Mechanisms of cooperation and competition of two-species transport in narrow nanochannels

Flow of particles of two different species through a narrow channel with solely two discrete spatial positions is analyzed with respect to the species' capability to cooperate or compete for transport. The origin of the latter arises from particle-channel and interparticle interactions within the channel, i.e., blocking the position of a particle, and its interaction with its neighbors in the channel. The variety of occupation options within the channel defines the state space. The transition dynamics within is considered as a continuous Markov process, which, in contrast to mean-field approaches, conserves explicitly spatial correlations. A strong repulsive interaction between particles of the same kind and a very attractive empty channel imply a strong entanglement of transport of both species. In the limiting case of perfect coupling, flows in state space are restricted to a cyclic subspace, where they become all equivalent in the steady state. In particular, this implies equal particle flows of the two species. Entanglement of transport implies that the species mutually exert entropic forces on each other. For parallel directed concentration gradients this implies that the species' ability to cooperate increases with the degree of entanglement. Thus, the gradient of one species reciprocally induces a higher flow of the other species when compared to that in its absence. The opposite holds for antiparallel gradients where species mutually hamper their transport. For a sufficient strong coupling, the species under the influence of the stronger concentration gradient drives the other against its gradient, i.e., the positive mixing entropy production of the driving species becomes the motor for the negative mixing entropy production of the driven one. The degree of effectiveness by which negative entropy production emerges at the cost of positive entropy production increases with the coupling strength. This becomes evident from location and connectivity of the sources of entropy production in state space.

Figure 1

The nine-dimensional state space $\mathbf{\Sigma }=\left\{\mathbf{\sigma }\right|\mathbf{\sigma }=({\sigma }_2,{\sigma }_1),{\sigma }_i=0,A,B\}$, with transition rates between states. Right: A sketch of the channel connecting the two baths.

Figure 2

Flows and occupation probabilities in state space. Different energetic levels of the empty channel state and channel states occupied by two particles of the same species are considered. For simplicity, all are set equal to $\mathrm{\Delta }E=E_{00}=E_{AA}=E_{BB}$, i.e., the coupling strength quantifying entanglement of transport of the two species [see Eq. (18)]; $\mathrm{\Delta }E=0$ (a), $\mathrm{\Delta }E=7$ (b), and $\mathrm{\Delta }E=15$ (c). Values for probabilities are color coded (see bar). So are flow values, which were normalized to that with the maximum magnitude. In addition, flow magnitude is coded by the thickness of the arrows, which indicate the flow direction. Particle concentrations of species $A$ in the right (1)and left (2) bath are $c_1^{\left(A\right)}=10$ and $c_2^{\left(A\right)}=0.1$, respectively. $k_{+}^{\left(A\right)}$ and $k_{-}^{\left(A\right)}$ were set equal to 1. Concentrations of $B$ were set equal in both baths, and jump-in rates were chosen to be $k_{+}^{\left(B\right)}{c}_i^{\left(B\right)}=e^1\times 0.1, i=1,2$ and jump-out $k_{-}^{\left(B\right)}=e^{-1}$. This choice of rates implied a moderate attractive particle-channel interaction $\mathrm{\Delta }{\mathrm{\Phi }}^{\left(B\right)}=-ln(k_{+}^{\left(B\right)}/k_{-}^{\left(B\right)})=-2$ for species $B$ compared with that of $A, \mathrm{\Delta }{\mathrm{\Phi }}^{\left(A\right)}=-ln(k_{+}^{\left(A\right)}/k_{-}^{\left(A\right)})=0$. Note that with increasing $\mathrm{\Delta }E$, flow is mainly present on the cyclic state space CS (yellow color) (19).

Figure 3

Particle flow of species $A$ (dashed lines) and $B$ (solid lines) through the channel as a function of the concentration gradient of $A$. Particle concentration in bath 1 is elevated and that in bath 2 is held constant ($k_{+}^{\left(A\right)}{c}_2^{\left(A\right)}=0.1$). Concentrations of $B$ are equal in both baths. These and the constants $k_{+},k_{-}$ for both species are identical with that in Fig. 2. The interaction between the species, i.e., entanglement of respective transports, is varied by the coupling strength $\mathrm{\Delta }E$ of Eq. (18). The gray line gives the flow $J^{\mathrm{cs}}$ in the cyclic state space CS (19) and is obtained from Eq. (21). Note that the flows of $A$ and $B$ converge towards this flow with increasing $\mathrm{\Delta }E$.

Figure 4

Phase diagram showing cooperation, promotion, and competition of two-species particle transport as a function of their coupling strength, quantified by $\mathrm{\Delta }E$ [Eq. (18)]; $\mathrm{\Delta }E=0$ (a), $\mathrm{\Delta }E=2$ (b), $\mathrm{\Delta }E=3$ (c), and $\mathrm{\Delta }E=4$ (d). Concentrations in the left bath (2) were held constant at $k_{+}^{\left(A\right)}{c}_2^{\left(A\right)}=0.1$ and $k_{+}^{\left(B\right)}{c}_2^{\left(B\right)}=e^1\times 0.1$, whereas concentrations in the right bath (1) were elevated. The other parameters are identical with those in Fig. 2. Pink denotes cooperation with profit for both, turquoise competition at the cost of both, violet shows that species $B$ is promoted by $A$ at the cost of the latter and vice versa in blue (see text).

Figure 5

Effect of opposing concentration gradient of species $A$ and $B$ on respective flows as a function of the coupling strength $\mathrm{\Delta }E$. Concentrations and jump-in and -out rates for species $A$ are that from Fig. 2, i.e., the concentration gradient is directed from bath 1 to bath 2, with $c_1^{\left(A\right)}/c_2^{\left(A\right)}=100$. Concentration of $B$ in bath 1 is held constant at $k_{+}^{\left(B\right)}{c}_1^{\left(B\right)}=e^1\times 0.1$, and $c_2^{\left(B\right)}$ is increased. The stronger the coupling the higher must be the oppositely directed gradient of $B$ to make its flow cease.

Figure 6

Entropy production (left panel), entropy production related to particle exchange (mid panel), and flow with occupation probability (right panel) in state space as a function of the coupling strength $\mathrm{\Delta }E$ [see Eq. (18)]. $\mathrm{\Delta }E=5$ (a), $\mathrm{\Delta }E=10$ (b), and $\mathrm{\Delta }E=20$ (c). The values of entropy production are normalized to their maximum magnitude ($\to {\dot{S}}_{\mathbf{\sigma },\varsigma }/\text{Max}\left(\right|{\dot{S}}_{\mathbf{\sigma },\varsigma }\left|\right)$ and color coded from $-1$ to +1 (left color bar). Coding of flow and occupation probability (0 to +1, right color bar) is identical with that in Fig. 2. Concentration gradients of $A$ and $B$ are opposite, $c_1^{\left(A\right)}/c_2^{\left(A\right)}=100,c_2^{\left(B\right)}/c_1^{\left(B\right)}=50$, with $k_{+}^{\left(A\right)}{c}_2^{\left(A\right)}=0.1,k_{-}^{\left(A\right)}=1$ and $k_{+}^{\left(B\right)}{c}_1^{\left(B\right)}=e^1\times 0.1,k_{-}^{\left(B\right)}=e^{-1}\times 1$, the latter implying a moderate attractive particle-channel interaction ${\mathrm{\Phi }}^{\left(B\right)}=-ln(k_{+}^{\left(B\right)}/k_{-}^{\left(B\right)})=-2$. Note that for loose coupling (a) flows of both species between the baths follow the direction of their gradient, i.e., they are opposite, $\left(0A\right)\to \left(A0\right)$ vs. $\left(0B\right)\leftarrow \left(B0\right)$. A strong coupling (c) almost restrains the transition dynamics of the system to the CS (19) in which directions of particle flows are parallel $\left(0A\right)\to \left(A0\right)$ and $\left(0B\right)\to \left(B0\right)$.

Figure 7

(a) Negative mixing entropy production of species $B, {\dot{S}}^{\left(B\right)}=\mathrm{\Delta }{\mu }^{\left(B\right)}{J}^{\left(B\right)}$. The negative sign is due to transport of $B$ in an antiparallel direction to its concentration gradient. (b) The corresponding efficiency which is defined by the negative entropy production of $B$ per positive entropy production related to transport of $A$. The concentration gradient of $A$ is directed from bath 1 (right) to bath 2 (left), $c_1^{\left(A\right)}/c_2^{\left(A\right)}=100$, with $k_{+}^{\left(A\right)}{c}_2^{\left(A\right)}=0.1$. Particle concentration of $B$ in bath 1 is set to $k_{+}^{\left(B\right)}{c}_1^{\left(B\right)}=e^1\times 0.1$ and that in bath 2 is elevated about this level to induce an opposing gradient to that of species $A$. Jump-out rates at the channel ends of both species are as in Fig. 6. Various coupling strengths $\mathrm{\Delta }E$ are considered. With maximum coupling strength the efficiency curve (orange line) approaches ${\dot{S}}^{\left(B\right)}/{\dot{S}}^{\left(A\right)}\approx \mathrm{\Delta }{\mu }^{\left(B\right)}/\mathrm{\Delta }{\mu }^{\left(A\right)}=ln(c_1^{\left(B\right)}/c_2^{\left(B\right)})/ln(c_1^{\left(A\right)}/c_2^{\left(A\right)})={log}_{10}(c_1^{\left(B\right)}/c_2^{\left(B\right)})/{log}_{10}\left(100\right)=1/2{log}_{10}(c_1^{\left(B\right)}/c_2^{\left(B\right)})$ as predicted by Eq. (30).