Functional Role for Transporter Isoforms in Optimizing Membrane Transport

Abstract

Quantitative analysis of carrier parameters demonstrates that with decreasing substrate concentration the optimal strength of substrate-carrier interaction which maximizes the flux across the membrane increases and requires less fine-tuning than at higher concentrations of the substrate.

Many of the nutrients that a cell needs for its functioning, such as sugars, amino acids, nucleotides, or organic bases, require specialized transporters to cross the cell membrane (1–3). The rapid growth of available information characterized as “transporter explosion” by Uhl and Hartig (4), has led to creation of the transporter classification system, with division of all transporters into channels and carriers. Channel proteins are mainly considered to function as selective pores which do not need conformational rearrangements at each substrate translocation event. Their optimization principles were addressed in several recent articles (see e.g., (5–9)). This Letter is focused on optimization of carrier-facilitated transport. A carrier transfers substrates via a mechanism that includes at least four steps: i), Binding of the substrate to the carrier on one side of the membrane; ii), carrier conformational change leading to substrate transition to the other side; iii), dissociation of the substrate, and iv), return of the carrier to its initial conformation/position in the membrane (1–3).

The diagram describing these steps of transport in the case of a passive uniporter carrier is shown in Fig. 1.

Figure 1.

Schematic representation of the simplest model of carrier-facilitated exchange of substrate molecules, S, between sides 1 and 2 of the membrane (12,13). The model assumes that the unloaded transporter jumps between the two conformations in which it can bind substrate molecules on sides 1 and 2, with the jump rates α and β. Similar rates of the conformational transitions of the ES-complex (loaded transporter), $E{S}^{\left(1\right)}\rightleftarrows E{S}^{\left(2\right)}$, are α′ and β′. Formation and dissociation of the ES-complex, $E+S\rightleftarrows ES$, are characterized by rate constants k_b and k_d, which in general can be different on the two sides of the membrane, where the substrate concentrations, respectively, are $C_S^{\left(1\right)}$ and $C_S^{\left(2\right)}$.

In the human genome there are 43 distinct families of transport systems that comprise >300 isoforms of individual solute carriers (10). Although the majority of these transport systems is responsible for uptake of specific substrates, a substantial number of transporters are used for uptake of the same solute, and often have an overlapping expression of multiple isoforms that exists in the same cell type. So, the naturally arising question (see, e.g., (11)) is: Why are there so many transporter isoforms?

Here we offer a possible answer by analyzing the carrier-facilitated transport described by the kinetic scheme in Fig. 1 (see also (12,13)) with the focus on the optimal efficiency of the transporter. Analytical expressions are derived for the optimal values of i), the dissociation rate constant, and ii), the ratio of the forward and backward rates of the carrier conformational transitions, which maximize the flux.

We demonstrate that at lower substrate concentrations stronger substrate binding is required, and that the deviations from optimal interaction become more critical as the substrate concentration increases, i.e., higher concentrations necessitate more precise tuning. Thus, uniporters designed to transport the same molecule in the same cell have to be optimized with different amino-acid sequences, with one gene coding for a uniporter protein that functions most efficiently at high solute concentrations, whereas another gene is coding for the one that is most efficient at low concentrations. Although quantitative analysis of optimization of carrier-facilitated transport was conducted almost 30 years ago for the liquid membranes of extraction technology (14), to the best of our knowledge it has never been applied to biological carriers.

Usually it is assumed that the flux of the substrate molecules across the membrane is controlled by the conformational dynamics of the transporter. This implies fast equilibration between loaded ES and unloaded E states of the transporter on both sides of the membrane. We relax this assumption and consider loading and unloading of the transporter and its conformational dynamics on an equal footing. Consider a membrane containing N transporters assuming that $C_S^{\left(2\right)}$ = 0. Let the number of transporters in each of the four states (see Fig. 1) at time t be $N_E^{\left(1\right)}\left(t\right)$, $N_E^{\left(2\right)}\left(t\right)$, $N_{ES}^{\left(1\right)}\left(t\right)$, and $N_{ES}^{\left(2\right)}\left(t\right)$, respectively, i.e., $N_E^{\left(1\right)}\left(t\right)+N_E^{\left(2\right)}\left(t\right)+N_{ES}^{\left(1\right)}\left(t\right)+N_{ES}^{\left(2\right)}\left(t\right)=N$. The rate equations determining variations of these numbers with time are (see notations in Fig. 1)

$$\begin{array}{l}\frac{d{N}_E^{\left(1\right)}\left(t\right)}{dt}=\beta N_E^{\left(2\right)}\left(t\right)-\left(\alpha +k_b^{\left(1\right)}{C}_S^{\left(1\right)}\right)N_E^{\left(1\right)}\left(t\right)+k_d^{\left(1\right)}{N}_{ES}^{\left(1\right)}\left(t\right),\\ \frac{d{N}_{ES}^{\left(1\right)}\left(t\right)}{dt}=k_b^{\left(1\right)}{C}_S^{\left(1\right)}{N}_E^{\left(1\right)}\left(t\right)-\left({\alpha }^{\prime }+k_d^{\left(1\right)}\right)N_{ES}^{\left(1\right)}\left(t\right)+{\beta }^{\prime }{N}_{ES}^{\left(2\right)}\left(t\right),\\ \frac{d{N}_E^{\left(2\right)}\left(t\right)}{dt}=k_d^{\left(2\right)}{N}_{ES}^{\left(2\right)}\left(t\right)-\beta N_E^{\left(2\right)}\left(t\right)+\alpha N_E^{\left(1\right)}\left(t\right),\\ \frac{d{N}_{ES}^{\left(2\right)}\left(t\right)}{dt}={\alpha }^{\prime }{N}_{ES}^{\left(1\right)}\left(t\right)-\left({\beta }^{\prime }+k_d^{\left(2\right)}\right)N_{ES}^{\left(2\right)}\left(t\right).\end{array}$$ (1)

To demonstrate our major findings we limit ourselves to the case when the rate constants of the conformational transitions are independent of whether the transporter is loaded or not, α = α′, β = β′, and the dissociation rate constants on both sides of the membrane are the same, $k_d^{\left(1\right)}=k_d^{\left(2\right)}=k_d$.

The maximum rate of substrate binding to the transporters is achieved when $N_E^{\left(1\right)}=N$ (i.e., all transporters are empty, and they are on side 1 of the membrane). Using this rate, $k_b^{\left(1\right)}{C}_S^{\left(1\right)}N$, as a scale, we can write the steady-state flux, J, of the substrate molecules across the membrane as

$$J=F\left(\alpha ,\beta ,k_d|k_b^{\left(1\right)}{C}_S^{\left(1\right)}\right)k_b^{\left(1\right)}{C}_S^{\left(1\right)}N,$$ (2)

where factor $F\left(\alpha ,\beta ,k_d|k_b^{\left(1\right)}{C}_S^{\left(1\right)}\right)$ can be interpreted as the transporter efficiency. The flux is a monotonic function of the product $k_b^{\left(1\right)}{C}_S^{\left(1\right)}$, whereas it is nonmonotonic with respect to k_d, α, and β. With this in mind, we split the arguments of the transporter efficiency into two groups separated by a vertical line.

Solving Eq. 1 in steady-state, we can write the efficiency in the form

$$F=\frac{\alpha \beta }{\left(\alpha +\beta \right)\left[{\left(\sqrt{k_d}-\sqrt{\frac{\beta k_b^{\left(1\right)}{C}_S^{\left(1\right)}}{k_d}}\right)}^2+\alpha +{\left(\sqrt{\beta }+\sqrt{k_b^{\left(1\right)}{C}_S^{\left(1\right)}}\right)}^2\right]},$$ (3)

analysis of which shows that the efficiency reaches its maximum at $k_d^{\ast }=\sqrt{\beta k_b^{\left(1\right)}{C}_S^{\left(1\right)}}$. Interestingly enough, k^∗_d being proportional to $\sqrt{\beta }$ is independent of α. Fig. 2 presents the transporter efficiency at equal jump rates, α = β, as a function of the normalized dissociation constant k_d/β for five values of the normalized concentration $k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta$. In similarity to our earlier results on optimization of channel-facilitated transport (6), optimization of the carrier-facilitated transport at lower substrate concentrations requires not only stronger substrate binding to the transporter protein, but also is less sensitive to the tuning. Indeed, the optimization curve for $k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta$ = 0.0001 is much wider than that for $k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta$ = 0.1.

Figure 2.

Transporter efficiency as a function of normalized dissociation constant k_d/β, Eq. 3 with α = β, for five values of the normalized concentration, $k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta$ = 0.0001, 0.003, 0.1, 1, and 10, from top to bottom.

In the more general case, when α ≠ β, function $F\left(\alpha ,\beta ,k_d|k_b^{\left(1\right)}{C}_S^{\left(1\right)}\right)$ optimized with respect to k_d can be further optimized with respect to the ratio of the conformational jump rates, ν = α/β,

$$F\left(\alpha ,\beta ,k_d^{\ast }|k_b^{\left(1\right)}{C}_S^{\left(1\right)}\right)=\frac{\nu }{\left(1+\nu \right)\left[\nu +{\left(1+\sqrt{k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta }\right)}^2\right]}.$$ (4)

This function is monotonic in β and nonmonotonic in ν as shown in Fig. 3. The function vanishes as ν → 0 and ν → ∞ and has a maximum in-between. The optimal value of the jump rate ratio, ν = ν^∗, that maximizes the transporter efficiency in Eq. 4 is

$${\nu }^{\ast }=1+\sqrt{k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta }.$$ (5)

Thus, the optimal forward jump rate is always higher than the backward jump rate, α^∗ = ν^∗ β > β. This agrees with a number of experimental observations where such a phenomenon is called “accelerated exchange” (15).

Figure 3.

Transporter efficiency as a function of the ratio of conformational jump rates ν = α/β, Eq. 4 for four values of the normalized concentration, $k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta$ = 0.0001, 0.1, 1, and 10, from top to bottom.

The transporter efficiency optimized with respect to both k_d and ν is given by

$$F\left({\alpha }^{\ast },\beta ,k_d^{\ast }|k_b^{\left(1\right)}{C}_S^{\left(1\right)}\right)=\frac{1}{{\left(2+\sqrt{k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta }\right)}^2}.$$ (6)

Respectively, the maximum flux of the substrate molecules across the membrane containing N transporters is

$$J_{\max }=\frac{k_b^{\left(1\right)}{C}_S^{\left(1\right)}N}{{\left(2+\sqrt{k_b^{\left(1\right)}{C}_S^{\left(1\right)}/\beta }\right)}^2}.$$ (7)

The optimized flux monotonically increases with β from zero at β = 0 to $k_b^{\left(1\right)}{C}_S^{\left(1\right)}N/4$ as β → ∞. Respectively, the optimized transporter efficiency, Eq. 6, increases from zero to its maximum value 1/4. Qualitatively similar nonmonotonic behavior of the steady-state flux can be obtained using different simplifying assumptions; for example, in the case of symmetric transporter, β′ = α′ and β = α. Thus, the results obtained above are of general nature.

The existence of multiple transporter isoforms that carry the same molecule is well documented for almost any important substrate (1–3,10,11). Although this variety of isoforms may seem redundant and, in principle, could be explained by the lack of strong evolutionary pressures to decrease the size of the genome, our analysis offers a different possibility. We have demonstrated that transporter efficiency is fine-tuned to specific ranges of substrate concentration, so that different isoforms might be tailored accordingly to adjust their amino-acid composition for the optimal strength of substrate/transporter interactions and the transition rates between different conformations.