Teaching from classic papers: Hill's model of muscle contraction

A. V. Hill's 1938 paper "The heat of shortening and the dynamic constants of muscle" is an enduring classic, presenting detailed methods, meticulous experiments, and the model of muscle contraction that now bears Hill's name. Pairing a simulation based on Hill's model with a reading of his paper allows students to follow his thought process to discover key principles of muscle physiology and gain insight into how to develop quantitative models of physiological processes. In this article, the experience of the author using this approach in a graduate biomedical engineering course is outlined, along with suggestions for adapting this approach to other audiences.     

THE SARCOTUBULAR SYSTEM OF FROG SKELETAL MUSCLE : A Morphological and Biochemical Study

In the frog skeletal muscle cell a well defined and highly organized system of tubular elements is located in the sarcoplasm between the myofibrils. The sarcoplasmic component is called the sarcotubular system. By means of differential centrifugation it has been possible to isolate from the frog muscle homogenate a fraction composed of small vesicles, tubules, and particles. This fraction is without cytochrome oxidase activity, which is localized in the mitochondrial membranes. This indicates that the structural components of this fraction do not derive from the mitochondrial fragmentation, but probably from the sarcotubular system. This fraction, called sarcotubular fraction, has a Mg⁺⁺-stimulated ATPase activity which differs from that of muscle mitochondria in that it is 3 to 4 times higher on the protein basis as compared with the mitochondrial ATPase, and is inhibited by Ca⁺⁺ and by deoxycholate like the Kielley and Meyerhof ATPase. We therefore conclude that the "granules" of the Kielley and Meyerhof ATPase, which were shown to have a relaxing effect, are fragments of the sarcotubular system. The isolated sarcotubular fraction has a high RNA content and demonstrable activity in incorporating labeled amino acids, even in the absence of added supernatant.     

Energetics of Shortening Muscles in Twitches and Tetanic Contractions : I. A Reinvestigation of Hill''s Concept of the Shortening Heat

Shortening heat was defined by Hill as the "difference between heat produced when shortening occurs and that produced in a similar contraction without shortening." For the tetanus the "similar contraction" was an isometric one at or near l_o. By contrast, in a twitch the "similar contraction" was one in which only activation heat was produced. The applicability of Hill's concept of the shortening heat has been reexamined in both the twitch and tetanus of Rana pipiens semitendinosus muscles. Results of this investigation confirm the existence of an extra heat production accompanying shortening in the twitch and tetanus. In both cases, this shortening heat was proportional to distance shortened and relative afterload. However, at a given afterload the amount of shortening heat produced per distance shortened was greater in the twitch than the tetanus. This difference suggests that the base lines or "similar contractions" employed for the twitch and tetanus are not equivalent. The discrepancy is not remedied by utilizing in the tetanus the activation heat as the myothermic baseline and suggests that some heat producing factor(s) has been omitted in Hill's formulation of the shortening heat. Finally, the existence of Hill's feedback heat, an energy liberation associated with the presence of tension during mechanical relaxation, was not confirmed. This result strongly indicates that relaxation is energetically passive.     

The biphasic force-velocity relationship in whole rat skeletal muscle in situ.

Edman has reported that the force-velocity relationship (FVR) departs from Hill's classic hyperbola near 0.80 of measured isometric force (J Physiol 404: 301-321, 1988). The purpose of this study was to investigate the biphasic nature of the FVR in the rested state and after some recovery from fatigue in the rat medial gastrocnemius muscle in situ. Force-velocity characteristics were determined before and during recovery from fatigue induced by intermittent stimulation at 170 Hz for 100 ms each second for 6 min. Force-velocity data were obtained for isotonic contractions with 100 ms of 200-Hz stimulation, including several measurements with loads above 0.80 of measured isometric force. The force-velocity data obtained in this study were fit well by a double-hyperbolic equation. A departure from Hill's classic hyperbola was found at 0.88+/-0.01 of measured isometric force, which is higher than the approximately 0.80 reported by Edman et al. for isolated frog fibers. After 45 min of recovery, maximum shortening velocity was 86+/-2% of prefatigue, but neither curvature nor predicted isometric force was significantly different from prefatigue. The location of the departure from Hill's classic hyperbola was not different after this recovery from the fatiguing contractions. Including an isometric point in the data set will not yield the same values for maximal velocity and the degree of curvature as would be obtained using the double hyperbola approach. Data up to 0.88 of measured isometric force can be used to fit data to the Hill equation.     

Interaction of factors determining oxygen uptake at the onset of exercise.

Considerable debate surrounds the issue of whether the rate of adaptation of skeletal muscle O2 consumption (QO2) at the onset of exercise is limited by 1) the inertia of intrinsic cellular metabolic signals and enzyme activation or 2) the availability of O2 to the mitochondria, as determined by an extrinsic inertia of convective and diffusive O2 transport mechanisms. This review critically examines evidence for both hypotheses and clarifies important limitations in the experimental and theoretical approaches to this issue. A review of biochemical evidence suggests that a given respiratory rate is a function of the net drive of phosphorylation potential and redox potential and cellular mitochondrial PO2 (PmitoO2). Changes in both phosphorylation and redox potential are determined by intrinsic metabolic inertia. PmitoO2 is determined by the extrinsic inertia of both convective and diffusive O2 transport mechanisms during the adaptation to exercise and the rate of mitochondrial O2 utilization. In a number of exercise conditions, PmitoO2 appears to be within a range capable of modulating muscle metabolism. Within this context, adjustments in the phosphate energy state of the cell would serve as a cytosolic "transducer," linking ATP consumption with mitochondrial ATP production and, therefore, O2 consumption. The availability of reducing equivalents and O2 would modulate the rate of adaptation of QO2.     

The Meyerhof quotient and the synthesis of glycogen from lactate in frog and rabbit muscle. A reinvestigation

1. The conversion of lactate into glycogen was demonstrated in frog sartorius muscle in oxygen. The rates and amounts are highest when lactate is added to the bathing medium and are dependent on lactate and CO(2) concentration, as well as pH. The glycogen content of a resting muscle can be doubled in 4h at 24 degrees C. 2. Sartorius muscle, recovering aerobically in liquid paraffin from a period of anoxia, converts preformed lactate into glycogen at a lower rate and in smaller amounts than when lactate is added in an aqueous medium. The lower rates are similar to those Meyerhof found under the same conditions, after correction for temperature; they can be attributed partly to low muscle pH and partly to the limited amounts of lactate present. 3. Rabbit psoas muscle also shows the ability to convert added lactate into glycogen under aerobic conditions. The rates are low and similar to those in frog sartorius muscle recovering from anoxia. 4. The present experiments yield a Meyerhof quotient of 6.2, compared with Meyerhof's value of 4-5. However, these values are not significantly different from one another. 5. It is suggested that the glycogen coefficient, i.e. mol of glycogen formed/mol of lactate disappearing, is a more reliable way of assessing the resynthetic mechanism than the original quotient, i.e. mol of lactate disappearing/mol of lactate oxidized. The found coefficient is 0.419+/-0.024.     

A Phenomenological Theory of Muscular Contraction: I. Rate Equations at a Given Length Based on Irreversible Thermodynamics

A phenomenological theory for contracting muscle based on irreversible thermodynamics and the sliding filament theory is developed. The individual cross bridges, considered as subunits, are viewed as linear energy converters with constant transport coefficients. With this view of the subunits, phenomenological equations applicable to the whole muscle are obtained. The transport coefficients are shown to be a function of a single parameter which is the number of activated cross bridges at any instant. By requiring Hill's force-velocity relation (1) to be satisfied, the response of the muscle is related to the number of activated cross bridges. The resulting theory differs significantly from the theory developed by Caplan (2) and a comparison of the theories is presented. The theory is shown to correlate well with the heat data of Woledge (3) for a tortoise muscle and gives a value of Y (ratio of chemical affinity to enthalpy of reaction) equal to 0.945. The comparison of the theory with Hill's frog muscle data (1) and (4) is also encouraging. In part II of this series, length variations are considered and the resulting theoretical predictions are shown to be consistent with experimental data.     

Sercarzian immunology--In memoriam. Eli E. Sercarz, 1934-2009

During his long career as a principal investigator and educator, Eli Sercarz trained over 100 scientists. He is best known for developing hen egg white lysozyme (HEL) as a model antigen for immunologic studies. Working in his model system Eli furthered our understanding of antigen processing and immunologic tolerance. His work established important concepts of how the immune system recognizes antigenic determinants processed from whole protein antigens; specifically he developed the concepts of immunodominance and crypticity. Later in his career he focused more on autoimmunity using a variety of established animal models to develop theories on how T cells can circumvent tolerance induction and how an autoreactive immune response can evolve over time. His theory of "determinant spreading" is one of the cornerstones of our modern understanding of autoimmunity. This review covers Eli's entire scientific career outlining his many seminal discoveries.     

Analysis of diffusion delay in a layered medium. Application to heat measurements from muscle.

An analysis is presented of diffusional delays in one-dimensional heat flow through a medium consisting of several layers of different materials. The model specifically addresses the measurement of heat production by muscle, but diffusion of solute or conduction of charge through a layered medium will obey the same equations. The model consists of a semi-infinite medium, the muscle, in which heat production is spacially uniform but time varying. The heat diffuses through layers of solution and insulation to the center of the thermal element where heat flow is zero. Using Laplace transforms, transfer functions are derived for the temperature change in the center of the thermopile as a function of the temperature at any interface between differing materials or as a function of heat production in the muscle. From these transfer functions, approximate analytical expressions are derived for the time constants which scale the early and late changes in the central temperature. We find that the earliest temperature changes are limited by the diffusivities of the materials, whereas the approach to steady state depends on the total heat capacity of the system and the diffusivity of muscle. Hill (1937) analyzed a similar geometry by modeling the layered medium as a homogeneous system with an equivalent half thickness. We show that his analysis was accurate for the materials in his system. In general, however, and specifically with regard to modern thermopiles, a homogeneous approximation will lead to significant errors.(ABSTRACT TRUNCATED AT 250 WORDS)     

Partial activation of thin filaments in resting frog skeletal muscle fibers

X-ray patterns from frog skeletal muscles at rest show a series of relatively weak meridional reflections which may be indexed as the 5, 7, 9, 11 and 13 orders of the repeat period of about 2 x 385 A. According to the model of the thin filaments structure, suggested by V. V. Lednev and G. M. Frank (1977), this period is specific for the activated or "switched on" state of the actin--containing filaments. At the same time, according to the generally accepted model (suggested in 1972), the axial repeat period of the thin filament structure is approximately equal to 385 A and does not depend on the functional state of the muscle. The existence of the repeat period of about 2 x 385 A in the thin filaments of a resting muscle suggests that even at rest the thin filaments of vertebrate skeletal muscle are not completely inhibited. It may be suggested that partial activation of the thin filaments in a resting muscle is the result of formation of long life rigorlike crossbridges, the existence of which was postulated by D. K. Hill in 1968 on the basis of his studies on resting tension in the frog skeletal muscle.     

Effect of voluntary exercise on H2O2 release by subsarcolemmal and intermyofibrillar mitochondria

Previous data have demonstrated that, to handle the oxidative stress encountered with training at high intensity, skeletal muscle relies on an increase in mitochondrial biogenesis, a reduced H(2)O(2) production, and an enhancement of antioxidant enzymes. In the present study, we evaluated the influence of voluntary running on mitochondrial O(2) consumption and H(2)O(2) production by intermyofibrillar mitochondria (IFM) and subsarcolemmal mitochondria (SSM) isolated from oxidative muscles in conjunction with the determination of antioxidant capacities. When mitochondria are incubated with succinate as substrate, both maximal (state 3) and resting (state 4) O(2) consumption were significantly lower in SSM than in IFM populations. Mitochondrial H(2)O(2) release per unit of O(2) consumed was 2-fold higher in SSM than in IFM. Inhibition of H(2)O(2) formation by rotenone suggests that complex I of the electron transport chain is likely the major physiological H(2)O(2)-generating system. In Lou/C rats (an inbred strain of rats of Wistar origin), neither O(2) consumption nor H(2)O(2) release by IFM and SSM were affected by long-term, voluntary wheel training. In contrast, glutathione peroxidase and catalase activity were significantly increased despite no change in oxidative capacities with long-term, voluntary exercise. Furthermore, chronic exercise enhanced heat shock protein 72 accumulation within skeletal muscle. It is concluded that the antioxidant status of muscle can be significantly improved by prolonged wheel exercise without necessitating an increase in mitochondrial oxidative capacities.     

Early contributions of Russian stress and exercise physiologists.

In Russia, the free development of scientific ideas was suppressed in 1950 as a result of the actions of the Joint Session of the Academy of Sciences and the Academy of Medical Sciences of the USSR. Hans Selye's theory on the general adaptation syndrome was considered unscientific. From 1956 to 1958, Pjotr Anokhin and Pjotr Gorizontov presented definitive arguments for having the theory accepted by scientists while the significance of hormones in adaptation became a topic of endocrine studies (Boris Aleshin, Igor Eskin, Vassily Komissarenko, Samuel Leites, and Michael Kolpakov). Later, Felius Meerson made essential contributions to the adaptive significance of protein synthesis and stress-limiting systems. The area of exercise physiology dealing with acute and chronic adaptation to strong physiological stressors was founded by Leon Orbeli and developed by Aleksei Krestovnikov. Significant contributors to this area were Vladimir Farfel, Nikolai Yakovlev, and Nikolai Zimkin. Although the majority of their publications have remained unknown outside of Russia, it is interesting that many of their results have been "rediscovered" by others. Yakovlev also deserves recognition because he was among the founders of contemporary exercise biochemistry and because his research has provided the foundation for current investigations. Several generations of young scientists have been inspired by the above-mentioned Russian scientists. Today, however, the research activities of scientists are no longer limited by political pressures but by financial resources instead.     

Terminology for contractions of muscles during shortening, while isometric, and during lengthening.

Communication among scientists must be clear and concise to avoid ambiguity and misinterpretations. The selection of words must be based on accepted definitions. The fields of biomechanics, muscle physiology, and exercise science have had a particularly difficult time with terminology, arising from the complexity of muscle contractions and by the use of inappropriate terminology by scientists. The dictionary definition of the verb "contract," specifically for the case of muscle, is "to undergo an increase in tension, or force, and become shorter." Under all circumstances, an activated muscle generates force, but an activated muscle generating force does not invariably shorten! During the 1920s and 1930s, investigators recognized that the interaction between the force generated by the muscle and the load on the muscle results in either shortening, no length change (isometric), or lengthening of the muscle. The recognition that muscles perform three different types of "contractions" required that contraction be redefined as "to undergo activation and generate force." Modifiers of contraction are then needed to clarify the lack of movement or the directionality of movement. Despite the contradiction, for 75 years the lack of movement has been termed an "isometric contraction." The directionality of the movement is then best described by the adjectives "shortening" and "lengthening." The definitions of "concentric" as "having the same center" and of "eccentric" as "not having the same center" are consistent with hypertrophy, or remodeling of the heart muscle, but are inappropriate to describe the contractions of skeletal muscles.     

A Concept of Educational Activity for Schoolchildren

About two years ago, on a cool Southern California day, Vasili Davydov addressed a group of social scientists at the University of California, San Diego. He began his talk with a paradox. He had come, he said, to tell us about educational activity. He promised to exhibit principles that promote educational activity, and applied programs deriving from those principles. Then he laughed. "But you'll never see educational activity in the school," he said, and laughed again.     

Creation of first malaria vaccine raises troubling questions about "intellectual racism". Interview by Kirsteen MacLeod.

Some of the problems caused by malaria, which places a huge roadblock in front of economic progress in the Third World, may be solved by a new vaccine created by Dr. Manuel Patarroyo, a Columbian physician and researcher. "Imagine how things would be if Canadians had malaria," he says. "Episodes last 10 days, then there are 10 days of recovering. This leaves only 10 days each month in which to do some productive work. Then imagine killing the population of Toronto each year, and you can see the huge toll in terms of the number of yearly deaths globally from malaria." His discovery also raises the issue of "intellectual racism" because of criticism of Patarroyo''s methods by Western scientists. Patarroyo, meanwhile, turned down a $60-million offer for his vaccine, and instead donated the patent to the World Health Organization.     

Contractile Characteristics of Mimosa pudica L

The force-velocity characteristics of the primary pulvinus of Mimosa pudica have been determined using a new polytonic measurement technique. The contractile characteristics were determined from a modified form of Hill's equation (Hill, A. V. 1938. Proc. Roy. Soc. London B126: 136-195) describing the physiological contractile behavior of animal muscle. The values of the resulting Hill's constants were found to be remarkably similar to those of intact animal muscle and reconstituted contractile collagen.     

Oxygen cost of exercise hyperpnea: implications for performance.

We addressed two questions concerned with the metabolic cost and performance of respiratory muscles in healthy young subjects during exercise: 1) does exercise hyperpnea ever attain a "critical useful level"? and 2) is the work of breathing (WV) at maximum O2 uptake (VO2max) fatiguing to the respiratory muscles? During progressive exercise to maximum, we measured tidal expiratory flow-volume and transpulmonary pressure- (Ptp) volume loops. At rest, subjects mimicked their maximum and moderate exercise Ptp-volume loops, and we measured the O2 cost of the hyperpnea (VO2RM) and the length of time subjects could maintain reproduction of their maximum exercise loop. At maximum exercise, the O2 cost of ventilation (VE) averaged 10 +/- 0.7% of the VO2max. In subjects who used most of their maximum reserve for expiratory flow and for inspiratory muscle pressure development during maximum exercise, the VO2RM required 13-15% of VO2max. The O2 cost of increasing VE from one work rate to the next rose from 8% of the increase in total body VO2 (VO2T) during moderate exercise to 39 +/- 10% in the transition from heavy to maximum exercise; but in only one case of extreme hyperventilation, combined with a plateauing of the VO2T, did the increase in VO2RM equal the increase in VO2T. All subjects were able to voluntarily mimic maximum exercise WV for 3-10 times longer than the duration of the maximum exercise. We conclude that the O2 cost of exercise hyperpnea is a significant fraction of the total VO2max but is not sufficient to cause a critical level of "useful" hyperpnea to be achieved in healthy subjects.(ABSTRACT TRUNCATED AT 250 WORDS)     

Changes of Volume in Striated Muscle

The record of change in volume, obtained from a frog sartoriusmuscle mounted isometrically at reference length shows a characteristictwo-phased change. Five msec alter the start of stimulus pulseof 4 msec an increase in volume begins. This increase peaksat about 7 msec and a decrease in volume begins. This decreaseis at a minimum at about 23 msec, and the volume then returnsto its pre-twitch level. (The temperature is 24°C.) A smallvariation, averaging less than 10%, is seen in the above parameterswhen records from different sartorius muscles are compared.(The time to the minimum of the decrease is slightly more variable,showing as much as 40% variation among different sartorius muscles.) A rise in tension begins approximately at the start of the increasein volume. Tension reaches a peak at about 33 msec after thestart of the stimulus. (This is well alter the minimum of thevolume-decrease.) The record of change in volume obtained from muscles contractingisometrically at lengths well below reference length is characteristicallydifferent from that obtained from muscles contracting isometricallyat reference length. "Slack" muscles, upon stimulation, showa large, rapid decrease in volume. This is probably similarto the decrease described by Ernst in his studies on "tensionless"muscles. This large, rapid decrease in volume is not seen ifthe muscle is stimulated isometrically at reference length.It is likely that this effect of length of muscle on the recordof volumechange explains the difference in the results obtainedby Ernst and co-workers on one hand and Meyerhof and his co-workerson the other. (Meyerhof's work appears to have been done withthe muscle held at or near reference length.) The magnitudeof the initial decrease in volume varies with length of themuscle (below reference length). It is maximal at "slack" or"end-free" lengths and decreases as the muscle is held nearreference length. The initial decrease in volume at these lengthsis abruptly terminated by a rapid volume-increase. The magnitudeof this increase is also a function of the muscle length. Followingthis increase, the volume-record returns to its initial level.     

The bioenergetic mechanisms of homeothermy

A number of experiments on isolated muscles and animals allowed to conclude that the amount of energetic expenditures and heat production per 1 unit of muscular work (or power) was not a biological constant as proposed by A. Hill. This value can be changed in the organism, and this allows to increase heat production and homoiotermy in certain environment. The work/heat production ratio was shown to be regulated for all kinds of biological work in organism. The data obtained and the calculations change our notions of energetics and of principles of homoiothermy.     

Force regulation by Ca2+ in skinned single cardiac myocytes of frog.

Atrial and ventricular myocytes 200 to 300 microm long containing one to five myofibrils are isolated from frog hearts. After a cell is caught and held between two suction micropipettes the surface membrane is destroyed by briefly jetting relaxing solution containing 0.05% Triton X-100 on it from a third micropipette. Jetting buffered Ca2+ from other pipettes produces sustained contractions that relax completely on cessation. The pCa/force relationship is determined at 20 degrees C by perfusing a closely spaced sequence of pCa concentrations (pCa = -log[Ca2+]) past the skinned myocyte. At each step in the pCa series quick release of the myocyte length defines the tension baseline and quick restretch allows the kinetics of the return to steady tension to be observed. The pCa/force data fit to the Hill equation for atrial and ventricular myocytes yield, respectively, a pK (curve midpoint) of 5.86 +/- 0.03 (mean +/- SE.; n = 7) and 5.87 +/- 0.02 (n = 18) and an nH (slope) of 4.3 +/- 0.34 and 5.1 +/- 0.35. These slopes are about double those reported previously, suggesting that the cooperativity of Ca2+ activation in frog cardiac myofibrils is as strong as in fast skeletal muscle. The shape of the pCa/force relationship differs from that usually reported for skeletal muscle in that it closely follows the ideal fitted Hill plot with a single slope while that of skeletal muscle appears steeper in the lower than in the upper half. The rate of tension redevelopment following release restretch protocol increases with Ca2+ >10-fold and continues to rise after Ca2+ activated tension saturates. This finding provides support for a strong kinetic mechanism of force regulation by Ca2+ in frog cardiac muscle, at variance with previous reports on mammalian heart muscle. The maximum rate of tension redevelopment following restretch is approximately twofold faster for atrial than for ventricular myocytes, in accord with the idea that the intrinsic speed of the contractile proteins is faster in atrial than in ventricular myocardium.