A Sliding-Filament Cross-Bridge Ensemble Model
of Muscle Contraction for Mechanical Transients
J. E. WOOD* AND R. W. MANN+
Depurtmenr o/ Mechunicul Engineering,
Mussrrchuwtrs Imritute of Technoloa, Cambridge
Received 9 January
I981; revised -70 Mq
Massachusetts
02139
I981
ABSTRACT
A high-efficiency
mechanochemical
oped via the formalism
of statistical
model of contraction
for striated muscle is develmechanics. The myosin cross-bridges
of the half-
sarcomere ensemble cycle through five biochemical states. The structural components
of the
sliding-filament
system include extensible myosin S2-units and indefinite arrays of equivalent actin sites. ATP ligand exchanges provide the far-from-equilibrium
thermodynamic
driving potential for cross-bridge
cycling. The strain-dependent
rate constants
obey the
self-consistency
requirements
of detailed balance. The mathematical
solution for the linear
time-varying
system of equations of cross-bridge
biokinetics employs discrete-time
state
transition
matrices. Computer
simulations
describe the time evolution of ultrastructural
(macromolecular)
phenomena
and their ensemble (macroscopic)
averages. Modest fidelity
for mechanical transients of muscle under many varied protocols, including conditions of
fatique. is obtained between the simulations and their experimental
counterpart.
I.
INTRODUCTION
The macroscopic performance of activated skeletal muscle derives from
the parallel and serial summation of small (macromolecular)
thermodynamic
systems. These systems accomplish mechanochemical
transductions
through
complete biochemical cycles driven by a single far-from-equilibrium
substrate.
Upon these physiological premises, we develop herein a quantitative model of
contraction which shows the viability of such a formulation at predicting
some of the rich spectrum of known muscle behavior.
The formal integration
of thermodynamics,
statistical mechanics, and
biochemical kinetics as applied to the sliding-filament
model for striated
muscle contraction has been developed in two major papers by Hill [ 11, 121.
*Research
’ Whitaker
Associate.
Professor of Biomedical
MATHEMATICAL
BIOSCIENCES
Engineering
57:21 l-263
OElsevier North Holland, Inc., 1981
52 Vanderbilt Ave., New York, NY 10017
211
(1981)
0025-5564/81/100211+53$02.75
212
J. E. WOOD
AND
R. W. MANN
the development
of the general formalism as it applies to our quantitative muscle model will not be repeated herein. We instead proceed immediately to build a model based upon a specific state diagram, using a
specific set of rate constants. For supplemental applications of the formalism,
the reader is referred to the work of Hill et al. [13-161.
Notwithstanding
the prolific contributions of Hill et al., the ultrastructural
modeling herein has a recognizable lineage, with the most recent and direct
ideological antecedents commencing with the oft-cited model of A. F. Huxley
(1957) [ 171, followed by the “flip’‘-type model of Huxley and Simmons [ 18,
191, the latter mechanism being subsequently adapted by Julian, Sollins, and
Sollins [22] to a cross-bridge cycle incorporating attachment and detachment.
The mathematical development for what is here to be denoted generically
as Model AM7 is specific for a half-sarcomere ensemble of cross-bridges
which cycle reversibly through five biochemical states (two detached, three
attached) while interacting with an indefinite array of equivalent actin sites
(Figure 2), and while exchanging ATP ligands with the sarcoplasmic milieu.
No attempt is made to rigorously compare or defend the merits of Model
AM7, or the class of models of which AM7 is a member (statisticalmechanical, biokinetic formulation of a sliding-filament
type model), against
other models or model types (such as electrostatic). Rather, the model serves
as an example: one showing the feasibility of unifying the observable
mechanics of striated muscle within the framework of a single integrated
physiochemical formalism, using a single set of parameters.
The simulations presented in this paper deal exclusively with contraction
characteristics of the model at maximal activation (i.e., all actin sites are
available for binding) and at maximal filament overlap (i.e., the plateau
region of operation). Results for time-varying activation (to simulate neuromuscular coupling) and for time-varying overlap (to simulate “creep”) are
not presented herein.
Thus,
II.
THEORY
AND DETAILS
FOR A HALF-SARCOMERE
The following derivations are for an ensemble of cross-bridges (CBS)
within the structural unit of a half-sarcomere (HS). However, the mathematical results are independent of the actual number of cross-bridges within the
ensemble, since all computed CB-dependent macroproperties
are normalized
with respect to the time-invariant ensemble size.
Our choice of nomenclature,
though defined herein when necessary, is in
most cases taken to be the same as Hill’s [ 11, 121.
A.
STATE DIAGRAM
Model AM7 is based upon the “minimal” diagram, Figure 1. This diagram
accounts for the minimal number of basic processes needed in the dominant
ENSEMBLE
MODEL
OF MUSCLE
213
CONTRACTION
IT
2
4
FIG. I. “Minimal” biochemical state diagram for the homogeneous
actin-myosin-ATP
system. All transitions are reversible. Heavy path (counter-clockwise)
is presumed dominant
cycle for the homogeneous
system, and for the structural
system at isometric and during
shortening,
when ATP is far from equilibrium
with respect to ADP and P,. Model AM7
transitions
A=actin;
presumed
are isomorphic
with this dominant
cycle, states 1 through 5. Abbreviations:
M=myosin;
T=ATP;
D=ADP;
and P=P,. Note that products D and P are
to act as a single species.
actin-myosin-ATP
cycle. Specifically, the five processes are: attachment,
desorption of ADP+P (denoted as DP, or more simply as D), binding of
ATP (T), detachment, and hydrolysis of ATP (T-D).
Of the nine transitions
shown in Figure 1, only the five transitions along the heavy path (counterclockwise) are considered to be important in the construction of the AM7
state transition matrix (Section 1I.D). The transitions MDP-M,
MT-M,
AM-M,
and AMDP++ AMT of the diagram are assumed to have minor if
not negligible fluxes (during shortening processes), and are thus eliminated
along with state M.
Further reduction of the dominant (AM7) cycle is possible in a systematic
and rigorous manner, consonant with the requirements
of statistical mechanics, as detailed by Hill [12, Appendix II]. But recent biochemical work
[25, I] indicates that the actual paths which describe physiological cross-bridge
activity will most likely have more than five states in the dominant cycle,
with more than one “high-probability”
cycle. Under test-tube (homogenized)
conditions, such alternate cycles, which may include “refractory” states, will
exist predominantly
among detached states. However, for the structural
system (see below; Figure 2), conditions
of sustained stretching of the
half-sarcomere will “force” dormant cycles involving attached states to become kinetically important-these
cycles being otherwise insignificant under
isometric or shortening conditions (Wood, unpublished work). In any case,
214
J. E. WOOD
AND
R. W. MANN
whether one uses a reduced or an expanded version of the minimal diagram,
the methods of state-space analysis, as used below, are still applicable.
B. STRUCTURALCONSIDERATIONS
Figure 1 applies to the “homogeneous”
system (i.e., a solution of Sl-units
+F-actin+T+D+P+Mg++
+Na+, etc.) whose rate constants are constant
for given species concentrations.
In contrast, the “structural”
system (i.e.,
interdigitating
hexagonal arrays of actin and myosin filaments) introduces a
new parameter, the subensemble coordinate x, which, neglecting variations of
the interfilament
spacing, locates the actin site relative to some reference or
fixation point of the cross-bridge. The molecular strain force of the small
thermodynamic
system of actin-myosin-ATP
will, for attached states, depend
on x. Consequently,
the energy level of the complex will be x-dependent,
and, by detailed balance, the rate-constant
pairs which interface with an
attached state will necessarily be functions of X. For the minimal diagram as
applied to a single site [Ill, the cross-bridge populations of all six states thus
become x-dependent.
Introducing
a linear array of equally spaced (by an interval 6) and
equivalent (no azimuthal dependence) sites expands the dominant cycle of
Figure 1 to the state diagram of Figure 2-there
being a topologically
-3
-2
-,
1
5,
1
+1
+2
A
\
,:\ \Q,
-
\
52
FIG. 2. Schematic.
‘1
AM7 structurochemical
state diagram
M
(at x) for a single myosin
cross-bridge
interacting
with an indefinite array of equivalent actin sites. The cross-bridge
head (labeled Sl) is referred to as the SI-unit. The helical structure (labeled S2) is referred
to as the S2-unit. The term “cross-bridge”
refers to a myosin projection (Sl +S2) whether
attached to an actin site or not. QM is the partition function of the SZ-unit strain energy,
which derives from the S2 stiffness K,. Note that here we take D=D.P.
ENSEMBLE
MODEL
OF MUSCLE
215
CONTRACTION
similar, but kinetically different diagram for each subensemble (of relative
size dx/6) betwen x and x+dx, on the interval --~/~Gxs
+6/2 (periodicity of sites). The complexities of modeling cross-bridges with multiple heads
which compete for the same set of sites are obviated by assuming singleheaded CBS which operate independently
of neighboring CBS.
The arrayed sites are sequentially labeled with integer index m (see Figure
3), where m =0 is assigned to that site nearest the cross-bridge in its
“unstrained”
configuration (i.e., that site nearest the maximum of the freefluctuation distribution of state MD). And x =O is assigned to that subensemble for which the structural strain energy in attached state AMD at m =O
is minimum, i.e., (x, m)=(O,O) coincides with the energy minimum of state
AMD. Thus, detached CBS in state MD of subensemble x may fluctuate
freely, but when they do attach to site m (state AMD), they necessarily attach
at x + m8. To shorten notation, we define the scalar xm = x + ma. Then, the
minimum-strain group of CBS in state MD will attach at xm = 0. CBS in state
MT at x =0 will also attach (state AMT) to site m =O at xm =O. However,
their strain energy will not be a minimum. Only by way of relative sliding
movement of the filaments can a CB “diffuse” from one x-subensemble
to
another.
In theory, the state diagram should extend over all values of m (i.e.,
indefinitely,
disregarding any mechanical limitations of the cross-bridge),
___--
-
M-E
FIG. 3. Schematic (scaled). Model AM7-2 cross-bridge (at x on site m=O) in each of
its three possible attached configurations.
Q,+, gives the probability
of attachment
MDAMD at a particularx
and m. Abbreviations:
A=actin;
M=myosin;
T=ATP;
D=ADP.P.
The horizontal dashed line at bottom is the centerline of the myosin filament core.
216
J. E. WOOD
AND
R. W. MANN
thus requiring an infinite number of state equations to be solved simultaneously at each x. However, we assume that the SZunit, which is a double
a-helix some 500 A in length, behaves as a Hookian element (with stiffness
K,) throughout displacements of 125 A or so: a strain of about 25%. In such
a case, the partition function Q,,,, of the S2 strain energy A, [see Equation
(26)] will be Gaussian in shape (see Figure 2, dashed bell-shaped curve) with
standard deviation a given by 02=kT/KM,
where kT is the Boltzmann
energy. With u we can estimate the width of the isometric distributions, since
the probability distribution of fluctuations for a free (detached and unbiased)
cross-bridge will be proportional to Qu (see [ 111). As examples, K,,,, = 10e9
dyn/A yields u = 60 A, while K, = IO-’ dyn/A (AM7-2 value) gives u 220
A and 6a = 120 A (compare with the width of the t =O.O- distributions in
simulation 3, Figure 10). Thus, with an actin monomer spacing of 6 =55 A,
the isometric distributions, with K, = lo-* dyn/A, become rapidly insignificant for 1m I> 1. With no significant loss of accuracy (compared to an infinite
set of sites), and in view of the very real constraints on cross-bridge mobility,
we can restrict the number of sites to a finite set. This then enables an exact
closed-form solution for the isometric distributions (Section 1I.C. 1). During
some experimental
procedures, such as rapid isotonic shortening,
some
cross-bridges will be “dragged” into positions with high site numbers (such as
m = -6, state AMT, at V,,). In such cases, the CB probability functions
will take on significant values over an xm range considerably wider than at
isometric (see simulation 3, Figure IO; compare distributions
at t >50 msec
against t = 0.0).
Concerning
other structural components,
the modeling herein ussumes
that the Sl-unit, the light-meromyosin
(LMM) core of the myosin filaments,
and the actin monomers (G-actin) and filaments (F-actin) are themselves
essentially inelastic (zero shearing, bending, or stretching compliance), and
are therefore not significant contributors to the compliance of skeletal muscle
as observed in quick release (or stretch) experiments (see the linear Tl curves
of simulation
9, Figure 16). By contrast, the Sl-S2 covalent linkage is
assumed to offer negligible resistance (infinite compliance) to articulations.
Thus, for Model AM7-2 the S2-unit is left not only as the major contributor,
but as the only contributor
to measurable cross-bridge, and hence, gross
muscle compliance.
Consequently,
it is the only element of the model
begetting the structural energy surface (see Section ILE, and Figure 6), and
thus the only mechanical element whose strain regulates the thermodynamic
rates of the cross-bridge state transitions. The details of the myofilament
lattice structure, which is otherwise hexagonal, are inconsequential
to our
formulation of cross-bridge kinetics. The only implied ultrastructural
requirements of the ensemble are that the subensembles be of equal density and that
the CBS act independently.
So, with one simple macromolecular
mechanism
and a minimum of degrees of freedom, we can proceed to examine the limits
ENSEMBLE
MODEL
OF MUSCLE
and capabilities
of this
chanochemical data.
C.
STATE-SPACE
mechanism
EQUATIONS
217
CONTRACTION
at
reproducing
OF ENSEMBLE
experimental
me-
KINETICS
The equations for cross-bridge biokinetics follow immediately from the
state diagram of Figure 2. For each detached state there will be a state
equation at every x within the interval --6/2~xG
+6/2. For each attuched
state, there will be an equation for each site m at x on the same -f6/2
domain.
Notationally,
we let k,,(xm)
denote the first-order rate constant for
transition i-j at xm. For transitions involving a ligand adsorption, we let
k:,(xm) denote a pseudo-first-order
(PFO) rate constant, again for i-j at
xm. Rate constants between detached states will not be functions of subensemble coordinate x, even though the state probabilities, in our case p ,( t, x)
and ps( t, x), are functions of X. Otherwise, a transition will involve an
attached state and will thus be a function of xm. A more detailed treatment
of the individual rate constants for AM7 is given in Section 1I.F.
As a specific example for a detached state, the partial differential equation
for the state probability density function p ,( t, x) is:
+
2 [b(xm)dt, xm)]+k~,p~(t,x),(1)
m
where the sum on m is over all significant sites (see Section 1I.B and, for
example, Figure 3, where m = - 3, - 2, - 1,O,+ 1, + 2). We denote by 9 the
total-derivative operator
(2)
This operator is analogous to the substantial (or material) derivative of fluid
dynamics. The second term of oi is the subensemble “mixing” term due to
relative filament motion, with the velocity of contraction given by t) = -dx/dt
(t) is positive for shortening).
For an attached-state example, we write the equation for pz( t, xm):
(3)
there being an equation
for every site m at x.
ENSEMBLE
MODEL
OF MUSCLE
219
CONTRACTION
where the xm-dependent submatrices and subvectors are repeated for each of
the a values of m. The individual elements of the subvectors of C(x) are
given by
k(xm)=k,(xm),
cJxm)=
k4,(xm),
The 3 X 3 tridiagonal submatrix which repeats along the diagonal of C(x) for
each value of m is denoted by C( xm), and will also be used in Section 1I.D. 1.
Its elements are
(9)
For the reader unfamiliar with the cross-bridge ensemble class of models
being discussed, it should be noted that Model AM7 does not incorporate
any damping (viscous) elements such as might be attributable to the movement of the myofilaments
through the sarcoplasmic fluid. All apparent
damping characteristics of the model, even the third-order transient effects
(see the simulations of Section III.B), are a result of the first-order (or PFO)
biochemical rate constants, albeit on at least a fifth-order system. There are
no rate constants in our formulation which are dependent on the slidingfilament velocity. Moreover, as seen in Section II.F, all rate constants satisfy
detailed balance at all times.
1. Steady Isometric Distributions.
For maximally activated, isometric,
steady-state conditions, qfi=O. Equation (5) then yields the isometric state
probabilities for detached statep,(x)
and for the three attached states on the
a sites m at x:
PO(x)=-k’(x)B(x).
The corresponding
(at x) probability
normalization
condition
~Y(x)=l-
for state
(10)
1
is then
~((P~(xm)+p:(xm)+p,O(xm))+p,O(x)
given by the
J. E. WOOD AND R. W. MANN
220
The steady isometric distributions
are seen as the t =O.O distributions
13).
D.
TRANSIENT
ANALYSIS
for the artached states of Model AM7-2
of simulations 3 and 6 (Figures 10 and
OF STATES
When isometric boundary conditions are applied to the half-sarcomere
system, our system of partial differential equations (5) becomes a linear
system of ordinary first-order differential equations with time-invariant
rate
constants. Such systems have a classical closed-form solution expressible in
terms of a state transition matrix:
(12)
&(t,x)=exp[Qx)r]
This time-dependent
function yields
filament sliding) for any time t >O:
the transient
distributions
(with no
(13)
However, the structurochemical
state diagram of Figure 2 consists of
multiple closed loops (xm-dependent
cycles) which incorporate PFO (liganddependent) rate constants operating at nonequilibrium
conditions. Hence the
system matrix A(x) admits the possibility of having complex eigenvalues with
negative real parts [IO]. The analytical solution in such cases is dealt with
adequately in control-theory
texts [23] and is thus not pursued herein. But
more particularly, we can avoid such complications
and at the same time
make considerable
gains in computational
efficiency by introducing
an
approximating
formulation of the system equations (5). Whatever small loss
of accuracy (of order 2% during transients) is incurred in this procedure has
proven to be an acceptable tradeoff for an order-of-magnitude
reduction in
computational
requirements. We achieve this reduction by breaking the state
diagram into two parts-one
dealing with the kinetics of the attached states,
the other with the kinetics of the detached states. Our analysis is detailed
below.
We refer to our analysis of the kinetics of the attached states of Model
AM7 as an approximate/discrete
formulation. This formulation assumes, as
an approximation,
that the detached-state
populations p, and ps remain
constant over a small time interval At = T (this will also be the discrete
interval at which the distributions are shifted when there is relative filament
motion). Under such an assumption, the normalization condition will not, in
general, hold strictly throughout
T. So, rather than taking one of the
detached states as dependent and substituting its normalization
equivalent
into the set of 1+3a state equations selected to be independent,
the dependent detached state is instead updated (“corrected”) via the normalization
ENSEMBLE
MODEL
OF MUSCLE
CONTRACTION
221
condition at the end of every time step T. Moreover, p, and ps, by being
constant and “unnormalized”
over T, act as constant chemical potential
cross-bridge reservoirs, essentially uncoupling the otherwise uninterrupted
reciprocation via detached states of cross-bridges attached at different m sites
within the same x-subensemble. This momentary (over 7’) elimination of the
cross-coupling kinetics between different sites allows a simplification of the
analysis, and thereby a considerable reduction in the computational
requirements of the system by virtue of the much smaller state-transition
matrices
needed [ +( T, xm) versus &( T, x); see below].
1. Attached-State Analysis. We begin our approximating procedure by
reformulating
the exact state equations (5). We now express, at every
coordinate xm, the exact time-continuous
system equations (including filament sliding) for attached states in the linear form
ol)P(t,xm)=C(xm)P(t,xm)+B(xm)P(t,x)
(14)
where
and where
The matrix C(xm) is the same 3 X 3 submatrix of Section 1I.C [Equation (9)].
If, for the moment, we introduce a condition of stationary filaments, the
“mixing” terms of 9 vanish. The exact solution to this time-continuous
but
isometric system can then be discretized in time for every xm (see [23, p.
3401).
If we introduce the approximation that the “control vector” of detached
states P( t, x) is constant over the (k + I)th interval T, we obtain the
computer-implemented
recursive relationship for attached states,
P((k+1)T,xm)=9(T,xm)[P(kT,xm)-Po(kT,xm)]+Po(kT,xm),
(15)
where, at the end of the k th time interval (after the k th shift, if necessary;
see
ENSEMBLE MODEL OF MUSCLE CONTRACTION
223
The attached transition matrix +( T, xm) gives only the time evolution of
the cross-bridge kinetics at one site (m) within a subensemble (x) due to
thermodynamic
transitions between attached states (with the detached states
acting as constant flux sources), and does not include the continuous translocations of the distributions required with relative filament motion. If, for the
purposes of digital simulation, the distributions (both attached and detached)
are shifted instantaneously
at equally spaced time intervals T (as above), then
the transition matrix +(T, xm) need be computed only once (at the beginning of a simulation) for each site m at x (see Section 1II.B).
2. Detached-State Analysis.
The approximate solution (for stationary
filaments) is completed by computing the detached-state populations at each
x. We used
~s((k+l)T,x)=~,(T,x)[~,(kT,x)-~p,O(kT,x)]+~,O(kT,x),
(21)
where
&(T,x)=exp[c,(x)T],
G(x)= m
p;(kT,x)=-
b,(kT, x>
c5(x)
’
The state p,((k + l)T, x) (MD) is then computed
condition (4) evaluated at x and (k + 1)T.
using the normalization
3. Shifting of the Distributions.
The “isometric” solution presented above
[Equations (14)-(21)] for the approximate/discrete
analysis can be extended
to simulations involving relative filament motion. The necessary “mixing” of
the subensembles is accomplished by simply shifting by As (the HS contraction velocity V= - As/T)
the (k + l)T distributions computed above (all
states). The newly shifted (k + 1)T attached distributions, operated on by the
stationary transition matrices +(T, xm), in turn determine the as yet unshifted (k +2)T attached distributions.
Likewise for the detached distributions. The new (k t2)T distributions (all states) are then shifted in accord
with the extant half-sarcomere boundary conditions. This procedure is repeated throughout the simulation.
It should be noted that attached states are shifted linearly along the
xm-axis, whereas the detached states are shifted circularly about only the
x-axis. Specifically, because of the periodicity of the m sites, the p,( t, x) for
detached states are shifted such that they are mapped back onto the interval
J.
224
What goes out one end of the interval must come back in
the other end.
E.
FREE-ENERGY
LEVELS
AND
SURFACES
When the CB ensemble is viewed from its aggregate, x-averaged, dynamic
output, differing molecular mechanisms of tension generation which yet yield
identical state energy functions become dynamically indistinguishable
(assuming the same set of forward rate constants) and thus mechanistically
opaque. That is, the state energy (and thus force) functions and rate constants as seen by a cross-bridge during its stochastic walk-and
not the
molecular basis of tension per se-are
the eventual determinants of muscle
performance. But to be explicit in constructing the energy functions for each
state, rather than merely stating them without foundation,
we choose a
molecular model whereby the principal strain energy of an attached state is
incurred in the S2-unit owing to discrete changes in the angular orientation
of the Sl-unit and/or to relative motion of the filaments (see Figure 3).
Strain contributions
due to bending of the Sl-S2 linkage are considered of
smaller order, and thus neglected. We further assume, as a minor geometrical
simplification,
that the S2-unit is parallel to the actin filament. One then
obtains a single, straightforward relation between the site coordinate xm, the
structural strain coordinate s (which is taken to be the actual S2 distortion
away from the position of minimum S2 strain, and thus has dimensions of
length), and the individual CB contribution
to ensemble tension.
The linearity of the experimentally observed Tl curve (the tension change
synchronous with a length step) for skeletal muscle [7, 81 translates, for the
assumed ultrastructure
of Model AM7, into a stretchable S2 element with
constant elastic modulus K,. A CB in attached state i (i=2,3,4)
with 52
strain s, will then generate a force
F] =KMs,.
(22)
The strain s, is related to xm by the general expression
s,=(x+mi3+h,)=xm+h,,
where h, is the S2 strain in state i when (xm) = (00) = 0.
The standard thermodynamic
relations between force F,, Helmholtz
energy A,, and partition function Q, are given by
(23)
free
ENSEMBLE MODEL OF MUSCLE CONTRACTION
By integrating Equation (22), one then obtains
energy function for attached states at xm,
225
the general form of the
A,(xm)=AP+~K,(x+ms+h,)2
=A; ++K,s,Z
(25)
=A:+A,(s,),
where A: is the energy minimum for state i, and the function AM(s,) is the
additive contribution
of the myosin structural strain energy. The structural
free-energy levels (the “well” minima at each xm; see Figure 6) for attached
states will thus be parabolic in form (see Figure 5) and thus strictly defined,
each parabola deriving from the same macromolecular
source (the SZunit).
The partition function for attached states may then be written as
Qi(xm)=exp[-A,(xm)/kT]
=exp[ -Aj;lkT]
exp[ -A,,.,(s,)/kT]
=Qi’Q,(s,>.
(26)
Since detached states generate zero force, their energy levels A:, being
independent of x, are constant, as are their partition functions Qy.
For transitions involving an ATP (T) or ADP.P (D) ligand exchange, the
energy minimum for the states with bound ligand need to be “corrected” for
ligand by the appropriate
chemical potential. We adopt the “priming”
procedure of Hill [ 11, p. 2781. As an example, consider the transition
AM *AMT. The energy minimum for the state AMT becomes At’ = Ai - pT,
while for the rigor state AM the minimum remains unchanged. The energies
to be compared for this transition at xm then yield a potential difference of
Al(xm)-A,(xm)=[(A&)+A,(s,)]-[A:+A,(s,)].
Likewise, for transition
AMD-AM,
(27)
we have
A?(xm)-A;(xm)=[AP+A,(s,)]-[(A~-a,)+A,(s,)]. (28)
The isomeric hydrolysis transition MT-MD,
the attach-detach
transition
MD++AMD, and the detach-attach
transition AMT-MT
do not involve
ligand exchanges with the sarcoplasmic solution, and thus do not need to be
corrected.
The partitioning of the forward-reverse rate constant pairs for transitions
between attached states is dependent
on the resultant interstate energy
J. E. WOOD AND R. W. MANN
226
surface-this
being the sum of the “homogeneous”
surface and the “structural” surface. Our choice of energy surfaces for attached transitions resemble the type proposed by Huxley and Simmons [ 18, 19]-to wit, stable states
defined by narrow energy wells which bound a “flat” (or slightly convex)
homogeneous
reaction surface; the parabolic strain-energy
surface thence
superimposed [see Figure 6(b)]. The intrinsic thermodynamic stability of such
high-specificity states-in
comparison with the energetically less defined, and
thus more labile, activated complexes (these being some intermediate configuration of ligand binding and Sl angular displacement) which exist between
attached “states’‘-not
only defines the time scale of the diagram (Fig. l),
but effectively makes the Sl-actin bond(s), at least within the bounds of the
energy wells, inelastic with respect to angular displacements
of the crossbridge. It is noted however that the combination
of a flat (or nearly flat)
homogeneous surface and a parabolic-segment
structural surface is prone to
the creation of “phantom” states for some values of xm (see [ 131, and Section
II.F.2 and Figure 6(b) herein). We shall assume these “new” states to be
liable intermediates, and neglect their existence.
As we intend to show, Model AM7-2 serves as one example of a
cross-bridge model, which, despite incorporating
an elastic element (in our
case the SZ-unit) whose compliance characteristics (K,) are independent of
the biochemical state of the cross-bridge, nevertheless mimics with reasonable
fidelity the established experimental
data of the muscle literature. This
realistic performance is achieved notwithstanding
the constraints which such
an element imposes on the form of the structural free-energy surface.
(Eisenberg and Hill [2, p. 641 argue against the viability of such an independent elastic element. For more discussion on this point, see Section 1II.C
herein.)
F.
RATE CONSTANTS
The notation herein is that of Hill [l l] (Q,, X,, py, cr, etc.; see definitions
below), except that we take k,, (rather than Hill’s a,,) to denote rate
constants. We also use Hill’s priming procedure for partition functions,
which follows directly from the priming procedure for free energies (Section
1I.E). But as seen below, we differ from Hill and Simmons [15, p. 961 in the
priming procedure for rate constants.
For attached transitions AMD- AM and AM-AMT
of the AM7 cycle,
we shall need, respectively,
Q;(xm)=exp[-A;(xm)/kT]=exp[-Az(xm)/kT]exp[p,/kT]
=Q,(xm)h,t
(29)
Q;(xm)=exp[-A’,(xm)/kT]=exp[-A,(xm)/k7’]exp[/.+/kT]
=Q,(xm)bv
(30)
ENSEMBLE
MODEL
OF MUSCLE
221
CONTRACTION
where h =exp(p/kT)
is the absolute activity of ligand chemical
and where, for example, Q,(xm)= Q,“Q,(s,)
[as per Equation
The two pseudo-first-order
(PFO) rate constants of the
corresponding
to the adsorption transitions AM- AMD and
respectively, are given by
potential p,
(26)].
AM7 cycle,
AM- AMT
k;z(xm)=k;,(xm)cD,
(31)
k;,(xm)=G(xm)c,,
(32)
these being the product of a second-order rate constant (k:)
and a ligand
of
concentration.
In particular, cn and cr are the molar concentrations
ADP.P and ATP, respectively, surrounding the cross-bridges. In a manner
analogous to the “corrected” (primed) partition functions, the PFO (primed)
rate constants are directly proportional to a ligand property.
Using the PFO rate constant for ATP adsorption (k&), the equilibrium
reaction rate k$ nac
be referred to the nonequilibrium
rate with
k;Pq( xm) = kfy;c )
(33)
.
T
Then, for an allied pair of model parameters k&(--h
and cr/c;
(see
Section III.A), the equilibrium parameter k$( -h
is located via Equation
(33). This then allows one to specify any new ratio cr./c+ to obtain the
corresponding
new PFO adsorption rate parameter k;4(-h
A similar
relation for the adsorption of D (rate k
is obtainable if one desires to vary
(c,/GJ.
Via detailed balance, the ratio of reciprocal rate constants for each
transition i-j of the biochemical state diagram at xm can be equated to the
ratio of corresponding state partition functions as in the Eyring [4] theory. In
general, we have
kl,(xm)Q,(~m)=k,,(x~I)e,(xm).(341
We list here for reference the five fundamental
complete cycle of AM7 at xm (Figure 2):
k12(xm)Q2(xm)
relations
<Q_“/Q”,Q,<s
K,:(x~)--k2,~xm~
=e,=
2
needed
for the
)
,
z
t
Qdxm>
=----(3%)
Q;( xm) =(Qg/QzO)[QM(s3)/Q~(sZ)1/AD,
K23(xm)E
k;?( xm)
G(xm) QG(xm)
(35c)
K34(x’+=k4,(xm)
=e,(xm)=(Q40/Q30)[Qnr(~4)/QM(Sj)lhT1
kdxm)
(354
K,,
+
15
Q5
=<Qb’Q:‘,.
228
J. E. WOOD
AND R. W. MANN
We note, as a check, that at any xm,
K,z(xm)K23(xm)K34(xm)K,,(xm)K,,
= 2
=exp[ -1.
(36)
As discussed further in Section IIIA, pr - bo is the potential energy which
drives the cross-bridges through the AM7 cycle.
The specific partitioning of the above ratios, as pertains to Model AM7-2,
is detailed in the following subsections. Discussed further in Sections III.A,C
is the motivation underlying some particulars, both numerical and structural,
on our choice of rate constants (see Figure 7).
1. Transition MD ++AMD. The partitioning of the rate constants of the
main attachment transition (MD- AMD) of the dominant cycle was chosen
as follows:
xm<-h,,
h,h)=k%Q,dd
(374
k,,(xm) =G 9
(37b)
where kp2 and ki, are the “unstrained” rates, and are not necessarily equal to
the “unstructured”
(homogeneous solution) rates. The structural rates at zero
(minimum) S2 strain then satisfy the condition
xm= -h,,
(38)
energy
By specifying ky2, ki, can then be calculated. An xm-dependent
profile which could yield the above partitioning is shown in Figure 6(a).
On the domain xma -h, we take the “compromise” form
x,2-h2,
k12(xm)=k?~Qii2(s2)
(394
k2,(xm)=%/Q,k?(s2).
(39b)
partitioning
enhances the attachment probability of those CBS with a
tensile (positive) strain, thus increasing the half-sarcomere power output at
speed.
It is noted in Figure 7 that we truncate k,,(xm)
at kzax (to facilitate
smooth stretching). In such case, the partitioning reverts to the form
This
(404
k,,(xm)=kt$“,
k12(xm)=kiY$QM(s2).
I
(4Ob)
229
2. Transition AMD-AM.
The partitioning herein of the rate constants
for conformational
(“flip”) transitions between attached states is predicated
on the Huxley-Simmons
energy surface, as discussed in Section 1I.E. We
employ the treatment of Julian et al. [22], this being an extension of the
Huxley-Simmons
theory such that the strain-energy term may manifest itself
in the expression of either the forward or the reverse rate constant, depending
on coordinate xm during the transition [see Figure 6(b)].
The transition AMD++AM (2-3) in the forward (dominant) direction
presumes the desorption of products D.P simultaneous
with an angular
change of the Sl-unit. The strain of the S2-unit thereby increases by h, -h 2
instantaneously
(relative to the time scale of the diagram).
At the “midpoint” coordinate, defined as
the S2 strain energy in the state AMD equals that in AM, though the elastic
energy in state 2 at hz3 derives from compression, while that in state 3 at hz3
derives from tension. This midpoint, which incidently is not the midpoint of
the reaction coordinate, is unique and serves as the breakpoint
in the
partitioning.
For values of xm less than h,, the energy barrier which the
cross-bridge “sees” in the AMD-AM
direction is simply that of the homogeneous energy well for state 2. Likewise, for xm greater than hz3, CBS
coming out of state 3 in the AM-AMD
direction will see only the constant
depth of their energy well, and will thus transition at an xm-independent
rate. Conversely, the partitioning is such that when a transitioning CB sees a
net increase in strain energy, it must necessarily surmount the entirety of that
structural energy “barrier” before coming to reside in the quasiequilibrium
well of the neighboring state. For reasons mentioned in Section II.E, it is in
the neighborhood
of the midpoint h,, that “phantom”
states, which we
disregard, may appear.
On the domain xm<h,, we have the resulting partition
xmQh,,,
k,,( xm) =k&
(424
(42b)
where the equistrain rate constants k& and kg,‘, which may approximate
homogeneous rates, satisfy the condition,
_
xm=h2,,
the
(43)
230
J. E. WOOD
AND
R. W. MANN
_
On the domain xm 2 h 23 we have
xmah,,
,
k,,(xm)=k:,Q,(s,>/Q,(~*)~
(444
ki2( xm) =k&‘.
(ab)
For our model, k,,( -h 2) is taken as the input parameter governing the
rates of the transition 2-3 and (via detailed balance) the transition 3 -2.
The equistrain rates are then obtained with the conversions
k~,=k,,(-h,>/Q,(S32)r
(454
k:','=k,,(-h2)Q~'/Q3QM(S~2),
(45b)
where s 32=h3-hZ.
3. Transition AM-AMT.
partitioning for the attached
to be similar to those of
processes of ATP adsorption
amount ha-h,.
As in the previous section,
On the domain
xm<h,,
xm<h,,
,
The energy surfaces and the manner of
(“flip”) transition AM-AMT
(5 “6) are taken
AMD++AM.
Here, AM- AMT involves the
concurrent with an increase in S2 strain by the
we define a “midpoint”
coordinate,
we have chosen the partitioning
k&( xm) = k&‘,
k,,(xm)=k~,Q~(s~)/QM(s,),
_
and on the domain xm 2 h 34, we have
xm>17h34,
k;,(xm>=k~~‘Q,(s,>/Q,(s,),
k43(xm)=k%.
Analogous to transition AMD-AM,
we take as a model input parameter
ki4( - h3), with which we can then obtain the equistrain rates,
(494
(49b)
where s 43=h4-h3.
ENSEMBLE MODEL OF MUSCLE CONTRACTION
231
4. Trans
AMT-MT.
During shortening of the half-sarcomere, the
principal mode of CB detachment is via AMT- MT (4- 5). This transition
(and MT - AMT’) is unaffected by the myofibrillar ATP concentration.
OnthedomainxmG-h,,
where the S2 is in compression, we take
k,,(xm)=
k& =constant,
xmg-h
(50a)
k,,(xm)=
ki5$QM(s4).
5
Of all model parameters, k4> has the greatest influence on the maximum
steady no-load shortening velocity (V,,). Increasing k4< in effect diminishes
the internal resistance of the ensemble to contraction, thus increasing I/.
On the domain xrna -h
where the S2 (in the state AMT) is in tension,
we take
xrna
-h
k,,( xm)=kG
=constant,
(5la)
(5lb)
5. Trans
MT-MD.
The isomeric transition MT-MD
(5-l)the enzymatic hydrolysis of ATP by myosin-is
a detached transition and
thus has rates (per CB) independent of any value of xm. If the forward rate
constant k5, is specified, then the reverse (anabolic) rate is given by detailed
balance as simply
k,, =ks,Qi'/Qb
III.
COMPUTER
(52)
SIMULATIONS
In the interest of physiological realism, the formalism for an indefinite
array of equivalent actin sites [12] was adopted for Model AM7. This
formulation, on a f
array of sites, enables an exact closed-form matrix
solution for the steady isometric distributions (see Section II.C.l)-a
property not explicitly obtainable from cross-bridge models which assume distributed attachment and a continuum of sites. The capability to start the records
from the exact isometric distributions at any level of activation facilitates the
simulation procedures and minimizes computational costs.
A.
PARA METERS
The following parameters are for Model AM7, version 2. These parameters were chosen not only to produce the desired mechanochemical
trends in
_I.E. WOOD AND R. W. MANN
232
the simulations which follow, but also to replicate reasonably some twenty
other protocols and output quantities (such as F,, S,, V,,, etc.).
No attempt was made to use the “homogeneous”
(test-tube system of
S 1-units and F-actin) biochemical rate constants as found in the literature for
some of the transitions included in the AM7 cycle. Nor is any attempt made
herein to ascribe atomic-level mechanisms to the homogeneous
rate constants. Instead, they are taken to be purely phenomenological,
the entire set
of rate constants (in concert with the nine energy-level parameters A,, h,, and
K,) for AM7-2 being chosen in accord with the effect they produce in
unison on the ensemble-averaged
mechanochemical
performance of the halfsarcomere. No optimization algorithm, other than trial and error, was implemented in seeking an “optimal” set of parameters. Hence, the parameters
presented do not necessarily represent an optimal set, nor the only set which
might reasonably simulate the currently available data. They are simply a set
which represents a good compromise for the performance attributes sought.
Since the transition fluxes between states are the product of the rate
constants k,, and the state probabilities p,, parameters which effect the
isometric probability distributions, such as the h,, will consequently have an
influence on the “rates” and “extents” of the observed macroscopic transient
phenomena. Thus rate constants alone do not determine the gross transient
time courses. So hence no one parameter (rate constants included) could be
altered without affecting in some way, however small, all of the various traces
and isometric output specifications. All results are a synergism of all parameters.
We begin our quantification
by constructing the AM7 model nexus, the
strain-dependent
basic free-energy levels.
The temperature of operation for the isothermal system is
T=277”K
Boltzmann’s
Boltzmann’s
constant
(4°C).
(53)
k=1.38054X10-RdynA/QK.
(54)
is
energy for our system is then
kT=3.82410X
The stiffness of the myosin SZunit
K, = 10.0X
lop6 dynA.
is taken to be
10
-’ dyn/A.
(56)
coordinates of the energy minima for attached states are
given by the -h,, where h, is the SZstrain in state i when attached at
(xm)=(OO):
h, =O.O,
h, =76.5 A,
(57)
h, = 100.0 A.
We do not concern ourselves with the absolute energy levels of a crossbridge state, nor the absolute values of the chemical potentials (or equivalently, the concentrations)
of the sarcoplasmic moieties. Instead, we define
the nonequilibrium
operating point relatiue to some point of systemic equilibrium, the ratios cT/cF and c,,/cb defining the degree of deviation from
equilibrium. The extant chemical potentials of T and D, relative to some pair
of equilibrium potentials, can then be related to the above concentration
ratios by
iTln(
c,/c;),
(5ga)
Apn = pu - &, = kTln( c,/c;),
(58b)
where we must have & -pb,
and where, from the equilibrium constant for
ATP hydrolysis, c; defines cb, or vice versa. However, c’f and ch do not
need to be made explicit, since they are implied in the values assigned to the
set of PFO equilibrium rate constants. Likewise, the nonequilibrium
(extant)
pair cr and cn are implied in the values assigned to the nonequilibrium
PFO
rate constants [see Section II.F, in particular Equations (3 l)-(32)].
In particular, we define the physiological operating point to be
CD/C& = 1,
CT/c;
=
109.
(594
(59b)
Thus, the energy drop for one complete cycle of Model AM7-2 (physiological, in the T-D
direction; see Figure 4) at any xm will be equal to the
negative of the per-cross-bridge thermodynamic driving potential:
CT/G
plT-pun=kTln----=20.723kT.
CD/C;)
(60)
Taking A,,=A
p’=O.O (at cn/ce n-- 1) as the reference energy level, we
have the self-extending
set of free-energy minima for Model AM7-2 (at
234
ENSEMBLE MODEL OF MUSCLE CONTRACTION
A:,--[ + 2.0
0.0
- 1.0
-11.0
+
+
-ln(c,/c;)]kT=
-ln( c,,/c&)] kT=
-ln(c,/c;)]kT=
]kT=
5.723-ln(c,/c;)]kT=
2.0
-In(cT/c’+)]kT=
235
f2.0
0.0
- 1.0
kT,
kT,
kT,
- 11.0 kT,
- 15.0 kT,
- 18.723kT,
(61)
It is seen that the equilibrium set of minima associated with the implied
1 and cr./c;= 1. Oth
equilibrium sets
(G> c&) pair is given when co/&=
are given whenever cr/c$=cn/c$.
In fact, whenever the sarcoplasmic T and
D are in equilibrium (l+=p,,),
the indefinite nonequilibrium
set of energy
levels becomes a closed set, with the net energy drop for any complete cycle
equal to zero (see Figure 4). In such case, the net CB cycling flux and the net
ensemble tension are also zero (see simulation 14, Figure 21) and thus the
potential of the muscle for doing positive work is zero.
This completes the specification of the parameters necessary to construct
the o
c nstant
energy functions for the detached states and the paro
ba l
energy
functions for the attached states (see Figure 5).
The parameters necessary to completely specify the xm-dependent
rate
constants are given as
ki’,
max
k
21
=0.030 msec- ’
=O.lOO msec-’
k,,(-h
msec-’
k&(-h
1.400 msec -I
at cT/c;=
10’.
(62)
k,
=0.375
k+45
=
0.050 msec- ’
k
=
0.500 msec- ’
51
msec- ’
These values, in conjunction
with the basic free-energy levels (Section
1I.E.) and the partitioned state transitions (Section ILF), complete the description of the a pr
rate constants. The complete set of xm-dependent
rate constants is plotted in Figure 7.
236
J. E. WOOD
AND R. W. MANN
MT
MD
-2:o
---20
I
-2ho
-1:o
I
-100
1
I
+50
401
I
+100
I
+150
xm
FIG. 5. Free-energy
levels versus xm for the five states of Model AM7-2. The energy
scale is in units of kT, with T=277”K.
ATP is at its defined physiologic operating point
[Equations
(591. All energy-state
functions
have been implicitly
corrected
for ligand
[although the primes have been omitted, the energy minima shown match those of Equation
(61)]. Superscript
c denotes that the state is also a member of the equilibrium
set:
c~/c.$=c.~/c&=
I. The heavy pathway represents a possible stochastic walk for a crossbridge during steady shortening.
The only necessary
spacing:
structural
parameter
6=55 A.
It is noted
formulation
regarding
actin is the intrasite
(63)
that the intrafilament
spacing does not enter into the model
as an identifiable parameter. Our structural interpretation
of the
ENSEMBLE
MODEL
OF MUSCLE
AMD
237
CONTRACTION
AMD
,o--c
-A”3
AM
FIG. 6.
MD -AMD
shown,
(a) Sections through a possible free-energy surface for attach-detach
transition
(I -2). Homogeneous profile = -.
Structural profile = __.
-,
As
k,,( um)=constant.
(b) Typical section through transition
AMD++AM
(2-3).
Homogeneous
profile= -_
S2 strain energy profile = - - - -. Total (superpositioned)
energy profile = __.
-.
Note the existence of a phantom state. Note also that the
reaction coordinate 5 and the Sl angular displacement
coordinate 0 are in parallel. Similar
sections would he found for transition AM++AMT.
small thermodynamic
Figure 3.
cross-bridge
system is shown in the scaled schematic of
recursive approximate-discrete
state-transition-matrix
solution to the
equations of cross-bridge kinetics for Model AM7 (Section II.D.1) was
implemented (with FORTRAN IV) on a digital computer. A constant time step
of Tz0.25 msec was used for all simulations. Although the distributions were
shifted instantaneously
by some amount As at each discrete time t= kT (this
kT should not be confused with Boltzmann’s energy), the half-sarcomere
velocity was calculated as though the shift had proceeded uniformily over the
k th interval, that is, V= -As/T.
The x-axis (-6/2~:x~
+6/2) was divided into increments of Ax=5 A.
6~55 A thus generates 11 subensembles of equal relative size Ax/6 centered
at x*=-25,
-20 ,..., -5,0,+5
,..., t25 A. The discretized xm-axis is then
generated with x*m F-X* + m8, with
spanning the expected range of sites
with significant populations for the simulation (see Section 1I.B).
For a half-sarcomere
ensemble with N cross-bridges, the total force
generated by the ensemble, on an x-axis divided into n equal segments
240
J. E. WOOD
In general, the ensemble
fraction of CBS in state i is given by
p,(t)=+
where, for detached
we must have
AND R. W. MANN
2 xP,(t,x*m),
x’ m
state fractions,
(71)
m=O. Of course, because N is constant,
(i=1,2,3,4,5).
Z&)=1
Prior to each set of simulations, a logical sequence of preliminary computations was necessary. These were, at euch discrete (x*m) coordinate:
(a) Evaluation of the ten rate constants for the five state cycle of AM7
(Section 1I.F).
(b) Determination
of the eigenvalues of submatrix C( x*m) [Equation (9)].
(c) Computation
of the corresponding
matrix
of eigenvectors,
E(x*m) [Equation (17)].
(d) Computation
of the transition
matrix +(r, x*m) [Equation (19)],
which is then stored for the stepwise iterations of the simulations [Equation
(15)l.
AN of the simulations shown in Figures 8-21 were performed with Model
AM7-2. It has the following performance specifications per fully activated
half-sarcomere at maximum filament overlap:
Isometric
force (ensemble
mean):
&=3.895X
Isometric
stiffness (ensemble
IO-‘dyn/CB.
mean):
K, =6.988X
Isometric
10 -9 dyn/ACB.
mean strain of attached
ATP hydrolysis
rate (ensemble
7; =4.013X
Maximum
unloaded
(74)
mean):
10e3 ATP/msecCB.
contraction
(at cr/cq=
(75)
velocity (at Fz0.0):
Vmax= 18.398 A/msecHS.
ATP driving potential
(73)
CBS:
s, =55.74 A.
Isometric
(72)
(76)
109):
Apr. =20.723kT.
(77)
ENSEMBLE
MODEL
OF MUSCLE
CONTRACTION
241
1.0
r
-5
0.6
-
-4
0.6
-
-3
K
V
-2
.
0.2
-
0.0
-
-
0.0
0.2
0.4
0.6
0.6
1.0
P
1
-0
F I F,,
FIG. 8. Simulurion I. Plot of velocity (V), stiffness (K), and mechanical power (P)
versus force (F) for steady isotonic shortening.
K and F have been normalized
by their
respective isometric values K, and F,. Velocity is normalized by Vmar. Power is relative to
J:A,ur
(the isometric net energy consumption
rate). Compare
of Ferenczi, Goldman, and Simmons [S, Figure I], and compare
of Julian and Sollins [21, Figure 61.
V with experimental
K with experimental
results
results
Power index:
F,Vm/,,/Apr,.?;
Maximum
=22.53.
isotonic true efficiency (at F/F,
(78)
~0.35):
17max= ( ~v/‘A~,J;-)~~=~O%.
Minimum
(79)
isotonic stiffness (at
0.2498.
Isometric
state probabilities
(ensemble
fractions):
# =0.2583,
.
p,” =0.2439,
State probabilities
(ensemble
jg =0.0430,
j; =0.3737,
fractions)
$:=0.0811.
(81)
at vmax:
ji, =0.6878,
p2 = 0.0250,
(80)
ps =0.1377,
p3 = 0.0603,
j4 =0.0892.
(82)
a
2
I
04.
-
-400
-500‘
-
-
-300
-200
-100 -
o-
S&u&ion
4. Simulations
records. Distance of shortening per half-sarcomere
with a second force drop of 0.2& at 1~20
relative to isometric.
Compare
with records of
numbers are loads
and Nolan [24, Figure 81.
Identifying
Podolsky
msec.
( D,,) versus time, for isotonic contractions.
FIG. 1 I
t - msec
ENSEMBLE MODEL OF MUSCLE CONTRACTION
245
246
J. E. WOOD
I
t = o.o-
.(.
t = 20.5
AND R. W. MANN
msec
.,
-100
0
-50
xm
t = o.o+
t - 0.25
1
t = 28.5
t = 60.0
T
FIG. 13. S/mrrlcrt~~~ 6. Probability
distributions
of attached states at selected times for
a -60-k
isometric transient (see corresponding
force record. simulation
5. Figure 12).
Chronology
of events: t =O.O-,
isometric distributions:
I =O.O+, length step of -60 A is
applied to half-sarcomere
(TI-point);
t =0.25 msec, j? attains its maximum value for the
transient;
f ~5.0 msec, T2-point occurs and j., attains its maximum value; t =20.5 msec,
phase-3 recovery
tension attains minimum
(T3-point):
t ~28.5 msec. the number of
attached cross-bridges
is a minimum (i.e., & + & + p4 = min):
msec, an intermediate
stage of phase-4 tension recovery.
ENSEMBLE MODEL OF MUSCLE CONTRACTION
247
ENSEMBLE
MODEL
OF MUSCLE
CONTRACTION
249
FIG. 16. Simulution 9. Curves of Tl and T2 (see [ 19, Figures 5, 14; 8, Figure 131). Set A
derives from simulation
5 (Figure 12: isometric,
V=O.O, TO= 1.0). Set B derives from
simulation 8 (Figure 15: isovelocity, V=2.7350 A/msecHS,
TO=O.4).
t= 10.0 msec, not shown). This phenomenon is a result of the oscillationduring the shortening phase of its sinusoidal displacement cycle-shifting
CBS in attached states 2 and 3 to xm positions with lower (less positive) S2
strains, and thus to positions with lower resistance to transitions in the
direction of the higher-tension-producing
states 3 and 4. Because of the
strong energetic asymmetry (Az’>Ai >A:‘) provided by the physiologic
concentration
of ATP (at equilibrium, the order would be Ai’>A;‘>A!;
see
Figure 4), and the energy-well structure of the attached states (Section ILE),
the CBS are not “drawn” back with increasing rates to the lower-tension
states during the lengthening phase of the cycle. The result is a net increase of
CB flux to the higher-tension states even though the length oscillation itself is
symmetrical in time.
One of the most dramatic illustrations of attached CBS transitioning
en
masse
to their next state in the dominant
cycle is provided by quick
250
N
FIG. 18.
Simulation
II. Simulation
records.
t -
Relative
msec
force versus
time, for instanta-
compared
with the analogous
the steady force level eventually
experimental
generated
records
velocity. These records
may be
of Julian and Sollins [2 I, Figure 51.
by the prescribed
r=O.O msec. Length steps of -30,
-40,
- 50, and - 55 A/HS were applied at f= 10.0
msec. followed by length ramps of f2, +4, t8, and + 16 &‘msecHS
respectively. The
step sizes were chosen to drop the force at r= 10.0 msec to a value approximately
equal to
neous length steps, followed by length ramps, with a superimposed
sinusoidal
length
oscillation. The length oscillation, 5 A (peak-to-peak
per HS) at 500 Hz, was activated at
,
0
100
0.0
was continued.
Compare
FIG. 19.
Shortening was stopped
100
Simulation records. Continuation
of simulation I I (Figure 18).
for I a 100.0 msec, while the sinusoidal length oscillation
with Julian and Sollins [2l, Figure 51.
msec
ENSEMBLE
MODEL
1.0
OF MUSCLE
Ouick
253
CONTRACTION
Stretch
\s=+3oii
3
1,
2
4
0.0
5
ty
0
30
20
10
l.O-
Quick
40
50
60
40
50
60
40
50
60
\S=-608
Release
PI
0.5-
r
o.oL
l.O-
0
lsotomc
10
20
30
Contractton
F = 0.0
4
0.5
-
o.ol.O-
lsoveloctty
P,
Contraction
v = vlll,x
.
0.5
-
o.o30
20
t
-
msec
Simuhlion
13. State probability
fractions versus time. Records correspond
to:
FIG. 20.
simulation 5 (.S = +30 A, not shown); simulation 5 (Figure 12, S = -60 k); simulation 2
(Figure 9, F =O.O); and simulation IO ( V = Vmax, not shown). The & records for the isotonic
and isovelocity contractions
are truncated when they match, to three significant digits, their
respective steady-state values. Note also that although the isotonic ( F ~0.0) and isovelocity
( v = V,,) contractions
go through dramatically
different transients, they yet both achieve
set of steady-state
CB distributions
[implying that the operating
coordinate
(F, V)=(O, V,,&
is mathematically
both stationary and unique].
the same
J. E. WOOD
254
AND R. W. MANN
0.6
0.4
0.2
0.0
FIG. 21.
Sinruluriorr
14. Plot of steady
isometric
values of: force (F,);
mean strain
of
attached cross-bridges
(S,); ATPase rate (J:);
and the population,fraction
of the rigor
state (,$ (AM)): all versus log,,( cT/c$).
( cT/cf)
is the ratio, actual to equilibrium,
of
myofibrillar
ATP concentrations.
ADP concentration
is fixed at co/ch=
1 [as in Equation
(61)]. f;,, S,. and J: are normalized
with respect to their values at the physiological
operating point for AM7-2 (c,/c;=
109. co/~ b= I). The dashed portions of the curves are
where the computer solution was algorithmically
singular (though not necessarily theoretically singular). However. it can be shown rigorously for this class of models (see [ 1I, p.
2931) that the net ensemble force (and hence the mean cross-bridge strain) must vanish when
I here, since D was not varied);
ATP is in equilibrium with its products ( cT/c ;=co/c&=
thus the legitimate
extrapolation
of F, and S, to zero.
shortening steps applied to the isometric half-sarcomere (see simulation 5,
Figure 12). For each shortening step, all of the SZ-units of the attached CBS
are instantaneously
shortened by the same amount (this is equivalent to a
uniform shift of the distributions;
see simulation 6, Figure 13). This decrease
in CB strain decreases the forward-transition
energy barrier for CBS in state
2 at xm>h,,
(see Section II.F.2) and for CBS in state 3 at xrnaK3, (Section
II.F.3), thus increasing their transition propensity for states 3 and 4 respectively. The ever-present homogeneous affinities A: -A:’ and A!’ -A; are thus
255
transduced at increasing rates into mechanical force as the CBS transition to
higher-tension attached states. This rapid CB flux to states of higher positive
strain corresponds to the very early tension rise (1 to 3 msec) following the
Tl point. As seen in simulation 6 (Figure 13), the probability distributions
corresponding
to simulation 5 (- 60 A) show that within 0.25 msec of the
applied step, virtually all of the CBS which were in state 2 (AMD) at
isometric have transitioned
to the rigor state 3 (AM). And, as seen from
Figure 5, the free-energy drop from state 3 to 4 for most of the CBS, after
th
length shift, is not as great as the preceding drop (at the same xm) from state
2 to 3; hence the additional 5 msec until the number of CBS in state 4 (AMT’)
achieves a maximum.
Other examples of tension recovery in response to length steps, as accomplished by CBS transitioning en masse
to states of lower energy yet higher
tension, are seen in simulations 7, 8, and 11 (Figures 14, 15, and 18).
On computer simulations, in contrast to the laboratory (inertial) setting,
all mechanical changes in sarcomere (or fiber) length and force can be
performed instantaneously.
To this extent, the computer records represent
the limiting case of what is to be expected experimentally.
As a notable
example, quick length steps performed experimentally require about 0.2 msec
to complete. But in the course of what amounts to an extremely fast length
ramp (though not so designed) there is an appreciable increase of cross-bridge
flux in the direction of tension recovery. Thus the “true” Tl point is never
observed; the resulting experimental TI data curve shows increasing deviation from linearity for progressively larger steps (see [7]). The apparent
curvature is partially attributable to the recovery transitions (predominantly
AMD-AM
for AM7-2) which occur during the small but finite time step
ass&ated with the length step. By contrast, the simulation counterpart is the
straight Tl line(s) of simulation 9, shown in Figure 16 [the linear Tl line(s)
being a consequence of K, =constant].
But there exists still another possible source of discrepancy between the
model performance and the experimental records-namely,
differences in the
uniformity of filament overlap within the two sarcomere populations (model
and experimental). The computer simulations herein are for a single halfsarcomere, or equivalently, a series of half-sarcomeres all of identical length,
assumed to be operating at the plateau region of overlap. The experimental
records for their physical counterpart are for a minimum of a single fiber
(lo3 to lo4 sarcomeres in series), within which, with significant likelihood,
the sarcomeres (or half-sarcomeres) will not all be of identical length. Even
for fiber preparations designed to operate within the plateau region, it can be
expected that inside the real fiber some of the sarcomeres will be operating
slightly outside of this region. So even though all sarcomeres cormected in
series will see the same absolute force, phase differences in overlap will in
256
J. E. WOOD
AND
R. W. MANN
general produce phase differences in efct
ensemble sizes and thus phase
differences in the force per cross-bridge per effective half-sarcomere ensemble. For many of these serial ensembles, the effective mean force (EMF)
which they “see” will not match their isometric-producing
effective mean
force (IEMF), which is
of overlap. This mismatch (determined by
the ratio EMF/IEMF
for each HS) will generate subtle asynchronous
transient activity within the interconnected
sarcomere population (this activity, when it becomes unstable, is the origin of the well-known “creep”
phenomenon
of striated muscle; see [20]). Such activity, particularly
for
fibers having transient-producing
boundary conditions, will in general cause
filtering, or smoothing, of the concatenated
dynamics of real sarcomere
populations
at some of the critical points of the transient records. These
phase-generated
smoothing effects should of course become more pronounced for asynchronously
innervated muscle bundles (the normal physiological mode of skeletal-muscle recruitment),
whose resultant force is the
superposition of phase differences in overlap lengths na d
calcium activation
na ,d
in the case of fatigue, ATP concentrations.
Not so marked as the Tl/T2
responses, but equally ubiquitous, is the
early tension recovery as observed in the responses to the applied length
ramps of simulation 10 (isovelocity shortening), shown in Figure 17. As was
the case for length steps, whenever the sarcomere is shortening, the S2 units
of the attached CBS, irrespective of the attached biochemical state, are
obliged to shorten accordingly. On the whole, the CBS, being moved steadily
in the direction of decreasing positive force
(although not necessarily decreas“see” monotonically
diminishing impedance in the direcing strain energy),
tion of tension-restoring
transitions. This is reflected in the monotonically
increasing rates (k,, and k
at which CBS will flip forward as they are
shifted towards more negative xm. However, the tension recovery phenomena for length ramps, appearing as the first “bulge” in the early phase of each
trace (simulation
IO), is less pronounced
than for the isometric transients
(simulation 5, Figure 12). Two reasons can be given for this difference, both
attributable
to the more gradual shifting process of the length ramp. First,
the CBS, instead of accumulating in one state momentarily (such as AMT for
the larger isometric steps, e.g. -80 A), have more time from the onset of the
ramp to complete their cycles (including detachments and reattachments).
Second, any tension gains due to flipping transitions
are always being
annulled by the steady shortening.
Constant subisometric loads are another set of HS boundary conditions
which will cause a general shifting of the CBS to positions which are more
energetically favored to propel them in the direction of ATP adsorption and
hydrolysis. But although the load may be constant, the resultant shifting
(shortening) rate will in general not be constant except at steady state. This is
seen in the length traces of simulation 2 (Figure 9). Interestingly, for isotonic
257
loads just somewhat less than isometric, there may even be a reversal of
shifting, the sarcomere actually lengthening momentarily before achieving its
steady-state shortening rate (see simulation 2, trace 0.8). This reversal phenomenon is precipitated by the length shift associated with the initial force
drop (from isometric to subisometric at t =O.O). Just as for the isometric
transient, CBS in states 2 and 3 will then be more inclined to make it to state
4. Further shortening, synchronous with the quantum force increases associated with the earliest flip transitions, further expedites these same transitions
-the influx of CBS to state 4 via forward flips ( ki4) exceeding the outflux
via detachments
(kJ5) and reverse flips (k4s). But this “autocatalytic”
process is finite (sincepi andjt are finite, and attachment k,, is rate-limiting),
and eventually, as the CBS bearing the greatest tension continue to detach
(kd5), the burden of the load is transferred more and more to the few
remaining CBS of states 2 and 3. So, about 2-3 msec after the population of
state 4 has peaked and is on the decline (detachments predominantly),
the
attached S2s (and thus the HS) are compelled to lengthen slowly in order that
their combined forces may continue to match the constant imposed subisometric load. Reattachments
eventually bring stability to the situation, with
steady contraction being the end result.
When the applied load is zero, the maximum physiological velocity of
steady contraction (V,,) can be achieved. (During the initial phases of some
isotonic transients, velocities greater than V,, are observed, as in trace 0.0 of
simulation 2, Figure 9.) At V,,, all attached CBS in compression exert at
any instant a total force equal and opposite to the total force generated by all
of the CBS whose S2 is in tension (the net AM7-2 HS ensemble force being
zero). It is the ongoing net flux of CBS driven by ATP in the direction of
higher tension-producing
states which obligates the unloaded myofilaments
to slide past each other (in the direction of lower tension-producing
xm-values)
at a rate which exactly annuls the aggregate of the internal strains. Equal
subensemble densities (Ax/s)
assure ultimate stationarity of the distributions wherein the rates, stochastic patterns, and thus fluxes at which CBS
trunsition through the structural diagram (and thus generate tension) are
exactly offset by the rate at which the same distributions
are shifted to
positions of lower (zero) tension. Theoretically, and thus computationally,
the half-sarcomere
can be driven faster than V,,, resulting in a steady
negative tension output. (In simulations 10 and 11 (Figures 17 and 18), the
HS is length-driven
slower than Vmax, yet transient negative tensions are
observed. Also, for the larger length steps of simulations 5 and 8 (Figures 12
and 15), compressive HS forces are momentarily generated.) However, for
real fiber systems, driving the fiber into compression, even though fully
activated, will result in buckling of the specimen.
Also at V,,, it is noted that the number of CBS in state 4 (p4 = 0.0892 at
V,, for AM7-2), which is the attached state with the lowest energy minimum
258
J. E.
WOOD
AND
R. W. MANN
(at physiologic ATP), exceeds the number of CBS in the other two attached
states combined ( p2 + p3 = 0.0250 + 0.0603 = 0.0853 at V,,). This is in contrast to the isometric distributions wherein state 4 is the least populated of
the three attached states [see Equation
(Sl)]. This “inversion” of the distributions at or near V,, is due to the increased average rate constants for
transitions AMD ---tAM and AM- AMT which the highly shifted CBS see,
these rates being much greater than the detachment rate out of state 4
(AMT-MT).
The detachment rate constant k,, is thus rate-limiting for the
forward-cycling attached CBS, although for the overall cycle the attachment
rate k,, is rate-limiting, thus retaining the majority (69%) of the ensemble
CBS in detached state 1 at V,,,. By contrast, transition AM-AMT
(in the
large) appears rate-limiting
at isometric (AM-AMT
is completely ratelimiting at equilibrium, as shown by PI: in simulation 14, Figure 21).
The steady-state cause-and-effect
relation between the generation of ensemble force and the HS velocity of contraction
is symmetrical, that is,
reversible. If one imposes a certain force (load) on the HS model, or on real
muscle, a certain single-valued velocity is ultimately attained (neglecting
microscopic velocity fluctuations due to CB noise within the HS ensemble).
Conversely, if one prescribes a certain velocity of shortening (or lengthening),
a certain unique force is obtained (neglecting force fluctuations). Necessarily,
the same steady-state force-velocity relationship is generated whether one is
conducting length-controlled
or force-controlled
experiments (as in simulation 1, Figure 8). This is because those CB distributions which are stationary
in time inherently define both the ensemble force (F) and the velocity of
shifting (V). When plotted over the shortening domain, these stationary
(F, V) point pairs yield a hyperbolic (actually, near-hyperbolic
for AM7-2)
force-velocity relationship. This form, in a nontrivial way, is a consequence of
the rate-constant
formulations
and the SZunit elastic modulus (K, =
constant). However, as seen from muscle models by others, rate constants do
not in fact have to satisfy detailed balance in order to generate an F-V
hyperbola. Thus, to produce only a hyperbolic F-V relationship is not a
stringent test of a model.
During a force-controlled experiment, the total ensemble force at all times
matches the load, with the CB cycling fluxes eventually settling in on that
pattern which achieves the match with smooth and steady shifting. Conversely, prescribing the shifting rate (length-controlled)
in effect prescribes
the rate at which the CBS encounter new xm, and thus new rate constants.
This “forcing” of the CBS into the rate constants (for either shortening or
lengthening)
eventually results in a stationary distribution
of CBS, which
generates a net force as indicated by the steady-state force-velocity relationship.
ENSEMBLE
MODEL
OF MUSCLE
CONTRACTION
259
But in the process of arriving at the same steady-state endpoint having
started from common initial conditions, a force-controlled
protocol and a
length-controlled
protocol will in general generate different patterns of
transient CB dynamics. An example of this difference is graphically shown in
simulations 13c and 13d (Figure 20). Although the state population fractions
go through dramatically different transient patterns for the two protocols,
they achieve the same steady-state values. Moreover, it is noted that the
steady-state values are achieved more quickly, and with fewer oscillations, for
the isovelocity contraction than for the isotonic contraction. But this should
be expected, since the interplay between the elastic S2-units and the inherent
damped feedback mechanism of the cyclical rate constants is of higher order
and complexity than simple steady shifting. We can thus say that the length
oscillations observed in the Huxley-Simmons force-clamp records [ 19, Figure
21 are not experimental artifacts, but indeed have their origin in the kinetics
of cycling cross-bridges.
Perhaps not so expected is the unimodal rise in isometric tension (F,)
predicted by Model AMl-2 as one lowers the ATP concentration ratio cr/cq
from the normal physiological operating point ( cT/c; = 10’) towards equilibrium (see simulation 14, Figure 21). This would be the direction with body
fatigue, and ultimately death (at equilibrium). The increased tension is simply
a result of more CBS moving into the rigor configuration (as indicated bypy).
But although the isometric tension may increase (peaking at about
log,,(c,/c;)=6.7),
the ability of the muscle to recover tension in response
to shortening steps is severely retarded, as diagnosed by the slower isometric
cycling rate (J,“). Likewise, the ability to contract isotonically is drastically
curtailed, the unloaded contraction velocity being virtually zero for ATP
concentration
ratios as high as 105. Of course, at equilibrium (cr/c$ = l),
both the generated isometric force and the mean (attached) CB strain (S,)
must be zero, with the CBS achieving a Boltzmann distribution, wherein, for
AM7-2, 99 + %I of the CBS reside in the rigor state (AM, the lowest-energy
state at equilibrium). To the other extreme, hyperphyisologic concentrations
of ATP (ratios greater than 10’ for AM7-2) will increase the contraction
velocity (as indicated by J,“), but only at the expense of a decreased isometric
force output, as more CBS will reside in the detached states (the attachment
transition MD- AMD again becoming rate-limiting,
but in this case at
isometric).
The cross-bridge mechanism of Model AM7 is in contrast to models
whose sole mode of compliance derives from angular displacement(s) of the
attached myosin head(s) concomitant with relative motion of the filaments
(see [2, 31). Rather than have the Sl take on a few highly specific and thus
discrete orientations as in Model AM7 (which has three distinct CB orienta-
260
_I.
tions as shown in Figure 3), the Sl for this alternate model, as a necessary
consequence of the corollary assumption that the S2 is completely inelastic,
will take on a continuum of orientations. The ligand reaction coordinate .$
and the angular displacement coordinate 8 then become orthogonal, rather
than parallel as in our system [see Figure 6b]. Also, the homogeneous and
structural surfaces become one and the same, rather than superimposed as is
the case with an independent
elastic element. Consequently,
the forcegenerating mechanism of the angular-displacement
model, being analytically
the derivative (with respective to either x or 8) of the covalent actin-myosin
displacement
potentials, allows considerable freedom of definition for the
purposes of modeling unless one adheres strictly to the molecular details of
the actin-myosin
bond structure. This level of explicitness however awaits
further biochemical
analysis. Moreover, the xm-dependence
of the rate
constants becomes less obvious. All of this is not to say that a valid and
equally accurate model of the mechanochemistry
of striated muscle could not
eventually be made with a CB mechanism of this type, but only that the
formulation appears somewhat more conjectural and no less complex than
the single structural degree of freedom (KM) from which Model AM7
derives, via the K,-dependent
rate constants, its predictive powers.
Admittedly, for our model, the homogeneous reaction surfaces between
attached states could have been formed other than flat (at co/c;
= 1,
cr/ct; = 109). But since transitions along the reaction coordinate are taken
to be instantaneous,
even though parallel to displacements of the S2 strain,
any modifications
to the homogeneous
transition surface would have no
effect on the tension generated by any given CB, and only a small effect on
the partitioning,
which would thus cause only a small change in state
populations and kinetics. So the simplicity of the flat homogeneous surfaces
was retained not only because they facilitate the energy-level descriptions,
and hence the rate-constant partitioning, but because they also minimize the
number of degrees of freedom with which to fit the data, thus placing the
burden of demonstration
more squarely on the particular choice of CB
mechanism. In conjunction with an independent elastic element, any higherorder refinement of the homogeneous surface at this stage of our biological
understanding
of the underlying ultrastructural
mechanics of skeletal muscle
does not seem warranted.
However, for models which assume a compliant Sl-actin bond and an
inelastic S2, the homogeneous
surface is their starting point and must
necessarily be given some more manifold description in the direction of both
the reaction coordinate
and the angular-displacement
coordinate.
But
whatever energy surface is chosen along the displacement
(mechanical)
coordinate, a minimal mechanical requirement of the ensemble would seem
to be the ability of the model to yield a linear, or near-linear, Tl-curve. This
ENSEMBLE
MODEL
OF MUSCLE
CONTRACTION
261
could be satisfied by using energy levels identical to those of AM7-2 (i.e.,
parabolic functions of xm), although the molecular origin of the levels for the
two models would be very different (elastic SZ-unit versus elastic Sl-actin
bond). If in fact the experimental Tl-curve is still linear for stretch steps in
excess of 100 A/HS, say 150-200 A/HS, this might discount the inelastic-S2
model, since the Sl-actin bond will have definite limits of rotation (at most
2 90”) and thus definite limits of axial accommodation before the bond must
necessarily break (assuming all other elements are inelastic).
Most likely the compliance of the actual in uivo actin-Sl-S2 complex will
be some combination
of S2 strain, Sl-S2 linkage strain, Sl bending, and
Sl-actin bond displacements,
each having a different, even nonconstant,
elastic modulus. Longitudinal strains induced in the polymeric double helix
of globular actin and the LMM core of the myosin filaments should be
relatively inconsequential
to the mechanical dynamics of skeletal muscle,
with the steady-state performance being particularly unaffected. The sarcolemma membrane
and sarcoplasmic
reticulum
structures,
though not
accounted for in Model AM7-2, will undoubtedly yield contributions to gross
elastance, albeit small for the fiber lengths encountered
during normal
articulations of the skeletal members.
The next step towards upgrading the physiological realism of the modeling
would be to implement the formalism for indefinite arrays of equivalent
groups of sites (EGS); see [12]. However, this additional complexity might
not be warranted at this time, given the present uncertainties in the physiochemical nature of the cross-bridges and their mechanochemisms.
Yet, with
the EGS formalism, there will be fewer eligible sites (in comparison to AM7,
where a/f sites are equivalent, and thus equally eligible) by virtue of the
azimuthal dependence of the actin sites as seen by a passing CB. This
intrinsic constraint
on the rate of occurrence of any given attachment
transition
might automatically
satisfy the experimental
observations
of
Haselgrove and Huxley [9]. They concluded that more than half of all
ensemble CBS are in detached states at isometric. For AM7-2, only 30% of
the ensemble CBS are in detached states at isometric. By adopting the ESG
formalism, this could possibly be raised to the 50% level indicated by x-ray
diffraction studies. However, changes in AM7-2 CB stiffness and energetics
would have to be made to maintain the present AM7-2 per-CB tensions,
power outputs, and efficiencies, thus entailing a new set of model parameters
and, of course, a new round of modeling.
The National Institute of Neurological and Communicative Disorders and
Stroke provided the primary funding for the work contained herein through
National Research Service Award #5-F32-NS05345-PHY
(issued to J.E. W.,
and sponsored by R. W.M. and Dr. F. J. Julian). Supplemental funding was
262
J. E. WOOD
AND
R. W. MANN
provided by the Germeshausen Professorship, and the Whitaker Professorship of
Biomedical Engineering, both conferred on R. W.M. Simulations for AM7-2
where paid for by University of Utah Summer Research Grant #I 7082.
As implied in our model designation, there were six precursors to AM7.
Models AM3 through AM6 were covered under the above receipts. Additional
computation time for A M6 was made available courtesy of Dr. C. H. Suh of the
University of Colorado. Models A Ml and A M2 received support predominantly
from Biomedical Research Training Grant #.5-TO1 -GM-02136 administered by
the Massachusetts Institute of Technology under the auspices of the National
Institute of Generat Medical Sciences. Dr. Fred J. Julian of the Boston
Biomedical Research Institute also provided support for investigations on Model
AM2 through U.S. Public Health Service Research Grant HL-16606, American
Heart Association Grant-in-Aid 73-699, and a grant from the Muscular Dystrophy Associations of America. Moreover, special gratitude is extended to Dr.
Julian for many productive hours of discussion, and for his encouragements since
the inception of this work.
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E. Eisenberg
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12
T. L. Hill, Theoretical
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13
T. L. Hill and E. Eisenberg,
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systems. Btochenustrv 15: 1629- 1635 (1976).
T. L. Hill. E. Eisenberg, Y-D. Chen, and R. .I. Podolsky. Some self-consistent
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Bmphys. J. 15:335-372 (1975).
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