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Can muscle fibers be stable on the descending limbs of their sarcomere length-tension relations?
J. Biomethanics. Vol. 30, Nos. I l/l?. pp 1 I:%! IS?. 1997 (8 1997 Elsewrr Saence Ltd. All r\ght\ reserved Printed m Great Brltam onr!-9?90!97 Sl7 no + .on
PII: SOO21-9290(97)00079-l
TECHNICAL
NOTE
CAN MUSCLE FIBERS BE STABLE ON THE DESCENDING LIMBS OF THEIR SARCOMERE LENGTH-TENSION RELATIONS? George I. Zahalak* Department
of Mechanical
Engineering,
Washington
University.
St. Louis,
MO
63130,
U.S.A.
Abstract-The authors of a paper published in this journal (Allinger etal. (1996) J. Biomechunics 29,627T633) have concluded, based on a theoretical analysis, that muscle fibers may be stable even if some of their constituent sarcomeres have lengths falling on the descending limbs of their static length-tension relations. This note reconsiders the question of fiber stability and concludes that fibers are unstable if at least one of their sarcomeres is on the descending limb. A positive ‘short-range’ sarcomere stiffness has no effect on fiber stability, and a positive ‘effective stiffness’ of the whole fiber does not imply sarcomere stability. :i 1997 Elsevier Science Ltd. All rights reserved Keywords:
Muscle
fiber;
Stability;
Length-tension
relation;
INTRODUCTION
As it has long been a truism within the muscle mechanics community that the answer to the question posed in the title of this paper is in the negative, the reader may wonder why further discussion is warranted. The following analysis is motivated by a paper published in this journal (Allinger et al., 1996) that questions the conventional wisdom on this topic. These authors, referred to as ‘AEH’ in the sequel, present a concise theoretical analysis of the static stability of a muscle fiber, modeled as a chain of sarcomeres, and reach the following conclusions: (1) the fiber is unstable if more than one of its sarcomeres is on the descending limb of the sarcomere length-tension relation, (2) the fiber can be stable if one of its sarcomeres is on the descending limb, provided that the negative stiffness of this sarcomere is too large, (3) high positive values of the so-called ‘short-range’ sarcomere stiffness, measured in rapid length perturbations, can stabilize the fiber, and (4) a positive ‘effective stiffness’ of the whole fiber, measured by stretching or shortening the fully activated fiber. implies sarcomere stability. While the first of these conclusions is undoubtedly true, I hope it will be clear from the following calculations that a fiber will be unstable if any one of its sarcomeres is on the descending limb (i.e. has a negative static stiffness). regardless of the magnitudes of the ‘short range’ or ‘effective’ stiffness.
Descending
limb.
state of the fiber is one in which all the generalized coordinates (i.e. sarcomere lengths) maintain constant, time-independent values. This equilibrium state. and by association the fiber itself, is said to be unstable if in response to an arbitrarily small disturbance the values of the generalized coordinates deviate progressively from those at equilibrium; otherwise the equilibrium state, and the fiber, is called stable. Considerable insight can be gained by first examining a radically simplified fiber consisting of just two sarcomeres. As described above. for purposes of static analysis each sarcomere is represented by a spring whose stiffness corresponds to the local (constant) slope of the sarcomere length-tension relation, and may be either positive or negative. The potential energy of this system, springs plus load. is I/(x,,
x2) = +k,(x,
- XT)” + fkZ(X,
- P[(Xl
- xfy
+ x2) - (XT + x9)],
(1)
where x7 and x: represent the sarcomere lengths at which the sarcomere forces vanish [see Fig. l(b)]. (Note that potential energy is always indeterminate to within a constant.) We can rewrite it in terms of the stretches, y, = (x, - x7), as V(y1. At equilibrium g
ANALYSIS
We must begin by clearly defining the system whose stability is under investigation. Figure l(a) shows a muscle fiber, modeled as a chain of n sarcomeres, and loaded by an external force P; in the absence of inertial effects each sarcomere generates a contractile force equal to P. This mechanical system has n degrees of freedom, which may be conveniently taken as the sarcomere lengths x1, x2, , x,. In a strict sense this model is more representative of a fibril than a fiber, but if one assumes that the sarcomeres of parallel fibrils are in register and are uniformly activated, then the model can serve for the fiber as a whole. Following AEH it is assumed that the length-tension characteristics of each sarcomere can be represented by piecewise linear functions, [Fig. 2(a)]. Thus on each linear segment a sarcomere has a constant stiffness, positive or negative, and under quasi-static conditions the fiber can be represented as a chain of springs [Fig. 1 (b)]. An equilibrium
yz) = fky:
the potential
.‘I
+ fk,y: energy
= ki yi - P = 0
- P(.vt
(2)
+ ~2).
is stationary
implying
and
g = k,YL oy,
and
yy’ = Piliz.
that
- p = 0,
(3)
or y:” = P/k, This equilibrium is stable a minimum. That is
if, in addition CYV
dV=T$dyi+$p
2i, j aYi
dJ’j
to being
dy,dyj
for any set of stretch increments (dy,, dy,), matrix (8’IV/8yiayj) must be positive definite.
(4) stationary,
+ ... > 0
V is
(5)
which implies that the But in the present case
(6) which
obviously
is positive k,>O
Received in final form 31 July 1997. * Address correspondence: Dr G. I. Zahalak, Washington University, Department of Mechanical Engineering, St. Louis, MO 63130, U.S.A.
proving that the unconstrained stiffnesses are positive. 1179
definite and
if and only
if
k,>O,
fiber is unstable
(7) unless both sarcomere
TechnIcal Note
(1111
so
y;‘=
Yk,,(k
I + k>)
and
!‘I(e1 =
Yh2;(i,
+ /\2).
II 1)
and stability requires
I q2n.I
(12)
A-
Fig 1. A muscle fiber modeled as a chain of sarcomeres, and loaded by a force P. (b) Representation of the fiber as a chain of linear elastic springs for static analysis. The fiber is unstable if at least one sarcomere has a negative static stiffness.(c) Representation of the fiber as a chain of viscoelastic elements for dynamic analysis. The masses attached to the dampers. m;, can be regarded as arbitrarily small. The fiber cannot be stabilized by high ‘short-range’ stiffness.
Thus, as found by AEH, the constrained isometric fiber may be internally stable even if one of the sarcomere stiffnessesis negative. so long as the sum (k, + k2) is positive. But, as the preceding unconstrained fiber analysis shows. such a fiber could not support a static force in the absence of an isometric constraint. and therefore should be classified as unstable. To examine the effect of ‘short-range’ sarcomere stiffness(Ford et (II., 1977; Flitney and Hirst, 1978) we must consider the dynumic stability of the fiber, because the force increment defining this stiffnessis measured in a very rapid length perturbation, and decays quickly when the sarcomere is held isometric at the perturbed length. An appropriate model is one where the sarcomeres are represented by linearized Hill models [Fig. 2(b)]. For slow stretches the static stiffness of the ith sarcomere is ki, as before, but in a rapid stretch or shortening it becomes the short-range stiffness(k, + k:) because the damper does not move; these short-range stiffnesses may all be taken as positive. In addition, as dynamic system response is now under consideration, lumped sarcomere masses,rylr and rn2, have been added to the model. This two-sarcomere fiber can be described formally as a linear sixth-order state-variable control system, k = Ax + Bu.
. ,‘X,
x; 4
with the state vector xT = (F,, F2, el, v2, .x1, x2), where the u, = gi are the sarcomere velocities and the Fi are the sarcomere forces. The control vector is uT = (XT, x3, P). The dynamic stability of this system is determined by its characteristic equation, Det(ctI - A) = 0, where 1 is the identity matrix (Dorf, 1992). This characteristic equation can be shown to be a sixth-order polynomial in the eigenvalue (or, stability root) s(
* x
~8’ + [z;’
+ ~;‘]a’
+ [m,‘m;‘(kl +
r---
(13)
x,+-x,-
[m;‘m;*k,7;‘(k,
+ m~‘m;‘7;‘7;‘kIkz
(b) Fig. 2. (a) Piecewise linear approximation of sarcomere static length-tension characteristics at constant maximal activation. The static stiffnesses are ki, and may be positive or negative. A typical ‘short-range’ stiffness,k, + k;, is indicated on the descending limb. (b) A viscoelastic two-sarcomere fiber model for dynamic analysis, with massesmi, stiffnesseski and k$, damping constants ci. This ‘fiber’ is unstable if either kl or k2 is negative, regardless of the values of k; and k;.
+ CT;%;’
+ k;)(kz
+ m;‘(k,
+ k;) + + k;) =
7;‘7;‘(m,-‘k,
+ n~;‘m;‘k27~1(kl
0.
+ k’,) + m;‘(kz
+ k’&x’
+ m;‘kJ]x’ + k;)]or (15)
where 7i = rJki are the time constants of the system. According to the standard Routh-Hurwitz criterion of control theory (Dorf, 1992) this system will be dynamically unstable if one or more of the coefficients in equation (15) are negative. Assuming that the short-range stiffnesses (ki + ki) are positive, as well as the sarcomere time constants ti and masses mr, it can be concluded that the system is unstable if either k, or kz is negative. But this is exactly the same conclusion reached previously through a static stability analysis of the unconstrained fiber. The preceding calculations demonstrate that the stability of the fiber is completely determined by the static stiffnesses of the sarcorneres. It having been assumed that they are positive, the short-range st@nesses, (k; + ki), have no effect on stability.
In their analysis AEH considered only the stability of the constrained isometric fiber, in which case the number of degrees of freedom is reduced by one because of the constraint ~1,+ yz = Y = Const.
6%
For the n-sarcomere fiber the potential energy of the system,in terms of sarcomere stretches, is
(16)
Technical For
the unconstrained
fiber equilibrium
EY -=kiy,-P=O, r:yi and stability
requires
requires so
that
yj”=P/ki,
that
Qt=$
momenta are denoted by pt, the motion by Hamilton’s equations
p+-g+Q,.
I
where the Hamiltonian N = T + V is the sum potential energies. the latter being
k,>O
for
i=l,...,n.
i: yi = Y = Const. I=, This equation may be used to reduce the number by one and write the potential energy as V(,,.....!,.,l=~k,(Y-~~ii)2+f~~kiri-PY.
and
(19)
and the dissipative forces Qr can be expressed dissipation function Qi = - (ciF/&j,), where
as gradients
of a Rayleigh
08) (20) of degrees of freedom
(21)
which
(22) Setting the right-hand side of equation (22) equal to zero produces a set of linear equations for the determination of the equilibrium state, (y!‘. . $‘). Differentiating this equation again yields the elements of the stability matrix
of this matrix
of the kinetic
Obviously the if and only if
So, as in the two-sarcomere case, stability of the unconstrained nsarcomere fiber requires that all the sarcomere stiffnesses be positive. For the n-sarcomere fiber subjected to an isometric constraint the length remains constant, so that
The determinant
1181
the corresponding generalized of the model fiber is governed (17)
be positive definite, where 6, is the Kronecker b-symbol. matrix corresponding to Eq. (18) will be positive-definite
from
Note
is
(24) and the same formula holds for any principal minor of rank m if n is replaced by m. A necessary and sufficient condition that a matrix be positive-definite is that all its principal minors be positive (Meirovich, 1970, p. 233). Obviously, the stability matrix will be positive-definite if all the stiffnesses are positive. If one of the stiffnesses, which may without loss of generality be taken to be k,, is negative, then repeated application of equation (24) show that the stability matrix will be positive-definite only if ki > 0 for i > 2. (This is a necessary, but not sufficient, conditions.) More specifically, it can be deduced from equation (24) that positive definiteness of the stability matrix requires that k, > (- kl) for i 2 2. A corollary of this last result is that the constrained isometric fiber cannot be stable if more than one sarcomere stiffness is negative. AEH also arrived at this conclusion using somewhat different arguments. What effect will the short-range stiffnesses have on the stability of an n-sarcomere fiber? As the short-range stiffnesses are transient this question needs to be addressed again by a dynamic analysis. For a system with many degrees of freedom the RouthhHurwitz criterion used for the two-sarcomere fiber is impractical. In this case a concise analysis can be carried out by the methods of Hamiltonian mechanics, but to make the fiber model a non-degenerate canonical (Hamiltonian) system masses must be attached to the dampers [Fig. l(c)]. As the following conclusions regarding fiber stability are independent of the magnitudes of these added masses, we may conceive of them as arbitrarily small. It is convenient in this case to take for generalized coordinates the positions of all the masses with respect to fixed laboratory coordinates, rather than sarcomere stretches. It is even more convenient to use displacements from the static equilibrium positions (which may be computed from equation (22)) rather than the mass positions themselves. If these displacement generalized coordinates are denoted by qi and
In the above expressions we define 4 _ i = q. = 0. There is one generalized velocity, dZn-i, missing from the dissipation function, but F is obviously a positive-definite function of all the other velocities (assuming that the damping constants, c,, are all positive). The dissipation rate, IQ& = - 2F, is a negative semi-definite function of the velocities, but the phase-space manifold along which the dissipation rate vanishes, Gi + *“- i = 0, clearly cannot contain any trajectory of the system except the equilibrium point qi = 0. pi = 0. This last characteristic identifies our fiber model as a system with peruasiue damping (Meirovich, 1970). The stability of a system with pervasive damping is completely determined by the behavior of its Hamiltonian at the equilibrium point (Meirovich, 1970, Theorems 6.9.5 and 6.9.6). If H is positive-definite the system is stable, whereas if H can take on negative values for small displacements from equilibrium then the system is unstable. In our specific case, for positive masses, in, and m). T is a positive-definite function of the generalized momenta, so that H is or is not a positivedefinite function of position in phase space depending on whether V is or is not a positive-definite function of the generalized coordinates, 4;. Clearly, as equation (27) has the form of a sum-of-squares. V will be a positive-definite function of the qi if all the stiffnesses, ki and k;, are positive: but if any one of these stiffnesses is negative then I’, and therefore H, can take on negative values for certain coordinate combinations, and the fiber is unstable according to the theorems cited. In particulur. ifmy one of the static stiJinesses. k,, is negutioe the fiber will be dynamicull!~ unstuble regardless of the ~ulues of the k: and thrrqfore ofrhe short-range stiflnesses, k, + ki. Thus although a statically stable fiber having all the k, > 0 could be destabilized by one negative short-range stiffness, the converse is not true: a statically unstable fiber with at least one k, < 0 cannot be stabilized by high positive short-range stiffnesses.
DISCUSSION It can be concluded on the basis of the preceding calculations that (1) an unconstrained muscle fiber will be unstable if just one of its sarcomeres is on the descending limb of its length-tension curve (i.e. has negative stiffness), and (2) the transient short-range stiffness has no effect on stability. These conclusions disagree with the conclusions of AEH, who also suggested that a positive ‘effective stiffness’, measured as a static force increase following a stretch of a whole fiber. could be interpreted to imply that the fiber’s sarcomeres were stable. This latter argument can hardly be accepted because several investigators have offered arguments and calculations to prove that such increases of tension on stretch, or corresponding decreases on shortening, are precisely the result of internal sarcomere instability and redistribution (Morgan, 1990. 1994; Edman et al., 1993). That sarcomeres are not uniform within a fiber, and exhibit movement and internal redistribution even during nominally isometric contractions. are experimentally established facts (Julian and Morgan, 1979; Lieber and Baskin, 1983; Brown and Hill, 1991) and not matters of conjecture. AEH also present a simple thought experiment involving two combs with interlocking teeth, to argue that a mechanical system can have, in one sense, a negative stiffness and yet be stable. This example is not a valid analogy of muscle. The basic problem is that the teeth of the combs remain interlocked after a length perturbation. whereas the
Techmcal
11x2
cross-bridges of muscle break and re-form less strained bonds with actin -thus rapidly dissipating the force increment produced by the length perturbation (and causing the short-range stiffness to decay). If the teeth of the combs could slip past each other and re-engage other teeth with less strain, then the system of combs would be unstable-~ just as the muscle fiber is. The calculations I have presented confirm the validity of the conventional view of sarcomere instability. Although it complicates modeling, internal sarcomere redistribution during normal function appears to be a physiological reality that affects significantly the macroscopic mechanical properties of muscle. Of course, all the preceding discussion has been for isolated fibers (with ‘parallel’ elasticity sufficiently low to exhibit a clear descending limb of the length-tension relation). Sufficiently stiffexternal elastic constraints can always stabilize a fiber. Thus an isolated single fiber that is unstable due to negative sarcomere stiffness may well be stable if it is activated within a muscle where it is surrounded by restraining passive tissue, including other un-activated muscle fibers. But if the surrounding contractile tissue were also activated then presumably much of the stabilizing restraint would disappear. An adequate understanding of these issues of inter-fiber interaction will require further analysis. ilcknoM;ledgements----This 9318631.
work
was supported
by NSF
Grant
BES-
REFERENCES
Allinger, T. L., Epstein, M. and Herzog, W. (1996) Stability fibers on the descending limb of the force-length relation. ical consideration. Journal of Biomechanics 29, 627-633.
of muscle A theoret-
Note Brown. L. M. and Hill, L. (1991 I Some observations on variation\ in filament overlap in tetanized muscle fibers stretched durmg a tetanus. detected in the electron microscope after rapid fixation. Joirr& 11; Muscle Rrwmh and Cdl Mutter 12, I71 182. Dorf, R. C. (1992) Modern Corriro/ S,~sirms. Addison-Wesley. Reading, MA, U.S.A. Edman. K. A. P., Caputo, C. and Lou. E‘. (1993) Depression of tetamc force induced by loaded shortening of frog muscle fibers .Ioicrrrtr/ of Physiology 466, 53.5 552. Flitney, F. W. and Hirst, D. G. (1978) Cross-bridge detachment and sarcomere ‘give’ during stretch of active frog’s muscle. Jourd of Phvsiologp 276, 449-465. Ford. L. E., Huxley, A. F. and Simmons. R. M. (1977) Tension responses to sudden length change in stimulated from muscle fibers near slack length. Journnl of Physiology 269, 441-515. Julian, F. J. and Morgan, D. L. (1979) The effect on tension of nonuniform distribution of length changes applied to frog muscle fibers. Journal of Phvsiolouv 293. 319-392. Lieber, R. L. and Baskin, R. J. (1983) Intersarcomere dynamics of single muscle fibers during fixed-end tetani. Journui of Gearraf Pbysiolog) 82, 347364. Meirovich, L. (I 970) Metliods of’ Ana@ica/ Dynamics. McGraw-Hill. New York. Morgan, D. L. (1990) New insights into the behavior of muscle during active lengthening. Biophysi& Journal 57, 209-221. Morgan, D. L. (1994) An explanation for residual increased tension in striated muscle after stretch during contraction. E.xperimenral Physiology 79, 831-838.
PII: SOO21-9290(97)00079-l
TECHNICAL
NOTE
CAN MUSCLE FIBERS BE STABLE ON THE DESCENDING LIMBS OF THEIR SARCOMERE LENGTH-TENSION RELATIONS? George I. Zahalak* Department
of Mechanical
Engineering,
Washington
University.
St. Louis,
MO
63130,
U.S.A.
Abstract-The authors of a paper published in this journal (Allinger etal. (1996) J. Biomechunics 29,627T633) have concluded, based on a theoretical analysis, that muscle fibers may be stable even if some of their constituent sarcomeres have lengths falling on the descending limbs of their static length-tension relations. This note reconsiders the question of fiber stability and concludes that fibers are unstable if at least one of their sarcomeres is on the descending limb. A positive ‘short-range’ sarcomere stiffness has no effect on fiber stability, and a positive ‘effective stiffness’ of the whole fiber does not imply sarcomere stability. :i 1997 Elsevier Science Ltd. All rights reserved Keywords:
Muscle
fiber;
Stability;
Length-tension
relation;
INTRODUCTION
As it has long been a truism within the muscle mechanics community that the answer to the question posed in the title of this paper is in the negative, the reader may wonder why further discussion is warranted. The following analysis is motivated by a paper published in this journal (Allinger et al., 1996) that questions the conventional wisdom on this topic. These authors, referred to as ‘AEH’ in the sequel, present a concise theoretical analysis of the static stability of a muscle fiber, modeled as a chain of sarcomeres, and reach the following conclusions: (1) the fiber is unstable if more than one of its sarcomeres is on the descending limb of the sarcomere length-tension relation, (2) the fiber can be stable if one of its sarcomeres is on the descending limb, provided that the negative stiffness of this sarcomere is too large, (3) high positive values of the so-called ‘short-range’ sarcomere stiffness, measured in rapid length perturbations, can stabilize the fiber, and (4) a positive ‘effective stiffness’ of the whole fiber, measured by stretching or shortening the fully activated fiber. implies sarcomere stability. While the first of these conclusions is undoubtedly true, I hope it will be clear from the following calculations that a fiber will be unstable if any one of its sarcomeres is on the descending limb (i.e. has a negative static stiffness). regardless of the magnitudes of the ‘short range’ or ‘effective’ stiffness.
Descending
limb.
state of the fiber is one in which all the generalized coordinates (i.e. sarcomere lengths) maintain constant, time-independent values. This equilibrium state. and by association the fiber itself, is said to be unstable if in response to an arbitrarily small disturbance the values of the generalized coordinates deviate progressively from those at equilibrium; otherwise the equilibrium state, and the fiber, is called stable. Considerable insight can be gained by first examining a radically simplified fiber consisting of just two sarcomeres. As described above. for purposes of static analysis each sarcomere is represented by a spring whose stiffness corresponds to the local (constant) slope of the sarcomere length-tension relation, and may be either positive or negative. The potential energy of this system, springs plus load. is I/(x,,
x2) = +k,(x,
- XT)” + fkZ(X,
- P[(Xl
- xfy
+ x2) - (XT + x9)],
(1)
where x7 and x: represent the sarcomere lengths at which the sarcomere forces vanish [see Fig. l(b)]. (Note that potential energy is always indeterminate to within a constant.) We can rewrite it in terms of the stretches, y, = (x, - x7), as V(y1. At equilibrium g
ANALYSIS
We must begin by clearly defining the system whose stability is under investigation. Figure l(a) shows a muscle fiber, modeled as a chain of n sarcomeres, and loaded by an external force P; in the absence of inertial effects each sarcomere generates a contractile force equal to P. This mechanical system has n degrees of freedom, which may be conveniently taken as the sarcomere lengths x1, x2, , x,. In a strict sense this model is more representative of a fibril than a fiber, but if one assumes that the sarcomeres of parallel fibrils are in register and are uniformly activated, then the model can serve for the fiber as a whole. Following AEH it is assumed that the length-tension characteristics of each sarcomere can be represented by piecewise linear functions, [Fig. 2(a)]. Thus on each linear segment a sarcomere has a constant stiffness, positive or negative, and under quasi-static conditions the fiber can be represented as a chain of springs [Fig. 1 (b)]. An equilibrium
yz) = fky:
the potential
.‘I
+ fk,y: energy
= ki yi - P = 0
- P(.vt
(2)
+ ~2).
is stationary
implying
and
g = k,YL oy,
and
yy’ = Piliz.
that
- p = 0,
(3)
or y:” = P/k, This equilibrium is stable a minimum. That is
if, in addition CYV
dV=T$dyi+$p
2i, j aYi
dJ’j
to being
dy,dyj
for any set of stretch increments (dy,, dy,), matrix (8’IV/8yiayj) must be positive definite.
(4) stationary,
+ ... > 0
V is
(5)
which implies that the But in the present case
(6) which
obviously
is positive k,>O
Received in final form 31 July 1997. * Address correspondence: Dr G. I. Zahalak, Washington University, Department of Mechanical Engineering, St. Louis, MO 63130, U.S.A.
proving that the unconstrained stiffnesses are positive. 1179
definite and
if and only
if
k,>O,
fiber is unstable
(7) unless both sarcomere
TechnIcal Note
(1111
so
y;‘=
Yk,,(k
I + k>)
and
!‘I(e1 =
Yh2;(i,
+ /\2).
II 1)
and stability requires
I q2n.I
(12)
A-
Fig 1. A muscle fiber modeled as a chain of sarcomeres, and loaded by a force P. (b) Representation of the fiber as a chain of linear elastic springs for static analysis. The fiber is unstable if at least one sarcomere has a negative static stiffness.(c) Representation of the fiber as a chain of viscoelastic elements for dynamic analysis. The masses attached to the dampers. m;, can be regarded as arbitrarily small. The fiber cannot be stabilized by high ‘short-range’ stiffness.
Thus, as found by AEH, the constrained isometric fiber may be internally stable even if one of the sarcomere stiffnessesis negative. so long as the sum (k, + k2) is positive. But, as the preceding unconstrained fiber analysis shows. such a fiber could not support a static force in the absence of an isometric constraint. and therefore should be classified as unstable. To examine the effect of ‘short-range’ sarcomere stiffness(Ford et (II., 1977; Flitney and Hirst, 1978) we must consider the dynumic stability of the fiber, because the force increment defining this stiffnessis measured in a very rapid length perturbation, and decays quickly when the sarcomere is held isometric at the perturbed length. An appropriate model is one where the sarcomeres are represented by linearized Hill models [Fig. 2(b)]. For slow stretches the static stiffness of the ith sarcomere is ki, as before, but in a rapid stretch or shortening it becomes the short-range stiffness(k, + k:) because the damper does not move; these short-range stiffnesses may all be taken as positive. In addition, as dynamic system response is now under consideration, lumped sarcomere masses,rylr and rn2, have been added to the model. This two-sarcomere fiber can be described formally as a linear sixth-order state-variable control system, k = Ax + Bu.
. ,‘X,
x; 4
with the state vector xT = (F,, F2, el, v2, .x1, x2), where the u, = gi are the sarcomere velocities and the Fi are the sarcomere forces. The control vector is uT = (XT, x3, P). The dynamic stability of this system is determined by its characteristic equation, Det(ctI - A) = 0, where 1 is the identity matrix (Dorf, 1992). This characteristic equation can be shown to be a sixth-order polynomial in the eigenvalue (or, stability root) s(
* x
~8’ + [z;’
+ ~;‘]a’
+ [m,‘m;‘(kl +
r---
(13)
x,+-x,-
[m;‘m;*k,7;‘(k,
+ m~‘m;‘7;‘7;‘kIkz
(b) Fig. 2. (a) Piecewise linear approximation of sarcomere static length-tension characteristics at constant maximal activation. The static stiffnesses are ki, and may be positive or negative. A typical ‘short-range’ stiffness,k, + k;, is indicated on the descending limb. (b) A viscoelastic two-sarcomere fiber model for dynamic analysis, with massesmi, stiffnesseski and k$, damping constants ci. This ‘fiber’ is unstable if either kl or k2 is negative, regardless of the values of k; and k;.
+ CT;%;’
+ k;)(kz
+ m;‘(k,
+ k;) + + k;) =
7;‘7;‘(m,-‘k,
+ n~;‘m;‘k27~1(kl
0.
+ k’,) + m;‘(kz
+ k’&x’
+ m;‘kJ]x’ + k;)]or (15)
where 7i = rJki are the time constants of the system. According to the standard Routh-Hurwitz criterion of control theory (Dorf, 1992) this system will be dynamically unstable if one or more of the coefficients in equation (15) are negative. Assuming that the short-range stiffnesses (ki + ki) are positive, as well as the sarcomere time constants ti and masses mr, it can be concluded that the system is unstable if either k, or kz is negative. But this is exactly the same conclusion reached previously through a static stability analysis of the unconstrained fiber. The preceding calculations demonstrate that the stability of the fiber is completely determined by the static stiffnesses of the sarcorneres. It having been assumed that they are positive, the short-range st@nesses, (k; + ki), have no effect on stability.
In their analysis AEH considered only the stability of the constrained isometric fiber, in which case the number of degrees of freedom is reduced by one because of the constraint ~1,+ yz = Y = Const.
6%
For the n-sarcomere fiber the potential energy of the system,in terms of sarcomere stretches, is
(16)
Technical For
the unconstrained
fiber equilibrium
EY -=kiy,-P=O, r:yi and stability
requires
requires so
that
yj”=P/ki,
that
Qt=$
momenta are denoted by pt, the motion by Hamilton’s equations
p+-g+Q,.
I
where the Hamiltonian N = T + V is the sum potential energies. the latter being
k,>O
for
i=l,...,n.
i: yi = Y = Const. I=, This equation may be used to reduce the number by one and write the potential energy as V(,,.....!,.,l=~k,(Y-~~ii)2+f~~kiri-PY.
and
(19)
and the dissipative forces Qr can be expressed dissipation function Qi = - (ciF/&j,), where
as gradients
of a Rayleigh
08) (20) of degrees of freedom
(21)
which
(22) Setting the right-hand side of equation (22) equal to zero produces a set of linear equations for the determination of the equilibrium state, (y!‘. . $‘). Differentiating this equation again yields the elements of the stability matrix
of this matrix
of the kinetic
Obviously the if and only if
So, as in the two-sarcomere case, stability of the unconstrained nsarcomere fiber requires that all the sarcomere stiffnesses be positive. For the n-sarcomere fiber subjected to an isometric constraint the length remains constant, so that
The determinant
1181
the corresponding generalized of the model fiber is governed (17)
be positive definite, where 6, is the Kronecker b-symbol. matrix corresponding to Eq. (18) will be positive-definite
from
Note
is
(24) and the same formula holds for any principal minor of rank m if n is replaced by m. A necessary and sufficient condition that a matrix be positive-definite is that all its principal minors be positive (Meirovich, 1970, p. 233). Obviously, the stability matrix will be positive-definite if all the stiffnesses are positive. If one of the stiffnesses, which may without loss of generality be taken to be k,, is negative, then repeated application of equation (24) show that the stability matrix will be positive-definite only if ki > 0 for i > 2. (This is a necessary, but not sufficient, conditions.) More specifically, it can be deduced from equation (24) that positive definiteness of the stability matrix requires that k, > (- kl) for i 2 2. A corollary of this last result is that the constrained isometric fiber cannot be stable if more than one sarcomere stiffness is negative. AEH also arrived at this conclusion using somewhat different arguments. What effect will the short-range stiffnesses have on the stability of an n-sarcomere fiber? As the short-range stiffnesses are transient this question needs to be addressed again by a dynamic analysis. For a system with many degrees of freedom the RouthhHurwitz criterion used for the two-sarcomere fiber is impractical. In this case a concise analysis can be carried out by the methods of Hamiltonian mechanics, but to make the fiber model a non-degenerate canonical (Hamiltonian) system masses must be attached to the dampers [Fig. l(c)]. As the following conclusions regarding fiber stability are independent of the magnitudes of these added masses, we may conceive of them as arbitrarily small. It is convenient in this case to take for generalized coordinates the positions of all the masses with respect to fixed laboratory coordinates, rather than sarcomere stretches. It is even more convenient to use displacements from the static equilibrium positions (which may be computed from equation (22)) rather than the mass positions themselves. If these displacement generalized coordinates are denoted by qi and
In the above expressions we define 4 _ i = q. = 0. There is one generalized velocity, dZn-i, missing from the dissipation function, but F is obviously a positive-definite function of all the other velocities (assuming that the damping constants, c,, are all positive). The dissipation rate, IQ& = - 2F, is a negative semi-definite function of the velocities, but the phase-space manifold along which the dissipation rate vanishes, Gi + *“- i = 0, clearly cannot contain any trajectory of the system except the equilibrium point qi = 0. pi = 0. This last characteristic identifies our fiber model as a system with peruasiue damping (Meirovich, 1970). The stability of a system with pervasive damping is completely determined by the behavior of its Hamiltonian at the equilibrium point (Meirovich, 1970, Theorems 6.9.5 and 6.9.6). If H is positive-definite the system is stable, whereas if H can take on negative values for small displacements from equilibrium then the system is unstable. In our specific case, for positive masses, in, and m). T is a positive-definite function of the generalized momenta, so that H is or is not a positivedefinite function of position in phase space depending on whether V is or is not a positive-definite function of the generalized coordinates, 4;. Clearly, as equation (27) has the form of a sum-of-squares. V will be a positive-definite function of the qi if all the stiffnesses, ki and k;, are positive: but if any one of these stiffnesses is negative then I’, and therefore H, can take on negative values for certain coordinate combinations, and the fiber is unstable according to the theorems cited. In particulur. ifmy one of the static stiJinesses. k,, is negutioe the fiber will be dynamicull!~ unstuble regardless of the ~ulues of the k: and thrrqfore ofrhe short-range stiflnesses, k, + ki. Thus although a statically stable fiber having all the k, > 0 could be destabilized by one negative short-range stiffness, the converse is not true: a statically unstable fiber with at least one k, < 0 cannot be stabilized by high positive short-range stiffnesses.
DISCUSSION It can be concluded on the basis of the preceding calculations that (1) an unconstrained muscle fiber will be unstable if just one of its sarcomeres is on the descending limb of its length-tension curve (i.e. has negative stiffness), and (2) the transient short-range stiffness has no effect on stability. These conclusions disagree with the conclusions of AEH, who also suggested that a positive ‘effective stiffness’, measured as a static force increase following a stretch of a whole fiber. could be interpreted to imply that the fiber’s sarcomeres were stable. This latter argument can hardly be accepted because several investigators have offered arguments and calculations to prove that such increases of tension on stretch, or corresponding decreases on shortening, are precisely the result of internal sarcomere instability and redistribution (Morgan, 1990. 1994; Edman et al., 1993). That sarcomeres are not uniform within a fiber, and exhibit movement and internal redistribution even during nominally isometric contractions. are experimentally established facts (Julian and Morgan, 1979; Lieber and Baskin, 1983; Brown and Hill, 1991) and not matters of conjecture. AEH also present a simple thought experiment involving two combs with interlocking teeth, to argue that a mechanical system can have, in one sense, a negative stiffness and yet be stable. This example is not a valid analogy of muscle. The basic problem is that the teeth of the combs remain interlocked after a length perturbation. whereas the
Techmcal
11x2
cross-bridges of muscle break and re-form less strained bonds with actin -thus rapidly dissipating the force increment produced by the length perturbation (and causing the short-range stiffness to decay). If the teeth of the combs could slip past each other and re-engage other teeth with less strain, then the system of combs would be unstable-~ just as the muscle fiber is. The calculations I have presented confirm the validity of the conventional view of sarcomere instability. Although it complicates modeling, internal sarcomere redistribution during normal function appears to be a physiological reality that affects significantly the macroscopic mechanical properties of muscle. Of course, all the preceding discussion has been for isolated fibers (with ‘parallel’ elasticity sufficiently low to exhibit a clear descending limb of the length-tension relation). Sufficiently stiffexternal elastic constraints can always stabilize a fiber. Thus an isolated single fiber that is unstable due to negative sarcomere stiffness may well be stable if it is activated within a muscle where it is surrounded by restraining passive tissue, including other un-activated muscle fibers. But if the surrounding contractile tissue were also activated then presumably much of the stabilizing restraint would disappear. An adequate understanding of these issues of inter-fiber interaction will require further analysis. ilcknoM;ledgements----This 9318631.
work
was supported
by NSF
Grant
BES-
REFERENCES
Allinger, T. L., Epstein, M. and Herzog, W. (1996) Stability fibers on the descending limb of the force-length relation. ical consideration. Journal of Biomechanics 29, 627-633.
of muscle A theoret-
Note Brown. L. M. and Hill, L. (1991 I Some observations on variation\ in filament overlap in tetanized muscle fibers stretched durmg a tetanus. detected in the electron microscope after rapid fixation. Joirr& 11; Muscle Rrwmh and Cdl Mutter 12, I71 182. Dorf, R. C. (1992) Modern Corriro/ S,~sirms. Addison-Wesley. Reading, MA, U.S.A. Edman. K. A. P., Caputo, C. and Lou. E‘. (1993) Depression of tetamc force induced by loaded shortening of frog muscle fibers .Ioicrrrtr/ of Physiology 466, 53.5 552. Flitney, F. W. and Hirst, D. G. (1978) Cross-bridge detachment and sarcomere ‘give’ during stretch of active frog’s muscle. Jourd of Phvsiologp 276, 449-465. Ford. L. E., Huxley, A. F. and Simmons. R. M. (1977) Tension responses to sudden length change in stimulated from muscle fibers near slack length. Journnl of Physiology 269, 441-515. Julian, F. J. and Morgan, D. L. (1979) The effect on tension of nonuniform distribution of length changes applied to frog muscle fibers. Journal of Phvsiolouv 293. 319-392. Lieber, R. L. and Baskin, R. J. (1983) Intersarcomere dynamics of single muscle fibers during fixed-end tetani. Journui of Gearraf Pbysiolog) 82, 347364. Meirovich, L. (I 970) Metliods of’ Ana@ica/ Dynamics. McGraw-Hill. New York. Morgan, D. L. (1990) New insights into the behavior of muscle during active lengthening. Biophysi& Journal 57, 209-221. Morgan, D. L. (1994) An explanation for residual increased tension in striated muscle after stretch during contraction. E.xperimenral Physiology 79, 831-838.