**4.1 Pancreas**

*Sugar Intake - Risks and Benefits and the Global Diabetes Epidemic*

During fasting periods, glucose levels in the blood decrease causing inhibition of insulin production in the pancreas by the action of hormones known as catecholamines [4]. Consequently, α-cells in the pancreas are stimulated to produce glucagon hormone that acts antagonistically to insulin. Glucagon makes a function on the different hepatocyte receptors triggering both the action of the phosphorylase enzyme and the glycogenolysis process. Glycogenolysis is the process in which glycogen is converted into glucose to increase blood glucose levels and recover the lack of glucose, setting its concentrations in the desired levels [5]. This is symbol-

*Diabetes Mellitus* is a condition appearing when the glucose homeostasis is broken, that is, plasma glucose levels are no longer maintained at desired levels. This is mainly due to a deficit in the production of insulin from the pancreatic β-cells or

Although some organs need fatty acids to carry out their metabolic processes, most tissues in the human body use glucose as their main source of energy. Good glucose utilization depends on keeping blood glucose levels within range at all times and on the proper functioning of the glucose homeostatic mechanism. Several complementary physiological processes are involved in the glucose homeostatic mechanism. The gastrointestinal tract is responsible to produce and absorb glucose, the liver carries out biochemical reactions such as glycogenolysis, glycolysis, and gluconeogenesis, the kidneys filter, reabsorb, and in some cases excrete glucose,

**124**

**Figure 1.**

ized in **Figure 1** by the minus sign.

*The glucose homeostasis in the human body.*

from a resistance to the action of the produced insulin.

**4. Main organs involved in glucose homeostasis**

The pancreas is a special organ because has both endocrine and exocrine functions. Exocrine functions consist of the production and secretion of digestive enzymes whereas endocrine functions include production and secretion of hormones. This chapter is primarily focused in the endocrine function given the crucial role on glucose homeostasis. Endocrine component of the pancreas consists of clustered cells forming the so-called islets of Langerhans. Islets of Langerhans are small island-shaped structures within exocrine pancreatic tissue representing only 1–2% of the entire organ [6]. The pancreatic islet endocrine cells include five different types that produce and release important hormones directly into the bloodstream: α -cells produce glucagon, β -cells produce amylin-, C-peptide, and insulin [7], γ -cells produce pancreatic polypeptide (PP) [8], δ -cells produce somatostatin [7], and ε -cells produce ghrelin [9]. Two of the pancreatic hormones play an essential role in the regulation of the blood glucose levels are insulin, which acts to lower it, and glucagon, which acts to raise it [10]. The balanced antagonistic action between them, maintain the glucose concentrations within the narrow range of 4–6 *mM* (70 to 110 *mg dL* / ) [6]. However, both hormones are inhibited by somatostatin [11]. Production and secretion of the hormones by pancreatic cells are stimulated by external signals such as nutrients intake, fasting, or stress. Blood glucose levels decrease during periods of rest such as sleep, between meals, or during fasting periods. In these cases, pancreatic α -cells release glucagon to drive glycogenolysis and gluconeogenesis processes. Unlike, in postprandial state, i.e., after a meal ingestion, insulin is released from β -cells in the pancreas to reduce blood glucose levels via glycogenesis [12–14]. Insulin is released on demand but is produced and stored in large, dense-core vesicles that are recruited near the plasma membrane into the β -cells in the islets of Langerhans after stimulation so that insulin is readily available to upcoming stimuli [15]. Glucose is the main signal to release insulin from the pancreas, but free fatty acids and amino acids can increase glucose-induced insulin secretion through the so-called incretin effect. As before mentioned, the incretin effect is originated in the intestinal tract (mainly duodenum) once the food is ingested.

Insulin is a protein made up of 51 amino acids and when produced, it is first synthesized as a single polypeptide known as preproinsulin. Preproinsulin is an insulin gene encoded in 110 amino acids that are then processed into proinsulin. Proinsulin undergoes maturation into active insulin through the action of two different types of cells. One of them cleaves at 2 positions, releasing insulin and a fragment known as C-peptide [16], in an equimolar ratio, into the bloodstream.

Insulin is released from β-cells in the pancreas in two phases, first one is triggered in response to glucose levels and second one is triggered independently of sugar. Glucose and insulin in the bloodstream work together to avoid glucose from going out of range. Thus, Glucose is removed from the circulation thanks to the ability of insulin to cause insulin-dependent tissues to take up glucose [17–19]. Additionally, insulin promotes lipogenesis [20, 21], and the incorporation of amino acids into proteins [22] when it is in high concentrations. Different at low concentrations, which produce lipolysis in adipocytes, releasing free fatty acids by stimulating the use of lipids over glucose to satisfy energy needs at rest [23]. The release of insulin from β-cells is tightly regulated and exactly satisfies the metabolic demand for caloric nutrients in the body [16, 23]. Regarding C-peptide, it has been important to follow some insulin states that are difficult to measure [24].

#### **4.2 Liver**

The liver is perhaps considered the main blood glucose regulating organ in the human body because it functions in two different ways: controlling the rate of glucose absorption from the portal system and producing glucose from non-carbohydrate precursors or glycogen. As a curious fact, the liver is the only organ being irrigated by venous and arterial blood simultaneously. Venous irrigation comes from the portal system, provides the 75% of the blood supply, and carries blood rich in nutrients that were absorbed from the small intestine through enterocytes and hormones that were released by the pancreas. On the other hand, 25% of the remaining hepatic blood supply is arterial supply and is oxygen-rich blood coming from the aorta [4]. Blood from terminal branches of the hepatic artery and portal vein at the periphery of lobules is emptied into low-pressure vascular channels called sinusoids. Sinusoids are lined with endothelial cells and flanked circumferentially by plates of parenchymal cells-hepatocytes allowing the exchange of nutrients and oxygen between the blood and the hepatic cells [25]. Millions of sinusoids made up the lobules in the liver. Hepatocytes take up nutrients from blood in the sinusoid and once carry on all metabolic functions, return the substances resulting from the biochemical reactions to the blood via hepatic vein.

As mentioned earlier, the liver is a key organ in maintaining glucose concentrations in the desired range over both post-absorptive and postprandial states1 . In the liver, four biochemical processes regarding glucose metabolism take place: glucose production from glycogen (glycogenolysis) and from non-carbohydrate precursors (gluconeogenesis), glucose consumption during the postprandial state (glucolysis), and glucose storage from the formation of glycogen (glycogenesis). Glucose phosphorylation (formation of glycogen) and dephosphorylation (formation of glucose from glycogen) occurs through the action of insulin and glucagon, respectively. Hepatocytes express dozens of enzymes that alternately turn on and off depending on whether blood glucose levels are rising or falling outside the normal range [26]. In the post-absorptive state, the human body is under fasting and the body must rely initially on stored glycogen to supply with glucose to the central nervous system and simultaneously regulate plasma glucose concentrations. If the fast is prolonged, the glycogen stores end, and the glucose dosage in the liver depends only on gluconeogenesis. On the other hand, after an ingested meal, i.e., in the postprandial state, absorbed nutrients enter the liver first from hepatic portal vein. Consequently, glycogen concentrations in the hepatocytes are restored by taking up a portion of the ingested glucose, minimizing the fluctuations of glycemia. In this case, gluconeogenesis is also occurring at a constant rate but the glucose output generated from glycogenolysis is suppressed. These result in a net switch from hepatic glucose output to hepatic glucose uptake [27].

Hepatic gluconeogenesis occurs by the action of additional groups of enzymes that are activated to start synthesizing glucose out of such precursors as amino acids and non-hexose carbohydrates such as glutamine, alanine, lactate and glycerol. Otherwise, the suppression of the glycogenolysis during the post-absorptive period and the activation of the glycogen synthesis during the postprandial period are mainly driven by stimulation of insulin secretion and suppression of glucagon secretion.

**127**

also occur within the renal tubules [5].

*Main Organs Involved in Glucose Metabolism DOI: http://dx.doi.org/10.5772/intechopen.94585*

reenters the circulation [34].

**4.3 Kidneys**

is the synthesis of excess glucose into fatty acids [36].

In addition to being the primary site of glucose utilization during the postprandial period and glucose dosing during the post-absorption period, the liver is the primary site of clearance of insulin in the human body [28, 29]. Although the kidneys are the main site of extrasplanchnic insulin clearance, with additional contributions resulting from uptake and degradation by peripheral insulin-sensitive tissues, i.e., skeletal muscle and adipose tissue, the liver is the main organ responsible for clearance of exogenous and in particular endogenous insulin [30]. Insulin clearance from the liver is a dynamic process that can be modified within a few days under conditions of changing energy and, in particular, carbohydrate intake and before major changes in basal insulin secretion [31]. However, during first-pass transit near to 50% of the portal insulin is removed in the liver [32]. Removal of insulin from circulation does not imply the immediate destruction of the hormone [33]. A significant amount of receptor-bound insulin is released from the cell and

Hepatic glucose uptake is maximally stimulated by conditions that mimic the postprandial state, such as portal venous hyperglycemia and hyperinsulinemia [35]. Once glucose reaches the hepatocytes, it is phosphorylated to glucose 6-phosphate to synthesize glycogen, among other metabolic pathways. The ability of the liver to store glycogen is limited, and when glycogen concentrations reach maximum capacity, the hepatocytes initiate a process known as lipogenesis. Lipogenesis

In conclusion, during short periods of fasting, glycogenolysis is the predominant source of glucose released into the bloodstream. However, during prolonged periods of fasting, the glycogen store is gradually depleted and glycogenolysis decreases as glycogen stores are depleted. So, gluconeogenesis becomes the predominant source of glucose for the human body. This unique ability of the human

The kidneys are two bean-shaped organs that are primarily engaged in filtering the blood and excreting waste. Filtration is about cleaning the blood to send it back into circulation, maintaining an overall fluid balance, creating hormones that help make red blood cells, promoting bone health, and regulating blood pressure [37]. Recent studies have demonstrated that kidneys also play a central role in glucose homeostasis through utilization of glucose, glucose production, and glucose filtration and reabsorption via sodium glucose co-transporters (SGLTs) and glucose transporters (GLUT-2). Moreover, the kidneys are an important site of insulin clearance from the systemic circulation, removing approximately 50% of peripheral insulin [34]. The kidneys have a super-specialized microscopic structural and functional unit called the nephron. Nephrons have the ability to distribute all functions in each of their parts. For example, the glomerulus is a network of small blood vessels known as capillaries located within Bowman's capsule. Blood is filtered across the glomerular capillaries into Bowman's space. These capillaries are multiple branches of the afferent arteriole but then converge at the efferent arteriole to exit the glomerulus and surround the renal tubules, including the proximal convoluted tubule, the proximal rectus tubule, the loop of Henle, the distal convoluted tubule, and the collecting ducts. Urine continually forms within the tubules to be excreted with waste products. Reabsorption, secretion, chemical reactions, and excretion

The release of glucose occurs predominantly in the renal cortex, while the utilization of glucose is limited to the renal medulla. For this reason, the kidneys

liver to store and release glucose is crucial to supporting periods of fasting.

1 Postprandial state is the time frame after a meal or food intake. Postabsorptive state is the period following absorption of nutrients from the digestive tract, that is, is the time when enterocytes stop providing nutrients to the hepatic portal circulation. Fasting is the willing abstinence or reduction from some or all food, drink, or both, for a long period of time (˷ 8 hours).

#### *Main Organs Involved in Glucose Metabolism DOI: http://dx.doi.org/10.5772/intechopen.94585*

*Sugar Intake - Risks and Benefits and the Global Diabetes Epidemic*

biochemical reactions to the blood via hepatic vein.

from hepatic glucose output to hepatic glucose uptake [27].

suppression of glucagon secretion.

some or all food, drink, or both, for a long period of time (˷

Hepatic gluconeogenesis occurs by the action of additional groups of enzymes that are activated to start synthesizing glucose out of such precursors as amino acids and non-hexose carbohydrates such as glutamine, alanine, lactate and glycerol. Otherwise, the suppression of the glycogenolysis during the post-absorptive period and the activation of the glycogen synthesis during the postprandial period are mainly driven by stimulation of insulin secretion and

1 Postprandial state is the time frame after a meal or food intake. Postabsorptive state is the period following absorption of nutrients from the digestive tract, that is, is the time when enterocytes stop providing nutrients to the hepatic portal circulation. Fasting is the willing abstinence or reduction from

8 hours).

The liver is perhaps considered the main blood glucose regulating organ in the human body because it functions in two different ways: controlling the rate of glucose absorption from the portal system and producing glucose from non-carbohydrate precursors or glycogen. As a curious fact, the liver is the only organ being irrigated by venous and arterial blood simultaneously. Venous irrigation comes from the portal system, provides the 75% of the blood supply, and carries blood rich in nutrients that were absorbed from the small intestine through enterocytes and hormones that were released by the pancreas. On the other hand, 25% of the remaining hepatic blood supply is arterial supply and is oxygen-rich blood coming from the aorta [4]. Blood from terminal branches of the hepatic artery and portal vein at the periphery of lobules is emptied into low-pressure vascular channels called sinusoids. Sinusoids are lined with endothelial cells and flanked circumferentially by plates of parenchymal cells-hepatocytes allowing the exchange of nutrients and oxygen between the blood and the hepatic cells [25]. Millions of sinusoids made up the lobules in the liver. Hepatocytes take up nutrients from blood in the sinusoid and once carry on all metabolic functions, return the substances resulting from the

As mentioned earlier, the liver is a key organ in maintaining glucose concentra-

. In

tions in the desired range over both post-absorptive and postprandial states1

the liver, four biochemical processes regarding glucose metabolism take place: glucose production from glycogen (glycogenolysis) and from non-carbohydrate precursors (gluconeogenesis), glucose consumption during the postprandial state (glucolysis), and glucose storage from the formation of glycogen (glycogenesis). Glucose phosphorylation (formation of glycogen) and dephosphorylation (formation of glucose from glycogen) occurs through the action of insulin and glucagon, respectively. Hepatocytes express dozens of enzymes that alternately turn on and off depending on whether blood glucose levels are rising or falling outside the normal range [26]. In the post-absorptive state, the human body is under fasting and the body must rely initially on stored glycogen to supply with glucose to the central nervous system and simultaneously regulate plasma glucose concentrations. If the fast is prolonged, the glycogen stores end, and the glucose dosage in the liver depends only on gluconeogenesis. On the other hand, after an ingested meal, i.e., in the postprandial state, absorbed nutrients enter the liver first from hepatic portal vein. Consequently, glycogen concentrations in the hepatocytes are restored by taking up a portion of the ingested glucose, minimizing the fluctuations of glycemia. In this case, gluconeogenesis is also occurring at a constant rate but the glucose output generated from glycogenolysis is suppressed. These result in a net switch

**4.2 Liver**

**126**

In addition to being the primary site of glucose utilization during the postprandial period and glucose dosing during the post-absorption period, the liver is the primary site of clearance of insulin in the human body [28, 29]. Although the kidneys are the main site of extrasplanchnic insulin clearance, with additional contributions resulting from uptake and degradation by peripheral insulin-sensitive tissues, i.e., skeletal muscle and adipose tissue, the liver is the main organ responsible for clearance of exogenous and in particular endogenous insulin [30]. Insulin clearance from the liver is a dynamic process that can be modified within a few days under conditions of changing energy and, in particular, carbohydrate intake and before major changes in basal insulin secretion [31]. However, during first-pass transit near to 50% of the portal insulin is removed in the liver [32]. Removal of insulin from circulation does not imply the immediate destruction of the hormone [33]. A significant amount of receptor-bound insulin is released from the cell and reenters the circulation [34].

Hepatic glucose uptake is maximally stimulated by conditions that mimic the postprandial state, such as portal venous hyperglycemia and hyperinsulinemia [35]. Once glucose reaches the hepatocytes, it is phosphorylated to glucose 6-phosphate to synthesize glycogen, among other metabolic pathways. The ability of the liver to store glycogen is limited, and when glycogen concentrations reach maximum capacity, the hepatocytes initiate a process known as lipogenesis. Lipogenesis is the synthesis of excess glucose into fatty acids [36].

In conclusion, during short periods of fasting, glycogenolysis is the predominant source of glucose released into the bloodstream. However, during prolonged periods of fasting, the glycogen store is gradually depleted and glycogenolysis decreases as glycogen stores are depleted. So, gluconeogenesis becomes the predominant source of glucose for the human body. This unique ability of the human liver to store and release glucose is crucial to supporting periods of fasting.

#### **4.3 Kidneys**

The kidneys are two bean-shaped organs that are primarily engaged in filtering the blood and excreting waste. Filtration is about cleaning the blood to send it back into circulation, maintaining an overall fluid balance, creating hormones that help make red blood cells, promoting bone health, and regulating blood pressure [37]. Recent studies have demonstrated that kidneys also play a central role in glucose homeostasis through utilization of glucose, glucose production, and glucose filtration and reabsorption via sodium glucose co-transporters (SGLTs) and glucose transporters (GLUT-2). Moreover, the kidneys are an important site of insulin clearance from the systemic circulation, removing approximately 50% of peripheral insulin [34].

The kidneys have a super-specialized microscopic structural and functional unit called the nephron. Nephrons have the ability to distribute all functions in each of their parts. For example, the glomerulus is a network of small blood vessels known as capillaries located within Bowman's capsule. Blood is filtered across the glomerular capillaries into Bowman's space. These capillaries are multiple branches of the afferent arteriole but then converge at the efferent arteriole to exit the glomerulus and surround the renal tubules, including the proximal convoluted tubule, the proximal rectus tubule, the loop of Henle, the distal convoluted tubule, and the collecting ducts. Urine continually forms within the tubules to be excreted with waste products. Reabsorption, secretion, chemical reactions, and excretion also occur within the renal tubules [5].

The release of glucose occurs predominantly in the renal cortex, while the utilization of glucose is limited to the renal medulla. For this reason, the kidneys can be considered as two separate organs [38–42]. The renal medulla has an appreciable glucose phosphorylation capacity and, therefore, the ability to accumulate glycogen [42]. However, the kidney medulla consumes glucose anaerobically due to its low oxygen tension and low levels of oxidative enzymes, limiting the ability to produce glucose from glycogen. Consequently, lactate is the main metabolic end product of glucose taken up at the renal medulla, unlike carbon dioxide (*CO*2 ) and water that are the end products of glucose uptake of aerobic energy requirements. In contrast, the renal cortex does not have appreciable glycogen stores [43] because has little glucose phosphorylation capacity but has a high level of oxidative enzymes like 6-phosphatase. Consequently, this part of the kidney does not take up and use much glucose, with oxidation of free fatty acids acting as the main source of energy [44]. Therefore, it is likely that glucose release by the normal kidney is primarily due to gluconeogenesis, that is, the synthesis of glucose-6-phosphate from non-carbohydrate precursors such as glutamine, lactate, alanine, glycerol, etc. [45], being glutamine the substrate with more specificity in the kidney but lactate the most abundant.

In addition, to its function both in the use and in the production of glucose, the kidneys contribute to the regulation of glucose in the blood by filtering and reabsorbing glucose. The glomeruli filter glucose once it reaches the kidneys, with other substances such as precursors and insulin, into the proximal tubules, where all the glucose is reabsorbed through the glucose transporting proteins present in the cell membranes within the proximal tubules [46], rendering the urine virtually glucose free. Before being reabsorbed, gluconeogenesis and glucose uptake occur. Glucose production is suppressed by insulin [45] or stimulated by non-carbohydrate precursors [41, 47]. An interesting fact is that GLUT-2 glucose transporters are independent of insulin and for that reason, the kidneys can continue their physiological functions even in states of insulin deficiency [23].

As before mentioned, gluconeogenesis in the human body is mainly carried out by the liver and the kidneys. In the post-absorptive state, both liver and kidneys release glucose into the circulation in comparable amounts [48]. However, in the postprandial state, although overall endogenous glucose release decreases substantially, renal gluconeogenesis increases by approximately twice liver gluconeogenesis. In this sense, the hepatic and renal glucose release into the circulation in the post-absorptive state correspond to the 25–30% and 20–25% of total glucose, respectively, while in postprandial state, hepatic gluconeogenesis is reduced by ∼ 80% and the release of glucose molecules generated via this pathway decreases as these molecules are largely directed into the formation of hepatic glycogen. As a consequence of these changes, renal gluconeogenesis increases accounts for ∼ 60% of postprandial endogenous glucose release [49].

#### **4.4 Gastrointestinal tract**

The gastrointestinal (GI) tract is an organ system, consisting of the mouth, esophagus, stomach, and intestines, where humans ingest food, digest it to extract and absorb energy and nutrients, and expel the remaining waste as feces. However, the literature on glucose homeostasis includes the gastrointestinal tract as a complete organ without taking into account the physiological functions and glucose consumption of the stomach and small intestine as separate organs involved in glucose metabolism.

Meal is ingested through mouth and enters in the stomach to be mixed. The rate at which nutrients pass from the stomach to the duodenum, i.e., crossing the pyloric valve, is known as the gastric emptying rate and is a key determinant of

**129**

brain diseases.

*Main Organs Involved in Glucose Metabolism DOI: http://dx.doi.org/10.5772/intechopen.94585*

tory hormones of glucose including amylin from

appropriate tissue stores [52–54].

glucose in the bloodstream.

**4.5 Brain**

postprandial glucose flow. In the fed state, glucose homeostasis becomes more complex as the gastrointestinal tract becomes a second source of exogenous glucose. Marked and rapid changes in glucose flux occur as a result of the considerable inflow of meal-derived glucose into the circulation [50]. The delivery of nutrients from the gastrointestinal tract occurs through an important rate limiting mechanical step in the form of gastric emptying rate: the rate at which the pylorus allows small boluses of gastric content to pass into the duodenum for downstream absorption. Importantly, neither insulin nor glucagon has direct effects on gastric emptying and exogenous glucose diffusion from the gastrointestinal tract [51]. However, the influx of glucose is accompanied by secretion of several other regula-

glucose-dependent inhibitory peptide (GIP), glucagon-like peptide-1 (GLP-1), and cholecystokinin (CCK) from endocrine cells in the small intestine. Endocrine cells in the small intestine collectively influence glucose homeostasis via several mechanisms of action including regulation of insulin and glucagon responses, as well as the modulation of nutrient passage from the gastrointestinal tract to

A key contribution of the GI tract on glucose homeostasis is the incretin effect. This physiological response came from the observation that an oral glucose load results in an increased insulin response compared to the response seen when intravenous glucose administration replicates the same changes in plasma glucose

response is observed as a result of a signal passed from the gut. The two hormones responsible for this effect are GIP and GLP-1. Both GIP, secreted from enteroendocrine K-cells in the proximal small bowel, and GLP-1, secreted from enteroendocrine L-cells in the distal ileum and colon, have a strong insulinotropic effect [57]. Additionally, GLP-1 inhibits postprandial glucagon secretion in a glucosedependent manner, slows gastric emptying, and reduces food intake, contributing to postprandial glucose regulation [58]. Regarding the role of the stomach in the metabolism of glucose, the stomach must consume glucose to generate the energy necessary to mechanically carry out the digestion process. Although the consumption of glucose in the stomach is relatively low, it can affect the concentration of

The human brain depends on glucose as its main source of energy; neurons have the highest energy demand [59] of all types of cells in the human body, requiring continuous delivery of glucose from blood. Glucose metabolism provides the fuel for physiological brain function through the generation of ATP, the foundation for neuronal and non-neuronal cellular maintenance, as well as the generation of neurotransmittersTherefore, tight regulation of glucose metabolism is critical to brain physiology. In this sense, the alteration of glucose metabolism in the brain is the basis of several diseases that affect both the brain and the entire organism. Glucose is required in the brain to provide the precursors of neurotransmitter synthesis and ATP to fuel their actions. Additionally, glucose is important for the brain's energy demands unrelated to signaling. Cellular compartmentalization of glucose transport and metabolism are closely related to local regulation of blood flow, and glucose-sensing neurons govern the brain–body nutrient axis. Glucose metabolism is connected to cell death pathways by the glucose-metabolizing enzymes [60]. Thus, disruption glucose delivery pathways and metabolism leads to debilitating

[55, 56]. In other words, when glucose is ingested orally, an augmented

β

β-cell

#### *Main Organs Involved in Glucose Metabolism DOI: http://dx.doi.org/10.5772/intechopen.94585*

*Sugar Intake - Risks and Benefits and the Global Diabetes Epidemic*

kidney but lactate the most abundant.

ological functions even in states of insulin deficiency [23].

of postprandial endogenous glucose release [49].

**4.4 Gastrointestinal tract**

glucose metabolism.

can be considered as two separate organs [38–42]. The renal medulla has an appreciable glucose phosphorylation capacity and, therefore, the ability to accumulate glycogen [42]. However, the kidney medulla consumes glucose anaerobically due to its low oxygen tension and low levels of oxidative enzymes, limiting the ability to produce glucose from glycogen. Consequently, lactate is the main metabolic end product of glucose taken up at the renal medulla, unlike carbon dioxide (*CO*2 ) and water that are the end products of glucose uptake of aerobic energy requirements. In contrast, the renal cortex does not have appreciable glycogen stores [43] because

has little glucose phosphorylation capacity but has a high level of oxidative enzymes like 6-phosphatase. Consequently, this part of the kidney does not take up and use much glucose, with oxidation of free fatty acids acting as the main source of energy [44]. Therefore, it is likely that glucose release by the normal kidney is primarily due to gluconeogenesis, that is, the synthesis of glucose-6-phosphate from non-carbohydrate precursors such as glutamine, lactate, alanine, glycerol, etc. [45], being glutamine the substrate with more specificity in the

In addition, to its function both in the use and in the production of glucose, the kidneys contribute to the regulation of glucose in the blood by filtering and reabsorbing glucose. The glomeruli filter glucose once it reaches the kidneys, with other substances such as precursors and insulin, into the proximal tubules, where all the glucose is reabsorbed through the glucose transporting proteins present in the cell membranes within the proximal tubules [46], rendering the urine virtually glucose free. Before being reabsorbed, gluconeogenesis and glucose uptake occur. Glucose production is suppressed by insulin [45] or stimulated by non-carbohydrate precursors [41, 47]. An interesting fact is that GLUT-2 glucose transporters are independent of insulin and for that reason, the kidneys can continue their physi-

As before mentioned, gluconeogenesis in the human body is mainly carried out by the liver and the kidneys. In the post-absorptive state, both liver and kidneys release glucose into the circulation in comparable amounts [48]. However, in the postprandial state, although overall endogenous glucose release decreases substantially, renal gluconeogenesis increases by approximately twice liver gluconeogenesis. In this sense, the hepatic and renal glucose release into the circulation in the post-absorptive state correspond to the 25–30% and 20–25% of total glucose, respectively, while in postprandial state, hepatic gluconeogenesis is reduced by ∼ 80% and the release of glucose molecules generated via this pathway decreases as these molecules are largely directed into the formation of hepatic glycogen. As a consequence of these changes, renal gluconeogenesis increases accounts for ∼ 60%

The gastrointestinal (GI) tract is an organ system, consisting of the mouth, esophagus, stomach, and intestines, where humans ingest food, digest it to extract and absorb energy and nutrients, and expel the remaining waste as feces. However, the literature on glucose homeostasis includes the gastrointestinal tract as a complete organ without taking into account the physiological functions and glucose consumption of the stomach and small intestine as separate organs involved in

Meal is ingested through mouth and enters in the stomach to be mixed. The rate

at which nutrients pass from the stomach to the duodenum, i.e., crossing the pyloric valve, is known as the gastric emptying rate and is a key determinant of

**128**

postprandial glucose flow. In the fed state, glucose homeostasis becomes more complex as the gastrointestinal tract becomes a second source of exogenous glucose. Marked and rapid changes in glucose flux occur as a result of the considerable inflow of meal-derived glucose into the circulation [50]. The delivery of nutrients from the gastrointestinal tract occurs through an important rate limiting mechanical step in the form of gastric emptying rate: the rate at which the pylorus allows small boluses of gastric content to pass into the duodenum for downstream absorption. Importantly, neither insulin nor glucagon has direct effects on gastric emptying and exogenous glucose diffusion from the gastrointestinal tract [51]. However, the influx of glucose is accompanied by secretion of several other regulatory hormones of glucose including amylin from β -cells in the pancreas and glucose-dependent inhibitory peptide (GIP), glucagon-like peptide-1 (GLP-1), and cholecystokinin (CCK) from endocrine cells in the small intestine. Endocrine cells in the small intestine collectively influence glucose homeostasis via several mechanisms of action including regulation of insulin and glucagon responses, as well as the modulation of nutrient passage from the gastrointestinal tract to appropriate tissue stores [52–54].

A key contribution of the GI tract on glucose homeostasis is the incretin effect. This physiological response came from the observation that an oral glucose load results in an increased insulin response compared to the response seen when intravenous glucose administration replicates the same changes in plasma glucose [55, 56]. In other words, when glucose is ingested orally, an augmented β -cell response is observed as a result of a signal passed from the gut. The two hormones responsible for this effect are GIP and GLP-1. Both GIP, secreted from enteroendocrine K-cells in the proximal small bowel, and GLP-1, secreted from enteroendocrine L-cells in the distal ileum and colon, have a strong insulinotropic effect [57]. Additionally, GLP-1 inhibits postprandial glucagon secretion in a glucosedependent manner, slows gastric emptying, and reduces food intake, contributing to postprandial glucose regulation [58]. Regarding the role of the stomach in the metabolism of glucose, the stomach must consume glucose to generate the energy necessary to mechanically carry out the digestion process. Although the consumption of glucose in the stomach is relatively low, it can affect the concentration of glucose in the bloodstream.

#### **4.5 Brain**

The human brain depends on glucose as its main source of energy; neurons have the highest energy demand [59] of all types of cells in the human body, requiring continuous delivery of glucose from blood. Glucose metabolism provides the fuel for physiological brain function through the generation of ATP, the foundation for neuronal and non-neuronal cellular maintenance, as well as the generation of neurotransmittersTherefore, tight regulation of glucose metabolism is critical to brain physiology. In this sense, the alteration of glucose metabolism in the brain is the basis of several diseases that affect both the brain and the entire organism. Glucose is required in the brain to provide the precursors of neurotransmitter synthesis and ATP to fuel their actions. Additionally, glucose is important for the brain's energy demands unrelated to signaling. Cellular compartmentalization of glucose transport and metabolism are closely related to local regulation of blood flow, and glucose-sensing neurons govern the brain–body nutrient axis. Glucose metabolism is connected to cell death pathways by the glucose-metabolizing enzymes [60]. Thus, disruption glucose delivery pathways and metabolism leads to debilitating brain diseases.

The brain uses about 120 g of glucose per day - 60-70% of the body's total glucose metabolism. The brain has little stored glucose and has no additional sources of stored energy. Brain function begins to become seriously affected when glucose levels fall below ~ 40 / *mg dL*. Glucose levels significantly below this can lead to permanent damage and death. The brain cannot use fatty acids for energy (fatty acids do not cross the blood–brain barrier of the neurons), but ketone bodies can enter the brain and be used for energy in hypoglycemic conditions. In this sense, the brain can only use glucose, or, under conditions of starvation, ketone bodies (acetoacetate and hydroxybutyrate) for energy.


*3.2.1 Trialkyltin fluorides*

*Enhanced Oil Recovery Processes - New Technologies*

applications [83].

molecular weight.

weight (Mw: 980,000 g.mol−1

In propane and butane, PDMS was soluble at pressures close to the vapour pressures of propane and butane, while in ethane, it required high pressure (much greater than its vapour pressure) to attain solubility. For example, at 333 K and 2 wt% of PDMS in ethane, propane, and butane, the cloud point pressures were obtained to be equal to 18, 2.56, and 0.92 MPa, respectively. Furthermore, they found PDMS to be an effective thickener in propane and butane but an ineffective thickener in ethane. For example, at 333 K and at a concentration of 2 wt%, PDMS achieved viscosity increases of 1.2-fold in ethane, 2-fold in propane, and 4-fold in butane. It was also found to be a better thickener at high pressure (62 MPa). Overall, high molecular weight PDMS is not a viable thickener of NGL for EOR

In comparison to the results obtained by Heller et al. for 2.2 wt% of PAO (poly(1-pentene), poly (1-hexene), and P-1-D) in butane at 298 K, DRA increases butane viscosity substantially at even lower concentrations (0.5 wt%), because of the extremely high molecular weight of DRA, regardless the relatively high concentration of hexane added to the system. Although PDMS has a higher molecular

(fourfold) than the viscosity obtained by low molecular weight PAO (fivefold). These observations indicate that the increase in gas viscosity depends on several factors and not solely on the molecular weight of the additives. Additional factors that can influence the viscosity-enhancing ability of an additive include the nature of additives and the solvent, the concentration of additives, the molecular weight distribution of the additives, and the type of intermolecular interactions among the additives and the solvent [84, 85]. These chemical additives (PDMS and PAO) have different chemical structures. PAO has a carbon-carbon backbone with atactic molecular structure of mostly uniform head-to-tail connections with some headto-head-type connections in the structure [86]. On the other hand, PDMS has silicone-oxygen backbone and more flexible molecules than P-1-D molecules. Hence,

PDMS can have lower steric hindrance and greater bond angle (143o

mass. For example, the viscosity of PAO at 1000 g.mol−1

**3.2 Small-molecule self-associating thickeners**

gases are discussed in the following section.

C-C-C) to rotate around the Si-O bond [87]. Furthermore, the effect of molecular structure and polymer molecular weight on viscosity has been studied by Zolper et al. [87] who found that similar viscosity can be obtained for different molecular

ity of PDMS at around 10,000 g/mol. This was attributed to the additional attractive intermolecular forces between the polymers with increasing branches, which leads PAO to having higher viscosity indices than PDMS. Therefore, the effect of PAO on butane viscosity could be attributed to the structure of the polymer. These effects are more pronounced in improving the solvent viscosity than the PAO

Similarly to studies for CO2, low molecular weight self-associating compounds have been studied as thickening agents of light alkane gases for gas mobility control and hydraulic fracturing purposes [54, 82, 88, 89]. Ideally, small-molecule compounds need two processes to attain dissolution and viscosity enhancement [88, 89]. The first is a high-pressure heating cycle, which disrupts the intermolecular association to enhance the dissolution. The second process is the cooling cycle to re-establish the intermolecular association necessary for viscosity enhancement. However, some small-molecule compounds do not require this two-step process to attain dissolution and viscosity enhancement in NGL [89]. Previous studies on the application of small-molecule compounds to increase the viscosity of light alkane

) than PAO, PDMS offers a lower relative viscosity

vs. 110° for

is equivalent to the viscos-

**80**

Dunn and Oldfield first reported on the use of tri-n-butyl tin fluoride (TBTF) as a direct thickener of non-polar solvents including carbon tetrachloride and n-propane [89]. **Figure 8** illustrates the association mechanism of tributyltin fluoride [90], where a linear polymeric chain of penta-coordinate tin atoms are linked by fluorine atoms. TBTF is a white powder with a melting point of 544 K [88, 89]. The three butyl arms attached to the tin atom enhances the solubility of TBTF in a hydrocarbon solvent, while the intermolecular association formed among the tin and fluorine atoms induces the viscosity-enhancing effect. It has been found that TBTF is soluble in organic liquids and light alkanes under stirring for several minutes without requiring a heating and/or a cooling cycle [89]. TBTF is an effective thickener for intermediate hydrocarbon components. Dandge et al. [54] found TBTF to be capable of improving the viscosity of propane and butane. For instance, at concentrations of 0.13–0.15 wt% at 298 K, it increased the viscosity of these components by 2–10-fold at 8.3 MPa. In addition, they also found that TBTF was only partially soluble in ethane and with no measurable viscosity change [79]. Later on, Enick and co-workers confirmed the ability of TBTF to thicken propane and butane liquids at 298 K at concentrations of 0.2–5 wt% [56]. Other trialkyltin fluorides have been tested in hydrocarbon solvents. Tripropyltin fluoride (TPTF) is not soluble in propane and butane, because the propyl arms are too short to induce the dissolution of TPTF in these solvents [91]. Therefore, it confirms that the solubility of trialkyltin fluoride in n-alkane increases as the number of carbon atoms in n-alkyl arms (R) increases [54]. However, at equivalent mass concentration, TBTF in n-hexane or n-butane has shown to outperform in viscosity enhancement compared with other solvents [54]. For example, at a concentration of 10 g/L and 310 K, TBTF increases the viscosity of n-hexane by 750-fold (from 0.265 cP to 196 cP), while the tetrachloroethylene viscosity is enhanced by 380-fold (124.45 cP) [54].

A recent study has tested the solubility and viscosity enhancement ability for dilute concentrations (>1 wt%) of TBTF in ethane, propane, and butane at high pressures (38–64 MPa) and high temperatures (298–373 K) [82, 88, 89]. TBTF is soluble in propane and butane above the corresponding vapour pressure of these components, while in ethane, TBTF is soluble at much higher pressures than the ethane vapour pressure. In addition, it was observed that the relative viscosity of TBTF in NGL components increases slightly with increasing pressure at all temperatures and TBTF concentrations. Increasing the pressure does not affect the self-assembly of the supramolecular structure; it only affects the solvent strength which has a less significant effect on the solution viscosity. Furthermore, as temperature increases, the intermolecular association between the tin and fluoride

**Figure 8.** *Association mechanism of tributyltin fluoride [89].*

molecules diminish, leading to a significant decrease in the viscosity enhancement in all light alkane components. For example, with 1 wt% concentration of TBTF in ethane at 298 K and 62 MPa, the achieved relative viscosity is 90, and it drops to 75 at 313 K. The relative viscosity significantly drops further to 20, 6, and 1.5 at 333, 353, and 373 K, respectively [82] (**Table 2**).

### *3.2.2 Hydroxyaluminum di-2-ethylhexanoate (HAD2EH)*

A mixture of aluminium disoap and gasoline liquid is heated to high temperatures (368–373 K) to promote its dissolution by dismantling the intermolecular associations between the aluminium disoaps. Then it is cooled down to allow self-assembly of the disoap molecules, whereby the viscosity of the solution is enhanced significantly [88, 89]. Enick and co-workers [62] studied a single aluminium salt, referred to as hydroxyaluminum di-2-ethylhexanoate (HAD2EH). **Figure 9** depicted the association mechanism of HAD2EH. They found HAD2EH to exhibit a remarkable solubility in light hydrocarbon gases such as propane and butane. It is also capable of thickening these components at dilute concentrations. For example, at 293 K, HAD2EH concentrations in the range of 0.2–1 wt% were capable of increasing the viscosity of the solution by 10–100-fold as tested in a high-pressure falling-ball cylinder viscometer. However, the solution formed was hazy, due to a portion of the HAD2EH molecules forming solid fibres in both liquid propane and butane at high pressures.

**Figure 9.** *Association mechanism of HAD2EH molecules [89].*

**Figure 10.** *Molecular structure of phosphate di-/monoester, phosphonic acid ester, and dialkyl phosphinic acid [89].*

**Figure 11.**

*Chelation mechanism and micellar structure of phosphate ester/metal ion complex [96].*

### *Direct Gas Thickener DOI: http://dx.doi.org/10.5772/intechopen.88083*

*Enhanced Oil Recovery Processes - New Technologies*

353, and 373 K, respectively [82] (**Table 2**).

*3.2.2 Hydroxyaluminum di-2-ethylhexanoate (HAD2EH)*

molecules diminish, leading to a significant decrease in the viscosity enhancement in all light alkane components. For example, with 1 wt% concentration of TBTF in ethane at 298 K and 62 MPa, the achieved relative viscosity is 90, and it drops to 75 at 313 K. The relative viscosity significantly drops further to 20, 6, and 1.5 at 333,

A mixture of aluminium disoap and gasoline liquid is heated to high temperatures (368–373 K) to promote its dissolution by dismantling the intermolecular associations between the aluminium disoaps. Then it is cooled down to allow self-assembly of the disoap molecules, whereby the viscosity of the solution is enhanced significantly [88, 89]. Enick and co-workers [62] studied a single aluminium salt, referred to as hydroxyaluminum di-2-ethylhexanoate (HAD2EH). **Figure 9** depicted the association mechanism of HAD2EH. They found HAD2EH to exhibit a remarkable solubility in light hydrocarbon gases such as propane and butane. It is also capable of thickening these components at dilute concentrations. For example, at 293 K, HAD2EH concentrations in the range of 0.2–1 wt% were capable of increasing the viscosity of the solution by 10–100-fold as tested in a high-pressure falling-ball cylinder viscometer. However, the solution formed was hazy, due to a portion of the HAD2EH molecules

forming solid fibres in both liquid propane and butane at high pressures.

*Chelation mechanism and micellar structure of phosphate ester/metal ion complex [96].*

*Molecular structure of phosphate di-/monoester, phosphonic acid ester, and dialkyl phosphinic acid [89].*

**82**

**Figure 11.**

**Figure 10.**

**Figure 9.**

*Association mechanism of HAD2EH molecules [89].*

Dhuwe et al. [82, 89] examined blends of HAD2EH in several NGL (i.e. ethane, propane, and butane) individually under a range of pressures (34–62 MPa) and temperatures (298–273 K). HAD2EH was soluble in propane and butane while insoluble in ethane. At a temperature of 298 K, HAD2EH was insoluble in all light alkanes. Heating to 373 K, high pressure, and stirring were required to attain dissolution, followed by cooling to a temperature above 313 K to obtain a single phase solution. If the solution is cooled down to a temperature of 298 K, HAD2EH precipitated in both propane and ethane. Accordingly, it was observed that HAD2EH was an effective thickener in butane and propane at temperatures as low as 313 K. For example, at a concentration of 0.5 wt% HAD2EH and temperatures of 333–373 K, butane viscosity increases by 15–19-fold, while propane is thickened by 2–3-fold [82, 89].


kinetic parameters: R0 ¼ βλb

this new context.

daðaþbÞ

YðtÞ ¼ ∑ ∞ i¼n0 yi

where YðtÞ ¼ ∑∞

214 Prion - An Overview

Figure 2. Nucleated polymerization model of prion dynamics. This demonstrates the key steps in the nucleated polymerization model (NPM) of prion aggregate dynamics. (The description of the kinetic parameters is in the text.) This model is characterized by prion aggregates below a critical size n0 ¼ 2 (the nucleus size) resolving to protein monomers in the normal folded state.

In 1999 Masel, Jensen and Nowak conducted an extensive analysis of the NPM [43]. In particular, they sought to link experimental observations on the time to appearance of prion disease symptoms with the kinetic parameters of the NPM. Among other contributions, Masel and colleagues determined a viable range of minimal nucleus sizes, n0. Overall, there was remarkable consistency between parameters predicted from different experimental data sets analyzed providing support at the time for this mathematical formulation. In addition, Masel et al. [43] (and then Greer and colleagues with a generalization [44]) demonstrated that the dynamics of aggregates under the NPM are consistent with the long-incubation time observed for prion phenotypes. If prion disease begins with the introduction of a small amount of prion protein (in the form of aggregates) those aggregates will first have to increase in size until there are enough fragmentation sites to permit aggregate amplification through fragmentation.

In early twenty-first century, mathematicians continued formalizing the NPM. Prüss and colleagues [45] demonstrated that the prion phenotypes were globally asymptotically stable and not merely locally stable, through deriving a Lyapunov function. Engler et al. [46] analyzed the well-posedness of the generalization of the NPM where aggregate sizes were continuous, instead of discrete. As such, rather than an infinite system of ordinary differential equations, the system consisted of a single ODE for protein in the normal configuration and a PDE specifying the distribution of aggregate sizes. While this formulation departs from the physically discrete nature of aggregates, in the limit of large aggregate sizes these formalisms are provably equivalent [47] and the use of PDEs permits a wider array of mathematical techniques. Most notably, the continuous relaxation on aggregate sizes has permitted determination of the explicit asymptotic density [44, 46]. (In comparison, the asymptotic density for the aggregate model with discrete aggregate sizes, while first approximated in 2003 by Pöschel et al. [48], was derived only recently by Davis and Sindi and required special functions [49].)

While today mathematical models of prion aggregate dynamics have been formulated under many more general kinetic assumptions (see [47, 50–52] for example) most of these models have been compared to only in vitro aggregation studies. For yeast, in vivo comparisons have been made for the Sup35/[PSI+ ] prion system [53–55], but linking experimental outcomes uniquely to specific kinetic parameters remains challenging.

Nucleation. As mentioned in the previous section, the rate-limiting step in prion phenotypes is thought to be the time to the appearance of stable nucleus, that is, an aggregate of misfolded protein that persists stably. (It is typically thought that this nucleus corresponds to a misfolded aggregate of a minimal stable size [42, 43].) The self-assembly of particles into aggregates is fundamental to many physical, chemical and biological processes. Such a process is referred to in statistical physics as nucleation and mathematical models of nucleation have been studied for nearly a century [56]. In contrast to other biochemical models of protein aggregation, the spontaneous appearance of a prion nucleus is thought to be rare [57, 58]. As such, mathematical models of prion appearance are often framed as first-passage processes; that is, these models focus on determining the amount of time until a critical event occurs, in this case the appearance of a prion nucleus.

One of the earliest models of self-assembly of particles was proposed by in 1916 by Smoluchowski [59]. He considered the evolution of the density of clusters of discrete particle sizes under the assumption that clusters of any size could join together (coagulation). In 1935, Becker and Döring introduced kinetic equations for a similar process but where clusters could only change in size through monomer addition or removal [60]. More generally, models of particle self-assembly are distinguished by their associated set of biochemical equations governing the evolution of cluster sizes. As such, the problem of prion nucleus appearance can be framed as: given a set of biochemical equations governing misfolded protein aggregate formation, determine the time it takes for a critical sized nucleus to form [61].

disease symptoms with the kinetic parameters of the NPM. Among other contributions, Masel and colleagues determined a viable range of minimal nucleus sizes, n0. Overall, there was remarkable consistency between parameters predicted from different experimental data sets analyzed providing support at the time for this mathematical formulation. In addition, Masel et al. [43] (and then Greer and colleagues with a generalization [44]) demonstrated that the dynamics of aggregates under the NPM are consistent with the long-incubation time observed for prion phenotypes. If prion disease begins with the introduction of a small amount of prion protein (in the form of aggregates) those aggregates will first have to increase in size until there are enough fragmentation sites to permit aggregate amplification through fragmentation.

In early twenty-first century, mathematicians continued formalizing the NPM. Prüss and colleagues [45] demonstrated that the prion phenotypes were globally asymptotically stable and not merely locally stable, through deriving a Lyapunov function. Engler et al. [46] analyzed the well-posedness of the generalization of the NPM where aggregate sizes were continuous, instead of discrete. As such, rather than an infinite system of ordinary differential equations, the system consisted of a single ODE for protein in the normal configuration and a PDE specifying the distribution of aggregate sizes. While this formulation departs from the physically discrete nature of aggregates, in the limit of large aggregate sizes these formalisms are provably equivalent [47] and the use of PDEs permits a wider array of mathematical techniques. Most notably, the continuous relaxation on aggregate sizes has permitted determination of the explicit asymptotic density [44, 46]. (In comparison, the asymptotic density for the aggregate model with discrete aggregate sizes, while first approximated in 2003 by Pöschel et al. [48], was derived only recently by Davis and Sindi and required special functions [49].) While today mathematical models of prion aggregate dynamics have been formulated under many more general kinetic assumptions (see [47, 50–52] for example) most of these models have been compared to only in vitro aggregation studies. For yeast, in vivo comparisons have

Nucleation. As mentioned in the previous section, the rate-limiting step in prion phenotypes is thought to be the time to the appearance of stable nucleus, that is, an aggregate of misfolded protein that persists stably. (It is typically thought that this nucleus corresponds to a misfolded aggregate of a minimal stable size [42, 43].) The self-assembly of particles into aggregates is fundamental to many physical, chemical and biological processes. Such a process is referred to in statistical physics as nucleation and mathematical models of nucleation have been studied for nearly a century [56]. In contrast to other biochemical models of protein aggregation, the spontaneous appearance of a prion nucleus is thought to be rare [57, 58]. As such, mathematical models of prion appearance are often framed as first-passage processes; that is, these models focus on determining the amount of time until a critical event occurs, in this case the

One of the earliest models of self-assembly of particles was proposed by in 1916 by Smoluchowski [59]. He considered the evolution of the density of clusters of discrete particle sizes under the assumption that clusters of any size could join together (coagulation). In 1935, Becker and Döring introduced kinetic equations for a similar process but where clusters could

] prion system [53–55], but linking experimental outcomes

been made for the Sup35/[PSI+

216 Prion - An Overview

appearance of a prion nucleus.

uniquely to specific kinetic parameters remains challenging.

Broadly speaking, two mathematical formulations have been used to describe the time to nucleus formation: deterministic and stochastic. In a deterministic mathematical model, the predictions or model output is always the same for a given input. In such a formulation, the Law of Mass Action is used to convert the set of biochemical equations to a system of ordinary differential equations (ODEs) [62]. For the standard aggregation processes, like the Becker-Döring process, systems of ODEs have been extensively studied [60, 63, 64]. In these ODEs, the mathematical model output is a continuously varying quantity approximating the concentration or number of aggregates of each possible size. The time to nucleation would then be specified as the time at which the value associated with the critical nucleus size exceeds a threshold value. When the number of total proteins present is large, a deterministic formulation describes the dynamics well; however, when the number of proteins is small, random effects begin to dominate and to capture these effects a stochastic formulation is required [65]. (We note that for in vitro experiments of prion aggregation, when the concentration of proteins far exceeds physiological settings, deterministic models have proven to be consistent with observed quantities [66, 67].)

Stochastic mathematical models allow for the possibility of the same input to produce different output. In this case, the state of the system is given not as a deterministic quantity, but a random variable that can take on different values [68]. For example, given a coin with two sides (heads and tails), the number of times a coin must be flipped until heads appears is a random variable; one might attain heads on the first try or require many trials before heads appears. Because the observed output can change, the quantity of interest is not the specific output but rather its properties. To continue our example, we might wish to know either what the mean (average) number of flips will be required from a fair coin to produce heads and possibly the variance in that quantity. Alternatively, we might wish to know the probability associated with observing any possible outcome (i.e., what is the probability we flip the coin k times before observing a head); this corresponds to a probability density function. For our example of the coin, the number of flips required before heads appears is given as a geometric probability distribution. That is, the probability that k coin flips are required before the first heads is observed is given by: ð1−pÞ k−1 p where p is the probability of heads on any given trial. For all but simple systems, such as our coin flip example, it is not possible to obtain an explicit formula for our random variable in question. As such, an increasingly sophisticated set of mathematical and computational tools have been employed to aid in such processes.

We note that for nucleation problems, we are interested not in the state of our protein molecules at any particular time, but the first-arrival time of the nucleus. That is, the time at which the first aggregate of minimal stable size appears. Below we refer to misfolded protein aggregates smaller than the critical nucleus as proto-nuclei and any aggregate larger than the nucleus size as a propagon. (We note this is consistent with the definition of a propagon as being a prion aggregate capable of transmitting the prion phenotype upon transmission to an environment with normally folded protein [33, 69].)

We will first frame this problem as a continuous-time stochastic process and then discuss how statistical properties of the first-arrival time may be computed. (For a detailed discussion of stochastic processes and first arrival times in biological systems, refer to [61, 68].) For simplicity, let us assume that our system consists of a total number of m molecules of our protein in question and that this number remains constant (i.e., no synthesis or degradation). In this case, if we observe the system at any particular time the state of the system consists of the number of protein aggregates of each possible size. If niðtÞ is the total number of aggregates of each size i at time t (i ¼ 1 corresponds to monomer), then for all time we have:

$$\sum\_{i=1}^{m} \text{i } n\_i(t) = m. \tag{10}$$

We use Ω to denote set of all possible molecular configurations and observe that the size of this space increases exponentially in m but, because there is no synthesis or degradation, Ω is finite. We distinguish between two sets of configurations in Ω; A: those configurations with only proto-nuclei and AC ¼ Ω: those where the system has at least one propagon. While not required, it is often assumed that the system begins in the all-monomer state (i.e., n1ð0Þ ¼ m, niðiÞ ¼ 0 for all i > 1) [61, 70]. At any given time, aggregates of any size may increase or decrease in time through dynamics such as monomer addition, fragmentation and/or coagulation as allowed by the biochemical assumptions (for example, the Smoluchowski [59] or Becker-Döring [60] assumptions). Refer to Figure 3 for a visualization of the stochastic model. The first-arrival time is the time that the stochastic process reaches any configuration in AC.

Figure 3. Stochastic model for prion nucleus appearance. The rate-limiting step in the appearance of prion phenotypes is thought to be the waiting-time until the appearance of a nucleus, an aggregate of misfolded prion protein that exceeds a critical size. In a stochastic formulation, the number of prion subunits of each size is tracked in time. As detailed in the text, we are considering a reduced system where the total number of protein molecules (m) remains fixed in time.

There are three methods for computing first-arrival times and their associated moments (mean, variance, etc.) in this stochastic formulation. First, these quantities may be defined directly by analyzing solutions of the chemical master equation (CME). The CME is the first-order linear differential equation that describes the time-evolution of the probability of the system to occupy any particular configuration [68, 71–74]. Because the size of the state-space is exponential in the number of monomers, the CME is computationally intractable for all but very simple systems. Second, computational simulations representing individual realizations of the nucleation process are generated in silico; the mean and moments of the first-passage times are then calculated from these empirical results [75, 76]. While Monte Carlo approaches are typically easy to code and highly parallelizable, they suffer from slow convergence and in the case of rare events, like nucleation, individual realizations may take arbitrarily long to terminate [77]. Third, heuristics may be used to simplify the dynamics in particular regimes. For example, in a series of studies Chou and colleagues [65, 78–80] approximated the mean first-arrival time to a critical nucleus for the Becker-Döring model (only monomer growth or detachment) in for two parameter regimes (strong growth and weak growth) by computing the arrival time for the dominant pathway from the all-monomer state to the appearance of the first aggregate of a minimal stable size. While dominant path approaches are readily apparent for some models of aggregation, like Becker-Döring, they are difficult to determine for more general sets of reactions. Further, because these results rely on particular parameter combinations, the approximations pose challenges to parameter inference—where we want to determine the kinetic parameters that best match available data.

being a prion aggregate capable of transmitting the prion phenotype upon transmission to an

We will first frame this problem as a continuous-time stochastic process and then discuss how statistical properties of the first-arrival time may be computed. (For a detailed discussion of stochastic processes and first arrival times in biological systems, refer to [61, 68].) For simplicity, let us assume that our system consists of a total number of m molecules of our protein in question and that this number remains constant (i.e., no synthesis or degradation). In this case, if we observe the system at any particular time the state of the system consists of the number of protein aggregates of each possible size. If niðtÞ is the total number of aggregates of each size i

We use Ω to denote set of all possible molecular configurations and observe that the size of this space increases exponentially in m but, because there is no synthesis or degradation, Ω is finite. We distinguish between two sets of configurations in Ω; A: those configurations with only proto-nuclei and AC ¼ Ω: those where the system has at least one propagon. While not required, it is often assumed that the system begins in the all-monomer state (i.e., n1ð0Þ ¼ m, niðiÞ ¼ 0 for all i > 1) [61, 70]. At any given time, aggregates of any size may increase or decrease in time through dynamics such as monomer addition, fragmentation and/or coagulation as allowed by the biochemical assumptions (for example, the Smoluchowski [59] or Becker-Döring [60] assumptions). Refer to Figure 3 for a visualization of the stochastic model. The first-arrival time is the time that the stochastic process reaches any configuration in AC.

Figure 3. Stochastic model for prion nucleus appearance. The rate-limiting step in the appearance of prion phenotypes is thought to be the waiting-time until the appearance of a nucleus, an aggregate of misfolded prion protein that exceeds a critical size. In a stochastic formulation, the number of prion subunits of each size is tracked in time. As detailed in the text, we are considering a reduced system where the total number of protein molecules (m) remains fixed in time.

i niðtÞ ¼ m: (10)

environment with normally folded protein [33, 69].)

218 Prion - An Overview

at time t (i ¼ 1 corresponds to monomer), then for all time we have:

∑ m i¼1

> We note that beyond the mathematical challenges in modeling nucleation there remain many practical challenges. Experimentally, it is typically not possible to separate the spontaneous appearance of a propagon, an initial infectious aggregate, from the prion phenotype itself. Finally, critical events in the underlying biochemical kinetics of nucleus formation in prion disease are unknown. In our formulation above, we described protein subunits, but the protein itself is only capable of aggregating when in a particular conformational state. As such, an accurate predictive model of spontaneous nucleation must also include a model of protein misfolding. While it is clear that particular prion variants (distinct conformations) are favored under particular experimental conditions [81], the connection between nucleus formation and conformation has yet to be fully explored. As such, nucleation remains a challenge on experimental, mathematical and computational fronts.

## 4. Present state and challenges in prion disease modeling

As described above, a combination of mathematical and experimental studies over the past few decades have led to the formulation of a protein-only form of inheritance associated with prion phenotypes. We first summarize our present knowledge and then remark on present day studies and challenges that remain in modeling prion disease. Today we believe that prion phenotypes are established through two distinct phases, nucleation and amplification. Once an initial nucleus—prion aggregate above a critical size—is introduced to a host, four steps are required for successful in vivo propagation of prion phenotypes (see Figure 4). First, normal folded protein must be continuously created. Second, aggregates of the misfolded form of the protein act as templates by converting normally folded protein to the same misfolded state. Third, the total number of templates increases through fragmentation where a single aggregate is split into two (or more) smaller aggregates. Finally, misfolded protein must spread through other cells. For yeast, this transfer of misfolded protein occurs through cell division where for mammals this likely involves extracellular diffusion [33].

Figure 4. In vivo prion propagation. The infectious agent of prion disease is aggregates of misfolded proteins (squares). Four steps are essential to stable propagation of the prion form of the protein: (1) new normally folded protein must be created, (2) prion aggregates act as templates to convert normally folded protein to the misfolded state, (3) prion aggregates are fragmented into smaller aggregates each of which may be capable of acting as a template, and (4) prion infectious units must spread through other cells. In the case of yeast prions, this transmission occurs through the normal process of cell division while for mammalian disease transmission corresponds to diffusion and transport through tissue.

However, beyond this basic understanding remain many challenges, both mathematical and biological. We briefly outline some open questions in prion biology we believe are amenable to interdisciplinary approaches.

Consideration of the cellular environment. Prion phenotypes are established through protein misfolding, but protein misfolding itself is not rare. Eukaryotic cells have developed a complex network of molecular chaperones and protein degradation factors that act continuously to identify and clear misfolded proteins [33]. As such, understanding the in vivo propagation of prion phenotypes requires considering the environment in which they appear. In the case of yeast prions, the molecular chaperone Hsp104 has been shown to be essential for the propagation of [PSI+ ] prion phenotype in yeast. While comparatively few mathematical studies have considered the role of Hsp104 as enzyme catalyzing fragmentation [55, 82], the results from these studies have resolved previously unsupported results on shifts in aggregate size distributions and (as we will discuss further below) rates of loss (curing) of specific [PSI+ ] strain phenotypes.

Spread of prion aggregates. A major open question in prion biology amenable to mathematical analysis is how prion aggregates spread between cells either through cell division (yeast prion phenotypes) or within a mammalian tissue. Although this question was first explored when Nowak and colleagues [42] presented simulations of their NPM where prion aggregates could move between distinct cells in a population and when Payne and Krakauer [83] considered prions spreading in tissue as a traveling wave, our understanding of the spread of prion aggregates is still incomplete. Because the long incubation period and complicated physical domains involved in mammalian prion disease will pose significant experimental and mathematical challenges, yeast prion phenotypes may provide a useful tool in this question. While stochastic models have been developed for yeast, which link cellular levels of prion aggregates with computational simulations [55], formulations amenable to analytical treatment need to be developed to allow for a more systematic characterization of the spread of prion aggregates.

Reversing prion phenotypes. Mammalian prion diseases remain untreatable and ultimately fatal and as such the identification of clinical treatments and methods of early detection remain important scientific and technical challenges. Of particular challenge may be that drugs which act to promote aggregate fragmentation, with the goal of causing all aggregate to drop below the critical nucleus size, may in fact promote prion amplification at low doses when they would merely act to accelerate the exponential growth of prion aggregates [84].

A promising avenue toward finding approaches to manage prion diseases in mammals, might be to more clearly understand the biochemical processes responsible for a number of reversible prion phenotypes in yeast [33]. As was demonstrated by Derdowski et al. [55], a combination of enzyme-limited fragmentation and aggregate-size transmission bias appeared to be responsible for the observed natural rates of curing for the [PSI+ ] weak phenotype. In addition, for yeast treatment with GdnHCl has been shown to significantly slow aggregate fragmentation leading to a natural reversal of prion phenotypes by dilution of the aggregate during cell division [69, 85–88].

Further, biological and mathematical researchers should consider mechanisms of curing prion disease through studying mutations in prion proteins, which are known to slow or halt the disease progression. Such mutations are known to exist in mammals [89] and yeast [90].

Evolvability of prions. While prions were originally implicated in mammalian disease, the fact that they persist as a number of harmless heritable phenotypes in yeast raises intriguing questions about how and why prions may have evolved [57, 91, 92]. Because the yeast phenotype [PSI+ ] is associated with a decreased efficiency in stop-codon recognition, it is thought to serve as an evolutionary capacitor by promoting the generation of novel transcripts [57]. More recently, a combination of mathematical and experimental studies have demonstrated that smaller [PSI+ ] aggregates still retain function associated with the normal Sup35 conformation offering the possibility this system evolved to tune stop-codon recognition [54]. While we are still far from understanding the forces behind prion phenotype evolution, the evidence continues to mount for possible beneficial examples of prion-like mechanisms [93].

## 5. Conclusion

folded protein must be continuously created. Second, aggregates of the misfolded form of the protein act as templates by converting normally folded protein to the same misfolded state. Third, the total number of templates increases through fragmentation where a single aggregate is split into two (or more) smaller aggregates. Finally, misfolded protein must spread through other cells. For yeast, this transfer of misfolded protein occurs through cell division where for

However, beyond this basic understanding remain many challenges, both mathematical and biological. We briefly outline some open questions in prion biology we believe are amenable to

Figure 4. In vivo prion propagation. The infectious agent of prion disease is aggregates of misfolded proteins (squares). Four steps are essential to stable propagation of the prion form of the protein: (1) new normally folded protein must be created, (2) prion aggregates act as templates to convert normally folded protein to the misfolded state, (3) prion aggregates are fragmented into smaller aggregates each of which may be capable of acting as a template, and (4) prion infectious units must spread through other cells. In the case of yeast prions, this transmission occurs through the normal process of cell division while for mammalian disease transmission corresponds to diffusion and transport through tissue.

Consideration of the cellular environment. Prion phenotypes are established through protein misfolding, but protein misfolding itself is not rare. Eukaryotic cells have developed a complex network of molecular chaperones and protein degradation factors that act continuously to identify and clear misfolded proteins [33]. As such, understanding the in vivo propagation of prion phenotypes requires considering the environment in which they appear. In the case of yeast prions, the molecular chaperone Hsp104 has been shown to be essential for the propaga-

considered the role of Hsp104 as enzyme catalyzing fragmentation [55, 82], the results from these studies have resolved previously unsupported results on shifts in aggregate size distributions and (as we will discuss further below) rates of loss (curing) of specific [PSI+

Spread of prion aggregates. A major open question in prion biology amenable to mathematical analysis is how prion aggregates spread between cells either through cell division (yeast prion phenotypes) or within a mammalian tissue. Although this question was first explored

] prion phenotype in yeast. While comparatively few mathematical studies have

] strain

mammals this likely involves extracellular diffusion [33].

interdisciplinary approaches.

tion of [PSI+

220 Prion - An Overview

phenotypes.

Many questions remain about prion phenotypes and it is essential once again for scientists with different backgrounds to utilize their disciplinary expertise and methods to address these questions. As we have discussed, two critical points in the history of prion disease came from researchers that were not primarily biologists, namely the mathematician Griffith [4] and the veterinarian Hadlow [25]. If the past is any predictor, future studies in prion phenotypes will continue to benefit from an interdisciplinary approach.

## Author details

Suzanne S. Sindi

Address all correspondence to<EMAIL_ADDRESS>
Applied Mathematics, University of California, Merced, California, United States of America

## References

[13] May G., The sheep: its wool, races, breeding, nutrition, and use, as well as its diseases: First Edition in 2 vol, Eduard Trewendt, Breslau, 1868.

researchers that were not primarily biologists, namely the mathematician Griffith [4] and the veterinarian Hadlow [25]. If the past is any predictor, future studies in prion phenotypes will

Applied Mathematics, University of California, Merced, California, United States of America

[1] Jungck, J.R., Ten equations that changed biology: mathematics in problem-solving biol-

[2] Reed, M.C., Mathematical biology is good for mathematics. Notices of the AMS, 2015. 62

[3] Webb, T., et al., Age of onset and death in inherited prion disease are heritable. American Journal of Medical Genetics Part B: Neuropsychiatric Genetics, 2009. 150(4): 496–501.

[5] Prusiner, S.B., Novel proteinaceous infectious particles cause scrapie. Science, 1982. 216

[6] Tuite, M.F. and T.R. Serio, The prion hypothesis: from biological anomaly to basic regulatory mechanism. Nature Reviews Molecular Cell Biology, 2010. 11(12): 823–33.

[7] Schneider, K., et al., The early history of the transmissible spongiform encephalopathies

[8] Poser, C.M., Notes on the history of the prion diseases. Part II. Clinical Neurology and

[9] Poser, C.M., Notes on the history of the prion diseases. Part I. Clinical Neurology and

[10] Goldfarb, L.G., Kuru: the old epidemic in a new mirror. Microbes and Infection, 2002. 4

[11] Brown, D.R., Neurodegeneration and Prion Disease. 2005: Springer Science & Business

[12] Leopoldt J.G., Useful Introduction to Agriculture Based on Experience. volume 5, Chris-

[4] Griffith, J.S., Self-replication and scrapie. Nature, 1967. 215(5105): 1043–4.

exemplified by scrapie. Brain Research Bulletin, 2008. 77(6): 343–55.

continue to benefit from an interdisciplinary approach.

Address all correspondence to<EMAIL_ADDRESS>
ogy curricula. Bioscene, 1997. 23(1): 11–36.

Neurosurgery, 2002. 104(2): 77–86.

Neurosurgery, 2002. 104(1): 1–9.

Media, New York, NY, USA.

tian Friedrich Günthern, Glogau, Berlin, 1759.

Author details

222 Prion - An Overview

Suzanne S. Sindi

References

(10): 1172–6.

(4542): 136–44.

(8): 875–82.

[48] Pöschel, T., N.V. Brilliantov and C. Frömmel, Kinetics of prion growth. Biophysical Journal, 2003. 85(6): 3460–74.

[31] Wickner, R.B., [URE3] as an altered URE2 protein: evidence for a prion analog in Saccha-

[32] Cox, B., Cytoplasmic inheritance. Prion-like factors in yeast. Current Biology, 1994. 4(8):

[33] Sindi, S.S. and T.R. Serio, Prion dynamics and the quest for the genetic determinant in protein-only inheritance. Current Opinion in Microbiology, 2009. 12(6): 623–30.

[34] Zhang, Z., et al., De novo generation of infectious prions with bacterially expressed

[35] Cohen, S.I., et al., From macroscopic measurements to microscopic mechanisms of pro-

[36] Morris, A.M., M.A. Watzky and R.G. Finke, Protein aggregation kinetics, mechanism and curve-fitting: a review of the literature. Biochimica et Biophysica Acta (BBA)-Proteins and

[37] Eigen, M., Prionics or the kinetic basis of prion diseases. Biophysical Chemistry, 1996. 63

[39] Caughey, B., et al., Aggregates of scrapie-associated prion protein induce the cell-free conversion of protease-sensitive prion protein to the protease-resistant state. Chemistry &

[40] Come, J.H., P.E. Fraser and P.T. Lansbury, Jr., A kinetic model for amyloid formation in the prion diseases: importance of seeding. Proceedings of the National Academy of

[41] Lansbury, P.T., Jr. and B. Caughey, The chemistry of scrapie infection: implications of the

[42] Nowak, M.A., et al., Prion infection dynamics. Integrative Biology Issues News and

[43] Masel, J., V.A. Jansen and M.A. Nowak, Quantifying the kinetic parameters of prion

[44] Greer, M.L., L. Pujo-Menjouet and G.F. Webb, A mathematical analysis of the dynamics

[45] Prüss, J., et al., Analysis of a model for the dynamics of prions. Discrete and Continuous

[46] Engler, J., J. Pruss and G.F. Webb, Analysis of a model for the dynamics of prions II.

[47] Doumic, M., et al., Scaling limit of a discrete prion dynamics model. Communications in

of prion proliferation. Journal of Theoretical Biology, 2006. 242(3): 598–606.

Journal of Mathematical Analysis and Applications, 2006. 324(1): 98–117.

Sciences of the United States of America, 1993. 90(13): 5959–63.

'ice 9' metaphor. Chemistry & Biology, 1995. 2(1): 1–5.

replication. Biophysical Chemistry, 1999. 77(2): 139–52.

Dynamical Systems–Series B, 2006. 6(1): 225–35.

Mathematical Sciences, 2009. 7(4): 839–65.

[38] Prusiner, S.B., Molecular biology of prion diseases. Science, 1991. 252(5012): 1515–22.

recombinant prion protein. The FASEB Journal, 2013. 27(12): 4768–75.

tein aggregation. Journal of Molecular Biology, 2012. 421(2): 160–71.

romyces cerevisiae. Science, 1994. 264(5158): 566–9.

Proteomics, 2009. 1794(3): 375–97.

Biology, 1995. 2(12): 807–17.

Reviews, 1998. 1(1): 3–15.

744–8.

224 Prion - An Overview

(1): A1–18.

[82] Davis, J.K. and S.S. Sindi, A mathematical model of the dynamics of prion aggregates with chaperone-mediated fragmentation. Journal of Mathematical Biology, 2016. 72(6): 1555–78.

[65] D'Orsogna, M., G. Lakatos and T. Chou, Stochastic self-assembly of incommensurate

[66] Knowles, T.P., et al., An analytical solution to the kinetics of breakable filament assembly.

[67] Xue, W.-F., S.W. Homans and S.E. Radford, Systematic analysis of nucleation-dependent polymerization reveals new insights into the mechanism of amyloid self-assembly. Proceedings of the National Academy of Sciencesof the United States of America, 2008. 105

[68] Allen, L., An Introduction to Stochastic Processes with Applications to Biology 2003:

[69] Byrne, L.J., et al., The number and transmission of [PSI+] prion seeds (Propagons) in the

[70] Davis, J.K. and S.S. Sindi, Initial condition of stochastic self-assembly. Physical Review E,

[71] Van Kampen, N.G., Stochastic Processes in Physics and Chemistry. Vol. 1. 1992: Elsevier.

[74] Van Kampen, N.G., Stochastic Processes in Physics and Chemistry. 1981, Amsterdam:

[75] Gillespie, D.T., Stochastic simulation of chemical kinetics. Annual Review of Physical

[76] Gillespie, D., Exact stochastic simulation of coupled chemical reactions. The Journal of

[77] Hammersley, J., Monte Carlo Methods. 2013: Springer Science & Business Media. New

[78] Yvinec, R., M.R. D'Orsogna and T. Chou, First passage times in homogeneous nucleation

[79] D'Orsogna, M.R., Zhao, B., Berenji, B., & Chou, T. Combinatoric analysis of heterogeneous stochastic self-assembly. The Journal of Chemical Physics, 2013, 139(12), 121918.

[80] D'Orsogna, M.R., Q. Lei and T. Chou, First assembly times and equilibration in stochastic coagulation-fragmentation. The Journal of Chemical Physics, 2015. 143(1): 014112. [81] Sharma, J. and S.W. Liebman, Exploring the basis of [PIN(+)] variant differences in [PSI

and self-assembly. The Journal of Chemical Physics, 2012. 137(24): 244107.

(+)] induction. Journal of Molecular Biology, 2013. 425(17): 3046–59.

[72] Durrett, R., Essentials of Stochastic Processes. 1999: Springer, New York, NY, USA.

clusters. The Journal of Chemical Physics, 2012. 136(8): 084110.

yeast Saccharomyces cerevisiae. PLoS One, 2009. 4(3): e4670.

[73] Doob, J.L., Stochastic Processes. Vol. 101. 1953: New York: Wiley.

Science, 2009. 326(5959): 1533–7.

Pearson/Prentice Hall. Boca Raton, FL, USA.

North-Holland Publishing Company. 419 p.

Physical Chemistry, 1977. 81(25): 2430–61.

(26): 8926–31.

226 Prion - An Overview

2016. 93(2): 022109.

Amsterdam, The Netherlands.

Chemistry, 2007. 58: 35–55.

York, NY, USA.

## *Edited by Yusuf Tutar*

Protein aggregation causes malfunction in several biochemical processes. Genetic and spontaneous formations of these transmissible spongiform encephalopathies are fatal to humans and animals. Conformational change of normal form of the protein to misfolded form causes its accumulation. The misfolded infectious protein agent forms the pathogenesis of the disease. This book presents pathology of the disease along with current knowledge of the structure-activity mechanism in the first two sections. Dyshomeostasis of metals is implicated in the pathogenesis of prions, and this influence is discussed further to understand the prion mechanism. Genetic resistance and immunobiology of the disease are elaborated in the following section. Finally, a computational study on the dynamics of the prion propagation provides a structural basis of the mechanism.

Photo by Molekuul / iStock

Prion - An Overview

Prion

An Overview

*Edited by Yusuf Tutar*


*3.4. Curvature Forms*

Given our connection 1-form *ω*, we can proceed in two ways: the first consists in taking a map called the *horizontal projection* and in defining the curvature as this projection applied on the exterior derivative of *ω*. In this way, we naturally see that curvature measures the displacement of the commutator of two vectors from being horizontal.

We will proceed in a different way though. We will define the curvature through a *structure equation*.

**Definition 14.** *Given ω* ∈ Ω1 (*P*, g)*, a principal connection* 1*-form, the* 2*-form* Ω ∈ Ω2 *G* (*P*, g) *satisfies the following:*

$$
\Omega = d\omega + \frac{1}{2} [\omega \wedge \omega] \tag{23}
$$

*whic is called curvature* 2*-form.*

In Equation (23), [*ω* ∧ *ω*] denotes the bilinear operation on the Lie algebra g called differential Lie bracket. It is defined as follows:

$$[\omega \wedge \eta](\iota, \upsilon) = \frac{1}{2} \{ [\omega(\iota), \eta(\upsilon)] - [\omega(\upsilon), \eta(\iota)] \},\tag{24}$$

where *u* and *v* are vector fields.

It follows straightforwardly that, if we take two general horizontal vector fields *u*, *v* ∈ Γ(*HE*) and we use the ordinary formula12 for the exterior derivative of a 1-form *dω*(*u*, *v*) = *uω*(*v*) − *vω*(*u*) − *ω*([*u*, *v*]), since *ω*(*u*) = *ω*(*v*) = 0, we get

$$
\Omega(u, v) = -\omega([u, v]).\tag{25}
$$

We see that Ω measures how the commutator of two horizontal vector fields is far from being horizontal as well.

12 Here, we regard *ω*(*u*) as a function *ω*(*u*) : *P* → g belonging to the algebra of smooth functions to g, C ∞(*P*, g).


*2.1. Patients*

Between October 2014 and September 2019, infants admitted to the neonatal intensive care unit at Tuebingen University Children's Hospital site were evaluated for their eligibility to participate in this study. Inclusion criteria were (i) gestational age (GA) at birth ≤34 weeks (w), (ii) postmenstrual age at study ≤38 w, (iii) persistent AOP, and (iv) frequent apneas, expressed as an apnea score ≥5 on a rating scale used in the unit [11], despite treatment with hospital pharmacy-produced ca ffeine base and cf-NCPAP. The apnea score assigned points depending on the response of the nursing sta ff to events involving bradycardia (heart rate < 80/min) or hypoxemia (SpO2 < 80%). If no intervention was needed, one point was given; two points if tactile stimulation was applied or a manual inspiration via the ventilator, and three points if the baby had to be turned over in preparation for bagging; eight points were given if the baby was felt to briefly need positive pressure ventilation to recover. The respective points were added up. Exclusion criteria were severe congenital malformations, neuromuscular conditions or chromosomal abnormalities, the presence of symptomatic apnea (e.g., due to sepsis, hypoglycemia or cerebral hemorrhage), or the lack of written informed parental consent.