Diffusion of molecules through nanopores under confinement: Time-scale bridging and crowding effects via Markov state model

Abstract

Passive transport of molecules through nanopores is characterized by the interaction of molecules with pore internal walls and by a general crowding effect due to the constricted size of the nanopore itself, which limits the presence of molecules in its interior. The molecule–pore interaction is treated within the diffusion approximation by introducing the potential of mean force and the local diffusion coefficient for a correct statistical description. The crowding effect can be handled within the Markov state model approximation. By combining the two methods, one can deal with complex free energy surfaces taking into account crowding effects. We recapitulate the equations bridging the two models to calculate passive currents assuming a limited occupancy of the nanopore in a wide range of molecular concentrations. Several simple models are analyzed to clarify the consequences of the model. Eventually, a biologically relevant case of transport of an antibiotic molecule through a bacterial porin is used to draw conclusions (i) on the effects of crowding on transport of small molecules through biological channels, and (ii) to demonstrate its importance for modelling of cellular transport.

The Optimal Permeation of Cyclic Boronates to Cross the Outer Membrane via the Porin Pathway

“…By using the electrophysiology of a single OmpF channel reconstituted in an artificial lipid bilayer membrane [ 27 ], one can, in principle, determine the average dwell time, , when the ionic current through the pore is blocked (or reduced) due to the presence of the molecule inside. As reported in detail in [ 40 ], we have the following: and we can estimate the dwell time for antibiotics as being well below 1 μs. These fast events were not observed experimentally for vaborbactam (M. Winterhalter private communication) using standard single-channel single-molecule electrophysiology techniques.…”

“…At a low substrate concentration, one can neglect the interaction between the substrate molecules. Then, diffusion is described by the linear 1D Smoluchowski equation, and the diffusion current is proportional to the substrate concentration gradient, as given by a Kramers-type integral formula [40]. As performed previously [30], the high substrate concentration condition was treated with a two-state Markov model [40]; we used value of 1 nm 2 /ns for the effective diffusion constant.…”

“…By using the electrophysiology of a single OmpF channel reconstituted in an artificial lipid bilayer membrane [27], one can, in principle, determine the average dwell time, τ b , when the ionic current through the pore is blocked (or reduced) due to the presence of the molecule inside. As reported in detail in [40], we have the following:…”

“…At a low substrate concentration, one can neglect the interaction between the substrate molecules. Then, diffusion is described by the linear 1D Smoluchowski equation, and the diffusion current is proportional to the substrate concentration gradient, as given by a Kramers-type integral formula [ 40 ]. As performed previously [ 30 ], the high substrate concentration condition was treated with a two-state Markov model [ 40 ]; we used value of 1 nm /ns for the effective diffusion constant.…”

“…In the linear regime (at low concentration limit), the translocation diffusive current is proportional to the concentration gradient (Fick’s law) and is the same from cis to trans and from trans to cis given the same concentration gradient. At high concentration gradients, the saturated (maximum) translocation current in Figure 9 b is equal to the rate of the Markov state model [ 40 ]. By using the electrophysiology of a single OmpF channel reconstituted in an artificial lipid bilayer membrane [ 27 ], one can, in principle, determine the average dwell time, , when the ionic current through the pore is blocked (or reduced) due to the presence of the molecule inside.…”