# Protein overexpression can induce the elongation of cell membrane
nanodomains

Julie Corneta, Pascal Preirab, Laurence Saloméb, Frédéric Daumasb,
Bernard Laganec, Nicolas Destainvillea, Manoel Manghia∗, Fabrice
Dumasb${\dagger}$
aLaboratoire de Physique Théorique, Université Toulouse III - Paul Sabatier,
CNRS, Toulouse, France
bInstitut de Pharmacologie et de Biologie Structurale (IPBS), Université de
Toulouse,
CNRS, Université Toulouse III - Paul Sabatier (UPS), Toulouse, France
cInfinity, Université de Toulouse III - Paul Sabatier, CNRS, Inserm, Toulouse,
France

<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

In cell membranes, proteins and lipids are organized into sub-micrometric
nanodomains of varying size, shape and composition, performing specific
functions. Despite their biological importance, the detailed morphology of
these nanodomains remains unknown. Not only can they hardly be observed by
conventional microscopy due to their small size, but there is no full
consensus on the theoretical models to describe their structuring and their
shapes. Here, we use a combination of analytical calculations and Monte Carlo
simulations based upon a model coupling membrane composition and shape to show
that increasing protein concentration leads to an elongation of membrane
nanodomains. The results are corroborated by Single Particle Tracking
measurements on HIV receptors, whose level of expression in the membrane of
specifically designed living cells can be tuned. These findings highlight that
protein abundance can modulate nanodomain shape and potentially their
biological function. Beyond biomembranes, this meso-patterning mechanism is of
relevance in several soft-matter systems because it relies on generic physical
arguments.

## Introduction

The plasma membrane forms a hydrophobic barrier to separate the interior from
the exterior of cells and a two-dimensional fluid matrix for proteins.
Membrane components self-organize spontaneously to maximize molecular
interactions in order to reach the lowest free energy state. The first attempt
to describe membrane organization by Singer and Nicholson is now 50 years old
[1], and many updates have been proposed since then [2, 3, 4, 5, 6, 7].
Lipid–lipid, protein–protein and lipid–protein interactions lead to the
formation of domains in which these constituents are unevenly distributed.
This structuring is complicated by the interaction with the protein scaffold
of the inner membrane surface and the involvement of membrane associated
elements [8]. Among the huge diversity of membrane domains, the most studied
ones are certainly lipid rafts described as heterogeneous, dynamic, and short-
lived cholesterol- and sphingolipid-enriched membrane nanodomains (10-200 nm)
that are in a liquid ordered phase [9]. Beyond rafts, it is now widely
understood that nanoscale clustering is a common feature of membrane proteins.
This results in a cell membrane with composition and physical properties that
are different from average membrane properties. A number of studies have
reported the involvement of nano-domains in biological functions such as
signal transduction [10], regulation of membrane trafficking [11], immune
signalling [7, 8], or infectious processes [12, 13, 14, 15, 16].

Over the past 20 years, super-resolution microscopy techniques among which
Single Particle Tracking (SPT) have revealed that these nanodomains have
dimensions ranging from a few to several hundred nanometers [18, 17]. SPT
allows to follow the dynamics of individual molecules in living cells with
unique accuracy over dozens of seconds [for review see [19, 20, 18]]. While
the behavior of a single trajectory might be stochastic, the statistical
analysis of many trajectories provides insight on properties such as membrane
receptor activation, assembly/dissociation of signalling clusters, or protein
reorganization due to virus interactions [21, 23, 25, 24, 22, 26].

Figure 1: Example of oddly shaped MOR trajectories acquired by SPT on NRK
fibroblasts. Trajectories have been acquired in the absence of ligand (a-b) or
in the presence of 1 $\mu$M of agonist ligand DAMGO (c-h). Out of 100
trajectories without ligand, 2 are curled (2 %) whereas in presence of DAMGO
it increases to 9 over 60 (15 %).

SPT permits to classify trajectories in different categories than can be
correlated to various physiological events. For instance, proteins that have
no specific interaction with any membrane structure present a Brownian motion,
those interacting with cytoskeleton filaments display a directional motion and
those that have an affinity for micro-domains present a permanently or
transiently confined diffusion [27]. For many years, in different studies, we
have observed some oddly shaped confined SPT trajectories that we could not
correlate to physical or physiological events. We give an illustrative example
of these atypical trajectories that mostly presented an elongated or curved
horseshoe shape in Fig. 1. The measurements were obtained by tracking the Mu
Opioid Receptor (MOR) at basal state or in the presence of an agonist named
DAMGO. Interestingly, we have observed that the proportion of these elongated
and/or curled trajectories seemed to increase in the presence of the agonist
as shown in Fig. 1. It is known that the binding of ligands induces
conformational changes of GPCRs [28] and can alter the oligomeric arrangement
of receptors [29]. More recently, Civciristov and co-authors stated that
“DAMGO or morphine induces the assembly of different protein-interaction
networks” [30]. This might induce a local increase of protein concentration
and reshape membrane domains, in agreement with the observation that some
nanodomains show more elongated contours when membrane proteins are
overexpressed [31]. In the same idea, Merklingler et al. revealed by super-
resolution microscopy that increasing the expression level of syntaxin induces
an increase of the local protein concentration and elongation of the
nanodomains in which it is confined [compare their Fig. 5 and 5S in [32]].
Taken together, these observations led us to hypothesise that the increased
local concentration of membrane proteins can induce a modification of the
shape of the domains that enclose them.

To verify this hypothesis, we adopt in parallel theoretical and experimental
approaches.

Figure 2: Examples of SPT trajectories of CCR5 receptors confined into
nanodomains at the surface of Affinofile cells (see text). More roundish
nanodomains are observed when the proteins have a low expression level (left)
and more elongated ones when the proteins are overexpressed (right).

In our theoretical framework, membrane meso-patterning ensues from the
competition between (i) short-range attractive forces promoting a condensed
phase of proteins and (ii) an additional mechanism, related to the coupling
between the membrane composition and its shape (curvature), making too large
domains unstable in the favor of smaller ones in equilibrium [33, 34, 35]. We
performed mesoscale numerical simulations [36], where a vesicle made of two
species having different spontaneous curvatures is simulated with a Monte
Carlo (MC) algorithm coupling the composition and the membrane elasticity, see
Supplemental Material (SM). Below the critical temperature, it leads to the
formation of nanodomains, instead of a macrophase separation. As a function of
the parameter values, we observe the formation of more or less elongated
nanodomains. These simulations are guided by analytical calculations
indicating the range of physical parameters entering the numerical model for
which the elongation mechanism occurs. In these calculations, we use a
simplified model of pairwise mid-range repulsive forces because it has been
demonstrated that the coupling between composition and curvature plays the
same role as such a pairwise repulsive force, from a statistical physics point
of view [37]: both mechanisms make large domains unstable because their energy
grows faster than their size. Since we have also an expertise in the study of
the dynamics of HIV receptors, we have in parallel performed SPT experiments
(Fig. 2) on three transmembrane proteins involved in HIV-1 infection: CD4,
CCR5 and CXCR4 [38, 12]. It has been shown that their distribution at the
surface of target cells is heterogeneous [39, 40, 41, 42, 43], and proposed
that membrane nanodomains, by concentrating the receptors, would be the
preferential sites of virus entry into cells [45, 44, 12]. We have used the
293-Affinofile cell line to record receptor trajectories at different
expression levels. These cells, in which CD4 and CCR5 expression can be
independently induced [46], were further transduced to stably express low or
high level of CXCR4. We have recorded the trajectories of these proteins at
different expression levels. Our observations reveal that an elongation of
membrane nanodomains occurs upon protein overexpression consistently with our
theoretical findings.

## Materials and Methods

### Cell culture and transfection

Affinofile cells, provided by Dr. B. Lee, Mont Sinaï hospital, New York, NY,
USA, were maintained in Dulbecco’s modified Eagle’s medium with 10% dialysed
fetal calf serum supplemented with 50$\mu$g/ml blasticidin (D10F/B). 24-well
plates were seeded with $1.2\times 10^{5}$ Affinofile cells/well, and
expression of CD4 and CCR5 was induced the following day at 37∘C with various
concentrations of minocycline and/or ponasterone, respectively, for 18h [46].

Stable expression of CXCR4 was performed through lentiviral transduction of
293-Affinofile cells with the pTRIP deltaU3 lentiviral vector encoding the
CXCR4 sequence. Lentiviral particles were prepared by transient cotransfection
of HEK 293T cells with the pTRIP lentiviral vector [a gift from P. Charneau,
Institut Pasteur, Paris, originally described in [47]] encoding the CXCR4
sequence, the p8.71 encapsidation plasmid and a plasmid encoding the VSV
envelope glycoprotein G (pVSVG) (at a 2/2/1 ratio). Transfection was performed
using a standard calcium phosphate precipitation method. Forty-eight hours
later, lentiviral particles were harvested and quantified using a HIV capsid
protein ELISA kit. Then, 293-Affinofile cells were inoculated by different
amounts of lentiviral particles so as to obtain cells with different
transduction efficiencies. From the different sets of transduced cell
populations, cell clones expressing low or high levels of CXCR4 were further
selected using serial dilutions in 96-well culture plates. During all these
experiments, CXCR4 expression was monitored by flow cytometry using the anti-
CXCR4 mAb 12G5 labeled with phycoerythrin.

### CD4, CCR5 and CXCR4 quantification

The number of CD4, CCR5 and CXCR4 proteins per cell, as displayed in Table 1,
have been estimated by quantitative fluorescence-activated cytometry (qFACS)
using Quantibrite PE calibration beads (BD quantibrite beads cat. number
340495). BD Quantibrite PE is a set of beads covalently conjugated with four
levels of phycoerythrin (PE). By running a BD Quantibrite PE tube at the same
instrument settings as the assay, the fluorescence intensity can be converted
into the number of PE molecules (i.e. antibody) bound per cell [48]. 100
$\mu$L of the Affinofile cells in suspension (1$\times 10^{5}$) have been
incubated with 3 $\mu$L of phycoerythrin-conjugated anti-CD4, anti-CCR5 or
anti-CXR4 for 60 min at 4∘C in the dark and in the presence of the ATPase
inhibitor sodium azide (0.1%) to avoid internalization. Cells were then washed
twice with PBS, resuspended in 100 $\mu$L of PBS and immediately analyzed by
flow cytometry without fixation. All experiments have been carried out in
triplicate.

number of proteins | low | high
---|---|---
CD4 | $2800\pm 300$ | $100000\pm 13000$
CCR5 | $3600\pm 800$ | $125000\pm 8000$
CXCR4 | $15000\pm 2000$ | $120000\pm 7000$
Table 1:  Average number of proteins per cell in the low and high expression
levels for the three types of HIV receptors. Low expression of CD4 and CCR5
conditions correspond to cells that were not induced by minocycline and/or
ponasterone. High CD4 and CCR5 expression conditions correspond to those
induced with 5 ng/ml minocycline and/or 1 $\mu$M ponasterone, respectively.

### Single Particle Tracking experiments

To be able to acquire long time trajectories we have used antibodies that
present higher affinities than Fab fragments. Since CD4, CCR5 and CXCR4 were
known to interact with each other, we have used antibody clones that had been
validated in the literature and described as not to interfere in those
interactions (when the information was available): T21/8 for CCR5 [49], OKT4
for CD4 [50] and 12G5 for CXCR4. Cells were plated on coverslips previously
incubated with 0.1 mg/ml poly-L-lysine for 5 min. For a homogenous attachment
of the cells, a gentle centrifugation (50 g, 7 min) was performed. Proteins
(CD4, CCR5 or CXCR4) were labeled for 15 min with 0.03 nM biotinylated
antibodies coupled to 0.3 nM fluorescent (655 nm) streptavidin-coated Quantum
Dots (QD) (Molecular Probes, LifeTechnologies). This antibody/QD ratio of 1:10
ensures that no more than one antibody is bound per particle to prevent that
one particle interacts with multiple receptors on the cell surface [51].
Tracking and observations were performed at room temperature on an Axioplan 2
microscope (Zeiss) equipped with a Cascade II 512 EM-CCD camera (Roper
Scientific) operating at a 25 Hz acquisition frequency. The fluorescent
nanoparticles were illuminated with an X CITE 120 light source containing a
metal halide vapor short arc lamp (Exfo), and observed through a fluar
100$\times$/1.30 oil UV objective associated to a 1.6$\times$ multiplier tube
lens in front of the camera. Acquired video durations go from 30 s to 80 s.
The maximum duration of measurements of a slide has been fixed to 30 min in
order to avoid cellular stress. In these conditions, 1 to maximum 4
trajectories could be acquired on a same cell.

As any experimental measurement, SPT suffers from a small, but finite
localization error. To experimentally bound the average error made during the
position determination, we immobilized quantum dots on a glass coverslip and
embedded them in a 15% polyacrylamide gel, recorded 80 s videos and determined
the positions of each particle in each image. The pointing error was defined
in $x$ and $y$ as the standard deviation of the position distribution of the
immobilized quantum dots. We obtained a value of 7 nm in both $x$ and $y$
coordinates (Fig. S5). The thermal drift of the microscope is de facto taken
into account by this measurement.

### Single Particle Tracking trajectory analysis

The trajectories analysis is based on the calculation of the Mean Square
Displacement (MSD) :

${\rm MSD}(s)=\langle[\mathbf{r}(t+s)-\mathbf{r}(t)]^{2}\rangle$ (1)

where $\mathbf{r}$ is the 2D tracked particle position and the average is
taken over the successive frames at successive times $t$. It characterizes the
displacement during frames separated by a duration $s$. MSD plots are a useful
tool, having been used for several decades to characterize the diffusional
behavior of tracked objects in SPT experiments [21, 54, 24, 52, 53]. They
allow us to classify the diffusive behaviors of trajectories or parts of
trajectories into different categories namely random, confined, walking
confined or directed diffusion [27], according to the mathematical expression
of the MSD. To do so, we systematically determined, by using a non-linear
least mean-square regression algorithm based on a $\chi^{2}$ test, which
equation gave the best fit of the MSD versus time plots [21] between:

  * •

free diffusion in 2 dimensions, with diffusion coefficient $D$

${\rm MSD}(s)=4Ds$ (2)

  * •

diffusion confined in a domain

${\rm MSD}(s)\simeq 2\Delta\mathbf{r}^{2}[1-\exp(-s/\tau)]$ (3)

where the relaxation time $\tau=\Delta\mathbf{r}^{2}/(2D)$;

  * •

directed diffusion

${\rm MSD}(s)=4Ds+v^{2}s^{2}$ (4)

where $v$ is the drift or transport velocity;

  * •

walking confined diffusion

${\rm MSD}(s)=A\left[1-\exp\left(-4D_{\rm micro}s/A\right)\right]+4D_{\rm
macro}s$ (5)

where $A$ is the characteristic area of the confined region, $D_{\rm macro}$
is the long time diffusion and $D_{\rm micro}$ is the diffusion coefficient
inside the domain.

It has to be noticed that no walking confined diffusion trajectories
(diffusion in a confined domain, itself diffusing) have been observed here.
For more details, Fig. 2 provides examples extracted from the SPT trajectories
studied in this work.

### Confinement index

Figure 3: Top: Example of use of the confinement index $\Lambda$ on one of our
SPT experimental trajectories, in function of time. It is calculated over a
sliding time window of duration $\delta t$ given by the color code. By the end
of the trajectory, the index becomes larger than the threshold 4 (dashed
line), indicating a marked confinement zone [52]. The colored line below the
plots represents the duration of the confinement. Middle: the so-obtained
transient confinement zone is represented in red on the SPT trajectory which
is split into two parts, presumably a free random walk and the confined
transient confinement zone. Bottom: This is confirmed by inspection of the MSD
plots, the first one is linear and the second one is typical of diffusion
confined in a nanodomain. The time is in s on all axes.

In Ref. [52], some of us have developed and characterized a tool to detect
transient confinement in single-molecule membrane trajectories. Basically, it
consists in identifying, along a trajectory, sets of $n\gg 1$ successive
positions $\mathbf{r}_{i}$ [of duration $\delta t(n)$], the variance
$\Delta\mathbf{r}^{2}(n)$ of which is significantly smaller than what it would
be if the diffusion were free, $\Delta\mathbf{r}^{2}(n)\ll 4D\delta t(n)$. The
index $\Lambda=D\delta t(n)/\Delta\mathbf{r}^{2}(n)$ must be large enough
(larger than 4 in practice), for a long enough duration, to rule out the
possibility of statistical fluctuations of a non-confined trajectory and to
ascertain confined diffusion. The diffusion coefficient $D$, that can depend
on the membrane region where the tracked molecule diffuses, must be monitored
in real time with the help of a local MSD plot. This tool has been shown to be
robust and to detect transient confinement zones with a good accuracy. A
comprehensive example is given in Fig. 3. Transient confinement zones that are
isolated through this procedure are considered as independent measurements.

### Reagents and antibodies

Minocycline (Sigma Aldrich, St. Louis, MO, USA) was dissolved in dimethyl
sulfoxide to generate a stock concentration of 1 mg/mL. Ponasterone
(Invitrogen, Carlsbad, CA, USA) was dissolved in 100% ethanol to generate a
stock of 1 mM. Blasticidin HCl (Invitrogen, Carlsbad, CA, USA) was dissolved
in sterile water to generate a stock solution of 5 mg/mL. For protein
quantification the following phycoerythrin-conjugated IgGs antibodies have
been used: anti-CD4 (OKT4, Biolegend), anti-CCR5 (2D7, BD Biosciences, San
Jose, CA, USA), anti-CXCR4 (12G5, Biolegend). For SPT experiments the
following biotinylated antibodies have been used: anti-CD4 (OKT4, Biolegend,
Ref. 317406), anti-CCR5 (T21/8, Biolegend), anti-CXCR4 (12G5, Biolegend).

## Results

### Elongated domains are more stable than circular ones above a certain size

We first study analytically the shape of one membrane nanodomain in
thermodynamical equilibrium. Its stability comes from the action of two
competitive interactions. A short-range attraction between components of the
same nature and/or repulsion between components of different nature [55, 56]
gives rise to a line tension at the boundary between two phases. A mid-range
repulsive interaction due to the coupling between membrane composition and
membrane elasticity prevents macrophase separation [57]. We study for which
range of the physical parameters an elongated shape is more stable than a
circular one. The model considers generically an elliptic nanodomain of semi-
axes $r_{0}a$ and $r_{0}/a$, with $a\geq 1$, so that the ellipse has area
$A=\pi r_{0}^{2}$. Its aspect ratio AR, defined as the ratio of major to minor
axes, is ${\rm AR}=a^{2}$. The total energy of the system is $E_{\rm
tot}(a)=E_{\rm bulk}+E_{\rm rep}+E_{\rm line}$ where $E_{\rm bulk}\propto
r_{0}^{2}$ is the cohesive energy of the nanodomain due to inter-molecular
short-range forces, and is simply proportional to its area. $E_{\rm bulk}$
does not depend on $a$ and will be skipped in the following calculations. The
repulsion energy between the membrane components inside the nanodomain is
supposed to be pairwise and to have a finite range $\xi$:

$E_{\rm rep}=E_{0}\int_{A\times
A}\rho(\mathbf{r})\,\varphi\left(\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{\xi}\right)\rho(\mathbf{r}^{\prime}){\rm
d}\mathbf{r}\,{\rm d}\mathbf{r}^{\prime}$ (6)

where $\varphi$ is the interaction potential, and $\rho(\mathbf{r})$ the
particle density inside the nanodomain. The parameter $E_{0}$ sets the
strength of the repulsion. In the case where this repulsive energy comes from
the coupling with the membrane curvature, the molecules in the nanodomain
induce a spontaneous curvature $C_{0}$ different from the average one (assumed
to be 0 for simplicity sake). It has been shown that the screening length is
$\xi=\sqrt{\kappa/\sigma}$ and that $E_{0}\propto\sigma C_{0}^{2}$ in the low
tension limit [37, 36]. Here $\kappa$ and $\sigma$ are respectively the
membrane bending modulus and surface tension. In principle, the function
$\varphi$ decays exponentially at long distances, being for example a Bessel
function. In order to get an analytically tractable model, we assume it to be
the Gaussian
$\varphi({|\mathbf{r}-\mathbf{r}^{\prime}|}/{\xi})=\exp[-(\mathbf{r}-\mathbf{r}^{\prime})^{2}/(2\xi^{2})]$,
which is sufficient at the scaling level. We also suppose that
$\rho(\mathbf{r})=\frac{1}{2}\exp\left[-\frac{1}{2}\left(\frac{x^{2}}{(r_{0}a)^{2}}+\frac{(ay)^{2}}{r_{0}^{2}}\right)\right]$
is a Gaussian density with $\mathbf{r}=(x,y)$. The prefactor $1/2$ ensures
$\int_{\mathbb{R}^{2}}\rho(\mathbf{r}){\rm d}^{2}\mathbf{r}=\pi r_{0}^{2}$.
Introducing the dimensionless repulsion length $\ell=\xi/r_{0}$, one gets
$E_{\rm rep}(a)=\pi^{2}E_{0}r_{0}^{4}f(a)$ where

$f(a)=\frac{\ell^{2}}{\sqrt{\ell^{2}+2a^{2}}\sqrt{\ell^{2}+2/a^{2}}}$ (7)

Figure 4: The nanodomain adimensional energy $E_{\rm tot}/(\pi r_{0}\lambda)$
as a function of the ellipse aspect ratio parameter $a$, for $\varepsilon=10$
and $\ell=0.5$, 1, 2, 4, 8, 12 and 16, from bottom to top.

The line energy reads $E_{\rm line}=\lambda P$ where $\lambda$ is the line
tension. The circumference $P$ of an ellipse of semi-axes $\alpha$ and $\beta$
is given by the elliptic function. However, a very good approximation by
Ramanujan is
$P\simeq\pi\left[3(\alpha+\beta)-\sqrt{(3\alpha+\beta)(\alpha+3\beta)}\right]$,
thus $E_{\rm line}(a)=\pi\lambda r_{0}g(a)$ with

$g(a)=\left[3\left(a+\frac{1}{a}\right)-\sqrt{\left(3a+\frac{1}{a}\right)\left(a+\frac{3}{a}\right)}\right].$
(8)

Fig. 4 shows how $E_{\rm tot}/(\pi r_{0}\lambda)=\pi\varepsilon f(a)+g(a)$
behaves in function of $a$ for various values of $\ell$, where we have
introduced the new dimensionless parameter

$\varepsilon=\frac{E_{0}r_{0}^{3}}{\lambda}$ (9)

measuring the relative strengths of repulsion and line energies. One observes
that there is a range of values of $\ell$ for which $a=1$ is not an energy
minimum. It implies that there exist values of the repulsion length $\xi$ for
which the most stable nanodomain shape is an ellipse ($a>1$), and not a disc
($a=1$). Their relative stability can be addressed by examining the behavior
of $E_{\rm tot}$ close to $a=1$. Expanding $f$ and $g$ at order 2 reads
111Using the exact elliptic function leads to the same expansion at order 2
for $g$. $f(a)\simeq f(1)-\frac{4\ell^{2}}{(2+\ell^{2})^{3}}(a-1)^{2}$ and
$g(a)\simeq g(1)+\frac{3}{2}(a-1)^{2}$. Introducing
$A_{\ell}=4\ell^{2}/(2+\ell^{2})^{3}$, it follows that $\frac{1}{2}\frac{{\rm
d}^{2}E_{\rm tot}}{{\rm
d}a^{2}}=-\pi^{2}E_{0}r_{0}^{4}A_{\ell}+\frac{3}{2}\pi\lambda r_{0}$. Ellipses
are stable when $\frac{{\rm d}^{2}E_{\rm tot}}{{\rm d}a^{2}}<0$, i.e.
$A_{\ell}>\frac{3}{2\pi\varepsilon}$; $A_{\ell}$ has a maximum
$A^{*}=\frac{8}{27}$ at $\ell^{*}=2$. There exists a region of stability of
ellipses iff $A^{*}>3/(2\pi\varepsilon)$, i.e. $\varepsilon>\frac{81}{16\pi}$.

Since the numerical prefactors in our expressions come from the choices of
repulsive potential Gaussian shape $\varphi$ and Gaussian density profiles in
the nanodomain, we simplify the principal results as follows: (i) Elliptic
domains are stable for $E_{0}r_{0}^{3}>\lambda$, i.e. for large enough domain
radius $r_{0}$, strong enough repulsion strength $E_{0}$ or weak enough line
tension $\lambda$. (ii) If this condition is satisfied, $\ell=\xi/r_{0}$ must
belong to an interval $[\ell^{*}_{i},\ell^{*}_{s}]$ (see Fig. S8) distributed
around the maximum abscissa $\ell^{*}$ of order unity to stabilize ellipses
with respect to discs. If the repulsion range is too short as compared to the
domain radius, repulsion is not strong enough to destabilize circular domains.
If it is too long, the gain in elongating the domain cannot compensate the
line energy cost. (iii) The stability of ellipses requires that the repulsion
range $\xi$ and the cluster radius $r_{0}$ are on the same order of magnitude.

In the context of protein nanodomains, $E_{0}\approx\sigma C_{0}^{2}$ and we
use the realistic values $\sigma=10^{-4}$ J/m2, $C_{0}=0.05$ nm-1, $\lambda=1$
pN [35], and $r_{0}=200$ nm (see below). Then
$\varepsilon=E_{0}r_{0}^{3}/\lambda>1$ and $\pi r_{0}\lambda\simeq 150k_{B}T$
(where $k_{B}T$ is the thermal energy at room temperature) 222For a small
range of values of $\ell\gtrsim\ell^{*}_{s}$, two local minima of $E_{\rm
tot}$ coexist, at $a=1$ and $a>1$, for example for $\ell=12$ in Fig. 4. Since
minima are well pronounced with these values (the energetic barrier is $\gg
k_{B}T$), one may find coexistence of discs and ellipses.. Hence our scaling
law shows that for realistic parameter values, circular domains can become
unstable even at the sub-micrometric scale in the cell membrane context. It is
difficult to go beyond this rough estimate because the lipid and protein
composition of the domains in which the receptors evolve is largely unknown at
the current stage of knowledge. Thus the parameters $\lambda$ and $E_{0}$
cannot be precisely quantified.

### Numerical simulations confirm that domains on a vesicle elongate when
curving species concentration increases

Figure 5: Top: Simulation aspect ratio probability distributions of A (red)
domains for vesicles with $\bar{\phi}=0.05$ (red), $\bar{\phi}=0.20$ (green)
and $\bar{\phi}=0.35$ (blue). Other parameters are $c_{1}=8.0$,
$\tilde{\sigma}=300$ and $\tilde{J}_{I}=0.5$. Whereas the $\bar{\phi}=0.05$
and $\bar{\phi}=0.2$ distributions are close, the $\bar{\phi}=0.35$ one is
significantly different from the $\bar{\phi}=0.20$ one ($p$-value below the
computer accuracy, KS statistical test, see SM). Bottom: simulations snapshots
of the corresponding vesicles with the given values of $\bar{\phi}$.

To confirm and illustrate these results, we perform MC simulations, where we
also observe elongated nanodomains, in particular when increasing
concentration of the component forming the domains. We use a vesicle model,
the discretized version of a continuous biphasic membrane model, developed in
Ref. [36] (see also SM): a lattice-gas model, with Ising interaction parameter
$J_{I}$, describes the binary mixture of two species A and B. It is coupled to
a discretized Helfrich model accounting for the membrane elasticity, where the
local spontaneous curvature $C$ depends on the composition. In these
simulations, the species A can be considered as a phase containing the
membrane proteins of interest and/or particular lipids. It has a spontaneous
curvature $C_{\rm A}$ different from the majority phase (species B with
$C_{\rm B}=2/R<C_{\rm A}$ where $R$ is the average radius of the vesicle). We
chose $c_{1}=R(C_{\rm A}-C_{\rm B})=8$ 333The range of parameters of interest
is restricted: we want to study rather small and numerous domains, which
implies that $c_{1}=R(C_{\rm A}-C_{\rm B})$ has to be large enough [36]. On
the other hand, if $c_{1}$ is too large the domains get as small as the
lattice spacing and their AR cannot be determined accurately.. Besides, we
observe in the experimental domains as shown in Fig. 2 that boundary
fluctuations are small. This implies that the line tension $\lambda$ of the
domain boundary has to be high enough and thus that the interaction parameter
$\tilde{J}_{I}\equiv J_{I}/(k_{B}T)$ is significantly larger than its critical
value $\tilde{J}_{I,c}=\ln(3)/4\simeq 0.27$ for an hexagonal lattice.
Therefore we focus on the value $\tilde{J}_{I}=0.5$. We then run simulations
with a typical value $\kappa=20~{}k_{B}T$ and a dimensionless surface tension
$\tilde{\sigma}\equiv\sigma R^{2}/(k_{B}T)=300$, which corresponds to quasi-
spherical vesicles [58], and study rather low A-species concentration
$\bar{\phi}=0.05$ and 0.20 versus a higher one $\bar{\phi}=0.35$. We run long
simulations, up to $3\times 10^{10}$ MC steps on 2562 vertices to have good
enough statistical sampling [36]. To measure the AR of domains lying on a
quasi-spherical surface, we project each of them onto the plane tangent to the
average sphere at the domain center of mass and to compute domain covariance
matrix with its in-plane coordinates $(x,y)$. The AR is then simply the ratio
of the square roots of its two eigenvalues (see SM). Fig. 5 shows obtained AR
distributions. We note that the curves intersect at $\mathrm{AR}_{0}\simeq 2$.
Domains with an $\mathrm{AR}\leq 2$ (respectively $\mathrm{AR}>2$) are thus
called roundish (resp. elongated). The increase of concentration does not have
any significant effet on the AR distribution at low concentrations, between
$\bar{\phi}=0.05$ and 0.20. By contrast, when $\bar{\phi}$ grows from 0.20 to
0.35, one observes an increase in the proportion of elongated domains from
$28\%$ to $40\%$. In the SM, we also measure the typical cluster sizes. As
expected [36], the domains for $\bar{\phi}=0.35$ have a larger typical size
than the ones at $\bar{\phi}=0.2$. The average cluster size growing when
$\bar{\phi}$ is increased, more clusters fulfil the condition
$r_{0}^{3}>\lambda/E_{0}$ and therefore become elongated as predicted by the
analytical model. Now we estimate the parameter $\varepsilon$. We get
$\lambda\sim 0.01$ pN from the value of $J_{I}$ and a vesicle radius $R=10$
$\mu$m 444In Ref. [59], it is explained how renormalization group methods
allow one to relate $\lambda$ to $R$ and $J_{I}$. In fact, we chose a value of
the vesicle radius $R=10$ $\mu$m for illustration sake, but the dimensionless
value of $\varepsilon$ does not depend significantly on this value owing to
the way the different parameters entering $\varepsilon=\sigma
C_{0}^{2}r_{0}^{3}/\lambda$ scale with $R$: $\sigma\propto R^{-2}$,
$C_{0}\propto R^{-1}$, $r_{0}\propto R$ and $\lambda\propto R^{-1}$.. The
surface tension has been estimated to be $\sigma\sim 10^{-8}$ J/m2 with those
parameters and the domain curvature is $C_{\rm A}=(c_{1}+2)/R\sim 1$ $\mu$m-1
[59]. The observed domain radius $r_{0}$ is about $2$ $\mu$m. It follows that
$\varepsilon={E_{0}r_{0}^{3}}/{\lambda}\approx 10$, consistently larger than
1, and $\pi r_{0}\lambda\simeq 30k_{B}T$. In the SM, we are led to similar
conclusions by exploring a second numerical model where proteins are
represented as point-like objets [60] (Figs. S10 and S13).

### HIV receptor nano-domains tend to elongate under overexpression

Our laboratory has long been interested in the early mechanisms of HIV
infection and in the role of membrane domains in this process. To study the
influence of protein overexpression on the shape of the domains, we have
chosen to take advantage of the 293-Affinofile cell lines developed by
Johnston et al. [46]. This inducible cell line was originally engineered to
manipulate the CD4 and CCR5 expression levels over a range covering that found
on primary HIV-1 target cells. The proteins CD4 and CCR5 can be simultaneously
and independently regulated with the help of variable concentrations of
minocycline and ponasterone, respectively (Fig. S1). We have stably
transfected this cell line to express CXCR4, the third membrane protein that
can also be involved in HIV infection. We have established two stable cell
lines, one expressing a low number ($15000\pm 2000$ per cell) of CXCR4 and one
expressing a high number ($120000\pm 7000$) of this protein (Fig. S2). Thanks
to this, we were able to generate cells presenting any possible combination of
protein expression at their surface (Table 1, Figs. S1 and S2). Note that this
model is relevant regarding HIV infection since, whatever the expression level
of the three proteins, the cells can be infected by HIV-1 viruses (see Fig.
S3) [61, 62].

We have performed SPT experiments to track these different proteins in
different conditions. The collected data give the positions of the tracked
proteins every 40 ms for $\geq 30$ s. 544 individual trajectories have been
acquired with different expression levels of each protein. 35% of the
trajectories showed a free diffusion over the entire duration of the
measurement and 65% showed confined diffusion (either permanently or
transiently). For each of the latter trajectories, we have isolated the
confinement zones thanks to the confinement index $\Lambda$ of Ref. [52] (see
Materials and Methods). The duration of the confined trajectories is at least
4 times larger, and in general much longer, than the diffusion time to explore
the whole confinement zone (Fig. S4). Indeed by construction, the confinement
index detects a transient confinement zone because the spatial extent of the
trajectory is significantly smaller than it would be expected for free
diffusion. We have then measured the sizes and shapes of these confinement
zones. If we again define the radius $r_{0}$ through the ellipse area $A=\pi
r_{0}^{2}$ (see analytical model), we find the typical values $r_{0}\simeq
150$ nm (respectively 200 nm) in the low (resp. high) expression state (Table
S1).

Figure 6: Experimental aspect ratio probability distributions when all
proteins have a low (blue, $n_{\rm low}=165$ measurements in 113 cells
acquired in 21 independent experiments) or high (red, $n_{\rm high}=317$
measurements in 211 cells acquired in 38 independent experiments) expression
level ($p$-value $<10^{-7}$ for KS statistical test, see SM). Inset: fractions
of elongated nanodomains in both conditions. Error bars are standard errors of
the mean.

In Fig. 6 we compare the AR distributions of the three proteins that we have
pooled in the context where the three proteins have simultaneously either low
or high expression levels, one of them being tracked (the individual
distributions before pooling are shown in Fig. 7). We observe a significant
increase in the proportion of elongated versus rounded domains. In the SM, we
also explain that some confinement zones with $\mathrm{AR}\leq 2$ are however
classified as elongated because they are in fact curled or coiled elongated
nanodomains. They appear as peaks in Fig. 6, to be compared to Fig. S6 (where
the distribution without curled or coiled nanodomains is also shown). In
overexpressed conditions, the AR distribution is significantly shifted towards
higher values and the proportion of elongated nanodomains raises from 47% to
61%.

Figure 7: Same as Fig. 6 for the three types of proteins tracked separately,
when they all have low (blue) and high (red) expression levels. (a) $p\simeq
2\times 10^{-4}$, $n_{\rm low}=52$ measurements in 29 cells acquired in 7
independent experiments, $n_{\rm high}=111$ measurements in 83 cells acquired
in 13 independent experiments, (b) $p\simeq 0.002$, $n_{\rm low}=21$
measurements in 15 cells acquired in 4 independent experiments, $n_{\rm
high}=131$ measurements in 88 cells acquired in 15 independent experiments,
and (c) $p\simeq 0.001$, $n_{\rm low}=92$ measurements in 69 cells acquired in
10 independent experiments, $n_{\rm high}=75$ measurements in 47 cells
acquired in 9 independent experiments. Protein name is written in lowercase
letters when expressed at low level and in uppercase letters when
overexpressed. The protein that has been tracked is underlined. Error bars on
histograms are standard errors of the mean (s.e.m.). The $p$-values are not
calculated with these error bars but with the full distributions via a
Kolmogorov-Smirnov statistical test (see below). In (d), the cumulative
distribution of (a).  Figure 8: Same as Fig. 6 for different conditions [(a)
$p\approx 9\times 10^{-4}$, $n_{\rm low}$, $n_{\rm high}=29$ measurements in
16 cells acquired in 5 independent experiments, and (b) $p\approx 0.03$,
$n_{\rm low}$ and $n_{\rm high}=33$ measurements in 19 cells acquired in 6
independent experiments. Protein name is written in lowercase letters when
expressed at low level and in uppercase letters when overexpressed. The
protein that has been tracked is underlined.

Then, we have compared the behavior of each protein individually in a low or
high expression context. In the Figures displayed below, we use the following
notations for the experimental conditions: when a protein is expressed at a
low expression level, its name is written in lowercase letters, when a protein
is overexpressed, its name is written in uppercase letters. The protein that
has been tracked is underlined. We also observe a significant increase in the
proportion of elongated versus rounded domains regardless of the protein being
considered (Fig. 7). Alternatively, the cumulative distribution of panel (a)
shown in panel (d) displays a shift to higher AR values upon overexpression.
To go further, we have focused on the single-spanning CD4 and seven-spanning
CCR5 receptors that have been abundantly studied in the literature [44, 63,
64, 65]. We have analyzed changes of the shape of the nanodomains of each of
these proteins when it is the only one to be overexpressed. In Fig. 8, we
observe that overexpression of the sole CCR5 is accompanied by a strong
increase of the proportion of elongated nanodomains (33% to 76%) while for the
CD4, we have observed a slight increase of the proportion of elongated
nanodomains (46% to 52%), however statistically significant.

## Discussion

This work thus combines theoretical and experimental approaches to show that
increasing the concentration of the minority phase can lead to a noticeable
elongation of nanodomains in membranes, as already observed experimentally
[32] or in numerical simulations [66]. However, to our knowledge, this effect
had never been quantified so far. A simple physical mechanism in thermodynamic
equilibrium can be put forward to explain why elongated nanodomains are more
stable than roundish ones under favorable circumstances. When $\bar{\phi}$
grows, nanodomains become more and more numerous with a growing typical size
$r_{0}$ (see Fig. S11) [67, 36]. However, too large domains are intrinsically
unstable, because of the effective long-range repulsion due, for example, to
membrane deformation induced by the spontaneous curvature of the domain
constituents. One way of dealing with this instability is to generate more
elongated nanodomains above a critical size, in which the repulsive energy (of
magnitude $E_{0}$) is lower, at the price, however, of a higher line energy,
proportional to the line tension $\lambda$. More quantitatively, we propose a
scaling argument, which writes $E_{0}r_{0}^{3}>\lambda$, predicting when
circular domains become unstable at the benefit of elongated ones, provided
that the typical domain size $r_{0}$ must be comparable to the range $\xi$ of
the repulsion. In our SPT experiments, we have measured $r_{0}\approx 200$ nm
when receptors are overexpressed which indeed corresponds to the realistic
value $\xi=\sqrt{\kappa/\sigma}\sim 100$ nm in cells [35].

Pointing out the analogy between experimental and numerical nanodomain
morphologies implicitly assumes that a curvature-composition coupling
mechanism is at play in the case of HIV receptror-containing nano-domains, at
least for CCR5 and CXCR4, two class-A G-Protein Coupled Receptors (GPCRs). The
spontaneous curvature induced by class-A GPCRs has very recently been
investigated in detail in live cells [68]. A spontaneous curvature of about
0.04 nm-1 is deduced, presumably related to the crystal structures of those
GPCRs that reveal their transmembrane part to be up-down asymmetric across the
bilayer. This spontaneous curvature is typically in the range of values that
can promote sub-micrometric nanodomains, as expected [35].

To our knowledge, no such measurements have been performed on CCR5 or CXCR4.
But they belong to the same class A [38] and thus share structural
similarities with those of Ref. [68]. The effect of overexpression is less
marked for CD4. CD4 has only one transmembrane segment and its impact on the
local membrane curvature should be less important than for CCR5 which has
seven transmembrane segments.

The present work then shows that the accumulation of proteins into nanodomains
can conduct to a change of their morphology. This could probably be
extrapolated to other membrane proteins since such “untypical” nanodomain
shape has already been observed with other proteins in different cell types
without being explained so far. However, this mechanism should not be confused
with the one studied for example in Ref. [69]. There, very anisotropic
proteins adsorbed on the membrane, such as BAR domains, tend to organize in
linear structures with a strong order due to strong interactions. In these
structures, each protein has a fixed position and cannot diffuse easily,
contrary to what we observe here. In our case, the domains are likely much
more disordered, with a liquid-like order. Here the proteins under study may
also induce a moderate anisotropic curvature, but anisotropy is rapidly
averaged out due to thermal fluctuations and fast rotational diffusion [70].

Domain elongation thus reveals local accumulation of specific proteins in the
cell membrane. Such an effect, influencing biological processes as HIV entry
[71, 27], can be revealed by SPT thanks to its unique performances. From a
soft-matter physics viewpoint [72], we conjecture further that elongation of
nanodomains is the signature of the transition between 2D hexagonal and
lamellar phases, where the minority phase transforms into parallel stripes
[73]. Indeed, in phase diagrams ensuing from approximate calculations, a
region of coexistence between these two phases was identified [74] that might
contain the elongated nanodomain stability region, as numerical simulations
suggest it [66, 75]. Refined calculations will be necessary to get a full
understanding of meso-patterning in the future.

## Acknowledgments

We are indebted to Blandine Doligez for her sound advice on non-parametric
statistical tests and to Thorsten Lang for indicating us the existence of
elongated nanodomains observed by STED under overexpression of syntaxin. We
thank Samuel Tranier for his help in the determination of the diameter of CD4,
CCR5 and CXCR4 proteins.

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See pages - of DomainShapeSI_2drev.pdf


# A Wiener process perspective on local intrinsic dimension estimation methods

Piotr Tempczyk  Institute of Informatics, UW  NASK  PL4AI  Łukasz Garncarek
PL4AI  Snowflake  Dominik Filipiak  Adam Mickiewicz University, Poznań
Perelyn  University of Innsbruck  Adam Kurpisz  PL4AI  BFH Bern Business
School  ETH Zurich

###### Abstract

Local intrinsic dimension (LID) estimation methods have received a lot of
attention in recent years thanks to the progress in deep neural networks and
generative modeling. In opposition to old non-parametric methods, new methods
use generative models to approximate diffused dataset density and scale the
methods to high-dimensional datasets like images. In this paper, we
investigate the recent state-of-the-art parametric LID estimation methods from
the perspective of the Wiener process. We explore how these methods behave
when their assumptions are not met. We give an extended mathematical
description of those methods and their error as a function of the probability
density of the data.

##  1 Introduction

LID estimation has gained increasing attention in recent years as part of the
fast-growing field of topological data analysis. It was able to progress from
non-parametric methods to parametric ones thanks to the latest progress in the
field of generative modeling. LID estimation methods are algorithms for
estimating the manifold’s dimension from which the data point $x$ was sampled.
In those methods, we assume that the data lie on the union of one or more
manifolds that may be of different dimensions.

The estimation of intrinsic dimension is a critical issue in data analysis and
machine learning [Ansuini et al., 2019, Li et al., 2018, Rubenstein et al.,
2018] and was investigated in relation to dimensionality reduction and
clustering [Vapnik, 2013, Kleindessner and Luxburg, 2015, Camastra and
Staiano, 2016], analyzing the training and representation learning processes
within deep neural networks [Li et al., 2018, Ansuini et al., 2019, Pope et
al., 2020, Loaiza-Ganem et al., 2024], verifying the union of manifolds
hypothesis for images Brown et al. [2022], and used to improve out-of-
distribution detection algorithm Kamkari et al. [2024a].

Before Tempczyk et al. [2022], two approaches were presented to solve the
intrinsic dimension estimation problem, both based on non-parametric methods.
The first approach employs global methods such as PCA and its variants; see,
e.g., Fukunaga and Olsen [1971], Minka [2000], Fan et al. [2010]. These
methods are known to suffer from issues related to a manifold curvature and
non-uniformity of the data distribution. They also assumed that data lies on a
single manifold of constant dimension, so dimensionality is the same for all
$x$. The second approach is based on local methods that explore the geometric
properties of neighborhoods Johnsson et al. [2014], Levina and Bickel [2004]
calculating some statistics using points from the neighborhood of $x$.
Although all aforementioned methods perform reasonably well for a small number
of dimensions, the higher dimensionality affects their performance Tempczyk et
al. [2022], Campadelli et al. [2015], Camastra and Staiano [2016].

Very recently, a new approach emerged for the local methods, which is based on
a two-step procedure. In the first step, the dataset is perturbed with some
noise, and in the second step, its dimensionality is estimated using various
techniques. These include generative models to analyze changes in density
Tempczyk et al. [2022], Kamkari et al. [2024a] or in singular values of the
Jacobian Horvat and Pfister [2022, 2024] for different noise magnitudes, and
analyzing rank of scores from diffusion model as mentioned in Stanczuk et al.
[2024].

Although the new algorithms provide state-of-the-art performance for large and
real-life data sets, it is crucial for their performance that adding noise to
the data set is an injective process, which means that the resulting densities
after adding noise uniquely identify the original distribution of the data
set. Otherwise, it is impossible to neither uniquely reverse the process nor
analyze the original structure of the data set. One example in the theoretical
model that does not admit such a problem is a flat manifold with a uniform
distribution of data points. In such a case those algorithms perform best, as
showed in experimental results. In other cases, a hypothetical line of attack
would be to decrease the magnitude of noise added to the data set so that the
manifold is approximately flat and uniform in the scale of the noise
magnitude. This, however, is not possible in practice for several reasons. One
is that data is often a set of discrete points (especially in the case of
audio and visual data), and considering the noise of much smaller magnitude
than the minimum distance between points does not lead to any meaningful
process. Moreover, neural net training is done with finite precision and the
stochastic gradient descent procedure introduces noise to the density
estimates, so very small changes in density cannot be observed in practice,
which may lead to poor quality estimates of LID.

In this paper, we point out and exploit the fact that adding Gaussian noise of
varying magnitudes can be seen as studying the evolution of the Wiener process
describing the diffusion of particles (points of the dataset) in space. This
point of view enables us to employ Fick’s Second Law of Diffusion to eliminate
time derivatives from mathematical descriptions of state-of-the-art LID
algorithms Tempczyk et al. [2022], Kamkari et al. [2024a], and replace them
with spatial derivatives. Such considerations can be taken into account in the
second step of the considered algorithms, leading to more accurate results.

#### Contribution

  1. 1.

We explore the first step of existing algorithms in the language of Wiener
process and calculate important cases of diffusion from lower-dimensional
manifolds with non-uniform probability density into higher dimensional ambient
space.

  2. 2.

We recognize and define new categories (isolated and holistic algorithms) for
the Wiener process based parametric LID estimation algorithm family and
categorize existing algorithms accordingly.

  3. 3.

We derive closed-form expressions for crucial parameters used in two state-of-
the-art isolated LID estimation algorithms as a function of on-manifold
density and manifold dimensionality, which can be viewed as closed-form
expression for deviation from flat manifold with uniform distribution case for
them.

##  2 Related work

The review of non-parametric methods for local and global intrinsic dimension
estimation can be found in Campadelli et al. [2015], Camastra and Staiano
[2016], and a broad comparison of those methods on bigger datasets (in terms
of dimensionality) can be found in Tempczyk et al. [2022].

Although we do not analyze non-parametric methods in this paper, the
interesting and worth mentioning recent work on non-parametric LID estimation
is Bailey et al. [2022]. Authors explore the connection of LID to other well-
known measures for complexity: entropy and statistical divergences and develop
new analytical expressions for these quantities in terms of LID. The same
authors in their follow-up paper Bailey et al. [2023] establish the
relationships for cumulative Shannon entropy, entropy power, Bregman
formulation of cumulative Kullback–Leibler divergence, and generalized Tsallis
entropy variants, and propose four new estimators of LID, based on nearest
neighbor distances.

###  2.1 Parametric methods for LID estimation

During the last few years many parametric methods for estimating LID emerged.
In Zheng et al. [2022] the authors prove that VAE are capable of recovering
the correct manifold dimension and demonstrate methods to learn manifolds of
varying dimension across the data sample. Authors of Yeats et al. [2023]
connects the adversarial vulnerability of score models with the geometry of
the underlying manifold they capture. They show that minimizing the Dirichlet
energy of learned score maps boosts their robustness while revealing LID of
the underlying data manifold.

#### Wiener process based algorithms

Among parametric methods, there is a group of algorithms, that have one thing
in common: they simulate a Wiener process on a dataset and directly use some
properties of time evolving density to estimate LID. We can divide those
algorithms into three groups:

  1. 1.

LIDL Tempczyk et al. [2022] and its efficient and most accurate implementation
using diffusion models called FLIPD Kamkari et al. [2024b]. These algorithms
use the rate of change of the probability density at point $x$ during the
Wiener process to estimate LID at $x$. For small diffusion times $t$ the
logarithm of a density is a linear function of a logarithm of $t$, and the
proportionality coefficient is equal $d-D$ for small $t$, where $d$ is
manifold density and $D$ is ambient space dimensionality. Authors of FLIPD
show, that their algorithm is efficient and our experiments show, that is more
scalable than algorithms from the second group (and with high probability more
accurate, but there is a lack of comparison between FLIPD and methods from
second group) and more accurate than algorithms from the third group, which
led us to the conclusion that at this moment this family of algorithms is
state-of-the-art in LID estimation.

  2. 2.

ID-NF Horvat and Pfister [2022] (using normalizing flows), and the follow-up
paper Horvat and Pfister [2024] ID-DM (using diffusion models) algorithms
analyze how a singular values of a Jacobian of a function transforming a
standard normal distribution into a diffused dataset at $x$ evolves during the
Wiener process. Authors use an observation, that when transforming a
$d$-dimensional manifold into a $D$-dimensional Gaussian we have to expand
space more in the off-manifold directions, especially for small diffusion
times $t$.

  3. 3.

The third group, for now, is composed of one algorithm: Stanczuk et al.
[2024]. We will call this algorithm NB – Normal Bundle – following the
convention from Kamkari et al. [2024b]. Authors of NB make an observation,
that for small diffusion times $t$ the score function from the Score-based
diffusion model Song et al. [2020] lies in the normal space to the manifold at
$x$. The procedure is the following: calculate the score function for a bunch
of diffused versions of $x$ for a fixed diffusion time $t$, stack them as
column vectors into a matrix $S$, and estimate the rank of $S$, which is then
used as an estimate of $D-d$.

##  3 Isolated and holistic algorithms

In this work are going to take a closer look at the Wiener process based
algorithms described in the last paragraph of Section 2. Although all of those
algorithms apply a Wiener process to the dataset during their first phase,
when looking at their second phase, we can divide them into two groups:
isolated (LIDL, FLIPD, NB) and holistic (ID-NF, ID-DM). The intermediate
results of the first group used to calculate LID use only (assuming that a
generative model approximates diffused data distribution $\rho_{t}$ perfectly)
information about the shape of the data probability density function (without
normalization constant). Their estimate depends on the gradient of the
logarithm of $\rho_{t}$ around $x$ (NB) or $d\log\rho_{t}/d\log t$ at $x$
(LIDL, FLIPD). Because in practice $\rho_{t}$ is a function of the values of
original data distribution $p_{S}$ in a ball of radius $r\approx 4\sqrt{t}$
around $x$ in the data space. What the values of $p_{S}$ outside this ball are
do not matter in practice for isolated algorithms because the diffused
particles in the Wiener process can travel longer distances than $r$ with very
low probability (less than $3.7\cdot 10^{-5}$). When we add some new data
points to the dataset far away from $x$, it doesn’t change the shape of $\psi$
close to $x$ (this operation only changes a normalization constant, which
becomes 0 after taking a logarithm of $\psi$ and taking a derivative).

The holistic algorithms work quite differently. In the case of ID-NF and ID-
DM, they calculate singular values of the Jacobian of the function
$\zeta:\mathrm{R}^{N}\rightarrow\mathrm{R}^{N}$ transforming $\rho_{t}$ into a
standard normal distribution. This $\zeta$ function strongly depends on the
entire shape of $\rho_{t}$. When we change $\rho_{t}$ far away from $x$ we
don’t change the shape of $\rho_{t}$ close to $x$. The same is not true for
$\zeta(x)$. When we add many data points to the dataset far away from $x$ and
we keep the latent distribution fixed we have to change the way we compress
and stretch the space to match the new distribution. This property makes the
analysis of the behavior of ID-NF and ID-DM much harder (maybe even
impossible) as a function of density evolution around $x$.

To give an illustrative example of this behavior let’s imagine that our
dataset is one-dimensional and consists of 10K points sampled from
$\mathcal{N}\left(0,1\right)$. Because we are transforming it into
$\mathcal{N}\left(0,1\right)$, $\zeta$ is an identity function. Now let’s add
to the dataset another 10K points sampled from
$\mathcal{N}\left(100,1\right)$. Now we have to stretch our space in some
areas to transform one density into another and our $\zeta$ function (and its
Jacobian) is far from being an identity function.

This approach can be extended for new algorithms in the future, and similar
analysis can be carried out for the new algorithms from isolated group if only
some intermediate results used to calculate LID can be expressed in the
analytical form.

##  4 Wiener process perspective on LID estimation

Wiener process is a stochastic process modeling particle diffusion. Its
increments over disjoint time intervals are independent and normally
distributed, with variance proportional to time increments. Since in the
machine learning community the term _diffusion_ is already overloaded, we will
stick to Wiener process when speaking of particle diffusion process.

In this section we present a new perspective on perturbing datasets, unifying
the approaches seen in the algorithms presented in Tempczyk et al. [2022],
Stanczuk et al. [2024], Horvat and Pfister [2022, 2024], Kamkari et al.
[2024b]. As already mentioned, all these algorithms consist of two stages, the
first of which amounts to perturbing the dataset with normally distributed
random noise of fixed variance $t$. In the second stage, each of the
algorithms utilizes the behavior of the perturbed density in the neighborhood
of a fixed point under changes of the noise variance.

The first phase of each algorithm can be interpreted as applying the Wiener
process to the points in the dataset. Afterward, the resulting set of points
is used to train a generative model (depending on the chosen algorithm) to
estimate the distribution of the dataset undergoing the Wiener process at time
$t$. From the point of view of differential equations, the distribution
density function of the diffused dataset is described by Fick’s Second Law of
Diffusion.

######  Fick’s Second Law of Diffusion.

Let $\rho_{t}:\mathbb{R}^{D}\mapsto\mathbb{R}$ denote the probability density
function modeling particles undergoing diffusion at time $t$. Then $\rho_{t}$
satisfies the differential equation

$\frac{d}{dt}\rho_{t}=C\Delta\rho_{t},$ (1)

where $C\in\mathbb{R}$, and $\Delta$ stands for the standard Laplacian in
$\mathbb{R}^{D}$.

Now, let us get to the specifics. Given a dataset embedded in
$\mathbb{R}^{D}$, we assume that it has been drawn from some latent union of
submanifolds $S$ endowed with a probability measure $p_{S}$ (which can be
naturally treated as probability measure on $\mathbb{R}^{D}$). The goal of
Local Intrinsic Dimension estimation is to find out the dimension of $S$ at
any point of the dataset.

To model the Wiener process with initial distribution $p_{S}$ (which is not a
function on $\mathbb{R}^{D}$), let us first define

$\phi_{t}^{D}(x)=(2\pi t)^{-D/2}e^{-\left\lVert{x}\right\rVert^{2}/2t}.$ (2)

This is the density of normal distribution on $\mathbb{R}^{D}$ with covariance
matrix $tI$. It is the fundamental solution of the differential equation given
by Fick’s second Law of Diffusion (1) with $C=1/2$. In our context, this means
that the convolution

$\rho_{t}=p_{S}*\phi_{t}$ (3)

is the solution of (1) for $t>0$ and hence it describes the Wiener process
starting from the initial probability distribution $p_{S}$.

To limit the complexity introduced by curvature, we will consider from now on
only flat manifolds. This means that, without loss of generality, we may
assume that $S$ is the first factor in product decomposition
$\mathbb{R}^{D}=\mathbb{R}^{d}\times\mathbb{R}^{D-d}$. We will denote the
coordinates of $\mathbb{R}^{d}$ and $\mathbb{R}^{D-d}$ by $x$ and $y$,
respectively. If we moreover assume that $p_{S}$, now a probability
distribution on $\mathbb{R}^{d}$, has a density
$\psi\colon\mathbb{R}^{d}\to\mathbb{R}$, we can separate variables in (3),
obtaining

$\rho_{t}(x,y)=(\psi*\phi_{t}^{d})(x)\phi_{t}^{D-d}(y)$ (4)

for $(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{D-d}$.

The decomposition $\mathbb{R}^{D}=\mathbb{R}^{d}\times\mathbb{R}^{D-d}$ gives
rise to the decomposition of the Laplacian on $\mathbb{R}^{D}$ into Laplacians
on the factors, namely $\Delta=\Delta_{x}+\Delta_{y}$, by simply grouping the
derivatives with respect to $x_{i}$ and $y_{i}$ coordinates.

######  Lemma 4.1.

For $t>0$ and $x\in\mathbb{R}^{d}$ we have

$\Delta\rho_{t}(x,0)=\frac{d-D}{t}\rho_{t}(x,0)+(2\pi
t)^{(d-D)/2}\Delta_{x}(\psi*\phi_{t}^{d})(x).$ (5)

###### Proof.

By using the product decomposition (4), we get

$\Delta\rho_{t}(x,y)=(\psi*\phi_{t}^{d})(x)\Delta_{y}\phi_{t}^{D-d}(y)+\Delta_{x}(\psi*\phi_{t}^{d})(x)\phi_{t}^{D-d}(y)$
(6)

Since $\phi_{t}^{D-d}(0)=(2\pi t)^{(d-D)/2}$, we obtain

$\Delta_{x}(\psi*\phi_{t}^{d})(x)\phi_{t}^{D-d}(0)=(2\pi
t)^{(d-D)/2}\Delta_{x}(\psi*\phi_{t}^{d})(x),$ (7)

which corresponds to the second term of (5).

To derive the first term, we note that a direct computation yields the
following.

$\Delta_{y}\phi_{t}^{D-d}(y)=\phi_{t}^{D-d}(y)\left(\frac{\left\lVert{y}\right\rVert^{2}}{t^{2}}+\frac{d-D}{t}\right).$
(8)

Evaluating this at $y=0$ provides the desired result. ∎

Let us stop here for a moment to discuss the expression
$\Delta_{x}(\psi*\phi_{t}^{d})$. When applying a differential operator to a
convolution, under certain regularity conditions we can move it to either of
the factors, which can turn out to be quite helpful. As it will turn out in
the examples, $\Delta_{x}\psi*\phi_{t}^{d}$ will be the most desired form of
$\Delta_{x}(\psi*\phi_{t}^{d})$.

More precisely, given a convolution $f*g$, and a $k$-th order differential
operator $L$, the following conditions guarantee that $L(f*g)=Lf*g$
(conditions for the other equality follow from commutativity of convolution):

  1. 1.

All partial derivatives of $f$ up to order $k$ (including $0$-th order, i.e.
$f$ itself) exist and are bounded.

  2. 2.

The function $g$ is integrable, i.e. $\int\lvert{g}\rvert<\infty$.

The second condition is clearly satisfied by densities of probability
distributions, and $\phi^{d}_{t}$ satisfies the first condition. Hence, for
any $\psi$ we can write

$\Delta_{x}(\psi*\phi_{t}^{d})=\psi*\Delta_{x}\phi_{t}^{d}.$ (9)

The second equality however requires additional assumptions;

$\Delta_{x}(\psi*\phi_{t}^{d})=\Delta_{x}\psi*\phi_{t}^{d},$ (10)

holds if $\psi$ is bounded and has bounded partial derivatives up to order
$2$.

##  5 From Wiener process to LID estimation in LIDL

We are going to use results from the last section to analyze how the first
family of algorithms [Tempczyk et al., 2022, Kamkari et al., 2024b] behaves
for some particular cases. Because the main contribution from authors of
Kamkari et al. [2024b] is an substantial improvement on the side of density
estimation, from the theoretical perspective when dealing with perfect density
estimators, both algorithms estimate the same quantity and give the same
results. This is the reason from now on we will be analyzing LIDL because we
want to analyze the aspects of those implementations that do not depend on the
problems with density estimation itself.

Given a point $x\in S$ and a set of times $t_{1},\dots,t_{n}$, LIDL estimates
the linear regression coefficient $\alpha$ of the set of points
$(\log\delta_{i},\log\rho_{t_{i}}(x))$, where $\delta_{i}=\sqrt{t_{i}}$. It is
proved in Tempczyk et al. [2022] that

$\log\rho_{t}(x)=(d-D)\log\sqrt{t}+O(1),$ (11)

and therefore $\alpha\approx d-D$. The authors show that if $t$ is small
enough, this estimate is accurate.

This procedure can be seen as approximating the asymptotic slope of the
parametric curve $(\log\sqrt{t},\log\rho_{t}(x))$, or in other words, the
graph of $s\mapsto\log\rho_{e^{2s}}(x)$ for $s\to-\infty$. Another approach
would be to study the derivative of this function. Let us define

$\beta_{t}(x)=\frac{2t}{\rho_{t}(x)}\frac{d}{dt}\rho_{t}(x)=\frac{t}{\rho_{t}(x)}\Delta\rho_{t}(x),$
(12)

where the last equality comes from the diffusion equation (1). Moreover,
denote the asymptotic slope of the aforementioned curve by

$\beta(x)=\lim_{s\to-\infty}\frac{d}{ds}\log\rho_{e^{2s}}(x)=\lim_{t\to
0^{+}}\beta_{t}(x).$ (13)

The next proposition will enable us to conclude that indeed these two
approaches are equivalent.

######  Proposition 5.1.

Given a strictly positive differentiable function $f\colon(0,a)\to(0,\infty)$
and a positive real number $\alpha>0$, the following conditions are
equivalent.

  1. 1.

The function $f$ explodes at $0$ like $t^{-\alpha}$, i.e. for some positive
constants $c,C>0$ one has $c<t^{\alpha}f(t)<C$ for some $\epsilon>0$ and
$t\in(0,\epsilon)$.

  2. 2.

$\log f(t)=-\alpha\log t+O(1)$.

  3. 3.

$\lim_{t\to 0^{+}}{\log f(t)}/{\log t}=-\alpha$.

  4. 4.

$\lim_{t\to 0^{+}}{tf^{\prime}(t)}/{f(t)}=-\alpha$.

###### Proof.

All limits will be taken with $t\to 0$, and we will omit this from notation.

If $t<t^{\alpha}f(t)<C$, then $\log(t^{\alpha}f(t))=O(1)$. Rearranging leads
to point 2. Going further, dividing by $\log x$ and observing that $\lim
O(1)/\log t=0$ yields point 3.

Now, assume condition 3. Since $\lim\log t=-\infty$, for the whole expression
to tent to $-\alpha<0$, we need $\lim\log f(t)=\infty$. In this case however
we may apply the de l’Hospital rule giving rise to point 4.

It remains to show the final implication. Let $g(t)=t^{\alpha}f(t)>0$ be the
function we want to bound. Substituting $f(t)=t^{-\alpha}g(t)$ into
$\lim_{t\to 0}{tf^{\prime}(t)}/{f(t)}=-\alpha$, we obtain

$\lim\frac{t(-\alpha
t^{-\alpha-1}g(t)+t^{-\alpha}g^{\prime}(t))}{t^{-\alpha}g(t)}=\lim\left(-\alpha+\frac{g^{\prime}(t)}{g(t)}\right)=-\alpha,$
(14)

amounting to $\lim(\log g(t))^{\prime}=0$. But this means that for some
$\epsilon>0$, the derivative $(\log g(t))^{\prime}$ is bounded on
$(0,\epsilon)$, implying that $\log g(t)$ is Lipschitz and therefore bounded
on this interval. This yields the desired estimates. ∎

Now, by putting $f(t)=\rho_{t}(x)$, we can see that the last condition of
Proposition 5.1 holds with $\alpha=-\beta(x)/2$. Thus, the equivalent
condition 2. yields the following equivalence of both presented approaches.

######  Corollary 5.2.

For $t$ near $0$ the following estimate holds

$\log\rho_{t}(x)=\beta(x)\log{\sqrt{t}}+O(1).$ (15)

As a consequence, the estimation of Local Intrinsic Dimension using LIDL can
be achieved by computing $\beta(x)$, yielding $d=D+\beta(x)$.

The next proposition provides an elegant expression for $\beta_{t}(x)$, and
consequently for $\beta(x)$, expressed in terms of the density $\psi$ on
$\mathbb{R}^{d}$.

######  Proposition 5.3.

For $t>0$ and $x\in S=\mathbb{R}^{d}\subseteq\mathbb{R}^{D}$ we have

$\beta_{t}(x)=d-D+\frac{\Delta_{x}(\psi*\phi_{t}^{d})(x)}{\psi*\phi_{t}^{d}(x)}\cdot
t.$ (16)

###### Proof.

Since $\rho_{t}$ is the solution of the diffusion equation on $\mathbb{R}^{D}$
with $C=1/2$, we can replace the time derivative with $\Delta/2$, obtaining by
Lemma 4.1

$\beta_{t}(x)=\frac{2t}{\rho_{t}(x)}\frac{d}{dt}\rho_{t}(x)=\frac{t}{\rho_{t}(x)}\Delta\rho_{t}(x)=d-D+\frac{t(2\pi
t)^{(d-D)/2}\Delta_{x}(\psi*\phi_{t}^{d})(x)}{\rho_{t}(x)},$ (17)

where in the second term $(2\pi t)^{(d-D)/2}$ cancels out after expanding
$\rho_{t}$ in the denominator. ∎

###  5.1 Examples

From the theoretical considerations in Tempczyk et al. [2022] it follows, that
$\beta(x)=d-D$ if $\psi$ is sufficiently regular and positive near $x$. In
other words, in this case

$\lim_{t\to 0^{+}}\frac{\psi*\Delta\phi_{t}^{d}(x)}{\psi*\phi_{t}^{d}(x)}\cdot
t=0.$ (18)

In this section, we will try to obtain this conclusion directly in a few
special cases by analyzing the behavior of $\beta_{t}(x)$.

#### The “uniform distribution” on Euclidean space

Clearly, there is no such thing as the uniform distribution on
$\mathbb{R}^{d}$. However, from a purely theoretical viewpoint, in our
differential equation approach we don’t need the assumption of $\phi$ being a
probability density; it could be any function. And since constant functions
are usually the simplest examples, we will now investigate what happens if we
put $\psi(x)\equiv 1$ on the whole $\mathbb{R}^{d}$ space.

Using Proposition 5.3 and the fact that $\psi$ has bounded derivatives, this
case leaves us with

$\beta_{t}(x)=d-D+\frac{\Delta_{x}\psi*\phi_{t}^{d}(x)}{\psi*\phi_{t}^{d}(x)}\cdot
t=d-D,$ (19)

since $\Delta_{x}\psi\equiv 0$. This expression is constant in $t$, and in
particular its limit at $0$ is $\beta(x)=d-D$. In this case LIDL estimator is
not biased for all $t>0$.

#### Normal distribution

Let us now consider the normal distribution on $\mathbb{R}^{d}$ with
covariance matrix
$\Sigma=\mathop{\operatorname{diag}}(\sigma_{1}^{2},\dots,\sigma_{d}^{2})$,
and denote its density function by $\psi$. The convolution $\psi*\phi_{t}^{d}$
is the density of the normal distribution with covariance matrix $\Sigma+tI$.
If we simplify notation be putting $\phi_{i}=\phi_{\sigma_{i}^{2}+t}^{1}$, we
get

$\psi*\phi_{t}^{d}(x)=\prod_{i=1}^{d}\phi_{i}(x_{i}).$ (20)

To compute the Laplacian of this convolution, note that

$\psi*\phi_{t}^{d}(x)=\frac{\psi*\phi_{t}^{d}(x)}{\phi_{i}(x_{i})}\cdot\phi_{i}(x_{i}),$
(21)

where the first factor does not depend of $x_{i}$, and therefore

$\frac{\partial^{2}(\psi*\phi_{t}^{d})}{\partial
x_{i}^{2}}(x)=\psi(x)*\phi_{t}^{d}(x)\cdot\frac{1}{\phi_{i}(x_{i})}\frac{\partial^{2}\phi_{i}}{\partial
x_{i}^{2}}(x_{i}),$ (22)

leading to

$\beta_{t}(x)=d-D+t\frac{\Delta(\psi*\phi_{t}^{d})(x)}{\psi*\phi_{t}^{d}(x)}=d-D+t\sum_{i=1}^{d}\frac{1}{\phi_{i}(x_{i})}\frac{\partial^{2}\phi_{i}}{\partial
x_{i}^{2}}(x_{i})=d-D+t\sum_{i=1}^{d}\frac{x^{2}_{i}-(\sigma_{i}^{2}+t)}{(\sigma_{i}^{2}+t)^{2}}.$
(23)

It is easy to see that the second derivatives of $\phi_{i}$ are continuous in
$t>-\sigma_{i}^{2}$, so the sum in the above expression has finite limit for
$t\to 0$, and therefore $\beta(x)=d-D$.

In the special case where $\Sigma=\sigma^{2}I$, these calculations simplify
further, as $\psi*\phi_{t}^{d}=\phi_{\sigma^{2}+t}^{d}$, and since

$\Delta_{x}\phi_{\sigma^{2}+t}^{d}(x)=\left(\frac{\left\lVert{x}\right\rVert^{2}}{(\sigma^{2}+t)^{2}}-\frac{d}{\sigma^{2}+t}\right)\phi_{\sigma^{2}+t}^{d}(x),$
(24)

we have

$\beta_{t}(x)=d-D+\left(\frac{\left\lVert{x}\right\rVert^{2}}{(\sigma^{2}+t)^{2}}-\frac{d}{\sigma^{2}+t}\right)t.$
(25)

(a) Example of the bias of a LIDL estimate for different points from
$\mathcal{N}(0,1)$ and for different values of $t$. This plot recreates a
numerical calculations presented in Figure 4 from Tempczyk et al. [2022].

(b) Plot of a LIDL estimate as a function of time for 3-dimensional Gaussian
distribution with $\Sigma=\mathop{\operatorname{diag}}(1,10^{-6},10^{-12})$
and for three different points $x=(0,0,\sigma_{3})$, which represents a
different distance from 0.

Figure 1: Deviations from desired LID estimates for Gaussian distributions.

These results express analytically experimental observations from LIDL and
FLIPD papers, which we show in Fig. 1. We can also observe, that if we move to
the regions of very low probability for a Gaussian, it generates very high
positive bias, which may highly overestimate the true LID. Luckily, most of
the points in our dataset comes from the region of high probability, but we
should be less certain of the estimates for points from low probability
regions.

#### Arbitrary distribution with sufficiently nice density

By this point is should already be clear what a nice density is. We want to be
able to use the equality
$\Delta_{x}(\psi*\phi_{t}^{d})=\Delta_{x}\psi*\phi_{t}^{d}$. In order to do
so, we need $\psi$ to be bounded, twice differentiable, and have bounded first
and second-order partial derivatives. Moreover, we will require $\psi$ to have
_continuous_ second-order partial derivatives. This is not very restrictive,
and many distributions satisfy these properties, including the normal
distribution or more generally, mixtures of Gaussians.

In this case, we have

$\beta_{t}(x)=d-D+\frac{\Delta_{x}\psi*\phi_{t}^{d}(x)}{\psi*\phi_{t}^{d}(x)}\cdot
t,$ (26)

however this time $\Delta_{x}\psi$ is some arbitrary continuous function.
Being differentiable, $\psi$ is also continuous, and we can use the general
fact that for a bounded continuous function, $f$ on $\mathbb{R}^{d}$ one has

$\lim_{t\to 0^{+}}f*\phi_{t}^{d}(x)=f(x).$ (27)

This gives us, for $x$ such that $\psi(x)>0$,

$\lim_{t\to 0^{+}}\frac{\Delta_{x}\psi*\phi_{t}^{d}}{\psi*\phi_{t}^{d}}\cdot
t=\frac{\Delta_{x}\psi(x)}{\psi(x)}\lim_{t\to 0^{+}}t=0,$ (28)

and again $\beta(x)=d-D$. We have proven that in this case $\beta$ yields a
correct estimate of dimension, circumventing all the complexities of the
proofs in Tempczyk et al. [2022].

It is worth noting, that when $\Delta_{x}\psi=0$, the estimate is accurate. It
is the case for the aforementioned ”uniform distribution” on $\mathbb{R}^{d}$,
but it is also true if locally the density is a linear function of $x$. In
Fig. 1(a) we can observe, that for $x\approx\pm 1$ and small values of $t$,
the estimate is accurate.

##  6 Conclusions

In this work, we have outlined a new perspective for Wiener process based
algorithms for LID estimation and shown some results for LIDL, in which we
find an analytical description for the phenomena observed in experiments. The
future directions from here are: analyzing the uniform distribution on a line
segment, adding curvature to this analysis, and an attempt to analyze an NB
algorithm in the same way as LIDL.

##  7 Acknowledgements

Some experiments were performed using the Entropy cluster at the Institute of
Informatics, University of Warsaw, funded by NVIDIA, Intel, the Polish
National Science Center grant UMO2017/26/E/ST6/00622, and ERC Start- ing Grant
TOTAL. The rest of the experiments were performed using the GUŚLARZ 9000
workstation at the Polish Lab for AI (PL4AI).

We want to thank Marek Cygan and Adam Goliński for their overall support
during this project.

#### CRediT Author Statement

#### Piotr Tempczyk

(35% of the work): Conceptualization, Methodology, Software, Validation,
Formal analysis, Investigation, Data Curation, Writing - Original Draft,
Writing - Review & Editing, Visualization, Supervision, Project
administration.

#### Adam Kurpisz

(30% of the work): Conceptualization, Methodology, Validation, Formal
analysis, Investigation, Writing - Original Draft, Writing - Review & Editing,
Visualization, Supervision.

#### Łukasz Garncarek

(25% of the work): Conceptualization, Methodology, Formal analysis,
Investigation, Writing - Original Draft, Writing - Review & Editing.

#### Dominik Filipiak

(10% of the work): Software, Data Curation.

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