i pH oscillations and bistability predicted by a model of a trienzymatic system that produces an anti-depressant Senior Thesis Presented to The Faculty of the School of Arts and Sciences Brandeis University Department of Biochemistry Dr. Irving R. Epstein, Advisor by Zhiheng Wang April 2017 ii Copyright by Zhiheng Wang © 2017 iii Acknowledgements I would like to first thank Dr. Irving Epstein for giving me priceless opportunity to work in his lab. His wisdom and advice enabled me to conduct this research project and finish my senior honor thesis. I would also like to thank my mentor Dr. Zulma Jiménez for her patience and support. She provided me opportunity to start this project and guided me throughout the process. I would also like to thank all the lab members in the Nonlinear Dynamics group at Brandeis University for providing me help and friendly environment for the past three years. Finally, I would like to thank my family and friends for their trusts and love. iv Abstract pH oscillations and bistability predicted by a model of a trienzymatic system that produces an anti-depressant A thesis presented to the Department of Biochemistry Brandeis University Waltham, Massachuetts By Zhiheng Wang A controlled drug delivery system is typically designed to deliver the drug at a controlled rate; its benefits including improve patients’ compliance compare to traditional formulations as well as not disrupting circadian rhythms if set on a 24-hour period. Some delivery systems exhibit the potential of producing drug in-situ without pre-loading the drug. Because of the availability of methylated creatine in human muscle, we use creatine-creatinase system as the source of an anti-depressant sarcosine. Sarcosine (N-methylglycine) is an exogenous amino acid that acts as a glycine transporter inhibitor. It can enhance glycine concentration around NMDA receptors; as a result, modulates glutamatergic transmission function, which is usually found impaired in patients with depressant and schizophrenia. Here we aim to establish the basis for a pH Sensitive Drug Delivery System (PSDDS) with characteristics of both biocompatibility and rhythmic delivery of the antidepressant. The system will have an enzymatic network as building block which is capable of inducing pH oscillations and sarcosine production. In this work, we carry out numerical simulations using a model based on the kinetics of the enzymatic reactions. The results of the simulations generate sets of the initial conditions necessary for the system to produce the rhythmic behavior mentioned above. Our enzymatic network is composed of creatinase-urease-sarcosine oxidase. Urea produced by v substrate- enzyme creatine-creatinase will be hydrolyzed by urease to produce ammonia which participates in the pH oscillation. Furthermore, the concentration of sarcosine is controlled by the addition of sarcosine oxidase. Our results show that pH oscillations with a pH variation of 5 units are obtained by introducing an inhibitory effect on the enzyme urease. Urease inhibition adds a negative feedback which depends on the concentration of OH‾. We also found that the magnitude of the pH variation is related to the concentration of the enzymes and substrates added to the reaction through of a constant influx. vi Table of Contents Lists of Tables VII Lists of Figures VIII 1. Introduction 1 2. Background 2 2.1 Oscillations in Nature 2 2.2 Characteristics of Oscillations 6 2.3 pH Oscillators 10 2.4 Enzymes 15 3. Methods and Models 19 3.1 Model 19 3.2 Simulations 25 4. Results and Discussion 27 4.1 Urea-urease system 28 4.2 Creatine-creatinase-urease system 36 4.3 Creatine-creatinase-urease-sarcosine oxidase system 40 5. Conclusion 44 6. Appendix 46 7. References 48 vii Lists of Tables Table 1 Rate constants for urea-urease reactions 21 Table 2 Rate constants for creatinase 23 Table 3 Rate constants for sarcosine oxidase 23 Table 4 Initial conditions (ics) of the system 24 Table 5 Inflow concentrations of substrates and enzymes for trienzymatic model 25 Table 6 Enzymes constants used for kinetic constants calculations 46 Table 7 Calculated enzyme kinetic constants 46 viii Lists of Figures 2.1 Physiological rhythms 2 2.2 External stimuli entrains the brain activity 3 2.3 Phosophofructokinase (PFK) reaction 4 2.4 Two false interpretations of chemical oscillations 4 2.5 Perturbation diagram of system with excitability 7 2.6 Effect of perturbations on refractory period 7 2.7 Steady state prediction plot 8 2.8 Cross-shaped phase diagram 9 2.9 Reversible and irreversible hysteresis 10 2.10 One-substrate pH oscillator 11 2.11 Two-substrate pH oscillator 12 2.12 Schematic drawing of Ca2+ pulses 13 2.13 Therapeutic effects for conventional formulation and controlled release drug delivery 14 2.14 Typical bell-shaped curve of urease 15 2.15 Reactions in the trienzymatic system 17 3.1 An example of odes15s used in numerical simulations 25 4.1 Schematic drawing of the enzymatic system 28 4.2 Simulated phase diagram of urea-urease with inflow of acetic acid (adapted) 29 4.3 Simulated phase diagram of urea-urease with inflow of acetic acid 30 4.4 Simulated phase diagram of urea-urease with inflow of sulfuric acid (adapted) 30 4.5 Simulated phase diagram of urea-urease with inflow of sulfuric acid 31 4.6 Simulated phase diagram of urea-urease with inflow of acetic acid (low k7) 33 ix 4.7 Simulated phase diagram of urea-urease with inflow of sulfuric acid (low k7) 33 4.8 Simulated phase diagram of urea-urease with inflow of acetic acid (large k7) 34 4.9 Simulated phase diagram of urea-urease with inflow of sulfuric acid (large k7) 34 4.10 Simulated phase diagram of urea-urease with inflow of acetic acid (low acid + large k7) 35 4.11 Simulated phase diagram of urea-urease with inflow of sulfuric acid (low acid + large k7) 36 4.12 Examples of oscillations found in the model with inflow of acetic acid 36 4.13 Examples of oscillations found in the model with inflow of sulfuric acid 37 4.14 Simulated phase diagram of creatine-creatinase-urease with inflow of acetic acid 39 4.15 Simulated phase diagram of creatine-creatinase-urease with inflow of sulfuric acid 39 4.16 Simulated phase diagram of creatine-creatinase-urease with inflow of acetic acid (large k7) 40 4.17 Simulated phase diagram of creatine-creatinase-urease with inflow of sulfuric acid (large k7) 41 4.18 Sarcosine concentration versus time 42 4.19 Sarcosine concentration versus time with addition of inhibitory effect 43 1 1. Introduction Designing pH oscillators is of particular interest as it can provide both insights into complex dynamic behaviors and can be coupled with a pH sensitive hydrogel as a drug delivery device. Typical pH oscillators usually are composed of inorganic species such as halogen and sulfites which are too toxic to incorporate with hydrogels. In order to generate a more biocompatible pH oscillator, enzymes are obvious candidates as they participate many biological oscillating reactions. Here, we want to build an enzyme-based pH oscillator that generates and delivers a drug at a controlled rate. Sarcosine is a newer drug to treat patients with depression, which can be synthesized by the enzyme creatinase. The latter produces also urea, which combined with urease can be used as the base for the pH oscillator. Since creatine can be found in human muscles, human muscle will be considered as a free source of substrate for the enzyme creatinase. In this project, we study the dynamical behavior of the enzymatic system by simulating the kinetics of a system composed of creatine-creatinase-urease-sarcosine oxidase. We look for the initial conditions required for the enzymatic system to produce sarcosine and to induce temporal fluctuation of the pH. We hope this study can provide new insights in designing enzymatic oscillating systems. In section 2, we offer background on enzymatic pH oscillators, including a brief review of oscillating chemistry history and how to use empirical methods to find pH oscillators. Section 3 discusses the computational method we used here, as well as the differential equations solved. Tables of enzyme constants and other conditions can be found as well. Results of numerical simulations are discussed in section 4. In this section, also the effect of the addition of an inhibitor for the enzyme urease is discussed. In the last chapter, conclusions and suggestions for future work are offered. 2 2. Background 2.1 Oscillations in Nature 2.1.1 Biological rhythms Physiological rhythms are results of nonlinear dynamics. They are central to our lives as they constitute many of our bodily processes; for instance, heartbeats and sleeping cycles that occur daily keep us alive. Figure 2.1 shows some examples of complex physiological rhythms. These physiological rhythms are not strictly periodic but show some fluctuations in their dynamics. Disruption of physiological rhythms can lead to diseases; however, extremely regular rhythms are sometimes also considered pathological. (1) Diseases associated with dysregulation of physiological rhythms are termed ‘dynamical diseases’, examples are tumors, asthma, hypertension, osteoarthritis, which are hard to tackle with traditional drug formulations. (2) As one representative of physiological rhythms, circadian rhythms are molecular circuits that allow organisms to temporally coordinate a plethora of processes, with a rhythm close to 24h, optimizing cellular function in synchrony with daily environmental cycles. Circadian rhythms have been widely observed in prokaryotes, fungi, algae, plants and mammals. (1) It is critical for organisms to have such oscillating molecular mechanisms, for the Figure 2.1 Physiological rhythms. a). White blood count of a patient with cyclical purpose of maintaining homeostasis as well as adaption neutropenia.