Teaching from classic papers: Hill's model of muscle contraction
http://www.100md.com
《生理教育》
Abstract
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.
Key words: striated muscle; energetics; heat production; force-velocity; education;
modeling; simulation; MATLAB
ALL GRADUATE STUDENTS in our department are required to take Advanced Quantitative
Physiology, a team-taught collection of month-long modules on mathematical modeling of
various cellular and organ systems. As part of this course, I decided to teach a module on
basic muscle physiology using two classic papers as readings: A. V. Hill's "The heat of
shortening and the dynamic constants of muscle" (1) and A. F. Huxley's "Muscle structure
and theories of contraction" (2). The objectives of this module are to 1) review the basic
physiology of striated muscle contraction; 2) utilize two elegant, classic papers as
examples of the thought processes involved in developing a model from physiological data;
and 3) contrast the Hill and Huxley models as examples of more phenomenological and more
mechanistic models, respectively, of the same physiological process.
Taking a discovery learning approach, I ask students to use a simple simulation to
conduct the experiments Hill described in his paper and then fit Hill's and Huxley's models
to their simulated data. While fitting of Huxley's model to experimental data requires a
reasonably strong background in mathematics, a simulation-based approach to understanding
Hill's thought process and model could be employed in teaching physiology to a range of
audiences. This article therefore presents my simple simulation and illustrates how this
simulation can be used together with Hill's classic paper to help students with varying
levels of mathematical expertise develop an inquiry-based understanding of Hill-type muscle
models and related concepts in muscle physiology.
Approach
This section outlines how I presented Hill's paper and related material in our Advanced
Quantitative Physiology course, to give the reader some context. The specifics unavoidably
depend heavily on the background of students in the course. Therefore, in Teaching Points,
I tried to abstract more general advice for adapting this approach to undergraduate,
graduate, and medical student audiences.
Lectures.
The muscle physiology module of our course includes one 3-h lecture each week for 3 wk.
Although the biomedical engineering graduate students taking this course come from a
variety of undergraduate majors, they are required to have taken an introductory physiology
course or to take our own undergraduate physiology course concurrently. Our course
therefore focuses more on modeling than on teaching the details of the physiology. I use
the 3 h of lecture in the first week to review the basics of striated muscle physiology:
structure and organization, excitation-contraction coupling, twitch and tetanic responses,
definitions of preload and afterload, force-length and force-velocity relationships,
energetics of contraction, and basic differences between cardiac and skeletal muscle.
Students are assigned to read Hill's paper before the second 3-h session, which focuses on
Hill's paper (1) and particularly on his model of muscle contraction. The third week is
devoted to Huxley's 1957 paper (2), which will not be discussed further here for the
reasons explained above.
Hill's paper.
A. V. Hill's paper "The heat of shortening and the dynamic constants of muscle" (1) is
a wonderful classic from a bygone era, 60 pages of detailed methods, experiments, and
modeling representing years of work. In the first of three sections, Hill outlines the
design and construction of his experimental system, with detailed circuit diagrams, the
complete equations for a Wheatstone bridge amplifier, instructions on how to build a
thermopile, and more. (For many students, it may be appropriate to skip or skim this
section, but for an engineering audience, this is elegant and historically interesting
work.) The second section presents the results of a series of experiments on mechanics and
heat production in frog skeletal muscle. The final section presents the classic two-
component Hill model with a contractile and an elastic element in series, develops the
appropriate equations, and shows that this model explains many of the key experimental
observations presented earlier in the paper. The only caution with regard to this paper is
its length: while it is not too much to ask of students in a graduate course, for
undergraduates or medical students selected excerpts will probably be more appropriate.
Simulation.
My simple MATLAB simulation is provided in the APPENDIX.1 This is by no means a
research-grade simulation; it correctly solves for the forces predicted by Hill's model
assuming length is prescribed as an input and a tetanizing stimulus is applied beginning at
the start of the simulation, but does not simulate other situations in which Hill's model
might be of interest, such as contractions against a constant afterload. By choosing an
arbitrary but reasonable rate of baseline heat liberation, it also produces heat tracings
consistent with Hill's figures from the 1938 paper. It is written so that with minor
modifications it could be transferred to any programming language: rather than using
MATLAB's built-in equation solvers, this simulation solves Hill's model by simply stepping
through the length and time inputs and using the calculated rate of force rise at each step
to project the force at the next. Many of our students have never written a program like
this to directly solve a simple set of equations; for them, one of the secondary goals of
this module is to write their own version of my simulation. For a general audience, the
only important point is that this method only works well for a sufficiently large number of
steps. All of the results shown here were obtained by breaking a period of 5 s into 1,000
steps (Table 1).
In lectures, I introduce the model by demonstrating how to use the simulation to
predict the response of Hill's model to isometric tetanus (Table 1 and Fig. 1A) and a step
change in length (Fig. 1B). We graph and discuss the results as a way of exploring the
implications of two features of Hill's model: 1) force is the same in the contractile and
elastic elements because they are in series and 2) force in the elastic element is
proportional to stretch. In equations,
where L is muscle length, Lce and Lse are the lengths of the contractile and series
elastic elements, P is muscle force, and is the spring constant for the series elastic
element. During isometric contraction, total length remains constant and the contractile
element can only shorten by stretching the elastic element; to stretch the elastic element
further and further, the contractile element must generate more and more force (in Fig. 1A,
as Lce decreases and Lse increases, P rises).
DISCOVERY LEARNING QUESTIONS
1. Draw the expected length-vs.-time curves for the contractile and series elastic
elements in Hill's model during the development of an isometric tetanus. Use the simulation
to obtain the actual responses and comment on the agreement or disagreement with your
prediction.
2. Repeat the cycle of prediction, simulation, and comparison for a step release from
isometric tetanus and for a constant-velocity release from isometric tetanus.
If the activated model is then subjected to a sudden change in length, the time
derivatives of Eq. 1, A and B, are of interest:
where vce is the shortening velocity of the contractile element. For a sudden step
decrease in length (dL/dt < 0), the fact that the contractile element has a limited maximal
shortening velocity means that the drop is absorbed primarily by the elastic element, with
a resulting immediate drop in force; force then recovers as the contractile element
restretches the elastic element (Fig. 1B).
The remainder of the student work with the simulation is discovery learning, where
students work with the simulation to discover for themselves some of the basic principles
identified by Hill that motivated his model. The students work through a homework
assignment that asks them to follow in Hill's footsteps to perform simulated versions of
his experiments and determine the Hill constants a and b as well as the spring constant for
the series elastic element , the maximum velocity of unloaded shortening vmax, and the
maximal isometric tetanic force P0. Some aspects of this homework are described below; the
entire homework and solution set are available from the author on request.
Homework problems.
First, I ask the students to simulate releases of different rates and durations from an
initial fully developed tetanus and explore how the rate and amount of heat liberation
depend on the rate and distance of shortening. Through simulations with an appropriate
choice of length inputs, students can easily verify Hill's critical finding: that the
amount of excess heat liberated as a result of shortening depends only on the amount
shortened, not on the rate
DISCOVERY LEARNING QUESTIONS
1. Simulate releases of different rates and durations from an initial fully developed
tetanus and explore how the rate and amount of heat liberation depend on the rate and
distance of shortening. Read part II, section a (p. 157–161), of Dr. Hill's paper (1) and
compare your findings with his conclusions.
Next, I ask students to design and conduct appropriate experiments to determine the
constants Hill termed a and b in his paper. Hill defined constant a as the slope of the
relationship between the amount of shortening x and the amount of associated excess heat
(Hill's Fig. 10E):
Students can easily repeat his approach to obtain a value for a (Fig. 3). Next, Hill
considered the excess energy released during shortening. The mechanical work performed is
force times distance; the additional energy released beyond that of an isometric
contraction is therefore the mechanical work plus the extra heat liberated:
Hill found experimentally that if he plotted the rate of excess energy liberation for
muscles shortening at a constant load (P and a constant) against the amount of load, he
obtained a straight line; he defined constant b as the slope of the relationship between
the excess energy rate and steady-state force P (Hill's Fig. 11):
With the use of data from the same set of simulations they used to obtain a, students
can repeat this part of his analysis as well (Fig. 3C). Equation 5 contains an extra
constant, c, which represents the y-intercept of the plot of (P + a)v against P (Fig. 3C).
Because the excess heat of shortening must be zero when there is no shortening (during an
isometric contraction), c must equal bP0, where P0 is the force generated by an isometric
contraction. Incorporating this fact and rearranging the equations yields the famous Hill
equation, presented as Eqs. 1 and 2 in his paper:
This last equation describes the classic hyperbolic force-velocity relationship of
muscle, but was discovered as described here based on considerations of energy liberation.
DISCOVERY LEARNING QUESTIONS
1. With the use of the simulation provided, perform appropriate releases from isometric
tetanus and determine Hill constant a as the slope of the relationship between the amount
of shortening and the associated excess heat, as in Hill's Fig. 10E (1).
2. Next, use the same data to determine Hill constant b from the slope of the
relationship between steady-state force P and the energy rate, (P + a)v, as in Hill's Fig.
11 (1).
3. Next, try an alternate approach to determine the Hill constants by performing a
series of constant velocity releases from tetanus and fitting the resulting force-velocity
data to an appropriate equation.
4. Compare the values for the constants you obtained using the two methods outlined
above. Propose an explanation for any differences.
To conduct these simulations, students must wrestle with some of the same experimental
design issues that would face them if it were possible to have them perform these
experiments in the laboratory. For example, to obtain the most accurate value for constant
b, it is best to obtain data at the widest possible range of shortening velocities and
associated forces. However, very rapid shortening velocities can only be maintained briefly
before the muscle length reaches unreasonable values and the simulation breaks down, and
over such short times the force may not reach a true steady-state value associated with the
imposed velocity. In addition, the simulation as written adds some Gaussian noise to the
computed force and heat outputs, so that students must average multiple trials or come up
with other strategies to handle the presence of noise.
While there is teaching value in having students repeat Hill's thought process and derive
the Hill constants as he originally described, Hill constants are rarely obtained in this
way today. It is much more common to simply fit force-velocity data to obtain these
constants. Therefore, I also ask students to simulate constant-velocity releases at a
number of different velocities, find the steady-state force associated with each velocity,
and fit the force-velocity data to the equation (P + a)(v + b) = k, where k is a constant
(Fig. 3D). Because most simple graphing programs do not have this equation as a built-in
feature, this provides an opportunity to discuss curve fitting approaches. In this case,
three or more force-velocity data pairs can be used to construct a system of equations that
is linear in the constants a, b, and (k – a x b), easily converted to matrix form and
solved in MATLAB.
For the remainder of the homework, students write and work with their own
implementation of Hill's equations. Because of this, for the earlier parts of the homework,
they do not have access to my original MATLAB program but only to an executable version
created with the MATLAB command "pcode." This has the added advantage that the Hill
constants in the model are unknowns from the students' point of view. I can vary them from
year to year and compare the values determined by the students with the true values in the
simulation to help me assess their work. Once students write their own simulations using
constants from Hill's paper and verify that their simulations give appropriate responses
for the development of an isometric tetanus, recovery from a step decrease in length, and
constant-velocity shortening, I ask them to modify Hill's model and their implementation to
improve agreement with some experimental data that is not well explained by Hill's original
model. There are several possible challenges of this nature that can be assigned. For
example, Jewell and Wilkie (3) showed that force recovery after a step decrease in length
follows a different time course than predicted by Hill's model, and students can adjust the
model to improve the fit by modeling the series elastic element as nonlinear rather than
linear.
Teaching Points
The approach outlined in the previous section was developed for a graduate biomedical
engineering course and may not be appropriate for courses with a different target audience.
On the basis of my experience teaching physiology to undergraduate biology students,
graduate biomedical engineering students, and medical students, I have tried to suggest
ideas that may be more appropriate for each of these audiences in the text below and in the
DISCOVERY LEARNING QUESTIONS that accompany each figure. The degree of difficulty of these
exercises increases from Fig. 1 to Fig. 3, and the figures themselves serve as the
instructor "answer key" for the questions.
Medical physiology courses.
Of the properties explored by Hill in his paper, medical students are typically
interested primarily in the force-velocity relationship. The simulation introduced here
could be used as an exploratory tool for students to simulate releases from tetanus at
different velocities, plot the resulting steady-state forces against the prescribed release
velocities, and thereby "discover" the force-velocity relationship. However, Hill's model
in itself does not provide insight into the mechanisms that underlie the force-velocity
relationship; this relationship is specified for the contractile element in his model,
based on his empirical finding that the energy rate (P + a)v is linearly related to force
P. Because the density of medical school physiology courses also generally discourages
reference to primary sources, the approach outlined here will probably be of least interest
to those teaching medical physiology.
Undergraduate and graduate physiology courses.
Once a physiology course begins to consider energetics and heat production in muscle in
addition to force-length and force-velocity relationships, Hill's model and paper are of
much more importance and interest. In this case, the interaction between the contractile
and series elastic elements in Hill's model must be understood, as must the concept that
there are two types of work performed by the contractile element during variously loaded
contractions: external work (Px) of the type familiar from physics and internal work (ax),
which appears as excess heat production.
In these situations, a number of uses of the simulation presented here may be
appropriate. Before ever reading Hill's paper or learning about his model, students might
be asked to prescribe a range of inputs to the simulation, explore the force and heat
responses, and propose possible explanations for the results they see. Students may well
articulate that the force response to a step decrease in length has an immediate component
and a slower recovery component or discover on their own that shortening produces an extra
amount of heat that depends only on the distance shortened. Each of these insights and the
effort to explain them will help prepare students to better understand Hill's model once it
is introduced.
After Hill's model has been introduced, a series of simulations focusing on helping
students understand the series interaction between the elements may be appropriate.
Suggested exercises are provided under the heading DISCOVERY LEARNING QUESTIONS in Fig. 1.
Asking students to first predict the response of the elastic and contractile elements for
the cases of isometric force development, a step decrease in muscle length, and constant
velocity shortening and then simulate these cases and compare the outputs Lce and Lse to
their predictions will force them to think carefully about how the two elements interact.
Students usually find the concept that the contractile element stretches the series element
during isometric contraction relatively easy to grasp. However, they often guess that the
two elements will absorb a step change in length equally, not anticipating the fact that
the maximal shortening velocity of the contractile element limits its response. During
various shortening protocols, they often predict changes in force and series elastic
element length that do not correspond, but one or two failures of this type can be used to
remind them of the behavior of simple springs (familiar from physics) and cement the
concept that force and elastic element length must change in parallel.
Both undergraduate and graduate students can easily discover for themselves Hill's
critical observations that the amount of excess heat associated with shortening depends
only on the distance shortened (see suggested DISCOVERY LEARNING QUESTIONS in Fig. 2) and
that the rate of excess energy liberation is linearly proportional to force (Fig. 3C).
Depending on the intended degree of difficulty, these exercises can be combined with a
derivation of Hill's equation in lecture or with reading of the derivation in Hill's paper
(as outlined in Homework problems) to recreate Hill's thought process in originally
discovering this equation.
Finally, at the graduate level, I consider the additional effort to repeat Hill's
procedure for fitting his model to data (see suggested DISCOVERY LEARNING QUESTIONS in Fig.
3) very valuable. Fitting a model to data is a critical skill for all science and
engineering graduate students, and they should be aware of the range of methods available
to them for doing so, from simple linear regression to more complicated nonlinear fits.
Because the simulation as written adds random (Gaussian) noise to the heat and force
outputs, students will find that their values for a and b will not match the "true" values
incorporated in the simulation. They can be asked to generate and test several ideas for
reducing the impact of noise on their estimates. Typically, in a first attempt, they will
have used data from just three or four simulated releases, and they will have determined
the "excess heat" by comparing individual heat values at a single time point (e.g., the
final heat value at 5 s). Many of them realize that multiple trials are usual in
experimentation and may propose to overcome the effects of noise by rerunning the same
simulation several times and averaging the values they obtain for a and b. They may not
realize that simulating more different releases (as shown in Fig. 3) rather than
repetitions of the same releases may be a more efficient approach. In actual experiments,
time, expense, or other practical limits may restrict the number of trials that can be
performed, making it essential to extract the best possible information from each trial.
However, students almost never propose linear fits to the individual heat tracings to
reduce noise in the estimate of excess shortening heat for each release; this, in fact, was
the approach Hill used (see Fig. 10 in Hill's paper).
There are two aspects of the approach described here that may appear particularly
daunting to physiology students and instructors. First, particularly in undergraduate
courses, students may be discouraged by the effort required to become familiar enough with
MATLAB to construct appropriate inputs to my simulation and graph the outputs. We want our
engineering students to become proficient with MATLAB and modules such as this provide
opportunities for practice. Understanding that this may be an unwanted distraction in a
physiology course, I have written a simple graphical interface that allows students to
choose from three example length inputs (isometric, step change in length, or constant
velocity release) or to outline their own length inputs by simply clicking with the mouse
on a length-time plot. The selected length input is then passed to my simulation, and the
results are graphed automatically. Students can use the MATLAB Array Editor window to view
the resulting input or output arrays and copy values for graphing or further analysis.
Instructors who choose to use this option should be able get students up and running with
the simulation after a 10- to 15-min introduction to the MATLAB environment and the
graphical interface.
The second potentially intimidating aspect of the exercise as employed in our course is
asking students to implement Hill's model themselves. It is tempting to dismiss this as too
difficult for students who do not have strong math backgrounds, and I did not focus on this
aspect in outlining teaching points here. However, I contend that it is exactly these
students who most need to overcome their fear of generating and using models as needed. A
number of software packages (including Simulink, available as a package with MATLAB) now
allow students to construct a model simply by connecting elements such as a differentiator
or an integrator graphically, and this exercise can be liberating for students who believed
themselves incapable of implementing models involving differential equations. In fact, even
writing a program to solve a simple differential equation is often easier than integrating
that equation analytically, as Hill does in his paper, and learning how to solve simple
equations computationally could provide graduate physiology students with a new and very
powerful tool.
Undergraduate and graduate engineering/mathematics courses.
The approach outlined in this article is by design most appropriate for engineering
audiences. Undergraduates should be comfortable using simulations to explore features of a
model as outlined above and implementing a simple model such as this graphically. Graduate
students should be facile with a range of implementation options, with fitting models to
data and with modifying simple models to improve agreement with data as discussed
previously.
Conclusions
A. V. Hill's classic paper "The heat of shortening and the dynamic constants of muscle"
(1) is a wonderful example not only of careful experimentation but also of the thought
process required to generate a model from experimental observations. I attempted to engage
students in thinking about and understanding relevant aspects of muscle physiology through
a simple simulation that allows them to reproduce his observations and parts of his thought
process. This approach is outlined here in the hope that it will be of use to others who
teach muscle physiology to a range of audiences.
APPENDIX: SIMULATED RESPONSE OF HILL MODEL TO PRESCRIBED LENGTH INPUT
function [P,H,Lse,Lce] = hill(L,t)
% Function hill accepts length & time inputs L(n), t(n) and
% computes following outputs assuming tetanizing
% impulse starts at first time point:
% P(n) - force
% H(n) - heat
% Lse(n) - series elastic element length
% Lce(n) - contractile element length
% Inputs and outputs are column vectors that must all
% have same length n.
% Establish constants (Hill 1938 p.174, mean data at
% 0°C: a = 399*0.098, b = 0.331)
% Note that because Hill reports force with units of
% force/unit area and lengths in unitless fractions of
% muscle length, force, and heat all have units of
% force/area.
% Initialize arrays
% General solver for prescribed length input to Hill model
end
% Add some noise if desired for more realistic output
Acknowledgments
The author acknowledges past mentors and teachers of cardiac physiology for inspiration
and students from the Advanced Quantitative Physiology course at Columbia University for
feedback to the continuing development of this module. The original publisher, the Royal
Society, has graciously granted permission to post Dr. Hill's article in the APS Archive of
Teaching Resources as a supplement to this article. Dr. Hill's paper is also available
through JSTOR (http://www.jstor.org/), a subscription service available at many
institutions.
Footnotes
1 Three MATLAB files containing the simulation, demonstrations of its use, and
graphical interface for creating length inputs are available from
http://advanphysiol.physiology.org/cgi/content/full/00072.2005/DC1 or from the author's
website at http://www.columbia.edu/jh553/protocols/programs.html.
Received for publication October 20, 2005. Accepted for publication February 5, 2006.
REFERENCES
Hill AV. The heat of shortening and the dynamic constants of muscle. Proc R Soc London
B Biol Sci 126: 136–195, 1938.
Huxley AF. Muscle structure and theories of contraction. Prog Biophys Biophys Chem 7: 255–
318, 1957.
Jewell BR and Wilkie DR. An analysis of the mechanical components in frog's striated
muscle. J Physiol 143: 515–540, 1958.
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.
Key words: striated muscle; energetics; heat production; force-velocity; education;
modeling; simulation; MATLAB
ALL GRADUATE STUDENTS in our department are required to take Advanced Quantitative
Physiology, a team-taught collection of month-long modules on mathematical modeling of
various cellular and organ systems. As part of this course, I decided to teach a module on
basic muscle physiology using two classic papers as readings: A. V. Hill's "The heat of
shortening and the dynamic constants of muscle" (1) and A. F. Huxley's "Muscle structure
and theories of contraction" (2). The objectives of this module are to 1) review the basic
physiology of striated muscle contraction; 2) utilize two elegant, classic papers as
examples of the thought processes involved in developing a model from physiological data;
and 3) contrast the Hill and Huxley models as examples of more phenomenological and more
mechanistic models, respectively, of the same physiological process.
Taking a discovery learning approach, I ask students to use a simple simulation to
conduct the experiments Hill described in his paper and then fit Hill's and Huxley's models
to their simulated data. While fitting of Huxley's model to experimental data requires a
reasonably strong background in mathematics, a simulation-based approach to understanding
Hill's thought process and model could be employed in teaching physiology to a range of
audiences. This article therefore presents my simple simulation and illustrates how this
simulation can be used together with Hill's classic paper to help students with varying
levels of mathematical expertise develop an inquiry-based understanding of Hill-type muscle
models and related concepts in muscle physiology.
Approach
This section outlines how I presented Hill's paper and related material in our Advanced
Quantitative Physiology course, to give the reader some context. The specifics unavoidably
depend heavily on the background of students in the course. Therefore, in Teaching Points,
I tried to abstract more general advice for adapting this approach to undergraduate,
graduate, and medical student audiences.
Lectures.
The muscle physiology module of our course includes one 3-h lecture each week for 3 wk.
Although the biomedical engineering graduate students taking this course come from a
variety of undergraduate majors, they are required to have taken an introductory physiology
course or to take our own undergraduate physiology course concurrently. Our course
therefore focuses more on modeling than on teaching the details of the physiology. I use
the 3 h of lecture in the first week to review the basics of striated muscle physiology:
structure and organization, excitation-contraction coupling, twitch and tetanic responses,
definitions of preload and afterload, force-length and force-velocity relationships,
energetics of contraction, and basic differences between cardiac and skeletal muscle.
Students are assigned to read Hill's paper before the second 3-h session, which focuses on
Hill's paper (1) and particularly on his model of muscle contraction. The third week is
devoted to Huxley's 1957 paper (2), which will not be discussed further here for the
reasons explained above.
Hill's paper.
A. V. Hill's paper "The heat of shortening and the dynamic constants of muscle" (1) is
a wonderful classic from a bygone era, 60 pages of detailed methods, experiments, and
modeling representing years of work. In the first of three sections, Hill outlines the
design and construction of his experimental system, with detailed circuit diagrams, the
complete equations for a Wheatstone bridge amplifier, instructions on how to build a
thermopile, and more. (For many students, it may be appropriate to skip or skim this
section, but for an engineering audience, this is elegant and historically interesting
work.) The second section presents the results of a series of experiments on mechanics and
heat production in frog skeletal muscle. The final section presents the classic two-
component Hill model with a contractile and an elastic element in series, develops the
appropriate equations, and shows that this model explains many of the key experimental
observations presented earlier in the paper. The only caution with regard to this paper is
its length: while it is not too much to ask of students in a graduate course, for
undergraduates or medical students selected excerpts will probably be more appropriate.
Simulation.
My simple MATLAB simulation is provided in the APPENDIX.1 This is by no means a
research-grade simulation; it correctly solves for the forces predicted by Hill's model
assuming length is prescribed as an input and a tetanizing stimulus is applied beginning at
the start of the simulation, but does not simulate other situations in which Hill's model
might be of interest, such as contractions against a constant afterload. By choosing an
arbitrary but reasonable rate of baseline heat liberation, it also produces heat tracings
consistent with Hill's figures from the 1938 paper. It is written so that with minor
modifications it could be transferred to any programming language: rather than using
MATLAB's built-in equation solvers, this simulation solves Hill's model by simply stepping
through the length and time inputs and using the calculated rate of force rise at each step
to project the force at the next. Many of our students have never written a program like
this to directly solve a simple set of equations; for them, one of the secondary goals of
this module is to write their own version of my simulation. For a general audience, the
only important point is that this method only works well for a sufficiently large number of
steps. All of the results shown here were obtained by breaking a period of 5 s into 1,000
steps (Table 1).
In lectures, I introduce the model by demonstrating how to use the simulation to
predict the response of Hill's model to isometric tetanus (Table 1 and Fig. 1A) and a step
change in length (Fig. 1B). We graph and discuss the results as a way of exploring the
implications of two features of Hill's model: 1) force is the same in the contractile and
elastic elements because they are in series and 2) force in the elastic element is
proportional to stretch. In equations,
where L is muscle length, Lce and Lse are the lengths of the contractile and series
elastic elements, P is muscle force, and is the spring constant for the series elastic
element. During isometric contraction, total length remains constant and the contractile
element can only shorten by stretching the elastic element; to stretch the elastic element
further and further, the contractile element must generate more and more force (in Fig. 1A,
as Lce decreases and Lse increases, P rises).
DISCOVERY LEARNING QUESTIONS
1. Draw the expected length-vs.-time curves for the contractile and series elastic
elements in Hill's model during the development of an isometric tetanus. Use the simulation
to obtain the actual responses and comment on the agreement or disagreement with your
prediction.
2. Repeat the cycle of prediction, simulation, and comparison for a step release from
isometric tetanus and for a constant-velocity release from isometric tetanus.
If the activated model is then subjected to a sudden change in length, the time
derivatives of Eq. 1, A and B, are of interest:
where vce is the shortening velocity of the contractile element. For a sudden step
decrease in length (dL/dt < 0), the fact that the contractile element has a limited maximal
shortening velocity means that the drop is absorbed primarily by the elastic element, with
a resulting immediate drop in force; force then recovers as the contractile element
restretches the elastic element (Fig. 1B).
The remainder of the student work with the simulation is discovery learning, where
students work with the simulation to discover for themselves some of the basic principles
identified by Hill that motivated his model. The students work through a homework
assignment that asks them to follow in Hill's footsteps to perform simulated versions of
his experiments and determine the Hill constants a and b as well as the spring constant for
the series elastic element , the maximum velocity of unloaded shortening vmax, and the
maximal isometric tetanic force P0. Some aspects of this homework are described below; the
entire homework and solution set are available from the author on request.
Homework problems.
First, I ask the students to simulate releases of different rates and durations from an
initial fully developed tetanus and explore how the rate and amount of heat liberation
depend on the rate and distance of shortening. Through simulations with an appropriate
choice of length inputs, students can easily verify Hill's critical finding: that the
amount of excess heat liberated as a result of shortening depends only on the amount
shortened, not on the rate
DISCOVERY LEARNING QUESTIONS
1. Simulate releases of different rates and durations from an initial fully developed
tetanus and explore how the rate and amount of heat liberation depend on the rate and
distance of shortening. Read part II, section a (p. 157–161), of Dr. Hill's paper (1) and
compare your findings with his conclusions.
Next, I ask students to design and conduct appropriate experiments to determine the
constants Hill termed a and b in his paper. Hill defined constant a as the slope of the
relationship between the amount of shortening x and the amount of associated excess heat
(Hill's Fig. 10E):
Students can easily repeat his approach to obtain a value for a (Fig. 3). Next, Hill
considered the excess energy released during shortening. The mechanical work performed is
force times distance; the additional energy released beyond that of an isometric
contraction is therefore the mechanical work plus the extra heat liberated:
Hill found experimentally that if he plotted the rate of excess energy liberation for
muscles shortening at a constant load (P and a constant) against the amount of load, he
obtained a straight line; he defined constant b as the slope of the relationship between
the excess energy rate and steady-state force P (Hill's Fig. 11):
With the use of data from the same set of simulations they used to obtain a, students
can repeat this part of his analysis as well (Fig. 3C). Equation 5 contains an extra
constant, c, which represents the y-intercept of the plot of (P + a)v against P (Fig. 3C).
Because the excess heat of shortening must be zero when there is no shortening (during an
isometric contraction), c must equal bP0, where P0 is the force generated by an isometric
contraction. Incorporating this fact and rearranging the equations yields the famous Hill
equation, presented as Eqs. 1 and 2 in his paper:
This last equation describes the classic hyperbolic force-velocity relationship of
muscle, but was discovered as described here based on considerations of energy liberation.
DISCOVERY LEARNING QUESTIONS
1. With the use of the simulation provided, perform appropriate releases from isometric
tetanus and determine Hill constant a as the slope of the relationship between the amount
of shortening and the associated excess heat, as in Hill's Fig. 10E (1).
2. Next, use the same data to determine Hill constant b from the slope of the
relationship between steady-state force P and the energy rate, (P + a)v, as in Hill's Fig.
11 (1).
3. Next, try an alternate approach to determine the Hill constants by performing a
series of constant velocity releases from tetanus and fitting the resulting force-velocity
data to an appropriate equation.
4. Compare the values for the constants you obtained using the two methods outlined
above. Propose an explanation for any differences.
To conduct these simulations, students must wrestle with some of the same experimental
design issues that would face them if it were possible to have them perform these
experiments in the laboratory. For example, to obtain the most accurate value for constant
b, it is best to obtain data at the widest possible range of shortening velocities and
associated forces. However, very rapid shortening velocities can only be maintained briefly
before the muscle length reaches unreasonable values and the simulation breaks down, and
over such short times the force may not reach a true steady-state value associated with the
imposed velocity. In addition, the simulation as written adds some Gaussian noise to the
computed force and heat outputs, so that students must average multiple trials or come up
with other strategies to handle the presence of noise.
While there is teaching value in having students repeat Hill's thought process and derive
the Hill constants as he originally described, Hill constants are rarely obtained in this
way today. It is much more common to simply fit force-velocity data to obtain these
constants. Therefore, I also ask students to simulate constant-velocity releases at a
number of different velocities, find the steady-state force associated with each velocity,
and fit the force-velocity data to the equation (P + a)(v + b) = k, where k is a constant
(Fig. 3D). Because most simple graphing programs do not have this equation as a built-in
feature, this provides an opportunity to discuss curve fitting approaches. In this case,
three or more force-velocity data pairs can be used to construct a system of equations that
is linear in the constants a, b, and (k – a x b), easily converted to matrix form and
solved in MATLAB.
For the remainder of the homework, students write and work with their own
implementation of Hill's equations. Because of this, for the earlier parts of the homework,
they do not have access to my original MATLAB program but only to an executable version
created with the MATLAB command "pcode." This has the added advantage that the Hill
constants in the model are unknowns from the students' point of view. I can vary them from
year to year and compare the values determined by the students with the true values in the
simulation to help me assess their work. Once students write their own simulations using
constants from Hill's paper and verify that their simulations give appropriate responses
for the development of an isometric tetanus, recovery from a step decrease in length, and
constant-velocity shortening, I ask them to modify Hill's model and their implementation to
improve agreement with some experimental data that is not well explained by Hill's original
model. There are several possible challenges of this nature that can be assigned. For
example, Jewell and Wilkie (3) showed that force recovery after a step decrease in length
follows a different time course than predicted by Hill's model, and students can adjust the
model to improve the fit by modeling the series elastic element as nonlinear rather than
linear.
Teaching Points
The approach outlined in the previous section was developed for a graduate biomedical
engineering course and may not be appropriate for courses with a different target audience.
On the basis of my experience teaching physiology to undergraduate biology students,
graduate biomedical engineering students, and medical students, I have tried to suggest
ideas that may be more appropriate for each of these audiences in the text below and in the
DISCOVERY LEARNING QUESTIONS that accompany each figure. The degree of difficulty of these
exercises increases from Fig. 1 to Fig. 3, and the figures themselves serve as the
instructor "answer key" for the questions.
Medical physiology courses.
Of the properties explored by Hill in his paper, medical students are typically
interested primarily in the force-velocity relationship. The simulation introduced here
could be used as an exploratory tool for students to simulate releases from tetanus at
different velocities, plot the resulting steady-state forces against the prescribed release
velocities, and thereby "discover" the force-velocity relationship. However, Hill's model
in itself does not provide insight into the mechanisms that underlie the force-velocity
relationship; this relationship is specified for the contractile element in his model,
based on his empirical finding that the energy rate (P + a)v is linearly related to force
P. Because the density of medical school physiology courses also generally discourages
reference to primary sources, the approach outlined here will probably be of least interest
to those teaching medical physiology.
Undergraduate and graduate physiology courses.
Once a physiology course begins to consider energetics and heat production in muscle in
addition to force-length and force-velocity relationships, Hill's model and paper are of
much more importance and interest. In this case, the interaction between the contractile
and series elastic elements in Hill's model must be understood, as must the concept that
there are two types of work performed by the contractile element during variously loaded
contractions: external work (Px) of the type familiar from physics and internal work (ax),
which appears as excess heat production.
In these situations, a number of uses of the simulation presented here may be
appropriate. Before ever reading Hill's paper or learning about his model, students might
be asked to prescribe a range of inputs to the simulation, explore the force and heat
responses, and propose possible explanations for the results they see. Students may well
articulate that the force response to a step decrease in length has an immediate component
and a slower recovery component or discover on their own that shortening produces an extra
amount of heat that depends only on the distance shortened. Each of these insights and the
effort to explain them will help prepare students to better understand Hill's model once it
is introduced.
After Hill's model has been introduced, a series of simulations focusing on helping
students understand the series interaction between the elements may be appropriate.
Suggested exercises are provided under the heading DISCOVERY LEARNING QUESTIONS in Fig. 1.
Asking students to first predict the response of the elastic and contractile elements for
the cases of isometric force development, a step decrease in muscle length, and constant
velocity shortening and then simulate these cases and compare the outputs Lce and Lse to
their predictions will force them to think carefully about how the two elements interact.
Students usually find the concept that the contractile element stretches the series element
during isometric contraction relatively easy to grasp. However, they often guess that the
two elements will absorb a step change in length equally, not anticipating the fact that
the maximal shortening velocity of the contractile element limits its response. During
various shortening protocols, they often predict changes in force and series elastic
element length that do not correspond, but one or two failures of this type can be used to
remind them of the behavior of simple springs (familiar from physics) and cement the
concept that force and elastic element length must change in parallel.
Both undergraduate and graduate students can easily discover for themselves Hill's
critical observations that the amount of excess heat associated with shortening depends
only on the distance shortened (see suggested DISCOVERY LEARNING QUESTIONS in Fig. 2) and
that the rate of excess energy liberation is linearly proportional to force (Fig. 3C).
Depending on the intended degree of difficulty, these exercises can be combined with a
derivation of Hill's equation in lecture or with reading of the derivation in Hill's paper
(as outlined in Homework problems) to recreate Hill's thought process in originally
discovering this equation.
Finally, at the graduate level, I consider the additional effort to repeat Hill's
procedure for fitting his model to data (see suggested DISCOVERY LEARNING QUESTIONS in Fig.
3) very valuable. Fitting a model to data is a critical skill for all science and
engineering graduate students, and they should be aware of the range of methods available
to them for doing so, from simple linear regression to more complicated nonlinear fits.
Because the simulation as written adds random (Gaussian) noise to the heat and force
outputs, students will find that their values for a and b will not match the "true" values
incorporated in the simulation. They can be asked to generate and test several ideas for
reducing the impact of noise on their estimates. Typically, in a first attempt, they will
have used data from just three or four simulated releases, and they will have determined
the "excess heat" by comparing individual heat values at a single time point (e.g., the
final heat value at 5 s). Many of them realize that multiple trials are usual in
experimentation and may propose to overcome the effects of noise by rerunning the same
simulation several times and averaging the values they obtain for a and b. They may not
realize that simulating more different releases (as shown in Fig. 3) rather than
repetitions of the same releases may be a more efficient approach. In actual experiments,
time, expense, or other practical limits may restrict the number of trials that can be
performed, making it essential to extract the best possible information from each trial.
However, students almost never propose linear fits to the individual heat tracings to
reduce noise in the estimate of excess shortening heat for each release; this, in fact, was
the approach Hill used (see Fig. 10 in Hill's paper).
There are two aspects of the approach described here that may appear particularly
daunting to physiology students and instructors. First, particularly in undergraduate
courses, students may be discouraged by the effort required to become familiar enough with
MATLAB to construct appropriate inputs to my simulation and graph the outputs. We want our
engineering students to become proficient with MATLAB and modules such as this provide
opportunities for practice. Understanding that this may be an unwanted distraction in a
physiology course, I have written a simple graphical interface that allows students to
choose from three example length inputs (isometric, step change in length, or constant
velocity release) or to outline their own length inputs by simply clicking with the mouse
on a length-time plot. The selected length input is then passed to my simulation, and the
results are graphed automatically. Students can use the MATLAB Array Editor window to view
the resulting input or output arrays and copy values for graphing or further analysis.
Instructors who choose to use this option should be able get students up and running with
the simulation after a 10- to 15-min introduction to the MATLAB environment and the
graphical interface.
The second potentially intimidating aspect of the exercise as employed in our course is
asking students to implement Hill's model themselves. It is tempting to dismiss this as too
difficult for students who do not have strong math backgrounds, and I did not focus on this
aspect in outlining teaching points here. However, I contend that it is exactly these
students who most need to overcome their fear of generating and using models as needed. A
number of software packages (including Simulink, available as a package with MATLAB) now
allow students to construct a model simply by connecting elements such as a differentiator
or an integrator graphically, and this exercise can be liberating for students who believed
themselves incapable of implementing models involving differential equations. In fact, even
writing a program to solve a simple differential equation is often easier than integrating
that equation analytically, as Hill does in his paper, and learning how to solve simple
equations computationally could provide graduate physiology students with a new and very
powerful tool.
Undergraduate and graduate engineering/mathematics courses.
The approach outlined in this article is by design most appropriate for engineering
audiences. Undergraduates should be comfortable using simulations to explore features of a
model as outlined above and implementing a simple model such as this graphically. Graduate
students should be facile with a range of implementation options, with fitting models to
data and with modifying simple models to improve agreement with data as discussed
previously.
Conclusions
A. V. Hill's classic paper "The heat of shortening and the dynamic constants of muscle"
(1) is a wonderful example not only of careful experimentation but also of the thought
process required to generate a model from experimental observations. I attempted to engage
students in thinking about and understanding relevant aspects of muscle physiology through
a simple simulation that allows them to reproduce his observations and parts of his thought
process. This approach is outlined here in the hope that it will be of use to others who
teach muscle physiology to a range of audiences.
APPENDIX: SIMULATED RESPONSE OF HILL MODEL TO PRESCRIBED LENGTH INPUT
function [P,H,Lse,Lce] = hill(L,t)
% Function hill accepts length & time inputs L(n), t(n) and
% computes following outputs assuming tetanizing
% impulse starts at first time point:
% P(n) - force
% H(n) - heat
% Lse(n) - series elastic element length
% Lce(n) - contractile element length
% Inputs and outputs are column vectors that must all
% have same length n.
% Establish constants (Hill 1938 p.174, mean data at
% 0°C: a = 399*0.098, b = 0.331)
% Note that because Hill reports force with units of
% force/unit area and lengths in unitless fractions of
% muscle length, force, and heat all have units of
% force/area.
% Initialize arrays
% General solver for prescribed length input to Hill model
end
% Add some noise if desired for more realistic output
Acknowledgments
The author acknowledges past mentors and teachers of cardiac physiology for inspiration
and students from the Advanced Quantitative Physiology course at Columbia University for
feedback to the continuing development of this module. The original publisher, the Royal
Society, has graciously granted permission to post Dr. Hill's article in the APS Archive of
Teaching Resources as a supplement to this article. Dr. Hill's paper is also available
through JSTOR (http://www.jstor.org/), a subscription service available at many
institutions.
Footnotes
1 Three MATLAB files containing the simulation, demonstrations of its use, and
graphical interface for creating length inputs are available from
http://advanphysiol.physiology.org/cgi/content/full/00072.2005/DC1 or from the author's
website at http://www.columbia.edu/jh553/protocols/programs.html.
Received for publication October 20, 2005. Accepted for publication February 5, 2006.
REFERENCES
Hill AV. The heat of shortening and the dynamic constants of muscle. Proc R Soc London
B Biol Sci 126: 136–195, 1938.
Huxley AF. Muscle structure and theories of contraction. Prog Biophys Biophys Chem 7: 255–
318, 1957.
Jewell BR and Wilkie DR. An analysis of the mechanical components in frog's striated
muscle. J Physiol 143: 515–540, 1958.