Reaction-Diffusion Processes in Muscle Metabolism Parametric and Sensitivity Analysis Santosh Kumar Dasika
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2010 Reaction-Diffusion Processes in Muscle Metabolism Parametric and Sensitivity Analysis Santosh Kumar Dasika
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COLLEGE OF ENGINEERING
REACTION-DIFFUSION PROCESSES IN MUSCLE METABOLISM -
PARAMETRIC AND SENSITIVITY ANALYSIS
By
SANTOSH KUMAR DASIKA
A Dissertation submitted to the Department of Chemical and Biomedical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Degree Awarded: Fall Semester, 2010
Copyright © 2010 Santosh Kumar Dasika All rights reserved. The members of the committee approve the dissertation of Santosh K. Dasika defended on 20th Oct 2010.
______Dr. Bruce R. Locke Professor Directing Dissertation
______Dr. Bryant Chase University Representative
______Dr. Stephen T. Kinsey Committee Member
______Dr. Teng Ma Committee Member
______Dr. Samuel C. Grant Committee Member
Approved:
______Bruce R. Locke, Chair, Department of Chemical and Biomedical Engineering
______Ching-Jen Chen, Dean, Dean, College of Engineering
The Graduate School has verified and approved the above-named committee members.
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Dedicated to Guruji
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ACKNOWLEDGEMENTS
I am grateful to have this opportunity to acknowledge all those who have assisted, guided and supported me in my studies leading to this thesis. Firstly, I am grateful to Dr. Bruce Locke for giving me this wonderful opportunity to work on this project and for his constant guidance, support and motivation throughout the contribution of my research work. I would also like to thank Dr. Stephen Kinsey for his valuable comments and insights into the biological aspects of the project.
I would also express my sincere thanks to Dr. Bryant Chase, Dr. Teng Ma and Dr.
Samuel Grant for serving as the committee members. I would like to thank Dr. Sachin Shanbhag
, School of Computational Science, Florida State University, for his valuable help in Monte
Carlo simulations, Dr. Mark Sussman, Dept. of Mathematics, Florida State University, for his help in solving the reaction diffusion equations using Finite Difference scheme. Dr. Milen
Kostov, Dept. of Chemical and Biomedical Engineering, Florida State University, for sponsoring a general account at the High Performance Computing, Florida State University, and Mr. Dan
Voss of High Performance Computing, Florida State University for his support in executing jobs at the HPC clusters.
I would also like to thank National Science Foundation grant for financial support.
Last, but not the least, I would like to thank my family and friends, specially my mom
Ms. Vijaya Lakshmi, dad Mr. Prabhakar Rao, sister and brother in law, and my dearest friend
Jaswanth, for their constant support and encouragement throughout my PhD. I would like to take this opportunity to thank my Guruji (whom I consider the almighty) for his blessings.
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TABLE OF CONTENTS
LIST OF FIGURES……………………………………………………………………………. viii
LIST OF TABLES…………………………………………………………………………….. xiii
ABSTRACT…………………………………………………………………………………… xiv
Chapter 1 INTRODUCTION ...... 1
1.1 Introduction ...... 1
1.2. Description of muscle ...... 2
1.3. Muscle physiology ...... 4
1.3.1. Mitochondrial distribution ...... 6
1.3.2. Cell mitochondrial distribution during growth ...... 9
1.4. Objective ...... 10
1.5. Organization of the text ...... 11
Chapter 2 LITERATURE REVIEW ...... 12
2.1 Introduction ...... 12
2.2. Saks group ...... 13
2.3. Kinsey and Locke‟s group ...... 18
2.4. Modeling O2 transport...... 21
Chapter 3 EFFECTS OF VARIOUS PARAMETERS ON INTRA-CELLULAR
DIFFUSION LIMITATION OF METABOLITES ...... 24
v
3.1. Introduction ...... 24
3.2. Mathematical methods ...... 27
3.2.1. Mitochondrial rate law ...... 28
3.2.1.1. Rate law for mitochondrial oxidative phosphorylation ...... 28
3.2.1.2. Reaction diffusion model ...... 29
3.2.1.3. Volume averaging ...... 32
3.2.1.3.1. Boundary conditions ...... 37
3.2.1.4. Effectiveness factor ...... 38
3.3. Solution technique ...... 40
3.4. Results ...... 43
3.4.1. Mitochondrial rate law ...... 43
3.4.2. Concentration Profiles and Effectiveness Factors ...... 46
3.4.2.1. Effects of Km,ATPase and Boundary O2 concentration ...... 51
3.5. Discussion ...... 53
3.5.1. Numerical issues ...... 53
3.5.2. Hill coefficient ...... 56
3.5.3. Comparison of model with experimental data ...... 57
3.6. Conclusions ...... 59
Chapter 4 EFFECTS OF MB AND CK ON INTRA-CELLULAR METABOLITE
DIFFUSION LIMITATION ...... 62
vi
4.1. Introduction ...... 62
4.2. Modeling methods and formulation ...... 67
4.3. Computational Results and Discussion ...... 69
4.3.1. Experimental data comparison ...... 69
4.3.2. Effect of mitochondrial volume fraction and boundary O2 concentration ...... 77
4.4. Conclusions ...... 80
Chapter 5 SENSITIVITY ANALYSIS ...... 82
5.1. Introduction ...... 82
5.2. Mathematical methods ...... 85
5.3. Results and discussion ...... 88
Chapter 6 : CONCLUSIONS AND FUTURE WORK ...... 102
6.1. Conclusions ...... 102
6.2. Future work ...... 103
A. ATP flux vs. O2 for [Pi] = 5 mM ...... 108
References………………………………………………………………………………...... 113
Bibliographical Sketch………………………………………………………………………. 132
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LIST OF FIGURES
CHAPTER 1
Figure 1.1: Mitochondrial distribution in muscle fibers type 1 ...... 7
Figure 1.2: Mitochondrial distribution in muscle fibers type 2 ...... 8
Figure 1.3: Mitochondrial distribution in muscle fibers type 3 ...... 8
Figure 1.4: Mitochondrial distribution in muscle fibers type 4 ...... 9
CHAPTER 2
Figure 2.1: Schematic of Wong model (Wong, 2006) ...... 22
CHAPTER 3
Figure 3.1: Volume averaging ...... 34
Figure 3.2: Steady state concentrations and rate profiles for a case that leads to substantial elevation in Pi and sizable gradients in [O2] and Rmito (L = 50 µM, Vmax,ATPase = 25 mM/min and
0 Km,ATPase = 0.15 mM for εmito = 0.1, and O2 = 35.1 µM. In these plots, X refers to the space, where, X = 0 refers to the center of the cell while X = 1 refers to the periphery of the cell ...... 47
Figure 3.3: Influence of diffusion length, andVmaxATP ase on
0 Km,ATPase = 0.15 mM, kmt = 1100 µ/s and O2 = 35.1 µM ...... 49
Figure 3.4: Concentration profiles for
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Figure 3.5: Influence of diffusion on the interaction between the effectiveness factor,VmaxATPase,
0 and length for εmito = 0.1, Km,ATPase = 0.15 mM and O2 = 35.1 µM (A): < RATPase > vs L and
Vmax,ATPase (B): < RATPase > without diffusion limitations vs. L and Vmax,ATPase (C): vs. L and
Vmax,ATPase (D): vs. L and RATPase ...... 51
0 Figure 3.6: Influence of Km,ATPase and O2 on the interaction of the effectiveness factor, length
0 and Vmax,ATPase for εmito = 0.1. Km,ATPase has a small effect on η, while lower O2 leads to a large
0 0 reduction in the surface with a high O2 . (A): vs L and RATPase, Km,ATPase = 0.15 mM, O2 =
0 35.1 µM, (B): vs L and RATPase, Km,ATPase = 5 mM, O2 = 35.1 µM, (C): vs L and RATPase,
0 0 Km,ATPase = 0.15 mM, O2 = 7.85 µM (D): vs L and RATPase, Km,ATPase = 5 mM, O2 = 7.85 µM
...... 52
Figure 3.7: Plots based on ATPase rate; the results were inaccurate although the MATLAB code was free of errors, A. η vs. Vmax and L, B. η vs. ATPase rate and L, C. ATPase with finite diffusion rate vs. Vmax and L, and D. ATPase rate without diffusion vs. Vmax and L ...... 54
Figure 3.8: Solution is the point where ATPase rate and mitochondrial rate intersect; marginal error in [ATP] would have a significant error in the ATPase rate...... 56
Figure 3.9: Effect of Hill's coefficient; no visible effect was observed by varying the Hill‟s coefficient ...... 56
Figure 3.10: Effect of
0.430.
ix mitochondria assuming 22.4 l O2 per mole of O2, an ATP/O ratio of 2.5, and an intracellular water content that was 70% of wet mass. In cases where direct measurements were unavailable,
3 O2 consumption in ml/min/cm of mitochondrial volume 3 for mammals (39), and 1 for fishes (
Burpee, et al, 2010)...... 59
CHAPTER 4
Figure 4.1: Effect of
Figure 4.2: [ATP] vs. L for the case without Mb & CK rates(blue +), with Mb (broken red lines), with CK (black +), and with Mb & CK rates (solid yellow line) for εmito = 0.25, L = 25
0 µm, Km,ATPase = 0.15 mM, O2 = 35.1 µM, and (A) Vmax,ATPase = 25 mM/min; [ATP] is higher in presence of CK due to facilitated diffusion role of CK, the facilitated diffusion effect of CK is more predominant for higher ATP demand...... 72
Figure 4.3: [O2] vs. L for the case without Mb & CK rates (blue +), with Mb (broken red lines), with CK (black +), and with Mb & CK rates (solid yellow line) for εmito = 0.25, L = 25 µm,
0 Km,ATPase = 0.15 mM, O2 = 35.1 µM, (A) Vmax,ATPase = 25 mM/min (B) Vmax,ATPase = 75 mM/min; Mb does not have significant effect I enhancing O2...... 72
Figure 4.4: [RATPase] vs. L for the case without Mb & CK rates (lower blue triangle),with Mb
(broken red lines), with CK (upper black triangle), and with Mb & CK rates (solid yellow line)
0 for εmito = 0.25, L = 25 µm, Km,ATPase = 0.15 mM, O2 = 35.1 µM, (A) Vmax,ATPase = 25 mM/min,
(B) Vmax,ATPase = 75 mM/min; significant drop in RATPase can be observed for the case without the
CK reaction for Vmax,ATPase = 75 mM/min where as the RATPase does not drop due to the facilitated diffusion due to CK rate...... 73
x
Figure 4.5:
With Mb, (C) With CK, and (D) With Mb & CK; the
Vmax,ATPase 5 mM/min when CK reaction is absent implying CK facilitates the diffusion of ATP.
...... 74
Figure 4.6:
With CK, (C) With Mb, and (D) With Mb & CK; the
Mb and CK at low Vmax,ATPase. Mb and CK have no significant effect on
...... 75
Figure 4.7: ɳ vs. L and for
Without Mb and CK (surface) and with CK (mesh), (C) Without Mb and CK (surface) and with
Mb and CK (mesh); sharper drop in ɳ in the presence of CK reaction can be observed. However, the cells would operate under reaction control regime with the presence of CK reaction, where otherwise the cells would operate under diffusion control regime...... 76
Figure 4.8: Ratio of (A) ɳ with Mb vs. without Mb and CK, (B) ɳ with CK vs. without Mb and
CK, (C) ɳ with Mb and CK vs. without Mb and CK, vs. L and
ɳ ratio can be observed in the presence of CK, and the extreme (large L of 150 µm and Vmax,ATPase of 15 mM/min), where the cell operates under diffusion limitation. The effect of Mb reaction is minimal compared to CK reaction, and the combined effect is the sum of effects individual rates.
...... 77
Figure 4.9: ratio of ɳ vs. L and for
0 0 Mb and CK for (A) εmito = 0.1 and O2 = 35.1 µM, (B) εmito = 0.1 and O2 = 7.85 µM, (C) εmito =
0 0 0.45 and O2 = 35.1 µM, (D) εmito = 0.45 and O2 = 7.85 µM; the effect of Mb and CK decreases with increase in εmito and boundary [O2]...... 79
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Figure 4.10: Ratio of ɳ vs. L and for
0 without Mb and CK for εmito = 0.01 and O2 = 35.1 µM; Mb and CK‟s facilitated diffusion role is predominant at low εmito...... 80
CHAPTER 5
Figure 5.1: Schematic of various conditions effecting diffusion/reaction control ...... 84
Figure 5.2: (A)
0 Vmax,ATPase, O2 and εmito are the most sensitive parameters. The solid red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B represent the SC
0 for
Figure 5.3: (A)
0 with respect to all parameters, vs. Vmax,ATPase; Vmax,ATPase, O2 and εmito are the most sensitive parameters. The cell is limited by diffusion for Vmax,ATPase greater than 25 mM/min. The solid red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig.
0 B represent the SC for
0 Figure 5.4: (A) η (B) SC with respect to all parameters, vs. Vmax,ATPase; Vmax,ATPase, O2 , and εmito control the diffusion limitation in the cell. The solid red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B represent the SC for
0 respect to Vmax,ATPase, Km,ATPase, KmtO2, O2 , L, and εmito respectively...... 92
Figure 5.5: plots showing η vs. other parameters – (A) Kmt,O2, (B) Km,ATPase, (C) εmito, and (D)
0 O2 , which led to the conclusion that the cell operates under reaction control regime as long as
0 Vmax,ATPase < 25mM/min, O2 ≥ 35.1 µM, and εmito ≥ 0.1...... 93
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Figure 5.6: η vs. L for (A) Vmax,ATPase = 5 mM/min and (B) Vmax,ATPase = 25 mM/min; η drops steeply with increase in Vmax,ATPase and L; however afore mentioned conditions are applicable for most cells...... 94
Figure 5.7: η vs. εmito and Vmax,ATPase for (A) O2 = 7.85 µM, (B) O2 = 15 µM, (C) O2 = 25 µM, and (D) O2 = 35.1 µM; the range of values for parameters under which the cell is not limited by diffusion increases with increase in boundary O2 concentration. The η in the region in red, orange, yellow, green, cyan ranges from 1 – 0.9, 0.9 – 0.8, 0.8 – 0.7, 0.7 – 0.6, 0.6 – 0.5 respectively, while the η in the region in blue is less than 0.5...... 95
0 Figure 5.8: SC for
(C) εmito, and (D) L; diffusion of O2 is the limiting step ...... 98
Figure 5.9: SC for
0 O2 , (C) εmito, and (D) L; diffusion of O2 is the limiting step...... 99
APPENDIX A
Figure A1: Plot of ATP flux vs. A) ADP for [Pi] = 5 mM and [O2] = 50 µM and B) O2 for
[ADP] = 0.13 mM and [Pi] = 5 mM…………………………………………………….……... 107
Figure A2: ATP flux vs. O2 as substrate varying Pi for [ADP] = 0.13 mM, from simulated data and from expression (Eq. 5)…………………………………………………………………… 107
APPENDIX B
Figure B1: (A)
0 Figure B2: (A)
Figure B3: (A)
Figure B4: (A)
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LIST OF TABLES
CHAPTER 5
Table 5.1:
Table 5.2: SC for all concentrations and rates with respect to diffusion constants for all parameters; transport of O2 from cell periphery to the mitochondria may be the limiting step ... 97
Table 5.3: Concentrations and rates, and SC with respect to all parameters with all parameters
0 with Mb and CK for hypothetical case of O2 = 7.85 µM, εmito = 0.01, L = 300 µm, and
Vmax,ATPase = 100 mM/min, where Mb and CK are expected to have significant effect, along with concentrations and rates, and SC with respect to all parameters with all parameters without Mb
0 and CK for hypothetical case of O2 = 35.1 µM, εmito = 0.1, L = 50 µm, and Vmax,ATPase = 25
0 mM/min; Vmax,ATPase, O2 , and εmito are the most sensitive parameters even in the presence of Mb and CK, highlighted in bold...... 100
APPENDIX A
Table A1: Normalized sensitivity coefficients………………………………………………... 105
Table A2: Values for the fitting constants…………………………………………………….. 106
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ABSTRACT
Although diffusion is an important phenomenon for intra-cellular metabolite transfer, it is seldom considered while modeling cellular energetics. Weisz (1973) hypothesized that most biological and biochemical systems would most efficiently operate at the edge of the effectiveness factor curve where (1) they are not diffusion limited, but (2) further increases of reaction rate and capacity would lead to diffusion limitations. Hence it is of importance to study the factors that may affect intra-cellular metabolite diffusion limitations. The present study focuses on analysis of the influence of various parameters on intra-cellular metabolite diffusion limitations. In order to do so, a simplified mitochondrial rate law was developed as a function of
ADP, O2, and Pi. A reaction-diffusion model was developed to include ADP, ATP, O2, creatine kinase (CK), phosphor-creatine (PCr), myoglobin (Mb), Mb-O2 complex (MbO2), and Pi as metabolites. The volume averaging technique was performed to account for contributions from all the mitochondria in the cell and the effectiveness factor (η) was computed. Simulations showed that muscle fibers are not limited by diffusion up to certain combinations of ATP turnover rate and fiber size, after which the fibers start to be limited by diffusion for any further increase in ATP turnover rate and fiber size. Also, as the mitochondrial volume fraction increases, the cell can sustain higher ATP turnover rates without diffusion limitation.
Comparison of model analysis with experimental data revealed that none of the fibers were strongly limited by diffusion. However, while some fibers were near substantial diffusion limitation, most were well within the domain of reaction-control of aerobic metabolic rate. This may constitute a safety factor in muscle that provides a level of protection from diffusion constraints under conditions such as hypoxia.
xv
Myoglobin (Mb) and creatine kinase (CK) have been proposed to facilitate the diffusion of O2 and ADP/ATP, respectively, although the conditions where Mb functions as a facilitator are subject to debate. Nevertheless, it is important to establish the significance and to quantify the effects of Mb and Ck in enhancing metabolite diffusion of metabolites when the cell is not limited by diffusion. The aim of the present work was to determine the conditions (parameter space) where the Mb and CK can affect the intra-cellular metabolite diffusion limitations. To do so the reaction-diffusion model was modified for three cases including a) the effects of Mb alone, 2) CK alone, and 3) without Mb and CK, and η was computed for all cases. The effect of
Mb on enhancing intra-cellular diffusion was found to be smaller compared to that of CK at moderate cell size and ATP turnover rates. However, both Mb and CK can have significant effects in enhancing diffusion for larger cells with high ATP demand, and also for cells with low mitochondrial volume fractions (εmito) and low O2 concentrations. Comparison of the case with
Mb and CK and the case without Mb and CK with experimental data revealed that Mb and CK do not significantly affect the η except in the regions very close to the transition to diffusion limitations which are just beyond the range of biological observations.
Sensitivity analysis was performed to identify the key parameters that control intra- cellular metabolite diffusion limitations. It was observed that when the cell is reaction-controlled, the cell is limited by ATP turnover rate, represented by Vmax,ATPase, while when the cell is limited by diffusion, oxygen supply and transport limits ATP production by the mitochondria. The diffusion constant for O2 was the most sensitive of the diffusion constants, implying that the transport of O2 to mitochondria is the limiting step when the cell is limited by diffusion and hence the cell may be prone to hypoxia when the cell is limited by diffusion. To avoid hypoxia, the cell may have adapted itself in such a way as to avoid diffusion limitations, and this may be
xvi the reason most cells are reaction controlled while some operate at the edge of diffusion limitation.
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CHAPTER 1
INTRODUCTION
1.1 Introduction
Skeletal muscle is unique in its capability to change its energy expenditure. ATP is the universal currency of energy in muscle and transition from rest to exercise increases energy demand more than 100-fold and is a major challenge to cellular energetics (Sahlin et al, 1998).
Despite large fluctuations in energy demand muscle ATP remains practically constant and demonstrates a remarkable precision of the system in adjusting the rate of the ATP generating processes to the demand. ATP is efficiently synthesized by aerobic metabolism of oxidative phosphorylation by mitochondria and diffuses to the sites of ATP consumption. Limitation in energy supply is a classical hypothesis of muscle fatigue (Sahlin et al, 1998). Intra-cellular metabolite diffusion limitation may limit the ATP production by mitochondria, and may ultimately lead to muscle fatigue. The following text briefly describes the muscle physiology.
Muscles are comprised of tissues and tissues are made of cells. Cells are the basic units of life. Though measured in micro-meters, cells are sophisticated chemical reactors. The human body mainly comprises of ten organ systems that perform physiological functions such as motion and respiration, and these organs are comprised of repeated functional units such as tissues
(Palsson, 2004). It has been observed that of all the organs in the human body the muscle tissues are highly organized (Lieber and Friden, 2000). This organized structure of muscle fibers makes them ideal models to study the relationship between the cell structure and metabolism taking
1
place in the cell, hence this has attracted the attention of many researchers. Muscles and bones comprise what is known as the musculoskeletal system. All the muscles in the human body show biochemical specialization which allows them to perform their particular physiological functions.
The major functions of muscles include body movements such as walking, breathing; providing support and posture for the body; and producing heat so as to maintain body temperature The following text describes the structure and components of a muscle.
1.2. Description of muscle
The muscles are surrounded by a sheet of fibrous connective tissue known as fascia.
Superficial fascia separates muscle from skin and functions to provide a pathway for nerves and blood vessels, stores fat, insulates, and protects muscles from trauma while deep fascia lines the body wall and limbs and holds muscles with similar functions together and allows free movement of muscles, carries nerves, blood vessels, and lymph vessels, and fills spaces between muscles. Tendons and aponeuroses are extensions of connective tissue beyond muscle cells that attach muscle to bone or other muscle. Tendon (synovial) sheaths enclose certain tendons and allow them to slide back and forth more easily. An aponeurosis is a tendon that extends as a broad, flat layer. Each muscle consists of fascicles or group of cells known as muscle fibers or myofibers mainly due to the shape of these cells which are thread-like with range of diameter between 5 – 100 µm and length between 1 mm to several centimeters (Sciote and Morris, 2000).
The human body consists of more than 600 different muscles and they account for approximately 40 percent of body weight (Everett, 1998). Essentially, the muscle tissues can be classified based on the morphological and functional characteristics into three categories as skeletal, smooth, and cardiac muscles (Unqueira and Carneiro, 2005). Smooth muscles are found
2
near the walls of hollow organs such as arteries and veins. They consist of cells with single centrally located nucleus and the cells are spindle-shaped. These muscles are involuntary in nature meaning they are not under conscious control and are controlled only by the nervous system (Rizzo, 2001). The skeletal muscles are attached to the bones and are responsible for the movement of the body. They consist of long thin and multi-nuclei cells that are striated meaning alternating light bands of the thin protein filaments of actin and dark bands of the thick protein filaments of myosin are visible when viewed under the microscope. They are voluntary muscles and are under the control of the central nervous system (Rizzo, 2001). Cardiac muscles can be found only in the heart. They are striated as skeletal muscles and are uni-nucleated as smooth muscles. They are voluntary in nature and are controlled by the central nervous system. The cells of cardiac muscle are cylindrical in shape with branches that connect to other cardiac cells and are much shorter than both skeletal or smooth muscle cells. These muscles cause contraction or beating of the heart and responsible for pumping blood through the body (Rizzo, 2001).
Organelles in the cells are structures within the cells such as mitochondria, nucleus.
Eukaryote cells mainly consist of nucleus, cytosol, cytoplasm, golgi apparatus, and mitochondria
(Farabee, 2006). The description of these components is well documented and can be found elsewhere (Farabee, 2006). As mentioned above, skeletal muscle cells, owing to their thread like shape, are known as myofibers or muscle fibers. They have components similar to other cells. since the muscle cells were studied extensively, the components of the muscle cells have different names such as the muscle fibre cytoplasm known as sacroplasm. One of the major differences between the muscle fiber and other eukaryote cells arises from the fact that the muscle fiber sacroplasm is filled with rod-like substance known as myofibril, which runs in parallel to the longitudnal axis of the muscle fiber. The myofibrils consist of two types of
3
filaments, also known as myofilaments: thin filaments of actin, which are about 7 nm in diameter; and thick filaments of myosin about 15 nm in diameter (Cooper, 2000). The muscle fibers also consist of sacromeres, and are made of the myofibrils. The muscle contraction, resulting in the change of muscle length and transforming the tension associated with the change in length into energy, is thought to be the result of interaction between the two types of myofilaments in the sacromeres according to the sliding filament theory (Cooper, 2000). In the muscle contraction phase, the sacromere consumes the chemical energy produced by converting
ATP to ADP, and this energy is converted into different forms by the muscle contraction such as heat.
1.3. Muscle physiology
It is well known that the transition from rest to exercise in the skeletal and cardiac muscles is accompanied by the increase in rate of cellular ATP hydrolysis. Also, bio-energetic pathways are regulated such that ATP is synthesized at rates that match hydrolysis rates (Suarez,
2003). Failure for energy supply or ATP synthesis to meet the energy demand or ATP hydrolysis may lead to muscle fatigue (Sahlin et al, 1998). Hence there is a need for efficient supply of ATP to meet the need of ATP for hydrolysis. It is well known that the mitochondria is an efficient source for ATP, producing ATP through the aerobic process of oxidative phosphorylation, in the process, consuming oxygen (O2) delivered to the cell through the capillaries. This makes O2 one of the important elements in human life. Muscles that have evolved to operate at high frequencies for extended periods require high capacities for aerobic ATP synthesis and, therefore, high volume fraction of mitochondria. In the flight muscles of hummingbirds and insects, mitochondrial volume densities are as high as 35% and 45%, respectively (Suarez, 2003).
4
Although oxygen is essential for human existance, prolonged exposure to O2 of concentration greater than 21% is harmful to the human body. Such an exposure of oxygen concentration greater than 21% at 1 atm pressure for as little as 6 hrs would result in chest soreness, cough, and sore throat and prolonged exposure leads to lung damage (Halliwell and
Cross, 1994). Also, excess concentration of O2 leads to the production of Reactive Oxygen
Species (ROS). O2 is partially reduced in the mitochondrial respiratory chain to form a superoxide, which is a radical, also known as ROS. They are comprised mainly of a group of
. highly reactive species such as hydrogen peroxide (H2O2), hydroxyl radical (OH ), and
. superoxide anion (O2 ) (Rho et al, 2005). The majority of ROS are thought to be produced in the mitochondria (Lenaz, 2001) and are derived from metabolism of molecular O2 (Halliwell, 1999).
Cells possess ROS scavangers or antioxidants such as catalase, glutathione peroxidase, and superoxide dismutase (Kim et al, 2001) to protect against ROS. The imbalance between the antioxidant defensive forces and the ROS is termed oxidative stress (Aruoma, 1998). Though
ROS assist in programmed cell death or apoptosis (Suzuki et al, 1997; Tan et al, 1998), excessive concentrations of ROS have delirious effects such as DNA damage and enhanced aging of cells (Bennett, 2001; Cooke et al, 2003). ROS have been identified as root cause for some of the diseases such as Alzheimer‟s disease (Christen, 2000; Butterfield and Lauderback,
2002; Markesbery, 1997), and arthritis (Hadjigogos, 2003; Da Sylva et al, 2005; Loeser, 2006).
Although the skeletal muscle cells are more exposed to oxidative stress, they are thought to be well equipped with anti-oxidants to withstand oxidative stress (Clanton et al, 1999). Also, as with the effect of higher concentrations of O2, the effects of lower concentrations of O2 or hypoxia are drastic. The O2 supply to the cells through hemoglobin in the blood is hampered by reactants like carbon monoxide (CO), which reacts with hemoglobin and hinders the transport of
5
O2, and cyanides which react with O2 resulting in lower supply of O2 to the cells. The reduction of O2 supply to the cells hampers the production of ATP through oxidative phosphorylation, which is an efficient way of ATP production compared to the anerobic process of glycolysis
(Webster et al, 1999). This results in improper cell death and ultimately results in tumors and cancer being developed (Ohkuwa et al, 2003). The central nervous system is more susceptable to hypoxia compared to other body organs (Ohkuwa et al, 2003).
1.3.1. Mitochondrial distribution
Human cells contain up to 5000 mitochondria (Perkins and Frey, 2000). The arrangement of mitochondria varies with the type of muscle. Since ATP is primarily produced in mitochondria and transported to the myofibrils of the cell, the distribution of mitochondria is an important aspect in cellular structure. Hence the distribution of mitochondria in the cell has attracted the attention of many researchers. The transport of metabolites from and to the mitochondria depends on both the mitochondrial volume fraction as well as mitochondrial spatial distribution within fibers (Kayar et al, 1988). The number of mitochondria varies depending on the type of muscle. Cardiac and slow twitch skeletal muscles have high mitochondrial content, fast glycolytic skeletal muscle fibers have low mitochondrial content, but fast oxidative fibers have intermediate mitochondrial content (Milner et al, 2000). It was concluded from the experiments conducted on the cardiac and skeletal muscle fibers of adult rats that in the cardiac muscle fibers the mitochondria are distributed everywhere within the cell and they occupy up to
35% of the volume of the cell whereas in the skeletal muscle fibers, there are fewer mitochondria that are arranged in pairs and found close to the myofibrils (Bossen et al, 1978; Saks et al, 2004;
Vendelin et al, 2005). It was concluded from the experiments performed on horse skeletal muscle
6
tissue that the mitochondrial volume density was maximum near the capillaries while there was a sharp decline in the mitochondrial volume density at the center (Kayar et al, 1988). One of the possible reasons for this conclusion accounts for the fact that the O2 is transported to the mitochondria through the capillaries. Studies on the skeletal muscles of adult blue crabs reported that mitochondria are found at the periphery of the fiber (Boyle et al, 2003).
Mitochondria are typically uniformly distributed in small white muscle fibers (Boyle et al, 2003). In the present study, the small muscle fibers are assumed to be spherical in shape with
100 µm diameter. Fig. 1.1 shows a cell in which mitochondria are homogeneously distributed.
For simplicity, they are classified as type 1 cell models in the present study. We focus on these cells in the present study.
Cell
Mitochondria
Figure 1.1: Mitochondrial distribution in muscle fibers type 1
In large white muscle fibers mitochondria are distributed at the periphery of the cell
(Boyle et al, 2003). In the present study, the large muscle fibers are assumed to be spherical in shape with 600 µm diameter. Fig. 1.2 shows a cell in which mitochondria are predominantly found at the periphery of the cell. They are classified as type 2 cell models in the present study.
7
Cell
Mitochondria
Figure 1.2: Mitochondrial distribution in muscle fibers type 2
Mitochondria are distributed closely to the capillaries in the skeletal muscle fibers of horse (Kayar et al, 1988). Fig. 1.3 shows a cell in which mitochondria are predominantly found at the periphery of the cell. In the present study, the large muscle fibers are assumed to be spherical in shape with 60 µm diameter (Kayar et al, 1988). They are classified as type 3 cell models in the present study.
Capillary
Cell
Mitochondria
Figure 1.3: Mitochondrial distribution in muscle fibers type 3
In the dark muscle fibers in which the mitochondrial sizes are comparable to that of the cells the mitochondria are clustered at the center of the cell. In the present study, the dark muscle fibers are assumed to be spherical in shape with 36 µm diameter. Fig. 1.4 shows a cell in which
8
mitochondria are predominantly found at the center of the cell. They are classified as type 4 cell models in the present study.
Cell
Mitochondria
Figure 1.4: Mitochondrial distribution in muscle fibers type 4
1.3.2. Cell mitochondrial distribution during growth
As mentioned previously, muscle fibers are cylindrical in shape. They range between 10
– 100 µm along the radial axis (Johnson et al, 2004). Dimensions of the cells exceeding this range are thought to be comprimising the ATP synthesis through the aerobic metabolism of oxidative phosphorylation, which relies on the O2 flux across the membrane (Kim et al, 1998). It has been reported that the muscle fibers of adult blue crabs exceed this range and the dimensions of white muscle fibers of blue crabs often exceed 500 µm (Johnson et al, 2004). An interesting feature of these white muscle fibers is that the distribution of mitochondria varies as a function of fiber size. In juveniles, mitochondria are uniformly distributed throughout the fibers and the population is equally divided between subsarcolemmal and intermyofibrillar fractions. However, in white fibers from adults, mitochondria are largely subsarcolemmal (Boyle et al, 2003). Thus, in large fibers, there is a cylinder of oxidative potential around the periphery of the cell whereas the inner core of the fiber has limited aerobic capacity. This developmental redistribution of mitochondria dramatically increases intracellular diffusion distances between mitochondria in
9
large fibers, more so than would be expected from increases in fiber diameter alone. Since contraction in these fibers is anaerobically powered and relies on endogenous fuels, large cell size should not impact this process. However, the small surface area to volume ratios (SA:V) and intracellular diffusion limitations associated with large fiber size would be expected to affect aerobic metabolism (Johnson, 2004).
1.4. Objective
The aerobic metabolism of oxidative phosphorylation plays a significant role in dictating the cellular dimension (Kim et al, 1998). Hence a clear understanding of the effects of intracellular metabolite diffusion on the aerobic metabolism would help in understanding the factors that dictate fiber dimension. Hence the aim of the present study is to undestand the effects of fiber size and other factors such as mitochondrial volume fraction, ATP demand, and boundary O2 concentration on the intra-cellular diffusion limitation of metabolites. The present study focusses on skeletal muscles and does not deal with cells during growth. Rather, the present study analyzes the role of metabolite diffusion in various sizes of muscle fibers namely large muscle fibers, which may exceed >500 µm in diameter, and small muscle fibers which fall in the range of 10-100 µm. The present study focusses on the type 1 fibers, in which mitochondria are homogeneously distributed.
As ATP is efficiently produced in mitochondria using the aerobic metabolism of oxidative phosphorylation, where O2 is consumed, the aim of the present work is to incorporate
O2 transport in the reaction diffusion model. In previous work the role of inter-mitochondrial diffusion of high energy phosphate metabolites was studied with relatively simple ADP- dependent Michaelis-Menton expressions for ATP production at the mitochondria (Locke and
10
Kinsey, 2008) and the role of O2 transport was assessed utilizing empirical and highly simplified expressions for mitochondrial production of ATP as functions of ADP, O2, and inorganic phosphate (Pi) (Hardy et al, 2009; Jimenez et al, 2008). The present work focuses on utilizing a more detailed model of oxidative phosphorylation in the mitochondria within the framework of a reaction-diffusion analysis of intracellular high energy phosphate metabolism in order to assess the role of diffusion in constraining rates of aerobic ATP turnover. Volume averaging (Whitaker,
1998) technique will be employed to account for contribution from all mitochondria. As creatine kinase (CK) and myoglobin (Mb) are known to facilitate diffusion of ATP/ADP and O2, the present study aims at understanding the effects of Mb and CK on the intra cellular metabolite diffusion limitations. Finally, the parameters influencing diffusion limitations will be identified by performing sensitivity analysis (Varma et al, 1999). By identifying key parameters, one can obtain a valuable insight into the constraints on the cells needed to avoid being limited by diffusion.
1.5. Organization of the text
The present text is arranged as follows. Chapter 2 deals with the literature review of the reaction diffusion models available for muscle metabolic function. Chapter 3 deals with the effects of fiber size and other parameters on the intra-cellular metabolite diffusion limitation,
Chapter 4 deals with the roles of creatine kinase (CK) and myoglobin (Mb) on the intra-cellular metabolite diffusion limitations while chapter 5 deals with sensitivity analysis to determine the important parameter(s) that may have significant effect on the intra-cellular metabolite diffusion limitation and defining the limits in parameter space where reaction control transitions into diffusion control. The conclusions of the present work are presented in Chapter 6.
11
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
The crustaceans, which include many sea animals such as crabs, lobsters, and fish, have muscle fibers that invaginate to create subdivisions such that no portion of the contractile proteins are more than 20–50 µm from the membrane that is continuous with the sarcolemma
(Selverston, 1967). These invaginations increase the surface area of the fiber and function to permit all parts of the muscle to contract at the same time, a function analogous to the mammalian muscle (Boyle et al, 2003). Hence, studying the factors underlying the growth of these crustaceans would assist in understanding the factors that influence the cell structure during growth of human muscle fibers. It has been observed that most muscle cells have diameters within the range of 10 µm – 100 µm (Russell et al, 2000). Experiments performed on the blue crabs showed that while the muscle fiber diameter in juvenile crabs measures <60 µm, those for the adult crabs measure >600 µm (Boyle et al, 2003). One of the reasons for this behavior can be attributed to the fact that the diffusion distance between mitochondria increases as the fiber size increases, and the diffusion coefficients decrease 2- fold from small muscle fibers to large muscle fibers (Boyle et al, 2003).
Performing experiments in order to study the conditions that govern the fiber size can be both time consuming and expensive. An alternative approach to aquiring knowledge is by developing mathematical models based on the reactions and diffusion of the substrates within the cell. An addded benefit of modeling is the testing some of conditions which cannot be tested
12
experimtentally. This gives a valuable insight into understanding the system. A mathematical model should be simple enough so it would be easier to solve and analyze the model, yet at the same time be adequately described to mimic and account for the physics of the problem. There have been many research groups which have been actively performing experiments on various aspects of the cell while some groups have focussed mainly on developing mathematical models.
The Saks group and Drs. Kinsey and Locke‟s group are some of the groups that are actively involved in developing reaction diffusion models for muscle fibers. The Saks group has focussed mainly on the cardiac cells and Drs. Kinsey and Locke‟s group mainly focussed on developing models for skeletal muscle cells in crustaceans. In the following text the step by step development of the models by these groups is presented.
2.2. Saks group
The Saks group has been one of the active groups in developing 1-D and 2-D reaction diffusion models to describe cellular metabolism in muscle. These models were based on the compartmentalization of the cells. In other words, the metabolites react within these compartments and then diffuse to other compartments within the cell. These models are analyzed in detail in the following text.
The initial models by the Saks group mainly concentrated on estimating the Creatine
Kinase (CK) enzyme activity in the cardiac cell. In their initial papers of the series (Aliev and
Saks, 1993, 1994) they considered Adenosine Di-Phosphate (ADP), Adenosine Tri-Phosphate
(ATP), Creatine (Cr), and Phospho-Creatine (PCr) as the substrates in their model. They hypothesised that a functional coupling exists between mitochondrial Adenine Nucleotide
Translocase (ANTase) enzyme, located in the Inter-Membrane space of mitochondria, and CK.
13
The ANT reaction is responsible for transfer of ATP and ADP from and to the mitochondrial matrix. They hypothesised that the ATP is transfered by the ANTase to the active site where Cr reacts forming PCr and ADP ATP Cr ADP PCr and this ADP is transfered back to the matrix by the ANTase. They assumed a rapid equilibrium of CK reaction and that the distribution of the CK enzyme in free state and in complex form can be expressed using the probabilites of adenine nucleotide fluxes from and to mitochondria (Aliev and Saks, 1993, 1994).
Hence their model was based on the probability distribution of the enzymes along with the enzyme kinetics. They considered three compartments within the cell: the mitochondrial matrix, mitochondrial IM space, and the mitochondrial outer membrane. It was assumed that the oxidative phosphorylation, which takes place in the mitochondrial matrix, provides stable ADP and ATP concentrations. The other significant assumptions include no diffusion barriers for ADP and ATP from and to the IM space and the outer membrane space, and close proximity between
CK and ANT. A Pascal progam was coded to solve for the various kinetic constants to quantitatively express the coupling of oxidative phosphorylation and CK reactions. One of the major drawback in the model was the assumption of no diffusion barrier for ADP and ATP between IM space and outer membrane of the cell as the existance many diffusion barriers is well documented.
In their extension to a 1-D reaction-diffusion model of cells, they introduced the concept of compartmentalized energy transfer within the cell and aimed at studying quantitatively the compartmentalized energy metabolism using the 1-D reaction diffusion model of the cardiac cells (Saks and Aliev, 1996; Aliev and Saks, 1997). The model (Saks and Aliev, 1996) mainly consists of the myofibril region wherein the substrates diffuse and reactions in the bulk take place. In this model they included inorganic phosphate (Pi) as a substrate along with ADP, ATP,
14
Cr, and PCr. One of the significant developments in this model was the inclusion of permeability of mitochondrial outer membrane to ANTases. They assumed the cell to comprise of three compartments: the mitochondrial matrix, the mitochondrial IM space, and myofibrils. One of the important assumptions in the model is the rapid equilibrium reaction of ANT. According to this assumption, the ATP produced by the ATP synthase in the mitochondrial matrix is exported to the IM space by the ANTase. Also, the myofibril was assumed to be symmetric in order to consider symmetric boundary condition for myofibrils. Using the probability approach, they expressed the net rate of coupling of ATP in mitochondrial matrix by the ANTase to the CK. In this model {(Saks and Aliev, 1996; Aliev and Saks, 1997)}, they non-dimensionalized the system of equations and solved using finite difference approach and Runge-Kutta iteration scheme for the substrate concentrations as a function of time and space. One of the interesting aspects of the modeling is the assumption that the substrates react within the respective compartments and diffuse within the compartments with the aid of ANTase-CK coupling. This model was used to compare the experimental data obtained for determination of the energy fluxes in the isolated hearts of transgenic mice with knocked out creatine kinase isoenzymes
(Aliev et al, 1998).
To further analyze the factors that regulate the oxidative phosphorylation in the cardiac cells, the Saks group extended the 1-D model developed previously (Aliev and Saks, 1993,1994), which consisted only of transverse dimension, by including the longitudinal direction (Vendelin et al, 2000). The concept of Intra-Cellular Energy Units (ICEU) was introduced (Saks et al,
2001) according to which it was hypothesized that the muscle fiber subcellular structures such as mitochondria and sacromeres are organized as discrete functional complexes, termed as ICEU. In this model (Saks et al, 2001), the three compartments considered were the same as the 1-D
15
models, namely: the mitochondrial matrix, the mitochondrial IM space, and myofibrils. However the significant difference in this model is the consideration of myofibrils as a 2-D rectangular system of transverse length of 1.2 µm and longitudinal length of 1.0 µm. In addition to ATP,
ADP, Cr, PCr, and Pi, AMP is included as a metabolite in the model. Along with the CK reaction, they included the Argenine Kinase (AK) reaction, which takes place in the mitochondrial IM space. The spatial non-homogenious nature of myofibrillar CK, and ATPase was accounted for by including discontinous functions in the model for ATPase and CK. In addition, the Korzeniewski‟s model for isolated mitochondria (Korzeniewski, 1998) was used to describe the mitochondrial oxidative phosphorylation and the mitochondrial martix was considered as a zero-ordered system. The mitochondrial IM space is considered as a 1-D system where in the metabolites are transported from the mitochondrial matrix to the myofibrils. A finite element approach was employed to solve for the 2-D continuity equations for the various substrates.
This model was modified later (Saks et al, 2003). In this modified model, the cardiac cell was considered to be cylindrical in shape with a diameter of 20 µm. The mitochondria of diameter 1µm were considered to be distributed randomly such that they occupy one-fourth of the volume of the cell. This model consisted of the compartments: the mitochondrial matrix, and mitochondrial IM space, which are distributed randomly within the cell and are considered in the boundary conditions, and the cytosol, which is considered as the continuous medium. In this medium the substrates diffuse and the CK reaction, the AK reaction, and the hydrolysis of ATP take place. The fluxes for substrates between cytoplasm and mitochondrion are expressed as the product of a permeability factor times the difference between the substrate concentration in the mitochondrial IM space and the concentration in the bulk. These fluxes were employed as the
16
boundary conditions in the model. The constants in the model were considered from the experimental data available. The complete system of partial differential equations was solved using finite element approach.
Summarizing the contribution of the Saks group to the mathematical modeling of the cardiac muscle cells, they used the constants obtained from the experimental data. They compartmentalized the cell mainly into three compartments and they hypothesized that the functional coupling exists between CK and ANTase, which regulates the mitochondrial oxidative phosphorylation. One of the important assumptions in their modeling is the non-equilibrium of the CK reaction in the cell. The consequence of this assumption is that their models could not replicate the experimental data when the cells are not in resting state. Another important assumption in their models is the rapid-equilibrium of ANT reaction. According to this assumption, the amount of ATP synthesized by the ATPase is transported rapidly to the IM space.
Though their models could mimic the actual cell kinetics by using appropriate choice of parameters, one of the difficulties in employing their model is the number of variables involved, which makes it computationally intensive to model and employ for our purposes of analyzing the effects of diffusion of substrates on the cell growth. One of the reasons for their model being computationally intensive is due to the fact that they solve all the reactions within the mitochondria simultaneously. A model which considers mitochondria as a boundary, instead of using mitochondria as a part of model and solving for the mitochondrial oxidative phosphorylation equations, is desired as it reduces the computational burden. Also, this model
(Vendelin et al, 2000) does not take into account the mitochondrial distribution in the cell.
17
2.3. Kinsey and Locke’s group
Drs. Kinsey and Locke‟s group is another group actively involved in developing reaction diffusion model for skeletal muscle cells. The Hubley‟s model (Hubley et al, 1997) was the initial model of the series. The 1-D reaction diffusion model was employed to study the effects of temperature on the intracellular diffusion and effects on the mitochondrial density. This model mainly consisted of ATP, ADP, Cr, and PCr as substrates and the inorganic phosphate (Pi) concentration was computed as the sum of initial Pi concentration, and changes in ADP and Cr concentrations. It is well known that ATP hydrolyzes in the cytosol of the cell forming ADP and releasing energy. The Hubley model considered the forward CK reaction, assuming the reverse
CK reaction to attain rapid equilibrium, and myosin ATPase reaction and the anaerobic glycolytic reactions as the reactions taking place in the bulk or the cytosolic region of the cell.
The diffusion length was derived from the mean free spacing between mitochondrial clusters in the gold fish muscles. Constant concentrations equal to the initial concentrations of the substrates were assumed at the mitochondrial boundary and symmetric boundary conditions were assumed at the center of the diffusion distance. To mimic the metabolic effects of swimming, the model was run as a series of loops. Each loop was considered to last for half of the tailbeat cycle, the duration of each loop was dictated by the tailbeat frequency, a 50% duty cycle was employed and the myosin ATPase was activated only during odd numbered loops. The model was solved using an IMSL subroutine in FORTRAN. The sensitivity of the model to the parameters used was performed by varying the ATP produced by glycolytic process. It was observed that the ATP production within the cell does not vary with the temperature. An advantage with the model is that it is simple as it involves fewer substrates and hence it will be easier to program and analyze
18
the model. One of the drawbacks with the model concerns with constant substrates being produced at the mitochondria.
It is well known that the mitochondrial ATP synthesis by oxidative phosphorylation depends on the available ADP and other conditions. The Hubley model was extended to study the effects of diffusion of various substrates on the fiber size (Pathi, 2005), (Jordan, 2005). This model considered ATP, ADP, Pi, Arginine Phosphate (AP), and Arginine (Arg) as substrates, and the basal metabolism, the Arginine Kinase reaction, and myosin ATPase reaction as phase reactions. Other significant changes include considering the mitochondrial ATP production by oxidative phosphorylation through a Michaelis Menten type of kineticsThe system of partial differential equations was solved using COMSOL Multiphysics, Burlington, MA (previously known as FEMLAB). The results from the mathematical model were then compared with the data obtained from the experiments performed on blue crabs (Kinsey et al, 2005). The model results reproduced the experimental data on phosphate concentration during the recovery phase of the muscle activity. One of the significant finding from the study was that the mitochondrial reaction rate used could reproduce the experimental data but the used rate does not have any impact on limiting the diffusion of aerobic flux in large fibers (Kinsey et al, 2005). They employed this model (Kinsey et al, 2005) with parameters adjusted to comply with blue crab dark levator fibers to test whether the observed aerobic rates of AP resynthesis were fast enough to be limited by intracellular metabolite diffusion (Hardy et al, 2006). They concluded that high
ATP turnover rates may lead to diffusion limitation in muscle even when diffusion distances are short, as in the subdivided dark fibers (Hardy et al, 2006). The reaction diffusion model in
Kinsey et al (2005) was then applied with parameters adjusted to comply with fish white fibers
19
(Nyack et al, 2007). They concluded that the recovery rate in these fibers is primarily limited by low mitochondrial density (Nyack et al, 2007).
Locke and Kinsey (2008) described in detail on computing the effectiveness factor (η), which is a measure of diffusion limitation. They concluded that most muscle fibers are not substantially limited by diffusion but many are on the brink of rather substantial diffusion limitation. However they used simplified kinetics for mitochondrial ATP production, did not take into account the mitochondrial volume fraction in the cell and did not include O2 transport in their modeling. In their subsequent work on tail-flipping crustaceans (Jiminez et al, 2008), they included O2 transport in their modeling. Also, they considered mitochondria to be homogeneously distributed within the cell, and O2 delivered from cell periphery to mitochondria diffusion with a linear driving force. They computed η and found that diffusion limitation was minimal under most conditions and concluded that diffusion might act on the evolution of fiber design but usually does not directly limit aerobic flux. In their subsequent work (Hardy et al,
2009), the considered two distinct populations of mitochondria, the inter-membrane (IM) space mitochondria, which is uniformly distributed within the region, and sub-sacrolemma (SS) mitochondria, which is clustered at the periphery of the cell. They considered the SS mitochondria as the boundary condition and computed η for various types of mitochondrial distribution. They observed that larger cells can sustain high ATP turnover rates when the mitochondria are clustered at the periphery. They concluded that changes in fiber structure during development are a response to diffusion constraints, and they ameliorate many of the consequences of hypertrophic growth.
20
2.4. Modeling O2 transport
Around 70% of O2 delivered to the cells is consumed by the mitochondria in oxidative phosphorylation (Trimarchi et al, 2000). Also, O2 transport to mitochondria limits the maximum
O2 consumption rate under conditions of high ATP demand. (Wagner, 2008). Hence oxygen plays an important role in ATP oxidative phosphorylation in mitochondria, although most of the models do not include O2 while modeling cellular energetics. Although Jiminez et al (2008) and
Hardy et al (2009) included O2 dynamics while computing ATP production by mitochondria, they utilized a simplified first order reaction rate for O2 consumption. The pathway for oxygen transport to tissue begins at the red blood cells, also known as erythrocyte. Erythrocytes in the peripheral circulation release oxygen by diffusion into the surrounding plasma with a flux related to the local PO2 gradient. The oxygen then diffuses through the plasma, across the capillary wall, across the interstitium, into muscle cells and to the mitochondria, where most of the oxygen is consumed in the process of oxidative phosphorylation resulting in the production of ATP
(Eggleton et al, 2000). The role O2 plays in the aerobic metabolism makes it important to study the transport of O2 to mitochondria and has attracted the attention of many researchers. There have been many mathematical models describing the O2 transport to mitochondria (McGuire and
Secomb, 2001), (Roy and Popel, 1996). The mathematical model developed by Wong (2006) is considered below as it forms the base for the modeling work of this dissertation and this model focused mainly on analysis of the effects on the O2 delivery due to myoglobin (Mb) (further literature review of the oxygen transport with myoglobin in given in Chapter 5.
Fig. 2.1 shows the schematic of the 1-D reaction diffusion model for oxygen delivery dynamics. The main aim of the model was to study the role of Mb on the oxygen transport in the
21
muscles. Mb, O2, and Mb-bound O2 complex were considered as the substrates. As O2 is transported to the cell through Mb via facilitated diffusion, and O2 forms a complex with Mb, a reversible reaction was considered as the reaction in the bulk phase. The distance between the capillary and the muscle fiber was considered to be the diffusion distance. No flux boundary conditions were assumed for Mb and Mb-O2 complex at both the boundaries since no reaction takes place at the boundary. A constant supply of O2 was assumed at the capillary boundary, while a first ordered pulsed reaction was assumed to take place at the other boundary. Also, the
Michaelis-Menten type of kinetics was assumed at the muscle fiber boundary. The continuity equations were then solved using FEMLAB (now COMSOL Inc, Burlington, MA, USA). One of the important results of the study was that myoglobin-O2 reaction facilitated O2 diffusion at low levels of O2 consumption. It was concluded that the first order kinetics for O2-Mb reaction attained steady state rapidly and increase in the diffusion length resulted in decrease of O2 concentration at the reacting boundary. This suggests that higher concentrations of O2 are available for mitochondria that are closer to the capillaries.
Figure 2.1: Schematic of Wong model (Wong, 2006)
22
As described in the text above, there have been several models that describe the ATP transport in the cell. Also, there has been considerable work on O2 transport to mitochondria. To our knowledge there has been no reaction diffusion model for cellular energetic which considers
O2 along with ATP. Considering the significance of O2 in the ATP production through aerobic metabolism of oxidative phosphorylation, it is important to include O2 in modeling the cellular energetic along with ATP. The present study includes O2 as a substrate along with the ATP and analyzes the effects of intracellular diffusion limitations on the skeletal muscle fiber dimension growth.
23
CHAPTER 3
EFFECTS OF VARIOUS PARAMETERS ON INTRA-CELLULAR
DIFFUSION LIMITATION OF METABOLITES
(Based upon and adapted from Dasika SK, Kinsey ST, Locke BR, Reaction- diffusion constraints in living tissue: Effectiveness factors in skeletal muscle design,
in press (BIT10-517.R1(22926)), Biotechnology and Bioengineering. Copyright
[2010 and John Wiley & Sons, Inc.]. This material is reproduced with permission of
John Wiley & Sons, Inc.)
3.1. Introduction
In 1973 Weisz proposed that biological systems, from enzymes and enzyme complexes, to organelles, cells, tissues, and complete organisms, function within constraints imposed by chemical reaction and diffusion (Weisz, 1973). He introduced and applied the classical engineering concept of effectiveness factor that had been well known and applied to heterogeneous catalysis (Aris, 1975; Langmuir, 1908; Thiele, 1939; Zeldovich, 1939). Weisz suggested that such biological and biochemical systems would most efficiently operate at the edge of the effectiveness factor curve where (1) they are not diffusion limited, but (2) further increases of reaction rate and capacity would lead to diffusion limitations. That is, cells, like chemical reactors, are most efficient where catalytic activity is as high as it can be without significant limitations by diffusion. This concept of effectiveness factor has been well known and used by bioengineers for many systems including immobilized enzymes (Fink et al, 1973;
24
Shimizu and Inoue, 1983; Wadiak and Carbonell, 1975), bacterial cells growing in biofilms
(Bhaskar and Bhamidimarri, 1991; Buffiere et al, 1995; Vos et al, 1990; Zhang and Bishop,
1994), cells growing in bioreactors for tissue engineering and many other applications (Haugh and Schneider, 2006). With the increasing interest of bioengineers in tissues and organs, both in vivo and for cellular, metabolic, and tissue engineering, we are interested in testing Weisz‟s hypothesis for non-diseased living tissue.
Diffusion is one of the important phenomena by which metabolite transport occurs within the cell. However most work on energy metabolism in skeletal muscle has focused only on the catalytic aspects of cellular enzyme systems, neglecting substrate diffusive fluxes, based largely on the reasoning that intracellular diffusion distances and fiber dimensions tend to be modest
(Russell et al, 2000). However, a purely kinetic analysis may be insufficient to explain many cellular processes (reviewed in Suarez, 2003). The principal hurdle to understanding the role of diffusion and metabolic organization is that most experimental measurements are volume- averaged over the entire cell, and it is difficult to observe localized concentration gradients.
However, several studies that employed reaction-diffusion mathematical models of aerobic metabolism in muscle have found evidence of intracellular gradients (Mainwood and Rakusan,
1982; Meyer et al, 1984; Hubley et al, 1997; Aliev and Saks, 1997; Kemp et al, 1998; Vendelin et al, 2000; Saks et al, 2003). While these studies implicate diffusion limitation of aerobic flux, they are almost exclusively focused on a narrow range of muscle fiber designs in mammals.
Therefore, we are currently lacking a broad-based, quantitative synthesis of reaction-diffusion processes that takes advantage of the wide diversity of muscle types in the animal kingdom.
25
Quantitative analysis of cellular energy metabolism is critical to understanding both cell function and cell design. For example, an accurate knowledge of the rate of ATP turnover in skeletal muscle as a function of fiber size, mitochondrial density and distribution, and oxygen supply is important to assess changes in muscle as it responds to aging, growth, and metabolic diseases. In order to develop such an approach a model of the mitochondrial ATP production by oxidative phosphorylation should be coupled to a sufficiently detailed description of the sarcoplasmic ATP consumption and the rate of extacellular O2 supply. We have considered in previous work the role of inter-mitochondrial diffusion of high energy phosphate metabolites where relatively simple ADP-dependent Michaelis-Menton expressions for ATP production at the mitochondria were utilized (Locke and Kinsey, 2008). We have also considered the role of
O2 transport utilizing empirical and highly simplified expressions for mitochondrial production of ATP as functions of ADP, O2, and inorganic phosphate (Pi) (Hardy et al, 2009, Jimenez et al,
2008). The present work focuses on utilizing a more detailed model of oxidative phosphorylation in the mitochondria within the framework of a reaction-diffusion analysis of intracellular high energy phosphate metabolism in order to assess the role of diffusion constraints on cell design.
A number of approaches, including the flux based method, flux-balance analysis, and non-equilibrium thermodynamics, have been employed to develop mathematical models for mitochondrial ATP production (Beard, 2005; Korzeniewski, 1996; Korzeniewski, 1998;
Korzeniewski, 2001; Korzeniewski and Zoladz, 2001; Saks et al. 2003; Stucki, 1991). Because of the robust nature of the previously developed non-equilibrium thermodynamics-based model for isolated mitochondria (Beard, 2005) we will focus on modifying and utilizing that model to predict ATP formation in terms of the supply of the key substrates, ADP, Pi, and O2.
26
The overall goal of the present work is to determine the effects of diffusion on energy metabolism in a broad range of skeletal muscle fibers through determination of the effectiveness factor (i.e., the ratio of observed reaction rate to the reaction rate where diffusion is infinitely fast) (Locke and Kinsey, 2008; Weisz, 1973). In order to do so, we first develop, based upon the published model of oxidative phosphorylation in the mitochondria (Beard, 2005), a reaction rate law for mitochondrial ATP production as a function of ADP, Pi, and O2 concentrations at the external membrane of the mitochondria. This mitochondrial rate law is then employed as a boundary condition for the reaction diffusion model of the metabolic reactions occurring in the sarcoplasm. Using spatial homogenization techniques (i.e., the method of volume averaging) the reaction-diffusion model is simplified and the resulting system of mass balances is solved to determine the intracellular concentrations of ATP, ADP, Pi, Creatine (Cr), Phospho-Creatine
(PCr), Myoglobin (Mb), Myogloin-O2 complex (MbO2) and O2, the reaction rates, and the corresponding effectiveness factors. The model output, including ATP turnover rate, is thereafter evaluated for a range of cell sizes (spanning 5 to 50 µm in radius), mitochondrial densities (10, and 45% of cell volume), and O2 supply concentrations at the capillaries (35.1 µM and 7.85 µM) and these results are used with data for a broad range of animals to predict diffusion limitation using the effectiveness factor.
3.2. Mathematical methods
The following steps were involved in developing the reaction diffusion model. These steps are described below.
1. Developing mitochondrial rate law
2. Incorporating O2 in the reaction diffusion model
27
3. Performing volume averaging
4. Computing effectiveness factor
3.2.1. Mitochondrial rate law
Most of the cellular models considered a Michaelis Menten reaction rate with ADP as substrate
for the mitochondrial rate (Pathi, 2005; Locke and Kinsey, 2008; Kinsey et al, 2005). ATP
synthesis by aerobic metabolism requires O2 and hence an accurate description of mitochondrial
ATP synthesis should include O2 as a substrate. In this regard, the Korzeniewski (1998) group
and the Saks group implemented a detailed mitochondrial oxidative phosphorylation model
(Aliev et al, 1998; Saks et al, 2001) as a boundary condition in their model. While the approach
to using the detailed mitochondrial model is more appropriate, it would be computationally more
intensive. Hence a simplified mitochondrial rate law for ATP synthesis by aerobic metabolism of
oxidative phosphorylation is necessary.
3.2.1.1. Rate law for mitochondrial oxidative phosphorylation
A model for isolated mitochondria was developed that describes the mitochondrial oxidative
phosphorylation in terms of oxygen, ADP, inorganic phospate supplied to the mitochdondria.
This model is based upon that developed by Dr. Beard (Beard, 2005). The following steps were
performed in order to develop a rate law for mitochondria as a function of ADP, O2, and Pi.
These steps are described in detail in (Dasika, 2007).
Reproducing results published in the Beard‟s paper (Beard, 2005).
Performing sensitivity analysis to identify parameters that have dominating influence on
the system
28
Modifying these parameters such the model results are in close agreement with the
physical data results available in literature
Performing simulations using the modified Beard‟s model and collecting the simulated
data
Performing Monte Carlo Simulated Annealing simulations to determine the constants in
the rate law.
3.2.1.2. Reaction diffusion model
The present study assumes that mitochondria are homogeneously distributed throughout the cell volume. Some of our previous models (Kinsey et al, 2005; Pathi, 2005; Jordan, 2005;
Hardy et al, 2006) did not include O2 because the focus was on inter-mitochondrial ATP reactions and diffusion. However, we have also used simplified rate laws that included O2
(Hardy et al, 2009; Jimenez et al, 2008). The present work seeks to incorporate a more accurate description of mitochondrial oxidative phosphorylation to predict the concentrations of ATP,
ADP, Pi, Cr, PCr, Mb, MbO2, and O2 and the corresponding overall ATP turnover rates. It is assumed that the O2 delivered to the cell is consumed only by mitochondrial oxidative phosphorylation (Wittenberg and Wittenberg, 1989). The continuity equation for a given species
„A‟ (ATP, ADP, O2, Mb, Cr, PCr, MbO2 or Pi) in the inter mitochondrial space is given by
(3.1)
where, is the concentration of species A in the inter-mitochondrial space, t is the time, is the diffusion constant in the inter-mitochondrial space, and is the inter-mitochondrial
myofibrilar or “bulk” phase reaction term for species A. ATP consumption in skeletal muscle is
29
dominated by the myosin ATPase and the sarcoplasmic reticulum Ca2+ ATPase during steady state contraction in aerobic muscle, and a variety of other ATPases during the slow aerobic recovery following burst contraction in anaerobic muscle. In the present study, the myofibrillar
ATPase reaction rate (RATPase) describes ATP demand during both steady-state contraction in aerobic fibers and post-contractile recovery in anaerobic fibers is described by a Michaelis
Menten expression
(3.2)
where is the non-dimensional ATP concentration, and and are the
Michaelis – Menten constants associated with the ATPase reaction. Due to reaction stoichiometry the ADP and Pi reaction rates are related by .
The creatine kinase reaction is modeled according to the following equations (Morrison and James, 1965)
(3.3)
Where
(3.4)
The CK reaction is defined in the direction of ATP formation. The CK reaction rates for various metabolites are related as
(3.5)
30
The Mb reaction is defined as
(3.6)
The Mb reaction is modeled using a simple kinetic expression as
(3.7)
where, k1,Mb and k2,Mb represent the rate constants for forward and backward reactions for the formation of myoglobin-oxygen complex respectively. The myoglobin reaction rates for various species are related as
(3.8)
At the mitochondrial boundary, the fluxes are balanced by the net mitochondrial reactions for each participating species by
(3.9)
where, is the mitochondrial production or consumption rate for species A (this is the function determined by the SA method discussed elsewhere in this manuscript) and is the unit normal vector. The mitochondrial reaction rates for the various species are related through reaction stoichiometry by
(3.10)
31
3.2.1.3. Volume averaging
The mitochondria can be found everywhere in the cell (Vendelin et al, 2005). In skeletal muscle fibers of adult crabs, most of the mitochondrial clusters can be found in the periphery of the cell
(Boyle et al, 2003). It is well known that O2 is delivered by the capillaries to the periphery of the cell. In order to consider the O2 to all mitochondria and substrate contribution from mitochondria, and to account for the cell size rather than the inter-mitochondrial spacing as the diffusion distance, volume averaging technique needs to be applied. The previous cellular models (eg. Pathi, 2005) considered the inter-mitochondrial distance as the diffusion distance.
Considering the cell size as the diffusion distance provides us with a freedom of neglecting the mitochondrial distribution as the mitochondrial distribution changes with the cell size and hence may not accurately depict the cell size. This technique was introduced by Professor Stephen
Whitaker (Whitaker, 1967) mainly to analyze transport and reaction in multiphase systems. This technique had been applied in wide variety of areas such as reaction-diffusion problems in porous catalysts (Ryan, 1981); application to cell growth in polymer scaffold (Galban and Locke,
1999); mathematical modeling of gas-solid reactions in corona reactors (Grymonpre et al, 1998); and cell growth model in 3-D constructs (Pathi, 2005). The aim of the volume averaging approach is to represent a system of partial differential equations for different phases in the system into system of equations represented by averaged concentration. The following text briefly explains the technique of volume averaging.
In the present study we assume mitochondria to be uniformly distributed within cytosolic region of the cell. This makes the cell as a two phase system: 1) the bulk phase or the cytosolic region of the cell and 2) the mitochondrial phase, where majority of ATP is effectively produced.
32
Fig. 3.1 shows the cellular system consisting of cytosolic phase, represented by c-phase, and the mitochondrial phase, represented by m-phase. In the averaging volume or control volume the volume of bulk phase is represented as Vc while the mitochondrial volume is represented as Vm.
The total control volume is V = Vm+Vc. The mitochondrial volume fraction and bulk phase volume fraction are represented by εm and εc respectively.
The system of equations is defined as
(3.11)
Where, represents the concentration of metabolite A in the γ phase, represents the
diffusion constant for metabolite A in the γ phase, and represents the homogeneous
reactions for metabolite A. In the above system of equations, it is assumed that the metabolites are dilute, i.e., ), where is the mole fraction of species A. The following boundary
conditions are considered for the present study. At the mitochondrial boundary, the flux for the species is equal to the mitochondrial rate.
For ATP, (3.12)
for ADP, and Pi, (3.13)
For O2, (3.14)
33
Cell Mitochondria
ι
Control Volume κ-phase
L γ-phase
L: diameter of the cell
r0 : diameter of the control volume ι : dia eter of the itocho dria
Mitochondrial surface reactions r 0
Figure 3.1: Volume averaging
Where, is the unit normal vector directed from γ-phase to the κ-phase,
, is the mitochondrial rate law developed previously.
The species equilibrium weighted concentration in the control volume is given by the following expression
(3.15)
In the above equation (eq. 3.15), represents the intrinsic phase average concentration of species A in the phase i and represents the volume fraction of phase i. The superficial volume averaged concentration of species is defined as
34
(3.16)
Also, the intrinsic volume average is for species A in phase i is calculated as
(3.17)
And the intrinsic and superficial volume averaged concentrations are related as
. The local spatial deviation in concentration for species A in phase i is given by Gray‟ s
decomposition as
(3.18)
1. Using the spatial averaging theorem,
(3.19) ηγκ γκ
2. Ignoring any local spatial effects in the diffusion constants,
3. Ignoring the bulk phase reactions,
4. Assuming constant mitochondrial volume fraction
5. Integrating eq. 1, substituting for eq.s 3.15 – 3.19 results in
η γ (3.20) γκ γκ
where, η γ γκ γκ
Or
35
η η γ (3.21) γκ γκ γκ γκ
Using the length constraint results in
η η (3.22) γκ γκ γκ γκ
Since the variations in the bulk- phase volume fraction, are neglected, the term
η can be neglected. Hence eq. (3.20) reduces to γκ γκ
η γ (3.23) γκ γκ
Obtaining solution to the eq. (3.23) constitutes the closure problem. Analysis of the case of Michaelis-Menten kinetics boundary conditions was also reported (Wood et al, 2007). The volume averaged species continuity equations, simplified to one-dimensional rectangular
Cartesian coordinates is
(3.24)
36
where, represents the volume averaged concentration of species i, is
the effective diffusion constant for species i (Whitaker, 1999), the volume fraction of
mitochondria in the muscle cell, is the volume averaged bulk phase reaction (i.e., the
myofibrillar ATPase reaction rate in the present study), is the volume averaged
effective mitochondrial reaction rate, and x is the spatial position. The stoichiometric coefficient of 0.25 was obtained from (Beard, 2005) by plotting the ratio of the complex IV flux to F1F0-
ATPase flux vs the substrates. This ratio was close to 4 in all cases.
The volume averaged effective reaction rates are given by the following equations (Wood et al, 2007)
(3.25)
3.2.1.3.1. Boundary conditions
The domain for solution of system of equations (Eq. 3.24) is the radius of the cell, L.
Hence the following boundary conditions are employed. At the center of the cell, i.e., at X = 0, the no flux BC is assumed for all the species, i.e.,
(3.26)
At the cell periphery, i.e., at X = 1, no-flux is assumed for ATP, ADP, and Pi, where
, while O2 is assumed to be delivered into the cell via capillaries at the cell
periphery and across the cell membrane. Hence the following equation is used for the O2 flux at the boundary
37
(3.27)
Where, is the boundary O2 concentration, is the volume averaged O2 concentration,
and Kmt is a mass transport coefficient that reflects transport across the layer comprised of the capillary wall, interstitial space, and the muscle fiber membrane. Eqs. 3.24 – 3.27 describe the volume averaged 1-D reaction diffusion model.
While some metabolites can leak across the cell membrane the change in metabolite concentration due to leakage is expected to be insignificant. For example, it has long been established using 31P-NMR in contracting muscle that high energy phosphate molecule concentrations change with changes in work load, but that the total pool of phosphorylated metabolites is constant and the pre- and post-contractile concentrations are equivalent (Meyer et al, 1982; Kushmerick and Meyer, 1985; Kushmerick et al, 1992).
3.2.1.4. Effectiveness factor
The effectiveness factor (η) is defined as the ratio of average reaction rate with diffusion to the reaction rate diffusion is not limited (Weisz, 1973). Effectiveness factors near 1 imply that the reaction rate is not limited by diffusion and the smaller the effectiveness factor the larger the limitation by diffusion. In order to compute the effectiveness factor, the concentrations determined by the solution of Eq. 3.24 were used to determine the reaction rate in the presence of finite diffusion rates and this solution was combined with the results for the case without diffusion limitations, as described in previous work (Hardy et al., 2009; Jimenez et al., 2008;
Locke and Kinsey, 2008). The system of equations when diffusion is not limited is described below.
38
The system of equations for the case with finite diffusion is given in eq. 3.24. For any species A, the volume averaged continuity equation is represented as
(3.28)
Where, represesnts the volume averaged concentration of species A and represents the volume averaged reaction rate for species A. Integrating both sides of eq. 3.28 under the limits from X = 0 (center of the cell) to X = 1 (periphery of the cell), we have
(3.29)
Where, is the average concentration of species A, and
represents the averaged rates for species A, given by . It is assumed that the
reaction is fast enough that the reaction is not limited by diffusion. In the above derivation the no- flux boundary condition is applied at both the boundaries and is applicable to all species except O2. For O2, the no flux BC is applied at the center of the cell while eq. 3.27 is applied at the periphery of the cell. After integrating and applying limits for various metabolites, we obtain the following system of equations for the case without any diffusion limitations.
(3.30)
39
We are interested in the steady state solution. Hence simplifying the system of eq. 3.30, we have
(3.31)
Where, AD0 and AP0 are constants, and , , are the ATP, ADP, and Pi concentrations for the case without diffusion limitations respectively. In the present study we used AD0 = ADP0 + ATP0 and AP0 = ATP0 + Pi0, where the subscript 0 represents the initial concentration for species. The solution to system of algebraic equations (eq. 3.31) gives the rates and concentrations for the case without diffusion limitations.
3.3. Solution technique
The system of equations (eq. 3.24 – 3.27) is solved using the Strang splitting technique wherein the Laplacian part is solved using a Backward-Time-Center-Space {BTCS) finite difference technique while the reaction terms are solved using Forward-Euler technique
(Strikwerda, 2004). The steady state solutions were then averaged over the length domain to obtain the averaged concentrations. Some numerical difficulties were encountered for the case of low volume fraction, low Km, and low boundary O2 concentration, , and in these cases, determination of the effectiveness factor was more accurately computed using the mitochondrial reaction term than the myofibrial ATPase rate. The numerical technique is described in brief below.
40
In order to solve the system of equations numerically, the continuous differential terms can be replaced with the discrete difference terms as follows
(3.32)
Where, is the concentration of species A at the nth time step, and mth point in space, k
= Δt is the time step size. This kind of differencing scheme is known as forward differencing.
Similarly, the spatial term can be replaced with the discrete term as
(3.32)
where, k = ΔX is the space step size. This kind of differencing scheme is known as central differencing.
In order to solve the system of equations in eq. (3.24), we employ the Backward-Time-
Center-Space (BTCS) technique. The main reason for using this scheme is that this scheme is unconditionally stable meaning the solution is not restricted to a time or space step condition.
Also, the eq. (3.24) is solved using Strang splitting, where in the PDE part is solved first and then the bulk phase reactions are solved using the Forward Euler technique described later. In the
BTCS, the following difference equations are used
(3.34)
Where, represents the solution at the (n+1)th time instant, and represents the solution at the n th time instant. Rearranging the eq. (3.34), we have
41
(3.35)
Where, . The solution from eq. (3.34) serves as the intermediate solution, which will
then be used to solve the bulk reactions part. To denote the intermediate solution, the star (*) is used. The spatial domain is discretized as .
For ADP, Pi, Cr, PCr, Mb, MbO2, and ATP, we have the following BC‟s at m = 0 (center of the cell) and m = M (periphery of the cell)
(3.36)
Or
(3.37)
Similarly, at X = 1, (3.38)
For O2, we have at X = 0, . At the cell periphery (X = 1), we have the following
BC
(3.39)
Solving for , we have
(3.40)
where, (3.41)
42
The system of equations in eq. (3.34) can be represented as , where A is the tri-diagonal matrix and and are the vectors of length M-1. The elements of matrix
A are constant and hence the solution can then be obtained as . Once we have the intermediate solution, the solution at the (n+1)th time instant can be computed based on nth time instant using the forward Euler technique as follows
(3.42)
Or
(3.43)
A MATLAB program (MATLAB v.7.0, Boston, MA) was written to solve the system of equations and compute the effectiveness factor. We encountered some numerical issues while solving the system of equations, which will be discussed below.
3.4. Results
3.4.1. Mitochondrial rate law
The results from our MATLAB program to simulate the isolated mitochondrial oxidative phosphorylation reactions were consistent with the results published in the literature. The normalized concentration profiles (not shown here) for NADH were plotted vs. Pi to compare with the published results and the sum of the squares of the error between the output from our code and the original published code was computed and found to be 6X10-10, well below acceptable limits for error.
43
Fig. A1 A and B (Appendix A) show sample plots of ATP flux as functions of ADP and
O2, respectively, using the isolated mitochondrial model, neglecting the adenylate kinase and creatine kinase reactions as described above. The results clearly show typical Michaelis-Menton kinetics for the ATP production rate, however further analysis of the model was needed in order to match the apparent Vmax and Km for this model with experimental values from the literature for isolated mitochondria. When all parameters were set as in published work for cardiac muscle
(Beard, 2005) the Km values did not match experimental results for skeletal muscle mitochondria. Table A1 (Appendix A) shows the results from the sensitivity analysis and the four most sensitive parameters (with normalized sensitivity coefficients greater than 4x10-6) are highlighted in red. The parameters with S.C. values higher than 4x10-6 were assumed to be the most important for control of the ATP concentration in the IM space. Although the cut-off value of 4x10-6 was set arbitrarily, other computations, not shown here, were performed by increasing and decreasing the value of each parameter by a factor of 100 and showed no significant effect on the Vmax and Km values of parameters with sensitivity coefficients less than
4x10-6.
The parameters kPiHt, xPiHt, xANT, and xC3, (Table A1, Appendix A) have the largest effects on the ATP production rate, and the apparent Vmax and Km values. xPiHt and kPiHt are the parameters associated with the Pi transport flux from the mitochondrial matrix to the IM space. xANT, and xC3 are the adjustable parameters for adenine nucleotide translocator (ANT) and complex III fluxes, respectively (Beard, 2005). The parameters kPiHt, xPiHt, xANT, and xC3 were modified to 120, 0.95, 0.86, and 0.0055, so that the resulting apparent Vmax and Km values were close to reported values. Apparent Vmax and Km values for O2 are 2.95 mM/min and 2.36 µM respectively, and for ADP are 3.41 mM/min and 20.46 µM respectively, which are consistent
44
with measurements in skeletal muscle mitochondria (Vmax = 3.3 ± 0.30 mM/min for O2)
(Tonkonogi and Sahlin, 1997), (Km = 2 ± 0.5 µM for O2) (Wiedemann and Kunz, 1998), (Vmax =
6.0 ± 3.18 mM/min and Km = 35 ± 15 µM for ADP) (Vicini and Kushmerick, 2000).
A variety of algebraic functions for the rate law were tested using the simulated annealing methodology and the final expression was chosen based on the sum of the squares of the differences between the ATP flux from the simulated data and the algebraic expression. Eq. 3.44 represents the system with good accuracy and with the lowest cost function. Table A2 (Appendix
A) lists the fitting constants used in Eq. 3.44 along with their values and units.
(3.44)
In the following text, „simulated data‟ refers to results obtained from computation of the system of ordinary differential equations from Beard‟s (2005) model with modified parameters as discussed above, while „data from expression‟ refers to the results obtained using Eq. 3.44.
Fig. A2 (Appendix A) shows sample plots of ATP flux from the simulated data, the data from the expression (Eq. 3.44), and the absolute difference between the simulated data and data from the expression for the case with O2 as substrate. These results show that the proposed algebraic function generally agrees with the results of the solution of the ordinary differential equation model. The final cost obtained using Eq. 3.44 with the fitting constants shown in Table 2 was
1.88x10-6. This cost function was based on 3000 data points. The average error between the simulated data and the expression associated with each data point was 2.51 x 10-5 M/s or, approximately 25 µM/s, which represents the accuracy of the developed expression. The plots for the other cases are not shown here. These results show that out of the 24 cases of different
ADP, O2, and Pi concentrations computed, the expression given by eq. 3.44 represent 22 cases
45
with high accuracy and the expression was somewhat less accurate at significantly lower concentrations of ADP with a maximum error is 35 µM/s.
3.4.2. Concentration Profiles and Effectiveness Factors
To illustrate spatial changes in concentrations and reaction rates, the aforementioned reaction diffusion model (Eqs. 3.24 – 3.27) were solved for the case of L = 50 µm, Vmax,ATPase =
0 25 mM/min, O2 = 35.1 µM, and Km,ATPase = 0.15 mM. This length scale is typical of a juvenile fish (or a mammal). Fig. 3.2 shows the plots of the steady state concentration profiles of ATP,
ADP, Pi, and O2, and the ATPase rate and mitochondrial rate as functions of the non-dimensional spatial position for this case. As the ATP is consumed, ADP and Pi are produced as shown in
Fig. 3.2A, and there is a significantly higher ATP concentration at the cell periphery (X =1).
Further, most of the O2 delivered to the cell is consumed close to the periphery of the cell (Fig.
3.2B). Although the O2 (Fig. 3.2B) and subsequent mitochondrial production of ATP (Fig. 3.2C) drop to near zero at the center of the fiber, neither the ATP concentration (Fig. 3.2A) nor the
ATPase rate (Fig. 3.2D) drop to zero throughout the cell. This is because of the relatively low metabolic demand causing the ATP formed by the mitochondria near the boundary to be sufficient to supply the entire fiber. Similar graphs were plotted for the case using the same
Vmax,ATPase of 25 mM/min, but with L = 150 µm, which simulates a large white fiber from an adult fish. In this case the intracellular concentration and reaction rate gradients were much less substantial than in Fig. 3.2. These results are consistent with the shift in mitochondrial distribution toward the periphery of the fiber that occurs as fibers increase in size during muscle hypertrophic growth (Boyle et al, 2003; Hardy et al, 2009; Nyack et al, 2007). Similar results were observed for different combinations of length scales and ATP turnover rates, where the O2
46
concentration gradient becomes steeper with increases in diffusion distance and aerobic metabolic rate.
A B
CD
Figure 3.2: Steady state concentrations and rate profiles for a case that leads to substantial elevation in Pi and sizable gradients in [O2] and Rmito (L = 50 µM, Vmax,ATPase = 25 mM/min and Km,ATPase = 0.15 mM for εmito = 0.1,
0 and O2 = 35.1 µM. In these plots, X refers to the space, where, X = 0 refers to the center of the cell while X = 1
refers to the periphery of the cell
(A) ATP, ADP, Pi vs X; ADP and Pi concentrations increase as ATP is depleted, but gradients are minimal.
(B) O2 vs X; most of the O2 delivered to the cell is consumed at the periphery of the fiber.
(C) Mitochondrial rate vs X; drop in mitochondrial ATP production is caused by the depletion of O2.
(D) Myofibrillar ATPase rate vs X; despite almost no mitochondrial ATP production (Rmito) in the fiber core,
ATP diffusion from the active mitochondria near the fiber periphery is sufficient to supply the myofibrillar
ATPases.
47
In the present study, the diffusion distance, L, the volume fraction of mitochondria, ,
the Michaelis-Menten kinetic constants for myofibrillar ATPase, Vmax,ATPase, and Km,ATPase , and the boundary O2 concentrations, were varied. The diffusion distance (i.e., cell radius) was
varied from 5 µm (typical insect flight muscle) to 50 µm and the Vmax,ATPase was varied from 1 mM/min to 75 mM/min. We used two different mitochondrial fractional volumes ( = 0.1
and 0.45), two O2 boundary concentrations ( and ) and two ATPase
Km values (Km,ATPase = 0.15 mM and 5 mM). The system of equations (Eqs. 3.24) was solved for each Vmax,ATPase and L and the resulting concentrations and rates were averaged over the length domain to determine the averaged concentrations of species and the averaged mitochondrial rate and myofibrillar ATPase rate (referred to as ATPase rate, RATPase, in the following text). The averaged rates and concentrations are denoted by „< >‟. For example, the averaged ATPase rate is denoted by „
3.3D becomes essentially independent of L. At higher ATPase rate or high Vmax,ATPase the ATP production is limited by O2 supply. Hence the
48
L. We increased the Kmt to five times the actual value to understand if the
L was due to the O2 supply limitation.
A B
CD
Figure 3.3: Influence of diffusion length, andVmaxATP ase on
mM, kmt = 1100 µ/s and .
(A):
and Vmax,ATPase, (D):
Fig. 3.4 shows the plots of
Though the
49
A B
CD
Figure 3.4: Concentration profiles for
of L and Vmax,ATPase for kmt = 5500 µ/s (five times the value used in all other calculations) with all other
parameters the same as Fig. 3.3. A.
with diffusion, and D.
Fig. 3.5 shows the corresponding plots of
L for the same parameters as used in Fig. 3.3. As Vmax,ATPase increases, the ATP demand increases until the supply of ATP becomes limited due to the diffusion constraints and the
ATPase rate levels off (Fig. 3.5A). In contrast, for the case without diffusion limitations, the
ATPase rate is linearly dependent on Vmax,ATPase and independent of L (Fig. 3.5B). The effectiveness factor decreases with increases in Vmax,ATPase and L (Fig. 3.5C), as expected based on the similar decrease in
50
experimentally, the η is plotted as a function of
A B
CD
Figure 3.5: Influence of diffusion on the interaction between the effectiveness factor,VmaxATPase, and length for
, Km,ATPase = 0.15 mM and . (A): < RATPase > vs L and Vmax,ATPase (B): < RATPase >
without diffusion limitations vs. L and V max,ATPase (C): vs. L and Vmax,ATPase (D): vs. L and RATPase
3.4.2.1. Effects of Km,ATPase and Boundary O2 concentration
Fig. 3.6 shows the effects of Km,ATPase and boundary O2 concentrations on η with L and
Vmax,ATPase for = 0.1. Comparisons of Fig. 3.6A with Fig. 3.6C and Fig. 3.6B with Fig. 3.6D
show significant effects of O2 supply on the transition from reaction control to diffusion control.
As the boundary O2 concentration decreases, the mitochondrial ATP production rate becomes limited by the supply of O2 and hence the ATPase rate becomes diffusion limited at lower
Vmax,ATPase and L. This results in a drop in η at relatively low ATPase rates whereas a higher
51
boundary O2 concentration results in production of more ATP, resulting in a drop in η at relatively higher ATPase rates. Comparison of Fig. 3.6A with Fig. 3.6B and Fig. 3.6C with 3.6D shows that Km,ATPase has a smaller effect on η than does O2 on the transition from reaction to diffusion control.
A B
CD
Figure 3.6: Influence of Km,ATPase and on the interaction of the effectiveness factor, length and Vmax,ATPase
for . Km,ATPase has a small effect on η, while lower leads to a large reduction in the surface
with a high . (A): vs L and RATPase, Km,ATPase = 0.15 mM, (B): vs L and RATPase, Km,ATPase
= 5 mM, (C): vs L and RATPase, Km,ATPase = 0.15 mM, (D): vs L and RATPase,
Km,ATPase = 5 mM,
52
3.5. Discussion
The tendency of fibers to be nearly diffusion limited may imply that there is positive selective pressure for fibers to be as large as possible. Johnston et al, (2003, 2004) proposed that in some fishes this may be a means of reducing costs of muscle basal metabolic rate, since large fibers have a lower surface area per volume of cell over which to maintain ionic gradients. This may be particularly important in muscle, since it makes up a large fraction of animal body mass.
However, an implication of fibers being nearly diffusion limited is that they are more sensitive to challenges such as hypoxia. A reduction in blood O2 concentration shifts the position of the effectiveness factor surface, leading to a lowering of for a given rate of ATP turnover and fiber size (Fig. 3.6). Therefore, fibers that are subject to periodic or chronic reductions in O2 might be expected to have a combination of fiber size and aerobic capacity that make them less likely to become diffusion limited.
3.5.1. Numerical issues
The numerical solution technique for the system of equations (Eq. 3.24) was discussed above. We employed BTCS Finite Difference scheme to solve the Laplacian part of the reaction diffusion system, which is an implicit scheme, i.e., an unconditionally stable method. The factor
for the solution to be stable if an explicit finite difference scheme were to be
employed (Strikwerda, 2004). However, this condition can be relaxed by employing an implicit scheme such as BTCS (Strikwerda, 2004), implying that a larger time step can be used to get a stable solution. Although we obtained a stable solution by employing an implicit scheme, we had some numerical issued with the solution.
53
We computed η based on the averaged ATPase rates. Fig. 3.7 shows the plots for η based on ATPase rates. Fig. 3.7C shows that there exists a peak value for ATPase rate after which the
ATPase rate with finite diffusion drops. However based on the ATPase rate function, we expected rather a plateau behavior for ATPase rate. This difference in the behavior for ATPase rate was the reason for a skewed η surface plot (Fig. 3.7B). We were interested in the steady state solution, where at steady state the ATP sink (ATPase rate) should be equal to the ATP source
(mitochondrial rate). However the ATPase rate and the mitochondrial rate did not match at steady state. Also, the ATPase rate showed a peak kind of behavior. We discuss the reason for this numerical issue below.
A B
C D
Figure 3.7: Plots based on ATPase rate; the results were inaccurate although the MATLAB code was free of
errors, A. η vs. Vmax and L, B. η vs. ATPase rate and L, C. ATPase with finite diffusion rate vs. Vmax and L,
and D. ATPase rate without diffusion vs. Vmax and L
54
As an example, we solved for the ATP concentration for the case without diffusion limitations with the following parameters , , Km,ATPase = 0.15 mM. We varied the ATP concentration and plotted the ATPase rate and mitochondrial rate. The solution to the system is the point where both the rates intersect. Fig. 3.8 shows the plots of mitochondrial and ATPase rates as functions of ATP concentration. It can be observed in this case that the mitochondrial rate is very flat in comparison to the ATPase rate near the point of intersection of the two curves. Hence a small numerical error in the computation at this point based upon the
ATPase rate would have a significant error in the actual rate. We believe that this is the reason for the erroneous ATPase rate. To combat this issue we used the averaged mitochondrial rate to compute the η in all our plots. Another possible solution to this numerical issue is to use a finer time step, which implies more number of iterations, and as a result, more amount of computation time.
55
Figure 3.8: Solution is the point where ATPase rate and mitochondrial rate intersect; marginal error in
[ATP] would have a significant error in the ATPase rate.
3.5.2. Hill coefficient
While analysis of Beard‟s (2005) model of oxidative phosphorylation gave an effective
Hill coefficient for ADP near 1, previous work indicates that the Hill Coefficient for oxidative phosphorylation dependence on ADP is between 2 and 2.5 (Jeneson et al, 1996; Jeneson et al,
2000; Kushmerick, 1998; Jeneson et al, 2009). We did not observe any effect of changing A in
0 Equation (3.42) to 2.5 on the computed η surface for the case with εmito = 0.1, O2 = 35.1µM, and
Km,ATPase = 0.15 mM. Fig. 3.9 shows the plots for η vs.
2009)). No visible effect was observed by varying the Hill coefficient.
Case with Mb & CK, A = 1.03
A
Case with Mb & CK, A = 2.5
B
Figure 3.9: Effect of Hill's coefficient; no visible effect was observed by varying the Hill’s coefficient
56
While large variations in reaction rate parameters such as Vmax or apparent Vmax (or Vmit in Equation 3.43 in this case) values can substantially affect the rate and hence the effectiveness factors, variations of the Hill coefficient are expected to have small effects on η. The role of such Hill coefficients is to change the curvature of the reaction rate law where higher values of A would lead to steeper and more S-shaped rate function; our previous work (Locke and Kinsey,
2008) has shown that steeper rate laws, i.e. with smaller Km, lead to slightly steeper drops in the effectiveness factor near the transition from reaction controlled to diffusion controlled, but do not change significantly the maximum rate where this transition occurs. Future work is necessary to reconcile Beard‟s (2005) model, which gives an effective Hill Coefficient for ADP near 1, with the references cited above, which point to more cooperativity in this reaction. It is interesting to point out that the effective Hill Coefficient for Beard‟s (2005) model of inorganic phosphate is
2.5. It should also be emphasized that the rationale to use Beard‟s (2005) model as a base for mitochondrial oxidation is the need to incorporate the dependency on oxygen concentration, which is not included in the references cited above.
3.5.3. Comparison of model with experimental data
Fig. 3.10 shows the effect of
57
edge of the effectiveness factor curve where (1) they are not diffusion limited, but (2) further increases of reaction rate and capacity would lead to diffusion limitations. A number of fibers, regardless of aerobic capacity, tend to reside in positions on the surface that are near a significant decrease in , which is similar to findings in our prior work (Jimenez et al., 2008; Kinsey et al.,
2007) and consistent with the second part of Weisz‟s hypothesis. However, this is not the case for all fibers, many of which are safely in the reaction-control range of the effectiveness factor surface. Thus, based on the present analysis, we must reject the second hypothesis that fibers are generally near the brink of diffusion limitation.
It should be noted that the experimental ATP turnover rates were calculated conservatively, yielding relatively low rates for each muscle type. Further, the experimental data are plotted on the surface with a mito that is greater than (most cases) or equal to the mitochondrial fractional volume in the tissue. That is, the position of data on the surface yields an effectiveness factor that is higher than would be seen if a relatively lower mito were used (data not shown). In addition, the data in Fig. 3.10 were plotted under the high boundary O2 condition, and tissue hypoxia will lead to a dramatic reduction in for many fibers (Fig. 3.8). Finally, more detailed kinetic expressions may lead to a greater sensitivity of aerobic metabolic rate to diffusion. On the other hand, some of the simplifications in our analyses would be expected to make fibers appear more sensitive to diffusion than they are in vivo. For instance, our model assumed a uniform distribution of mitochondria, but we have previously shown that distributional shifts in mitochondria toward the fiber periphery during hypertrophic growth lead to a higher and enhanced ATP turnover rates (Hardy et al, 2009). Thus, fibers may use changes in the distribution of mitochondria as a means of preserving rates of aerobic ATP flux
58
without incurring diffusion limitation (current work is underway to address mitochondrial spatial redistributions).
Fish Mammal Fish Mammal
A B
Figure 3.10: Effect of
literature for a variety of muscle types. Two mitochondrial fractional volumes were used to generate the surfaces, where (A) model and experimental data ranged from 0.024 to 0.109, and (B) model mito
= 0.45 and experimental data ranged from 0.110 to 0.430.
phosphagen recovery measurements in muscle or from measurements of O2 consumption in tissue, isolated fibers or isolated mitochondria assuming 22.4 l O2 per mole of O2, an ATP/O ratio of 2.5, and an intracellular
water content that was 70% of wet mass. In cases where direct measurements were unavailable,
was calculated from the mitochondrial volume density assuming a sustainable rate of O2 consumption in
ml/min/cm3 of mitochondrial volume 3 for mammals (39), and 1 for fishes ( Burpee, et al, 2010).
3.6. Conclusions
The main aim of the present work was to determine the interaction of metabolite and O2 diffusion with key fiber design features such as fiber size, mitochondrial volume density, O2 concentration, and ATP turnover rates. To do so, we developed a simplified algebraic expression for the mitochondrial ATP production rate as a function of ADP, Pi, and O2 concentration. Such
59
an expression can be used to rapidly assess the effects of metabolic concentrations on ATP flux and was employed as a boundary condition in a reaction-diffusion model of the inter- mitochondrial metabolic reactions in the muscle cell. Beard‟s model (Beard, 2005) of mitochondrial oxidative phosphorylation was utilized and a sensitivity analysis was developed to determine the important model parameters that have dominating effects on the ATP flux. These parameters were modified in order to determine apparent Vmax and Km values for ATP formation as functions of O2 and ADP that closely match experimental values for isolated mitochondria available in the literature. Simulations were conducted and data fitting was performed based on the simulated annealing approach to develop an algebraic function that describes the numerical solution of the system of 17 ordinary differential equations for mitochondrial oxidative phosphorylation. The mitochondrial rate law developed in the present study predicted ATP production rate for a wide range of ADP, O2, and Pi concentrations with high accuracy while the accuracy was somewhat lower for cases of significantly low ADP concentration.
The reaction diffusion model was formulated by use of the method of volume averaging to account for all the mitochondria, and the solutions were used to compute the effectiveness factor for various cases. The parameter space delineating the diffusion limitations of aerobic metabolic rates by metabolites and O2 was determined from the model. The model results were compared with the experimental data indicating that most of the muscle fibers analyzed were not strongly limited by diffusion, but were reaction-controlled and often near the brink of substantial diffusion limitations. The fact that most fibers had a combination of reaction rates and fiber sizes that place them well within the reaction control region of the effectiveness factor surface may represent a safety factor that protects the muscle from diffusion limitation that may arise under conditions such as hypoxia. These results provide insight that might be useful in future
60
bioengineering applications to build artificial muscle or muscle replacement tissue or to treat and assess muscle disease.
61
CHAPTER 4
EFFECTS OF MB AND CK ON INTRA-CELLULAR METABOLITE
DIFFUSION LIMITATION
(Based upon and adapted from Dasika SK, Locke BR, Kinsey ST, Effects of Mb and
CK on intra-cellular metabolite diffusion limitation. Biotechnology and
Bioengineering (under preparation).
4.1. Introduction
In aerobic muscle fibers ATP is primarily produced by the aerobic metabolism of oxidative phosphorylation at the mitochondria, which then diffuses to cellular ATPases and is available to meet the energy demand of the cell. However, with the increased workload the demand of ATP increases. It has been proposed that due to larger size of the ADP molecule,
ADP diffusion to and from the mitochondria maybe restricted (Vendelin et al, 2005; Jacobus,
1985), because of which the ATP synthesis by mitochondrial oxidative phosphorylation maybe limited by the availability of ADP. This is a bigger concern when the ATP demand increases and the ATP demand may not be met. Also, the higher ADP concentrations can regulate the movement of mitochondria, implying that the mitochondria cluster near the sites of ATP demand and consumption (Hollenbeck, 1996), implying that the diffusion distance between the ATP production and consumption would be critical for adequate energy supply. The ATP produced in the mitochondrial matrix is accumulated at the cristae, which diffuses to the sites of ATP consumption. The cristae arrangement increases the capacity of mitochondrial ATP production
62
without occupying additional intracellular space. However, it creates difficulties in ATP export from the mitochondrial intracristal space, as diffusional flux requires a significant concentration gradient. (Dzeja and Terzic, 2003). To overcome this limitation there needs to be a phosphotransfer system capable of accelerating ATP/ADP transport thereby providing direct access to ATP in the mitochondrial matrix (Dzeja and Terzic, 2003). Therefore, an alternative source of ATP supply may be necessary for the fiber to sustain the ATP demand. This may be the reason the cell has developed phosphagen systems. The phosphagens are primarily found in vertebrates in the form of creatine (Cr) and phosphocreatine (PCr) where as in non-vertebrates, phosphagens are found in other forms such as arginine (Ar) and arginine phosphate (AP)
Ellington and Kinsey, 1998, Ellington, 2001). The lack of the phosphogen systems would hinder efficient intracellular communication (Dzeja and Terzic, 2003).
An extensive review of the functions of Cr-PCr system in cellular metabolism can be found elsewhere (Wyss and Kaddurah-Daouk, 2000; Walliman et al, 1992; Ingwall, 2004;
Andres et al, 2008; Balaban, 2009). Some of the functions of the Cr/PCr system include (i) spatial energy buffering, where PCr acts as a transporter of high energy phosphates from the sites of ATP production to those of consumption, (ii) proton buffering, and (iii) preventing the rise of intracellular ADP concentration, thus preventing the inactivation of other cellular kinases
(Walliman et al, 1992). In slow-twitch skeletal muscles, where there is a chronically elevated need for ATP during sustained contractions, the PCr shuttle hypothesis was proposed according to which two distinct CK isoenzymes are present, one at the ATP production site and one at the
ATP utilization site (Bessman and Geiger, 1981). The CK isoenzymes in the mitochondria convert Cr to PCr liberating ADP and the PCr diffuses through the cytosol to the ATP consumption sites, where the CK isoenzymes at the consumption site convert PCr to Cr
63
liberating ATP, which can then be consumed. This theory is well documented (Wyss and
Kaddurah-Daouk, 2000). However, in fast-twitch skeletal muscles, the Creatine Kinase (CK) reaction will be in near equilibrium owing to the large pool of PCr available for hydrolysis for immediate ATP synthesis (Wyss and Kaddurah-Daouk, 2000). Though the CK reaction is near equilibrium, under higher ATP demand the displacement of equilibrium in CK reaction may be transmitted in one cellular locale through a near equilibrium over macroscopic distances in the form of waves in which the incoming flux may be transmitted in either direction instantaneously
(Dzeja and Terzic, 2003).
O2 plays a key role in regulating changes in ATP turnover. The Mb facilitated diffusion of O2 was first proposed by Wittenberg in 1959. According to him, Mb adjacent to the cell membrane picks up O2, traverses the cytosol by translational diffusion to unload O2 in the vicinity of mitochondria, and finally diffuses back to the cell membrane in the deoxygenated state (Wittenberg, 1959). Since then, much of the research on Mb has focused on the facilitated diffusion role of Mb. Apart from facilitating diffusion of O2, Mb has been proposed to function as temporary O2 reservoir (reviewed in Jurgens et al, 2000; Papadopoulos et al, 2001; Ordway and Garry, 2004) and as a scavenger of NO (Flogel et al, 2001; Frauenfelder et al, 2001; Rassaf et al, 2007; Kreutzer and Jue, 2004). Numerous reviews have been made ever since Wittenberg
(1959) proposed the role of Mb in facilitating O2 diffusion with some supporting the extent of role (Wittenberg and Wittenberg, 1989; Conley et al, 2000), while many contradicting the same
(Gros et al, 2010; Jurgens et al, 2000).
During periods of high metabolic demand, Mb is an essential protein in the delivery of oxygen from the erythrocyte to mitochondria (Wittenberg and Wittenberg, 1989). However,
64
experiments on mice without Mb, generated by gene-knockout technology showed that Mb is not required to meet the metabolic requirements of pregnancy or exercise in a terrestrial mammal
(Garry et al, 1998). Experiments on knockout mice suggested that Mb is necessary to support cardiac function during development although adaptive responses evoked in some animals can fully compensate for the defect in cellular oxygen transport resulting from the loss of Mb
(Meeson et al, 2001). However, these animals appear to have both cellular and cardiovascular adaptations that might compensate for the lack of Mb (Godecke et al, 1999).
A key element of the Mb-facilitated diffusion theory involves the translational diffusion of Mb in the cell. It has been suggested that the relatively low diffusion coefficient of Mb in the cell cannot support a significant role for Mb in facilitating O2 diffusion, as Mb cannot compete effectively with free O2 diffusion (Lin et al, 2007a). For terrestrial mammals, Mb can contribute to the O2 flux only if intracellular PO2 falls significantly, especially with increased energy demand (Lin et al, 2007a). Also, there is no conclusive experimental evidence for Mb facilitating diffusion of O2 (reviewed in Jurgens et al, 2000). Jurgens et al (1994) concluded that the in vivo diffusion coefficient for Mb (DMb) is 10 times less than that in dilute Mb solutions, which implies the over-estimation of the role of Mb facilitated O2 diffusion. They recalculated the Mb facilitated diffusion flux using the DMb they evaluated and concluded that the contribution from
Mb facilitated diffusion to overall O2 transport is insignificant (Jurgens et al, 1994). The Jurgens group (Endeward et al, 2010) based on the mathematical model studies on the modified Krogh‟s cylinder model concluded that Mb plays no significant role during diastole. McGuire and
Secomb (2001), based on their theoretical studies on O2 consumption observed minor overall effects of Mb facilitated O2 diffusion on the O2 transport. They observed that the diffusivity of
Mb in muscle is two orders of magnitude less than the diffusivity of oxygen as the reason for
65
relatively insignificant role of Mb facilitated O2 diffusion (McGuire and Secomb, 2001).
Papadopoulos et al (2001) determined the radial and longitudinal diffusivity of Mb in skeletal and cardiac muscle cells using FRAP technique and concluded that the translational diffusion of
Mb is very slow within muscle cells, Mb diffusion in muscle cells is isotropic, and the MbO2 gradients in heart muscle may be relatively small and, thus, do not predict a major role of MbO2 diffusion in intracellular O2 transport in the heart.
However, high Mb content will compensate for slow diffusivity of MbO2 (reviewed in
Conley et al, 2000). Fibers with O2 diffusion limitation tend to have higher Mb content than those without O2 diffusion limitation, suggesting the need for facilitated diffusion by Mb (Conley et al, 2000). Furthermore, Mb knockout mice maintain aerobic function, indicating that there is no link between Mb and oxidative phosphorylation (Conley et al, 2000). Gros et al (2010) reviewed the role of Mb facilitated diffusion of O2 and concluded that the role of Mb becomes significant when the PO2 falls below 1.7 mmHg (Lin et al, 2007a; Lin et al, 2007b) (1.7 mmHg being the equipoise diffusion). This result is in agreement with Wyman‟s (1966) work, where the author concluded that Mb may be responsible for a substantial part of the transport of oxygen in muscle, especially at low partial pressures of oxygen, although the author notes that Mb- facilitated O2 diffusion vanishes at full oxygen saturation of Mb where the gradient of Mb-O2 saturation becomes zero. This suggests that the facilitated diffusion role of Mb may be important under hypoxia because of the need to increase the PO2 gradient across the sarcolemma to promote
O2 flux into the cell as well as to enhance intracellular transport of O2 to the mitochondria.
Hypoxia may be prevented by increases in Mb concentration as Mb facilitates oxygen transport in muscle fibers and may reduce extracellular oxygen tension preventing hypoxic cores (Murray,
1974; Terrados et al, 1990; Wittenberg and Wittenberg, 2003). Bekedam et al (2009) evaluated
66
[Mb] in chronic heart failure (CHF) patients and concluded that Mb plays an important role in facilitating O2 transport to mitochondria in CHF patients where oxygen extraction from the blood by skeletal muscle is increased (Bekedam et al, 2009). They hypothesized that an increase of the
[Mb] can improve O2 extraction, which may improve exercise tolerance. As the role of Mb as facilitator of O2 diffusion is well accepted, the present study considers Mb‟s role as a facilitator for O2 diffusion.
Computational and experimental studies on skeletal muscle fibers revealed that while most fibers are not limited by diffusion, some fibers operate on the brink of diffusion limitations
(Chapter 3; Locke and Kinsey, 2008, Jiminez et al, 2008, Dasika et al, 2010). The diffusion limitations become more predominant as the fiber size increases (Jiminez et al, 2008). It was observed that as the fiber size and the ATP demand or the ATP turnover rate increase, the muscle fiber may be limited by metabolite diffusion. Since the Cr/PCr system and the Mb system may facilitate the diffusion of the metabolites, the presence of these components may result in the muscle fiber operating without diffusion limitations. It should be noted that when the fiber is not limited by diffusion, no significant enhancement in metabolite diffusion due to Mb and CK may be observed as diffusion is not the limiting step. Hence the aim of the present study is to analyze the effects of CK and Mb reactions on the diffusion limitations of the skeletal muscle fibers. The reaction-diffusion model used in our previous study (Dasika et al, 2010) was modified to analyze the effects of the CK and myoglobin reactions.
4.2. Modeling methods and formulation
General aspects of the reaction-diffusion model used here are described in previous work
(Chapter 3; Dasika et al, 2010). In this model, the metabolites ATP, ADP, Pi, O2, Mb,
67
Myoglobin-bound O2 (MbO2), Cr, and PCr were included. Briefly, a simplified rate law was developed for mitochondrial ATP production as a function of ADP, Pi, and O2 from Beard‟s model (Beard, 2005). The development of mitochondrial rate law is described in detail in
(Dasika, 2007). The only reactive sink for O2 delivered to the fiber was consumption by the mitochondria. The effectiveness factor (ɳ) was computed from the ratio of the ATPase rate for the case with diffusion constraints to that with diffusion limitations. The value of ɳ ranges from 1
(reaction rate is not limited by diffusion) to 0 (reaction is completely limited by diffusion).
In the following text Mb reaction refers to myoglobin-O2 binding reaction. To analyze the effects of the CK and Mb reactions, the reaction-diffusion model (Dasika et al, 2010) was modified for three cases to include CK alone, Mb alone, and a model without Mb and CK. This was done as follows. In order to study the effects of CK alone, the Mb O2 binding reaction was set to zero. In this case we included the metabolites ATP, ADP, Pi, O2, Cr, and PCr. Similarly to analyze the effects of Mb alone, we set the CK reaction to zero and included the metabolites
ATP, ADP, Pi, O2, Mb, and MbO2. We set the CK and Mb rates to zero and included the metabolites ATP, ADP, Pi, and in order to determine the effects of absence of these reactions.
The steady-state concentrations of metabolites were computed for all the cases and the spatially averaged ATPase rates were computed. Similar computations were performed with case without diffusion limitations as described in (Chapter 3; Locke and Kinsey, 2008, Dasika et al, 2010). To determine the overall effects of the Mb and CK reactions, L and Vmax,ATPase were varied and the averaged (represented by „< >‟) concentrations and reaction rates were determined for each set of parameters. In the case of experimental data comparison, the experimental data were placed on the η surface using given lengths and ATPase rates.
68
4.3. Computational Results and Discussion
4.3.1. Experimental data comparison
Most of the muscle fibers are not limited by diffusion while some fibers operate at the edge of diffusion limitations (Chapter 3; Dasika et al, 2010). Therefore Mb and CK will have no significant effect in enhancing metabolite diffusion as diffusion in not a limiting step. Fig. 4.1 shows the plots for ɳ with Mb and CK and without Mb and CK as a function of
Mb and CK for both the volume fractions.
Fish Mammal Fish Mammal
102 A 102 B
Fish Mammal Fish Mammal
CD
Figure 4.1: Effect of
(A) εmito = 0.1, case without Mb and CK, (B) εmito = 0.1, case with Mb and CK, (C) εmito = 0.45, case
without Mb and CK, (D) εmito = 0.45, case with Mb and CK
69
from measurements of O2 consumption in tissue, isolated fibers or isolated mitochondria assuming
22.4 l O2 per mole of O2, an ATP/O ratio of 2.5, and an intracellular water content that was 70% of
wet mass. In cases where direct measurements were unavailable,
3 mitochondrial volume density assuming a sustainable rate of O2 consumption in ml/min/cm of
mitochondrial volume 3 for mammals (39), and 1 for fishes (S.T.K., unpublished results). Most of the
species operate near the edge of diffusion limitation. The effect of CK is undetectable.
The facilitated effects of Mb and CK reactions are expected to have effect when the cell is limited by diffusion. However no detectable effect of Mb and CK was observed even at the extremes, i.e., for high ATPase rate and larger size cells, wherein the cells are limited by diffusion (Fig. 4.1). Higher mitochondrial volume fraction combined with smaller cell sizes may be the contributing factors for this effect (or no effect) in the presence of Mb and CK.
To illustrate the effects of Mb and CK reactions on spatial changes in concentrations and reaction rates, the reaction-diffusion models were solved for mitochondrial volume fraction εmito
0 = 0.25, L = 25 µm, Vmax,ATPase = 25 mM/min and Vmax,ATPase = 75 mM/min, O2 = 35.1 µM, and
Km,ATPase = 0.15 mM. The [ATP] is expected to be higher in the presence of CK as CK facilitates the diffusion of ADP/ATP. However, when diffusion is not a limiting step, the effect of facilitated diffusion of Mb and CK are insignificant. Fig. 4.2 shows the spatial variation of the steady-state ATP concentration for L = 25 µm for all four models for Vmax,ATPase = 25 mM/min and Vmax,ATPase = 75 mM/min. X = 0 corresponds to the center of the cell while X = 1 corresponds to the periphery of the cell. As expected, the [ATP] is slightly higher in the presence of CK. Although the [ATP] is higher in the presence of CK due to the facilitated diffusion role of CK, the effects of Mb and CK are minimal for Vmax,ATPase = 25 mM/min due to (1) shorter diffusion distances and (2) relatively lower ATP demand. However, for higher Vmax,ATPase, or
70
higher ATP demand, diffusion is expected to be a limiting step, due to which, the facilitated diffusion role of CK is more predominant and the [ATP] is higher in the presence of CK. This can be observed from Fig. 4.2B, which is a plot of the spatial variation of [ATP] for Vmax,ATPase =
75 mM/min. The Mb-O2 binding reaction has no effect on ATP as might be expected under normoxic conditions (Zhang et al, 1999, Jurgens et al, 2000, Beard et al, 2003). Even though O2 is delivered from the capillaries to the mitochondria via diffusion, the O2 delivery is not limited by the diffusion distance (Zhang et al, 1999), which implies that the facilitated diffusion effects due to Mb may not play a significant role in O2 delivery under these conditions.
Fig. 4.3 shows the plots of steady-state spatial variation of [O2] for Vmax,ATPase = 25 mM/min and Vmax,ATPase = 75 mM/min. The steady-state [O2] is slightly higher, although insignificant, in the presence of CK. This result is consistent with the study of Roman et al. (2002) on MM-CK knockout mice, where they concluded that the O2 consumption kinetics are faster in the absence of CK, which suggests an increased O2 consumption, as a result, lower [O2] in the absence of
CK. However, it must be noted that the knockout mice lacking CK develop many adaptive responses to deal with poor ATP buffering capacity, an increase in mitochondrial volume density being one of them. This may be the reason for elevated O2 consumption in the absence of CK in the knockout mice.
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Figure 4.2: [ATP] vs. L for the case without Mb & CK rates(blue +), with Mb (broken red lines), with CK
0 (black +), and with Mb & CK rates (solid yellow line) for εmito = 0.25, L = 25 µm, Km,ATPase = 0.15 mM, O2 =
35.1 µM, and (A) Vmax,ATPase = 25 mM/min; [ATP] is higher in presence of CK due to facilitated diffusion role
of CK, the facilitated diffusion effect of CK is more predominant for higher ATP demand.
A B
Figure 4.3: [O2] vs. L for the case without Mb & CK rates (blue +), with Mb (broken red lines), with CK
0 (black +), and with Mb & CK rates (solid yellow line) for εmito = 0.25, L = 25 µm, Km,ATPase = 0.15 mM, O2 =
35.1 µM, (A) Vmax,ATPase = 25 mM/min (B) Vmax,ATPase = 75 mM/min; Mb does not have significant effect I
enhancing O2.
Fig. 4.4 shows the plots of spatial variation of steady state RATPase for all models for the previously mentioned conditions. The RATPase is higher in the presence of CK. At steady state, the
ATP supply equals to the demand implying that larger amounts of ATP are produced to meet higher ATP demands due to the feedback from the Cr/PCr shuttle confirming the facilitated diffusion role of CK. Again, Mb has no detectable effect on RATPase under these conditions. This result is in agreement with Wu and Beard‟s (2009) theoretical studies on cardiac myocytes, where they found relatively insignificant role of Mb on the cellular energetics under normoxic conditions, although they did not consider the facilitated diffusion role of Mb, rather they
72
considered the O2 storage role of Mb by fixing the total [Mb] and computing the [O2] based on the Mb saturation level (Wu and Beard, 2009).
A B
Figure 4.4: [RATPase] vs. L for the case without Mb & CK rates (lower blue triangle),with Mb (broken red lines), with CK (upper black triangle), and with Mb & CK rates (solid yellow line) for εmito = 0.25, L = 25 µm,
0 Km,ATPase = 0.15 mM, O2 = 35.1 µM, (A) Vmax,ATPase = 25 mM/min, (B) Vmax,ATPase = 75 mM/min; significant drop
in RATPase can be observed for the case without the CK reaction for Vmax,ATPase = 75 mM/min where as the
RATPase does not drop due to the facilitated diffusion due to CK rate.
To determine the overall effects of Mb and CK on the diffusion limitations of metabolites we varied L from 25 – 150 µm and Vmax,ATPase from 1 – 15 mM/min, and computed the averaged concentrations and rates. We considered these conditions in order to evaluate the hypothesis that
Mb and CK are expected to enhance diffusion when diffusion is a limiting step, although a cell as large as 150 µm in radius may not contain Mb. The
CK due to the CK facilitated diffusion of ATP/ADP. Fig. 4.5 shows plots of
CK.
73
Figure 4.5:
With CK, and (D) With Mb & CK; the
reaction is absent implying CK facilitates the diffusion of ATP.
Fig. 4.6 shows the plots of
Again, there is no significant effect of the presence of Mb whereas the
74
A B
CD
Figure 4.6:
With Mb, and (D) With Mb & CK; the
Vmax,ATPase. Mb and CK have no significant effect on
The
However, as the cell size increases, due to the metabolite diffusion limitations within the cell the
75
demand (Fig. 4.7B and 4.7C). This result is consistent with the previous study by Locke and
Kinsey (2008), where they observed significant effects of phosphagen kinase for larger cells with high ATP demand. However, they used linear reaction rate kinetics in their study and did not consider the role of oxygen. The ratios of ɳ with CK alone, Mb alone, and with Mb and CK were computed to determine the effects of Mb and CK quantitatively.
A B
C
Figure 4.7: ɳ vs. L and for
and CK (surface) and with CK (mesh), (C) Without Mb and CK (surface) and with Mb and CK (mesh);
sharper drop in ɳ in the presence of CK reaction can be observed. However, the cells would operate under
reaction control regime with the presence of CK reaction, where otherwise the cells would operate under
diffusion control regime.
Fig. 4.8 shows ratio plots as a function of L and
76
larger effect on enhancement of diffusion of metabolites compared to Mb. These results (Fig.
4.8) imply that under normoxic conditions the presence of CK and Mb will have significant effects for large cells with high ATP demand.
A B
C
Figure 4.8: Ratio of (A) ɳ with Mb vs. without Mb and CK, (B) ɳ with CK vs. without Mb and CK, (C) ɳ
with Mb and CK vs. without Mb and CK, vs. L and
the combined effect is the sum of effects individual rates.
4.3.2. Effect of mitochondrial volume fraction and boundary O2 concentration
To analyze the effects of Mb and CK on diffusion with varying mitochondrial volume fraction and boundary O2 concentration, we used εmtio = 0.01, 0.1, and0.45. We used two
0 different boundary O2 concentrations: O2 = 35.1 µM (PO2 = 20 Torr) representing the normoxic
0 conditions and O2 = 7.85 µM (PO2 = 4.5 Torr) representing the hypoxic conditions. . With an
77
increase in εmtio, the number of mitochondria in the cell increases resulting in larger amounts of
ATP produced by mitochondria and shorter diffusion distances for ATP/ADP between sites of
ATP production and ATP consumption. Thus, the effects CK and Mb are expected to be lower as the εmtio increases. Fig. 4.9 shows plots of ratio of η in the presence of Mb and CK to that when these reactions are absent for εmtio of 0.1 and 0.45. The Mb and CK have significant effects for low volume fraction (εmito = 0.01, Fig. 4.10), where the maximum ratio is 3, while the maximum ratio of ɳ decreases with an increase in εmtio (maximum ratio is 1.7 for εmtio = 0.1, Fig. 4.9A and
1.6 for εmtio = 0.45, Fig. 4.9C). Again, the effects of Mb and CK are predominant for larger cells with higher ATP demand.
As the boundary O2 concentration increases the amount of O2 available for ATP production increases, while the diffusion distance for ATP/ADP decreases with increase in εmtio.
Therefore, the effects of Mb and CK decrease with increases in εmtio and boundary O2 concentration. Conversely, Mb and CK have larger effects for lower εmtio and boundary O2 concentrations. Fig. 4.9A and 4.9B show the plots of the effectiveness factor ratio for high and low boundary O2 respectively for εmtio = 0.1 while Fig. 4.9C and 4.9D show the plots of the ratio for high and low boundary O2 respectively for εmtio = 0.45. The effects of Mb and CK increase with decrease in boundary O2 concentration and εmtio. Again, these effects are predominant for large cell sizes with high ATP demand.
78
100 A 100 B
100 100 CD
Figure 4.9: ratio of ɳ vs. L and for
0 0 0 for (A) εmito = 0.1 and O2 = 35.1 µM, (B) εmito = 0.1 and O2 = 7.85 µM, (C) εmito = 0.45 and O2 = 35.1 µM, (D)
0 εmito = 0.45 and O2 = 7.85 µM; the effect of Mb and CK decreases with increase in εmito and boundary [O2].
79
Figure 4.10: Ratio of ɳ vs. L and for
0 CK for εmito = 0.01 and O2 = 35.1 µM; Mb and CK’s facilitated diffusion role is predominant at low εmito.
4.4. Conclusions
The main aim of the present study was to analyze the effects of Mb and CK on the reaction-diffusion limitations of oxidative metabolism. To do so, the previously developed reaction-diffusion model was modified for three cases to include Mb alone, CK alone, and to exclude Mb and CK. Experimental data reveals that Mb and CK have play insignificant role in enhancing diffusion in biologically relevant range. To analyze where Mb and CK might have an effect, hypothetically large cells were considered. Although cells as large as 150 µm in radius may not contain Mb, this analysis provides with an insight to conditions where the presence of
Mb and CK are important. The facilitated diffusion effects of Mb and CK are significant for large cells with high ATP demand, where these cells are limited by diffusion. The effect of CK in enhancing diffusion is larger compared to that of Mb. As the mitochondrial volume fraction
80
and boundary O2 increase, the effects of Mb and CK on enhancing metabolite diffusion decrease.
Mb and CK have significant effects on the diffusion limitation of metabolites at the extreme limit of cell size and ATP demand, which are near or out of biological relevance.
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CHAPTER 5
SENSITIVITY ANALYSIS
(Based upon and adapted from Dasika SK, Locke BR, Kinsey ST, (Based upon and
adapted from Dasika SK, Locke BR, Kinsey ST, Effects of Mb and CK on intra-cellular
metabolite diffusion limitation. Biotechnology and Bioengineering (submitted).
5.1. Introduction
Diffusion is one of the important phenomena for intra-cellular metabolite transport. For instance, in aerobic metabolism ATP is transported via diffusion from mitochondria to the sites of ATP consumption, while ADP is transported via diffusion to mitochondria for ATP synthesis.
O2 diffuses from the capillaries of the cell to mitochondria. Mammalian muscle fibers typically range between 10 - 100 µm in diameter (Russell et al, 2000) and fibers exceeding the maximum size are thought to comprise aerobic metabolism (Mainwood and Rakusan, 1982; Meyer, 1988;
Hubley et al., 1997; Boyle et al., 2003; Johnson et al., 2004; Kinsey et al., 2005; Hardy et al.,
2006). While most of the cells are not limited by diffusion, larger size cells with high ATP demand may operate at the edge of diffusion limitation (Chapters 3 & 4; Kinsey et al, 2005;
Hardy et al, 2006; Nyack et al, 2007; Kinsey et al, 2007; Locke and Kinsey, 2008; Jiminez et al,
2009; Dasika et al, 2010).
In our previous studies we computed the ɳ (Chapters 3 & 4; Dasika et al, 2010) and examined the various factors affecting metabolite diffusion limitations in the skeletal muscle fibers. These parameters included fiber radius (L), Vmax and Km for ATPase, boundary O2
0 concentration (O2 ), and mitochondrial volume fraction (εmito) (Chapter 3; Dasika et al, 2010).
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We observed that large cells with high ATP demand, represented by Vmax,ATPase, may operate at the edge of diffusion limitations (Dasika et al, 2010). However, other parameters such as εmito,
0 and O2 play an important role in the cell operating under diffusion limitations. Also, under identical conditions, the averaged ATP concentration (represented by
Dasika et al, 2010), implying intra-cellular metabolite diffusion may be the limiting step and cannot be ignored. Fig. 5.1 shows the schematic of the effects of various parameters on the diffusion/reaction limitations in skeletal muscle fibers. When the fibers are limited by reaction, they may be limited by the ATP demand or by the mitochondrial capacity whereas when the fibers are limited by diffusion, they can be limited either by the system properties such as the mitochondrial volume fraction and fiber size, or due to external factors such as membrane transport coefficient, or the boundary oxygen concentration, or due to the intra-cellular transport of various metabolites. Therefore the aim of the present chapter is to identify key parameters that have the largest effects on diffusion limitations, define a range of parameters under which the cell is not limited by diffusion, and to identify the key metabolite, the diffusion of which, may be the limiting step when the cell is limited by diffusion.
Sensitivity analysis (SA) (Brock et al, 1998; Varma et al, 1999) is one of the techniques to quantitatively determine the effects of parameters on a defined function. The sensitivity of any species concentration, for example, with respect to a given parameter is defined as the slope of the concentration with respect to the parameter, where a higher slope indicates a greater sensitivity of the model to the parameter value. The sensitivity coefficient (SC) of any species concentration Ci with respect to a parameter ɸj determines how the species concentration Ci will be affected by perturbations in the parameter ɸj. A negative SC would imply that increasing the
83
parameter ɸj would result in decreases in Ci. The higher the magnitude of SC the more sensitive the parameter ɸj is to the species Ci. Metabolic control analysis (MCA), in essence, is a typical sensitivity analysis of a dynamical system. MCA was first presented by presented in comprehensive forms by (Kacser & Burns, 1973) and (Heinrich & Rapoport, 1974), and Reder
(1988) presented a landmark paper on the formulation of control analysis. Detailed reviews of
MCA have been published (Fell, 1992; Liao and Delgado, 1993; Wildermuth, 2000; Cascante et al, 2002; Wright et al, 2008; Moreno-Sanchez et al, 2008). Most of the systems in which
SA/MCA were applied previously ignored diffusion in their studies while some studies
(atmospheric reaction-diffusion models, intracellular enzyme models, and groundwater reaction- diffusion models) included diffusion while computing SC (Rabitz, et al, 1983; Reuven et al,
1986; Yang et al, 1997; Kholodenko et al, 2000; Tebes-Stevens and Valocchi, 2000).
Figure 5.1: Schematic of various conditions effecting diffusion/reaction control
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5.2. Mathematical methods
The general aspects of the reaction-diffusion model are described in Chapter 3 and
Dasika et al (2010). In brief, we utilized the metabolites ATP, ADP, O2, and Pi with two reactions including the ATPase reaction and the mitochondrial reaction. The averaged species continuity equation for any component i is given by
(5.1)
where, represents the volume averaged concentration of species i, represents the reaction rates, X represents the non-dimensional distance, and t represents time. The control coefficient or sensitivity coefficient (SC) for species i, with respect to parameter , si,j, are computed by
(5.2)
We performed sensitivity analysis with respect to the parameters Vmax,ATPase, Km,ATPase,
0 Kmt,O2, O2 , L, and εmito. The SC s(i,j) is of limited applicability in its original form. The parameters and the various output quantities of a model may have different units. For example, rate coefficients belonging to reactions of different orders have different units. In such cases, the elements of the SC are incomparable. The usual solution to this problem is the introduction of normalized sensitivity coefficients. The normalized SC are computed by
(5.3)
To compute the SC for all species with respect to all parameters, eq. 5.2 was substituted into eq. 5.1. An example calculation is shown below.
85
To compute the SC for ATP with respect to Vmax,ATPase,
(5.4)
where, the subscript T represents , V represents Vmax,ATPase, and
(5.5)
(5.6)
where, the subscripts in the terms , , and represent the SC for ADP, O2, and Pi with
respect to Vmax,ATPase, respectively. Since the initial conditions (IC) and boundary conditions (BC) for all species except for O2 are constant, applying eq. 5.2 to initial and boundary conditions for
SC for ADP, ATP, and Pi with respect to all parameters result in zero IC and BC. The IC for SC for O2 w.r.t all parameters is zero. Similarly, at center of the cell, i.e., at X = 0, SC for O2 w.r.t all parameters is zero. However, we assumed O2 to be delivered to the cell via capillary according the following expression
(5.7)
where, is the membrane transfer coefficient. Applying eq. 5.2 to eq. 5.7 for all parameters
86
(5.8)
The system of equations (eq. 5.1 - 5.8) was then solved simultaneously in MATLAB
V.7.9 (MATLAB, Cambridge, MA, USA) and the steady state solution was computed. η is computed as the ratio of averaged ATPase rate with finite diffusion to the rate calculated when diffusion is not limiting
(5.9)
Where, and represent averaged ATPase rates with finite diffusion and
without diffusion limitations, respectively. The SC for η with respect to and parameter is computed by
(5.10)
where, and are the SC for and with respect to parameter ,
respectively. The SC for all concentrations, rates, and η are then normalized according to eq. 5.3.
87
5.3. Results and discussion
We defined the following parametric values as the reference case: Vmax,ATPase = 25
0 mM/min, Km,ATPase = 0.15 mM, Kmt,O2 = 1100 µm/s, O2 = 35.1 µM, L = 50 µM, and εmito= 0.1.
This case was chosen because the effectiveness factor is just over the edge where diffusion limitations become important. Table 5.1 shows the values for the
SC for these with respect to all parameters. As we increase the Vmax,ATPase, the
(Dasika et al, 2010), hence we would expect a negative SC with respect to Vmax,ATPase. Similarly, as the cell size grows larger, the
0 negative SC with respect to L. Similarly, as Km,ATPase, Kmt,O2, O2 , L, and εmito increase,
0 increases, hence gives positive SC with respect to as Km,ATPase, Kmt,O2, O2 , L, and εmito. From these results, it is evident that Vmax,ATPase is the most sensitive parameter. Our previous studies on
η for skeletal muscle fibers indicated that Vmax,ATPase and L may be the most sensitive parameters
(Dasika et al, 2010). However the SCs with respect each parameter(Table 5.1) indicate that
0 although L is an important parameter, O2 and εmito have larger effects compared to L. This
0 implies that for a given cell size, Vmax,ATPase, O2 , and εmito control the
0 Similarly, the
0 expect positive SC with respect to Vmax,ATPase, Kmt,O2, O2 , and εmito, and negative SC with respect
0 to Km,ATPase and L. Although L is an important parameter, Vmax,ATPase, O2 , and εmito are the parameters which have the largest effects. These parameters control the
0 Vmax,ATPase determines the ATP demand while O2 and εmito regulate the ATP production by the aerobic metabolism of oxidative phosphorylation. Similar analysis can be applied to η, which
88
0 implies that Vmax,ATPase, O2 , and εmito control and thus determine the regions where the cell operates under reaction control or diffusion control.
Table 5.1:
0 Vmax,ATPase Km,ATPase Kmt,O2 O2 L εmito
1100 35.1 25 mM/min 0.15 mM µm/s µM 50 µm 0.1
1.597
η 0.897 -0.028 0.000 0.002 0.020 -0.002 0.014
To determine a limit on these parameters SC were evaluated by varying each parameter individually while keeping others constant and computing concentrations, rates, η, and SC.
However, in the present study we concentrated on varying Vmax,ATPase as it is relatively more sensitive to other parameters. As Vmax,ATPase increases, the ATP demand increases, resulting in the drop of
25 mM/min (Fig. 5.2A), and above this range most of the ATP is depleted. Since
However, for any Vmax,ATPase greater than 25 mM/min, as most of the
89
SC with respect to Vmax,ATPase decreases in magnitude for any Vmax,ATPase greater than 25 mM/min. Fig. 5.2B is the plot of SC for
0 plot (Fig. 5.2B) that the SC with respect to Vmax,ATPase, O2 and εmito are higher than those of the other parameters, implying that for Vmax,ATPase higher than 25 mM/min, the
0 higher εmito or mitochondria, implying that the cell is limited by O2 and εmito for Vmax,ATPase greater than 25 mM/min.
A B
0 Figure 5.2: (A)
εmito are the most sensitive parameters. The solid red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B represent the SC for
0 KmtO2, O2 , L, and εmito respectively.
Fig. 5.3 shows the plot of
Vmax,ATPase.
However, for any Vmax,ATPase greater than 25 mM/min, the
90
not increase (Fig. 5.3A) while the
Vmax,ATPase greater than 25 mM/min. Fig. 5.3B shows the SC‟s for
Vmax,ATPase higher than 25 mM/min, the
0 in O2 and εmito. This implies that larger ATP demand could be met by supplying more ATP by oxidative phosphorylation with more mitochondria represented by εmito, and by supplying more
0 O2, suggesting that the
A B
Figure 5.3: (A)
0 parameters, vs. Vmax,ATPase; Vmax,ATPase, O2 and εmito are the most sensitive parameters. The cell is limited by
diffusion for Vmax,ATPase greater than 25 mM/min. The solid red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B represent the SC for
0 Km,ATPase, KmtO2, O2 , L, and εmito respectively.
Because the
91
higher than 25 mM/min, which may be due to the diffusion limitations within the cell. Hence it may be concluded that the cell operates under reaction control for Vmax,ATPase lower than 25 mM/min while the cell may be limited by diffusion for Vmax,ATPase higher than 25 mM/min. This can be observed in Fig. 5.4, which is the plot of η and SC for η with respect to all parameters vs.
Vmax,ATPase. We assume that the cell is not limited by diffusion for η > 0.9. Hence the cell starts to operate under diffusion control regime for Vmax,ATPase higher than 25 mM/min.
A B
0 Figure 5.4: (A) η (B) SC with respect to all parameters, vs. Vmax,ATPase; Vmax,ATPase, O2 , and εmito control the
diffusion limitation in the cell. The solid red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B represent the SC for
0 O2 , L, and εmito respectively.
Other parameters were varied, one at a time, and the concentrations, rates, η and SC were computed and plotted. The SC were plotted similar to Figs. 5.2 – 5.4 (data included in appendix
0 B). They follow similar trends highlighting Vmax,ATPase, O2 , and εmito as the parameters with the highest sensitivity coefficients. Fig. 5.5 shows the plots of η vs. other parameters. The cell may operate under reaction control regime if εmito is greater than 0.1, Km,ATPase greater than 0.05 mM,
92
0 O2 greater than 35.1 µM, and O2 membrane transfer coefficient greater than 1100 µm/s. Under the conditions mentioned above, the
0 Vmax,ATPase, O2 , and εmito are the parameters with the highest sensitivity coefficients, it may be
0 concluded that for any cell smaller than 100 µm, if
A B
C D
0 Figure 5.5: plots showing η vs. other parameters – (A) Kmt,O2, (B) Km,ATPase, (C) εmito, and (D) O2 , which led to
0 the conclusion that the cell operates under reaction control regime as long as Vmax,ATPase < 25mM/min, O2 ≥
35.1 µM, and εmito ≥ 0.1.
The key result of the present study is the determination of the set of conditions that define the transition from reaction to diffusion control. In our previous studies we observed that the η drops for higher Vmax,ATPase and larger cell sizes (Dasika et al, 2010). However, L was found to
93
be a less sensitive parameter in the present study. Although this result does not alter the conclusion of our previous work, this result was surprising considering that most of the larger size cells would be limited by the diffusion. In order to determine the underlying reason behind this result, we computed η while varying L. Fig. 5.6 shows the plot of η vs. L for two different
Vmax,ATPase. Although cells as large as 400 µm in radius may not exist, we varied L to 400 µm to analyze the effect of extreme values of L. As the Vmax,ATPase increases, the η drops steeply.
However, for cells smaller than 50 µm in radius, the η is close to 1, implying that the cells smaller than 50 µm in radius would be reaction controlled. As most cells are smaller than 100
µm, the previously mentioned set of parametric values, if met, would ensure that the cell is reaction controlled. Moderately larger size cells would operate under reaction control for lower
Vmax,ATPase or lower ATP demand. This is shown in Fig. 5.6A, which is the plot of η vs. L for
Vmax,ATPase = 5 mM/min. This may be the reason most of larger size cells may not have high ATP demand to avoid being operated under diffusion limited region.
A B
Figure 5.6: η vs. L for (A) Vmax,ATPase = 5 mM/min and (B) Vmax,ATPase = 25 mM/min; η drops steeply with
increase in Vmax,ATPase and L; however afore mentioned conditions are applicable for most cells.
94
One of the aims of the present study was to define a region of parameters under which a cell is not diffusion limited. To do so, we utilized the three parameters with the largest sensitivity coefficients, computed η, and identified the region where η is closer to 1. We varied εmito and
Vmax,ATPase and computed η for different O2 concentrations. Fig. 5.7 shows the contour plot of η as a function of εmito and Vmax,ATPase for four different O2 concentrations. The region in red is the region where η is greater than or equal to 0.9 (reaction control). The reaction control region broadens as the boundary O2 concentration increases. Because the mitochondrial volume fraction is an intrinsic property of the cell, the plots in Fig. 5.7 suggest that the cell can avoid operating under diffusion limitation for higher ATP demand or high Vmax,ATPase if the cell is not limited by the O2 supply.
A B
C D
Figure 5.7: η vs. εmito and Vmax,ATPase for (A) O2 = 7.85 µM, (B) O2 = 15 µM, (C) O2 = 25 µM, and (D) O2 = 35.1
µM; the range of values for parameters under which the cell is not limited by diffusion increases with
95
increase in boundary O2 concentration. The η in the region in red, orange, yellow, green, cyan ranges from 1
– 0.9, 0.9 – 0.8, 0.8 – 0.7, 0.7 – 0.6, 0.6 – 0.5 respectively, while the η in the region in blue is less than 0.5.
The next issue that arises from this conclusion is to determine which metabolite caused the cell to be limited by diffusion. In the present study we computed the SC for η with respect to all parameters and found the important parameters that control diffusion limitations. However, in the above we did not determine which metabolite, e.g., oxygen or ATP, in particular is limiting.
As mentioned above, the domain in which the cell operates under reaction control broadens with increases in boundary O2 concentration, which suggests that the O2 diffusion may be the limiting step. To do so, we computed the SC for all concentrations and rates with respect to the diffusion constants for each species.
As DADP (diffusion constant for ADP) increases, the
Similarly, as DATP increases, more ATP is available for hydrolysis, hence the
96
Table 5.2: SC for all concentrations and rates with respect to diffusion constants for all parameters;
transport of O2 from cell periphery to the mitochondria may be the limiting step
DADP DATP DO2 DPi 133 122 1000 281 µm2/s µm2/s µm2/s µm2/s
ADP -0.015 0.093 -0.329 -7.96E-08 ATP 0.064 -0.41 1.446 3.50E-07
O2 -0.013 -0.006 0.302 -7.29E-08
Pi -0.009 0.055 -0.194 -4.69E-08
RATPase 0.01 0.004 0.234 5.39E-08
These results show that the SC with respect to DO2 is higher compared to the others, implying that the diffusion of O2 controls ATP synthesis by aerobic metabolism of oxidative phosphorylation. This result supports our hypothesis that O2 delivery may be the limiting step.
We were interested in determining the limiting step under extreme conditions, where a cell is
0 limited by diffusion. We varied Vmax,ATPase, Km,ATPase, Kmt,O2, O2 , L, and εmito, one parameter at a time, and computed the SC for
97
negative, as expected. However, it is not zero (Fig. 5.8A-D). As the DATP increases more ATP is consumed and we would expect further drop in
A B
C D
0 Figure 5.8: SC for
L; diffusion of O2 is the limiting step
The SC for
As expected, when the cell is not limited by diffusion, the SC with respect to the diffusion constants is close to zero, meaning they are not sensitive (Fig. 5.9A - D). However, under the conditions where the cell is limited by diffusion, the SC with respect to diffusion constant for O2 is higher compared to other metabolites, highlighting that increasing DO2 increases
98
diffusion limitations. This may be the reason the cell may be susceptible to hypoxia under high
ATP demand conditions due to the limitations in O2 delivery to mitochondria for ATP synthesis.
A B
C D
0 Figure 5.9: SC for
and (D) L; diffusion of O2 is the limiting step
We ignored the creatine kinase (CK) reaction and the myoglobin-O2 (Mb) binding reaction in the above analysis. Since CK and Mb have some effect on η at extreme conditions i.e., for larger size cells with significantly higher ATP demand or high Vmax,ATPase (Chapter 4;
Locke and Kinsey, 2008) we also determined the effects of Mb and CK under normal conditions, i.e., for moderate cell sizes with not so high Vmax,ATPase and observed minimal effect of these reactions, as expected. Also in order to reduce computational time, we ignored these reactions in
0 the above calculations. However under extreme conditions such as significantly low O2 and
εmito, for larger size cells with high ATP demand or high Vmax,ATPase, Mb and CK have significant
99
effects on enhancing diffusion of metabolites as shown in Chapter 4. To check if the results in the absence of Mb and CK are comparable to those in their presence under extreme conditions,
0 we computed the SC for a hypothetical case of O2 = 7.85 µM, εmito = 0.01, L = 300 µm, and
Vmax,ATPase = 100 mM/min, which fall in the range where Mb and CK were found to have significant effects on the effectiveness factor, although such a cell is outside of the biologically relevant range.. Although under these extreme conditions Mb and CK enhance the diffusion of metabolites, the ranking of the sensitive parameters does not change. Table 5.3 shows the
0 with respect to all parameters for the base case are presented in table 5.3 Vmax,ATPase, O2 , and
εmito are the most sensitive parameters for
Although the SC for
0 value for Km,ATPase, and the Km,ATPase does not vary significantly. Therefore, the Vmax,ATPase, O2 , and εmito are the most sensitive parameters for
Vmax,ATPase (Fig. 5.2A). Therefore
Table 5.3: Concentrations and rates, and SC with respect to all parameters with all parameters with Mb and
0 CK for hypothetical case of O2 = 7.85 µM, εmito = 0.01, L = 300 µm, and Vmax,ATPase = 100 mM/min, where Mb and CK are expected to have significant effect, along with concentrations and rates, and SC with respect to all
0 parameters with all parameters without Mb and CK for hypothetical case of O2 = 35.1 µM, εmito = 0.1, L = 50
0 µm, and Vmax,ATPase = 25 mM/min; Vmax,ATPase, O2 , and εmito are the most sensitive parameters even in the
presence of Mb and CK, highlighted in bold.
100
0 conc/rates Vmax,ATPase Km,ATPase Kmt,O2 O2 L εmito 100 mM/min 0.15 mM 1100 µm/s 7.85 µM 300 µm 0.01
mM
0 conc/rates Vmax,ATPase Km,ATPase Kmt,O2 O2 L εmito 25 mM/min 0.15 mM 1100 µm/s 35.1 µM 50 µm 0.1
In conclusion, the present study determined the key parameters that control diffusion limitations in skeletal muscle fibers. To do so we computed the sensitivity coefficients for the
0 rates, concentrations, and η with respect to the parameters Vmax,ATPase, Km,ATPase, Kmt,O2, O2 , L,
0 and εmito. Vmax,ATPase, O2 , and εmito are the parameters with the highest sensitivity that control diffusion limitations. When the cell is reaction-controlled, it can either be limited by the mitochondrial capacity or myosin demand (Vmax,ATPase) while when the cell is diffusion limited,
0 the cell can be limited by ATP supply or by O2 , and εmito. We computed the SC with respect to the diffusion constants of all metabolites to determine which species is limiting when the cell operates under diffusion control. We found that the cell is limited by DO2, implying that when the cell is diffusion controlled, it is limited by the supply of O2 to mitochondria, supporting the notion that the cell is susceptible to hypoxia under higher ATP demand conditions.
101
CHAPTER 6
CONCLUSIONS AND FUTURE WORK
6.1. Conclusions
The aim of the present study was to analyze the effects of various parameters on the intra- cellular metabolite diffusion limitations. To do so, we developed a simplified rate law for mitochondrial ATP production as a function of ADP, O2, and Pi, incorporated O2 transport in the reaction-diffusion model, employed volume averaging technique to account for contribution from all mitochondria, and computed η, which is a measure of diffusion limitations. Larger size
0 cells with high ATP turnover rates may be limited by diffusion. Also, as the εmito and O2 decreased, the larger size cells may be limited by diffusion even for moderate ATP turnover rates. Comparison of model analysis with experimental data revealed that none of the fibers were strongly limited by diffusion. However, while some fibers were near substantial diffusion limitation, most were well within the domain of reaction-control of aerobic metabolic rate. This may constitute a safety factor in muscle that provides a level of protection from diffusion constraints under conditions such as hypoxia.
The role of CK and Mb as facilitators of diffusion is well known although there is an on- going debate as to the extent of Mb‟s role in facilitated diffusion. The facilitated diffusion role of
Mb and CK may not be significant when the cell is not limited by diffusion. Hence, the present study aimed at evaluating the facilitated diffusion roles of Mb and CK in enhancing the diffusion of metabolites. To do so the reaction diffusion model was modified to include the CK reaction alone, the Mb O2 binding reaction alone, and to exclude both reactions, along with the ATPase
102
and mitochondrial ATP synthesis reactions. Under normoxic conditions and in larger size cells with higher ATP demand, CK plays a significant role in enhancing diffusion compared to Mb.
Mb and CK do not have significant effects in enhancing the diffusion of metabolites for moderate size cells (smaller than 50 µm in radius) under normoxic conditions. However, they
0 play significant roles in enhancing diffusion at low O2 and εmito, where the cell may be limited by diffusion. Comparing experimental data with the results with cases that include or remove Mb and CK revealed that Mb and CK do not play significant roles in enhancing diffusion in most of the fibers that operate in the reaction controlled regime.
Sensitivity analysis was performed to determine the key parameters that control diffusion and reaction limitations in skeletal muscle fibers. To do so the sensitivity coefficients for the
0 rates, concentrations, and η with respect to the parameters Vmax,ATPase, Km,ATPase, Kmt,O2, O2 , L,
0 and εmito were computed. Vmax,ATPase, O2 , and εmito are the parameters with the highest sensitivities that control diffusion limitations. Furthermore, when the cell is reaction controlled, the cell is limited by the ATP demand reflected by Vmax,ATPase while when the cell is diffusion
0 limited, the cell is limited by ATP supply or O2 , and εmito. We computed the SC with respect to the diffusion constants of all metabolites to determine where the transport of each species is limited and found that the cell is limited by DO2, implying that when the cell is diffusion controlled, it is limited by the supply of O2 to mitochondria, supporting the notion that the cell is susceptible to hypoxia under higher ATP demand conditions.
6.2. Future work
The sacro plasmic (SR) or Ca2+ and actomyosin (AM) ATPases predominantly consume
ATP in the cell (Jeneson et al, 2000). In the present study we lumped all the ATPases and
103
considered a simple Michaelis Menten kinetics that accounts for all ATPase rates. However , more detailed kinetic expressions for the ATPases may lead to a greater sensitivity of aerobic metabolic rate to diffusion. Cooke and Pate (1985) proposed a pseudo Michaelis-Menten kinetics rate expression for SR ATPase. This rate expression together with the Michaelis-Menten kinetics for AM ATPase would account for a detailed expression for ATPases. On the other hand, some of the simplifications in the present analyses would be expected to make fibers appear more sensitive to diffusion than they are in vivo. For instance, the reaction diffusion model in the present study assumed a uniform distribution of mitochondria, but that distributional shifts in mitochondria toward the fiber periphery during hypertrophic growth would lead to a higher and enhanced ATP turnover rates (Hardy et al., 2009). It would be interesting to study the effects of different types of mitochondrial distributions on intra-cellular metabolite diffusion limitations as the mitochondrial distribution alters with the hypertrophic growth of the cell, suggesting that the cell may adapt itself to avoid being limited by diffusion.
104
APPENDIX A
SUPPLEMENTARY TABLES AND FIGURES IN CHAPTER 3
Table A1: Normalized sensitivity coefficients. The parameters xC3, xPiHt, kPiHt, and xANT, which have
dominating effects on ATP production, are highlighted in bold.
Parameter Value S.C.
(Beard, 2005)
R 4.253 3.88E-06
xDH 0.1099 3.25E-06
xC1 0.54088 2.1E-06
xC3 0.14483 5.51E-06
xC4 2.27E-05 3.99E-07
xF1 154.82 6.33E-08
xANT 0.010723 5.91E-05
xPiHt 374420 1.004464
kPiHt 0.000729 -0.19193
xKH 31775000 0
xH1e 250.02 -1.2E-06
kPi,1 0.000139 -3.2E-06
kPi,2 0.000624 2.99E-06
105
Table A2: Values for the fitting constants in the reaction rate law for mitochondrial oxidative
phosphorylation as given in Eq. (18).
Fitting Value Unit
Constants
-1 Vmit 0.095 M s
K1 1 unitless
-4 B K2 10 M
-4 C K3 10 M
-3 A K4 3.86*10 M
A 1.034 unitless
B 2.532 unitless
C 0.8468 unitless
106
A B
Figure A1: Plot of ATP flux vs. A) ADP for [Pi] = 5 mM and [O2] = 50 µM and B) O2 for [ADP] = 0.13 mM
and [Pi] = 5 mM.
A B
CD
Figure A2: ATP flux vs. O2 as substrate varying Pi for [ADP] = 0.13 mM, from simulated data and from expression (Eq. 5); the developed empirical rate law mimics the simulated data with appreciable accuracy
(the absolute error between the simulated data and the model result is close to zero).
107
A. ATP flux vs. O2 for [Pi] = 5 mM
B. ATP flux vs. O2 for [Pi] = 10 mM
C. ATP flux vs. O2 for [Pi] = 20 mM
D. ATP flux vs. O2 for [Pi] = 25 mM
108
APPENDIX B
SUPPLEMENTARY FIGURES IN CHAPTER 5
AB
CD
Figure B1: (A)
respect to all parameters, (C)
0 with respect to all parameters, vs. Km,ATPase; Vmax,ATPase, O2 and εmito are the most sensitive parameters. The solid red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B
0 represent the SC for
109
AB
CD
Figure B2: (A)
respect to all parameters, (C)
0 0 with respect to all parameters, vs. O2 ; Vmax,ATPase, O2 and εmito are the most sensitive parameters. The solid
red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B
0 represent the SC for
110
AB
CD
Figure B3: (A)
respect to all parameters, (C)
0 with respect to all parameters, vs. Kmt,O2; Vmax,ATPase, O2 and εmito are the most sensitive parameters. The solid
red line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B
0 represent the SC for
111
AB
CD
Figure B4: (A)
respect to all parameters, (C)
0 with respect to all parameters, vs. L; Vmax,ATPase, O2 and εmito are the most sensitive parameters. The solid red
line, solid blue line, solid green line, solid black line, solid magenta line, broken red line in Fig. B represent
0 the SC for
112
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BIBLIOGRAPHICAL SKETCH
Santosh Kumar Dasika did his under graduate in Chemical Engineering at the University
College of Technology, Osmania University, Hyderabad, Andhra Pradesh, India. Upon graduation, he did his Masters in Control Systems Engineering at the West Virginia University
Institute of Technology, Montgomery, WV. Upon completion, he joined Florida State University and started working with Dr. Bruce Locke, where he did his Master‟s and PhD in Chemical
Engineering.
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