# Transport Reversal during Heteroexchange: A Kinetic Study

Authors: V. Makarov; L. Kucheryavykh; Y. Kucheryavykh; A. Rivera; M. J. Eaton; S. N. Skatchkov; M. Inyushin
Journal: Journal of Biophysics
 (2013)

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## Abstract

It is known that secondary transporters, which utilize transmembrane ionic gradients to drive their substrates up a concentration gradient, can reverse the uptake and instead release their substrates. Unfortunately, theMichaelis-Menten kinetic scheme, which is popular in transporter studies, does not include transporter reversal, and it completely neglects the possibility of equilibrium between the substrate concentrations on both sides of the membrane. We have developed a complex two-substrate kinetic model that includes transport reversal. This model allows us to construct analytical formulas allowing the calculation of a “heteroexchange” and “transacceleration” using standard Michaelis coefficients for respective substrates. This approach can help to understand how glial and other cells accumulate substrates without synthesis and are able to release such substrates and gliotransmitters.


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## Body

## 1. Introduction

Unlike “primary” or ATP dependent transporters that create the major ionic gradients of K/Na/H and Cl/CO2 ions across cellular membranes harnessing the energy reserved in ATP, the “secondary transporters” utilize the energy available from transmembrane ionic and/or pH gradients and membrane potential to drive their substrates up a steep concentration gradient. Transporters on neurons and astrocytes clearing neurotransmitters from the synaptic cleft and extracellular space mainly belong to different “secondary transporters” families. Recently, it has been shown that astrocytes and other glial cells accumulate monoamines [1] and polyamines [2, 3] while lacking the enzymes for their synthesis [1, 4–6]. One among many known representatives of the “secondary transporters” that utilize the transmembrane ionic gradients and membrane potential is the family of organic cation transporters (OCT). These transporters take up different mono- and polyamines [7], and cells expressing such transporters also release these substrates using possibly two pathways: (i) large pores and (ii) transport reversal. Here we analyze one of transport reversal mechanisms.Energy Calculations. Experimentally, it has been shown that secondary transporters can reverse their uptake releasing their substrates instead [8–10]. Energy based calculations were introduced to analyze the conditions for substrate release or uptake for this kind of transporter [11, 12]. It was established that substrate transport depends on the energy balance of coupled transport of the substrate and simultaneously transported ions (see Appendix A). Most secondary transporters could be reversed by membrane potential and by changes in the principal ion gradients and substrate concentrations. Experimentally, the reversal was shown for the glutamate transporters (for the review see [13]), GABA (reviewed by [12]), and for glial organic cation transporters [14, 15]. Being reversed, electrogenic transporters usually change the direction of the net ion flow. We summarize the energy balance study, introduced by Rudnick [11] in Appendix A. This analysis only studies one substrate uptake/release by a secondary transporter.Michaelis-Menten Scheme. The kinetic concept based on the Michaelis-Menten scheme proved very useful for transporter mediated substrate uptake and inhibition [16, 17]. This kinetic model predicts saturability and specificity of secondary transporters in many cases, and atypical transport kinetics can be explained by multiple binding sites [18]. We have summarized this classic concept in Appendix B. Unfortunately, as one can see, the Michaelis-Menten model does not include transporter reversal, and it completely neglects the reversal constant (see Appendix B). A more complex transporter kinetic model is needed to predict quantitatively at least the following well-established experimental observations.(1)
It has been shown that one transporter substrate can release another one already accumulated inside the cell. Sometimes this is called “heteroexchange.” For example, dopamine, tyramine, and amphetamine, which are substrates for the neuronal dopamine transporter (DAT), can release the substrate named N-methyl-4-phenylpyridinium (MPP) through DAT [19], with releasing ability of these substances correlated with the elicited coupled transport current. Also, it was shown that L-glutamate and its transportable analogs (substrates for EAATs) specifically release L-aspartate (another EAAT substrate) through this transporter and can be blocked by nontransportable analogs [20].(2)
A special term was coined for the release of the (tracer) substrate by the same substrate, a process named “transacceleration.” While the phenomenon is not kinetically different from the “heteroexchange” described in the previous paragraph, it is well established experimentally (see, e.g., [21]). As new transporter models arise (e.g., a channel-transporter model [22]), it might be important to get this phenomenon explained by a purely thermodynamic model, not by using kinematic assumptions.Here we present kinetic algorithms that more accurately explain the behavior of a secondary transporter pumping two substrates simultaneously; it predicts transporter reversal by the application of an additional second substrate to the transporter already in equilibrium with the first substrate, “transacceleration” and other interactions.


## 2. Results

We modified the Michaelis-Menten kinetic model to include two different transportable substrates and also additional elementary steps, characterized by their kinetic coefficients, which are necessary for the transporter not only to uptake but also to release substrates. The model is presented in AppendixC by relations 1–8. This model can be considered as a system of kinetic equations describing the dynamics of the model (C.2)–(C.11). A general solution for this scheme is difficult to obtain analytically. But some particularly interesting cases can be resolved (see Appendices C.1, C.2, and C.3), and we are presenting them below.


### 2.1. Equilibrium Conditions for Both Substrates (See AppendixC.1)

Practically, the initial concentrations of substratesS1 and S2 are considered known (i.e., S10 and S20), and then substrate concentration can be measured in the outside solution (x1 and x2 in our notation for this section of Appendix C). In this way, we tried to reduce all equations to measurable parameters.It follows from relationships(C.30) (Appendix C) that at fixed concentration of S10 (initial concentration of first substrate S1) and variable concentration of S20 (different initial concentrations of S2), the equilibrium concentration of x1 (the S1 substrate outside) increases with increasing S20 (the effect of S1 substrate releasing from the cell), and similarly, the equilibrium concentration of x1 decreases with decreasing S20 (effect of S1 substrate transport inside of the cell). The same behavior follows from (C.30) for the equilibrium concentration of x2 at fixed concentration of S20 and variable concentration of S10. These respective dependencies are shown in Figure 1.Figure 1
Dependence ofx1, x2 on S20 at S10=1.5 a.u., K11=1.5, K12=1.8,  K21=1.3, K22=2.Conclusions of  AppendixC.1
(1)
Effect of substrate beingreleased in case of competition in the two-substrate system can be observed, if at equilibrium condition most of the transporter is coupled by both substrates of interest: T0≫y.
(2)
Efficiency of the substratereleasing process is dependent on equilibrium constant values describing processes of substrate-transporter intermediate complex formation.
(3)
Correct sign of the square root term in relations(C.30) is defined by the conditions of
(1)S10≥x1≥0,S20≥x2≥0.
(4)
Relationships(C.30) can be used for analysis of equilibrium substrate concentration dependence on initial substrate concentrations.
(5)
The relationship of(2)x2=S10+S20-αx1β
can be used to determine parameters α and β, if concentrations of x1 and x2 can be simultaneously measured as functions of S10 and S20, the initial concentrations of the first and second substrates.


### 2.2. Two-Substrate System Dynamics at the Initial Time

Transporter velocities (transport rates) can be determined if (similar to Michaelis-Menten scheme) there is no equilibrium between substrate concentrations inside and outside and processes of the type(3)S1′+T⟶(S1T),S2′+T⟶(S2T)
can be neglected. We also assume there are the initial conditions where S1 and S2 are added to the external solution, thus x1=S10 and x2=S20. In that case (see Appendix C.2), (1)
(4)υx1=k12K1S10[T0]1+K1S10+K2S20,(5)υx2=k22K2S20[T0]1+K1S10+K2S20
are analogous to the Michaelis-Menten formulation for a two-substrate system. If S20=0, we obtain the exact Michaelis-Menten formula for the first substrate velocity, and if S10=0, we obtain the exact Michaelis-Menten formula for the second substrate velocity. Note also that the term k12[T0] can be interpreted as V1max, and k22[T0] as V2max.(2)
The constantsk12, KM1, k22, and KM2 can be determined experimentally similar to the standard procedures used in the Michaelis formulation. There are some important equations:
(i)
ifS10≫KM,2 and S10≫αS20:
(6)υ1,s=k12[T0],the velocity at maximum,V1max,υx2=k22αS20[T0]S10.(ii)
IfS10≪αS20 and αS20≫KM,1:
(7)υx1′=dx1dt=k12S10[T0]αS20,υ2,s′=k22[T0],the velocity at maximum,V2maxυ1,sυ2,s=k12S10[T0]αS20k22αS20[T0]S10=k12k22[T0]2(seeAppendixC).


### 2.3. Effect of the Equilibrium Reverse Bias for a First Substrate When a Second One is Added to the System

If previously the equilibrium was established for a first substrate between outside concentration of the substrate and the inside concentration, the addition of a second substrate will produce a reverse bias (equilibrium shift). In the beginning, at initial time, some of the transporter molecules in the outside bind to the second substrate while inside there is still no second substrate. That means the availability of outside transporter for a first substrate becomes reduced. Thus equilibrium for a first substrate starts to break down; that is, the velocity of first substrate transport to outside (release) becomes bigger than its transport to the inside. At initial times during the start of the process and far from equilibrium for a second transporter, (8) allows the calculation of the velocity of the first substrate release due to transport reversal (see Appendix C.3):
(8)υx1=A[1-KM,2S20+KM,2],
where KM,2 is Michaelis constant for a second substrate and S20 is initial concentration of a second substrate,
(9)A=k11′S10K0+1y0,y0=[T0](K11K12+1)K11S10+K11K12+1=[T0]KSS10+1,
where
(10)KS=K11(K11K12+1).
Thus, finally we have
(11)A=k11′S10K0+1[T0]KSS10+1.
Taking into consideration the relation (8) we have calculated the dependence of the velocity of first substrate release on the concentration of a second substrate, at initial times after it was added to the system. Functional dependence (8) is represented in Figure 2.Figure 2It can be seen from (8) and Figure 2 that the velocity of reversed transport (release) of a first substrate is 0 if S20=0, because there is equilibrium between the velocities of inward and outward flow of the first substrate through the transporter. Thus, the “net” velocity, is equal to zero. Also, from (8) and as seen in Figure 2, with increase of a second substrate concentration, when S20≫KM,2, the velocity of a first substrate release becomes saturated and can be calculated as
(12)υx1,S=A=k11′S10K0+1[T0]KSS10+1.
Release velocity depends on the first substrate concentration S10, and at given value of S10 the value of A is a constant. Thus, (8) for the velocity at half maximal value at a certain concentration of second substrate S20 can be written as
(13)υx1=A[1-KM,2S20,1/2+KM,2]=A2,
and because of this equation it can be calculated as
(14)KM,2=S20,1/2.
There is similarity between the formula of velocity of transporter reversal due to second substrate addition and Michaelis-like formulas for the velocity of substrate uptake.The formula that predicts the velocity of substrate uptake (see AppendixC(C.54) or Appendix B(B.10)) can be written as
(15)υx2=k22S20[T0]S20+KM,2,
where the maximum velocity is represented by
(16)υx2,max=k22[T0].
Thus, for the half maximal velocity,
(17)υx2,max2=k22S20,1/2[T0]S20,1/2+KM,2=k22[T0]2.
Thus we can write
(18)KM,2=S20,1/2.
To say in plain words, the Michaelis constant for a second substrate can be determined in two ways: (i) from the standard Michaelis formulas at transport velocity measurements for the second substrate, or (ii) from the release velocity measurements of a first substrate, from our formula, where a second substrate produces release of the first one.In the most important case, ifKSS10≫1 (8)
(19)AS=k11′K0+1[T0]KS=Const.,υx1,S01=AS[1-KM,2S20+KM,2],
then AS can be interpreted as the release force for a first substrate after the addition of a second one. In the case of KSS10≪1, the release force can be written as
(20)A(S10)≈ASS10=k11′K0+1[T0]KSS10;
that is, in this case the release force for a first substrate after the addition of a second one has a linear dependence on the first substrate concentration.


## 2.1. Equilibrium Conditions for Both Substrates (See AppendixC.1)

Practically, the initial concentrations of substratesS1 and S2 are considered known (i.e., S10 and S20), and then substrate concentration can be measured in the outside solution (x1 and x2 in our notation for this section of Appendix C). In this way, we tried to reduce all equations to measurable parameters.It follows from relationships(C.30) (Appendix C) that at fixed concentration of S10 (initial concentration of first substrate S1) and variable concentration of S20 (different initial concentrations of S2), the equilibrium concentration of x1 (the S1 substrate outside) increases with increasing S20 (the effect of S1 substrate releasing from the cell), and similarly, the equilibrium concentration of x1 decreases with decreasing S20 (effect of S1 substrate transport inside of the cell). The same behavior follows from (C.30) for the equilibrium concentration of x2 at fixed concentration of S20 and variable concentration of S10. These respective dependencies are shown in Figure 1.Figure 1
Dependence ofx1, x2 on S20 at S10=1.5 a.u., K11=1.5, K12=1.8,  K21=1.3, K22=2.Conclusions of  AppendixC.1
(1)
Effect of substrate beingreleased in case of competition in the two-substrate system can be observed, if at equilibrium condition most of the transporter is coupled by both substrates of interest: T0≫y.
(2)
Efficiency of the substratereleasing process is dependent on equilibrium constant values describing processes of substrate-transporter intermediate complex formation.
(3)
Correct sign of the square root term in relations(C.30) is defined by the conditions of
(1)S10≥x1≥0,S20≥x2≥0.
(4)
Relationships(C.30) can be used for analysis of equilibrium substrate concentration dependence on initial substrate concentrations.
(5)
The relationship of(2)x2=S10+S20-αx1β
can be used to determine parameters α and β, if concentrations of x1 and x2 can be simultaneously measured as functions of S10 and S20, the initial concentrations of the first and second substrates.


## 2.2. Two-Substrate System Dynamics at the Initial Time

Transporter velocities (transport rates) can be determined if (similar to Michaelis-Menten scheme) there is no equilibrium between substrate concentrations inside and outside and processes of the type(3)S1′+T⟶(S1T),S2′+T⟶(S2T)
can be neglected. We also assume there are the initial conditions where S1 and S2 are added to the external solution, thus x1=S10 and x2=S20. In that case (see Appendix C.2), (1)
(4)υx1=k12K1S10[T0]1+K1S10+K2S20,(5)υx2=k22K2S20[T0]1+K1S10+K2S20
are analogous to the Michaelis-Menten formulation for a two-substrate system. If S20=0, we obtain the exact Michaelis-Menten formula for the first substrate velocity, and if S10=0, we obtain the exact Michaelis-Menten formula for the second substrate velocity. Note also that the term k12[T0] can be interpreted as V1max, and k22[T0] as V2max.(2)
The constantsk12, KM1, k22, and KM2 can be determined experimentally similar to the standard procedures used in the Michaelis formulation. There are some important equations:
(i)
ifS10≫KM,2 and S10≫αS20:
(6)υ1,s=k12[T0],the velocity at maximum,V1max,υx2=k22αS20[T0]S10.(ii)
IfS10≪αS20 and αS20≫KM,1:
(7)υx1′=dx1dt=k12S10[T0]αS20,υ2,s′=k22[T0],the velocity at maximum,V2maxυ1,sυ2,s=k12S10[T0]αS20k22αS20[T0]S10=k12k22[T0]2(seeAppendixC).


## 2.3. Effect of the Equilibrium Reverse Bias for a First Substrate When a Second One is Added to the System

If previously the equilibrium was established for a first substrate between outside concentration of the substrate and the inside concentration, the addition of a second substrate will produce a reverse bias (equilibrium shift). In the beginning, at initial time, some of the transporter molecules in the outside bind to the second substrate while inside there is still no second substrate. That means the availability of outside transporter for a first substrate becomes reduced. Thus equilibrium for a first substrate starts to break down; that is, the velocity of first substrate transport to outside (release) becomes bigger than its transport to the inside. At initial times during the start of the process and far from equilibrium for a second transporter, (8) allows the calculation of the velocity of the first substrate release due to transport reversal (see Appendix C.3):
(8)υx1=A[1-KM,2S20+KM,2],
where KM,2 is Michaelis constant for a second substrate and S20 is initial concentration of a second substrate,
(9)A=k11′S10K0+1y0,y0=[T0](K11K12+1)K11S10+K11K12+1=[T0]KSS10+1,
where
(10)KS=K11(K11K12+1).
Thus, finally we have
(11)A=k11′S10K0+1[T0]KSS10+1.
Taking into consideration the relation (8) we have calculated the dependence of the velocity of first substrate release on the concentration of a second substrate, at initial times after it was added to the system. Functional dependence (8) is represented in Figure 2.Figure 2It can be seen from (8) and Figure 2 that the velocity of reversed transport (release) of a first substrate is 0 if S20=0, because there is equilibrium between the velocities of inward and outward flow of the first substrate through the transporter. Thus, the “net” velocity, is equal to zero. Also, from (8) and as seen in Figure 2, with increase of a second substrate concentration, when S20≫KM,2, the velocity of a first substrate release becomes saturated and can be calculated as
(12)υx1,S=A=k11′S10K0+1[T0]KSS10+1.
Release velocity depends on the first substrate concentration S10, and at given value of S10 the value of A is a constant. Thus, (8) for the velocity at half maximal value at a certain concentration of second substrate S20 can be written as
(13)υx1=A[1-KM,2S20,1/2+KM,2]=A2,
and because of this equation it can be calculated as
(14)KM,2=S20,1/2.
There is similarity between the formula of velocity of transporter reversal due to second substrate addition and Michaelis-like formulas for the velocity of substrate uptake.The formula that predicts the velocity of substrate uptake (see AppendixC(C.54) or Appendix B(B.10)) can be written as
(15)υx2=k22S20[T0]S20+KM,2,
where the maximum velocity is represented by
(16)υx2,max=k22[T0].
Thus, for the half maximal velocity,
(17)υx2,max2=k22S20,1/2[T0]S20,1/2+KM,2=k22[T0]2.
Thus we can write
(18)KM,2=S20,1/2.
To say in plain words, the Michaelis constant for a second substrate can be determined in two ways: (i) from the standard Michaelis formulas at transport velocity measurements for the second substrate, or (ii) from the release velocity measurements of a first substrate, from our formula, where a second substrate produces release of the first one.In the most important case, ifKSS10≫1 (8)
(19)AS=k11′K0+1[T0]KS=Const.,υx1,S01=AS[1-KM,2S20+KM,2],
then AS can be interpreted as the release force for a first substrate after the addition of a second one. In the case of KSS10≪1, the release force can be written as
(20)A(S10)≈ASS10=k11′K0+1[T0]KSS10;
that is, in this case the release force for a first substrate after the addition of a second one has a linear dependence on the first substrate concentration.


## 3. Discussion and Conclusions

We have studied the extended kinetic model for a secondary transporter simultaneously dealing with two substrates, which includes direct (outside-in) and reverse transport (inside-out). The model was solved in different equilibrium conditions (see AppendicesC.1, C.2, and C.3). We have shown that when both substrates are in equilibrium, addition of one of them leads to reequilibrium and release of the second substrate (Appendix C.1). This was emphasized in Appendix C.3, when the system was studied for conditions where a first substrate is in equilibrium (inside-outside concentrations) and a second one is just added and is far from equilibrium. This situation is of a special interest as it has been studied experimentally [19, 20]. Also, this is what probably happens when methamphetamine, ephedrine, or other similar substances induce dopamine (and other monoamine) release from monoamine neurons primarily via membrane transporters, reversing the dopamine transporter (DAT), norepinephrine transporter (NET), and/or serotonin transporter (SERT) [23–27] and also reversing VMAT vesicular transport [28]. In addition, it has been recently shown that astrocytes and other glial cells accumulate polyamines [2, 3] while lacking the enzymes for their synthesis [4–6], and OCT type of transporters (that are expressed in glia) take up different polyamines [7]. Polyamines are released in brain from glial cells, but the mechanisms of such release are unknown [29].Actually, as we understand now from formula (8) it can be ANY transportable substrate. This formula allows us to classify experimental measurements of a “heteroexchange” related substrate release for substrate-transporter pairs, using standard Michaelis coefficients.The special term for the release of the (tracer) substrate by the same substrate, a process named “transacceleration,” can be explained by changes in equilibrium according to formula (8). There is no fundamental thermodynamic difference if the system has two chemically distinct substrates for the same transporter or there are radiolabelled and unlabelled chemically similar substrates. Thus, a new added substrate produces the release of a similar tracer substrate (labelled, e.g., with radioactive isotope) by equilibrium shift as shown in Appendix C.3.We also have shown that if we assume both substrates are far away from equilibrium, and transporter reversal can be neglected (AppendixC.2), the formulas for the uptake velocity of both substrates become the same as in the Michaelis-Menten scheme (see Appendix C.2, (C.53) and (C.54)), with the respective inhibitory coefficients.We suggest that formula (8) will be especially useful in the study of polyspecific transporters with known multiple substrates, such as the organic cation transporters (OCT) that participate in the transport of different monoamines [30], as well as polyamines [7].



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