# Quantitative evaluation of methods
to analyze motion changes in single-particle experiments

Gorka Muñoz-Gil  Institute for Theoretical Physics, University of Innsbruck,
Technikerstr. 21a, A-6020 Innsbruck, Austria  Harshith Bachimanchi  Department
of Physics, University of Gothenburg, Origovägen 6B, SE-41296 Gothenburg,
Sweden  Jesús Pineda  Department of Physics, University of Gothenburg,
Origovägen 6B, SE-41296 Gothenburg, Sweden  Benjamin Midtvedt  Department of
Physics, University of Gothenburg, Origovägen 6B, SE-41296 Gothenburg, Sweden
Maciej Lewenstein  ICFO – Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860
Castelldefels (Barcelona), Spain  ICREA, Pg. Lluís Companys 23, 08010
Barcelona, Spain  Ralf Metzler  Institute for Physics & Astronomy, University
of Potsdam, Karl-Liebknecht-Str 24/25, D-14476 Potsdam-Golm, Germany  Asia
Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea  Diego
Krapf  Department of Electrical and Computer Engineering, Colorado State
University, Fort Collins, Colorado 80523, USA  Giovanni Volpe  Department of
Physics, University of Gothenburg, Origovägen 6B, SE-41296 Gothenburg, Sweden
Carlo Manzo<EMAIL_ADDRESS>Facultat de Ciències, Tecnologia i
Enginyeries, Universitat de Vic – Universitat Central de Catalunya (UVic-UCC),
C. de la Laura,13, 08500 Vic, Barcelona, Spain  Institut de Recerca i
Innovació en Ciències de la Vida i de la Salut a la Catalunya Central (IRIS-
CC), 08500 Vic, Barcelona, Spain

###### Abstract

The analysis of live-cell single-molecule imaging experiments can reveal
valuable information about the heterogeneity of transport processes and
interactions between cell components. These characteristics are seen as motion
changes in the particle trajectories. Despite the existence of multiple
approaches to carry out this type of analysis, no objective assessment of
these methods has been performed so far. Here, we have designed a competition
to characterize and rank the performance of these methods when analyzing the
dynamic behavior of single molecules. To run this competition, we have
implemented a software library to simulate realistic data corresponding to
widespread diffusion and interaction models, both in the form of trajectories
and videos obtained in typical experimental conditions. The competition will
constitute the first assessment of these methods, provide insights into the
current limits of the field, foster the development of new approaches, and
guide researchers to identify optimal tools for analyzing their experiments.

random walk; heterogeneous diffusion; anomalous diffusion; trajectory
analysis; time series; single-particle tracking; stochastic systems; deep
learning; changepoint analysis

## Introduction

Figure 1: Rationale for the challenge organization. a, The interactions of
biomolecules in complex environments, such as the cell membrane, regulate
physiological processes in living systems. These interactions produce changes
in molecular motion that can be used as a proxy to measure interaction
parameters. b-c, Time-lapse single-molecule imaging allows us to visualize
these processes with high spatiotemporal resolution (b) and, in combination
with single-particle tracking methods, provide trajectories of individual
molecules (c). d-e, Analytical methods can be applied to imaging data, either
raw (b) or processed in the form of trajectories (c), to infer interaction
kinetics and quantify their dynamic properties at the ensemble (e.g.,
probability distributions, d) or single-trajectory level (e.g., changepoints,
e).

Physiological processes occurring in living cells rely on encounters and
interactions between molecules. Archetypal examples include gene regulation,
transduction of biological signals, and protein delivery to specific
locations. All these processes involve the active or passive transport of
biomolecules in highly complex, time-varying, and far-from-equilibrium
environments, such as the cell membrane (Fig. 1(a)). One of the most powerful
tools to study these transport phenomena is the combination of live-cell
single-molecule imaging with single-particle tracking [1, 2] because it can
provide the time and location when and where single events take place (Fig.
1(b-c)). Alternative ensemble methods (e.g., fluorescence correlation
spectroscopy or fluorescence recovery after photobleaching [3]) usually
provide limited information because they lose track of crucial information
when averaging out spatial and temporal fluctuations.

Methods for single-molecule imaging and single-particle tracking have seen
tremendous progress in the last decade, in terms of both experimental
acquisition and data analysis [1, 2, 4, 5]. The abundance of experimental
single-particle trajectories, encompassing molecules, protein complexes,
vesicles, and organelles, has led to the development of numerous methods
dedicated to the reliable detection of changes in their motion patterns (as
summarized in Table 1). These changes serve as valuable indicators for the
occurrence of interactions within the system. For instance, diffusing
particles may exhibit variations in diffusion coefficients (due to processes
like dimerization, ligand binding, or conformational changes) or shifts in
their mode of motion (attributed to transient immobilization or confinement at
specific scaffolding sites) (Fig. 1(a)) [6]. These interactions can also
result in deviations from standard Brownian motion, as characterized by
Einstein’s free diffusion model, which includes a linear mean-squared
displacement (MSD) and a Gaussian distribution of displacements [7]. This is
the case, e.g., of spatiotemporal heterogeneities producing transient
subdiffusion at specific timescales [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19]. Other mechanisms can instead produce asymptotically-anomalous diffusion
[20, 21, 22, 2]. Anomalous diffusion compatible with models such as fractional
Brownian motion [23, 24, 25, 26, 27, 28], continuous-time random walk [29,
30], scaled Brownian motion [31], and Lévy walk [32] has been observed for
telomers, macromolecular complexes, proteins, and organelles in living cells.
Several approaches have been recently proposed to detect and quantify these
behaviors [33, 34], also involving machine-learning techniques [35, 36, 37,
38, 39, 40, 41].

To gain insights into the performance of methods to detect anomalous diffusion
from individual trajectories, in 2021 we successfully ran the $1^{\rm st}$
AnDi Challenge [42]. The discussion that developed between members of diverse
research communities working on biology, microscopy, single-particle tracking,
and anomalous diffusion (including experimentalists, theoreticians, data
analysts, and computer scientists) emphasized the necessity for deeper
insights into biologically relevant phenomena. First, there is a need to
evaluate methods to determine the switch between different diffusive
behaviors, as often observed in experiments. Second, it is necessary to assess
the methods’ crosstalk in detecting inherent anomalous diffusion from
nonlinearity in the MSD due to motion constraints or heterogeneity. Third,
there is a need to determine whether the bottleneck of the analysis process is
at the level of the analysis of the single trajectories or associated with
their extraction from the experimental videos. These needs shaped the design
of the $2^{\rm nd}$ AnDi Challenge, defining its scope with a focus on
characterizing and ranking the performance of methods that analyze changes of
dynamic behavior. While we have retained the name of the $1^{\rm st}$ AnDi
Challenge to build upon its already-established community, we would like to
emphasize that the main focus of this $2^{\rm nd}$ AnDi Challenge is on
revealing heterogeneity rather than anomalous diffusion. In the simulated
datasets, anomalous diffusion will either emerge from heterogeneity itself or
be intentionally introduced for evaluation purposes.

There is a multitude of methods that have been designed to identify and
characterize heterogeneous diffusion (Table 1). They can be classified based
on the heterogeneity they aim to identify or on the kind of analysis they
perform. We consider three heterogeneity classes that these methods aim to
identify: (i) changes in the value of the diffusion coefficient $D$; (ii)
changes in the anomalous diffusion exponent $\alpha$ (often classified as
subdiffusion, diffusion, or superdiffusion); and (iii) changes in the
phenomenological behavior associated with interactions with the environment
(often classified as immobilization, confinement, (free) diffusion, and
directed motion). While changes in the diffusion coefficient and in the
phenomenological behavior have been widely reported, the exploration of
changes in the anomalous diffusion exponent is a more recent development [43,
44, 45, 46], which is attracting increasing interest also from the theoretical
point of view [47, 48, 49, 50]. The introduction of new methods for data
analysis, as promoted by the Challenge, could push the performance for
detecting subtle changes in these diffusion properties in systems where they
have so far been overlooked. Along this line, it must be pointed out that the
traditional analysis based on the calculation of the scaling exponent of the
mean-squared displacement (MSD) can create some ambiguity between the last two
classes. Just to provide an example, a particle performing Brownian diffusion
in a confined region has an exponent $\alpha=1$ in terms of the generating
motion, but its MSD features a horizontal asymptote at long times,
corresponding to $\alpha=0$. In the following, we will refer to the exponent
$\alpha$ as the characteristic feature of the generating motion.

From the analysis point of view, we can identify two classes of methods: (i)
ensemble methods, meant to determine characteristic features out of an
ensemble of trajectories (Fig. 1(d)) and (ii) single-trajectory methods, meant
to identify changepoint (CP) locations through trajectory segmentation (Fig.
1(e)). While most available methods rely on the analysis of trajectories
obtained from video processing [51], recent advances in computer vision have
led to methods capable of directly extracting information from raw movies
without requiring the explicit extraction of trajectories [52, 53]. Each
method has its own set of advantages and disadvantages, and its performance
may depend on the specific problem under consideration. However, there is
currently no universally accepted gold standard for determining which method
to use to address each specific problem.

To cater to these more advanced needs, we decided to run an open competition
as the $2^{\rm nd}$ Anomalous Diffusion (AnDi) Challenge. The rationale
described above shaped the scope of the challenge, defining the choice of the
datasets and the design of the tasks. To rely on objective ground truth, we
will assess the methods’ performance on simulated datasets inspired by models
of diffusion and interactions documented in biological systems. Datasets will
describe particles undergoing fractional Brownian motion (FBM, [54]) with
piecewise-constant parameters. FBM-type motion has been widely observed in
biological systems by means of microrheology, a technique that uses large
tracer particles as probes to study the properties of the environment [55].
Anomalous diffusion compatible with FBM has also been reported for telomers
and macromolecular complexes in living cells [56, 23, 26, 24, 27, 25, 28, 20].
Beyond this evidence, in the context of the Challenge, FBM serves as a tool to
enable the tuning of diffusion parameters. The combination of parameter values
and interaction models might produce situations that do not correspond to
previously documented biological scenarios but will be valuable to test the
methods’ performance in a wide range of conditions. In biological experiments,
other kinds of motion and even non-Gaussian behavior have been reported [21].
However, the choice of FBM does not limit the generality of the Challenge
since other models of diffusion and non-Gaussian behavior can be obtained by
properly tuning the parameters of the simulations. Datasets provided for the
last phase of the competition will also include actual experiments for their
comparative analysis with the Challenge methods (these data will not be used
for the ranking).

The standard and straightforward approach in live-cell single-molecule imaging
primarily captures information related to lateral motion. In cases involving
flat membranes or isotropic systems, employing 2D imaging and tracking
techniques suffices for obtaining accurate motion-related parameters. However,
when dealing with motion on non-flat surfaces or within anisotropic 3D
environments, relying solely on 2D projections can result in critical
information being overlooked, potentially leading to the misinterpretation of
diffusion coefficients or the appearance of apparent anomalous diffusion
effects [57, 58]. Consequently, drawing definitive conclusions under such
circumstances should be avoided or approached with caution. To study motion
occurring in 3D space, it is advisable to employ 3D tracking methods, such as
off-focus imaging (i.e., the analysis of ring patterns in the defocused point
spread function) [59], interference/holographic approaches [60], multifocus
imaging [61], or point spread function engineering [62]. Although more
challenging, these methods can measure also the motion along the axial
dimension, facilitating a more accurate characterization. For the purposes of
the Challenge, we have chosen to concentrate on studying changes in diffusion
behavior occurring within a 2D context, driven by particle interactions of
various types.

While this challenge focuses on data from biological systems, the use of
regime-switching detection and trajectory segmentation extends well beyond the
domain of living cells. Particularly interesting applications also include,
e.g., the analysis of biomedical signals [63], speech [64], traffic flows
[65], seismic signals [66], econometrics [67, 68], ecology [69], and river
flows [70].

## Methods

### Datasets and ground truth

In order to benchmark the different methods on data with known ground truth,
we rely on numerical simulations. We developed the andi-datasets Python
package [71] to generate the required datasets to train and evaluate the
various methods. Details about available functions can be found in the hosting
repository [71].

Particle motion is simulated according to fractional Brownian motion (FBM,
[54]), a model that reproduces Brownian and anomalous diffusion processes by
tuning the correlation of the increments through the Hurst exponent $H$. FBM
is a Gaussian process with a covariance function

${\rm E}[B_{H}(t)B_{H}(s)]=K\left(t^{2H}+s^{2H}-|t-s|^{2H}\right),$ (1)

where ${\rm E}[\cdot]$ denotes the expected value and $K$ is a constant with
units ${\rm length^{2}\cdot time^{-2H}}$. In order to generalize FBM in two
dimensions (2D), a trajectory ${\bf R}(t)$ is represented as ${\bf
R}(t)=\\{X(t),Y(t)\\}$, where $X(t)$ and $Y(t)$ are independent FBM along the
$x$ and $y$ axes, respectively [33]. The anomalous diffusion exponent is
related to the Hurst exponent as $\alpha=2H$ [54], and the MSD for an
unconstrained FBM in 2D scales with time $t$ as

${\rm MSD}(t)=4Kt^{\alpha}.$ (2)

When $\alpha=1$, FBM reverts to Brownian motion and $K$ corresponds to the
diffusion coefficient $D$. FBM can thus produce subdiffusion for $0<H<1/2$
($0<\alpha<1$), Brownian diffusion for $H=1/2$ ($\alpha=1$), and
superdiffusion for $1/2<H<1$ ($1<\alpha<2$).

We will consider the following physical models of motion and interactions(Fig.
2(a)):

  * •

Single-state model (SSM) — Particles diffusing according to a single diffusion
state, as observed for some lipids in the plasma membrane [14, 15, 72]. This
model also serves as a negative control to assess the false positive rate of
detecting diffusion changes.

  * •

Multi-state model (MSM) — Particles diffusing according to a time-dependent
multi-state (2 or more) model of diffusion undergoing transient changes of $K$
and/or $\alpha$. Examples of changes of $K$ have been observed in proteins as
induced by, e.g., allosteric changes or ligand binding [73, 74, 75, 76].

  * •

Dimerization model (DIM) — Particles diffusing according to a 2-state model of
diffusion, with transient changes of $K$ and/or $\alpha$ induced by encounters
with other diffusing particles. Examples of changes of $K$ have been observed
in protein dimerization and protein-protein interactions [77, 78, 79, 80, 81].

  * •

Transient-confinement model (TCM) — Particles diffusing according to a space-
dependent 2-state model of diffusion, observed for example in proteins being
transiently confined in regions where diffusion properties might change, e.g.,
the confinement induced by clathrin-coated pits on the cell membrane [82]. In
the limit of a high density of trapping regions, this model reproduces the
picket-and-fence model used to describe the effect of the actin cytoskeleton
on transmembrane proteins [9, 83].

  * •

Quenched-trap model (QTM) — Particles diffusing according to a space-dependent
2-state model of diffusion, representing proteins being transiently
immobilized at specific locations as induced by binding to immobile
structures, such as cytoskeleton-induced molecular pinning [84, 17].

While the interaction mechanisms producing the heterogeneous diffusion are
inspired by biological scenarios, some of the combinations of diffusion
parameters and models lead to situations that may not correspond to previously
documented biological contexts. Nevertheless, this approach holds substantial
value as it enables the comprehensive assessment of method performance across
a broad spectrum of conditions.

In the simulations, each dynamic state is characterized by a distribution of
values for the parameters $K$ and $\alpha$. For each trajectory, the values of
$K$ and $\alpha$ for each state are randomly drawn from Gaussian distributions
with bounds $\alpha\in(0,2)$ and $K\in[10^{-12},10^{6}]$ pixel2/frameα. The
interaction distance and the radius of confinement or trapping have constant
values across each experiment. Simulations are provided in generalized units
(i.e., pixels and frames) that can be rescaled to meaningful temporal and
spatial scales.

A detailed description of the simulation procedure is presented in Appendix A.
Table 2 summarizes all the parameter values used to generate the Pilot Dataset
that can be downloaded at
https://drive.google.com/drive/folders/1R41I5Y8ynXdXbUFLWOtQXx1nkUh3bWw3?usp=share_link.
The Pilot Dataset provides typical numerical experiments that will be made
available to the participants in the different phases of the challenge.

Datasets provided for the last phase of the competition will also include
results from actual experiments that have reported the occurrence of
heterogeneous diffusion but for which the ground truth is not established
beyond any doubt. Therefore, these data will not contribute to the challenge
scoring. Nevertheless, the predictions provided by different methods will be
comparatively analyzed and discussed also with respect to the conclusions
reached in the original publications. Together with the quantitative results
obtained from simulations, these analyses will assess the applicability of the
methods to real-world experimental data.

### Competition design

To enable the assessment of the performance of previously established methods
while fostering the development of new approaches and the participation from
diverse disciplines, the challenge is organized along two tracks:

  * •

Track 1 — based on the analysis of raw videos.

  * •

Track 2 — based on the analysis of trajectories.

For each track, datasets are provided according to a hierarchical structure
(Fig. 2(b-c)) that includes:

  * •

Experiment — A given biological scenario defined by a model of interactions
and a set of parameters describing the dynamic interplay of the particles and
the environment.

  * •

Field of view (FOV) — A region of the sample where the recording takes place.
Particles within the same FOV can undergo interactions among themselves and/or
with the environment.

  * •

Video (Track 1 only) — Videos corresponding to each FOV.

  * •

Trajectory (Track 2 only) — Trajectory corresponding to the motion of an
individual particle.

For both tracks, all the particles used in the simulations are
provided/visualized (i.e., full labeling conditions). The effect of blinking
or photobleaching is not taken into account.

Figure 2: Physical models of interaction and structure of the simulated
datasets. a, Examples of 2-dimensional trajectories undergoing interactions
inducing changes in their motion. From left to right: _single-state model_
(SSM) without changes of diffusion; _multi-state model_ (MSM) with time-
dependent changes between different diffusive states (red and blue);
_dimerization model_ (DIM) where a particle (red) selectively interacts with
another particle (gray) and the two transiently co-diffuse with a different
motion (blue trajectory); _transient-confinement model_ (TCM) where a particle
diffuses inside (blue) and outside (red) compartments with osmotic boundaries
(gray area); _quenched-trap model_ (QTM) where a particle is transiently
immobilized (blue) at specific loci through interactions with static features
of the environment (gray areas). b An experiment (left panel) consists of
simulations performed according to one of the models of interactions described
in a (here shown a TCM experiment), with a set of parameters describing the
dynamic interplay of the particles and/or the environment. From the same
experiment, several fields of view (FOVs) are selected. Particles within the
same FOV (right panel) diffuse and undergo interactions among themselves
and/or with the environment (gray areas) that affect their trajectories. c,
Time traces of the coordinates of exemplary trajectories from the experiment
depicted in b displaying changes of diffusion properties at specific times
(dashed vertical lines). For the challenge, the motion analysis can be either
performed directly from the video recording of the FOV (Track 1), or from
detected trajectories linking the coordinates of individual particles at
different times (Track 2).

In each track, participants can compete in two different tasks, as typically
done in the analysis of experimental data:

  * •

Ensemble Task — Ensemble-level predictions providing for each experimental
condition the model used to simulate the experiment, the number of states, and
the fraction of time spent in each state. For each identified state,
participants should determine the mean and standard deviation of the
distribution of the generalized diffusion coefficients $K$, and the mean and
standard deviation of the distribution of the anomalous diffusion exponent
$\alpha$ corresponding to the underlying motion.

  * •

Single-trajectory Task — Trajectory-level predictions providing for each
trajectory a list of $M$ inner CPs delimiting $M+1$ segments with different
dynamic behavior. For each segment, participants should identify the
generalized diffusion coefficient $K$, the anomalous diffusion exponent
$\alpha$ corresponding to the underlying motion, and an identifier of the kind
of constraint imposed by the environment ($0=$ immobile, $1=$ confined, $2=$
free (unconstrained), $3=$ directed ($\alpha\leq 1.9$)). For Track 1,
predictions must be provided for a subset of particles (in the following, we
will refer to them as VIP, very important particles) identified through a
label map of the first frame of the movie. For Track 2, predictions must be
provided for all trajectories in the FOV.

For each task, several metrics will be evaluated (see section Scoring and
evaluations). Participants can provide partial submissions, e.g., including
predictions for a limited subset of experiments or for specific parameters.
For ranking purposes of the challenge, missing predictions will be scored with
the worst possible value of the corresponding metric.

### Competition timeline

We plan to run the 2nd AnDi Challenge as a time-limited competition from
December 1, 2023, to July 15, 2024 (the competition dates may be shifted
depending on the duration of the Stage 1 review). The competition will be
hosted on the Codalab platform and will be divided into three phases, namely
Development, Validation, and Challenge.

The Development Phase (December 1, 2023, to April 30, 2024) is intended for
the participants to set up their methods, test them, and familiarize
themselves with the datasets and the scoring platform. An unlabeled dataset
will be available and the public leaderboard will show scores obtained on this
dataset. The code to generate further data will also be available.

The Validation Phase (May 1, 2024, to June 30, 2024) will be a test of the
actual final challenge. A new dataset will be provided and the public
leaderboard will show the scores for each task.

The Challenge Phase (July 1, 2024, to June 15, 2024) will be the final stage
of the competition. Only teams in the top 10 ranking of each task will be
allowed to participate in this phase. A new dataset will be provided and the
number of submissions will be limited to 1 per day. Results obtained by other
participants will not be publicly disclosed, as the leaderboard will be made
publicly available only after the deadline.

The analysis of the final results will be performed from July to October 2024.
If Stage 1 review is successful, the manuscript will be submitted by the end
of October 2024. Members of the teams in the top-5 ranking of each task will
be invited to contribute to the article as authors.

The results of the challenge will be discussed with the participants and other
experts from the field during the $2^{\rm nd}$ Anomalous Diffusion Workshop
that will be held in December 2024.

### Dataset structure

The datasets used in the challenge (Fig. 3) include different experiments,
each contained in a folder labeled with a sequential number (EXP_[exp number])
and corresponding to a specific model and a fixed set of parameters. The
information about the model and the parameters is unknown to challenge
participants. Each experiment folder contains a list of files labeled with a
sequential number (FOV_[fov number]) associated with 30 FOVs. Each FOV reports
data from a variable number of particles diffusing on a $128\times 128$ pixel2
area.

Figure 3: Structure of the dataset. Datasets for each track are contained in
the respective directories, namely, Track 1 and Track 2. These include a
folder for each experiment labeled with a sequential number (EXP_[exp
number]). Each experiment folder contains a list of 30 files labeled with a
sequential number (FOV_[fov number]) associated with different non-overlapping
FOVs. For Track 1, the files consist of 200-frame videos and an additional map
corresponding to the segmentation of VIP particles at the first frame. For
Track 2, the files contain a table whose columns correspond to the trajectory
index, time step, $x$-coordinate, and $y$-coordinate. Figure 4: Examples of
challenge datasets. a, b, Frames of exemplary videos (FOVs) provided for Track
1 corresponding to the different diffusion models and reproducing low (a) and
high number density (b).  c, d, The upper rows show plots of the trajectories
for exemplary FOVs provided for Track 2 corresponding to the different
diffusion models and different sets of parameters exhibiting evident (c) or
subtle changes (d) of diffusion, i.e., data in c reproduce conditions were
diffusion changes are easier to identify as compared to d. For example, for
the SSM, all trajectories in c have the same generalized diffusion
coefficient, whereas in d they have a broad distribution of $K$. For the MSM,
trajectories in c undergo large changes of diffusivity and show long dwell
times in each state, as compared to d. For the DIM, trajectories in c undergo
large changes of anomalous diffusion exponent and show long dwell times in
each state, as compared to d. For the TCM, trajectories cannot leave the
confinement zone once they get inside c, whereas in d the compartments have a
finite transmittance. For the QTM, the dwell times in the traps are much
longer in c than in d. The lower rows show time traces of the $x$-coordinate
of representative trajectories. The different colors indicate different
diffusive states.

For Track 1 (Fig. 3(a)), the coordinates of particles in the same FOV are used
to generate 200-frame videos as a series of 8-bit images in the multi-tiff
format using Deeptrack 2.1 [5]. Noise is added to the synthetic images to
account for background fluorescence and shot noise. A map corresponding to the
segmentation of VIP particles at the first frame for which CPs and diffusion
parameters must be detected is also provided as a tiff file. Connected
components of the map are labeled with unique integer values that correspond
to the particle index.

For Track 2 (Fig. 3(a)), we provide a csv file for each FOV containing a table
whose columns contain trajectory index, time step, x-coordinate, and
y-coordinate. Coordinates of simulated trajectories are corrupted with
Gaussian noise corresponding to finite (subpixel) localization precision. The
trajectories have a maximum length of 200 frames.

Besides localization precision, motion blur can introduce a significant
contribution to noise, in particular if the camera frame rate is slow compared
to particle motion [85]. However, this aspect will not be included in the
Challenge datasets since it would introduce complexities in the definition of
the ground truth that could detract from the focus of the work. Nevertheless,
the simulation software incorporates the capability to introduce the effect of
motion blur both in videos and trajectories.

Exemplary data for all the models are shown in Fig. 4. Files in different
tracks labeled with the same experiment and the FOV index (e.g.,
Track_1/EXP_4/FOV_3.tiff and Track_2/EXP_4/FOV_3.csv) include simulations
obtained with the same set of dynamics parameters but do not correspond to the
motion of the same set of particles.

###  .1 Scoring and evaluation

The performance of the methods will be evaluated using specific metrics for
each Task. For ranking purposes in the challenge, composite metrics will be
used, as described below.

####  .1.1 Ensemble Tasks

Participation in an ensemble task requires predictions of the type of model
used for simulating each experiment, the number of states $S$ of the model,
and the parameters of each state. The type of model will be simply evaluated
as correct or wrong. The prediction of the number of states will be assessed
by measuring the difference with the ground truth. For both the generalized
diffusion coefficient and the anomalous diffusion exponent, predictions must
include the mean, the standard deviation, and the relative weight of each
state. From these values, we will compute the associated multi-modal
distributions $P_{\alpha}$ and $P_{D}$. The similarity of these distributions
to the ground-truth distributions $Q_{\alpha}$ and $Q_{D}$ will be assessed by
means of the first Wasserstein distance ($W_{1}$),

$W_{1}(P,Q)=\int_{\mathrm{supp}(Q)}|\mathrm{CDF}_{P}(x)-\mathrm{CDF}_{Q}(x)|dx,$
(3)

where $\mathrm{CDF}_{Q}$ is the cumulative distribution function of the
distribution $Q$ and $\mathrm{supp}(Q)$ is the support ($\alpha\in(0,2)$ and
$K\in[10^{-12},10^{6}]\,\ {\rm pixel^{2}\cdot frame^{-\alpha}}$).

####  .1.2 Single-trajectory Tasks

Participation in a single-trajectory task requires predictions of the $M$ CPs
and the dynamic properties, i.e., the generalized diffusion coefficient $K$,
the anomalous exponent $\alpha$, and diffusive-type identifiers of the
resulting $M$$+1$ segments. Different metrics will be used to evaluate the
methods’ performance.

#####  CP detection metrics

Following Ref. [51], given a ground-truth CP at locations $t_{({\rm GT}),i}$
and a predicted CP at locations $t_{({\rm P}),j}$, we define the gated
absolute distance:

$d_{i,j}=\min(|t_{({\rm GT}),i}-t_{({\rm P}),j}|,\varepsilon_{\mathrm{CP}}),$
(4)

where $\varepsilon_{\mathrm{CP}}$ is used as a fixed maximum penalty for CPs
located more than $\varepsilon_{\mathrm{CP}}$ apart. For a set of $M$GT
ground-truth CPs and $M$P predicted CPs, we can solve a rectangular assignment
problem using the Hungarian algorithm [86] by minimizing the sum of distances
between paired CPs:

$d_{\rm CP}=\min_{\rm paired\>CP}\Big{(}\sum d_{i,j}\Big{)}.$ (5)

The distance $d_{\rm CP}$ allows to define a pairing metrics:

$\alpha_{\rm CP}=1-\frac{d_{\rm CP}}{d_{\rm CP}^{\rm max}},$ (6)

where $d_{\rm CP}^{\rm max}=M_{\rm GT}$ $\varepsilon_{\mathrm{CP}}$ is the
distance associated with having all predicted CPs unpaired or at a distance
larger than $\varepsilon_{\mathrm{CP}}$ from all ground-truth CP. The metric
$\alpha_{\rm CP}$ is bound in $[0,1]$, taking a value of 1 if all ground-truth
and predicted CPs are matching exactly. Similarly, we define a CP localization
metric:

$\beta_{\rm CP}=\frac{\displaystyle{d_{\rm CP}^{\rm max}-d_{\rm
CP}}}{\displaystyle{d_{\rm CP}^{\rm max}+\overline{d_{\rm CP}}}},$ (7)

where $\overline{d_{\rm CP}}$ is the distance associated with having all
unassigned predicted CPs at a distance larger than $\varepsilon_{\mathrm{CP}}$
from all ground-truth CPs. This metric measures the presence of spurious CPs
and is bound in $[0,\alpha_{\rm CP}]$, taking value $\alpha_{\rm CP}$ if no
spurious CPs are present. We also calculate the number of true positive (TP),
i.e., the paired true and predicted CPs with a distance smaller than
$\varepsilon_{\mathrm{CP}}$ . Spurious predictions, i.e., not associated with
any ground truth or having a distance larger than $\varepsilon_{\mathrm{CP}}$
are counted as false positive (FP). Ground truth CPs not having an associated
prediction at a distance shorter than $\varepsilon_{\mathrm{CP}}$ are
considered false negative (FN). Given an experiment containing $N$
trajectories, we compute the overall number of TP, FP, and FN. We then use
these values to calculate the Jaccard similarity coefficient (JSC) over the
whole experiment as:

${\rm JSC}=\frac{{\rm TP}}{{\rm TP}+{\rm FN}+{\rm FP}}.$ (8)

For the predicted CPs classified as TP, we also compute the root mean square
error (RMSE), defined as:

${\rm RMSE}=\sqrt{\frac{1}{N}\sum_{\begin{subarray}{c}{\rm paired\>CP}\\\
d_{i,j}<\varepsilon_{\mathrm{CP}}\end{subarray}}\Big{(}t_{({\rm
GT}),i}-t_{({\rm P}),j}\Big{)}^{2}}.$ (9)

##### Metrics for the estimation of dynamic properties

For the evaluation of methods performance for the estimation of the dynamic
properties, we first follow a procedure similar to the one described above for
the pairing of the CPs. Predicted CPs are used to define the predicted
trajectory segments. We define a distance between predicted and ground-truth
segments based on the JSC calculated with respect to their temporal support,
where time points at which predicted and ground-truth segments overlap are
considered as TP, predicted time points not corresponding to the ground truth
as FP, and ground-truth time points not predicted as FN. The Hungarian
algorithm is used to pair segments by maximizing the sum of the JSC. Only
paired segments are used to calculate metrics assessing methods performance
for the estimation of dynamics properties. For the generalized diffusion
coefficient $K$, we use the mean squared logarithmic error (MSLE) defined as:

${\rm MSLE}=\frac{1}{N}\sum_{\begin{subarray}{c}{\rm paired}\\\ {\rm
segments}\end{subarray}}{\Big{(}\log(K_{{\rm(GT)},i}+1)-\log(K_{{\rm(P)},j}+1)\Big{)}^{2}}.$
(10)

For the anomalous diffusion exponents $\alpha$, we use the mean absolute error
(MAE):

${\rm MAE}_{\alpha}=\frac{1}{N}\sum_{\begin{subarray}{c}{\rm paired}\\\ {\rm
segments}\end{subarray}}{|\alpha_{{\rm(GT)},i}-\alpha_{{\rm(P)},j}|},$ (11)

where $N$ is the total number of paired segments in the experiment,
$\alpha_{{\rm(GT)},i}$ and $\alpha_{{\rm(P)},j}$ represent the ground-truth
and predicted values of the anomalous exponent of paired segments,
respectively. For the classification of the type of diffusion, we use the
F1-score:

${\rm F_{1}}=\frac{2{\rm TP_{\rm c}}}{2{\rm TP_{\rm c}}+{\rm FP_{\rm c}}+{\rm
FN_{\rm c}}},$ (12)

where TPc, FPc, and FNc represent true positives, false positives, and false
negatives with respect to segment classification. The metric is calculated as
a micro-average, which aggregates the contributions of all classes to compute
the average metric and is generally preferable when class imbalance is
present.

###  .2 Metrics for challenge ranking

For ranking purposes, we will use the mean reciprocal rank (MRR) as a summary
statistic for the overall evaluation of software performance [42]:

${\rm MRR}=\frac{1}{N}\cdot\sum_{i=1}^{N}\frac{1}{{\rm rank}_{\rm M_{i}}},$
(13)

where ${\rm rank}_{\rm M_{i}}$ corresponds to the position in an ordered list
based on the value of the corresponding metrics ${\rm M_{i}}$.

For Task 1, the metrics involved in the calculation will be the F1-score of
the model, the MAE of the distributions of $K$ and $\alpha$. For Task 2, the
JSC and the RMSE of CPs, the MSLE of $K$, and the MAE of $\alpha$.

## Data availability

A Pilot Dataset is available for download at
https://drive.google.com/drive/folders/1R41I5Y8ynXdXbUFLWOtQXx1nkUh3bWw3?usp=share_link

## Code availability

All software used for the Challenge is available at
https://github.com/AnDiChallenge/andi_datasets.

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###### Acknowledgements.

G.M-G. acknowledges support from the European Union (ERC, QuantAI, Project No.
10105529) and the Austrian Science Fund (FWF) through the SFB BeyondC F7102.
M.L. acknowledges support from: ERC AdG NOQIA; Ministerio de Ciencia y
Innovación Agencia Estatal de Investigaciones
(PGC2018-097027-B-I00/10.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan Nacional FIDEUA
PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, QUANTERA DYNAMITE
PCI2022-132919, Proyectos de I+D+I “Retos Colaboración” QUSPIN
RTC2019-007196-7); MICIIN with funding from European Union
NextGenerationEU(PRTR-C17.I1) and by Generalitat de Catalunya; Fundació
Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund
FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT \
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2022-1-0042); EU Horizon 2020
FET-OPEN OPTOlogic (Grant No 899794); EU Horizon Europe Program (Grant
Agreement 101080086 — NeQST), National Science Centre, Poland (Symfonia Grant
No. 2016/20/W/ST4/00314); ICFO Internal “QuantumGaudi” project; European
Union’s Horizon 2020 research and innovation program under the Marie-
Skłodowska-Curie grant agreement No 101029393 (STREDCH) and No 847648 (“La
Caixa” Junior Leaders fellowships ID100010434: LCF/BQ/PI19/11690013,
LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013). Views and
opinions expressed in this work are, however, those of the author(s) only and
do not necessarily reflect those of the European Union, European Climate,
Infrastructure and Environment Executive Agency (CINEA), nor any other
granting authority. Neither the European Union nor any granting authority can
be held responsible for them. R.M. acknowledges DFG grant ME 1535/12-1. D.K.
acknowledges funding from the National Science Foundation grant 2102832. G.V.,
H.B., J.P., and B.M. acknowledge funding from ERC StG ComplexSwimmers. C.M.
acknowledges support through grant RYC-2015-17896 funded by
MCIN/AEI/10.13039/501100011033 and “ESF Investing in your future”, grants
BFU2017-85693-R and PID2021-125386NB-I00 funded by
MCIN/AEI/10.13039/501100011033/ and FEDER “ERDF A way of making Europe”.

## Competing interest

The authors declare no competing interests.

Method | Output level | Type of Approach | Diffusion parameters | Diffusion classes
---|---|---|---|---
Hidden Markov method with measurement noise propagation [87, 88] | Single trajectory | Stats | $D$ | confined/Brownian
Hidden Markov method and simulations [43] | Single trajectory | Stats |  $D$, $\alpha$ |
Supervised trajectory segmentation algorithm with supported vector machine [89] | Single trajectory | ML |  | confined/drift/directed
Deep learning and moment scaling spectrum [90] | Single trajectory | ML + Stats | $D$ | immobile/Brownian (fast/slow)
Piecewise linear approximation [91] | Single trajectory | Stats | velocity |
Detection of velocity and diffusion coefficient[92, 93] | Single trajectory | Stats |  $D$, velocity | Brownian/directed
Back-propagation neural network [94] | Single trajectory | ML |  | confined/directed
Divide-and-conquer moment scaling spectrum [95] | Single trajectory | Stats |  $D$, $\alpha$, confinement radius | immobile/confined/Brownian/directed
Local convex hull [96] | Single trajectory | Stats |  $D$,$\alpha$ |
Recurrence analysis [97] | Single trajectory | Stats |  $D$,$\alpha$ |
Gamma mixture and a hidden Markov method [98] | Single trajectory | Stats + ML | $D$ |
Superstatistical framework [99] | Single trajectory | Stats | activity, persistence |
Probability distribution of square displacements [100] | Ensemble | Stats |  $D$, $\alpha$ | immobile/confined/Brownian/directed
Probability of transient confinement zones [101] | Single trajectory | Stats |  | confined/Brownian
Probability of jumps and transient confinement zones [102] | Single trajectory | Stats |  | confined/Brownian + jumps
Packing coefficient [103] | Single trajectory | Stats |  | immobile/confined/Brownian
Nonparametric Bayesian inference [104] | Ensemble | Stats + ML | $D$ |
Displacement distribution and autocorrelation of squared displacements [105] | Ensemble | Stats | $D$ |
Hierarchical Dirichlet process modeling [106] | Single trajectory | Stats | force |
Sliding window of time-averaged MSD [107] | Single trajectory | Stats |  $D$,$\alpha$ |
Random forest [108] | Single trajectory | Stats |  | confined/Brownian/directed/anomalous
Gyration quantification and a Bayesian statistics [109] | Single trajectory | Stats | $D$ | fast/slow
Variational Bayesian treatment of hidden Markov method [110] | Ensemble | Stats | $D$ |
Bayesian model selection of hidden Markov method [111] | Single trajectory | Stats | $D$ | Brownian/directed
Measurement of anomalous diffusion using recurrent neural networks [37] | Single trajectory | ML | $\alpha$ |
Pointwise diffusion properties with transformers [45] | Single trajectory | ML |  $D$ and $\alpha$ |
Graph-neural network with attention [40] | Single trajectory & Ensemble | ML |  $D$, $\alpha$ |
Recurrent neural network-based autoencoders [112] | Single trajectory | ML |  $D$, $\alpha$ |
Semantic segmentation with convolutional networks [113] | Single trajectory | ML | $\alpha$ |
Table 1:  Methods for the analysis of heterogeneous diffusion. Methods for the
analysis of heterogeneous diffusion are classified based on the level of
output provided (single-trajectory methods vs. ensemble methods) and on the
type of approach (machine learning (ML), classical statistics (Stats), or a
combination of the two). We also indicate the diffusion parameters that each
method estimates and the diffusion classes that each method uses to classify
the trajectories. Only methods accounting for switches of diffusive behavior
within the same trajectory are included.

Experiment | Model | Numb. of states | State | K (pixel2/frameα) |  $\alpha$ distrib. | Diffusion class | Model-specific parameters
---|---|---|---|---|---|---|---
$\mu$ | $\sigma$ | $\mu$ | $\sigma$
0 | SSM | 1 | 1 | 1 | 0.01 | 0.5 | 0.01 | 2 |  \- $\left(\begin{array}[]{cc}1&1\\\ 1&1\end{array}\right)$
1 | SSM | 1 | 1 | 0.1 | 0.01 | 1.9 | 0.01 | 3 |  \- $\left(\begin{array}[]{cc}1&1\\\ 1&1\end{array}\right)$
2 | MSM | 2 | 1 | $1$ | 0.01 | 1.5 | 0.01 | 2 | $M=\left(\begin{array}[]{cc}0.99&0.01\\\ 0.01&0.99\end{array}\right)$
2 | $0.05$ | 0.01 | 0.5 | 0.01 | 2
3 | MSM | 3 | 1 | $1$ | $0.01$ | 1.5 | $0.01$ | 2 | $M=\left(\begin{array}[]{ccc}0.98&0.01&0.01\\\ 0.01&0.98&0.01\\\ 0.01&0.01&0.98\end{array}\right)$
2 | 0.5 | 0.01 | 0.5 | 0.01 | 2
3 | 0.01 | 0.01 | 0.75 | 0.01 | 2
4 | QTM | 2 | 1 | 1 | 0.01 | 0.8 | 0.01 | 2 | $r_{t}=0.6$, $N_{t}=300$, $P_{b}=1$, $P_{u}=0.01$
2 | 0 | 0 | 0 | 0 | 0
5 | QTM | 2 | 1 | 1 | 0.01 | 1.5 | 0.01 | 2 | $r_{t}=1.0$, $N_{t}=150$, $P_{b}=1$, $P_{u}=0.05$
2 | 0 | 0 | 0 | 0 | 0
6 | DIM | 2 | 1 | 1 | 0.01 | 1.2 | 0.01 | 2 | $r=0.6$, $N=100$, $P_{b}=1$, $P_{u}=0.01$
2 | 1 | 0.01 | 0.8 | 0.01 | 2
7 | DIM | 2 | 1 | 1 | 0.01 | 1.2 | 0.01 | 2 | $r=1.0$, $N=80$, $P_{b}=1$, $P_{u}=0.01$
2 | 3 | 0.01 | 0.5 | 0.01 | 2
8 | TCM | 2 | 1 | $1$ | 0.01 | 0.8 | 0.01 | 2 | $r_{c}=5$, $N_{c}=30$, $T=0.1$
2 | $1$ | 0.01 | 0.4 | 0.01 | 1
9 | TCM | 2 | 1 | $1$ | 0.01 | 1 | 0.01 | 2 | $r_{c}=10$, $N_{c}=30$, $T=0.0$
2 | $0.1$ | 0.01 | 1 | 0.01 | 1
Table 2:  Parameters of the pilot dataset. Example of a dataset composed of 10
experiments reproducing the models of diffusion employed in the $2^{\rm nd}$
Anomalous Diffusion Challenge: _single-state model_ (SSM); _multi-state model_
(MSM); _dimerization model_ (DIM); _transient-confinement model_ (TCM);
_quenched-trap model_ (QTM). The diffusion class correspond to $0=$ immobile,
$1=$ confined, $2=$ free (unconstrained), $3=$ directed. The parameters
specific to each model are: for MSM, the transition matrix $M$; for QTM, the
trap radius $r_{\rm t}$, the number of traps $N_{\rm t}$, the probability of
trapping $P_{\rm b}$ and untrapping $P_{\rm u}$; for DIM, the interactions
radius $r$, the number of particles $N$, the probability of binding $P_{\rm
b}$ and unbinding $P_{\rm u}$; for TCM, the compartment radius $r_{\rm c}$,
the number of compartments $N_{\rm c}$, and the boundary transmittance $T$.
Simulations are provided in generalized units (i.e., pixels and frames) that
can be rescaled to meaningful temporal and spatial scales. See Appendix A for
further details. For all experiments, we simulated $N=100$ particles
(Experiment 7 has $N=80$) in a box of size $L=230$ pixel with a FOV size
$L_{FOV}=128$ pixel, and a maximum trajectory length of $200$ frames. For
Track 1, movies were rendered using a $\mathrm{FWHM}_{\rm PSF}=2.1$ pixel and
a $\mathrm{SNR}=7.1$. For Track 2, trajectories were corrupted with Gaussian
localization noise with $\sigma_{N}=0.12$ pixel.

##  Appendix A Simulations of diffusion and interaction models

Trajectories are simulated according to a 2-dimension fractional Brownian
motion (FBM) [54]. FBM is a continuous-time Gaussian process $B_{H}(t)$ with
stationary increments and a covariance function
$E[B_{H}(t)B_{H}(s)]=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H})$, where $H$
represents the Hurst exponent and is related to the anomalous diffusion
exponent $\alpha$ as $H=\alpha/2$ [54]. The FBM features three regimes: one
where the increments are positively correlated ($1/2<H<1$, i.e., $1<\alpha<2$,
superdiffusive); one where the increments are negatively correlated
($0<H<1/2$, i.e., $0<\alpha<1$, subdiffusive); and one where the increments
are uncorrelated ($H=1/2$, i.e., $\alpha=1$, diffusive Brownian motion).

The models included in the challenge describe trajectories where diffusion
properties are piecewise constant along segments of varying duration $T_{\rm
s}$ and undergo sudden changes. To obtain a trajectory segment of length
$T_{\rm s}$ with given anomalous diffusion exponent $\alpha$ and generalized
diffusion coefficient $K$, a set of $T_{\rm s}-1$ displacements for each
dimension are sampled from a fractional Gaussian noise generator [114]. The
displacements are then standardized to have variance $\sigma^{2}=2K\Delta t$,
where $\Delta t$ is the sampling time.

Simulations are performed considering particles diffusing in a square box of
size $L$ with reflecting boundary conditions. However, to avoid boundary
effects, the fields of view used for the challenge datasets correspond to a
square region of size $L_{\rm FOV}\ll L$ within the central part of the
original box (Fig. 2(b)).

For Track 1, trajectory coordinates are used as sub-pixel localizations of
individual particles to simulate movie frames as in single-molecule
fluorescence experiments [5]. Each particle has a random intensity $I_{i}$
that corresponds to the total number of photons collected by the detector.
$I_{i}$ is drawn from a uniform distribution in the interval $[I_{\rm
min},I_{\rm max}]$ and fluctuates over time according to a normal distribution
with mean $I_{i}$ and standard deviation $\sigma_{I}$. Each particle is
rendered as a diffraction-limited spot using an Airy disk as a point-spread
function (PSF) with full width at half maximum ${\rm FWHM_{\rm PSF}}=2.1$ px.
A constant background of $I_{\rm bg}=100$ counts is added to each frame.
Images are corrupted with Poisson noise.

For Track 2, trajectory coordinates are corrupted with noise from a Gaussian
distribution with zero mean and standard deviation $\sigma_{N}$ to take into
account the finite localization precision obtained in tracking experiments.

All the models share a set of parameters required for the simulations that are
described here. Model-specific parameters are defined when describing the
details of the models in the following sections.

  * •

$[K_{1},K_{2},\dots,K_{n}]$: average values of the (Gaussian) distribution of
the generalized diffusion coefficient for each of the $n$ diffusive states
considered in a given experiment, with support $[10^{-12},10^{6}]$ ${\rm
pixel^{2}\cdot frame^{-\alpha}}$.

  * •

$[\sigma_{K_{1}},\sigma_{K_{2}},\dots,\sigma_{K_{n}}]$: standard deviations of
the (Gaussian) distribution of the generalized diffusion coefficient for each
of the $n$ diffusive states considered in a given experiment. If not provided,
it is considered to be equal to 0 (i.e., the distribution is
$\delta(K-K_{i})$).

  * •

$[\alpha_{1},\alpha_{2},\dots,\alpha_{n}]$: average values of the (Gaussian)
distribution of the anomalous diffusion exponent for each of the $n$ diffusive
states considered in a given experiment, with support $(0,2)$.

  * •

$[\sigma_{\alpha_{1}},\sigma_{\alpha_{2}},\dots,\sigma_{\alpha_{n}}]$:
standard deviations of the (Gaussian) distribution of the anomalous diffusion
exponent for each the $n$ diffusive states considered in a given experiment.
If not provided, it is considered to be equal to 0 (i.e., the distribution is
$\delta(\alpha-\alpha_{i})$).

  * •

$L$: size of the box where trajectories are simulated with reflecting boundary
conditions.

  * •

$L_{\rm FOV}$: size of the box defining the FOV used for the challenge
datasets. The same particles can enter and exit the FOV over time but, for
evaluation purposes, they will be considered as generating different
trajectories.

  * •

$\Delta t$: sampling time at which the original motion of the particle is
tracked. For the challenge datasets, we consider $\Delta t=1$.

  * •

$T$: duration of the recording over each FOV, given as the number of time
steps $\Delta t$. It also corresponds to the maximum trajectory duration. For
the challenge, we set $T=500$;

  * •

$T_{\rm min}$: minimum duration of a trajectory to be included in the dataset.
For the challenge, we use $T=20$;

  * •

$I_{\rm bg}$ (Track 1): background level of noise (counts) used in the
simulation of videos.

  * •

${\rm FWHM_{\rm PSF}}$ (Track 1): full width at half maximum in pixels of the
point-spread function used to render fluorescent particles.

  * •

$I_{\rm tot}$ (Track 1): mean value in counts of the total fluorescence
collected for the detected particles.

  * •

$\sigma_{\rm tot}$ (Track 1): standard deviation in counts of the distribution
of total fluorescence collected for the detected particles.

  * •

$I_{\rm peak}$ (Track 1): mean value in counts of the peak fluorescence
collected for the detected particles. Can be calculated as $I_{\rm
peak}=I_{\rm tot}\frac{4\ln{2}}{\pi{\rm FWHM_{\rm PSF}}^{2}}$

  * •

${\rm SNR}$ (Track 1): typical signal-to-noise ratio of the movies, calculated
as the average peak intensity over the standard deviation of the noise [51]
and thus equal to

$SNR=\frac{I_{\rm peak}}{\sqrt{I_{\rm peak}+I_{\rm bg}}}.$ (14)

  * •

$\sigma_{N}$ (Track 2): standard deviation of the Gaussian localization noise
used to corrupt trajectory coordinates.

  * •

$t_{\rm min}$: minimum distance between changepoints, corresponding to the
minimum amount of time that a particle spends in a state. Shorter segments are
eliminated by smoothing the time trace of the state label using a majority
filter with a window of 5 steps. For the challenge, we set $t_{\rm min}=3$.;

A schematic representation of each of the models presented below is shown in
Fig. 2(a).

###  A.1 Model 1 - Single-state model (SSM)

This model simply corresponds to particles diffusing according to FBM with
constant generalized diffusion coefficient $K$ and anomalous diffusion
exponent $\alpha$. For each trajectory, a value of $K$ and a value of $\alpha$
are sampled from the corresponding distribution. Data corresponding to these
models are necessary to establish the false positive rate of the methods
toward the detection of changes of diffusion properties.

###  A.2 Model 2 - Multi-state model (MSM)

The multi-state model is a Markov model describing particles undergoing FBM
whose diffusion properties can change at random times. The number of states
$S$ is fixed for a given experiment as are the parameters defining the
distributions of $K$ and $\alpha$ for each state. For each trajectory, $S$
values of $\alpha$ and $S$ values of $K$ are sampled from the distribution of
the corresponding states, i.e., one per state. At every time step, a diffusing
particle has a given probability to undergo a change in one of its diffusive
parameters (either $\alpha$ or $K$). The probability of switching is given by
a transition matrix $M$. Namely, $M_{i,j}$ is the probability of switching
from state $i$ to state $j$ at each time step. In the same sense, $M_{i,i}$ is
the probability of remaining in state $i$. The residence time in a given state
$i$ can be directly calculated from the previous probability as

$\tau_{i}=\frac{1}{\sum_{j\neq i}M_{ij}}=\frac{1}{1-M_{ii}}.$ (15)

Model 2 (MSM) parameters • $M$: transition matrix between diffusive states.

###  A.3 Model 3 - Dimerization (DIM)

This model considers the case in which dimerization, i.e., the transient
binding of two particles, may occur and produce changes in the diffusion
properties of both particles. In particular, we consider the case of $N$
circular particles of radius $r$. For each trajectory, a value of $\alpha$ and
a value of $K$ are sampled from the corresponding distributions associated
with the monomeric state. If two particles are at a distance $d<2r$, then they
have a probability $P_{\rm b}$ of binding. The two particles forming a dimer
move with equal displacements, according to a generalized diffusion
coefficient $K$ and an anomalous diffusion exponent $\alpha$ drawn from the
distributions associated with the dimeric state. At each time step, the dimer
has a probability $P_{\rm b}$ of breaking its bond, freeing the two particles
to go back to their original motion parameters. The particles cannot form any
new dimer until taking a new step. Only dimers are allowed and subsequent hits
with other particles will not affect either the particles or the dimers.

Model 3 (DIM) parameters • $N$: number of diffusing particles in the box of
size $L$. • $r$: interaction radius, corresponding to the radius of the
diffusing particles. • $P_{\rm b}$: probability that two particles bind to
form a dimer in each time step. For this to happen, the particles must be at a
distance $d<2r$. • $P_{\rm u}$: probability that a dimer breaks up at each
time step so that the two particles go back to diffusing independently.

###  A.4 Model 4 - Transient-confinement model (TCM)

This model considers an environment with $N_{\rm c}$ circular compartments of
radius $r_{\rm c}$. The compartments are distributed randomly throughout the
environment such that they do not overlap. We consider that the compartments
are osmotic, i.e., a particle reaching their boundary from the exterior has a
probability 1 of entering them, but a particle reaching the boundary from the
interior of a compartment has a probability $T$ of exiting it (and $1-T$ of
being reflected back to the interior of the compartment). The diffusion inside
and outside the compartment is different, hence defining two diffusive states.
For each trajectory, two values of $\alpha$ and two values of $K$ are sampled
from the corresponding distributions, corresponding to the motion outside and
inside the compartments.

Model 4 (TCM) parameters • $N_{\rm c}$: number of compartments in the box of
size $L$. • $r_{\rm c}$: radius of the compartments. • $T$: transmittance of
the boundary. Probability that a particle reaching the boundary from inside
the compartment exits the compartment.

###  A.5 Model 5 - Quenched-trap model (QTM)

This model considers the diffusion of particles in an environment with $N_{\rm
t}$ immobile traps of radius $r_{\rm t}$. The values of $\alpha$ and $K$ are
sampled for each trajectory from the corresponding distributions and define
its unrestrained motion. A particle that enters the domain defined by a trap
has a probability $P_{\rm b}$ of binding to the trap and, hence, getting
temporarily immobilized ($K=0$, $\alpha=0$). At each time step, a trapped
particle has a probability $P_{\rm u}$ of unbinding and being released from
the trap, going back to its unrestrained motion. A particle cannot be trapped
again until taking a new step.

Model 5 (QTM) parameters • $N_{\rm t}$: number of traps in the box of size
$L$. • $r_{\rm t}$: radius of the traps. • $P_{\rm b}$: probability that a
particle binds to a trap and gets immobilized. For that to happen, a particle
must be at a distance $d<r_{t}$ from the trap. • $P_{\rm u}$: probability that
a trapped particle unbinds from a trap and starts diffusing independently at
each time $\delta t$.

## List of acronyms

RMSE

    root mean square error
MAE

    mean absolute error
MSLE

    mean squared logarithmic error
JSC

    Jaccard similarity coefficient
FP

    false positive
TP

    true positive
FN

    false negative
CP

    changepoint
FOV

    Field of view
SSM

    Single-state model
MSM

    Multi-state model
DIM

    Dimerization model
TCM

    Transient-confinement model
QTM

    Quenched-trap model
  *[CP]: changepoint
  *[SSM]: Single-state model
  *[MSM]: Multi-state model
  *[DIM]: Dimerization model
  *[TCM]: Transient-confinement model
  *[QTM]: Quenched-trap model
  *[FOV]: Field of view
  *[TP]: true positive
  *[FP]: false positive
  *[FN]: false negative
  *[JSC]: Jaccard similarity coefficient
  *[RMSE]: root mean square error
  *[MSLE]: mean squared logarithmic error
  *[MAE]: mean absolute error