Low affinity uniporter carrier proteins can increase net substrate uptake rate by reducing efflux

Many organisms have several similar transporters with different affinities for the same substrate. Typically, high-affinity transporters are expressed when substrate is scarce and low-affinity ones when it is abundant. The benefit of using low instead of high-affinity transporters remains unclear, especially when additional nutrient sensors are present. Here, we investigate two hypotheses. It was previously hypothesized that there is a trade-off between the affinity and the catalytic efficiency of transporters, and we find some but no definitive support for it. Additionally, we propose that for uptake by facilitated diffusion, at saturating substrate concentrations, lowering the affinity enhances the net uptake rate by reducing substrate efflux. As a consequence, there exists an optimal, external-substrate-concentration dependent transporter affinity. A computational model of Saccharomyces cerevisiae glycolysis shows that using the low affinity HXT3 transporter instead of the high affinity HXT6 enhances the steady-state flux by 36%. We tried to test this hypothesis with yeast strains expressing a single glucose transporter modified to have either a high or a low affinity. However, due to the intimate link between glucose perception and metabolism, direct experimental proof for this hypothesis remained inconclusive. Still, our theoretical results provide a novel reason for the presence of low-affinity transport systems.


Derivation of the rate equation of symmetric transport in terms of first order rate constants
The transporter can be in any of four states, the binding-site facing outward, with and without substrate bound, es e and e e , respectively, and inward facing with and without substrate bound, es i and e i . Assuming that binding and unbinding is much faster than the movement of the binding site over the membrane, we can use the quasi-steady state approximation for the fraction of carriers that have substrate bound to them, both inside and outside, where K D is the substrate-transporter dissociation constant and e x,tot = es x + e x is the total number of transporters with their binding site facing the x site of the membrane (i.e. x = e or x = i). By definition, in steady state e x,tot is constant. This gives rise to the equality k 2 es e + k 4 e e = k 2 es i + k 4 e i . Defining and solving the steady state condition gives an expression for the total amount of outward and inward facing carriers, normalized to the total amount of transporters: The net uptake rate is than given by: Filling in the σ s, the term within the square brackets reduces to and the denominator to Hence, the rate equation in terms of the first order rate constants is given by Defining the macroscopic kinetic parameters In order to find the optimal K M , K opt M , we simply take the derivative of v with respect to K M and set that to zero. Since we have dv dK M = −e tot k cat K opt M is found by solving The physical (i.e. positive) solution to this quadric equation is In comparison, the reversible Michaelis-Menten rate equation does not have an optimal affinity. Reducing the K M will always increase the rate.
where the macroscopic kinetic parameters, k cat , K M,e , K M,i and α, can be expressed in terms of the first order rate constants.
Furthermore, the first order rate equations are related through the equilibrium constant. Since we are considering facilitated diffusion, no free energy dissipation is coupled to the transport process,i.e. K eq = 1, and we have: This poses a constraint on the first order rate constants. Practically, this means that a mutation that affects e.g. the strength of extracellular substrate to carrier binding must also affect some of the other steps in the transport cycle (e.g. intracellular substrate release). Combined with the complicated dependency of the macroscopic parameters on the first order rate constants, an analytical approach is unfeasible. We therefore employed a parameter sampling approach to gauge to what extent our conclusions about the rate-affinity trade-off and the substrate efflux hypothesis are valid for this more general rate equation.

The rate-affinity trade-off
As in the main text, the parameter sampling approach gives a mixed picture about the theoretical underpinnings of the rateaffinity trade-off. Figure S1 shows scatterplots of k cat versus affinity (defined as 1/K M,e ) for randomly sampled sets of first order rate constants, with a number of different constraints assumed for some of these. In the absence of any constraints, there is a clear negative correlation between the k cat and the affinity (figure S1A). However, this is not a true Pareto-front, as it appears as though there is always a possibility that the k cat is enhanced without reducing the affinity (or vice verse). The fact that not the whole k cat − 1/K M space is filled is due to the finite numbers of samples rather than due to a true constraint. On the other hand, if we assume that there is a (biophysical) limitation on the rate of substrate-transporter binding (k 1 f ), (e.g. the diffusion limit), we do find a true Pareto front (figure S1B), the location of which depends on the actual maximal k 1 f -value. However, this conclusion does not hold if other rate constants are assumed to have some biophysical limit, as shown by the examples of restricted k 2 f (figure S1C) or a restricted substrate-transporter dissociation constant K D,e (≡ k 1r /k 1 f , figure S1D). All in all, there are reasonable theoretical arguments to be made for a rate-affinity trade-off, but the logic is not water-tight.

Enhanced uptake by reduced efflux
In the analysis in the main text above we made the biologically motivated, simplifying assumption that the transporter is symmetric. However, our reasoning does not critically dependent on this symmetry, since it is a general property of this scheme that the substrate and product bind to different states of the transporter. Moreover, since all first order rate are interdependent through constraint (S15), K M,i and K M,e are expected to be correlated. To test this, we randomly sampled all first order rate constants from a log-normal distribution and rescaled them such that K eq = 1 and k cat = 1 (for details, cf. Appendix ). These parameters were used to calculate the K M,e , K M,i and the net steady state uptake rate J under conditions of high an low external substrate (s e = 100 and s e = 1, respectively). The results are depicted in figure S2. Indeed, K M,e and K M,i are correlated, albeit not strongly (Spearman correlation = 0.64, Figure S2A). More importantly, however, there appears to be an s e -dependent optimal affinity (figure S2B). Furthermore, the set of parameters that has the highest J under low substrate conditions, performs relatively poorly under high substrate conditions (large, light red dot), and vice verse (blue dot).

Parameter sampling procedure
Sets of parameters were constructed by drawing the first order rate constant randomly from a log-normal distributions. This was done in a way such that constraint (S15) is satisfied. These parameter sets were used to calculate the steady state uptake rate (given by equation (S13)) and macroscopic kinetic parameters (given by equation (S14)). The parameter sampling was performed in Wolfram Mathematica 9.0 using the functions RandomVariate and LogNormalDistribution, which has the probability density function (PDF):

Rate affinity trade-off
To generate the data depicted in figure S1, for each subfigure 10 000 parameter sets were constructed. Each parameter set was constructed in the following way: • Two sets of four numbers, X ≡ {x 1 , x 2 , x 3 , x 4 } and Y ≡ {y 1 , y 2 , y 3 , y 4 } were randomly drawn from a log-normal distribution given by the PDF (S16) with µ = 0 and σ = 2.
• For figure S1A, there are no restrictions on individual rate constants. The set of forward rate constants, K f ≡ {k 1 f , k 2 f , k 3 f , k 4 f } is just given by the first set, K f = X . To get the reverse rate constants K r ≡ {k 1r , k 2r , k 3r , k 4r }, Y needs by be rescaled by a factor This is to ensure that K eq = 1. Hence, K r = a · Y .
• For figure S1B we also need to ensure that k 1 f is restricted to some constant value c. We set the first order rate forward constants to K f = c x 1 · X and K r = c x 1 · a · Y . Similarly, for figure S1C we need to restrict k 2 f to c. We use K f = c x 2 · X and K r = c x 2 · a · Y • For figure S1D, the K D,e is fixed to a constant value c. Here, we used for the forward rate constant simply K f = X . Since K D,e ≡ k 1r /k 1 f , we set k 1r = c · x 1 . Defining β = x 2 x 3 x 4 y 2 y 2 y 4 1 c 1/3 and setting {k 2r , k 3r , k 4r } = β · {y 2 , y 3 , y 4 } additionally ensures K eq = 1.

Enhanced uptake by non-symmetric low affinity transporters
Figures S2 A and B are constructed from the same 10000 parameter sets. Each set was constructed such that k cat = 1 and K eq = 1 as follows: • Two sets of four numbers, X ≡ {x 1 , x 2 , x 3 , x 4 } and Y ≡ {y 1 , y 2 , y 3 , y 4 } were randomly drawn from a log-normal distribution given by the by the PDF (S16) with µ = 0 and σ = 2.
• The set Y is obtained by rescaling Y by a factor • Both sets are rescaled by a factor 1/k cat , as defined in equation (S7a) with k i f → x i and k ir → y i ,i.e.
This rescaling is done to ensure that k cat = 1 for each parameter set. The first order rate constants are thus K f = 1 k cat X and K r = ã k cat Y .    Figure S3. Effect of substrate efflux is V max -dependent. Steady-state glycolytic flux relative to the transporters, J glycolysis /V max,GLT as a function of K M,GLT for different values of V max,GLT . A lower V max reduces the intracellular glucose concentration. As a consequence, the effect of substrate efflux is less pronounces (the steady-state flux is closer to the V max ) and the optimal transporter affinity is higher. Due to this effect, the reduced-efflux hypothesis can only be tested using transporters that are comparible both in k cat and V max . The same model as in Fiugure 2B was used for these simulations, with [Glucose] = 110mM. Figure S4. Repair of auxotrophies in the EBY.VW4000 strain. A PCR amplification of the LEU2, HIS3 and TRP1 genes with sequences overlapping either the adjacent cassettes or the integration site. B In vivo assembly of the marker cassettes and integration in the CAN1 locus of EBY.VW4000.