A mixed population of competing TASEPs with a shared reservoir of particles

The totally asymmetric simple exclusion process (TASEP) has become a paradigm in non-equilibrium statistical physics, due to its rich phenomenology and wide applicability [1, 2, 3, 4, 5, 6]. The majority of this work has focused on the case of a single TASEP with a fixed entry rate (corresponding to infinite availability of particles). However, many physical and biological situations involve multiple lattices with different parameters, whose dynamics is coupled via competition for a finite pool of particles. In this paper, we present a simple mean-field theoretical framework to study such scenarios, which can be used for an arbitrary number of lattices and for any distribution of lattice parameters. We show that non-trivial behavior, including extension of the shock phase to a finite region of parameter space, and “buffering” of individual currents to changes in the reservoir particle number, can emerge from the competition between lattices for particles. We further show that in these systems, information on the whole system can be extracted from measurements of the behaviour of an individual lattice subtype.

The standard TASEP is a stochastic process which describes the collective movement of particles along a one-dimensional lattice composed of $L$ sites. On the lattice, a particle hops stochastically from site $i$ to $i+1$, provided that site $i+1$ is empty. Particles enter the first site (if unoccupied) and leave the last site with fixed rates. This model shows three distinct phases controlled by the boundary rates: the low density (LD) phase occurs when the entry rate is limiting, the high density (HD) phase when then exit rate is limiting, and the maximal current (MC) phase when both entry and exit rates are large enough that the current is limited only by the hopping rate along the lattice. Exact steady-state solutions for the density of particles on the lattice and the current are known [7, 8, 9].

First introduced as a model of protein synthesis [10, 11], the TASEP nowadays provides a generic model for transport processes, with applications including the production of mRNA [12, 13] and protein [14, 15] in biological cells, the motion of motor proteins along cytoskeletal filaments [16, 17, 18, 19, 20], the transport of vesicles in fungal hyphae [21], collective insect motion [22] and traffic flow problems [23]. Most of these studies have focused on particle motion on a single TASEP track (including tracks with multiple lanes, e.g. [24]); in reality, however, one often has multiple tracks, which may have different lengths, hopping and boundary rates, and which compete for a finite number of particles. For example, in traffic dynamics vehicles may distribute themselves across different roads, while in biological cells different mRNA molecules compete for the cellular protein production machinery [25]. Our aim in this paper is to provide a simple, intuitive and generic framework to study such problems.

This work is not the first to consider the effects of a finite reservoir of particles, or of competition for particles between several TASEPs. Recently, Adams et al [26] and Cook and Zia [27] studied a single TASEP with a finite reservoir of particles, using Monte Carlo simulations, mean-field theory and both simple and generalised domain-wall theories. Cook et al [28] extended this work, using Monte Carlo simulations and domain-wall theory to describe several TASEPs of the same or different lengths, sharing the same particle reservoir. This work revealed interesting effects due to the reservoir-induced coupling between TASEPs. In particular, a regime can exist in which reservoir density is independent of the total particle number. The domain-wall approach is powerful because it can incorporate fluctuations and thus describe accurately the density profiles on the lattices. However, this approach is difficult to generalize to many lattices with arbitrary distributions of entry and exit rates. Here, we take a much simpler, mean-field approach. Our approach does not provide information on fluctuations and cannot predict density profiles. However, it yields a simple way to calculate phase diagrams, and is easy to generalize to arbitrary populations of lattices.

In section II, we describe our mean-field approach and present relations for the partitioning of particles between the reservoir and lattices. We apply this methodology to the case of a single TASEP coupled to a finite reservoir in section III.1 and to a population of identical lattices in section III.2. As a result of the competition for particles, a new region of the phase diagram emerges, in which the LD and HD phases coexist; the size of this region increases as the particle availability decreases. Next, in section IV, we consider a system with two different types of lattices and we find that an LD-HD phase transition on one lattice type causes the behaviour of the other lattice type to become independent of the particle number. In section V, we generalize to the case of mixed populations of lattices with arbitrary distributions of boundary rates. Here we find that an LD-HD phase transition on any lattice type “buffers” all other lattices to changes in the particle number, and we show that the entire distribution of lattice parameters can be extracted from the current on a single lattice subtype. Finally we present our conclusions in section VI.

We begin our discussion by recalling the behavior of a single TASEP with fixed hopping rate $\gamma$, entry rate $\tilde{\alpha}$ and exit rate $\tilde{\beta}$ [1, 2, 3, 4, 5]. For the purposes of this paper we discuss only the mean-field results for the steady-state average particle density (number of particles per unit length) and current on the lattice, which become exact in the limit of a large lattice. We first define rescaled dimensionless entry and exit parameters $\alpha:=\tilde{\alpha}/\gamma$ and $\beta:=\tilde{\beta}/\gamma$ (i.e. we use the hopping rate $\gamma$ to define the unit of time). The (mean-field) density $\rho$ of particles on the lattice takes one of three values, depending on $\alpha$ and $\beta$, as summarized in Table 1. The current $J$ is related to the density by $J=\gamma\rho(1-\rho)$. In the low density (LD) phase, where $\alpha<\beta$ and $\alpha<1/2$, the behavior of the system is dictated by the entry rate and $\rho=\alpha$. In the high density (HD) phase, where $\alpha>\beta$ and $\beta<1/2$, the exit rate is limiting, particles queue on the lattice, and $\rho=1-\beta$. When both entry and exit rates are large, $\alpha,\beta\geqslant 1/2$, neither rate is limiting and the system is in the maximal current (MC) phase, for which $\rho=1/2$. The transition from either LD or HD to the MC phase is continuous in the density, but the transition between the LD and HD phases is discontinuous in the density. On the HD-LD transition line, where $\alpha=\beta$ and $\alpha,\beta<1/2$, there is “coexistence” between the LD and HD phases: part of the lattice takes on the density of the LD phase while the remaining part takes on the density of the HD phase, with a well-defined boundary between these domains, called shock. This scenario is known as a shock phase (SP) [2].

In this paper, we consider the case shown in Figure 1, in which multiple lattices with different lengths, entry and exit parameters compete for a finite number of particles. The reservoir of free particles is assumed to have a finite volume $V$ ¹¹ 1 This scenario is typical of biochemical situations (e.g. motor proteins on cytoskeletal filaments) in which both particles and tracks are confined in a finite volume, and the rate of particle binding events per track is proportional to the free particle concentration. When a particle exits a lattice, it immediately enters the reservoir, from where it is free to enter any other (or the same) lattice. We assume that free particles are well-mixed (not correlated) and homogeneously distributed within the reservoir. Since particles are conserved, the total number of particles $N$, can be written as

where $N_{r}$ denotes the number of particles in the reservoir and $N_{l}$ the number of particles on the lattices. In turn, $N_{l}$ can be found by summing over the lattices:

where $M$ is the total number of lattices, $\rho_{j}$ is the density of particles on lattice $j$ and $L_{j}$ is the length of lattice $j$.

The entry rate of particles onto any given lattice depends on the density $N_{r}/V$ of particles in the reservoir. Since the reservoir is well-mixed, we expect this rate to be linear in the particle density, so that an individual TASEP $j$ experiences an injection rate $\alpha_{j}$ given by

where $\alpha_{0,j}$ is the “intrinsic” affinity of lattice $j$ for particles (with units of volume/time). This relation implies that the rate of particle binding events per lattice is proportional to the free particle concentration: this is expected to be true as long as the reservoir does not become too crowded with particles.

In the limit of a large total number $N$ of particles ($N\to\infty$ at constant $N/V$), $N_{r}=N-N_{l}\gg N_{l}$, so that $N_{r}/V$ will tend to the total particle concentration $N/V$, and we recover the standard TASEP with constant entry and exit rates. Since the maximum value of $N_{l}$ is $\sum_{j}L_{j}=LM$ (denoting $L:=\langle L_{j}\rangle$ as the average over all lattices), the standard TASEP limit is approached when $N\gg LM$. This means that the finite reservoir plays an important role only for $N/LM\sim 1$ or smaller. Following previous work [26, 27, 28], the exit rate from lattice $j$ is assumed to be independent of the reservoir, being simply given by its “intrinsic” exit rate $\beta_{j}$.

Within the mixed population of TASEPs, the phase behavior of a given lattice $j$ is completely determined by its ($N_{r}$-dependent) effective entry rate $\alpha_{j}$ and exit rate $\beta_{j}$, according to the rules for a single TASEP, as listed in Table 1. For any given value of the reservoir particle number $N_{r}$ (assuming fixed $V$), subpopulations of lattices will be in the LD, HD and MC phases, depending on their values of $\alpha_{j}$ and $\beta_{j}$ ²² 2 In this work, we use a different convention from previous authors [26, 27, 28] to define the phases of the single- and multi-track TASEP with finite resources. These authors use a saturating function for the injection rate $\alpha$ and label the phase of the system according to the phase that would be obtained in the infinite reservoir limit (i.e. at the saturation value of the injection rate). We instead define the phase of the lattices as a reservoir density-dependent variable, determined by the value $\alpha(N_{r}/V)$ of the injection rate.. We can therefore write the total number of particles on the lattices as

where $N_{LD}$, $N_{HD}$ and $N_{MC}$ are the total numbers of particles on lattices in the LD, HD and MC phases, respectively. The density of particles on each lattice depends on which phase it is in, according to the relations in Table 1. Equation (4) can therefore be rewritten as

Here, we explicitly note that $N_{LD}$ depends on the reservoir particle number $N_{r}$ through the dependence of the effective injection rate $\alpha_{j}$ on $N_{r}$. The three sums in Eq. (5) are over lattices in the LD, HD and MC phases: because $\alpha_{j}$ depends on $N_{r}$, changes in the reservoir particle number will drive phase transitions on the lattices, so that these three sums will encompass different lattice subpopulations for different values of $N_{r}$. We can then combine Eqs. (1) and (5) to express the relation between the reservoir particle number $N_{r}$ and the total particle number $N$:

Equation (6) forms the basis of our theoretical approach; in the remainder of the paper we explore its implications, first for some simple cases and then for more complex scenarios.

We first consider the case of a homogeneous population of lattices. Our system thus contains $M$ identical lattices of length $L$, intrinsic injection rate $\alpha_{0}$ and exit rate $\beta$, coupled to a finite pool of $N$ particles. Since the lattices are identical, in our mean-field approach each lattice experiences the same effective injection rate $\alpha=\alpha_{0}N_{r}/V$. This implies that all the lattices are in the same phase; we refer to this as the “global phase” of the system. The relation between $N$ and $N_{r}$ is now given by:

where the first, second and third relations refer to the global LD, HD and MC phases, respectively.

We now move on to the more relevant case, where the lattices are not all identical. Here, we find that the coupling between lattices induced by the finite reservoir has interesting and non-trivial effects. We first consider a population composed of two different types of lattices: $M^{(1)}$ lattices with intrinsic injection rate $\alpha_{0}^{(1)}$ and $M^{(2)}$ lattices with intrinsic injection rate $\alpha_{0}^{(2)}>\alpha_{0}^{(1)}$. Note that here we introduce a new notation: we use upper indexes in brackets (e.g., $\alpha_{0}^{(i)}$) to indicate properties shared by all lattices in the same subpopulation, as opposed to lower indexes (e.g., $\alpha_{0,j}$) which we used to denote the properties of individual lattices. For the sake of simplicity we suppose that all the lattices have the same length $L$ and exit rate $\beta$. Our methodology can easily be extended to the case of different $L$ and $\beta$ (see Section V).

Because the two lattice subpopulations have different values of $\alpha_{0}$, they will undergo the LD-HD or LD-MC phase transition at different values of the reservoir particle number $N_{r}$. For the same value of $N_{r}$, the two subpopulations of lattices can therefore be in different phases. The total number of particles can be expressed in terms of the particle densities $\rho^{(1)}$ and $\rho^{(2)}$ on the type-1 and type-2 lattices:

The densities $\rho^{(i)}$ ($i=1,2$) depend on the phase of the lattice: if $\beta<1/2$ one has

while if $\beta\geqslant 1/2$ the densities are

We can express Eq. (9) in a compact way by making use of the Heaviside function (defined as $\theta(z)=1$ for $z\geqslant 0$ and $\theta(z)=0$ otherwise). For $\beta<1/2$ this results in:

which can be rearranged to give:

For the case where $\beta>1/2$ we could write an equivalent equation, using instead the densities and boundary conditions appropriate for the LD to MC phase transition. It turns out however that this is exactly equivalent to Eq. (13), but with $\beta$ replaced by $1/2$. Equation (13) can therefore be used to describe the full behavior of the system, for any value of $\beta$, with the proviso that for $\beta>1/2$, we simply set $\beta=1/2$ in the equation.

From Equation (13) we can obtain the physical relevant relation $N_{r}(N)$ by inversion (as in Sec. III B). $N_{r}(N)$ is plotted in Figure 5, for the case $\beta<1/2$: the two $\theta$-steps of Equation (13) appear as two plateaus, corresponding to two regimes where the reservoir is buffered due to the emergence of a shock phase (SP) on one lattice subtype. In the case $\beta\geqslant 1/2$ we do not obtain this effect, because the particle density on a lattice is continuous across the LD to MC transition. Figure 5 also shows the results of kinetic Monte Carlo simulations for a multi-TASEP system coupled to a finite reservoir, with the same parameter set. The agreement between the mean-field solution and the simulations is very good; the slight discrepancy around the phase transitions is due to finite size effects (the mean-field solution becomes exact only in the limit $L\to\infty$).

Equation (13) also provides a simple way to determine the phase boundaries for this system; these occur as the discontinuities of the Heaviside step functions are approached from above and below – i.e. at $N_{r}\to\left(V\beta/\alpha_{0}^{(2)}\right)^{\pm}$ and $N_{r}\to\left(V\beta/\alpha_{0}^{(1)}\right)^{\pm}$. Appendix .2 gives explicit forms for these phase boundaries.

Figure 6 shows that the current per lattice and the particle density on the individual lattices exhibit a remarkable dependence on the total particle number $N$. For the range of values of $N$ over which lattice type 2 is in the SP, the current and density on the lattices of type 1 remain constant, even though these lattices are far away from a phase transition. This buffering effect occurs because the reservoir particle number remains constant while any lattice subtype is in the SP; this fixes $\alpha$, and hence the current and density, of all lattices which are in the LD-phase (see Table 1). Hence, lattices of type 1 are affected by the phase transition on lattice type 2, through the coupling to the finite particle reservoir. Note that while lattice type 2 is in the SP, its current remains constant, but its density increases linearly with $N$. This reflects the fact that, in the “buffering regime”, the SP lattices absorb particles as $N$ increases, keeping $N_{r}$ constant. The same observation applies to lattice type 1 in the second buffering region.

To conclude this section, we investigate the phase behavior of this system as a function of the numbers $M^{(1)}$ and $M^{(2)}$ of lattices in the two subpopulations. The parameter space ($M^{(1)}$, $M^{(2)}$) is likely to be relevant experimentally: for example, in intracellular transport problems, $M^{(1)}$ and $M^{(2)}$ might represent the number of cytoskeletal filaments of two different types along which motor proteins can travel. Figure 7 shows the phase diagram of the system in the ($M^{(1)}$, $M^{(2)}$) plane for (a) $\beta<1/2$ and (b) $\beta\geqslant 1/2$. It is clear that the competition for particles plays an important role: the phase behavior of each lattice subpopulation depends strongly on the size of the other subpopulation.

We now formulate the concepts of Sections II-IV in a general way, to allow us to describe a mixed lattice population with an arbitrary distribution of parameter values. As in the preceding discussion – see Eqs. (1) and (6) – our strategy is to express the total number of particles $N=N_{r}+N_{LD}+N_{HD}+N_{MC}$ as a function of the boundary parameters of the lattices.

The total number of lattices is denoted by $M$ and the normalized distribution of boundary rates and lengths (i.e. their relative frequencies) for the mixed lattice population is denoted by $P(\alpha_{0},\beta,L)$. Such a continuous distribution could be appropriate in situations where lattice parameter values are sensitive to small environmental changes, or where the total number of lattices is very large. We can then express the total number of particles on lattices in the LD, HD and MC phases as:

where the integration limits are determined by the constraints on $\alpha=\alpha_{0}N_{r}/V$ and $\beta$ for the LD, HD and MC regions, respectively, as defined in Table 1. Note that the MC case can be recovered as a special case of the HD with $\beta=1/2$. Therefore, we can limit our analysis to $\beta\leqslant 1/2$, i.e. assuming $P(\beta>1/2)=0$ ⁴⁴ 4 We can restrict our study to the LD and HD phases by redefining the distribution of parameters as follows: $P_{new}(\alpha_{o},\beta=1/2,L):=\int_{1/2}^{\infty}d\beta\;P(\alpha_{o},\beta,L)$, $P_{new}(\alpha_{o},\beta>1/2,L):=0$, and $P_{new}(\alpha_{o},\beta<1/2,L):=P(\alpha_{0},\beta<1/2,L)$. From now on, for the sake of simplicity, in the text we refer to $P_{new}(\alpha_{o},\beta,L)$ as to just $P(\alpha_{o},\beta,L)$..

We also assume that the lengths of the lattices are independent of the boundary rates, such that $P(\alpha_{0},\beta,L)=P(\alpha_{0},\beta)P(L)$. This allows us to perform the integration over $L$ in Eq. (14), yielding

where $\langle L\rangle$ is the average lattice length and we have inserted the limits of integration as detailed in Table 1. Note that since $P(\beta>1/2)=0$ we are free to choose the upper limit of the $\beta$-integration as infinity. This simplifies our further calculations.

After adding and subtracting the term $\langle L\rangle M\int_{0}^{\infty}d\beta\int_{V\beta/N_{r}}^{\infty}d\alpha_{0}\frac{\alpha_{0}N_{r}}{V}\,P(\alpha_{0},\beta)$ and inserting the expressions for the subpopulations into the particle conservation equation, we arrive at the result

where $\langle{\alpha_{0}}\rangle=\int_{0}^{\infty}d\beta\int_{0}^{\infty}d\alpha_{0}\,\alpha_{0}P(\alpha_{0},\beta)$ is the average value of $\alpha_{0}$. Equation (16) is a generalisation of Eqs. (6) and (13) to a continuous distribution of parameters.

In this paper, we have presented a mean-field theoretical framework to study systems in which multiple TASEPs with different parameters compete for a common pool of particles. We expect this approach to be useful in modelling a wide range of systems, from control of gene expression in biological cells to traffic flow problems. Previous work has addressed the effects of a finite particle reservoir on TASEP dynamics, for single lattices  [26, 27] and for multiple lattices with equal boundary rates [28]); here, we extend this work to mixed populations of lattices with an arbitrarily complex distribution of parameters.

Our theoretical approach, presented in Section II, is based on combining the equation for conservation of the total number of particles with the mean-field results for the standard TASEP. This approach provides a simple way to deal with mixed populations of lattices. Although our mean-field theory does not provide information on fluctuations or on density profiles, it nevertheless reveals interesting phenomena which emerge from the competition for particles in a mixed multi-TASEP system and provides a method to calculate the full phase diagram. These phenomena arise because the finite reservoir effectively couples all the lattices, so that a phase transition on one lattice influences the behavior of the others. Although in this work we used the mean-field TASEP results, one could easily incorporate into the same framework exact or simulated relations for the particle density as a function of the entry and exit rates.

A key observation which emerges from our work is that for a mixed population of lattices coupled to a finite reservoir of particles, any lattice subtype which enters the SP absorbs all further particles added to the system, buffering the particle reservoir and making the currents and densities of all other lattices insensitive to changes in the total particle number $N$, even though these lattices may be far from their phase boundaries. This effect is specific to systems where the lattices have different intrinsic entry and exit rates: if $\alpha_{0}$ and $\beta$ are the same for all lattices (and if lattices are large enough to neglect finite size effects), phase transitions occur on all lattices at the same critical particle reservoir number and the current does not show an additional plateau as a function of $N$. This was the case in previous work [26], which considered a mixture of lattice with different lengths but identical boundary rates. In these conditions, a plateau in the current as a function of $N$ is due only to finite size effects.

The physical mechanism underlying the buffering of the system to changes in $N$ is attributable to lattices undergoing the LD-HD transition that “soak up”changes in the reservoir particle number $N_{r}$. As they undergo this transition, lattices enter the shock phase, in which a queue of particles forms at the end of the lattice. For shock phase lattices, the particle entry rate $\alpha=\alpha_{0}N_{r}/V$ is fixed by the exit rate $\beta$ ($\alpha=\beta$): thus the number of particles on a shock phase lattice adjusts to compensate for changes in $N_{r}$. In the phase diagram for the standard TASEP model, the shock phase occurs only on the line separating the HD and LD phases, where $\alpha=\beta$; in the case of a finite reservoir of particles, however, the shock phase occupies a finite region of the $(\alpha_{0},\beta)$ phase diagram. This is because the same entry rate $\alpha$ can be achieved over a range of $N$, $N_{r}$ being set by the position of the domain wall.

An interesting analogy can be drawn between this phenomenon and first order phase separations in (equilibrium) thermodynamics. The plateau in $N_{r}(N)$ which arises in our models is a direct consequence of the discontinuity in the particle density as a lattice undergoes the LD-HD phase transition. Similarly, a first order phase transition such as the boiling of water involves a discontinuity in the entropy, which is associated with latent heat: during the transition, the temperature remains constant even though further heat energy is constantly being supplied. In this analogy, heat plays the role of $N$ in our models while temperature plays the role of the reservoir particle number $N_{r}$.

We also show in this paper that the coupling between lattices induced by a finite particle reservoir makes it possible (under some circumstances) to extract the entire distribution of lattice parameters from measurements of the reservoir density, or indeed the current carried by a single lattice subtype, as a function of the total particle number $N$. It will be interesting to explore the feasibility of this approach for extracting information from real, noisy, experimental data.

The multi-track TASEP with finite particle reservoir studied here bears a similarity to previous work on TASEPs on closed networks [29]. In particular, a system of $M$ lattices with a common reservoir can be mapped onto a network topology formed by $M$ rings having a unique common site  [30]. The common site plays an analogous role to a particle reservoir. However, in that problem, in contrast to the one studied here, both the exit and entry rates depend on the occupation of the common site.

A key priority for future work must be to explore ways to include the effects of fluctuations, which are neglected in our mean-field approach. Previous work has shown that interesting fluctuation-driven effects, including localisation of domain boundaries, can occur in TASEPs with finite particle number [27]; extending this work to complex mixtures of TASEP is likely to prove fruitful. Another promising avenue may be to study how the behavior of systems of the type studied here changes with changes in their volume: this should prove relevant when modelling transport or protein production dynamics in growing biological cells.

In summary, we have presented a simple and intuitive mean-field theoretical framework for studying multi-TASEP problems with finite reservoir of particles. Our method has allowed us to show that interesting physical phenomena emerge from the competition for particles among non-identical lattices, including buffering of the system to changes in the total particle number. This approach should prove a versatile tool for studying a wide variety of “real-world” problems [31] involving competition among complex populations of transport processes.