AGENT-BASED SIMULATION OF MOLECULAR PROCESSES

An Application to Actin-polymerisation

Stefan Pauleweit

Institute of Computer Science, Dept. of Systems Biology & Bioinformatics, University of Rostock, 18051 Rostock, Germany

J. Barbara Nebe

Center for Biomedical Research, Dept. of Cell Biology, University of Rostock, 18057 Rostock, Germany

Olaf Wolkenhauer

Institute of Computer Science, Dept. of Systems Biology & Bioinformatics, University of Rostock, 18051 Rostock, Germany

Keywords:

Agent-based model, Actin polymerisation, Arp2/3, Systems biology.

Abstract:

Agent-based modelling is widely used in ecology, economics and the social sciences. For the life science it

is an increasingly used technology. Here we use agent-based modelling to simulate the formation of actin

ﬁlaments, which is a major part in the cytoskeleton of the cell and plays a role in a number of cell functions.

We present in this paper three models with different levels of detail and show the potential of agent-based

models in systems biology by comparing the simulations to already published results.

1 INTRODUCTION

Agent-based simulations are a promising application

emerging in life sciences (Merelli et al., 2007). Ap-

plications of agent-based technologies in systems bi-

ology include studies in which each cell is modelled

as an agent (Thorne et al., 2007). Examples include

bacterial chemotaxis (Emonet et al., 2005), the phe-

nomenon where cells direct their movements in re-

sponse to external signals, models of epidermal tissue

(Grabe and Neuber, 2005), the formation of a 3D skin

epithelium (Sun et al., 2009) or a hybrid model, and

combination of agent-based simulations and differen-

tial equations to analyse the cell response to epider-

mal growth factors (Walker et al., 2006). Moreover,

agent-based models for intracellular interactions rep-

resenting the carbohydrate oxidation cell metabolism

(Corradini et al., 2005), the cell cycle (S¨utterlin et al.,

2009), the NF-κB signalling pathway (Pogson et al.,

2008) and molecular self-organisation, with the focus

on packing rigid molecules (Troisi et al., 2005), have

been proposed.

Actin polymerisation is a molecular process that

generates long ﬁlaments with a barbed and a pointed

end from single actin molecules that become part of

the cytoskeleton. The cytoskeleton provides the phys-

ical structure and shape of cells, as well as plays

an important role in a number of cell functions, in-

cluding cell motility (Cooper, 1991; Pantaloni et al.,

2001), endocytosis (Galletta et al., 2010), or cell

division (Pelham and Chang, 2002). Understand-

ing of actin organisation has important implications

for practical medical applications, including the de-

velopment of new topographies for implant surfaces

(Matschegewski et al., 2010; Nebe et al., 2007).

Here we focus on the spatial and time dependent

simulation of actin polymerisation. The literature de-

scribes a number of models analysing the cell motil-

ity driven by actin ﬁlaments, using partial differen-

tial equations (Mogilner and Edelstein-Keshet, 2002).

Another study used Brownian dynamics to analyse

the self-assembly process of actin and the dynamics

of long ﬁlaments (Guo et al., 2009). The distribution

of the length of actin ﬁlaments inside a cell was anal-

ysed with a discrete and continuous model (Edelstein-

Keshet and Ermentrout, 1998). Different models us-

ing stochastic Π-calculus as a representative of pro-

cess algebra, have also been published (Cardelli et al.,

2009).

In this paper we describe simulations using

276

Pauleweit S., Barbara Nebe J. and Wolkenhauer O..

AGENT-BASED SIMULATION OF MOLECULAR PROCESSES - An Application to Actin-polymerisation.

DOI: 10.5220/0003599702760282

In Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2011), pages

276-282

ISBN: 978-989-8425-78-2

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

an agent-based approach with communicating X-

machines (Gheorghe et al., 2005), implemented in

a software called Flexible Large-scale Agent-based

Modelling Environment (FLAME) (Kiran et al.,

2008). This allows us to analyse the spatial and time

dependent behaviour during the composition of the

ﬁlament structure by free actin with a high degree of

physical realism. The outline of the paper is as fol-

lows. Section 2 explains the three models in detail.

Section 3 discusses the output of the models and Sec-

tion 4 sums up the conclusions and gives a brief out-

look for further studies.

2 AGENT-BASED SIMULATION

An agent is formally deﬁned as an ﬁnite-state ma-

chine. Because the ﬁnite-state machine model is

too restrictive for general system speciﬁcation, an

extension with a memory, the so called X-machine

promise a better implementation (Holcombe, 1988).

If a system contains more than one agent, the par-

ticular X-machines must be able to communicate to-

gether and this leads to a communication X-machine

system (Gheorghe et al., 2005). This concept is im-

plemented in the software named Flexible Large-scale

Agent-based Modelling Environment (FLAME) (Ki-

ran et al., 2008).

Using the actin model generated by X-ray analysis

(Oda et al., 2009) we ﬁx the size of one molecule to

A×50

A. The dimension of the molecule is shown

in Figure 1. Each agent contains an identiﬁcation

Figure 1: The physical size of the actin molecule determines

the size of an agent in the simulation. For the two dimen-

sional simulation the width and height of an actin agent is

set to 50

A×50

A (Oda et al., 2009).

number and two binding sides to connect to another

agent, namely bottom-bound (BB) and top-bound

(TB) and can switch between three different states

(free, bottom-bound, fully bound). A free binding

side is denoted with the constant −1. As long as both

binding sides are marked with −1 (free), the agent is

randomly rotating and moving around in a distance of

1–200

A, which is an approximation for the compu-

tational expensive calculation of Brownian dynamics.

If a molecule binds to another, then the identiﬁcation

number of the counterpart is stored in the BB (respec-

tively TB) variable; the agent becomes immobilised

and its rotation will be adapted. The precondition for

binding is, that one of the agents is already bound

(bottom-bound). This leads to the condition that at

least one agent has to be stuck in the beginning of

the simulation. This is done by initialising one agent

with BB = 0. If a free agent binds to an already bound

one, the second becomes then fully bound (BB 6= −1,

TB 6= −1). The whole schema of the actin–actin in-

teractions is also shown in Figure 2. The polymerisa-

Figure 2: An actin-agent can be in three different states.

A free molecule can bind to an already bound initial agent

(BB = 0). The already bound actin-agent then become fully

bound. Then the third actin-agent can bind to the second

actin-agent which become fully bound and so on.

tion of actin ﬁlaments is characterised by a 70

◦

an-

gle branching on several positions mediated by the

Arp2/3 protein (Stossel et al., 2006). To simulate this

branching process, a new agent with a third binding

side was implemented. The orientation of the branch-

ing side to the left or right was set randomly. This

agent is restricted to bind only to actin-agents, so that

a Arp2/3-Arp2/3 combination is prohibited. In Figure

3 the scheme for the interactions and state changes is

illustrated. Similar to the actin-agent, the size of this

agent was determined from published measurements

(Robinson et al., 2001). Figure 4 shows the approxi-

mated dimensions of Arp2/3. An agent-based model

has to include the reaction kinetic in a reasonable way.

Due to the nature of spatial simulations with individ-

ual molecules, this may be done by an interaction vol-

ume, which deﬁnes a reaction zone around a particu-

lar agent (see Figure 5). Andrews and Bray (2004)

AGENT-BASED SIMULATION OF MOLECULAR PROCESSES - An Application to Actin-polymerisation

277

Figure 3: In addition to the actin-agent (Figure 2), the simu-

lation was extended with a second type of agent for Arp2/3.

This agent can bind to an bottom-bound actin-agent. Then

another actin-agent can bind to the top-binding side of the

Arp2/3-agent, the next actin-agent to the middle binding

side and the Arp2/3-agent becomes fully bound.

Figure 4: The physical size of Arp2/3 determines the size

of the agents in the simulation (Robinson et al., 2001).

Figure 5: The interaction boundary (dashed circle) deﬁnes

the reaction volume around an agent. If a second agent en-

ters this area, the reaction takes place.

developed an algorithm to determine this volume, but

considered more detailed interactions. Another way

is described by Pogson et al. (2006) where the inter-

action radius r is calculated by:

r =

3k∆t

4πN

where k is the kinetic rate constant, ∆t the discrete

time interval and N

is Avogadro’s constant (6.022×

). The rate constant for actin-actin assembly was

determined with 11.6µM

−1

(Fujiwara et al., 2007)

and leads to a radius of 0.166

A for ∆t = 1s. If two

or more agents enter the interaction volume at the

same time step, the closest molecule to the reaction

molecule assembles to it, if two or more have the same

distance, one will be chosen by chance.

To compare our results with the simulation of

Cardelli et al. (2009), we used the same number of

1200 free actin agents and 30 Arp2/3 agents. Cardelli

uses this number of agents to simulate a concentration

of 1200 µM. For concentration values in a spatial sim-

ulations, it is necessary to calculate the volume of the

environment:

Actin

= N

×V × c [1/mol × l × mol/l]

V = 1200/(N

· 1200× 10

−6

)

V = 1.66× 10

−18

l = 1.66× 10

−21

where n

Actin

is the number of molecules, N

again Avogadro’s constant, c is the concentration of

molecules and V is the volume. Assuming the envi-

ronment as a cube, the length of a side is approxi-

mately 1184.0

For a simulation including the dissolving of actin

from a ﬁlament, we used the rate constant of 5.4 s

−1

for ADP-actin at the barbed end from the literature

(Fujiwara et al., 2007). Only agents with a free top-

bound (in case of Arp2/3 also middle bound) can be

released from the ﬁlament. To avoid an instant re-

coupling, the molecule will be moved to outside the

interaction boundary.

Our present agent-based simulation takes place in

a 2D environment, so that we have to introduce a fac-

toring constant of 100 for the radius, the dissolving

rate constant and the size of the environment, follow-

ing the paper of Cardelli et al. (2009).

3 RESULTS

3.1 Actin-Actin Interactions

Figure 6 shows the time plot of the growth of one ﬁl-

ament. The curve shows in the beginning a linear in-

crease (see inset of Figure 6), but later becomes loga-

rithmic. After 390 seconds 50 agents were integrated

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

278

in the ﬁlament, which corresponds to a ﬁlament of

length 0.25µm. A length of 1µm is reached after 1882

seconds and at the end of one hour, 240 agents form

a ﬁlament with a length of 1.2µm. In agreement with

Figure 6: The time plot shows the result of the simulation

for a simple actin polymerisation with 1200 agents and a

time step ∆t = 1s. Inset: Linear slope of the binding process

in the beginning.

published measurements (Fujiwara et al., 2007), the

increase in length of actin is linear in the beginning

of the simulation. The logarithmic curve on can be

explained by the decreased number of free molecules

and the spatial phenomena, by which the simulated

ﬁlament is growing close to the boundary of the en-

vironment. The number of reachable free molecules

close to this boundary is then much lower. The dif-

ference in the speed of elongation is related to two

reasons:

1. Actin ﬁlaments can growth on both side, whereas

the simulation allows only the growth at the

barbed end.

2. Actin can build small motile fragments, which

then elongate the ﬁlament (Stossel et al., 2006).

This increases the speed of polymerisation signif-

icantly.

3.2 Branching Process

Adding a new agent for the Arp2/3 protein, we simu-

lated the actin polymerisation with the branching pro-

cess. To visualise this, Figure 7 shows a snapshot

of the spatial distribution at the end of one hour. In

this simulation an overall ﬁlament length of 1µm was

reached after 578 seconds. At the end of one hour,

nearly all agents were involved in the ﬁlament struc-

ture, 1121 actin agents were fully bound. Addition-

ally 28 Arp2/3 agents are fully bound, two of them

Figure 7: The ﬁgure (cropped for better illustration) shows

the end result of the simulation for one hour with 1200

actin-agents and 30 Arp2/3-agents. The black points mimic

the free actin, the blue the agents bind to the ﬁlamental

structure.

Figure 8: The time plot shows the result of the branching

simulation for actin of 1200 actin-agents, 30 Arp2/3-agents

and a time step ∆t = 1s.

had an open binding side. Figures 8 and 9 show

the time plots for the actin and arp agents respec-

tively. Both time curves are sigmoidal with an inﬂec-

tion point around 1300 seconds.

In contrast to the ﬁlament formation, solely with

actin, the branching process accelerate the elongation

signiﬁcantly. The snapshot in Figure 7 shows the spa-

tial consideration and is in good agreement with pre-

viously published simulations (Cardelli et al., 2009,

Figure 17).

3.3 Disassembly Process

To model the disassembly of actin and Arp2/3

molecules from the ﬁlamental structure, we added a

AGENT-BASED SIMULATION OF MOLECULAR PROCESSES - An Application to Actin-polymerisation

279

Figure 9: The time plot shows the result of the branching

simulation for Arp2/3 with 1200 actin agents, 30 Arp2/3

agents and a time step ∆t = 1s.

new probability for each agent.

After introducing this new variable, the assembly of

Figure 10: The time plot shows the result of the branching

simulation, including the disassembly process, for actin of

1200 actin-agents, 30 Arp2/3-agents and a time step ∆t =

1s.

the actin ﬁlament slowed down. As shown in Fig-

ure 10, the assembly of 200 molecules and therefore

an overall length of 1µm is reached after 760 sec-

onds. After one hour, the ﬁlament contained 717 fully

bounded actin-agents and is branched out at 14 differ-

ent positions (see also Figure 11). This model shows

therefore a comparable time progression to the simu-

lation of Cardelli et al. (2009) , especially for Arp2/3,

although our ﬁlamental growth is somewhat slower.

Figure 11: The time plot shows the result of the branching

simulation, including the disassembly process, for actin of

1200 actin-agents, 30 Arp2/3-agents and a time step ∆t =

1s.

4 CONCLUSIONS AND

OUTLOOK

Instead of the commonly used rate equations to simu-

late intracellular molecular processes, we introduced

an agent-based approach. This allowed us to over-

come some restrictions imposed by differential equa-

tion models, more precisely any number and any dis-

tribution, as well as spatial behaviour of molecules

can be easily modelled. Our model simulates actin

polymerisation, an important key player for different

cell functions.

The spatial outcome of our model is compara-

ble to alternative models of Cardelli et al. (2009), us-

ing the stochastic Π-calculus. Because the FLAME-

frameworkproduces XML-ﬁles for each time step, we

are also able to create an animated version for track-

ing the ﬁlament formation (not shown here). Addi-

tionally a time dependent analysis of the behaviour of

the single molecules and the ﬁlaments can be done.

The limit in using the agent-based approach is only

given by computational purposes.

Our overall aim is the development of a biophysi-

cal realistic model for actin polymerisation in human

cells. The advantage of our approach is the possibil-

ity to to extend the simulation to a massive number

of molecules with the aid of the parallelised FLAME

software version and, more important, the easy imple-

mentation of external inﬂuences. This should enable

us to analyse observed phenomena of actin cluster-

ing on titan pillar surface structures (Matschegewski

et al., 2010) with applications to implant technolo-

gies. This interesting issue makes it necessary to in-

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

280

clude more proteins like capping proteins which stop

the elongation of the ﬁlament (Pollard and Cooper,

1986).

ACKNOWLEDGEMENTS

We are grateful for ﬁnancial support of the research

training school “Welisa”, which is founded by the

German Research Foundation (DFG 1505/1). Fur-

thermore the authors are thankful for the helpful ad-

vice of Prof. Mike Holcombe and Mark Burkitt from

the University of Shefﬁeld.

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