Kinetic Modelling of Transport Inhibition by Substrates in ABC Importers

Abstract Prokaryotic ATP-binding cassette (ABC) importers require a substrate-binding protein (SBP) for the capture and delivery of the cognate substrate to the transmembrane domain (TMD) of the transporter. Various biochemical compounds have been identified that bind to the SBP but are not transported. The mechanistic basis for the “non-cognate” substrates not being transported differs. Some non-cognate substrates fail to trigger the appropriate conformational change in the SBP, resulting in loss of affinity for the TMD or the inability to allosterically activate transport. In another mechanism, the SBP cannot release the bound non-cognate substrate. Here, we used rate equations to derive the steady-state transport rate of cognate substrates of an ABC importer and investigated how non-cognate substrates influence this rate. We found that under limiting non-cognate substrate concentrations, the transport rate remains unaltered for each of the mechanisms. In contrast, at saturating substrate and SBP concentrations, the effect of the non-cognate substrate depends heavily on the respective mechanism. For instance, the transport rate becomes zero when the non-cognate substrate cannot be released by the SBP. Yet it remains unaffected when substrate release is possible but the SBP cannot dock onto the TMDs. Our work shows how the different mechanisms of substrate inhibition impact the transport kinetics, which is relevant for understanding and manipulating solute fluxes and hence the propagation of cells in nutritionally complex milieus.


Introduction
ATP-binding cassette (ABC) transporters form a large family of membrane transport proteins [1][2][3] with a common domain architecture. The translocator unit is composed of two transmembrane domains (TMDs), which form the translocation pathway for the substrate, and two cytoplasmic nucleotide-binding domains (NBDs), which bind and hydrolyse ATP. ABC transporters use the energy of ATP hydrolysis to transport various compounds, such as sugars, amino acids, vitamins, compatible solutes, metal ions, antibiotics, lipids, polypeptides, and many others 1 . Some ABC transporters are specific for a single compound 4 , whereas others have broad substrate specificity and are able to transport multiple compounds 5,6 . The proposed transport mechanism of ABC transporters is based on the alternating access model 7,8 , in which the translocator switches between inward-and outward-facing conformations to expose a substrate binding cavity on alternate sides of the membrane.
ABC importers are typical of prokaryotic organisms but recently a number of proteins with the 'exporter fold' have been discovered that mediate cellular uptake, both in prokaryotic and eukaryotic cells 9,10 . Both Type I and II ABC importers require a soluble extracellular or periplasmic substrate-binding protein (SBP) for their function 11,12 . In Gram-positive bacteria and archaea, the SBPs are attached to the membrane via a lipid-or protein anchor, or are directly fused to the TMDs 13,14 . In Gram-negative bacteria, most SBPs are present as soluble protein in the periplasmic space, but some are lipid-anchored 15 or tethered to the TMD of the ABC transporter 16 , which is analogous to the surface association of SBPs in Gram-positive bacteria and archaea. The SBP binds the transported substrate, i.e., the cognate substrate, for delivery to the translocator unit. SBPs share a domain architecture, consisting of two rigid subdomains connected by a flexible hinge region 11 . Binding of a cognate substrate brings the two subdomains together, thereby switching the SBP from an open to a closed conformation [17][18][19] . Different closed conformations can be formed with different cognate substrates 20 . These closed conformations productively interact with the translocator and activate transport and the ATPase activity [21][22][23] .
Various biological compounds have been identified that bind to SBPs but are not transported by ABC importers 6,[23][24][25][26][27][28][29] . These non-transported molecules are termed herein non-cognate substrates, whereas the molecules taken up are named cognate substrates. To date, non-cognate cognate substrates can act as inhibitors of cognate substrate transport and thereby severely affect the transport rate in vitro and in vivo 20,25,28,29 . Note that this type of inhibition is distinct from trans-inhibition that exist in some ABC importers 31,32 . In the trans-inhibition mechanism the importer contains additional regulatory domains fused to the NBDs. Here, the activity of the transporter is regulated by the transported cognate substrate 31,32 . For instance, the transport of cognate L-and D-methionine by the E. coli MetINQ system is regulated by the internal pool of Lmethionine 33 .
Many non-cognate substrates induce a conformational change in the SBP that is distinct from that of cognate substrates 20,34,35 . For instance, the SBP MalE can bind maltotetraitol, βcyclodextrin and maltotetraose with high affinity, but only maltotriose is transported 25 .
Maltotriose induces closing of MalE, whereas maltotetraitol and β-cyclodextrin trigger the formation of different conformational states 20,34,35 . In other SBPs, the non-cognate substrate does not induce a conformational change within the SBP 20 . These observations provided possible explanations why the non-cognate substrate cannot be transported: the SBP-non-cognate substrate complex fails to interact with the translocator (model A; Figure 1) or it docks onto the translocator but fails to activate ATPase activity and transport (model B; Figure 1).
In the Mn 2+ importer PsaBCA of S. pneumoniae, a third mechanism of substrate inhibition is prevalent 20,30 . Here, both cognate Mn 2+ and non-cognate Zn 2+ induce highly similar closing of the SBP PsaA. However, the PsaA-Zn 2+ complex is so stable that PsaA cannot open and release the metal ion to the PsaBC unit (model C; Figure 1) 20,26,30,36 . Reversible and irreversible substrate binding has also been shown for YtgABC 37 . Thus, the rate of transport is not only influenced by the conformational state of the substrate-bound SBP but also by the substrate release kinetics.
In this paper, we used rate equations to analyse how non-cognate substrates influence the kinetics of transport. We considered three inhibition mechanisms of the non-cognate substrate that we here describe as model A, B and C. The most drastic differences between the models are found at saturating cognate and non-cognate substrate and saturating SBP concentrations. Then, the transport rate is not influenced by the non-cognate substrate in model A, the transport rate is partially reduced in model B, and transport is completely inhibited in model C. These findings show that the influence of the non-cognate substrate on the rate of cognate substrate transport depends crucially on its effect on the SBP.

Model description
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We modelled the transport cycle of an ABC importer by a minimal mathematical model based on available biochemical and structural data [1][2][3] . We focus on Type I ABC importers, as they are structurally and mechanistically different from other ABC importers 38 . We constructed four reaction schemes that model how a cognate substrate is transported, and how a non-cognate substrate interacts with the importer but is not transported (Figure 1). These models are termed model 0, A, B and C. The transport of cognate substrate is modelled in the absence of non-cognate substrate (model 0) and in its presence (model A, B and C). The difference between models A, B and C relates solely to how the non-cognate substrate is bound by the SBP and how the noncognate substrate-SBP complex interacts with the translocator. The models consist of different states X i , which are connected via rate constants k j .
First, we describe the common steps of model 0, A, B and C, that are, the steps that involve the transport of cognate substrate ( Figure 1). The first step is the reversible binding of cognate substrate X 6 to the open conformation of the SBP X 1 with an association and dissociation rate constant k 1 and k 2 , respectively. Binding of the cognate substrate induces immediate closing of the SBP 19 . The substrate-free (open) and substrate-bound (closed) states of the SBP are denoted by X 1 and X 2 , respectively. In our models, we ignore any other SBP states, such as a substrate-free closed or a substrate-bound open conformation, as these states represent only a very small fraction of the total SBP population and/or short-lived 19,20,[39][40][41][42] . Next, the substrate-bound SBP docks onto the inward-facing conformation of the translocator with an association rate constant k 3 . The formed complex is denoted X 3 and the free translocator X 5 . Once docked on the translocator, the SBP can either undock (with a rate constant k 4 ) or transfer the cognate substrate to the TMDs (rate constant k 5 ). In this latter step, the SBP has to open and release the substrate into the TMD cavity of the outward-facing conformation 43 . The formed complex is denoted X 4 . In the final step, ATP hydrolysis triggers formation of the inward-facing conformation and release of the cognate substrate into the cytoplasm. We assume that all these processes occur with rate constant k 6 in an irreversible process. By making the step with rate constant k 5 and k 6 irreversible, the transporter can only function in the import direction and it can only hydrolyse and not synthesise ATP. In our models we do not consider the proposed non-canonical mechanism of transport, in which SBP docking would precede substrate binding 44,45 .
Next, we describe the additional steps of model A, B and C describing how a non-cognate substrate binds to the SBP and how the non-cognate substrate-SBP complex interacts with the importer (Figure 1). In model A (Figure 1), the non-cognate substrate X 7 binds to the open conformation of the SBP X 1 with an association and dissociation rate constant k 7 and k 8 , respectively. The SBP with a non-cognate substrate bound is denoted by X 8 . In model A, it is assumed that the non-cognate substrate does not trigger the correct conformational change in the J o u r n a l P r e -p r o o f Journal Pre-proof SBP 20,34,35 so that the SBP has no affinity for the translocator. Thus, the key characteristic of model A is that transport fails because the non-cognate substrate-SBP complex cannot dock onto the translocator. Model A is exemplified by the action of non-cognate substrates arginine and lysine of amino-acid importer GlnPQ, as these compounds leave the structure of SBD1 in the open conformation 20 .
In model B (Figure 1), the non-cognate substrate is bound reversibly by the SBP with an association and dissociation rate constant k 7 and k 8 , respectively. Contrary to model A, the noncognate substrate-bound SBP can dock on to the translocator with an association and dissociation rate constant k 9 and k 10 , respectively. The transport in model B, however, fails because the noncognate substrate-SBP complex does not activate the translocator. We have previously shown that many non-cognate substrates induce a conformational change in the SBP that is distinct from those induced by cognate substrates 20 After the complex between the non-cognate substrate and the SBP has been formed, it can dock onto the translocator with an association rate constant k 9 . Next, the SBP can only undock again (with a rate constant k 10 ) but it cannot open and transfer the substrate to the translocator.

Comparing models
By using the law of mass action for each step of the reaction mechanism, we can formulate the equations that describe the time evolution of the concentrations of state X j (see Supplementary Information for details). We calculated the steady-state transport rate of a single translocator of model ( ∈ {0, , , }): J o u r n a l P r e -p r o o f Journal Pre-proof where 4 is the steady-state concentration of state X 4 of model and is the total translocator concentration. For model A, B and C, we calculated the steady-state transport rate relative to model 0: where ∈ { , , }. The value is indicative for the amount of inhibition by the non-cognate substrate: the transport rate of cognate substrate is not influenced by the presence of non-cognate substrate when = 1, transport is completely inhibited when = 0, and transport occurs with a reduced rate when 0 < < 1. Formally, and are functions of the rate constants, the total non-cognate substrate concentration , the total cognate substrate concentration , the total SBP concentration , and the total translocator concentration . However, for notational convenience we will omit this explicit dependence throughout this paper.
To compare the steady-state transport rates for the different models, we numerically solved the steady-state concentrations for a particular set of parameter values. We chose parameters that reflect known cases and typical assumptions and conditions of Type I importers. First, was set to 20 µM and to 1 µM, so that the SBP to translocator ratio is 20:1 [46][47][48][49] . The rate constants k 1 and k 7 were set to 10 s -1 µM -1 and k 2 and k 8 to 10 s -1 , thereby fixing the cognate and non-cognate dissociation constant K D to 1 µM 21,29 . The rate constants k 3 and k 9 were set to 1 s -1 µM -1 and k 4 to 10 s -1 and k 10 to 20 s -1 , thereby fixing the K D between the SBP and the translocator to 10 µM when the SBP has a cognate substrate bound and to 20 µM when a non-cognate substrate is bound 48,[50][51][52] . We chose k 5 and k 2 to be equal, because both steps involve the opening of the SBP and release of substrate. Finally, k 6 was set to 4 s -1 , so that the maximal turnover rate is k 5 k 6 /(k 5 +k 6 )  3 s -1 14 .
Unless stated otherwise, we used these rate constants throughout this paper. Figure 2 shows for a total cognate substrate concentration between 0 and 60 µM and in the presence and absence of 15 µM non-cognate substrate. We observe that 0 increases with cognate substrate and approaches a maximum at high concentrations. This behaviour is also commonly observed experimentally 21  To put it more formally, Figure 2 shows that ≥ ≥ irrespective of the precise cognate substrate concentration.

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To analyse if this conclusion depends on the particular choice of model parameters, such as the total SBP concentration or the rate constants, we compared a large set of , and values, which were calculated with random model parameters (details in Supplementary Information).
The model parameters (i.e., the rate constants and , , and ) were randomly drawn from a broad distribution. In total, 8·10 4 random model parameter combinations were used to calculate , and . In Figure 3, the histograms for the resulting ( , ), ( , ) and ( , ) pairs are shown. We observe that the amount of inhibition in model A is always less than or equal to model B and C ( ≥ and ≥ ), and that the inhibition in model B is always less than or equal to model C ( ≥ ) (Figure 3). Therefore, we conclude that: ≥ ≥ irrespective of the rate constants or protein and substrate concentrations. Secondly, we see with certain model parameter combinations that transport is not influenced by the presence of non-cognate substrate ( = 1).
Thirdly, with certain model parameter combinations, drastic differences are observed between model A, B and C, e.g., = 1 and = 0. These cases will be analysed in more detail in the next sections.
For simplicity, we treated many of the downstream steps of the transport cycle as a single step.
To rule out that the conclusions depend on this oversimplification, we also considered several alternative model topologies, in which an additional step is included after substrate transfer to the TMDs ( Figure S1a-c), the cognate substrate is translocated across the membrane before the SBP undocks ( Figure S2a-c) or the SBP undocks before the cognate substrate is translocated ( Figure   S3a-c). Random parameter combinations were simulated and we find that the conclusions for the models of Figure 1 are also valid for the alternative model topologies (compare Figure 3 with Figure S1d, Figure S2d and Figure S3d). This suggests that our conclusions are not strictly model dependent.

High substrate concentration
Next, we analysed the situation that the substrates are available to the SBP at saturating concentrations (i.e. ≫ and ≫ ) and the SBP is present in large excess of the translocator concentration (i.e. ≫ ). The first condition is important for ABC importers in bacteria that are (transiently) exposed to high nutrient concentrations such as gut microbiota and enterobacteriaceae [46][47][48][49]53 . In a subset of ABC transporters (abundantly present in firmicutes), multiple SBPs are directly linked to the importer, giving rise to more than one SBP per transporter complex 13 and J o u r n a l P r e -p r o o f Journal Pre-proof = 0 (8) By using Eq. 6, 7 and 8, we calculated the transport rate at high substrate concentrations and with a total SBP and translocator concentration of 25 and 0.5 µM, respectively. We see that the rates calculated with Eq. 6, 7 and 8 are in good agreement with the numerical solution, in which no approximations are made (Figure 5a).
To gain more insight in the amount of inhibition for each non-cognate interaction mechanism, we determined the transport rate relative to 0 , From Eq. 9, 10 and 11 we conclude that transport still occurs when the non-cognate substrate binds reversibly to the SBP (model A and B). In contrast, irreversible binding (model C) completely inhibits transport under these conditions (see also Figure 5a). The interpretation of this result is simple. When the non-cognate substrate concentration is higher than the SBP Next, we analyse how and (as given by Eq. 9 and 10, respectively) depend on the total SBP concentration; a variable that might be adjusted by the cell 49 . In Figure 5b, is shown for different SBP concentrations. We observe that the amount of inhibition decreases with increasing J o u r n a l P r e -p r o o f Journal Pre-proof SBP concentration (Figure 5b). When the translocator becomes saturated with SBP (Figure 5c), then the following limit is obtained = 1 (12) This means that transport becomes insensitive to non-cognate substrate. This conclusion is valid irrespective of the rate constants and the substrate, SBP and translocator concentration, as long as the substrates and SBPs are both present at saturating levels.
By using Eq. 10 we calculated for different SBP concentrations (Figure 5d). We see that Eq. 13 and 14 show that the extent of inhibition depends on the rate constants and the ratio ⁄ .
Thus, in the presence of a high SBP concentration, the non-cognate substrate inhibits transport in model B, but not in model A. In conclusion, the different non-cognate interaction mechanisms have a radically different influence on the inhibition of transport, ranging from a complete inhibition in model C to a complete preservation of transport in model A.

Discussion
ABC importers constitute major uptake pathways of prokaryotes, and Type I and Type II importers require an extra-cytoplasmic SBP for function 11 [56][57][58] , although pockets through which the substrate passes in the transition of the TMD from outward-to inward-facing must be present. The binding pockets within the TMDs have been linked to the regulation of transport 59,60 , however, we believe that further molecular details are required to model these interaction mechanisms. For instance, it is unknown how a trapped non-cognate substrate is removed from the binding pockets of the TMDs and how the importer resets to its resting state.
Here, we used rate equations to model different non-cognate SBP-interaction mechanisms and analysed their effect on the steady-state transport rate. We conclude that when the same substrate, SBP and translocator concentrations and the common set of rate constants are used, a hierarchy in the extent of inhibition exists among the models (Figures 3). More specifically, inhibition is most severe when the non-cognate substrate binds irreversibly to the SBP (model C; Figure 1).
Inhibition is less prominent, when the binding is reversible and the SBP with a non-cognate substrate bound can dock onto the translocator (model B; Figure 1). When the binding is reversible, but the SBP with a non-cognate substrate bound cannot dock (model A; Figure 1), then the extent of inhibition is the least of all three mechanisms. In model A only a fraction of the total SBP population is effectively taken out by the binding of non-cognate substrate. In model B, the non-cognate substrate-bound SBP can dock onto the translocator, so only a fraction of both the SBP and translocator population is effectively taken out by the non-cognate substrate. This explains why transport in model B is always slower than in model A. In model C, the non-cognate substrate binds irreversibly, and more SBPs have a non-cognate substrate bound than in model B.
Therefore, the SBPs that can effectively participate in transport is reduced even further in model C when compared to model B.
Analytical results were obtained in the presence of low and high substrate concentrations (Section 'Low substrate concentration' and 'High substrate concentration', respectively). We observe that transport in model A, B and C is not influenced by the non-cognate substrate when both cognate and non-cognate substrates are present at low concentrations ( Figure 4). When the non-cognate substrate concentration is well below the SBP and translocator concentration, then these protein concentrations can only be changed by an amount that is smaller than the noncognate substrate concentration, which in this limit is negligible when compared to the total protein concentrations. The conclusion holds irrespective of the rate constants. Thus, it should even apply when different cognate and/or non-cognate substrates are compared. For example, the non-cognate substrates arginine and lysine have in common that they do not trigger closing of SBD1 and both inhibit glutamine and asparagine transport via GlnPQ 20 . However, the K D of J o u r n a l P r e -p r o o f arginine binding by SBD1 is more than one order of magnitude lower than that of lysine 20 . This implies that also their association (k 7 ) and/or dissociation (k 8 ) rate constants are different, because K D = k 8 / k 7 . Since Eq. 3 is independent of these rate constants, their effect on transport is the same, i.e., transport of glutamine and asparagine by GlnPQ is not inhibited at low concentrations of arginine and lysine. These predictions can be verified experimentally, by performing uptake assays at substrate concentrations that are below the SBP and translocator concentration.
Contrary to the inhibition at low substrate concentrations, the different non-cognate interaction mechanisms inhibit transport completely differently in the limit that the non-cognate and cognate substrates and SBPs are present at saturating concentrations (see Eq. 11, 12 and 13 In conclusion, the different mechanisms of substrate inhibition have strongly varying impact on the transport kinetics of ABC importers, which might explain how prokaryotes maintain efficient uptake in chemical diverse external environments and might contribute to the development of effective inhibitors against SBPs of pathogenic bacteria.

Methods
The system of nonlinear equations was numerically solved with the software package MATLAB Rate constants are denoted above the arrows and the states X j are depicted as cartoon. The cognate and non-cognate substrates are shown in green and red, respectively. The SBP is depicted in light grey and the translocator in dark grey. Details of the models are described in the main text.       , where L and l are the total non-cognate and cognate substrates concentrations, respectively. The total SBP ( ) and total translocator ( ) concentrations are 25 and 0.5 µM, respectively. The continuous line denotes the numerical solution with no approximations made and the points are the transport rate calculated with Eq. 6, 7 and 8 as indicated. The normalized steady-state transport rate as a function of the relative non-cognate substrate concentration ( + ) ⁄ and various SBP concentrations for model A (b) and B (d) calculated with Eq. 9 and 10. Relative population of the free translocator state 5 ⁄ (c) and 5 ⁄ (e) as function of the relative non-cognate substrate concentration and various SBP concentrations. In all the calculations of this figure the total substrate (l + L) was equal to 100 mM. The rate constants were used as described in the Section 'Model description'.