A joint model of the contractile system of striated muscle (preliminary calculations)

Values of the variables of a model of the muscle contractile apparatus were calculated on the basis of a previously described theoretical system JMCSM (i.e. the Joint Model of the Contractile System of Muscle) and after introducing some necessary simplifying presuppositions. The calculations made it possible to compare the model with the original biological system and provided data needed for the subsequent stage of simulation experiments. The properties of the model are apparently not inconsistent with the properties of the original biological system. The basic equation of the theoretical model could be expressed in the form analogous to Hill's equation; the procedure by which the equations were derived and the interpretation of their parameters are mutually independent. The model also suggests a different interpretation of the dependence of the heat of shortening on other quantities of the system. It is necessary to consider these calculations and their results as preliminary with regard to the simplifying presuppositions and to the possible inaccuracies of the estimations of input data.     

Can Quick Release Experiments Reveal the Muscle Structure? A Bionic Approach

The goal of this study was to understand the macroscopic mechanical structure and function of biological muscle with respect to its dynamic role in the contraction.A recently published muscle model,deriving the hyperbolic force-velocity relation from first-order mechanical principles,predicts different force-velocity operating points for different load situations.With anew approach,this model could be simplified and thus,transferred into a numerical simulation and a hardware experiment.Two types of quick release experiments were performed in simulation and with the hardware setup,which represent two extreme cases of the contraction dynamics:against a constant force (isotonic) and against an inertial mass.Both experiments revealed hyperbolic or hyperbolic-like force-velocity relations.Interestingly,the analytical model not only predicts these extreme cases,but also additionally all contraction states in between.It was possible to validate these predictions with the numerical model and the hardware experiment.These results prove that the origin of the hyperbolic force-velocity relation can be mechanically explained on a macroscopic level by the dynamical interaction of three mechanical elements.The implications for the interpretation of biological muscle experiments and the realization of muscle-like bionic actuators are discussed.     

Mechanical control of the rising phase of contraction of frog skeletal and cardiac muscle

The effect of shortening on contractile activity was studied in experiments in which shortening during the rising phase of an isotonic contraction was suddenly stopped. At the same muscle length and the same time after stimulation the rise in tension was much faster, if preceded by shortening, than during an isometric contraction, demonstrating an increase in contractile activity. In this experiment the rate of tension rise determined in various phases of contraction was proportional to the rate of isotonic shortening at the same time after stimulation. Therefore, the time course of the isotonic rising phase could be derived from the tension rise after shortening. The rate of isotonic shortening was found to be unrelated to the tension generated at various lengths and to correspond closely to the activation process induced by shortening. The length response explains differences between isotonic and isometric contractions with regard to energy release (Fenn effect) and time relations. These results extend previous work which showed that shortening during later phases of a twitch prolongs, while lengthening abbreviates contraction. Thus the length responses, which have been called shortening activation and lengthening deactivation, control activity throughout an isotonic twitch.     

A phenomenological model for estimating metabolic energy consumption in muscle contraction

A phenomenological model for muscle energy consumption was developed and used in conjunction with a simple Hill-type model for muscle contraction. The model was used to address two questions. First, can an empirical model of muscle energetics accurately represent the total energetic behavior of frog muscle in isometric, isotonic, and isokinetic contractions? And second, how does such a model perform in a large-scale, multiple-muscle model of human walking? Four simulations were conducted with frog sartorius muscle under full excitation: an isometric contraction, a set of isotonic contractions with the muscle shortening a constant distance under various applied loads, a set of isotonic contractions with the muscle shortening over various distances under a constant load, and an isokinetic contraction in lengthening. The model calculations were evaluated against results of similar thermal in vitro experiments performed on frog sartorius muscle. The energetics model was then incorporated into a large-scale, multiple-muscle model of the human body for the purpose of predicting energy consumption during normal walking. The total energy estimated by the model accurately reflected the observed experimental behavior of frog muscle for an isometric contraction. The model also accurately reproduced the experimental behavior of frog muscle heat production under isotonic shortening and isokinetic lengthening conditions. The estimated rate of metabolic energy consumption for walking was 29% higher than the value typically obtained from gait measurements.     

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.     

Rheology of myocardium. The relation between force, velocity, sarcomere length and activation in rat cardiac muscle

The relations between force, shortening velocity and sarcomere length (F-V-SL) during cardiac contraction, underlie Starling's Law of the Heart. F-V-SL were investigated in isolated, intact and skinned trabeculae and myocytes from rat heart. SL and V were measured with laser diffraction techniques; F was measured with a silicon strain gauge. The "ascending" F-SL relation appeared to result from both length dependent sensitivity of the contractile system to activator calcium ions and the presence of restoring forces (Fr), residing in the collagen skeleton of the muscle. Fr increased exponentially with decreasing SL below slack length to 25% of maximal twitch force (Ft) at SL = 1.60 microns. V was inversely proportional to the load and attained a maximum at zero load (Vo). Vo increased with factors that increased F: [Ca++], SL, and time during the twitch. Vo reached a maximum and remained constant (13.5 microns/s) when F attained or exceeded 50% of its maximum value. Viscous force in the passive muscle increased with V to a maximum of 4% of Ft at V = 40 microns/s. The relation between Vo and these factors could be predicted by a model of contraction in which the measured visco-elastic properties of myocardium were incorporated, while the truly unloaded maximal velocity of sarcomere shortening was assumed to be independent of the level of activation of the contractile filaments. A model of the cardiac cycle which explains the relation between Frank's and Starling's laws is presented.     

A simulation model of African Anopheles ecology and population dynamics for the analysis of malaria transmission

Background

Malaria is one of the oldest and deadliest infectious diseases in humans. Many mathematical models of malaria have been developed during the past century, and applied to potential interventions. However, malaria remains uncontrolled and is increasing in many areas, as are vector and parasite resistance to insecticides and drugs.

Methods

This study presents a simulation model of African malaria vectors. This individual-based model incorporates current knowledge of the mechanisms underlying Anopheles population dynamics and their relations to the environment. One of its main strengths is that it is based on both biological and environmental variables.

Results

The model made it possible to structure existing knowledge, assembled in a comprehensive review of the literature, and also pointed out important aspects of basic Anopheles biology about which knowledge is lacking. One simulation showed several patterns similar to those seen in the field, and made it possible to examine different analyses and hypotheses for these patterns; sensitivity analyses on temperature, moisture, predation and preliminary investigations of nutrient competition were also conducted.

Conclusions

Although based on some mathematical formulae and parameters, this new tool has been developed in order to be as explicit as possible, transparent in use, close to reality and amenable to direct use by field workers. It allows a better understanding of the mechanisms underlying Anopheles population dynamics in general and also a better understanding of the dynamics in specific local geographic environments. It points out many important areas for new investigations that will be critical to effective, efficient, sustainable interventions.     

A diverse view of protein dynamics from NMR studies of HIV-1 protease flaps

Internal motion in proteins fulfills a multitude of roles in biological processes. NMR spectroscopy has been applied to elucidate protein dynamics at the atomic level, albeit at a low resolution, and is often complemented by molecular dynamics simulation. However, it is critical to justify the consistency between simulation results and conclusions often drawn from multiple experiments in which uncertainties arising from assumed motional models may not be explicitly evaluated. To understand the role of the flaps of HIV-1 protease dimer in substrate recognition and protease function, many molecular dynamics simulations have been performed. The simulations have resulted in various proposed models of the flap dynamics, some of which are more consistent than others with our working model previously derived from experiments. However, using the working model to discriminate among the simulation results is not straightforward because the working model was derived from a combination of NMR experiments and crystal structure data. In this study, we use the NMR chemical shifts and relaxation data of the protease "monomer" rather than structural data to narrow down the possible conformations of the flaps of the "dimer". For the first time, we show that the tips of the flaps in the unliganded protease dimer interact with each other in solution. Accordingly, we discuss the consistency of the simulations with the model derived from all experimental data.     

Regional timing of myocardial shortening is related to prestretch from atrial contraction: assessment by high temporal resolution MRI tagging in humans

Earlier studies have shown substantial nonuniformity in normal left ventricular (LV) myocardial function concerning both the degree of shortening and timing of shortening. We hypothesized that nonuniform LV function may be related to nonuniform prestretch induced by atrial contraction. Eleven healthy human subjects were studied using MRI myocardial tagging and strain analysis. The amount of circumferential prestretch was assessed in 30 LV segments. Prestretch was defined as the difference in strain between end diastole (at ECG R wave) and diastasis. Furthermore, both the degree of shortening (quantified as peak circumferential shortening, peak systolic shortening rate, and amount of postsystolic shortening) and timing of shortening (quantified as the onset time of shortening and time to peak shortening) were assessed. LV prestretch was found to be nonuniform, with the highest values in the lateral wall. The amount of segmental prestretch correlated significantly with peak shortening (r = 0.79), peak shortening rate (r = 0.50), amount of postsystolic shortening (r = 0.67), onset time of shortening (r = -0.57), and time to peak shortening (r = 0.71) (P < 0.001 for each of these relations). These relations may be explained by regional differences in wall stress or by a regional Frank-Starling effect. The correlation between timing of shortening and prestretch demonstrates that mechanical timing is not determined by electrical phenomena alone. In conclusion, regional variation in LV function correlates with the nonuniform prestretch from atrial contraction.     

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.     

Isokinetic plantar flexion: experimental results and model calculations

In isokinetic experiments on human subjects, conducted to determine moments that can be exerted about a joint at different angular velocities, joint rotation starts as soon as the moment increases above the resting level. This contraction history differs from the one in experiments on isolated muscle, where the force is allowed to increase to an isometric level before shortening is initiated. The purpose of the present study was to determine the influence of contraction history on plantar flexing moments found during maximal voluntary plantar flexion on an isokinetic dynamometer. In ten subjects, plantar flexing moments were measured as a function of ankle angle at different angular velocities. They were also calculated using a model of the muscle-tendon complex of the human triceps surae. The model incorporates elastic tendinous tissue in series with muscle fibers. The input of the model consists of time histories of active state (the force generating capacity of contractile elements) and shortening velocity of the muscle-tendon complex. Different time courses of active state were offered at fixed length of the muscle-tendon complex. The time course yielding a close match between the calculated rise of plantar flexing moment and the rise measured during fixed angle contractions was used to calculate moment-angle curves for isokinetic plantar flexion. The active state value reached when a peak occurred in calculated moment-angle curves was found to be lower if the angular velocity was made higher. Comparing measured and calculated results, it was concluded that moment-angular velocity diagrams determined in studies of isokinetic plantar flexion in human subjects reflect not only the influence of shortening velocity of contractile elements on the force which can be produced by plantar flexors.     

On the processivity of DNA replication

In this paper we describe the nature and importance of processive enzymatic reactions in biological processes. A model is set up to describe the processive synthetic process in DNA replication, and experiments are presented to define and test the model, using the components of the T4 phage-coded five-protein (in vitro) DNA replication system of Alberts. Nossal and coworkers. These experiments are performed either with a homogeneous oligo dT-poly dA primer-template system, or with a natural primer-template system using phage M13 DNA. The results are used to define some molecular aspects of the microscopic "processivity cycle".     

Lattice simulations of aggregation funnels for protein folding.

A computer model of protein aggregation competing with productive folding is proposed. Our model adapts techniques from lattice Monte Carlo studies of protein folding to the problem of aggregation. However, rather than starting with a single string of residues, we allow independently folding strings to undergo collisions and consider their interactions in different orientations. We first present some background into the nature and significance of protein aggregation and the use of lattice Monte Carlo simulations in understanding other aspects of protein folding. The results of a series of simulation experiments involving simple versions of the model illustrate the importance of considering aggregation in simulations of protein folding and provide some preliminary understanding of the characteristics of the model. Finally, we discuss the value of the model in general and of our particular design decisions and experiments. We conclude that computer simulation techniques developed to study protein folding can provide insights into protein aggregation, and that a better understanding of aggregation may in turn provide new insights into and constraints on the more general protein folding problem.     

Shortening velocity and myosin heavy chains of developing rabbit muscle fibers

The regulation of vertebrate muscle contraction with respect to the role of the different subunits of myosin remains somewhat uncertain. One approach to gaining a better understanding of the molecular basis of contraction is to study developing muscle which undergoes changes in myosin isozyme composition and contractile properties during the normal course of maturation. The present study utilizes single fibers from psoas muscles of rabbits at several ages as a model system for fast-twitch muscle development. This approach eliminates the inherent problems of interpreting results from studies on whole muscles which usually contain heterogeneous fiber types with respect to contractile properties and isoenzyme composition. Maximum velocity of shortening and tension-generating ability of individual fibers were measured and the myosin heavy chain composition of the same fibers was examined using an ultrasensitive sodium dodecyl sulfate-polyacrylamide gel system. The results indicate that 1) with regard to contractile properties, there is a transitional period from slow to fast shortening velocities within the first postnatal month; 2) a strong, positive correlation exists between the speed of shortening and tension-generating ability of individual postnatal day 7 fibers, suggesting that as more myosin is incorporated in these developing fibers it is of the fast type; and 3) there is a wide variation in maximum velocity of shortening among postnatal day 7 psoas fibers which is also a time when a mixture of heavy chain isoforms characterizes the myosin composition of single muscle fibers.     

Positive work as a function of eccentric load in maximal leg extension movements

Negative and positive work performed during leg extension movements of 53 well trained subjects was measured with the help of a special dynamometer. The subjects performed four maximal push off trials against five different loads (25-105 kg): two two-legged extensions from a squatting position (SM) with a knee angle of 70 degrees and two trials with a preliminary counter movement (CM) but with the same extension range as in the SM. Positive work differed only by about 4% between CM and SM in spite of large differences in initial forces at the onset of concentric contraction. Based on simulations, it is suggested that in CM the advantage of stored elastic energy can almost completely be nullified by the disadvantage of a limited shortening distance of the contractile elements. It is hypothesised that elastic energy in CM can only cause considerable extra work during concentric contraction compared to the maximal positive work done in SM if the total range of lengthening and shortening of the muscle(s) involved is larger in CM than in SM.     

Responses of the spinal α-motoneurone-Renshaw cell system to various differentially distributed segmental afferent and descending inputs

A previously presented multi-loop model of the mammalian spinal -motoneurone-Renshaw cell system was extended to incorporate different physiological input patterns: Ia fibres from primary muscle spindle endings, spinal input systems descending in the ventral quadrant and from the nucleus ruber. The main goal of the computer simulation calculations was to present a number of dynamic input-output relations between these inputs which are distributed inhomogenously to different types of -MNs (that is, S-, FR-, and FF-type MNs) and the outputs of pools of the latter, for the purpose of experimental testing. The main outcome was that the phase relations of the outputs of the different types of MNs depend very much on the overall strength of recurrent inhibition, such that small changes of this strength, which appears to be small anyway, can significantly alter these phase relations. Since this strength is alterable through descending and segmental afferent inputs, this provides a physiological means of phase-decorrelation although it is unlikely to put the discharges of different MN types totally out of phase (by about 180°). Also, the inhomogeneity of recurrent inhibition would help to prevent a strong phase separation of this kind. Yet a decorrelation at the microscopic level could help suppress physiological tremor.     

A multisegmental cross-bridge kinetics model of the myofibril

Striated muscle is a mechanical system that develops force and generates power in serving vital activities in the body. Striated muscle is a complex biological system; a single mammalian muscle fibre contains up to hundred or even more myofibrils in parallel connected via an inter-myofibril filament network. In one single myofibril thousands of sarcomeres are lined up as a series of linear motors. We recently demonstrated that half-sarcomeres (hS) in a single myofibril operate non-uniformly. We outline a mathematical framework based on cross-bridge kinetics for the simulation of the force response and length change of individual hS in a myofibril. The model describes the muscle myofibril in contraction experiments under various conditions. The myofibril is modeled as a multisegmental mechanical system of hS models, which have active and viscoelastic properties. In the first approach, a two-state cross-bridge formalism relates the hS force to the chemical kinetics of ATP hydrolysis, as first described by Huxley [1957. Muscle structure and theories of contraction. Prog. Biophys. Mol. Biol. 7, 255-318]. Two possible types of biological variability are introduced and modeled. Numerical simulations of a myofibril composed of four to eight hS show a non-uniform hS length distribution and complex internal dynamics upon activation. We demonstrate that the steady-state approximation holds only in restricted time zones during activation. Simulations of myofibril contraction experiments that reproduce the classic steady-state force-length and force-velocity relationships, strictly constrained or “clamped” in either end-held isometric or isotonic contraction conditions, reveal a small but conspicuous effect of hS dynamics on force.     

Analysis of pattern recognition by man using detection experiments

This paper addresses the problem of analyzing biological pattern recognition systems. As no complete analysis is possible due to limited observability, the theoretical part of the paper examines some principles of construction for recognition systems. The relations between measurable and characteristic variables of these systems are described. The results of the study are:

1. Human recognition systems can always be described by a model consisting of an analyzer (F _A) and a linear classifier.
2. The linearity of the classifier places no limits on the universal validity of the model. The principle of organization of such a system may be put into effect in many different ways.
3. The analyzer function F _A determines the transformation of external patterns into their internal representations. For the experiments described in this paper, F _A can be approximated by a filtering operation and a transformation of features (contour line filter).
4. Narrow band filtering (comb filter) in the space frequency domain is inadequate for pattern recognition because noise of different bandwidths and mean frequencies affects sinusoidal gratings differently. This excludes the use of a Fourier analyzer.
5. The relations between the measurable variables, which are the probabilities of detection (P _D curves), and the characteristic variables of the recognition system are established analytically.
6. The probability of detection not only depends on signal energy but also on signal structure. This would not be the case in a simple matched filter system.
7. The differing probabilities of error in multiple detection experiments show that the interference is pattern specific and the bandwidth (steepness of the P _D curves) is different for the different sets of patterns.
8. The distance between the reference vectors in feature space can be determined from the internal representation of the patterns defined by the model. Through multiple detection experiments it is possible to determine not only the relative distances between the patterns but also their absolute position in feature space.

     

A mathematical model for estimating muscle tension in vivo during esophageal bolus transport

We present a model of esophageal wall muscle mechanics during bolus transport with which the active and "passive" components of circular muscle tension are separately extracted from concurrent manometric and videofluoroscopic data. Local differential equations of motion are integrated across the esophageal wall to yield global equations of equilibrium which relate total tension within the esophageal wall to intraluminal pressure and wall geometry. To quantify the "passive" (i.e. inactive) length-tension relationships, the model equations are applied to a region of the esophagus in which active muscle contraction is physiologically inhibited. Combining the global equations with space-time-resolved intraluminal pressure measured manometrically and videofluoroscopic geometry data, the passive model is used to separate active and "passive" components of esophageal muscle tension during bolus transport. The model is of general applicability to probe basic muscle mechanics including the space-time stimulation of circular muscle, the relationship between longitudinal muscle tension and longitudinal muscle shortening, and the contribution of the collagen matrix surrounding muscle fibers to passive tension during normal human esophageal bolus transport and in pathology. Example calculations of normal esophageal function are given where active tone is found to extend only over a short intrabolus segment near the bolus tail and segmental regions of active muscle squeeze are demonstrated.     

Predicting muscle force generation during fast-starts for the common carp Cyprinus carpio

Muscle contractile properties have been characterised for white myotomal muscle from the common carp Cyprinus carpio at 10, 15, and 20 °C. The time course of muscle force development was measured when one, two, or three stimuli were delivered at the onset of constant velocity shortening. As the shortening velocity increased several parameters decreased including the maximum force, the time course for the contraction and the relative duration of the deactivation compared to the activation. The maximum force and the relative rates of activation to deactivation for the contraction were relatively independent of temperature, whereas the duration of the contraction decreased with increasing temperature. A predictive model was developed which was based on fitting a modified Weibull distribution to these observations. The model was used to interpolate the expected contractile forces during cyclic length-changes. Measured and predicted values for force and power during such cyclic work-loop experiments showed an excellent agreement over the range of shortening regimes typically found during swimming behaviours. However, the predicted force was overestimated during the deactivation phase of the contractions when the shortening velocities exceeded those found during swimming. Accepted: 25 May 1999