Transport of ions,1, 2 proteins,3 antibiotics4, 5 and other macromolecular solutes through channels and pores is ubiquitous in nature. In particular channel-facilitated diffusion relies on optimized binding sites for the transported particles inside the channel.6 Well characterized examples include membrane channels such as maltoporins7, 8 or aquaglyceroporin9 found in abundance in bacterial membranes. This has been confirmed by (i) ex situ crystallographic structure studies,10, 11 (ii) indirect ionic current measurements through protein channels reconstituted into planar lipid bilayers12–14 and (iii) molecular dynamics simulations.9 These results suggest that organisms can maximize nutrient uptake driven by diffusion by favoring intimate interaction between the protein channel and the translocating species. This is counterintuitive since a strong binding site implies a long residence time in the channel. However, a few theoretical studies have independently rationalized such findings by considering the transport of particles through a channel using a continuum diffusion model based on the Smoluchowski equation,15 discrete stochastic models16, 17 or a generalized macroscopic Fick's diffusion law,18 all demonstrating that an attractive potential in the channel may enhance the particle flux. Remarkably, one intriguing approach predicts a maximum in the diffusive current with respect to the binding potential.15

Testing the validity of these models with an in situ tunable model system on the micro scale resembling a protein channel and its transported species would open the way to a deeper understanding of the general physical mechanism and could ultimately help in the investigation of more complicated processes such as the flux of antibiotics across bacterial membrane proteins.19 In order to mimic channel-facilitated membrane transport, we combine microfluidics,20 particle tracking21 and holographic optical tweezers (HOTs)22 to create a fully controlled and tunable environment to study passive diffusion.

Our experimental model system mimics facilitated membrane transport, using 450 nm polystyrene colloidal particles as translocating species and a microfluidic chip constituted of two macroscopic baths separated by a polydimethylsiloxane (PDMS) barrier and connected via a sub-micrometer channel23 as the protein channel. Since the channel cross section is close to the particle dimension we can assume quasi one-dimensional (1D) diffusion along the channel length (upper scheme in **Figure** **1**).

We mimic facilitated transport by creating in the channel a binding site for the diffusing particles with extended optical line traps generated by HOTs. Our approach offers the unique possibility to control the potential landscape without affecting the channel geometry (lower scheme in Figure 1). We use particle tracking based on digital video microscopy to measure the translocation probability, the average lifetime and the diffusion current through the channel. The single-particle occupation probability distribution along the channel length allows the determination of the potential energy generated by the optical line trap. We demonstrate that the average lifetime increases, as expected, upon coupling a line trap in the channel but at the same time the translocation probability is significantly elevated. Combined, this leads to a maximum in the diffusive current, a factor of three higher than for channels without binding potential.

A microfluidic chip was designed and fabricated through multilevel lithography23 to mimic biological membrane channels: a semi-cylindrical channel of length 4 μm and radius 0.75 μm is in equilibrium with two macroscopic reservoirs filled with 450 nm polystyrene particles dispersed in a 5 mM KCl solution. Particles mainly explore the regions closer to the channel entrance, up to 300 nm inside the channel (scheme in **Figure** **2** and Video 1). It is noteworthy to observe that without any line trap the channel is either empty (with a probability *p _{e}* = 0.94) or populated by a single particle (

*p*= 0.06) while it is only rarely occupied by two or more particles at the same time (

_{1}*p*= 0.001). Due to the dimensions of the channel and particle we assume 1D diffusion along the channel length. Eventually some particles reach the central part of the channel (micrograph in Figure 2 and Video 2). This can be visualized by plotting the number of times

_{>1}*d(x)*we find a particle at position

*x*in the channel, (histogram in Figure 2, see methods for details) as measured by tracking 342 independent particle trajectories (see example trace in Video 3): the entrances of the channel are explored by the particles three times more often with respect to its central part. From our results we determine the particle concentration in the channel to be around (0.017 ± 0.011) μm

^{−3}which is close to the particle concentration in the bulk (0.01 μm

^{−3}). Overall the average translocation probability and diffusion current are

*p*= (0.07 ± 0.05) and

_{Tr}*J*= (7 ± 3) h

^{−1}, respectively. This is in very good agreement with a simple diffusion model predicting

*J*6.7 h

_{Pr}=^{−1}(see Methods), using the measured bulk and channel diffusion coefficients of 0.88 μm

^{2}/s and 0.25 μm

^{2}/s, respectively, and a volumetric particle concentration of 0.01 μm

^{−3}.

It was suggested7 that living systems enhance the transport of particles through their membrane channels by developing specific binding sites. Here, we used HOTs to create a range of potential landscapes with optical line trap of length, *λ,* fixed at 3 μm. We can easily tune the potential depth using the laser power, *P* (**Figure** **3**a), thus mimicking binding sites of different strengths. Our line trap creates an attractive potential for the diffusing particles (Figure 1). Following the diffusion over several hours, we generated histograms of the particle distribution along the channel (Figure 3b) which, in contrast to the case of free diffusion (Figure 2), leads to an increase in the particle population of the channel upon increasing laser power (top panel in Figure 3b). Importantly in the case of the highest laser power the probability to find the channel empty decreases to *p _{e}* = 0.74, while finding more than one particle (

*p*) increases up to 0.09.

_{>1}Moreover while the counts in the outer regions do not significantly change, the probability to find a particle in the central part for *P =* 25 mW is more than 30 times higher. This clearly shows that the attractive potential facilitates particles reaching the centre of the channel. We use the position distribution to determine the potential profile *u(x)* by using the Boltzmann distribution (see Methods). The profiles of the trap (bottom panel in Figure 3b) show a slight increase of the effective trap length, while the average depth *U* linearly increases with the laser power, as expected (Figure 3c).21

The presence of the binding site forces the incoming particles to explore the channel for longer times compared to free-diffusion. This increases the translocation probability *p _{Tr}* up to a value of (0.11 ± 0.02) for a potential well of depth around 1.5 k

_{B}T (

**Figure**

**4**a). At the same time, the lifetime

*τ*increases with

*U*(Figure 4b) as predicted by the model described by Equation (1)24 where

*τ*is defined as the average of the time differences between the instants in which each single particle enters

*t*and leaves the channel

_{In}*t*. For the maximum value of

_{Out}*U*= ∼2.7k

_{B}T,

*τ*is around 30 s and thus 15 times larger compared to the free-diffusion regime.

We can not only measure the potential depth, the translocation probability and the lifetime but follow each particle individually. This allows for determining the diffusion current *J* as a function of *U* as shown in Figure 4c. This indicates a clear maximum: *J* increases up to an average value of 18 h^{−1} for *U* = 1.5 k_{B}T and then decreases to 7 per hour for the deepest well with *U =* 2.7 k_{B}T (Figure 4c). Even for our simple attractive potential, there is an optimal *U* enhancing *J* by a factor of three. The maximum is explained by considering that at *U =* 0, particles attempt to enter but are not likely to reach the centre of the channel, while at 0 < *U <* 1.5 k_{B}T, the channel is populated but *τ* is only slightly increased, leading to an enhanced transport. However, for *U >* 1.5 k_{B}T, the particles are trapped in the channel for longer times effectively decreasing *J*. Berezhkovskii and Bezrukov15 developed a model for *J* (see Equation (2)) which predicts a maximum in very good agreement with our data (solid line in Figure 4c).

The increase in *J* is a consequence of the change in the translocation probability, *p _{Tr}*(|

*x*|,

*U*). Figure 4d shows

*p*(

_{Tr}*U =*1.5 k

_{B}T

*)/p*(

_{Tr}*U =*0) as a function of position |

*x*|

*.*For

*U*= 0 particles will escape to the original reservoir in 75% of the cases, while at

*U*= 1.5 k

_{B}T more particles reach the channel centre at |

*x*|

*=*0 and may translocate.

The presence of one particle in the channel does not exclude other particles from entering. Figure 4 shows our complete experimental results allowing the presence of more than one particle. The model assumes that only a single particle can be in the channel. For this case, we extract *p _{Tr}* by ignoring any further particle attempting to enter. We find that

*p*increases up to a value of 0.11 for the deepest potential well as expected while the dependence of diffusion current and lifetime remains unchanged (see Figure S1 in Supporting Information).

_{Tr}We have shown here that optimizing the transport through a channel is possible using a tunable binding potential. Our experimental model system sheds light on the general physical principles governing diffusive currents and suggests a pathway to enhance particle flux. This generic finding in our simplified model system suggests a pathway to a deeper understanding of more complicated mechanisms ongoing in living systems such as drug uptake where molecular interactions and solvent properties play a major role.4, 5

The wide range of available colloidal particles, in conjunction with the presented experimental approach will allow for testing a variety of transport processes through quasi 1D structures25, 26 relevant for catalysis, osmosis and particle separation27, 28 or systems based on entropic barriers to control transport.29 Other relevant examples are the recently introduced nanopores mimicking transport through the nuclear pore complex.30–32

In summary we presented a synthetic mimic of facilitated membrane transport by exploiting building blocks such as colloidal particles, microfluidics equipped with a sub-micrometer channel and holographic optical tweezers. We characterized the particle exchange between the bulk and the channel in terms of translocation probability, lifetime and particle flux. We mimicked the interaction between the transported species and the channel by coupling an optical line trap and mapped the energy potential by measuring the position dependent probability distribution. The average lifetime and the translocation probability increase in the presence of the optical line trap indicating that diffusive flux through channels can be enhanced with an optimized potential. Our approach opens the way for improving the design of biological and synthetic nanochannels or -pores for optimized transport.