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Self-organized fractal architectures driven by motility-dependent chemotactic feedback
Subhashree Subhrasmita Khuntia,1Debasish Chaudhuri,2, 3 , ∗and Abhishek Chaudhuri1, †
1Department of Physical Sciences, Indian Institute of Science Education and Research Mohali,
Sector 81, Knowledge City, S. A. S. Nagar, Manauli PO 140306, India
2Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India
3Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India
(Dated: April 24, 2025)
Complex spatial patterns in biological systems often arise through self-organization without a
central coordination, guided by local interactions and chemical signaling. In this study, we explore
how motility-dependent chemical deposition and concentration-sensitive feedback can give rise to
fractal-like networks, using a minimal agent-based model. Agents deposit chemicals only while
moving, and their future motion is biased by local chemical gradients. This interaction generates
a rich variety of self-organized structures resembling those seen in processes like early vasculoge-
nesis and epithelial cell dispersal. We identify a diverse phase diagram governed by the rates of
chemical deposition and decay, revealing transitions from uniform distributions to sparse and dense
networks, and ultimately to full phase separation. At low chemical decay rates, agents form stable,
system-spanning networks; further reduction leads to re-entry into a uniform state. A continuum
model capturing the co-evolution of agent density and chemical fields confirms these transitions and
reveals how linear stability criteria determine the observed phases. At low chemical concentrations,
diffusion dominates and promotes fractal growth, while higher concentrations favor nucleation and
compact clustering. These findings unify a range of biological phenomena—such as chemotaxis,
tissue remodeling, and self-generated gradient navigation—within a simple, physically grounded
framework. Our results also offer insights into designing artificial systems with emergent collective
behavior, including robotic swarms or synthetic active matter.
I. INTRODUCTION
The spatial architecture of living systems emerges
through self-organization without a central coordina-
tion [1–3], often, with information propagating via self-
generated chemical signals [4–7]. These systems display
large-scale patterns, from shell and neural structures [8–
11] to ecological distributions [12,13]. Communication
among components is crucial for such coordinated pat-
terning, enabling adaptive responses to environmental
shifts [14,15]. Signal dynamics and adaptation, in turn,
limit communication efficacy [16]. A key aspect of self-
organization is the modification of the environment by
motile agents through localized signals, influencing sub-
sequent behaviors [17,18]. This manifests in quorum
sensing, biofilm formation, ant trails, neural fascicula-
tion, and pedestrian tracks, generating spatiotemporal
structures that reflect adaptive functionality and reveal
underlying local rules [19–26].
Such self-organisation should, in principle, extend to
eukaryotic chemotaxis, essential for wound healing, de-
velopment, and cancer metastasis [27,28]. However, pre-
dicting cellular chemotactic responses remains challeng-
ing due to insufficient information about chemoattrac-
tant sources and their interaction with the cell environ-
ments [29]. Often seen as a passive process where cells
follow external gradients, chemotaxis fails to account for
the active role cells play in shaping these patterns [30–33],
∗For correspondence: debc@iopb.res.in
†For correspondence: abhishek@iisermohali.ac.in
thus overlooking important physiological mechanisms.
For instance, cells like melanoma and Dictyostelium not
only respond to chemoattractants but also actively de-
grade them (e.g., lysophosphatidic acid and cAMP),
modulating the very gradients they navigate [34,35].
Similarly, epithelial cells at low densities form branched,
network-like structures in response to growth factor limi-
tation [36], suggestive of an active, self-patterning mech-
anism. Such structures are further seen during the early
stages of vasculogenesis in embryonic development [37–
41]. Despite these observations, the minimal ingredi-
ents necessary to produce such structures—particularly
fractal-like networks—remain unclear. In this study, we
explore how such behavior-mediated changes can give rise
to fractal network-like structures within a chemotactic
framework. Despite the distinct physiological properties
and underlying causes of these processes, our findings
show that similar structures can emerge naturally from
chemotactic behaviours, providing a unified framework
for understanding the development of spatial patterns in
living systems.
We investigate an agent-based model where agents
act as excluded volume diffusers depositing chemicals
along their paths. These chemical trails influence their
subsequent movements, guiding them towards areas of
higher chemical concentration [42,43]. In natural sys-
tems, however, the sensitivity of agents to concentra-
tion changes is typically limited. To account for this,
we incorporate concentration-dependent sensitivity into
our model [44]. As the chemical concentration exceeds
a critical threshold, agents’ bias toward higher concen-
trations plateaus. Chemical dynamics in the model in-
arXiv:2504.16539v1 [physics.bio-ph] 23 Apr 2025
2
clude both evaporation and deposition processes. Evap-
oration is stochastic with a decay rate β, while de-
position, characterized by αand local chemical activ-
ity, depends on motility to optimize resource utiliza-
tion [45,46]. Specifically, agents release chemicals
solely in motion, ensuring resource efficiency, ceasing
when movement halts due to crowding-induced repulsion.
These factors, concentration-limited responsiveness and
movement-dependent deposition, strongly influence the
system’s self-organization.
II. RESULTS
The details of the model is outlined in the Methods
section with the help of Fig.1(a). Consider a uniform
distribution of Nexcluded-volume agents, moving ran-
domly like particles in a gas. In the presence of a signifi-
cant self-generated chemical landscape, agents are drawn
to areas of higher chemical concentration through a pos-
itive feedback loop. Over time, this feedback loop gives
rise to spatial inhomogeneities in both agent distribu-
tion and chemical concentration. As the chemical field
strengthens, agents are increasingly drawn toward re-
gions of higher concentration, leading to clustering in-
stabilities reminiscent of the Keller-Segel (KS) model of
chemotaxis. In the KS framework, such instabilities typ-
ically emerge when chemical sensitivity or local concen-
tration exceeds a critical threshold.
To explore how this feedback mechanism governs emer-
gent structure, we systematically vary the chemical de-
position rate (α) and decay rate (β) and examine the
resulting morphologies.
A. Re-entrant transition and system spanning
network
The morphology of the system evolves dramatically
with changing chemical parameters, illustrated through
typical particle configurations and the chemical profile
in Fig. 1(b) and (c) respectively. At high decay rates
(large β), rapid evaporation prevents long-range chemi-
cal accumulation, and agents remain homogeneously dis-
tributed (HP phase) with minor, transient clustering
(fourth columns of Fig. 1(b) and (c), Supplementary
Movie S1). As βdecreases, the slower decay allows lo-
cal concentrations to build up, reinforcing aggregation
via chemotactic feedback. This initiates the formation
of sparse networks (SN phase, third column of Fig. 1(b)
and (c), Supplementary Movie S2), which transition into
dense, branching structures as decay is further reduced
(columns 1 and 2 in row 3 of Fig. 1(b) and (c), Sup-
plementary Movie S3). At low βand moderate α, sharp
chemical gradients develop, stabilizing intricate, system-
spanning fractal networks (Fig. 1d). These networks are
stabilized by directional chemical bias and spatial exclu-
sion, which prevent collapse into compact clusters. At
higher deposition rates (α), the network structures col-
lapse into a large, dense aggregate marked as a single
cluster phase (SC, row 1, column 2 and row 2, column
1 in Fig. 1(b,c) Supplementary Movie S4). However,
at very low β, the system transitions again: chemical
buildup becomes so strong that agent sensitivity satu-
rates, weakening the chemotactic bias, resulting in a re-
entrant homogeneous phase (RHP, Supplementary Movie
S1), where particles become effectively insensitive to the
uniform chemical field (Fig. 1(b),(c), First row, first col-
umn). Thus, the model reveals a non-monotonic transi-
tion as βdecreases: the system moves from a disordered
homogeneous phase to networks and clusters, and then
returns to a second homogeneous state at extreme pa-
rameter values. This re-entrant behavior arises not from
the depletion of signals, but from the loss of responsive-
ness due to oversaturation — an effect particularly rele-
vant in biological systems where chemotactic sensitivity
is dynamically regulated.
B. Phase diagram
Figure 2(a) presents the phase diagram and a heatmap
of the fractal dimension df, computed via the correla-
tion dimension method [47,48] (see SI, Fig. S5). Lower
dfindicates network-like states, while df≈2 corre-
sponds to homogeneous or compact cluster phases. Phase
boundaries, shown as visual guides, separate five regimes
(right to left): homogeneous phase (HP), sparse network
(SN), dense network (DN), system-spanning cluster (SC),
and re-entrant homogeneous phase (RHP). Representa-
tive points are evaluated and described in the diagram.
The visually distinct phases in particle configurations and
chemical profiles [Fig. 1(b,c)] are quantitatively charac-
terized in Sec. II C. The instability in homogeneous phase
is analyzed via mean-field dynamics in Sec. II D.
C. Cluster size
We analyze the scaled cluster size distribution P(¯n),
where ¯n=n/N, to distinguish homogeneous (HP &
RHP) from clustered states (SN & SC), using a clus-
tering algorithm with cutoff separation rc= 2σ. Fig-
ure 2(b) distinguishes the SN phase from the HP phase.
In SN, P(¯n) is bimodal — an approximately exponential
decay for small clusters coexists with a delta-like peak
near ¯n= 1, indicating an infinite cluster. In contrast,
the HP phase shows an approximately homogeneous par-
ticle distribution, with P(¯n)∼¯n−νexp(−¯n/¯n∗) [49,50],
where ν≈3/2 and ¯n∗is the typical cluster size (see SI,
Fig. S3, and Supplementary Movie S1). The power-law
correction reflects giant number fluctuations (GNF) due
to transient microclusters (see SI, Fig. S4). A similar
P(¯n) signature marks the re-entrant SC–RHP transition
in Fig. 2(c).
To identify transitions between clustered phases (SN
3
3
1
1
2
2
3 4
FIG. 1. (a) Schematic of agent-based rules: (a-i) A focal particle (red-shaded circle) senses chemical concentration from
surrounding grid points (gray boxes) within a hexagonal neighborhood. The movement direction (blue arrow) is determined
by a weighted sum of local chemical gradients and Gaussian noise (width π/4). (a-ii) If a trial move is rejected, no chemical is
deposited. (a-iii) If accepted, the particle moves and deposits chemical at the new location (darker gray grid). Move lengths
are drawn uniformly from [0, σ]. (b, c) Particle configurations (b) and chemical profiles (c) shown across α–βspace. Rows
1–3 correspond to α= 1,50,100, while columns 2–4 have β= 0.008,0.1,0.5. Column 1 uses β= 0.002,0.004,0.0005 for rows
1–3, respectively. At high β, transient micro-clusters form in the homogeneous phase (HP, column 4, figures b14,b24 ,b34).
Lower βyields sparse network (SN, column 3, figures b13,b23 ,b33) dense network (DN, b31,b32 ,b22), single clusters (SC, b21 ,
b22), and re-entrant homogeneous phase (RHP, b11). The colorbar represents the concentration of the chemical in units of α
β
(d) Chemical profile at α= 5.0, β= 0.008 shows efficient, system-spanning fractal clusters with low chemical cost.
and DN), we use the mean cluster size ⟨¯n⟩and its stan-
dard deviation, ∆¯n(Fig. 2(d)). As βdecreases within
an intermediate range, ⟨¯n⟩rises and saturates at 1, indi-
cating the emergence of a system-spanning cluster. No-
tably, ∆¯npeaks near the SN–DN transition. In addition
to fractal dimension, df, DN and SC phases are further
distinguished by the scaled convex hull area s(cluster
area relative to the simulation box; see SI, Fig. S6). In
Fig. 2(e), sdrops sharply from 1 as βdecreases, signal-
ing the DN–SC transition. With further decrease in β,s
rises again as the spanning cluster grows.
D. Continuum Description and Linear Stability
Analysis
In this section, we develop an effective continuum
model starting from the microscopic dynamics. The time
evolution of the k-th particle at position rkcoupled to
the chemical field c(r, t), is described by:
˙rk=−µ∇kU+µ∇kf(c) + p2DTηk(t)
˙c=αh({r,˙
r})−βc (1)
Here, U=Pk<k′u(|rk−r′
k|) represents the exclu-
sion interaction between the walkers, modeled by the
Weeks-Chandler-Andersen (WCA) potential [51,52] in
the simulation. The function f(c) = c/(1 + c) defines
how the particle motion is influenced by the local chem-
ical concentration. Translational diffusion, with coeffi-
cient DT, is driven by uncorrelated Gaussian noise satis-
fying ⟨ηk(t)⟩= 0 and ⟨ηk(t)ηk′(t′)⟩=δkk ′δ(t−t′). The
motility-dependent deposition rate h({r,˙
r}) decreases in
dense regions where volume exclusion hinders particle
motion.
In the absence of chemical coupling, local repulsion
in an excluded volume system can be approximated as
4
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
FIG. 2. (a) Phase diagram showing structural transitions with αand β. Background indicates fractal dimension df. At high
β, particles form a homogeneous phase (HP, ◁) with no stable clusters (b). A second homogeneous phase (RHP, ▷) appears at
lowest β(c). Aggregates emerge in between, separated from RHP by a linear stability boundary obtained using Eq. 3(solid
line). Decreasing βfrom HP leads to a sparse network (SN, •), then a fractal network (DN, ⋄), and finally dense clusters (SC,
□); dashed lines mark phase boundaries. (b, c) Cluster size distributions P(¯n) at selected α,βvalues. (d) Mean cluster size
and its standard deviation, with a peak in the deviation marking the SN–DN transition. (e) A drop in the area spanned by
clusters ssignals the DN–SC transition; srises at low βas SC approaches the RHP phase.
FIG. 3. Typical configurations during quench from a homogeneous to SC phase at α= 100.0, β= 0.008 are shown in (a–d)
at times t= 8 ×104, 1.6×105, 8 ×105, and 3 ×106. The lower panel (f–i) shows the evolution from a compact initial state
at α= 5.0, β= 0.008 corresponding to DN phase, over times t= 103, 104, 105, and 105. (j) shows particle and chemical
density profiles, ρand c, along the dashed diagonal in (f). (e) presents similar profiles for a configuration like (f), but at
α= 100.0, β= 0.008, corresponding to the SC phase in the upper panel (Supplementary movie S5). The fitted lines follow
ρ(x) = ρi
2[1 + tanh(x/w)] and c(x) = ρ(x)[ρc−ρ(x)] in units of α/β, where ρi= 0.46, w= 2.0, ρc= 2.712 in (e) and ρi= 0.84,
w= 4.0, ρc= 1.002 in (j), respectively.
Vint =Rdxdx′ρ(x)λδ(x−x′)ρ(x′), enhancing collective
diffusivity to D=DT+ 2λ. The coupled evolution of
particle density and chemical concentration is:
˙ρ=−∇ · [ρχf′(c)∇c−D∇ρ]
˙c=αρ(ρc−ρ)−βc. (2)
The functional dependence ρ(ρ−ρc) captures the de-
crease in the rate of chemical deposition with reduced
motility at high density controlled by ρc(see SI; Fig.S1
and S2).
We conduct a linear stability analysis around the ho-
mogeneous state (ρ0, c0), where c0=ρ0(ρc−ρ0)α
β. Since
the trace of the stability matrix satisfies T r(M)<0,
instability arises when Det(M) = 0, resulting in the
quadratic equation c2
0+ (2 −χ′)c0+ 1 = 0, with χ′=
(ρc−2ρ0)χ
(ρc−ρ0)D. The solutions C±=χ′−2±√χ′2−4χ′
2define the
range of α/β values within which the homogeneous state
5
becomes unstable towards pattern formation:
C−
ρ0(ρc−ρ0)<α
β<C+
ρ0(ρc−ρ0).(3)
This condition aligns with the observed pattern forma-
tion at intermediate βfor a fixed α, including the re-
entrance to the homogeneous phase in Fig. 2(a) and the
KS instability.
At low chemical concentrations, diffusion dominates
over attraction, leading to a homogeneous phase. As con-
centration increases, particles begin clustering. This sup-
ports the first bound conceptually. However, unlike the
linear stability analysis (Eq. 3), the transition boundary
in our simulations is independent of α.
As walkers begin to cluster, only those at the edges
deposit chemical. At high β, rapid evaporation weakens
chemical binding, causing the clusters to quickly disperse.
This explains the transient clustering and giant number
fluctuations observed in HP (see SI). The second bound,
shown in Fig. 2a, marks the phase boundary between
the RHP and SC phases at very high chemical concen-
trations. However, the KS instability does not directly
capture the fractal nature of the DN phase.
E. Quench Dynamics
To explore the dynamics leading to the SC and DN
phases, we perform two quench simulations (Supplemen-
tary movie S5). Quenching to the SC phase from a homo-
geneous state shows nucleation and growth (Fig. 3(a–d)),
with mean cluster size scaling as ⟨¯n⟩ ∼ t1/3(see SI,
Fig. S7). In contrast, quenching to the DN phase from
a compact state yields a network structure. The com-
pact distribution in Fig. 3(f) shows a density profile along
the dashed diagonal as ρ(x) = ρi
2[1 + tanh(x/w)], with
ρias the interior density and wthe interface width.
In steady state, the corresponding chemical profile is
c(x) = ρ(x)[ρc−ρ(x)] (in units of α/β), peaking near the
interface and obeying ρc<2ρi; see Fig. 3(j). The diffu-
sive flux is maximal along the diagonal, with a value √2
times that along the xor yaxes. At low chemical deposi-
tion—as in Fig. 3(f–i)—this flux dominates early dynam-
ics, ejecting particles along the diagonal. The resulting
chemical peak stabilizes the emerging thin layer, forming
the first finger-like pattern. This process repeats at new
corners, progressively generating a dense network (see SI;
Fig. S1 and S8).
At higher chemical deposition rates (ρc>2ρi), the
local peak in c(x) vanishes (Fig. 3(e)), and the chem-
ical closely tracks the density, enhancing its stability.
This regime aligns with the nucleation-growth behavior
in Fig. 3(a–d), where the DN instability is absent, yield-
ing steady compact bands as in Fig. 3(d).
F. Mean squared displacement
The evolution of network structures is linked to walker
dynamics. Here, we focus on the mean squared dis-
placement (MSD) of each walker, averaged over all par-
ticles (Fig. 4). The MSD exhibits diffusive scaling at
long times across all cases, though the slope in the diffu-
sive regime — indicating the walker’s diffusivity — varies
with different parameter sets. Additionally, we observe
non-monotonic behaviour in the MSD for high deposi-
tion (Fig. 4(b)). The effective diffusivity, Deff, is shown
in Fig. 4(c) for two αvalues as βchanges. For the lower
deposition rate (α= 5 in Fig. 4(c)), Deff decreases as
βdecreases and asymptotically approaches zero. Re-
duced evaporation increases the chemical bias, leading
to walker aggregation and low diffusivity. At high depo-
sition (α= 50), MSD and Deff vary non-monotonically
with β: diffusivity first decreases, then rises as βis fur-
ther reduced. This re-entrant behavior is due to the
reduced chemical response at higher concentrations, en-
abling more random movement.
These trends align with the phase diagram (Fig. 2):
at low α,Deff decreases from HP to DN as βdrops. At
high α, a re-entrant transition appears — Deff falls in
DN, rises in SC, and reaches high values in RHP at very
low β, resembling the HP phase.
III. DISCUSSION
We have presented a minimal agent-based model to
explore how motility-dependent chemical deposition and
concentration-limited responsiveness can lead to com-
plex self-organized patterns in active systems. By incor-
porating local interactions, feedback between movement
and chemical signaling, and excluded volume effects, our
model generates a rich phase diagram, including tran-
sitions between homogeneous states, network-like mor-
phologies, dense clusters, and re-entrant homogeneity.
A central outcome of our model is the emergence of
fractal, system-spanning networks that closely resemble
patterns seen in biological processes such as vasculoge-
nesis and epithelial tissue remodelling. These networks
appear when agents deposit chemicals only while moving
and respond to chemical gradients with a saturating sen-
sitivity. At intermediate values of chemical deposition
and decay, agents form extended, branching structures
that are stable yet highly dynamic. At very low decay
rates, chemical accumulation drives full phase separation
and large aggregate formation. Interestingly, a re-entrant
transition back to a homogeneous state is observed at ex-
tremely low decay rates, where the sensitivity of agents
becomes effectively saturated.
This framework connects well with observations in vas-
culogenesis and angiogenesis [37,53] mediating blood
vessel formation, essential for wound healing and tu-
mour growth. In vasculogenesis, endothelial cells (ECs),
which line blood vessels, organise into solid cords that
6
FIG. 4. (a–b) Log-log plots of mean squared displacement for α= 1 and 100 at various βvalues. (c) Effective diffusivity Deff,
scaled by the bare diffusivity D0(at α= 0), versus βfor α= 5 and 50.
remodel into a vascular network. Previous studies have
attributed vascular network formation to contact inhibi-
tion in cellular Potts models, where cell compressibility
leads to effective pressure within aggregates migrating
toward a chemoattractant [38–41]. When cultured on
Matrigel, a mimic of the extracellular matrix, ECs form
honeycomb-like patterns with cords surrounding empty
spaces, suggesting that ECs have intrinsic patterning
abilities that allow them to organize without relying on
external morphogen cues. They produce and respond
to various chemoattractants such as VEGF-A, SDF-1,
FGF-2, Slit-2, and semaphorins, which regulate their be-
haviour during vascular development [54,55]. ECs also
express VE-cadherin, which modulates their motility and
cohesion [56,57]. The reduced motility of ECs in re-
sponse to high local chemoattractant concentrations may
mimic the mechanisms captured in our model, leading to
spontaneous organization into vascular-like structures.
Similar principles may apply to epithelial-
mesenchymal transition (EMT), where epithelial
cells break contact and migrate individually or in
groups [58,59]. In vitro, epithelial cell clusters disperse
in response to growth factors like EGF (epidermal
growth factor) or HGF (hepatocyte growth factor) [60].
A partial EMT generates leader cells with enhanced
motility that guides follower cells while maintaining
some cell-cell adhesion [61]. At low densities and re-
duced EGF, non-transformed mammary epithelial cells
form multicellular clusters with branched, fractal-like
morphologies [36], which only occur when proliferation
and motility are suppressed by growth factor limitation.
Interestingly, the EGF receptors transmit stimulatory
signals to the cytoplasm and also endocytose and
degrade ligand EGF, enabling breast cancer cells to mi-
grate by generating self-created EGF gradients [62]. Our
results suggest that the observed fractal-like architecture
may be explained by motility-limited aggregation, where
chemical feedback stabilizes low-density, highly branched
morphologies. While previous explanations relied on
analogies to non-living systems (e.g., diffusion-limited
aggregation), our model offers a biologically plausible
mechanism grounded in active motility and local signal
degradation.
Chemotaxis and chemokinesis are two common migra-
tory behaviours in motile cells; the former involves di-
rected movement along a chemical gradient, while the
latter reflects speed changes based on chemical distribu-
tion [63–65]. These behaviors may coexist, as in Myx-
ococcus xanthus. Chemokinesis alone can lead to cell
accumulation in low-motility zones, such as neutrophils
near immune complexes. Similarly, self-propelled Pt/Au
rods in opposing hydrogen peroxide and salt gradients
accumulate in salt-rich, peroxide-poor regions due to
chemokinesis [66]. We believe that inverse chemokinesis,
where motility controls chemical distribution, as high-
lighted in our study, can broaden the current understand-
ing of migration dynamics in both natural and engineered
contexts.
Finally, our findings have implications for synthetic
active matter and artificial swarm intelligence, where
minimal local interactions can generate coordinated be-
haviour on a larger scale [67]. Microfluidic maze experi-
ments [31], or programmable colloidal systems [68], can
be used to test the predictions of our model and explore
efficient navigation, aggregation, or dispersion under a
minimal set of rules.
In summary, our work shows how biologically inspired
local feedback mechanisms between motion and signaling
can produce a wide array of collective structures. Though
abstract by design, the model captures key ingredients
relevant to both natural and artificial systems, offering a
versatile platform to study emergent spatial organization
in chemically interacting active agents.
METHODS
We employ a hybrid Monte Carlo simulation with N
agents guided by a self-generated chemical field. In this
auto-chemotactic model, agents move in continuous time
and space, while the chemical field - serving as spatial
memory - is discretized on a triangular lattice (Fig. 1(a)).
7
Each agent updates its position as
r(t+ 1) = r(t) + dr (cos θ(t),sin θ(t)),
where dr is sampled uniformly from (0, v0∆t), with
v0∆t= 1 fixed to represent the walker body size across
all simulations.
The orientation θ(t) is determined by the local chemi-
cal environment, as illustrated in Fig. 1(ai):
θ=⟨θc⟩+ ∆θ,
where
⟨θc⟩=
6
X
i=1
P(θi|ci)θi,
and P(θ|c) = c
1+cis the probability of moving along di-
rection θgiven local concentration c. This biases motion
toward higher concentrations, saturating at large c. The
average direction ⟨θc⟩(blue arrow in Fig. 1(ai)) is per-
turbed by a random angle ∆θdrawn from a Gaussian
distribution with zero mean and standard deviation π
4
(depicted as an arc in the figure), forming a chemical
cone with axis ⟨θc⟩and half-angle π/4. To account for
excluded volume, the walker attempts a step ∆rin di-
rection eat speed v0with a certain probability:
P(r−→ r+∆r) = e−∆U
kBT
with ∆U≡U(r+ ∆r,r′)−U(r,r′), where, U(r,r′) rep-
resents the Weeks-Chandler-Anderson (WCA) potential
U(r≡ |r−r′|) = (4ϵ1
r12 −1
r6+1
4,if r < 21/6
0,otherwise .
The chemical can evaporate stochastically with a rate β.
This leads to the chemical dynamics following:
dc(r, t)
dt =α(t)X
i
δ(r,ri)−βc(r, t).
The deposition of the chemical is directly coupled to the
particle dynamics. The rate α(t) of the chemical deposi-
tion at time tdepends on the agent’s motility state and
is defined as
α(t) = (α, if dr(t)= 0
0,otherwise.
This rule links chemical deposition to agent motility:
immobile particles, blocked by repulsion, do not deposit
chemicals. Meanwhile, deposited chemicals evaporate
over time (Fig. 1(aii)). Conversely, if a particle can
move, it deposits an amount α dt on each surrounding
chemical grid point (Fig. 1(aiii)). Thus, in aggregates,
mostly particles at the interface, where movement is
possible, actively contribute to chemical deposition.
AUTHOR CONTRIBUTION
AC and DC designed the work. SSK performed the
simulations and calculations. All the authors wrote the
paper.
DATA AVAILABILITY
All relevant data can be found within the article and
the supplementary information.
COMPETING INTERESTS
There are no competing interests.
ACKNOWLEDGMENT
DC acknowledges support from the Department of
Atomic Energy (OM no. 1603/2/2020/IoP/R&D-
II/15028) and an ICTS-TIFR Associateship. AC ac-
knowledges support from the Indo-German grant (IC-
12025(22)/1/2023-ICD-DBT). Numerical computations
were performed using the HPC facility at IISER Mohali,
SAMKHYA (IOP Bhubaneswar), and PARAM Smriti.
SSK acknowledges IOP Bhubaneswar for its hospitality
during part of the research.
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1
Supplemental Materials: Self-organized fractal architectures driven by
motility-dependent chemotactic feedback
I. LINEAR STABILITY ANALYSIS
In this section, we perform the linear stability analysis of the coupled equations for the evolution of particle density
and chemical concentration (Eq. (2) in the main text). The equations are given as:
˙ρ=−∇ · [ρχf′(c)∇c−D∇ρ]
˙c=αρ(ρc−ρ)−β c (S1)
This analysis is performed about the homogeneous state (ρ0, c0). The homogeneous chemical concentration is obtained
by putting ˙c= 0, giving c0=ρ0(ρc−ρ0)α/β. Substituting ρ=ρ0+δρ, c =c0+δc in Eq. S1 where (δρ, δc) denote
perturbations around the homogeneous state, we get
∂δρ
∂t =−∇ · [χρ0f′(c)∇δc −D∇δρ]
=−χρ0f′(c)∇2δc +D∇2δρ
∂δc
∂t =α(ρc−2ρ0)δρ −βδc +Dc∇2δc
(S2)
Expanding in Fourier modes, δρ =δρ0eiq.r+ωt , δc =δc0eiq.r+ωt we get,
ωδρ0
δc0=A B
C Dδρ0
δc0(S3)
where
A=−Dq2,B=q2χρ0f′(c0),
C=α(ρc−2ρ0),D=−β.
The characteristic equation is ω2−ωTr + Det = 0 with Tr = (A+D) and Det = AD − BC. It has solutions
ω±=1
2[Tr ±√∆].(S4)
where ∆ = Tr2−4 Det.Note that Tr <0. Therefore, instability arises when Det <0:
Det = AD − BC <0
q2(βD −α(ρc−2ρ0)χρ0f′(c0)) <0
βD −α(ρc−2ρ0)χρ0f′(c0)<0
β
αD−α(ρc−2ρ0)χρ0
(1 + c0)2<0 (S5)
As noted earlier, ˙c= 0, gives α/β =c0/ρ0(ρc−ρ0). The instability condition then becomes
c2
0+ (2 −χ′)c0+ 1 <0 (S6)
where χ′=(ρc−2ρ0)χ
(ρc−ρ0)D. Solving Eq. S6 gives
C−< c0< C+
i.e. C−<αρ0(ρc−ρ0)
β< C+(S7)
where C±=χ′−2±√χ′2−4χ′
2. It is assumed that χ′>4 for the existence of real solutions. Note that C±denotes the
range of values of α/β where the homogeneous state becomes unstable, leading to pattern formation.
2
-4 -2 0 2 4
-2.5
-2
-1.5
-1
-0.5
010-5
-4 -2 0 2 4
-1.5
-1
-0.5
010-5
-4 -2 0 2 4
0
0.2
0.4
0.6
0.8
1
-4 -2 0 2 4
0
0.2
0.4
0.6
0.8
1
-4 -2 0 2 4
-5
0
5
10-5
-4 -2 0 2 4
0
0.2
0.4
0.6
0.8
1
FIG. S1. Time evolution of particle density ρand associated currents from numerical integration of Eq. S1 for: (a) Jρ=
−D∇ρwithout chemical, (b) Jc=χρ∇f(c) at a constant chemical deposition, and (c) Jcusing density-dependent deposition
αρ(1 −ρ/ρc). The initial density profile is ρ(x) = 1
2(1 −tanh(x) ), with chemical fields set to their respective steady states. In
(b), chemotactic drift competes with diffusion, resulting in slower spreading. In (c), density-dependent deposition drives front
propagation. Density evolution for (a), (b), and (c) is shown in (d), (e), and (f), respectively. Profiles are shown at times 0,
20τ, and 100τ, as indicated in the figures. Parameters: D= 0.01, χ= 1.0, and α/β = 0,2,20 for (a–c).
II. NUMERICAL INTEGRATION
In our model, in the main text, we used a mobility dependent chemical deposition. To compare it with simpler
models, where deposition does not depend on mobility, here we perform numerical integrations of the relevant equations
for the evolution of the particle density, ρand the chemical c. For this purpose, we use two different chemical evolution
equations - the first is as in Eq. S1, while the second set of equations is given as:
˙ρ=−∇ · [ρχf′(c)∇c−D∇ρ]
˙c=αρ −β c. (S8)
For this purpose, we start with the same initial density profile ρ(x)=(ρi/2)(1 −tanh (x/w)), with ρi= 0.95 and
w= 1. To integrate the equations in one dimension, we used a finite difference scheme with σ, τ as units of length
and time, respectively. The discretization length, dx = 0.005σand the time interval between each step are chosen to
be dt = 0.0002τ, keeping Courant-Friedrichs-Lewy (CFL) condition for stability into consideration.
In Fig. S1(a) we look at the diffusive current Jρ=−D∇ρfor α
β= 0 at three different time points. This serves as
the control. As expected, the dynamics is purely diffusive and shows a smooth density profile with particles moving
from x < 0 to x > 0 as time progresses (Fig. S1(d)). Next, we solve Eq. S8 for α
β= 2. In this case the chemical
deposition is independent of the mobility of the particles. There is a competition between the diffusive flux Jρand the
chemical drive Jc=χρ∇f(c). The spatial variation of Jcis plotted in Fig. S1(b) at three different time points. The
resultant density profile now shows a much slower time evolution (see Fig. S1(e)) as compared to the purely diffusive
case (α= 0, Fig. S1(d)).
Finally, we look at the case where the deposition is mobility dependent and solve Eq. S1 for α
β= 20. The spatial
variation of Jcis now very different from that of Fig. S1(b). Due to the density dependence, deposition starts
decreasing with increase in density near ρ= 0.5 leading to a maximum in Jc. For x < 0, there is a positive chemical
3
drive as shown in Fig. S1(c). This drive pushes particles from ρ > 0.5 towards the lower dense region and results in
a local peak near ρ≈0.5 as shown in Fig.S1(f ). Further, this peak is stabilized by the chemical and also the front
propagates towards x > 0 as time progresses shown in Fig.S1(f).
We can therefore conclude that mobility dependent chemical deposition serves as a key factor for the instability
towards network formation. In Fig. S2, we show the simulation data for chemical concentration versus particle density
at steady state at two distinct regions of the phase diagram: (a) DN phase and (b) SC phase. In both the phases,
the corresponding fit to a functional dependence of the chemical concentration of the form c=aρ −bρ2shows a good
agreement. Note that this functional form is as expected from Eq. S1 with ˙c= 0.
FIG. S2. Plot of local particle density versus chemical concentration (scaled by α
β) at steady state. Panel (a) shows the fractal
network phase (DN, α= 5, β= 0.008), while (b) shows the aggregate SC phase (α= 100, β= 0.008). The solid line fits the
non-monotonic function f(ρ) = aρ −bρ2, with (a, b) = (0.56 ±0.02,0.38 ±0.03) in (a) and (4.2±0.3,3.9±0.5) in (b). Thus
ρc=a/b is 1.47 in (a) and 1.07 in (b).
4
III. CLUSTER SIZE DISTRIBUTION
102
10-4
10-2
100
102
10-4
10-2
100
102
10-4
10-2
100
102
10-4
10-2
100
102
10-4
10-2
100
102
10-4
10-2
100
102
10-4
10-2
100
102
10-4
10-2
100
102
10-4
10-2
100
FIG. S3. (a-i) Cluster size distribution, P(n) at α= 75.0. The data is fitted with the functional form: P(n)∼exp(−n/n∗)/nν.
We have fixed ν≈1.5 as in [49,50]. From (a) to (i), the fitting parameter 1/n∗is given by 0.067,0, 0.167, 0.05, 0.028, 0.025,
0.01, 0, 0.011. The vanishing 1/n∗in (b) and (h) marks the phase boundaries of the homogeneous phase.
In Fig. S3, we show the variation of the cluster size distribution, P(n) with βat an intermediate αvalue. In the
top row, we show the variation of the distribution as we move from the RHP phase to the SC phase. In Fig. S3(a)
we see a distribution of the form P(n)∼exp(−n/n∗)/nνcorresponding to the RHP phase. In the SC phase, the
distribution in addition to this functional form shows a large aggregate near n≈Nas expected. At the boundary
between the RHP and SC phase, the distribution shows a power-law dependence.
In the middle row, we move across values of βcorresponding to the boundary between SC and DN phase (Fig. S3(d)),
deep in the DN phase (Fig. S3(e)) and at the boundary between the DN and SN phase (Fig. S3(f)). All these plots
show a large aggregate near n≈Nwith increasing spread of the distribution.
In the bottom row, we show the variation of the distribution as we go from the SN phase to the HP phase. Inside
the SN phase (Fig. S3(g)), there is a spread in the distribution, indicating the formation of a small network of clusters.
Again, near the boundary between the SN and HP phase we see a power-law distribution (Fig. S3(h)). With increasing
β, we enter the homogeneous HP phase with an exponential distribution of cluster sizes (Fig. S3(i)).
The distributions clearly indicate homogeneous phases at two regimes, one at high β(Fig. S3(a)) corresponding to
5
FIG. S4. Plot of giant number fluctuations, ⟨δN2
b⟩=⟨N2
b⟩ − ⟨Nb⟩2versus ⟨Nb⟩, where Nbdenotes the number of particles
and ⟨· · · ⟩ denotes averages over different boxes of linear dimension ≈2,4,5,8,10,16,20,32,40 in units of rc.⟨δN 2
b⟩ ∼ ⟨Nb⟩γ
with γ∼1.2 suggesting giant number fluctuation in HP.
HP phase and another at very low β(Fig. S3(b)), corresponding to RHP phase. However, there are clear differences
between the HP and RHP phases. This is due to the presence of giant number fluctuations in HP (Fig. S4), which is
absent in RHP. As a result, the exponential cluster size distribution in HP requires a necessary algebraic correction
in the form n−νmultiplying exp(−n/n∗). This algebraic correction is not essential to describe RHP (Fig. S3).
FIG. S5. Evolution of the fractal dimension dffollowing a quench from a compact configuration to steady state at α= 25.0 for
two decay rates: β= 0.002 (SC phase) and β= 0.008 (in DN phase). For the evolution of pattern to network of filamentous
elements, dfdecreases toward 1, while during the evolution to homogeneous structure, dfremains close to 2.
IV. FRACTAL CHARACTERIZATION
We quantify the fractal nature of the network phase and also distinguish the non-network structure using the
concept of correlation dimension [47,48], which calculates the total number of pairs of particles which have a distance
between them that is less than a certain distance. For this purpose, a particle, i, is chosen, and Ni(r), the number of
particles within a distance, raround i, is calculated. Ni(r) is averaged over all the particles to obtain the correlation
dimension, C(r). For a fractal, this is expected to show a power law behavior with the exponent giving the fractal
6
dimension df:
C(r)∼rdf.(S9)
In Fig. S5, we show the time evolution of the fractal dimension obtained from Eq. S9 for two values of βcorresponding
to the DN and SC phases, starting as before from a quenched system. In the quenched state, the particles are packed
in a square box with df= 2, the system dimension. In the SC phase, dffluctuates about the same, indicating a
homogeneous aggregate. In the DN phase, dfdecreases sharply with time before saturating at a lower value ∼1.4,
characterising a network structure.
V. SYSTEM SPAN
The system span, scan be used to estimate the area covered by a system of connected particles. In Fig. S6 we
show the system span, s, for the SC phase in Fig. S6(a) and the DN phase in Fig. S6(b), which we calculate using
the convex hull method. We start with the quenched system, i.e., all particles clustered at the centre of the simulation
box. After 106steps of evolution, we measure the span of this cluster. We use the convhull function in MATLAB,
which uses the Qhull algorithm[69]. This gives a polygon marked in red as in the Fig. S6, and sis obtained by
calculating the area of the polygon, Ascaled by the simulation box area: s=A/LxLy.
FIG. S6. System span, sof the (a) SC phase and (b) DN phase using the convex hull method. sis lower in the SC phase
and has a high value in the DN phase, quantifying the system spanning network structure.
VI. GROWTH LAWS
In Fig.S7(a) we show mean cluster size growth laws. A quench to the SC phase shows growth law ⟨¯n⟩ ∼ t1/z
consistent with z≈3 for the conserved dynamics (for t > 104). In contrast, a quench to the DN phase, despite
showing initial consistency with such conserved dynamics, quickly crossovers to a much faster exponential growth
⟨n⟩ ≈ nsat −n0exp(−t/t∗) where nsat ≈1474, n0≈1665 and t∗≈46530. Arguably, t∗will grow with system size.
VII. PERTURBATION STUDY
We started with a state where the particles were arranged uniformly with 0 < y < 50, but with a small sinusoidal
perturbation at the centre, x≈50, as shown in Fig. S8(a,e). The system is then allowed to evolve at parameter values
corresponding to the SC phase (α= 25, β = 0.002) and the DN phase (α= 25, β = 0.008). The time evolution of the
chemical configurations are presented in the top (b-d) and bottom (f-h) panels in Fig. S8. The evolution is different
for both cases. In the SC phase, the perturbation is suppressed in time with the formation of a macrocluster. In the
DN phase, the perturbation grows as time progresses. This is consistent with the analytical picture presented before.
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FIG. S7. Evolution of mean cluster size, ⟨n(t)⟩scaled with the total number of particles(N), ⟨¯n⟩=⟨n(t)⟩
Naveraged over 20
initial configurations.The purple line represents ¯nsat −¯n0exp(−t/t∗) with ¯nsat ≈0.98,¯n0≈1.2 and t∗≈46530
FIG. S8. Evolution of the perturbed state. (a,e) present the initial chemical configuration with the concentration shown in
the colorbar between them. (b-d) show the evolution of the chemical configurations of the SC phase at α= 25, β = 0.002 and
DN phase in (f-h) at α= 25, β = 0.008. Configurations are at timesteps=1,4000,8000,16000 for columns 1 −4 respectively.
The colorbar indicates the chemical concentration scaled with α
β.A separate colorbar for chemical concentration is used for (a)
and (e). It is seen that the perturbation is suppressed in the SC phase whereas it grows in the DN phase.
VIII. PAIR CORRELATION
In the DN phase, the pair correlation decays in an exponential manner. This behavior is similar to what is seen in
an equilibrium fluid. It suggests that the structure is relatively uniform and lacks sharp features.
In contrast, the SC phase shows a different trend. Here, the pair correlation decays linearly. This matches the
prediction of Porod’s law. It indicates the presence of sharp interfaces between regions.
IX. SUPPLEMENTARY MOVIES
Movie-1: RHP and HP phase at α= 100.0, β = 0.002 and α= 100.0, β = 0.5 respectively.
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FIG. S9. Pair correlation function, ˜g(r) = g(r)−1 at β= 0.008 for two different phases SC phase at α= 100.0 and DN phase
at α= 1.0. The functions used to fit are mentioned in the plot. For the linear function ˜g0= 1.35, r∗= 28.29; and exponential
function has ˜g0= 3.56, r∗= 1.55
Movie-2: SN phase at α= 1.0, β = 0.1
Movie-3: DN phase at α= 0.01, β = 0.001
Movie-4: SC phase at α= 100.0, β = 0.008
Movie-5: Quench Dynamics corresponding to Fig. 3 in the main text
