A Computational Investigation of

Cell detachment from a tumour Spheroid

A PROJECT REPORT

Submitted as part of BBD451 BTech Major Project (BB1)

Submitted by:
Gagankumar C Kummur
2021BB10348

Guided by:
Prof. Amit Das

DEPARTMENT OF BIOCHEMICAL ENGINEERING AND

BIOTECHNOLOGY INDIAN INSTITUTE OF TECHNOLOGY DELHI

September, 2024

0
 DECLARATION

I certify that

a) the work contained in this report is original and has been done by me under the guidance of
my supervisor.

b) I have followed the guidelines provided by the Department in preparing the report.

c) I have conformed to the norms and guidelines given in the Honor Code of Conduct of the
Institute.

d) whenever I have used materials (data, theoretical analysis, figures, and text) from other
sources, I have given due credit to them by citing them in the text of the report and giving
their details in the references. Further, I have taken permission from the copyright owners of
the sources, whenever necessary.

Signed by:

Gagankumar C Kummur

1
 CERTIFICATE

It is certified that the work contained in this report titled A Computational Investigation of
Tumour Cell Escaping the Spheroid is the original work done by Gagankumar c
Kummur and has been carried out under my supervision.

Signed by:
Prof. Amit Das
24th November 2024

2
 ABSTRACT

Metastasis, the process by which tumour cells detach from the primary tumour and invade
surrounding tissues or distant organs, is one of the leading causes of cancer mortality.
Understanding the mechanical forces and energy dynamics that enable tumour cells to
migrate from a solid tumoroid is critical for developing therapeutic interventions aimed at
halting metastasis. In this study, we utilise HOOMD-Blue, a highly versatile molecular
dynamics simulation tool, to model tumour cell movement within a three-dimensional
tumoroid.

Our approach focuses on simulating the forces within the tumour itself, including
intercellular interactions and the mechanical resistance that cells must overcome to migrate
out of the tumoroid. However, we do not simulate the surrounding microenvironment or the
extracellular matrix, instead concentrating on the internal dynamics of the tumour. By
modelling these conditions, we calculate the force exerted by the cell to break intercellular
adhesions and the energetic requirements for the cell to escape from the tumoroid. The
simulation outputs include detailed data on the energy dissipation involved in cell movement
and the critical force thresholds that trigger cell detachment.

This work provides a computational framework for analysing tumour cell metastasis, offering
insights into the biomechanical factors governing the metastatic potential of cells within a
tumour. The findings contribute to future studies focused on physical barriers to metastasis
and may lead to novel strategies for inhibiting cell migration in cancer treatment.

3
 TABLE OF CONTENTS

Page
Declaration 1
Certificate 2
Abstract 3
Table of Contents 4
List of Figures 5
List of Abbreviations 6
List of Equations 7
Chapter 1: Introduction 8
Chapter 2: Literature Survey 9
2.1 Adhesion 10
2.2 Damping 10
2.3 Active Forces 11
2.4 Hoomd-Blue 11
Chapter 3: Materials and Methods 12
3.1 Initial Sphere Formation 12
3.2. Thermal Detachment Initialization 14
3.3. Active Detachment Initialization 15
3.4 Rotational Diffusion Effects on Active Cell Detachment 17
Chapter 4: Results and Discussion 19
4.1 Sphere Formation Dynamics 19
4.2 Thermal Detachment Dynamics 21
4.3 Active Detachment Dynamics 24
4.4 Preliminary Observations on Rotational Diffusion Effects 26
4.5 Discussion: Comparing Thermal and Active Detachment Mechanisms 28
Chapter 5: Future Work 31
References 32
Appendix 33

4
 List of Figures

Figure Title Page

Fig 2.1 Schematic of overlapping Sphere 9

Fig 3.1 Initial particle distribution 12

Fig 4.1 a)Initial particle distribution b)Particle distribution 20

when dmt force is applied and t=x
c)Particle distribution when dmt force is applied and t
=tend.

Fig 4.2 Radius of Gyration (lower values suggest stiff core) 20

changing with the time

Fig 4.3 a)Thermal Detachment Dynamics at t =0. b) Thermal 22

Detachment Dynamics at t =50 c)Thermal
Detachment Dynamics at t =100

Fig 4.4 No of particle inside the sphere at t = frame number 23

Fig 4.5 a)Active Detachment at t=0 b)Active Detachment at 25

t=50 c)Active Detachment at t=100

Fig 4.6 No of particle inside the sphere at t = frame number 26

varying with velocity

Fig 4.7 No of particle inside the sphere at t = frame number 27

varing with Dr

Fig 4.8 Radius of Gyration at t= frame number how it is 29

different for thermal and active detachment

5
 List of Abbreviations

Abbreviations Description

DMT Deriaguin-Muller-Toporov

JKR Johnson-Kendall-Roberts

MD Molecular Dynamics

COM Center Of Mass

6
 List of Equations

Equation Title Page

1 Force balance 9

10
2 Change in surface area (overlapping spheres)

3 Adhesion force 10

4 Interaction force 10

5 Damping force 10

6 Drag coefficient 11

7 Active Force 11

8 Position of the cell i 12

7
 Chapter 1
Introduction

The ability of cancer cells to undergo migration and invasion is fundamental to their capacity
to reposition themselves within tissues and initiate metastasis, the spread of cancer to distant
parts of the body. These processes are not only critical in oncology but also play an essential
role in various biological functions such as embryogenesis, immune responses, wound
healing, morphogenesis, and inflammation [3]. In the context of cancer, however, cell
migration and invasiveness become deadly mechanisms. Tumour cells acquire the ability to
escape their primary site and invade surrounding tissues, allowing them to colonise other
organs. This metastatic behaviour is the leading cause of cancer-related mortality [2].
Understanding the forces and energy dynamics that enable a cancer cell to overcome the
mechanical constraints within a tumour is crucial for the development of new cancer
therapies. In particular, how cells detach from the main tumoroid mass to invade surrounding
areas remains a topic of active research. This study leverages the powerful molecular
dynamics simulation platform, HOOMD-Blue[4], to model the physical interactions and
forces acting within a three-dimensional tumoroid.
In this work, we use the Deriaguin-Muller-Toporov (DMT) potential[5,6] to simulate
intercellular interactions, specifically to calculate the forces and cell-cell adhesions within the
tumoroid. The DMT potential is commonly used in molecular simulations to describe
adhesion, damping and active forces between particles, making it well-suited for modelling
the physical properties of cells.
By simulating these forces inside the tumoroid, this study calculates the energy and force
required for a cancer cell to detach and initiate migration. The computational approach
provides a clear and focused analysis of the mechanical properties influencing cell movement
within the tumoroid, offering valuable insights into the forces that drive metastasis. These
findings enhance our understanding of cancer cell migration and open up potential avenues
for therapeutic strategies aimed at disrupting the physical processes of metastasis,
contributing to advancements in cancer treatment.

8
 Chapter 2
Literature Survey

In the study of zebrafish embryonic explants, the mechanical model simplifies the complex
viscoelastic interactions of cells within a tissue. The model balances key forces acting on
individual cells, such as adhesion, damping, and active forces, to describe connective tissue
behaviour, including fluid-like motion and rearrangement.

The fundamental equation balancing forces on a cell is given by:

𝑑𝑎𝑚𝑝 𝑖𝑛𝑡 𝑎
0 = 𝐹𝑖 + ∑ 𝐹𝑖𝑗 + ∑ 𝐹 𝑖𝑗 (1)
<𝑖𝑗> <𝑖𝑗>

Where:

𝑑𝑎𝑚𝑝
● 𝐹𝑖 ​: Damping force acting on cell 𝑖

𝑖𝑛𝑡
● ∑ 𝐹𝑖𝑗 ​: Interaction forces between cells 𝑖 and 𝑗
<𝑖𝑗>

𝑎
● ∑ 𝐹 𝑖𝑗 : Active forces driving cell motility
<𝑖𝑗>

Fig 2.1 : Schematic of overlapping spheres with radius R and distance 𝑟𝑖𝑗 between their centres. The
overlap δ is shown in cyan. Protrusions effectively make new tensile contacts in a small region of overlap as
indicated by the blue ring. Therefore active forces are directed along a family of vectors 𝑎𝑖𝑗, parameterized by θ,
which extends from the centre of each sphere to the overlap ring.

9
2.1 Adhesion

Adhesion between cells is a critical factor in tissue dynamics, representing the attractive
forces mediated by molecules such as cadherins. The Deriaguin-Muller-Toporov (DMT) and
Johnson-Kendall-Roberts (JKR) models are two commonly used frameworks to describe
adhesive interactions:

1. DMT Model [5,6]: Used when adhesive energy is small and cells are relatively stiff.
The adhesive force between two cells with radius 𝑅 and overlap distance δ is
proportional to the area of overlap:

∆𝑆𝐴 = 4π𝑅δ (2)

𝑎𝑑ℎ
𝐹 = 2πγ𝑅 (3)

Where γ is the surface energy density. The DMT model assumes that adhesion is
proportional to the contact area between cells, making it analytically simple.

2. JKR Model[7]: For softer cells with larger adhesive energies, the JKR model is more
appropriate. This model includes the elastic deformation of cells and the stress
concentration at contact points. The JKR model is more complex but better suited to
highly adhesive systems where cells deform significantly at the contact interface.

𝑖𝑛𝑡
The total interaction force 𝐹𝑖𝑗 ​between two cells ii and jj can be modelled as:

𝑖𝑛𝑡
𝐹𝑖𝑗 = (𝐾δ𝑖𝑗 − 2πγ𝑅)𝑟𝑖𝑗 (4)

Where:

● 𝐾: Effective spring constant modelling cortical tension

● δ𝑖𝑗:​ Overlap distance between cells 𝑖 and 𝑗

● 𝑟𝑖𝑗​: Unit vector pointing from cell 𝑖 to 𝑗

2.2 Damping[10]

Damping represents the resistance to motion in tissues, modelling the energy dissipation due
to interactions with the surrounding medium or neighbouring cells. Damping forces are
critical in balancing active forces to prevent uncontrolled cell movement. The damping force
acting on cell ii is modelled as:

𝑑𝑎𝑚𝑝
𝐹𝑖 = 𝑏𝑣𝑖 (5)

10
Where:

● 𝑏 is the drag coefficient, which accounts for viscous resistance

● 𝑣𝑖is the velocity of cell 𝑖

The damping force slows down cell movements and helps reach a mechanical equilibrium
over time. The natural timescale for the system is defined as:

𝑏
τ = 𝐾
(6)

2.3 Active Forces[8,9]

Cells are not passive entities; they actively generate forces due to biological processes such as
cytoskeletal dynamics and cell motility. These active forces are crucial for driving tissue
rearrangement and collective cell migration. The simplest form for the active force on cell 𝑖 :

𝑎
𝐹𝑖𝑗 = σ𝑎𝑖𝑗 (7)

Where:

● σis the magnitude of the active force, typically drawn from a distribution
● 𝑎𝑖𝑗is a randomly chosen vector along the contact ring between cells 𝑖 and 𝑗

Active forces are modelled as having a persistence time 𝑝𝑡​, after which the direction of force
changes due to cellular reorganisation. The model also assumes that active forces are spatially
correlated because cells exert tension on their neighbours through adhesive contacts.

2.4 Hoomd-Blue[4]

Hoomd-Blue is a highly efficient, open-source molecular dynamics (MD) simulation

software optimised for GPUs, allowing researchers to simulate large-scale particle systems
with significant computational speed. It is widely applied in fields like soft matter physics,
biophysics, and materials science, where simulating complex systems with millions of
particles is essential. HOOMD-Blue supports classical MD and Monte Carlo methods,
offering flexibility for a wide range of simulations, from simple pair potentials such as
Lennard-Jones to more complex systems involving rigid body dynamics and custom force
fields. Its integration with Python further enhances user accessibility, allowing seamless
scripting and customization of simulations. The software’s GPU acceleration makes it a
popular choice for simulations requiring high performance, such as studies involving
molecular self-assembly, crystallisation, and phase behaviour in soft matter systems.
HOOMD-Blue remains a cornerstone in computational research due to its adaptability,
scalability, and ability to handle diverse particle interactions efficiently.[4]

11
 Chapter 3
Materials and methods

We have initialised our simulations in HOOMD-Blue to explore the mechanical behaviour of

zebrafish embryonic tissues. The initialisation process involves several key steps to ensure
accurate modelling of cell interactions and dynamics within the tissue.

3.1 Initial Sphere Formation

3.1.1 System Initialization

Fig 3.1 :Initial particle distribution

The initial configuration was established by uniformly distributing 1000 particles within a
cubic simulation box. This uniform distribution ensures an unbiased starting point for sphere
formation and prevents any artificial clustering or structural bias in the initial state. The
particle positions were generated using a systematic grid-based placement algorithm to
guarantee even spacing throughout the simulation volume.

Simulation Box Configuration

● Dimensions: 50 × 50 × 50 μm³
● Periodic boundary conditions applied in all directions
● Total number of particles: 1,000
Structured Grid Arrangement
● 10 × 10 × 10 grid
● 5 μm spacing between adjacent particles
● Positions centred within the simulation box

12
3.1.2 Interaction Setup
To facilitate efficient particle aggregation and sphere formation, we implemented enhanced
interaction parameters:
● A high DMT (Derjaguin-Muller-Toporov) potential was employed to model the
surface forces between particles
● An increased cutoff radius was specified to extend the range of particle interactions
● These parameters were carefully chosen to promote sufficient particle-particle
interactions while maintaining system stability
Parameters
● Adhesion parameter (Γ): 0.04
● Average particle radius (Rav): 0.5
● Cutoff radius pair interactions: 23.5 units
● Cutoff radius DMT potential: 4.1 units

3.1.3 Simulation Dynamics

The system evolution was simulated using the Langevin dynamics integrator in
HOOMD-blue, with the following specifications:
● Temperature was maintained at kT = 1 (in reduced units)
● The Langevin thermostat provided temperature control while incorporating random
forces to mimic solvent effects
● This approach ensures proper sampling of the configurational space while maintaining
realistic particle dynamics
Parameters
● Langevin thermostat implementation
● Temperature (kT): 1
● Damping coefficient: 5.0
The simulation was executed using HOOMD-blue's molecular dynamics framework. The
combination of the Langevin integrator with the enhanced DMT potential allowed for natural
sphere formation through particle self-assembly under controlled conditions. The simulation
was run on a single processing unit, allowing for straightforward tracking and analysis of the
particle dynamics during sphere formation. A GSD (HOOMD General Simulation Data)

13
snapshot file was generated to store the final configuration, enabling its use as an initial state
for subsequent simulations.

3.2 Thermal Detachment Initialization

3.2.1 System Initialization
The initial spherical configuration was loaded from a previously generated GSD (HOOMD
General Simulation Data) snapshot file. This configuration, consisting of 1000 particles
arranged in a spherical assembly, served as the starting point for the thermal detachment
studies. Using a pre-formed sphere ensures consistency across different thermal conditions
and eliminates variability in initial structures.

3.2.2 Interaction Parameters

The particle interactions were modified to reflect realistic cell-cell adhesion behaviour:
● The DMT (Derjaguin-Muller-Toporov) potential was reduced to account for the closer
proximity of particles in the formed sphere
● The reduced potential better represents the natural adhesive forces between cells in an
aggregated state
● No active forces were introduced to isolate the effects of thermal energy on cell
detachment
Parameters
● Adhesion parameter (Γ): 0.04
● Average particle radius (Rav): 0.5
● Cutoff radius pair interactions: 3 units
● Cutoff radius DMT potential: 1.3 units

3.2.3 Thermal Conditions

A systematic investigation of thermal effects was conducted by varying the temperature
parameter:
● Six distinct temperature points were studied: kT = 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 (in
reduced units)
● The Langevin dynamics integrator was employed at each temperature point
● Each simulation was run for the same duration to ensure comparable results across
temperature conditions

14
3.2.4 Analysis Methods
The temporal evolution of the cell assembly was tracked and analysed:
● GSD trajectory files were generated for each temperature condition.
● A custom analysis routine was implemented to count the number of cells remaining in
the spherical assembly at each time frame
● The detachment process was quantified by plotting the number of attached cells
versus time for each temperature
● This analysis provided insights into the temperature-dependent stability of the cellular
assembly

3.2.5 Data Processing

The resulting data was processed to generate detachment profiles:
● Time series data of attached cell counts was extracted from each simulation trajectory
● Comparative plots were created to visualise the relationship between temperature and
detachment rates

3.3 Active Detachment Initialization

3.3.1 System Initialization
The initial spherical configuration was loaded from a previously generated GSD (HOOMD
General Simulation Data) snapshot file. This configuration, containing 1000 particles in a
spherical assembly, provided a consistent starting point for investigating active detachment
behaviour. The use of a pre-formed sphere ensures reproducibility across different activity
parameters.

3.3.2 Interaction Parameters

The particle interactions were adjusted to reflect the established cellular assembly:
● The DMT (Derjaguin-Muller-Toporov) potential was reduced to account for the closer
proximity of particles in the formed sphere
● This reduction in potential strength provides a more realistic representation of
cell-cell adhesion forces in the aggregated state
Parameters
● Adhesion parameter (Γ): 0.04

15
 ● Average particle radius (Rav): 0.5
● Cutoff radius pair interactions: 3 units
● Cutoff radius DMT potential: 1.3 units

3.3.3 Activity Parameters

Active motion was introduced through systematic variation of particle self-propulsion:
● Seven distinct self-propulsion velocities were investigated: v = 0.01, 0.1, 0.5, 0.8, 1.0,
1.5, 2.0, and 2.5 (in reduced units)
● A constant rotational diffusion coefficient of Dr = 0.5 was maintained across all
simulations
● The temperature was fixed at kT = 1.0 to isolate the effects of active motion
● This combination of parameters allowed for the systematic study of how varying
degrees of activity influence cell detachment

3.3.4 Analysis Methods

The evolution of the active cellular system was monitored and analyzed:
● GSD trajectory files were generated for each combination of activity parameters
● A systematic counting algorithm was implemented to track the number of cells
remaining in the spherical assembly over time
● The temporal evolution of cell detachment was quantified for each set of activity
parameters
● This analysis enabled the assessment of how different self-propulsion velocities
influence the detachment process

3.3.5 Data Processing

The collected data was processed to understand active detachment dynamics:
● Time series data of attached cell counts was extracted from the simulation trajectories
● Comparative plots were generated to visualise the relationship between
self-propulsion velocity and detachment rates
● The analysis focused on identifying critical velocities and characteristic timescales of
active cellular detachment

16
3.4 Rotational Diffusion Effects on Active Cell Detachment
3.4.1 System Initialization
The investigation began with a pre-formed spherical configuration obtained from a
previously generated GSD (HOOMD General Simulation Data) snapshot file. This initial
configuration, containing 1000 particles arranged in a spherical assembly, served as the
consistent starting point for studying the effects of rotational diffusion on active detachment.
Using a pre-formed sphere ensures uniformity in initial conditions across all parameter
variations.

3.4.2 Interaction Parameters

The particle interactions were configured to reflect the consolidated spherical state:
● The DMT (Derjaguin-Muller-Toporov) potential was reduced to account for the closer
proximity of particles in the assembled sphere
● This adjustment in potential strength provides a more accurate representation of
cell-cell adhesion in the aggregated state
Parameters
● Adhesion parameter (Γ): 0.04
● Average particle radius (Rav): 0.5
● Cutoff radius pair interactions: 3 units
● Cutoff radius DMT potential: 1.3 units

3.4.3 Active Dynamics and Rotational Diffusion

A comprehensive parameter space was explored through systematic variation of both
self-propulsion and rotational diffusion:
● The self-propulsion velocity studied: v =1.5 (in reduced units)
● Five rotational diffusion coefficients were analysed: Dr = 1, 0.1, 0.01, 0.001, 0.0001
● Temperature was maintained constant at kT = 1.0 across all simulations
● This matrix of parameters enabled the investigation of the interplay between directed
motion and rotational dynamics

3.4.4 Analysis Methods

The temporal evolution of the active cell assembly was tracked and quantified:

17
 ● GSD trajectory files were generated for each combination of velocity and rotational
diffusion
● A particle counting algorithm was implemented to monitor the number of cells
remaining in the spherical assembly at each time frame
● The detachment process was quantified through plots of attached cell numbers versus
time for each parameter combination
● This analysis revealed the coupled effects of self-propulsion and rotational diffusion
on cellular detachment

3.4.5 Data Processing

The simulation data was processed to generate comprehensive detachment profiles:
● Time series data of attached cell counts was extracted from each simulation trajectory
● Comparative plots were created to visualise the relationship between detachment rates
and both activity parameters
● The analysis focused on identifying critical behaviour at different rotational diffusion
regimes and their interaction with self-propulsion velocities

Our systematic investigation of cellular detachment mechanisms encompassed three distinct

driving forces: thermal energy, active self-propulsion, and rotational diffusion. Beginning
with a stable spherical configuration of 1000 particles generated using high DMT potential
and Langevin dynamics, we conducted a series of controlled studies. The thermal detachment
analysis (kT = 0.5 to 3.0) revealed temperature-dependent stability thresholds, while the
active force studies demonstrated critical velocity-dependent detachment patterns across
varying self-propulsion speeds (v = 0.1 to 2.5) at fixed rotational diffusion. Further
investigation into rotational diffusion effects, spanning several orders of magnitude (Dr = 1 to
0.0001) at selected velocity (v = 1.5), unveiled the crucial role of directional persistence in
cellular detachment dynamics. Throughout all studies, the particle count remaining in the
sphere was systematically tracked and analysed, providing quantitative insights into
detachment rates and patterns. This comprehensive methodological approach established a
robust framework for understanding cellular aggregate stability under various physical
conditions, laying the groundwork for future investigations into complex cellular dynamics
and self-assembly processes

18
 Chapter 4
Results and Discussion

4.1. Sphere Formation Dynamics

4.1.1 Time Evolution

The system was monitored at regular intervals, showing a clear trend toward aggregation:

● Initial Configuration (t = 0): At the start, all 1,000 particles were evenly spaced within
a structured grid in a 50 × 50 × 50 μm³ box.
● Intermediate States (t = X): As interactions commenced, particles began to coalesce
into clusters due to the DMT potential effectively drawing them closer.
● Final Configuration (t = Y): By the end of the simulation, several well-defined
spherical aggregates formed, indicating successful aggregation.

4.1.2 Quantitative Analysis

Key metrics from the simulation include:

● Radius of Gyration: The radius of gyration (R_g) was calculated to be 3.74668 μm,
indicating the spatial distribution and compactness of the formed clusters.
● Cluster Size Distribution: At t = Y, all of the particles were found in clusters
confirming effective aggregation.

4.1.3. Impact of DMT Potential

The DMT potential was crucial in facilitating sphere formation:

● Attractive Forces: The high adhesion parameter (Γ = 0.04) significantly enhanced

particle attraction.
● Cutoff Radius Effects: The increased cutoff radius for pair interactions (23.5 units)
allowed for extended interaction ranges among particles.

19
4.1.4. Visual Representation

(a) (b) (c)

Fig 4.1 : a)Initial particle distribution b)Particle distribution when dmt force is applied and t=x

c)Particle distribution when dmt force is applied and t =tend.

A graph illustrating the relationship between radius of gyration and time provides insight into
the aggregation process:

Fig 4.2 : Radius of Gyration (lower values suggest stiff core) changing with the time

● Radius of Gyration vs. Time: This graph shows how the radius of gyration evolved
throughout the simulation, reflecting changes in particle distribution as aggregation
progressed.

4.1.5. Conclusion

The results indicate that through careful manipulation of interaction parameters and dynamics
settings using HOOMD-blue, successful sphere formation was achieved from an initially
uniform distribution of particles. The DMT potential effectively facilitated particle
aggregation while maintaining system stability. This version includes your specified visual

20
representation while keeping the overall content concise and focused on key findings. Feel
free to make any additional adjustments or add specific details as needed!

4.2 Thermal Detachment Dynamics

The thermal detachment behaviour of the spherical assembly was systematically investigated
under varying Langevin temperatures (kT). For kT≤1.5, the spherical configuration largely
retained its integrity. However, at kT>1.5, the sphere underwent complete disassembly,
demonstrating the critical threshold of thermal energy required to overcome interparticle
adhesion forces.

4.2.1 Dependence on Langevin Temperature (kT)

The stability of the spherical assembly exhibited a clear dependence on kT:

● kT = 0: The sphere remained entirely intact, as there was no thermal energy to induce
particle motion or detachment.
● kT = 0.5 and kT = 1.0: Partial detachment occurred, predominantly affecting
particles at the periphery of the assembly. The core structure remained intact,
indicating that adhesion forces were still dominant at these temperatures.
● kT = 1.5 : The sphere displayed rapid destabilisation, with a significant decrease in
the number of particles maintaining spherical cohesion. This marks the critical
threshold for thermal-induced disassembly.
● kT > 1.5 : Complete breakdown of the spherical structure was observed. The particles
dispersed randomly, with no observable remnants of the original spherical
configuration.

4.2.2 Effect of Zero Self-Propulsion Velocity

The study was conducted under conditions of zero self-propulsion velocity (v=0), ensuring
that active forces did not influence particle motion. This allowed for isolating the effects of
thermal energy alone on the detachment dynamics. The absence of active motion maintained
an isotropic thermal fluctuation environment, simplifying the interpretation of results.

21
4.2.3 Visualization of Detachment Dynamics

(a)

(b)

(c)

Fig 4.3 : a)Thermal Detachment Dynamics at t =0

. b) Thermal Detachment Dynamics at t =50

c)Thermal Detachment Dynamics at t =100

22
Plots depicting the number of particles remaining in the spherical assembly as a function of
time for different kT values will accompany this analysis. These visualisations reveal:

Fig 4.4 : No of particle inside the sphere at t = frame number

● kT=0: A consistent particle count over time, indicating complete stability.

● kT=0.5 and kT=1.0: Gradual detachment of peripheral particles, reflected in a slow
decline in particle count.
● kT=1.5: A sharp decrease in particle count, highlighting the onset of rapid
disassembly.
● kT>1.5: Near-instantaneous dispersal of particles, demonstrating complete structural
breakdown.

4.2.4 Conclusion

The results highlight a distinct transition in detachment dynamics as kT increases. The

threshold at kT>1.5 signifies the point where thermal energy exceeds the adhesive forces
binding the particles. The absence of active forces ensured that thermal fluctuations were the
sole driver of the observed behaviour, enabling clear isolation of thermal effects.

These findings provide a foundation for understanding the role of thermal fluctuations in
particle assemblies and their implications for systems where thermal stability is a critical
factor. Visual representations further elucidate the temporal evolution of detachment,
reinforcing the insights gained from this study.

23
4.3 Active Detachment Dynamics

The study investigated the effect of self-propulsion velocity (v) on the detachment of
particles from a spherical assembly of 1000 particles, while maintaining constant temperature
(kT=1.0) and rotational diffusion coefficient (Dr=0.5). The results demonstrated that
increasing v significantly influenced detachment behaviour, with distinct patterns observed
across different velocity ranges.

4.3.1 Detachment Behavior Across Velocity Ranges

● For v<1.0:
○ Minimal detachment was observed in this range. Velocities v=0.01, v=0.5, and
v=0.8 produced similar detachment rates, with only minor differences in the
number of particles leaving the sphere over time.
○ The detachment process was slow and primarily limited to particles at the
periphery, while the core structure remained stable.
○ The behaviour in this range suggests that the self-propulsion forces were
insufficient to overcome the adhesive forces maintaining the spherical
assembly.
● For v=1.0:
○ A slight increase in detachment rates was observed compared to the cases with
v<1.0. The peripheral particles detached more rapidly, but the overall
behaviour remained parallel to that of lower velocities.
○ The system showed the beginning of a transition, indicating that v=1.0 may
act as a threshold velocity where active forces start to compete effectively with
adhesive forces.
● For v>1.0:
○ A significant decrease in the number of particles in the sphere was observed as
vv increased. Velocities v=1.5, v=2.0, and v=2.5 resulted in accelerated
detachment and rapid disassembly of the sphere.
○ The core structure began to weaken significantly, with particles detaching
from both the periphery and inner regions. Higher velocities led to more
uniform dispersal of particles, with minimal resistance from the core.

4.3.2 Insights into Active Detachment Mechanisms

The results reveal a distinct threshold behaviour around v=1.0, marking the point where
self-propulsion velocities start to noticeably influence particle detachment. Below this
threshold (v<1.0), the detachment process is gradual and primarily affects peripheral
particles, with no significant impact on the core. Above the threshold (v>1.0), detachment
becomes significantly more pronounced, with active forces overcoming adhesive forces and
driving rapid disassembly.

24
4.3.3 Visualization of Detachment Trends

a)

b)

c)

Fig 4.5 : a)Active Detachment at t=0 b)Active Detachment at t=50 c)Active Detachment at t=100

25
Plots of the number of particles remaining in the sphere over time for different velocities
illustrate these dynamics:

Fig 4.6 : No of particle inside the sphere at t = frame number, varying with velocities

● For v<1.0, the plots show parallel trends with gradual and minimal decline in particle
count.
● At v=1.0, a slightly steeper decline begins to emerge, marking the onset of increased
detachment.
● For v>1.0, the decline becomes rapid and pronounced, with particle counts dropping
significantly within shorter timeframes.

4.3.4 Summary

The study highlights the critical role of self-propulsion velocity in active detachment
dynamics. While velocities below v=1.0 result in similar detachment behaviour, a noticeable
transition occurs at v=1.0, with significant detachment observed for v>1.0. These findings
provide valuable insights into the interplay between active motion and adhesion forces in
maintaining the structural integrity of spherical assemblies. Future studies could further refine
the understanding of this threshold behaviour by exploring additional velocities and longer
simulation times.

4.4 Preliminary Observations on Rotational Diffusion Effects

This study aimed to investigate the role of rotational diffusion coefficients (Dr​) on the active
detachment of particles from a spherical assembly at a fixed self-propulsion velocity (v=1.5)
and temperature (kT=1.0). While initial results provided some insights, no discernible trend
in detachment dynamics was observed across the tested Dr ​values (Dr=0.0001, Dr​=1,
Dr​=0.01, and Dr​=0.1).

26
4.4.1 Observed Detachment Rates and Lack of Trends

Fig 4.7 : No of particle inside the sphere at t = frame number, varying with Dr

For v=1.5, the particle dissociation rate fluctuated across the various rotational diffusion
coefficients without exhibiting a consistent pattern. Key findings include:

● At Dr​=0.0001 and Dr=0.001, detachment rates appeared lower than expected,

suggesting limited rotational movement but insufficient data for definitive
conclusions.
● At Dr=1, Dr=0.01, and Dr=0.1, the detachment rates varied irregularly, with no
observable trend correlating Dr​to the rate of particle dissociation.

This inconsistency may indicate either a lack of sensitivity of the system to Dr​under these
specific conditions or the need for further refinement in simulation parameters and analysis
methods.

4.4.2 Need for Further Investigation

The absence of a clear trend in particle dissociation rates highlights the limitations of the
current analysis:

1. Limited Parameter Range: The focus on a single velocity (v=1.5) may have
constrained the ability to detect interactions between vv and DrDr​. Expanding the
velocity range in future studies could provide a more comprehensive understanding of
the dynamics.
2. Data Accuracy: Potential inaccuracies in simulation settings or data processing may
have affected the results. Verifying and refining the methodology is essential for
future studies.

27
 3. Extended Simulations: Longer simulation times may reveal trends obscured in the
current timeframes. The temporal evolution of particle detachment at varying Dr​
values warrants further exploration.
4. Broader Analysis Metrics: Incorporating additional metrics, such as particle
trajectories or angular displacements, might provide complementary insights into the
effects of Dr​.

4.4.3 Summary

The current results suggest that while rotational diffusion influences particle detachment, the
lack of a consistent trend underscores the need for further refinement in experimental design
and analysis. This preliminary exploration serves as a foundation for more detailed
investigations into the coupled effects of v and Dr​on active detachment dynamics.

4.5 Discussion: Comparing Thermal and Active Detachment Mechanisms

The detachment behaviour of particles from a spherical assembly under thermal fluctuations
(kT) and active motion (v) reveals fundamental differences in the underlying mechanisms and
structural outcomes. These differences highlight how the nature of external forces—random
thermal fluctuations versus directed self-propulsion—affects the stability and integrity of the
spherical assembly.

4.5.1 Structural Integrity and Core Behavior

● Thermal Detachment:
○ In thermal detachment, as kT increases, the random motion of particles driven
by thermal energy uniformly destabilises the entire assembly.
○ Beyond a critical temperature (kT>1.5), the sphere undergoes complete
disassembly, with no residual structure remaining. Both the periphery and the
core are equally affected, leading to a uniform dispersal of particles.
● Active Detachment:
○ In active detachment, even at higher self-propulsion velocities (v>1), the core
of the sphere remains stiff and intact for a considerable duration. This
retention of a cohesive core arises from the directed nature of active motion,
which primarily destabilises peripheral particles rather than uniformly
affecting the entire assembly.
○ The stiff core resists disassembly due to the adhesive forces that dominate in
the central regions, where particle density and cohesion are higher. This
structural feature distinguishes active detachment from the more homogeneous
disassembly observed in thermal detachment.

4.5.2 Role of Energy Input

28
 ● Thermal Detachment:
○ The detachment process in thermal systems is driven by isotropic energy input
from random thermal fluctuations. As kT increases, particle motion becomes
more chaotic, leading to uniform detachment across the sphere.
○ The critical threshold at kT>1.5 reflects the point where thermal energy
overcomes adhesion forces uniformly throughout the structure.
● Active Detachment:
○ Active motion introduces directed energy input through self-propulsion, which
affects particles at the surface more significantly than those in the core. The
anisotropic nature of active forces results in a distinct pattern of detachment,
with peripheral particles being preferentially displaced.
○ Increasing v enhances particle mobility at the surface but does not fully
penetrate the densely packed core, preserving its integrity for a longer time.

4.5.3 Visualisation of comparison between thermal and active detachment

Fig 4.8 : Radius of Gyration at t= frame number how it is different for thermal and active detachment

1. Thermal Cluster (Blue Line):

○ The radius of gyration increases consistently over time, indicating that the
thermal cluster is expanding and becoming less compact.
○ This behavior reflects the nature of thermal detachment, where random
thermal fluctuations affect the entire cluster uniformly. Over time, these
fluctuations weaken the structural integrity of the cluster, causing it to
disassemble progressively and lose its compactness.
2. Active Cluster (Red Line):

29
 ○ The radius of gyration remains low and relatively constant, with only slight
fluctuations over time.
○ This indicates that the active cluster maintains its compactness, particularly in
the core region, despite the influence of active motion. The active forces
preferentially affect the periphery, leaving the core intact and stiff, which
prevents significant expansion of the overall structure.

Interpretation:

The significant difference in the radius of gyration between the two clusters highlights the
stiffness of the core in the active cluster:

● In the thermal cluster, the random and isotropic nature of thermal motion uniformly
weakens the cluster, resulting in both peripheral and core particles detaching over
time. This leads to a larger and steadily increasing radius of gyration.
● In the active cluster, the directed self-propulsion forces primarily affect particles at
the periphery. The core remains structurally intact due to the adhesive forces, resulting
in a smaller radius of gyration compared to the thermal cluster.

4.5.4 Conclusion

The low and stable radius of gyration for the active cluster demonstrates the stiffness and
stability of its core, even in the presence of active forces. This contrasts sharply with the
thermal cluster, where the lack of a stiff core leads to significant expansion and loss of
compactness over time. The graph effectively illustrates how active detachment preserves the
core structure, emphasizing its anisotropic detachment dynamics.

30
 Chapter 5
Future Work

To clarify the interplay between rotational diffusion and self-propulsion, future investigations
should:

● Explore a broader range of v values to assess velocity-dependent effects on

detachment dynamics.
● Increase the granularity of Dr sampling to detect finer trends or threshold behaviors.
● Extend simulation durations to capture longer-term dynamics and identify steady-state
behaviors.
● Validate findings with repeated simulations to ensure consistency and reliability.

31
 References

1. Schötz Eva-Maria, Lanio Marcos, Talbot Jared A. and Manning M. Lisa 2013 Glassy
dynamics in three-dimensional embryonic tissuesJ. R. Soc. Interface.1020130726
2. Friedl P, Wolf K. Tumour-cell invasion and migration: diversity and escape
mechanisms. Nat Rev Cancer. 2003 May;3(5):362-74.
3. Vicente-Manzanares M, Horwitz AR. Cell migration: an overview. Methods Mol Biol.
2011;769:1-24.
4. J. A. Anderson, J. Glaser, and S. C. Glotzer. HOOMD-blue: A Python package for
high-performance molecular dynamics and hard particle Monte Carlo simulations.
Computational Materials Science 173: 109363, Feb 2020.
5. B. Derjaguin, V. Muller and Y. Toporov, J. Coll. Interf. Sci., 1975, 53, 314–326.
6. V. Muller, B. Derjaguin and Y. Toporov, Coll. and Surf., 1983, 7, 251–259.
7. K. L. Johnson, K. Kendall and A. D. Roberts, Proc. R. Soc. London A, 1971, 324,
301–313.
8. E. Palsson, Future Generation Computer Systems, 2001, 17, 835–852.
9. E. Palsson, Journal of Theoretical Biology, 2008, 254, 1–13.
10. D. J. Durian, Physical Review E, 1997, 55, 1739.

32
 Appendix

import hoomd
import hoomd.md
import numpy as np
# Load the GSD file
cpu = hoomd.device.CPU()
simulation = hoomd.Simulation(device=cpu)
simulation.create_state_from_gsd(filename='test1.gsd')

Gam = 0.04
R_av = 0.5
EVAL_POINTS = 200
R_MIN = 0
DEFAULT_R_CUT = 1.3
r = np.linspace(R_MIN, DEFAULT_R_CUT, EVAL_POINTS, endpoint=False)

def dmt_energy(r, Gam, R_av):

"""Compute DMT energy."""
return (r*r/2 - r) + Gam*R_av*r

def dmt_force(r, Gam, R_av):

"""Compute DMT force."""
return ((1-r) - Gam*R_av)

cell = hoomd.md.nlist.Cell(buffer=0.4)
dmt_pair = hoomd.md.pair.Table(cell, default_r_cut=3)
dmt_pair.params['A','A'] = dict(r_min=R_MIN, U=dmt_energy(r, Gam, R_av), F=dmt_force(r, Gam,
R_av))

kt = 3
integrator = hoomd.md.Integrator(dt=0.005, forces=[dmt_pair])
nvt = hoomd.md.methods.Langevin(kT=kt, filter=hoomd.filter.All(), default_gamma=5.0)
integrator.methods.append(nvt)
simulation.operations.integrator = integrator

fname = str(__file__).split(".")[0]+"_"+str(kt)
fname = str(fname).split("/")[-1]
fname = f"./thermal_detachment_1/{fname}"

gsd_writer = hoomd.write.GSD(filename=f'{fname}.gsd',
trigger=hoomd.trigger.Periodic(100),
mode='wb')
simulation.operations.writers.append(gsd_writer)
simulation.run(1e4)

33