Multiple phase transitions in a system of exclusion processes with limited reservoirs of particles and fuel carriers

Article (PDF Available)inJournal of Statistical Mechanics Theory and Experiment 2012(03) · January 2012with16 Reads
DOI: 10.1088/1742-5468/2012/03/P03002 · Source: arXiv
Abstract
The TASEP is a paradigmatic model from non-equilibrium statistical physics, which describes particles hopping along a lattice of discrete sites. The TASEP is applicable to a broad range of different transport systems, but does not consider the fact that in many such systems the availability of resources required for the transport is limited. In this paper we extend the TASEP to include the effect of a limited number of two different fundamental transport resources: the hopping particles, and the "fuel carriers", which provide the energy required to drive the system away from equilibrium. As as consequence, the system's dynamics are substantially affected: a "limited resources" regime emerges, where the current is limited by the rate of refuelling, and the usual coexistence line between low and high particle density opens into a broad region on the phase plane. Due to the combination of a limited amount of both resources, multiple phase transitions are possible when increasing the exit rate beta for a fixed entry rate alpha. This is a new feature that can only be obtained by the inclusion of both kinds of limited resources. We also show that the fluctuations in particle density in the LD and HD phases are unaffected by fluctuations in the number of loaded fuel carriers, except by the fact that when these fuel resources become limited, the particle hopping rate is severely reduced.

Figures

arXiv:1201.2855v1 [cond-mat.stat-mech] 13 Jan 2012
Multiple phase transitions in a system of exclusion
processes with limited reservoirs of particles and
fuel carriers
Chris A Brackley
1
, Luca Ciandrini
1
and M Carmen
Romano
1,2
1
Institute for Complex Systems and Mathematical Biology, SUPA, University
of Aberdeen, Aberdeen, AB24 3UE, United Kingdom
2
Institute of Medical Sciences, Foresterhill, University of Aberdeen, Aberdeen,
AB25 2ZD, United Ki ngdom
E-mail: cab@chrisbrackley.co.uk, l.ciandrini@abdn.ac.uk
PACS numbers : 05.60.-k, 05.40.-a,02.50.Ey
Abstract. The TASEP is a paradigmatic model from non-equilibrium statistical
physics, which describes particles hopping along a lattice of discrete si tes. The
TASEP is applicable to a broad range of different transport systems, but does not
consider the fact that in many such systems the availability of resources required
for the transport is limited. In this paper we extend the TASEP to include
the effect of a limited number of two different fundamental transport resources:
the hopping particles, and the “fuel carriers”, which provide the energy required
to drive the system away from equilibrium. As as consequence, the system’s
dynamics are substantially affected: a “limited resources” regime emerges, where
the current is limited by the rate of refuelling, and the usual coexistence line
between low and high particle density opens into a broad region on the phase
plane. Due to the combination of a limited amount of both resources, multiple
phase transitions are possible when increasing the exit rate β for a fixed entry
rate α. This is a new feature that can only be obtained by the inclusion of both
kinds of limited resources. We also show that the fluctuations in particle density
in the LD and HD phases are unaffected by fluctuations in the number of loaded
fuel carriers, except by the fact that when these fuel resources become limi ted,
the particle hopping rate is severely reduced.
Keywords: driven diffusive systems (theory), stochastic processes (theory)
Multiple phase transitions in a TASEP with limited particles and fuel carriers 2
1. Introduction
The totally asymmetric simple exclusion process (TASEP) is one of the fundamental
models of non-equilibrium statistical mechanics [1, 2, 3]. Essentially a driven diffusion
model, it has many applications in physics and beyond, including traffic models [4],
the movement of molecular motors in biological systems [5], and protein synthesis in
messenger RNA (mRNA) trans lation [6, 7]. It also belongs to the same universality
class as some surface growth models [8]. In this paper we study a constrained TASEP
where finite resources are shared among several lattices. By nite resources we mean
a constrained numbe r of both particles and fuel carriers”, whose role is to provide
the energy needed to the movement of the particles. Molecular motors requiring ATP
or GTP molecules are an example of s uch systems occurring in nature. In this paper
we introduce a new model which includes the finite availability of both re sources, in
contrast to previous works where the effect of having a finite number of a single type
of resource was studied in isolation [9, 10, 11, 12, 13, 14]. As a result, multiple phase
transitions can occur when varying one of the fundamental parameters of the model
–the exit rate β while keeping the rest of the parameters constant: the sys tem can
go from a high density r e gime, to a shock phase, then to a high density phase again,
visit the shock phase once mor e, and finally reach a low density phase. This is a novel
effect that emerges only by combining both limited resources . We use a mean-eld
approach and verify our re sults by means of Monte Carlo s imulations.
In its most simple form, the TASEP consists of a 1D lattice of L sites upon which
particles can sit, se e figure 1(a). Each site can be occupied only by one particle at
a time, and particles move from site to site in one direction (say rightward) with
a hopping rate k. Since particles cannot pass each other, movement requires tha t
the downstream site is vacant. A s ystem with open boundaries, as we shall conside r
here, can display rich dy namics with multiple boundary induced phases [15]. Particles
are allowed to hop onto the lattice with rate α at one e nd, and off of the lattice
with rate β at the other. For a system with constant internal hopping rate k it is
possible to solve the steady-state of the system exactly [16, 2, 17, 18], whilst the
full relaxation dynamics have be en solved using matrix methods [19, 2 0]. Mean-field
methods have also been extensively used [16], since they are easily tractable a nd yield
a good appr oximation in many cases. There has been much extension of this simple
model, for example varia ble hopping rates (site or pa rticle dependent) [21, 22, 23],
extended particles which cover more than one site [6, 2 4], branching lattices [25, 26],
particles which have multiple internal states [27, 28, 29], as well as a TASEP with a
constrained reservoir of particles [9, 10, 11, 12].
For the standard TASEP we denote the o ccupation of the ith lattice site n
i
= 1 if
the site contains a particle and n
i
= 0 otherw ise. The system is characterised by the
steady-state particle current J (the r ate at which particles pass any given point on
the lattice), and the mean site occupancy (often called density) ρ
i
= hn
i
i, where · ·i
denotes average over realisations of the system (which we assume is ergodic, so this is
equivale nt to a time average). The average density is therefore given by ρ = L
1
P
i
ρ
i
.
There are four possible phases depending on the values of α and β: the entry limited
or low density (LD) phase, the exit limited or high density (HD) phase, a maximal
current (MC) phase wher e the current depends only on the internal hopping rate, and
a mixed LD-HD or shock phase (SP). A mean-field approach [16] (which turns out to
be exact in the L limit) can be used to calculate J and ρ
i
for given α and β.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 3
Figure 1. Schematic diagrams describing the various systems. (a) The TASEP
with open boundaries its most simple form. Particles enter at fixed rate α, hop
at fixed rate k and leave with rate β. (b) A finite number of fuel carriers can be
introduced. When a particle hops, fuel from one carrier is used. (c) Complete
model wi th finite fuel carrier and finite particles. The entry rate depends on the
number of particles in the reservoir, and the hopping rate depends on the number
of loaded fuel carriers.
The density in the bulk (far from the ends o f the system) is given by
(LD) ρ
LD
= α/k for α < β, α < k/2 ,
(HD) ρ
HD
= 1 β/k for β < α, β < k/2 ,
(MC) ρ
MC
= 1/2 for β, α k/2 ,
(1)
and the current is always given by J = kρ(1 ρ). The SP, which occurs for
α = β < k/2, presents an HD region on the right of the lattice a nd an LD region
on the left, separated by a boundary which diffuses fr eely through the lattice. This
has often been described using a domain wall (DW) theor y [30]. Due to the fr ee
diffusion of the DW a time average of the density in SP gives ρ
SP
= 1/2, but the
current depends on the density in either the LD or HD regions o f the lattice, i.e.
J
SP
= α(1 α/k) = β(1 β/k).
In this paper we consider several TASEPs which share a c ommon finite pool of
both particles and fuel carriers. The entry rate, which is the sa me for each TASEP,
depe nds on the availability of particles in a common pool (i.e., particles which are not
involved with any lattice). A model describing several TASEPs sharing a common
pool of particles has been introduced and thoroughly studied in [9, 10, 11], where
the authors use the DW theory along with known exact results. In this paper we
use an alterna tive recent mean-field (MF) approach that allows us to simplify the
calculations [12]. Importantly, we combine this with a model for a finite pool of fuel
carriers [13, 14] which, as noted above, can be viewed as carriers that provide the
Multiple phase transitions in a TASEP with limited particles and fuel carriers 4
energy which drives the motion, i.e. allowing the particles to hop. Although we
consider a fixed number of fuel carriers, we suppose that it takes a finite time to
“refuel” them w ith their cargo once it has been used (figure 1 (b) shows a s chematic
representation of this model). We show that novel effects arise when both types of
limited resource are considered, e.g. multiple phase transitions ca n occur when varying
the exit rate β. The outline of the paper is as follows: in sec tion 2 we summarise the
previous results for the two models s e parately, before describing in section 3 a mean-
field mo de l for a system with b oth a finite pool of particles and fuel carriers which are
refuelled at a finite rate (figure 1(c)). We then interpret the mean-field model results
and compare them with results from Monte Carlo simulations. Fina lly, in section 4
we analyse the effect of both limited resources on the fluctuations in the number of
particles on the lattice.
2. Finite resources - Review of previous results
We first introduce and describe a system containing multiple TASEPs in which each
lattice s hares the s ame r eservoir of particles; then we present the concept of fuel-
carriers and the effect of a finite rate of refuelling on the exclusion pr ocess dynamics.
2.1. Finite number of particles
In this work we analyse a s ystem of M identical lattices of length L. The total number
of available particles is N, while the number of free particles in the reservoir is N
r
.
Since the lattices are identical and experience the same injection and depletion rates,
we observe the same phase for each. We can write the total number of particles as
N = N
r
+ LMρ, (2)
where ρ is the density on each lattice. The entry rate of the M lattices depends on
the numb e r of free particles via a saturating function
α = α
0
tanh
N
r
N
= α
0
tanh
N LMρ
N
, (3)
where the constant α
0
gives the e ntry rate in the limit N
r
and is an intrinsic
property of the lattices . Without loss of ge ne rality, we fix the normalisation factor
N
to be LM/2, i.e. the tota l number of particles used if all the lattices were in the
MC phase.
Throughout this paper we define the different phases according to the values of α
and β and the resulting density ρ, following [12]. With this choice of nomenclature we
solve e quations for α in terms of α
0
and N. Since the densities in each phase are the
same as those in the sta ndard TASEP (equations (1)), for a given set of parameters
(α
0
, β, N ) we find the resulting α which determines the phas e ; e.g. if α < β and
α < k/2 the system will be in the LD pha se. By substituting equation (3) into these
inequalities, we get a re presentation of the different phases on the α
0
β plane.
As a c onsequence of having a finite number of particles, we encounter different
regimes for small, mid-range and large values of N. We show typical phase diagrams
Equation (3) is consistent wi th the function used in [9, 10, 11], and is relevant, e.g. for the
application to protein synthesis .
Multiple phase transitions in a TASEP with limited particles and fuel carriers 5
Figure 2. Phase diagrams for a TASEP of length L = 500 with a finite number
of (a) N = 225, (b) N = 475, and (c) N = 600 particles (infinite amount of l oaded
fuel carriers). For s mall N there are not enough particles to support the HD and
MC phases, and the SP (coexistence) line opens into a region. For N > LM/2 all
four phases can be obtained. If N is increased further the HD phase grows at the
expense of a shrinking SP.
for these reg imes in figure 2. If N < LM/2, then the HD and MC phases no longer
exist there are too few particles to support the high density or maximal current
phases. Instead, ther e are only two phases: the LD phase and the SP (figure 2(a)). As
described in the previous section, the latter occurs when the entry and exit rates are
equal, i.e., α = β, and there is coexistence between an LD region and an HD regio n;
since α depends on both α
0
and β (through its depe ndenc e on the bulk density), the
line opens into a region on the α
0
-β phase plane. That is to say, the condition α = β
is fulfilled for a certain range of α
0
[12]. If N = LM/ 2, then the lattices can support
an MC phase, and for N > LM/2 there are enough particles for an HD phase to exist
(figure 2(b)). As N is increased, the size o f the HD phase on the α
0
-β plane increases,
at the cost of reducing the size of the SP phase (figure 2(c)). For N LM the SP
phase reduces to a line and we recover the original unconstra ine d TASEP.
In the unconstrained TASEP within the SP, the LD and HD regions of the system
are separated by a domain wall (DW) which can diffuse freely across the lattice.
However, if there is a finite number of particles, in the case of a single lattice the DW
is pinned to one position [9, 10] (actually the DW fluctuates about its mean position
like a noisy damped oscillator). This is be c ause if the DW were to move to the right
this would increase the number of free particles, increasing the entry rate and therefore
driving the DW leftwards. Similar ly if the DW moves to the left the number o f free
particles decreases, decreasing the entry rate and driving the DW rightwards. The
opening of the SP line into a region on the α
0
-β pla ne is possible because a different
mean p osition for the DW corr esponds to a different value of α
0
, while keeping α = β.
Hence, the system can maintain α = β for different values of α
0
. As detailed in [11], if
more than one TASEP is in contact with the same pool of particles, the DW on each
lattice once again performs a random walk; there is however a pinning of the total
number of particles on all lattices.
2.2. Finite reloading time for fuel carriers
In many of the systems that can be described by a driven lattice ga s, the
energy required fo r the advancement of the particles is obtained from some kind
of finite resourc e. For instance, molecular motors consume ATP molecules, and
ribosome movement on mRNAs during protein synthesis requires aminoacylated tRNA
Multiple phase transitions in a TASEP with limited particles and fuel carriers 6
complexes and GTP. The latter case has re cently been described in [13, 14, 31], and
here we briefly review those results, before in the next sectio n combining this with the
a finite pool of particles model.
We consider a finite numb e r
¯
T of fuel c arriers, T of which are carrying fuel. Every
time a pa rticle moves, the fuel from one of the loaded ca rriers is used, and hence, T
is reduced by one. The hopping rate of pa rticles depends on the availability of loa de d
fuel carrie rs, and the empty carriers are refuelled at a ra te V . For simplicity, the
hopping rate is taken to be directly proportional to the loaded fuel carriers, i.e.
k = aT, (4)
where a is a constant. Moreover, the rate of refuelling is taken to depend on the
number
¯
T T of unloaded fuel carriers as
V =
V
0
(
¯
T T )
b +
¯
T T
,
which has the form of the well known Michaelis-Menten equation in biochemistry.
The recharging rate is therefore a saturating function of the number of empty carriers
(
¯
T T ) with maximum value V
0
and saturation determined by the constant b. Any
saturating function will give the same qualitative res ults, but the above fo rmulation
allows for a straightforward analytical treatment [13, 14] §.
In this section we describe a c ollection of M identical TASEPs with a finite
number of fuel carrie rs (but no constraints on the number of particles); following the
common mean-eld trea tment [16] the particle density on the ith site of each lattice
is given by
1
dt
= α(1 ρ
1
) kρ
1
(1 ρ
2
),
i
dt
= kρ
i1
(1 ρ
i
) kρ
i
(1 ρ
i+1
), i = 2, ...L 1,
L
dt
= kρ
L1
(1 ρ
L
) βρ
L
.
(5)
The inclusion of a finite pool of fuel carriers leads to the additional equation
dT
dt
=
V
0
(
¯
T T )
b +
¯
T T
(L1)M
X
j=1
kρ
j
(1 ρ
j
), (6)
where the sum is over all of the L 1 sites which use fuel carriers on each of the M
lattices. We assume that the particles do not require a fuel carrier to leave the Lth
site, i.e. the exit rate β is constant. In the steady-state we identify the term under
the sum in (6) as the particle current, and using equation (4) we find
k = a
¯
T
abJ(L 1)M
V
0
J(L 1)M
, (7)
i.e., the hopping rate is now itself a function of the current. Following [13, 14], upon
solving eq uations (5) in the steady-state we find the four phases as in the original
TASEP, but now the current and density are given as follows
(LD) J
LD
= J (α), ρ
LD
= D(α) for α < α
and α < β,
(HD) J
HD
= J (β), ρ
HD
= 1 D(β) for β < α
and β < α,
(MC) J
MC
= α
/2, ρ
MC
= 1/2 for α, β α
,
(SP) J
SP
= J (α), ρ
SP
= 1/2 for α = β < α
,
(8)
§ An alternative model would be to have a hopping rate which is a saturating function of
¯
T , and
then have a constant refuelling rate. This would give qualitatively similar behaviour to the present
definitions. Our choice is most applicable to protein synthesis, i.e. refuelling due to an enzymatic
reaction.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 7
where
J (α) =
1
2
α
1
α
a(
¯
T + b)
+
¯
T
¯
T + b
V
0
L
M
s
¯
T
¯
T + b
V
0
L
M
+ α
1
α
a(
¯
T + b)

2
4α(a
¯
T α)
a(
¯
T + b)
V
0
L
M
,
D(α) =
1
2α
α
1 +
α
a(
¯
T + b)
¯
T
¯
T + b
V
0
L
M
+
s
¯
T
¯
T + b
V
0
L
M
+ α
1
α
a(
¯
T + b)

2
4α(a
¯
T α)
a(
¯
T + b)
V
0
L
M
,
and
α
=
a
4
(
¯
T + b) +
V
0
L
M
s
a
4
(
¯
T + b) +
V
0
L
M
2
a
¯
T
V
0
L
M
,
with L
= L 1. The behaviour of these functions as α and β are varied dep ends
on the parameters a, b and V
0
. By considering the steady-state of equatio n (6) and
noticing that the maximal value that the recharging rate can possibly have is equal to
V
0
, we note that the particle current is limited from above by V
0
/L
M. Hence , ther e
are substantially two different cases: (i) if V
0
/L
M 1, the recharging rate is very fast
and the particle current is not influenced by it; we recover the r esults of the original
TASEP; (ii) if in co ntrast V
0
/L
M 1, the recharging o f the fuel carriers can limit
the value of the particle current. Figure 3 shows the current for sets of parameters
corres ponding to each case. In ca se (ii) (figure 3(b)), J (α) shows a sharp change from
increasing with α, to almost independent of α (though we note that the derivative of
J (α) remains continuous). The value of J (α) is severely reduced compared to the
one obtained for case (i) (see figure 3(a )). We r efer to the regime w he re the current
appears independent of α as a limited resources (LR) regime, s inc e the rate at which
fuel is used by the particles approaches the rate at which fuel c arriers ar e re loaded.
Thus the pool of loaded carriers becomes depleted and the hopping rate k reduces. For
some choices of a and b, the LR regime exists within each of the phases (LD, HD and
MC). In the LR re gime within the LD phase, the sensitivity of the current to changes
in α or β is greatly r educed, whilst the sensitivity of the density is greatly increased.
In the MC phas e, the current is greatly reduced in the LR regime compared to that
in case (i). For further details see [13, 14].
The onset of the LR regime depends particularly on the value of the three
quantities a
¯
T , V
0
/LM and b/
¯
T , the former two controlling at what value of α or β the
onset will occur, and the latter controlling the sharpness of the change in behaviour.
In the rest of this pa per we choose b such that there is a sharp onset of LR, a nd take
V
0
as the control parameter for the fuel carriers, fixing the other parameters. This
choice not only gives the most interesting dynamics, but it has also been shown to
be the biological relevant regime in the context of protein synthesis [13, 14]. When
the onset of LR is sharp we can estimate the value of α or β at which this oc curs
by equating the rate of fuel carrier use (appr oximately αL
M for small α in LD and
βL
M for small β in HD) and the maximum recharging rate. This gives
α
LR
= β
LR
¯
T V
0
(
¯
T + b)L
M
. (9)
Multiple phase transitions in a TASEP with limited particles and fuel carriers 8
Figure 3. Plots showing the current as a function of α for different parameters.
In (a) V
0
= 300 s
1
, and in (b) V
0
= 20 s
1
. In both cases a = 2 × 10
4
,
¯
T = 5000, and b = 50. Solid lines show the current in the LD phase, J
LD
= J (α),
and dashed lines the current in the MC phase, J
MC
= α
/2. The dotted line is at
α
, where there is a transi tion from LDMC. The dot-dashed line in (b) shows
the value of α
LD
as estimated in equation (9).
Figure 4. Steady-state fuelling level T /
¯
T as a function of rate V
0
for a model
with no constraints on the number of particles (section 2.2). Other parameters are
L = 500, α = 0.5 s
1
, β = 0.1 s
1
, a = 2 × 10
4
s
1
, b = 50, and
¯
T = 5000. For
small V
0
the number of loaded fuel carriers is depleted; for large V
0
the carriers are
practically always f ully loaded, and we recover the original unconstrained TASEP.
This value is repres ented in figure 3(b) by a dot-dashed vertical line. As it is shown
there, the estimation predicts q uite accurately the onset of the LR regime.
3. Constrained reservoir of particles and finite refuelling rate
A much more realistic model for natural processes such as biological transport has to
include the finite availability of both par ticle s and fuel carriers. As we show later
in this section, it is only when combining the two schemes discussed above that
we can se e emerging novel effects, such as multiple phase transitions. Analogous
to the de pendence of the entry rate α on the number of pa rticles N , the s teady-
state proportion of loaded fuel carriers is a saturating function of V
0
(see figure 4).
Therefore, by regarding N and V
0
as control parameters, we can va ry the number of
available particles and loaded carriers respectively; in both cases a saturating function
of the resource determines the dynamics.
The quantities of interest are the particle current J, the number of loaded fuel
carriers T and the particle density ρ, which is linked to the number of free particles
by equation (2). The expressions for J a nd ρ given in equations (8) still hold; however
α is no longer a control parameter, and it can be eliminated using equation (3). This
is the case in all phases exce pt in the SP: since the DW cannot move freely on the
lattice, the mean density now depends on the size of the LD and HD regions. We
Multiple phase transitions in a TASEP with limited particles and fuel carriers 9
calculate the average value ρ
SP
using equation (3); using the fact that in this pha se
α = β leads to
ρ
SP
=
N
LM
1
N
N
tanh
1
β
α
0

. (10)
The hopping rate k (and therefore T ) in each phase can be found by substituting the
appropriate equation for the current J in (7).
We now turn to the pro ble m of finding the boundaries between the different phases
as functions of α
0
, β, N and V
0
i.e., eliminating α. Our aim is to draw the α
0
-β
phase plane for any given values of V
0
and N . As noted in previous sections, due to
finite particles, the SP line o pens into a region, and the HD and MC phases do not
exist if N < LM/2. Each phase boundary can be written in terms of either α
0
as a
function of β, or vice versa. We now consider the boundaries between e ach phase in
turn, cons ulting figure 2 as an ansatz for the arrangement of the phases .
(i) MC/LD phase boundary. The MC phas e can exist if N > LM/2. If we consider
starting in the MC phase with large β and reducing α
0
, we will cros s into the LD
phase when α = α
. Using (3) this gives an equation for the boundary in the
α
0
-β plane, where
α
0
= α
coth
N
N
LM
2N
for β α
and N > LM/2 .
Note that α
depe nds on a, b,
¯
T , V
0
, L and M. Hence the MC/LD boundary is
a vertical line on the α
0
-β phase plane.
(ii) MC/HD phase boundary. If we now consider starting in the MC phase with large
α
0
and r educing β, then we cross into the HD phase when β = α
, i.e., in the
α
0
-β plane this boundary is given by the horizontal line
β = α
for α
0
α
coth
N
N
LM
2N
and N > LM/2 .
(iii) HD/SP phase boundary. Here we consider moving from the HD to the SP. For
the system to be in the HD phase requires β < α
, N > LM/2 and β < α. Using
equation (3) in the latter inequality gives β < α
0
tanh(N
r
/N
); using equation
(2) and the density in HD phase gives the following equation for the b oundary
α
0
= β coth
N
N
LM
N
(1 D(β))
for β < α
and N > LM/2 .(11)
Hence, the HD/SP phase boundary is a curved line on the α
0
-β phase plane.
(iv) LD/SP phase boundary. Finally we consider starting the the LD phase and
moving to the SP. In LD we require α < β and α < α
. Finding an expression
for thes e inequa lities poses some difficulty, since by using equation (3) and the
density in the LD phase we obtain
α = α
0
tanh
N
N
LM
N
D(α)
,
an equation which cannot be so lved analytically to find α as a function of α
0
.
Instead we solve this numerically, setting α = β (which o ccurs at the SP) to find
β as a function of α
0
. This gives ano ther curved line on the α
0
-β phase plane.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 10
Figure 5. Colour on-line. Phase diagrams in the α
0
-β plane at different values
of V
0
and N. White lines show the phase boundaries as determined by the mean-
field model of section 3. Colour maps show the steady-state particle density ρ
from M onte Carlo simulations. Parameters used are V
0
= 20 or 300 s
1
, and
N = 200, 450, or 600. Other parameters are L = 500, M = 1, a = 2 × 10
4
s
1
,
b = 50, and
¯
T = 5000. The dotted line in (f) shows the line of constant α
0
used
in figure 8.
We can then construct the phase plane by plotting the phase boundaries (β as a
function of α
0
) for any given values of N and V
0
. Unless otherwise stated, throughout
the rest of this paper we use parameters
¯
T = 5000 so that
¯
T LM and hence, it is
always the refuelling which is the limiting process, and not the total number of fuel
carriers. This represents re alistic scenarios in biological transport processes, such as
protein synthesis. We set the time scale of the system by choosing a = 2 × 10
4
s
1
,
such that the maximum hopping rate is k = 1 s
1
. A value of b = 50 then gives a
sharp onse t of LR as shown in figure 3(b). The phase diagram boundaries calculated
using the mea n-field approach are shown in figure 5 using white lines.
To test the validity of the mean-field results derived above we perform simulations
using a continuous time Monte Carlo method [32]. The le ngth of the Monte Carlo time
step is chosen from an exponential distributio n, such that the events occur according
to a Poisson process, with a single event occurring at each step. Possible events are
the movement of a particle (either on to, along, or off of the lattice) or the refuelling of
a fuel carrier. The event which occurs is chos e n stochastically from the set of particles
which have a vacancy to their rig ht and the set of empty fuel carriers. Particles are
chosen with a probability such that they move with a rate k, and empty fuel car riers
are chosen with a probability such that they are recharged with rate V ; after each
event T is updated accordingly. To remove any transient effects associated with the
initial condition we disregard the first 5 × 1 0
6
time s teps. Assuming that the system
is ergodic we average currents and densities over at least a further 4 × 10
7
steps.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 11
Figure 6. Col our on-line. Colour maps showing Monte Carlo results for
the steady-state fuelling level T /
¯
T at different values of α
0
and β for sm all
V
0
= 20 s
1
. White lines show phase boundaries as predicted by the mean-
field model. (a) Small N = 200; (b) mi d-range N = 450; (c) large N = 600. For
these parameters we have α
LR
= β
LR
0.040 from (9). Other parameters are as
figure 5.
In figure 5 we prese nt a ser ies of phase planes for different values of N and V
0
.
We show the pa rticle density ρ o btained from Monte Carlo simulations as a colour
map with the phas e boundaries from the mean-field model overlaid (white lines ). Note
that the mean-field model very closely predic ts the boundaries. Here we show data
for M = 1 lattices, but the plots look the same for M > 1 with appropriately scaled
parameters. Introducing more lattices doe s not change the macroscopic behaviour,
but changes the microscopic behaviour for the SP (see section 3.1).
In figur e 5 we note that for small V
0
the phase diagrams look broadly similar
to the larg e V
0
case, but as we expect from [13, 14], the phase transitions occur at
much smaller values of α
0
and β. We also obtain a limited r esources (LR) regime
within each of the phases. This can seen in figure 6, which shows colour plots for the
proportion of charged fuel carriers (T /
¯
T ) for different values of N , with small V
0
. The
onset of the LR regime can b e c le arly seen as T/
¯
T decreases dramatically over a small
range of α
0
or β. For large V
0
the refuelling is so quick that T/
¯
T is constant through
all phases, i.e. we recover the res ults for a TASEP with a finite pool of particles, but
no constraints on the fuel carriers, since thes e are refuelled almost as soon as they are
used (data not shown). For clarity, throughout the rest of the pa per when we refer
to LR or limited resources, we spec ific ally mean the regime where the pool of loaded
fuel carriers has become depleted.
Crucially, the presence of the LR regime also alters the shape of the phase
boundaries; a noticeable “kink” can be see n in the LD/SP and HD/SP phase
boundaries at the p oint of LR onset. Particularly strik ing is the s hape of the HD/SP
boundary for the large N small V
0
case, and we exa mine this in deta il in section 3 .2.
3.1. The Shock Phase
As noted above, due to the finite number of particles, the coexistence line present
in the original TASEP the shock phase opens out into a region on the phas e
plane. We examine the behaviour in this phase by locating the domain wall (DW)
which separates the regions of LD and HD, and examining how the position of this
is affected by the finite number of fuel carriers and how it changes at different values
of α
0
and β. The introduction of a finite numbe r of particles also gives a change
from a DW which can wander freely along the lattice, to one where the wandering is
constrained by the presence of the reservoir. This is most easily explained in the case
Multiple phase transitions in a TASEP with limited particles and fuel carriers 12
of M = 1 lattices, where the DW is on average fixed in position. A fluctuation which
leads to movement of the DW away from its mea n position will change the number of
particles in the reservoir; this in turn changes the entry rate α, a cting like a restoring
force on the DW. In actual fact the DW executes Gaussian fluctuations about its
mean value, and we discuss fluctua tions further in section 4. If multiple lattice are
introduced (M > 1), then as in [11] the DW is not pinned, but rather it is the sum of
the DW position on all lattices which executes Gaussian fluctuations about a mean.
We focus on the M = 1 ca se for the rest of this section.
The relative mean position x [0, 1] of the pinned DW (where xL gives the site
at which it is located) can be estimated fro m the mean-field model by approximating
the density in the SP as follows
ρ
SP
=
LD
+ (1 x)ρ
HD
.
Since x is always selected such as to maintain the condition α = β, we can use equation
(10) a nd the densities from (8) to show that
x =
1
1 2D(β)
1 D(β)
N
LM
1
N
N
tanh
1
β
α
0

.
We also note that the difference between ρ
LD
and ρ
HD
decreases as β increases, i.e.,
the “height” of the wall decreases.
As it is the most interesting ca se, we focus on parameters where the SP has the
largest area on the phase diagram, namely the mid-range N cases, i.e, figures 5(b)
and (e). In figure 7 we show plots for the mean position of the DW as a function of
β, for large and small values of V
0
. Also s hown for each case is the density in the HD
region of the lattice (to the right of the DW) as a function of β, which will aid in the
following dis cussion.
For lar ge V
0
(figures 7 (a) and (c)) there is a monotonic increase in x with β. A
larger va lue of β req uires that more particles be present in the reser voir in order to
achieve α = β. At sma ll β, x is approximately constant with β; this is because the
decrease in density on the HD side of the la ttice is a sufficient release of particles to
maintain α = β. Due to the saturating form of the function α(N
r
) (equation (3)), for
larger va lue s of N
r
a greater increase in N
r
is required to give the same increase in
α. So for larger values of β, the change in the HD density as β increases no-longe r
releases sufficient particles to keep α = β; the DW also must move towards the right
such that there is a steep increase of x with β.
In the small V
0
case there is an LR regime within the SP which results in an
interesting dependence of x on β (figure. 7(b)); in contrast to the large V
0
value case,
x does not increase monotonically with β. We c an understand this behaviour by again
considering the density in the HD region of the lattice. We note that for the sma ll V
0
case, ρ
HD
changes differently with β depending on whether the system is in the LR
regime or not, and that the maximum in x at β 0.04 s
1
corres ponds to the ons e t of
the LR regime. For β < 0.04 s
1
we see from figure 7(d) that, increasing β results in
a decrease in the density in the HD region and ther e fore a release of particles to the
reservoir and an increase in α. However the decrease of ρ
HD
with β is not eno ugh to
maintain α = β. The DW must also move rightwards, i.e. there is a n initial increas e
of x with β. After the onset of LR, β > 0 .04 s
1
, figure 7(d) shows that ρ
HD
decreases
much more quickly with increasing β. So now the density on the HD side of the DW
decreases much more r apidly as β incre ases. The resulting release of particles would
be too gre at to mainta in α = β if the wall did not also move leftwards x decreases
again.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 13
Figure 7. Plots (a) and (b) show the relative mean DW position for systems in
the SP with V
0
= 300 s
1
and V
0
= 20 s
1
resp ectively. In both cases N = 450;
in (a) α = 0.6 s
1
, and in (b) α = 0.1125 s
1
. In (b) the onset of the LR
regime is at β 0.04 s
1
. Plots (c) and (d) show how the density in the HD
phase ρ
HD
= 1 D(β) varies with β, again for V
0
= 300 s
1
and V
0
= 20 s
1
resp ectively. In the SP this is the density to the right of the DW. In (d), initiall y
the density decreases s lowly w ith increasing β; at the onset of LR the rate of
density decrease becomes more severe - a change in the behaviour not seen in a
model wi th an infinite number of fuel carries.
Deeper within the LR regime figure 7(d) we have the same situation as before:
due to the saturating function α(N
r
), for lar ge N
r
we need a gre ater increase in N
r
to
give the same increase in α. The wall has to move rightward as β increa ses in order
to release enough pa rticles to mainta in α = β.
3.2. Multiple Phase Transitions
By combining the effects of both types of limited resources, we obtain a novel phase
diagram (figure 5(f) and figure 6(c)). There is an unusual kink shape in the phase
boundary be tween the HD and SP regimes, given by equation (11). As can be seen in
figure 6(c), this is at the point where the sys tem enters the limited resources regime.
It is possible to dr aw a vertical line at constant α
0
through the phase diagram (dashed
line in figure 5(f)), which cuts throug h the phases HDSPHDSPLD a s β
increases, i.e., by varying only one parameter we can go from the HD phase through
a transition to SP, and then a transition back to HD, etc. Figure 8 shows how the
quantities ρ, T /
¯
T , J, x (where applicable) and α, vary along this line of constant
α
0
; we show both Monte Carlo results and the prediction of the mean-field model.
The mean-field mo del p e rforms well deep within each phase, but begins to show s ome
discrepancy near the phas e boundaries; we discuss this further below.
We label the phases shown in figure 8 with roman numerals I–V and explain each
in turn. These are that phases which are crossed by the dashed line in figure 5(f).
Phase I At very small values of β we have α β, so the system is in the HD
phase. Again consider ing how ρ
HD
varies with β, from figure 7(d) we see that
for small β the s lope is small,
HD
/dβ 1. A decreasing density means an
Multiple phase transitions in a TASEP with limited particles and fuel carriers 14
increasing number of free particles, i.e., dN
r
/dβ 1; the hyperbolic tangent form
of equation (3) means that α incre ases with β but at a very low rate (dα/dβ 1,
see figure figure 8(d)) k.
Phase II We arrive at phase II as follows: in phase I we started with very sma ll
values of β such that α β. By increasing β, particles are freed and therefo re,
α also incre ases. However, dα/dβ 1 in phase I, and hence, we eventua lly reach
α = β, and there is a transition to an SP phase I I. Here we have coexistence of
both LD and HD separated by a DW.
The current and therefore the fuel carrie r use rate increases with β through
phase I, and initially in phas e II (figure 8(b)). About half way through phase II
(β 0.04 s
1
) the s ystem enters the LR regime (see crosses in figure 8(a)).
From figure 7(d) we know that in the first half of phase II, the density in the HD
part of the system decreases slowly with β
HD
/dβ 1. The cor responding
increase in the number of free particles would not be enough to keep α = β, so
the DW also moves rightward, i.e. there is an initial increase in x in phase II.
At the onset of LR the slope of the c urve in figure 7(d) gets steeper, i.e.,
HD
/dβ 1. As β is increased further (in the second half of phase II) the DW
must move leftwards again in o rder to keep α = β; i.e. after initially increasing
with β, x then de c reases as LR onsets, as shown in figure 8(c) (see inset).
Phase III Once the DW reaches the le ftmos t side of the lattice the system can no
longer maintain the condition α = β, so there is a phase transition and we re-enter
the HD phase. Further increase of β incr eases the numb er of free particles N
r
,
however since α is a saturating function of N
r
, dα/dβ begins to decrease. That
is, as β increases through phase III the slope of α(β) gets sha llower (figure 8(d)).
Phase IV If we keep increasing β, we reach β = α, and then the system enter s the SP
for a second time. From figure 7(d) we see that deep within the LR regime ρ
HD
again varies slowly with β, and the difference between the LD and HD densities
is small; therefore in this s e cond SP, changes in the density in the two regions
of the lattice would not significantly change the number of free particles. Thus
rapid variation of the DW position is required as β increases in order to maintain
α = β.
Phase V Once the DW reaches the rightmost edge of the lattice, the system ca n no
longer maintain the condition α = β by moving the wall, and there is a transition
to the LD phase.
The above description accounts for the changes in DW position, current and
density predicted by the mean-field theory, but a s we noted prev iously there is some
discrepancy with the Monte Carlo results, particularly near the transitions. This is due
to the fact that our mean-field model assumes that the density is constant thr oughout
the lattice, when in fact there is some change near the edges [7]. Also the mean-field
treatment ignores correlations in the density which occur near the DW. These edge
effects become less significant as L is increased, and therefore the discrepancy between
the mean-field and simulation results reduces (da ta not shown).
We also note that there is some difficulty in determining the existence and
position of the domain wall. In figure 8(c) we define DW position from simulations by
considering the mean particle de nsity at each lattice site ρ
i
; we define the existence
of a DW if for any pair of adjacent lattice sites i, i + 1 the density cuts through 0.5.
k In the model with finite resources α depends on β, in contrast to the standard TASEP.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 15
Figure 8. Colour on-line. Plot s howing (a) how the density ρ (points) and
fuelling level T /
¯
T (crosses), (b) the current J, (c) the mean DW position x, and
(d) the entry rate α as given by equation (3), vary with β for smal l V
0
and large N
at a fixed value of α
0
= 0.08 s
1
. Points show Monte Car lo results and lines the
prediction from the mean-field model. The system passes through several phases
and we indicate with dotted lines the positions of the boundaries as predicted by
the mean-field model. We label each phase with roman numerals I-V. The inset
in plot (c) shows a zoom around the SP phase II. The dashed l ine in (d) shows
α = β.
Then, the position of the DW is g iven by lattice site i. The difficulty arises in the fact
that this c an also occur near the edges of the system when it is not in the SP. This
explains why it appears that there are DWs when the system is not in the SP we
are actua lly detecting the decrease in the density at the edge of the system.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 16
4. Fluctuations
In this section we examine fluctuations in the density of particles on the la ttice,
focusing on the case of M = 1 lattice. We obtain power spectra for the fluctuations
in density by taking the average o f the Fourier transform of several different time
series. These time series are ge ne rated by recording the densities at time intervals
of 125 s. Since our simulation method does not advance time in regular steps,
the time intervals are not exactly uniform; therefore we use c ubic spline fitting to
obtain a time series at regular intervals. Defining ρ(t) as the instantaneous density
at time t, the power spectrum is given by I(ω) = h|FT [ρ(t) ρ] |
2
i, where the angled
brackets denote average over different simulation runs, and FT[f(t)](ω) is the discrete
Fourier transform of the time series f(t). As before ρ denotes the time average density.
From now on, we use the fo llowing notation: if we mean the instantaneous value of a
quantity at time t, then explicit time dependence is indicated; symbols without time
depe ndence denote the time average of the quantity.
4.1. LD and HD Phase
A good approximation to the power spectrum of the density fluctuations in the LD
phase for the original TASEP (no particle or fuel carriers constraints) can be found
via a continuum description, with fluctuations travelling through the lattice with a
velocity v and an effective diffusion coefficient D [33]. At low frequencies (ω v
2
/D)
the power spectrum shows oscillations, or dips in power, at unit multiple of 2πv/L,
i.e., at frequencies co rresponding to the length of time it takes a fluctuation to traverse
the entire lattice. These oscillations are damped for ω > v
3
/DL, after which I ω
2
;
for large frequencies I ω
3/2
. For the case of finite re-fuelling, but no particle
constraints (i.e. the N limit), we obtain a similar power spectrum . As shown in
figure 9(a), we have oscillations with dips at multiples of 2πv/L (with v k(1 2 ρ ),
see [33]); at larger values of α, when the system is in the LR regime, the oscillations
are severely damped, figure 9(b). This is because the hopping rate k is highly reduced
in the LR regime, meaning that v is reduced, and hence, the damping occurs at
much lower freq uenc ies. What remains o f the dips is still clearly visible in the power
spectrum at low multiples of 2πv/L, implying that the density fluctuations are not
largely a ffected by the fluctuations in the hopping rate in this regime. The I ω
2
for mid- ranged ω and I ω
2/3
for large ω re lationships are maintained even for
parameters such that the mean number of loaded fuel carriers T is small, c onditions
under which one might have though that fluctuations in T would become important
(data not shown). We find that actually fluctuations in loaded carrier levels become
small when the mean value is small.
+
In summary, just as in the original TASEP, when there are constrained fuel
carriers the oscillations in the power spectrum of the number of particles on the lattice
are damped for ω > v
3
/DL. In the original TASEP this is most noticeable at larger
values of α, which give large values of ρ, and therefore small v. With constrained fuel
carriers small v is obtained in the LR regime due to the reduction in the value of k,
i.e., at much smaller values of α.
Due to particle-hole symmetry (which is maintained in the present model), our understanding of
fluctuations in the LD phase can also be applied to the HD phase.
+
In fact we find that T h(T (t) T )
2
i for small T in the LR. The probability of finding T loaded
fuel carriers appears to be very close to a Poisson distribution in this regime, but further discussion
of this is beyond the scope of the present work.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 17
Figure 9. Plots showing power spectra of the total density from simulations of
a system with finite fuel carriers with V
0
= 20 s
1
, but no constraints on the
number of particles (N ). Plot (a) is for α = 0.01 s
1
, (b) for α = 0.04 s
1
,
and (c) α = 0.06 s
1
; in the latter two plots the system is in the LR regime, and
we note that the oscillations have been damped out (see text). The dashed line
has a slope ω
2
, and the dotted lines in each plot are at ω = 2πv/L and 4πv/L.
These spectra were obtained by averaging the Fourier tr ansforms of 500 different
time series.
Figure 10. Colour on-line. Pl ots comparing power spectra of density fluctuations
for systems with a l arge number of particles (black) and a small number of particles
(red). (a) Systems not in the LR regime, with V
0
= 300 s
1
. Black lines show
results for a large number of particles: N = 600 and α
0
= 0.4 s
1
. Red l ines show
results for a small number of particles: N = 220 and α
0
= 2.83 s
1
. (b) Systems
in the LR regime, with V
0
= 20 s
1
. Again black lines show results for a large
number of particles: N = 600 and α
0
= 0.07 s
1
. Red lines show results for a
small number of particles: N = 220 and α
0
= 0.65 s
1
. The different values of
α
0
used in each case are chosen such that the systems we compare have the same
mean density and m ean hopping rate.
Density fluctuations have also previously been studied for the TASEP with
constrained particles [34]. The effect in the LD re gime is to suppress the fluctuations.
This is due to the feedback effect of the particle pool which stabilises the de nsity
(an increase in density leads to a decrease in entry rate). The effect is reduced fo r
higher frequencies, as these correspond to short time scales over which the feedback
from the particle pool has less influence, i.e. for time scales shorter than the
time it takes a fluctuation to traverse the entire lattice. Turning to the present
model with constrained par ticle s and fuel carriers, figure 10 shows power spectra
for systems with different total numb e rs of particles, both when there are no limited
fuel resources (figure 10(a)) and when there are (figure 10(b)). No te that to fairly
compare fluctuations from two simulations the mean densities and hopping rates (and
therefore positions of the “dips” in the power spectrum) must be the same; we therefo re
choose different va lue s of α
0
to give the same value of v in each case. Again we see
little qualitative difference between the non-LR and LR ca ses, i.e., fluctuations in the
number of loaded fuel carriers have little effect on the fluctuations in the density.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 18
4.2. Shock Phase
We now consider fluctua tions in the density, and in the position of the DW, when the
system is in the SP. We ag ain consider a single lattice (M = 1), and focus on the case
of small V
0
, and a mid-range value of N, corresponding to figures 5(e) and 6(b), for
which the SP has the largest area on the α
0
-β plane.
To locate the DW we average the occupation of each site over a short time
window of length τ, so as to average over short time microscopic fluctuations in the
occupation [25]. The value of τ is chosen short enough so as to probe the movement
of the DW on mesoscopic time scales rather than probing only the mean-field de nsity
profile. Following the approach of [25] we choose for τ the smallest value which gives
exactly one micro-domain wall (µDW) in the averaged profile. A µDW is defined as
any point where the density crosses 0.5 (from above or below) b etween one lattice site
and the next.
We show in figure 11 the behaviour of the DW at different values of α
0
and
β. Figures 11(a) and (b) show the long time mean density profile for parameters
for outside and just within the LR re gime respectively. Also shown is the mean
over the mesoscopic time τ , i.e. a snapshot of the density profile from which
the “instanta ne ous” DW position can be found. We note that in the LR regime
(figure 11(b)) there are larger fluctuations in the τ averaged profile than for non-LR,
and s hape of the DW in the long time average profile is wider. The τ averaging
method works well for small values of β; however for larger β when the sys tem is in
the LR regime, determination of the position of the shock is much more difficult. This
is because, due to the limited ava ilability of the fuel carriers, the particles move more
slowly the density fluctuations in the regions to the left and rig ht of the DW e xist
on time scales similar to tha t of the movement of the wall. Any value of τ which will
average out the microscopic fluctuations, will also average out the movement of the
wall. This problem is co mpounded by the fact that the difference between the mean
density on either side of the DW decreases with increasing β. Therefore we can only
accurately measure the time course of the DW position just inside the LR regime;
deep within that r egime, we can only measure the mean wall position.
Figures 11(c)-(d), (e) a nd (f) show respectively typical time courses, normalised
histograms of the wall positions, and the correlation function of the time course defined
as
C(t
) =
hδx(t)δx(t
)i
t
hδx(t)
2
i
t
, (12)
where δx(t) = x(t) x, and · ·i
t
denotes average over time. The correlation
functions are approximately exponential, and the examples shown have correlation
time 2.3 × 10
3
s (β = 0.007 s
1
) and 3.6 × 10
3
s (β = 0.049 s
1
- the LR regime). We
find that whilst the time scale over which the DW moves is comparable in both cases,
the width of the distribution is much wider in the LR case.
For the SP, previous studies [33, 34] have treated dens ity fluctuatio ns analytically
by making the approximation that any fluctuations travel quickly along the lattice and
are absorbed by the DW. That is to say, any fluctuation can be treated as a movement
of the DW, and so only fluctuations in the rate at which particles move onto the i = 1
and off o f the i = L sites need be c onsidered. The fluctuations can then be described
using a simple Langevin equation, leading to a power spectra I (ω
2
+ γ
2
)
1
, where
the constant γ represents the restoring force which localises the DW.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 19
Figure 11. Colour on-line. Simulation data showing the behaviour of the domain
wall in the SP for smal l V
0
and mid-range N corresponding to figure 5(e), for
α
0
= 0.1125 s
1
and various values of β. Plots (a)-(b) show density profiles, and
(c)-(d) the DW position time courses for β = 0.007 s
1
(black) and β = 0.049 s
1
(red). The latter corresponds to a system which is in the LR regime. For the
density profiles long time averages ar e shown with heavy lines, and mesoscopic
time averages (fr om which we determine the instantaneous wall positions) with
light l ines. The mesoscopic averaging time was τ = 500 s. Plot (e) shows DW
posi tion histograms (normalised) for the same two values of β, and plot (f) shows
the time correlation function from (12).
Figure 12. Plot showing power spectra of density fluctuations from simulations of
systems in the SP with α
0
= 0.1 s
1
and values of β = 0.01, 0.02, 0.03, 0.04, 0.05
and 0.06 s
1
. The onset of the LR regime is at β 0.04 s
1
, so the top 2 curves
are for systems with LR. Other parameters are L = 500, V
0
= 20 s
1
and N = 450.
The dashed line shows the slope ω
2
.
Multiple phase transitions in a TASEP with limited particles and fuel carriers 20
Figure 13. Plot showing<