Multispecies model with interconversion, chipping, and injection

Motivated by the phenomenology of transport through the Golgi apparatus of cells, we study a multispecies model with boundary injection of one species of particle, interconversion between the different species of particle, and driven diffusive movement of particles through the system by chipping of a single particle from a site. The model is analyzed in one dimension using equations for particle currents. It is found that, depending on the rates of various processes and the asymmetry in the hopping, the system may exist either in a steady phase, in which the average mass at each site attains a time-independent value, or in a “growing” phase, in which the total mass grows indefinitely in time, even in a finite system. The growing phases have interesting spatial structure. In particular, we find phases in which some spatial regions of the system have a constant average mass, while other regions show unbounded growth.

Figure 1

Illustration of the allowed moves with $A$ (white) and $B$ (black) particles: Injection of $A$ particles at rate $a$ at site 1. $A\to B$ and $B\to A$ conversions at rates $u$ and $v$, respectively (see site 4). An $A$ particle can hop rightward (e.g., from site 5 to 6) at rate $p_R$ and leftward (to site 4) at rate $p_L$. Similarly, a $B$ particle (from site 7) can hop to sites 8 or 6 at rates $q_R$ and $q_L$, respectively. Stack movement corresponds to all particles on a given site (here site 2) collectively hopping to the left or right neighbor (site 1 or 3) at rate $D_L$ or $D_R$, respectively.

Figure 2

Schematic depiction of a phase with pileups of $B$ for (a) fully asymmetric ($\gamma =1$), (b) fully symmetric ($\gamma =1/2$) hopping. Dashed lines show $\langle m^B\rangle$ as a function of $x$ at two time instants $t_1$ (small dashes) and $t_2$ (big dashes), respectively, such that $0\ll t_1<t_2$. The two lines overlap in regions of steady state (i.e., $\langle m^B\rangle$ remains constant in time); the $t_2$ line lies above the $t_1$ line in regions of $B$ pileups ($\langle m_B\rangle$ increases with time). Insets: Occupation probability $s^B$ as a function of $x$. While $s^B=1$ in regions of $B$ pileups, $s^B<1$ in regions of steady state. For $\gamma =1/2$, the region of pileups ($x_1<x<x_2$) exists between regions of steady state near each boundary; for $\gamma =1$, the entire region to the right ($x>x_1$) of the steady state region has $B$ pileups.

Figure 3

Schematic diagram showing variation of the particle currents $j^A$, $j^B$, and $j$ with distance $x$ from left boundary for $\gamma =1$ for (a) a system in steady phase, (b) a system with a steady state region and a $B$ pileup region. In a region of steady state, total particle current $j$ does not change with $x$. In a region of pileups, $j\left(x\right)$ decreases with $x.$

Figure 4

In the absence of interconversion ($u=v=0$), the occupation probability profiles $s_0^A$ and $s_0^B$ (dashed lines) are flat, being $a/w$ and 0, respectively. When $u\ne 0$, $v\ne 0$, there is a net interconversion from $A$ to $B$ at each site, pushing $s^A$ down from the value $s_0^A$ and pushing up $s^B$ from the value $s_0^B$. Here, $s^A$ and $s^B$ (solid lines) are occupation probabilities with interconversion.

Figure 5

Time evolution on the first site can be mapped to random walk in 2D $m^A-m^B$ space. (a) A possible random walk in $m^A-m^B$ space. Allowed moves are enumerated in the upper right corner of the figure along with the corresponding rates at which they occur. (b) Trails in $m^A-m^B$ space corresponding to various scenarios. Trail in solid line shows steady state, trail (in dashed line) directed along $m^A$ axes shows pileup of $A$, trail (in dashed line) along $m^B$ axes shows pileup of $B$, and fourth trail (dotted-dashed line) shows pileup of both $A$ and $B$.

Figure 6

$v-u$ phase diagram for the first site when $A$ and $B$ hopping rates are equal: (a) $p=q,p/a=0.2$, (b) $p=q,p/a=0.8$.

Figure 7

Occupation probabilities $s_i^A$ and $s_i^B$ as a function of site number $i$ in a steady phase for $\gamma =1$, $a=1$, $u=0.5$, $v=0.25$, $p=2$, $q=1$. Data points are obtained from Monte Carlo simulations, and solid lines are plots of the analytical results in Eq. (7). $s^A$ and $s^B$ approach their asymptotic values over a length $l_D$ defined in Eq. (8).

Figure 8

Each site can be thought of as having $A$ and $B$ compartments. Exchange of particles takes place between compartments at a given site, i.e., through interconversion currents (represented by vertical arrows). Exchange of particles takes place between sites through hopping currents (represented by horizontal arrows).

Figure 9

Results of Monte Carlo simulations for $a=1.3$, $u=1$, $v=0.33$, $p=0.86$, $q=1$. Snapshots of the lattice showing $m^B$ profile at $t_1=200000$ (squares) and $t_2=300000$ (circles) ($t$ in units of 500 MC Steps). Inset: Semilog plot of $m_i^B\left(t_2\right)-m_i^B\left(t_1\right)$ vs $i$. Solid line shows the analytical prediction.

Figure 10

Occupation probabilities $s_i^A$ and $s_i^B$ as a function of site number $i$ (a) in a steady phase (with $a=1.3$, $u=0.02$, $v=0.01$, $p=2.2$, $q=0.5$), and (b) in a growing phase (with $a=1.0$, $u=0.0136$, $v=0$, $p=2.2$, $q=0.3$). Data points obtained from Monte Carlo simulations seem to agree well with analytical predictions (solid lines).

Figure 11

$s_i^A$ and $s_i^B$ for $a=1.2$, $u=0.2$, $v=0.1$, $p=2.2$, $q=0.7$, and $\gamma =0.7$. For these parameters, $\tilde{\eta }=103$ and $\tilde{\xi }=8.4$. Solid lines are plots of Eq. (14) with $x$ replaced by $i/L$.

Figure 12

Results of Monte Carlo simulations for a phase with pileups of both $B$ and $C$ ($\gamma =1$). Parameters: $a=5$, $w=6$, $q=2$, $r=0.3$, $u=0.7$, $v=0.2$, $k=0.65$, $l=0.62$. Data points show $m_i^B$ and $m_i^C$ at two time intervals $t_1=5\times {10}^6$ and $t_2={10}^7$. Pileups of $C$ occur earlier in the lattice (site 13 onward) than pileups of $B$ (site 15 onward). Inset: $s_i^A$, $s_i^B$, $s_i^C$ at long times.