Optimizing the Substrate Uptake Rate of Solute Carriers

The diversity in solute carriers arose from evolutionary pressure. Here, we surmised that the adaptive search for optimizing the rate of substrate translocation was also shaped by the ambient extracellular and intracellular concentrations of substrate and co-substrate(s). We explored possible solutions by employing kinetic models, which were based on analytical expressions of the substrate uptake rate, that is, as a function of the microscopic rate constants used to parameterize the transport cycle. We obtained the defining terms for five reaction schemes with identical transport stoichiometry (i.e., Na+: substrate = 2:1). We then utilized an optimization algorithm to find the set of numeric values for the microscopic rate constants, which provided the largest value for the substrate uptake rate: The same optimized rate was achieved by different sets of numerical values for the microscopic rate constants. An in-depth analysis of these sets provided the following insights: (i) In the presence of a low extracellular substrate concentration, a transporter can only cycle at a high rate, if it has low values for both, the Michaelis–Menten constant (KM) for substrate and the maximal substrate uptake rate (Vmax). (ii) The opposite is true for a transporter operating at high extracellular substrate concentrations. (iii) Random order of substrate and co-substrate binding is superior to sequential order, if a transporter is to maintain a high rate of substrate uptake in the presence of accumulating intracellular substrate. Our kinetic models provide a framework to understand how and why the transport cycles of closely related transporters differ.


INTRODUCTION
Cellular membranes are diffusion barriers for polar solutes. Uptake of these solutes into a cell or a subcellular compartment is, therefore, contingent on solute carriers (SLC). For this reason, SLCs are vital for many physiological functions. The latter include cellular uptake of nutrients and extrusion of toxic compounds from the interior of a cell (Hediger et al., 2004;Omote et al., 2006;Sano et al., 2020). In addition, SLCs are involved in higher order functions, such as neurotransmission (e.g., reuptake of neurotransmitters subsequent to their vesicular release; Rudnick and Sandtner, 2019;Bhat et al., 2021). Many of the SLCs can harvest the energy contained in the transmembrane ion gradients to drive uphill transport of their substrate against an opposing substrate gradient (Mitchell, 1979). These are termed concentrative or secondary active transporters, which either work as symporters or antiporters (Jennigs, 2018). Another class of SLCs only facilitates passive diffusion of a polar solute by providing an Frontiers in Physiology | www.frontiersin.org 2 February 2022 | Volume 13 | Article 817886 aqueous pathway, via which the solute can enter or leave the cell. The latter are termed facilitating or equilibrative transporters. Both, the concentrative and the equilibrative SLC operate by the alternate access mechanism (Jardetzky, 1966) which entails the following sequence of events: Extracellular substrate first binds to the transporter in its outward-facing conformation. On substrate binding the transporter rearranges to adopt the inward-facing conformation. From there the substrate is released into the cytosol. Subsequent to this, the carrier rearranges again to return to the substrate-free outward-facing conformation. From this point on, this series of reactions can repeat all over. Substrate uptake by a solute carrier is, therefore, a process, which encompasses several partial reactions. These include conformational change and binding/unbinding reactions of substrate and (co)-substrates to and from the transporter. These partial reactions form a closed loop, which is also referred to as the transport cycle.
We have recently described an approach to kinetic modeling of a solute carrier, which allows for deriving analytical expressions for its functional descriptors (Schicker et al., 2021). These include the K M and the V max for substrate uptake, the rate of basal substrate release from the interior of the cell, etc. The corresponding terms express these descriptors as a function of the microscopic rate constants used to parameterize the kinetic model. In the present study, we derived the defining terms for the substrate uptake rate of a sodium symporter for five different reactions scheme, which all adhere to the same transporter stoichiometry (Na + :substrate = 2:1). The rationale for obtaining these analytical terms was as follows: The substrate uptake rate is the only functional descriptor of a transporter, for which compelling arguments can be made that it has been optimized (i.e., maximized) by evolution. These are: the magnitude of solute flux through SLCs into a cell or a cell organelle is determined by the number of transporter units expressed on the cell or organelle surface and the substrate uptake rate (i.e., turnover rate) of the individual transporters. Accordingly, to maintain a substrate flux, which is commensurate with the physiological needs, the cell can either increase the number of transporters or the rate of substrate turnover. The former is associated with two problems: (i) Protein synthesis is energetically costly (Millward and Garlick, 1976;Waterlow et al., 1978;Siems et al., 1984) and (ii) additional transporters occupy space in the membrane. Membranes cannot be infinitely crowded by transmembrane proteins (Bar-Even et al., 2011). Having to have fewer transporters, thus, increases the energy efficiency of a cell/organism. This is expected to improve fitness at conditions in which nutrients are scarce. Accordingly, the substrate uptake rate of a solute carrier fulfills all criteria of a trait subject to evolutionary selection.
We emulated the evolutionary pressure on the substrate uptake rate by relying on an optimization algorithm. This searched for the set of microscopic rate constants, which returned the largest value for the substrate uptake rate at given intra-and extracellular concentrations of Na + and substrate. The resulting sets of values provided by the optimization algorithm were subjected to an in-depth analysis. This analysis showed how a solute carrier must adjust its operation to cycle at a high rate at the various conditions/challenges, which it may encounter.

Numerical Simulations
Time-dependent changes in state occupancies of the model in Figure 1A were evaluated by numerical integration of the resulting system of differential equations using the Systems Biology Toolbox (Schmidt and Jirstrand, 2006) and MATLAB 2018a (MathWorks, Natick, MA, United States).

Optimization of the Substrate Uptake Rate
Explicit expressions for the substrate uptake rate were derived as described previously (Burtscher et al., 2019;Schicker et al., 2021). Numerical sets of values for the microscopic rate constants maximizing these expressions were generated by a simulated annealing algorithm (Metropolis et al., 1953;Tsallis and Stariolo, 1996). In brief, in an initial step, a set of values for the microscopic rate constants was randomly drawn from independent normal distributions, centered at chosen start values with SDs of the same size. The substrate uptake rate for the drawn set was then calculated and compared with the substrate uptake rate calculated from the original set (i.e., the start values). The probability of accepting the new set was: prob. = exp.[−(−current Value + best Value)/T(iter)], with T being a temperature parameter, which was chosen to decrease exponentially with the number of iterations. If accepted, the new set was used for the next iteration, if not, the old set was retained. This procedure was repeated for 5,000 iterations. We safeguarded against trapping in a local maximum by reinitiating the algorithm with the maximum T value increased by 10% using the best parameter set found in the first round of iterations. If this yielded a better overall value for the substrate uptake rate, the next run was reinitiated with the original T. Otherwise T was increased by additional 10%. This procedure was repeated until reheating was unsuccessful in obtaining a better set of values for 10 times. On completion, the algorithm reported the best parameter set (i.e., the set of values which gave the largest substrate uptake rate).
In the optimization of the sequential binding order schemes, we kept the detailed balance constraint, by allowing the algorithm to vary all microscopic rate constants except one. This rate constant was then calculated from the other rate constants such that detailed balance was maintained. In the case of the random order binding scheme, it was necessary to calculate three rate constants because of the larger number of loops. Each set of microscopic rate constants was also evaluated for adherence to the other imposed constraints (e.g., diffusion limit for the association rates of substrate and co-substrate). Only if a set of values complied with the imposed constraints it was passed on to the annealing algorithm.

Kinetic Models of SLC Can Predict
Substrate Turnover Rates Figure 1A shows the reaction scheme of a hypothetical symporter: In each cycle, the transporter translocates one substrate molecule through the membrane together with two Na + ions. We selected FIGURE 1 | Optimization of secondary active transporters operating in a sequential binding mode for translocation of substrate at a low and high concentration.
(A) Reaction scheme of a Na + symporter in the sequential binding mode ("first in, first out") referred to as NaSNaS: The apo-outward-facing transporter (To) first binds two Na + ions (To2Na). On substrate binding, (To2NaS) the transporter rearranges to adopt the inward-facing conformation (Ti2NaS). Subsequent to the release of its cargo into the cytosol (TiNa2S → TiS → Ti) the substrate-free inward-facing transporter (Ti) undergoes a conformational change upon which the transporter returns to To. (B) Simulated substrate uptake rate of the symporter operating according to the scheme outlined in (A) after application of .1 μM (green trace) and 100 μM (magenta trace) of substrate (S out ). For this simulation the microscopic rate constants (i.e., k on Na + out and k flip-in S) were parameterized with the values for the optimized transporter T1 shown in Table 1. The symbols in green and in magenta show the substrate uptake rates computed for T1 with the analytical expression (see supplement) for .1 μM and 100 μM S out , respectively. The uptake rates obtained with the two different approaches were identical. (C) Substrate uptake rates of (Continued)  Wright et al., 2011) and phosphate transporters (PiT-1/SLC20A1 and PiT-2/SLC20A2; Forster et al., 2013) operate with this stoichiometry. For the sake of simplicity, we assumed binding of the two sodium ions to occur in a single reaction. In Figure 1B, we used this model to predict the rate of substrate uptake through the transporter by assuming that two different concentrations of extracellular substrate (S out ) were applied, that is, .1 μM (magenta line in Figure 1B) and 100 μM (blue line in Figure 1B). As seen, on exposure of the cell to the substrate, the substrate uptake rate rose. The rise was large on application of 100 μM S out and small on application of .1 μM S out . In the simulation, the substrate was removed after 60 s upon which the substrate uptake rate dropped to zero. The data in Figure 1B were obtained by numerically solving the system of differential equations underlying the kinetic model. An alternative approach to compute the substrate uptake rate relies on deriving its defining analytical term. For the sake of space, we show the term in the supplement. Figure 1B also displays the substrate uptake rates obtained by this second approach in the presence of .1 μM (open circle in magenta) and 100 μM S out (open circle in blue). It is evident that the substrate uptake rates predicted by the two methods were identical.

Maximizing the Substrate Turnover Rate
The extracellular concentration of a substrate (S out ) is a given quantity, that is, it is typically not subject to control by a single cell. Accordingly, SLC, which are tasked with transporting a substrate into the interior of a cell, must adjust their operation to the substrate concentration they encounter. The substrate concentration surrounding a cell can therefore, be assumed to exert evolutionary pressure. To emulate optimization of the substrate uptake rate by evolution, we maximized this rate, utilizing its defining function. For this purpose, we employed an optimization algorithm, which can approximate global minima/maxima of a function (i.e., simulated annealing-for details see the method section). The optimization algorithm can find the set of numeric values for the microscopic rate constants, which returns the largest value for the substrate uptake rate at given intra-and extracellular concentrations of Na + and substrate.
We note that microscopic rate constants are a priori not mathematically constrained: They can assume values between zero and infinity. If they are permitted to vary across the entire mathematically possible range, the optimized substrate uptake rate will also adopt values between zero and infinity. It is a futile and meaningless exercise to maximize a function, for which it is known that no maximum exists. Fortunately, however, there are limits to the values of the microscopic rate constants. For instance, the association rates of co-substrates and substrate cannot be larger than the diffusion limit, which therefore imposes an upper limit on these rates. Likewise, the dissociation rates of substrate and co-substrate must also have an upper limit, because raising the dissociation rate constant results in affinity loss, which, when substantial, prevents the co-substrate and the substrate from interacting with the transporter in their physiological concentration ranges. It is also clear that a conformational change cannot occur with infinite velocity. We selected 1,000 s −1 as the upper limit for conformational transition rates. The choice of this value was based on information obtained from the literature (Zhang et al., 2007;Schicker et al., 2011;Hasenhuetl et al., 2018;Erdem et al., 2019). Another necessary constraint was to ensure that every set of optimized values complied with the rule of microscopic reversibility: The product of the rates in the forward direction in a loop must equal the product of the rates in the opposite direction. The combined constraints reshape the parameter space of the function, such that it harbors critical points, which do not exist in the unconstrained parameter space.
We performed ten optimization runs in which we assumed that S out was .1 μM and 100 μM ( Figure 1C). In all runs, we set the extra-and intracellular Na + concentration to 150 mM and 10 mM and the intracellular substrate concentration (S in ) to zero. The data points show the substrate uptake rates, to which the optimization algorithm converged when S out was set to .1 μM (left column) and 100 μM (right column). The rates were low at .1 μM S out (7.44 s −1 ± .08 s −1 ) and high at 100 μM S out (431.40 s −1 ± .66 s −1 ). We emphasize that each point in Figure 1C represents a unique set of numeric values for the microscopic rate constants. In Table 1, we show the ten sets, which we obtained from the optimization runs where S out was set to .1 μM and 100 μM. Although the values of the microscopic rate constants differed between sets, they all gave essentially FIGURE 1 | transporters optimized for .1 μM S out (green open circles) and for 100 μM S out (magenta open circles). The data points in the graph show the substrate uptake rates obtained from 10 optimization runs. The corresponding sets of numeric values for the microscopic rate constants are listed in Table 1. (D) The curves represent the substrate uptake rate as a function of S out of transporters optimized for .1 μM S out (in green) and 100 μM S out (in magenta). The data are means from the 10 independently optimized transporters (T 1 -T 10 and T 11 -T 20 ), error bars indicate S.D. (E) The data in panel (D) were plotted as normalized values (V max = 1). (F) Plotted are the K D values for binding of the substrate to the outward-facing conformation of the transporter and the corresponding K M values of transporters optimized for .1 μM and 100 μM S out , respectively. The coefficients of variation for the K D s were: .43 and .094 and for the K M s: .094 and .03 at .1 μM and 100 μM S out , respectively. (G) Substrate uptake rate as a function of out Na + of transporters optimized for .1 μM S out (in green) and for 100 μM S out (in magenta). The concentration dependence for out Na + differed considerably between transporters optimized for the same S out . (H) The same data as in (G) but normalized (I) Plotted are the K D values for Na + binding to the outward-facing conformation of the transporter and the corresponding K M values of transporters optimized for .1 μM S out and for 100 μM S out . The variation in the K D values was much less than the variation in the K M values. The coefficients of variation for the K D s were: .17 and .06 and for the K M s: 1.81 and 1.12 at .1 μM and 100 μM S out , respectively. (J) Shown are the K D values of substrate binding to the inward-facing conformation of transporters optimized for .1 μM and 100 μM S out , respectively (coefficients of variation: .47 and .36). (K) Plotted are the K D values of Na + binding to the inward-facing conformation of transporters optimized for .1 μM S out and for 100 μM S out (coefficients of variation: .43 and .2). (L) Plotted are the optimized substrate uptake rates as a function of the S out , for which they were optimized. Each data point is the means ± SD of the substrate uptake rate obtained from 10 optimization runs. The optimized rate rose upon increase of S out . It leveled out at 500 s −1 because of the constraints imposed in the optimization. the same substrate uptake rate when optimized for the same substrate concentration. We therefore conclude that the optimized rate can be realized by different sets of numeric values for the microscopic rate constants. Because of the large differences in these values, each set can be viewed to define a transporter with an individual phenotype. For this reason, we will from here on treat the term "transporter" and "a set of optimized values" as a synonymous description.

Analysis of Optimized Transporters
The observations summarized in Table 1 warranted further scrutiny. They suggest that not all reactions, which a transporter undergoes, require the same extent of fine-tuning to support a high substrate uptake rate. We analyzed the transporters in Table 1 to understand, which reactions in the transport cycle do and do not require precise adjustment. Accordingly, we examined for each individual transporter, the values for a collection of descriptors of transporter function: (i) These included descriptors, which can be computed with the kinetic model but also obtained experimentally (i.e., V max and the K M for substrate/co-substrate) and (ii) descriptors, which can only be extracted from the kinetic model (i.e., K D s for substrate and co-substrate to the outward-and inward-facing conformation of the transporter). The rationale was as follows: If the values of a descriptor are all similar for transporters optimized for the same substrate concentration, we can conclude that fine-tuning of the reactions, which affect this descriptor, is essential for the realization of the optimized rate. Conversely, this is not the case, if the values are vastly different.
In Figure 1D, we plotted the substrate uptake rate as a function of S out for the transporters in Table 1: The V max values were low and high for the group of transporters, which were optimized for .1 μM and 100 μM S out , respectively. Within the two groups of transporters, the V max values varied, but the magnitude of this variation was small: V max of transporter optimized for .1 μM S out and 100 μM S out , were 78.40 ± 6.61 s −1 and 478.38 ± 1.58 s −1 , respectively. In Figure 1E, the data were normalized to maximum velocity, because differences in the apparent affinity of the optimized transporters for the substrate can be more readily appreciated in this representation. It is evident that the transporters, which were optimized for .1 μM S out displayed a higher apparent affinity for the substrate than those optimized for 100 μM S out . The K M values within groups fell into narrow ranges, that is, .95 ± .08 μM and 11.1 ± .31 μM for transporters optimized for .1 μM and 100 μM S out , respectively. Our analysis therefore indicates that transporters, must have low K M and low V max values to support a high substrate uptake rate when S out is low, but a high K M and a high V max value when S out is high.
We computed the K D values for substrate binding to the outward-facing conformation of the individual transporters ( Figure 1F): It is evident that the K D values for the substrate were low and high for transporters, which were optimized for .1 μM and 100 μM S out , respectively (.1 μM S out : .24 μM ± .03 μM; 100 μM S out : 4.72 μM ± .42 μM). It also evident that the K D values were smaller and that they covered a larger range than the corresponding K M values ( Figure 1F). Inspection of Table 1 shows that the rate constants, which govern Na + binding to the outward-facing state, are subject to large variations in transporters 1-10; in contrast, these rate constants differed substantially less in transporters 11-20, which were optimized to cope with 100 μM S out . We illustrated the resulting difference in apparent affinity for Na + by plotting the absolute ( Figure 1G) and normalized substrate uptake rate ( Figure 1H) of the optimized transporters as function of the Na + concentration: for those transporters, which were optimized for .1 μM S out the apparent affinity for Na + varied over five orders of magnitude (blue curves in Figures 1G,H). In contrast, for transporters optimized for 100 μM S out , the apparent affinity for Na + fell into a narrow range (magenta curve in Figures 1G,H). Thus, if the extracellular substrate concentration is low, the Na + binding reaction does not need to be fine-tuned to obtain optimal rates. However, at a higher substrate concentration, the Na + binding reaction is subject to stringent constraints. This observation can be rationalized by taking into account that, in the optimization, Na + was assumed to be present at a high concentration (i.e., 150 mM). At this concentration Na + binding is unlikely to become rate limiting for substrate transport if S out is low. At a higher substrate concentration, the apparent association rate of the substrate (k app ) is expected to increase. In this scenario, Na + binding becomes rate limiting, if it occurs at too low a rate.
We compared the K D values for Na + binding to the outwardfacing conformation of the transporter t the corresponding K M values ( Figure 1I): Within each group, the variation in the K D values was small, that is, K D = 25.10 ± 3.91 mM and 79.21 ± 3.30 mM for transporters optimized for S out .1 μM and S out 100 μM, respectively. This contrasted with the large variation in the corresponding K M values for Na + seen for transporters optimized for .1 μM S out . This discrepancy can be explained as follows: The K D values are determined by the ratio of the dissociation and association rates for Na + but not by the absolute values of these rates. Conversely, given that the K D values and the K M values for Na + were found to differ, it is safe to conclude that K M values are highly dependent on the absolute values of these rates. These observations therefore imply that the affinity for Na + -rather than the velocity of Na + binding to the transporter is subject to precise adjustment for supporting optimal uptake rates at low substrate concentrations.
Figures 1J,K summarize the K D values for binding of substrate and of Na + to the inward-facing conformation of the transporter, respectively. The coefficients of variation in K D values of the inward-facing state were larger by a factor of 2 to 4 than the corresponding K D values of the outward-facing state: K D S in was 7.46 μM ± 3.4 μM and 701 μM ± 155 μM for transporters optimized for .1 μM S out and for 100 μM S out , respectively. Likewise, the K Na D in + was 8.81 mM ± 3.63 mM and 6.46 mM ± .70 mM for transporters optimized for .1 μM and 100 μM S out , respectively. The high K D S in at 100 μM S out was dictated by the low substrate affinity for the outward-facing state of transporters, which had been optimized for this condition, and the requirement to maintain microscopic reversibility. However, the larger variation in K D S in and K Na D in + indicates that the reactions, which define the substrate and co-substrate affinities for the inward-facing conformation, do not require as stringent an adjustment as those, which define the corresponding affinities to the outwardfacing conformation. Finally, we surveyed transporter optimization over a large range of extracellular substrate concentration (1 nM to 10 mM; Figure 1L): The resulting optimized substrate uptake rate of the transporters increased as a function of S out but leveled off at an uptake rate of about 500 s −1 . This upper limit reflect the constraint imposed by the boundary conditions of the optimization (i.e., a diffusion-limited k on and an upper limit of 1,000 s −1 for the rate of conformational transitions, see above). It was 249.4, 499.4, and 997.8 s −1 when the upper limit of the rate of conformational transitions was set to 500, 1,000, and 2,000 s −1 , respectively.

Testing Different Modes of Transport
The reaction scheme in Figure 1A describes a hypothetical symporter, which binds co-substrate and substrate in a sequential order: The two Na + ions are the first to bind when the transporter adopts the outward-facing conformation and the first to dissociate upon conversion of the transporter to the inward-facing conformation. In the subsequent description, we will refer to this reaction scheme as the NaSNaS scheme. This notation lists from left to right the order of the binding/unbinding events starting at the outward-facing apo-state (To) in clockwise direction. In Figures 2A-C, we show the reaction schemes for the three (sequential) alternatives. According to our notation, we refer to these as NaSSNa, SNaNaS and SNaSNa schemes.
In addition, we also examined the reaction scheme of a transporter, in which the two Na + ions and the substrate are allowed to bind in random order ( Figure 2D). We explored the impact of these five reaction schemes (Figures 1A, 2A-D) on the optimized substrate uptake rates by raising S out from .1 ( Figure 2E) to 30 ( Figure 2F) and 100 μM ( Figure 2G) and by conducting 10 optimization runs for each reaction scheme. As evident from Figure 2E, the substrate uptake rate was roughly the same for all schemes, when S out was low (i.e., .1 μM). However, when optimized for a higher S out , the schemes differed in the magnitude of the optimized substrate uptake rates, which they were able to support. The rank order was as follows: NaSSNa < SNaSNa < SNaSNa < SNaNaS = random. The rank order was the same for transporters optimized for 30 μM and 100 μM S out , (cf. Figures 2F,G). Thus, random order of substrate and co-substrate binding and SNaNaS are best suited to support a large substrate uptake rate.

Raising the Intracellular Substrate Concentration
Due to the way SLC operate, the intracellular concentration of the substrate and the substrate uptake rate are inversely correlated. This can be explained as follows: The substrate must be released into the cytosol to complete a full cycle. As the intracellular substrate concentration (S in ) increases progressively during uptake, rebinding of the substrate to the inward-facing conformation occurs at a more frequent rate. This hampers progression through the transport cycle and thus reduces the substrate uptake rate.
Here we propose that-similar to the extracellular substrate concentration (S out )-S in can also exert an evolutionary pressure on the operation of a solute carrier. Depending on the physiological context, transporters may encounter intracellular concentrations of their cognate substrate, which range from low to high levels. . Na + is the first to bind when the transporter adopts the outward-facing conformation and the last to dissociate from the inward-facing conformation. (B) SNaSNa scheme (sequential). Substrate is the first to bind to the outward-facing conformation and the first to dissociate from the inward-facing conformation. (C) SNaNaS scheme (sequential). Substrate is the first to bind to the outward-facing conformation and the last to dissociate from the inward-facing conformation (D). Random binding order scheme. Na + and substrate bind in random order. (E) Plotted is the optimized substrate uptake rate for all schemes of transporter optimized for .1 μM S out . At this low substrate concentration the optimized rate is approximately the same for all schemes. (F) Shown is the optimized substrate uptake rate for the different schemes optimized for 30 μM S out . The height of the optimized rate differed between schemes (G) the same as in (F) but of transporter optimized for 100 μM S out . The rank order of the optimized rates among schemes was the same as in (F).  (Yaffe et al., 2018). Accordingly, under physiological conditions, monoamine transporters for dopamine (DAT/SLC6A3), norepinephrine (NET/SLC6A2) and (SERT/ SLC6A4) are unlikely to encounter high intracellular concentrations of their substrate. Conversely, the creatine transporter-1 (SLC6A8) must maintain substrate influx in the presence of millimolar intracellular creatine (Snow and Murphy, 2001). It is clear that a solute carrier, which must support a high substrate uptake rate against high S in , must adjust its operation differently than a transporter, which does not need to overcome the hurdle imposed by frequent rebinding of the substrate to the inwardfacing conformation.

Frontiers in
We first optimized the substrate uptake rate of a transporter operating according to the NaSNaS scheme (illustrated in Figure 1A) by performing 20 optimization runs each, where S out was 30 μM and S in was set at 0, .1, 1, and 5 mM. It is evident from Figure 3A that the optimized substrate uptake rate decreased by raising S in . This is in line with the inverse correlation of S in and the substrate uptake rate discussed above. Figure 3B illustrates the range of K D values for substrate binding to the inward-facing conformation of the transporters, which had been optimized to cope with different concentrations of S in : K D values of substrate for the inward-facing conformation increased as S in was raised. This was to be expected, because lowering the intracellular affinity for substrate reduces the extent by which S in can rebind, and it thus allows for a higher substrate uptake rate in the presence of S in .
We then compared the uptake rate of transporters optimized for 30 μM S out and 0 mM or 1 mM S in over a large range of extracellular substrate concentration. As can be seen from Figure 3C, it was inevitable that transporters optimized in the presence of 1 mM S in had negative uptake rates at low extracellular substrate concentrations, that is, the transporters cycled in the backward rather than the forward mode and hence mediated substrate efflux from the cell. It is also clear that the presence of 1 mM S in reduced the maximum achievable uptake rate V max in the forward transport mode and shifted the K M . Because K M and K D differ (cf. Figure 1F), we examined the range of K D values for substrate binding to the outward-facing conformation of transporters optimized for 0 and 1 mM S in ; these are illustrated together with the corresponding K M values in Figure 3D: Both the K D values and the K M values for substrate increased, if the transporter had to cope with a high intracellular substrate concentration. The variation in these parameters was low (coefficient of variations = .070 and .025 for K D and K M , respectively, of transporters optimized in the presence of 30 μM S out and 1 mM S in ). We conclude that the decrease in the apparent (K M ) and the true affinity (K D ) for the substrate, is required to allow for rapid cycling of the transporters in the presence of high S in .
Finally, we examined how the selective pressure exerted by high intracellular substrate affected the affinity of the transporters to the co-substrate ion. As can be seen from Figure 3E, many The curves show the substrate uptake rate as a function of S out of transporters optimized for 0 mM (green) and 1 mM S in (magenta). In the presence of S in , the substrate uptake rate assumed negative values when S out was low. In this range of S out , the transporters cycled in the reverse direction. In the presence of 1 mM S in, V max was reduced.
Frontiers in Physiology | www.frontiersin.org 9 February 2022 | Volume 13 | Article 817886 different solutions emerged: On average, transporters optimized to cope with 1 mM S in (magenta lines in Figure 3E) required higher extracellular Na + concentrations to support substrate uptake than those optimized in the absence of intracellular substrate (green lines in Figure 3E). However, in both groups, the optimized transporters displayed highly variable responses to Na out + . This is reflected in the large range of the K M values for Na + (Figure 3F). In contrast, the K D of Na + for binding the outward-facing conformation did not vary to any substantial extent. This observation is consistent with our conclusion from Figure 1I, namely, that the constraint is imposed by the true affinity for Na + rather than by its association rate (see above).
We also computed K D s for Na + binding to the inward-facing conformation of the transporters optimized for 0 mM and 1 mM S in : There was an overlap in the range of K D values for Na in + ( Figure 3G). In addition, their variation was larger than that of K D values for Na iout + (cf. Figures 3F,G). Hence, we conclude that Na + binding to the inward-facing conformation need not be stringently adjusted to allow for high substrate uptake rates.

Binding Order Affects the Magnitude of the Optimized Substrate Uptake Rate Also in the Presence of S in
We next addressed the question, if the binding order of substrate and co-substrate determined the cycle rate of a transporter challenged with high concentrations of intracellular substrate. Optimization runs were carried out with S out = 30 μM and S in = 1 mM for all five schemes (cf. Figures 1A, 2A). It is evident from Figure 4A that these schemes differed in the magnitude of the optimized substrate uptake rates, which they were able to support. The rank order was NaSSNa < NaSNaS < SNaSNa < SNaNaS = random. This rank order differed from that observed in the absence of S in (cf. Figure 2F). At constant S out = 30 μM, we also varied the internal substrate concentration by lowering to S in 100 μM and raising it to the point, where net uptake rat was zero (Figure 4B). For all schemes we found an optimized substrate uptake rate of zero when S in was 6.75 mM. This was to be expected because at the chosen concentrations of Na + (i.e., 150 mM Na out + and 10 mM Na in + ) the concentrative power (S in /S out ) of the transporter is 225. This numerical value is identical for all schemes, because they are governed by the same transport stoichiometry. Accordingly, the transporters cannot further cycle productively in a forward direction, when S in becomes 225 times larger than S out (30 μM * 225 = 6.75 mM). For all schemes, we extracted the K D of substrate binding to the inward-facing conformation of the transporters optimized at varying S in ( Figure 4C). In all instances, this K D increased with increasing S in . However, the magnitude of the drop in affinity depended on the reaction scheme. For several secondary active transporters, binding of substrate and co-substrate was shown to occur in a cooperative manner: The apparent substrate affinity for the transporter depended on the concentration of the co-substrate (Meinild and Forster, 2012;Perez et al., 2014;Hasenhuetl et al., 2018;Erdem et al., 2019). It was low and high when the concentration of the co-substrate was low and high, respectively. Thus, the substrate can bind with higher affinity to transporters, when they are bound to the co-substrate (e.g., Na + ). In this way, the concentration of the co-substrate determines the abundance of high and low affinity states for the substrate. Under physiological conditions, the co-substrate concentration is lower on the intracellular than on the extracellular side. Accordingly, cooperative binding is predicted to promote the forward cycling mode by reducing the substrate affinity to the inward-facing conformation. In fact, the drop in intracellular affinity resulting from cooperative binding is a requirement for maintaining a large substrate uptake rate at high S in (Erdem et al., 2019). Notably, cooperative binding is contingent on a random binding order for substrate and co-substrate. For this reason cooperative binding can only be assessed in the random binding order scheme. Based on this consideration, a rise in S in is predicted to increase the extent of cooperativity. We verified this prediction in optimization runs and extracted the cooperativity for transporters optimized at 30 μM S out and 0 mM S in or 1 mM S in by calculating the ratio K D S in in the absence of bound Na + / K D Si n in the presence of bound Na + . It is evident from Figure 4D that there is a large range of optimized solutions, but on average cooperativity was more pronounced at higher S in .

Substrate Selectivity Increases the Substrate Uptake Rate
Evolution also optimized SLCs for substrate specificity. Selective transporters presumably arose from unselective ancestors. As a starting point, we posited that an unselective SLC must display low affinity for the various substrates: It is difficult to envisage a substrate binding site, which can provide strong bonding interactions to accommodate many distinct molecular scaffolds. We also assumed that, in the evolutionary trajectory from an unselective to a specific transporter, an increase in substrate specificity ought to translate in higher uptake rates for the substrate. Accordingly, in the optimization, we modeled an unselective solute carrier as a transporter, which had a low (true) affinity (i.e., a high K D ) for substrate by implementing a constraint, which prevented the substrate K D from dropping below a userdefined arbitrary value (e.g., 10 mM). With this constraint in place, the optimization algorithm only returned sets of values for the microscopic rate constants, which defined transporters with a high K D for substrate. In the subsequent description, we refer to such sets as unselective transporters. In contrast, sets generated in optimization runs, in which the K D for substrate was not constrained, are referred to as selective transporters. For the optimizations summarized in Figure 5, we employed the random binding order scheme, we assumed zero-trans conditions and the presence of 1 μM S out . Figure 5A shows the FIGURE 3 | transporters optimized for 0 mM S in and 1 mM S in . The dependence on the Na + concentration was highly variable between transporters optimized for the same S in . (F) Plotted are the K D values for Na + binding to the outward-facing conformation of the transporter and the corresponding K M values of transporters optimized for 0 mM S in and for 1 mM S in . The variation in the K M values was larger than the variation in the K D values. The coefficients of variation for the K D s were .041 and .033 and for the K M s 2.03 and .97 at .1 μM and 100 μM S out , respectively. At high S in both the K M and the K D values for Na + increased. (G) KDs for Na + for the inward facing conformation of the transporters optimized for 0 mM and 1 mM S in , respectively.
Frontiers in Physiology | www.frontiersin.org result for 20 selective and unselective transporters (K D ≥ 10 mM): It is evident that the optimized substrate uptake rat of the unselective transporters (magenta symbols, Figure 5A) was lower by about three orders of magnitude than that of the selective SLCs (green symbols, Figure 5A). In Figure 5B, we examined the Michaelis-Menten kinetics of the substrate uptake rate of these optimized transporters. Figure 5C shows the same data normalized to V max to illustrate the distribution of K M . The V max values of the unselective transporters were lower than those of the selective SLCs but they varied over about orders of magnitude ( Figure 5D). Similarly, the K M values of the unselective transporters, which were consistently higher than those of the selective SLCs, were again highly variable. In Figure 5E, we show optimized rates as a function of the concentration of substrate, for which the rates were optimized. Displayed in this plot are the data for selective transporters (unconstrained substrate K D ) and unselective A B C D FIGURE 4 | Binding order of co-substrate and substrate affects the optimized substrate uptake rate, when S in is high. (A) Shown are the substrate uptake rates for transporters, which operate according to the reaction schemes illustrated in Figures 1A, 2A-D and which were optimized for 30 μM S out and 1 mM S in . The optimized substrate uptake rate differed between reaction schemes. The data are the rates obtained from 20 optimization runs. (B) Plotted are the substrate uptake rates for all reaction schemes optimized for 0 mM, .1 mM, 1 mM, 5 mM, and 6.75 mM S in , respectively. The data are means ± SD of the substrate uptake rates obtained from 20 optimization runs. At 6.75 mM S in , the substrate uptake rate was zero for all reaction schemes. This is due to the fact that the concentrative power is defined by the transport stoichiometry (Na + : substrate = 2:1) and the Na + gradient rather than the reaction scheme: With a gradient of 150 mM external to 10 mM internal Na + , the concentrative power of the transporters is 225, which yields 6.75 mM S i at 30 μM S out (C) Shown are K D values for the substrate binding to the inward-facing conformation for all reaction schemes of transporters optimized to 0 mM, .1 mM, 1 mM, and 5 mM S in , respectively. While the absolute values of this K D differed between schemes, they all rose with increasing S in . (D) The random binding order scheme, allows for Na + and substrate to bind in a cooperative manner: The affinity for substrate is high when Na + is bound and low in its absence. Cooperativity was defined as the ratio of the K D of substrate binding to the inward-facing conformation in the absence and presence of Na + . Transporters optimized for 1 mM S in displayed larger cooperativity values than those optimized for 0 mM S in (p < .0001; Wilcoxon signed rank test).
transporters (constrained at K D > .1 mM, 1 mM, and 10 mM). It is evident that at the various [S out ] tested the selective transporters had larger substrate uptake rates than the unselective ones. These results confirm that SLCs can raise their transport capacity by becoming more specific for their cognate substrates. We, therefore, consider it plausible that specific SLCs arose from ancestors, which were unselective and that this transformation was driven by the need to support high substrate uptake rates. Conversely, there are transporters, which are under evolutionary pressure to remain unselective, because they support the disposition of xenobiotics. This is exemplified by members of the SLC22 family, which recognize diverse substrates to mediate disposition of drugs and xenobiotics: Both organic cation (OCT1-3/SLC22A1-3) and anion transporters (OAT1-3/ SLC22A6-8) translocate most of their substrates with K M values in the high μM range (VanWert et al., 2010;Motohashi and Inui, 2013). This is despite the fact that, in most instances, they are faced with substrate concentrations in the low micromolar range. However, we find this in good agreement with our results, which showed that the unselective transporters that we optimized for 1 μM S out , displayed K Ms in the submillimolar range (see Figure 5E). It is worth noting that the SLC22 family also encompasses members, which have a narrow substrate specificity; these have K M value in the low micromolar range.

DISCUSSION
Solute carriers have a long evolutionary history: More than 50% of SLC subfamilies, which are present in the human genome, are also found in prokaryotes (Hoglund et al., 2011). Eukaryotic transporters have longer N-and C-termini than bacterial transporters. This presumably reflects evolutionary adaptation to the increase in complexity: The N-and C-termini harbor site for posttranslational modifications (e.g., phosphorylation by protein kinases) and docking sites for the protein machinery required for trafficking between cellular compartments (Chiba et al., 2014). The evolutionary history also suggests that individual SLC subfamilies expanded and contracted during phylogenesis (Caveney et al., 2006; The circles in pink, light blue, and dark blue are the optimized rates of unselective transporters, for which the substrate K D was not allowed to drop below .1 mM, 1 mM, and 10 mM, respectively (n = 10 for each S out ). The optimized rates of the unselective transporters were lower than that of the selective ones.

A B
FIGURE 6 | Relation between the K M and the substrate turnover rate of transporters in the SLC6 family. (A) Plotted are data obtained from literature of eight members of the SLC6 family. The line in the graph is a linear fit to the data points (r 2 = .915). It is evident that in the real world a positive correlation exists between the K M and the turnover rate. (B) Relation between K M and V max for the various substrates of the Lumbricus terrestris monoamine transporter. The data were taken from Caveney et al. (2006). The line in the graph is a linear fit to the data points (r 2 = .981). Denecke et al., 2020). Expansion was not only driven by the adaptation to new substrates but also by the requirements to optimize concentrative power and uptake rate: Phosphate transporters of the SLC34 subfamily differ in their concentrative power and in their electrogenicity (Forster et al., 2013). Similarly, the three closely related monoamine transporters provide different solution to the trade-off between harvesting the membrane potential and maintaining constant uptake at variable voltage . Here, we explored how the concentration of substrate, which a solute carrier encounters on both, the extra-and intracellular side can exert evolutionary pressure on the operating mode of a transporter. Our approach relied on analytical expressions for descriptors of transporter function (i.e., K M and V max of substrate transport) as a function of the microscopic rate constants, which parameterize the kinetic models of SLC. Arguably, the outcome of evolutionary adaptation must maximize substrate uptake rate at the prevailing conditions. Accordingly, our optimization algorithm searched for the microscopic rate constants, which yielded the largest possible value for the substrate uptake rate. The pertinent insights can be summarized as follows: (i) low extracellular substrate concentrations select for transporters, which have low K M and V max . Only this combination allows for a high rate in the transport cycle, but there is a surprisingly broad range of microscopic rate constants, which support this solution. (ii) In contrast, a transporter operating at high extracellular substrate concentrations has a substantially more restricted parameter space and maintains a high uptake rate only if it has a high K M and a high V max for substrate. (iii) Random order of substrate and co-substrate binding is superior to all possible sequential orders, if a transporter is to maintain a high rate of substrate uptake in the presence of accumulating intracellular substrate, because it allows for cooperative binding.
Solute carriers have long been known to fall into two categories, that is, high-affinity-low-capacity transporters and low affinity-high capacity transporters. It is important to note, however, that there is not any relation between the K M value and the V max value, which a priori dictates that these two parameters must move in the same direction. We examined the relation between turnover rates and K M in the SLC6 family, because turnover rates have been determined with high precision by electrophysiological recordings (Bicho and Grewer, 2005;Erdem et al., 2019;Bhat et al., 2021;Shi et al., 2021) and the individual steps of the transport cycle have been analyzed in detail. In addition, the K M values for cognate substrate span more than two orders of magnitude. It is evident from Figure 6A that there is a good correlation (r 2 = .915) between turnover rate and K M . Similarly, the monoamine transporter of the earthworm Lumbricus terrestris can translocate several substrates albeit with substantial differences in V max and K M : The K M for norepinephrine is 20-fold higher than for tyramine (Caveney et al., 2006). Again, there is a remarkable correlation (r 2 = .981) between uptake velocity and K M (Figure 6B). It is, therefore, safe to conclude that the existing dichotomy-in the real world and in our data sets-is a consequence of the optimization of the substrate uptake rates.
Similarly, our optimization algorithm required boundary conditions to identify realistic maxima. Table 2 provides a compilation of substrate turnover rates reported for a collection of SLC. The list includes carriers, which cycle at a rate of 3 s −1 as well as such that cycle at a rate of about 700 s −1 . These rates are reasonably close to those, which the optimization algorithm returned, that is, about 500 s −1 and 4 s −1 for S out 100 μM and .1 μM, respectively. This confirms that the constraints, which we imposed in the optimization, were realistic. Importantly, our analysis establishes a relation between the turnover rate of the transporter and the substrate concentration for which it was optimized. Thus, the substrate turnover rate allows for inferring the concentration range, in which a candidate solute carrier operates under physiological conditions.
We subjected transporters to a selection by optimization at low substrate concentration but at high concentrations of Na + co-substrate Nevertheless, some of the solutions produced hypothetical transporters, which were optimized to operate at very low Na + concentrations (cf. Figures 1G,H). While these are obviously of little benefit to a multicellular organism with homeostatic control of extracellular ion composition, these transporters are optimally adapted to support nutrient uptake of a unicellular organism invading ecological niches with low ambient salt concentrations. Bacterial transport is thought to rely mainly on the proton motive force. However, there are several examples of bacterial Na + -dependent symporters (Wilson and Ding, 2001). In addition, our analysis showed that K M values for substrate and co-substrate differed substantially from the K D values for their binding to the outward-facing conformation (cf. Figures 1F,I). This highlights the fact that experimentally determined K M values for substrate and co-substrate are not necessarily adequate measures of true affinity (i.e., k off /k on ). Transporters are forced to adjust ratios of binding and unbinding reactions, which keep the optimum K D values in a narrow range at both, low and high substrate concentrations. In contrast, the initiation of the transport cycle is limited by the apparent on rates of substrate and the co-substrate ion(s). It is worth noting that all solutions resulted in transporters, which bound substrate at a rate close to the diffusion-imposed limit (cf. Table 1). As a consequence, this allows for the emergence of high-affinity/ low-capacity SLC, which translocate substrate effectively at very low co-substrate concentrations, because large variations in k on for Na + are tolerated.
Substrate accumulation on the intracellular side also exerts a selective pressure. It is low, if the transporter operates in a relay with another transporter, which sequesters the substrate, or with an enzyme, which modifies the substrate: The monoamine transporters for dopamine (DAT/SLC6A3), norepinephrine (NET/ SLC6A2), and (SERT/SLC6A4) need not cope with rising intracellular level of their cognate substrate because of the action of vesicular monoamine transporters (vMAT1/SLC18A1 & vMAT2/SLC18A2) which shuffle cytosolic monoamines into vesicles (Hou and Matherly, 2014;Sitte and Freissmuth, 2015). Similarly, the reduced folate carrier (RFC/SLC19A1) and the proton-coupled folate transporter (PCFT/SLC46A4) are unlikely to face inhibition by accumulation of intracellular folate, because it is converted to polyglutamylated folate by folylpolyglutamate synthase (Raz et al., 2016). In fact, it has been argued that, in several instances, transporters and metabolizing enzymes are spatially organized to promote sustained influx of substrate: Direct or indirect tethering of enzymes to SLCs creates membrane transport metabolons, which effectively lower [S in ] to preclude inhibition on the intracellular side (Moraes and Reithmeier, 2012). In contrast, the creatine transporter-1 (CrT-1/SLC6A8) must maintain the forward transporter mode, although intracellular concentrations are in the range of 5-7 mM (Wyss and Kaddurah-Daouk, 2000). Our analysis allows for understanding how the adaptation to high intracellular substrate is achieved. It is obvious that the transporter can only progress in the forward cycle mode, if the inward-facing conformation of the transporter has a low affinity for the substrate, because a low affinity precludes rebinding of the substrate. However, to afford a lower substrate affinity to the inward-facing conformation, the K M for substrate must increase and, as a corollary, the true affinities of the substrate and the Na + ions for the outward-facing must decrease. In this context, it is worth mentioning that the three monoamine transporter (NET/SLC6A2, DAT/SLC6A3, and SERT/SLC6A4) display high K M for substrate and Na + , that is, in the low micromolar and millimolar range, respectively, (Bulling et al., 2012;Li et al., 2017;Bhat et al., 2021). In contrast, the glycine transporter-1 (GlyT1/SLC6A9) and CrT1/SLC6A8 feature about 3-20-fold lower apparent affinities for their cognate substrate and co-substrate Na + (Boehm et al., 2003;Erdem et al., 2019). This is consistent with the fact that these transporters support influx of their cognate substrates in the presence of millimolar concentrations of S in , while the monoamine transporters do not. Thus, the solutions, which were explored by our optimization algorithm, reflect parameter space visited during the evolutionary adaptation of transporters.
Our study has several limitations: (i) We restricted our analysis to sodium symporters with a 2 Na + and 1 substrate stoichiometry. It is evident, however, that the approach can be extended to any other stoichiometry of symport. Our approach is also applicable to antiporters although incorporating the k on and k off of the counter-transported solute substantially increases the parameter space and thus the computational effort. (ii) We did not examine the impact of allosteric regulation, which is a major computational challenge. It is clear, however, that evolution also selected for transporters for allostery: DAT/SLC6A3, for instance, harbors an allosteric Zn 2+ binding site, while its next relatives NET/SLC6A2 and SERT/SLC6A4 do not (Li et al., 2017). (iii) While modeling the evolutionary trajectory to substrate specificity, we assumed that an ancestral unselective transporter gave rise to a specific transporter. This need not be the case. In fact, two rounds of genome duplication events are thought to have occurred after the split of vertebrates from cephalochordates (Wolfe, 2001;Simakov et al., 2020). Duplicated transporters have two possible evolutionary fates: They can be subject to inactivating mutations and lost. Alternatively, they can accumulate mutations, which initially relax their substrate specificity and eventually evolve by further mutations into transporters with novel substrate specificity. We also did not explore the structural basis for evolution in substrate affinity. It is worth noting that solute carries can achieve the substrate translocation step by three different mechanisms, that is, based on a rocker switch, a rocking bundle and a sliding elevator (Diallinas, 2021). In transporters, which operate via a sliding elevator mechanism, the substrate binding site must reside in the elevator, which moves along the scaffold domain in a direction perpendicular to the membrane plane and thus translocates the substrate. Surprisingly, in the uric acid-xanthine permease UapA of Aspergillus nidulans, which operates via a sliding elevator mechanism, selectivity can be relaxed by several mutations, which are in the scaffold domain and thus outside of the binding site proper (Diallinas, 2021). Two non-mutually exclusive interpretations have been provided for the effect of the mutations: They may eliminate a selectivity filter, which restricts access to the binding site (Vlanti et al., 2006), and/or they may relax a brake and thus facilitate triggering of the elevator movement (Diallinas, 2021). In transporters, which operate by a rocker switch or a rocking bundle mechanism, the binding site is more deeply buried in the hydrophobic core of the protein than in those operating as sliding elevators. Accordingly, it is readily conceivable that restrictions can be imposed on access to the binding site. In fact, the selective binding to the closely related DAT/SLC6A3 and SERT/SLC6A4 is determined by the association rate constant rather than the dissociation rate constant (Hasenhuetl et al., 2015); access to the binding site is limited-at least in part-by the extracellular loops (Esendir et al., 2021). Exchanging the N-terminus of DAT/SLC6A3 with that of SERT/ SLC6A4 reduces the K M of the resulting chimaera for dopamine (Sweeney et al., 2017). Taken together, these observations suggest that changes in transporter selectivity and uptake rates can be brought about by mutations of many residues, which are not necessarily confined to the substrate binding site. During evolution the sequential impact of mutations is likely to change microscopic rate constants along trajectories as diverse as explored by our computational optimization. A long-term challenge is to delineate such an evolutionary path through the landscape of possible microscopic rate constants by sequential mutagenesis. This should be gratifying, because it is expected shed light on the evolutionary history underlying transporter diversity.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.