Mathematical Modelling of Energy Generation in the Mitochondria
Chapter One
WORK OBJECTIVES:
- To develop a mathematical model for energy generation in the mitochondria based on the physics that would provide insights into the operation in the mitochondria and limiting mechanisms. We hope that, with the model, it will be possible for an individual to predict the behavior of he mitochondria under wide range of operating conditions.
CHAPTER TWO
LITERATURE REVIEW
In 1997 Gerhard Magnus and Joel Keizer published a minimal mathematical model to describe Ca2+ handling by mitochondria in the pancreatic beta-cell. Their kinetic model includes six transport mechanisms in the inner mitochondrial membrane, including proton pumping via respiration and proton pumping by the F0F1-ATPase, a proton leak, adenine nucleotide exchange, the Ca2+ uniporter and Na+/Ca2+ exchange. In their 1998 model, Gerhard Magnus and Joel Keizer continued their development of a kinetic model of bursting electrical activity in the pancreatic beta-cell. . Their minimal model of Ca2+ handling is expanded to include, amongst other factors, the glucose dependence of the rate of production of mitochondrial reducing equivalents. The basis of their model is that when exposed to a threshold concentration of glucose, pancreatic beta-cells from a wide range of species exhibit a complicated pattern of electrical activity. Bursts of action potential spikes (the “active” phase) are observed, separated by a “silent” phase of membrane repolarisation. And at even higher glucose concentrations, continuous action potentials are seen. This electrical activity has two important physiological correlates: increased cytosolic Ca2+ concentration ([Ca2+]i) and increased rate of insulin secretion during the active phase. It is generally accepted that the rise in [Ca2+]i plays a major role in insulin secretion and that the action potential spikes during a burst are responsible for the rise in [Ca2+]i. Because the glucose signal for insulin secretion operates via metabolism rather than through a plasma membrane-bound receptor, the details of how glucose stimulates electrical activity have been difficult to resolve. However, experimental data and theoretical models have begun to investigate the phenomenon.
Cortassa, M. et al (2003) presented an integrated thermokinetic model describing control of cardiac mitochondrial bioenergetics. The model describes the tricarboxylic acid (TCA) cycle, oxidative phosphorylation, and mitochondrial Ca2+ handling. The kinetic component of the model includes effectors of the TCA cycle enzymes regulating production of NADH and FADH2, which in turn are used by the electron transport chain to establish a proton motive force (ΔμH), driving the F1F0-ATPase. In addition, mitochondrial matrix Ca2+, determined by Ca2+ uniporter and Na+/Ca2+ exchanger activities, regulates activity of the TCA cycle enzymes isocitrate dehydrogenase and α-ketoglutarate dehydrogenase. The model is described by twelve ordinary differential equations for the time rate of change of mitochondrial membrane potential (ΔΨm), and matrix concentrations of Ca2+, NADH, ADP, and TCA cycle intermediates. It is used to predict the response of mitochondria to changes in substrate delivery, metabolic inhibition, the rate of adenine nucleotide exchange, and Ca2+. The model is able to reproduce, qualitatively and semi-quantitatively, experimental data concerning mitochondrial bioenergetics, Ca2+ dynamics, and respiratory control. Significant increases in oxygen consumption (VO2), proton efflux, NADH, and ATP synthesis, in response to an increase in cytoplasmic Ca2+, are obtained when the Ca2+-sensitive dehydrogenases are the main rate-controlling steps of respiratory flux. These responses diminished when control is shifted downstream (e.g., the respiratory chain or adenine nucleotide translocator). The time-dependent behavior of the model, under conditions simulating an increase in workload, closely reproduces experimentally observed mitochondrial NADH dynamics in heart trabeculae subjected to changes in pacing frequency. The steady-state and time-dependent behavior of the model support the hypothesis that mitochondrial matrix Ca2+ plays an important role in matching energy supply with demand in cardiac myocytes.
Again Cortassa, S. et al in 2004 described a unique mitochondrial oscillator that depends on oxidative phosphorylation, reactive oxygen species (ROS), and mitochondrial inner membrane ion channels. Cell-wide synchronized oscillations in mitochondrial membrane potential (ΔΨm), NADH, and ROS production was previously described in isolated cardiomyocytes in their earlier model published in 2003, where it was hypothesized that the balance between superoxide anion efflux through inner membrane anion channels and the intracellular ROS scavenging capacity play a key role in the oscillatory mechanism.In this present work, the hypothesis was formally tested using a computational model of mitochondrial energetics and Ca2+ handling including mitochondrial ROS production, cytoplasmic ROS scavenging, and ROS activation of inner membrane anion flux. The mathematical model reproduces the period and phase of the observed oscillations in ΔΨm, NADH, and ROS. Moreover, they experimentally verified model predictions that the period of the oscillator can be modulated by altering the concentration of ROS scavengers or the rate of oxidative phosphorylation, and that the redox state of the glutathione pool oscillates. In addition to its role in cellular dysfunction during metabolic stress, the period of the oscillator was shown to span a wide range, from milliseconds to hours, suggesting that it may also be a mechanism for physiological timekeeping and/or redox signaling.
CHAPTER THREE
Cellular respiration
Cellular respiration is the set of the metabolic reactions and processes that take place in the cells of organisms to convert biochemical energy from nutrients into adenosine triphosphate (ATP), and then release waste products. The reactions involved in respiration are catabolic reactions, which break large molecules into smaller ones, releasing energy in the process as they break high-energy bonds. Respiration is one of the key ways cell gains useful energy to fuel cellular activity.
Chemically, cellular respiration is considered an exothermic redox reaction. The overall reaction is broken into many smaller ones when it occurs in the body, most of which are redox reactions themselves.
Nutrients that are commonly used by animal and plant cells in respiration include sugar, amino acids and fatty acids, and a common oxidizing agent (electron acceptor) is molecular oxygen (O2). Bacteria and archaea can also be lithotrophs and these organisms may respire using a broad range of inorganic molecules as electron donors and acceptors, such as sulfur, metal ions, methane or hydrogen. Organisms that use oxygen as a final electron acceptor in respiration are described as aerobic, while those that do not are referred to as anaerobic [Campbell , et al 2006].
The energy released in respiration is used to synthesize ATP to store this energy. The energy stored in ATP can then be used to drive processes requiring energy, including biosynthesis, locomotion or transportation of molecules across cell membranes.
CHAPTER FOUR
MODEL FORMULATION
The first stage of glucose metabolism in eukaryotic cells is glycolysis which takes place in the cytoplasm of a cell. During glycolysis, some molecules of ATP and NADH are produced but its primary output is pyruvate.
The pyruvate is transported into the mitochondria where it is rapidly oxidized and decarboxylated by the “pyruvate dehydrogenase complex PDH”.
The output of the PDH complex are molecule of CO2, a molecule of NADH and acetyl coenzyme A ( acetyl coA ).
The acetyl coA enters the citric acid cycle where more NADH and FADH2 is produced by additional dehydrogenases.
CHAPTER FIVE
DISCUSSION
Our modelled equations evaluate the dynamics of energy demand and production in an individual mitochondria knowing full well that energy metabolism in one mitochondria varies from that of another mitochondria in an organism.
The constants in the solution of our modelled equation are the intrinsic factors of the organism involve, which is a measure of its energy demand, physiological/nutritional state, environmental factors, and every other factor that directly or otherwise influence the organisms energy metabolism. If this equation is to be applied in a population (separate living organisms),then the constants will vary between organisms. Hence, making them variables and not constants for all organisms.
In figure 1, we observe the rate of oxygen consumption in the body as the membrane potential increase. We find out that the oxygen consumption remained approximately the same as the membrane potential increases, but started decreasing sharply at some points. The point where we observed decrease in the oxygen consumption draws to the fact that most of the electrons transferred during the electron transporting are not used to reduce oxygen to water (H2O). This is because electrons will leak as in the case of the formation of reactive oxygen species (ROS). Secondly, the leakage in protein makes it such that it is not all the electrons that are transferred (donated) to oxygen that can be stabilized (that is, lead to the formation of water). Because the electron transport chain is not perfect Physically, one would say that from the point of decrease in the oxygen consumption, that electrons began to leak and reacting oxygen species formation began leading to oxidative stress (fever). We also observed that the rate of oxygen consumption can never be zero though we must note that it can be very small.
NADH is produced in mitochondria during the T.C.A cyclic reaction. In figure 2, the graph of the relationship between the mitochondria NADH production and the membrane potential we observed that the NADH production remain approximately the same as the membrane potential increases but started increasing algebraically at some point with in crease in membrane potential. Physically, it implies that increases in the membrane potential increases the reaction rate of the T.C.A cycle. Secondly, it draws to the fact that most of the Nelectrons transferred during the electron transporting are used to reduce oxygen to water (H2O).
In fig 3, we have electron efflux against membrane potentials, for ,where we observed a similarity with figure 3 above. This is true because the electron that passes through the respiratory chain are contained in the NADH produced during the T.C.A. cycle reaction activities.
Fig 4. Illustrated the energy production as a function of the protomotive force for RAT = 0.38Mm, t = 0.5s, Bo = 10mv, = 160V. It is true that energy production is continuous because both the ATPase and the translocator activities are continuous; but from the graph it is seen at some point that energy production increases asymptotically. This is true because the mitochondria energy production depends on the body energy demand. The point of asymptotic increase reflects equilibrium. A situation where ATPase activities equals the translocator activities.
Fig 5, shows the rate of the adenine nucleotide trans-locates as a function of the ADP/ATP ratio for various membrane potential . we observed that increase in the ratio of ADP/ATP decreases the activities of the adenine nucleotide transporter. This is so because if ATP is not phosphorylated by the ATP synthase little amount of ATP will be available for the translocator to transport out thereby reducing it functionality. This condition may be obtained when the body is feeling cool, therefore, stopping or reducing FIFO activity and favouring heat generations.
Fig6. Shows Energy generation with its dependence on time when Translocator activities is > ATPase activities. The graph reveals that energy production increases as time increases. In this case, the process of energy generation will be spontaneous i.e. the process will be fast. It eventually signals that the body requires instant energy as in the case of exercise.
The graph of Translocator activities is < ATPase activities, if we were to consider will be opposite of graph 6 above. In this case, the process of energy generation will be slowed down by the accumulated ATP in the mitochondria and thereby inhibiting ATP production
Conclusion:
The mitochondria generate Biological energy for the metabolic activities of the body through a process called oxidative phosphorylation. A process which generate energy far more than glycolysis does. Glycolysis initiate the process of energy production but produces little energy. Oxidative phosphorus generate energy by the active passage of electron which are stored in the electron carried NADH through the respiratory chain which is embedded in the inner membrane of the mitochondria. In this work, we have argued in our model that the rate at which energy is produced in the mitochondria is proportional to the sum/difference of the rate of the ATP syntheses reaction rate and the reaction of the adenine nucleotide translocation reaction. It is sum if the reaction rate of ATPpase reaction favours energy production and difference if it does not.
Finally, since it is the long term goal to develop a model that will be possible for an individual to predict the behaviour of the mitochondria under certain conditions, our model simulation in one study focused largely on the response of mitochondria NADH production to increase in the membrane potentials and oxygen consumption rates to increase in membrane potentials . Here, it is seen that increase in membrane potential decreases the oxygen consumption rate and in another sense increases the rate of NADH production and electron efflux. In another study our simulation focused largely on the mitochondria energy production with dependence on the force in the mitochondria which translocate proton from the matrix to the inter membrane space (P.M.F) . The simulation shows that increase in PMF enhances energy production. Lastly, our simulation focused on the response of the adenine necleotide translocator as the ADP/ATP increase.
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