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

**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.

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**ABSTRACT:**We introduce a mean-field theoretical framework to describe multiple totally asymmetric simple exclusion processes (TASEPs) with different lattice lengths and entry and exit rates, competing for a finite reservoir of particles. We present relations for the partitioning of particles between the reservoir and the lattices: These relations allow us to show that competition for particles can have nontrivial effects on the phase behavior of individual lattices. For a system with nonidentical lattices, we find that when a subset of lattices undergoes a phase transition from low to high density, the entire set of lattice currents becomes independent of total particle number. We generalize our approach to systems with a continuous distribution of lattice parameters, for which we demonstrate that measurements of the current carried by a single lattice type can be used to extract the entire distribution of lattice parameters. Our approach applies to populations of TASEPs with any distribution of lattice parameters and could easily be extended beyond the mean-field case.Physical Review E 01/2012; 85(1 Pt 1):011142. · 2.31 Impact Factor - SourceAvailable from: Jordan Brankov[Show abstract] [Hide abstract]

**ABSTRACT:**We report here results on the study of the totally asymmetric simple exclusion process, defined on an open network, consisting of head and tail simple-chain segments with a double-chain section inserted in between. Results of numerical simulations for relatively short chains reveal an interesting feature of the network. When the current through the system takes its maximum value, a simple translation of the double-chain section forward or backward along the network leads to a sharp change in the shape of the density profiles in the parallel chains, thus affecting the total number of particles in that part of the network. In the symmetric case of equal injection and ejection rates α=β>1/2 and equal lengths of the head and tail sections, the density profiles in the two parallel chains are almost linear, characteristic of the coexistence line (shock phase). Upon moving the section forward (backward), their shape changes to the one typical for the high- (low-) density phases of a simple chain. The total bulk density of particles in a section with a large number of parallel chains is evaluated too. The observed effect might have interesting implications for the traffic flow control as well as for biological transport processes in living cells. An explanation of this phenomenon is offered in terms of a finite-size dependence of the effective injection and ejection rates at the ends of the double-chain section.Physical Review E 06/2013; 87(6-1):062116. · 2.31 Impact Factor

Page 1

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 Brackley1, Luca Ciandrini1and M Carmen

Romano1,2

1Institute for Complex Systems and Mathematical Biology, SUPA, University

of Aberdeen, Aberdeen, AB24 3UE, United Kingdom

2Institute of Medical Sciences, Foresterhill, University of Aberdeen, Aberdeen,

AB25 2ZD, United Kingdom

E-mail: cab@chrisbrackley.co.uk, l.ciandrini@abdn.ac.uk

PACS numbers: 05.60.-k,05.40.-a,02.50.Ey

Abstract.

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.

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 limited,

the particle hopping rate is severely reduced.

The TASEP is a paradigmatic model from non-equilibrium statistical

As as consequence, the system’s

Keywords: driven diffusive systems (theory), stochastic processes (theory)

Page 2

Multiple phase transitions in a TASEP with limited particles and fuel carriers2

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) translation [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 finite resources we mean

a constrained number 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 such systems occurring in nature. In this paper

we introduce a new model which includes the finite availability of both resources, 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 system can

go from a high density regime, to a shock phase, then to a high density phase again,

visit the shock phase once more, and finally reach a low density phase. This is a novel

effect that emerges only by combining both limited resources. We use a mean-field

approach and verify our results by means of Monte Carlo simulations.

In its most simple form, the TASEP consists of a 1D lattice of L sites upon which

particles can sit, see 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 that

the downstream site is vacant. A system with open boundaries, as we shall consider

here, can display rich dynamics with multiple boundary induced phases [15]. Particles

are allowed to hop onto the lattice with rate α at one end, 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 been solved using matrix methods [19, 20]. Mean-field

methods have also been extensively used [16], since they are easily tractable and yield

a good approximation in many cases. There has been much extension of this simple

model, for example variable hopping rates (site or particle dependent) [21, 22, 23],

extended particles which cover more than one site [6, 24], 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 occupation of the ith lattice site ni= 1 if

the site contains a particle and ni= 0 otherwise. The system is characterised by the

steady-state particle current J (the rate at which particles pass any given point on

the lattice), and the mean site occupancy (often called density) ρi= ?ni?, where ?···?

denotes average over realisations of the system (which we assume is ergodic, so this is

equivalent to a time average). The average density is therefore given by ρ = L−1?

or low density (LD) phase, the exit limited or high density (HD) phase, a maximal

current (MC) phase where 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 ρifor given α and β.

iρi.

There are four possible phases depending on the values of α and β: the entry limited

Page 3

Multiple phase transitions in a TASEP with limited particles and fuel carriers3

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 with 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 of the system) is given by

(LD)

(HD)

(MC)

ρLD= α/k

ρHD= 1 − β/k

ρMC= 1/2

for α < β,α < k/2 ,

for β < α,β < k/2 ,

for β,α ≥ k/2 ,

(1)

and the current is always given by J = kρ(1 − ρ).

α = β < k/2, presents an HD region on the right of the lattice and an LD region

on the left, separated by a boundary which diffuses freely through the lattice. This

has often been described using a domain wall (DW) theory [30]. Due to the free

diffusion of the DW a time average of the density in SP gives ρSP = 1/2, but the

current depends on the density in either the LD or HD regions of the lattice, i.e.

JSP= α(1 − α/k) = β(1 − β/k).

In this paper we consider several TASEPs which share a common finite pool of

both particles and fuel carriers. The entry rate, which is the same for each TASEP,

depends 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 alternative 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

The SP, which occurs for

Page 4

Multiple phase transitions in a TASEP with limited particles and fuel carriers4

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 with their cargo once it has been used (figure 1(b) shows a schematic

representation of this model). We show that novel effects arise when both types of

limited resource are considered, e.g. multiple phase transitions can occur when varying

the exit rate β. The outline of the paper is as follows: in section 2 we summarise the

previous results for the two models separately, before describing in section 3 a mean-

field model for a system with both 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. Finally, 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 shares the same reservoir of particles; then we present the concept of fuel-

carriers and the effect of a finite rate of refuelling on the exclusion process dynamics.

2.1. Finite number of particles

In this work we analyse a system 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 Nr.

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 = Nr+ LMρ, (2)

where ρ is the density on each lattice. The entry rate of the M lattices depends on

the number of free particles via a saturating function

α = α0tanh

?Nr

?N − LMρ

N∗

?

= α0tanh

N∗

?

, (3)

where the constant α0gives the entry rate in the limit Nr→ ∞ and is an intrinsic

property of the lattices ‡ . Without loss of generality, we fix the normalisation factor

N∗to be LM/2, i.e. the total 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 equations for α in terms of α0and N. Since the densities in each phase are the

same as those in the standard TASEP (equations (1)), for a given set of parameters

(α0, β, N) we find the resulting α which determines the phase; e.g. if α < β and

α < k/2 the system will be in the LD phase. By substituting equation (3) into these

inequalities, we get a representation of the different phases on the α0–β plane.

As a consequence 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 with the function used in [9, 10, 11], and is relevant, e.g.

application to protein synthesis.

for the

Page 5

Multiple phase transitions in a TASEP with limited particles and fuel carriers5

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 loaded

fuel carriers). For small 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 regimes 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, there 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 region;

since α depends on both α0and β (through its dependence on the bulk density), the

line opens into a region on the α0-β phase plane. That is to say, the condition α = β

is fulfilled for a certain range of α0[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 of the HD phase on the α0-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 unconstrained 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 because 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. Similarly if the DW moves to the left the number of free

particles decreases, decreasing the entry rate and driving the DW rightwards. The

opening of the SP line into a region on the α0-β plane is possible because a different

mean position for the DW corresponds to a different value of α0, while keeping α = β.

Hence, the system can maintain α = β for different values of α0. 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 gas, the

energy required for the advancement of the particles is obtained from some kind

of finite resource. For instance, molecular motors consume ATP molecules, and

ribosome movement on mRNAs during protein synthesis requires aminoacylated tRNA

Page 6

Multiple phase transitions in a TASEP with limited particles and fuel carriers6

complexes and GTP. The latter case has recently been described in [13, 14, 31], and

here we briefly review those results, before in the next section combining this with the

a finite pool of particles model.

We consider a finite number¯T of fuel carriers, T of which are carrying fuel. Every

time a particle moves, the fuel from one of the loaded carriers is used, and hence, T

is reduced by one. The hopping rate of particles depends on the availability of loaded

fuel carriers, and the empty carriers are refuelled at a rate 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 =V0(¯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 V0and saturation determined by the constant b. Any

saturating function will give the same qualitative results, but the above formulation

allows for a straightforward analytical treatment [13, 14] §.

In this section we describe a collection of M identical TASEPs with a finite

number of fuel carriers (but no constraints on the number of particles); following the

common mean-field treatment [16] the particle density on the ith site of each lattice

is given by

dρ1

dt

dρi

dt

dρL

dt

=

=

=

α(1 − ρ1) − kρ1(1 − ρ2),

kρi−1(1 − ρi) − kρi(1 − ρi+1),

kρL−1(1 − ρL) − βρL.

i = 2,...L − 1,

(5)

The inclusion of a finite pool of fuel carriers leads to the additional equation

dT

dt=V0(¯T − T)

b +¯T − T

−

(L−1)M

?

j=1

kρj(1 − ρj), (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

V0− J(L − 1)M,

(7)

i.e., the hopping rate is now itself a function of the current. Following [13, 14], upon

solving equations (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)

(HD)

(MC)

(SP)

JLD= J(α),

JHD= J(β),

JMC= α∗/2,

JSP= J(α),

ρLD= D(α)

ρHD= 1 − D(β)

ρMC= 1/2

ρSP= 1/2

for α < α∗and α < β,

for β < α∗and β < α,

for α,β ≥ α∗,

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.

Page 7

Multiple phase transitions in a TASEP with limited particles and fuel carriers7

where

J(α) =1

2

?

α

?

1 −

α

a(¯T + b)

??

?

+

¯T

¯T + b

V0

L′M

−

¯T

¯T + b

V0

L′M+ α

?

1 −

α

a(¯T + b)

??2

−4α(a¯T − α)

a(¯T + b)

V0

L′M

,

D(α) =

1

2α

?

α

?

1 +

α

a(¯T + b)

??

?

V0

L′M+ α

−

¯T

¯T + b

V0

L′M

+

¯T

¯T + b

?

1 −

α

a(¯T + b)

??2

−4α(a¯T − α)

a(¯T + b)

V0

L′M

,

and

α∗=a

4(¯T + b) +

V0

L′M−

??a

4(¯T + b) +

V0

L′M

?2

− a¯T

V0

L′M,

with L′= L − 1. The behaviour of these functions as α and β are varied depends

on the parameters a, b and V0. By considering the steady-state of equation (6) and

noticing that the maximal value that the recharging rate can possibly have is equal to

V0, we note that the particle current is limited from above by V0/L′M. Hence, there

are substantially two different cases: (i) if V0/L′M ≫ 1, the recharging rate is very fast

and the particle current is not influenced by it; we recover the results of the original

TASEP; (ii) if in contrast V0/L′M ≪ 1, the recharging of the fuel carriers can limit

the value of the particle current. Figure 3 shows the current for sets of parameters

corresponding to each case. In case (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 refer to the regime where the current

appears independent of α as a limited resources (LR) regime, since the rate at which

fuel is used by the particles approaches the rate at which fuel carriers are reloaded.

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 regime within the LD phase, the sensitivity of the current to changes

in α or β is greatly reduced, whilst the sensitivity of the density is greatly increased.

In the MC phase, 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, V0/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 paper we choose b such that there is a sharp onset of LR, and take

V0 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 occurs

by equating the rate of fuel carrier use (approximately αL′M for small α in LD and

βL′M for small β in HD) and the maximum recharging rate. This gives

¯TV0

(¯T + b)L′M.

αLR= βLR≈

(9)

Page 8

Multiple phase transitions in a TASEP with limited particles and fuel carriers8

Figure 3. Plots showing the current as a function of α for different parameters.

In (a) V0 = 300 s−1, and in (b) V0 = 20 s−1. In both cases a = 2 × 10−4,

¯T = 5000, and b = 50. Solid lines show the current in the LD phase, JLD= J(α),

and dashed lines the current in the MC phase, JMC= α∗/2. The dotted line is at

α∗, where there is a transition from LD→MC. The dot-dashed line in (b) shows

the value of αLDas estimated in equation (9).

Figure 4. Steady-state fuelling level T/¯T as a function of rate V0 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 × 10−4s−1, b = 50, and¯ T = 5000. For

small V0the number of loaded fuel carriers is depleted; for large V0the carriers are

practically always fully loaded, and we recover the original unconstrained TASEP.

This value is represented in figure 3(b) by a dot-dashed vertical line. As it is shown

there, the estimation predicts quite 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 particles and fuel carriers. As we show later

in this section, it is only when combining the two schemes discussed above that

we can see emerging novel effects, such as multiple phase transitions. Analogous

to the dependence of the entry rate α on the number of particles N, the steady-

state proportion of loaded fuel carriers is a saturating function of V0 (see figure 4).

Therefore, by regarding N and V0as control parameters, we can vary 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 and 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 except 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

Page 9

Multiple phase transitions in a TASEP with limited particles and fuel carriers9

calculate the average value ρSPusing equation (3); using the fact that in this phase

α = β leads to

ρSP=

N

LM

?

1 −N∗

N

tanh−1

?β

α0

??

. (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 problem of finding the boundaries between the different phases

as functions of α0, β, N and V0— i.e., eliminating α. Our aim is to draw the α0-β

phase plane for any given values of V0and N. As noted in previous sections, due to

finite particles, the SP line opens 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 α0as a

function of β, or vice versa. We now consider the boundaries between each phase in

turn, consulting figure 2 as an ansatz for the arrangement of the phases.

(i) MC/LD phase boundary. The MC phase can exist if N > LM/2. If we consider

starting in the MC phase with large β and reducing α0, we will cross into the LD

phase when α = α∗. Using (3) this gives an equation for the boundary in the

α0-β plane, where

?N

Note that α∗depends on a, b,¯T, V0, L and M. Hence the MC/LD boundary is

a vertical line on the α0-β phase plane.

(ii) MC/HD phase boundary. If we now consider starting in the MC phase with large

α0 and reducing β, then we cross into the HD phase when β = α∗, i.e., in the

α0-β plane this boundary is given by the horizontal line

α0= α∗coth

N∗−LM

2N∗

?

for β ≥ α∗and N > LM/2 .

β = α∗for α0≥ 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 < α0tanh(Nr/N∗); using equation

(2) and the density in HD phase gives the following equation for the boundary

α0= 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 α0-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 these inequalities poses some difficulty, since by using equation (3) and the

density in the LD phase we obtain

α = α0tanh

?N

N∗−LM

N∗D(α)

?

,

an equation which cannot be solved analytically to find α as a function of α0.

Instead we solve this numerically, setting α = β (which occurs at the SP) to find

β as a function of α0. This gives another curved line on the α0-β phase plane.

Page 10

Multiple phase transitions in a TASEP with limited particles and fuel carriers10

Figure 5. Colour on-line. Phase diagrams in the α0-β plane at different values

of V0and 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 Monte Carlo simulations. Parameters used are V0 = 20 or 300 s−1, and

N = 200, 450, or 600. Other parameters are L = 500, M = 1, a = 2 × 10−4s−1,

b = 50, and¯T = 5000. The dotted line in (f) shows the line of constant α0 used

in figure 8.

We can then construct the phase plane by plotting the phase boundaries (β as a

function of α0) for any given values of N and V0. 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 realistic scenarios in biological transport processes, such as

protein synthesis. We set the time scale of the system by choosing a = 2 × 10−4s−1,

such that the maximum hopping rate is k = 1 s−1. A value of b = 50 then gives a

sharp onset of LR as shown in figure 3(b). The phase diagram boundaries calculated

using the mean-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 length of the Monte Carlo time

step is chosen from an exponential distribution, 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 chosen stochastically from the set of particles

which have a vacancy to their right and the set of empty fuel carriers. Particles are

chosen with a probability such that they move with a rate k, and empty fuel carriers

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 × 106time steps. Assuming that the system

is ergodic we average currents and densities over at least a further 4 × 107steps.

Page 11

Multiple phase transitions in a TASEP with limited particles and fuel carriers11

Figure 6.

the steady-state fuelling level T/¯ T at different values of α0 and β for small

V0 = 20 s−1. White lines show phase boundaries as predicted by the mean-

field model. (a) Small N = 200; (b) mid-range N = 450; (c) large N = 600. For

these parameters we have αLR= βLR≈ 0.040 from (9). Other parameters are as

figure 5.

Colour on-line. Colour maps showing Monte Carlo results for

In figure 5 we present a series of phase planes for different values of N and V0.

We show the particle density ρ obtained from Monte Carlo simulations as a colour

map with the phase boundaries from the mean-field model overlaid (white lines). Note

that the mean-field model very closely predicts 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 does not change the macroscopic behaviour,

but changes the microscopic behaviour for the SP (see section 3.1).

In figure 5 we note that for small V0 the phase diagrams look broadly similar

to the large V0case, but as we expect from [13, 14], the phase transitions occur at

much smaller values of α0 and β. We also obtain a limited resources (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 V0. The

onset of the LR regime can be clearly seen as T/¯T decreases dramatically over a small

range of α0or β. For large V0the refuelling is so quick that T/¯T is constant through

all phases, i.e. we recover the results for a TASEP with a finite pool of particles, but

no constraints on the fuel carriers, since these are refuelled almost as soon as they are

used (data not shown). For clarity, throughout the rest of the paper when we refer

to LR or limited resources, we specifically 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 seen in the LD/SP and HD/SP phase

boundaries at the point of LR onset. Particularly striking is the shape of the HD/SP

boundary for the large N small V0case, and we examine this in detail 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 phase

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 α0 and β. The introduction of a finite number 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

Page 12

Multiple phase transitions in a TASEP with limited particles and fuel carriers12

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 mean position will change the number of

particles in the reservoir; this in turn changes the entry rate α, acting like a restoring

force on the DW. In actual fact the DW executes Gaussian fluctuations about its

mean value, and we discuss fluctuations 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 case 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 from the mean-field model by approximating

the density in the SP as follows

ρSP= xρLD+ (1 − x)ρHD.

Since x is always selected such as to maintain the condition α = β, we can use equation

(10) and the densities from (8) to show that

1

1 − 2D(β)

We also note that the difference between ρLDand ρHDdecreases as β increases, i.e.,

the “height” of the wall decreases.

As it is the most interesting case, 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 V0. Also shown 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 discussion.

For large V0(figures 7 (a) and (c)) there is a monotonic increase in x with β. A

larger value of β requires that more particles be present in the reservoir in order to

achieve α = β. At small β, x is approximately constant with β; this is because the

decrease in density on the HD side of the lattice is a sufficient release of particles to

maintain α = β. Due to the saturating form of the function α(Nr) (equation (3)), for

larger values of Nr a greater increase in Nr 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-longer

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 V0 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 V0value case,

x does not increase monotonically with β. We can understand this behaviour by again

considering the density in the HD region of the lattice. We note that for the small V0

case, ρHDchanges 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−1corresponds to the onset of

the LR regime. For β < 0.04 s−1we see from figure 7(d) that, increasing β results in

a decrease in the density in the HD region – and therefore a release of particles to the

reservoir and an increase in α. However the decrease of ρHDwith β is not enough to

maintain α = β. The DW must also move rightwards, i.e. there is an initial increase

of x with β. After the onset of LR, β > 0.04 s−1, figure 7(d) shows that ρHDdecreases

much more quickly with increasing β. So now the density on the HD side of the DW

decreases much more rapidly as β increases. The resulting release of particles would

be too great to maintain α = β if the wall did not also move leftwards – x decreases

again.

x =

?

1 − D(β) −

N

LM

?

1 −N∗

N

tanh−1

?β

α0

???

.

Page 13

Multiple phase transitions in a TASEP with limited particles and fuel carriers13

Figure 7. Plots (a) and (b) show the relative mean DW position for systems in

the SP with V0= 300 s−1and V0= 20 s−1respectively. 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 ρHD= 1 − D(β) varies with β, again for V0 = 300 s−1and V0 = 20 s−1

respectively. In the SP this is the density to the right of the DW. In (d), initially

the density decreases slowly with 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 with 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(Nr), for large Nrwe need a greater increase in Nrto

give the same increase in α. The wall has to move rightward as β increases in order

to release enough particles to maintain 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 between the HD and SP regimes, given by equation (11). As can be seen in

figure 6(c), this is at the point where the system enters the limited resources regime.

It is possible to draw a vertical line at constant α0through the phase diagram (dashed

line in figure 5(f)), which cuts through the phases HD→SP→HD→SP→LD as 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

α0; we show both Monte Carlo results and the prediction of the mean-field model.

The mean-field model performs well deep within each phase, but begins to show some

discrepancy near the phase 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 considering how ρHD varies with β, from figure 7(d) we see that

for small β the slope is small, dρHD/dβ ∼ −1. A decreasing density means an

Page 14

Multiple phase transitions in a TASEP with limited particles and fuel carriers14

increasing number of free particles, i.e., dNr/dβ ∼ 1; the hyperbolic tangent form

of equation (3) means that α increases with β but at a very low rate (dα/dβ ≪ 1,

see figure figure 8(d)) ?.

Phase II We arrive at phase II as follows: in phase I we started with very small

values of β such that α ≫ β. By increasing β, particles are freed and therefore,

α also increases. However, dα/dβ ≪ 1 in phase I, and hence, we eventually reach

α = β, and there is a transition to an SP – phase II. Here we have coexistence of

both LD and HD separated by a DW.

The current — and therefore the fuel carrier use rate — increases with β through

phase I, and initially in phase II (figure 8(b)). About half way through phase II

(β ≈ 0.04 s−1) the system 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 — dρHD/dβ ∼ −1. The corresponding

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 curve in figure 7(d) gets steeper, i.e.,

dρHD/dβ ≪ −1. As β is increased further (in the second half of phase II) the DW

must move leftwards again in order to keep α = β; i.e. after initially increasing

with β, x then decreases as LR onsets, as shown in figure 8(c) (see inset).

Phase III Once the DW reaches the leftmost 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 increases the number of free particles Nr,

however since α is a saturating function of Nr, dα/dβ begins to decrease. That

is, as β increases through phase III the slope of α(β) gets shallower (figure 8(d)).

Phase IV If we keep increasing β, we reach β = α, and then the system enters the SP

for a second time. From figure 7(d) we see that deep within the LR regime ρHD

again varies slowly with β, and the difference between the LD and HD densities

is small; therefore in this second 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 can 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 as we noted previously 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 throughout

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 (data 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 density at each lattice site ρi; we define the existence

of a DW if for any pair of adjacent lattice sites i,i + 1 the density cuts through 0.5.

? In the model with finite resources α depends on β, in contrast to the standard TASEP.

Page 15

Multiple phase transitions in a TASEP with limited particles and fuel carriers15

Figure 8.

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 small V0and large N

at a fixed value of α0= 0.08 s−1. Points show Monte Carlo 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 line in (d) shows

α = β.

Colour on-line.Plot showing (a) how the density ρ (points) and

Then, the position of the DW is given by lattice site i. The difficulty arises in the fact

that this can 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 actually detecting the decrease in the density at the edge of the system.

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- Available from Luca Ciandrini · Jun 3, 2014
- Available from ArXiv