The Lattice Model Of Particle Transport Through Nano-channels

When membranes channels are very narrow so that their average diametersrnare comparable to the average size of the permeating particles, then two particlesrncan’t pass each other inside the channel. These molecular-sized channelsrnare usually referred to as nano-channels. The research work presentedrnin this thesis deals with the modeling of nano-channels, description of theirrntransport properties and the channel gating mechanisms. As particles permeaternthrough nano-channels, they interact with multiple weakly attractive andrnrepulsive sites on the wall of the nano-channel, where the attractive sites act asrnpotential wells and the repulsive sites act as potential barriers. In this thesis,rnthe potential energy landscape of the internal wall of the entire length of thernnano-channel is modeled as a spatial distribution of potential wells and potentialrnbarriers. Moreover, we considered the potential wells as the lattice sites ofrna one dimensional lattice with s lattice sites and particles permeating throughrnnano-channels spend more time in the vicinity of these potential well sites. Wernassumed that between these potential wells, there are particle repelling sites orrnpotential barriers that must be hopped by the permeating particles. Thus, thernheight of the potential barrier between the potential wells or lattice sites controlsrnthe rate of particle transport through nano-channels. The phenomenon ofrnparticle transport or diffusion through nano-channels is modeled as hoppingrnof particles between nearest neighbor potential wells across the intervening potentialrnbarriers.rnThe occupation numbers that denote the lattice sites in the emitter and collectorrnbaths are e and c and the channel sites are i, where i = 1, 2, ..., s withrns denoting the total number of lattice sites. The average value of these internalrnand external occupation numbers lies in the range between zero and one. Inrnthis work, we employ the discrete or lattice model of particle transport throughrnnano-channels, which employs theMaster equation formulation of the rate equationsrnthat describe the time rate of change of the average values of the occupationrnnumbers associated with each lattice site. The effects of the potentialrnbarriers between the potential wells are taken into account via the forward andrnbackward transition rates. These transition rates describe the rate of forwardrnand backward hopping of a particle sitting at a given potential well across thernintervening potential barriers.rnIn the first chapter,we have proposed a simple lattice model of nano-channelsrnand simple expressions for the total rate at which particle are injected into andrnout of the nano-channel at the two end sites. In chapter two, starting fromrnthe Master equation of the rate equations and employing the steady state assumptionrnand the mean filed approximation of the effective transition ratesrnfor both the internal and external sites, we have derived a general expressionrnof the average particle flux through a nano-channel at steady state. The expressionrnof the average particle flux is a multi variable function. That is, J =rnJ(ne, nc, e, c, rne, rnc, 1, ..., s, rn1, ..., rns, k+rne , k−rnc , k−rn1 , ..., k−rns , k+rn1 , ..., k+rns ), where rni = 1−rn i. The nano-channel is characterized by the channel parameters 1, ..., s, rn1, ..., rns,rnk−rn1 , ..., k−rns , k+rn1 , ..., k+rns . The effects of the emitter bath is taken into account viarnthe bath parameters ne, e, rne, k+rne and that of the collector bath via the bath parametersrnnc, c, rnc, k−rnc . In chapter three and four, we have presented a detailedrndescription of the properties of a nano-channel with one and two internal potentialrnwells, respectively.rnOur results show that the average particle flux through a nano-channel asrna function of n, for the specified values of the other parameters, is a rapidlyrnincreasing function of n for small and intermediate values. When n = 0,rnthe average flux has a minimum value and as n increases to large values, thernaverage particle flux through a nano-channel with both one and two internalrnpotential wells increases slowly and finally enters into the saturation region inrnwhich it no longer increase as the concentration gradient increases. For mostrncases, the range of the concentration gradient in which the average particle fluxrnbecomes saturated lies between 1.0 M and 10 M for the specified values of thernother parameters. Moreover, the average particle flux approaches more rapidlyrnxxxvrnto its saturation value when the external potential well Ve is occupied and thernexternal potential well Vc is empty. The average particle flux as a function of n,rnthrough a nano-channel with a single internal potential well is greater than thernaverage particle flux through a nano-channel with two internal potential wells,rnthat is J1( n| e, c) > J2( n| e, c).rnMoreover, the average particle flux through a nano-channel, with one andrntwo internal potential wells, as a function of the external occupation numberrn e, for the specified values of the other parameters, is a rapidly increasing functionrnof e for small values of e, usually e 0.2 and after that it becomes saturated.rnWhen e = 0, the optimized particle flux assumes a minimum value andrnthe nano-channel is in the inactivated state. When e = 1 the nano-channelrnbecomes fully activated and as a result the average flux approaches a maximumrnpositive value. Thus, e acts as a gating variable that controls the direction andrnmagnitude of particle transport through a nano-channel. Moreover, the averagernparticle flux as a function of c, for the specified values of the other parameters,rnapproaches to a maximum positive value as c ! 0 and decreases as c increasesrnand finally as c ! 1, the flux decreases to a small negative value. When c = 0,rnthe average particle flux has a maximum positive value and the nano-channelrnis in the fully activated state and when c = 1 the nano-channel becomes fullyrninactivated and as a result the flux approaches a small negative value. Therefore,rn c also acts as a gating variable that controls the magnitude and directionrnof particle transport through a nano-channel.rnIn this thesis, we have also shown that the average values of the occupationrnnumbers of the internal potential wells as a function of n, for the specifiedrnvalues of the other parameters, are rapidly increasing functions of n for smallrnand intermediate values. When n ! 0, the average occupation numbers havernsmall positive values and as n increases to large values, the average occupationrnnumbers of the internal potential wells of a nano-channel with both onernand two internal potential wells increase slowly and finally approach the saturationrnregion in which they no longer increase as the concentration gradient increases.rnFor most cases, the range of the concentration gradient in which thesernaverage occupation numbers become saturated lies between 1.0 M and 10 Mrnfor the specified values of the other parameters. Moreover, the average occupaxxxvirntion numbers approach more rapidly to the saturation value when the externalrnpotentialwells Ve and Vc are occupied. For a nano-channel with two internal potentialrnwells, the average occupation number as functions of n of the internalrnpotential well V1 is greater than the average occupation number of the internalrnpotential well V2, that is 12( n| e, c) > 22( n| e, c).rnThe average value of the occupation number of the internal potential wellrnV1 increases more rapidly as e increases and slowly as c increases. Similarly,rnthe average value of the occupation number of the internal potential well V2 increasesrnmore rapidly as c increases and slowly as e increases. As the averagernvalue of the occupation number of V1 increases, the average value of the occupationrnnumber of V2 also increases, and vice versa. The average value of thernoccupation number of V1 increases as the values of the normalized potentialrnbarriers u−rn1 , u+1 and u+2 increase and as u+e , u−rn2 and u−c decrease. Similarly, thernaverage value of the occupation number of V2 increases as the values of the normalizedrnpotential barriers u−rn2 , u+2 and u−rn1 increase and as u+e , u+1 and u−c decrease

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