Entropy facilitated active transport

TL;DR

This paper proposes an entropy-based interpretation of active ion transport, linking pore geometry to driving force and Gibbs energy changes, with significant entropic contributions observed in various proteins.

Contribution

It introduces a novel entropy-facilitated perspective on active transport, connecting pore geometry to thermodynamic driving forces.

Findings

Entropic contribution from pore geometry is significant in several proteins.

Pore geometry influences the Gibbs energy and affinity variations.

The approach offers a new way to interpret active transport mechanisms.

Abstract

We show how active transport of ions can be interpreted as an entropy facilitated process. In this interpretation, the pore geometry through which substrates are transported can give rise to a driving force. This gives a direct link between the geometry and the changes in Gibbs energy required. Quantifying the size of this effect for several proteins we find that the entropic contribution from the pore geometry is significant and we discuss how the effect can be used to interpret variations in the affinity at the binding site.

Equations

$$\rho (x)={\rho }_0\frac{A(x)}{A_0}$$

$$\rho (x)={\rho }_0\exp \left(\frac{\Delta S_{\text{pore}}(x)}{R}\right)$$

$$\Delta S_{\text{pore}}=R\ln \frac{A(x)}{A_0}$$

$$F_{\text{ent}}=T\frac{\partial \Delta S_{\text{pore}}}{\partial x}=RT\frac{1}{A(x)}\frac{\partial A(x)}{\partial x}$$

$$\Delta S=R\ln \frac{c_{\text{out}}{A}_{\text{in}}}{c_{\text{in}}{A}_{\text{0}}}$$

$$J_{\text{Ca}}=-L\left[1-\left(\frac{c_{\text{out}}{A}_{\text{in}}}{c_{\text{in}}{A}_{\text{0}}}\right)\exp \left(-\frac{\Delta H}{RT}\right)\right]$$

$$J_{\text{Ca}}=L\left(\frac{c_{\text{out}}{A}_{\text{in}}}{c_{\text{in}}{A}_{\text{0}}}\right)\exp \left(-\frac{\Delta H}{RT}\right)$$

$$J_{\text{Ca}}=L\left(\frac{c_{\text{out}}{A}_{\text{in}}}{c_{\text{in}}{A}_0}\right)$$

$$-T\Delta S=RT\ln \left[\left(\frac{c_{\text{in}}}{c_{\text{out}}}\right){\left(\frac{r_0-a}{r_{\text{in}}-a}\right)}^2\right]$$

Full text

Active transport is of major importance in biology; meaning transport of a compound against its chemical potential, driven by a chemical reaction. A notable example is the large P-type ATPase protein family which functions as ion or lipid pumps, crucial for a wide range of processes in almost all forms of life [1]. The P-type ATPases share a common topology and their operation can be described using a Post-Albers cycle with four key conformations: E1, E1P, EP2 and E2 [2]. The transitions E1P$\leftrightarrow$E2P and E2$\leftrightarrow$E1 in the catalytic cycle are associated with large conformational changes [2] and it is clear that the energy released by ATP hydrolysis drives the enzyme between these states.

One may ask about the meaning of the conformational changes. In particular, the formation of a wide funnel-shaped outlet channel can be observed in several P-type ATPases [3, 4, 5, 6] when the enzyme changes from the E1P state to the E2P state. The occluded ion is then eventually exposed to the lumen; but why does the ion leave its binding site when the chemical potential is so much larger in the direction of transport than in the opposite direction? Can this be influenced by the observed pore geometries? How can the change in Gibbs energy by the chemical reaction, which may take place relatively far away from the binding site (see e.g. Ref. [7]), be transferred and used at the ion binding site? These questions, which have been of major interest since the discovery of the pumps, are still not fully answered and this is the topic we discuss here.

We shall argue that active transport can be better understood by shifting focus from an energy to an entropy barrier in a specific part of the translocation process. When the outlet channel is formed in the E2P state, the shape of this channel directly facilitates the transport of ions by allowing for an entropy increase, or equivalently, a chemical potential decrease in the E2P state as we will demonstrate. The idea that the actual pore geometry may play a role in active transport, stems from previous studies which have shown how entropic barriers can play a major role for separation purposes [8, 9, 10].

To exemplify the impact of the pore geometry, we will take the Ca2+ transporting Ca2+-ATPase of sarcoplasmic reticulum (SR) but we note that the ideas presented may apply more generally. For the Ca2+-ATPase, we illustrate the variation of the chemical potential as ions are transported from the cytosol to the lumen of the SR in Fig. 1. It is known that the binding of Ca2+ is fast [11] and at equilibrium, the chemical potential of Ca2+ in the E1 state, $\mu_{\text{Ca2.E1}}$, is equal to the chemical potential in the cytosol, $\mu_{\text{out}}$. During operation of the pump, there is probably a small difference between $\mu_{\text{out}}$ and $\mu_{\text{Ca2.E1}}$ but the chemical potential in the final state, $\mu_{\text{in}}$, is clearly larger than $\mu_{\text{out}}$ as depicted in Fig. 1. There is a large uncertainty related to the chemical potential of Ca2+ when the enzyme is in the state E2P, $\mu_{\text{Ca2.E2P}}$. In Fig 1 it is assumed to be close to $\mu_{\text{in}}$ (which is the lower bound resulting in a positive Ca2+ flux), enabling the ion to pass to the lumen. The variation in binding energy inherent in this picture has been attributed to the enthalpic part of the chemical potential, as the enthalpy gives a measure of the bond strength: Repulsive forces can raise the chemical potential $\mu_{\text{Ca2.E2P}}$, which results in a low affinity binding site. In such a situation the ions can move spontaneously to the lumen. Alternatively, this change in chemical potential can be attributed to entropic effects. The leap from $\mu_{\text{Ca2.E1}}$ to $\mu_{\text{Ca2.E2P}}$ is due to the hydrolysis of ATP and it can arise from increasing the enthalpy of the ion, or by lowering the entropy of the ion. The latter can be brought about by a change in the actual shape of the channel where the ion is transported. A particle moving in a pore with a varying cross-sectional area will be subjected to entropic forces due to the change in area and this can in fact facilitate the transport.

The outlet channels observed in the P-type ATPases are narrow but widens towards the exit of the channel, as illustrated in Fig 2. We assume that the ion transport is effectively one-dimensional, directed along a spatial coordinate $x$. The ion density along the transport direction may be approximated by [8]

[TABLE]

where $\rho_{0}$ and $A_{0}$ are the ion density and cross-sectional area at a reference point, say $x=0$. The size of the particles has been shown to influence the transport properties [12] and here, both $A(x)$ and $A_{0}$ measures the available area, e.g. for a circular geometry: $A(x)=\pi\left(r(x)-a\right)^{2}$ where $r(x)$ is the radius of the pore at position $x$ and $a$ the radius of the ion. The density given by Eq. 1 can also be viewed as the canonical distribution function

[TABLE]

where we have introduced the change in entropy, $\Delta S_{\text{pore}}(x)$, associated with the change in cross-sectional area. From these equations, we can conclude that change in the entropy during ion motion along the pore is directly related to the cross sectional area

[TABLE]

and that the position dependent entropy along the ion trajectory is giving rise to a (thermodynamic) force of entropic nature, $F_{\text{ent}}$, acting on the ions

[TABLE]

The direction of the force directly depends on the slope of the channel. For the conical structure shown in Fig. 2 the entropic force is positive, and will facilitate translocation in direction of the wider opening.

The overall entropy change ($\Delta S$) for the ion translocation will thus have two contributions: the normal contribution from the ion concentration difference ($\Delta S_{\text{Ca}}=-R\ln(c_{\text{in}}/c_{\text{out}})=-\Delta\mu_{\text{Ca}}/T$), and a special contribution from the pore shape. Per mol of ion we have $\Delta S=\Delta S_{\text{Ca}}+\Delta S_{\text{pore}}$ and by introducing Eq. 3 for $x=1$, we obtain

[TABLE]

The entropic force depends on the concentration ratio, but also on the ratio of the cross-sectional areas at the two sides of the channel. A large $c_{\text{in}}/c_{\text{out}}$ ratio may be counteracted by a smaller $A_{0}/A_{\text{in}}$ ratio, meaning that translocation may take place against the gradient in concentration. In other words, it may be facilitated by entropic forces induced by the formation of a conical pore and the larger the ratio $A_{\text{in}}/A_{0}$, the larger is the co-acting entropic force.

The translocation rate, $J_{\text{Ca}}$, can be shown to be

[TABLE]

where $L$ is the backward reaction rate, and $\Delta H$ is the change in enthalpy. The equation holds when the activity coefficients inside and outside the membrane are the same. In the outset $\Delta H$ is unknown. In the case that the pump operates far from equilibrium the last term in the parenthesis in Eq. 6 dominates, giving

[TABLE]

The equation gives an Arrhenius behavior of the current through the dependence on $\Delta H$. Assuming that the entropy change alone drives the translocation, we have

[TABLE]

where the flux is given by the ratio of concentrations and cross-sectional areas.

Assuming a circular cross-sectional area, the entropic driving force is,

[TABLE]

where $r_{0}$ is the radius at the start of the channel, $r_{\text{in}}$ at the exit of the channel and $a$ the radius of the ion. In Fig. 3 we show the entropic contribution to the driving force for several enzymes. As the results show, the entropic contribution can be sizable. It is not sufficient to explain active transport by itself for all the enzymes as can be seen by the results for the Ca2+-ATPase. In this case, the entropic contribution is on the order of $10$ kJ/mol ion transported at $37^{\circ}\text{C}$. This accounts for $40\%$ of the chemical potential difference which indicate that other contributions, for instance enthalpic effects, may also be important for this protein.

For the Na+/K+-ATPase we find that, due to the narrowness of the pore, the entropic contribution ($20$ kJ/mol) is relatively large compared to the other enzymes. This is sufficient to overcome the concentration ratio of Na+ at physiological conditions, however, in this case, the Na+ ions are transported against the resting membrane potential. Assuming a resting potential of $-60$ mV and a concentration ratio on the order of $10$, the required Gibbs energy is $12$ kJ/mol ion transported. This means that the entropic force estimated in Fig. 3 is still sufficient for this enzyme. The entropic contribution for the two other enzymes is similar to the Ca2+-ATPase. We have neglected the possibility of complexing the ions with other species such as water which could increase the effective radius and the entropic force.

We have shown that the actual pore geometry observed in several transport enzymes may give rise to an entropic force, facilitating transport. This entropic effect can be used, together with enthalpic effects, to explain how conformational changes influences the binding site, allowing transport of compounds against their concentration gradients. In particular, the entropic force gives a direct link between the energy released by a reaction at the active site and changes in the chemical potential at the binding site: By modifying the conformation and the pore geometry, a low affinity binding site can be created.

Author contributions

JMR, AL, DB, and SK developed the theory, performed the analysis, and wrote the manuscript. JMR initiated the work.

Acknowledgements

The Norwegian University of Science and Technology is thanked for supporting the stay of JMR.