Bistability in Human Dihydrofolate Reductase Catalysis

Transcription

1 Bistability in Human Dihydrofolate Reductase Catalysis Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Yongjia Fan, B.A. Graduate Program in Chemical Engineering The Ohio State University 2010 Thesis Committee: Dr. Martin Feinberg, Advisor Dr. Shang-Tian Yang

2 Copyright by Yongjia Fan 2010

3 Abstract Bistability is the existence of two stable steady states consistent with the same operating conditions. The mechanism for bistability is generally presumed to lie in apparent feedback. However, as suggested by mathematical analysis, bistability can be found in certain enzyme reactions without any apparent feedback. One example is the human dihydrofolate reductase (DHFR) reaction. Simulations indicated that the DHFR reaction with a mechanism without apparent feedback has the capacity for bistability in certain open-reactor configurations. A primary objective of this thesis is to explore the existence of bistability in the human DHFR reaction in simulations under different operating conditions. Simulations showed that the human DHFR reaction exhibited bistability in the context of a partially open continuous flow stirred tank reactor (CFSTR) in which DHFR was trapped within a membrane compartment. Simulations on the dynamics of the human DHFR reaction when coupled with the other two enzyme reactions in the thymidylate synthesis pathway in which DHFR works indicated that the DHFR reaction could still exhibit bistability, and the bistability was propagated along the pathway. However, it was observed in exploratory experiments that the membranes used in a laboratory realization of the same partially open CFSTR hindered the mass transfer of the ii

4 substrate and product. Additional simulations were performed with the human DHFR reaction in a partially open CFSTR, taking account of the hindrance effect. In this case, the human DHFR reaction not only had the capacity for bistability but also exhibited periodic oscillations. As an alternative to the partially open CFSTR, simulations were performed in which the human DHFR reaction was carried out in a classical fully open CFSTR. The simulation indicated that the human DHFR reaction in a fully open CFSTR also had the capacity for bistability. In the same fully open CFSTR, and in the presence of an extra feed stream of methotrexate, which is an inhibitor of DHFR, the human DHFR reaction exhibited bistability with respect to the changes in methotrexate concentration. Another objective of this thesis is to lay the foundation for experimental verification of the existence of bistability in the human DHFR reaction in a partially open CFSTR, a classical fully open CFSTR, and a partially open CFSTR with DHFR tethered to chitin beads. Preliminary experiments with a partially open CFSTR encountered many challenges mainly due to the usage of small molecular weight cutoff membranes. It turned out to be iii

5 difficult for the DHFR-in-solution reaction to reach a steady state within the partially open CFSTR. Another plan was to use human DHFR tethered to chitin beads in a partially open CFSTR with large molecular weight cutoff membranes. Genetically modified human DHFR with a chitin binding domain, was successfully immobilized onto chitin beads. However, the rate constants of immobilized human DHFR might deviate from those of free human DHFR. Therefore, kinetics studies of the immobilized human DHFR should be considered in preparation for any flow experiments with DHFR tethered to chitin beads in the future. The third plan was to perform the human DHFR reaction in a classical fully open CFSTR. Experiments showed that the fully open CFSTR could reach a steady state quickly. However, there remain concerns about whether the two distinct stable steady states found in simulations would be sufficiently distinct as to be convincingly detected with currently available instruments. iv

6 Dedication Dedicated to my husband Yuan Wen and my parents v

7 Acknowledgments My sincere appreciation and gratitude first goes to my academic advisor Dr. Martin Feinberg for his guidance, advice and support throughout my graduate studies and researches at The Ohio State University. It has been such a blessing and rewarding experience to work with him. There is a Chinese proverb saying that once a teacher, always a father. I m not a perfect student, but I would like to respect him as my father for a life time. I also want to extend my sincere appreciation to Dr. Shang-Tian Yang for serving on my qualifier exam and thesis committee, allowing me to share lab spaces and equipment with his students, and giving me great help and advice on my graduate studies, and Dr. David Wood for serving on my qualifier exam committee and giving me insightful advice on my research project. My appreciation also goes to my group-mates. Dr. Yangzhong Tang who has graduated and currently works at Harvard Medical School helped and taught me a lot in the early stage of my research. Ms. Haixia Ji has always been very nice and helpful to me and has treated me like her sister. Mr. Daniel Knight helped me a lot on criticizing my thesis draft. vi

8 I would also like to thank the members from Dr. Shang-Tian Yang s and Dr. Jeffery Chalmers group, who shared their lab spaces and equipment with me. Special thanks go to Jake Elmer from Dr. Andre Palmer s group for helping me on the enzyme immobilization experiments. I am grateful to Chemical and Biomolecular Engineering Department at The Ohio State University and United States National Science Foundation for their financial supports of my graduate studies. Finally, I would like to express my gratitude and appreciation to my husband Yuan and my parents for their endless and selfless support and encouragement. I would like to thank God for His guidance and blessings to help me to go through every day and everything, especially during the difficult times. vii

9 Vita April 21, Born in Baotou, China B.S. Pharmaceutical Engineering Zhejiang University Hangzhou, China University Fellowship The Ohio State University Columbus, OH present Graduate Research Associate, Department of Chemical and Biomolecular Engineering, The Ohio State University Columbus, OH FIELDS OF STUDY Major Field: Chemical and Biomolecular Engineering viii

10 Table of Contents Abstract... ii Dedication... v Acknowledgments... vi Vita... viii Table of Contents... ix List of Tables... xi List of Figures... xii Chapter 1: Background Bistability in Biological Systems Mechanisms behind Bistability Dihydrofolate Reductase and the Thymidylate Synthesis Pathway... 6 Chapter 2: Simulations of Human DHFR Catalysis, the Thymidylate Synthesis Pathway, and Methotrexate Inhibition XPPAUT Introduction Human DHFR Catalysis in a Partially Open CFSTR Three-Enzyme Cycle Reactions of the Thymidylate Synthesis Pathway in a Partially Open CFSTR ix

11 2.4 Human DHFR Catalysis in a Partially Open CFSTR Taking Account of Hindrance Effects Human DHFR Catalysis in a Classical CFSTR Human DHFR Catalysis in the Presence of Methotrexate in a Classical CFSTR.. 40 Chapter 3: Problems in Eliciting Bistability Experimentally: Some Preliminary Exploration Human DHFR Catalysis in a Partially Open CFSTR Materials and Methods Experimental Procedures and Results Human DHFR Immobilization on Chitin Beads Materials Experimental Procedures and Results Human DHFR Reaction in a Classical CFSTR Chapter 4: Future Work More Simulations and Experiments on Human DHFR Reaction in a Classical CFSTR Flow Experiments with the Catalysis of Immobilized Human DHFR on Chitin Beads in a Partially Open CFSTR References Appendix A: *.ODE Files for the Simulations in Chapter x

12 List of Tables Table 1. Comparison between 5 kd and 10 kd (PLC 10) Millipore Ultracel PLC membranes Table 2. Comparison among different carrier proteins on keeping DHFR activity xi

13 List of Figures Figure 1. A detailed mechanism and rate constants for human DHFR catalysis... 7 Figure 2. Computed bistability and switch-like behavior in a hypothetical DHFR experiment... 8 Figure 3. Thymidylate synthesis pathway Figure 4. Diagram of a partially open CFSTR used in this virtual human DHFR reaction Figure 5. Bifurcation diagram of steady-state H4F concentration vs. total DHFR concentration for the human DHFR reaction in a partially open CFSTR Figure 6. Bifurcation diagram of steady-state H4F concentration vs. feed mixture flow rate for the human DHFR reaction in a partially open CFSTR Figure 7. Order sequential mechanism of human TS catalysis Figure 8. Random sequential mechanism of human SHMT catalysis Figure 9. Bifurcation diagram of steady-state H4F concentration vs. total DHFR concentration for the thymidylate synthesis pathway Figure 10. Bifurcation diagram of steady-state CH2H4F concentration vs. total DHFR concentration for the thymidylate synthesis pathway Figure 11. Bifurcation diagram of steady-state H2F concentration vs. total DHFR concentration for the thymidylate synthesis pathway xii

14 Figure 12. Rejection profiles of Millipore Ultracel PLC membranes Figure 13. Bistability and periodic oscillation of steady-state concentration of H4F vs. total DHFR concentration when taking account of hindrance effects Figure 14. Time-course changes of NADPH concentration within the reactor chamber when taking account of hindrance effects with three different DHFR concentrations Figure 15. Virtual DHFR reaction in a classical CFSTR Figure 16. Bifurcation diagram of steady-state H4F concentration vs. DHFR concentration in feed stream for the human DHFR reaction in a classical CFSTR Figure 17. Bifurcation diagram of steady-state H4F concentration vs. substrate solution flow rate for the human DHFR reaction in a classical CFSTR Figure 18. Bifurcation diagram of steady-state H4F concentration vs. flow rate of DHFR feed steam for the human DHFR reaction in a classical CFSTR Figure 19. Molecular structures of H2F and methotrexate Figure 20. Kinetic scheme of methotrexate inhibiting human DHFR Figure 21. Bifurcation diagram of steady-state H4F concentration vs. concentration of methotrexate in the feed solution Figure 22. Time-course changes of H4F concentration with three different feed concentrations of methotrexate Figure 23. Effects of three different concentrations of methotrexate on the bifurcation diagram of steady-state H4F concentration vs. DHFR concentration in feed solution xiii

15 Figure 24. Exploded diagram of CFSTR drawn by Paul Green Figure 25. Partially open CFSTR with a membrane compartment Figure 26. Absorbance spectrum of 75 μm NADPH in Appleman s buffer Figure μm NADPH retention by 5 kd membrane with an initial NADPH concentration = 75 μm Figure μm NADPH retention by 5 kd membrane with an initial NADPH concentration = 300 μm Figure 29. Depiction of immobilized human DHFR on a chitin bead Figure 30. Picture of a PAGE gel for DHFR immobilization experiment Figure 31. TGA result of 100 μl wet chitin beads tethered human DHFR Figure 32. A simple classical CFSTR Figure 33. UV absorbance changes of the effluent solution with respect to time xiv

16 Chapter 1: Background 1.1 Bistability in Biological Systems Over the past several decades, molecular biology has uncovered many facts about individual components and their interactions in biological systems, such as protein structure and function, DNA transcription and RNA translation. However, those facts alone are not enough to gain a comprehensive understanding of biological systems. Because of their extreme complexity, biological systems such as metabolic, signaling, and regulatory pathways often contain large and complicated networks of chemical reactions. Many dynamics and behaviors of biological systems, like multiple steady states, oscillations, hysteresis, and robustness, arise at the systems level, and are difficult to explore or study based on the knowledge of individual cellular functions alone. Therefore, a system-level understanding of network structure and function becomes a significant theme in biological science [1-3]. Bistability, the capacity to achieve two distinctively different steady states in response to a single set of external signal inputs, is a ubiquitous and important system-level phenomenon of biological systems [4, 5]. Bistability seems essential for a variety of cell operations, such as gene regulation [6, 7], cell cycle regulation [8, 9], bio-signal 1

17 transduction [10, 11], cell differentiation [3, 12, 13], cell fate determination [14, 15], and much more. There are now many well-studied examples of how bistability plays a role in cellular processes. One of the most classic bistability examples is the gene regulation of the lac operon in E. Coli. The lac operon is composed of three structural genes, and regulates sugar metabolism in E. Coli. The E. Coli culture can switch between two growth modes rapidly and completely, depending on different concentrations of glucose and lactose [4, 16]. Besides the lac operon, bistability was also observed and studied in other gene regulation pathways. The production of a secondary metabolite in Streptomyces coelicolor, a polyketide, exhibited bistability in response to a threshold concentration of butyrolactone under the regulation of a pair of genes [17]. Under the regulation of the transcription of gene comk, Bacillus subtilis could switch between two distinct physiological states, competent or non-competent, in response to different environmental stresses [18]. The mitogen-activated protein kinase (MAPK) cascade, which is an intracellular protein network that regulates many cellular functions, is an example of a well-characterized bistable signaling pathway. The MAPK cascade functions as a bistable switch in response to the stimulus of an extracellular-signal-regulated kinase (ERK) [19-21]. Bistability also plays an essential role in cellular differentiation in mammalian cells. For instance, the differentiation of human HL60 (human promyeloctic leukemia) cells into neutrophils can exhibit an all-or-none switch-like bistable behavior when monitoring the expression level of a differentiation marker called Mac-1 [22]. One of the most 2

18 famous and intensively studied bistable cell fate determination examples is found in Xenopus oocyte maturation. In response to changes in the concentration of the steroid hormone progesterone, Xenopus oocyte maturation displays bistability phenomena, i.e., it is an all-or-none maturation rather than a continuous process according to different progesterone concentrations [14, 23]. Many more examples of bistable biological systems can be found in several review papers [24, 25]. 1.2 Mechanisms behind Bistability Before the recognition of bistability in lac operon in the 1950s [26], less attention was paid to the chemical dynamics in biological systems [24]. However, with the development of molecular biology, more and more dynamic behaviors of biological systems were discovered, such as multiple steady states, oscillations, biological clocks, etc. Researchers realized that cells and organisms contained complex nonlinear dynamic networks rather than simple linear chemical reactions. Meanwhile, many mathematical models were constructed and many theoretical studies were conducted to unveil the mechanisms behind the observed dynamical behaviors. Early theoretical and experimental studies about the mechanism of bistability convinced researchers that bistability was to be expected only in biological systems with overt feedback. Some theorists presented some simple mathematical models to argue that bistability was based on feedback loops [27-29]. Meanwhile, the limited biological bistability examples recognized by experimentalists were all in the biological systems 3

19 with some sort of feedback, such as the bistable lac operon [26] and gene regulations in bacteria [30, 31]. Later on, with the development of both molecular biology and computer technology, studies on bistable biological systems and their mechanisms blossomed. Many theoretical studies still argued that one of the minimal requirements for bistability is a positive feedback loop or a double-negative feedback loop (another necessary requirement is some type of so-called ultrasensitivity, which can filter small stimuli from the feedback loop and keep the system at a stable off-state) [23, 32-34]. For example, the mechanism behind the well studied bistable system, lac operon in E. Coli., was proved by mathematical modeling that a positive feedback loop played important role in the system [4, 35]. A series of papers were published on the MAPK signaling pathway in Xenopus oocyte maturation, in which the bistable maturation response to the progesterone concentration level was demonstrated experimentally, and then was suggested to be attributed to positive feedbacks and ultrasensitivity by modeling [23, 25, 36]. Bistable comk gene regulation in Bacillus subtilis was also proved to require positive feedback loops [18]. In a paper written by Brandman et al, fourteen natural systems which could exhibit bistability were confirmed to contain multiple feedback loops [37]. Under the influence of the feedback theory, positive feedback loops or equivalent double negative feedback loops became a design principle for synthetic bistable regulation networks [38, 39]. However, it might not be true that feedback loops are necessary for bistability; it could arise in the absence of explicit feedback loops. A couple of recent papers were published 4

20 to show theoretically that positive feedback loops in the two site mitogen-activated protein kinase pathway were not necessary for its bistability. Instead, its bistability arose from the multisite phosphorylation and dephosphorylation reactions [40-42]. Also, in a simple enzyme catalysis experiment, multiple steady states of residual sucrose concentrations were achieved when sucrose was hydrolyzed by invertase in a continuous stirred tank reactor [43]. Craciun, Tang, and Feinberg presented a theorem in their recent PNAS paper [44] that could draw connections between reaction network structure and the capacity for bistability from a unique perspective. The theorem assumed mass action kinetics for all chemical reactions. Its assertion was based on the graphs and structures associated to a reaction network and did not depend on reaction parameter values, which are hard to know in many cases. The theorem indicated that, for a mass-action network, its species reaction graph, named and defined by the authors in their paper, must satisfy stringent conditions in order for the corresponding differential equations to have the capacity for more than one steady state. Some simple and ordinary enzyme-driven reactions without any apparent feedback loops satisfied those conditions and did indeed have the capacity for bistability. Many biological systems in cells and organisms are essentially chemical reaction networks. Their behaviors can often be described and predicted by a set of chemical reactions taking place in an aqueous solution within membrane-bounded compartments. 5

21 Therefore, the theorem mentioned above may have great applications in dynamics studies of biological systems. It can provide a general and subtle method to draw connections between biological system structures and the capacity for bistability. In their paper, Craciun et al. examined a published mechanism (with rate constants in [45]) of human dihydrofolate reductase (DHFR) catalysis, and showed with modeling that bistability could be achieved in a partially open continuous flow stirred tank reactor (CFSTR) when human DHFR converted dihydrofolic acid (H2F) to tetrahydrofolic acid (H4F). This reaction is a simple two-substrate two-product enzyme catalysis reaction without any apparent feedback loops. This will be discussed in more detail in the next section. 1.3 Dihydrofolate Reductase and the Thymidylate Synthesis Pathway DHFR is a ubiquitous and important enzyme in all eukaryotic and prokaryotic cells. DHFR plays a crucial role in the maintenance of adequate cellular levels of H4F and its derivatives, which are essential for the synthesis of purine nucleotides and thymidylate, which are building blocks for DNA synthesis [46-48]. Therefore, as a catalyst for the only reaction producing H4F in human body, human DHFR became an important target of anticancer drugs, and was studied intensively [47, 49, 50]. Methotrexate, which is an analogue of H2F and can competitively inhibits DHFR reaction, has for a long time been one of the most commonly used anti-cancer chemotherapy drugs [47, 50]. DHFR catalyzes the reduction of H2F to H4F by nicotinamide adenine dinucleotide phosphate (NADPH). The overall reaction is shown in equation (1). 6

22 H2F + NADPH + H + H4F + NADP + (1) For human DHFR, Appleman et al. [45] proposed a mechanism at the level of elementary reactions for the overall reaction (1), and measured or inferred rate constants for all the elementary reactions (measured at 20 C, ph = 7.65). The kinetic scheme is shown in Figure 1. First and second order rate constants are in units of sec -1 and μm -1 sec -1, respectively. Although it is a simple two-substrate-two-product reaction, simulations in [44] indicated that it could exhibit bistability in a simple idealized partially open CFSTR, in which DHFR was entrapped by membranes within the reactor chamber. The simulations were based on the rate constants shown in Figure 1. Figure 1. A detailed mechanism and rate constants for human DHFR catalysis. E = human DHFR; H2F = dihydrofolic acid; H4F = tetrahydrofolic acid; NH = NADPH; N = NADP + ; EX = X bound to human DHFR. This figure is taken from [44]. 7

23 As described in [44], consider a 3.5 ml CFSTR, in which human DHFR is trapped between two membranes which fully retain human DHFR molecules but allow all other species to pass through freely. A solution containing 100 µm H2F and 400 µm NADPH is fed continuously through the inlet at a constant flow rate of F ml/min, and the mixture of both products and unreacted substrates are removed from the CFSTR also at F ml/min. Computer simulations indicate that for values of F between ml/min and ml/min, there are three steady states, two of them stable. A plot of steady state concentration of H4F vs. F is shown in Figure 2. Figure 2. Computed bistability and switch-like behavior in a hypothetical DHFR experiment [44] The theory presented in [44] and its simulated DHFR experiment are very important for at least two reasons. First, it is important for biologists to be able to identify enzyme- 8

24 driven reactions that have the capacity to exhibit multiple steady states, especially in simple and ordinary reactions, since such identification is critical and necessary to correctly interpret experimental results. Second, DHFR is a very important enzyme in biological organisms and is also a classical anti-cancer drug target. Comprehensive understanding of DHFR dynamics is necessary in the study of anti-cancer drugs that target at DHFR. DHFR does not work alone in physiological conditions. Instead, it works in a three enzyme cycle called the thymidylate synthesis pathway, as shown in Figure 3. The other two enzymes in the cycle are thymidylate synthase (TS) and serine hydroxymethyltransferase (SHMT). SHMT is the enzyme following DHFR in the thymidylate synthesis pathway. It catalyzes the conversion from H4F to 5,10-methylene- H4F, in which L-serine donates a carbon and is converted to glycine [51]. TS is in charge of transferring a methyl-group from 5,10-methylene-H4F to deoxyuridine monophosphate (dump) thereby generating H2F and deoxythymidine monophosphate (dtmp) [52, 53]. The thymidylate synthesis pathway as a whole is crucial for cell proliferation. TS and DHFR exist on the same single polypeptide in cells, with the DHFR domain on the amino terminus and the TS domain on the carboxy terminus [54, 55]. Their sequential reactions are the key factor in maintaining the cellular H4F level, which is essential for DNA synthesis. Also, TS catalysis generates dtmp, which will be phosphorylated later to 9

25 thymidine triphosphate, which is also used in DNA synthesis. Therefore, TS, like DHFR, is also regarded as an anti-cancer drug target. The SHMT catalysis reaction is regarded as the largest intracellular provider of one-carbon units, which are necessary substrates in many biosynthetic reactions [56]. Figure 3. Thymidylate synthesis pathway [54] The bistability of DHFR catalysis is of great interest by itself, as mentioned before, but also because that bistability might generate increasingly interesting dynamics by coupling with the action of the other two enzymes in the thymidylate synthesis pathway. Therefore, kinetic simulations on the whole pathway are also of great interest. The Michaelis- Menten kinetic parameters of recombinant human TS and recombinant human SHMT 10

26 were found in reference [57] and [51], respectively. With these kinetic parameters, simulations on the dynamics of the whole thymidylate synthesis pathway become possible. If human DHFR dynamics have the capability of performing bistability, the bistable property may be passed along the cycle and affect the other two enzyme reactions in the thymidylate synthesis pathway. 11

27 Chapter 2: Simulations of Human DHFR Catalysis, the Thymidylate Synthesis Pathway, and Methotrexate Inhibition 2.1 XPPAUT Introduction The theorem presented in Craciun, Tang and Feinberg s PNAS paper [44] can discriminate correctly between those mass action networks that might exhibit multiple steady states and those that certainly cannot. However, it is not a direct tool for exploring the existence of multiple steady states in the networks which have the potential capacity for multiple steady states. For those networks that satisfy the necessary conditions required by the theorem to preclude multiple steady states, it can explicitly draw a conclusion that the reaction network cannot admit more than one positive steady state, no matter what (positive) values the rate constants, effluent coefficients, or species supply rates take. [44]. However, for those networks that violate either one or both of the theorem s required conditions, the theorem provides no information about the networks. For networks for which the theorem does not preclude bistability, simulations and computations are needed to explore for which parameter values (if any) the multiple steady states exist, and what the nature of the steady state locus looks like. XPPAUT is a software tool with a graphical interface that can simulate, animate, and analyze dynamic systems [58, 59]. XPPAUT works well and fast on solving differential 12

28 equations, graphing the solutions, and constructing bifurcation diagrams which are of great interest in bistability studies. Although the bifurcation diagram of the hypothetical human DHFR reaction in Figure 2 was not constructed by XPPAUT, XPPAUT has since been used to probe the influence of parameter values on the existence of multiple steady states in a more systematic way. 2.2 Human DHFR Catalysis in a Partially Open CFSTR Based on the kinetic studies of recombinant human DHFR by Appleman et al. [45], the mechanism of human DHFR catalysis is simple and ordinary: two substrates, NADPH and H2F, are catalyzed by DHFR and converted to NADP + and H4F. There is no particular order in the sequence of substrate H2F or NADPH binding to DHFR to form the ternary complex DHFR NADPH H2F [45, 60]. The thirteen reversible reactions in Figure 1 merely represent binding and unbinding between enzyme and substrates or products, binding and unbinding between substrates or products and enzyme-substrate complexes or enzyme-product complexes, and the chemical transformation of substrates to products [44]. A similar virtual experiment as in [44] was considered as follows: In a CFSTR, human DHFR molecules were trapped between two ideal membranes, which can completely retain DHFR molecules within the reactor chamber but allow all other small molecules (substrates, products, etc.) to pass freely. A solution containing H2F and NADPH was fed continuously through the inlet at a constant flow rate. The reacted mixture, containing 13

29 H2F, NADPH, H4F, and NADP +, was drawn from the outlet. Perfect mixing was assumed. The diagram of a virtual CFSTR is shown in Figure 4. H2F, NADPH DHFR Membrane Stirrer H4F, NADP +, H2F, NADPH Figure 4. Diagram of a partially open CFSTR used in this virtual human DHFR reaction Based on the rate constants of Appleman et al. [45] and anticipated experimental conditions and constraints, one XPPAUT simulation was performed for a partially open CFSTR with a volume of 5.6 ml, a feed solution flow rate of 0.2 ml/min, and a feed composition of 400 μm NADPH and 100 μm H2F. For total DHFR concentration in the reactor chamber greater than μm or less than μm, there was a unique steady-state composition within the CFSTR. However, for intermediate values of total DHFR concentration, there were three different steady states, two of them stable, and one unstable. An XPPAUT-generated diagram of steady-state concentration of H4F vs. total DHFR concentration is shown in Figure 5 (The *.ode file used in XPPAUT to generate Figure 5 containing the ordinary differential equations governing the species concentrations and parameter values is in Appendix A. Also, all other *.ode files that were used to generate bifurcation diagrams in XPPAUT can be also found in Appendix 14

30 A). Note the very narrow range of DHFR concentrations in which multiple steady states exist and also the dramatic change of steady-state H4F concentration as the total DHFR concentration varies from μm to μm. Figure 5. Bifurcation diagram of steady-state H4F concentration vs. total DHFR concentration for the human DHFR reaction in a partially open CFSTR. Reactor volume = 5.6 ml, feed mixture flow rate = 0.2 ml/min. In feed mixture, H2F concentration = 100 μm, NADPH concentration = 400 μm. Solid line indicates stable steady states, while dashed line indicates unstable steady states. In the same CFSTR and under the same operating conditions, except that the total DHFR concentration was held fixed at μm and the feed mixture flow rate was varied, another XPPAUT simulation was performed. For feed mixture flow rates greater than ml/min or less than ml/min, there was a unique steady-state composition within the CFSTR. However, for values of flow rate in between, there were three 15

31 different steady states, two of them stable, and one unstable. An XPPAUT-generated diagram of steady-state concentration of H4F vs. feed mixture flow rate is shown in Figure 6. Here again, note the very narrow range of flow rates for which there are multiple steady states, and note also the sharp switch-like change in steady-state H4F concentration as the flow rate changes from ml/min to ml/min. Figure 6. Bifurcation diagram of steady-state H4F concentration vs. feed mixture flow rate for the human DHFR reaction in a partially open CFSTR. Reactor volume = 5.6ml, total DHFR concentration in reactor chamber = μm. In feed mixture, H2F concentration = 100 μm, NADPH concentration = 400 μm. Solid line indicates stable steady states, while dashed line indicates unstable steady states. 16

32 2.3 Three-Enzyme Cycle Reactions of the Thymidylate Synthesis Pathway in a Partially Open CFSTR As mentioned in Section 1.3, the thymidylate synthesis pathway is an important metabolism pathway in the human body and crucial for cell proliferation. The pathway is composed of three enzymes: TS, DHFR, and SHMT, which are closely connected and work in a cycle as seen in Figure 3. Therefore, the dynamic studies of the three-enzyme cycle as a whole are of great interest. Moreover, if human DHFR catalysis does have the capacity for bistability as indicated in Section 2.2, the bistable DHFR reaction may engender interesting dynamics in the whole cycle when coupled with the actions of the other two enzymes. The rate constants of all elementary reactions like those of human DHFR catalysis in reference [45] are not available in the literature for either human TS catalysis or human SHMT catalysis. However, the Michaelis-Menten kinetic parameters of recombinant human TS and recombinant human SHMT are available in the references [52] and [51], respectively, making the simulations on the dynamics of the three-enzyme cycle reactions in thymidylate synthesis pathway possible. TS is the enzyme before DHFR in the thymidylate synthesis pathway. It catalyzes the methylation of dump by 5,10-methylene-H4F (CH2H4F) to produce dtmp and H2F. The overall reaction of TS catalysis is shown in equation (2). CH2H4F + dump H2F + dtmp (2) 17

33 As proven experimentally, the mechanism of TS catalysis is ordered sequential binding [61, 62]. That is to say, TS must bind to the substrate dump first to form the TS dump complex, and then substrate CH2H4F is able to bind to the binary complex. The Michaelis-Menten kinetic scheme for the reaction is shown in Figure 7. Figure 7. Order sequential mechanism of human TS catalysis The differential equation representing the formation rate of product dtmp or H2F is as follows (The brackets represent the concentration of the species): d[dtmp ] d[h2f] k cat *[TS dump H4 F] (3) dt dt The equilibrium dissociation constant for the binary complex TS dump (K 1 m), the steady-state Michaelis-Menten constant for the formation of the ternary complex TS dump H4F (K 2 m), and the TS mass balance are, respectively, K 1 [TS][dUMP ] m, [TS dump] [CH2H4F ][TS dump ] K 2 m (4) [TS dump CH 2H4F] [ TS total ] [ TS] [TS dump] [TS dump CH2H4F] (5) After substituting equations (4) and (5) into (3) and rearranging it, the product formation rate function which was used in XPPAUT simulations of TS catalysis was as follows: 18

34 d[dtmp ] dt d[h2f] dt K 1 m K k cat 2 m [TS K total 2 m ][dump ][CH2H4F ] [dump ] [dump ][CH2H4F ] (6) Steady-state Michaelis-Menten kinetic parameters of recombinant human TS expressed in E. Coli. were determined in reference [52] ( measured at 25 C). As labeled in Figure 7, K 1 m = 3.4 µm, K 2 m = 8 µm, and k cat = 1.0 s -1 [52]. SHMT is the enzyme after DHFR in the thymidylate synthesis pathway. It catalyzes the conversion of H4F to CH2H4F and serine to glycine. The overall reaction is shown in equation (7). H4F + Serine CH2H4F + Glycine (7) The mechanism of SHMT catalysis is random sequential binding [63, 64], which means there is no particular order in the binding of substrate H4F or serine to SHMT. The Michaelis-Menten kinetic scheme for SHMT catalysis is shown in Figure 8. 19

35 Figure 8. Random sequential mechanism of human SHMT catalysis The differential equation representing the formation rate of product glycine or CH2H4F is as follows: d[glycine dt ] d[ch2h4f ] k cat *[SHMT serin e H4F ] (8) dt The equilibrium dissociation constants for the binary complex SHMT serine (K 1 m) and SHMT H4F (K 2 m), the steady-state Michaelis-Menten constants for the formation of the ternary complex TS dump H4F (K 3 m and K 4 m), and the SHMT mass balance are, respectively, [SHMT ][serine ] K 1 m, [SHMT serine] K 2 m [SHMT ][H4F] [SHMT H4F] ][SHMT H4F ] K 3 [serine m, [SHMT serin e H4F] e] K 4 [H4F][SHMT serin m (9) [SHMT serin e H4F] [ SHMT ] [SHMT] [SHMT serine] [SHMT H4F] [SHMT serine H4F] total (10) 20

36 After substituting equations (9) and (10) into (8) and rearranging it, the product formation rate function which was used in XPPAUT simulations of SHMT catalysis was as follows: d[glycine dt ] d[ch2h4f ] dt K 1 m K 4 m k K cat 4 m [SHMT total 3 m [serine ] K ][serine ][H4F] [H4F] [serine ][H4F] (11) The steady state Michaelis-Menten kinetic rate constants of recombinant human cytosolic SHMT was measured in reference [51] at 30 C. As labeled in Figure 8, K 1 m = 300 µm, K 2 m = 40 µm, K 3 m = 100 µm, K 4 m = 20 µm, and k cat = 575 min -1. The rate constants of the three enzyme reactions were measured at different temperatures: DHFR reaction at 20 C, TS reaction at 25 C, and SHMT at 30 C. As is known, rate constant is a function of temperature. Therefore, rate constant values will be affected by the measurement temperatures. However, rate constants of all three enzymes measured at the same temperature were not available in literature. Since the differences among the three temperatures under which the rate constants were measured are relatively small, they were used together in a same simulation. Nevertheless, errors in simulation results that might be caused by the temperature differences should be kept in mind. With kinetic parameters of all three enzyme reactions in the thymidylate synthesis pathway, a virtual partially open CFSTR experiment, similar to the one shown in Figure 4, was considered as follows: in the CFSTR chamber, DHFR, TS, SHMT molecules were trapped between two ideal membranes, which can completely retain enzyme molecules but allow all other small molecules (substrates, products, etc.) to pass freely. A solution 21

37 containing H2F, NADPH, dump, and serine was fed continuously through the inlet at a constant flow rate. The mixture of all products and unreacted substrates was drawn from the outlet. Perfect mixing was assumed. In order to be consistent with the operating conditions of the DHFR catalysis in Section 2.2, XPPAUT simulations on the whole thymidylate synthesis pathway were conducted with the following parameters: the volume of the CFSTR was 5.6 ml, the feed solution was composed of 100 μm H2F, 400 μm NADPH, 400 μm serine, and 400 μm dump, the flow rate of the feed stream was 0.2 ml/min, and the concentrations of TS and SHMT in the reactor chamber were both 0.06 μm. Simulations indicated that the steady-state concentration of H4F displayed a bistable response to the changes in total DHFR concentration as seen in Figure 9. Compared with the simulation result when only DHFR catalysis was considered in Figure 5, the bistable range within which bistability happened was shifted and expanded in Figure 9. For values of total DHFR concentration greater than μm and less than μm, there were three different steady states, two of them stable, and one unstable. However, when only DHFR catalysis was considered, the bistable range was between μm and μm, which was much smaller. Meanwhile, the steady-state concentrations of H4F in Figure 9 were much lower than those in Figure 5. Also, the differences between two steady states corresponding to a same total DHFR concentration within the bistable range were much smaller than those in Figure 9. It seemed that the bifurcation diagram in Figure 9 was expanded and shifted in the horizontal direction, compressed in the vertical 22

38 Steady state H4F concentration (μm) direction, and shifted towards the much lower steady-state H4F concentration direction compared to the one in Figure 5. This result should be because that human DHFR reaction was coupled with the other two reactions, and H4F was consumed by the reactions along the pathway. Total DHFR concentration (μm) Figure 9. Bifurcation diagram of steady-state H4F concentration vs. total DHFR concentration for the thymidylate synthesis pathway. Reactor volume = 5.6 ml, feed mixture flow rate = 0.2 ml/min. In feed stream, H2F concentration = 100 μm, NADPH concentration = 400 μm, serine concentration = 400 μm, dump concentration = 400 μm. In the reactor chamber, TS concentration = SHMT concentration = 0.06 μm. Solid line indicates stable steady states, while dashed line indicates unstable steady states. 23

39 Simulations also indicated that the bistability of the DHFR reaction was passed along the thymidylate synthesis pathway and affected the dynamics of the other two enzyme reactions. Steady-state concentrations of CH2H4F and H2F also had the capacity for bistability as total DHFR concentration varied, shown in Figure 10 and Figure 11, respectively. Note that the total DHFR concentration ranges within which bistability happened for three enzymes were the same: between μm and μm. The three enzyme reactions displayed bistable responses to the changes of total DHFR concentration simultaneously. Considering Figure 9 and 10 together, it is interesting to note that of a 100 μm H2F stream coming in to the reactor chamber, about 90% of H2F is in the form of CH2H4F when there is sufficient DHFR (sufficient DHFR means that the DHFR concentration is bigger than μm, as seen in Figure 9 and 10), instead of the form of H4F when just considering the DHFR reaction in Section

40 Steady state CH2H4F concentration (μm) Total DHFR concentration (μm) Figure 10. Bifurcation diagram of steady-state CH2H4F concentration vs. total DHFR concentration for the thymidylate synthesis pathway. Reactor volume = 5.6ml, feed mixture flow rate = 0.2 ml/min. In feed stream, H2F concentration = 100 μm, NADPH concentration = 400 μm, serine concentration = 400 μm, dump concentration = 400 μm. In the reactor chamber, TS concentration = SHMT concentration = 0.06 μm. Solid line indicates stable steady states, while dashed line indicates unstable steady states. 25

41 Steady state H2F concentration (μm) Total DHFR concentration (μm) Figure 11. Bifurcation diagram of steady-state H2F concentration vs. total DHFR concentration for the thymidylate synthesis pathway. Reactor volume = 5.6 ml, feed mixture flow rate = 0.2 ml/min. In feed stream, H2F concentration = 100 μm, NADPH concentration = 400 μm, serine concentration = 400 μm, dump concentration = 400 μm. In the reactor chamber, TS concentration = SHMT concentration = 0.06 μm. Solid line indicates stable steady states, while dashed line indicates unstable steady states. 2.4 Human DHFR Catalysis in a Partially Open CFSTR Taking Account of Hindrance Effects In the simulations described above, the membranes used in the partially open CFSTR are ideal, which means they can completely retain enzyme molecules while allowing all other small molecules, including substrates and products, to pass freely. However, under real experimental conditions and constraints, this kind of ideal membrane may not exist. Based on experimental observation and measurements which will be discussed in Section 26

42 3.3.2, it is almost impossible to find a membrane which can completely discriminate between human DHFR molecules and substrate or product molecules, especially under the effect of the concentration polarization phenomena (discussed in detail in Section 3.3.2). A membrane which can fully retain human DHFR molecules could also partially retain the passage of substrate and product molecules. Therefore, this hindrance effect should be considered in XPPAUT simulations of the reactions that happen in the partially open CFSTR, so the simulation results can correctly reflect the real situations taking place in the CFSTR. The reasons for the hindrance effect of the membrane to substrates and products are three-fold. First, membrane pore sizes normally are not exactly uniform and have wide rejection ranges. Membranes used in later experiments were ultrafiltration membranes, which separate molecules by their sizes (molecular weights). Membrane ratings (molecular weight cut-offs) are based on retention (rejection) rather than on passage properties [65]. In other words, membrane rating only focuses on the size of the largest molecule that can pass through the membrane. For molecules smaller than the molecular weight cut-off of a membrane, they may not exhibit free passage due to the existence of smaller pores in the membrane. Second, the molecular weight differences between human DHFR molecules and the substrate or product molecules are not big enough to allow an ideal separation between them. Human DHFR has a molecular weight of approximately 25 kd. The molecular weight of NADPH/NADP + and H2F/H4F are / Da and / Da, respectively. Figure 12 shows the rejection profiles of Millipore 27

43 Ultracel PLC membranes which were used in the experiments to be descried later in Chapter 3. From the figure one can tell that all membranes can retain molecules smaller than their molecular weight cut-offs to some extent. The two membranes PLCCC (5kD cut-off) and PLC10 (10 kd cut-off), which can retain over 95% human DHFR molecules (molecular weight labeled in red) can also retain NADPH/NADP + and H2F/H4F at some level. Third, the concentration polarization phenomena caused by dead-end ultrafiltration and high liquid pressure further reduces the flux rates of small solutes through membranes. Based on experimental observations and measurements, concentration polarization may become severe when low molecular weight cut-off membranes are used, and become the main reason for the hindrance effect. This will be discussed in Section in detail. 28

44 Figure 12. Rejection profiles of Millipore Ultracel PLC membranes [66]. Labels refer to different molecular weight cutoffs as follows: PLCCC = 5 kd, PLC10 = 10 kd (tight retention), PLCGC = 10 kd, PLCTK = 30 kd, PLCHK = 100 kd, PLCMK = 300 kd, PLCXK = 1000 kd. The molecular weight of human DHFR was labeled in red. For the purpose of modeling the partially open CFSTR, while taking account of membrane hindrance effects to small molecules, the concept of the hindrance factor is introduced here. The hindrance effect of a membrane with respect to a particular solute (substrate, product or enzyme) can be represented by a hindrance factor λ i for the i th species, defined by C pi i (12) Cci 29

45 C pi is the concentration of the solute in the permeate side, and C ci is the concentration of the solute in the concentrated side. λ i has a value between 0 and 1. A value of zero indicates complete retention of species i by the membrane, and a value of one indicates free passage of species i through the membrane. The hindrance factor is expected to be affected by the material and molecular weight cut-off of the membrane used in experiment, the feed mixture flow rate, the concentrations of different species in the effluent mixture, and the concentrations of the enzyme or any other large molecules that are retained within the reactor chamber. Based on the experimentally measured hindrance factors, simulations on new dynamical effects that hindrance might cause become possible. The detailed experimental measurement procedures of hindrance factors will be discussed in Section When the Millipore Ultracel PLC 5KD molecular weight cut-off membrane was used with a feed mixture flow rate = 0.2 ml/min, reactor volume = 5.6 ml, a feed concentration of NADPH = 150 μm, and a feed concentration of H2F = 110 μm, hindrance factors were measured experimentally. The values are as follows: λ H2F = λ H4F = 0.9, λ NADPH = λ NADP+ = 0.5. With these operating conditions and initial conditions of NADPH concentration = 150 μm, H2F concentration = 110 μm, simulations of human DHFR catalysis in a partially open CFSTR taking account of the hindrance effects were performed. It turned out that the human DHFR reaction not only had the capacity for bistability, but also periodic oscillation, as shown in Figure 13. In that figure, at each DHFR concentration the circles represent the upper and lower values of the H4F concentration realized in an 30

46 oscillation; open circles represent unstable periodic solutions and filled circles represent stable periodic solutions. The kinetic behavior of the system turned out to be very interesting. There are three different states in the region when DHFR concentration is between μm and μm or between μm and μm: one stable steady state, an unstable periodic oscillatory state, and a stable periodic oscillatory state. This can be regarded as another type of bistability between a stable fixed steady state and a stable oscillation. When total DHFR concentration is greater than μm and less than μm, there are four different states: three unstable steady states and a stable oscillatory state. When total DHFR concentration is less than μm or greater than μm, there is only one steady state. 31

47 Steady state H4F concentration (μm) Total DHFR Concentration (μm) Figure 13. Bistability and periodic oscillation of steady-state concentration of H4F when taking account of hindrance effects. Reactor volume = 5.6ml, feed flow rate = 0.2 ml/min. In the feed stream, H2F concentration = 110 μm, NADPH concentration = 400 μm. Initial conditions: NADPH concentration = 150 μm, H2F concentration = 110 μm. Hindrance factors: λ H2F = λ H4F = 0.9, λ NADPH = λ NADP+ = 0.5. Figure 14 shows the changes of H4F concentration with respect to time in three virtual experiments with different total DHFR concentrations, all corresponding to the same initial condition within the reactor. Note that it might take a very long time, or be impossible for the system to reach a fixed steady state when taking the hindrance effect into account. 32

48 NADPH concentration in the reactor chamber (μm) [DHFR] = 7.83 nm [DHFR] = 7.86 nm [DHFR] = 7.89 nm Time (sec.) Figure 14. Time-course changes of NADPH concentration within the reactor chamber when taking account of hindrance effects with three different DHFR concentrations. Reactor volume = 5.6ml, feed flow rate = 0.2 ml/min. In feed stream, H2F concentration = 110 μm, NADPH concentration = 400 μm. Initial conditions: NADPH concentration = NADP+ concentration = 150 μm, H2F concentration = H4F concentration = 110 μm. Hindrance factors: λ H2F = λ H4F = 0.9, λ NADPH = λ NADP+ = 0.5. The total DHFR concentrations in the reactor chamber were 7.83 nm (orange curve), 7.86 nm (red curve), and 7.89 nm (green curve), respectively. The simulation results bring an important message: the human DHFR-catalyzed reaction may exhibit abundant dynamic behaviors under different operating conditions and constraints. In a physiological environment, the substrates and products of the human DHFR reaction, NADPH, H2F, NADP+, and H4F, not only take part in DHFR catalysis, 33

49 but also interact with or participate in other metabolic processes. Suppose they are consumed by a first order reaction or produced by a zero th order reaction in other metabolic processes following a first order reaction mechanism, then the situation mathematically becomes quite similar to that when the hindrance effect is considered. In such cases, the human DHFR reaction might have the capacity for both bistability and oscillation. 2.5 Human DHFR Catalysis in a Classical CFSTR Because the hindrance effect of the membrane to small molecules and the resulting long transient time before the system reaches fixed steady states, which are far longer than what might be expected for the lifetime of the active enzyme in experiments, the intuitive partially open CFSTR with membrane compartments may not be a good choice to experimentally validate the existence of bistability in the human DHFR reaction. Initial experimental attempts encountered many difficulties and it was concluded that it was unrealistic to reach steady states in a partially open CFSTR containing ultrafiltration membranes within a practical time period. Initial experimental attempts and results of the human DHFR reaction in a partially open CFSTR will be discussed in detail in Section 3.1. A simple classical CFSTR may be a good alternative to the partially open CFSTR for achieving bistability in the human DHFR reaction (However, such experiments would require a far larger supply of enzyme.). Consider a virtual DHFR reaction in a classical 34

50 CFSTR as follows: two streams of solutions, one containing substrates (H2F and NADPH) and the other containing DHFR, are fed continuously into a classical CFSTR through its inlet. Both streams are fed at constant flow rates, which might be different from each other. The reacting mixture, containing products, residual substrates, enzyme, and all other binary or ternary enzyme complexes, are drawn from the outlet. Perfect mixing is assumed. A diagram of the classical CFSTR is shown in Figure 15. The DHFR stream is separate from the substrate stream to prevent reactions in the pipes. A new set of differential equations based on the elementary reaction rate constants in Figure 1 and new operating conditions of the classical CFSTR were used in new simulations on DHFR reactions in the classical CFSTR. H2F, NADPH DHFR Reacted mixture Stirrer Figure 15. Virtual DHFR reaction in a classical CFSTR Based on the dimensions of a real classical CFSTR that was constructed in anticipation of use in later experiments, an XPPAUT simulation was performed for a CFSTR with a volume of 1.15 ml, a substrate solution flow rate of 0.6 ml/min, a substrate solution 35

51 Steady state H4F concentration (μm) composition of 400 μm NADPH and 200 μm H2F, and a DHFR solution flow rate of 0.6 ml/min. An XPPAUT-generated diagram of steady-state concentration of H4F vs. total DHFR concentration in the feed stream is shown in Figure 16. For DHFR concentrations in the feed stream greater than μm or less than μm, there was a unique steady-state composition within the CFSTR. However, for intermediate values of total DHFR concentration, there were three different steady states, two of them stable, and one unstable. DHFR concentration in feed (μm) Figure 16. Bifurcation diagram of steady-state H4F concentration vs. DHFR concentration in feed stream for the human DHFR reaction in a classical CFSTR. Reactor volume = 1.15 ml, feed mixture flow rate = 0.6 ml/min, DHFR solution flow rate = 0.6 ml/min. In feed mixture, H2F concentration = 200 μm, NADPH concentration = 400 μm. Thick line indicates stable steady states, while thin line indicates unstable steady states. 36

52 Similarly, in the same classical CFSTR and under the same operating conditions, except that the flow rate of the DHFR feed stream was 0.2 ml/min, the DHFR concentration in the feed was held fixed at 0.08 μm, and the flow rate of substrates solution was varied, another XPPAUT simulation was performed. The steady-state concentration of H4F showed a bistable response to changes in substrates solution flow rate, as shown in Figure 17. Also in the same classical CFSTR and under the same operating conditions, except that the substrate solution flow rate and DHFR concentration in the feed stream were held fixed at 0.2 ml/min and 0.1 μm respectively, and the flow rate of DHFR feed stream varied, simulation showed that the steady-state concentration of H4F exhibited a bistable response to the changes of DHFR feed stream flow rate, as illustrated in Figure

53 Steady state H4F concentration (μm) Flow rate of substrate solution (ml/min) Figure 17. Bifurcation diagram of steady-state H4F concentration vs. substrate solution flow rate for the human DHFR reaction in a classical CFSTR. Reactor volume = 1.15 ml, DHFR feed solution flow rate = 0.6 ml/min, DHFR concentration in the feed stream = 0.1 μm. In substrate solution, H2F concentration = 200 μm, NADPH concentration = 400 μm. Thick line indicates stable steady states, while thin line indicates unstable steady states. 38

54 Steady state H4F concentration (μm) Flow rate of DHFR feed solution (ml/min) Figure 18. Bifurcation diagram of steady-state H4F concentration vs. flow rate of DHFR feed steam for the human DHFR reaction in a classical CFSTR. Reactor volume = 1.15 ml, substrate solution flow rate = 0.2 ml/min, DHFR concentration in the feed = 0.1 μm. In substrate solution, H2F concentration = 200 μm, NADPH concentration = 400 μm. Thick line indicates stable steady states, while thin line indicates unstable steady states. The simulations of human DHFR reaction in a classical CFSTR bring good news: a classical CFSTR is easier to be constructed compared to the partially open CFSTR with a membrane compartment. Also, a classical CFSTR avoids the concentration polarization problem, which is difficult to eliminate when membranes are used. However, experiments that take place in a classical CFSTR would require a far larger supply of human DHFR. 39

55 Fortunately, recombinant human DHFR can be made in laboratory in large supply following the protocol in reference [67, 68]. 2.6 Human DHFR Catalysis in the Presence of Methotrexate in a Classical CFSTR As mentioned in Section 1.3, methotrexate is an analogue of H2F and acts as an inhibitor of DHFR. Methotrexate can bind to DHFR firmly and inactivate it. Figure 19 shows the similarity in the molecular structures of H2F and methotrexate. Because it competitively inhibits DHFR function and thwarts the conversion of H2F to H4F, methotrexate is used as a popular cancer chemotherapy drug. If, in the absence of methotrexate, there can be a high conversion steady-state and a low conversion steady-state in the human DHFR reaction as proposed in [44] and in the simulations in Section 2.2, then dynamics of the human DHFR reaction in the presence of methotrexate could perhaps be more interesting and complicated than cancer scientists had supposed For this reason, studying human DHFR dynamics in the presence of methotrexate inhibition is of great significance. H2F Methotrexate Figure 19. Molecular structures of H2F and methotrexate. The backbones of the two molecules are the same. Differences in ligands of the two molecular structures are circled in red. 40

56 Appleman et al. studied the kinetics of methotrexate inhibiting human DHFR in reference [69]. The kinetic scheme of methotrexate inhibiting the human DHFR reaction is shown in Figure 20. Rate constants of all elementary reactions were measured or inferred by Appleman et al. under the same conditions as human DHFR catalysis in [45] (20 C, ph = 7.65). First and second order rate constants are in units of sec -1 and μm -1 sec -1. Figure 20. Kinetic scheme of methotrexate inhibiting DHFR. MTX = methotrexate, MTX* = isomer of methotrexate. First and second order rate constants are in units of sec -1 and μm -1 sec -1. With the methotrexate inhibition rate constants in Figure 20 and the human DHFR catalysis rate constants shown in Figure 1, dynamical simulations of human DHFR inhibited by methotrexate were performed using XPPAUT. Consider a classical CFSTR similar to the one discussed in Section 2.5: the volume of the CFSTR is 1.15 ml, three streams of solutions, containing substrates, DHFR, and methotrexate, respectively, were fed into the reactor continuously at constant flow rates. The reacted mixture was continuously drawn from the reactor. Perfect mixing was assumed. The operating conditions were as follows: flow rates of substrate solution, DHFR solution, and methotrexate solution were all at 0.2 ml/min. In the feed streams, NADPH concentration 41

57 Steady state H4F concentration (μm) = 400 μm, H2F concentration = 200 μm, and DHFR concentration = μm. The simulation result indicated that steady-state H4F concentration exhibited bistable behavior as the methotrexate concentration in the feed stream varied, shown in Figure 21. For methotrexate concentrations in the feed solution less than μm or greater than μm, there was only on steady state. However, when the methotrexate concentration in feed solution was between μm and μm, there were two stable steady states and one unstable steady state. Methotrexate concentration in feed solution (μm) Figure 21. Bifurcation diagram of steady-state H4F concentration vs. concentration of methotrexate in the feed solution. Limit points are labeled in red. Reactor volume = 1.15 ml, flow rates of substrates solution, DHFR solution, methotrexate solution = 0.2 ml/min. In substrates solution, H2F concentration = 200 μm, NADPH concentration = 400 μm. DHFR concentration in the feed solution = μm. Thick line indicates stable steady states, while thin line indicates unstable steady states. 42

58 The simulation result of human DHFR dynamics in the presence of methotrexate as shown in Figure 21 highlight an important message: the response of steady-state H4F concentration to changes in methotrexate concentration is not linear. As shown in Figure 21, steady-state H4F concentration may display a switch-like behavior when small changes of methotrexate concentration happen in the region near the limit points (labeled in red in Figure 21) of the bifurcation diagram. For example, if methotrexate concentration in feed solution increases from 0.45 μm to 0.47 μm (the right limit point in Figure 21 corresponding to methotrexate concentration = 0.46 μm), the steady state H4F concentration will dramatically drop from 94.5 μm to 74.2 μm. Similarly, if methotrexate concentration in feed solution decreases from 0.05 μm to 0.03 μm (the left limit point in Figure 21 corresponding to methotrexate concentration = 0.04 μm), the steady state H4F concentration will increase from 83.8 μm to 96.1 μm. Having the correct understanding for the dynamic behavior of this system is important and necessary for the study of methotrexate as an anti-cancer drug targeting DHFR. It will also help researchers to correctly interpret unexpected experimental results when studying the dynamics of human DHFR inhibition by methotrexate. With the same operating conditions as in Figure 21, four different time-course change curves of H4F concentration with three different methotrexate concentrations in the feed solutions were plotted in Figure 22. The methotrexate concentrations in the feed solutions were 0 μm (red curve), 0.3 μm (blue curve and green curve), and 0.5 μm (yellow line). The initial methotrexate concentrations within the CFSTR for the four curves were the 43

59 same as those in the feed streams. The initial concentrations of other species within the CFSTR for the red, green and yellow curves were exactly the same: H2F concentration = 200 μm, NADPH concentration = 400 μm, and DHFR concentration = μm. The initial conditions for the blue curve were the same as those for other three curves except that DHFR concentration = 0.1 μm. Note that a tiny difference in the initial DHFR concentrations can lead to two distinctly different steady states (blue curve and green curve). Figure 23 shows the effects of different feed methotrexate concentrations on the shape and position of the bifurcation diagram of the steady-state H4F concentration vs. DHFR concentration in the feed. The bifurcation diagram kept its original shape (when methotrexate concentration = 0 μm) and shifted to the left as methotrexate concentration increased. The dots in four different colors corresponded to the four different steady states in Figure 22. A good understanding of the bistability in human DHFR dynamics in the presence of methotrexate is crucial for experimentalists to correctly interpret experimental results. For example, if the feed methotrexate concentrations were both 0.3 μm, someone who saw only the low steady state in the experiment might think that the methotrexate was effective at blocking H4F production, but someone who saw only the high steady state come to a different conclusion. 44

60 H4F concentration (μm) [Methotrexate] = 0 µm [Methotrexate] = 0.3 µm [Methotrexate] = 0.3 µm [Methotrexate] = 0.5 µm Time (sec.) Figure 22. Time-course changes of H4F concentration with three different feed concentrations of methotrexate: 0 μm (red curve), 0.3 μm (blue curve and green curve), and 0.5 μm (yellow curve). Reactor volume = 1.15 ml, flow rates of substrates solution, DHFR solution, and methotrexate solution = 0.2 ml/min. In the substrate solution, H2F concentration = 200 μm, NADPH concentration = 400 μm. DHFR concentration in the feed solution = μm. The initial conditions for the red, green, and yellow curves are: H2F concentration = 200 μm, NADPH concentration = 400 μm. DHFR concentration = μm. The initial conditions for the blue curve are: H2F concentration = 200 μm, NADPH concentration = 400 μm. DHFR concentration = 0.1 μm. The initial concentrations of methotrexate in the CFSTR are the same as those in the feed solutions. 45

61 Steady state H4F concentration (μm) DHFR concentration in feed (ml/min) Figure 23. Effects of three different concentrations of methotrexate on the bifurcation diagram of steadystate H4F concentration vs. DHFR concentration in feed solution. Reactor volume = 1.15 ml, flow rates of substrates solution, DHFR solution, methotrexate solution = 0.2 ml/min. In the substrates solution, H2F concentration = 200 μm, NADPH concentration = 400 μm. DHFR concentration in the feed = μm. Thick line indicates stable steady states, while thin line indicates unstable steady states. 46

62 Chapter 3: Problems in Eliciting Bistability Experimentally: Some Preliminary Exploration Aside from simulations on dynamics of the human DHFR reaction, the thymidylate synthesis pathway, and methotrexate inhibition, another objective of this thesis is to lay the foundation for experimental verification of the existence of bistability in the human DHFR reaction. The first intuitive thought of a reasonable experiment was to construct a partially open CFSTR as described in Section 2.2. However, experimental observation showed that the concentration polarization problem (Section ) and the high liquid pressure exerted on the membrane caused by the low molecular weight cutoff membrane which was used to fully retain human DHFR molecules within the reactor chamber made it impossible for the partially open CFSTR containing free human DHFR to reach steady states within a practical time period (These problems are discussed in Section 3.1.2). Therefore, two alternative experimental plans were proposed to explore proper and practical experimental settings which may validate the existence of bistability in the human DHFR reaction. The first method utilized immobilized human DHFR tethered to chitin beads in a partially open CFSTR with very large molecular weight cutoff membranes (This is discussed in Section 3.2). The second studied the free human DHFR reaction in a classical CFSTR as described in Section

63 3.1 Human DHFR Catalysis in a Partially Open CFSTR Experiments were undertaken to elicit a response similar to the one described in the virtual experiments of Section 2.2. Ideally, experimental bifurcation diagrams of steadystate concentration of H4F vs. total DHFR concentration, and steady-state concentration of H4F vs. feed solution flow rate would be plotted in the same way as those in Figure 5 and Figure 6, respectively. However, as indicated by the curve in Figure 5, the multiple steady states are only predicted in a very narrow region of DHFR concentrations, between μm and μm. It would be too difficult to experimentally manipulate the DHFR concentration in such a narrow range. Nevertheless, it is reasonable to expect that a sharp switch-like change in the H4F steady-state concentration could be demonstrated in experiments in response to variations in the total DHFR concentration. That is to say, the steady-state concentration of H4F is predicted to jump from a moderate level to a significantly higher level within a very narrow range of DHFR concentrations. Similarly, as indicated by the curve in Figure 6, multiple steady states are only predicted in a very narrow region of feed mixture flow rates, between ml/min and ml/min. Accordingly, a sharp bistable switch-like curve of steady-state H4F concentration vs. feed mixture flow rate would be expected in experiment. Note that every point on the curves in Figure 5 and Figure 6 is at a steady state. Therefore, before achieving a switch-like curve of steady-state H4F concentration vs. either total DHFR concentration or feed mixture flow rate, a steady-state H4F concentration would have to be achieved within the membrane compartment of the partially open CFSTR with a 48

64 certain total DHFR concentration and a certain feed mixture flow rate in a single experiment Materials and Methods Materials and Devices Recombinant human DHFR was expressed and purified by Dr. Yangzhong Tang at Harvard Medical School according to references [67, 68]. H2F and NADPH were purchased from Sigma-Aldrich. Ultracel PLC membranes with 5 kd, 10 kd, and 30 kd molecular weight cutoffs were purchased from Millipore. The CFSTR was designed and fabricated by Paul Green at Ohio State University. The material of the CFSTR body is cast acrylic. An exploded diagram of the partially open CFSTR is shown in Figure 24, and a picture of the real CFSTR is shown in Figure 25. The pump used to continuously feed the CFSTR was a Shimadzu liquid chromatography pump. The detector used to monitor the real-time changes in light absorbance of the effluent solutions from the CFSTR was a Shimadzu UV/VIS photodiode array detector. A Shimadzu UV spectrophotometer was used in enzyme activity assays. 49

65 Screws Inlet part Porous support Membrane Membrane Porous support Outlet part Screws Figure 24. Exploded diagram of CFSTR drawn by Paul Green Figure 25. Partially open CFSTR with a membrane compartment 50

66 Experimental Conditions All reactions and measurements took place in the buffer solution (Appleman s buffer) described in reference [45]. The composition of the buffer was 50 mm Tris, 25 mm acetic acid, 25 mm MES, 100 mm NaCl, 25 mm β- mercaptoethanol and 0.02% sodium azide. The ph of the buffer was adjusted to 7.65 by adding 1 M NaOH. All reactions and measurements took place at room temperature (about 25 C, but uncontrolled in the preliminary experiments). Unless stated otherwise, 0.5 mg/ml casein was added in the CFSTR chamber as carrier protein in the DHFR reactions Analytical Methods NADPH has a characteristic absorbance at 340 nm, as seen in Figure 26, whereas NADP + does not. Therefore, conversion of NADPH to NADP + can be monitored by the light absorbance at 340 nm using UV detection. As reported in reference [70], the extinction coefficient for the DHFR reaction at 340nm is 12.3 mm -1 cm -1. With this extinction coefficient, conversion of NADPH to NADP + can be calculated according to the change in light absorbance at 340 nm. Knowing the stoichiometry of the overall DHFR reaction in equation (1), the amount of H4F generated at steady state should be equal to the amount of NADPH consumed. Therefore, steady-state H4F concentration can be calculated according to the change in light absorbance of the effluent solution from the CFSTR at 340 nm. 51

67 Light absorbance (mau) Wave length (nm) Figure 26. Absorbance spectrum of 75 μm NADPH in Appleman s buffer Enzyme Activity Assay The activity assay for human DHFR was performed as described in the Sigma Dihydrofolate Reductase Assay Kit [71]. The principle of the assay is based on reaction (1). The consumption of NADPH was calculated from the decrease in light absorbance at 340nm by a Shimadzu UV spectrophotometer. 52

68 3.1.2 Experimental Procedures and Results The first important requirement for establishing a successful continuous operation at steady state in the partially open CFSTR with a membrane compartment is the complete retention of the enzyme activity during the experiment. That is to say, the enzyme activity in the reactor chamber should be kept at a constant level throughout the flow experiment, so a steady state with respect to a certain enzyme concentration can be reached. Besides slow spontaneous enzyme activity loss at room temperature, there are two other reasons that may cause significant enzyme activity loss. First, if the molecular weight cutoff of the membrane is not proper, enzyme molecules could leak through the pores of the membrane, especially under high liquid pressure. Second, adsorption of enzyme molecules onto the inner walls of the reactor chamber could lead to enzyme inactivation Membrane Selection The material and molecular weight cutoff of the membrane used to trap the enzyme molecules were critical to the complete retention of enzyme molecules. Basic principles for choosing proper ultrafiltration membranes are summarized in references [72, 73]. The material of the membrane should be inert, strong, and interact with the solutes as little as possible. Regenerated cellulose is a popularly used membrane material, which has good mechanical strength and very low protein binding [74, 75]. The Millipore Ultracel PLC membrane, which is fabricated by casting regenerated cellulose onto a microporous polyethylene substrate and creates a uniform, ultra-low protein binding, and mechanically strong structure [65, 66], was used in the partially open CFSTR experiments. Molecular 53

69 weight cutoff is another important factor that contributes to the retention ability of a membrane. According to literature studies, the nominal molecular weight cutoff of a membrane should be no larger than half the molecular weight of the molecules that are to be retained, to retain at least 90% of said molecules [75, 76], and should be no larger than 1/4 to 1/5 of the molecular weight of the molecules that are to be retained for at least 99% retention [72, 75]. Obviously, the smaller the membrane s molecular weight cutoff, the higher retention capability it will have. However, the molecular weight cutoff of the membrane should not be too small, because this would require increased fluid pressure, which might break the membrane; more importantly, it could make the concentration polarization problem severe and cause other small solutes to accumulate near the membrane. Following the selection rules mentioned above, a 5 kd molecular weight cutoff membrane should be able to retain at least 99% human DHFR molecules which has a molecular weight of approximately 25 kd. However, a 5 kd membrane may partially block the passage of NADPH/NADP + and H2F/H4F molecules. As mentioned in Section 2.4, the molecular weight of NADPH/NADP + and H2F/H4F are / Da and / Da, respectively. According to the retention profiles of the Ultracel PLC membranes in Figure 12, one can tell that the rejection coefficient of the 5 kd membrane towards human DHFR is about 1, and the rejection coefficients towards NADPH/NADP + and H2F/H4F are about 0.25 and This means that 5 kd membrane can completely 54

70 retain human DHFR molecules but also partially block the passage of NADPH/NADP + and H2F/H4F molecules. Larger molecular weight cutoff membrane, such as a 10 kd molecular weight cutoff membrane, allows NADPH/NADP + and H2F/H4F molecules to pass more easily compared with a 5 kd membrane. However, the 10 kd membrane is not as effective as the 5 kd membrane in retaining human DHFR molecules. As seen in Figure 12, the rejection coefficients of the PLC10 10 kd membrane towards NADPH/NADP + and H2F/H4F are almost 0. This shows that the 10 kd membrane allows the substrates and products to pass freely, which is a presumption in the simulations in Section 2.2. Meanwhile, a 10 kd membrane requires a lower liquid pressure to achieve a desired flow rate compared to a 5 kd membrane, which reduces the force exerted on the membrane and helps to relieve the concentration polarization problem. Also, a 10 kd membrane allows flow reactions in the CFSTR to have higher flow rates, so the CFSTR can have a smaller residence time, expediting the process of reaching steady states. With a 5 kd membrane, the highest flow rate that the existing experimental CFSTR could afford was 0.25 ml/min, while the 10 kd membrane could operate at flow rates as high as 0.7 ml/min. However, a severe problem of the 10 kd membrane which make it a worse choice compared to a 5 kd membrane is that it can not completely retain human DHFR molecules within the reactor chamber: a problem that is compounded when it takes a long time for a flow experiment to complete. As seen in Figure 12, the rejection coefficient of the PLC10 10 kd membrane towards human DHFR is about Although the rejection 55

71 coefficient seems not too small, with the effect of a long reaction time and high liquid pressure, the activity loss of human DHFR in a 2 hour flow experiment with 10 kd membranes in both inlet and outlet ports is large. In experiments, the average total enzyme activity loss in a flow experiment with 10 kd membranes in both inlet and outlet ports was about 50% of the original activity. While with 30 kd membrane in the inlet port and 5 kd membrane in the outlet port, the total activity loss reduced to about 13%. Here, the total activity loss was a consequence of all factors that may cause activity loss: spontaneous activity loss, surface adsorption, and enzyme leakage through the membranes. Since enzymes used in the experiments were the same human DHFR and the materials of the two kinds of membranes were the same, it is reasonable to assume that the rate of spontaneous activity loss and the rate of activity loss due to surface adsorption were the same in both cases. Therefore, the differences in the total enzyme activity loss were most likely caused by enzyme leakage through the outlet 10 kd membrane. A comparison between the 5 kd and 10 kd membranes when used in the outlet ports is shown in Table 1. Table 1. Comparison between 5 kd and 10 kd (PLC 10) Millipore Ultracel PLC membranes Molecular weight cutoff Rejection coefficient towards human DHFR Rejection coefficient towards NADPH/NADP + Rejection coefficient towards H2F/H4F Highest affordable flow rate (ml/min) Total enzyme activity loss in 2 hrs 5 kd % 10 kd % 56

72 The partial rejection of the 5 kd membrane towards substrates and products could be mitigated by adding a higher concentration of substrates and products in the initial solution, which will be discussed in Section However, the large leakage of human DHFR molecules through 10 kd membrane is fatal to flow experiments in a CFSTR which requires the DHFR concentration be kept at a constant level during each experiment. Therefore, the 5 kd membrane is a better choice of the membrane used in the outlet port when compared with the 10 kd membrane Effects of Carrier Protein on Preventing Enzyme Activity Loss Enzymes may experience activity loss due to surface adsorption even if low protein binding materials are used [77]. To minimize the adsorption effect, a carrier protein is often added to an enzyme solution. Carrier proteins are inert proteins which do not interfere with enzyme reactions and help to occupy and saturate potential protein binding surfaces of the reaction containers. If a carrier protein is used in enzyme assays, the light absorbance of the carrier protein should not interfere with that of the target enzyme. The concentration of carrier protein in an enzyme reaction solution is usually much higher than that of the enzyme to make sure that potential binding surfaces are saturated. However, the concentration of carrier protein should not be too high in the current experimental situations, because a higher carrier protein concentration could cause a more severe concentration polarization problem. 57

73 In experiments, different kinds of carrier proteins were compared for their effects on preventing enzyme activity loss due to surface adsorption. Bovine serum albumin (BSA), human serum albumin (HSA) and casein were commonly used carrier proteins [77]. They all satisfy the requirements as carrier proteins for the human DHFR reactions. Human DHFR concentrations used in CFSTR flow experiments were on the order of 10-2 mg/ml. A ratio greater than ten to one of carrier protein to DHFR concentration was used in the reactor chamber solution to make sure that the inner surfaces were occupied and saturated by the carrier protein. Experiments comparing the effect of different kinds of carrier proteins on maintaining DHFR activity in the CFSTR were performed as follows: a solution containing 150 μm NADPH, 100 μm H2F, mg/ml DHFR and a certain concentration of a certain carrier protein was added to the reaction chamber, and the reaction was allowed to take place for 2 hours. There were no flows through the reaction chamber. That is to say, the CFSTR worked as a batch reactor in this situation. DHFR activity assays were performed on the reaction chamber solution at the beginning and the end of the batch reactions. Experimental results showed that, at room temperature, human DHFR lost 15% of its original activity after 2 hours when no carrier protein was added into the reaction chamber solution, compared to a 14% activity loss when 0.8 mg/ml BSA was added, a 12% activity loss when 0.5 mg/ml HSA was added, and a 5% activity loss when 0.5 mg/ml casein was added to the reaction chamber. Since all of their concentrations were much higher than that of DHFR, their effects as carrier proteins should not be affected by the small differences in their concentrations. Comparison among different carrier proteins on maintaining DHFR activity is shown in Table 2. Since 58

74 casein is the best in maintaining human DHFR activity among the three carrier proteins, unless stated otherwise, 0.5 mg/ml casein was added into the CFSTR chamber as a carrier protein in later flow experiments. Table 2. Comparison among different carrier proteins on keeping DHFR activity Carrier Protein BSA HSA Casein No additive Concentration 0.8 mg/ml 0.5 mg/ml 0.5 mg/ml N/A DHFR activity loss in 2 hrs 14% 12% 5% 15% Concentration polarization In a dead-end ultrafiltration process, pressure exerted on the solution causes solutes to flow toward the membrane surface. If the convective flux of the solutes exceeds the rate at which solutes pass through the membrane, the local concentration of the solutes near the membrane will be elevated. This phenomenon is called concentration polarization [34]. As mentioned in Section 2.4, the mechanism of the regenerated cellulose membranes used in the CFSTR to retain enzyme is size exclusion. Therefore, the macromolecules in the solution of the CFSTR chamber, including the carrier protein and DHFR, cannot pass the membrane but instead accumulate near the membrane wall under fluid pressure. When the concentrations of the macromolecules at the membrane surface and the fluid pressure exerted on the membrane are high enough, the accumulated macromolecules may form a gel-like layer [78] and act as another resistance to mass flow, and reduce the flux rates of small solutes. 59

75 Experimental results showed that NADPH molecules could be partially blocked by the 5 kd membrane. The setup of one experiment where NADPH passage was noticeably hindered by the 5 kd membrane is described as follows: a feed stream of 75 μm NADPH was continuously fed the flow system. A 5 kd membrane and a 30 kd membrane were used in the outlet and inlet ports of the CFSTR, respectively. (There should be no reverse flux in the CFSTR, and a higher molecular weight cutoff membrane in the inlet could help to reduce the liquid pressure inside the reactor.) As shown in Figure 27, an empty flow system with the liquid chromatography pump directly connected to the real-time UV detector was first baselined with Appleman s buffer solution for about 20 min. After t = 23 min, the empty pump-detector system was fed with 75 μm NADPH solution until it reached a steady state. The CFSTR whose reactor chamber was initially filled with a solution containing 75 μm NADPH and 0.5 mg/ml casein was mounted between the pump and the detector. The flow rate of the feed stream was set at 0.2 ml/min. It took some time for the feed stream to fully fill another two empty compartments of the CFSTR except the reactor chamber, and the first effluent from the CFSTR came out at t = 44 min. The irregular curve with peaks between 44 min and 54 min was due to bubbles in the reactor chamber. After degassing, the curve became smooth and showed the light absorbance changes of the effluent from the CFSTR. Note that the first effluent from the CFSTR had a much lower absorbance at 340 nm compared to that of the feed stream. This indicated that NADPH molecules were partially blocked by the 5kD membrane. As time went by, the absorbance of the effluent gradually increased towards the feed stream absorbance level. However, this process was slow and the absorbance of the effluent 60

76 UV adsorption at 340 nm (mau) solution was still lower than that of the feed stream even after two hours. Note that the hindrance effect of the 5 kd membrane on NADPH molecules made the effluent solution undergo a long transient process. Feed level Effluent level Time (min.) Figure μm NADPH retention by 5 kd membrane with an initial NADPH concentration = 75 μm The hindrance factor of the 5 kd membrane towards substrate and product molecules as defined in Section 2.4 were measured in experiments. In the above experiment, after the effluent absorbance reached a steady state, the light absorbance at 340 nm was measured for the solution in the CFSTR chamber and the effluent stream. Since the NADPH concentration of a solution is proportional to its light absorbance at 340 nm, λ NADPH could be calculated using equation (13): CpNADPH Absorbance of effluent solution at 340 nm NADPH (13) C Absorbance of CFSTR chamber solution at 340 nm cnadph 61

77 The hindrance factor of NADPH in the above experiment is 0.5. Because NADPH and NADP + have very similar molecular weights, it is reasonable to suppose that λ NADP+ = λ NADPH = 0.5. Under the same operating conditions, except that the initial concentration of H2F was 100 μm and the light absorbance was monitored at 282 nm, which is the characteristic wavelength of H2F, λ H2F and λ H4F were measured and calculated as follows: λ H2F = λ H4F = 0.9. Based on the definition of the hindrance factor, one possible way to mitigate the long transient process before the effluent solution reaches a steady state is to put an initial solution with a concentration of C ci = C pi / λ i in the reactor chamber before each experiment. In theory, after the effluent reaches the steady state, the concentration of a certain solute in the effluent should equal to that of the feed mixture solution. Therefore, C pi in this equation can be substituted by the feed concentration of species i. This method would help to speed the effluent concentration to reach a steady state. Experimental results showed that when a solution with a composition of C ci = C pi / λ i was used as the initial conditions, the transient process before reaching a steady state was greatly shortened. In one experiment similar to the one in Figure 27, a feed stream of 150 μm NADPH was continuously fed the flow system, and a 5 kd membrane and a 30 kd membrane were used in the outlet and inlet ports, respectively. As shown in Figure 28, an empty flow system with the liquid chromatography pump directly connected to the UV detector was first baselined by Appleman s buffer solution for about 35 min (the small spark at t = 15min was due to bubbles in the system). After t = 35 min, the empty pumpdetector system was fed with 150 μm NADPH solution until it reached a steady state. 62

78 Then the system was baselined back to zero absorbance level (this level lasted for a very short time, so that in Figure 28 it was only a dot). The CFSTR whose reactor chamber was initially filled with a solution containing 300 μm NADPH (hindrance factor for NADPH = 0.5) and 0.5 mg/ml casein was mounted between the pump and the detector. The flow rate of the feed stream was set at 0.2 ml/min. It took some time for the feed stream to fully fill all the parts of the CFSTR, and the first effluent from the CFSTR came out at t = 65 min. Note that the absorbance of the first effluent from the CFSTR was higher than the feed level, but the high absorbance of the effluent disappeared quickly after t = 90 min, and kept at a constant level which was a little bit lower than the feed level thereafter. This might be due to an initial lack of macromolecules accumulated near the membrane and therefore less concentration polarization phenomenon at the beginning, but the blockage effect taking dominance relatively quickly. The experimental result indicated that a higher initial NADPH concentration of 300 μm helped the effluent to reach a steady state much faster than when an initial NADPH concentration equal to that of the feed was used. 63

79 Figure μm NADPH retention by 5 kd membrane with an initial NADPH concentration = 300 μm Although using an initial composition of higher concentration of the substrate could effectively mitigate the long transient process experienced by the substrate solution, the simulations in Section 2.4 indicated that the dynamics of the human DHFR reaction in a partially open CFSTR taking account of hindrance effects would become very complicated and make the system experience long transient process before reaching 64

80 steady states or enter periodic oscillations. This situation contradicted the proposed experimental plan which intended to experimentally validate the existence of bistability in a partially open CFSTR when human DHFR catalyzes the conversion of H2F to H4F. This conclusion was confirmed in experiments: good steady states were never achieved in a partially open CFSTR with a 5 kd membrane. Therefore, other experimental designs which can keep human DHFR activity at a constant level as well as avoiding hindrance effects should be considered. 3.2 Human DHFR Immobilization on Chitin Beads Because of the enzyme leakage through the 10 kd membrane and the concentration polarization caused by the 5 kd membrane, as mentioned in Section 3.1, it appears that it will be extremely difficult to elicit the dynamical effect predicted in [44] by means of the partially open enzyme-in-solution CFSTR originally envisioned. An alternative route might involve immobilization of recombinant human DHFR (with an added chitin binding domain, CBD) on micron-sized beads, which would then be suspended in solution within the CFSTR chamber. The beads and tethered DHFR molecules could then be well-retained by very large pore size membranes at the feed and effluent ports. As a result, higher flow rates could be reached at acceptable pressure, which in turn would help to speed up the process of reaching steady states. More importantly, the concentration polarization effect could be mitigated. 65

81 Although bead immobilization has clear advantages, the biggest concern about the immobilized enzyme is that it may change the kinetics of the enzyme to some extent. The reasons for the kinetics to change are as follows: the structure and orientation of the enzyme molecules may be changed during the immobilization process [42]; insertion of the CBD into DHFR might change the catalytic activity of DHFR; and the immobilized enzyme molecules work in a different microenvironment, as compared to the free enzyme in bulk solution [42]. It should be kept in mind, though, that the bistability effects simulated in [44] were based upon kinetics measured by Appleman et al [45] in solution, and it is a concern that the tethering of DHFR to beads would lead to kinetic departures from the differential equations upon which the simulations were based. Nevertheless, if the predicted effects could in fact be realized in a bead-based experiment, then that would lend credence to the original solution-kinetics simulation. As suggested by Professor David Wood, human DHFR molecules which were genetically modified to facilitate attachment were immobilized on chitin beads. Chitin is an insoluble linear β-1,4-linked homopolymer of N-acetylglucosamine (GlcNAc) [79, 80]. It offers a number of desirable characteristics which make chitin a popularly used supporting material in enzyme immobilization: biocompatibility, physiological inertness, nontoxicity, and hydrophilicity [80]. The chitin binding domain (CBD) is a small peptide consisting of about 52 amino acids, which has strong affinity to chitin beads. Using recombinant DNA technology, the CBD gene can be genetically fused to the human DHFR gene. After expression in E. Coli. strains, the CBD-DHFR fusion protein can be specifically 66

82 immobilized onto chitin beads through CBD from crude cell-free extract. A depiction of immobilized human DHFR on chitin beads is shown in Figure 29. CBD Human DHFR Chitin Bead Figure 29. Depiction of immobilized human DHFR on a chitin bead Materials The chitin beads were purchased from New England Biolabs. They have a molecular weight in the range of 750,000 Da. The sizes of the chitin beads are approximately μm in diameter. The membranes used in the CFSTR to retain the immobilized DHFR were AcetatePlus membranes purchased from Fisher Scientific with pore diameters of 0.22 μm. The E. Coli. strain which contains vectors that can express CBD-DHFR was prepared by Professor David Wood s group. 67

83 3.2.2 Experimental Procedures and Results E. Coli. Strain Culture and CBD-DHFR Expression Frozen E. Coli. cells which contained vectors that can express CBD-DHFR were first inoculated into 8 ml Lysogeny Broth (LB) medium containing 0.1 mg/ml ampicillin. This starter culture was incubated at 37 C overnight and then was centrifuged at 3000 g for 5 min. The supernatant was discarded and the precipitate was dissolved in 100 ml Terrific Broth (TB) medium containing 0.1 mg/ml ampicillin. The initial optical density at 600 nm (OD 600 ) of the 100 ml culture was measured. After the OD 600 of the 100 ml culture reached about 0.9, 0.5 mm IPTG was added into the culture to induce the expression of CBD-DHFR at 30 C for 4 hours. After induction, the culture was centrifuged at 3716 g for 30 min. The supernatant was discarded and the precipitated cells were stored at -80 C for future use Cell Lysis and CBD-DHFR Purification The cells prepared in Section were weighed and resuspended in lysis buffer in a ratio of 4 ml lysis buffer per gram of cell. The lysis buffer contained 20 mm Tris, 2mM MgCl 2, 250 mm NaCl, 0.5% Triton X, and 0.1% β- mercaptoethanol, and 25 μm DNAse. The cells were lysed on ice by sonication for 8 min. After sonication, cells were kept on ice and shaken for one hour. The cells were lysed completely after this step and were centrifuged at 3220 g for 30 min, after which the precipitate was discarded. A 10 μl sample of the supernatant (cell lysate) was taken and saved for later electrophoresis. The remaining cell lysate was mixed well with chitin beads to allow CBD-DHFR to bind onto 68

84 the chitin beads. The chitin beads were washed by 50 ml of Appleman s buffer 3 times. Two 10 μl samples of the supernatant were taken after the first wash and the last wash. Since chitin beads can only bind with CBD-DHFR, rather than any other proteins in E. Coli. cells, the CBD-DHFR immobilization and purification was completed at the same time. 50 μl of chitin beads tethered DHFR were boiled in a 95 C water bath for 5 min to separate CBD-DHFR from the beads. A 10 μl sample of the supernatant of the boiled chitin beads which contained denatured CBD-DHFR was taken for electrophoresis Poly Acrylamide Gel Electrophoresis (PAGE) PAGE is a common technique used to separate proteins by their molecular weights and is a good qualitative indication of the purity of protein samples. It can also provide an estimate of the protein concentration in a sample. PAGE gel was prepared according to the protocol provided by Jake Elmer. Before running a PAGE gel, the four 10 μl samples were mixed with 10 μl Lamelli buffer and were incubated in a 95 C water bath for 5 min. 10 μl of each sample were loaded into wells on the gel in the following order: from left to right, lane 1 = cell lysate, lane 2 = supernatant after the first wash of the beads, lane 3 = supernatant after the third wash of the beads, lane 4 = supernatant from the boiled beads, and lane 5 = molecular weight ladder. After the electrophoresis device was hooked up, the PAGE gel was run at 110 V for 75 min. The gel was then taken out and stained in Coomassie Blue and shaken overnight. The gel was destained with a destaining buffer and finally the bands on the PAGE gel were visualized and photographed under UV light, as shown in Figure 30. Lane 1 had a much thicker band at the position of about 27 kd 69

85 compared with other bands in the same lane, which indicates that CBD-DHFR expression level in the cells was high, and the concentration of CBD-DHFR was higher than any other protein in the E. Coli cells. Lane 2 and lane 3 indicated that all other proteins in E. Coli. cells were removed completely after the beads were washed 3 times (some faint protein bands in lane 2 after the first wash and no protein bands in lane 3 after the third wash). Lane 4 had a single band at the molecular weight of about 27 kd. It indicated that CBD-DHFR was successfully immobilized onto chitin beads and was pure. Lane 5 showed the molecular weight ladder. In conclusion, the DHFR immobilization experiment was successful and pure immobilized DHFR was ready for further experiments. Figure 30. Picture of a PAGE gel for DHFR immobilization experiment 70

86 Activity Assay of Immobilized Human DHFR The principle of the activity assay of immobilized human DHFR is the same as described in Section Because chitin beads precipitate quickly in solution, the activity assay had to take place in a 3 ml beaker with a magnetic stirrer on a stir table rather than in a cuvette. Activity of immobilized DHFR was calculated based on the light absorbance changes at 340 nm before and after the DHFR reaction. Experimental results showed that the activity of immobilized DHFR was unit per ml of wet beads. Here, the unit is defined as follows: one unit of enzyme will convert 1.0 μm of H2F to H4F in one minute at ph 6.75 at a certain temperature. The dry weight per ml of wet beads was measured using thermal gravimetric analysis (TGA). TGA is a testing method performed on samples to determine characteristics of a material by measuring changes in weight in relation to changes in temperatures. Since TGA can determine the moisture content of a material, it was used to determine the dry weight of chitin beads tethered human DHFR per ml of wet beads. In one TGA experiment, 100 μl wet beads, which weighed mg before experiment, were analyzed. The TGA result is shown in Figure 30. The total weight of the wet beads was normalized to 100% as seen from the vertical axis in the beginning. The beads were then heated gradually and the real-time weight changes were monitored with respect to temperature. As seen in Figure 30, the wet beads continuously lost weight until 220 C. This weight loss was due to the evaporation of the free water and bound water in the wet beads. After all water evaporated, the weight of the bead stayed constant, which was the 71

87 result of a steady molecular structure of chitin. As the temperature increased above 300 C, the beads lost weight again but in a very slow rate. This weight loss was possibly because of a slow degradation of chitin. The dry weight of the wet chitin beads was the result of wet weight multiplied by the weight percentage when the wet beads lost all the free and bound water, which was equal to mg * 5% = mg. Therefore, enzyme activity per mg dry beads was then calculated to be unit per mg dry beads. Figure 31. TGA result of 100 μl wet chitin beads tethered human DHFR In conclusion, the human DHFR immobilization onto chitin beads through CBD was proved to be successful experimentally. The next step was to perform flow experiments in a partially open CFSTR with very large molecular weight cutoff membranes. However, since the free human DHFR reaction in a classical CFSTR is considered to be an easier 72

88 alternative of the partially open CFSTR reaction in 3.1 than the immobilized DHFR reaction which needs to take account of the variations in DHFR kinetics due to the genetic modification and immobilization, DHFR reactions in a classical CFSTR became the first priority. Therefore, flow experiments with immobilized human DHFR were put aside as a part of the future work. 3.3 Human DHFR Reaction in a Classical CFSTR As indicated in the simulations in Section 2.5, the human DHFR reaction in a classical CFSTR may display bistability when plotting steady-state H4F concentration vs. DHFR concentration in the feed, or DHFR solution flow rate, or substrate solution flow rate. A simple classical CFSTR constructed according to the one in Figure 15 is shown in Figure 32. The volume of the CFSTR was measured as 1.15 ml. Experimental results showed that the DHFR reaction could reach a steady state quickly in the classical CFSTR. One experiment was performed as follows: a mg/ml DHFR solution was fed into the CFSTR at 0.2 ml/min. A substrate solution containing 200 μm NADPH and 100 μm H2F was also continuously fed into the CFSTR, and the reacted mixture was drawn from the outlet. As shown in Figure 33, the empty pump-detector system was first baselined with Appleman s buffer. At t = 37 min, the system was fed with the substrate solution until it reached a steady state. At t = 46 min, the CFSTR was connected between the pump and the detector, and the DHFR and substrate streams were continuously fed into the CFSTR, both at 0.2 ml/min. As seen from the figure, the 73

89 effluent reached a steady state at about t = 72 min. That is to say, it took the CFSTR about 25 min to reach a steady state when substrate solution flow rate was 0.2 ml/min. After maintaining the system at this steady state for a while, the substrate solution flow rate was changed to 0.15 ml/min, while the DHFR solution flow rate was left unaltered at 0.2 ml. The effluent absorbance level began to drop which indicated that the NADPH concentration within the reactor decreased. After 38min, the effluent reached a lower steady state at about t = 110 min. It is reasonable to expect the system to take longer to reach a steady state when the flow rate is lower. Outlet Substrate inlet DHFR inlet Rubber plug Magnetic stirrer Plastic tube Figure 32. A simple classical CFSTR 74

90 UV absorbance at 340 nm (mau) Initial feed level Substrate flow rate = 0.2 ml/min Substrate flow rate = 0.15 ml/min Effluent level Time (min.) Figure 33. UV absorbance changes of the effluent solution with respect to time. In the substrate solution, NADPH concentration = 200 μm, H2F concentration = 100μM. DHFR solution flow rate = 0.2 ml/min. DHFR concentration in the feed solution = mg/min. As stated in the beginning of Section 3.1, achieving a steady-state H4F concentration vs. either DHFR concentration or feed solution flow rate within the reactor is a prerequisite of achieving a switch-like steady-state H4F curve. This prerequisite has been achieved in the classical CFSTR. The next step is to conduct flow experiments in the CFSTR 75