Transport on a Lattice with Dynamical Defects

0.6 MB·English

by  Francesco Turci

Transport on a Lattice with Dynamical Defects
Francesco Turci†,1 Andrea Parmeggiani,2 Estelle Pitard,1 M. Carmen Romano,3,4 and Luca Ciandrini†,3,∗
1Laboratoire Charles Coulomb Universit´e Montpellier II and CNRS, 34095 Montpellier, France
2Laboratoire de Dynamique des Interactions Membranaires Normales et Pathologiques,
UMR 5235 CNRS and Universit´e Montpellier II, 34095 Montpellier Cedex 5, France
3SUPA, Institute for Complex Systems and Mathematical Biology,
King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
4Institute of Medical Sciences, Foresterhill, University of Aberdeen, Aberdeen AB25 2ZD, United Kingdom
2
1 Many transport processes in nature take place on substrates, often considered as unidimensional
0 lanes. These unidimensional substrates are typically non-static: affected by a fluctuating environ-
2 ment, they can undergo conformational changes. This is particularly true in biological cells, where
the state of the substrate is often coupled to the active motion of macromolecular complexes, such
g
as motor proteins on microtubules or ribosomes on mRNAs, causing new interesting phenomena.
u
Inspiredbybiologicalprocessessuchasproteinsynthesisbyribosomesandmotorproteintransport,
A
we introduce the concept of localized dynamical sites coupled to a driven lattice gas dynamics. We

2 investigate the phenomenology of transport in the presence of dynamical defects and find a novel
2 regimecharacterizedbyanintermittentcurrentandsubjecttoseverefinite-sizeeffects. Ourresults
demonstrate the impact of the regulatory role of the dynamical defects in transport, not only in

] biology but also in more general contexts.
h
c
e
Spatial organisation and order is a fundamental char- tice with dynamical defects, i.e. sites whose features are
m
acteristic exhibited by many complex systems. The high dynamically coupled with the presence of particles. To
-
t degree of coordination and organisation, particularly in do that, we use an approach based on a paradigmatic
a
living systems, is often achieved by forms of molecular model in non-equilibrium statistical physics, namely the
t
s transport, also crucial for material biosynthesis [1]. totallyasymmetricsimpleexclusionprocess(TASEP)[5].
.
t Transport in several biological processes is carried out We define as dynamical defects sites with two possible
a
m on complex substrates of relative small sizes, such as mi- conformational states that depend on their particle oc-
crotubule filaments or mRNA strands [1]. These sub- cupancy. In the presence of dynamical defects, a new
-
d strates are not rigid, and can either actively or passively phenomenology appears. The current-density relation
n change their conformation in time. Remarkably, the dy- presents a plateau characterised by an intermittent flow
o
namicalfeaturesofthesubstratesaremostlycoupledand of particles that cannot be estimated by standard mean-
c
regulated by their interplay with the transport process. field arguments. Remarkably, in this new regime we find
[
In the case of protein synthesis, for example, ribosomes large finite-size effects that strongly affect the transport
2
canencounterregionsofthemRNAstrandthatobstruct characteristics.
v
the path and interfere with their movement. Those local
4 The Model - ThestandardTASEPisamodelofparti-
0 regions act like a dynamical switch that obstruct, when cles moving unidirectionally on a one-dimensional dis-
8 folded,orallow,whenunfolded,thepassageofribosomes.
crete lattice with a fixed hopping rate γ. Steric in-
1 Importantly their folding-unfolding dynamics is strongly
teraction excludes that more than one particle can oc-
.
7 coupled to the presence of ribosomes: a secondary struc- cupy the same site, and overtaking is not allowed. The
0 ture cannot fold until the ribosome has moved forward.
average TASEP current of particles J as a function
2 Thetrafficdynamicsishighlyenrichedduetothecou-
of particle density ρ is given by the parabolic relation
1
pling between the state of the substrate and the parti-
: JTASEP(ρ) = γρ(1−ρ) in the large lattice limit. The
v cles, and despite its primary importance to the physics
presence of one or more defects in the lattice, defined as
i
X of transport phenomena, it has been usually neglected “slow” sites with hopping rates smaller than γ, leads to
in the literature, with the exception of some particular
r a reduction of the maximal current flow [6] and confers
a cases related to microtubules dynamics [2]. Importantly,
a truncated parabola shape to the current-density rela-
the phenomenology goes beyond of biological contexts;
tion J(ρ). The defects considered so far in literature,
substratedynamicsandtheirbottlenecksareindeedcou-
however, are static since hopping rates associated to the
pled with the flow of carriers in many other disparate
slow sites are constant.
systems, suchasintelligenttrafficlights[3]orpedestrian
The periodic boundary case illustrates well the main
traffic [4]. An extensive theory capable of studying the
phenomenology of the model. We consider a ring of L
emergent effects of coupling between substrate and par-
sites on which N ≤L particles are allowed to hop in one
ticles transport is therefore necessary.
direction,fixingthedensityatρ=N/L. Aregionofsize
InthisLetterwepresentandstudytransportonalat-
d represents the dynamical defect and has an intrinsic
2
lag between the passage of two consecutive particles at a
given site [9]. Each regime exhibits a different slope in a
log-linear scale defining characteristic timescales. Three
main dynamical regimes appear: (i) TASEP-like regime:
when u > f > γ or u > γ > f, due to the rapid open-
ing,thecurrentisnotaffectedbythedefect. Thesystem
therefore behaves like a homogeneous TASEP and the
only relevant timescale is 1/γ (white region in Fig. 1b);
(ii) Static defect-like regime: when f > u > γ the dy-
namicaldefectactslikeastaticone,allowingthepassage
of particles in a way that can be described with a single
FIG. 1. (a) Mechanism for a 1-site dynamical defect: the effectiverateq <γ(top-rightgrayregioninFig.1b),sim-
site s (grey square) closes or opens with rates f and u, re- ilar to the model in [6]; (iii) Intermittent regime: when
spectively. Particles hopping at rate γ cannot enter closed γ >f >utherearetwostronglyseparatedtimescales,in
regions. Moreover, if a particle occupies the site s, the clos- contrast to the former cases. The short timescale 1/γ is
ingofthatsiteisforbidden. (b)Sketchofthephasediagram
the time separation between the passage of two consec-
ofthesysteminthef/γandu/γspace. Thedifferentregimes
utive particles during an opening event, while the long
are separated by dashed lines.
timescale 1/u corresponds to the time intervals during
which the defect is closed and obstructs the passage of
particles(blackregioninFig.1b). Thisisanovelregime
caused by the presence of the dynamical defect. The re-
two-state dynamics coupled to the presence of particles. maining cases (γ >u>f and f >γ >u) are crossovers
The defect can pass from the closed state to the open regimes between the intermittent and the homogeneous
state with rate u. The inverse process occurs with rate or static-defect TASEP-like behaviors, respectively.
f only when all d sites of the defect are empty. Our Strikingly, numerical simulations also show that the sys-
results indicate that the single-site defect d=1, Fig. 1a, temissubjecttoseverefinite-sizeeffects. Ifnototherwise
exhibitsthemainfeaturesofthemodel. Thereforeinthis specified,thefollowingresultsareintendedtobevalidin
Letter we focus, for sake of simplicity, on this case. The the limit of large systems. We will come back to the
model proposed here can be applied to describe further finite-size effects in the last section.
systems; forinstance, junctionsintransportnetworks[7]
Mean-field approaches – To compute the current-
can be thought of as particular dynamical defects with
density relationship we use a finite-segment mean-field
influx dependent dynamics.
(FSMF) approach [10]: the dynamics within the defect
Numerical Simulations - We have performed is treated exactly while the injection and exit rates in
continuous-time Monte Carlo simulations based on the and from the dynamical defect are given by effective val-
Gillespie algorithm [8]. We have studied the model in a ues. By imposing the current continuity and assuming
large parameter space taking several lattice sizes L (up that the system is split in HD and LD phases separated
to 4000) and varying the rates γ,u,f in order to explore by a dynamical defect at site s, the densities have to be
all the different dynamical phases sketched in Fig. 1b. coupled as ρ =1−ρ =1−ρ (ρ denotes the den-
HD LD s s
Similarly to other cases involving static defects or lo- sity of the defect site and we assume that the site s is in
calized blockages in 1D systems, our model shows a theLDphase). WethenmodelbothHDandLDpartsas
reduction of the current with respect to the homoge- openTASEPswitheffectiveentryandexitrates[7],with
neous TASEP; the current-density relation is a trun- α(cid:48) =γ(1−ρ )beingtheinjectionrateandβ(cid:48) =γ(1−ρ )
s s
cated parabola with a constant plateau value, see Fig. the depletion rate to and from the defect, respectively.
S1 [9]. In large systems, when the opening rate u de- We solve the related master equation in the steady-
creases with respect to the other typical time scales, the state [9] and compute, as a function of ρ ,f,u and γ,
s
current-densityplateaulowerstosmallerandsmallerval- the probability to find a particle on site s, which is by
ues, occupying a wider interval of densities and merging definition equal to the normalized density ρ . One then
s
withtheTASEPparabolaγρ(1−ρ)onlyattheextremi- obtains an expression for ρ as a function of all other
s
ties(forρ→0andρ→1). Asin[6],clustersofparticles parameters [9]. The plateau current is then given by
formbeforethedefectsothattheaveragedensityprofiles J = γρ (1−ρ ). This procedure can be extended
plateau s s
are usually characterized by a sharp separation between to larger defects (d>1).
a High Density (HD) and a Low Density (LD) phase. To validate the FSMF theory, in Fig. 2a we show the
Dependingontherelationshipamongtheparametersγ,u relativedifference∆J/J =(J−J )/J betweensim-
FSMF
and f we obtain different dynamical regimes, for which ulations and the FSMF in all the different regimes of
wedeterminethecharacteristictimescalesviathecompu- the phase diagram. Data are taken in the middle of
tationoftheprobabilitydistributionfunctionofthetime
3
!J L = 500 # = 1 $= 0.5 !tf" L=500 # = 1 f = 0.1 tions (Fig. 2b).
J 0 25 !00.. 54= 0.6 f(11- ! ) "MJ0F.3 L! =n=1 0o50H0n 5,e0 ef n=t c1fl e,1 0,o$0 w=a 0n.5p10i0an0stseersmtihttreonutghdytnhaemdiecfsecistefostraabltiismheedosfootrhdaert
0.3
1/u and then a large current J =γρ˜(1−ρ˜) flows for
open
−0.1 20 0.1
a time (cid:104)t (cid:105). This gives an average plateau current
"iMF0 f
J
−0.2 f=0.01, FSMF u/f
01.01 15(cid:2)(cid:4) 0.1 JiMF =γρ˜(1−ρ˜)1−ρ˜+u/f (1)
100
−0.3 0
10−6 10−4 10−2 1 102 10−2 1 102 104 106 10−4 whi1c0−h2 we1refer to as intermittent mean-field current
u/f a) u/f b) u
(iMF). Figure 3a shows that deep in the intermittent
FIG. 2. (color online) (a) Relative difference ∆J/J = (J − regime(whereρ˜∼0.5)thisapproachprovidesasubstan-
J )/J between simulations and FSMF approach. Cir- tial improvement to predict the current in the plateau,
FSMF
cles and diamonds show the crossover from the intermittent compared to the FSMF.
regime to the TASEP-like regime, while squares and trian-
Finite-sizeeffects–Asanticipated,thesystempresents
glesshowthecrossoverfromanintermediate-nonintermittent
strong finite-size effects, remarkably pronounced in the
regime to the slow-site like regime, see Fig.1b. (b) Average
closingtimes(cid:104)t (cid:105)(inseconds)fordifferentdensities. Vertical intermittent regime; here the current-density relation of
f
dashed lines represent the boundaries between the different small lattices becomes asymmetric (Fig. 3a) showing a
regimes crossed (see, dotted-dashed line in the inset). reduction of the current for small densities and enhance-
ment for large densities. Figure 3a shows that the value
of J in the plateau is not affected by the system size.
However, the plateau disappears when systems with just
the plateau of the current-density relation (ρ = 0.5),
a few sites are considered (Fig. 3b).
also to avoid deviations due to finite-size effects, which
will be discussed in the last section. This analysis re-
J J
vealsthattheFSMFapproximationisnotappropriatein L = 200 1000 iMF !"(1-")
500 FSMF 4 iMF
theintermittentregime(circlesanddiamondsinFig.2a,
u = 0.07
u/f < 1), while it gives reasonably good results in the 0.05
3
0.02
other regimes, in particular in the TASEP-like and the 0.5 0.01
slow-site-like regimes.
2
Intermittent Dynamics– We now turn our attention
to the intermittent regime, where the FSMF fails; we 1
start by analysing the average time between consecutive
opening and closing events (cid:104)t (cid:105), Fig. 2b. When the cou- 0 0
f 0 0.2 0.4 0.6 0.8 1.0 0 0.5 1.0
pling between the conformation of the defect and the ! a) " b)
presence of particles is weak, i.e. u is the largest rate,
the average time scales as (cid:104)t (cid:105) ∼ f−1. But when the FIG.3. (coloronline)(a)Intheintermittentregimetruncated
f
parabolasarefoundonlyinthelargeLlimit: smallersystems
coupling is strong, i.e. γ is the largest rate, the former
showcurrentreduction(enhancement)atlow(high)densities.
expression is corrected as (cid:104)t (cid:105) = [f(1−ρ˜)]−1, where ρ˜
f TheFSMFcalculationisunabletopredicttheplateauvalue,
is the probability to find a particle on the defect site
while the iMF does. The parameter used are f = 1s−1,u =
given that the site s is open. This probability can be 0.01s−1 and γ = 100s−1. (b) For very small systems (here
approximated in some limiting cases: in the TASEP-like L = 16) the iMF (dashed lines) correctly approximates the
regime, u is the fastest rate and hence the defect is al- numerical J(ρ) relation (f =1s−1,γ =100s−1).
most always open. In this case ρ˜ ∼ ρ, and the current
is very close to the one predicted by the pure TASEP. Such a behavior can be rationalized as follows: while
In contrast, in the intermittent regime J(ρ) exhibits a inthethermodynamiclimittheprobabilitytofindapar-
plateau [9], within which the opening-closing dynamics ticle on the defect site, when open, is well approximated
is independent of the total density. In this case particles byρ˜∼0.5,inverysmallsystems(e.g. L=16inFig.3b)
are blocked for long times behind the closed defect that the unstable HD-LD interface moving from the dynam-
occasionallyopens, thenallowingforacollectivepassage ical defect quickly relaxes over the whole system to the
of particles, as confirmed by kymographs or space-time homogeneous TASEP density. Therefore, when the de-
snapshots of the system [9]. Just after the opening of fect allows the passage of particles, all the sites can be
the defect, ρ˜can be estimated from a simple mean-field considered to be identical and the density on the defect
approach: dρ˜/dt=γρ (1−ρ˜)−γρ˜(1−ρ ), where ρ and ρ˜ ∼ ρ. Thus, identifying ρ˜ with ρ in Eq. (1) gives the
l r l
ρ denote the density of particles before and after the quantitative good agreement shown in Fig. 3b. More-
r
defect, respectively. Approximating ρ by 1 and ρ by 0, over, this reasoning allows determining the lower bound-
l r
one gets ρ˜ = 0.5 in accordance with numerical simula- ary for the J (ρ) relation for very small system sizes L,
L
4
theupperboundaryJ (ρ)beingthetruncatedparabola cess (TASEP), a prototypic model of transport in non-
∞
profile predicted by the iMF theory. For intermediate equilibriumphysics. Inthisframeworkwehaveprovided
system sizes L the situation is again different. Remark- originalmean-fieldargumentsthatallowreproducingand
ably,thecurrentforρ>0.5isenhancedcomparedtothe rationalizing the rich phenomenology of the model. We
the current obtained in the thermodynamic limit, while have thus unveiled a novel dynamical regime, character-
it is reduced for ρ < 0.5 (Fig. 3a). This effect can be ized by an intermittent current of particles and induced
understood by analyzing the relaxation dynamics after by the local interaction and competition between par-
an opening event by integrating the system of L coupled ticle motion and the defect dynamics. Importantly, we
ODEs describing the occupancy of each site in the lat- have found that a particle-lattice interaction triggers se-
tice [9]. verefinite-sizeeffectsthathaveacounterintuitivestrong
impact on transport. For different system densities, the
Intermittenceandrelatedfinite-sizeeffectshavestrong
smallsizeofthelatticeunexpectedlyreducesorenhances
consequences on the density profiles too. While, similar
the flow of particles, inducing an asymmetric current-
to the static-defect case, outside the intermittent regime
density profile.
there is a sharp phase separation between the HD-LD
Given the small size of biological substrates, the
beforeandafterthedynamicaldefect(Fig.4a),thepres-
physics of the new intermittent regime is highly rele-
enceofintermittenceinducesrelevantboundaryeffectsin
vant to understand protein synthesis and motor protein
the density profiles (Fig. 4b). This is a signature of the
transport. Bursts of gene expression have been often re-
transient relaxation of the HD-LD interface during the
ported[11]andourresultsprovideaphysicalmechanism
time (cid:104)t (cid:105). Such strong correlations are associated to the
f
to explain the contribution of the translation process to
non-stationary dynamics of finite systems, and are par-
them. Moreover, intermittent behavior can strongly in-
ticularly evident at densities for which current reduction
fluence motor protein current fluctuations that can be
(or enhancement) occurs.
measured in state of the art experiments [12]. In the
u = 0.01 , ! = 1 u = 0.01 , ! = 100 biological context of mRNA translation, our results on
1-! -› finite-size effects would correspond to an increase in pro-
s
tein production for small mRNA strands compared to
0.98 0.98
longer strands; interestingly, highly expressed proteins
)
x constituents of ribosomes are short [13].
!( ! = 0.8 ! = 0.8
0.7 0.7 The model presented here can be extended by in-
0.02 0.4 0.02 0.4
cluding, for example, larger interaction sites or differ-
0.1 0.1
!s -› ent boundary conditions. However, the observed phe-
nomenology does not change: if the defect is extended
0 250 500 250 500
(d > 1) the finite-size effects are even more pronounced
sites a) sites b)
thaninthed=1case[14],andlatticeswithopenbound-
FIG. 4. (color online) Density profiles for a lattice with L= aries, more common than lattices with periodic bound-
500,f =1s−1,u=0.01s−1. (a) For γ =1s−1 there is a clear aryconditionsinpracticalapplications,presentthesame
HD/LD separation with a sharp domain wall, in agreement characteristics, although edge effects can be relevant if
with the FSMF (horizontal dotted lines). (b) In presence of the defect is moved close to the boundaries [14].
intermittence(γ =100s−1)longercorrelationsareestablished
and simulations deviate from the FSMF prediction. We are grateful to I. Stansfield for bringing to our at-
tention this research topic, and to N. Kern, I. Neri and
C. A. Brackley for valuable discussions. FT is supported
Conclusion - In this Letter we have introduced the by the French Ministry of Research, EP by CNRS and
concept of transport on a lattice in the presence of lo- PHC no 19404QJ, LC by a SULSA studentship, MCR
cal interactions between particles and the substrate, also byBBSRC(BB/F00513/X1,BB/G010722)andSULSA,
referred to as dynamical defects. This concept is key to APbytheUniversityofMontpellier2ScientificCouncil.
understand natural transport phenomena occurring on
substrates with a fluctuating environment. It allows un-
derstandingfundamentalbiologicalprocessessuchaspro-
teinsynthesisorintracellulartraffic,wherethelocalcon-
formation ofthe substrate can deeplyinfluence the char- ∗ [email protected]; †denotes equal contribution.
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