Kinetic ratchet effect as a non-equilibrium design principle for selective channels
KKinetic Ratchet Effect as a Non-equilibrium Design Principle for Selective Channels
Chase Slowey and Zhiyue Lu ∗ Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3290 (Dated: February 25, 2021)In living cells, ion channels passively allow for ions to flow through as the concentration gradientrelax to thermal equilibrium. Most ion channels are selective, only allowing one type of ion togo through while blocking another. One salient example is KcsA, which allows for larger K + ionsthrough but blocks the smaller Na + ions. This counter-intuitive selectivity has been explained by twodistinct theories that both focus on equilibrium properties: particle-channel affinity and particle-solvent affinity. However, ion channels operate far from equilibrium. By constructing minimalkinetic models of channels, we discover a ubiquitous kinetic ratchet effect as a non-equilibriummechanism to explain such selectivity. We find that a multi-site channel kinetically couples thecompeting flows of two types of particles, where one particle’s flow could suppress or even invert theflow of another type. At the inversion point (transition between the ratchet and dud modes), thechannel achieves infinite selectivity. We have applied our theory to obtain general design principlesof artificial selective channels. I. INTRODUCTION
In biology, ion transportation, and more specifically,the selective transportation of K + ions through the pas-sive channel KcsA has been studied extensively. [1–9]Interestingly, the passive channel allows for large ions(K + ) to go through while blocking smaller ones (Na + ),all without significantly impeding the flow of the largerions. The explanations for the selectivity have been clas-sified [3] into 3 main hypotheses: “snug-fit hypothesis”,[10–12]“field-strength hypothesis”[13, 14], and “over-coordination hypothesis”[15–18]. While the boundariesbetween those hypotheses are not clearly drawn, we be-lieve that they are derived from two main dogmas – oneemphasizes the binding affinity between the cations andthe channel’s residue, and the other emphasizes the dehy-dration of the cations (i.e., the solvation free energy). Wewill briefly revisit the two dogmas and argue that neitherof these are sufficient in describing the non-equilibriumtransportation properties found in ion channels.One dogma focuses on the binding affinity between anion and the binding sites within the channel. This ex-planation attributes the channel’s selective preference oftransporting K + to the fact that it has a greater bindingaffinity with the sites of the channel than Na + . [10–13, 19, 20] In contrast, the other dogma focuses on anion’s binding affinity with the water molecules in thesolution. In this explanation, the channel’s preferencefor transporting K + is attributed to the observation thatNa + has a higher dehydration energy (i.e., a lower sol-vation free energy). [12, 14, 21, 22] Numerous computa-tional and experimental works have been performed toverify hypothesises derived from one or both of the twodogmas. [3, 7, 21, 23, 24]Note that both mechanisms mentioned above attemptto address a non-equilibrium phenomenon with intu- ∗ [email protected] itions from equilibrium thermodynamics, where the (non-equilibrium ) transportation rate of an ion is problemat-ically associated with either the ion’s (equilibrium) bind-ing affinity with water molecules or the ion’s (equilib-rium) binding affinity with the channel. With the re-cent development of stochastic and non-equilibrium ther-modynamics, it has been demonstrated that equilibriumthermodynamic quantities such as binding affinities orfree energies, typically defined by the natural logarithmsof equilibrium Boltzmann distributions, are not capableof fully describing a system’s non-equilibrium kinetics[25–38]. For a channel at thermal equilibrium, where theconcentrations of ions on both sides are balanced, thebinding affinities of ions with the channel and the bathfully determine the state of the system. However, when achannel is driven out of equilibrium by the concentrationgradient, the steady-state probability, kinetics, and thetransportation of ions through the channel are beyondthe scope of equilibrium thermodynamics.By modeling the kinetics of ion channels with stochas-tic thermodynamics, we revisit the selectivity of ion chan-nels and discover a novel kinetic ratchet effect that allowsfor infinitely high selectivity without dramatically im-peding the flow in the channel. Specifically, the kineticratchet effect couples the non-equilibrium flows of twotypes of particles and can utilize the flow of one type ofparticle to actively transport the other against the con-centration gradient. This anti-gradient transportationfunctions as a chemical motor, resembling a Maxwell’sdemon. [39–42] In this work, we demonstrate that we canuse this new mechanism as a general design principle ofartificial channels to achieve highly selective transporta-tion.The paper is structured as follows. In section II wefirst examine an oversimplified model of a single-site ionchannel maintained at a non-equilibrium steady state,and demonstrate that the single-site-channel selectivityonly depends on the dehydration process (solvation freeenergy) rather than the binding affinities between par-ticles and the channel. In section III we introduce the a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b stochastic dynamics model of a multi-site ion channel.The multi-site channel demonstrate a novel ratchet effectand a dynamical phase transition between a dud phaseand a ratchet phase. In the dud phase, both types ofparticles transport down their concentration gradients.Whereas in the ratchet phase, the flow of one type ofparticle kinetically drives the other type into an activetransportation against the gradient. We find that thechannel is highly selective within and at the vicinity ofthe ratchet phase. We also find that it is infinitely se-lective at the dud-ratchet phase transition line, totallyblocking the flow of one type of particle while allowingfor the other to go through. In section IV we construct aminimal model where the energy landscape of the chan-nel for both particles is purely flat where binding affin-ity, solvation free energy for both particles are set tozero (i.e., flat-multi-site channel). This minimal modeldemonstrates that the dud-ratchet transition is ubiqui-tous for multi-site channels. Finally in section V we ap-ply perturbations to the flat-multi-site channel. We findthat compared with the binding affinities between par-ticles and channel, the particles’ solvation free energieshave a more prominent control over the high selectivityand low resistance of a multi-site channel. Our simplemodels ignores the charge of the particles and thus ourresult can be applied to the general design of artificialselective channels. II. SINGLE-SITE CHANNEL
Let us start by considering a simple channel consistingof a single binding site that can hold up to one particle.The channel sits between two chemical baths – mixturesof two types of particles (‘A’ and ‘B’). The concentra-tion gradients of ‘A’ and ‘B’ are kept constant in oppos-ing directions, and thus the channel is kept at a non-equilibrium steady state. We model the dynamics of thechannel as a Markovian process with possible transitionsshown in Fig. 1. In our model the channel can be foundin three possible microscopic states (micro-states). No-tice that a transition between micro-states in the single-site channel is bipartite, as it can be associated to theexchange of particles with either bath (sketched as lightblue and light green shades).We describe the state of the channel by the probabil-ity distribution over its micro-states, (cid:126)p . Its dynamics isMarkovian, following the bipartite master equation, d(cid:126)pdt = ˆ R bi · (cid:126)p (1)where ˆ R bi ≡ ˆ R l + ˆ R r is the bipartite probability transi-tion rate matrix. ˆ R l/r is the probability transition ratematrix corresponding to transition events due to the par-ticle exchange with the left/right bath.The entries of the transition rate matrix ˆ R l/r are de-fined by the temperature, energy change and the parti-cle’s concentration via the Arrhenius law. [43] For exam- R R R R
12 R R R R A B
FIG. 1: Single-site ion channel has three possiblestates, empty (1), occupied by a particle ‘A’ (2) or oc-cupied by a particle ‘B’ (3). There are 8 possible tran-sitions that could occur for the channel’s state, wherefour of them correspond to the particle exchange withthe left bath (shown as the light blue shade), and theother four with the right bath (shown as the light greenshade). The transition from state j to i caused by theexchange of a particle with the l/r bath is stochasticwith a probability rate denoted by the matrix element R ij,l/r ple, R ,l represents the probability rate of particle ‘A’entering the empty channel from the left bath (i.e., tran-sition from state 1 to state 2). The full set of transitionrates are listed below: R ,l = [ A ] l · e − β ( E ˜ b,A − F sol,A ) R ,r = [ A ] r · e − β ( E ˜ b,A − F sol,A ) R ,l = R ,r = e − β ( E ˜ b,A − E s,A ) R ,l = [ B ] l · e − β ( E ˜ b,B − F sol,B ) R ,r = [ B ] r · e − β ( E ˜ b,B − F sol,B ) R ,l = R ,r = e − β ( E ˜ b,B − E s,B ) (2)where [ X ] l,r represents the concentration of particle ‘X’in the left/right bath, E s,X is the interaction energy be-tween particle ‘X’ and the site, E ˜ b,X is the barrier energyfor the particle’s entering and leaving a bath, and F sol,X is the solvation free energy of the single particle in thebath. To preserve normalization, the diagonal elementsof the bipartite rate matrix ˆ R is chosen such that eachcolumn of the matrix sums to zero: R ii = − N (cid:88) k (cid:54) = i ( R ki ) , for all i (3)where N = 3 is the total number of micro-states ofthe channel. For convenience, we have chosen the unitsystem such that the inverse temperature is unity β =( k B T ) − = 1.Given the master equation Eqn. 1, we can solve forthe non-equilibrium steady state of the channel at anygiven concentration gradient. At the steady state theprobability to find the channel in any given micro-stateis denoted by (cid:126)p ss satisfying ˆ R · (cid:126)p ss = 0. By analyticallysolving for the steady state probability, we can furthersolve for the steady-state transportation rate of particle‘A’ and ‘B’ through the channel, J A and J B (see SI).Throughout this paper, we choose the convention thatthe current is positive if the particle flows from the leftbath to the right bath.Given a selective channel that transports ‘A’ whileblocking the flow of ‘B’, we define the channel’s selec-tivity for ‘A’ as the log ratio of the currents for ‘A’ and‘B’: ξ A = ln (cid:12)(cid:12)(cid:12)(cid:12) J A J B (cid:12)(cid:12)(cid:12)(cid:12) (4)Here, we obtain an analytical solution of the steady stateselectivity as (see SI), ξ A = ln | [ A ] l − [ A ] r | · e − E ˜ b,A + E ˜ b,B + F sol,A − F sol,B | [ B ] l − [ B ] r | (5)This minimal model provides us with a simple testingtool to illustrate the limitations of the intuitions derivedfrom equilibrium thermodynamics. The binding affin-ity dogma claims that a channel prefers to transport theparticles with higher binding affinity (‘A’) because ‘A’spends more time in the channel compared to ‘B’. We findin Eqn. 5 that such equilibrium binding-preference doesnot grant a higher transportation rate when the system isdriven out of thermal equilibrium. Rather, the selectiv-ity of a single-site channel is independent of the bindingaffinity between the particle and the channel. Moreover,the single-site selectivity is determined by both the con-centration gradient and the particle’s dehydration energy(solvation free energy). To further examine the selectiv-ity of complex ion channels, we focus on multi-site ionchannels for the remainder of the paper. III. DYNAMICAL PHASE TRANSITION INMULTI-SITE CHANNEL
It has been shown that the KcsA channel contains 4binding sites, where each site can hold up to one K + orNa + ion. Here we construct a simple Markov model for n -site channel and examine it’s non-equilibrium trans-portation of two competing particles ‘A’ and ‘B’. We dis-cover that the competition between two types of parti-cles through a multi-site channel leads into a dynamicalphase transition between two phases of operation: thedud mode and the ratchet mode.For an n -site channel, each site can be found in oneof three states: empty, filled with an ‘A’, or filled with a‘B’. Thus the channel can take N = 3 n different states.For simplicity, we assume that the particles are neu-tral and that they do not interact with each other exceptfor the same-site exclusion. We characterize the inter-action between a particle and the channel by the energylandscape sketched in Fig. 3. ⌧
Free Energy of a Single ParticlePosition in Ion Channel AB 𝐸 !",$ 𝐸 ",$ 𝐸 %,$ 𝐹 %&’,$ FIG. 3: Energy landscapes experienced by a given par-ticle (‘A’ or ‘B’) through the channel. We denote thesolvation free energy of a single particle ‘X’ in eitherbath by F sol,X and the binding energy between ‘X’ andany given site by E s,X . The energy barrier for particle‘X’ to enter/exit the channel is denoted by E ˜ b,X , andthe barrier for hopping between neighboring sites is de-noted by E b,X . We assume that the E s,X and E b,X areuniform among all sites.Using this general landscape, we utilize the ArrheniusLaw to define the transition probability rates from onemicro-state to another.[43] Examples of the micro-statesof a 5-state channel are (A A A A A), (B B B B B), (AA B), or ( ). The transitions between micro-statesare classified into three types of events:(1) ‘X’ hops from one site to an empty neighbor site: R ik = e − β ( E b,X − E s,X ) (6)(2) ‘X’ in the left/right bath enters an empty left/rightedge site: R ik = [ X ] l/r · e − β ( E ˜ b,X − F sol,X ) (7)(3) ‘X’ hops from the left/right edge site to theleft/right bath: R ik = e − β ( E ˜ b,X − E s,X ) (8)For convenience, we have chosen the unit system suchthat the inverse temperature is unity β = ( k B T ) − = 1.With these transition rates, we can write down themaster equation: d(cid:126)pdt = ˆ R · (cid:126)p (9)where (cid:126)p is the probability vector that consists of the prob-abilities of all N = 3 n micro-states of the channel.We numerically compute the steady state probabilitiesand the steady state currents for particles ‘A and ‘B’.This allows us to obtain the steady state selectivity ofthe channel. (See SI.) Our numerical results (Fig. 4) areobtained for a 3-site channel computed at varying choicesof parameters E s,A and F sol,A . All other parameters arefixed at [ A ] l = 1, [ A ] r = 1000, [ B ] l = 1000, [ B ] r = 100, E s,B = F sol,B = -4.0, E b,A = E ˜ b,A = -2.2, E b,B = E ˜ b,B = -1.2. -8 -6 -4 -2 0 F sol,A -4-3.5-3-2.5-2-1.5-1-0.50 E s , A Ratchet Dud (a) Phase Diagram -8 -6 -4 -2 0 F sol,A -4-3.5-3-2.5-2-1.5-1-0.50 E s , A -4-2024681012 ξ (b) Efficiency Map FIG. 4: We identify two distinct dynamical phases ofthe channel at the non-equilibrium steady state. In the‘ratchet’ phase, the current of particle ‘B’ flows fromlow concentration to high concentration, driven by aratchet effect induced by the flow of ‘A’. In the ‘dud’phase, both particle ‘A’ and ‘B’ flow from their highconcentration side to their low concentration side. Onthe right panel, we illustrate the channel’s selectivity( ξ A ). Notice that at the phase boundary, ξ A reaches in-finity as a result of J B = 0. Near the ratchet mode, thecurrent of J B is significantly suppressed by J A , achiev-ing high selectivity for particle A .As shown in Fig. 4(a), by varying the energy land-scape, we have discovered two distinct dynamical phasesof the channel’s behavior – “ratchet mode” and “dudmode”, distinguished by the direction of the flow for par-ticle ‘B’. In the dud mode, both ‘A’ and ‘B’ flow fromhigh concentration toward low concentration. However, in the ratchet mode, without any external input of en-ergy, the flow of particle B is inverted – flowing fromits low concentration side toward the high concentrationside. We argue that this Maxwell’s-Demon-like effect is aresult of the ratchet coupling of the flows of ‘A’ and ‘B’.In the ratchet mode, A’s down-gradient flow dominatesthe multi-site channel and creates a bias toward the leftbath, that inverts the down-gradient transportation of Binside the channel. This inversion leads to a negative en-tropy production for particle ‘B’ that appears to violatethe second law of thermodynamics. However, when theentropy production of particle ‘A’ is considered, the sys-tem has a positive total entropy production and thus thesecond law of thermodynamics is restored. We carefullyexamine such an effect with a minimal model in the nextsection.At the dynamical phase transition, the flow of ‘B’ isinverted J B = 0, while J A remains non-zero. Thus,the channel achieves an infinitely high selectivity ξ A = ln | J A /J B | = ln | J A / | → ∞ at such dynamical phasetransition. At the transition line, the channel allows forparticle ‘A’ to go through while completely blocking theflow of particle ‘B’. Away from the transition line intothe interior and the vicinity of the ratchet mode, we findthat the channel still achieves a high selectivity. This isillustrated by Fig. 4(b) where the high selectivity regionis shown in red and yellow ξ A >
7. Our result suggeststhat the ratchet effect provides us with a kinetic mech-anism that allows one type of particle to travel acrossthe channel while suppressing or inverting the flow of theother type of particle. We will examine such mechanismsas design principles of selective channels in the next twosections.
IV. RATCHET EFFECT AS A CHEMICALMOTOR
To examine the dynamical phase transition of selec-tive channels, let us consider the minimal model by re-moving the unnecessary features of the energy landscape.Consider an n-site channel transporting two competingtypes of particles, ‘A’ and ‘B’. For simplicity, we requirethat the channel’s energy landscape for both types ofparticles be flat: E s,A = E s,B = 0, E b,A = E b,B = 0.The particles’ interaction with the solvent is also “flat”: F sol,A = F sol,B = 0, and E ˜ b,A = E ˜ b,B = 0.In Fig. 5, we illustrate a kinetic mechanism to explainthe dynamical phase transition with the help of two imag-inary “gears”. Here we consider the A particles withinthe channel as teeth of an imaginary gear rotating to-wards the right, whereas the B particles represent teethof another gear driven toward the left. For the sake ofthe argument, let us consider the concentration gradi-ent of two particles maintained in opposing directions.In this case, the two gears push against each other viateeth-teeth interactions. Because of the stochastic parti-cle hopping, the gear teeth can have defects that change BAAA B
Left Bath Right Bath µ A
FIG. 5: The dynamics of ion channel occupancy il-lustrated by the interaction between two imaginarystochastic gears. The teeth of the two gear representsparticles A or B within the channel. Unlike conven-tional gears, the teeth pattern of both gears can changefreely unless the teeth bumps into one another. Thegears are driven by the chemical potential differencesbetween the left and right baths. The chemical poten-tials are determined by the concentrations of the parti-cles, e.g., µ A,l = − β − ln[ A ] l .over time. Such defects allow for the two gears to slipagainst each other, allowing for two types of particles tosimultaneously travel down the opposing gradients, as isseen in the dud mode. In the ratchet mode, however,the two “gears” are better coupled with each other andthus one gear dominates the motion and the other gearis forced to “rotate” with the dominant gear.By numerically computing the steady state currentsof these channels at various ranges of concentration gra-dients, we obtain a phase diagram in Fig. 6. Here wedefine 4 modes of the channel. (1) Same-direction-dud is a mode where both ‘A’ and ‘B’ flow from theirhigh concentration to the low concentration side and suchflows are in the same direction. (2)
Ratchet-A mode iswhere the flow of ‘B’ pushes ‘A’ against the gradient of‘A’. (3)
Ratchet-B mode is where the flow of ‘A’ pushes‘B’ against the gradient of ‘B’. (4)
Opposite-direction-dud is a mode where ‘A’ and ‘B’ transport in oppositedirections and both of them travel from their high con-centration to low concentration side. Notice that in mode(1), the gradients of ‘A’ and ‘B’ are in the same direction,whereas in modes (2), (3), and (4), the gradients of ‘A’and ‘B’ are in opposite directions.In the ratchet-B mode, ‘B’ particles are transportedagainst their own gradient, from low concentration tohigh. At the steady state, the total entropy of ‘B’ par-ticles in the left and right baths decrease at a constantrate: ˙ S B = J B ln [ B ] l [ B ] r < S A = J A ln [ A ] l [ A ] r (11)and ˙ S tot = ˙ S A + ˙ S B ≥ η B = − ˙ S B ˙ S A ≤ η A = − ˙ S A ˙ S B ≤
1. Note that in Fig. 6, we havedenoted efficiency as η in general, but they correspondto either η A or η B depending on the mode of operation. (4)(1) (4)(1)(3) (3)(2)(2) η (a) Efficiency of 3-site channel (1) (1)(4) (4)(3) (2) (2) (3) η (b) Efficiency of 5-site channel FIG. 6: Phase diagrams of minimal model of multi-sitechannels resembles that of a Maxwell’s Demon[40, 42].As we vary [ B ] r and [ A ] l , we have kept [ B ] r + [ B ] l =[ A ] r + [ A ] l = 1000 constant.In Fig. 6, one can compare the phase diagrams and theefficiencies of a 3-site and a 5-site channel. We find thatthe longer the channel, the more prominent the ratcheteffect. This can be explained by the “gear” mechanismillustrated in Fig. 5. As channel length increases, moreteeth from the two gears can be expected to interact witheach other, reducing the chance of gear slippage (sup-pressing the dud mode). This can be rationalized by thefact that as the coupling between the flow of ‘A’ and ‘B’is stronger in a longer channel. The channel’s highestefficiency and the range of parameters that supports aratchet effect is larger for the 5-site channel comparedwith the 3-site channel. For each channel, we find thatthe highest efficiency is achieved at the “linear-responseregime” where both concentration gradients for ‘A’ and‘B’ approaches zero. At the center of each diagram,where the gradients for ‘A’ and ‘B’ are exactly equal tozero, the system is at a state of thermal equilibrium and˙ S tot = ˙ S A = ˙ S B = 0. (a) 3-site channel (b) 5-site channel FIG. 7: Illustration of ξ A for multi-site channels(color bar) characterizing the high selectivity regionin light blue and yellow. Note that infinite selectivity isachieved along the center of the yellow regions. ξ (a) 3-site channel ξ (b) 5-site channel FIG. 8: Illustration of the normalized currents of ‘A’( ˜ J A ) (color bar) characterizing the extra resistance ofthe flow of ‘A’ caused by particle ‘B’.The performance of a selective channel consists of twocharacteristics, the selectivity ( ξ A ) and the extra resis-tance caused by the competing particles. Let us firstlook at the selectivity in Fig. 7. Here we show thata longer channel can achieve a higher selectivity. Notethat a channel is highly selective (in light blue and yel-low) within and at the vicinity of the ratchet mode region.The channel achieves an infinite selectivity at the ratchet-dud transition. Even though the ratchet effect enhancesthe selectivity, it causes extra resistance for the flow ofthe preferred particle ‘A’. To characterize the resistanceof selective channels, let us introduce the normalized cur-rent: ˜ J A = J A J diff A (14)where J diff A is the diffusion limit current of particle A, defined as the steady state of current ‘A’ through a flat-landscape multi-site channel in the absence of particle‘B’. In Fig. 8, we have shown the normalized currentin the ratchet modes for both 3-site and 5-site channelsand find that the current of particle ‘A’ can achieve upto approximately 1 / / V. PERTURBATION TO THE MINIMALMULTI-SITE MODEL
Among the many theories developed around the se-lective KcsA channel, the binding-affinity dogma andthe solvation-free-energy dogma have left us two choicesin optimizing the design of artificial channels that arehighly selective – focusing on particle-site binding affin-ity, E s,X , or focusing on the solvation free energies, F sol,X . Here, we apply two types of perturbations tothe minimal multi-site channel model (flat landscape) toexamine which parameter plays a more important role inthe channel’s performance. An optimal channel shouldachieve both high selectivity and low resistance.In Figs. 9 and the SI figures, we compare the numeri-cal result of the unperturbed model (flat landscape) withthe two types of perturbations to E s,X and F sol,X respec-tively.One perturbation (to E s,X ) focuses on the distinc-tion between the two particle’s binding affinities withthe channel. (See Fig. 9 b,e.) By decreasing the bind-ing affinity of particle ‘A’ with the channel’s sites fromunperturbed E s,A = 0 (Fig. 9 a,d) to E s,A = − F sol,X ) focuses on the sol-vation free energy difference between two types of parti-cles (Fig. 9 c,f). By decreasing the solvation free energyof particle ‘B’ from unperturbed F sol,B = 0 (Fig. 9 a,d) to F sol,B = − E s,X , the perturbation to F sol,X more signifi-cantly enhances the ratchet effect (Fig. 9 c,f). It is shownthat perturbing the solvation free energy can significantlyincrease the selectivity of the channel (Fig. S1 c,f) with-out trading in the current for particle ‘A’. This can beshown by the increased normalized current in Fig. S2 c,f,which indicates that the enhancement of the selectivitydoes not negatively impact the flow through the channel.By comparing the two types of perturbations, we con-clude that both the binding affinities and the dehydrationenergy (solvation free energies) affect the channel’s selec- s,A = F sol,B = 0 E s,A = -1, F sol,B = 0 E s,A = 0, F sol,B = -1(a) (b)(d) (f)(c)(e) FIG. 9: Perturbation to the phase diagrams for the 3-site and 5-site channels. The color map illustrates the steady-state entropy efficiency ξ A and ξ B . Panels a, b, and c are obtained for a 3-site channel. Panels d, e, and f are ob-tained for a 5-site channel. Panels a and d are obtained in the unperturbed model from section IV (with completelyflat energy landscape). Panels b and e are obtained by the perturbation E s,A = −
1, whereas panels c and f are ob-tained by the perturbation F sol,B = −
1. Compared to the perturbation to E s,A , the perturbation to F sol,B leads toa larger increase to the phase diagram range of the ratchet effect.tivity. However, the difference in particles’ solvation freeenergy is a more promising parameter for enhancing theratchet effect and thus improving the channel’s selectiv-ity. Additionally, the selectivity enhancement by alteringthe solvation free energy does not cause a trade-off in thechannel’s rate of flow, whereas that of the binding affin-ity enhances the selectivity by trading in the flow rate ofthe preferred particles. VI. CONCLUSION
In this work, we find a new kinetic ratchet mechanismto explain the non-equilibrium selective transportationof particles through channels. For a multi-site channel,the ratchet effect couples the competing flows of differ-ent types of particles, allowing for the dominant particle’sflow to suppress or even invert the flow of the other typeof particle. At the inversion (dynamical phase transitionbetween dud and ratchet modes), the channel achievesan infinite selectivity. The ratchet mode resembles aMaxwell’s demon, as the inverted flow of particle ‘B’(against its gradient) enriches particle ‘B’ in its high con-centration bath, which appears to violate the second lawof thermodynamics. However, when the entropy produc-tion the dominant particle flow ‘A’ is included, the total entropy production remains non-negative and the secondlaw of thermodynamics is restored.The ratchet effect is ubiquitous to multi-site channelseven when the particles do not interact with each other(except for same site exclusion). Thus it can be uti-lized to guide the design of artificial selective channels.The ratchet effect is more prominent for longer channels,granting a higher selectivity and a lower flow rate. Fora channel with a fixed number of sites, we find that in-creasing the distinction between two particle’s bindingaffinities with the channel’s sites plays a minimal role inimproving the channel’s performance – trading the par-ticle’s flow rate for selectivity. In contrast, by increasingthe particles difference in their solvation free energy, theselectivity of the channel is significantly magnified with-out any trading in the flow rate of the preferred particles.
VII. ACKNOWLEDGMENTS
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