# White paper: The Helix Pathogenicity Prediction Platform

Bas Vroling  These authors contributed equally towards this work Bio-Prodict,
Nijmegen, the Netherlands  Centre for Molecular and Biomolecular Informatics
(CMBI),
Radboud University Medical Center, Nijmegen, the Netherlands  Correspondence:
<EMAIL_ADDRESS>Stephan Heijl These authors contributed equally
towards this work Bio-Prodict, Nijmegen, the Netherlands

###### Abstract

In this white paper we introduce Helix, an AI based solution for missense
pathogenicity prediction. With recent advances in the sequencing of human
genomes, massive amounts of genetic data have become available. This has
shifted the burden of labor for genetic diagnostics and research from the
gathering of data to its interpretation. Helix presents a state of the art
platform for pathogenicity prediction of human missense variants. In addition
to offering best-in-class predictive performance, Helix offers a platform that
allows researchers to analyze and interpret variants in depth that can be
accessed at helixlabs.ai.

##  1 Introduction

Millions of human genomes and exomes have been sequenced, but their clinical
application remains limited due to the difficulty of distinguishing rare
disease-causing variants from benign genetic variation. Without clear lines of
evidence that show that a missense variant is either benign or neutral,
variants are deemed ‘variants of uncertain significance’, or VUS. The large
number and rarity of these VUSes presents a formidable obstacle, as the
interpretation of these variants is a difficult and time-consuming task,
thereby severely limiting the actionability of diagnostic sequencing.
Furthermore, the large amount of VUSes hinder the effective implementation of
applications that require large amounts of genotype-phenotype associations
such as disease association studies and patient stratification in drug
development programs.

To deal with this ever-growing problem, a large number of computational tools
have been developed that aim to predict the pathogenicity of novel variants.
These tools are typically based on limited data describing evolutionary
conservation and constraints, sometimes combined with the physicochemical
characteristics of the amino acids and information about the structural
domains in which these residues reside. The emergence of many ‘meta-
predictors’ (or ‘ensemble predictors’; predictors that take predictions of
other predictors as inputs) has led to an increase in overall predictive
performance. However, despite obvious progress, these tools leave room for
improvement, especially in a clinical context.

At the core of missense variant prediction lies the problem of understanding
the role of proteins and their constituent amino acids. A good understanding
of this relation between sequence, structure and function is the ultimate goal
that enables insight in the impact of missense variants.

Recently, advances have been made in closing this sequence-function gap by
artificial intelligence (AI) methods, where contextual language models have
been developed that can provide information about how proteins are formed,
shaped and function based on protein sequences alone[1]. Trained on billions
of protein sequences, these models contain a wealth of information and can be
applied to a wide range of predictive tasks with just one protein sequence as
input. However, the predictive power of models solely based on these protein
representations on explicit protein prediction tasks still fall short of
predictive methods that include evolutionary information in the form of
multiple sequence alignments (MSAs) or information derived thereof.

This evolutionary information (e.g., amino acid conservation patterns) can be
used because functional and structural requirements lead to specific selective
pressures on proteins. In turn, identifying these evolutionary constraints
provides data and insights that can be used for missense variant effect
prediction and interpretation.

The ability to detect and interpret these evolutionary signals is dependent on
the quality of the data that is used. The application of deep and high-quality
multiple sequence alignments integrated with protein structure data is
commonplace in a number of fields that require detailed understanding of the
role of individual amino acids. Protein optimization pipelines and drug
development programs often critically depend on the availability of deep
annotations and integration of protein sequence and structure data. Such high-
quality protein annotation data has not been available at scale, and the lack
of such data at large scale has hampered its application for missense effect
prediction.

We present Helix, a missense variant effect predictor built on a resource of
deep integrated protein data, in which over 30,000 protein family structure-
based MSAs, over 50,000 protein structures and $\sim$60,000 human targeted
sequence-based MSAs are combined to provide deep annotations for all human
proteins at the level of individual amino acids. Helix incorporates this data
in combination with Contextual Language Model representations that contain
implicit information about how proteins are formed, shaped and function.

The features used by Helix describe proteins and their constraints at multiple
levels. This includes extensive MSA-derived metrics describing selective
pressures that act on proteins as a whole, on networks of residues required
for specific functions and on individual positions. This information is
integrated with features describing structural aspects, e.g. secondary
structure, flexibility and solvent-accessible surfaces, both at the level of
individual protein structures, as well as in the form of aggregated data
across protein domains and families. Additionally, Helix uses descriptors of
gene mutation tolerance, gene occurrence throughout the evolutionary tree as
well as data describing protein-protein interactions. Best in class
classification systems trained on this data are combined with state of the art
AI methodologies to produce a predictor that offers the best of both worlds.

Helix was trained on a large set of well-annotated variants, using a strict
training regime where circularity is actively avoided. We benchmark the
performance of Helix using 10-fold cross validation datasets and two
previously published, clinically relevant datasets. We show that Helix
consistently outperforms existing predictors.

Predictions, however, are just that. For the effective use of these
predictions more context is often required. To this end, Helix provides high-
quality predictions together with detailed variant reports that contain
references to relevant scientific literature, information about evolutionary
constraints, underlying data quality assessment and interactive protein
structures.

##  2 Methods

Helix is built on top of the proprietary 3DM platform. 3DM is a protein data
and analytics platform that collects, combines and integrates protein data for
protein (super-)families. Examples of data included for every protein family
are protein sequences and structures, SNP data from whole genome sequencing
studies, scientific literature and patents, and data derived thereof. Built
around uniquely deep and high quality structure based multiple sequence
alignments, the 3DM platform offers data and tooling that enable researchers
to efficiently investigate the relation between protein sequence, structure
and function. Details about the methodology behind 3DM can be found in Kuipers
et al.[2].

Currently there are over 30.000 3DM protein family systems available. These
are built using templates selected from a) the SCOP classifications of protein
structures, and b) by clustering the structures in the PDB. Overall, these
cover the complete structural space, and serve as an extremely rich data
source with which to annotate all human protein sequences. In short, this
resource contains high-quality MSA data integrated with protein structure
data, scientific literature, SNP data, etc for all parts of sequence space
that can be mapped to an existing protein structure. In terms of human
proteins, this resource can be used to annotate any part of a human protein
that can be aligned with a (not necessarily human) protein structure.

###  2.1 Annotating the human proteome

To enable predictions for the complete human proteome, it is necessary that
every individual amino acid is annotated. Since there are many (parts of)
proteins for which no structure information is available and therefore cannot
be annotated with the data present in the 3DM platform, full-length sequence
based alignments are a necessity to ensure full sequence coverage. Therefore,
Helix bases predictions not just on a single alignment - where possible
multiple alignments with different depths are used to provide optimal
predictive power. All positions in the human exome are covered by full length,
sequence-based multiple sequence alignments (MSAs). In addition, (PFAM)
domains are extracted and covered by deeper, more specific alignments. In
areas for which a mapping to a protein structure can be made, the deep
structure-based alignments (and associated data) from the 3DM platform are
used. In typical sequence based MSAs there is a very distinct tradeoff between
coverage, alignment depth and alignment quality. In contrast, by using
multiple MSAs the requirement for full sequence coverage is fulfilled, while
maximising alignment quality and depth at the same time.

Figure 1: Illustration of the principles behind using multiple alignments for
annotating a single human protein. Every single position is covered by the
full length alignment, while select areas are enhanced by using domain or
structure-based alignments.

####  2.1.1 Features

The features used in our tabular data models are hand curated and include a
large number of descriptors of evolutionary pressure that are extracted from
multiple sequence alignments. These include several indicators of evolutionary
pressure, acting on proteins as a whole, on networks of residues required for
specific functions and on individual positions. Metrics include measures of
conservation and change, entropy metrics, correlated behaviour and where
available, extensive structural data (e.g. solvent exposure, electrostatic
interactions, hydrogen bonding) that describe the structural aspects of
individual amino acids. The feature set is expanded with a set of indicators
of vulnerability on the gene level. Minor allele frequency is explicitly not
used as a feature, as this would introduce a large amount of circularity and
predictive performance on rare variants would be very limited as a result.
Predictions from other classifiers are not used as a feature, meaning that
Helix is built from the ground up without suffering from restrictions on the
training data.

####  2.1.2 Datasets and data preprocessing

The reference variant dataset used in this whitepaper was composed of variants
from ClinVar[3] and gnomAD[4] and a dataset maintained by the Dutch genome
diagnostic laboratories (VKGL)[5]. ClinVar variants were included if the
Clinvar review status was one star or higher, excluding variants with
conflicting interpretations. ClinVar variants with “Benign” and “Likely
benign” interpretations were included and labeled as benign variants, whereas
those variants with “Pathogenic” and “Likely pathogenic” interpretations were
included and labeled as pathogenic variants. gnomAD variants were selected and
labeled benign if the minor allele frequency exceeded 0.1%.

###  2.2 Machine learning

10-fold cross validation was used to train and analyze model performance. In
this process, the training data is divided into 10 partitions, where
repeatedly 9 partitions are used for training and one for validation. The
details of how this partitioning of the data takes place has important
consequences for model performance and generalizability to real-world
applications.

Most practical implementations of the 10-fold cross validation technique use a
random split, where the selection of variants that end up in the training or
validation set for each fold is performed completely random, irrespective of
gene or residue positions. As we have shown in previous research[6] this leads
to issues with circularity and artificially increases validation scores. In
order to mitigate inflated scores, we use a gene split strategy, where
variants present in training and validation sets are separated by gene; i.e.
all variants for any given gene are either present in the training set or in
the validation set. Variants present in the test sets (BRCA1[7], Clinical[8])
were excluded from the training/validation data and exclusively used for
testing.

####  2.2.1 Ensemble prediction

Using a variety of models to predict pathogenicity and combining their
predictions into an ensemble allows for better predictive performance than
using a single classifier[9]. This process is used to combine all our models,
while restricting the training and test set combination to prohibit any
contamination between datasets.

Models used in the ensemble prediction include state of the art Gradient
Boosting classifiers, custom neural networks that perform well on tabular data
and proprietary contextual language models. The latter are trained on a
multitude of subtasks to infuse the model with external data related to the
pathogenicity prediction problem.

##  3 Results

The primary metric that was used for performance assessment is Matthew’s
Correlation Coefficient. MCC is robust in the face of imbalanced datasets and
will report low scores for naive results. Comparisons are made with a number
of commonly used baseline, as well as newer predictors. The former includes
SIFT[10], CADD[11] and PolyPhen2[12], while the latter consists of REVEL[13]
and VEST4[14].

Figure 2 shows the Helix performance on the validation set. Helix predictions
were obtained by using 10-fold cross validation. Prediction scores were
obtained by testing on unseen genes, a more stringent way of testing compared
to the evaluation of the other tools listed. To minimize training bias effects
and ensure an as fair as possible comparison, variants were excluded from this
analysis if present in the HGMD[15] dataset or the PolyPhen HumDiv andHumVar
training sets.

Figure 2: Helix performance on the 10-fold cross validation set. This set
covers over 400.000 variants in proteins the model has not seen before.

Two datasets that more accurately reflect variants that might require
classification in a clinical setting were selected to independently assess and
compare Helix predictions (Figure 3 and Figure 4). Variants present in these
sets were excluded from the Helix training data. Overall, these datasets are
composed of variants that are clinically relevant, have been reviewed
recently, and where there is very limited overlap with training datasets of
existing predictive tools.

The Clinical dataset (n=1729) contains variants from the Deciphering
Developmental Disorders (DDD) study together with manually classified clinical
variants and variants identified in Amish individuals. Figure shows

Figure 3: Helix performance on Clinical dataset.

The BRCA1 dataset (n=1605) contains experimentally assayed BRCA1 variants
using saturation genome editing.

Figure 4: Helix performance on BRCA1 dataset.

###  3.1 Reports

Helix predictions can be accessed through the web application, where reports
for every single variant in the human exome can be found. Reports annotate
variants with structural information, literature and prediction data, with the
ultimate goal of providing domain experts the data and information they need
to make the best decisions.

Figure 5: A sample report produced by Helix.

####  3.1.1 Variant information

WIthin the reports information is presented about the individual variant.
Known associated phenotypes for the gene are described, together with relevant
information about the affected gene and links to external sources (e.g.
ClinVar, gnomAD, UCSC, Ensembl). Furthermore, information about the
conservation for the wildtype and variant type residue is presented, together
with structural information (Figure 5 A, B). Literature associated with the
variant is listed. Importantly, this is not limited to papers describing the
exact variant but extends to variants described in homologous proteins (Figure
5, C).

####  3.1.2 Prediction

The variant report provides details on various aspects of the predictions
(Figure 5, D). Agreement between the different sub-predictors in the ensemble
is shown, where a high agreement generally indicates more certainty in the
prediction.

A diverse set of metrics is used to present an intuitive score indicating the
quality of the data underlying the prediction. This score is an indication of
the amount of confidence one should place on a given prediction. No matter how
good the predictive model is, if the data that is provided to the model is of
limited quality (i.e. small MSAs, no protein structures available), one should
be careful in placing too much confidence in these predictions.

In addition, the report offers a visual indication of the factors that are
estimated to contribute most to the individual predictions. Features are
grouped into a few categories to simplify interpretation.

##  4 Discussion

Helix is built on a unique resource of integrated protein data, and represents
a state-of-the-art predictive tool that outperforms currently available
predictors by a large margin. However, Helix is by no means ‘finished’; many
challenges remain open for the problem of predicting the effects of missense
variants. In fact, despite offering impressive predictive performance, Helix
is just getting started. Novel challenges lie in problem domains such as
accurate modeling of protein-protein interactions and the integration of
pathway information to be able to assess downstream effects of variants within
the context of larger systems.

Helix offers an interactive platform that allows researchers to investigate
both the variants as the predictions in great detail. The variant reports, in
which variants and their associated predictions are presented in a coherent
fashion to add context and interpretability, offer an important first step to
help researchers make sense of all that data.

In the future, Helix will continue to integrate novel technologies and
performance increasing features. Helix is supported by a longstanding
bioinformatics company that is dedicated to its further development. As the AI
field continues to provide new solutions that touch the variant effect
prediction space, we envision including these technologies with even better
performance and interpretability as a result.

##  5 Acknowledgments

The research leading to these results has received funding from the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 634935 (BRIDGES), No 635595 (CarbaZymes) and No 685778 (VirusX).

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Deformation profiles and microscopic dynamics of complex fluids during
oscillatory shear experiments
Paolo Edera,a Matteo Brizioli,a Giuliano Zanchetta,a George Petekidis,b Fabio
Giavazzi,a and Roberto Cerbino∗a,c

Oscillatory shear tests are widely used in rheology to characterize the linear
and non-linear mechanical response of complex fluids, including the yielding
transition. There is an increasing urge to acquire detailed knowledge of the
deformation field that is effectively present across the sample during these
tests; at the same time, there is mounting evidence that the macroscopic
rheological response depends on the elusive microscopic behavior of the
material constituents. Here we employ a strain-controlled shear-cell with
transparent walls to visualize and quantify the dynamics of tracers embedded
in various cyclically sheared complex fluids, ranging from almost-ideal
elastic to yield stress fluids. For each sample, we use image correlation
processing to measure the macroscopic deformation field, and echo-Differential
Dynamic Microscopy to probe the microscopic irreversible sample dynamics in
reciprocal space; finally, we devise a simple scheme to spatially map the
rearrangements in the sheared sample, once again without tracking the tracers.
For the yield stress sample, we obtain a wave-vector dependent
characterization of shear-induced diffusion across the yielding transition,
which is accompanied by a three-order-of-magnitude speed-up of the dynamics
and by a transition from localized, intermittent rearrangements to a more
spatially homogeneous and temporally uniform activity. Our tracking free
approach is intrinsically multi-scale, can successfully discriminate between
different types of dynamics, and can be automated to minimize user
intervention. Applications are many, as it can be translated to other imaging
modes, including fluorescence, and can be used with sub-resolution tracers and
even without tracers, for samples that provide intrinsic optical contrast.

##

††footnotetext: a Dipartimento di Biotecnologie Mediche e Medicina
Traslazionale, Università degli Studi di Milano, via F.lli Cervi 93, 20090
Segrate, Italy††footnotetext: b FORTH/IESL and Department of Materials Science
and Technology, University of Crete, 71110 Heraklion, Greece††footnotetext: c
University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna,
Austria††footnotetext: † Electronic Supplementary Information (ESI) available:
[details of any supplementary information available should be included here].
See DOI: 10.1039/cXsm00000x/

##  1 Introduction

As compared with molecular hard solids - such as steel, wood or bones - soft
materials made of micron-sized constituents promptly deform with relatively
small mechanical loads: we easily spread butter on a slice of bread,
mindlessly distribute toothpaste on our teeth, effortlessly rub our faces with
cream, smoothly tickle our palate with a chocolate mousse, and readily pull
our cheeks with a little pinch. Beyond having practically ubiquitous
consequences in our lives, this softness opens up unique possibilities when
doing experimental research on soft matter. A paradigmatic example is passive
microrheology, in which the linear moduli of a soft material are probed by
embedded microscopic tracer particles that operate as micro-scale rheometers
powered by tiny thermal fluctuations 1, 2. Despite being a technique that has
had enormous success, passive microrheology is unable to provide complete
rheological information in all cases of interest. For instance, if the storage
modulus of the material is too large or if one is interested in the non-linear
rheological properties, it is often preferred to recur to active
microrheology, in which at least one of the embedded particles is driven by an
external force obtained, for instance, via optical or magnetic trapping 1, 3,
4, 5. Passive and active microrheology techniques are very powerful tools for
the characterization of soft materials, as they can access high frequencies
and enable a space-resolved study of the rheological properties of complex,
heterogeneous systems 3. However, the fact that the energy is injected at a
local, particle scale makes them unsuitable to study and predict the complex
failure modes observed in soft materials when they are mechanically perturbed
at the macroscopic scale.

More insightful information on the microscopic mechanisms underlying
plasticity in soft materials can be obtained by imposing a controlled
macroscopic deformation to the sample - for instance by displacing one or more
of the confining surfaces of a shear rheometer - while monitoring over time
the structural rearrangements occurring at the microscale. Rearrangements are
typically monitored by accurate tracking of embedded tracer particles,
performed in direct (rheo-microscopy or rheo-imaging) or in reciprocal space
(rheo-scattering), very often but not necessarily by optical means 6, 7, 8, 9.
These rearrangements may be partly or entirely reversible, with the
irreversible component becoming dominant for increasing shear strain across
the transition to non-linearity 10, 11, 12, 13, 14. This kind of experiments
provides thus very useful insights on how the microscopic structure and
dynamics determine the material rheology, which can be used to design enhanced
materials, e.g. with a longer lifetime, controlled relaxation, improved
ductility or emerging memory 15.

A variety of shear rheology tests is compatible with the simultaneous
characterization of the tracer particles’ motion. The latter can be generally
split into the sum of affine and non-affine displacements 16; the relative
importance of these two contributions depends on the type of test, and on the
scheme used to sample the particles’ configuration in space and time. For
instance, during Small/Medium/Large Amplitude Oscillatory Shear
(SAOS/MAOS/LAOS) rheology experiments, the so-called echo protocol enables one
sampling the microscopic configuration at each deformation cycle, by obtaining
a stroboscopic movie of the particles configurations either in direct or in
reciprocal space. In this way, the affine contribution is suppressed and one
can focus on the non-affine irreversible particle rearrangements.

Most of the initial experiments with the echo protocol were performed with
Diffusing Wave Spectroscopy (DWS)10, and Dynamic Light Scattering (DLS)17.
Being based on the collection of multiply scattered light, DWS is very
sensitive to a single, small length scale, whereas DLS is intrinsically a
multi-scale technique, at the expense of an increased experimental complexity
18. More recently, microscopy has also been employed to track particle motion
in oscillatory shear experiments with the echo protocol 12.

If one is interested in going beyond the study of irreversible non-affine
particle displacements typically obtained with the echo protocol, imaging and
microscopy offer an important advantage over scattering experiments: the
affine particle motion that characterizes the intra-cycle particle dynamics
can be readily removed from the particle trajectories, whereas more caution is
needed in scattering experiments 19. However, the information extracted from
scattering experiments is characterized by an enviable statistical robustness
of the obtained information. For these reasons, it would be interesting to
have both techniques available during oscillatory shear experiments.

A possible compromise is represented by Digital Fourier Microscopy (DFM)20, a
family of techniques based on collecting in direct space temporal sequences of
images of the sample, which are subsequently spatially Fourier transformed to
obtain scattering information in the reciprocal space as a function of the
wave-vector $\mathbf{q}$. The most popular DFM technique is Differential
Dynamic Microscopy (DDM)21, 22, 23, 24, which relies on subtracting images
acquired at different times, a feature that appears well suited to echo
experiments: when two images acquired at an integer multiple of the shear
oscillation period are subtracted, only the irreversible particle
displacements are captured and contribute to the final result. At the same
time, one can take advantage of the availability of the direct space movies to
remove the rigid translations associated to the affine intra-cycle particle
dynamics25, a step that may benefit from the large amount of image
registration algorithms available26.

The potential of DDM in combination with shear rheology experiments was
recently suggested in two companion articles by Aime and Cipelletti 6, 19, in
which the authors performed a critical comparison with DLS for probing shear-
induced rearrangements in Fourier space. In both cases, the intermediate
scattering functions for $\mathbf{q}$ parallel to the direction of the applied
shear (shear direction) are mostly dominated by the relaxation due to the
affine particle displacements, whereas in the perpendicular direction
(vorticity direction) only the non-affine particle rearrangements are probed.
The main advantage of DDM over DLS is that the limited spatial coherence of
illumination typical of bright-field DDM experiments results in a finite,
$q$-dependent depth-of-focus, which can be used to probe slices of the sample
perpendicular to the optical axis 22. This capability is surely appealing for
those cases where the deformation field deviates from a simple homogeneous
profile, such as in the presence of wall slip, shear banding, or edge fracture
27. All these different phenomena have the common effect of making the actual
deformation field inhomogeneous and difficult to predict, causing problems in
the interpretation of the results of both rheological and DLS experiments.
Indeed, while in a rheological experiment, the characterization of the
material constitutive equation for the shear stress $\sigma(\gamma)$, requires
the accurate knowledge of the actual shear strain $\gamma$ of the sample, in
DLS experiments, where the signal is the result of an average over the entire
scattering volume, it is almost impossible to discern the contributions coming
from different portions of a macroscopically inhomogeneous sample. Aime and
Cipelletti have clearly and convincingly showed that a proper DDM
characterization of the affine displacements is possible, providing that an
instrument calibration of the $q$-dependent depth-of-focus is performed 19.
However, the full potential of combining DDM analysis with an echo protocol,
where the effect of affine displacements is minimized, remains yet unexplored.

In this work, we perform oscillatory shear experiments with a strain-
controlled cell, characterized by a parallel-sliding-plates geometry. During
the sample oscillation, we monitor the dynamics of tracer particles by using
bright-field DDM in conjunction with the echo protocol (echo-DDM). The depth-
of-focus is tuned and optimized by varying the numerical aperture of the
microscope condenser so that, even if the deformation field is not homogeneous
across the whole gap, it is possible to selectively image narrow "slices" of
the sample, where the shear can be assumed to be approximately constant (see
Fig.1). We apply this optimization procedure to the imaging of various
samples, some of which presenting heterogeneity of the deformation field in
the direction of the velocity gradient (gradient direction). We perform a
preliminary accurate characterization of the deformation field across the
whole gap in standard samples with pure elastic solid or pure viscous liquid
response, retrieving the expected linear deformation profile. By applying the
same protocol to more complex soft matter samples (a viscoelastic fluid and a
yield stress material), we demonstrate that our approach can also effectively
probe deviations from such ideal profile, due for example to wall slip or
shear banding. Quantification of these effects is key to correctly relate the
microscopic irreversible dynamics to the local macroscopic deformation in echo
experiments.

Once obtained a reliable characterization of the local strain for all our
samples, we use echo-DDM to probe the presence of a shear-induced non-affine
rearrangements. The elastic solid, characterized by the absence of a true
irreversible non-affine dynamics, serves as a reference to estimate the
instrumental limits; as expected, Newtonian and Maxwell liquids do not exhibit
any appreciable variation in non-affine activity as a function of the shear
amplitude; by contrast, in the yield stress sample, we find intermittent,
small scale displacements of the tracers for small strain amplitudes, and
shear-induced diffusion 28, 29, 30, 31 for amplitudes larger than the one
corresponding to the crossover point of the viscoelastic moduli. All the
results obtained with echo-DDM are compared and found in agreement with
particle tracking (PT) analysis of the tracer motion. Finally, we propose a
simple yet robust scheme to draw activity maps highlighting the rearrangements
that occur in the sheared sample.

The manuscript is organized as follows: in the Materials and Methods section
we describe the tested samples (Sect 2.1), the experimental set-up (Sect.2.2),
and the experimental protocols enabling the characterization of the
macroscopic deformation field (Sect.2.3) and of the irreversible non-affine
dynamics (Sect.2.4). The results obtained in the characterization of the
deformation profile and of the shear-induced rearrangements are presented and
discussed in Sect.3.1 and Sect.3.2, respectively, before presenting some
concluding comments in Sect.4.

##  2 Materials and Methods

###  2.1 Samples

We test our experimental approach on samples that are representative of
different classes of materials: an elastomer (Sylgard), a Newtonian fluid
(silicon Oil), a Maxwell fluid (worm-like micelles, WLM), and a simple yield
stress fluid (Carbopol). All the samples in our study are fairly transparent
and are thus seeded with tracer particles of known size. However, seeding is
not needed if the sample already provides a sufficient optical contrast,
because DDM can capture the dynamics of a variety of samples including
molecular mixtures 32, 33 and liquid crystals 34.

Sylgard 184 (Dow Corning) is as a two-component system made of a silicon oil
and a curing agent. A small fraction ($\phi\sim 0.1\%$) of sterically
stabilized PMMA particles of diameter $260$ nm are dispersed in the silicon
oil base before adding the curing agent. The sample is then mixed thoroughly,
centrifuged to get rid of bubbles, and finally loaded in the shear-cell before
the completion of the curing process, which takes about 48 h at room
temperature. The so-obtained sample is expected to behave like a perfectly
homogeneous elastic material over a wide range of shear strains35. The silicon
oil that we use is the viscous component described above in the preparation of
Sylgard. This sample behaves as a Newtonian viscous fluid with a shear
viscosity $\eta\simeq 5$ Pa s 36. Worm-like micelles are obtained by
dispersing Hexadecylpyridinium chloride monohydrate (CpyCl) surfactant in
water with Sodium Salicylate (NaSal) as a counter-ion37, 38. Both components
have been purchased from Sigma-Aldrich as powders. CpyCl is dispersed in
Milliq water at a concentration 200 mM and taken to $40\ ^{\circ}C$ for 30
minutes, until it turns to a clear solution. NaSal is dispersed in Milliq
water at a concentration of 120 mM. The two dispersions are slowly mixed in a
1:1 ratio and left on a rotating mixer overnight. The final molarities are 100
mM and 60 mM for CpyCl and NaSal, respectively. Polystyrene particles of 2
$\mu$m diameter (PS2, Microparticles GmbH) are dispersed at a volume fraction
$\phi=0.05\%$. The dispersion is initially mixed manually with a spatula and
left on a rotating mixer for one night. Carbopol 971 P NF has been purchased
from Lubrizol as a single component powder. The powder is dispersed in MilliQ
water to obtain a concentrated starting solution with concentration $c=5\%$
wt. The obtained dispersion has an acidic pH, that is neutralized by adding
few drops of NaOH 10 M to 10 ml of concentrated dispersion, while gently
stirring. During the process, the pH is controlled. After extensive mixing,
the neutralized dispersion is then diluted to obtain the sample at the desired
concentration $c=0.5\%$ wt. Eventually, PS2 tracer particles are dispersed at
a volume concentration $\phi=0.05\%$. Before the experiment, all samples are
centrifuged to get rid of the air bubbles.

###  2.2 Experimental Setup

####  2.2.1 Shear-cell

The samples are loaded between the two horizontal parallel plates of a
commercial strain-controlled shear-cell. With this shear-cell, a modified
version of the one described in Ref. 39 to enable bright-field imaging, we can
impose an oscillatory displacement of the top plate

$\Delta\mathbf{X}_{0}(t)=\text{ }A_{0}\sin{(2\pi\textrm{f}t)}\hat{u}_{x},$ (1)

while the bottom plate remains stationary. In our experiments, the oscillation
frequency is kept constant ($\textrm{f}=1$ Hz), while the amplitude $A_{0}$ is
varied in the range $[0.01-1]$ mm. The displacement of the top plate
identifies the shear direction, corresponding to the orientation of the
$x$-axis in Fig.1. The height of gap between the plates is 1 mm for Sylgard,
250 $\mu$m for Worm-like micelles, and 500 $\mu$m for the other samples. The
plates are standard glass slides whose surface was sandblasted to reduce wall
slip, with the exception of small a circular window of diameter 4 mm, which is
left clear to enable optical access to the sample. The orientation of the
plates is controlled by micrometer screws. The parallelism of the two plates
and the alignment of the moving plate along the shear direction are essential
to produce a simple shear deformation. The proper alignment of the top plate
is assessed by observing the plate surface under the microscope while it is
displaced along the shear direction and checking whether it remains in sharp
focus. As described in more detail in ESI, the angle between the two plates is
measured by means of a custom interferometric system, coupled to the rear port
of the microscope. The alignment accuracy than can be achieved is within
$\delta\theta\sim 10^{-3}\ rad$.

Due to a memory buffer register limitation, with the current version of the
shear-cell control software the length of an oscillatory experiment is limited
to $500$ cycles. Moreover, the shear-cell does not provide a real-time output
signal proportional to the position of the moving plate, which could have been
used to trigger a synchronized camera acquisition. Synchronization is a
crucial issue in our set-up and it will be discussed in detail in Sec. 2.4.3.

####  2.2.2 Microscope

The shear-cell described in the previous section is mounted on an inverted
microscope (Nikon Eclipse Ti-S), equipped with a 20x 0.45 NA objective. The
working distance of the objective (WD= 8 mm) is large enough to enable
scanning the whole gap between the two plates confining the sample. As
discussed also in the the ESI, the vertical resolution in the reconstructed
deformation field is determined by the depth-of-focus of the imaging system.
In our setup, we minimize the depth-of-focus (to about 20 $\mu$m, see ESI for
details) by keeping the condenser diaphragm almost completely open (LWD,
NA=0.52).

###  2.3 Deformation field

It is known that in several conditions of interest, the deformation field
within a sheared sample may be non-homogeneous even when a uniform stress is
applied 40. Two examples of deviation from homogeneity are shear banding and
wall slip. These phenomena affect a wide variety of samples 41 42 43 and while
they have an indisputable effect on the mechanical response and thus on the
interpretation of rheological data, it is not easy to pinpoint their presence
from purely mechanical measurements. In the following, we provide a simplified
description of these effects in the case of the parallel sliding plates
geometry, which is the one used in our experiments and is schematically
represented in Fig.1. A key advantage of this geometry, compared to other
common configurations like rotating plate-plate or rotating cone-plate, is
that it allows imposing a uniform strain at the confining surfaces while
ensuring easy optical access to the sample 6, 13, 12.

Ideally, when a uniform displacement $\Delta\mathbf{X}_{0}=A_{0}\hat{u}_{x}$
is imposed on the top plate, while the bottom one is fixed, an affine
displacement field is produced in the sample

$\Delta\mathbf{X}(z)=z\ \gamma_{0}\ \hat{u}_{x},$ (2)

where $z$ is the vertical coordinate (corresponding to the velocity gradient
direction), and has the origin of the axis at the bottom plate; $\hat{u}_{x}$
in the unit vector oriented along the $x$ (shear) direction and the ratio
$\gamma_{0}=A_{0}/h$ between the imposed displacement at the moving plate and
the gap width $h$ is the imposed shear strain. The displacement is uniform
over each horizontal plane, with no components along other directions.

Fig. 1: Effect of the depth-of-focus in optical microscopy. (a) Schematic
representation of a sheared sample in sliding parallel plates geometry, in the
shear-gradient ($x$-$z$) plane. Green arrows represent local displacements
with respect to the rest position, tracer particles are color-coded (from
black to white) according to their vertical position. The dotted line
corresponds to the intersection with the object plane of the imaging system,
whose optical axis coincides with the velocity gradient ($z$) direction. The
yellow shaded region, whose width coincides with the depth-of-focus $L_{f}$,
identifies the portion of the sample contributing to the image (c). Panel (b)
depicts the same sheared sample shown in panel (a), observed with a larger
depth-of-focus. The corresponding image, shown in panel (d), features a larger
number of particles and is characterized by a larger dispersion in the
particle displacements along the shear direction.

The corresponding strain, which is given by the spatial gradient of the
displacement field ${\nabla}\big{(}\Delta\mathbf{X}\big{)}$, is perfectly
homogeneous and the only non-zero component of the strain tensor corresponds
to the shear strain

$\mathbf{\gamma}=\frac{\partial{(\Delta\mathbf{X}\cdot\hat{u}_{x}})}{\partial
z}=\gamma_{0}\ .$ (3)

Deviations from this ideal case occur for example in case of wall slip, when
the adhesion between the sample and the confining plates is not sufficient to
prevent their relative motion. When wall slip occurs, in the absence of other
effects, like shear-banding, Eq.2 modifies to

$\Delta\mathbf{X}(z)=(z-z_{0})\ \gamma_{0}c\ \hat{u}_{x},$ (4)

where $z_{0}$ is the coordinate of the horizontal plane (outside the sample)
on which the extrapolated displacement profile would vanish, and $c<1$. In
this scenario, the strain field is still homogeneous, even if with the actual
shear strain $\gamma$ is smaller than the imposed one
$\gamma=c\gamma_{0}<\gamma_{0}$. Since the actual deformation is obtained
multiplying the imposed one by $c$, we define $c$ as the deformation-loss-
factor.
Another important example of deviation from homogeneity of the deformation
field is shear banding, corresponding to the separation of the sample into
macroscopic regions (bands) with different local shear strains 43, 44. As an
example, one can consider the simplistic case of a system separated in two
bands, in the absence of wall slip, which is described by the displacement
profile

$\Delta\mathbf{X}(z)=\begin{cases}\ z\ \gamma_{1}\ \hat{u}_{x}&\text{for
}z<z_{1}\\\ \Big{[}\ (z-z_{1})\ \gamma_{2}\ +z_{1}\
\gamma_{1}\Big{]}\hat{u}_{x}&\text{for }z>z_{1}\end{cases},$ (5)

where $\gamma_{1}$ and $\gamma_{2}$ are the local shear strains in the two
bands. Even in this simple case where just two bands are formed, many profiles
are compatible with a given imposed deformation $\gamma_{0}$, corresponding to
all possible choices of the parameters $z_{1}$ and $\gamma_{1}$ satisfying the
constraint $z_{1}\gamma_{1}\leq h\gamma_{0}$. Under these conditions, even if
the average shear strain is bound to be $\gamma_{0}$, the local deformation
can be very different, assuming in principle any value from $0$ to infinity,
which reinforces the importance of mapping the local strain.

In our bright-field microscopy experiments, we exploit the optical sectioning
capability resulting from the partial coherence of illumination to obtain an
accurate characterization of the displacement $\Delta\mathbf{X}(z)$ at
different heights $z$, and thus a faithful reconstruction of the actual shear
strain profile $\mathbf{\gamma}(z)$ within the sample.

####  2.3.1 Strain field under oscillatory shear

We impose an oscillatory displacement of the top plate described by Eq.1, with
imposed amplitude $A_{0}$ and shear strain $\gamma_{0}=A_{0}/h$. We then
perform observations of the sheared sample at different vertical positions
$z_{i}\in[0,h]$. For each height $z_{i}$, a stack of 200 images is acquired
with a frame rate $1/\delta t=100$ fps (see ESI).

We calculate the spatial cross-correlation function $c(\mathbf{r},t)$ of each
pair of consecutive frames collected at a given height

$c(\mathbf{r},t)=\Sigma_{\mathbf{x}}I(\mathbf{x},t+\delta t)\cdot
I(\mathbf{x}+\mathbf{r},t).$ (6)

The position $\mathbf{\bar{r}}(t)$ of the peak of $c(\mathbf{r},t)$ provides
an estimate for the rigid displacement occurred between the two images, taken
at times $t$ and $t+\delta t$, respectively. By summing up all contributions
from consecutive image pairs, we reconstruct the full displacement profile
$\Delta\mathbf{X}(z_{i},t)=\sum_{t^{\prime}\leq
t}\mathbf{\bar{r}}(t^{\prime})$ at the considered vertical position, as
schematically shown in Fig.2(a). Fitting of this profile to a sinusoidal
function enables extracting the corresponding amplitude $A(z_{i})$ (see
Fig.2(b)). We then estimate the local shear strain by numerically evaluating
the partial derivative of the displacement field along the vertical direction
$\gamma(z)=\partial{A_{0}(z)}/\partial z$. During an oscillatory shear
experiment, we perform repeated scans at different times to monitor a possible
time evolution of the deformation profile (further detains in the ESI).

Fig. 2: Determination of the local deformation field. (a) Displacement field
$\Delta\mathbf{X}(z_{i},t)=A(z_{i})\sin{(2\pi\textrm{f}t)}\hat{u}_{x}$ at
different vertical coordinates $z_{i}$ in a simulated oscillatory strain
experiment in the sliding parallel plates geometry in the absence of wall slip
and shear banding. From a sinusoidal fit of the displacement field at each
given height $z_{i}$ one can obtain the corresponding amplitude $A(z_{i})$
(b). The local shear strain can be then estimated as
$\gamma(z)=\partial_{z}A(z)$, which becomes z-independent for the example
considered in a).

###  2.4 Microscopic non-affine displacements

The emergence of both reversible and irreversible microscopic non-affine
displacements has been reported in simulations and experiments on sheared soft
solids, and it has been related to the transition from linear to non-linear
mechanical response, as well as to their yielding or flow behavior 45, 17, 31.
In such cases, even if the macroscopic displacement field $\Delta\mathbf{X}$
remains homogeneous (see Eq.2), the actual displacement field:

$\Delta\mathbf{x}(x,y,{z},t)=\Delta\mathbf{X}({z},t)+\Delta\mathbf{x}_{N.A.}(x,y,{z},t).$
(7)

includes also a non-affine contribution $\Delta\mathbf{x}_{N.A.}$ due to
rearrangements occurring on the scale of the elementary building blocks of the
material. In general, $\Delta\mathbf{x}_{N.A.}$ is inhomogeneous in the shear-
vorticity plane, and it is not necessarily oriented along the shear direction.
In the rest of Sect.2, after briefly reviewing the working principle of DDM,
we discuss how it can be adapted to capture and quantify irreversible non-
affine displacements in a periodically sheared sample.

####  2.4.1 Differential Dynamic Microscopy

The first step of the DDM analysis is the calculation of the difference
between two images acquired at times $t$ and $t+\Delta t$, respectively (see
Fig.3a-c). As proposed in Ref. 46, we then multiply the so-obtained image
difference with a windowing function before calculating its Fourier power
spectrum and successively average over all the results obtained with the same
time delay $\Delta t$ but different initial times $t$. The result of this
average is the image structure function (Fig.3d)

$D(\mathbf{q},\Delta t)=\langle|\hat{I}(\mathbf{q},t+\Delta
t)-\hat{I}(\mathbf{q},t)|^{2}\rangle,$ (8)

where $\hat{I}(\mathbf{q},t)=\text{FFT}\\{I(\mathbf{x},t)\\}$ is the Fast
Fourier Transform of the image at time $t$, $\mathbf{q}$ is the wavevector,
and the symbol $\langle\cdot\rangle$ indicates an average over the initial
time $t$. This calculation is iterated for all the time delays $\Delta t$ that
are needed to properly follow the sample relaxation.

Fig. 3: Differential Dynamic Microscopy. (a-c), from left to right: simulated
images of Brownian particles at different times $I(\mathbf{x},t)$ and
$I(\mathbf{x},t+\Delta t)$, and their difference. (d) Two-dimensional dynamic
structure function $D(\mathbf{q},\Delta t)$. (e) Azimuthally averaged
intermediate scattering function $f_{i}(q,\Delta t)$ obtained at different
wave vectors $q$. Fitting of the ISFs with a suitable model (continuous curves
in panel (e)) enables extracting a $q$-dependent relaxation rate $\Gamma(q)$
(panel (f)).

The interpretation of the results is based on the fact that
$D(\mathbf{q},\Delta t)$ can be written in terms of the real part
$f(\mathbf{q},\Delta t)$ of the intermediate scattering function (ISF), which
is the quantity encoding information on the relaxation of the density Fourier
modes, and whose squared amplitude is typically measured with DLS experiments:
21

$D(\mathbf{q},\Delta t)=a(\mathbf{q})[1-f(\mathbf{q},\Delta
t)]+b(\mathbf{q}).$ (9)

The amplitude term $a(\mathbf{q})$ is determined by the scattering properties
of the sample and by the microscope transfer function, whereas the additive
term $b(\mathbf{q})$ accounts for the noise in the detection chain 20. To
extract the ISF from the image structure function, $a(\mathbf{q})$ and
$b(\mathbf{q})$ need to be determined. Since $f(\mathbf{q},\Delta t\rightarrow
0)=1$, $b(\mathbf{q})$ could be in principle estimated as the limit for small
$\Delta t$ of $D(\mathbf{q},\Delta t)$. In practice, we fit a quadratic
function to the image structure function over a small-time interval close to
the origin (typically including the first five delay times) and we estimate
$b(\mathbf{q})$ as the intercept of the best fitting curve 47. Similarly,
since $f(\mathbf{q},\Delta t\rightarrow\infty)=0$, one could think to extract
$a(\mathbf{q})$ from the long-time plateau of $D(\mathbf{q},\Delta t)$.
Unfortunately, because of the finite acquisition time set by the experimental
constraints, in many cases we are not able to capture the full relaxation of
$f(\mathbf{q},\Delta t)$ for all wavevectors. Consequently, we decided to
adopt a different strategy 47. We identify a background image
$I_{0}(\mathbf{x})$ incorporating all the static, sample-independent
contributions to the images (see ESI for detail) and we estimate the amplitude
$a(\mathbf{q})$ directly from the power spectrum of the raw images, corrected
for the background
$a(\mathbf{q})=2\langle|\hat{I}(\mathbf{q},t)-\hat{I}_{0}(\mathbf{q})|^{2}\rangle-b(\mathbf{q})$.

Once $a(\mathbf{q})$ and $b(\mathbf{q})$ are known, Eq.9 can be inverted to
obtain the ISF. The sample dynamics can be then extracted, for example, by
assuming a specific model for the statistics of the particle displacements and
fitting it to the ISF. For a dispersion of identical, non-interacting Brownian
particles in a Newtonian fluid, the ISF is azimuthally symmetric and is given
by $f(q,\Delta t)=e^{-D_{0}q^{2}\Delta t}$, where $D_{0}$ is the diffusion
coefficient of the particles. In this case, $D_{0}$ can be estimated as a
coefficient of the $q$-dependent relaxation rate $\Gamma(q)=D_{0}q^{2}$,
obtained by fitting an exponential function $e^{-\Gamma(q)\Delta t}$ to the
ISF for different wavevectors (see Fig.3(e-f)). More generally, at least in
the case of identical, non-interacting particles, the IFS can be expressed in
terms of the Fourier transform of the probability distribution function of
particle displacements $P(\mathbf{r},\Delta t)$, i.e. the self part of the van
Hove function 48

$f(\mathbf{q},\Delta t)=\int d^{2}\mathbf{r}P(\mathbf{r},\Delta
t)e^{-j\mathbf{q}\cdot\mathbf{r}}.$ (10)

If the dynamics is anisotropic, it can be worth considering separately the
behavior of the ISF along specific directions in Fourier space. In this case,
the azimuthal average of $f(\mathbf{q},\Delta t)$ in the $\mathbf{q}$-plane
can be restricted to narrow angular sectors of aperture $\Delta\theta$
oriented along the direction of interest34. Under the same conditions under
which Eq.10 holds, we can write

$\langle|\Delta x(\Delta t)|^{2}\rangle=-\frac{\partial^{2}}{\partial
q_{x}^{2}}f(\mathbf{q},\Delta t)|_{\mathbf{q}=0},$ (11)

which enables estimating the mean square displacement along a selected
direction (say, the $x$-axis) directly from the low-$q$ behavior of the ISF
along the corresponding direction in the Fourier space.

####  2.4.2 Echo-Differential Dynamic Microscopy (Echo-DDM)

In the presence of an imposed oscillatory deformation, an effective strategy
to single out shear-induced irreversible rearrangements is to adopt a
stroboscopic or echo approach, where the microscopic configuration of the
system is probed at a sampling frequency that matches the oscillation
frequency. This approach to shear-induced dynamics has been previously
exploited in combination with both light scattering and microscopy10, 49, 17,
12. In the case of microscopy, one considers sequences of echo images
$I_{n}(\mathbf{x})$ taken at times $t_{n}=t_{0}+nT$, where $t_{0}$ is a time
offset, $T$ is the oscillation period and $n$ is an integer number comprised
between 1 and the total number of cycles $N$. This choice ensures that each
frame is captured in correspondence to the same realization of the macroscopic
displacement field and, as a consequence, any difference between the images
can be attributed to the occurrence of non-affine, irreversible displacements
within the sample. Since DDM is a very sensitive probe of the microscopic
dynamics, one could think to use it directly to extract the effective, shear-
induced dynamics present in the echo images. However, testing this idea
requires first minimizing a number of potential artifacts which, if not
adequately accounted for, can introduce spurious contributions to the echo
dynamics. These artifacts can arise for example from drifts or flows due to an
imperfect confinement of the sample or from a mismatch between the oscillation
frequency and the sampling frequency. In the following section, we present a
pre-processing procedure aimed at minimizing such effects and enabling robust
DDM analysis.

####  2.4.3 Correction of real and apparent drifts

As observed in previous work 12, the imperfect synchronization between
deformation and acquisition can lead to an apparent, phase-dependent
displacement in the echo images, possibly impairing the result of the
experiment (see Fig.4).

Fig. 4: Apparent displacement due to imperfect synchronization between
deformation and acquisition. In each panel, the continuous blue line
represents the displacement along the shear direction of the observed plane as
a function of time $\Delta X(t)=\sin{(2\pi\textrm{f}t)}$, the circles indicate
the position at the sampling times $t_{n}=(\varphi_{i}/2\pi+n)/\nu$, where
$\nu=\textrm{f}+\Delta\nu$ is the sampling frequency and $n$ is a positive
integer number. Circles of different colors correspond to different initial
phases $\varphi_{i}$. In each panel, a different value of the frequency
mismatch $\Delta\nu$ is considered $\Delta\nu/\textrm{f}=0,1\%$, and $10\%$,
respectively. If $\Delta\nu=0$ each sampling series always captures the system
in the same position. When $\Delta\nu\neq 0$ instead, an apparent displacement
is observed in each sampling series, the velocity of the apparent displacement
depending on both $\Delta\nu$ and on the initial phase $\varphi_{i}$.

A relatively simple solution for obtaining a good synchronization would be to
trigger the camera acquisition with a signal obtained from the shear-cell,
proportional to the displacement imposed on the upper plate. Unfortunately,
our shear-cell does not provide such a real-time output signal 39, which
forced us to act both at the instrumental level and at the image processing
level. The proposed solution encompasses the following steps: accurate
characterization of the oscillation frequency of the shear-cell, adoption of
the best-matching image acquisition frequency, measurement of the residual
drift in the collected images, and correction of the residual drift via image
registration. We now describe these corrective steps.

Characterization of the oscillation frequency. In our set-up, the shear-cell
and the camera are controlled by two different computers. The first step is to
characterize the frequency of the shear-cell in terms of the clock of the
camera’s computer. We observed that the actual frequency of oscillation is
systematically different from the nominal one in an amplitude-dependent
fashion, so that the actual frequency f turns out to be a function of the two
input parameters $\textrm{f}(\textrm{f}_{0},A_{0})$. This function for
$\textrm{f}_{0}=1\ Hz$, $A_{0}\ \in(10\ \mu m,1\ mm)$ is reported in ESI
(Fig.S2). The relative difference between the nominal and actual oscillation
frequency $(\textrm{f}-\textrm{f}_{0})/\textrm{f}_{0}$ is typically or the
order of $0.1\%$. This difference turns out to be significant, especially for
long experiments lasting some hundreds of oscillations. The experiment-to-
experiment variability is estimated with the root mean square deviation
measured for fixed input parameters $(\textrm{f}_{0},A_{0})$,
$\bar{\delta}\textrm{f}\sim 1\cdot 10^{-5}\text{Hz}$ (details of the frequency
characterization are given in ESI).

Frequency matching: To increase the statistics while keeping the duration of
the experiments fixed, it is convenient to acquire more than one image per
period. Thus the acquisition frequency $\nu$ has to meet the condition
$\nu=m_{\varphi}\textrm{f}$. The frequency $\nu$ is set indirectly by setting
the delay time between two consecutive images $\Delta t_{0}$, that can be
determined with a precision of $0.01\ ms$. This introduces a rounding error
$\delta\nu_{round}/\textrm{f}\sim 10^{-5}$ of the same order of magnitude as
the intrinsic shear frequency variability (see ESI for details). The parameter
$m_{\varphi}$ has been chosen in order to minimize the rounding error. Since
different shear amplitudes lead to a different actual frequency, the number of
images per period may vary between experiments at different amplitude, taking
values in the interval $m_{\varphi}\in[4,8]$.

Residual drift and registration: The residual frequency mismatch results in a
phase-dependent apparent drift (Fig.4) that is quite evident in long
acquisitions. By means of the same image cross-correlation algorithm used for
the determination of the displacement field (see Sect. 2.3.1), we characterize
the residual drift, and we remove its effect with a custom sub-pixel
registration algorithm implemented in the Fourier space and described in
detail in ESI. In the case of fluid samples, actual drifts, corresponding to
macroscopic flows, could also occur as a consequence of the loose lateral
confinement. With the aforementioned registration algorithm, we can
effectively remove these global contributions as well.

####  2.4.4 Summary of the echo strategy

In summary, our optimized strategy consists in acquiring a timed image
sequence $I(\mathbf{x},t)$ including exactly $m_{\varphi}$ images per period.
For each experiment at a given amplitude, we have $m_{\varphi}$ different
stacks of echo images, that we label with their initial phase
$\varphi_{m}=2\pi m/m_{\varphi}$, $m=0,1,...,(m_{\varphi}-1)$

$I_{n}(\mathbf{x}|\varphi_{m})=I(\mathbf{x},(n+\varphi_{m}/2\pi)T).$ (12)

Registration and computation of the dynamic structure function are then
sequentially performed on each stack separately by working at constant $m$ and
calculating the image structure function for various delay cycles indexed by
$n$. Finally, the dynamic structure functions obtained for different initial
phases are averaged to give a single dynamic structure function
$D(\mathbf{q},\Delta t)$, representative of the average response of the sample
within the cycle.

####  2.4.5 Echo-Particle Tracking

As a cross-check of the results obtained with echo-DDM, we also perform a
direct space analysis of the same image sequences, at least in those cases
where the size of the tracer particles is large enough to enable them to be
tracked individually. For each acquisition, echo image sequences are extracted
and pre-processed as described in Sect. 2.4.4. Images in each echo sequence
are convoluted with a Wiener kernel (with standard deviation of $2\mu$m) to
reduce noise and then analyzed with a customized version of the MATLAB
particle-tracking code developed by the Kilfoil group at the University of
Massachusetts 50 and available at https://github.com/dsseara/microrheology. We
extract the trajectories $[x^{(i)}(t),y^{(i)}(t)]$ of all particles (typically
$\sim 100$) in the field of view, and we evaluate the corresponding
displacement probability distribution functions (PDF) $P_{s}(\Delta x,\Delta
t)$ and $P_{v}(\Delta y,\Delta t)$, along the shear ($x$) and the vorticity
($y$) direction, respectively

$P_{s}(\Delta x,\Delta t)=\langle\delta[\Delta x-(x^{(i)}(t+\Delta
t)-x^{(i)}(t))]\rangle,$ $P_{v}(\Delta y,\Delta t)=\langle\delta[\Delta
y-(y^{(i)}(t+\Delta t)-y^{(i)}(t))]\rangle.$

In the above expressions, the average is performed over all the initial times
$t$, all the tracked particles $i$ and all the echo sequences corresponding to
different initial phases. From the same data, we also calculate the mean
square particle displacement (MSD) along the shear and the vorticity
direction, respectively

$MSD_{s}(\Delta t)=\langle|x^{(i)}(t+\Delta t)-x^{(i)}(t)|^{2}\rangle,$
$MSD_{v}(\Delta t)=\langle|y^{(i)}(t+\Delta t)-y^{(i)}(t)|^{2}\rangle.$

##  3 Results

###  3.1 Measurement of the deformation profiles

For each of the four samples, we perform a detailed characterization of the
deformation profile which is produced when an oscillatory shear deformation of
different amplitude is imposed.

####  3.1.1 Carbopol

In Fig.5(a), the results of vertical scans of the Carbopol sample for
different shear amplitudes in the interval $\gamma_{0}\in[10,100]\%$ are
shown. For each amplitude (labeled by a different color) two successive full
scans are performed. The excellent overlap between these pairs of measurements
indicates that the observed deformation profiles do not significantly evolve
over time and do not depend on the shear history. For all the imposed shear
amplitudes, the displacement profile is almost perfectly linear. Fitting of
each profile with Eq.4 enables estimating the corresponding deformation-loss-
factor $c=\gamma/\gamma_{0}$, which quantifies the discrepancy between the
imposed strain amplitude $\gamma_{0}$ and the measured one $\gamma$. In the
case of Carbopol, the deformation-loss-factor is almost constant $c\simeq 0.9$
(see Fig.5(c)), as can also be seen from the collapse of all the data onto a
single master curve (panel (b)), which is obtained by normalizing each
displacement profile by the displacement $A_{0}$ imposed on the top plate of
the shear-cell. The fact that the displacement profiles are linear for all the
investigated strain amplitudes indicates that the material response remains
homogeneous even in the transition region. This is a non-trivial result, as
the occurrence of inhomogeneous weakening, accompanied by the formation of
shear bands, is often observed for both hard materials and for other yield
stress fluids when driven out of the linear regime 43, 40, 51.

Fig. 5: Direct measurement of the deformation field. (a) Displacement profile
$A(z)$ in the Carbopol sample along the velocity gradient ($z$) direction for
different applied strain amplitudes $\gamma_{0}\in[10,100]\%$. Symbols of
different colors represent experimental data obtained at a different imposed
shear amplitude, while lines are best fitting curves with the model of Eq.4.
For each amplitude, two independent, steady-state measurements are performed.
In (b) the same profiles are rescaled with the amplitude $A_{0}$ of the
imposed displacement. (c) Deformation-loss-factor for different samples as a
function of the imposed shear strain $\gamma_{0}$. Symbols correspond to
Carbopol data. The color code is the same of panels (a-b).

####  3.1.2 Sylgard

For this sample, representative of an ideal elastomer, with nearly Hookean
solid response, we performed three successive scans for each amplitude in the
range $\gamma_{0}\in[1\%,70\%]$. Compared to Carbopol, the measured shear
strain is in even better agreement with the imposed one ($c\simeq 0.98)$, as
shown in Fig.5(c). The absence of wall slip is indicative of the strong
adhesion between the gel sample and the confining plates, which was obtained
by maintaining the sample in contact with the glass slides during the curing
process.

####  3.1.3 Silicon oil

Very similar results are also obtained for the Newtonian sample. The
reconstructed deformation profile is homogeneous, and $c\simeq 0.98$ (see
Fig.5(c)).

####  3.1.4 Worm-like micelles

Experiments performed with this sample reveal the presence of significant wall
slip, which becomes increasingly stronger at large shear amplitudes, as
indicated by the marked decrease of the deformation loss-factor as a function
of the imposed deformation amplitude (yellow line in Fig.5(c)). Interestingly,
for each amplitude, the reconstructed deformation profile is still
homogeneous, characterized by a well-defined shear strain $\gamma<\gamma_{0}$
(see ESI for more details).

###  3.2 Echo-DDM

Once a detailed characterization of the sample deformation field is available,
we proceed to quantify the shear-induced irreversible dynamics with the echo-
DDM approach introduced in Sect. 2.4.2. As pointed out in Sect. 2.4.3, a key
requirement for the accurate determination of the non-affine, irreversible
tracer displacements in the sample is the absence of spurious contributions
(such as drifts or external vibrations) that could mask the genuine signal.
Small imperfections or asymmetries in the setup may lead to a stress unbalance
which, in the absence of an effective lateral confinement, can induce a net,
albeit typically very small, sample flow. The relevance of this effect is
found to crucially depend on the rheological properties of the material and it
is particularly severe for low or no yield stress samples (see ESI for more
details). As shear-induced flows are often not perfectly homogeneous, they
cannot be completely removed with the registration procedure described in
Sect. 2.4.3. As a matter of fact, for both the silicon oil and the WLM
samples, the residual effect of shear-induced flows are found be large enough
to dominate the microscopic dynamics (see ESI ), suggesting that our shear
geometry is not optimal to measure irreversible tracer displacements in fluid-
like samples. We stress that the effect under discussion has no impact on the
results for the deformation profiles reported in Sect. 3.1, as the
displacement per period associated with shear-induced flows is typically at
least two orders of magnitude smaller than the amplitude of the affine
displacement (see Fig.S5).

We thus focus on the two samples that display a solid-like behavior at rest:
Sylgard and Carbopol. For these samples, all the echo acquisitions are
performed at the center of the cell: $z=h/2$. This allows to neglect boundary
effects (including the small wall slip) since the distance between the focal
plane and the sample edges is way larger than the depth-of-focus $h/2\gg
L_{f}$.

####  3.2.1 Sylgard

This sample displays an elastic, reversible mechanical response over the whole
explored range of shear amplitudes (up to $\gamma=80\%$) 35 and, as
demonstrated above, it is firmly attached to the confining plates. Based on
these premises, we do not expect to observe any genuine echo-DDM signal
associated to irreversible non-affine displacements. As a consequence, these
experiments can be thought of as a sort of calibration of the instrumental
"background noise". For Sylgard, the ISFs obtained from an echo-DDM experiment
with imposed shear amplitude $\gamma_{0}=75\%$ are shown in Fig.6.

Fig. 6: Echo-DDM analysis of Sylgard.  (a,b) Representative ISFs for different
$q$-values in the range $[0.15-2.2]\mu m^{-1}$, along the shear and the
vorticity directions, respectively. In the inset of panel (a), the same data
shown in the main panel are plotted as a function of the rescaled time delay
$q\Delta t$, displaying a nice collapse onto a single master curve (continuous
red line). The same curve is also reported for comparison in the inset of
panel (b), where the data shown in the corresponding panel are plotted as a
function of $q\Delta t$, collapsing also in this case onto a single curve
(continuous blue line).

As expected, the dynamics, along both the shear and the vorticity direction,
is extremely slow and only the very first part of the decay of the ISFs can be
observed. Nevertheless, when the ISFs are plotted as function of the rescaled
delay time $q\Delta t$, a nice collapse of all the curves is observed (insets
of Fig.6), suggesting the presence of a ballistic-like relaxation mechanism
20. Assuming an exponential-like decay $f(q,\Delta t)\simeq e^{-vq\Delta t}$,
we can get a rough, order-of-magnitude estimate of the characteristic
associated velocity along the shear direction $v_{s}\sim 3\cdot 10^{-4}$
$\mu$m/s.

We attribute this ultra-slow dynamics to the presence of an apparent relative
motion of the different planes contributing to the image, which is due to the
combination of imperfect synchronization and axial velocity gradient (see ESI
for details). Consistently with this interpretation, we found that $v_{b}$
compares fairly well with the characteristic velocity spread
$2\pi\delta\nu\gamma_{0}L_{f}\sim 5\cdot 10^{-4}$ expected in the presence of
a synchronization error $\delta\nu\sim 10^{-5}$ Hz (see Sect.2.4.3) and a
finite depth-of-focus $L_{f}\sim 10$ $\mu$m. Moreover, we observe that the
characteristic velocity related with the ballistic decay of the ISF along the
vorticity direction is markedly slower (by about a factor of three, see inset
of Fig.6(b)) compared to $v_{s}$. This is consistent with the presence in the
echo images of an apparent ballistic motion mainly directed along the shear
direction and marginally contributing to the dynamics in the perpendicular
direction due to the finite angular aperture of the bow-tie regions used to
calculate $f_{v}(q,\Delta t)$ (see Sect.2.4.1).

These results show that the residual effects of imperfect synchronization of
the echo dynamics are rather small, corresponding to characteristic
displacements of less than 1 nm per cycle. As it will be explicitly
demonstrated in the next subsection, as soon as a genuine contribution to the
dynamics, due to structural rearrangements, is present, the ballistic term
discussed in this subsection can be safely neglected.

####  3.2.2 Carbopol

This sample is a yield stress fluid. As such, it exhibits a transition from
solid to liquid, as the amplitude of the applied shear strain exceeds a
certain threshold. This threshold can be broadly identified with the cross-
over point of $G^{\prime}$ and $G^{\prime\prime}$ in a dynamic strain sweep
oscillatory rheological test, which we take here as an estimate of the sample
yield strain 52 (see ESI for details). We thus expect the Carbopol sample to
exhibit a dramatically different dynamics for two different shear amplitudes
$\gamma$, respectively below ($\gamma_{0}=20\%$) and above ($\gamma_{0}=80\%$)
the yield strain $\gamma_{C}\simeq 60\%$.

Fig. 7: Echo activity maps of the Carbopol sample. Panels (a-c) refer to an
echo experiment with imposed shear strain $\gamma_{0}=20\%$, while panels
(d-f) correspond to $\gamma_{0}=80\%$. (a) Representative bright-field image
of the tracers-seeded Carbopol sample. (b) Instantaneous activity map
capturing one of the rare, localized rearrangement events occurring in the
sheared sample for $\gamma_{0}=20\%$. (c) Integrated activity map outlining
the presence of few, localized "active regions" corresponding to displacing
tracers in an otherwise non-rearranging background. Panels (d-f) show the same
quantities of panels (a-c) for $\gamma_{0}=80\%$. In all panels the scale bar
corresponds to $35\mu m$.

Below the crossover point
When the imposed amplitude is small ($\gamma_{0}=20\%$), visual inspection of
the echo image sequences reveals that most of the tracers do not move
appreciably, while a few of them undergo rare, intermittent displacements (see
supplementary movie M1). The occurrence of these isolated rearrangement events
can be highlighted by considering instantaneous (Fig.7b) or integrated echo
activity maps (Fig.7c). Instantaneous echo activity map are obtained as the
square modulus of the difference between two consecutive images within an echo
sequence $|I_{n}(\mathbf{x})-I_{n+1}(\mathbf{x})|^{2}$, while the integrated
map corresponds to the time average of all the instantaneous maps in the
sequence, and is a proxy of the cumulative rearrangements occurred in the
sample during the whole experiment.

To be more quantitative, we apply echo-DDM by considering separately the
dynamics along the shear and the vorticity direction. As it can be seen from
Fig.8(a,b), for this amplitude we cannot capture the full relaxation of the
ISFs within our observation window. Nevertheless, as shown in the inset of
panels (a,b), all the ISFs obtained for $q\in[0.15-2.2]$ $mum^{-1}$ nicely
collapse onto a single master curve when plotted as a function of the rescaled
time delay $q^{2}\Delta t$. This indicates that even below yielding the system
undergoes an (extremely slow) diffusive-like relaxation, which results from
irreversible rearrangements. This is in agreement with the seminal
observations made with DWS in Ref. 10. Here, we have the advantage that we can
characterize this irreversible dynamics as a function of the wave-vector $q$
on a well-defined plane in the sample, while being able to simultaneously
measure the local strain. In order to measure the $q$-dependent relaxation
rates, we thus fitted the phenomenological model $f(q,\Delta t)=[1+\Delta
t\Gamma(q)]^{-1}$, to the experimental ISFs. According to Eq.11, this model
provides the Fickean scaling $\langle\Delta x^{2}(\Delta
t)\rangle=2D_{0}\Delta t$ for the 1D MSD when the DDM relaxation rates exhibit
a quadratic dependence $\Gamma(q)=D_{0}q^{2}$ on the wavevector. This is
exactly the scaling that we observe in both directions, with a moderate
deviation only for the smallest values of $q$ (Fig.8(c)). By fitting the
relaxation rates we obtain the estimates $D_{s}=(2.2\pm 0.2)\cdot 10^{-4}\mu
m^{2}/s$ and $D_{v}=(2.6\pm 0.2)\cdot 10^{-4}\mu m^{2}/s$ for the diffusion
coefficient associated with the shear-induced effective dynamics of the
tracers along the shear and the vorticity direction, respectively. Within the
experimental error, the shear-induced dynamics is isotropic $D_{s}\simeq
D_{v}$ with values for the diffusion coefficients about one order of magnitude
larger than the one associated with the thermal motion of the same tracers in
Carbopol under stationary conditions (see ESI for details). This indicates
that the slow diffusive-like relaxation we observe is due to small, rare,
microscopic rearrangements that are induced in the sample by the imposed shear
deformation, despite the fact that its amplitude is below the crossover point.

Fig. 8: Echo-DDM analysis of Carbopol.  Panels (a-c) refer to an echo
experiment with imposed shear strain $\gamma_{0}=20\%$, while panels (d-f)
correspond to $\gamma_{0}=80\%$. (a,b) Representative ISFs for different
$q$-values in the range $[0.15-2.2]\mu m^{-1}$, along the shear and the
vorticity directions, respectively. Continuous lines are best fitting curves
to the experimental data with the model $f(q,\Delta t)=[1+\Delta
t\Gamma(q)]^{-1}$, from which an estimate of the $q$-dependent relaxation rate
$\Gamma(q)$ is obtained. In the inset of each panel, the same data of the main
figure are plotted as a function of the rescaled time delay $q^{2}\Delta t$.
(c) $q$-dependent relaxation rates $\Gamma_{s}(q)$ (orange circles) and
$\Gamma_{v}(q)$ (blue circles), along the shear and vorticity directions,
respectively. Large circles correspond to $q$-values whose ISFs are shown in
panels (a-b), with the same colors. Dashed lines represent the best fits to
the relaxation rates with a quadratic model $\Gamma(q)=Dq^{2}$. (d-f) same of
(a-c) for imposed shear amplitude $\gamma_{0}=80\%$. In (f) the horizontal
black dashed line marks the maximum relaxation rate that can be reliably
measured, corresponding to the oscillation frequency $\nu=1Hz$ .

The results obtained with DDM on the echo image sequences are confirmed by PT
analysis of the same sequences. The estimated PDF of the tracer displacements
along the shear and the vorticity direction, is shown in Fig.9(a) and (b),
respectively. In agreement with DDM analysis, we do not find any statistically
significant difference between the shear and the vorticity direction,
confirming that the dynamics is substantially isotropic. Interestingly, for
all investigated time delays, the PDFs is not Gaussian, as one would expect
for ordinary diffusion, but displays exponential-like tails. Nevertheless, the
MSDs in the shear and vorticity directions, reported in panel (c), display a
clean linear dependence on the delay time, $MSD_{s}(\Delta t)\simeq
2D_{PT,v}\Delta t$, $MSD_{v}(\Delta t)\simeq 2D_{PT,v}\Delta t$, indicating
Fickean diffusion. The value of the associated diffusion coefficient is found
to be $D_{PT,s}=(1.14\pm 0.08)\cdot 10^{-4}\mu m^{2}/s$ and $D_{PT,v}=(1.24\pm
0.1)\cdot 10^{-4}\mu m^{2}/s$ respectively. The discrepancy (about a factor of
2) between the estimates for the diffusion coefficient obtained from DDM and
PT can be probably attributed to differences in the populations whose dynamics
is captured by the two techniques (DDM sees all the particles and PT only a
user-dependent selection).

Fig. 9: Echo-PT analysis of Carbopol. Panels (a-c) refer to an echo experiment
with imposed shear strain $\gamma_{0}=20\%$, while panels (d-f) correspond to
$\gamma_{0}=80\%$. (a,b) Probability density functions (PDFs) of particle
displacements $P_{s}(\Delta\mathbf{x},\Delta t)$ and
$P_{v}(\Delta\mathbf{x},\Delta t)$ for different time delays $\Delta t$ in the
range $[1,70]$ s evaluated along the shear and the vorticity direction,
respectively. (c) Symbols: mean square displacements of the particles along
the shear (red) and the vorticity (blue) direction, respectively. Large
circles correspond to the delay times whose PDFs are shown in panels (a-b),
with the same colors. The red (blue) dashed line is the best fitting curve to
the MSD along the shear (vorticity) direction with a linear model $MSD(\Delta
t)=2D\Delta t$. (d-f) same of (a-c) for imposed shear amplitude
$\gamma_{0}=80\%$.

Above the crossover point
When the sample is subjected to a large amplitude shear deformation
$\gamma_{0}=80\%$, exceeding the crossover threshold $\gamma_{C}$, a dramatic
change in the echo dynamics is observed. The amplitude of tracer displacements
increases by orders of magnitude and the shear-induced activity shows a much
larger degree of spatial homogeneity and temporal uniformity (see
supplementary movie M2). This can be also appreciated by inspecting the echo
activity maps reported in Fig.7(e-f), outlining the presence of plastic,
irreversible rearrangements distributed across the whole field of view. The
speed-up of the dynamics is well captured by echo-DDM analysis. In contrast
with the small deformation case, the obtained ISFs display an almost complete
temporal relaxation for all the investigated $q$-values, in both the shear and
the vorticity direction (see Fig.8(d-e)). By plotting the ISFs as a function
of the rescaled time delay $q^{2}\Delta t$, an excellent collapse of the ISFs
on a single master curve is observed, highlighting once again a diffusive-like
dynamics (inset of panels (d-e)). As it can be appreciated from Fig.8(d-e),
over the whole considered $q$-range, the decay of ISFs is very well captured
by the same one-parameter fitting model $f(q,\Delta t)=[1+\Delta
t\Gamma(q)]^{-{1}}$ used for the small deformation case. The obtained
$q$-dependent relaxation rates show a quadratic scaling with $q$ along both
directions (Fig.8f). Remarkably, in this case the dynamics is no longer
isotropic, displaying a faster relaxation along the shear direction. Indeed,
fitting the experimental relaxation rates to $\Gamma_{v}(q)=D_{v}q^{2}$ and
$\Gamma_{s}(q)=D_{s}q^{2}$ provides $D_{v}=(1.9\pm 0.3)\cdot 10^{-1}\mu
m^{2}s^{-1}$ and $D_{s}=(2.9\pm 0.3)\cdot 10^{-1}\mu m^{2}s^{-1}$,
respectively. Comparing the diffusion coefficients for $\gamma_{0}=20\%$ and
$\gamma_{0}=80\%$, we observe a $\sim 1000$-fold speed-up of the dynamics for
large amplitude deformation, which is a strong signature of shear-induced
diffusion. PT-based analysis of the same data fully corroborates the results
obtained with echo-DDM, and in this case the agreement with DDM is even better
than for small deformation. In particular, they exhibit a linear dependence on
$\Delta t$. From a linear fit $MSD=2D\Delta t$ of the MSDs along both the
investigated directions (Fig.9f) we obtain $D_{PT,v}=(1.7\pm 0.2)\cdot
10^{-1}\mu m^{2}s^{-1}$ and $D_{PT,s}=(2.8\pm 0.2)\cdot 10^{-1}\mu
m^{2}s^{-1}$, respectively, in fairly good agreement with the results obtained
from echo-DDM.

##  4 Conclusion

In this paper, we combined oscillatory deformation in a simple shear geometry
with bright-field optical microscopy. For four different samples, we
characterized the deformation field, finding stationary and homogeneous
deformation fields. We observed the presence of partial slip, and we measured
the actual local deformation. In echo measurements, we observed drifts of
different origin: "apparent" drifts due to an imperfect synchronization
between the shear and the acquisition frequencies, and "real" drifts due to
unavoidable instrumental imperfections and the lack of horizontal confinement.
The effect of drift on the microscopic dynamics is corrected with a custom
sub-pixel registration algorithm. Eventually, the DDM algorithm is applied to
echo movies. In the case of an "ideal" elastic solid, the applied procedure
shows, as expected, the absence of shear-induced-dynamics. In the fluid-like
samples, the dynamics is dominated by flows generated by small asymmetry in
the set-up and by the lack of lateral confinement of the samples. Although our
registration algorithm can significantly reduce the impact of these drifts,
their residual effect is still large enough to prevent the observation of any
additional shear-induced dynamics.

With Carbopol, we perform experiments by using two different strain
amplitudes, below and above the yield strain. The smaller amplitude causes a
spatially heterogeneous, temporally intermittent activity of the tracer
particles, which results in extremely slow, substantially isotropic shear-
induced diffusion. Above the yield strain, the shear-induced diffusive
relaxation becomes anisotropic and very fast, with an observed speed-up of
about $10^{3}$ compared to the previous case. Our results with Carbopol
represent the first clear observation of a truly diffusive dispersion relation
$\Gamma(q)\sim q^{2}$ in fluids that are yielding during oscillatory
experiments. Of note, this behavior contrasts with recent results obtained
with x-ray photon correlation spectroscopy 13, 53 in which a faster-than-
diffusive relaxation (closer to the behavior observed in colloidal
polycrystals 54) was found for a nanoemulsion and a colloidal gel; at the same
time, our results agree with the original interpretation provided for dense
emulsions 10 and colloidal glasses 17, 11, in both cases lacking an
experimental determination of the dispersion relation.

Beyond providing reliable and quantitative information about the shear-induced
microscopic rearrangements, our approach also enables characterizing the
deformation profiles across the sample and, in turn, spotting slippage, shear
bands or other unwanted effects. This is made possible by the partial
coherence of the illuminating light, which not only helps in identifying the
presence of the above-mentioned phenomena, but also allows imaging a selected
slice of the sample. In this work, we imaged its central part, where the
deformation is homogeneous, by excluding the sample boundaries where
deviations from homogeneity (mainly due to wall slip) possibly occur. In case
of shear-banding, the $z$-selectivity of the microscope can be exploited to
select zones where the macroscopic deformation field is approximately
homogeneous, an important advantage over traditional single and multiple light
scattering approaches. Another advantage of our approach is that it makes it
possible to combine real-space pre-processing (drift correction) and Fourier-
space analysis (Echo-DDM), which leads to a considerable simplification of the
data interpretation.

Extensions of our approach to other imaging modes (e.g. confocal fluorescence
microscopy) are easily within reach, as well as combination with a commercial
rheometer provided that the sample remains optically accessible. Finally, the
usual advantages (and disadvantages) of DDM over PT hold, including the
capability of DDM to probe the dynamics of sub-resolution particles 2.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We acknowledge useful discussions with Manuel Escobedo, Stefan Egelhaaf, and
Veronique Trappe.

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