed that eventually,
long strings extending along the preferred axis dominated the system at late times (Supplementary Movie 4 and 5).
The large extent of these strings is remarkable as they potentially provide a path for long-range transmission of
mechanical forces. While force measurements were not available for the experiments, we could probe this mechanism
in the simulations. Indeed, simulations confirmed that the strings act as hotspots for focusing active forces (Fig. 4h).
In summary, the high temporal and spatial resolution achieved in our experimental setup allowed us to describe
the ordering kinetics of endothelial cell layers within the framework of active nematic liquid crystals, highlighting
the governing role of topological excitation and explaining endothelial cell alignment as a dynamic transition with
cellular misalignment as an intermediate stage. The introduced unifying framework reconciles previous contradictory
observations reporting perpendicular or parallel endothelial cell alignment depending on experimental parameters [25–
27, 40–42]. The importance of the transient regime in endothelial cell alignment reveals the need for precise temporal
resolution in detecting changes in protein expression, exemplified by the previously observed transient upregulation
of JNK2 in bovine aortic endothelial cells [43]. The identified interplay of the activity-driven local nucleation of
defects and global ordering induced by the external driving field sets the spatiotemporal behavior of the endothelial
cell layers, which defines a phase diagram with distinct regions of tissue responses. This interplay gives rise to defects
that are not fully localized rather they are connected by string excitation in endothelial layers and are the biological
analogs of the pair-superfluid phase predicted for two-dimensional antiferromagnetic condensates under a magnetic
field [44]. In this context, string excitations have been identified in classic XY models of magnetism, which possess
both nematic and polar interactions [39, 44, 45] and are characterized by a linear interaction potential, in addition
to the normal logarithmic Coloumb potential between the defects. As such, the linear potential leads to an effective
line tension proportional to the length of the string binding the defect pairs together. The emergence of topological
excitations spanning multiple cell lengths during the ordering process underscores the need for precise, localized
measurements of protein expression and cellular responses at both local and global scales over time. The recent
identification of a nematic phase during embryogenesis across multiple species, along with the observed presence of
nematic defects during this process, highlights the potential significance of topological defects in development [46].
Our findings may set a benchmark for a wide range of biological phenomena in morphogenesis, tissue remodeling, and
disease progression, where spatiotemporally localized mechanical effects have consequences over long distances and
time scales.
ACKNOWLEDGEMENTS
We gratefully acknowledge financial support by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement no. 810104-PoInt). This research was conducted
within the Max Planck School Matter to Life supported by the German Federal Ministry of Education and Research in
collaboration with the Max Planck Society. This work was further supported by the Novo Nordisk Foundation grant
no. NNF18SA0035142, NERD grant no. NNF21OC0068687 (AD), the Villum Fonden Grant no. 29476 (AD), and
the European Union via the ERC-Starting Grant PhysCoMeT (AD) and the Horizon 2020 research and innovation
programme Marie Sklodowska-Curie grant no. 101029079-SIMMS (KT).
AUTHOR CONTRIBUTIONS
I.R. performed experiments with the contributions of D.R.V. and the support of V.P. under the supervision of A.R.B.
K.T. and performed the simulations under the supervision of A.D. A.R.B., and A.D. conceived the project. Data
Analysis was performed by I.R. and K.T. with the support of A.S. All authors participated in the writing of the
manuscript.
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COMPETING INTERESTS
The authors declare no competing interests.
I. MATERIALS AND METHODS
Primary human aortic endothelial cell culture: Primary human aortic endothelial cells (HAOEC) (PeloBiotech,
PB-CH-180-2011) were cultured in gelatine-coated T75 flasks (0.2 % gelatine in PBS incubated for 1h at 37 ◦C) using
Endothelial Cell Growth Medium Kit Classic (PeloBiotech, PB-MH-100-2190) from passage 5 up to passage 9. For
shear flow experiments, µ-slide I Luer ibiTreat (Channel height: 0.2mm) were incubated for 1h with 40µg/ml collagen
at 37 ◦C. After three washing steps with PBS, 50 µl of HAOECs at a seeding concentration of 3·106up to 6·106
cells/ml were seeded into the flow channel and left at 37 ◦C for 1 h to allow them to form first adhesions. Then, the
cell culture medium was added to the reservoirs, and the HAOECs were left overnight to adhere fully to the substrate.
The confluent cell layer resulted in 12 000 up to 40 000 cells per a field of view of 40 mm2. The cell culture medium,
PBS, and the flow channel slides were kept in the incubator one day before cell seeding to prevent bubble formation
in the flow channels.
Perfusion cell culture: To avoid bubble formation, the cell culture medium and perfusion set were kept in the
incubator one day before the flow experiment. The ibidi pump system and perfusion set (ibidi: orange) were prepared
according to the manufacturer’s instructions. After the pump system was calibrated, the flow channel was connected.
To allow the cells to adapt to the shear stress, a flow program slowly increased the flow velocity until reaching the
desired shear stress of 15 dyn/cm2, respectively 20 dyn/cm2after 1 h.
Live cell imaging: Phase contrast imaging was performed using a Leica DMi8 Thunder Imager and a 10x (0.32 NA)
air objective (Leica). To gather sufficient statistical data, we stitched high-resolution images over a scale of several
millimeters for up to 64 hours. The area of the resulting field of view was 40 mm2. The ibidi stage top incubation
system was adapted for live cell imaging with a custom-built lid to accommodate the tubing. The fluidic unit of the
pump system was kept in a nearby incubator at 37 ◦C and 5 % CO2. Through a small opening in the rear of the
incubator, the flow channel was mounted on the microscope stage while remaining connected to the pump system.
Images were acquired every 20 minutes over at least 2.5 days.
Characterization of nematic director field: We extracted the nematic field from the phase contrast images using
a custom Python3 script based on the method from reference [47] as previously described in [48]. We computed
the local nematic director n=t⊗tusing the tangent vector t= (Iy,−Ix)to the gradient of the intensity of the
image ∇I(x, y) = (Ix, Iy)within a box of dimensions L×L. From the local nematic director, we extracted the
eigenvalues n1> n2and the corresponding eigenvector e1to the biggest eigenvalue, which provided the nematic
director associated with the pixel in the center of the box. Here, we used a box of 130 µm×130 µm to compute the
nematic director for a pixel. The nematic director field was computed for each pixel, and the spacing between two
pixels is ≈5.2µm.
Detection of nematic defects: We obtained the Q- tensor from the nematic director field. The defect cores were
then identified as the local maxima, respectively minima, of the charge density q=1
4π(∂xQxk∂yQy k −∂xQyk ∂yQxk )
where |q|>0.1. Only every 5th pixel in the nematic director field was considered for defect detection.
Extraction of nearest neighbour distance: Defects were considered for kth nearest neighbor (knn) analysis if the
defects were present for at least 4 h. We used trackpy to obtain and filter the defect trajectories.[49] From the defect
trajectories, we constructed the knn-distance matrix, which we used to extract the knn distance for k= 0,1,2,3.
String visualization in experiments: The nematic director n= [cos(α),sin(α)] was defined both in orientation
region region 0≤α1≤π(first and second quadrant) and −π/2≤α2≤π/2(first and fourth quadrant). The parallel
and perpendicular strings were defined to go between two resolution points if the absolute angle difference |∆α1|or
|∆α2|was larger than π/2, respectively.
Length measurement for parallel and orthogonal strings: From the absolute angle difference obtained for
string visualization, a binary image was created by setting the points with |∆α1|or |∆α2|larger than π/2to one
and any other point to zero. Eventually, string length was measured by obtaining the skeletons from the segmented
features.
Correlation functions Taking the projection of the orientation parallel to the preferred axis δn (Ext. Data Fig. 2a)
allowed us to identify how far away the director was from the preferred axis. Using this, we observed how regions of
similar orientation coarsened by looking at the correlation functions Cδn⊥(∆r∥)(Ext. Data Fig. 2b) and Cδn⊥(∆r⊥)
(Ext. Data Fig. 2c), where we split up the correlations along the perpendicular ∆r⊥and parallel ∆r∥direction to
the preferred axis.
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Comparing the correlation functions along the axes perpendicular ∆r⊥and parallel ∆r∥to the preferred orientation
revealed that, as the system evolved towards an aligned state, the discrepancy in the decay behavior of the correlation
functions became more pronounced (Ext. Data Fig. 2d,e,g,h). This trend was observed in both experiments and
simulations.
We extracted a length scale ξfrom the correlation functions by determining the length when the correlation function
reached the value 1/ϵ. In experiments, the correlation along both axes became more persistent over time (Ext. Data
Fig. 2f). However, the perpendicular length xi⊥grew more monotonically, while the parallel length xi∥was strongly
connected to the multiple phases. In contrast, in simulations, only the parallel axis showed a stronger correlation in
the parallel direction, while the perpendicular axis remained unaffected (Ext. Data Fig. 2i). The cause for this is that
as the monolayer matures, the phenomenological properties of the cell layer changed slightly over time, increasing the
active length scale. This is not incorporated into the model; hence, the growth parallel set by the active length scale
didn’t change.
String visualization in simulations: The nematic director n= [cos(α),sin(α)] was defined both in orientation
region region 0≤α1≤π(first and second quadrant) and −π/2≤α2≤π/2(first and fourth quadrant). The parallel
and perpendicular strings were defined to go between two resolution points if the absolute angle difference |∆α1|or
|∆α2|was larger than π/2, respectively.
Simulations To complement the experiments, we simulated the cell layers as an active nematic film, where the
third-dimensional shear flow was incorporated as an aligning field. We solved the continuum equations using a hybrid
lattice Boltzmann approach [38]. We assumed that the active nematic film flows with collective velocity u and has
long-range orientational order described by the tensor order parameter Q, which mimics the cell alignment.
The dynamics of the orientational order parameter Qat each position r and time tare described by the Beris-Edwards
transport equation
∂t+u ·
∇
Q−
S= ΓQ
H. (1)
The co-rotation term
S=ξ
D+
Ω
Q+1
3
I+
Q+1
3
Iξ
D−
Ω−2ξ
Q+1
3
Itr
Q
Wdetermines the align-
ment of the cells in response to gradients in the velocity field, with Ωthe rotational part and Dis the extensional
part of the velocity gradient tensor W=
∇v =Ω+D. The alignment parameter ξis considered deep in the flow
aligning regime and set to ξ= 0.9as cells align strongly with themselves.
The molecular field H=−(δF
δQ−1
3ITr δF
δQ)is a functional derivative of free energy density F, describing the re-
laxation towards equilibrium at a rate ΓQ. The free energy consists of a Landau-De Gennes contribution, fLdG =
A0n1
21−ν
3tr Q2−ν
3tr Q3+ν
4tr Q22owith A0= 0.05, and Frank-Oseen deformation fFO =K
2
∇Q2
with
K= 0.02. We used ν= 2.55, which favors the isotropic state in the absence of activity or any aligning field. Lastly,
it contains a field strength fField =−ϵ0E·Q·Ewhere Eis a matrix setting the direction of the flow and ϵ0is the
strength of the field that induces nematic ordering along the flow axis.
The system also obeys the Navier-Stokes equations for the velocity field within the active film. Assuming constant
fluid mass density ρ(not active material concentration ϕ) leads to the incompressibility condition
∇ · u = 0.(2)
and
ρ∂t+u ·
∇u =−
∇p+
∇ ·
Π−γu, (3)
where pis the pressure and Πis the stress tensor which includes the standard viscous stress Πvisc = 2ηEfor film
viscosity η= 2/3. Furthermore, it contains the elastic stress due to the nematic nature of the order parameter
Πelastic = 2ξQ(Q:H)−ξH·Q−ξQ·H
−
∇Q:δF
δ
∇Q+Q·H−H·Q,(4)
where Q=Q+I/3. Lastly, the stress contains an active component
Πact =−ζQ.(5)
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We used ζ= 0.03 to induce weak nematic ordering in the absence of an aligning field.
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II. LIST OF SUPPLEMENTARY MOVIES
Supplementary Movie 1: Phase contrast images of HAOECs exposed to 20 dyn/cm2shear stress flowing from left
to right. The HAOECs align in the direction of the flow.
Supplementary Movie 2: Phase contrast images of HAOECs from Supplementary Movie 1 with an overlay of the
nematic director field and the topological defects as the cells reorient in the flow direction.
Supplementary Movie 3: Simulations of the nematic director field, colored by domain, were conducted with an
anisotropic field strength of ν= 0.25 along the horizontal axis and an activity set to ζ= 0.03.
Supplementary Movie 4: Phase contrast images of HAOECs with an overlay of the domains. The defects are
connected by red and green strings, indicating the energy intensive transition and the energy efficient transition
between the domains.
Supplementary Movie 5: Simulations of the nematic director, colored by domain, from Supplementary Movie 3,
with an overlay of defects that are connected by red and green strings indicating the energy intensive transition and
the energy efficient transition between the domains.
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III. EXTENDED DATA
Extended Data FIG. 1. (a) The major axis length of HAOECs during exposure to shear flow. (b) The minor axis length of
HAOECs during exposure to shear flow. (c) The ellipticity of HAOECs during exposure to shear flow. An ellipticity of 1
corresponds to a perfect circle.
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Extended Data FIG. 2. Correlation dynamics reveal anisotropic growth of similarly aligned regions. (a) Projections
of the nematic director from the preferred axis of orientation δn. (b) Anisotropic growth of correlation length parallel ∆r∥
and (c) perpendicular ∆r⊥to the preferred axis. (d) Time evolution of the perpendicular Cδn⊥(r⊥)and (e) parallel Cδn⊥(r∥)
correlation function for the experimental data, (f) with corresponding correlation lengths. (g-i) Corresponding simulation
figures to figures d-g.
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Extended Data FIG. 3. Coarsening dynamics of domains Domains start symmetrically spaced, but over time, they
coarsen anisotropically. The coarsening of the domains in the experimental data halts (t=11h to t=25) during the intermediate
disordering phase (II) and progresses during the initial ordering phase (I) (t=0 to t=10h) and the steady-state ordering phase
(III) (t=26 to t=64h), during which domains become more anisotropically shaped.
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