Cancer cells synthesize increased amount of fatty acids and protein building blocks to successfully divide and metastasize. To support their metabolism cancer cells require increased supply of metabolic substrates and nutrients. Plasma membrane transporters secure import of a wide range of substrates into the cytoplasm. Consistently, increased expression of several transporting proteins has been correlated to the increased metabolic activity of cancer cells and poor disease prognosis (Natecz, 2020; Sampedro-Núñez et al., 2020; Xu et al., 2020; Yamada et al., 2020).

There are two types of transporters, ATP biding cassette (ABC) and secondary active transporters. Secondary active transporters carry substrates across the plasma membrane using electrochemical gradient of ions. They bind the substrate on one side of the membrane followed by a conformational change allowing to release the substrate on the other side of the membrane (Boudker and Verdon, 2010). Several of these transporters have been shown to play an important role in cancer cells by increasing uptake of such substrates as glucose, glutamine, lactate, etc., and supporting cancer cell metabolism (Payen et al., 2017; Reckzeh et al., 2019; Scalise et al., 2020).

ABC transporters belong to a large family of transporters carrying several different substrates across the plasma membrane. To translocate substrates ABC transporters use the energy from ATP hydrolysis. They play an important role in keeping cellular homeostasis by regulating the level of several molecules including peptides, lipids, amino acids, or drugs (El-Awady et al., 2017; Neumann et al., 2017). Importantly, ABC transporters have also been shown to contribute to anti-cancer drug resistance presenting a major problem in finding an efficient anti-cancer therapy (Robey et al., 2018; Bock et al., 2019; Asif et al., 2020). These transporters carry out anti-cancer drugs efflux from cancer cells significantly decreasing the efficiency of anti-cancer therapies.

To be able to successfully overcome the problem of the increased transport of nutrients and metabolites and control drugs influx/efflux through transporters in cancer cells it is important to understand the exact mechanism of their structure and activity as well as their biological and physical features. This has been the objective of this article.

We have proposed a model that describes the trans-membrane transport and calculates the flow of substrates through the membrane. The model assumes that changes in transporter conformations induce forces that contribute to the translocation of substrates. Since entropy is a measure of conformations, the forces are referred to as entropic forces. These forces thus encode the information of the change in the available space of the substrates due to conformation changes (Zwanzig, 1992; Reguera and Rubí, 2001; Vazquez et al., 2008; Carusela and Rubi, 2017, 2018; Rubi, 2019). The computed translocation rates scale with the ratio between the time it takes for conformations to change and the time in which substrates diffuse through the transporter. This scaling behavior makes our model a general tool in the study of membrane transport.

We model a transporter as a small funnel-shaped motor whose structure changes over time by alternately opening and closing to the intra/extracellular medium, as represented in Figure 1. This form has been observed for example in P-glycoprotein whose structure is narrow at the cytoplasmic side, of about 9–25 Å in the middle, and wider at the extracellular surface (Loo and Clarke, 2001). The conical form has also been observed in pumps (Rubi et al., 2017) by means of crystallization experiments (Olesen et al., 2007).

This periodic movement makes it possible for substrates, such as aminoacids, ions, neurotransmitters, nutrients and different drugs, to overcome the potential barriers generated by interactions allowing them to pass to the other side of the plasma membrane.

Alternating gating increases transport efficiency with respect to that of diffusion. Transport of substrates through the membrane is the result of changes in its conformation. Primary active transporters couple substrate movements to a source of chemical energy, such as ATP hydrolysis. Secondary active transporters are driven by electrochemical gradients of ions. Transporters differ from ion channels in that their turnover rate is much slower than that of channels which is typically of the order of 10⁶s⁻¹. The rate of ABC transporters such as LeuT, MsbA, and of MFS secondary transporters frequently falls within the range (10⁻¹ − 10³)s⁻¹ (Ashcroft et al., 2009; Liu et al., 2018; Fitzgerald et al., 2019).

Following the protocol described in the previous section, we compute the transport rate Γ=V/L which is plotted in Figure 3 vs. ωL2/D. This quantity represents the ratio between the time in which the conformation of the transporter changes and the time that substrates take to diffuse through the transporter. The presence of resonant peaks at ωL2D~30, practically independent of the values of f_μ, reveals the occurrence of an amplification of the velocity of the substrates at a certain value of the oscillation frequency and therefore shows that transporters work under optimal transport conditions. The peaks become more pronounced when f_μ increases or equivalently the rates are higher when the concentration of substrates increases, as observed for example in experiments for LeuT transporters (Fitzgerald et al., 2019). The existence of an optimal frequency for transport of particles in microfluidic devices subjected to an external oscillatory potential was also found in Tan et al. (2017).

Considering transporters with length of the order of L₀ = 10nm and typical diffusion coefficients for membrane proteins of Eukaryotic cells, D0=(10-2-1)μm2/seg (Kaňa, 2013), we obtain values for Γ in the range of (10⁻¹ − 10³)seg⁻¹. Our model also provides values of the rates in ion channels. In this case, one would expect that since particles are lighter they move faster. Typical values of D0 for ions are in the range (10³ − 10⁴)μm²/seg, therefore Γ takes values between (10⁵ − 10⁶)seg⁻¹. These model predictions are in good agreement with data found in the literature for transport rates of slow and fast protein channels in membranes (Ashcroft et al., 2009). At the maximum opening configuration of the transporter, the entropic force changes from a value of the order of f_μ at the widest part, to about 20f_μ at the narrowness. Our results then show that transporters work in a regime dominated by entropic forces induced by conformational changes.

Membrane proteins transporters regulate the efflux/influx of substrates across plasma membrane of cancer cells and play a paramount role in the efficiency of cancer therapies as they regulate the retention of anticancer drugs in the cells. It is believed that the effectiveness of chemotherapy may be largely dependent on the activity of transporters. Knowing which is the intimate mechanism that makes possible the passage of substrates through the cell membrane is therefore a matter of vital importance.

In this article, we have proposed a computational model that analyses the activity of transporters. The model shows that changes in the transport configuration produced by the energy of ATP hydrolysis (ABC transporters) or by a electrochemical gradient of ions (secondary transporters) give rise to a force that makes the substrates able to overcome the potential barrier generated by the constrictions in order to pass to the other side of the membrane. Such a force results from the variation of the space that the substrates have to move within the transporter when its configuration changes, which generates an entropy gradient.

Transport rates are computed by following a simulation protocol in which we first obtain the time average of the velocity of the substrates which is subsequently averaged over a set of initial conditions of the position of a substrate. Our model reproduces experimental values of the rates for different ABC and secondary transporters and shows that they depend on substrate concentration, in accordance with data reported in recent experiments (Fitzgerald et al., 2019). It can also be used to compute translocation rates in ion channels showing that they are greater than for transporters, as observed in the experiments (Ashcroft et al., 2009). We have shown that the entropic force is greater than that produced by a mere concentration gradient and is directed in the opposite direction which shows that transport is active.

How readily substrates cross the membrane depends on the frequency of oscillation ω, which regulates the entropic force and therefore the changes in the conformation of the transporter, on the size of the substrates which is involved in D and on the size of the transporter L. The rates found scale with the combination of these quantities: ωL2/D which represents the ratio between the time in which conformations change and the time it takes for the substrates to diffuse. Our results could therefore be applied in general to membrane transport, to transport in microfluidic devices and peristaltic channels and to capture cellular heterogeneity beyond cancer cells (Pu et al., 2016; Łapińska et al., 2019; Rhia et al., 2020). Monitoring the entropic force by means of drugs could help to regulate transport activity leading to the design of better transport modulators that can be used in anti-cancer therapies.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

All authors have made a substantial contribution to the work and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.