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Mathematical modeling of maltose uptake system in E. coli using nanodisc fluorescence quenching data Hiller, Rebecca Marie 2013

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MATHEMATICAL MODELING OF MALTOSE UPTAKE SYSTEM IN E. COLI USING NANODISC FLUORESCENCE QUENCHING DATA  by Rebecca Marie Hiller B.Sc., B.A., Humboldt State University, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2013  c Rebecca Marie Hiller 2013  Abstract Recent data measured in nanodiscs conflicts with the standard theory of maltose transport in the MalE-MalFGK2 uptake system found in E. coli. Nanodisc fluorescence quenching data suggest an alternate pathway in which unliganded MalE binds the P-open transporter, facilitating maltose acquisition. Nanodisc data also indicate that MalE regulates maltose uptake at high concentrations. We analyzed four mathematical models of the maltose uptake system: the distinct standard and alternate models, and two integrated models. Nanodisc fluorescence quenching data and nonlinear regression analysis were used to fit equilibrium constants and kinetic rates. The flux through each pathway in an integrated model was calculated using asymptotic analysis and fit parameter values. We conclude that it is likely that transport occurs when liganded MalE associates to a P-open conformation of MalFGK2 , rather than binding to the P-closed transporter as suggested by the standard model. The standard pathway was calculated to be negative, i.e. to occur in reverse as a means of regulating maltose uptake at high concentration. This analysis conflicts with the standard model in which liganded MalE binds to a closed transporter and triggers an opening of the transporter proteins which in turn open the liganded MalE. The analysis also found that a relatively small amount of maltose transport may occur through the alternate pathway involving unliganded MalE.  ii  Contents  Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  ABC Transporters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Maltose Uptake System in E. Coli . . . . . . . . . . . . . . . . . . . . . . . . . .  2  1.3  Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.3.1  Supporting Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.3.2  Contradicting Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4  1.4  Alternate Model of Maltose Transport . . . . . . . . . . . . . . . . . . . . . . . .  4  1.5  Previous Modeling of Maltose Uptake System . . . . . . . . . . . . . . . . . . . .  5  iii  2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6  2.1  Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9  2.2  Alternate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9  2.3  Comprehensive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12  2.4  Full Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15  3 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1  Fluorescence Quenching Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 22  3.2  Equilibrium Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22  3.3  Time-Dependent Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22  3.4  ATPase Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25  3.5  Fit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25  3.6  Fit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25  4 Nonlinear Regression Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1  4.2  Equilibrium Data Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.1  Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27  4.1.2  Alternate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  4.1.3  Comprehensive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  4.1.4  Full Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  Time-Dependent Data Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.1  In the limit K1 → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  4.2.2  Fixed K1 , k1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  iv  4.3  ATPase Data Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  5 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1  Standard vs. Alternate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45  5.2  Alternate vs. Comprehensive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46  5.3  Comprehensive vs. Full . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47  6 Maltose Transport in Comprehensive Model . . . . . . . . . . . . . . . . . . 48 6.1  Analysis of Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48  6.2  Further ATPase Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52  7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55  A Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57  B Alternate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59  C Comprehensive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61  D Full Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66  E F-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69  F Flux Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70  v  List of Tables 2.1  Model states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  2.2  Additional Full model state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9  2.3  Kinetic rates and reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18  2.4  Additional kinetic parameters and reactions for the Full model . . . . . . . . . . 19  2.5  Equilibrium parameters and reactions . . . . . . . . . . . . . . . . . . . . . . . . 20  4.1  Equilibrium fit values for Standard and Alternate models . . . . . . . . . . . . . 30  4.2  Equilibrium fit to Comprehensive model, K1 = 0.01 . . . . . . . . . . . . . . . . . 34  4.3  Equilibrium fit to Comprehensive model, K1 = 0.1 . . . . . . . . . . . . . . . . . 35  4.4  Equilibrium fit to Comprehensive model, K1 = 1 . . . . . . . . . . . . . . . . . . 36  4.5  Equilibrium fit to Comprehensive model, K1 = 100 . . . . . . . . . . . . . . . . . 37  4.6  Equilibrium fit values for Comprehensive model, K1 → 0 . . . . . . . . . . . . . . 37  4.7  Equilibrium fit to Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  4.8  Time-dependent fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  5.1  SSR for equilibrium fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46  E.1 F-test values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69  vi  List of Figures 1.1  ABC transporter diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  MalE diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2  2.1  Standard model diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10  2.2  Alternative model diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13  2.3  Comprehensive model diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  2.4  Full model diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17  3.1  Equilibrium data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23  3.2  Time-dependent data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24  4.1  Equilibrium fit (Comprehensive vs. Alternate vs. Standard) . . . . . . . . . . . . 33  4.2  Equilibrium fit (Full vs. Comprehensive) . . . . . . . . . . . . . . . . . . . . . . . 38  4.3  Time-dependent fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44  6.1  Flux for K1 ≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51  6.2  Flux for K1 = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51  6.3  ATPase results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52  vii  Acknowledgements I am very grateful to the faculty and staff at the UBC, for all of their support during my graduate program. I especially want to thank my advisor, Dr. Eric Cytrynbaum, for all of the valuable insight and encouragement throughout this process. Many thanks to Dr. Franck Duong and Huan Bao in the Department of Biochemistry at UBC for involving us in your exciting research and providing the experimental data for this analysis. I also want to thank Dr. Dan Coombs for his time and contribution in reading my thesis, his excellent teaching, motivating research, and for always being so helpful. I have to thank Dr. Leah Keshet for being such an inspiration before and after I came to UBC, and for all of her support and advice. I owe special thanks to Kelly Paton, for all of the moral support and academic encouragement. I also want to thank Erin Moulding for her constant support and friendship since day one of the program. Special thanks to my parents, and parents-in-law, for all of your support with my education. I also cannot thank my husband enough for his endless support and confidence in my success, and for moving his life to Canada so I could attend UBC.  viii  Dedicated to Scott  ix  Chapter 1  Introduction 1.1  ABC Transporters  ATP binding cassette (ABC) transporters occur in both eukaryotic and prokaryotic organisms. ABC transporters consist of two ABC domains and two membrane-spanning (transmembrane) domains, as shown in Figure 1.1. The ABC domains are nucleotide binding domains, binding and hydrolyzing ATP to energize unidirectional substrate transport. Substrate transport occurs through a channel formed by the membrane-spanning domains [2]. The ABC transporter domains alternate between two conformations: P-closed (inward facing to the cytosol) and P-open (outward facing to the periplasm) [2].  Figure 1.1: Diagram of the P-closed (left) and P-Open (right) conformations of MalFGK2 , an ABC transporter with two membrane-spanning domains (MalF and MalG) and two ABC domains (K), also known as nucleotide binding domains.  1  ABC transporters facilitate the translocation of various substrates across cell or organelle membranes. Eukaryotic and bacterial ABC transporters mostly differ in both the type of substrate transported, and the direction of transport. Most eukaryotic ABC transporters export hydrophobic molecules from the cytoplasm. Bacterial ABC transporters, however, predominantly import nutrients into the cytoplasm. The import of these nutrients require specific binding proteins, which have been proposed to deliver the nutrients to the transporter [2]. Specifically, there is a class of binding protein-dependent transport systems primarily found in Gram-negative enteric bacteria. These transport systems are specific for sugars, amino acids, and ions [7]. In particular, we consider the binding-protein dependent maltose transport system in the bacteria E. Coli.  1.2  Maltose Uptake System in E. Coli  MalFGK2 is a bacterial ABC importer and binding protein-dependent transport system that specifically imports the nutrient maltose in Escherichia Coli. MalFGK2 is composed of 3 proteins: MalF and MalG (the transmembrane proteins) and MalK (the ATP-binding cassette dimer, also called the ATP interface), as seen in Figure 1.1. Maltose transport is dependent upon a substrate-binding protein, MalE, found in the periplasmic region of the bacterial cell [3]. MalE is a protein consisting of two symmetrical lobes that rotate toward each other to capture maltose with high-affinity, as shown in Figure 1.2. This is referred to as the closedliganded conformation of MalE [3]. MalE is found in the periplasm, and is known to be essential for efficient MalFGK2 import of maltose.  Figure 1.2: MalE reversibly binds to maltose to form closed-liganded MalE.  2  1.3  Standard Model  The standard model of maltose uptake characterizes MalE as a type of shuttle, delivering maltose to MalFGK2 . The association of maltose-bound MalE is assumed to trigger the conformation change from the P-closed transporter to the P-open transporter [3]. Maltose is subsequently delivered to the open transporter and MalE dissociates upon hydrolysis and transport to return to the periplasm. Recent experimental evidence instead suggests that unliganded MalE binds to the P-open conformation of MalFGK2 , stabilizing or activating the transporter [1].  1.3.1  Supporting Evidence  The standard model has been supported by crystallographic and biochemical analysis. An intermediate of the maltose uptake system has been crystallized which consists of intact MalFGK2 in complex with MalE, maltose, and ATP [4]. The maltose appears at the interface of the MalF-MalG subunits, about halfway into the lipid bilayer. The MalE is bound to the MalFGK2 in the open conformation, which has been argued to act as a cap on the transporter to ensure import of the maltose [4]. This intermediate was captured by blocking ATP hydrolysis with a mutation, such that the ATP-binding cassette dimer is captured in an ATP-bound, closed conformation [4]. This transition state is consistent with the standard model pathway of transport, although not uniquely so. The maltose transporter has also been crystallized in an intermediate step between the Pclosed and P-open conformations. It was found that interactions with maltose-bound MalE in the periplasm induces a partial closure of the MalK dimer in the cytoplasm and corresponding opening of the transmembrane domain, in the presence of nucleotides [5]. Furthermore, in the absence of MalE the transporter was crystallized in a P-closed resting state. This supports the assumption that liganded MalE triggers the conformation change to an open transporter state, however it was not shown that MalE without maltose does not induce the same intermediate state [5]. Additionally, an EPR spectroscopy analysis was used to conclude that ligand-bound MalE is needed for the closure of the MalK dimer in the presence of ATP [6].  3  1.3.2  Contradicting Evidence  There is experimental evidence which contradicts the assumptions of the standard model. A mutant MalFGK2 which transports lactose instead of maltose was found to still be dependent on MalE for activity, even though MalE does not bind lactose [7]. This suggests that although MalE is critical for transport it may not in fact act as a shuttle for the substrate. It was also determined that MalE can inhibit transport in excess when maltose is held at sub-stoichiometric levels [8]. The standard model would predict that an increased concentration of MalE would increase transport, as there would be a higher concentration of liganded Em to stimulate MalFGK2 to open. If liganded MalE were required to initiate maltose transport, it would be expected that closed-liganded MalE would have a high affinity for the transporter. However, it was reported to have a low affinity for the transporter, 50-100 µM [9]. This further suggests that the underlying assumptions of the standard model may conflict with experimental evidence. Nanodisc experiments have shown that the open, unliganded conformation of MalE has a high affinity for the open conformation of MalFGK2 . These experiments also showed that MalE with a high affinity for maltose has a low transport rate while MalE with a low affinity for maltose has a high transport rate [1]. An alternate pathway has been suggested to justify these results in which the unliganded MalE plays a critical role for transport.  1.4  Alternate Model of Maltose Transport  In contrast to the standard model, and in support of previous modeling suggesting that unliganded MalE does associate with the transporter [8], an alternate theory of maltose transport has been proposed [1]. In what we will refer to as the Alternate model, unliganded MalE binds with high affinity to MalFGK2 , thus increasing the affinity of the complex for maltose, to facilitate the acquisition of maltose. The Alternate model of maltose transport is supported by experimental data obtained using detergents, liposomes, and nanodisc experiments to investigate MalFGK2 [1]. Experimental data were used to conclude that transporter-bound MalE facilitates the acquisition of the sugar at low concentrations, while at high sugar concentrations the MalE can bind to maltose and dissociate [1]. This allosteric regulation of maltose therefore limits the overall rate to transport.  4  1.5  Previous Modeling of Maltose Uptake System  A basic standard model for the maltose uptake system has been proposed and analyzed [8]. The following reactions were included: M +E  Em where M is the substrate (maltose), E is the binding protein, and Em is the  closed, liganded binding protein. Em + A  D → A + E, where A is the MalFGK2 transporter, D is the complex formed when  MalGK2 binds to the closed liganded binding protein. These two reactions characterize the standard model. This model was taken a step further to allow for unliganded binding protein to bind to the transporter. This adds the reaction: A+E  G, where G is the MalFGK2 complex with the unliganded binding protein.  From this model it was concluded that (at least in the case of the maltose uptake system) it was likely that both the liganded and unliganded conformations of MalE associate with the transporter. A later extension of this model gave further evidence that both binding protein conformations bind to the transporter. The above model was fit to experimental data in which maltose transport was measured in the presence of constant maltose, but varying binding protein concentration [8]. This model analysis supports the theory that both liganded and unliganded MalE bind to the transporter. However, the model does not distinguish between the open and closed conformations of MalFGK2 , and therefore does not give insight into whether liganded MalE binds to the closed transporter to trigger a conformational change. This model analysis does not give insight into the role of unliganded MalE in transport.  5  Chapter 2  Mathematical Models We consider four biochemical reaction networks and their associated ordinary differential equation models for the maltose uptake system in E. coli. We refer to these four models as the Standard, Alternate, Comprehensive, and Full models. Table 2.2 gives the symbols used for concentrations of MalFGK2 transporter states, binding protein (MalE) states, and substrate (maltose) for the four models. We assume that the concentration of maltose in the periplasm is large relative to the concentration of transporter and binding protein. We therefore make the simplifying assumption that maltose concentration is approximately constant. All concentrations of transporter states and binding protein states are assumed to vary with time when systems are not at equilibrium. A further model assumption is that nucleotides are required to close the ATP interface resulting in a P-open transporter conformation [5]. Therefore in the absence of nucleotides, we assume that only P-closed (ATP interface open) transporter states are possible and that hydrolysis is inhibited. In the presence of AMP-PNP, a non-hydrolysable ATP analog, we assume that the ATP interface can close but cannot hydrolyze. This stabilizes the P-open conformation of the transporter. We consider two cases in the presence of AMP-PNP. The simpler case is that all ATP interfaces bind and remain closed in the presence of saturating AMP-PNP, in which case we assume that only P-open transporter states are possible. In the more general case we allow for both P-open and P-closed transporter states in the presence of AMP-PNP (only hydrolysis is inhibited). Both the Standard and Alternate models are nested within the Comprehensive model as 6  seen in Figure 2.3. The Comprehensive model is nested within the Full model shown in Figure 2.4.  7  Model  All  Standard  Illustration  Symbol  Definition  E  MalE  M  Maltose  Em  Closed, liganded MalE  A  P-closed transporter (open ATP interface)  F  P-open E-MalFGK2 complex with M associated  D  Liganded MalE bound to A  B  P-open transporter (closed ATP interface)  C  Unliganded MalE bound to B  Alternate  Table 2.1: The Comprehensive model consists of all concentration states in the above table.  8  Model  Illustration  Full  Symbol  Definition  G  Unliganded MalE bound to A  Table 2.2: The Full model consists of all concentration states in Table 2.1 and the above table.  2.1  Standard Model  The Standard Model consists of the following reactions: E+M  Ko  A + Em  EM  K3  D  K4  h  F →3 A + E + Min (standard pathway)  The Standard model includes transporter states A, D, and F as seen in Table 2.2 and Figure 2.1. The single pathway of maltose transport in this model is for a maltose-bound, closed-liganded MalE (Em ) to bind to the closed transporter resting state. This association instigates a conformational change which induces the opening of MalE. Maltose is subsequently released into the binding pocket of the transporter for hydrolysis and import. MalE then dissociates from the transporter and returns to the periplasm. Hydrolysis is assumed to occur at some constant rate h3 . MalE-maltose-FGK2 complex state (F) is the only P-open transporter state in the Standard model. For the sake of comparison to data (discussed later), in the absence of nucleotides we assume the forward rate k4+ = 0 (see Table 2.4 for kinetic rates), therefore the concentration F = 0 in the absence of nucleotides. In the presence of non-hydrolysable AMP-PNP, we assume both P-open states A and D, and P-closed state F are possible.  2.2  Alternate Model  The Alternate model consists of the following reactions: E+M  Ko  EM  9  E+M  Em  A  Figure 2.1: Standard model.  10  F  D  A  K1  h  B →1 A K2  B+E C +M F  βK2  h  C →2 A + E  βKo  h  F →3 A + E + Min (alternate pathway)  B + Em (open pathway or autoregulation step)  The Alternate model includes MalFGK2 states A, B, C, and F as shown in Table 2.2 and Figure 2.2. This model does not allow for MalE (liganded or unliganded) to bind to the closed transporter (A), and therefore does not include the standard pathway of maltose transport. The alternate pathway of maltose transport is for an unliganded MalE (E) to bind to the P-open conformation of MalFGK2 (B), forming a complex (C) which is assumed to be stabilized or increase the affinity of the transporter for maltose. Maltose can then subsequently bind to the complex and be imported to the cytosol via hydrolysis at a rate h3 . We will refer to this as the alternate pathway of maltose transport. We also allow for the MalE in complex F to bind maltose, acquire the closed conformation (Em ), and dissociate into the periplasm in an autoregulatory step. We allow this reaction to be reversible such that a maltose-bound MalE can bind to the open transporter (B), and consequently provide an additional pathway of maltose transport. This allows for the investigation of whether maltose-bound MalE may in fact bind to the open transporter in the presence of nucleotides. We refer to this route of maltose transport as the ‘open’ pathway. ATP hydrolysis is assumed to occur in the absence of MalE and maltose, as shown in nanodisc experiments [1]. In this case the P-open conformation (B) hydrolyzes ATP and returns to the P-closed conformation (A) at a constant rate h1 . It is also assumed that the MalE-complex C can hydrolyze at a constant rate h2 , causing the MalE to dissociate and the return of the transporter to the resting state A. Experiments have shown that the addition of MalE to MalFGK2 increased the ATPase activity 3-fold in nanodiscs and 4-fold in liposomes [1]. We assume that, in the presence of nucleotides, the forward rate from A to B, k1+ , is large compared with the backward rate k1− (see Table 2.4 for kinetic rates) , therefore the equilibrium constant K1 =  k1− k1+  << 1. This assumption is based on experimental evidence that AMP-PNP  (non-hydrolysable nucleotides) stabilizes the P-open conformation of MalFGK2 [1]. The transporter state A is the only P-closed state in the Alternate model. Therefore, in conditions without nucleotides no binding to the transporter can occur for the Alternate model 11  so we set k1+ = 0.  2.3  Comprehensive Model  The Comprehensivel model consists of the following reactions: E+M A  K1  Ko  h  B →1 A K2  B+E  A + Em C +M F  EM  βK2  h  C →2 A + E  K3  βKo  D  K4  h  F →3 A + E + Min (standard pathway)  h  F →3 A + E + Min (alternate pathway)  B + Em (open pathway or autoregulation step)  The Comprehensive model combines all reactions from the Standard and Alternate models, and includes transporter states A, B, C, D, and F , as seen in Table 2.2 and Figure 2.3. Equilibrium constants and kinetic rates for the Comprehensive model are found in Table 2.5 and Table 2.4 . This model allows for maltose transport to occur through the three pathways described previously and include the standard pathway, the alternate pathway (MalE binds to the open transporter first, maltose binds subsequently), and the open pathway in which Em binds to the open transporter (B). There are three P-open states in the Comprehensive model (B, C, F ) which do not occur in the absence of nucleotides. This results from our assumption that the forward rates k1+ , k4+ = 0 without nucleotides to close the interface. Therefore B = C = F = 0 in conditions without nucleotides. In the presence of nucleotides, we consider two cases for the Comprehensive model. In the first case, we assume that all transporter states can occur in the presence of non-hydrolysable nucleotides (AMP-PNP) such that the equilibrium parameter K1 in nonzero. In the second, more simplified case we assume that only P-open transporter states occur in the presence of saturating AMP-PNP, such that A = D = 0 in the presence of AMP-PNP (i.e. K1 → 0).  12  E+M  Em  A  B  Figure 2.2: The Alternative model.  13  C  F  E+M  Em  A  B  C  Figure 2.3: The Comprehensive model.  14  D  F  2.4  Full Model  The Full Model consists of the following reactions: E+M A  K1  Ko  h  B →1 A K2  B+E  A + Em C +M F  βK2  K3  βKo  K3 δ  G+M K4 δ β  h  C →2 A + E D  K4  h  F →3 A + E + Min (Standard pathway)  h  F →3 A + E + Min (Alternate pathway)  B + Em (Open pathway or autoregulation)  A+E  G  EM  G  Ko δ  D  C  The Full model adds one transporter state (G) and three reversible reactions to the Comprehensive model as seen in Table 2.2 and Figure 2.4. Equilibrium constants and kinetic rates for the Full model are found in Table 2.5 and Table 2.4. The Full model can be diagrammed as a cubed network of reactions. Detailed balance ensures that we only gain one additional equilibrium parameter from the Comprehensive model (δ), in addition to six new kinetic rates for each new forward/reverse reaction, as seen in Table 2.4. Transporter state G is unliganded MalE bound to P-closed MalFGK2 . This is an unlikely state but equilibrium fluorescence quenching data indicates that there is binding occurring between MalE and MalFGK2 in the absence of maltose and nucleotides [1], and so we consider state G in order to fit this experimental data. We allow for unliganded MalE to reversibly bind to the closed transporter and for maltose to subsequently bind to this complex, forming transporter state D (closed transporter in complex with MalE and maltose). The Full model also includes a reversible conformation change between open and closed transporter states with unliganded MalE bound. In the absence of nucleotides, it is assumed that B = C = F = 0 as in the Comprehensive 15  model. In the presence of AMP-PNP, we consider the stronger assumption that only P-open states occur (at equilibrium). Experimental data were insufficient to fit the larger Full model without this assumption.  16  E+M  Em  A  B  C  Figure 2.4: The Full model.  17  D  F  G  Model  Parameter  Reaction  Units  ko−  E + M ← Em  t−1  ko+  E + M → Em  c−1 t−1  k3−  A + Em ← D  t−1  k3+  A + Em → D  c−1 t−1  k4−  D←F  t−1  k4+  D→F  t−1  k1−  A←B  t−1  k1+  A→B  t−1  k2−  B+E ←C  t−1  k2+  B+E →C  c−1 t−1  β1 ko−  C +M ←F  t−1  βo ko+  C +M →F  c−1 t−1  β3 k2−  B + Em ← F  t−1  β2 k2+  B + Em → F  c−1 t−1  All  Standard  Alternate  Table 2.3: The Comprehensive model consists of all of the kinetic rates and reactions in the above table.  18  Model  Parameter  Reaction  Units  δ1 k3+  A→G  c−1 t−1  δ2 k3−  A←G  t−1  δ3 ko+  G→D  c−1 t−1  δ4 ko−  G←D  t−1  γ + k4+  G→C  t−1  γ − k4−  G←C  t−1  Full  Table 2.4: Additional forward and backward kinetic rates for the Full model. The Full model also consists of all parameters and reactions in Table 2.2.  19  Model  All  Standard  Parameter  Reaction  ko− ko+  E+M  K3 =  k3− k3+  A + Em  K4 =  k4− k4+  D  Full  K4  K3  c  D  c  F  d  k  K1 =  k1− k1+  A  K2 =  k2− k2+  B+E  β=  Em  t−1  h F → A + Min + E  kh  Alternate  Ko  Ko =  Units  β3 β2  =  β1 βo  K1  B  d K2  B + Em  c  C  βK2  F  βKo  d  C +M  h1  B →1 A  h  t−1  h2  C →2 A + E  h  t−1  h3  F →3 A + Min + E  h  t−1  δ=  δ1 δ2  =  δ4 δ3  =  γ−β γ+  A+E  K3 δ  G+M  δKo  D, G  K4 δ β  C  d  Table 2.5: Model parameters with c=concentration, t=time, and d=dimensionless. The Comprehensive model combines all parameters and reactions from the Standard and Alternate models. The Full model consists of all parameters and reactions shown in the above table.  20  Chapter 3  Experimental Data Several methods were used to study the maltose uptake system in E.coli, including detergents, liposomes, and nanodiscs [1]. Detergents are used to make membrane proteins soluble by forming detergent-protein-lipid micelles. Detergents have drawbacks for membrane protein stability and can interfere with many molecular techniques, including undesired partitioning of substrates and products [13]. Liposomes are most useful when compartmentalization of each side of the bilayer is needed [13]. However, liposomes are large, unstable, and difficult to prepare with precisely controlled size and stoichiometry. Because of the drawbacks associated with detergents and liposomes, nanodiscs are used as an alternative technique for studying membrane proteins. Nanodiscs are soluble nanoscale phospholipid bilayers which have been used to study the MalFGK2 transporter [1]. Nanodiscs can self-assemble integral membrane proteins and are therefore useful for understanding membrane protein function [13]. Nanodiscs allow for solubility of membrane proteins at the single molecule level. This is advantageous compared with liposomes or detergent micelles in terms of the size, stability, and access to both sides of the phospholipid bilayer domain. It also offers advantages in adding genetically modifiable features to the Nanodisc structure [13]. Fluorescence Quenching, specifically an electron-transfer based quenching reaction, was used as a measure of binding between MalFGK2 in nanodiscs (Nd-FGK2 ) and MalE. Quenching was measured at constant MalE concentration, with increasing concentration of Nd-FGK2 in the presence or absence of nucleotides and in the presence or absence of maltose. In the case where nucleotides were present, AMP-PNP was used as a non-hydrolysable ATP analog. This condition stabilizes the P-open state transporter [1].  21  The ATPase activity was measured at 37o C using Nd-FGK2 in the presence of ATP, in the presence or absence of MalE, and in the presence or absence of maltose [1]. The basal ATPase activity in the nanodisc was found to be approximately 10 fold higher than in proteoliposomes, however in both cases MalE increased the rate of ATP hydrolysis. There was a 3-fold increase in nanodiscs and a 4-fold increase in proteoliposomes. In the presence of maltose, an inhibition of the ATPase activity was observed in nanodiscs which was not observed in proteoliposomes. It was proposed that maltose could negatively affect the association of MalE with the transporter [1].  3.1  Fluorescence Quenching Experiments  All experimental data used for this analysis comes from experiments with MalFGK2 in nanodiscs. We will refer to MalFGK2 in nanodiscs as Nd-FGK2 . Fluorescence Quenching, specifically an electron-transfer based quenching reaction, was used as a measure of binding between MalFGK2 in nanodiscs (Nd-FGK2 ) and MalE. Equilibrium quenching was measured for a fixed MalE concentration, for a range of increasing concentration of Nd-FGK2 in the presence or absence of nucleotides and in the presence or absence of maltose. In the case where nucleotides were present, AMP-PNP was used as a non-hydrolysable ATP analog. This condition stabilizes the P-open state transporter [1].  3.2  Equilibrium Data  The equilibrium quenching was measured as a function of increasing Nd-FGK2 concentration for a fixed total MalE concentration of 20 nM. Experiments were done in the presence or absence of 1 mM of maltose, as shown in Figure 3.1.  3.3  Time-Dependent Data  Fluorescence quenching was measured as a function of time in minutes. The concentration of MalFGK2 was fixed at 90 nM, and the concentration of MalE at 20nM. We consider the experiment performed in the presence of AMP-PNP and in the absence of maltose as seen in Figure 3.2.  22  Figure 3.1: Equilibrium quenching data under 4 conditions: o= AMP-PNP (no maltose), + = AMP-PNP (with maltose), *=no nucleotides (no maltose), .=no nucleotides (with maltose). Tt is total concentration of Nd-FGK2 . This data appears in [1].  23  Figure 3.2: Time course quenching data in the presence of AMP-PNP to block hydrolysis, MalE (20 nM), and Nd-FGK2 (90 nM).  24  3.4  ATPase Experiments  Steady-state ATPase rates were measured for concentrations of MalE and Nd-FGK2 fixed at 2 µM in the presence of ATP. Experiments were performed under three conditions in nanodiscs: in the presence of MalE and maltose, in the presence of MalE only, and with only Nd-FGK2 .  3.5  Fit Methods  All models were fit to equilibrium fluorescence quenching data measured under hydrolysisblocking conditions, as well as to the steady-state ATPase data. The time-dependent data measured in the absence of maltose was used to fit the kinetic rate k2+ in the Comprehensive model, as well as to estimate the magnitude of the rate k1+ , the forward rate of the conformation change from P-closed to P-open transporter. Nonlinear regression and Matlab R were used to fit model parameters to experimental data and to generate confidence intervals on fit values. Estimates of equilibrium constants and hydrolysis rates of the maltose uptake system in nanodiscs were obtained for each model. Equilibrium fluorescence quenching data for Nd-FGK2 and MalE were used fit the equilibrium constants (Ki =  ki− ). ki+  ATPase data were used to solve for the hydrolysis rates h1 , h2 , h3 , using  fit equilibrium values. We assume that hydrolysis is the rate limiting step, and that we are at equilibrium under hydrolysis-blocking conditions. At equilibrium, the concentrations of each transporter state are found explicitly as a function of maltose, Ko (the affinity of MalE for maltose), and all other relevant equilibrium constants. To fit the quenching data we assume that in each model, the transporter states that have MalE bound represent quenching concentrations. We assume that the quenching data has been vertically scaled by some common unknown constant φ, which was fit in addition to all model parameters.  3.6  Fit Analysis  The Standard model and Alternate model have the same number of equilibrium constants being fit to the data. Both models are fit to three data sets (n=36), under conditions with AMP25  PNP (with and without maltose) and in the absence of nucleotides with maltose present. The sum of the squared residual error (SSR) is compared to determine which model is a better fit to the equilibrium data set. Neither the Standard or Alternate models have binding between MalE and MalFGK2 under conditions without nucleotides and without maltose, so this experimental data set is omitted from this comparison for simplicity. The Standard and Alternate model are both nested within the Comprehensive model. An F-test was used to determine whether the Comprehensive model is a significantly better fit to the three equilibrium data sets than the Standard or Alternate models. The Comprehensive model also has no fit to experimental data in the absence of both nucleotides and maltose, allowing for comparison of the Standard, Alternate, and Comprehensive models using only the three data sets described. In order to fit the fourth experimental equilibrium data set measured in the absence of both nucleotides and maltose, the Comprehensive and Full models are both fit to all four equilibrium data sets (n=48). The function quenching=0 was used for the Comprehensive model in the fourth data set in order to compare fits with the Full model. The Comprehensive model is nested within the Full model, and an F-test is used to determine whether the additional parameter in the Full model produces a significantly better fit to the four data sets. We consider in this case whether the extra data set measured in the absence of nucleotides and maltose is significant to the overall maltose uptake system.  26  Chapter 4  Nonlinear Regression Fit 4.1  Equilibrium Data Fit  4.1.1  Standard Model  In the absense of hydrolysis we have the following equalities at equilibrium for the standard model: D=  AEm K3 ,  F =  D K4 ,  E=  Em Ko M ,  A + D + F = Tt , E + D + F + Em = Et , where Tt =total [NdFGK2 ], Et =total [MalE]. From this system we obtain the following equilibrium concentration functions for each state: D = 0.5(( Eωt + where γ = F =  Ko M  Tt ω  +  K3 γ ) ω2  ±  + 1, ω = 1 +  ( Eωt +  Tt ω  +  K3 γ 2 ) ω2  1 K4 ,  D K4 ,  A = Tt − D − F , Em =  Eo −ωD , γ  and 27  − 4 Tωt E2 t )  Em Ko M .  E=  See Appendix A for full derivation and nondimensionalization. The nondimensional equations were fit to equilibrium quenching data for experiments for comparison to the CS model. Each data set contains twelve measurements (n=36). The data were measured under the following conditions: 1) no nucleotides with maltose, 2) AMP-PNP with maltose, and 3) AMP-PNP without maltose. The nondimensionalized versions of the functions below were used to fit each data set:  1. quenching=  D φ  2. quenching=  D+F φ  , ,  3. quenching= 0 was used for comparison to Alternate model fit, as no binding occurs between MalFGK2 and MalE for the Standard model.  The Standard model was fit for the equilibrium constants K3 and K4 , and the vertical scaling constant φ. Fit results are shown in Table 4.1 and in Figure 4.1.  4.1.2  Alternate Model  In the absense of hydrolysis we have the following equalities at equilibrium for the Alternate model: A = K1 B, BE = K2 C, BEm = βK2 F , βKo F = CM , EM = Ko Em , A + B + C + F = Tt , and E + Em + C + F = Et . From this system we obtain the following equilibrium functions for concentration of each state: 28  C = 0.5[( ψξ + ν2  Et ν  +  Tt ν )  ±  ( ψξ + ν2  where ψ = K2 (K1 + 1), ξ = 1 +  Et ν  M Ko ,  +  Tt 2 ν )  −  4Tt Et ] ν2  ν = 1 + σ, and σ =  M Ko β .  All other variables can be  expressed in terms of C as follows: E=  Et −νC ξ  Em =  EM Ko  , ,  ξ B = K2 C Et −νC ,  A = K1 B , and F = σC. See Appendix B for full derivation and nondimensionalization. The nondimensional equations were fit to equilibrium quenching data for the same three experiments for comparison to the Standard model and Comprehensive model. Each data set contains twelve measurements (n=36). The data were measured under the following conditions:1) no nucleotides with maltose, 2) AMP-PNP with maltose, and 3) AMP-PNP without maltose. The nondimensionalized versions of the functions below were used to fit each data set:  1. quenching = 0 was used for comparison to the Standard mode fitl as no binding occurs between MalFGK2 and MalE for the Alternate model , , 2. quenching= C+F φ 3. quenching= Cφ .  The Alternate model was fit for equilibrium parameters K2 and β, using the assumption that K1 << 1 in the presence of AMP-PNP, so that ψ = K2 . We do this to eliminate the need to fit K1 []. Evidence for this assumption comes from experiments indicating that the P-open conformation of the transporter is stabilized in the presence of non-hydrolyzable nucletides [1]. Stabilized P-open transporters would correspond to k1− << 1, where k1 is the backward rate from the P-open to P-closed transporter states: A ← B, and K1 = Figure 4.1 for fit results.  29  k1− . k1+  See Table 4.1 and  Model  Standard  Alternate  Parameter  Fit Value  95% Confidence Interval  K3  6.509832 µM  (-24.37, 37.39) µM  K4  0.059  (-0.2696, 0.3877)  φ  34.7222  (-20.123, 89.5674)  kh  0.7514 min−1  K2  44.94 nM  (30.702, 59.176) nM  β  14.3858  (9.7109, 19.0607)  φ  27.943  (26.2673, 29.6188)  h1  0.35 min−1  h2  1.2212 min−1  (1.2034, 1.2371) min−1  h3  0.9575 min−1  (0.8767, 1.0492) min−1  Table 4.1: Fit values for Standard and Alternate models. Hydrolysis rates were calculated using experimental values found in [1] and fit values above. The value of h1 has no dependence on fit values and therefore does not have a confidence interval.  30  4.1.3  Comprehensive Model  In the absense of hydrolysis we have the following equalities at equilibrium for the Comprehensive model: A = K1 B, BE = K2 C, BEm = βK2 F , βKo F = CM , D = K4 F , E + Em + C + D + F = Et , and A + B + C + D + F = Tt . From this system we obtain the following equilibrium functions for concentration of each state: + C = 0.5[( ψξ α2 where σ =  Eo α  M Ko β ,  +  To α)  ±  ξ = 1+  + ( ψξ α2  M Ko ,  Eo α  +  To 2 α)  −  4To Eo ] α2  α = 1 + σ(K4 + 1), and ψ = K2 (1 + K1 ). All other variables  can be expressed in terms of C as follows: F = σC, D = K4 F , Em =  EM Ko ,  E=  Eo −αC , ξ  B=  K2 Cξ Eo −αC .  and  See Appendix for full derivation and nondimensionalization. The nondimensional equations were fit to the same equilibrium quenching data for three experiments for comparison to the Alternate model. Each data set contains twelve measurements (n=36). The data were measured under the following conditions:1) no nucleotides with maltose, 2) AMP-PNP with maltose, and 3) AMP-PNP without maltose. The nondimensionalized versions of the functions below were 31  used to fit each data set:  1. quenching= D φ , , 2. quenching= C+D+F φ 3. quenching= Cφ .  The Comprehensive model was fit for equilibrium parameters K2 , K3 , and β and scaling constant φ. To allow for the possibility that K1 is not very small in the presense of nucleotides, and to determine the effect of the magnitude of K1 on model fits, the value of K1 was fixed at 0.01, 0.1, 1, and 100. The value of K4 was calculated using detailed balance. See Table ??, Table ??, and Figure 4.1 for fit results. We also fit equilibrium constants in the Comprehensive model using the simpler assumption that K1 → 0 in the presense of AMP-PNP (the ATP interface closes and is stabilized for all transporter present). See Appendix C for full derivation, and Table 4.6 for fit results.  4.1.4  Full Model  The concentrations of each state in the Full model at equilibrium were found using the following equalities and conservation equations: A = K1 B, BE = K2 C, BEm = βK2 F , βKo F = CM , D = K4 F , AE =  K3 δ G,  AEm = K3 D, E + Em + C + D + F + G = Et , A + B + C + D + F + G = Tt ,  32  Figure 4.1: Equilibrium quenching data under 4 conditions: o= AMP-PNP (no maltose), + = AMP-PNP (with maltose),*=no nucleotides (no maltose), .=no nucleotides (with maltose). The Standard model (upper left) has no fit to data in the absence of maltose. The Alternate model (upper right) has no fit in the absence of nucleotides. The Comprehensive model (lower) has no fit under conditions of no nucleotides when maltose is absent but does fit the data when maltose is present.  33  K1  Parameter  Fit Value  95% Confidence Interval  14.3966  (11.3702, 17.4231)  K2  44.49703 nM  (35.3802, 53.61188) nM  K4  0.00076413  (0.0005, 0.001)  K3  8.3833 µM  (5.4992, 11.2673) µM  φ  27.9428  (26.859, 29.0265)  h1  0.3535 min−1  h2  1.2212 min−1  (1.2072, 1.234) min−1  h3  0.9583 min−1  (0.8927, 1.0306) min−1  β  0.01  Table 4.2: Fit to nanodisc data for Comprehensive model parameters, for K1 = 0.01 . Hydrolysis rates were calculated using experimental values found in [1] and fit values above. The value of h1 has no dependence on fit values and therefore does not have a confidence interval.  34  K1  Parameter  Fit Value  95% Confidence Interval  14.4872  (11.4347, 17.5397)  K2  40.85636 nM  (32.48545, 49.22545) nM  K4  0.0071  (0.0044, 0.0097)  K3  8.3833 µM  (5.4992, 11.2673) µM  φ  27.9428  (26.859, 29.0265)  h1  0.3535 min−1  h2  1.2212 min−1  (1.2072, 1.234) min−1  h3  0.9643 min−1  (0.8964, 1.0392) min−1  β  0.1  Table 4.3: Fit to nanodisc data for Comprehensive model parameters, for K1 = 0.1. Hydrolysis rates were calculated using experimental values found in [1] and fit values above. The value of h1 has no dependence on fit values and therefore does not have a confidence interval.  35  K1  Parameter  Fit Value  95% Confidence Interval  14.9625  (11.766, 18.159)  K2  22.471 nM  (17.867, 27.074) nM  K4  0.0401  (0.0248, 0.0554)  K3  8.3833 µM  (5.4992, 11.2673) µM  φ  27.9428  (26.859, 29.0265)  h1  0.7 min−1  h2  1.2212 min−1  (1.2072, 1.234) min−1  h3  0.9959 min−1  0.9156, 1.084) min−1  β  1  Table 4.4: Fit to nanodisc data for Comprehensive model parameters, for K1 = 1 . Hydrolysis rates were calculated using experimental values found in [1] and fit values above. The value of h1 has no dependence on fit values and therefore does not have a confidence interval.  36  β  14.9625  (11.766, 18.159)  K2  0.44497 nM  (0.353802, 0.536119) nM  K4  0.0827  (0.0498, 0.1156)  K3  8.3833 µM  (5.4992, 11.2673) µM  φ  27.9428  (26.859, 29.0265)  h1  35.35 min−1  h2  1.2212 min−1  (1.2072, 1.234) min−1  h3  1.0367 min−1  ( 0.9392, 1.1431) min−1  100  Table 4.5: Fit to nanodisc data for Comprehensive model parameters, for K1 = 100. Hydrolysis rates were calculated using experimental values found in [1] and fit values above. The value of h1 has no dependence on fit values and therefore does not have a confidence interval.  K1  Parameter  Fit Value  95% Confidence Interval  14.3856  (11.3623, 17.4089)  K2  44.942 nM  (35.734, 54.148) nM  K3  8.3833 µM  (5.4992, 11.2673) µM  27.9428  (26.859, 29.0265)  β  All Values  φ  Table 4.6: Fit to Comprehensive model with the assumption that K1 → 0. The parameter K4 =  βK2 K1 K3  using detailed balance. The hydrolysis rate h1 is also a function of K1 as shown  in Section 4.3. As K1 → 0, h1 → 0.35 min−1 .  37  Figure 4.2: Equilibrium fit for Comprehensive model (top) and Full model (bottom). Equilibrium quenching data under 4 conditions: o= AMP-PNP (no maltose), + = AMP-PNP (with maltose), *=no nucleotides (no maltose), .=no nucleotides (with maltose).  38  F = σC, where σ = Em =  M Ko β  and  EM Ko .  We assume that in the absense of nucleotides, we only have P-closed transporter states: A + D + G = Tt , E + Em + D + G = Et . Then, G = 0.5[( Kξ23δγ + D=  M Ko δ G,  Et ξ  +  Tt ξ )  ±  where ξ = 1 +  ( Kξ23δγ +  Et ξ  +  Tt 2 ξ )  −  4Tt Et ], ξ2  and  M Ko δ .  See Appendix for full derivation and nondimensionalization. The nondimensional equations were fit to the equilibrium quenching data for all four experimental conditions for comparison to the Alternate model. Each data set contains twelve measurements (n=48). The data were measured under the following conditions:1) no nucleotides with maltose, 2) AMP-PNP with maltose, 3) AMP-PNP without maltose, and 4) no nucleotides without maltose. The nondimensionalized versions of the functions below were used to fit each data set:  , 1. quenching= D+G φ 2. quenching= C+F φ , 3. quenching= Cφ , 4. quenching= G φ.  The Full model was fit for equilibrium parameters K2 , K3 , β, δ, and scaling constant φ. Experimental data were insufficient to fit K1 , therefore the value of K1 was fixed at 0.01 and the value of K4 was calculated using detailed balance. See Table 4.7 and Figure 4.2 for fit results.  39  Parameter  Fit Value|K1 =0.01  β  14.3914  (11.5781, 17.2046)  δ  2.2197  (1.3961, 3.0433)  K2  44.888 nM  K4  0.000767477  (0.0005, 0.001)  K3  8.417206 µM  (5.713, 11.12141) µM  0.000118374  (0.0000887903, 0.00146) µM  K5 =  K3 δ  φ  95% Confidence Interval  (36.334, 53.44) nM  27.9523  (26.9445, 28.9601)  Table 4.7: Fit to Full model parameters. Here K4 again depends on K1 using detailed balance, and h1 depends on K1 .  4.2 4.2.1  Time-Dependent Data Fit In the limit K1 → 0  If we assume that AMP-PNP closes and stabilizes the ATP interface for all transporters present (K1 → 0), we can fit the value of k2+ using experimental time course of the quenching data in the presence of AMP-PNP (no maltose). The result relies upon our fit value of K2 = 44.942 nM, where K2 was also fit under the assumption that K1 → 0. In the absence of maltose and of P-closed MalFGK2 , we have the following system of ordinary differential equations B + C = Tt , C + E = Et , dB dt  = −k2+ BE + k2− C  dC dt  = k2+ BE − k2− C  40  = −k2+ BE + k2− C  dE dt  which can be simplified to give: = k2+ (Tt − C)(Et − C) − k2− C  dC dt  C(t) =  bk+ t 2 ) + bk+ t 1− c− e 2 c  c+ (1−e  ,  where, using Et = 20 nM, Tt = 90 nM: b=  (110 + K2 )2 − 7200  c+ =  110+K2 +b 2  c− =  110+K2 −b . 2  The fit to time course data gives k2+ = 0.0324 (nm)−1 (min)−1 , with 95% confidence interval (0.0289, 0.0359) and sum of squared residual error (SSR) 0.0055, using K2 = 44.942 nM. See Figure 4.3 for fit to time course data.  4.2.2  Fixed K1 , k1+  We now consider the less simplified case where both P-open and P-closed transporters can occur in the presence of AMP-PNP (K1 = 0). In the absence of maltose, and in the presence of AMP-PNP to block hydrolysis, we have the following system of ordinary differential equations for the Comprehensive model: dA dt  = −k1+ A + k1− B  dB dt  = k1+ A − k1− B − k2+ BE + k2− C  dC dt  = −k2− C + k2+ BE  A + B + C = Tt C + E = Et Substituting A = Tt − B − C, and E = Et − C into the above system, we obtain: dB dt  = k1+ (Tt − B − C) − k1− B − k2+ B(Et − C) + k2− C  41  dC dt  = −k2− C + k2+ B(Et − C).  We let b, c, and τ be nondimensional variables such that b =  B Et ,  c=  C Et ,  τ = tk1+ , κ =  k2+ , k1+  κ1 = K2 κ, κ2 = κEt : db dτ  = (ρ − b − c) − K1 b − κ2 b(1 − c) + κ1 c  dc dτ  = −κ1 c + κ2 b(1 − c)  We fit k2+ for a fixed magnitude of k1+ and K1 , and fit value K2 = 44.49703 nM. See Figure 4.3 and Table 4.8 for fit results. It was necessary to fix both k1+ and K1 rather than allowing the regression to fit the best values, as the one time-dependent quenching data set was only sufficient to fit k2+ and the quenching vertical scaling constant φ. For each fixed K1 , we observed how the magnitude of k1+ changes the fit value of k2+ and the associated SSR for the fit.  4.3  ATPase Data Fit  To calculate the hydrolysis rates in each model, the following functions were used where B, C, F are as written above. h1 B|Et =0,M =0 = R1 h1 B|M =0 + h2 C|M =0 = R2 h1 B + h2 C + h3 F = R3 where R1,2,3 are the experimental NdFGK2 ATPase rates measured in the presense of 1) No MalE and no maltose, 2) MalE without maltose, 3) MalE and maltose. Fit values of equilibrium parameters were used to calculate hydrolysis rates and their confidence intervals for each model.  42  K1  k1+ min−1 (fixed)  k2+ nm−1 min−1 (fit)  SSR  0.01  1  1.983  2.1638 ∗ 10−4  10  0.0927  5.1046 ∗ 10−5  102  0.0356  1.2705 ∗ 10−5  103  0.0327  1.3475 ∗ 10−5  104  0.0324  1.367 ∗ 10−5  1  3.6924  1.8126 ∗ 10−4  10  0.0932  4.9812 ∗ 10−5  102  0.0385  1.2669 ∗ 10−5  103  0.0356  1.3492 ∗ 10−5  104  0.0353  1.3672 ∗ 10−5  1  3.2416  6.057 ∗ 10−5  10  0.1110  3.3526 ∗ 10−5  102  0.0674  1.2850 ∗ 10−5  103  0.0645  1.3581 ∗ 10−5  104  0.0642  1.3681 ∗ 10−5  1  7.4885  2.0009 ∗ 10−5  10  3.4511  1.3791 ∗ 10−5  102  3.2609  1.3695 ∗ 10−5  103  3.2429  1.3693 ∗ 10−5  104  3.4686  1.422 ∗ 10−5  0.1  1  100  Table 4.8: Fit of k2+ for fixed K1 , k1+ . Values of k1+ < 1 gave fits to the data with higher SSR than for k1+ > 1. Best fits to the data occured for k1+ = 102 min−1 . Furthermore, the best fit occured for K1 = 0.1, k1+ = 102 min−1 , k2+ = 0.0385 nm−1 min−1 .  43  Figure 4.3: Time-dependent fit.  44  Chapter 5  Model Comparison To analyze the model fits to equilibrium quenching data, we compare the sum of squared residual errors (SSR) for the various models. We also determine whether the additional terms in the Comprehensive model and Full model are significant to the data when compared to their nested models. In our first comparison, we consider the fit to data sets from quenching experiments measured under three conditions: AMP-PNP only, AMP-PNP with maltose, and no nucleotides with maltose (n=36), as shown in Figure 4.1. The Standard, Alternate, and Comprehensive model fits are compared for this first data set. In our second comparison, we consider the Comprehensive model and Full model fits to data sets measured under all four experimental conditions, as shown in Figure 4.2. This includes quenching data measured in the absence of nucleotides and without maltose present, which is omitted from the first comparison as only the Full model has quenching under conditions without nucleotides and without maltose.  5.1  Standard vs. Alternate  Both the Alternate model and Standard model have two equilibrium constants fit to quenching data, therefore we simply compare the SSR for the model fits of these parameters. The Alternate model is a better fit than the Standard model as it has a lower SSR of 9.8918 ∗ 10−5 as seen in Table 5.1. This result is not surprising as the Standard model does not allow for the binding of unliganded MalE to the transporter, and is therefore insufficient to explain the quenching experiment performed in the presence of AMP-PNP, without maltose, in which strong quenching was observed. The Alternate model does not allow for the binding of MalE to maltose in the absence of nucleotides and therefore contradicts the quenching experimental  45  results performed in the absence of nucleotides. Since the equilibrium quenching measured was less strong in the absence of nucleotides compared with AMP-PNP as shown in Figure 3.1, the error contributed to the Alternate model for omitting this reaction was small compared with the error contributed to the Standard model for omitting a MalE-MalFGK2 quenching reaction. Model  Fit Parameters  SSR1  SSR2  Standard  K3 , K4  0.0063  0.0065  Alternate  K2 , β  9.8918 ∗ 10−5  3.2017 ∗ 10−4  Comprehensive  K2 , β, K3  4.0025 ∗ 10−5  2.67 ∗ 10−4  Full  K2 , β, K3 , δ  4.7432 ∗ 10−5  Table 5.1: Parameters fit to equilibrium nanodisc quenching data. SSR1 gives the sum of the squared residual error (SSR) for the model fit to 3 experiments in conditions of AMP-PNP (with and without maltose) and without nucleotides (with maltose), and SSR2 gives the SSR for model fits to all 4 experiments, including conditions without nucleotides (no maltose). The Full model has too many parameters to fit to only 3 experiments (therefore it has no SSR1 ).  5.2  Alternate vs. Comprehensive  Our analysis of the Standard and Alternate model fits above to the equilibrium data show that the Comprehensive model, which includes all reactions from both models, will better explain the data measured in all three conditions described above. Comparing fits for AMPPNP quenching experiments only, the Comprehensive model and Alternate model fit the data with the same SSR, indicating that the additional parameter in the Comprehensive model is not significant to the data when excluding data without nucleotides. However, when we include the additional data set from the no nucleotides experiment in the presence of maltose, the Comprehensive model is clearly a better fit as seen in Figure 4.1. We use an F-test to compare the SSR for the fits as seen in Appendix E and find there is a statistically significant advantage of the Comprehensive model over the Alternate model 46  in predicting the experimental quenching data for the three data sets. The parameter K3 , the equilibrium constant for binding liganded MalE to the closed transporter, is the additional equilibrium parameter in the Comprehensive model compared with the nested Alternate model. Equilibrium constant K3 is found to be significant to the data (p < 0.0001). See Appendix E for analysis. We conclude that the Comprehensive model is a better fit to the data than the Alternate model, and that K3 = 0 when comparing fits to experimental data measured in the presence of AMP-PNP, AMP-PNP (with maltose), and no nucleotides (with maltose).  5.3  Comprehensive vs. Full  We now compare the Full and Comprehensive models for the equilibrium parameter fits to all four quenching data sets as shown in Figure 4.2. See Table 5.1 for the SSR for each of these model fits. The Full model and Comprehensive model are fit to all 4 quenching experiments including the no nucleotides (no maltose) experiment. See Appendix E for analysis. We conclude that the Full model is a better fit to the data than the Comprehensive model when considering all four equilibrium experiments in the absence of nucleotides and in the presence of AMP-PNP. We conclude that the equilibrium parameter δ is nonzero in the reaction network of the maltose uptake system given by the Full model (p < 0.0001).  47  Chapter 6  Maltose Transport in Comprehensive Model 6.1  Analysis of Flux  Using nonlinear regression model fits we have shown that the Full and Comprehensive models provide superior fits to equilibrium quenching data compared with the Standard and Alternate models. Both the Full and Comprehensive models contain the following pathways for maltose transport: the standard, alternate, and open pathways. Along the standard pathway, liganded MalE binds to the P-closed transporter to initiate maltose transport. The open pathway of transport is initiated by the binding of liganded MalE to the P-open transporter. The reverse reaction has been described as an autoregulatory step through which transport is inhibited at high maltose concentration. The alternate pathway of transport is for unliganded MalE to selectively bind the P-open transporter forming a highaffinity complex, and for maltose to subsequently bind and be imported to the cytosol. We use experimental nanodisc data measured in the presence of ATP and maltose to analyze the flux through of each of these three pathways [1]. The Comprehensive model is used for this analysis as the available data are insufficient to estimate all of the additional kinetic parameters in the Full model shown in Table 2.4. The ordinary differential equation system for the Comprehensive model is analyzed using equilibrium constants K3 , K2 , β and hydrolysis rates h1 , h2 , and h3 , for the Comprehensive model as fit by nonlinear regression (see Chapter 4). The fit value for the binding rate of MalE to 48  P-open transporter k2+ from the time-dependent data as well as the estimate of the magnitude of k1+ were used for fixed K1 values. Kinetic rates k3+ (binding of P-closed transporter to liganded MalE), and k4+ (conformation change from P-closed to P-open with Em bound) are unknown. The binding rate of MalE to maltose, ko+ , has been measured experimentally at approximately 10−3 nm−1 min−1 [1]. The proportion of the equilibrium constant β applied to each of the associated backward rates, βo , β2 , as seen in Table 2.4 are unknown. We want to determine the dominant pathway of maltose transport in the Comprehensive model for various cases of our unknown forward rates. In order to solve the ODE system at steady state, we take an asymptotics approach. Using our fit hydrolysis rates, we have h1 < h3 < h2 , and we assume that the unknown forward rate k1+ is large such that h1 << k1+ . We estimated the magnitude of k1+ by choosing the fixed value with lowest SSR in the time-dependent fit (see Section 4.2.2). This analysis gives k1+ that is very large compared to h1 , for all fixed K1 values. We divide the steady state ODE system through by k1+ and regroup parameters. We let = then let γ =  ko+ , k1+  α=  k2+ , k1+  δ=  k3+ , k1+  ξ=  k4+ . k1+  h1 k1+  be a small parameter. We  We let each concentration A, B, C, D, F , E, Em be  written as an asymptotic solution, A = Ao + A1 , B = Bo + B1 , ...., Em = Emo + Em1 , and subsitute into the steady state ODE system. The resulting system and expressions for values of A, B, ..., Em , can be found in Appendix F. They depend on kj+ , j = 0, 1, 2, 3, 4, the kinetic binding rates, the solutions to the steady state system at equilibrium found in Appendix C, and on the equilibrium constants, Ki . We now calculate the flux through each pathway of the full model. We first label the fluxes as follows: f1  f2  f3  f5  f5  g3  f4  g3  g3  a A → B → C → F →, f3 = alternate pathway flux  A → D → F →s , f5 =standard pathway flux o B → F →, f4 = open pathway flux  g1  B→A g2  C→A At steady state we have the following system of equations, where g1 = h1 B, g2 = h2 C, g3 = h3 F , and g3 = g3 s + g3 a + g3 o : f1 = f4 + f2 + g1 49  f2 = f3 + g2 f3 + f4 + f5 = g3 . We can write the fluxes f2 , f4 as the following differences: f2 = k2+ BE − k2− C f4 = β2 k2+ BEm − β2 βk2− F Substitution of our asymptotic solutions gives the following approximation of the fluxes at steady state: f2 = k2+ (Bo + B1 )(Eo + E1 ) − k2− (Co + C1 ) f3 = f2 − g2 f4 = β2 k2+ (Bo + B1 )(Emo + Em1 ) − β2 βk2− (Fo + F1 ). f5 = g3 − f4 − (f2 − g2 ) The concentrations B1 , E1 , C1 , F1 , Em1 are found for fixed K1 = {0.01, 0.1, 1, 100} and for the fixed k1+ value found to have the lowest SSR as shown in Table 4.6. All related equilibrium fit values and k2+ fit value are used for each fixed K1 . The remaining unknowns for the flux calculation include k3+ , k4+ , βo , and β2 . The total maltose transport was calculated for fixed ko+ , βo and β2 , where: βo k +  C + M →o F , and the reverse F β2 k+  B + Em →2 F , and the reverse F  βo βko−  → C +M  β2 βk2−  → B + Em .  The parameters βo , β2 control the proportion of the equilibrium fit value β that is applied to the forward and backward rates for the above reactions. Thus we expect these two parameters to influence maltose flux through the alternate pathway (βo ) and the open pathway (β2 ). For example, fixing βo = β2 = 1 assumes that only the backward kinetic rates shown above are affected by the fit equilibrium parameter β. We consider all the fixed cases such that βo = 0.01, 1, 100 and β2 = 0.01, 1, 100, to analyze the effect on the maltose flux along each pathway. The direction and magnitude of the fluxes through each pathway were found to be relatively insensitive to the parameters k3+ and k4+ , and to the fixed values of βo and β2 . An experimental  50  value for ko+ , the forward binding rate of MalE to maltose, was used to compute the scaled flux through each pathway. The flux was not sensitive to values of ko+ greater than or equal to 10−4 nm−1 min−1 , and the experimental value of ko+ = 10−3 nm−1 min−1 falls within this range [1]. The standard pathway was negative for all fixed K1 values less than or equal to 1, as seen in Figure 6.1. The open pathway was the dominant pathway of maltose transport for fixed K1 ≤ 1, and the alternate pathway makes a relatively small contribution to overall transport. The standard pathway was positive and the dominant pathway of maltose transport for the fixed case of K1 = 100, as seen in Figure 6.2. For large K1 the open pathway is still positive, and the alternate pathway has a small negative magnitude.  Figure 6.1: The scaled flux in the Comprehensive model, for fixed K1 = 0.1 and ko+ = 10−3 nm−1 min−1 . For all fixed K1 ≤ 1, the open pathway (dark gray) was the dominant pathway, the standard pathway (light gray) was negative, and the alternate pathway (black) makes a small contribution to overall maltose transport.  Figure 6.2: The scaled flux in the Comprehensive model, for fixed K1 = 100 and ko+ = 10−3 nm−1 min−1 . For large K1 only, the standard model (light gray) was the dominant pathway of maltose transport. The alternate pathway (black) is negative for large K1 , and the open pathway (dark gray) is positive.  51  6.2  Further ATPase Results  Experiments have shown that MalE can inhibit transport in excess when maltose is held at sub-stoichiometric levels [7]. We replicate this result in the Comprehensive model as shown in Figure 6.3. Experiments have also shown that MalE increases the basal ATPase activity in NdFGK2 (MalFGK2 in nanodiscs) by 3-fold. These experiments also found that the in the presence of maltose, an inhibition of ATPase activity was observed in nanodiscs which was not observed in liposomes [1]. We replicate these results in the Comprehensive model as shown in Figure 6.3.  Figure 6.3: Comprehensive model results.(Top Left) Basal steady-state ATPase rate in the absence of MalE and in the absence of maltose. (Top Right) Total steady-state ATPase rate in the presence of 2 µM MalE, without maltose (black) and ATPase rate of unbound transporter state B → A only in the presence of 2 µM MalE (purple). (Bottom Left) Maltose transport inhibited for large concentrations of MalE, for fixed maltose concentration 2 µM. (Bottom Right) ATPase activity is inhibited in the presence of maltose, for fixed MalE concentration 2 µM.  52  Chapter 7  Discussion Experimental data in nanodiscs have shown that the standard model of maltose transport, in which liganded MalE binds to the closed transporter to trigger a conformation change and deliver maltose, is insufficient to explain all binding interactions observed between MalE and MalFGK2 in the absence of hydrolysis. Nonlinear regression fit analysis of four mathematical models has shown that models which include binding of unliganded MalE to the transporter produce significantly better fits to equilibrium data obtained from nanodisc experiments. Additionally, it was shown that omission of the standard pathway of maltose transport in the Alternate model did not result in a better fit to equilibrium data when compared with the Comprehensive model. This indicates that the Alternate model is insufficient to explain experimental binding data in the absence of nucleotides. The Full model was shown to be a significantly better fit to equilibrium nanodisc data than the three models nested within it, as it provides binding reactions for MalE and maltose under all experimental conditions considered. The nonlinear regression analysis confirms that a complex network of binding interactions can occur between the components of the maltose uptake system in E.coli when hydrolysis is blocked. The three possible pathways of maltose transport available within this network were analyzed using the Comprehensive model at steady state in the presence of ATP. The direction and quantity of maltose flux through each available pathway was computed. Results were relatively insensitive to values of the unknown parameters k3+ and k4+ , and to fixed values of βo and β2 . The standard pathway was negative for fixed values of the equilibrium constant between P-open and P-closed conformations of the transporter, K1 , less than or equal to 1. For large K1 , the standard pathway was positive and the major pathway of maltose  53  transport. The fit analysis to nanodisc time course data gave the best fit for fixed value K1 = 0.1. For K1 = 0.1, and for all fixed K1 ≤ 1, the open pathway was the major pathway of maltose transport. A relatively small flux of maltose was calculated along the alternate pathway. The negative flux along the standard pathway calculated in this range of small K1 could represent a means of regulating maltose uptake at high concentrations, as proposed [1]. In reverse, the standard pathway involves binding of MalE to the maltose in the transporter pocket and subsequent closing of the transporter. Liganded MalE subsequently dissociates from the Pclosed transporter, removing and regulating maltose import. The open pathway of maltose transport would rely on an affinity of MalE for the P-open transporter, as observed in nanodisc quenching experiments performed under hydrolysis blocking conditions [1]. Perhaps this high affinity encourages the MalE-MalFGK2 intermediate complex. The dominance of the open pathway indicates that the binding of ATP to the nucleotidebinding domain may close the transporter prior to the association of liganded MalE. We conclude that liganded MalE may selectively bind the P-open conformation of MalFGK2 in the presence of nucleotides to initiate maltose transport, and that the standard pathway may in fact occur in reverse as a mode of regulating maltose import at high concentrations. We also conclude that some small amount of maltose import may occur along the alternate pathway. For this analysis, an enhanced Michaelis-Menten formalism was used to fit ODE model parameters to experimental equilibrium data. Furthermore, the flux through the transport system was calculated using asymptotic analysis, fit parameter values, and experimentally measured values. Moreover, two apparently distinct models were compared and integrated, and the fluxes through potential transport pathways were calculated and compared. This type of analysis could be generalized to other biochemical transport systems to provide insight into detailed binding networks using biochemical data, and to distinguish between possible transport pathways in cases where available data may conflict with accepted models of biochemical transport.  54  Bibliography [1] Bao H, Duong F (2012) Discovery of an Auto-Regulation Mechanism for the Maltose ABC Transporter M alF GK2 . PLoS ONE 7(4): e34836. doi:10.1371/ journal.pone.0034836 [2] Locher K (2004) Structure and Mechanism of ABC transporters. Current Opinion in Structural Biology 14(4): 0959-440X. doi: 10.1016/j.sbi.2004.06.005 [3] Shilton B (2008) The dynamics of MBP MalFGK2 interaction: A prototype for binding protein dependent ABC-transporter systems. Biochimica et Biophysica Acta (BBA)Biomembranes 1778(9): 0005-2736. doi:10.1016/j.bbamem.2007.09.005 [4] Oldham M, Khare D, Florante Q, Davidson A, Chen J (2007) Crystal Structure of a catalytic intermediate of the maltose transporter. Nature 450: doi:10.1038/nature06264 [5] Oldham M, Chen J (2011) Crystal structure of the maltose transporter in a pretranslocation intermediate state. Science 332: 1202-1205. [6] Orelle C, Ayvaz T, Everly RM, Klug CS, Davidson AL (2008) Both maltose-binding protein and ATP are required for nucleotide-binding domain closure in the intact maltose ABC transporter. Proc Natl Acad Sci U S A 105: 12837-12842. [7] Merino G, Shuman HA (1997) Unliganded maltose-binding protein triggers lactose transport in an Escherichia coli mutant with an alteration in the matose transport system. J Bacteriol 179: 7687-7694 [8] Merino G, Boos W, Shuman HA, Bohl E (1995) The inhibition of maltose transport by the unliganded form of the maltose-binding protein of Escherichia coli: experimental findings and mathematicak treatment. J Theor Biol 177: 171-179. [9] Austermuhle MI, Hall JA, Klug CS, Davidson AL (2004) Maltose-binding protein is open in the catalytic transition state for ATP hydrolysis during maltose transport. J Biol Chem 279: 28243-28250.  55  [10] Bohl E, Shuman H, Boos W (1994) Mathematical Treatment of the Kinetics of Binding Protein Dependent Transport Systems Reveals that Both the Substrate Loaded and Unloaded Binding Proteins Interact with Membrane Components. J Theor Biol 172: 83-94. [11] Bohl E, Boos W (1997) Quantitative Analysis of Binding Protein-mediated ABC Transport Systems. J Theor Biol 186: 65-74. [12] Gould A, Telmer P, Shilton B (2010) Simulation of the Maltose Transporter ATPase by Unliganded Maltose Binding Protein. Biochemistry 49: 653-820. [13] Bayburt T, Sligar S (2010) Membrane Protein Assembly into Nanodiscs. FEBS Letters 548:1721-1727 [14] Marme N, Kneneyer JP, Sauer M, Wolfram J(2003) Inter- and intramolecular fluorescence quenching of organic dyes by tryptophan. Bioconjug Chem 14: 1133-1139.  56  Appendix A  Standard Model The concentrations of each state in the Standard model at equilibrium were found using the following equalities and conservation equations: D=  AEm K3 ,  F =  D K4 ,  E=  Em Ko M  A + D + F = To E + D + F + Em = Eo Now, A = To − D −  D K4  = To − D(1 +  1 K4 )  E + Em + D + F = Eo Em Ko M  + Em + D +  D K4  o Em ( K M + 1) + D(1 +  let γ =  Ko M  = Eo  1 K4 )  + 1, ω = 1 +  = Eo  1 K4 ,  then  γEm + ωD = Eo Em =  Eo −ωD γ  K3 D = AEm K3 D = (To − ωD)( Eo −ωD ) γ 57  K3 Dγ = (To − ωD)(Eo − ωD) K3 Dγ = To Eo − ωDEo − ωDTo + ω 2 D2 0 = To Eo − ωDEo − ωDTo + ω 2 D2 − K3 Dγ 0 = ω 2 D2 − D(ωEo + ωTo + K3 γ) + To Eo 0 = D2 − D( Eωo + D = 0.5(b ± where b =  √  Eo ω  To ω  +  K3 γ ) ω2  +  To Eo ω2  b2 − 4c)  +  To ω  +  K3 γ , ω2  c=  To Eo ω2  Nondimensional Standard Model Let D = dEo , A = aEo , F = f Eo , E = eEo , Em = jEo where d, a, f, e, and j are dimensionless variables. Let ρ = Then: d = 0.5( Ebo ±  b 2 Eo  − 4 Ec2 ) o  where: b Eo  =  1 ω  c Eo2  =  ρ . ω2  +  ρ ω  +  K3∗ γ ω2  a = ρ − ωd f=  d K4  j=  1−ωd γ  e=  jKo M  58  To Eo .  Appendix B  Alternate Model The concentrations of each state in the Alternate model at equilibrium were found using the following equalities and conservation equations: A = K1 B BE = K2 C BEm = βK2 F βKo F = CM EM = Ko Em A + B + C + F = Tt E + Em + C + F = Et E+  EM Ko  +C +  M Ko β C  = Eo  ξE + νC = Eo , where ξ = 1 + E=  M Ko ,  ν =1+σ  Eo −νC ξ  Em =  EM Ko  ξ B = K2 C Eo −νC  A = K1 B 59  F = σC Now A + B + C + F = To K1 B + B + C + σC = To B(K1 + 1) + C(1 + σ) = To Let ψ = K2 (K1 + 1), ψCξ Eo −νC  + νC = To  ψCξ + νC(Eo − νC) = To (Eo − νC) ψCξ + νEo C − ν 2 C 2 = To Eo − νTo C ψCξ + νEo C − ν 2 C 2 − To Eo + νTo C = 0 −ν 2 C 2 + C(ψξ + νEo + νTo ) − To Eo = 0 + C 2 − C( ψξ ν2  Eo ν  + C = 0.5[( ψξ ν2  +  Eo ν  To ν )  +  +  To ν )  To Eo ν2  ±  =0  ( ψξ + ν2  Eo ν  +  To 2 ν )  −  4To Eo ] ν2  Nondimensional Alternate Model Let C = cEo , B = bEo , A = aEo , F = f Eo , Em = jEo , E = eEo where c,b,a,f,j, and e are dimensionless variables. Let ρ =  To Eo ,  η=  ψ Eo .  Then: c = 0.5[( νηξ2 +  1 ν  + νρ ) ±  ( νηξ2 +  1 ν  + νρ )2 −  4ρ ] ν2  f = σc  60  Appendix C  Comprehensive Model The concentrations of each state in the Comprehensive model at equilibrium were found using the following equalities and conservation equations: A = K1 B BE = K2 C BEm = βK2 F βKo F = CM D = K4 F E + Em + C + D + F = Eo A + B + C + D + F = To F = σC, where σ = Em = E+  EM Ko  EM Ko  E(1 +  M Ko β  + C + K4 σC + σC = Eo  M Ko )  + C(1 + K4 σ + σ) = Eo  ξE + αC = Eo , where ξ = 1 + E=  M Ko ,  α = 1 + σ(K4 + 1)  Eo −αC ξ  61  B=  K2 Cξ Eo −αC  Now A + B + C + D + F = To K1 B + B + C + K4 σC + σC = To B(K1 + 1) + C(1 + K4 σ + σ) = To Let ψ = K2 (K1 + 1) ψCξ Eo −αC  + αC = To  ψCξ + αC(Eo − αC) = To (Eo − αC) ψCξ + αEo C − α2 C 2 = To Eo − αTo C ψCξ + αEo C − α2 C 2 − To Eo + αTo C = 0 −α2 C 2 + C(ψξ + αEo + αTo ) − To Eo = 0 + C 2 − C( ψξ α2  Eo α  + C = 0.5[( ψξ α2  +  Eo α  To α)  +  +  To α)  To Eo α2  ±  =0  ( ψξ + α2  Eo α  +  To 2 α)  −  4To Eo ] α2  Let κ = σ(1 + K4 ) Nondimensional Full Model Let C = cEo , B = bEo , A = aEo , F = f Eo , D = dEo Em = jEo , E = eEo where c,b,d,a,f,j, and e are dimensionless variables. Let ρ =  To Eo ,  η=  Then: c = 0.5[( αηξ2 +  1 α  + αρ ) ±  ( αηξ2 +  1 α  + αρ )2 −  4ρ ] α2  f = σc d = K4 σc No nucleotides derivation:  62  ψ Eo .  A=  K3 D Em  E=  Em Ko M  A + B + C + D + F = To K3 D Em  + 0 + 0 + D + 0 = To  K3 dEo jEo  + dEo = To  K3 d j  + dEo = To  K3∗ d j  + d = ρ, where K3∗ =  K3 Eo  E + Em + C + D + F = Eo Em Ko M  + Em + 0 + D + 0 = Eo  o Em ( K M + 1) + D = Eo o jEo ( K M + 1) + dEo = Eo o j( K M + 1) + d = 1, let  Ko M  +1=ω  jω + d = 1 j=  1−d ω  K3∗ ωd 1−d  +d=ρ  K3∗ ωd + d(1 − d) = ρ(1 − d) K3∗ ωd + d − d2 = ρ − ρd K3∗ ωd + d − d2 − ρ + ρd = 0 −d2 + d(K3∗ ω + 1 + ρ) − ρ = 0 d2 − d(K3∗ ω + 1 + ρ) + ρ = 0 d = 0.5((K3∗ ω + 1 + ρ) −  (K3∗ ω + 1 + ρ)2 − 4ρ)  Other Model with assumption that AMP-PNP closes interface  63  K4 , K1 → 0, therefore A → 0 and D → 0 1)AMP-PNP: B + C = To E + C = Eo , E = Eo − C K2 C E  + C = To  K2 C + EC = ETo K2 C + (Eo − C)C = (Eo − C)To K2 C + Eo C − C 2 = Eo To − To C −C 2 + C(K2 + Eo + To ) − Eo To = 0 C 2 − C(K2 + Eo + To ) + Eo To = 0 (cEo )2 − cEo (K2 + Eo + To ) + Eo To = 0 c2 − c(K2∗ + 1 + ρ) + ρ = 0 c = (K2∗ + 1 + ρ) −  b2 − 4ρ  2)AMP+Malt: B + C + F = To E + Em + C + F = Eo E+  EM Ko  E=  Eo −C(1+σ) γ  K2 C E  + C + σC = To  + C + σC = Eo  K2 C + C( Eo −C(1+σ) )(1 + σ) = To ( Eo −C(1+σ) ) γ γ K2 C +  Eo C(1+σ) γ  −  C 2 (1+σ)2 γ  =  To Eo γ  K2 C +  Eo C(1+σ) γ  −  C 2 (1+σ)2 γ  +  CTo (1+σ) γ  2  −C 2 (1+σ) + C(K2 + γ  Eo (1+σ) γ  +  −  CTo (1+σ) γ  −  To (1+σ) ) γ  To Eo γ  −  =0  To Eo γ  =0  C 2 (1 + σ)2 − C(K2 γ + Eo (1 + σ) + To (1 + σ)) + To Eo = 0  64  K2 γ C 2 − C( (1+σ) 2 +  Eo (1+σ)  +  To (1+σ) )  +  To Eo (1+σ)2  =0  Nondimensionalize: K2 γ (cEo )2 − cEo ( (1+σ) 2 + K∗γ  2 c2 − c( (1+σ) 2 +  K∗γ  1 (1+σ)  2 c = 0.5[( (1+σ) 2 +  quenching:  c+f φ  +  1 (1+σ)  =  Eo (1+σ)  +  ρ (1+σ) )  +  To (1+σ) )  +  ρ (1+σ) )  +  ρ (1+σ)2  −  To Eo (1+σ)2  =0  =0  b2 −  4ρ ] (1+σ)2  c(1+σ) φ  65  Appendix D  Full Model The concentrations of each state in the Full model at equilibrium were found using the following equalities and conservation equations: A = K1 B BE = K2 C BEm = βK2 F βKo F = CM D = K4 F AE =  K3 δ G  AEm = K3 D E + Em + C + D + F + G = Eo A + B + C + D + F + G = To F = σC, where σ = Em =  M Ko β  EM Ko  For the Full model we must use the stronger assumption that AMP-PNP stabilizes the P-open conformation in order to fit the data.  66  In the absense of nucleotides, we assume we have only P-closed states:  67  A + D + G = Tt E + Em + D + G = Et A=  K3 δE G  D=  M δKo G  K3 δE G  +  E+  EM Ko  E(1 +  M δKo G  +  M Ko )  + G = Tt  M δKo G  + G = Et  + G(1 +  M δKo )  = Et ,  Eγ + Gξ = Et , where ξ = 1 + E=  Et −ξG γ  K3 δE G  +  M δKo G  γ K3 δ Et −ξG G  M δKo ,  γ =1+  M Ko  + G = Tt  + Gξ = Tt  K3 γG + Gξδ(Et − ξG) = Tt δ(Et − ξG) K3 γG + GξδEt − ξ 2 G2 δ − Tt δEt + Tt δξG = 0 ξ 2 δG2 − G(K3 γ + ξδEt + Tt δξ) + Tt Et δ = 0 t +Tt δξ) + G2 − G (K3 γ+ξδE ξ2 δ  G = 0.5[( Kξ23δγ +  Et ξ  +  Tt ξ )  Tt Et δ ξ2 δ  =0  ( Kξ23δγ +  ±  Et ξ  +  Tt 2 ξ )  −  4Tt Et ] ξ2  Nondimensionalize: let g, d be nondimensional variables such that G = gEt , D = dEt and let ρ =  Tt Et ,  K3∗ =  K3 Et .  K3∗ γ ξ2 δ  1 ξ  g = 0.5[( d=  +  Then,  + ρξ ) ±  (  K3∗ γ ξ2 δ  +  1 ξ  + ρξ )2 −  4ρ ] ξ2  M δKo g  quenching:  g+d φ  =  M g(1+ δK ) o . φ  In the presence of AMP-PNP, we assume that we have only P-open states at equilibrium. Therefore in the presense of AMP-PNP the Full model is identical to the Comprehensive model. 68  Appendix E  F-test SSRAlternate  9.8918 ∗ 10−5  SSRComprehensive  2.6127 ∗ 10−4  SSRComprehensive  4.0025 ∗ 10−5  SSRF ull  4.7432 ∗ 10−5  n  36  n  48  F  47.085  F  193.8572  v1  1  v1  1  v2  32  v2  43  F0.00001,v1 ,v2  27.41602865  F0.00001,v1 ,v2  25.03720916  p  0.00000009  p  < 10−8  Table E.1: F-test values for comparing the Alternate and Comprehensive models (left) and for comparing the Comprehensive and Full models (right).  69  Appendix F  Flux Calculation The Comprehensive Model has the following ODE system, with β = dA dt  = −k1+ A + k1− B − k3+ AEm + k3− D + h1 B + h2 C + h3 F  dB dt  = k1+ A − k1− B − k2+ BE + k2− C − β2 k2+ BEm + β3 k2− F − h1 B  dC dt  = −k2− C + k2+ BE − βo ko+ CM + β1 ko− F − h2 C  dD dt  = k3+ Em A − k3− D − k4+ D + k4− F  dF dt  = β2 k2+ BEm − β3 k2− F + k4+ D − k4− F − h3 F + βo ko+ CM − β1 ko− F  dE dt  = −ko+ EM + ko− Em − k2+ BE + k2− C + h2 C + h3 F  dEm dt  = −k3+ Em A + k3− D − β2 k2+ BEm + β3 k2− F + ko+ EM − ko− Em  A + B + C + D + F = Tt E + Em + C + D + F = Et We now consider the steady state system with hydrolysis: 0 = −k1+ A + k1− B − k3+ AEm + k3− D + h1 B + h2 C + h3 F 0 = k1+ A − k1− B − k2+ BE + k2− C − β2 k2+ BEm + β3 k2− F − h1 B 0 = −k2− C + k2+ BE − βo ko+ CM + β1 ko− F − h2 C  70  β1 βo ,  β=  β3 β2 .  0 = k3+ Em A − k3− D − k4+ D + k4− F 0 = β2 k2+ BEm − β3 k2− F + k4+ D − k4− F − h3 F + βo ko+ CM − β1 ko− F 0 = −ko+ EM + ko− Em − k2+ BE + k2− C + h2 C + h3 F 0 = −k3+ Em A + k3− D − β2 k2+ BEm + β3 k2− F + ko+ EM − ko− Em A + B + C + D + F = Tt E + Em + C + D + F = Et Using our fit hydrolysis rates, we have h1 < h3 < h2 , and we assume that the unknown forward rate k1+ is large such that h1 << k1+ . Our previous analysis of k1+ in the time course quenching data supports this assumption that k1+ is large compared to h1 , for all fixed K1 values. We divide the steady state system through by k1+ and regroup parameters. We let =  h1 k1+  be a small parameter. We then let γ =  ko+ , k1+  α=  k2+ , k1+  δ =  k3+ , k1+  ξ =  k4+ , k1+  and have the  resulting steady state system (1): 0 = −A + K1 B − δEm A + K3 δD + B + ρ C + w F 0 = A − K1 B − αBE + αK2 C − αβ2 BEm + αββ2 K2 F − B 0 = −K2 αC + αBE − γβo CM + βo βKo γF − ρ C 0 = δEm A − K3 δD − ξD + K4 ξF 0 = αβ2 BEm − β2 βK2 αF + ξD − K4 ξF + γβo CM − βo βγKo F − w F 0 = −γEM + Ko γEm − αBE + K2 αC + ρ C + w F 0 = −δEm A + K3 δD − αβ2 BEm + β2 βK2 αF + γEM − γKo Em A + B + C + D + F = Tt E + Em + C + D + F = Et We let each concentration A, B, C, D, F, E, Em be written as an asymptotic solution, A = Ao + A1 , B = Bo + B1 , ...., Em = Emo + Em1 ,and subsitute into (1). We then consider the O(o) system: 0 = −Ao + K1 Bo − δEmo Ao + K3 δDo  71  0 = Ao − K1 Bo − αBo Eo + αK2 Co − αβ2 Bo Emo + β2 βαK2 Fo 0 = −K2 αCo + αBo Eo − γβo Co M + βo βKo γFo 0 = δEmo Ao − K3 δDo − ξDo + K4 ξFo 0 = αβ2 Bo Emo − β2 βK2 αFo + ξDo − K4 ξFo + βo γCo M − βo βγKo Fo 0 = −γEo M + Ko γEmo − αBo Eo + K2 αCo 0 = −δEmo Ao + K3 δDo − αβ2 Bo Emo + β2 βK2 αFo + γEo M − γKo Emo Ao + Bo + Co + Do + Fo = Tt Eo + Emo + Co + Do + Fo = Et The O(o) system is the quasi-steady state system, in this case it is equivalent to the system at equilibrium without hydrolysis. We have assumed that the hydrolysis rates are slow. We use our previously derived solutions to the equilibrium system: Ao , Bo , Co , Do , Fo , Eo , Emo = f (Tt , Et , M, Ki , β), Where Ki , i = 0, 1, 2, 3, 4, are the equilibrium ratios. We now consider the O(1) system: 0 = −A1 + K1 B1 − δ(Emo A1 + Em1 Ao ) + K3 δD1 + Bo + ρCo + wFo 0 = A1 − K1 B1 − α(Bo E1 + B1 Eo ) + αK2 C1 − αβ2 (Bo Em1 + B1 Emo ) + β2 βαK2 F1 − Bo 0 = −K2 αC1 + α(Bo E1 + B1 Eo ) − βo γC1 M + βo βKo γF1 − ρCo 0 = δ(Emo A1 + Em1 Ao ) − K3 δD1 − ξD1 + K4 ξF1 0 = αβ2 (Bo Em1 + B1 Emo ) − β2 βK2 αF1 + ξD1 − K4 ξF1 + βo γC1 M − βo βγKo F1 − wFo 0 = −γE1 M + Ko γEm1 − α(Bo E1 + B1 Eo ) + K2 αC1 + ρCo + wFo 0 = −δ(Emo A1 + Em1 Ao ) + K3 δD1 − αβ2 (Bo Em1 + B1 Emo ) + β2 βK2 αF1 + γE1 M − γKo Em1 A1 + B1 + C1 + D1 + F1 = 0 E1 + Em1 + C1 + D1 + F1 = 0 72  This system is linear, and can be reduced using the conservation equations. We let A1 = (−(B1 + C1 + D1 + F1 )), Em1 = (−(E1 + C1 + D1 + F1 )), and reduce the O(1) system to: B1 p1 + C1 p2 + D1 p3 + F1 p4 − Bo = 0 B1 p5 + C1 p6 + E1 p7 + F1 p8 − ρCo = 0 B1 p9 + C1 p10 + D1 p11 + E1 p12 + F1 p13 = 0 B1 p14 + C1 p15 + D1 p16 + E1 p17 + F1 p18 − wFo = 0 B1 p19 + C1 p20 + D1 p21 + E1 p22 + F1 p23 + ρCo + wFo = 0 where: p1 = −1 − K1 − αEo + αβ2 Emo p2 = −1 + αK2 + αβ2 Bo p3 = −1 + αβ2 Bo p4 = −1 + αβ2 Bo + β2 βαK2 p5 = αEo p6 = −K2 α − βo γM p7 = αBo p8 = βo βKo γ p9 = −δEmo p10 = −δEmo − δAo p11 = −δEmo − δAo − K3 δ − ξ p12 = −δAo p13 = −δEmo − δAo + K4 ξ p14 = αβ2 Emo  73  p15 = −αβ2 Bo + βo γM p16 = −αβ2 Bo + ξ p17 = −αβ2 Bo p18 = −αβ2 Bo − ββ2 K2 α − K4 ξ − βo βγKo p19 = −αEo p20 = −Ko γ + K2 α p21 = −Ko γ p22 = −γM − γKo − αBo p23 = −Ko γ And so we can solve the O(1) system: A1 , B1 ,C1 , D1 , F1 , E1 ,Em1 =f (Xo , kj+ , Ki , βo , β2 ) where kj+ , j = 0, 1, 2, 3, 4 are the unknown forward rates, and Xo , X = {A, B, C, D, E, F, Em } are the equilibrium concentrations. B1 = −(−((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p22 p6 + p20 p7 ) − p2 (p23 p7 − p22 p8 )) − (p4 (−p12 p6 + p10 p7 )−p2 (p13 p7 −p12 p8 ))(p21 p4 p7 −p3 (p23 p7 −p22 p8 )))(−(p16 p4 p7 −p3 (p18 p7 −p17 p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(Bo (p18 p7 − p17 p8 ) + p4 (Co p17 ρ − Fo p7 ω))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p17 p6 + p15 p7 ) − p2 (p18 p7 − p17 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))(−(p21 p4 p7 − p3 (p23 p7 − p22 p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(Bo (p23 p7 − p22 p8 ) + p4 (Co p22 ρ + p7 (Co r + Fo w)))))/(−((p11 p4 p7 −p3 (p13 p7 −p12 p8 ))(p4 (−p22 p6 +p20 p7 )−p2 (p23 p7 −p22 p8 ))−(p4 (−p12 p6 + p10 p7 )−p2 (p13 p7 −p12 p8 ))(p21 p4 p7 −p3 (p23 p7 −p22 p8 )))((p11 p4 p7 −p3 (p13 p7 −p12 p8 ))(p4 (−p17 p5 + p14 p7 ) − p1 (p18 p7 − p17 p8 )) − (p16 p4 p7 − p3 (p18 p7 − p17 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p17 p6 + p15 p7 ) − p2 (p18 p7 − p17 p8 )) − (p4 (−p12 p6 + p10 p7 )−p2 (p13 p7 −p12 p8 ))(p16 p4 p7 −p3 (p18 p7 −p17 p8 )))((p11 p4 p7 −p3 (p13 p7 −p12 p8 ))(p4 (−p22 p5 + p19 p7 )−p1 (p23 p7 −p22 p8 ))−(p21 p4 p7 −p3 (p23 p7 −p22 p8 ))(−p1 (p13 p7 −p12 p8 )+p4 (−p12 p5 +p7 p9 )))), C1 = −(−Bo p13 p16 p7 + Bo p11 p18 p7 + Bo p12 p16 p8 − Bo p11 p17 p8 − Co p13 p17 p3 ρ + Co p12 p18 p3 ρ − Co p12 p16 p4 ρ+Co p11 p17 p4 ρ+Fo p13 p3 p7 ω −Fo p11 p4 p7 ω −Fo p12 p3 p8 ω)/(p13 p17 p3 p6 −p12 p18 p3 p6 + p12 p16 p4 p6 − p11 p17 p4 p6 + p13 p16 p2 p7 − p11 p18 p2 p7 − p13 p15 p3 p7 + p10 p18 p3 p7 + p1 1p1 5p4 p7 − 74  p10 p16 p4 p7 −p12 p16 p2 p8 +p11 p17 p2 p8 +p12 p15 p3 p8 −p10 p17 p3 p8 )+(((p11 p4 p7 −p3 (p13 p7 −p12 p8 ))∗ (p4 (−p17 p5 + p14 p7 ) − p1 (p18 p7 − p17 p8 )) − (p16 p4 p7 − p3 (p18 p7 − p17 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 )))(−((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p22 p6 + p20 p7 ) − p2 (p23 p7 − p22 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p21 p4 p7 − p3 (p23 p7 − p22 p8 )))(−(p16 p4 p7 − p3 (p18 p7 − p17 p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(Bo (p18 p7 − p17 p8 ) + p4 (Co p17 ρ − Fo p7 ω))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p17 p6 + p15 p7 ) − p2 (p18 p7 − p17 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))(−(p21 p4 p7 − p3 (p23 p7 − p22 p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(Bo (p23 p7 − p22 p8 ) + p4 (Co p22 ρ + p7 (Co ρ + Fo ω))))))/(((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p17 p6 + p1 5p7 ) − p2 (p18 p7 − p17 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))(−((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p22 p6 + p20 p7 ) − p2 (p23 p7 − p22 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p21 p4 p7 −p3 (p23 p7 −p22 p8 )))((p11 p4 p7 −p3 (p13 p7 −p12 p8 ))(p4 (−p17 p5 +p14 p7 )−p1 (p18 p7 − p17 p8 )) − (p16 p4 p7 − p3 (p18 p7 − p17 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p17 p6 + p15 p7 ) − p2 (p18 p7 − p17 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p16 p4 p7 −p3 (p18 p7 −p17 p8 )))((p11 p4 p7 −p3 (p13 p7 −p12 p8 ))(p4 (−p22 p5 +p19 p7 )−p1 (p23 p7 − p22 p8 )) − (p21 p4 p7 − p3 (p23 p7 − p22 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))))), D1 = −(Bo p13 p17 p6 − Bo p12 p18 p6 − Bo p13 p15 p7 + Bo p10 p18 p7 + Bo p12 p15 p8 − Bo p10 p17 p8 − Co p13 p17 p2 ρ+Co p12 p18 p2 ρ−Co p12 p15 p4 r+Co p10 p17 p4 ρ+Fo p12 p4 p6 ω+Fo p13 p2 p7 ω−Fo p10 p4 p7 ω− Fo p12 p2 p8 ω)/(−p13 p17 p3 p6 + p12 p18 p3 p6 − p12 p16 p4 p6 + p11 p17 p4 p6 − p13 p16 p2 p7 + p11 p18 p2 p7 + p13 p15 p3 p7 − p10 p18 p3 p7 − p11 p15 p4 p7 + p10 p16 p4 p7 + p12 p16 p2 p8 − p11 p17 p2 p8 − p12 p15 p3 p8 + p10 p17 p3 p8 ) + ((−p13 p17 p2 p5 + p12 p18 p2 p5 − p12 p15 p4 p5 + p10 p17 p4 p5 + p1 p13 p17 p6 − p1 p12 p18 p6 + p12 p14 p4 p6 − p1 p13 p15 p7 + p1 p10 p18 p7 + p13 p14 p2 p7 − p10 p14 p4 p7 + p1 p12 p15 p8 − p1 p10 p17 p8 − p12 p14 p2 p8 − p17 p4 p6 p9 − p18 p2 p7 p9 + p15 p4 p7 p9 + p17 p2 p8 p9 )(−((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p22 p6 + p20 p7 ) − p2 (p23 p7 − p22 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p21 p4 p7 − p3 (p23 p7 − p22 p8 )))(−(p16 p4 p7 − p3 (p18 p7 − p17 p8 )) ∗ (Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (Bo (p18 p7 − p17 p8 ) + p4 (Co p17 ρ − Fo p7 ω))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p17 p6 + p15 p7 ) − p2 (p18 p7 − p17 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))(−(p21 p4 p7 − p3 (p2 3p7 − p2 2p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 −p12 p8 ))(Bo (p23 p7 −p22 p8 )+p4 (Co p22 ρ+p7 (Co ρ+Fo ω))))))/((p13 p17 p3 p6 −p12 p18 p3 p6 + p12 ∗ p16 p4 p6 − p11 p17 p4 p6 + p13 p16 p2 p7 − p11 p18 p2 p7 − p13 p15 p3 p7 + p10 p18 p3 p7 + p11 p15 p4 p7 − p10 p16 p4 p7 − p12 p16 p2 p8 + p11 p17 p2 p8 + p12 p15 p3 p8 − p10 p17 p3 p8 )(−((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p22 p6 + p20 p7 ) − p2 (p23 p7 − p22 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p21 p4 p7 − p3 (p23 p7 − p22 p8 )))((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p17 p5 + p14 p7 ) − p1 (p18 p7 − p17 p8 )) − (p16 p4 p7 − p3 (p18 p7 − p17 p8 )) ∗ (−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))∗(p4 (−p17 p6 +p15 p7 )−p2 (p18 p7 −p17 p8 ))−(p4 (−p12 p6 +p10 p7 )−p2 (p13 p7 −p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p22 p5 + p19 p7 ) − p1 (p23 p7 − p22 p8 )) − 75  (p21 p4 p7 − p3 (p23 p7 − p22 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))))), E1 = −(−Bo p13 p16 p6 + Bo p11 p18 p6 − Bo p11 p15 p8 + Bo p10 p16 p8 + Co p13 p16 p2 ρ − Co p11 p18 p2 ρ − Co p13 p15 p3 ρ+Co p10 p18 p3 ρ+Co p11 p15 p4 ρ−Co p10 p16 p4 ρ+Fo p13 p3 p6 ω−Fo p11 p4 p6 ω+Fo p11 p2 p8 ω− Fo p10 p3 p8 ω)/(−p13 p17 p3 p6 + p12 p18 p3 p6 − p12 p16 p4 p6 + p11 p17 p4 p6 − p13 p16 p2 p7 + p11 p18 p2 p7 + p13 p15 p3 p7 − p10 p18 p3 p7 − p11 p15 p4 p7 + p10 p16 p4 p7 + p12 p16 p2 p8 − p11 p17 p2 p8 − p12 p15 p3 p8 + p10 p17 p3 p8 ) + ((−p13 p16 p2 p5 + p11 p18 p2 p5 + p13 p15 p3 p5 − p10 p18 p3 p5 − p11 p15 p4 p5 + p10 p16 p4 p5 + p1 p13 p16 p6 − p1 p11 p18 p6 − p13 p14 p3 p6 + p11 p14 p4 p6 + p1 p11 p15 p8 − p1 p10 p16 p8 − p11 p14 p2 p8 + p10 p14 p3 p8 + p18 p3 p6 p9 − p16 p4 p6 p9 + p16 p2 p8 p9 − p15 p3 p8 p9 )(−((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p22 p6 + p20 p7 ) − p2 (p23 p7 − p22 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p21 p4 p7 − p3 (p23 p7 − p22 p8 )))(−(p16 p4 p7 − p3 (p18 p7 − p17 p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (Bo (p18 p7 − p17 p8 ) + p4 (Co p17 ρ − Fo p7 ω))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p17 p6 + p15 p7 ) − p2 (p18 p7 − p17 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))(−(p21 p4 p7 − p3 (p23 p7 − p22 p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 −p12 p8 ))(Bo (p23 p7 −p22 p8 )+p4 (Co p2 2ρ+p7 (Co ρ+Fo ω))))))/((−p13 p17 p3 p6 +p12 p18 p3 p6 − p12 p16 p4 p6 + p11 p17 p4 p6 − p13 p16 p2 p7 + p11 p18 p2 p7 + p13 p15 p3 p7 − p10 p18 p3 p7 − p11 p15 p4 p7 + p10 p16 p4 p7 + p12 p1 6p2 p8 − p11 p17 p2 p8 − p12 p15 p3 p8 + p10 p17 p3 p8 ) ∗ (−((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p22 p6 +p20 p7 )−p2 (p23 p7 −p22 p8 ))−(p4 (−p12 p6 +p10 p7 )−p2 (p13 p7 −p12 p8 ))(p21 p4 p7 − p3 (p23 p7 − p22 p8 ))) ∗ ((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p17 p5 + p14 p7 ) − p1 (p18 p7 − p17 p8 )) − (p16 p4 p7 − p3 (p18 p7 − p17 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))) + ((p1 1p4 p7 − p3 (p13 p7 − p12 p8 ))∗(p4 (−p17 p6 +p15 p7 )−p2 (p18 p7 −p17 p8 ))−(p4 (−p12 p6 +p10 p7 )−p2 (p13 p7 −p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p22 p5 + p19 p7 ) − p1 (p23 p7 − p22 p8 )) − (p21 p4 p7 − p3 (p23 p7 − p22 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))))), F1 = −(Bo p12 p16 p6 − Bo p11 p17 p6 + Bo p11 p15 p7 − Bo p10 p16 p7 − Co p12 p16 p2 ρ + Co p11 p17 p2 ρ + Co p12 p15 p3 ρ−Co p10 p17 p3 ρ−Fo p12 p3 p6 ω−Fo p11 p2 p7 ω+Fo p10 p3 p7 ω)/(−p13 p17 p3 p6 +p12 p18 p3 p6 − p12 p16 p4 p6 + p11 p17 p4 p6 − p13 p16 p2 p7 + p11 p18 p2 p7 + p13 p15 p3 p7 − p10 p18 p3 p7 − p11 p15 p4 p7 + p10 p16 p4 p7 + p12 p16 p2 p8 − p11 p17 p2 p8 − p12 p15 p3 p8 + p10 p17 p3 p8 ) + ((p12 p16 p2 p5 − p11 p1 7p2 p5 − p12 p15 p3 p5 + p10 p17 p3 p5 − p1 p12 p16 p6 + p1 p11 p17 p6 + p12 p14 p3 p6 − p1 p11 p15 p7 + p1 p10 p16 p7 + p11 p14 p2 p7 − p10 p14 p3 p7 − p1 7p3 p6 p9 − p16 p2 p7 p9 + p15 p3 p7 p9 )(−((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p22 p6 + p20 p7 ) − p2 (p23 p7 − p22 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p21 p4 p7 − p3 (p23 p7 − p22 p8 )))(−(p16 p4 p7 − p3 (p18 p7 − p17 p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(Bo (p18 p7 − p17 p8 ) + p4 (Co p17 ρ − Fo p7 ω))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p17 p6 + p15 p7 ) − p2 (p18 p7 − p17 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))(−(p21 p4 p7 − p3 (p23 p7 − p22 p8 ))(Bo (p13 p7 − p12 p8 ) + Co p12 p4 ρ) + (p11 p4 p7 − p3 (p13 p7 −p12 p8 ))(Bo (p23 p7 −p22 p8 )+p4 (Co p22 ρ+p7 (Co ρ+Fo ω))))))/((−p13 p17 p3 p6 +p12 p18 p3 p6 − p12 p16 p4 p6 + p11 p17 p4 p6 − p13 p16 p2 p7 + p11 p18 p2 p7 + p13 p15 p3 p7 − p10 p18 p3 p7 − p11 p15 p4 p7 +  76  p10 p16 p4 p7 + p12 p16 p2 p8 − p11 p17 p2 p8 − p12 p15 p3 p8 + p10 p17 p3 p8 )(−((p11 p4 p7 − p3 (p13 p7 − p12 p8 )) ∗ (p4 (−p22 p6 + p20 p7 ) − p2 (p23 p7 − p22 p8 )) − (p4 (−p12 p6 + p10 p7 ) − p2 (p13 p7 − p12 p8 ))(p21 p4 p7 − p3 (p23 p7 − p22 p8 )))((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p17 p5 + p14 p7 ) − p1 (p18 p7 − p17 p8 )) − (p16 p4 p7 − p3 (p18 p7 − p17 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))) + ((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p17 p6 +p15 p7 )−p2 (p18 p7 −p17 p8 ))−(p4 (−p12 p6 +p10 p7 )−p2 (p13 p7 −p12 p8 ))(p16 p4 p7 − p3 (p18 p7 − p17 p8 )))((p11 p4 p7 − p3 (p13 p7 − p12 p8 ))(p4 (−p22 p5 + p19 p7 ) − p1 (p23 p7 − p22 p8 )) − (p21 p4 p7 − p3 (p23 p7 − p22 p8 ))(−p1 (p13 p7 − p12 p8 ) + p4 (−p12 p5 + p7 p9 ))))). We now calculate the fluxes through each pathway of the full model. We first label the fluxes as follows: f1  f2  f3  f5  f5  g3  f4  g3  g3  A → B → C → F →c where f3 = CS pathway flux A → D → F →s where f5 =Standard Pathway flux B → F →a where f4 =Alternate Pathway flux g1  B→A g2  C→A We have the following system of equations, where g1 = h1 B, g2 = h2 C, g3 = h3 F . f1 = f4 + f2 + g1 f2 = f3 + g2 f3 + f4 + f5 = g3 Then all fluxes can be written in terms of fluxes f2 , f4 : f1 = g1 + f2 + f4 f3 = f2 − g2 f5 = g3 − f4 − f3 We can write the fluxes f2 , f4 as the following differences: f2 = k2+ BE − k2− C f4 = β2 k2+ BEm − β2 βk2− F 77  Substitution of our asymptotic solutions gives the following approximation of the fluxes at steady state: f2 = k2+ (Bo + B1 )(Eo + E1 ) − k2− (Co + C1 ) f4 = β2 k2+ (Bo + B1 )(Emo + Em1 ) − β2 βk2− (Fo + F1 ).  78  

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