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A Mathematical Model of Bovine Metabolism and Reproduction: Application to Feeding Strategies, Drug Administration and Experimental Design Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften am Fachbereich Mathematik und Informatik der Freien Universit¨atBerlin vorgelegt von Mohamed Omari Berlin, March 2019 Betreuer/Erstgutachter: Prof. Dr. Susanna R¨oblitz Department of Informatics, University of Bergen, Norway Zweitgutachter: PD Dr. Marcus Weber Konrad-Zuse-Zentrum f¨urInformationstechnik Berlin (ZIB) Datum der Disputation: 12. July 2019 Contents Introduction 3 1 General Background 9 1.1 Some Concepts of Mathematical Modelling in Systems Biology . .9 1.2 Numerical Simulations of Mathematical Models . 15 1.2.1 Numerical Integration of ODEs . 16 1.2.2 Sensitivity Analysis . 17 1.2.3 Biological Admissibility . 18 1.3 Some Concepts of Bayesian Inference and Information Theory . 20 1.3.1 Kernel Density Estimation . 20 1.3.2 Bayesian Formalism . 23 1.3.3 Monte Carlo Integration . 25 1.3.4 Information Content of Experimental Data . 26 1.3.5 Estimation of Mutual Information . 28 2 Model Development - MetRep Model 31 2.1 Bovine Metabolic Model . 32 2.1.1 Feed Intake . 33 2.1.2 Insulin and Glucagon . 34 2.1.3 Glucose Production and Storage in the Liver . 36 2.1.4 Glucose Utilization . 39 2.1.5 The System of Differential Equations . 40 2.2 Metabolic Reproductive Bovine Model (MetRep) . 42 2.2.1 IGF-1 and Insulin . 42 2.2.2 Lactation . 47 2.2.3 Reparametrization of the BovCycle Model . 47 3 Model Simulation - Feeding Strategies 49 3.1 Non-Lactating Cows . 49 3.1.1 Varying the Glucose Content in the DMI . 50 3.1.2 Acute Nutritional Restriction . 50 3.1.3 Chronic Nutritional Restriction . 51 3.2 Lactating Cows . 54 iii 3.2.1 Varying the Glucose Content in the DMI . 55 3.2.2 The Effect of Changing Glucose in the DMI on the Estrous Cycle . 56 3.2.3 The Effect of Changing Model Parameters on the Estrous Cycle 59 3.3 Discussion . 60 4 Model Application - Administration of Dexamethasone 65 4.1 Background . 65 4.2 Pharmacokinetic-Pharmacodynamic Model . 66 4.2.1 Pharmacokinetic Model . 66 4.2.2 Integrating the Pharmacokinetic Model into MetRep Model . 67 4.3 Simulation Results . 74 4.3.1 Pharmacokinetics of Dexamethasone . 74 4.3.2 Effects of Administrating a Single Dose of Dexamethasone in Non-Lactating Cows . 74 4.3.3 Effects of Administrating a Single Dose of Dexamethasone in Lactating Cows . 78 4.4 Discussion . 81 5 Model Application - Optimal Bayesian Experimental Design 85 5.1 Background . 85 5.2 Reducing Uncertainty in Model Parameters . 88 5.2.1 Formulation of the Experimental Design . 89 5.2.2 Results and Discussion . 90 5.3 Reducing Uncertainty in Model Prediction . 98 5.3.1 Formulation of the Experimental Design . 98 5.3.2 Results and Discussion . 99 Conclusion 105 Appendix 107 Bibliography 118 Zusammenfassung 139 Summary 141 Selbst¨andigkeitserkl¨arung 143 Danksagung 145 iv Dedicated to the memory of Niklas Werner Introduction Over the past 50 years, reproductive physiology in dairy industry has significantly changed to adapt to high milk production, [123, 33]. A few weeks after calving, modern high-yielding dairy cattle in intensive production systems give around 40 liters of milk per day. This is a high amount that comes at a cost. High-producing cows are highly susceptible to disease, show metabolic disorders and fertility prob- lems [33, 39, 123], see Fig 1. Early culling and smaller lifetime milk production are the consequences [221]. Countermeasures have already been taken, and the trend of breeding cows with ever increasing peak milk yield { prevalent for decades { may have come to an end. Optimizing lifetime milk production has proven to be more beneficial for both economic as well as environmental reasons [35]. Figure 1: Milk production and fertility in dairy cows. The inverse relationship between conception rate (CR%) and annual milk production of Holstein dairy cows in New York. Data from Butler et. al, [33]. The most critical time period for a cow's health and her future performance is the periparturient period and the period of early lactation [217, 218]. During that time, the cow mobilizes body reserves because of her inability to meet energy demands 3 4 Introduction solely from the feed energy consumed. This state is referred to as negative energy balance (NEB) [155], see Fig 2. Figure 2: Relationship between energy intake and requirements for a lac- tation in high producing dairy cows. This figure shows that during the first weeks of lactation, cows are in a negative energy balance and are using body energy reserves to meet their needs. A zero net energy balance (i.e., intake sufficient to meet requirements) was not achieved until a point in lactation where milk produc- tion decreases. Figure from Baumann et. al, [9]. For ruminants, the energy content in feed cannot be increased without limits due to the fermentative character of the digestive system [44]. High energetic feed with little fiber leads to an imbalance of microbes, rumen acidosis, and may even cause severe illness and death. Nevertheless, targeted feeding strategies are able to exten- uate the NEB and to ensure animal health and welfare [218, 173, 96]. A number of experimental and clinical studies were performed to examine the re- lationship between the metabolic status and the fertility of cows, both in quali- tative and in quantitative manners, e.g., [57, 167]. Reduced nutrition intake was observed to delay the onset of puberty in beef heifers [188, 50, 229], to change the growth pattern of the dominant follicles (maximal diameter, persistence, num- ber of follicular waves) [146], and to increase the period to conception postpartum Introduction 5 [227, 171, 182]. Studies in the postpartum period of dairy cows showed that the NEB is strongly correlated with low concentrations of glucose, insulin and IGF -1 in the blood [11, 30, 124]. Changes in the secretion of gonadotropins, caused by low glucose levels, lead to low FSH and LH concentrations [57, 33], whereby missing LH peaks cause anovulation [217], see Fig 3. Figure 3: Schematic illustration of the possible effect of NEB on the fate of dominant follicle. NEB is characterised by reduced concentrations of glucose, insulin and IGF -1 in the blood. This in turn reduces the pulsatile LH secretion as well as the ovarian LH sensitivity, which results ultimately in ovulation failure. On the other hand, it was reported that good feed management, e.g., nutritional manipulation that causes increased insulin [77], reduces the incidence of non-regular estrous cycles, often being associated with low average concentrations of insulin in the blood [75]. In general, the mechanisms that result in a reduced fertility are not completely un- derstood, although a close relationship to the glucose-insulin metabolism is widely supported. Therefore, to improve practical reproduction such as pregnancy rate along with milk production simultaneously, a systems biology approach can be adopted to seek better understanding of the mechanisms linking nutrition to fer- tility and milk production. Thus, as an attempt to contribute to the efforts per- formed for the improvement of fertility in dairy cows, this thesis focuses on the role of nutritional impacts in improving the reproductive performance in dairy cows. 6 Introduction Particularly, this thesis focuses on the investigation of the impact of glucose, as part of the feed and as one of the main energy sources of the body, on the estrous cycle dynamics in dairy cows. Thus, to be able to explore various feeding scenarios, the aim is to develop a mathematical model that represents metabolic processes as well as reproductive regulation. This model can allow us to analyse the impact of glucose originating from the feed on the reproductive hormones and the follicular development. In addition, the model can be used to simulate and design feeding and therapeutic strategies that may aid in reducing animal experiments. Outline In Chapter 1, a short overview of mathematical modelling in general is provided. Moreover, some basics of modelling approaches and numerical modelling techniques that are employed for developing and simulating a mathematical model in system biology are also introduced. The second part of Chapter 1 focuses on present- ing some technical tools borrowed from Bayesian inference and information theory. These tools are used for the simulation and application of the mathematical model of metabolic and reproductive regulation in Chapters 4 and 5. In Chapter 2, the development of a mathematical model, called the MetRep model throughout this thesis, which combines reproductive hormones and glucose-insulin dynamics within the body of the cow is described. The model is based on ordinary differential equations and relies on previously introduced models of the bovine es- trous cycle and the glucose-insulin dynamics. Necessary modifications and coupling mechanisms are thoroughly discussed. In Chapter 3, model simulation results for different nutritional regimes in lactat- ing and non-lactating dairy cows are examined and compared with experimental studies. In particular, depending on the amount of the dry matter and its glucose content, the model quantifies reproductive hormones and follicular development over time. In Chapter 4, an attempt to link the MetRep model to a pharmacokinetic model is performed. The coupling is based on mechanisms underlying homeostasis regulation by dexamethasone. In particular, the coupling takes into account the predominant role of dexamethasone in stimulating glucagon secretion, glycogenolysis and lipoly- sis and impairing the sensitivity of cells to insulin. The resulting pharmacokinetic- pharmacodynamic model simulates the effect of one single dose of dexamethasone on the physiological behaviour of the system, especially glucose metabolism. In Chapter 5, the MetRep model is used for simulating the design of experiments.

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