Simulation of The Contractile Behavior of An Isolated Cardiac Myocyte

By

Lena van der Stap and Patrick McNairnie

Abstract:

The computer simulation program was written for a PC platform in Visual Basic 6.0, an object-oriented programming language. The program allows the user to simply control the contraction cycle in terms of which sarcomeres or myofibrils in the cell are displayed and the number of time intervals in the contraction cycle to be displayed (up to 1000 one-millisecond intervals). The program shows the difference in the onset of activation of contraction of sarcomeres from the periphery of the cell to the cell’s center of mass. We used standard numerical methods to describe the mechanical behavior of cardiomyocyte contraction/relaxation and generated a set of data according to those equations. Using parameters and constants obtained from experiments performed in the lab or taken from the literature, sarcomeres or myofibrils are mapped as an isometric wire frame drawing in x,y,z space. An extruded hexagon best represents the space filling nature of myofibrils in a cardiomyocyte. We hope that a better understanding of the relationship between calcium regulation and contractility through scientific visualization can help improve treatment and prevention of heart disease.

Purpose of Model: To visualize and simulate the contractile behavior of an isolated heart cell as a bundle of myofibrils each of which is a chain of tandem sarcomeres that shorten and lengthen relative to the cell’s center of mass.

Introduction:

This computer simulation program is a simple, multi-component, and empirical way to visualize information about several different aspects of myocyte contraction. Data in the form of shortening curves, calcium transient curves, and charts comparing % shortening of control cells versus experimental cells give us somewhat of an idea of the effects inotropic agents have on contraction and relaxation. However, many subtleties about myocyte behavior under various conditions go unnoticed. For instance, since a cell’s properties are often intricately interdependent, an alteration of one of the many properties may result in unusual behavior not expected when properties are studied individually.

Background

Rees-Stealy Research Foundation Laboratory:

Although we are undergraduate students at San Diego State University, the bulk of our research on cardiac myocyte behavior is conducted at the Rees-Stealy Research Foundation (RSRF). By measuring heart cell contractility and intracellular calcium transients, and by understanding the electrophysiological aspects of contractility, particularly EC coupling, we are able to determine the effects of various pharmaceuticals or compounds on the ability of isolated cardiomyocytes to contract and relax properly. The major areas of research at the RSRF lab include clinical studies of heart performance in diseased states such as diabetic cardiomyopathy or heart dysfunction due to septic shock.

The Heart Cell: Overview of the Physiology of A Twitch Contraction:

Figure 1: Diagram of muscle architecture. Breakdown: muscle – cell – myofibril – sarcomere – myofilaments. Numbered cross-sections: 1 = actin array, 2 = myosin array, 3 = M line, 4 = actin/myosin array in overlapping cross-bridge region. Bio 336 web site at SDSU Home page (www.sdsu.edu)

The structure of a single, isolated heart cell has three levels to its contractile machinery. The sarcomere is the basic unit of contration. About fifty sarcomeres in tandem (end-to-end) make up a myofibril A bundle of 50 to100 myofibrils make up a fiber (or cell). Each sarcomere is made up of thick and thin filaments (mostly myosin and actin, respectively) that interact via crossbridges that extend from the myosin filaments. Six actin filaments are arranged in a hexoganal array around each myosin filament. Myosin filaments are arranged in trigonal arrays relative to each other. So, there is a 2:1 ratio of actin to myosin throughout the cell (see Figure 1). In addition to the contractile machinery, cellular structures such as mitochondria responsible for metabolism are wedged in between myofibrils. Two or more Nuclei make up about 2% of the cell’s volume and carry the genetic material. The sarcoplasmic reticulum(SR) makes up about 0.1% of the cell’s volume. It functions to store, release and resequester the free calcium involved in triggering contraction/relaxation. T-tubules (invaginations of the outer cell membrane) carry the electrical depolarization of the membrane toward the calcium storage area of the SR, effectively triggering the release of that calcium.

Heart cell contraction can be triggered either chemically or electrically.

Chemically, the release of activating neurotransmitters from the sympathetic nervous system regulates the entry of extra-cellular calcium into the cell through norepinephrine- or epinephrine-sensitive Ca++ ion channels (Lindemann, 1995). Since the Ca++ ion channels are also voltage sensitive, that is, they open and close in response to a change in the voltage difference across the membrane, contraction is also triggered electrically by cell membrane depolarization. Once trigger Ca++ enters the cell, it binds to ryanodine receptors on the sarcoplasmic reticulum thus releasing a large amount of stored calcium into each sarcomere in the cell’s myofibrils. This free calcium then binds to troponin causing the troponin-tropomyosin strands to shift into the groove between the double strands of the actin filament ( See Figure 2). This shift allows the myosin heads to bind to the actin filaments that surround them in a hexagonal array. Like many oars pushing a ship forward across a body of water, myosin filaments slide past actin chains, and the sarcomere contracts. Each individual sarcomere in each myofibril contracts in the same manner, but not at exactly the same time. A contraction propagates from the periphery of the cell inward toward its core, because T-tubules trigger peripheral sarcomere contraction first.

During relaxation, Ca++ is taken up by the longitudinal sarcoplasmic reticulum. As the free Ca++ in the cytosol is released from troponin and resequestered, the troponin-tropomyosin strands that line the actin filaments are shifted back towards the outside of the actin filament, and the myosin heads are released from the actin. As a result, the filaments in each sarcomere slide back to resting position, and the cell relaxes.

Figure 2: Actin/Myosin Interaction - Myosin heads as they bind to actin filaments at Troponin I sites along the Tropomyosin strands. (Chien, K.R., 1999)

Instrumentation and Techniques For Data Acquisition:

Several techniques and instruments are used to prepare cells and measure the various aspects of heart cell contractility. A Langendorff perfusion system is used to isolate individual cardiomyocytes. Once isolated, the cells are cultured as normal cells or treated with various compounds to mimic pathological conditions, such as diabetic cardiomyopathy or septic shock. Calcium ion flux is measured using a Photon Technology Inc. Ca++ ion ratio fluorescence system. Cell shortening is measured using a video edge detection system complete with image processing software.

Langendorff Perfusion Device

The Langendorff perfusion device is used to break down a large piece of human heart tissue or a small whole rat heart into individual cells. The tubing and solution wells are jacketed for thermo-regulation. If a whole heart is to be perfused, the aorta is stretched over a cannula through which the perfusate passes. Kreb’s solution is pumped through the cannula in flow that is retrograde from normal in order to perfuse the coronary arteries. Then, a solution containing the enzyme collagenase is pumped into the heart to break down the collagen that holds the cells together. The tissue is broken down further by mincing and agitating in collagenase solution.

Ca++ Ion Ratio Fluorescence System

The movement of Ca++ ions into and out of storage occurs respectively with each contraction and relaxation. It is tracked and recorded using a modified fluorescence microscope connected to detect the intensity of Indo-1, a fluorescent dye indicator which binds to the free calcium. The intensity of fluorescence from the Ca++ dye indicator is directly proportional to the concentration of free Ca++ inside the cell. The cells glow with increasing and then decreasing intensity as the Ca++ fluctuates out of and into the sarcoplasmic reticulum.

Image Processing via Edge Detection

Image processing via edge detection is used to measure the change in cell length with time. The isolated cardiomyocytes are placed in a physiological saline solution. Electric field stimulation is subsequently applied by inserting into the saline solution a pair of electrodes hooked up to a stimulator. For viewing, the paced cells are placed on the stage of an inverted phase contrast light microscope. The video image of the cell is aligned with a raster line that extends horizontally across the screen. Two cursors in the raster line detect the image contrast in the left and right edges of the cell. The space between the left and right edges is converted to voltage. MacLab/Chart software converts the voltage to length values. Ultimately, the length values are analyzed in terms contraction velocity, relaxation velocity, and % shortening.

Computer Simulation

The simulation program was written for a PC platform in Visual Basic 6.0, an object-oriented programming language. The program allows the user to simply control the contraction cycle in terms of which sarcomeres or myofibrils in the cell are displayed and the number of time intervals in the contraction cycle to be displayed (up to 1000 one-millisecond intervals). The program shows in terms of timing how each sarcomere’s behavior is distinctly different from the behavior of neighboring sarcomeres and how each sarcomere affects or is affected by its neighbors. The general strategy used in developing the simulation involves 1) the use of standard numerical methods to describe the mechanical behavior of cardiomyocyte contraction/relaxation, 2) the generation of a set of data according to those model equations using parameters and constants obtained from experiments performed in the lab or taken from the literature, and 3) to display this data in an animated graphic image of one contraction cycle.

Assumed Architecture of the Heart Cell:

The display is an animated, wire frame, isometric image that consists of a bundle of 61 myofibrils, or a total of 2989 sarcomeres. Based on examination of cells in cross-section, we have concluded that an extruded hexagon best represents the space filling nature of myofibrils in a cardiomyocyte (See Figure 3). We therefore reject the rod or extruded circle used by Campbell and Gerdes as a model for a myofibril (Campbell, 1987).

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Figure 3: Evidence of hexagonal shape of a myofibril cross-section. a) from b) from Cornell Med. College web site.

Cell dimensions were obtained both experimentally from our laboratory data and from the literature. The average sarcomere length was found to be about 1.8 – 1.9mm (Roos and Leung, 1987). Based on experimental data, the average cell length is about 100mm, the average diameter is about 10mm, and so with a resting sarcomere length of about 2mm, the average number of sarcomeres in a myofibril is 50. Also, the myofibrils must have a diameter of about 1mm (radius = 0.5mm). So, we packed the simulated myofibrils symmetrically in a space-filling manner so that the image was 9 hexagons in diameter, and the total number of simulated myofibrils came to 61. Also for symmetry, we used 49 sarcomeres in each myofibril instead of 50, so that the middle sarcomere (number 25) in the middle myofibril would be the cell’s center of mass around which it contracts when the cells are freely suspended in solution.

We used three different equations to describe the resting state, activation state, and the relaxation state of the cell as indicated by the shortening curve. For simplification, we made the shortening curve linear (See Figure 4). Because an actual shortening curve is almost completely linear in the activation region, the linear version of that part of the curve is quite accurate. This is because intrinsic load is easily overcome by the sudden release of calcium from the SR to initiate activation of the contraction cycle. However, the relaxation state is not so simple. In actuality, it is an exponential decay. Since this portion of the curve is less linear, more factors enter into the equation, and the factors affecting relaxation are much more significant. For instance, the velocity of relaxation depends on 1) the speed and efficiency by which calcium is resequestered via SERCA ( the calcium “pump”), 2) the resistance to compression of the sarcomeres’ internal structure, comparable to that of a compressed spring, and 3) the intrinsic load, having to push against and displace all other sarcomeres in the myofibril.

Resting State

Latency before onset of contraction Activation State

Relaxation State

x x

time time

Figure 4: Example of a shortening Example of shortening curve made

curve linear

Various other assumptions about the cardiomyocyte in general were made in order to simplify the simulation:

1. It is assumed myofibrils are not conjoined, but sarcomeres are.

2. It is also assumed there is no change in cell diameter due to constant volume as the cell contracts, hence no change in the diameter of each hexagon.

3. The presence of mitochondria, nuclei, and the sarcoplasmic reticulum was ignored because they take up such little space in the cell that their effect on contractility may be considered negligible under these simplified conditions.

4. Calcium movement is assumed to behave in an on/off manner as a square wave function. Note: This is to be changed to an exponential decay equation as work on this model progresses.

Animated Graphic Display:

Ultimately, the graphics display reduces the program’s results to an understandable form. Any combination of sarcomeres or myofibrils may be mapped as an isometric wire frame drawing in x, y, z space. The x and y coordinates are in the plane of the screen, while the z plane is represented by displacement. For example, when the x and y coordinate for the center of a hexagon representing one end of a sarcomere is calculated and drawn, an identical hexagon representing the other end of the sarcomere calculated and drawn but displaced to the right and up. The user may choose whether to view specific sarcomeres or myofibrils, or to view all sarcomeres and myofibrils. Once selected, the animation conveys the contractile behavior of the selected sarcomeres over the selected time intervals depending on where in the cell they are located.

GUI (Graphical User Interface):

The interface is simple, consisting of an Options menu with links to a Graphics Screen on which the animated image is displayed and a Myocyte Map.

Figure 5: Screen shot of the simulation program graphical user interface.

The Options Menu appears when the program is run. From here the user can select specific myofibrils and sarcomeres as well as the time interval over which to view them. In order to view the graphics screen the user must first select which type of longitudinal propagation is to occur (originating from an end or from a central region of the cell) and then calculate the data (click on CALC). Both the Graphics Screen and the Myocyte Map may be accessed from the “View” pull down menu. The Myocyte Map shows the numbering scheme used to designate the myofibrils and sarcomeres.

Parameters:

The following parameters were considered in designing the simulated cardiomyocyte:

• t = the independent variable, time, is divided into 1000 time intervals, each representing a millisecond.
• i = sarcomere number (position in a myofibril) is used to specify which sarcomere is selected, manipulated, or displayed.
• j = myofibril number specifies which myofibril is to be selected, manipulated, or displayed.
• x = the dependent variable, sarcomere length – x represents the instantaneous length found in sarcomere ( i ), found in myofibril ( j ), at any time ( t ). Therefore, length is a function of sarcomere position and time; x (i, j, t).
• x0 = initial resting length of each sarcomere – determined experimentally to be 1.8 – 1.9mm (Roos and Leung, 1987). We rounded to 2mm, since the average cell is about 100 mm long and each myofibril is composed of about 50 sarcomeres.
• xmin = the minimum sarcomere length allowed during contraction – determined to be 1.6 mm (relatively incompressible). Length is constrained by the compressible rigidity of the thick filaments. Therefore, a minimum sarcomere length must be accounted for.
• vshort = shortening velocity – speed of shortening once contraction is triggered by a “dump” of stored calcium from the SR into the cytosol of the cell. More specifically, vshort refers to the slope of the curve generated from shortening data, the change in length with respect to the change in time (dx/dt).
• vrelax = relaxation velocity – the speed of relaxation depends upon the speed at which calcium is resequestered by the calcium “pump” (SERCA) in the longitudinal region of the SR. vrelax refers to the slope of the lengthening part of the curve generated from shortening data. Furthermore, vrelax = vshort * -0.5, because relaxation is about half the speed of (takes about twice as long as) shortening. The negative sign indicates that relaxation occurs in the opposite direction as shortening.
• asdur = active state duration – the length of time a sarcomere is activated during contraction.
• propvel = propagation velocity – in spontaneous contractions, the speed of the spread of excitation of sarcomeres from one end of the cell to the other. Only occurs in damaged or dying cells. In normal cells extracellular calcium concentrations are much higher than intracellular calcium concentration. Consequently, when cells have “leaky membranes” a minute amount of calcium leaking in may trigger calcium release just large enough to activate contraction one or a few peripheral sarcomeres. “Leaky membranes” occur more frequently in the intercalated disc region (junction between two cells in tandem). As a result, a chain reaction of individual sarcomere contractions from sarcomere to adjacent sarcomere occurs. The wave propagates at a rate of 2ms per sarcomere (or 1mm /ms) from one end of the cell to the other along its axis. The propvel function allows the user to visualize asynchronous behavior such as that which causes arrhythmic heartbeats. Occasionally, calcium may leak out of the SR in the sarcomeres at the cell’s core. This would cause a spontaneous longitudinal propagation of contraction from the center of mass. We included the option of viewing a simulation of this type of contraction (See Figure 6: Options Menu).
• axialvel = axial velocity – speed of the axial spread of excitation from the peripheral sarcomeres toward the core sarcomeres. Occurs in spontaneously and non-spontaneously contracting cells.
• r = radius of each sarcomere – used to calculate, plot, and draw the hexagons and lines in the graphic display. Since each sarcomere has a resting length of 2mm and diameter of about 1um, then r = 0.5mm.

Numerical Methods:

We used Euler’s linear numerical methods to describe the mechanical behavior of cardiomyocyte contraction/relaxation. The program repeatedly updating the cell length with each increment of time by performing iterative calculations each one based on the previous result. For example, x(t) = x(t-1) + Dx, where t-1 is the length at the previous time increment. In other words, sarcomere length is updated from the just previous moment by a certain value for length change.

Changes/Improvements To Be Made:

The assumed design of the cell did not take into account that the cell’s volume remains constant during contraction, hence the diameter of each myofibril increases with contraction and decreases again with relaxation. However, we kept the program modular so that changes and improvements can be made without having to rewrite all of the code. Our simulation shows a completely uniform and symmetric contraction. In the future, we plan to take into account the non-uniform behavior of cells as a result of domains (groups of myofibrils with in the cell that are linked together and consequently shorten faster or slower than other domains in the cell (Roos, 1987; Roos and Taylor, 1989). These domains often do not start at the same value of x; they vary in length. Kinking of the cell is another non-uniformity caused by the existence of domains. Temperature’s effect on diffusion (particularly calcium diffusion out of and into the SR) is another variable yet to be considered. Also, changes in concentration of extra-cellular trigger calcium, changes in the behavior of voltage-sensitive gates (ion channels), and ideally, changes in sodium and potassium concentrations inside and outside the cell are among the prospective changes to the simulation. Another factor unaccounted for is the effect of mechanical (inertial) load on vrelax and vshort (Pietrabissa, 1991). There are no mechanical linkages between sarcomeres or myofibrils in our model.

Relevance:

The purpose of this simulation is to increase understanding of the fundamental driving forces of contraction by the ability to visualize a combination of different types of kinetic data together in one image. It may be used as demonstration tool for physicians and non-scientists. For scientists, it can be used to compare theoretical results with experimental data, or just to observe cell dynamics in as simple and straight forward a way as possible. Both the magnitude of the calcium release and the decay rate associated with resequestration of calcium by the longitudinal SR are affected by many drugs and toxins such as Lipopolysaccharide (LPS) found in the outer membrane of many types of gram-negative bacteria. LPS causes septic shock, a condition with symptoms including hypotension, fever, organ dysfunction, and ultimately heart failure. It causes 500,000 deaths per year in the U.S.A. Abnormalities in calcium regulation and left-ventricular relaxation often precede heart failure. We hope that a better understanding of the relationship between calcium regulation and contractility through scientific visualization can help improve treatment and prevention of heart disease.

Works Cited

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3. Lindemann, J. P., and Watanabe, A. M.: Mechanisms of adrenergic and cholinergic regulation of myocardial contractility. In Sperelakis, N. (ed.): Physiology and Pathophysiology of the Heart. 3rd ed. Boston, Kluwer Academic Publishers, 1995, pp. 467-494.

4. Pietrabissa, R., Montevecchi, F.M., Fumero,R., 1991. Mechanical characterization of a model of a multicomponent cardiac fiber. J Biomed Eng 13: 407-414.

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6. Roos, K.P., Leung, AF, 1987. Theorectical Fraunhofer light diffraction patterns calculated from three-dimensional sarcomere arrays imaged from isolated cardiac cells at rest. Biophys J 52: 329-341.

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