# Defect-mediated dynamics of coherent structures in active nematics

Mattia Serra<EMAIL_ADDRESS>University of California San Diego, Department of
Physics, CA 92093, USA Paulson School of Engineering and Applied Sciences,
Harvard University, Cambridge, MA 02138, USA  Linnea Lemma University of
California at Santa Barbara, Santa Barbara, CA 93111, USA Luca Giomi
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden,
Netherlands Zvonimir Dogic University of California at Santa Barbara, Santa
Barbara, CA 93111, USA L. Mahadevan<EMAIL_ADDRESS>Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Department of Organismic and Evolutionary Biology, Harvard University,
Cambridge, MA 02138, USA Department of Physics, Harvard University, Cambridge,
MA 02138, USA

###### Abstract

Active fluids, such as cytoskeletal filaments, bacterial colonies and
epithelial cell layers, exhibit distinctive orientational coherence, often
characterized by nematic order and topological defects. By contrast, little is
known about positional coherence – i.e., how a hidden dynamic skeleton
organizes the underlying chaotic motion – despite this being one of their most
prominent and experimentally accessible features. Using a combination of
dynamical systems theory, experiments on two-dimensional mixtures of
microtubules and kinesin and hydrodynamic simulations, we characterize
positional coherence in active nematics. These coherent structures can be
identified in the framework of Lagrangian dynamics as moving attractors and
repellers, which orchestrate complex motion. To understand the interaction of
positional and orientational coherence on the dynamics of defects, we then
analysed observations and simulations and see that +1/2 defects move and
deform the attractors, thus functioning as control centers for collective
motion. Additionally, we find that regions around isolated +1/2 defects
undergo high bending and low stretching/shearing deformations, consistent with
the local stress distribution. The stress is minimum at the defect, while high
differential stress along the defect orientation induces folding. Our work
offers a new perspective to describe self-organization in active fluids, with
potential applications to multicellular systems.

## Introduction

Many out-of-equilibrium systems, from bird flocks down to biofilms and the
cell cytoskeleton, consist of agents that consume energy and self-organize
into large-scale patterns and collectively moving structures 1; 2; 3; 4; 5; 6;
7, that are breathtaking in their beauty and complexity while being of
relevance to questions of embryonic development, wound healing and cancer 8;
9; 10; 11. Understanding the mechanisms that lead to these patterns and
characterizing the phases of active matter systems will unravel their
complexity, while also suggesting ways to mimic them using synthetic
materials, and eventually control and design active systems using external
fields.

Self-organized patterns are typically described in the language of Eulerian
flows, where the local velocity, pressure and orientation of the active
building blocks are treated as fields within a fixed control volume. Two-point
correlation functions, spectral densities and other quantities inspired by
studies of statistical steady states of turbulence in Newtonian fluids, are
typical outcomes of this approach 12; 13. Topological defects, i.e. localized
singularities in the orientation of the active building blocks 2, have also
been extensively studied and their dynamics is known to be inherently
entangled with the large-scale chaotic flow 13; 14; 15. But what is the
relation between the dynamics of defects and the large scale coherence in the
flow fields that are typically spatially heterogeneous and temporally
unsteady?

A natural framework to address this question is provided by the Lagrangian
description of fluid flow. By tracing the motion of passive particles in
unsteady flows, which may also include chaotic paths, one can often identify
robust skeletons, commonly referred to as Coherent Structures (CSs) 16; 17;
18, which shape trajectory patterns and reveal the organizing barriers to
material transport (see e.g. Refs. 19; 20). Here, we combine theoretical
concepts from nonlinear dynamics, active hydrodynamics simulations and
experiments on suspensions of microtubules and kinesin to unravel the CSs
underlying the chaotic flow of two-dimensional active nematics and their
relation to the dynamics of topological defects.

## Results

### Lagrangian deformations and coherent structures

Our theoretical and computational framework for kinematic analysis starts by
considering the velocity field $\mathbf{v}(\mathbf{x},t)$ of a planar active
nematic fluid, and the corresponding flow map

$\displaystyle\mathbf{F}_{t_{0}}^{t}(\mathbf{x}_{0})=\mathbf{x}_{0}+\int_{t_{0}}^{t}\mathbf{v}(\mathbf{F}_{t_{0}}^{\tau}(\mathbf{x}_{0}),\tau)\
d\tau\;.$ (1)

This evolves the initial position $\mathbf{x}_{0}$ of a virtual tracer
particle to the corresponding position
$\mathbf{F}_{t_{0}}^{t}(\mathbf{x}_{0})$ at time $t$, whereas the spatial
heterogeneity of $\mathbf{F}_{t_{0}}^{t}(\mathbf{x}_{0})$ describes Lagrangian
deformation of the nematic fluid over $[t_{0},t]$. Locally, a small fluid
patch can be stretched, sheared and folded. Stretching and shearing (Fig. 1)
are completely characterized by the right Cauchy-Green strain tensor field
$\mathbf{C}_{t_{0}}^{t}(\mathbf{x}_{0})$ 21 defined as

${\color[rgb]{0,0,0}\mathbf{C}_{t_{0}}^{t}(\mathbf{x}_{0})=\mathbf{\nabla
F}_{t_{0}}^{t}(\mathbf{x}_{0})^{\top}\cdot\mathbf{\nabla
F}_{t_{0}}^{t}(\mathbf{x}_{0})\;,}$ (2)

where $\mathbf{\nabla F}_{t_{0}}^{t}(\mathbf{x}_{0})$ is the Jacobian of the
flow map.

Figure 1: Lagrangian view of an active nematic fluid. Red curves demarcate
repelling and attracting regions in the flow identified by high foreword and
backward FTLE values. Repellers are based at the initial fluid configuration
while attractors at the final one. Initially close tracers that are at
opposite sides of a fw FTLE (${}_{f}\Lambda$) ridge will move far apart at
time $t$. Similarly, initially distant tracers are attracted to a bw FTLE
(${}_{b}\Lambda$) ridge at time $t$. An infinitesimal patch of nematic fluid
at $\mathbf{x}_{0}$ will get stretched and folded over $[t_{0},t]$. Different
fibers in this patch tend to align to a distinguished fiber
$\xi_{2}(\mathbf{x}_{0})$ during $[t_{0},t]$, and the curvature of the folded
patch is $\kappa_{t_{0}}^{t}(\mathbf{x}_{t},\mathbf{\xi}_{2})$.

Given a material patch consisting of straight fibers at the initial time (Fig.
1), and denoting by $\lambda_{1}\leq\lambda_{2}$ and
$\\{\bm{\xi}_{1},\bm{\xi}_{2}\\}$ the eigenvalues and the associated
orthonormal eigenvectors of $\mathbf{C}_{t_{0}}^{t}(\mathbf{x}_{0})$, one can
interpret $\bm{\xi}_{2}$ as the fiber ’s most stretched direction (by a factor
$\sqrt{\lambda_{2}}$) and $\bm{\xi}_{1}$ as the least stretched one (by a
factor $\sqrt{\lambda_{1}}$). In chaotic systems $\lambda_{2}$ usually grows
exponentially in time, i.e. $\lambda_{2}\sim e^{\sigma t}$, and it is
typically rescaled as

$\Lambda_{t_{0}}^{t}(\mathbf{x}_{0})=\frac{1}{|t-t_{0}|}\log\sqrt{\lambda_{2}(\mathbf{x}_{0})},$
(3)

which denotes the largest Finite-Time Lyapunov Exponent (FTLE). The maximal
stretching along $\bm{\xi}_{2}$ implies that, over time, other fibers align
with the local $\bm{\xi}_{2}$ direction, as illustrated in Fig. 1. This
alignment property follows from the corresponding asymptotic result 22.

By contrast, folding deformations determine variations in the curvature of
material fibers over time 23. This can be computed from the second order
spatial derivatives of $\mathbf{F}_{t_{0}}^{t}(\mathbf{x}_{0})$ and provides
additional information relative to stretching and shearing. Starting with an
infinitesimal patch of straight fibers at $\mathbf{x}_{0}$, from the
$\bm{\xi}_{2}$-alignment property follows that the most observable fluid
folding – i.e. what a colored patch would show – during $[t_{0},t]$ is the one
along the $\bm{\xi}_{2}$ fiber (Fig. 1). We compute this folding as

${\color[rgb]{0,0,0}\kappa_{t_{0}}^{t}({\bf
x}_{0},\bm{\xi}_{2})=\frac{[(\mathbf{\nabla}^{2}\mathbf{F}_{t_{0}}^{t}(\mathbf{x}_{0})\bm{\xi}_{2})\cdot\bm{\xi}_{2}]\cdot[\mathbf{\nabla_{\perp}F}_{t_{0}}^{t}(\mathbf{x}_{0})\cdot\bm{\xi}_{2}]}{\lambda_{2}^{3/2}}\;,}$
(4)

where
$(\nabla^{2}F_{t_{0}}^{t}(\mathbf{x}_{0})\mathbf{\xi}_{2})_{ij}=\tiny{\sum\limits_{k}}{F_{t_{0}}^{t}}_{i,jk}(\mathbf{x}_{0})\mathbf{\xi}_{2_{k}},\
i,j,k\in\\{1,2\\}$ and $\nabla_{\perp}=(-\partial_{y},\partial_{x})$, (See
Supplementary Information - SI, Section 1). The curvature
$\kappa_{t_{0}}^{t}(\mathbf{x}_{0},\bm{\xi}_{2})$ displays the Lagrangian
folding that occurred from $t_{0}$ to $t$ over the initial fluid
configuration. Analogously, the same folding can be displayed over the final
position by transporting $\kappa$ along trajectories (Fig. 1), namely:

$\kappa_{t_{0}}^{t}(\mathbf{x}_{t},\bm{\xi}_{2})=\kappa_{t_{0}}^{t}(\mathbf{F}_{t_{0}}^{t}(\mathbf{x}_{0}),\bm{\xi}_{2})\;.$
(5)

Large values of $\kappa_{t_{0}}^{t}(\mathbf{x}_{0},\bm{\xi}_{2})$ mark the
initial positions $\mathbf{x}_{0}$ of the nematic fluid that will undergo
large folding in the time interval $[t_{0},t]$, and, similarly
$\kappa_{t_{0}}^{t}(\mathbf{x}_{t},\bm{\xi}_{2})$ identifies the final
positions of nematic fluid that experienced small/large curvature changes. In
the SI Section 1, we provide general expressions of the material curvature and
its alternative formulation in terms of Eulerian quantities such as flow
vorticity, divergence and rate-of-strain tensor. Altogether, Eqs. (3-5)
completely quantify the maximal stretching and folding deformations of a
continuum moving under a given flow map $\mathbf{F}_{t_{0}}^{t}$. These
measures are model-independent (or agnostic to the mechanisms driving the
flow), hence applicable to experimental or computational velocity fields and
readily implement able.

The forward FTLE (fw FTLE or ${}_{f}\Lambda$), is a scalar field over the
initial particle positions $\mathbf{x}_{0}$ that quantifies the maximum local
deformation and identifies the location of maximum spatial separation of
initially close particles over the time interval $[t_{0},t]$. Similarly, the
backward FTLE (bw FTLE or ${}_{b}\Lambda$), defined over the final positions
$\mathbf{x}_{t}$, identifies the location of maximum spatial convergence of
initially distant particles over $[t_{0},t]$. Together they demarcate regions
of attraction (attracting CS) and repulsion (repelling CS) (Fig. 1), but do
not distinguish whether attraction or repulsion are caused by shear or normal
deformations 16; 17.

### Coherent structures organize particle motion

We now employ these analytical tools to analyze 2D microtubule based active
nematic liquid crystals assembled on a surfactant stabilized oil-water
interface 24. Large scale chaotic dynamics of these materials is collectively
driven by kinesin molecular motors that move along multiple filaments to
induce relative filaments sliding.

Using Particle Image Velocimetry (PIV), we reconstruct the velocity field of
autonomously flowing active nematics. From the velocity field, we compute
${}_{f}\Lambda$ and ${}_{b}\Lambda$ for different time scales $|T|=|t-t_{0}|$
using Eqs. (1-3). Figures 2a,b show the ${}_{f}\Lambda$ and ${}_{b}\Lambda$
fields for $|T|=100s$. As sketched in Fig. 1, particles are repelled from a
${}_{f}\Lambda$ ridge and attracted towards a ${}_{b}\Lambda$ ridge (covered
by magenta dots). Panel c shows the final position of a set of particles
initially released from a uniform grid and serves as a tracker of particles
motion. Movie1 shows the time evolution of the FTLE fields and particle
positions. Although fluid moves chaotically, there is an underlying hidden
coherent skeleton, captured by the FTLE fields, that dynamically organize
their motion while remaining invisible to the inspection of fluid tracers.

The FTLE also provides a $|T|-$dependent map of the stretching and shearing
Lagrangian deformation of the active continuum, with ridges that demarcate
sets of the fluid that will experience a distinguished deformation relative to
their neighbors. This Lagrangian deformation consistently integrates along
trajectories the separate contributions of viscous, elastic and active
stresses deforming the nematic fluid, and thus encodes a memory trace of the
nematodynamic field.

Figure 2: Dynamics of 2D microtubule based active nematics assembled on the
oil-water interface. The PIV velocity is reconstructed on a uniform $670\times
800\,\mu$m grid with spatial resolution $15.6\,\mu$m and temporal resolution
of $1$ frame per second. (a) ${}_{f}\Lambda$ field whose ridges signal
repelling CSs. White regions demarcate the set of particles that left the
domain where the velocity field is available. (b) ${}_{b}\Lambda$ field whose
ridges signal attracting CSs. Magenta dots represent the final ($t=200$)
position of nematogens started inside the magenta circle shown in panel a at
the initial time $t=100$. (c) Final position of nematogens started from a
uniform grid at the initial time. Particles started outside the magenta circle
are green. The time evolution is available as Movie1. (d-f) Fluorescence
recovery after photobleaching experiment in an active nematics. (d,f) Initial
and final configuration of the FRAP experiment. (e) ${}_{b}\Lambda_{148}^{0}$
along with advected particles at $t=148$, initialized in correspondence of the
photobleached regions at $t=0$. Movie2 shows panels d,e for increasing $t$.
Time is expressed in seconds (s). Colorbars encode attraction or repulsion
rates in s-1. In panels a-c, the ATP concentration is 250 $\mu$M. In panels
d-f, the ATP concentration is 18 $\mu$M.

To correlate these computed Lagrangian memory traces with direct observations
of the deformation patterns near ${}_{b}\Lambda$ ridges, we label regions of
microtubule based active nematics and observe their subsequent evolution (SI,
Section 5). This is achieved by photobleach ing nine circular regions of
radius $\approx 4\mu m$ (Fig. 2d). Using PIV data, we compute the
${}_{b}\Lambda$ field along with the position of Lagrangian tracers (magenta)
initialized at $t=0$ in correspondence of the photobleached regions ( Fig.
2e). Stripe-shaped ridges of ${}_{b}\Lambda_{148}^{0}$ reveal a horizontal
shear layer, along with regions of distinctly high attraction and Lagrangian
deformations. Our analysis predicts the evolution of advected and diffused
photobleached patches (Fig. 2f, Movie2). Overall, ${}_{b}\Lambda_{T}^{0}$
provides a $|T|-$dependent map of attraction and stretch or shear deformations
over the entire domain.

### Positive defects mediate attracting coherent structures

Having uncovered the invisible organizers of the flow fields using Lagrangian
coherent structures, we now turn to understand if and how they are related to
the visible dynamics of topological defects, well known to be correlated with
complex, large-scale nematodynamic flows 13; 15. As previously, we use
microtubule based 2D active nematics powered by kinesin motors. By assembling
slow nematics at 2 $\mu$M ATP concentration, we simultaneously measured both
the velocity $\mathbf{v}$ and the nematic director $\mathbf{n}$ fields (SI
Section 5).

Figure 3: Active suspension of microtubule bundles and kinesin at the water
oil interface with a 2$\mu M$ ATP concentration. (a) ${}_{b}\Lambda$ for
$|T|=200s$ along with the position of nematogens (magenta) attracted to a
${}_{b}\Lambda$ ridge and initially released from a circular blob. (b-c) Same
as (a) for larger $|T|$. Red arrows illustrate that +1/2 defects pull the
attracting ${}_{b}\Lambda$ ridges that in turn shape the Lagrangian motion of
nematogens. (d) Absolute Folding field
$|\kappa_{t_{0}}^{t}(\mathbf{x}_{t},\mathbf{\xi}_{2})|$ for $T=200s$, along
with topological defects and the director field at the current time $t_{0}+T$.
Defects are invariably located at regions of high folding-type and low
stretching or shearing-type Lagrangian deformation. Time is in s, the colorbar
in d encoding Lagrangian folding is in $1/{\mu m}$ while those encoding
attraction rates in s-1. Movie3 shows the time evolution of ${}_{f}\Lambda$
and ${}_{b}\Lambda$, along with particle motion.

We mark $+1/2$ disclinations with red dots and $-1/2$ disclinations with cyan
triangles, as shown in Fig. 1. Figures 3a-c show ${}_{b}\Lambda$ for
increasing time intervals, along with nematogens (magenta), initially released
from a circular blob and eventually attracted to a ${}_{b}\Lambda$ ridge.
Movie3 shows the time evolution of ${}_{f}\Lambda$ and ${}_{b}\Lambda$ along
with particle positions. The FTLE fields again uncover the organizers of fluid
motion, while remaining hidden to trajectory plots.

Along with ${}_{b}\Lambda$, Figs. 3a-c show the evolution of topological
defects, with red arrows indicating the direction of motion of positive
disclinations. These panels suggest that positive disinclination move and
deform ${}_{b}\Lambda$ ridges, which, in turns, direct particle motion.
Interestingly, $+1/2$ defects appear to be in regions of low Lagrangian
stretching or shearing deformation as quantified by the FTLE field. By
contrast, the Lagrangian folding measure is maximum at defects, as shown in
Fig. 3d, where the absolute folding field
$|\kappa_{t_{0}}^{t}(\mathbf{x}_{t},\bm{\xi}_{2})|$ is superimposed to the
nematic director $\mathbf{n}$ at the same of time of panel a.

To quantify our observations on deformations at defects and the correlation of
their dynamics with ${}_{b}\Lambda$, we turn to numerical simulations of an
incompressible ($\mathbf{\nabla}\cdot\mathbf{v}=0$) planar uniaxial active
nematic liquid crystal. We solve the nematodynamic equations

$\displaystyle\rho\frac{D{\bf
v}}{Dt}=-\mathbf{\nabla}p+\eta\mathbf{\nabla}^{2}\mathbf{v}+\mathbf{\nabla}\cdot(\mathbf{\sigma}^{e}+\mathbf{\sigma}^{a})\;,$
(6a) $\displaystyle\frac{D{\bf Q}}{Dt}=\lambda
S\mathbf{D}+\mathbf{Q}\mathbf{\Omega}-\mathbf{\Omega}\mathbf{Q}+\gamma^{-1}\mathbf{H}\;,$
(6b)

which can be derived from phenomenological arguments or microscopic models,
and capture typical experimental statistics (see e.g. Ref. 13). Here, $\rho$
and $\eta$ denote the density and viscosity of the nematic fluid,
$D/Dt=\partial_{t}+\mathbf{v}\cdot\mathbf{\nabla}$ is the material derivative,
$\lambda$ is the flow alignment parameter and $\gamma$ is the rotational
viscosity 25. In Eq. (6b), $\mathbf{Q}=S(\mathbf{n}\mathbf{n}-\mathbf{I}/2)$
denotes the nematic tensor, $0\leq S\leq 1$ the nematic order parameter,
$\mathbf{I}$ the identity tensor, $\mathbf{D}=[\mathbf{\nabla
v}+(\mathbf{\nabla v})^{\top}]/2$ is the symmetric and
$\mathbf{\Omega}=[\mathbf{\nabla v}-(\mathbf{\nabla v})^{\top}]/2$ the
antisymmetric part of the velocity gradient $\mathbf{\nabla v}$. Finally,
$\mathbf{H}=-\delta
F/{\delta\mathbf{Q}}=K\nabla^{2}\bm{Q}^{2}-(a_{2}+a_{4}|\bf Q|^{2}){\bf Q}$ is
the molecular tensor governing the relaxation dynamics of the nematic phase
defined as the variational derivative of the two-dimensional Landau-de Gennes
free energy $F=\int f\,dA$ with $f$ the free-energy density 25

$f=\frac{1}{2}K|\nabla{\bf Q}|^{2}+\frac{1}{2}a_{2}|{\bf
Q}|^{2}+\frac{1}{4}a_{4}|{\bf Q}|^{4}\;,$ (7)

where $|\cdot|$ denotes the Frobenius norm (i.e. $|{\bf
Q}|^{2}=Q_{ij}Q_{ij}$), $K$ is the orientational stiffness relating the
elastic free energy to spatial inhomogeneities in the configuration of the
nematic tensor and $a_{2}$ and $a_{4}$ are bulk moduli. Finally,
$\mathbf{\sigma}^{e}=-\lambda{\color[rgb]{0,0,0}S}\mathbf{H}+\mathbf{Q}\mathbf{H}-\mathbf{H}\mathbf{Q}$
denotes the elastic stress arising from a departure from the lowest free
energy configuration, and $\mathbf{\sigma}^{a}=\alpha\mathbf{Q}$ the
contractile ($\alpha>0$) or extensile ($\alpha<0$) active stress exerted by
the active particles along $\mathbf{n}$.

Since the typical Reynolds number of microtubules/kinesin suspensions varies
in the range $10^{-5}-10^{-3}$, depending on the ATP concentration, we
eliminate the convective derivative in Eq. (6a) and numerically integrate Eq.
(6) using finite differences on a $128\times 128$ collocated grid with
periodic boundary conditions. In all our simulations we set the parameter
values as follows: $\lambda=0.1$, $K=1$, $a_{2}=-1$, $a_{4}=2$, $\gamma=10$,
$\alpha=25$ and $L=5$, in previously defined rescaled units. This yields the
velocity field $\mathbf{v}$ along with the nematic tensor field $\mathbf{Q}$,
from which we identify topological defects (SI Section 2).

Figure 4: Lagrangian analysis of an active nematic fluid obeying eq. (6). (a)
${}_{b}\Lambda$ for $|T|=1.5$, along with the position of material particles
(magenta) attracted to a ${}_{b}\Lambda$ ridge and initially released from a
circular blob. (b-c) Same as (a) for larger $|T|$. Red arrows illustrate that
+1/2 defects pull the attracting ${}_{b}\Lambda$ ridges that, in turn, shape
particle motion. (d) Logarithm of the folding field modulus
$|\kappa_{t_{0}}^{t}(\mathbf{x}_{t},\mathbf{\xi}_{2})|$ for $T=1.5$ as in (a).
Movie4 shows the evolution of panels a,d for different $T$. Positive defects
are located at regions of high folding-type and low stretching or shearing-
type Lagrangian deformation.

Figures 4a-c show ${}_{b}\Lambda$ for different time intervals $|T|$, along
with the position of an initially circular set of particles (magenta) that are
attracted to the a ${}_{b}\Lambda$ ridge. Analogous to the analysis of the
experiments, we find that ${}_{b}\Lambda$ ridges are pulled (red arrows) and
shaped by moving Eulerian +1/2 disclinations while remaining insensitive to
-1/2 disclinations (Figs. 4a-c). In the SI Section 2, we quantify the
correlation between the evolution of ${}_{b}\Lambda$ and defects motion. We
first find a velocity field that transports and deforms ${}_{b}\Lambda$ over
increasing $T$, and evaluate it at disclinations. The ${}_{b}\Lambda$
evolution along with disclination velocities is available as Movie5. We then
compute the relative angle between the ${}_{b}\Lambda$ velocity at defects and
the disclination velocities. The mean and standard deviation of the relative
angle associated with positive disclinations are seven and five times smaller
compared those related to negative disclinations (SI Section 2).

We find this result nontrivial as ${}_{b}\Lambda$ is Lagrangian, i.e.,
contains information of particle trajectories, while disclinations are
Eulerian, hence agnostic to particle paths. This connection could provide a
quantitative framework to control the Lagrangian motion, mixing, and
deformation of active nematics by steering the position of Eulerian
disclinations. Finally, in the SI Section 3, we provide also an aggregate
measure of positional coherence by using the inverse of the broadly used
$H^{-1}$ mixing norm 26, and show how it decreases with increasing $T$ and
activity $\alpha$.

Figure 4d shows $\kappa_{t_{0}}^{t}(\mathbf{x}_{t},\bm{\xi}_{2})$ associated
with the $|T|=1.5$, as in Fig. 4a. Consistently with our experimental results
(Fig. 3) defects are preferentially located in regions of high Lagrangian
folding and low stretching. We obtain results similar to Fig. 4 for extensile
($\alpha<0$) active nematics, simulated using the same parameters listed above
and $\alpha=-25$. Movie6 shows the same as Movie4 for the extensile case.

### Stress gradients are maximal at positive defects

Motivated by the striking deformations associated with +1/2 defects, we
analyze the stress distribution using the simulation data (Fig. 5). We find
that both the deviatoric and isotropic total stresses are minimal at +1/2
defects, but have high stress gradients along the defect orientation (5a,b),
which induces folding deformation as sketched in Fig. 5d. We obtain similar
results for extensile active nematics (SFig. S4), where the folding direction
is towards the tail of the defect (SFig. S5b) as opposed to the head (Fig.
5d).

Figure 5: Stress and deformation around an isolated $+1/2$ defect (bottom
right in Figs. 4a,d) in contractile active nematics. (a) Maximum eigenvalue of
the total deviatoric stress $\mathbf{\sigma}^{D}$ in the proximity of a $+1/2$
defect (red cirlce). The leading eigenvector of $\mathbf{\sigma}^{D}$ is
marked by black lines and the the nematic director field by red lines (see
Figs. S3 for the separate viscous, elastic and active stress contributions).
(b) Pressure field normalized by the spatial maximum pressure in absolute
value completely characterizes the isotropic stress
$\mathbf{\sigma}^{I}=-p\mathbf{I}$. (c) Logarithm of the folding rate modulus
of the active nematic, computed from eq. (8). (d) Top: Sketch of the
deviatoric and isotropic stress distribution near an isolated $+1/2$ defect in
contractile active nematics. The arrow size is proportional to the stress
level, and blue marks the direction perpendicular to the defect orientation.
Bottom: sketch of the material deformation in correspondence the $+1/2$
defect. Figures S4 and S5b show equivalent analysis for the extensile case.

To bridge the gap between the Lagrangian deformations, which accounts for the
motion history of the nematic continuum, and Eulerian stresses based on an
instantaneous configuration, we derived an exact formula for the Eulerian
folding rate

$\dot{\kappa}(t,\mathbf{x},\mathbf{n})=[(\mathbf{\nabla
D}(\mathbf{x},t)\mathbf{n})\cdot\mathbf{n}]\cdot\mathbf{n_{\perp}}-\frac{\mathbf{\nabla}\omega(\mathbf{x},t)\cdot\mathbf{n}}{2}$
(8)

experienced by an infinitesimal patch of nematic fluid with orientation
$\mathbf{n}$ (SI Section 1). The folding rate can be computed from
$\mathbf{v},\ \mathbf{n}$, and arises from spatial heterogeneities of the
rate-of-strain tensor $\mathbf{D}$ and the vorticity $\omega$. If the nematic
continuum is an epithelium, for instance,
$\dot{\kappa}(t,\mathbf{x},\mathbf{n})$ measures the bending rate experienced
by the cell located at $\mathbf{x}$. Using eq. (8), we also find that
$\dot{\kappa}(t,\mathbf{x},\mathbf{n})$ is maximum in the vicinity of +1/2
defects (Fig. 5c), consistent with the corresponding stress distribution
(Figs. 5a,b) and Lagrangian folding (Fig. 4d).

It is interesting to compare these results with experimental findings that
suggest a biological functionality of topological defects in a biological
epithelial layer 10; 27; 28. The quantification of deformations and stress at
the defects is a key step in elucidating how mechanical stimuli are converted
into downstream biochemical signals. Positive defects with strength $+1/2$ in
monolayers of MDCK cells, for example, have been associated with sites of cell
apoptosis 27, with a possible explanation being high compressive stress at the
defect location. This hypothesis has been tested by correlating the isotropic
stress averaged over several ($6$) cell sizes in the neighborhood of a
topological defect during apoptosis (Fig. 3b in Ref. 27). Our findings,
however, show that bending deformations are dominant at positive
disclinations, suggesting there can be new relevant mechanisms at play at
topological defects in epithelia.

We observe a clear similarity of the extensile active nematic stress
distribution in SFig. S4h, with that experimentally measured in monolayers of
MDCK cells (Fig. 3a in Ref. 27) during apoptosis. The peculiar stress and
deformation distribution around +1/2 defects, together with the cells’ ability
to sensing curvature changes 29, may lead to uncovering novel feedback
mechanisms in epithelial dynamics. A natural next step is to correlate folding
deformation maps in epithelia with the corresponding YAP (Yes-associated
protein) distributions 30.

### Discussion

By combining concepts from nonlinear dynamics, experiments of two-dimensional
active nematics and active hydrodynamics simulations, we found that the motion
of active nematics is organized by hidden dynamic (time-dependent) attracting
and repelling CSs, whose motion is coupled to that of $+1/2$ topological
defects. As they move deform attracting CSs, which in turn regulate collective
motion. Furthermore, the Lagrangian time-scale dependent maps of stretching-
and folding-type deformations of a nematic continuum show that $+1/2$ defects
occur at locations of high bending and low stretching or shearing type
deformations. Motivated by this finding, we have discovered a characteristic
stress distribution around +1/2 defects: the stress is minimal at the defect,
but its large gradient along the defect’s orientation causes differential
stress that induces bending. The bending is towards the defect head (tail) for
contractile (extensile) extensile active nematics. Similar stress
distributions were measured experimentally in monolayers of MDCK (Madin Darby
canine kidney) cells 27.

More broadly, using only the measured velocity and nematic director, our
results provide a quantitative framework for assessing the motion, mixing and
deformation of active nematics. Emerging experimental evidence associated
biological functionality with topological defects of cells orientation 10,
actin fibers orientation 31, and the ability of cells to sense and react to
bending and stretching deformations 29. From this perspective, our approach
quantifies the stretching and folding deformations in a nematic continuum as
time-scale dependent maps with the ability to predict and perhaps eventually
control the flows. Investigating the correlation between the curvature and
stretching deformation maps in epithelial layers and other similar systems,
could elucidate how cells couple mechanical inputs to intracellular signals in
oriented active matter systems.

## Acknowledgements

We are grateful to Suraj Shankar and Nicola Molinari for helpful discussions.
This work is partially supported by the Schmidt Science Fellowship and the
Postdoc Mobility Fellowship from the Swiss National Foundation (M.S.), the
Netherlands Organization for Scientific Research (NWO/OCW) as part of the
Frontiers of Nanoscience program and the Vidi scheme (L.G.), and the
Department of Energy, Office of Basic Energy Sciences under Award No.
DESC0019733 (ZD), and the NSF Simons Center for Mathematical and Statistical
Analysis of Biology Award No. 1764269 (LM).

Supplemental Information

## S1. Lagrangian folding and Eulerian folding rates

The curvature of an infinitesimal material fiber at time $t$ starting from the
initial position $\mathbf{x}_{0}$, with an orientation $\theta$, and curvature
$\kappa_{0}$, due to the transport and deformation induced by the flow map
$\mathbf{F}_{t_{0}}^{t}(\mathbf{x}_{0})$, can be computed 23 as

$\displaystyle\kappa_{t_{0}}^{t}:\mathbb{R}^{2}\times\mathbb{S}^{1}\times\mathbb{R}\rightarrow$
$\displaystyle\ \mathbb{R},$ (1)
$\displaystyle\kappa_{t_{0}}^{t}(\mathbf{x}_{0},\theta,\kappa_{0})=$
$\displaystyle\frac{[(\mathbf{\nabla^{2}F}_{t_{0}}^{t}(\mathbf{x}_{0})\mathbf{e}_{\theta})\cdot\mathbf{e}_{\theta}]\cdot[\mathbf{\nabla_{\perp}F}_{t_{0}}^{t}(\mathbf{x}_{0})\cdot\mathbf{e}_{\theta}]}{(\mathbf{e}_{\theta}\cdot[\mathbf{C}_{t_{0}}^{t}(\mathbf{x}_{0})\cdot\mathbf{e}_{\theta}])^{3/2}}+\kappa_{0}\frac{\det[\mathbf{\nabla
F}_{t_{0}}^{t}(\mathbf{x}_{0})]}{(\mathbf{e}_{\theta}\cdot[\mathbf{C}_{t_{0}}^{t}(\mathbf{x}_{0})\cdot\mathbf{e}_{\theta}])^{3/2}},$

where $\mathbf{e}_{\theta}=[\cos\theta,\sin\theta]$ and
$(\nabla^{2}F_{t_{0}}^{t}(\mathbf{x}_{0})e_{\theta_{k}})_{ij}=\tiny{\sum\limits_{k}}{F_{t_{0}}^{t}}_{i,jk}(\mathbf{x}_{0})e_{\theta_{k}},\
i,j,k\in\\{1,2\\}$. By evaluating
$\kappa_{t_{0}}^{t}(\mathbf{x}_{0},\theta,\kappa_{0})$ for an initially
straight ($\kappa_{0}=0$) fiber aligned with the dominant eigenvector of
$\mathbf{C}_{t_{0}}^{t}$ ($\mathbf{e}_{\theta}=\mathbf{\xi}_{2}$), we obtain
Eq. (4) in the main text.

In the instantaneous limit ($t=t_{0}$), the material curvature rate of the
material fiber 23 is given by

$\displaystyle\frac{d\kappa_{t_{0}}^{t}(\mathbf{x}_{0},\theta,\kappa_{0})}{dt}|_{t=t_{0}}$
$\displaystyle={\dot{\kappa}}_{t_{0}}(\mathbf{x}_{0},\theta,\kappa_{0})$ (2)
$\displaystyle=[(\mathbf{\nabla
D}(\mathbf{x}_{0},t_{0})\mathbf{e}_{\theta})\cdot\mathbf{e}_{\theta}]\cdot{\mathbf{e}_{\theta}}_{\perp}-\frac{\mathbf{\nabla}\omega(\mathbf{x}_{0},t_{0})\cdot\mathbf{e}_{\theta}}{2}+\kappa_{0}\bigg{[}\mathbf{\nabla}\cdot\mathbf{v}(\mathbf{x}_{0},t_{0})-3\mathbf{e}_{\theta}\cdot[\mathbf{D}(\mathbf{x}_{0},t_{0})\cdot\mathbf{e}_{\theta}]\bigg{]},$

where $\mathbf{D}$ denotes the rate-of-strain tensor, $\omega$ the vorticity,
$\mathbf{\nabla}\cdot\mathbf{v}$ the divergence of the flow and $(\nabla
D(\mathbf{x}_{0},t_{0})e_{\theta_{k}})_{ij}=\tiny{\sum\limits_{k}}D_{ij,k}(\mathbf{x}_{0})e_{\theta_{k}},\
i,j,k\in\\{1,2\\}$. Integrating Eq. (2) along trajectories
$\mathbf{F}_{t_{0}}^{t}(\mathbf{x}_{0})$, provides an alternative formula to
compute Eq. (1) from know Eulerian quantities (see Eq. B1 in 23). Evaluating
eq. (2) along the current nematic director
($\mathbf{e}_{\theta}\equiv\mathbf{n}$), and assuming $\kappa_{0}\equiv 0$,
one can compute the instantaneous folding rate experienced by a nematic
continuum using only the velocity and the director fields inputs as

$\dot{\kappa}(t,\mathbf{x},\mathbf{n})=[(\mathbf{\nabla
D}(\mathbf{x},t)\mathbf{n})\cdot\mathbf{n}]\cdot\mathbf{n_{\perp}}-\frac{\mathbf{\nabla}\omega(\mathbf{x},t)\cdot\mathbf{n}}{2}.$
(3)

We note that eq. (3) allows quantifying the folding rate contribution coming
from spatial inhomogeneities of the rate of strain tensor and the vorticity.
To deploy these results in an experimental setting, consider a nematic
continuum that describes an epithelial tissue where $\mathbf{n}$ represents
the cell orientation field. Then, for instance, eq. (3) quantifies the
instantaneous bending rate of epithelial cells assuming that cells have
initially zero curvature.

## S2. Positive defects move and deform attracting Lagrangian coherent
structures

We quantify the influence of moving defects on the motion and deformation of
attracting Lagrangian Coherent Structures identified with the backward FTLE.
Denoting by $\mathbf{q}(\cdot,t):=[Q_{11}(\cdot,t),Q_{12}(\cdot,t)]^{\top}$
the vector containing the independent entries of the nematic tensor, and by
$\mathbf{x^{d}}(t):=\underset{\mathbf{x^{d}}(t)}{\arg}\mathbf{q}(\mathbf{x^{d}}(t),t)=0$
the time-$t$ position of disinclinations, we compute the disclination
velocities by Taylor-expanding the equation defining defect locations

$\underset{\mathbf{x^{d}}(t+\delta
t)}{\arg}{\mathbf{q}(\mathbf{x^{d}}(t+\delta t),t+\delta
t)}\approx\underset{\mathbf{x^{d}}(t)}{\arg}\mathbf{q}(\mathbf{x^{d}}(t),t)+[\mathbf{\nabla}\mathbf{q}(\mathbf{x^{d}}(t),t)\mathbf{v^{d}}(t)+\partial_{t}\mathbf{q}(\mathbf{x^{d}}(t),t)]\delta
t=0,$ (4)

and requiring the leading order term to vanish, i.e.

$\mathbf{v^{d}}(t)=-[\mathbf{\nabla}\mathbf{q}(\mathbf{x^{d}}(t),t)]^{-1}\partial_{t}\mathbf{q}(\mathbf{x^{d}}(t),t).$
(5)

By the implicit function theorem, $\mathbf{v^{d}}(t)$ exists whenever
$[\mathbf{\nabla}\mathbf{q}(\mathbf{x^{d}}(t),t)]$ is invertible. Following
32, we compute the index of the disclination at $\mathbf{x^{d}}$ as

$\text{ind}_{\text{d}}=\frac{1}{2\pi}\sum_{i=1}^{n}\Delta_{i},\quad\Delta_{i}=\phi_{i+1}-\phi_{i}-\pi\
\text{round}\bigg{(}\frac{\phi_{i+1}-\phi_{i}}{\pi}\bigg{)},\
\phi_{n+1}=\phi_{1},$ (6)

where $\phi_{i},\ i=1,...,4$ denotes the angle between the nematic director
and the horizontal axis at each of the grid points surrounding the defect.

To quantify the deformation of the FTLE field with respect to $|T|$, we use
Digital Particle Image Velocimetry to obtain a fictitious velocity field,
$\mathbf{v}_{FTLE}(\mathbf{x},T)$, that deforms the
$\text{FTLE}_{t_{0}}^{t_{0}-|T|}$ onto $\text{FTLE}_{t_{0+\Delta
T}}^{t_{0}-(|T|+\Delta T)}$ (Fig. S1a).

(a)

(b)

Figure S1: (a) Time intervals for backward-time FTLE computations. (b) Average
angle, in degrees, between a vector field describing the deformation of the bw
FTLE, and disclination velocities as a function of $|T|$. The bw FTLE
evolution along with disclination velocities is available as Movie5.

Evaluating such velocity field at the current location $\mathbf{x^{d}}_{i}(T)$
of defect $i$, we compute the relative angle between $\mathbf{v}_{FTLE}$ and
$\mathbf{v^{d}}$ as

$\mathbf{\varphi}_{i}(T)=\arccos{\frac{\mathbf{v}_{FTLE}(\mathbf{x^{d}}_{i}(T),T)\cdot\mathbf{v^{d}}_{i}(T)}{|\mathbf{v^{d}}_{i}(T)||\mathbf{v}_{FTLE}(\mathbf{x^{d}}_{i}(T),T)|}}.$
(7)

We note that $T$ automatically specify $t_{0}$ and vice-versa because the
initial time of our analysis $t_{A}$ is fixed. Fig. S1b shows the average
angle $<|\varphi|>$ between $\mathbf{v}_{FTLE}$ and positive and negative
disinclinations at each $|T|$, quantitatively confirming that positive
disinclinations move in directions similar to $\mathbf{v}_{FTLE}$ compared to
negative disinclinations. The overall angular distance between positive
disinclination and $\mathbf{v}_{FTLE}$ has a mean of $8.7^{\circ}$. By
contrast, for negative disinclinations the average misalignment is
$62^{\circ}$.

## S3. Positional coherence as a function of time and activity

(a)

(b)

Figure S2: (a) $H^{-1}$ mixing norm normalized by its initial value for the
activity value $\alpha=25$. The insets show the concentration fields at
different times computed solving Eq. (8) with initial distribution
$c_{0}(\mathbf{x})=\sin x$. The complete time evolution of the concentration
field is available as Movie7. (b) Same as (a) for different activity values.

We quantify how coherence varies with the time scale $T$ and the activity
parameter $\alpha$, by studying the stirring exerted by an incompressible
active nematic flow on a passive scalar $c(\mathbf{x},t)$. The passive scalar
evolves according to

$\partial_{t}c+\mathbf{v}\cdot\mathbf{\nabla}c=0$ (8)

with the initial condition $c(\mathbf{x},t_{0})=c_{0}(\mathbf{x})$. To measure
stirring, we use the $H^{-1}$ mixing norm, broadly adopted in fluid flows 26,
and defined as

$\displaystyle||c(\cdot,t)||^{2}_{H^{-1}}=|||\mathbf{\nabla}|^{-1}c(\cdot,t)||^{2}_{L^{2}}=\sum_{\mathbf{k}\neq
0}|\mathbf{k}|^{-2}|{\hat{c}}_{\mathbf{k}}(t)|^{2},$ (9)

where

$\hat{c}_{\mathbf{k}}(t)=\frac{1}{L}\int_{[0,L]^{2}}e^{-i\mathbf{k}\cdot\mathbf{x}}c(\mathbf{x},t)d\mathbf{x}$
(10)

are the Fourier coefficients of $c(\mathbf{x},t)$. The $H^{-1}$ measures the
variance of a low-pass-filtered image of the concentration field; the smaller
it is, the less coherence (more mixed) is the scalar field on large spatial
scales. In our analysis, $c_{0}(\mathbf{x})=\sin x$.

Fig. S2a shows the $H^{-1}$ mixing norm normalized by its initial value as a
function of $T$ for nematic flow analyzed in Fig.4, with activity value
$\alpha=25$. The insets show the concentration fields at three different
times, while the complete time evolution is available as Movie7. Fig. S2b
shows the normalized $H^{-1}$ mixing norm for three different values of
activity. As expected, spatial coherence decreases with increasing $T$, and
higher activity $\alpha$. Finally, we note that the activity values used here
are higher than those seen in typical biological nematic systems such as
epithelial, fibroblast and stem cells 10, where positional coherence is
present for larger $T$.

## S4. Stress at +1/2 defects

To quantify the correlation between the total stress and the location and
dynamics of topological defects, we consider the solution of the neamtodynamic
model (6) analyzed in Fig. 4. We first note that both the deviatoric viscous
stress $\mathbf{\sigma}^{v}$ and the elastic stress
$\mathbf{\sigma}^{e}=-\lambda
S\mathbf{H}+\mathbf{Q}\mathbf{H}-\mathbf{H}\mathbf{Q}$ are traceless; the
former follows from incompressibility, while the latter follows because
$\text{tr}[\mathbf{H}]=\partial F_{LdG}/\partial Q_{11}+\partial
F_{LdG}/\partial Q_{22}=0,\ Q_{11}=-Q_{22}$ and
$\text{tr}[\mathbf{Q}\mathbf{H}-\mathbf{H}\mathbf{Q}]=2\text{tr}[\text{skew}(\mathbf{Q}\mathbf{H})]=0$.
Finally, the active stress is also traceless because
$\text{tr}[\mathbf{\sigma}^{a}]=\alpha\text{tr}[\mathbf{Q}]=0$. Thus, the
maximum and minimum eigenvalues of the above stress tensors have equal
magnitude and opposite signs, hence providing a scalar representation of both
their maximum and minimum stress contributions. Figures S3a-d show the maximum
eigenvalue of $\mathbf{\sigma}^{v},\mathbf{\sigma}^{e},\mathbf{\sigma}^{a}$
and the total deviatoric stress
$\mathbf{\sigma}^{D}=\mathbf{\sigma}^{v}+\mathbf{\sigma}^{e}+\mathbf{\sigma}^{a}$,
along with the defects location and the nematic director field (red).
Surprisingly, and contrary to propositions in the literature 27, we find that
the maximum total deviatoric stress is minimum at defects (Fig. S3d and Fig.
S3g, which shows a zoomed version of the inset in Fig. S3d), meaning that the
defect locations are regions subject to minimum shear stress. The black arrows
in Fig. S3g represent the leading eigenvector field of $\mathbf{\sigma}^{D}$.

Because $\mathbf{\sigma}^{v},\mathbf{\sigma}^{e},\mathbf{\sigma}^{a}$ are
traceless, the only isotropic stress $\mathbf{\sigma}^{I}=-p\mathbf{I}$ is
given by the pressure. With our convention, positive pressure indicates
compressing isotropic stress. Following the numerical scheme in 13, we have
solved eq. (6) by using the streamfunction-vorticity formulation. To recover
the pressure, we solve the Poisson equation

$\mathbf{\nabla}^{2}p=\mathbf{\nabla}\cdot[\mathbf{\nabla}\cdot[\mathbf{\sigma}^{e}+\mathbf{\sigma}^{a}]],$
(11)

obtained by taking the divergence of eq. (6a), and using incompressibility.
Solving eq. (11), we determine the pressure distribution up to a constant that
will not affect the pressure topology. Here we set this constant such that the
spatial average of the pressure is zero. Figures S3e,h show the pressure field
normalized by the maximum pressure in absolute value. Interestingly, we find
that positive defects are typically located in regions of low isotropic
stress. A closer look at Figs. S3g,h, reveals that while deviatoric and
isotropic stresses are low at positive defects, the corresponding stress
gradients are high. In Fig. S5a, we show that the topology of both the
deviatoric and isotropic stresses induce differential stresses (blue),
perpendicular to the $+1/2$ defect orientation, that bends the active nematic
towards the head of the defect. This peculiar stress distribution leaves a
clear kinematic footprint in the deformation of the nematic medium, as shown
by the high values of Lagrangian folding at $+1/2$ defects (Fig. 4d), as well
as the Eulerian folding rate along $\mathbf{n}$, computed from eq. (3) and
shown in Figs. S3f,i.

Figure S3: Stress associated with a contractile active nematic ($\alpha>0$).
(a-d) Maximum eigenvalue of
$\mathbf{\sigma}^{v},\mathbf{\sigma}^{e},\mathbf{\sigma}^{a},\mathbf{\sigma}^{D}=\mathbf{\sigma}^{v}+\mathbf{\sigma}^{e}+\mathbf{\sigma}^{a}$,
along with the topological +1/2 (cirlces), -1/2 (triangles) and the nematic
director field in red. (e) Pressure field normalized by the spatial maximum
pressure in absolute value completely characterizes the isotropic stress
$\mathbf{\sigma}^{I}=-p\mathbf{I}$. (f) Logarithm of the folding rate modulus
of the active nematic, computed from eq. (3). (g-i) Zoomed view of the insets
in (d-f). The black direction field in (g) shows the leading eigenvector of
$\mathbf{\sigma}^{D}$. The time evolution of the above panels is available as
Movie8.

Figures S4 and S5b show the same analysis of Figs. S3 and S5a for extensile
active nematics, obtained by solving eq. (6) using the same parameters as the
contractile case and $\alpha=-25$.

It is worth pointing out that Fig. S4h is similar to the experimentally
measured isotropic stress within monolayers of MDCK (Madin Darby canine
kidney) cells, in the vicinity of $\pm 1/2$ nematic defects in the cell
orientation field 27. Indeed, Fig.3a of 27 shows that $+1/2$ defects are
located in a region of zero isotropic stress, and high isotropic stress
gradient along the orientation of the defect, precisely as in Fig. S4h. As in
the contractile case, at $+1/2$ defects the deviatoric stress is zero and has
a significant gradient along the defect’s orientation (Fig. S4g). Fig. S5b
shows the effect of the stress distribution around the $+1/2$ defects, which
induces high folding as confirmed both by the Lagrangian finite-time folding
(Fig. 4d) and the Eulerian folding rate (Fig. S4i). The peculiar stress
distribution of low stress and high stress gradients around $+1/2$ defects is
present in both contractile and extensile active nematics. In the former, the
stress gradient induces a folding deformation towards the head of the defect
(Fig. S5a), while in the latter towards the tail (Fig. S5b).

Figure S4: Stresses associated with an extensile active nematic ($\alpha=-25$)
following the analysis shown in Fig. S3. The time evolution of the above
panels is available as Movie9. Compare panel h with with Fig. 3a in 27, which
shows the experimentally measured isotropic stress within monolayers of MDCK
(Madin Darby canine kidney) cells, in the vicinity of $\pm 1/2$ nematic
defects in the cell orientation field.

(a)

(b)

Figure S5: Schematic of the local stress and deformation around a $+1/2$
defect in contractile (a) and extensile (b) active nematics. (a) Top: Sketch
of the deviatoric and isotropic stress distribution (consistent with Figs.
S3g,h) near a $+1/2$ defect. The arrow size is proportional to the stress
level, and blue marks the direction perpendicular to the defect orientation.
Bottom: sketch of the material deformation induced by the stress distribution
in neighborhood of a $+1/2$ defect. (b) Same as a for an extensile active
nematic ($\alpha<0$). Top: Sketch of the deviatoric and isotropic stress
distribution (consistent with Figs. S4g,h) near a $+1/2$ defect. Bottom:
folding deformation induced by the stress distribution near a $+1/2$ defect.

## S5. Experimental data

### Velocity and Orientation Fields

We prepared a microtubule-based active nematic at 1.4mM ATP doped with a small
fraction of Alexa-647 labeled MTs 33; 34 . We imaged the sample both using LC-
PolScope and epifluorscence microscopy. LC-PolScope provides a direct
measurement of the orientation field of the MTs 35. Particle image velocimetry
was used to find the velocity field from the fluorescence images. The LC-
PolScope and fluorescence images were taken sequentially within 2s of each
other; however, we treat the lag as negligible in the data as calculating the
velocity field coarse grains the data in time. Imaging was done on a Nikon Ti
Eclipse equipped with Andor Neo camera.

### Photobleaching

We prepared microtubule-based active nematic at 18$\mu$M ATP with Alexa-647
labeled MTs 33. It was important to use a low amount of ATP to slow the
dynamics so that the timescale of bleaching was faster than the movement of
the material. We included a small fraction of MTs labeled with Azide-DBCO-488
to simultaneously bleach regions and measure the velocity field of the
material. We used a Leica SP8 Confocal with a 20X NA 0.75 air objective to
bleach and image the sample. Since $\textrm{Image
Brightness}\propto\left(\textrm{NA}^{2}/M\right)^{2}$, bleaching is most
efficient at low magnification and high NA. To bleach, we decreased the range
over which the galvo-mirror scans by 20 times and turned a 633nm laser power
to its maximum. With this combination, we were able to bleach in under 5
seconds so that the distortions due to material movement were minimal. Using
Leica software, we were able to define regions of interest to bleach defined
shapes. To image the sample we reduced the 633nm laser power to 0.5% of its
maximum and simultaneously imaged with a 488nm laser.

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