CALCIUM MOVEMENT IN THE SARCOMERE AND ITS CONNECTION TO

MUSCLE CONTRACTION: A PILOT STUDY

Neil Goldsmith

A Thesis

Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

November 2008

Committee:

Dr. Lewis Fulcher, Advisor

Dr. Ronald Scherer

Dr. Donald Cooper

Dr. John Laird ii ABSTRACT

Dr. Lewis Fulcher, Advisor

The human body uses calcium as an activator for muscle contraction. A muscle contraction begins with the release of calcium from the terminal cisternae into the sarcomere. The interaction of calcium with myoplasmic proteins then causes the muscle to contract. A biophysical model of the sarcomere will be developed in order to use the model to connect chemical concentrations with force production by the muscle. Since the sarcomere is the base contractile unit of muscle, it should therefore be the appropriate starting point for such a model. The model includes calcium release, diffusion, binding, and uptake. Magnesium concentrations are also modeled as they compete for calcium binding sites on parvalbumin,

ATP, and troponin. The binding of calcium to troponin is of special importance because it results in unblocking of the actin sites. The actin will then interact with myosin in a multi-step process that is well understood but poorly quantified. This interaction leads to contraction of the sarcomere and thus the production of force. iii

Dedication

To my grandmother. Someone once told her I could do anything... and she never let me

forget it. iv

Acknowledgement

I gratefully and thankfully acknowledge the Physics department at Bowling Green State

University. I would like to personally thank Dr. Fulcher whose unending curiosity serves as

an example of what scientist should be. I would also like to thank Dr. Laird. He always

had time to listen and a wise word of advice to offer. v TABLE OF CONTENTS

Page

CHAPTER 1. INTRODUCTION ...... 1

CHAPTER 2. MUSCLE DATA ...... 16

2.1 Overviewofmuscletensiondata...... 16

2.1.1 Twitch measurements in the canine vocalis ...... 18

2.1.2 Tetanic contraction in the canine vocalis ...... 21

2.1.3 Twitch contraction in canine posterior cricoarytenoid muscle . . . . . 22

2.1.4 Twitch contraction in thyroarytenoid muscle ...... 24

2.2 Overviewofbiochemicaldata ...... 25

2.2.1 ATP utilization and its temperature dependence ...... 26

2.2.2 Effects of strain on actomyosin kinetics ...... 27

CHAPTER 3. SUMMARY OF EARLIER MODELS ...... 29

3.1 CannellandAllenmodel ...... 32

3.2 Baylor and Hollingworth’s 1998 model ...... 35

3.3 Baylor and Hollingworth’s 2007 model ...... 38

3.4 Shortenetal’s2007model ...... 42

CHAPTER 4. THE PRESENT MODEL ...... 44

4.1 Geometry ...... 44

4.2 SERCApump...... 46

4.3 Calciumrelease ...... 48

4.4 Diffusion...... 49

4.5 Kinetics ...... 50 vi

CHAPTER 5. RESULTS AND DISCUSSION ...... 52

5.1 Results...... 52

5.2 Towardsabettermodel ...... 59

REFERENCES ...... 63 vii LIST OF FIGURES

Figure Page

1.1 Muscleorganization...... 2

1.2 Exampletwitch ...... 4

1.3 Dualtwitch ...... 4

1.4 Steady frequency stimulus contraction ...... 5

1.5 Externalviewofsarcomeres ...... 7

1.6 Myosinpowerstroke ...... 8

1.7 Cannell and Allen’s model geometry ...... 10

1.8 Comparison of BH07 calcium release to Shorten et al.’s calcium release . . . 12

1.9 Force production kinetic scheme ...... 12

1.10 Shorten et al’s model vs. experimental data ...... 13

1.11 Titze’s predicted muscle tension curve vs. experimentaldata ...... 14

2.1 Example twitch with metric overlay ...... 18

2.2 50% relaxation time versus strain in the canine vocalis ...... 19

2.3 Force versus elongation in canine vocalis ...... 20

2.4 Mean contraction time for various species ...... 22

2.5 Temperatureeffectontwitch...... 23

2.6 Latencytimeofatwitch ...... 24

2.7 Twitch of a canine thyroarytenoid muscle ...... 25

2.8 Temperature dependence of resting force and ATP utilization...... 27

3.1 Sarcoplasmic calcium depletion ...... 39

3.2 SERCAkineticscheme ...... 40

4.1 SERCA models theoretical comparison ...... 47 viii

5.1 Surface graphs of calcium concentration gradient ...... 55

5.2 Spatially averaged calcium concentration during a twitch...... 57

5.3 Force producing crossbridge concentration for a twitch ...... 58

5.4 Force producing crossbridge concentration for multiple twitches ...... 59 ix LIST OF TABLES

Table Page

3.1 Half-sarcomeredimensions...... 32

3.2 Crank’s finite difference approximation equations...... 34

3.3 Reaction constants and concentrations of chemicals and proteins from Ref. [11]. 35

3.4 Diffusionconstants ...... 37

3.5 Constants and total concentrations of chemicals and proteins from Ref. [13] . 37

3.6 Concentration of myoplasmic constituents ...... 40

3.7 Rate constants for reactions shown in Fig. 3.2 and the troponin-calcium bind-

ingreaction ...... 41

3.8 Rateconstantsforforceproduction ...... 42

4.1 Half-sarcomere dimensions in present model ...... 45

4.2 Rate constants for reactions in the present model ...... 50 1 CHAPTER 1

INTRODUCTION

Figure 1.1 shows a top-down organizational view of skeletal muscle. The present study

will focus on the myofibril, an element of the muscle fiber, but some features at levels above

the myofibril are important to note. The tendons and to a lesser degree the epimysium,

perimysium, and blood vessels all add an elastic nature to the muscle. This elastic nature is

an important feature of the muscle because it allows the muscle to contract without moving

the limb to which it is attached. In the following pages data for isometric and isotonic muscle

contractions will be presented. Some care is needed to understand the isometric contraction

since the muscle length does not change. In this situation the muscle shortens while the

elastic elements are lengthened until the force of the contraction is balanced by the pull of

the elastic elements. This allows the contracting parts of the muscle to shorten without

moving the attached load. Since parts of the muscle shorten, it is not truly an isometric

contraction.

A second type of muscle contraction is carried out under isotonic conditions. In an

isotonic contraction the muscle contracts against a finite, constant load. In this situation

the muscle will develop tension equal to the load, the elastic elements will transfer that

force to the load, and the load will move. This would be the type of contraction where one

picks up a weight with a slow uniform motion. A sphincter muscle serves as an example of

an intermediate type of contraction. A sphincter is a ringlike muscle that serves to close

bodily passages through contraction. Sphincter muscles are not anchored to a load so they

do not perform an isotonic contraction, and they are required to reduce their size to close the

passage that passes through them, thus their contraction cannot be considered isometric.

Our understanding of a muscle that undergoes a contraction that is neither isotonic or isometric is important because the muscles of interest for this thesis are all skeletal muscles of the larynx and in voiced sounds they are required to perform in a manner that has elements 2

Figure 1.1: Organizational levels of skeletal muscle, taken from Raul654, Deglr6328, and

Rama [1]. of both isometric and isotonic contractions. Some examples are the lateral cricoarytenoid and the posterior cricoarytenoid. The lateral cricoarytenoid muscle rotates the arytenoid cartilage medially when it contracts, which relaxes and adducts the vocal folds [2]. The posterior cricoarytenoid is an antagonistic muscle to the lateral cricoarytenoid. The posterior cricoarytenoid rotates the vocal process of the arytenoid cartilage laterally and abducts the vocal folds [2]. In both of these cases the muscle creates a tension in a pseudo-isometric manner, until it has created enough force to move the load to which the muscle is attached, and then it contracts in a pseudo-isotonic fashion. Finally, when the muscle has moved the 3

attached load to the desired position it returns to a pseudo-isometric contraction. In the

first instance, the isometric contraction is not a true isometric contraction because the load

it is acting against is not infinite, and the muscle is shortening even though the load is not

moving. Once the muscle has generated sufficient force, then it will start to move the load.

As the load is moved it will become greater either by the stretching of an antagonist muscle

or the elongation of an attached elastic element such as a ligament or a tendon. This makes

the load variable, which means that the contraction is not isotonic. Once the force of the

load reaches the force of the muscle, the muscle returns to a state similar to an isometric

contraction. Again, it is not a true isometric contraction because the muscle will constantly

be tensing and relaxing instead of holding a steady length.

The force these laryngeal muscles exert in response to a stimulus has been measured.

The force has even been measured in vivo [3,4], which gives the observer the opportunity to witness a contraction that is neither truly isometric or isotonic. In the body the stimulus is an action potential, but the same response can be observed by the application of electricity to the muscle. This was discovered in 1756 by Leopoldo Marco Antonio Caldani [5,6]. An alternate source credits Luigi Galvani with the discovery in 1780 when he noticed that the limb of a dead frog would move in response to electricity [7]. Galvani’s find led to the art of corpse reanimation in Europe, and was quite the show for several years afterwards [7]. A single action potential or electrical pulse will elicit a physical response that looks similar to the force versus time curve in Fig. 1.2. The time from the onset of mechanical contraction to the maximum developed force is the contraction time, which is different for different muscles.

Muscles with a very short contraction time are called fast-twitch muscles, and those with considerably lomger contraction times are called slow-twitch muscles.

If a second action potential reaches the muscle before it has adequate time to relax, the muscle will exert a greater force than it had previously exerted. In Fig. 1.3 a small piece of muscle tissue was stimulated with two pulses 5 ms apart and then with two pulses 30 4

Figure 1.2: Representative graph of muscle in response to a single stimulus.

Figure 1.3: Summation of two twitches: (dotted trace) two pulses separated by 5 ms, (solid trace) two pulses separated by 30 ms, taken from Titze [8]. 5

ms apart. The trace of the 5 ms delay shows that greater force is produced than a simple

summation of the two individual twitches, suggesting the importance of nonlinear effects.

Peaks associated with the individual twitches can be seen in the 30 ms delay case. The height

of the first peak is approximately 0.18 N, while the summation of two twitches separated by

the 5 ms stimulus delay is greater than 0.5 N, more than double the force of a single twitch.

Figure 1.4: Muscular response to stimulation by a periodic sequence of pulses at frequencies between 50 - 90 Hz, taken from Titze [8].

If a muscle is stimulated at a frequency that is high enough, it will eventually achieve a limit cycle between the relaxation process and the contraction process. At this point the relaxation of muscle cells will be balanced by new contractions in response to the repeated stimulus. This can be seen in Fig. 1.4. It is clearest on the 50 Hz trace at any time beyond

0.15 s. This allows the muscle force to oscillate with a small amplitude about a steady force. 6

Also of note in Fig. 1.4 is a saturation effect with increasing frequency, that is, the 10 Hz increases in stimulation frequency are much less effective in increasing the force at higher frequencies than at lower frequencies.

A muscle fiber is made up of many myofibrils, and each myofibril is made up of many sarcomeres. The sarcomere is the smallest complete unit of muscle contraction and relax- ation. Within the sarcomere are proteins called actin and myosin. The interaction of these proteins is ultimately reponsible for the force produced by the muscle [9]. In Fig. 1.1 one notices that the actin and myosin are parts of long protein chains. The actin chain is called the thin filament, and the myosin chain is called the thick filament. The inhibition of the interaction between the thick and the thin filament is the process that relaxes the muscle.

This inhibition is accomplished by the formation of the troponin-tropomyosin complex [9], a combination of proteins that acts as a barrier between actin and myosin and will not allow them to interact [9].

The tropomyosin inhibition is removed when troponin binds to calcium [9]. At resting calcium levels within the sarcomere, most of the troponin is not bound to calcium and this leaves the tropomyosin in a position to inhibit the thin filament from binding to the thick filament. The calcium required to overcome this inhibition and thus cause a muscle contraction is stored within the sarcoplasmic reticulum and the terminal cisterna [9]. These are the membranes that surround the sarcomeres in Fig. 1.5. While they are considered different from each other, they are continuous and unless otherwise stated will be referred to as the sarcoplasmic reticulum. Two terminal cisterna are separated by a transverse tubule.

The terminal cisternae and the transverse tubule form a structure called the triad of the reticulum or the triadic junction. When an action potential travels down the transverse tubule it signals the terminal cisternae to release calcium into the extramyofibrillar space

[9]. The extramyofibrillar space is a small gap between the outer wall of the sarcomere and the region of the sarcomere containing troponin, and filaments of actin, and myosin, which is 7

Figure 1.5: Sarcomeres, myofibrils, and their surrounding structures, taken from Guyton and Hall [9].

called the myofibrillar space. Unless the distinction is necessary, the extramyofibrillar space and the myofibrillar space will be referred to as the myoplasmic space. The extramyofibrillar space promotes the diffusion of calcium by allowing it to diffuse laterally without the binding of troponin to slow its progress. The calcium will diffuse throughout the sarcomere and bind to troponin. Troponin then releases the barrier to interaction of the thin filament with the thick filament and the actin and myosin react with each other [9].

When a myosin head (Fig. 1.1) attaches to an actin molecule they form a crossbridge [9].

In Fig. 1.6 one sees the myosin head of the crossbridge rolling forward towards its hinge. This is the power stroke [9]. This motion drags the actin towards the center of the sarcomere and contracts the sarcomere. The myosin head can then bind to adenosine triphosphate 8

Figure 1.6: The conformational change in the myosin head that creates force in a muscle by a rotation, taken from Guyton and Hall [9].

(ATP) and release itself from the actin filament. The myosin head will then hydrolyze the

ATP into adenosine diphosphate (ADP) and a free phosphate group . The myosin uses the energy released in hydrolyzing ATP to return it to its pre-power stroke position so that it is ready to attach to the actin again. If the actin is available then it will repeat the process until the supply of ATP runs out, or the actin is sufficiently inhibited. This process of crossbridge cycling is known as the walk-along mechanism of muscle contraction [9]. While the basic steps are generally agreed upon, a specific kinetic scheme to accurately describe all the available data has proven elusive.

As the crossbridge cycling occurs and the muscle contracts, calcium buffers in the sar- comere are binding calcium and the Sarcolemmal Endoplasmic Reticulum Calcium ATPase pump, called the SERCA pump, removes calcium from the sarcomere to the sarcoplasmic reticulum [9]. Its operation is believed to remove two calcium ions with the hydrolysis of one ATP molecule. The SERCA pump and the buffers act quickly to lower the calcium con- centration back to near equilibrium levels. At equilibrium the action of the SERCA pump is balanced by a leak of calcium into the extramyofibrillar space. The source of this leak is uncertain but it is believed to be either calcium diffusion through the membrane separating the sarcoplasmic reticulum and the extramyofibrillar space or a backflow of calcium through 9

the SERCA pump [10]. In either case, the SERCA pump maintains a gradient across this

membrane of approximately five orders of magnitude.

Once the calcium is returned to the sarcoplasmic reticulum it diffuses to achieve equi- librium, and a buffer in the terminal cisterna binds to it, in order to store it until the next action potential initiates the process again. The buffer in the terminal cisterna is called calsequestrin [11]. The buffers in the sarcomere are phosphocreatine, ATP, parvalbumin, and two non-regulatory binding sites on troponin that are similar to parvalbumin. Some models use the transport of calcium into the sarcoplasmic reticulum by the SERCA pump as a buffer, since the calcium spends some time bound to the pump and during that time is in neither the myoplasm nor the sarcoplasm [12]. Not every model uses all of these buffers, and they certainly are not the only proteins that bind calcium within the sarcomere. The buffers listed above are those that play the most important roles in calcium regulation within the sarcomere.

In a 1984 paper, Cannell and Allen [11] created a computer model of the process previ- ously described. It was based on a half sarcomere of frog skeletal muscle. It assumed the full sarcomere had mirror symmetry around the M-line, the middle of the sarcomere. They also used a cylinder as a reasonable approximation to the shape of their half-sarcomere. This geometry is often used and is frequently a part of artists’ depictions of the sarcomere, as in

Figs. 1.1 and 1.5. Cannell and Allen further assume axial symmetry, or invariance under ro- tations about the central axis, which allows them to create a model that is a two-dimensional plane, with no reference to the angle about the central axis. This plane is divided into cells.

It has cells to represent the extramyofibrillar space, the myofibrillar space, the terminal cis- ternae, and the sarcoplasmic reticulum. All cells contain some concentration of calcium in the ionized state, Ca2+. Each cell of the myoplasmic space contains parvalbumin, and tro- ponin is present in only the myofibrillar space. Calsequestrin is present within the terminal cisternae and nowhere else. Crank’s finite difference approximation was used to handle the 10

diffusion of calcium [11]. This diffusion occured within the entire myoplasmic space and also

within the sarcoplasmic reticulum.

Figure 1.7: Geometry and divisions in the Cannell and Allen model [11].

The permeability of the membrane between the extramyofibrillar space from the terminal cisternae, denoted by P, in response to a stimulus was approximated by the function

−t/τon L −t/τoff P = Pmax[1 − e ] [e ], (1.1) where P has units of distance per time. In the Cannell and Allen version, L is equal to 1 and

−1 Pmax was 0.062 mm/s . Equation 1.1 is the general form that has been used by others. The uptake of the calcium by the SERCA pump, denoted by U in Fig. 1.7, was approximated by the first-order Hill equation, and it removed the calcium from the extramyofibrillar space into the adjacent sarcoplasmic reticulum cell.

Baylor and Hollingworth [13] recorded some improvements on Cannell and Allen’s

model. In their first paper, published in 1998, they showed the importance of ATP as a 11

buffer and transporter of calcium within the sarcomere. The diffusion of parvalbumin was

also considered, and the calcium release equation was altered to match the shorter duration

of the pulse in more recent data [13]. In a second paper, Baylor and Hollingworth [12] altered

the model to more accurately describe the contraction of fast-twitch mouse fibers. A time

delay, T, was added to the release function. The time variable t was replaced by t − T , and

at any time t < T the release was equal to zero. This parameter was added to account for a delay between the action potential reaching the muscle and the measured release of the calcium. Another difference was the location of the calcium release. In all previous works, the triad of the reticulum was located at the Z-line (see Figs. 1.5 and 1.7). In mammals this structure is offset from the Z-line by 0.5µm, and the new model incorporates this more realistic release point. The final important difference was the addition of a buffering SERCA pump. In all previous models the SERCA pump acted instantaneously to move a given number of moles of calcium from the extramyofibrillar space to the sarcoplasmic reticulum, and calcium never spent any time bound to the pump proteins. It is more realistic to believe that the SERCA pump would bind the calcium and for a certain time, and the calcium would not be present in any of the cells shown in Fig. 1.7 during that time. A kinetic scheme was offered to account for this buffering ability.

The models of calcium movement and regulation described above are important in any attempt to model a muscle, but thus far we have not considered the production of force.

This task was undertaken by Shorten et al. [14] to model fatigue in muscle contraction. In this model additional parameters are used to describe the calcium release, but they produce calcium transients similar to those produced by Baylor and Hollingworth [12], as shown in

Fig. 1.8. While differences are noted between the two functions used to release the calcium in the two models, it is unlikely that these differences are important for the kind of approximate treatment we develop.

Figure 1.9 is the kinetic scheme Shorten et al. [14] used to describe the force production 12

Figure 1.8: Calcium transient in the sarcomere, left graph is from the measurements of

Shorten et al. [14], right graph is from the calculation of Baylor and Hollingworth [12]. The

frequency of the stimulus is 67 Hz. The concentrations are micromolar.

Figure 1.9: A kinetic scheme to model force production in the sarcomere, taken from Shorten et al. [14]. 13

Figure 1.10: Production of tension in a muscle, taken from Shorten et al. [14]. within a sarcomere. In Fig. 1.10 one can see reasonable agreement between the model’s predicted tension and the experimental tension for one type of muscle. There are differences between the graphs, but they are close enough to support the use of the scheme of Fig. 1.9 as a foundation for the production of tension. An addition to the scheme which may achieve a better fit is the Fenn effect [15]. The Fenn effect describes the ability of a muscle to adjust its energy cost to the prevailing mechanical constraints. The energy required is supplied by the hydrolysis of ATP. The muscle seems to adjust the turnover rate of the crossbridges to match the function it is called to perform. This would have an effect on the transition in Fig. 1.9 from post power stroke to the detached state. If a muscle is shortening and exerting more force than its load, the crossbridges turn over at a greater rate because fewer crossbridges are required to produce force but more detached crossbridges are needed to perform the walk-along mechanism. If the muscle is exerting less force than is needed to move a load, the crossbridges will stay attached and as more crossbridges form, more force is produced. Once the muscle has formed enough crossbridges to overcome the opposing force, it will start to cycle the crossbridges and begin the walk-along mechanism. This effect is not accounted for 14 in the model of Shorten et al. [14].

Figure 1.11: Comparison of measured muscle tension with Titze’s parameterization, taken from Titze [8].

Our interest in the physiology of generating force by muscles began with the study of

Titze’s paper [8] concerned with neurophysiological sources of jitter in human speech. His approach was based on mathematical representations of individual muscle twitches, and it seemed reasonable that a better understanding of those twitches would furnish a more solid foundation for possible causes of jitter and shimmer. However, this undertaking proved too ambitious for us, and we decided to focus on constructing a reliable model of calcium movement in the sarcomere and its role in generating force. We believe that our model has great potential for applications, including the original goal of an explanation for jitter and shimmer.

In Fig. 1.11 one can see Titze’s [8] mathematical representation of muscle twitch. The curve appears to rise too quickly and decrease too slowly. Titze himself discussed this in his paper and offered ideas on how to improve the fit. Using his model he concluded that neurophysiological sources could be a major factor in the production of jitter and 15 shimmer and provided estimates for the percentage of jitter that could be attributed to neurophysiological sources [8]. Improvement on the basic force curve for a muscle would be able to offer a better estimate to the magnitude of neurologic sources role in jitter. One could undertake this by simply altering the mathematical representation with added parameters, as was suggested by Titze [8]. While this certainly could be successful, it would be limited in its applications. Limitations such as this are the impetus for us to develop a more in-depth and fundamental model based on the physiological factors that produce force.

In order to test the validity of our model, we will focus on the calcium transient and will attempt to make our calcium transient fit those measured by Baylor and Hollingworth [12].

Their experiments were done using a fluorescent dye and appear to be the most recent data on the subject. Attempts will be made to create an accurate model for a buffering SERCA pump. The SERCA pump is a saturable pump and is believed to be second order. This allows us to attempt a fit to the second-order Hill equation. Finally, the tension produced in response to a stimulus will be matched to the experimentally measured time courses of various other muscles. These muscles will all be laryngeal muscles but we expect our model to have a generality that makes it appropriate for a wide variety of muscles. 16 CHAPTER 2

MUSCLE DATA

2.1 Overview of muscle tension data

Theory and experimental data drive each other. If data are found that have no previous explanation, the theorist goes to work and comes up with a new idea to explain the scientific relationships underlying the data. The new theory is then able to make predictions that can be tested experimentally. Such progress in the interplay between experiment and theory drives our understanding of the physical world. A concrete example of this method is Ein- stein’s general theory of relativity. Einstein intuitively knew that a light ray was bent in a gravitational field, and yet the universal law of gravitation required that gravity act between two massive objects. Since light has no rest mass, the theory of gravity and the data on light were not consistent in some sense. To reconcile these two disparate ideas, Einstein created his general theory of relativity. With this theory he was able to make predictions about the angular separation of two stars that could be measured during a solar eclipse. During the next eclipse, Arthur Eddington made careful measurements of the locations of these stars and measurements supported Einstein’s theory. The experimental data that drive our the- oretical work are muscle data. Most of these data are in the form of tension versus time graphs. Many of the graphs have the same geometric shape, which is an indication that the same mechanism produces the force, the actomyosin cycle. The graphs differ in several details, and careful experimental work offers good metrics to quantify these differences.

The five measures that will be presented here are latency period, peak tension, time to peak tension, time between peak tension and half maximum tension, and tetanic contraction period. The latency period is the time between the onset of electrical activity in the muscle and the beginning of muscle contraction. Peak tension is simply the maximum force produced by a single twitch, and time to peak tension is the time it takes from the onset of mechanical 17 contraction to reach that maximum force. Time to half-maximum tension is the time it takes for the muscle to relax to half of its peak tension value. The tetanic contraction time is the amount of time it takes under a periodic stimulus to reach tetanus. In Fig. 1.4 one can see the muscle reaching tetanus when the tension levels off. The muscle stimulated at a frequency of 50 Hz reaches tetanus a short time after 150 ms and it took 9 stimulii, and thus 9 twitches, to get to that level.

Most skeletal muscles of the body are composed of a mixture of fast-twitch and slow- twitch muscle fibers, with still other fibers graduated between these two extremes [9]. The

fibers derive their names from the contraction time for each type of fiber. Fast-twitch

fibers are larger than slow-twitch fibers and are used for rapid development of a very strong contraction [9]. Slow-twitch muscle fibers have a smaller diameter and since muscle strength is primarily determined by muscle cross-sectional area, more of this type of fibers are required to match the strength of contraction of a muscle with predominantly fast-twitch fibers [9].

This decreased strength of each contraction is made up for by a longer relaxation time [9].

The relative amounts of the different muscle fibers in a given muscle work together to give a muscle its characteristic contraction. Furthermore, one can train a muscle to react more like either a fast- or slow-twitch muscle. A muscle will adjust as the demands placed on it change. Experiments have shown that the muscle’s contractile proteins in some smaller, more active muscles can be totally replaced in as little as two weeks [9]. These changes at the protein level may help explain how muscle types differ. One type of muscle may have a greater concentration of a certain protein or a different isoform of a protein that performs the same task but does it faster or slower than another isoform. An example of this comes from a paper written by Andruchov et al. [16], where they studied the effects of different isoforms of myosin on cross-bridge kinetics.

There are two other differences between muscle data that are not intrinsic to the muscle.

The temperature at which the data are collected has an effect on the resulting tension data. 18

Cooper et al. [4] took data that show a lower force produced and shorter contraction time

at higher temperatures for the canine posterior cricoarytenoid muscle. The temperature also

has an effect on the rate of ATP utilization, which plays a direct role in the dissociation of

the actomyosin complex into actin and myosin [17]. Another difference is the strain placed

on the muscle while contracting. The actomyosin kinetics are also altered as the strain on the

muscle is changed [18] and [19]. ATP utilization measurements [17], which were made using

fluorescent dyes, establish an additional feature of muscle contraction, that is, a shortening

of the muscle leads to a greater rate of ATP utilization. This is related to the Fenn Effect

[15] that was discussed in Chapter 1.

2.1.1 Twitch measurements in the canine vocalis

Alipour-Haghighi, Titze, and Durham [20] performed in vitro measurements on the twitch characteristics of the canine vocalis muscle. This muscle is part of the thyroarytenoid.

The characteristics they used to describe the twitch were the peak tension, the contraction time, and the 50% relaxation time, denoted by F, C, and R in Fig. 2.1.

Figure 2.1: Schematic representation of twitch response of a canine vocalis muscle, taken

from Alipour-Haghighi, Titze, and Durham [20].

A small piece of the muscle was held between a sample holder and an ergometer arm, 19 which allowed the adjustment of the strain on the muscle prior to stimulating the sample with parallel plate electrodes. After each pulse, the muscle was allowed to relax for 5 minutes. A time of this length is required since the muscle tension will relax on the order of 100 ms, but it may take several seconds to minutes for the calcium distribution to return to equilibrium.

The contraction times varied between 22 and 32 ms, with an average of 25.7 ms. The

50% relaxation time varied between 17 to 37 ms, with an average of 25.2 ms. Interestingly, they found no correlation between the strain placed on the muscle prior to stimulation and the resulting contraction time for the twitch, which led them to conclude that contraction time was independent of strain. An increase in contraction time was found to be associated with tissue from larger animals. The 50% relaxation time showed a slight increasing trend with strain as can be seen in Fig. 2.2. The active force was determined by how much of the

Figure 2.2: The 50% relaxation times versus strain for different canine vocalis muscles. Each letter represents a different canine, taken from Alipour-Haghighi, Titze, and Durham [20]. 20

total force after stimulation was due to passive force. The passive force was the tension in

the muscle before any stimulation. This would be due to elastic elements within the muscle

and had a direct correlation with the elongation of the muscle, as one might expect from an

elastic tissue. Upon stimulation, the muscle would develop greater tension, that is, the total

tension. In order to determine active force, the passive force was subtracted from the total

tension.

The total force minus the passive force gives the active force, or the force produced by the muscle in response to stimulation. Total tension, passive force, and active force curves are represented in Fig. 2.3. Active force decreases as the elongation of the muscle increases.

Figure 2.3: Force versus elongation in canine vocalis showing the relationship of the active

and passive forces, from Alipour-Haghighi, Titze, and Durham [20].

This decrease supports the sliding filament model of muscle contraction. As the elongation

increases, the overlap between the actin and the myosin filaments decreases, and thus the 21

potential number of force-producing cross-bridges is reduced.

2.1.2 Tetanic contraction in the canine vocalis

Alipour-Haghighi, Titze, and Perlman [21] studied the characteristics of the canine vocalis muscle in vitro under stimulation frequencies high enough so that the muscle was unable to relax before the next stimulation. Since a muscle relaxation time is about 100 ms, any stimulation frequency over 10 Hz should show an augmentation of the force. At these frequencies one will see a rise in the peak force up to a certain point where the force rounds off and creates a sigma-shaped curve. Such a curve can clearly be seen at 50 Hz in Fig. 1.4.

A sample of canine vocalis muscle was placed into an egometer that allowed Alipour-

Haghighi, Titze, and Perlman to control the strain on the muscle and observe how that strain affected the tetanic contraction time. The samples were then stimulated to elicit a tetanic response and the time for this response to develop was recorded. No mention was made of the actual rate of stimulation. This data was probably omitted because the result could be evaluated without it. They found that the time it takes for the muscle to reach tetanus increased with increasing strain. At resting length the muscle would have zero strain and the tetanic contraction times ranged from 100 to 200 ms. The strain of the muscle in vivo at any given time is unknown. It would depend on many factors, including the state of tension of any antagonist muscles. This fact taken together with the wide range of measured tetanic contraction times makes these data lack predictive power. What the data do accomplish is to point to a correlation between strain and tetanic contraction time. As strain increases, so does the tetanic contraction time. This serves as more of a qualitative measure than a quantitative measure.

These measurements were done under isometric conditions, essentially an infinite load.

Under isotonic conditions, a similar response is observed up to the point where the force is equal to the load. At that time the load will start to move, the muscle contracts, and the 22

tension stays constant. There is no explanation for why the muscle does not accelerate, but it

is an important feature of muscle contraction that should be addressed by model calculations.

A possible explanation is that the Fenn effect [15] alters the number of attached crossbridges

and thus the force produced. This could possibly be even more biophysically relevant than

the tetanic contraction time and therefore worth noting.

2.1.3 Twitch contraction in canine posterior cricoarytenoid mus-

cle

Cooper et al. [4] measured the dynamic properties of the canine posterior cricoaryteniod muscle in vivo. This differs significantly from the previous studies and allows one to see the dynamics of the muscle as it functions in the body. An interesting point noted by Alipour-

Haghighi, Titze, and Perlman’s paper on tetanic contraction was an increase in contraction time as the mass of the animal increased [21]. Cooper et al. [4] plotted the contraction time of the cricothryoid muscle as a function of the breath duration of a given species as shown in Fig. 2.4.

Figure 2.4: Mean contraction time for various species plotted versus the breath duration,

taken from Cooper et al. [4]. 23

Figure 2.5: The effect of temperature on contraction times and twitch strength, taken from

Cooper et al. [4]

The duration of the respiratory cycle is an exponential function of the animal mass. The plot shows that as the duration of the respiratory cycle increases, so does the contraction time, which connects the contraction time of the cricothyroid muscle to the mass of the animal.

Measurements of the twitch were made at different temperatures, and they showed that the muscle contraction time and peak tension varied with temperature. As the temperature increases, the contraction time and the peak tension decrease. The data are presented in

Fig. 2.5. These measurements were done on the same dog to eliminate any differences in contraction time due to the size of the dog. The mean value for the contraction time was determined to be 33.3 ms. Cooper et al. [4] also measured latency time, which was the delay between the onset of electrical activity in the muscle and the onset of a contraction. The average latency time was 3.3 ms. This latency period is partly responsible for the horizontal parts of the curves near t = 0 in Fig. 2.6. Since the horizontal section of the graph includes a conduction delay along the nerve as well as latency time, the value of the time derived 24

from this horizontal section will be larger than the latency period due to chemical processes

within the sarcomere.

Figure 2.6: The recurrent laryngeal nerve was stimulated at 0 ms and no contraction occurs

until a short time later, marked by a vertical dash, taken from Cooper et al. [4].

2.1.4 Twitch contraction in thyroarytenoid muscle

Measurements were done by two separate groups using different techniques on the canine thyroarytenoid muscle. Cooper, Pinczower, and Rice [3] measured the muscle in vivo and got a contraction time of around 12 ms 1 , as we estimated from Fig. 2.7. Alipour-Haghighi,

Titze, and Durham [20] measured the contraction time of this muscle in a different dog, using in-vitro techniques, and found a contraction time greater than 20 ms.

It is possible that the differences come from the in vitro versus in vivo techniques. It is also possible the canines varied in mass and this can affect the contraction time, as discussed

1D. S. Cooper, personal communication, 18 Aug 2008 25

Figure 2.7: Twitch of a canine thyroarytenoid muscle, taken from Cooper, Pinczower, and

Rice [3].

in connection with Fig. 2.4. This may not be able to explain the difference, because the

mass connection to the contraction time was established for a different muscle and has not

been established for the thyroarytenoid. Although the origin of this observed difference is

unclear, our model has the potential of giving insight into such differences.

2.2 Overview of biochemical data

It is necessary for a muscle to have ATP in order to relax, but not necessary to have it for the production of force. If a muscle contracts in the absence of ATP it will reach a given tension and stay at that tension. This happens when a human dies and the muscle uses up its last ATP sources, with no way to replenish them. This state is known as rigor mortis. In a live healthy muscle the ATP store is constantly being replenished. Of particular interest is the binding of ATP and its subsequent hydrolysis by myosin. In hydrolyzing the ATP, a myosin head is able to store some of the energy from the breaking of the phosphate bond and then release it during the power stroke. This creates a paradox in that ATP is necessary for relaxation but supplies the energy for contraction. That is why the ATP hydrolysis is coupled with the removal of calcium from the muscle and the subsequent inhibition of actin 26

by troponin to produce the relaxation of the muscle cell.

The ATP hydrolysis reaction is interesting and requires special comment due to its dependence on the mechanical state of the system [15]. When ATP is hydrolyzed, it is not

100% efficient in transferring energy. Thus, a temperature change in the muscle is a window into the amount of ATP being hydrolyzed at a given time. It was shown in 1923 by Wallace

O. Fenn that a muscle will increase its energy liberation in proportion to the amount of work done [15]. This requires the muscle to contract, lest no work will be done. Work is defined as a force applied over a distance and during an isometric contraction the load on the muscle does not move so no work can be done. This does not mean that during an isometric contraction, ATP is not hydrolyzed. In fact, ATP hydrolysis in an isometric contraction acted as a baseline for the measurements done by Fenn [15]. It was later shown that the temperature and strain played a role in establishing this baseline [15]. Thus, it would seem that an explanation of the muscular data and its dependence on temperature and strain are intimately linked to the cycling of the crossbridges.

2.2.1 ATP utilization and its temperature dependence

In Fig. 2.8 A, one can see that as temperature increases the resting force exerted by the

muscle increases in an apparently exponential manner. The resting force is the same force

that was referred to as passive force in Alipour-Haghighi, Titze, and Durham [20]. This same

relationship is also noted in Fig. 2.8 B where the resting rate of ATP utilization increases

with increasing temperature. Although the vertical axes of these two graphs differ in scale

and units, they appear to be behaving similarly. This may mean that as ATP utilization goes

up the force produced goes up. This appears to be at odds with the idea that ATP hydrolysis

is linked to the relaxation process in muscles. No explanation of these characteristics was

offered as the goal of the Hilber, Sun, and Irving’s [17] research was experimental. A well

developed model will potentially be able to explain these results. 27

Figure 2.8: Temperature dependence of resting force and ATP utilization, taken from Hilber,

Sun, and Irving [17].

2.2.2 Effects of strain on actomyosin kinetics

Siththanandan, Donnelly, and Ferenczi [18] measured the effects of strain on actomyosin kinetics. ATP was removed from a muscle sample and saturating levels of calcium were introduced to produce rigor within the muscle. This created an environment within the muscle in which the actomyosin complex could form and go through the power stroke, but no ATP was present to detach the myosin from the actin. A caged ATP compound was allowed to diffuse into the muscle so that upon excitation by laser, ATP would be liberated and cause a rapid relaxation that would not be diffusion-limited. In this way the kinetics of 28 the reaction could more clearly be measured. Two such runs were done at different tensions.

At greater initial tension, the rate of decrease in tension due to ATP detachment of cross bridges was greater, presumably because each crossbridge supported more force. It was also noted that crossbridge detachment rate increased with increasing tension. Siththanandan,

Donnelly, and Ferenczi [18] used their measurements to create a kinetic scheme for actomyosin cycling. These kinetics are not used in our model, but the relationship between strain and crossbridge kinetics is important in developing our model. 29 CHAPTER 3

SUMMARY OF EARLIER MODELS

Previous models of the sarcomere have focused primarily either on calcium diffusion and buffering or on tension production. There is surprisingly little published research linking the two in skeletal muscle. Considering the importance of calcium in the production of tension, it would seem important to develop a model that allows progress on both problems simulta- neously. Several factors operate to make the development of such a model less likely. Much of the muscle research being done focuses on cardiac muscles. The tension development research generally looks into the actomyosin system and uses phenomenological approxima- tions for any calcium considerations. The calcium regulation models avoid steps that do not directly impact the calcium. For instance, after troponin binds calcium no more attention is paid to the troponin except to track when it releases the calcium. Such simplifications are necessary as the computers used in the modeling have a finite ability to process information, and the more information one includes the greater the calculation time becomes. The influ- ence of such limitations is dwindling as computers become more powerful, and models have thus become more and more complex.

Cannell and Allen [11] created a model of calcium diffusion and regulation in frog skeletal

muscle, hereafter referred to as CA84 model. One of the motivations for the model was to

estimate the error in the measurement of myoplasmic calcium when an aequorin dye was

used. Aequorin is a fluorescent dye that has been widely used as an intracellular calcium

indicator but the relation between the light response and calcium concentration is nonlinear.

After the release of calcium into the sarcomere, a large calcium gradient in the myoplasm

develops. Thus, when the light signal from the whole preparation is collected, the resulting

light signal is not proportional to the mean concentration of calcium.

Baylor and Hollingworth [13] attempted to improve on this model by the use of data

from a fluorescent calcium indicator called furaptra. This model will be referred to as BH98. 30

They found the calcium release time to be much briefer than Cannell and Allen thought.

Both models have a peak calcium release at 1.7 ms, but the BH98 release controlled τon and

τoff (see Eqn. 1.1) in such a manner as to give a shorter pulse while retaining the same peak release time. A larger amount of calcium was also released in BH98 because of more recent data suggesting a larger calcium transient. This larger transient was somewhat offset by the addition of ATP as a buffer within the sarcomere. Diffusion of ATP and parvalbumin was considered, which helped to decrease the magnitude of the calcium gradient that would develop in the sarcomere in response to a stimulus. Baylor and Hollingworth also introduced a competition between calcium and magnesium for binding sites on parvalbumin, ATP, and the non-regulatory binding sites of troponin.

Baylor and Hollingworth [12] updated BH98 to adjust for differences between amphibian and mammalian skeletal muscle. This updated model will be called BH07. The main differ- ences between BH98 and BH07 are the location of the triadic junction and the waveform of the calcium release. In mammals the triadic junction is located approximately 0.5 µm from the end of the sarcomere, but it is located at the end of the sarcomere in the amphibian skeletal muscle. The release equation was altered in order to match the signal from furap- tra, similar to the adjustments made in their previous paper. The most dramatic change in

BH07 was the addition of a buffering SERCA pump. This was a logical addition because the SERCA pump has to bind calcium in order to transport it across the sarcoplasmic retic- ulum membrane. A SERCA pump binds two calcium ions before pumping them into the sarcoplasmic reticulum. This requires that time must pass between the binding of the first calcium and the binding of the second calcium and also when the calcium is transported across the membrane. The time necessary for the binding and transporting is time when the calcium is not present in either the myoplasm or the sarcoplasmic reticulum, it is bound to the SERCA pump in the membrane between them. A buffering SERCA pump readily accounts for the time when these calcium ions are not present. 31

The models discussed above all stop short of creating a physical force from the cross- bridge formation within the sarcomere. A model developed by Shorten et al. [14], hereafter referred to as SODS07, takes this step. Using a more complex model of calcium release, they were able to achieve calcium transients similar to those of Baylor and Hollingworth, depicted in Fig. 1.8. To produce force they used the kinetic scheme detailed in Fig. 1.9, where the post-power stroke concentration of attached crossbridges is proportional to force. This is one of the few papers that address the entire process of force production within the sarcomere, beginning with the calcium release. SODS07 force production will serve as a basis for force production in our model.

Before summarizing the previous models it is worthwhile to discuss some conventions used in describing them. The first symbol used is a closed bracket ([ ]). This will appear with a chemical symbol inside and should be read as the concentration of that chemical.

For instance [Ca2+] is the concentration of ionized calcium. All concentrations will be given in units of moles per liter, or molar (M). This should not be confused with lower case m which is reserved for units of meters. When the symbol for a chemical shows up without the brackets, such as Ca2+, its units will be moles.

Another convention that is used often in biochemical models is the Hill equation. This equation takes the basic form,

d[S] [S]nH = Vmax nH nH . (3.1) dt [S] + Km

The Hill equation is useful in describing the action of saturable pumps, such as the SERCA pump. The fractional part on the right-hand side of the Eqn. 3.1 approximates the saturation of the substrates at the binding site, and Vmax is a combination of the forward rate for that particular pump and maximum available concentration of binding sites on the pump. Each concentration is raised to the nH power, where nH is the Hill coefficient and is a measure of cooperativity. For instance, if a pump moves one atom at a time, then nH would equal one. If the pump moves two atoms at a time, then nH would equal two. Sometimes there 32

are other factors that can alter this value beyond how many atoms it pumps at a time, but

for now this approximation is adequate. The final constant is Km and is called the Michaelis constant. It is the concentration where the pump works at half of its maximum value.

3.1 Cannell and Allen model

Cannell and Allen’s 1984 model was designed to explore the intracellular processes that produce myoplasmic calcium concentrations during a single twitch and subsequently a rapid series of stimuli creating a tetanus [11]. They also wanted to measure the error that arises when using aequorin as a calcium indicator. These goals were achieved in CA84, but since it was based on an amphibian muscle, and we are interested in mammalian muscle, some ad- ditional considerations will be required. The methods described in CA84 have been adapted to each model that will be presented, attesting to the general nature of their framework.

Figure 1.7 shows the geometry used in CA84 to approximate the half-sarcomere. The

half-sarcomere was divided into 12 radial and 10 longitudinal elements for a total of 120 cells.

The sarcoplasmic reticulum made up 10 of these cells along the top row and the terminal

cistern was allocated a single cell. The extramyofibrillar space made up the 10 cells in the

second row and the remaining 100 cells were used to define the myofibrillar space. The

geometric parameters of the half-sarcomere are listed in Table 3.1.

Table 3.1: Half-sarcomere dimensions. Half-sarcomere dimensions Percent volume of compartments

Radius = 0.5µm Myofibrillar space = 85.0%

Length = 1.1µm Extramyofibrillar space = 6.0%

Volume = 0.86µm3 Terminal cisternae = 3.5%

Longitudinal SR = 5.5%

The release of calcium was calculated by multiplying Eqn. 1.1 by the area of the terminal 33

cistern and the calcium concentration difference between the terminal cisternae and the

adjacent extramyofibrillar space. This yields the equation, d[Ca2+] = P A([Ca2+] − [Ca2+] ), (3.2) dt tc x

with L = 1 in Eqn. 1.1 to determine P. Values chosen for A, Pmax, τon, and τoff were 0.54

2 9 mm mm x 10 liter of muscle , 0.062 s , 1 ms, and 5 ms, respectively [11]. The released calcium atoms went from the terminal cisternae cell into the adjacent extramyofibrillar cell. In Fig. 1.7

the terminal cisternae could be the the cell in the upper left-hand corner and the recipient

cell would be directly below it. The calcium was then allowed to diffuse radially through

the myofibrillar space and laterally through the sarcoplasmic reticulum and the myofibrillar

space. Thus the calcium concentration satisfies a diffusion equation of the form, ∂ [Ca2+]= D∇2[Ca2+] + source terms, (3.3) ∂t 2 −4 mm where D = 7 x10 s , the diffusion constant, and the source terms represent local concentration changes in the due to interaction with other proteins in the medium.

Crank [22] developed a numerical method for the solution of the diffusion equation where he approximated the derivatives in Eqn. 3.3 by finite differences. The spatial derivatives in the cylindrical geometry involved concentration differences in neighboring cells. For the ge- ometry of Fig. 1.7, these differences take the form of the six equations listed below, since it is important to treat the boundary cells differently from those in between. The lateral solid boundary cell (Eqn. 3.4), lateral central cell (Eqn. 3.5), lateral mirror boundary cell

(Eqn. 3.6), radial solid boundary cell (Eqn. 3.7), radial central cell (Eqn. 3.8), and radial mirror boundary cell (Eqn. 3.9) are the six types of cells that require individual equations.

A mirror boundary cell is one that is adjacent to the M-line for lateral diffusion and adjacent to the central axis of the cylinder for radial diffusion. A solid boundary cell is an extramy- ofibrillar cell for radial diffusion, and any cell adjacent to the Z-line for lateral diffusion. Let

2+ th [Ca ]v be the calcium conentration in the v cell. Then the derivatives of Eqn. 3.3 form the parts of the equations listed in Table 3.2. 34

Table 3.2: Crank’s finite difference approximation equations.

d[Ca2+] D v = ([Ca2+] − [Ca2+] ) (3.4) dt dx2 v+1 v 2+ d[Ca ]v D 2+ 2+ 2+ = ([Ca ] − − 2[Ca ] + [Ca ] ) (3.5) dt dx2 v 1 v v+1 2+ d[Ca ]v D 2+ 2+ = ([Ca ] − − [Ca ] ) (3.6) dt dx2 v 1 v d[Ca2+] 4D v = ([Ca2+] − [Ca2+] ) (3.7) dt dr2 v+1 v 2+ d[Ca ]v D 2+ 2+ 2+ = ((2v + 1)[Ca ] − 4v[Ca ] + (2v + 1)[Ca ] − ) (3.8) dt 2vdr2 v+1 v v 1 2+ d[Ca ]v D 2+ 2+ = ((2v − 1)[Ca ] − − (2v + 1)[Ca ] ) (3.9) dt 2vdr2 v 1 v

In addition to its release and diffusion, it is also important to consider the binding of calcium to proteins. The three proteins considered in CA84 are troponin, parvalbumin, and calsequestrin. Troponin has four binding sites for calcium. Two sites are regulatory and have a low affinity for calcium. The other two sites are high affinity sites that are similar to parvalbumin. Parvalbumin has two binding sites that will bind calcium or magnesium.

Calsequestrin only binds calcium and is confined to the terminal cistern. Calcium, and in some cases magnesium, was assumed to bind to each protein through the simple kinetic scheme,

Ca + S ←→kon SCa, (3.10) koff where S (short for substrate) denotes one of the proteins. The equation describing the change in concentration of the protein-substrate complex is

d[SCa] = k [Ca][S] − k [SCA] (3.11) dt on off

Table 3.3 lists the relevant constants controlling the chemical reactions and initial values for

the concentrations.

The final ingredient in CA84 was the inclusion of a SERCA pump to return the con- centration of calcium in the myoplasmic space to equilibrium. This pump is modeled by a 35

Table 3.3: Reaction constants and concentrations of chemicals and proteins from Ref. [11].

Protein Concentration (µM) On rate (M−1s−1) Off rate (s−1)

Troponinregulatory(totalsites) 70(140) 1.2x108 120

Parvalbumin and 940 Calcium

parvalbumin like sites 2.5 x 108 1

Magnesium

6.6 x 104 6

Calsequestrin 31000 240 0.2

Free metal concentrations

Resting myoplasmic [Ca2+] 0.06

Resting SR [Ca2+] 1500

Myoplasmic [Mg2+] 3300

mM first order Hill equation, meaning nH is equal to one, Vmax is 1.0 s and Km is 1.0 µM. Given these three constants and the calcium concentration in the extramyofibrillar space, the SERCA pump will work continuously. Thus, a counterflow of calcium is required to achieve equilibrium. This is modeled using a leak that takes the form of equation 3.2 with a constant permeability. At equilibrium, the amount of leak will match the amount of calcium pumped and the net flux across the membrane will be zero.

3.2 Baylor and Hollingworth’s 1998 model

BH98 added several features to CA84 in order to consider the role of ATP in the reg- ulation of myoplasmic calcium [13]. In BH98 the terminal cisternae and the sarcoplasmic reticulum were not considered. Equation 1.1 controlled the release of calcium into the ex- tramyofibrillar space at the same point as in CA84 but the calcium was allowed to come from a source that was not part of the model. Accordingly, the calcium removed by the SERCA 36

pump was only recorded as a loss in the myofibrillar space, but no gain was recorded else-

where.

Using data gathered from furaptra, Baylor and Hollingworth noted a distinctly briefer

time course for the release of calcium and a much greater peak concentration of myoplasmic

calcium. The constants in Eqn. 1.1, τon, τoff , and L were 1.50 ms, 0.63 ms, and 5. The

µM value of Pmax was chosen to give a peak release rate of 141 ms , although an exact value for this constant was not given. The spatially averaged peak concentration for one twitch was

approximately 18 µM, more than a factor of 2 larger than that of CA84.

The cylinder of the half-sarcomere in BH98 was longer than that for CA84. It had a length of 2µm. Baylor and Hollingworth also added a space near the M-line where no troponin was present. This was due to the length of the half-sarcomere being double that of the thin filament. They did not treat this as an extra space, but simply a space lacking troponin. The half-sarcomere of BH98 was divided into 6 horizontal elements and 3 radial elements. Diffusion was handled using an approximation from Fick’s law,

∆[Ca2+] Flux = −AD , (3.12) ∆x

where the flux measures the change in concentration per unit time, A is the area of the

boundary between two cells, D is the appropriate diffusion constant, ∆x denotes the center

to center distance between the two compartments, and ∆[Ca2+] is the difference between

the calcium concentrations in the adjacent compartments. A similar equation describes any

of the other molecules that diffuse within the sarcomere. Equation 3.12 performs the same

role in BH98 that Eqn. 3.3 does in CA84. More species were allowed to diffuse in BH98.

The diffusive species and their diffusion constants are given in Table 3.4. All calcium and

magnesium bound species are assumed to have the same diffusion constants as unbound

species.

For the sake of comparison, Table 3.5 is included to compare concentrations and con-

stants with CA84. Some of the differences can be explained by different volumes containing 37

Table 3.4: Diffusion constants Species Diffusion constant

2 −4 mm 10 s Free Ca2+ 3.0

Parvalbumin 0.15

ATP 1.4

Table 3.5: Constants and total concentrations of chemicals and proteins from Ref. [13] Protein Concentration (µM) On rate (M−1s−1) Off rate (s−1)

Troponin(regulatory) 240 0.885x108 115

Parvalbumin 1500 Calcium

0.417 x 108 0.5

Magnesium

0.00033 x 108 3

ATP 8000 Calcium

0.1364 x 108 30000

Magnesium

0.015 x 108 45 Free metal concentrations

Resting myoplasmic [Ca2+] 0.1

Myoplasmic [Mg2+] 1000 38

the species and some differences reflect recent data. A different temperature was also used.

The final difference between BH98 and CA84 was the SERCA pump. Baylor and

Hollingworth used a second-order Hill equation with a time-dependent function to turn

it on, that is, 2+ 2+ nH d[Ca ] −t Vmax[Ca ] − τ N ems = [1 e ] 2+ nH nH , (3.13) dt [Ca ]ems + Km

where N, τ , nH, and Km are equal to 10, 1 ms, 2, and 1 µM respectively. No reason was given for the function that turns the pump on other than to allow time for the calcium to

diffuse. The squaring of the concentration terms is reasonable because the SERCA pump is

believed to transport two calcium ions across the sarcoplamic membrane making it a second

order pump. This assumption is consistent with data from Lytton et al. [23], where the

Hill coefficient was reported as 2.1 ± .1. Baylor and Hollingworth also use a Vmax of 1.5

mM s which is larger than that used by Cannell and Allen, but not too different. Baylor and Hollingworth also used the same Michaelis constant as Cannell and Allen.

3.3 Baylor and Hollingworth’s 2007 model

Baylor and Hollingworth’s 2007 model (BH07) was actually two models with different geometry [12]. One model, the one of interest, had a length of 1.2 µm and a radius of 0.484

µm. The second model is longer with a smaller radius. We will only concern ourselves with the shorter model because it has the geometry closest to CA84. The release point was also adjusted to more closely match that of mammals. The triadic junction is located about 0.5 µm from the Z-line in mammals. This releases the calcium into a region of the extramyofibrillar space that allows it to diffuse in three directions as opposed to the two directions previously allowed in the CA84 and BH98. A consequence of this choice is a quicker diffusion of the calcium. Collectively, this increased freedom for diffusion of the unbound calcium, the diffusion of ATP and pavalbumin bound calcium, and the extra binding of calcium, one can see how the sarcomere is able to relieve the calcium gradient 39

that develops when a large amount of calcium is released into the myoplasmic space at a

point.

The release of calcium was handled as before in BH98 but when trying to stimulate the muscle a second time, it was noted that the release of calcium was far less. Since

BH98 and BH07 do not track the calcium in the sarcoplasmic reticulum, they were forced to introduce an independent parameter to describe this second release. The shortcoming of this ad hoc approach is that it only works for stimulation at 67 Hz and only for the fivefold stimulus that they considered. Using any other frequency or number of stimuli will require additional inputs from measurements of a muscle under those conditions. This is a major limitation of this model. The decrease in amount of calcium released as more stimulii come is a consequence of the fact that there must be a limited source of calcium in the sarcoplasmic reticulum, and it will decrease with each stimulus if adequate time is not available to restore equilibrium. This effect was noted in CA84, as we can see from Fig. 3.1.

Figure 3.1: The depletion of calcium in the sarcoplasmic reticulum in response to a single stimulus and a train of stimulii at 30 Hz, taken from Cannell and Allen [11].

Baylor and Hollingworth modelled a buffering SERCA pump to account for the interval of time during which the calcium is bound to the SERCA pump. This choice supersedes the limitations of the Hill approximations used in BH98 and CA84. BH07’s kinetic scheme for the SERCA pump also includes competition for binding sites on the pump from magnesium and hydrogen. Magnesium and hydrogen were assumed to stay at constant concentrations 40

Figure 3.2: A kinetic scheme for a buffering SERCA pump, taken from Baylor and Holling- worth [12]. throughout a muscle contraction. The kinetic scheme for BH07’s SERCA pump is presented in Fig. 3.2, where E is the SERCA pump. The concentration of each myoplasmic constituent is listed in Table 3.6.

Table 3.6: Concentration of myoplasmic constituents Constituent Concentration (µM)

resting [Ca2+] 0.050

resting [Mg2+] 1000

resting [H+] 0.1

troponin 120

SERCA pump 120

parvalbumin 750

ATP 8000

Table 3.7 lists the forward and backward rate constants for all reactions pertaining to the BH07 SERCA pump. The calcium and magnesium reactions with the pump all had time dependence determined by their individual rate constants. Hydrogen was assumed to react 41

several orders of magnitude quicker than calcium and magnesium. Accordingly, hydrogen

would return to equilibrium well before the next timestep in the simulation and so its binding

was considered to always be in equilibrium with the available binding sites on the SERCA

pump. This made hydrogen strictly concentration-dependent.

Table 3.7: Rate constants for reactions shown in Fig. 3.2 and the troponin-calcium binding

reaction Reaction Forward Reverse Ratio

M−1s−1 s−1 µM

Ca2+ + E ↽⇀ CaE 1.74 x 108 6.97 0.04

2+ 8 Ca + CaE ↽⇀ Ca2E 1.74 x 10 8.71 0.05

Mg2+ + E ↽⇀ MgE 8.71 x 104 4.36 50

2+ 4 Mg + MgE ↽⇀ Mg2E 8.71 x 10 87.1 1000 H+ + E ↽⇀ HE instantaneous instantaneous (pK=8)

+ H + HE ↽⇀ H2E instantaneous instantaneous (pK=8)

+ H + H2E ↽⇀ H3E instantaneous instantaneous (pK=6)

+ H + H3E ↽⇀ H4E instantaneous instantaneous (pK=5)

2+ Ca2E ↽⇀ E + (2 Ca ) 3.48M 0 0 Ca2+ + Trop ↽⇀ CaTrop 1.77 x 108 1544 8.72

2+ 8 Ca + CaTrop ↽⇀ Ca2Trop 0.885 x 10 17.1 0.194

Special note should be made on the third reaction from the bottom; Ca2E ↽⇀ E + (2 Ca2+) is the transport step where the SERCA pump releases the bound calcium into the sarcoplasmic reticulum. The units on this rate constant are different from other forward rate constants. Since it proceeds in the absence of any interaction with another chemical or protein, its units are s−1. The reactions in the two bottom rows describe the rate of reaction of calcium with the regulatory binding sites on troponin. One notices that the rate constants change when the troponin has bound one calcium. This is a refinement over previous models 42

that used the same rate constants for both regulatory sites

3.4 Shorten et al’s 2007 model

We only wish to focus on the force production part of Shorten et al.’s work [14], as much of their work involved complications unnecessary for our purposes. Figure 1.9 presents the kinetic scheme proposed by Shorten et al. The constants controlling this scheme are presented in Table 3.8.

Table 3.8: Rate constants for force production Parameter Definition Value (ms−1)

off k0 Troponin without calcium moving from unblocked to blocked 0.15

on kCa Troponin with 2 calcium bound moving from blocked to unblocked 0.15

off kCa Troponin with 2 calcium bound moving from unblocked to blocked 0.05

f0 rate of crossbridge attachment 1.5

fp rate of pre-power stroke crossbridge detachment 15

h0 forwardrateofpowerstroke 0.24

hp reverse rate of power stroke 0.18

g0 rate of post power stroke crossbridge detachment 0.12

A point that we may be able to improve in this scheme is an alteration of the rate of

post-power stroke crossbridge detachment. In a more complex scheme of crossbridge cycling

presented by Gordon, Regnier, and Homsher [24], there are three intermediates that occur

between the power stroke and crossbridge detachment. The first is a strain-sensitive forward

rate that is greater than 500 s−1 at low strain and between 3 and 10 s−1 at high strain.

The next is a release of ADP and is greater than 100 s−1. After this release the crossbridge

binds ATP at a rate termed “very rapid,” but no value was given and the final step is the

actual detachment which happens at greater than 1000 s−1. It is possible that the 0.12 ms−1 43

(120 s−1) detachment rate from Shorten et al. [14] is primarily a result of the rate constant controlling the ADP release. As strain increases, the strain sensitive constant decreases far enough that it becomes the limiting step. This could be incorporated using a variable rate of post-power stroke crossbridge detachment. 44 CHAPTER 4

THE PRESENT MODEL

Each model presented is made up of five basic components, and each component is a little model in and of itself. The five parts are the geometry, the calcium release, the diffusion of chemicals, the chemical interactions, and the SERCA pump which returns the model to equilibrium. These five mini-models work together to create a larger, more comprehensive model. Each of these five basic parts of the model will be presented in a section below. This compact organization should allow the reader an easy way to look up individual parts of the model. In our model, we try to stay as close to Baylor and Hollingworth’s 2007 model as possible. It is the most recent paper we have found on the subject and should offer the best numerical values for the parameters and the most reliable data. Additions were made in an attempt to allow the model to be able to reproduce tension vesus time data for a muscle.

Our model was created using the C++ programming language, and a half-second simulation of calcium movement and force production in the sarcomere takes approximately one-half hour to run on a 3.8 GHz PC.

4.1 Geometry

The geometry of our model follows that of Cannell and Allen’s 1984 model [11]. Cannell and Allen’s model contains cells to represent the sarcoplasmic reticulum, which is not present in the other models. The release point was altered so that the triadic junction was located approximately 0.5 µm from the Z-line, in accord with BH07 [12]. To allow for more exact geometry, the extramyofibrillar space was considered separate from, but adjacent to, the myofibrillar space. This allowed us to set its radius, and that radius would not change regardless of how many radial subdivisions we chose for the myofibrillar space. In the final model we allowed for 6 radial subdivisions of the myofibrillar space and one radial subdivision 45

of both the extramyofibrillar space and the sarcoplasmic reticulum.

The model contained 11 horizontal subdivisions. These were the same across all portions

of the sarcomere. This number was chosen to allow for a better approximation for the location

of the troponin. Our model length is 1.1 µm, and the actin filament, which is surrounded

by the troponin complex, extends 1.025 µm from the Z-line [25]. Therefore, all cells in the

myofribrillar space that are within 1 µm of the Z-line contain troponin, but the last row of

cells adjacent to the M-line do not.

Table 4.1: Half-sarcomere dimensions in present model Half-sarcomere dimensions Divisions

Radius = 0.5 µm Radial Subdivisions

Length = 1.1 µm Myofibrillar space 6

Volume = 0.86 µm3 Extramyofibrillar space 1

Terminal cisternae cell volume = 0.0301 µm3 Sarcoplasmic reticulum 1

Sarcplasmic reticulum cell volume = 0.00473 µm3 Horizontal Subdivisions

All spaces 11

Cell to adjacent cell distance

Radial

Myofibrillar space △rmy = 0.0768 µm Extramyofibrillar space △r = 0.0160 µm

Horizontal

All spaces △x = 0.10 µm

The terminal cisternae were modeled as one cell, embedded in the sarcoplasmic reticu- lum, but much larger in volume than a sarcoplasmic reticulum cell. This size difference is a consequence of the division of the sarcoplasmic reticulum into ten horizontal cells, eleven divisions minus the cell that is assigned to the terminal cisternae. The effects of this volume difference will be discussed in the section on diffusion. Since the triadic junction is moved 46

inward from the Z-line in our model, we are treating the two terminal cisternae as one cell.

In reality they are separated by the T-tubule system, but to model this would require un-

neccessary geometrical complexity in our model, which is not likely to contribute anything

essential. The parameters used in describing the geometry are listed in Table 4.1.

4.2 SERCA pump

We used the buffering SERCA pump modeled in BH07. Adjustments to the rate con- stants for the calcium binding and turnover rate have been altered to more closely approxi- mate the pump data from Lytton et al. [23]. The kinetics of our pump will have a common feature with other chemical reactions in that every chemical and protein presented in one of the following sections on kinetics will bind calcium or magnesium. In the general case the bound species then may, or may not, diffuse before releasing the bound chemical. The only way for a bound species to transport calcium to a different cell is through diffusion.

The SERCA pump will bind calcium and move that calcium to another space without the process of diffusion. This makes it fundamentally different from any other kinetic scheme included in our model.

Both rate constants for the binding of calcium by the proteins of the SERCA pump in the present model are 7 times greater than those in BH07. This brings the pump velocity versus negative logarithm of the calcium concentration (pCa) curve closer to the expected value for SERCA1a, which is the SERCA pump found in fast-twitch skeletal muscle [23].

A comparison of the Baylor and Hollingworth’s pump, the present model’s pump, and the expected SERCA pump in Fig. 4.1. The factor of seven was determined by setting the model’s equilibrium myoplasmic calcium concentration equal to the Michaelis constant, where the pump should be half saturated. The on rate for both steps of calcium binding to the pump were increased until a half-saturation was accomplished, at a factor of 7 increase to the on rates. The turnover rate of the SERCA pump was raised from the BH07 value of 3.48 s−1 to 47

Figure 4.1: Comparison of SERCA pump models with an ideal second-order saturable pump

4.2 s−1. This adjustment was needed to increase the pump speed so that it reached a correct maximum velocity of 1 mM/s [23].

In our model we use a leak through the sarcoplasmic membrane to balance the pumping action of the SERCA pump at equilibrium. This leak has been explained by some as a flux of calcium back through the pump instead of a leak through the membrane. Peinelt and

Apell [26] presented the SERCA kinetics quoted by Baylor and Hollingworth, but Baylor and

Hollingworth only use the portion of their kinetics that pertains to the myoplasmic side of the sarcoplasmic membrane. Peinelt and Apell went on to offer two schemes to model calcium binding on the sarcoplasmic side, but these are not utilized by Baylor and Hollingworth.

Binding on the sarcoplasmic side by the SERCA pump was also included in the scheme presented by Bartolommei et al. [27]. If one considers that the calcium concentration in the sarcoplasm is five orders of magnitude greater than in the myoplasm, it is reasonable that this binding on the sarcoplasmic side could be considerable. This would reduce the number of calcium binding sites available to cells in the extramyofibrillar space. In an attempt to stay close to the model of Baylor and Hollingworth, while still improving it, we increased the 48

on rates for binding calcium by the pump so that it would start out with more sites already

bound and thus fewer sites available to bind more calcium. The increase in the on rates is

an approximation to model the binding of calcium on the sarcoplasmic side without having

to include sarcoplasmic intermediates into our kinetic scheme.

These changes in the SERCA pump kinetics improve the fit of the SERCA pump to experimental values while still keeping the original BH07 scheme intact. We feel it is impor- tant not to stray too far from BH07 since most of our assumptions about the regulation of calcium concentrations come from their work. Furthermore, any changes in one component of the model will lead to changes in another component of the model. In this instance, our

SERCA pump velocity increases at a faster rate with lower calcium concentrations. On the whole, these facts lead to a considerably faster SERCA pump than BH07 modeled. Thus, we will need to adjust the calcium release function in order to achieve the same peak calcium transient at the same time as BH07.

4.3 Calcium release

With the updated SERCA pump, the BH07 calcium release made the spatially-averaged calcium peak 0.8 ms too soon. To achieve the same peak spatially averaged calcium transient at the same time as BH07, that is, 18 µM at 3.7 ms, the parameters in Eqn. 1.1 were adjusted.

Once a satisfactory release function was achieved, the difference between the terminal cisternae calcium concentration and the adjacent extramyofibrillar space calcium concentra- tion was recorded at 0.2 ms intervals. The value of the release function at every 0.2 ms interval was then divided by the difference to create a new set of data points, which when multiplied by this difference would give back the same function. Another curve fit was then done to create a gradient-dependent calcium release function. This step was done so that as the terminal cisternae calcium concentration depleted due to release, the release would decrease in proportion to it. This is in spirit to the approach of CA84, but the calcium 49 concentration in the terminal cisternae was treated as an independent variable in BH07, a limitation that we did not want to include. Equation 4.1 shows the final form of the release function used in our model,

2+ 2+ −(t+T )/τon L −(t+T )/τoff R = C([Ca ]TC − [Ca ]MY)[1 − e ] [e ], (4.1)

2+ 2+ where [Ca ]TC is the terminal cistern calcium concentration, and [Ca ]MY is the adja- cent extramyofibrillar space calcium concentration. The quantities L, C, τon, τoff , and T in Eqn. 4.1 are 4.8, 1.88 x 10−6, 30 ms, 0.4 ms, and 1.4 ms, respectively. For any time t < T the calcium release was taken to be zero. The release, R, is in moles and has to be divided by the volume of the extramyofibrillar cell into which it is released in order to get the appropriate increment in concentration. This allowed the model to keep track of concentrations instead of moles. The same process was used to determine the loss of calcium concentration by the terminal cisternae. The focus on concentration is required because the diffusion equation

(Eqn. 3.3) deals with concentrations.

4.4 Diffusion

Diffusion was considered by using Crank’s finite difference approximation to the deriva- tives of the diffusion equation [22]. This was chosen over Fick’s approximation because

Crank’s method is a numerical solution to the diffusion equation (Eqn. 3.3) using finite dif- ferences, which gives a detailed treatment of the space and time dependence of the diffusing ions, whereas Fick’s law only gives averages over certain volumes. The numerical solution will be more accurate, but will also have some inherent error. Fick’s law would bring the advantage of handling diffusion across incremental spaces of different size. Crank’s approxi- mation requires that radial subdivisions be evenly spaced, and this presents a challenge when one tries to include spaces of uneven radial displacements.

In our model this problem arises between the extramyofibrillar space and the myofibrillar 50

space. In Table 4.1, one can see that the myofibrillar radial increment is almost 5 times

greater than the extramyofibrillar radial displacement. To handle this situation, Eqn. 3.7 was

solved using an average of the two radial increments to calculate the change in concentration

between the two cells. The myofibrillar cell was then treated as a boundary cell and solved

accordingly.

4.5 Kinetics

The kinetics used in the present model are mostly the same as in BH07. In Baylor and

Hollingworth’s 1998 model, they accounted for magnesium binding to ATP, which was not present in their 2008 model. We have kept magnesium binding in our model. Since our model includes the sarcoplasmic reticulum, we also have included calsequestrin kinetics. Table 3.6 contains the concentrations of chemicals and proteins used in our model. The concentration of calsequestrin was 31 mM and the sarcoplasmic reticulum had a calcium concentration of

1.5 mM. Table 4.2 lists the reactions used in our model and rates that control those reactions.

Table 4.2: Rate constants for reactions in the present model Reaction Forward Reverse

M−1s−1 s−1

Ca2+ + Calseqeustrin ↽⇀ CaCalseqeustrin 240 0.2

Ca2+ + Parvalbumin ↽⇀ CaParvalbumin 0.417 x 108 0.5

Mg2+ + Parvalbumin ↽⇀ MgParvalbumin 0.00033 x 108 3

Ca2+ + ATP ↽⇀ CaATP 0.417x108 0.5

Mg2+ + ATP ↽⇀ MgATP 0.13 x 108 390

Ca2+ + Trop ↽⇀ CaTrop 1.77 x 108 1544

2+ 8 Ca + CaTrop ↽⇀ Ca2Trop 0.885 x 10 17.1 51

The method of force production was handled using the kinetic scheme presented in

Fig. 1.9. The constants from Table 3.8 were used in implementing this scheme. One will notice that the Fenn effect is not included in our model. This was attempted, but since it did not improve the results (in fact it made them worse) this idea was left out of our model.

More discussion on this point will be offered in the results and discussion section. 52 CHAPTER 5

RESULTS AND DISCUSSION

5.1 Results

A pilot study is designed to determine the feasibility of a full-scale treatment. In the present study we are trying to determine the possibility of creating an accurate model of muscle twitches using previous computer models of the sarcomere. If an accurate model could be developed then it would provide a good foundation for a study of jitter and shimmer.

The utility of a successful model would not be limited to the study of jitter and shimmer, although these topics were the primary motivation for our interest in this area. In fact, the broad scope of research where this model could be used suggests we should not try to model a specific muscle. Instead, we are attempting to develop a model that fits the general characteristics described in Chapter 2, without a focus on any one specific data set. If we are successful, then subsequent studies could use our framework and kinetics specific to their muscle of interest to specialize the model to their research. Conversely, a successful model may be able to determine kinetics if twitch data of a specific muscle is given.

That being said, at present the model cannot be used as a reliable model of muscle twitch. While the present model offers several important improvements over previous models, these improvements were not able to come together into a model capable of producing quantitative results that matched experimental data well. This conclusion is mainly based on the final force versus time curve in response to a single stimulus. Before we are able to discuss these results we need to discuss, more thoroughly, the small pieces of the model that combine to create the composite model. These smaller parts are the calcium release, SERCA pump, calcium diffusion and geometry, and the kinetics. A discussion of these is necessary to fully understand the force results from the model.

The geometry and the method of handling the calcium diffusion are dependent upon each 53 other. If one chooses all the cells of the model to have the same radial and axial displacements,

Crank’s finite difference approximation offers certain advantages. This is due to the fact that fewer calculations are needed to solve for the diffusion at a given time when compared with

Fick’s law. Using Crank’s method, a single cell requires two calculations, one for diffusion in the radial direction and one for diffusion in the axial direction. Fick’s law requires four such calculations, one for radial diffusion towards the outside of the cylinder, one for radial diffusion towards the center of the cylinder, and one for diffusion in each axial direction.

The downside of Crank’s method, as it pertains to our model, is the limitation presented by requirements on the number of radial divisions in such a model. The extramyofibrillar radial increment is so small that the total radius of the myoplasmic space would need to be divided into 30 elements to get a reasonable approximation to the size of the extramyofibrillar radial increment. While this is possible in the present model, it increases calculation time dramatically. Thirty radial divisions represent 24 more than the number used in our model, and each additional radial subdivision adds 11 cells in the axial direction that need to be calculated, for a total of 264 more cells that have to be considered. Each new cell has two equations to control its diffusion and 7 equations, at the minimum, to control the reactions that change the cell concentrations of their chemical contents. Multiplying the additional

264 cells by the 7 chemical equations creates an additional 1848 calculations. Multiply this by 2 for diffusion of each chemical in the radial and axial directions and the this brings the minimal increase in calculations to 3696. This alone would not be overwhelming, but these extra calculations are done once for every iteration through the program. The program will iterate 2000 times to create a half-millisecond of simulated time. Clearly, the addition of cells to the model has to be done judiciously, or the model would have a run time that would limit its usefulness.

The other option is the use of Fick’s law. This allows variable sizes for the cells, allowing for more customizable cell sizes. The trade-off for this advantage is a requirement of more 54 calculations for each cell. This increase in calculations would be more than compensated for by the decrease in calculations because of the smaller number of cells, leading to an overall decrease in the number of calculations. This would make Fick’s law a better choice if it were not for another consideration that arises with the size of the timestep. Since the diffusion equations for each cell are solved separately, a smaller timestep is required to keep the approximation from reducing the calcium concentration in the cell too much in a given timestep. This smaller timestep is also necessary because Fick’s law only gives the instantaneous flux. Taking this flux as the average flux over a given timestep is reasonable, assuming the timestep is small. The bigger the timestep, the more calcium diffused in a given timestep, and it is possible to exceed the amount of calcium available. Since flux is dependent on calcium concentrations, the greater the change in calcium concentrations during a timestep, the more Fick’s law becomes an unreliable way to solve for diffusion over time.

Two methods to deal with the challenge presented by the small extramyofibrillar space and calcium diffusion were examined. The size of the extramyofibrillar space was maintained in both. In the first, Fick’s law was used to handle the diffusion of calcium between the extramyofibrillar cell and the adjacent myofibrillar cell. This adjacent myofibrillar cell was then treated as a boundary and the myofibrillar space was divided into six radial divisions and diffusion within the myofibrillar space was handled by Crank’s method. In order to keep this method from diverging, the timestep had to be more than 100 times smaller than the value that was used in the final model. A 100-fold decrease in timestep would increase the number of calculations by 100-fold over the same simulated time period. The second method, and ultimately the method we selected, was described in the previous chapter. To examine the validity of this method, mass conservation was checked, and a contour map of the calcium concentration was produced. The present model showed a gain in calcium equal to less than 1% using a 100 Hz stimulation for 0.5 seconds. This extra mass was lost upon 55

return to equilibrium. The system did not retain this extra mass upon return to equilibrium.

The contour maps were examined in sequential time steps to see if the concentration gradient

Figure 5.1: Change in concentration gradient of calcium from 3.8 ms on the left to 4.8 ms

on the right

developed and diminished in a logical manner. A representative of these contour maps is

given in Fig. 5.1. Each line is an iso-concentration line and the concentration drops with

increasing distance from the calcium release point. The extramyofibrillar space is represented

by the cells across the top of the figures and one sees a greater range of concentrations moving

laterally outward from the calcium release. This is due to the lack of troponin binding in

this region. This can also be seen in the radial diffusion at the M-line.

An issue arose when handling the calcium release using the small extramyofibrillar space.

The calcium release was calculated as the number of moles moving out of the terminal cisternae and into the extramyofibrillar cell. The number of moles had to be divided by the volume to make it into a concentration change. The small volume of the extramyofibrillar cell made it important to keep the timestep small so as not to release too much calcium into the extramyofibrillar space. Too large a release would raise the concentration in the 56

extramyofibrillar space above the terminal cisternae. Additionally, if the difference between

the extramyofibrillar cell that receives the calcium and any adjacent myoplasmic cell is too

great, the concentration will not be numerically stable.

The calcium release was handled using the double exponential form from Baylor and

Hollingworth. This calcium release serves as a starting point and thus is an approximation.

It was set in a manner that allowed for the peak spatially averaged myoplasmic calcium concentration to be 18 µM at 3.7 ms. We assume the same restrictions when determining the parameters in Eq. 1.1. Different parameters were needed in order to get the same release with the incorporation of more chemical reactions and a different method of handling diffusion.

An additional consideration when adjusting the release was the depletion of sarcoplasmic reticulum calcium. After the first calcium release, an additional stimulus will release much less calcium. One possible reason for this is the aforementioned depletion. The release in our model does not deplete the sarcoplasmic reticulum enough, but does release considerbly more calcium than the original release of Baylor and Hollingworth.

Another departure from the methods of Baylor and Hollingworth was an adjustment of the SERCA pump kinetics. The reason and justification for this change was previously described but more needs to be said about the features of the pump velocity curve (Fig. 4.1).

At a pCa of 7, the expected pump velocity is considerably slower than the velocity of the pump in the present model. Conversely, at a pCa of 5, the theoretical pump works much faster than the pump used in the present model. This would affect the spatially averaged calcium concentration by increasing the the approach to equilibrium at any concentration less than 1 µM and slowing it at any concentrations above this. Figure 5.2 shows the spatially-averaged calcium concentration in the present model. The peak is at the correct magnitude and time but the concentration at larger times is considerably less than reported by Baylor and Hollingworth [12], who had a concentration at 25 ms of between 1 and 3 µM.

The present model gives a value at this time around 0.5 µM. 57

Figure 5.2: Single twitch spatially averaged calcium concentration in the myofibrillar space

The difference in concentrations at later times is important in bringing to light an issue with the kinetic scheme used to determine the force producing crossbridges. Figure 5.3 shows the concentrations of force-producing crossbridges in response to the calcium released during a single twitch. It peaks at 22.5 ms, has a latency period of 3.3 ms, and a half-relaxation time of 57.5 ms. To determine if the magnitude is reasonable, the calcium concentration was set to 1 mM. This high concentration saturates the troponin and thus allows for the greatest number of force producing crossbridges. The peak concentration in response to a twitch was divided by the maximal concentration at saturating calcium concentrations; this gave a value of 16%. If one examines Titze’s data (Fig. 1.3 and Fig. 1.4), the force of the muscle approaches a 1 N asymptote in Fig. 1.4. In Fig. 1.3, the force of the first twitch peaks at approximately 0.18 N. This would give a single-twitch force divided by the maximum force of 18%.

On the whole, the data on a single twitch matches favorably with several checkpoints.

The one major difference is in the half-relaxation time. The present model’s half-relaxation time is 20.5 ms greater than the largest published value. This is especially problematic when 58

Figure 5.3: Force-producing crossbridge (FPXB) concentration in response to calcium release during a twitch

considered in the light of the greater rate with which the present model moves calcium out of the myoplasmic space. An excellent model of the SERCA pump would pump calcium out slower at tenths of a micromolar concentrations. This would leave more calcium in the sar- comere for a longer time and accordingly, more calcium bound to troponin. The more calcium is bound to troponin, the greater is the concentration of force-producing crossbridges. This points to an issue with the kinetic scheme used to calculate the force-producing crossbridges, it does not relax fast enough.

The slower rate of relaxation in the model becomes a big issue when the calcium re- lease is repeated at a given frequency. Figure 5.4 shows the response to several stimulation rates, and its clear that our model is problematic. The concentration of force-producing crossbridges does not reproduce the features seen in Fig. 1.4 of Alipour-Haghighi, Titze, and

Perlman’s data [21]. Another issue is that the saturation level is the same for all frequen- cies of stimulation. The frequency only determines the time at which the force reaches the maximum level. We suspect that this discrepancy is a result of the kinetic scheme used to model the force production. 59

Figure 5.4: Force producing crossbridge concentration in response to a stimulus frequency

5.2 Towards a better model

An important consideration that needs to be addressed in future models is the temper-

ature dependence of the model. This dependence can be seen in two places, the diffusion

constant and the rate constants controlling the reactions. The diffusion constant is propor-

tional to the increase or decrease in absolute temperature, as can be seen in Eqn. 5.1, the

Stokes-Einstein relation, k T D = b , (5.1) γ where γ is a constant for a given species, kb is the Boltzmann constant, and T is the absolute temperature. Diffusion in this model is dealt with statistically, as is necessary since the number of molecules is simply too large to be followed on a computer. If one were to seek a more microsopic approach with atoms or molecules, the higher temperature would be due to incresed kinetic energy of each molecule. This would cause more collisions between atoms and molecules and thus more reactions per unit time, which would lead to an increase in the forward rate constants controlling those reactions. The increase in temperature also 60

increases the reverse rate as well as the forward rate constants. Without studying the

temperature dependence of an individual reaction, it would be difficult to even approximate

these effects. Temperature effects on a single set of reactions are probably small, but as

more reactions are considered the importance of temperature dependence would probably

grow. The temperature dependence of diffusion and kinetics is the likely cause of changing

characteristic twitches with change in temperature (See Fig. 2.5).

Of course, before temperature dependence may be addressed, an effective model needs

to be created. As previously stated, the present model is not free from inconsistencies and

would not be a reliable framework to build on. Our model needs to be improved before

additions can be made. One of the assets of the present model has proved to be a liability as

well. It includes the greatest number of reactions and most accurate geometric descriptions

of any model we have found in the literature. This is a positive in that we are able to see how

a change in one area affects another area of the model. The downside to this approach is the

difficulty of deciding what parts of our model are most problematic. It is most likely that

a solution to this overall problem will reside in multiple smaller studies aimed at isolating

individual problems.

The first of such studies should include a realistic geometry, correct calcium concen- trations, a calcium release function, and a method for handling diffusion. For this phase, getting the exact form of the release may not be important, since it is only a means to study diffusion. If one evenly divides the myoplasmic space and tracks the calcium concentration using Crank’s method to calculate it, you can get the concentrations at a given point at a given time. I suggest tracking the concentration at the center near the Z and M line and a centrally located cell. Next alter the geometry to accurately describe what is seen in the cell and try different methods of handling the diffusion. This will allow one to check concentra- tions at specific points at specific times and see if it matches Crank’s widely-used method.

Alternately, one could use Fick’s law as the foundation for diffusion calculations. In either 61 case, one should be able to verify the mass conservation and whether the method is capable of reproducing realistic patterns of diffusion.

Once an acceptable method for handling diffusion with a more intricate geometry is achieved, the focus should turn to the SERCA pump. At this point, the concentrations of magnesium and protons need to be added to the model. Then the model should be able to return to equilibrium after a calcium release. In order to test different models for the SERCA pump, a Gauss-Jordan elimination subroutine should be used to determine equilibrium conditions. This was done in the present model due to the large number of linear differential equations created in trying to use the buffering model of the SERCA pump. A possible, and simple, method for altering the SERCA pump would be an intermediate form of the pump that is open to binding on the saroplasmic side of the sarcoplasmic membrane.

This would allow it to bind calcium on the sarcoplasmic side and possibly release this calcium back into the myoplasm. Such a pump would be closer to the actual conditions between the sarcoplasm and the myoplasm than using a leak to balance pump activity. A protein-free membrane is highly impermeable to ions, thus the permeability is due to the proteins in the membrane and not a leak. These proteins are highly selective to the ions they interact with [28]. This explains why the SERCA pump would bind magnesium and protons, but not pump them across the membrane. That is why the movement of calcium across the membrane, in either direction, should be handled with a SERCA pump.

The issue that one would encounter when adding intermediates into a kinetic scheme would be the determination of the rate constants that control the formation and loss of the intermediates. Thus, it might be wise to simply use different published kinetic schemes until one that fits the theoretical pump adequately is observed. This possibility is the reason the

Gauss-Jordan elimination subroutine is so important. It will make the procedure for finding equilibrium concentrations described by the rate equations much more efficient.

Once suitable models for the SERCA pump and for the diffusion are completed, then 62 the buffers and troponin need to be programmed in. It is necessary not to try to solve the issue of force production in the absence of the buffers because they serve to help decrease the amount of free calcium, which in turn should help relax the muscle. This will also have an effect on the release functions. Unfortunately, the force production is so connected with the other processes of the program, it is impossible to remove it from them. This may again be a situation where determining the best force production kinetic scheme is a matter of testing all schemes until a satisfactory one is found. 63 REFERENCES

[1] Raul654, Deglr6328, and Rama, eds. “A top-down view of skeletal muscle.” Wikipedia. 11 Sept. 2005. 29 July 2008 http://en.wikipedia.org/wiki/image:skeletal muscle.jpg.

[2] Zemlin, W. R. (1997). Speech and Hearing Science: Anatomy and Physiology 4th ed. Massachusetts, Allyn & Bacon.

[3] Cooper, D. S., E. Pinczower and D. H. Rice (1993). “Thyroarytenoid Intramuscular Pressures.” Annals of Otology, Rhinology, and Laryngology 102(3):167-175.

[4] Cooper, D. S., M. S. Shindo, M. H. Hast, U. S. Sinha and D. H. Rice (1994). “Dynamic Properties of the Posterior Cricoarytenoid Muscle.” Annals of Otology, Rhinology, and Laryngology 103(12):937-944.

[5] Rothschuh, K. E. (1973). History of Physiology. New York, Krieger Publishing Co.

[6] Gillispie, C. C. (1971). Dictionary of Scientific Biography. New York, Scribner.

[7] Boese, A. (2007). Elephants on Acid and other Bizarre Experiments. New York, Harcourt, Inc.

[8] Titze, I. R. (1991). “A Model for Neurological Sources of Aperiodicity in Vocal Fold Vibration.” Journal of Speech and Hearing Research 34, 460-472.

[9] Guyton, A. C. and J. E. Hall (2001). Textbook of Medical Physiology. 10th ed. Philadelphia, W.B. Saunders Company.

[10] Higgins, E. R., M. B. Cannell, and J. Sneyd (2006).“A Buffering SERCA Pump in Models of Calcium Dynamics.” Biophysical Journal 91: 151-163.

[11] Cannell, M. B. and D. G. Allen (1984).“Model of Calcium Movements During Activation in the Sarcomere of Frog Skeletal Muscle.” Biophysical Journal 45: 913-925.

[12] Baylor, S. M., and S. Hollingworth (2007). “Simulation of Ca2+ Movements within the Sarcomere of Fast-Twtich Mouse Fibers Stimulated by Action Potentials.” Journal of General Physiology 130: 283-302.

[13] Baylor, S. M., and S. Hollingworth (1998). “Model of Sarcomeric Ca2+ Movements, Including ATP Ca2+ Binding and Diffusion, during Activation of Frog Skeletal Muscle.” Journal of General Physiology 112: 283-302. 64

[14] Shorten, P. R., P. O’Callaghan, J. B. Davidson and T. K. Soboleva (2007). “A Mathematical Model of Fatigue in Skeletal Muscle Force.” Journal of Muscle Research and Cell Motility 28: 293-313.

[15] Rall, J. A. (1982). “Sense and Nonsense about the Fenn Effect.” American Journal of Physiology - Heart and Circulatory Physiology 242(1): H1-H6.

[16] Andruchov, O., O. Andruchova, Y. Wang and S. Galler (2006). “Dependence of Cross-Bridge Kinetics on Myosin Light Chain Isoforms in Rabbit and Rat Skeletal Muscle Fibres.”Journal of Physiology 571: 231-246.

[17] Hilber K., Y. B. Sun and M. Irving (2001). “Effects of Sarcomere Length and Temperature on the Rate of ATP Utilisation by Rabbit Psoas Fibres.” Biophysical Journal 531(3): 771-780.

[18] Siththanandan, V. B., J. L. Donnelly and M. A. Ferenczi (2006). “Effect of Strain on Actomyosin Kinetics in Isometric Muscle Fibers.” Biophysical Journal 90: 3653-3665.

[19] Goldman, E. (1987). “Kinetics of the Actomyosin ATPase in Muscle Fibers.” Annual Review of Physiology 49: 637-654.

[20] Alipour-Haghighi, F., I. R. Titze and P. Durham (1987). “Twitch Response in the Canine Vocalis Muscle.” Journal of Speech and Hearing Research 30: 290-294.

[21] Alipour-Haghighi, F., I. R. Titze and A. L. Perlman (1989). “Tetanic Contraction in Vocal Fold Muscle.” Journal of Speech and Hearing Research 32: 226-231

[22] Crank, J. (1975). The Mathematics of Diffusion 2nd ed. New York, Oxford University Press.

[23] Lytton, J., M. Westlin, S. Burke, G. Shull, and D. MacLennan (1992). “Functional Comparisons Between Isoforms of the Sarcoplasmic or Endoplasmic Reticulum Family of Calcium Pumps.” Journal of Biological Chemistry 267(20), 14483-14489.

[24] Gordon, A., M. Regnier, and E. Homsher (1991). “Skeletal and Cardiac Muscle Contractile Activation: Tropomyosin ‘Rocks and Rolls’.” News in Physiological Sciences 16, 49-55.

[25] Peachy, L. D., R. H. Adrian, and S. R. Geiger (1983). Handbook of Physiology, Section 10: Skeletal Muscle Maryland, American Physiologica Society.

[26] Peinelt, C. and H. Apell (2002). “Kinetics of Ca2+, H+, and Mg2+ interaction with the Ion-Binding Sites of the SR Ca-ATPase.” Biophysical Journal 82, 170-181. 65

[27] Bartolommei, G., F. Tadini-Buoninsegni, M. Moncelli, and R. Guidelli (2008). “Electrogenic Steps of the ST Ca-ATPase Enzymatic Cycle and the effect of Curcumin.” Biochimica et Biophysica Acta 1778, 405-413.

[28] Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter (2008). Molecular Biology of the Cell: 5th ed. New York, Garland Science and Taylor & Francis Group.