Active transport and cluster formation on 2D networks

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by  Philip Greulich

Active transport and cluster formation on 2D networks
Philip Greulich and Ludger Santen
9
0 Fachrichtung Theoretische Physik, Universit¨at des Saarlandes, Saarbru¨cken, Germany
0
2

r Abstract. Weintroduceamodelforactivetransportoninhomogeneousnetworksem-
p
bedded in a diffusive environment and investigate the formation of particle clusters.
A
Inthepresenceofahard-coreinteraction,clustersizesexhibitanalgebraicallydecay-
4 ingdistributioninalargeparameterregime,indicatingtheexistenceofclustersonall
2 scales.Theresultsarecomparedwithadiffusionlimitedaggregationmodelandactive
transport on a regular network. For both models we observe aggregation of particles

] toclusterswhicharecharacterizedbyafinitesize-scaleiftherelevanttime-scalesand
h
particle densities are considered.
p
-
o PACS. 87.16.Uv Active transport processes – 45.70.Vn Granular models of complex
i systems
b
.
s
c 1 Introduction consisting of active stripes where particles per-
i
s form biased motion. They showed that though
y
movement of particles remains globally diffusive
h Activetransportprocessesplayanimportantrole
on long time-scales, the diffusion constant is en-
p for the functionality of social and biological sys-
hanced by the presence of the network.
[ tems. Examples are road traffic or active intra-

cellular transport of vesicles and organelles by Intransportsystemsconsideringstericexclu-
1
motor proteins that perform directed movement sioninteractionbetweenparticles,aggregationis
v
0 along microtubules or actin filaments. In recent a common phenomenon, manifesting in the for-
9 years stochastic systems of self-driven particles mation of jams. Jams can form in one dimen-
8 have been applied to model intracellular trans- sional systems with single tracks due to bound-
3 port of motor proteins in a number of works ary conditions (boundary induced phase transi-
4. [1,2,3,4,5,6,7].Whilethemicrotubulesusuallyar- tions [9]), induced by defects [10,11,12,13,14] or
0 range in an ordered pattern (e.g. a radial struc- theyemergespontaneouslyduetostochasticslow
9 ture in most mammalian cells, though longitu- downofvehiclesinhighwaytraffic[15,16].Intwo
0 dinal in neuronal axons), actin filaments often dimensional regular street networks, mutual in-
: form randomly structured undirected networks. terferenceofvehiclesatintersectionsleadtojam-
v
i Therefore the investigation of transport on net- ming [17].
X
works arises to be an interesting object of re- Studiesoftransportoninhomogeneoustopo-
r search. logical networks (graphs with nodes and edges,
a
Active transport on an undirected but reg- nodistances)revealedaninterestingphenomenol-
ular network has been investigated by Klumpp ogy. E.g. non-interacting particles performing a
et al. [8]. They studied the dynamic properties random walk exhibit an inhomogeneous density
of diffusive non-interacting particles on a lat- distribution on the nodes [18], while inclusion of
tice with an embedded regular square network an attractive zero range interaction even allows
2 Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks
the particles to aggregate and form a conden- riatodiscriminatebetweendifferentmicroscopic
sate, corresponding to nodes containing a finite dynamics.
fractionofparticles[19].Theseresultsshowthat
thestructureofatransportnetworkstronglyin-
fluences transport properties. In order to model 2 Network models
activetransportonactinfilamentnetworks,itis
therefore necessary to consider realistic network
In the following, we introduce stochastic mod-
structures.
els in order to study the influence of the net-
work structure on dynamical properties. While
For many biological processes, concentration
the presence of a network in a diffusive envi-
gradients are crucial. One example is the ag-
ronment without interaction between particles
gregation of proteins inside the cell or in the
merelyleadstotheenhancementofthediffusion
cell membrane. Clusters of aggregated proteins
process [8], we are rather interested in collec-
can be observed and characterized experimen-
tive patterns emerging due to interactions. Our
tally for example by fluorescence microscopy. In
simulations use stochastic dynamics in order to
some cases these clusters are essential for cell
integratethemany-particleMasterequation.At
functionality but they can also lead to dysfunc-
eachtimestepparticlesarerandomlychosenand
tionsorevenapoptosis.Inyeastcellmembranes
updated (random sequential update) according
forexampleoneobservestheaggregationofErd2p-
to the rules given in table 1 applying periodic
receptors which can promote the internalisation
boundaryconditions.Timestepsarenormalized
of toxins [20]. The existence of a quasi two-
so that on average each free particle performs
dimensionalirregularactinfilamentnetworkbe-
one diffusive step per time step ∆t.
neath the membrane [21,22] suggests a jamming
mechanism of vesicles prior exocytosis, resulting
inclustersofreceptorsonthemembranesurface.
2.1 Regular networks
The limits of resolution in optical microscopy
[23]donotallowthelocalizationofsinglerecep-
tors. By contrast the size of larger particle ag- As a the first example for active transport on
gregatescaninprinciplebegivenwithrelatively networks a discrete lattice gas model similiar
highprecision.Thereforeitisusefultorelatethe to the model investigated in [8] is considered.
cluster size distribution with microscopic trans- N ×N sites are arranged in a square lattice of
port mechanisms by means of theoretical mod- edge length Na where a is the lattice spacing.
elling. Each site can either be empty or occupied by
atmostoneparticle.Wedistinguishtheparticle
In this work we propose a model for active statesattached(A)anddetached(D).Thesystem
transportofextendedhard-coreparticles(corre- contains stripes of active sites that constitute a
sponding to vesicles) on a two-dimensional ran- regularsquaretransportnetworkwhereattached
domly disordered network embedded in a diffu- particles perform a directed motion. Detached
siveenvironment.Themodelismotivatedbyin- particles always move diffusively. The orienta-
tracellulartransportonsubmembranalnetworks, tion of stripes was chosen randomly with equal
wethereforeadaptthemodelparameterstothis probability. Attached Particles on active stripes
reference system. We check particle configura- can unbind becoming detached, while detached
tions in order to identify the formation of clus- particles can bind to stripes becoming attached.
tersandinvestigateclustersizedistributions.The Steps that would result in double occupation of
results are compared with a regular network in a site are prohibited if not state differently.
diffusiveenvironmentandadiffusivesystemwith- Comparedtothedynamicsofnon-interacting
outnetworkwhereattractiveparticle-particlein- self-driven particles qualitatively new features
teractions promote cluster formation similiar to arise due to the steric particle-particle interac-
a van der Waals gas. The main focus will be on tions at intersections of the network. Here we
robust properties of clusters that serve as crite- introduce an additional parameter, the blocking
Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks 3
ticles can perform directed paths along these fil-
aments.
2.2.1 General properties of the model
Themaincomponentsofourmodelarefilaments
and particles interacting via a spherical hard-
corepotentialrepresentedbyadiscofradiusr .
p
This hard core potential is implemented by can-
celling any steps that would result in an overlap
of discs. Filaments are represented by linear se-
Fig.1.Illustrationofthedynamicsinaregularnet- quences of nodes with a distance of d between
n
work. Dark grey discs are particles diffusing freely nodes.Theyaredirectedwitha(-)-end atwhich
with rate ω . Black discs represent attached par-
D they can shrink and a (+)-end at which new
ticles that can only step in the preferred direction
nodescanbegeneratedtoelongatethefilament.
of the active stripe they occupy (bold arrows) with
Particles can attach to nodes that are within a
ratep.Onfilamentsites(lightgrey),particlescanin-
distance less than d and perform steps to adja-
terchangebetweenattachedanddetachedstatewith b
cent nodes in the (+)-direction of the filament.
ratesω andω ,respectively.Crossedarrowsdenote
a d
steps that are inhibited due to the exclusion princi-
ple.
2.2.2 Dynamics of filaments and particles
The filament network is generated by stochas-
probability: If at least two particles are at sites
tic dynamics, updating network configurations
adjacenttoanintersectionsite,eachparticlemay
bytheprocessescommentedintable2andillus-
only access the intersection site with the proba-
trated in fig. 3. Since the model is motivated by
bility1−b(cf.figure1).Particlesonintersection
the dynamics of actin filaments some additional
sites retain their moving direction.
parameters originating from the biological pro-
The explicit rules for the particle dynamics
cesses in cells are included.
are displayed in table 1 and illustrated in figure
Thequantityρ introducedintable2rep-
1. We have chosen the default parameter values ARP
resents the density of free ARP2/3-complexes
analogous to [8] ω =1, p=0.5, ω =0.02 and
D d that serve as nucleation and branching seeds for
system size N =200. In [8] the attachment rate
filaments,whiletheactindensityρ corresponds
isequaltoone,whichcorrespondstoaneffective act
to the density of free nodes (actin monomers)
attachment rate ω = 0.25 if a particle is on
a constituting the filaments. Their initial values
an adjacent non-active site1. To be consistent
areρ0 andρ0 whichcorrespondstothecase
withthesubsequentcontinuousspacemodel,we ARP act
if all monomers are dissociated. The densities
choose b=1.
decrease with the growing filament network as
shown in fig. 4. After 5000 steps a stationary
actin density is reached. We assume that also
2.2 Inhomogeneous networks the structure of the network is stationary and
in qualitative agreement with a real actin net-
Generalising to continuous space and allowing work. We therefore stop network dynamics at
forarbitrarydirectionsandlengthsofactivestripesthis point.
we present a continuous model with randomly After construction of the network particles
generated linear filaments where hard-core par- obeying the exclusion principle are fed into the
systematrandompositions.Asmentionedabove,
1 Incontrastto[8],weallow forcrossingof active the particle positions are updated following a
stripes by diffusion. random sequential update scheme, whereby the
4 Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks
Process Particle state(s) Description Probability
Diffusion D Detached particles move to sites randomly chosen from ω =1
D
the four neighbors
Forward Step A Attached particles move to the next site in forward di- p=0.5
rection of filament
Attachment D→A Detached particles on filament sites becomes bound ω =0.25
a
Detachment A→D Attached particles become detached ω =0.02
d
Blocking D Forward movement of particles adjacent to intersection b=1
sites is inhibited if other particles occupy sites adjacent
to intersection
Table 1. Brief prescription of the dynamic processes in the square lattice model. Column 2 displays
the particle states “attached”(A) and “detached”(D). The right column displays the probability that the
respectiveprocessoccurswithinonetimestep.Numericalvaluesgivenintherightcolumnaredefaultvalues
which are used if not stated else and are chosen to fit the ones in [8].
Process Description Probability
Nucleation Initialization of filaments with arbitrary direction at an ω ρ ρ
n act ARP
arbitrarypointinthesystem.The(-)-endreceivesacap
inhibiting shrinking.
Branching Newfilamentsareinitializedatanexistingone(notnec- ω ρ2 ρ
b act ARP
essarily the (+)-end; angle between parent filament and
branch=70o[24].
Growth New nodes are generated at the (+)-ends of filaments . ω ρ
g act
Shrinking Nodes are removed at the (-)-end of filaments if the end ω
s
is not capped.
Uncapping Caps are removed. ω
u
Table 2. Dynamics of the filament network.
particle-particle as well as the interactions be- thediscretemodelintroducedinthelastsection
tweenparticlesandthegeneratedstaticnetwork relying on the model in [8].
areconsidered.Likeintheregularnetworkmodel
particlescanfreelydiffuseinthe’detached’state
andperformdirectedmovementinthe’attached’ 3 Results
state.Therulesoftheparticledynamicsarepre-
scribed in table 3 and illustrated in fig. 2. 3.1 Characterisation of Clusters
Although we do not consider a particular bi- Asalreadymentionedintheintroductionweaim
ological system, we choose parameters to fit the to relate the microscopic particle dynamics to
typicalorderofmagnitudeinrealvesiculartrans- the size distribution of their aggregates. In this
port.Ifnotstateddifferently,wewillusedefault section we discuss the definition of clusters for
parameters displayed in table 4 for our simu- the different model systems.
lations. The referenced works used experimen- Clusters are groups of particles that are con-
tal and modeling techniques to obtain the data nectedbyoverlappingneighborhoods.Wethere-
giveninthethirdcolumn.Forparticledynamics, fore introduce the λ−neighborhood of a parti-
we choose the parameters to be consistent with cle representing a disc of radius λ around the
Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks 5
Process Particle state(s) Description Probability
Diffusion D Detached particles move in a random direction. Step ω =1
D
widths are uniformly distributed between 0 and 2l
D
Step A Attached particles move to adjacent node in (+)- p
direction.
Attachment D→A Particles bind to nodes if their distance is less than d , ω
b a
becoming ’attached’
Detachment A→D Particles detach ω
d
Table 3. Particle dynamics. A=’attached’; D=’detached’
Parameter name Reference Reference Value model parameters
Filament dynamics:
nucleation rate ω [25] 8.7 ×10−5µM−2s−1 1.0×10−5lu−6tu−1
n
growth rate ω [25] 8.7µM−1s−1 0.25lu3tu−1
g
shrink rate ω [26] 4.2s−1 0.075tu−1
s
branch rate ω [25] 5.4 ×10−4µM−3s−1 0.0001lu9tu−1
b
uncap rate ω [25] 0.0018s−1 0.0001tu−1
u
actin density ρ0 [27] free actin: 0.1−1µM 2lu−3 1
act
ARP2/3 density ρ0 [28] 0.1µM 0.1lu−3
ARP
Particle dynamics:
particle radius r [29] 42.5nm (average) 0.5 lu
p
binding distance d [8] 1 site (50nm) 0.5 lu
b
node distance d [21] 36nm 0.36 lu
n
attachment ω [8] 1/4 of diffusive steps 0.25 tu−1
a
detachment ω [8] 0.8s−1 0.02tu−1
d
diffusive step l [8] 1 per time step 0.5 lu
D
step rate p [8] 20s−1 ⇒v=1µm/s 0.75tu−1
particle density ρ [30] 10-60 vesicles in bud (∼0.75µm radius) 0.04lu−3
Table 4. Default parameters of the model which are biologically motivated by transport of vesicles
by myosin on actin filaments. The referenced values are either based on experimental data or existing
models for intracellular transport [8] and filament dynamics [28]. Model parameters are chosen to be in
the order of magnitude of referenced values, fitted to time and space scale of the simulations. Length
scale: 1lu = 100nm = 2r ⇒1µM =0.6lu−3. Time scale: 1tu=∆t=0.025s. By default we consider
p
square systems of system length L=200lu.
1As shown in figure 4 the amount of overall actin is about ten-fold larger than free actin. Thus we
chose this factor to extrapolate the experimantal date to ρ0
act
center of the particle. A cluster is defined as a not stated differently we choose λ = 2r , which
p
set of particles included in a connected area of turns out to meet this condition (cf. fig 19).
λ−neighborhoods(cf.fig.5).Ifcontinuousspace In lattice models, static particle clusters are
variables are used, there exists no natural scale usually considered as as connected sets of ad-
which identifies two particles as neighbors. We jacent particles. However this definition is not
therefore have to specify the value of λ. In or- appropriate in this context since clusters move
der to extract relevant results, we fix λ so that by propagation of vacancies. Therefore we con-
qualitative results are robust on deviations. If sider particles separated by a single vacancy as
belonging to the same cluster.
6 Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks
Fig.2.Illustrationoftheparticledynamicsinanin-
Fig. 5. Illustration of particle clusters. Black
homogeneousnetwork.Darkgrey discs arefreepar-
discs represent particles, while grey discs are the λ-
ticles,blackdiscsrepresentparticlesattachedtofila-
neighborhoods of each particle. Connected grey ar-
mentssteppingtoadjacentnodes(distanced )with
n
eas are clusters; the size of a given cluster is the
ratep.Particlescanattachtofilamentswithrateω
a
number of particles on it.
if they are within the binding distance d and de-
b
tach with rate ω . Overlapping is inhibited due to
d
exclusion.
Our main interest is in ensemble and time
averages of cluster size distributions (CD) and
theirasymptoticbehaviour.CDsdisplaytherel-
ative frequency of cluster sizes emerging in the
system.Ifnotstateddifferentlyweaveragedover
50000timestepswithinindividualruns,evaluat-
ing cluster distributions in distances of 500 time
steps, taking an ensemble of 100 samples.
Fig. 3. Illustration of the filament dynamics. Fila-
mentsareimplementedassequencesofnodes(small Clustering also occurs for random particle
dots, corresponding to actin monomers) separated configurations. In fig. 6 and 7 we displayed clus-
byadistanced (shortbars).Filamentsegmentsare tersizedistributionsofrandomconfigurationsin
n
polarized,witha(+)-endwherenodesarecreatedto discrete and continuous space for different par-
elongate,anda(-)-endwherenodesdissociatecaus- ticle densities. Here the density ρ is the parti-
ing shrinking. cle number per area unit which corresponds to
(2r )2 in continuous space and one site in the
p
lattice model. If densities are not too large, the
formation of large clusters is impeded resulting
0.4 in an exponentially decaying cluster size distri-
bution. For high densities one observes a small
ρact peakattherightend.Atthesedensitiesclusters
0.2 spanning the whole system emerge. In order to
ruleoutthesekindsofrandomclusteringwewill
onlyconsiderdensitiesbelowtheregimeofspan-
0
0 5000 10000 ningclustersatrelevantscalesλ.Inthisworkwe
time steps
are interested in cluster formation mechanisms
Fig. 4. Density of free actin ρact in dependence beyond random clustering.
on time. After 5000 time steps a stationary state is
reached. Inthefollowing,wewillfocusonparticlecon-
figurations and cluster-size distributions in sev-
eral transport models.
Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks 7
(a) continuous space (a) continuous space
1 ρ=0.012 1
ρ=0.04 λ=2 r
quency 0.00.11 xρρex0===p002o..13.n8,2e 9pnetiaakl faitt 1~6 e0x0p0(-x/x0) quency 0.1 λλ==46 rrppp
relative fre0.00.000011 relative fre 0.00.0011
1e-05 10 20 30 0.00011 10 clust1e0r0 size 1000 10000
cluster size
(b) discrete space
(b) discrete space
10 ρ=0.01 d=0
ρ=0.03 y 0.1 d=1
uency 0.11 xρρex===p002o..13.n5,e 6pnetiaakl faitt 3~0 e0x0p0(-x/x0) quenc 0.01 d=2
q 0 e
relative fre0.00.000.000111 relative fr01.00e.00-000511
1e-05
1 10 100 1000 10000
10 20 30 40 cluster size
cluster size
Fig.7. Clustersizedistributionsofrandomparticle
Fig.6. Clustersizedistributionsofrandomparticle
configurations for different coarse graining scales. λ
configurationsindependenceontheparticledensity.
istheradiusoftheenvironmentasdefinedinsection
For low densities the CD decays fast with a short
3.1,whiledrepresentsthemaximumdistance(num-
size-scale. For large densities clusters on large size-
berofvacancies)allowedbetweentwoparticlescon-
scales and even such that span the whole system
nectingacluster.Oneobservesthatforlargecoarse
emerge (not visible in figure since on too large size-
graining scales clusters spanning the whole system
scale).
emerge.
3.2 Aggregation without network
where x,x(cid:48) are particle positions. This potential
can be implemented using a Metropolis accep-
Apossiblescenarioforclusteringofreceptorson
the cell surface is aggregation due to an attrac- tanceprobabilityp=min(e−β(V(xn+1)−V(xn)),1)
forastepfromx tox (ndenotesthetimein-
tiveinteractionbetweenproteinsdiffusinginthe n n+1
dex).Inthefollowingweusedimensionlessquan-
cell membrane. This process can be formulated
titiesandputβ =1.Thedefaultparametersare
as an equilibrium model consisting of diffusing
V = 2,d = 3.5r and particle density ρ =
hard-core particles (radius r ∼ 10nm; within 0 V p
p
0.04. Assuming a diffusion constant for mem-
the size-scale of membrane proteins) interacting
brane proteins D ≈0.0025µm2/s [31] we choose
via an attractive potential. We apply the par-
a time step corresponding to ∆t=0.02 seconds
ticle dynamics discussed in sec. 2.2 but do not
sothatonediffusivestepoflengthr =10nmis
consider filaments. In addition we introduce a p
performed per time step ∆t.
particle-particle interaction between the parti-
Infig.8typicalparticleconfigurationsatsev-
cles realized by a square well potential of the
eral runtimes are displayed, while in fig. 9 en-
form
semble averages of cluster size distributions are
(cid:26)−V for |x−x(cid:48)|≤d shown.Initialclusteringalreadyoccursonarather
V(x−x(cid:48))= 0 V (1)
0 for |x−x(cid:48)|>d small time-scale. Regarding the particle config-
V
8 Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks
urations we see that the number of clusters de-
creases with increasing runtime while the aver-
age size of remaining clusters increases. This is
due to diffusion and merging of existing clus-
ters after long times. Movement of large clus-
ters is strongly supressed, so that merging oc-
curs quite slowly. The coarsening process can
also be observed in the cluster size distribution.
We observe a characteristic scale for larger clus-
ters, manifesting in the emergence of a maxi-
mum,indicatingacharacteristicscaleforcluster
sizes. The dominant clusters always are within
the same size-scale which increases with time.
Since the cell membrane changes its struc-
turecontinuously,patternsarisingattime-scales
corresponding to a finite fraction of a cell cycle
cannot be assumed to be in a stationary state.
Computing time averages we therefore focus on
intermediatetimesandfixtheaveraginginterval
starting at 20000 time steps (corresponding to
∼7minutesinrealtime)afterrandominitialisa-
tion of particles and ending at 30000 time steps.
Thetimeintervalliesinthetransientregimefor
defaultparameters.Withinthisintervalwecom-
putedclustersizedistributions(timeandensem-
ble averages, 200 samples) for different param-
eter regimes and displayed them in figure 10.
One observes that for weak interaction no sig-
nificant clustering occurs manifesting in an ex-
ponentially decaying CD, while for strong inter-
action V , including the default parameters, a
0
maximumemergeshallmarkingtheformationof
clusters.
3.3 Directed transport on regular networks
In this section we examine features of particle
configurations and cluster distributions in the
model introduced in section 2.1, i.e. a regular
network of active stripes. As in the last section
we start time averaging after t = 20000 time
s
steps.Wecarefullycheckedthatastationarystate
has been reached at this point. (cf. fig. 12). As
time averaging interval we choose 50000 time
steps. In fig. 11 particle configurations for mod-
erateandhighdensitiesaredisplayed.Forparti-
cledensityρ=0.04oneobservessmallL-shaped
clusterscenteringatintersections.Forhigherden-
sities it appears that clusters that are becoming
Fig. 8. Configurations of particles (black discs
= particle neighborhoods with radius λ = 2r ),
p
exhibiting a mutual attractive interaction. Snap-
shots at different times for V = 2,d = 3.5, ρ =
0 V
0.04, L = 200, 1 timestep=ˆ0.025sec. One observes
that already at small times clusters form and for
longruntimestheythenumberofclustersdecreases,
while the average size of cluster increases.
Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks 9
1 (a)
100 time steps
y 1000 time steps 1
nc 0.1 10000 time steps V=1
ative freque 0.00.0011 100000 time steps e frequency 0.00.000.111 VVVV00000====1223..55
rel0.0001 ativ
el0.0001
50 100 150 r
cluster size
1e-05
20 40 60 80
Fig. 9. Cluster distributions in dependence on the cluster size
runtime in a system without network but attrac-
(b)
tivesquarewellinteractionpotential.Parametersare
V =3,d =3.5, ρ=0.04, L=200r , average over
0 V p
d=2.5
2to0w0arrudnss.laArgmeraxscimaluesm. establishes,thatmovesslowly equency 0.00.11 ddddVVVVV====3345.5
e fr 0.001
v
larger and merge with each other to form large ati0.0001
el
mesh-shapedclusters(cf.fig11(b)).However,in r
1e-05
this case clusters are hardly distinguishable and
50 100
not well separated which results in sensitive de- cluster size
pendence on the coarse graining scale (cf. fig.
(c)
14). In fig. 13 we plotted the cluster size distri-
1
butions averaged over time and 100 individual
ρ=0.0012
runs. Examining the cluster size distributions in y 0.1 ρ=0.004
nc ρ=0.012
fig. 13, one observes an exponential decay for ue 0.01 ρ=0.04
densities which are biologically relevant (cf. see eq
alsotheconfigurationinfigure11(a)).Therefore e fr 0.001
only small clusters emerge while large clusters ativ0.0001
are exponentially supressed. el 1e-05
r
For large densities ((cid:38) 0.1) the decay of the
1e-06
cluster size distribution becomes algebraic, indi- 50 100
cluster size
catingthatclustersonallsize-scalesexist.These
large clusters correspond to the ones generated Fig. 10. Plots of cluster distributions in the at-
bymergedsmallclustersasdisplayedinfig.11(b). tractive particles model at intermediate times (t =
20000−30000timesteps)independenceonthepo-
tentialdepthV (a),thepotentialwidthd (b)and
0 V
3.4 Inhomogeneous networks particle density ρ (c). One observes the transition
fromanexponentialdecay(nonclusteringphase)to
The filament growth dynamics described in sec- theformationofamaximum,correspondingtoclus-
tion 2.2 generate a network where single fila- tersatthissize-scale(condensation).Defaultparam-
ments have random length and direction. In or- eters are given in the text.
der to keep dynamics simple but retaining the
crucial features of disordered networks, we here
neglect branching and the dynamics of ARP2/3
toobtainanetworkofuncorrelatedfilamentori-
entations. The other processes are required to
obtain a disordered stationary network configu-
ration.
10 Philip Greulich, Ludger Santen: Active transport and cluster formation on 2D networks
(a) ρ=0.04 1
number of time steps=100
ncy 0.1 nnuummbbeerr ooff ttiimmee sstteeppss==110000000
e
u
q 0.01
e
e fr 0.001
v
ati
el0.0001
r
1e-05
10 20 30
cluster size
Fig. 12. Clustersizedistributionsinaregularnet-
workfordifferentruntimes(givenintimesteps).For
a given runtime, we chose the last 100 steps to per-
formthemeasurement,taking100samples.TheCD
does not change after 1000 time steps, indicating
that the system is in a stationary state.
(b) ρ=0.15
Theconfigurationforρ=0.04(fig.15)shows
that well separated compact clusters exhibiting
differentsizesemerge(seealsoscalinginfig.19),
while at ρ=0.008 no clusters are observed. In a
large parameter regime including the biological
relevantdefaultparameters(table4),theasymp-
totic decay of the CD is algebraic in contrast to
thepredominantexponentialbehaviouronareg-
ularnetwork.Foragivencombinationofnetwork
structureandparticledynamicsthepropertiesof
the CD are determined by the particle density,
while varying other parameters does not lead to
Fig. 11. Particleconfigurationsinaregularsquare
qualitative changes except in extreme regimes.
networkwithsystemlengthN =100sites.Theblack
In fig. 20 cluster size distributions of a regular
discs represent particle neighborhoods with radius
andinhomogeneousnetworkarecompared2.One
1.1sitessothatdiscsofparticleswithonevacancyor
observesthatclusteringissignificantlyenhanced
less in between overlap. One observes the formation
in the inhomogeneous network.
of small L-shaped clusters at intersection points for
moderate densities (a). For large densities clusters Due to the finite number of particles there is
merge forming cluster meshes on all size-scales (b). a cut-off at the upper end (e.g. in fig. 19). Fig.
18(a)showsthatforincreasingsystemsizeLthe
cut off regime tends to larger values, indicating
thatthisindeedisafinitesizeeffectandasymp-
3.4.1 Dynamics of interacting particles toticallyalgebraicbehaviourprevailsinthether-
modynamiclimit.Thoughtheexponentγ ofthe
algebraicdecayvariesfordifferentparticledensi-
We obtained particle configurations and cluster ties, the algebraic form is a robust feature. This
size distributions applying steric interactions, indicates that in the thermodynamic limit clus-
whichareshowninfigures15-19.Thetimeevolu- ters on all size-scales exist. In contrast to reg-
tionoftheclustersizedistribution(fig.17)shows
that a stationary state is reached after 10000 2 Differences in effective rates due to the different
time steps. Starting time averaging after 20000 spatialcharacterofthesystem(discreteandcontin-
timesteps(averaginginterval=50000timesteps) uous) are not significant since dependence on these
therefore captures the steady state dynamics. parameters is weak.