Introduction to Modeling Biological Cellular Control Systems

There are numerous cellular control systems in living organisms. These cellular systems are tightly controlled through numerous feedback control mechanisms so that cells in the living organisms can carry out numerous functions to survive. In this textbook, I select a number of cellular control systems that I have understood most to demonstrate how to model them mathematically in the setting of control theory.

In this chapter, we introduce preliminary systems theory. This includes controllability and observability of a system, stability of equilibrium points of a system, feedback control of a system, and parametric sensitivity of a system. These theories will be used to analyze biological cellular control systems.

Some molecular control mechanisms of blood glucose are described schematically in Fig. 4.1. Glucose comes from food and liver, and is utilized by brain and nerve cells (insulin-independent) via the glucose transporter 3 (GLUT3) or by tissue cells such as muscle, kidney, and fat cells (insulin-dependent) via the glucose transporter 4 (GLUT4). Glucose is transported into and out of liver cells by the concentrationdriven glucose transporter 2 (GLUT2), which is insulin-independent. In response to a low blood glucose level (<80 mg/dl or 4.4 mmol/L), α cells of the pancreas produce the hormone glucagon. The glucagon initiates a series of activations of kinases, and finally leads to the activation of the glycogen phosphorylase, which catalyzes the breakdown of glycogen into glucose. In addition, the series of activations of kinases also result in the inhibition of glycogen synthase and then stop the conversion of glucose to glycogen. In response to a high blood glucose level (> 120 mg/dl or 6.7 mmol/L), β cells of the pancreas secrete insulin. Insulin triggers a series of reactions to activate the glycogen synthase, which catalyzes the conversion of glucose into glycogen. Insulin also initiates a series of activations of kinases in tissue cells to lead to the redistribution of GLUT4 from intracellular storage sites to the plasma membrane. Once at the cell surface, GLUT4 transports glucose into the muscle or fat cells.

Yeast cells uptake calcium from their environment via Mid1p, Cch1p, and other unidentified transporters [7], and maintain a normal cytosolic Ca²⁺ level of 50 – 200 nM by means of a feedback control system [1, 16, 23]. The rise of cytosolic calcium activates calmodulin which in turn activates the serine/threonine phosphatase calcineurin (Fig. 5.1). The activated calcineurin de-phosphorylates Crz1p and suppresses the activity of Vcx1p. Activated Crz1p enters the nucleus and up-regulates the expression of PMR1 and PMC1 (for review, see [13]). Pmr1p pumps calcium ions into the organelle Golgi and possibly endoplasmic reticulum (ER).When the calcium concentrations in Golgi and ER exceed their resting levels of 300 μM [30] and 10 μM [1, 39], respectively, the calcium in ER and Golgi will be secreted along with the canonical secretory pathways. Pmc1p pumps calcium ions into vacuole, an organelle that stores excess ions and nutrients. While most calcium ions inside vacuoles form polyphosphate salts and are not re-usable, a small fraction of calcium ions can be channeled to the cytosol by Yvc1p. The total and free vacuolar calcium concentrations are 2mMand 30 μM, respectively [16]. When needed, Yvc1p channels calcium to the cytosol and contributes to the rise of cytosolic calcium concentration [14].

Various ions, such as Ca²⁺, Na⁺, and K⁺, enter and exit a cell through selective ion pumps and channels, such as potassium channels, voltage-gated sodium channels, voltage-gated calcium channels, sodium/calcium exchangers, and plasma membrane (PM) calcium ATPases, as demonstrated in Fig. 6.1. Calcium ions Ca²⁺ enter the cytosol through store-operated channels (SOC) and voltage-gated calcium channels (VGCC). The sarcoplasmic or endoplasmic reticulum Ca²⁺-ATPases (SERCA) pump Ca²⁺ from the cytosol into the endoplasmic reticulum (ER) and Ca²⁺ in ER are released to the cytosol through the inositol (1,4,5)-trisphosphate (IP₃)- and Ca²⁺-mediated inositol (1,4,5)-trisphosphate receptors (IP₃R). Ca²⁺ enter the mitochondrion through uniporters and exit through Ca²⁺/Na⁺ antiporters. Ca²⁺ exit the cytosol through plasma membrane Ca²⁺-ATPases (PMCA) and Ca²⁺/Na⁺ exchangers. Depletion of ER Ca²⁺ stores causes STIM1 to move to ER-PM junctions, bind to Orai1, and activate store-operated channels for Ca²⁺ entry [63]. Sodium ions Na⁺ enter the cytosol through sodium channels (NC) and Ca²⁺/Na⁺ exchangers, and exit through N⁺/K⁺ ATPases (NKA). Potassium ions K⁺ enter the cytosol through N⁺/K⁺ ATPases and exit through ATP-sensitive K⁺ channels (ATPKC), delayed rectifying K⁺ channels (DrKC), and calcium-activated K⁺ channels (CAKC). H⁺ ions in mitochondrion are ejected by the respiratory chain driven by the energy released from oxidation of NADP, which are produced from the tricarboxylic acid cycle (TAC, also called Krebs cycle).

The voltage-gated calcium channel is closed at the end of the cytosol side when the transmembrane voltage is at the resting voltage. If the membrane depolarization exceeds a threshold, most of the voltage-gated calcium channels stay shut, but some open, and calcium ions rush into the cell down their electrochemical gradient. The flow of positive charge inward on the calcium ions outweighs the outward flow of positive charge on potassium ions, so the cytosol is gaining positive charge and its voltages is moving in the positive direction.

Depletion of intracellular calcium stores such as the endoplasmic reticulum (ER) activates store-operated channels for Ca²⁺ entry across the plasma membrane (PM), as demonstrated in Fig. 8.1. This process is called the store-operated calcium entry (SOCE), a common and ubiquitous mechanism of regulating Ca²⁺ influx into cells [1, 7, 36, 38]. The best-studied store-operated channel (SOC) is the Ca²⁺ releaseactivated Ca²⁺ channel (CRAC) [5, 12, 13, 30, 38, 41, 53]. SOCE is a key feedback controller to stabilize ER Ca²⁺ and has been proposed as the main Ca²⁺ entry pathway in non-excitable cells [37, 38]. SOCE, originally known as capacitative calcium entry (CCE), was first proposed by Putney [42] and has been extensively studied later (for review, see Berridge [2, 3], Bird et al [4], Chakrabarti et al [6], Dirksen [9], Lewis [25], Parekh [37], Potier et al [39], Prakriya et al [40], Putney [43], and Shuttleworth et al [47]).

A mitochondrion is a membrane-enclosed organelle found in most eukaryotic cells. Mitochondria are sometimes described as “cellular power plants” because they generate most of the cell’s supply of adenosine triphosphate (ATP), used as a source of chemical energy. A mitochondrion contains outer and inner membranes composed of phospholipid bilayers and proteins. The space between the outer and inner membranes is called the inter-membrane space and the space within the inner membrane is called the matrix.

Phosphatidylinositol 4,5-bisphosphate (PIP₂) is the predominant (99%) phosphoinositide in mammalian cells [7]. PIP₂ is synthesized from phosphatidylinositol-4- phosphate (PIP) by PIP₂ synthases while PIP is synthesized from Phosphatidylinositol (PI) by PIP synthases. PIP₂ in cells is normally hydrolyzed by phospholipase C (PLC) to generate inositol 1,4,5-trisphosphate (IP₃) and diacylglycerol (DAG),which serve as second messengers for intracellular Ca²⁺ mobilization and PKC (protein kinase C) activation, respectively [4, 7]. Thus, PIP₂ plays important roles in PLCmediated cellular processes, such as glucose-stimulated insulin secretion [1], storeoperated calcium entry [2, 6], and sterol trafficking [5, 8]. Mathematical models for the process of phosphoinositide synthesis have been established (see, e.g., [3, 7]). In this chapter, we present the model developed by Xu et al [7] because of its simplicity.