Btdletin of Mathematical Biology, Vol. 41, pp. 813 828 Pergamon Press Ltd. 1979. Printed in Great Britain © Society for Mathematical Biology
0007-4985/79/1201-0813 $02.00/0
AN EXACT ANALYSIS OF T H E S L I D I N G M O T I O N IN C O N T R A C T I L E SYSTEM: A S I M P L I F I E D M O D E L O F M U S C L E SYSTEM
[ ] YOJI AIZAWA]"
Facult6 des Sciences, Universit6 Libre de Bruxelles HIROSHI SHIMIZU
Faculty of Pharmaceutical Sciences, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan
A system involving two kinds of sliding filaments is analysed with special attention to the actomyosin system. Rigorous results are obtained about the statistical effect originating from many active sites distributed on both filaments. It is necessary for the occurrence of smooth motion in sliding filament that the spatial periods of active sites on both filaments are relatively incommensurable, and that the number of active sites on each filament is large enough. Sufficient conditions for smooth contraction are derived under the assumption that both filaments are rigid; this is called rigid rod approximation in the present paper. The elastic mode of the filaments, during the sliding process, is analysed by perturbation theory based on the rigid rod approximation. A stochastic theory is briefly discussed in reference to the cooperative generation of contractile force, which is concerned in Hill's relation of muscle contraction.
i. Introduction. Ordered structures as well as ordered motions are often created in the system far from thermal equilibrium (Glansdorff and Prigogine, 1971; Nicolis and Prigogine, 1977). As the living bodies are always sustained in the nonequilibrium state, a variety of orders are observed in every part of an organism. It is essential for the bio-system that these local orders are unified into a global one so that each part of an organism may behave coherently and functionally with high efficiency. Among the many ordered functions in the living "~Present address: Department of Physics, Faculty of Sciences, University of Kyoto, Japan. 813
814
YOJI AIZAWA A N D H I R O S H I S H I M I Z U
bodies, motility is one of the most important phenomena. The motility of organisms is endowed with two types of coherent orders. One is the time periodic or oscillatory motion, which is typically observed in the ciliary system (Aiello and Sleigh, 1972; Gray, 1931; Aizawa, 1979), and the other is the temporally uniform motion. A typical example of the latter is the contractile motion in the striated muscle (Huxley, 1957; Huxley and Hanson, 1954). So far the mechanism o f the appearance of coherent motions in nonequilibrium state has not yet been well explained, at least from a theoretical point of view. The purpose of this paper is to make clear the cause for the appearance of coherent sliding motion in contractile systems. In the present paper we will show that coherent motion can be created in contractile system composed of sliding filaments, through a statistical effect. To extract this statistical effect in a mathematically rigorous manner, we consider the simple model system, which has a close connection with the actomyosin system in striated muscle. Our purpose, however, is not to analyse the real microscopic processes of actomyosin system in detail. At the present time there is a large amount of biochemical experimental data concerning molecular or microscopic processes in muscle, and we know that the structural change of bio-molecules plays an important role in the appearance of the coherent sliding motion. From a theoretical viewpoint, one of the most interesting problems is to construct the theory of these microscopic molecular processes. But in the present article, we wish to analyse the macroscopic description of the phenomena induced by a statistical (or many-body) effect in the actomyosin system. In other words, the purpose of the present article is to study the statistical physics of an idealized contractile system. Our starting point is the idea of two kinds of sliding filaments, an idea first proposed by H. E. Huxley and A. F. Huxley (Huxley, 1957; Huxley and Hansen, 1954). The fundamental idea of the sliding theory is supported not only by the structural analysis by electromicroscope and X-ray (Huxley and Brown, 1967; Elliot et al., 1967) but also by the bio-chemical discoveries of certain structural and functional proteins in the actomyosin system. Many results of physiological experiments have also been well explained, at least, qualitatively, in line with the sliding model (Huxley and Simmons, 1971). We will consider the following simplified model: There are two kinds of filaments, i.e., thin (actin) and thick (myosin) filaments, which are placed parallel to each other. On the myosin filament, many identical active sites occur at intervals of a certain distance 1M, and on the actin filament also other kinds of active sites occur at regular "intervals 1A. (In the present article, the active site on the myosin filament is called the "cross-bridge".) Both kinds of filaments are assumed to be long enough, so that there are many active sites on the filaments, and that all the active
AN EXACTANALYSISOF THE SLIDING MOTION
815
sites are fixed on each filament. The force is generated through the chemomechanical interaction between these two different kinds of active sites. There is no direct chemical interaction among active sites placed on the same filament but there is indirect interaction through mechanical motion of the filament. The motion takes place only in the one-dimensional direction parallel to the filaments. In the present paper, for the sake of convenience we will limit our discussion to the case where the system includes only one pair, an actin and a myosin filament. This limitation is not essential in our theory, and can easily be lifted when we consider the more realistic model of a sarcomere. Both filaments are soaked i n the protoplasmic solution which causes viscous damping to the sliding motion. The generation of the contractile force originates from the chemomechanical processes at the active sites on both filaments (Shimizu et al., 1976). We will assume that there exists a certain time-invariant force among two kinds of active sites. It is surmised that there are at least two reasons for which the sliding motion of the acto-myosin system is coherent and smooth. One originates from the statistical effect, which will be discussed in the present paper, another originates from the fact that the contractile force is not generated at random on each active site but is created synergetically in the same phase at almost every active site. (Nishiyama, 1976; Shimizu et al., 1976). Both causes are indispensable for the occurrence of the smooth sliding in acto-myosin system. The main motivation of the present paper is to analyse the first coherence origin exactly by our theoretical model mentioned above. In the analysis of Sections 2 and 3, we will take into account the second coherence origin beforehand by assuming that the generated force is independent of time. The close interrelation between these two coherence origins is discussed in "Section 4. The mathematical model is derived in Section 2. By means of the rigid rod approximation of both filaments, a reduced equation is derived in Section 3 and the small deviation from the rigid rod approximation is analysed. A generalization of our model is discussed in Section 4 in reference to Hill's relations.
2. Mathematical Model of Sliding Filaments. Since the sliding motion between both filaments is relative to each other, we will discuss the motion of a myosin filament relative to the actin filament in the present paper. As we do not take into account the internal degrees of freedom in each active site, the dynamical state of a myosin filament can be described by the ensemble of the coordinates {a~} (v= 1,2,...,N) of N cross-bridge heads. Here N denotes the number of cross-bridges on a myosin filament. The effect of the internal degrees of freedom of a cross-bridge will be discussed in Section 4., The equation of motion of the vth cross-bridge head in
816
VOJI AIZAWA A N D H I R O S H I S H I M I Z U
unloaded muscle is generally given by: d2a~ M~ dt 2 -
( d a ~ ) 7~f-+R~
+F~+V~({au} ),
(1)
where M~ is the effective mass of vth cross-bridge head (the mass of myosin molecule other than the cross-bridge is effectively included), Fv is the force caused by the activated site on the actin filament, V~ ({au}) is the mechanical interaction with the other cross-bridges through the myosin filament, 7v(da~/dt) is the viscous force at the vth cross-bridge, and R~ is the viscous force stemming from the part of the myosin filament connecting the vth and (v___1)th cross-bridge head. The interaction term between crossbridges V~ is very complicated and it is not easy to reduce to a simple form. Here we will introduce the following three generous assumptions in order to treat (1) without loss of the essentially significant mechanism in the sliding process: (I) The inertial term is neglected and only the steady state is discussed [Mv(dEa~/dt 2) = O] ; (I1) The interaction force depends only on the relative distance between the nearest neighboring cross-bridges, i.e., v+l
V~({au})= ~, V(au-au-x);
and
i I = "v
(III) The viscous force R~ is proportional to that of the vth cross-bridge head, i.e., R~w_yv(da~/dt ). By these assumptions (1) is rewritten as"
~. da~
~+ 1 =F~ + ~.. V(,(,,
au_ 1),
[d='v
(2)
The elastic mode of a myosin filament will be analysed in Section 3. As mentioned in Section 1, active actin sites are distributed periodically on the actin filament. When we write the spatial period as. 2r~kj 1 (=IA), the force acting on the vth cross-bridge F~ must satisfy the following periodic condition: F~ = F(av) =F(a~ + 2rckj 1).
(3)
where k A stands for the fundamental wave vector of force field F~. As
AN
EXACT
ANALYSIS
OF
THE
SLIDING
MOTION
817
assumed in the last section, all the cross-bridge heads are put in a timeinvariant force field. In addition, we assume that an actin filament is rigid. In other words, the external force is decomposed into a Fourier series, i.e., oo
F(a~)=
~
~._fl e ikA"av,
(4)
n~--oo
where//, =B.* (+ denotes complex conjugate) is satisfied because the force field F~ is real. The total force arising from the active actin site is given by (4), where //, is determined by the geometrical structure of a cross-bridge head and of an active actin site as well as by" the mechano-chemieal process of the force generation. The force F~ has two remarkable characteristics; the spatial localization of the force field, and the polarity of the force field. That is to say, the former means that the force field is localized sharply near the active actin site and the latter means that the potential function of the force field is distributed asymmetrically around the active site. From the second property of the force field, //040 is derived. This term is mainly responsible for the occurrence of the sliding motion. In general, / / , 4 0 is satisfied provided that the force field is localized sharply. All the parameters in (2)-(4) must be determined by the relevant experimental results. In the present article, however, we will not pursue the appropriate values of these parameters, since the purpose of this paper is to elucidate the mathematical framework of the sliding theory described by (1) (4). In this paper, (4) is only one assumption about the force field F~. Under the consideration mentioned above, the theoretical model of the contractile process is finally given by the following equation: v+l
dav= z f l , eikA"a~+ Z V(au-c~u-1), dt
n
(5)
/~:v
where the following transformation was made:
(~.)- 1 t---,t.
(6)
Equation (5) is rewritten as follows, ~.,AI/'~v
v+ 1
at = Zfl"(v)eikA"bv+ Z V ( b u - b u - , ) ,
where
(7)
818
YOJI AIZAWA
AND
HIROSHI
SHIMIZU
b. = a . - - 2~rkMl(V- 1), ft,(v) = ft, e i2."(k'/kM)~v- 1),
(8)
V(x),
V(x) =
and where 2rtk~t 1 (=IM) is the distance between two adjacent cross-bridge heads. Though (5) and (7) were derived under several simplifications of the real process of muscle contraction, the essential characteristics, which are responsible for the statistical effect of many cross-bridges, are included in the theoretical model obtained here. The generalization of (2) will be discussed in Section 4 in connection with the Hill's relation.
• 3. Analysis Based on the Rigid Rod Approximation. In this section we will first derive the analytic form of sliding motion of a myosin filament under the approximation that the filament is rigid. Secondly, the elastic mode of cross-bridges will be studied by taking into account small fluctuations in the position of cross-bridges around the fixed site on the rigid rod. (A) The sliding motion of myosin filament. By the rigid rod approximation, the motion of a myosin filament is described by a degree of freedom, which stands for the translational motion of the center of gravity of a myosin filament. By use of the boundary condition a, = al
+ 2 7 r k M 1 (v - -
1),
(9)
(5) is rewritten as" da v
dt - ~ f i , exp{inkAal +i2rcn(v-- 1)Z} n v+l
(10)
+ ~ V ( a u - a u_l), 11='¢
where Z =
kA/kM. By the summing over v, (10) can be written as" da 1
dt - ~fl"ei"k"l'
(11)
n
where ~n
N e i2rc(v-
fl" = N ,~=x
1)%
(12)
In deriving (11) we used the fact that the interactions among cross-bridges are internal forces, i.e.,
AN EXACT ANALYSIS OF THE SLIDING IVlOTION
V(a~-a~,)=O.
819
(13)
/1, V
Therefore, (5) and (7) become: da~ dt -- ~ fin exp {inkAa ~-- i2zcn(v -- 1)Z},
(14)
n
and
db, = ~fl, e~,k~b,_F(b~)" dt
(15)
,
In what follows, the solution of (14) and (15) is represented as av and b v, i.e.
f d t = f F ~ ) d b ~.
(16)
The force field F is considered the effective field which is obtained by taking into account the boundary conditions as well as the statistical effect arising from many cross-bridges. In other words, the effective force field F is obtained by the renormalization of the boundary conditions and the statistical effects. F is decomposed into the spatially uniform field (n=0) and the non-uniform components (n =p0). If we define the renormalization factor by ti,(= Ifl,/fl, I), the renormalization factor is given by
tin
__-
1 =~ ei2rm(v - - 1 ) . X
N v
=
lx/[l_cos(2rcnz)]
l[1-cos(2rmNz)],
(17)
where we have used (12). Equation (17) reveals the size effect determined by the geometrical structure of the acto-myosin system, namely, some Fourier components of the effective field F were omitted in the limiting case of N - 1 ~ 0 as follows: 1, for nz=integer lira o-,= N~ o0 0, for nz = irrational.
(18)
This gives the exact condition for the coherent sliding motion. That is to say, the effect of oscillating fields (n@0) decreases effectively in the order of 1IN when the number of cross-bridges N increases toinfinity, provided that Z is an irrational number, whereas the uniform contractile force remains invariant. From X-ray diffraction studies on striated muscle, (Huxley and Brown, 1967), it is known that the geometrical ratio k~4/kM(=Z) is about
820
YOJI AIZAWAAND HIROSHI SHIMIZU
1.16, which is not a simple ratio of integers but is surmised to be irrational. In the case that Z is not rational, the contraction speed db~/dt is also constant and the smooth sliding occurs in the limiting case of N - ~~ 0 , i.e., by =flot +by(O),
(19)
where b~(0) is the initial coordinate of vth cross-bridge. As shown in Appendix I, when N is finite as in real muscles, effects of oscillatory fields remain, and consequently the small oscillating behavior appears in the process of contraction. The oscillating term is a quantity of order N -1, as shown from (17). In the case when the effect of the oscillating field is strong enough in comparison with that of uniform field, the motion of the myosin filament will be locked spatially and no sliding motion can occur coherently. Equation (18) shows clearly not only the significant effect of the system size N in the process of muscle contraction but also the indispensable role of the structural parameter Z. (B) Elastic mode of cross-bridges. As mentioned in Section 2, the rigid rod approximation is a simplification of a real muscle system. We consider the small deviation from the rigid rod approximation by the following transformation; b~=b~+ec~. H e r e ~ is a small parameter. The linear variation equations for c~ are obtained from (7) and (15) to give.
dt
= Z {(fln(V)--fln) ei"kAb~+c'c'} n>0
+ • {inkAfln(v)ei.kAb~+c.c.}~c~ n>O v+l
+ ~, A~u(cu-cu-1)~,
(20)
It=v
where c~V(b v - bu)
,
(21)
and c.c. means the complex conjugate of the first term in each bracket { }. Here the higher order terms of o (e2) are neglected. The coefficients A~u defined above are the coefficients representing the rigidity of a myosin filament. Under the approximation that a myosin filament is almost rigid, all the coefficients A~, are assumed to be o(e-1). This is an assumption in the present paper, which should be verified by some experimental data, but is acceptable in a mathematical approach to the elasticity of a myosin
AN EXACT ANALYSIS O F T H E S L I D I N G M O T I O N
821
filament as a first approximation. If the present assumption fails, (20) must be improved by taking into account higher order effects of o(~°). We limit our discussion to the description by (20). The highest order term in the right hand side of (20) is o(e°), consequently the second term of r.h.s., which is of order o(e ~) can be neglected in the following discussion. After the approximation mentioned above, the dominant terms of (20) are obtained by putting e = 1; dc v
d-t- = ~ [(fl"(V)--fl")e*"k"b'+C'C'] +D(cv+l +c~-1 --2cv),
(22)
n>O
where Av, = D6v, u+ 1 (D = const.) are used. Equation (22) is the fundamental equation describing the elastic modes of myosin filament in the process of contraction, Suppose the case of smooth contraction ( N - 1 ~ 0 ) is observed approximately in a real system. In the case where kA/kM is irrational and ~ , ~ 0 (N-I--.0, n:p0), (22) is rewritten as follows: dc~
d r - = ~ [fl"(v)e"k'b'+C'C'] +D(c~+I +c~_, --2cv) n>O
= ~_, [flne"kAa,+C.C.]+O(cv+l +Cv_ 1-2c~).
(23)
n>0
The general solution of (23) is given by
cv =
Z[
F~(t-- t ' )g~,(t' )dt ' ],
Fvu(t)cu(O) +
(24)
3 where cv(0) is initial value of cv, and gu stands for
g.= Z
(25)
n>0
and as is shown in Appendix II:
Fvu(t )-=e -20' E/v+. ,(20 0 - I v .(2Ot)].
(26)
Here Ik is the kth modified Bessel function. If we neglect the effect of initial state, (24) is rewritten as
Fvu(t-- t )[fl, e"kA"~")+C.C.]dt ' ,
CV(o ~--
(27)
/t, n n>O
since each mode except for k = l
decays as e zkt, where 2 5 = - 4 D s i n 2 ( n ( k
822
YOJI AIZAWA A N D HIROSHI SHIMIZU
-1)/2N). The elastic mode is characterized by the Fourier spectrum, which is discussed in Appendix II. Our analysis in this section is based on a lowest order perturbation theory. We do not take into account the feedback effect of elasticity on the sliding motion. This feedback effect will be studied in a forthcoming paper dealing with the generalization of the present model. As mentioned in Section 1, the synchronous coordination of force generation is surmised to originate from the chemical process accompanying cooperative structural changes in the active sites. The structural change of a cross-bridge in rigor gives rise to the behavior of the elastic mode in addition to the dynamical interaction mentioned above. Consequently the power spectrum in real muscle fibers is more complicated than that obtained in Appendix II. The study of the elastic mode may be useful for better understanding of the synchronous mechanism in muscle contraction (Ford et al., 1977). 4. Generalization of Model and Discussion. In the previous section, we developed a dynamical theory of sliding filaments taking the many-body effect of cross-bridge into consideration. In the present section we will study a mathematical framework, in which the nonlinearity appearing in Hill's relation will be understood in terms of the coherent activation of each active site on both actin and myosin filaments. In the case of isotonic muscle contraction, the sliding velocity V is defined by the translational speed of a myosin filament, i.e., 1 ~
=
1 N
da v
ZF--T~fio--T.
(28)
v
Here ~ stands for the value in the case of smooth contraction (N l__,0), and T is the external load. In the derivation of (28), we u s e d the transformed variables in (6). If N is fixed at a finite value, the contractile velocity V has a small oscillation around a steady value depending on T (Appendix I). Equation (28) is the tension-velocity relation obtained by the dynamical model proposed in the previous section, and is quite different from Hill's relation, which is widely accepted in the description of the contracting process: V = b(To - T)(a + T ) - ~i
(29)
where a, b, and To are constants. Hill's relation, which has the same
AN EXACT ANALYSIS
OF THE
SLIDING
MOTION
823
physical background as the equation of motion (28), characterizes the dynamical behaviour of the sliding filaments. Here we would like to emphasize that the monotonically decreasing tendencies derived from (28) and (29) coincide with each other in the region 0 < T < To (or/~0), though details are different in both equations. As we will see in this section, the Hill's relation must be understood by taking into account the many-body effect of cross-bridges as well as the cooperative change of state of a cross-bridge head. Before we go on to the statistical understanding of the Hill's relation, we will briefly discuss the kinetic or phenomenological interpretation of Hill's relation in reference to the Tomita's theory. (Tomita, 1973). His theory was as follows. A head of cross-bridge may have two states, c~-state and fl-state. At the /%state, cross-bridge generates contractile force. After the hydrolysis of ATP, the cross-bridge releases the ADP molecules and returns into e-state. On the other hand, the sliding speed [V] decays due to viscosity as well as to energy dissipation caused by the external tension l,T]. These processes were described by the rate equations d/l-• dr = k ° - kll'ft] - k21-/~]l-V],
(30)
d[V] - dt
(31)
--
(]£3 + k 4 l , T ] ) l ' V ] + ks[/~][V],
where [/~] and [T] are the fl-state fraction of cross-bridges and the external tension, and each ki is a rate constant. F r o m these two equations Hill's relation (29) is derived under the condition of the isotonic case, i.e., d[V]/dt=dl,fl]/dt=O. Parameters in the Hill's relation are obtained as follows, a = k3/1,;4,
b=
kl/k
T o --=( k o k s -- k 1 k 3 ) / k l
2
k3.
(32)
From the dynamical viewpoint, (31) should be regarded as the equation of motion; the differential of velocity is proportional to the force. The first term of r.h.s, of (31) stands for the effective viscosity, and the nonlinear terms l,/~][V] in (30) and (31) represent the nonlinear effect caused by the positive feedback from the sliding motion. The nonlinear term of kgl,T]l,V] in (31) originates from the cooperative effect intermediated by the occurrence of tension in the actomyosin system. In view of the significant roles of tension, this effect is called "cooperative tension effect" in short. We will
824
YOJI AIZAWA .AND H I R O S H I S H I M I Z U
show that Hill's relation can be restored in our model proposed in Section 2 by taking into account this cooperative tension effect. We assume that a cross-bridge takes either the a- or fl-state stochastically in the same way as described in the Tomita's theory. Thus, the basic equation (2) must be rewritten as follows: 7. da ~ ~-=q~F.+
~+1 ~ V(av-a._O-T~,
(33)
/L=V
where T~ is external force acting on the vth cross-bridge arising from the tension, and q~ is the state variable defined by 1, for fl-state 0, for a-state,
q" =
(34)
which describes the attaching state of both filaments. In the case of q~ = 1, the vth cross-bridge creates the contractile force, and the state is called q-active in short. According to the activation of the B-state, the effective viscosity 7* must be different in the two states of the cross-bridge. Here we assume that the coefficient can take on either of two different values; =
+ A*-
(35)
where ~* and A* are constants, and ~ represents the following variable:
~ =
1, 0,
for active state for non-active state.
(36)
Generally speaking, the i-active state is not always equal to the q-active state described by (34). In a special case, however, both active states may be equal. Thus, if these two active states are identical, i.e., ¢ = q (=~0), the active state is considered as the//-state in Tomita's theory. If we define the sliding velocity [V-I by: (37) the velocity-tension relation, obtained from (33), is given by: IV] ~ * = ~q,F v-NIT], v
¥
(38)
AN EXACT ANALYSISOF THE SLIDING MOTION
825
where [T]=I/N~T~. In the derivation of this equation, (13) was used. Using the approximation of smooth sliding, i.e. F~flo, (34)-(38) lead to [-V](~7* + A*({)) = flo(t/) - IT],
(39)
where ( 4 ) and Q1) are mean values of { and tl defined by 1 N
(40) 1
u
The two kinds of variables, {~ and ~7~,are considered to be stochastic since: (1) the activation of actin and myosin molecules depends on the spatially non-uniform and random adsorption of ATP molecules; and (2) the crossbridge has an electrical charge which may oscillate randomly in the process of contraction as was mentioned in Section 3(B). In general, these two causes couple with each other, namely the chemical process of the contractile force generation may be affected by geometrical and mechanical states of actin and myosin molecules as well as by the contracting speed of the actin-myosin filament. In other words, it is reasonable to assume that (~) and (t/) are given functions not only of I T ] but also of IV] as follows: (~) = X([T]. [V]) (41) ( , ) = Y([T]. IV]). where X and Y describe the internal state of a cross-bridge, therefore, we cannot determine the functional forms of (41) in terms of such mechanical variables as {a,} used in (1). In order to know the time variation of (~) and (17) at the state of the smooth sliding contraction, we need adequate information about the process of mechano-chemical energy conversion in the actomyosin system; for example, a kinetic rate equation representing the time variation of (4), etc. For the present it is unavoidable to assume some appropriate functions X and Y. For the sake of convenience, assume ( 3 ) = ( q ) = (/?), and that the time evolution of (fl) is given by (30). The steady state value becomes: (fl)
-
k0 k I H- k2[V ]
( = (4)
= (q)),
(42)
826
YOJI AIZAWA AND HIROSHI
SHIMIZU
corresponding to (41). F r o m (39) and (42), we can derive a relation of Hill's type provided that the term of o([V]Z), which is a higher order small quantity, is neglected. However, it is not the main p u r p o s e of this section to pursue the apparent coincidence between Hill's relation and our theory, but to give a theoretical framework in which Hill's relation can be reasonably formulated by taking into account the equations of motion as well as the internal degrees of freedom of cross-bridges. In this paper we have discussed the dynamical behavior of sliding filaments ,with reference to a statistical effect of many active sites. T w o essentially significant effects were revealed in the present theory concerning the statistical mechanical problems in the contractile system: the first is the effect of system size given by (18) and (AI, 2), and the second is the "cooperative tension effect" represented by (41) and (42).
One of the present authors (Y. A.) thanks Dr. K. Nishiyama (Department of Biology, Faculty of Science, Kyushu University, Japan) and Prof. Y. M a t s u m o t o (Department of Physiology, Emory University School of Medicine, Atlanta, Georgia) for valuable discussion and for giving some important recent informations. This study was supported partly by a grant from the Ministry of Education. APPENDIX I OSCILLATORY CONTRACTION AND SPATIAL LOCKING OF A MYOSIN FILAMENT In order to make clear the physical meaning of (18), an exactly solvable model is analysed here. When fl,=O (except for n=0, _+1) is satisfied, (15) is rewritten as follows: dbv -dt
((AI, 1)
O~-t- O"lfl c o s ( k A b v ~- ~ ) '
where c~, fl, and ~b are appropriate real valued parameters defined from flo, fll and fl_ 1, and the renormalization factor al, is given by, ch -
1 sin(uNz) N sin(uz)
(AI, 2)
Equation (AI, 1) permits two quite different solutions depending on the value of the parameter alfl/a. That is to say, in the case of aiflO
where ctr) = fl" ~ / N ~ ~ o
~1 -2 "e'(k x m=~ / ~ ~ T J "
(AII, 7)
LITERATURE Aiello, E. and M. A. Sleigh. t972. "The Metachronical Wave of Lateral Cilia of Mytilus edulis." J. Cell Biol., 54, 493-506. Aizawa, Y. 1979. "A Mathematical Model of Ciliary Movement." In preparation. Elliott, G. F., J. Lowy and B. M. Millman. 1967. "Low-angle X-ray Diffraction Studies of Living Striated Muscle during Contraction." J. Molec. Biol., 25, 31-45. Ford, L. E., A. F. Huxley and R. M. Simmons. 1977. "Tension Responses to Sudden Length Change in Stimulated Frog Muscle Fibres Near Slack Length." J. Physiol., 269, 441 515. GTansdorff, P. and I. Prigogine. 1971. Thermodynamic Theory of Structure, Stability and Fluctuation. New York: Wiley. Gray, J. 1931. "The Mechanism of Ciliary Movement. Photographic Stroboscopic Analysis of Ciliary Movement." Proc. R. Soc., Lond., B107, 313 332. Huxley, A. F. 1957. "Muscle Structure and Theories of Contraction." Prog. Biophys. Chem., 7, 255 318. . 1971. "The Activation of Striated Muscle and its Mechanical Response." Proc. R. Sot., B178, 1-27. and R. M. Simmons. 1971. "Proposed Mechanism of Force Generation in Striated Muscle." Nature 233, 533-538. Huxley, H. E. and J. Hanson. 1954. "Changes in the Cross-striations of Muscle during Contraction and Stretch and their Structural Interpretation." Nature, 173, 973 976. - and W. Brown. 1967. "The Low-angle X-ray Diagram of Vertebrate Striated Muscle and its Behavior During Contraction and Rigor." J. Molec. Biol., 30, 383-434. Nicolis, G. and I. Prigogine. 1977. "Self-organization in Nonequilibrium System." New York: Wiley. Nishiyama, K. 1977. Private communication. Shimizu, H., T. Yamada, K. Nishiyama and M. Yano. 1976. "Energetic Enzyme Theory of Muscular Contraction. II. Relation Between Hill's Equation and Functions of Two-head Myosin." J. Theor. Biol., 63, 165-189. Tomita, K. 1973. "A Model of Muscle Contraction." In Synergetics (Ed. by H. Haken and B. G. Teubner), pp. 232-240. -
-
RECEIVED 3-28-78 REVISED
9-17-78
0007-4985/79/1201-0813 $02.00/0
AN EXACT ANALYSIS OF T H E S L I D I N G M O T I O N IN C O N T R A C T I L E SYSTEM: A S I M P L I F I E D M O D E L O F M U S C L E SYSTEM
[ ] YOJI AIZAWA]"
Facult6 des Sciences, Universit6 Libre de Bruxelles HIROSHI SHIMIZU
Faculty of Pharmaceutical Sciences, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan
A system involving two kinds of sliding filaments is analysed with special attention to the actomyosin system. Rigorous results are obtained about the statistical effect originating from many active sites distributed on both filaments. It is necessary for the occurrence of smooth motion in sliding filament that the spatial periods of active sites on both filaments are relatively incommensurable, and that the number of active sites on each filament is large enough. Sufficient conditions for smooth contraction are derived under the assumption that both filaments are rigid; this is called rigid rod approximation in the present paper. The elastic mode of the filaments, during the sliding process, is analysed by perturbation theory based on the rigid rod approximation. A stochastic theory is briefly discussed in reference to the cooperative generation of contractile force, which is concerned in Hill's relation of muscle contraction.
i. Introduction. Ordered structures as well as ordered motions are often created in the system far from thermal equilibrium (Glansdorff and Prigogine, 1971; Nicolis and Prigogine, 1977). As the living bodies are always sustained in the nonequilibrium state, a variety of orders are observed in every part of an organism. It is essential for the bio-system that these local orders are unified into a global one so that each part of an organism may behave coherently and functionally with high efficiency. Among the many ordered functions in the living "~Present address: Department of Physics, Faculty of Sciences, University of Kyoto, Japan. 813
814
YOJI AIZAWA A N D H I R O S H I S H I M I Z U
bodies, motility is one of the most important phenomena. The motility of organisms is endowed with two types of coherent orders. One is the time periodic or oscillatory motion, which is typically observed in the ciliary system (Aiello and Sleigh, 1972; Gray, 1931; Aizawa, 1979), and the other is the temporally uniform motion. A typical example of the latter is the contractile motion in the striated muscle (Huxley, 1957; Huxley and Hanson, 1954). So far the mechanism o f the appearance of coherent motions in nonequilibrium state has not yet been well explained, at least from a theoretical point of view. The purpose of this paper is to make clear the cause for the appearance of coherent sliding motion in contractile systems. In the present paper we will show that coherent motion can be created in contractile system composed of sliding filaments, through a statistical effect. To extract this statistical effect in a mathematically rigorous manner, we consider the simple model system, which has a close connection with the actomyosin system in striated muscle. Our purpose, however, is not to analyse the real microscopic processes of actomyosin system in detail. At the present time there is a large amount of biochemical experimental data concerning molecular or microscopic processes in muscle, and we know that the structural change of bio-molecules plays an important role in the appearance of the coherent sliding motion. From a theoretical viewpoint, one of the most interesting problems is to construct the theory of these microscopic molecular processes. But in the present article, we wish to analyse the macroscopic description of the phenomena induced by a statistical (or many-body) effect in the actomyosin system. In other words, the purpose of the present article is to study the statistical physics of an idealized contractile system. Our starting point is the idea of two kinds of sliding filaments, an idea first proposed by H. E. Huxley and A. F. Huxley (Huxley, 1957; Huxley and Hansen, 1954). The fundamental idea of the sliding theory is supported not only by the structural analysis by electromicroscope and X-ray (Huxley and Brown, 1967; Elliot et al., 1967) but also by the bio-chemical discoveries of certain structural and functional proteins in the actomyosin system. Many results of physiological experiments have also been well explained, at least, qualitatively, in line with the sliding model (Huxley and Simmons, 1971). We will consider the following simplified model: There are two kinds of filaments, i.e., thin (actin) and thick (myosin) filaments, which are placed parallel to each other. On the myosin filament, many identical active sites occur at intervals of a certain distance 1M, and on the actin filament also other kinds of active sites occur at regular "intervals 1A. (In the present article, the active site on the myosin filament is called the "cross-bridge".) Both kinds of filaments are assumed to be long enough, so that there are many active sites on the filaments, and that all the active
AN EXACTANALYSISOF THE SLIDING MOTION
815
sites are fixed on each filament. The force is generated through the chemomechanical interaction between these two different kinds of active sites. There is no direct chemical interaction among active sites placed on the same filament but there is indirect interaction through mechanical motion of the filament. The motion takes place only in the one-dimensional direction parallel to the filaments. In the present paper, for the sake of convenience we will limit our discussion to the case where the system includes only one pair, an actin and a myosin filament. This limitation is not essential in our theory, and can easily be lifted when we consider the more realistic model of a sarcomere. Both filaments are soaked i n the protoplasmic solution which causes viscous damping to the sliding motion. The generation of the contractile force originates from the chemomechanical processes at the active sites on both filaments (Shimizu et al., 1976). We will assume that there exists a certain time-invariant force among two kinds of active sites. It is surmised that there are at least two reasons for which the sliding motion of the acto-myosin system is coherent and smooth. One originates from the statistical effect, which will be discussed in the present paper, another originates from the fact that the contractile force is not generated at random on each active site but is created synergetically in the same phase at almost every active site. (Nishiyama, 1976; Shimizu et al., 1976). Both causes are indispensable for the occurrence of the smooth sliding in acto-myosin system. The main motivation of the present paper is to analyse the first coherence origin exactly by our theoretical model mentioned above. In the analysis of Sections 2 and 3, we will take into account the second coherence origin beforehand by assuming that the generated force is independent of time. The close interrelation between these two coherence origins is discussed in "Section 4. The mathematical model is derived in Section 2. By means of the rigid rod approximation of both filaments, a reduced equation is derived in Section 3 and the small deviation from the rigid rod approximation is analysed. A generalization of our model is discussed in Section 4 in reference to Hill's relations.
2. Mathematical Model of Sliding Filaments. Since the sliding motion between both filaments is relative to each other, we will discuss the motion of a myosin filament relative to the actin filament in the present paper. As we do not take into account the internal degrees of freedom in each active site, the dynamical state of a myosin filament can be described by the ensemble of the coordinates {a~} (v= 1,2,...,N) of N cross-bridge heads. Here N denotes the number of cross-bridges on a myosin filament. The effect of the internal degrees of freedom of a cross-bridge will be discussed in Section 4., The equation of motion of the vth cross-bridge head in
816
VOJI AIZAWA A N D H I R O S H I S H I M I Z U
unloaded muscle is generally given by: d2a~ M~ dt 2 -
( d a ~ ) 7~f-+R~
+F~+V~({au} ),
(1)
where M~ is the effective mass of vth cross-bridge head (the mass of myosin molecule other than the cross-bridge is effectively included), Fv is the force caused by the activated site on the actin filament, V~ ({au}) is the mechanical interaction with the other cross-bridges through the myosin filament, 7v(da~/dt) is the viscous force at the vth cross-bridge, and R~ is the viscous force stemming from the part of the myosin filament connecting the vth and (v___1)th cross-bridge head. The interaction term between crossbridges V~ is very complicated and it is not easy to reduce to a simple form. Here we will introduce the following three generous assumptions in order to treat (1) without loss of the essentially significant mechanism in the sliding process: (I) The inertial term is neglected and only the steady state is discussed [Mv(dEa~/dt 2) = O] ; (I1) The interaction force depends only on the relative distance between the nearest neighboring cross-bridges, i.e., v+l
V~({au})= ~, V(au-au-x);
and
i I = "v
(III) The viscous force R~ is proportional to that of the vth cross-bridge head, i.e., R~w_yv(da~/dt ). By these assumptions (1) is rewritten as"
~. da~
~+ 1 =F~ + ~.. V(,(,,
au_ 1),
[d='v
(2)
The elastic mode of a myosin filament will be analysed in Section 3. As mentioned in Section 1, active actin sites are distributed periodically on the actin filament. When we write the spatial period as. 2r~kj 1 (=IA), the force acting on the vth cross-bridge F~ must satisfy the following periodic condition: F~ = F(av) =F(a~ + 2rckj 1).
(3)
where k A stands for the fundamental wave vector of force field F~. As
AN
EXACT
ANALYSIS
OF
THE
SLIDING
MOTION
817
assumed in the last section, all the cross-bridge heads are put in a timeinvariant force field. In addition, we assume that an actin filament is rigid. In other words, the external force is decomposed into a Fourier series, i.e., oo
F(a~)=
~
~._fl e ikA"av,
(4)
n~--oo
where//, =B.* (+ denotes complex conjugate) is satisfied because the force field F~ is real. The total force arising from the active actin site is given by (4), where //, is determined by the geometrical structure of a cross-bridge head and of an active actin site as well as by" the mechano-chemieal process of the force generation. The force F~ has two remarkable characteristics; the spatial localization of the force field, and the polarity of the force field. That is to say, the former means that the force field is localized sharply near the active actin site and the latter means that the potential function of the force field is distributed asymmetrically around the active site. From the second property of the force field, //040 is derived. This term is mainly responsible for the occurrence of the sliding motion. In general, / / , 4 0 is satisfied provided that the force field is localized sharply. All the parameters in (2)-(4) must be determined by the relevant experimental results. In the present article, however, we will not pursue the appropriate values of these parameters, since the purpose of this paper is to elucidate the mathematical framework of the sliding theory described by (1) (4). In this paper, (4) is only one assumption about the force field F~. Under the consideration mentioned above, the theoretical model of the contractile process is finally given by the following equation: v+l
dav= z f l , eikA"a~+ Z V(au-c~u-1), dt
n
(5)
/~:v
where the following transformation was made:
(~.)- 1 t---,t.
(6)
Equation (5) is rewritten as follows, ~.,AI/'~v
v+ 1
at = Zfl"(v)eikA"bv+ Z V ( b u - b u - , ) ,
where
(7)
818
YOJI AIZAWA
AND
HIROSHI
SHIMIZU
b. = a . - - 2~rkMl(V- 1), ft,(v) = ft, e i2."(k'/kM)~v- 1),
(8)
V(x),
V(x) =
and where 2rtk~t 1 (=IM) is the distance between two adjacent cross-bridge heads. Though (5) and (7) were derived under several simplifications of the real process of muscle contraction, the essential characteristics, which are responsible for the statistical effect of many cross-bridges, are included in the theoretical model obtained here. The generalization of (2) will be discussed in Section 4 in connection with the Hill's relation.
• 3. Analysis Based on the Rigid Rod Approximation. In this section we will first derive the analytic form of sliding motion of a myosin filament under the approximation that the filament is rigid. Secondly, the elastic mode of cross-bridges will be studied by taking into account small fluctuations in the position of cross-bridges around the fixed site on the rigid rod. (A) The sliding motion of myosin filament. By the rigid rod approximation, the motion of a myosin filament is described by a degree of freedom, which stands for the translational motion of the center of gravity of a myosin filament. By use of the boundary condition a, = al
+ 2 7 r k M 1 (v - -
1),
(9)
(5) is rewritten as" da v
dt - ~ f i , exp{inkAal +i2rcn(v-- 1)Z} n v+l
(10)
+ ~ V ( a u - a u_l), 11='¢
where Z =
kA/kM. By the summing over v, (10) can be written as" da 1
dt - ~fl"ei"k"l'
(11)
n
where ~n
N e i2rc(v-
fl" = N ,~=x
1)%
(12)
In deriving (11) we used the fact that the interactions among cross-bridges are internal forces, i.e.,
AN EXACT ANALYSIS OF THE SLIDING IVlOTION
V(a~-a~,)=O.
819
(13)
/1, V
Therefore, (5) and (7) become: da~ dt -- ~ fin exp {inkAa ~-- i2zcn(v -- 1)Z},
(14)
n
and
db, = ~fl, e~,k~b,_F(b~)" dt
(15)
,
In what follows, the solution of (14) and (15) is represented as av and b v, i.e.
f d t = f F ~ ) d b ~.
(16)
The force field F is considered the effective field which is obtained by taking into account the boundary conditions as well as the statistical effect arising from many cross-bridges. In other words, the effective force field F is obtained by the renormalization of the boundary conditions and the statistical effects. F is decomposed into the spatially uniform field (n=0) and the non-uniform components (n =p0). If we define the renormalization factor by ti,(= Ifl,/fl, I), the renormalization factor is given by
tin
__-
1 =~ ei2rm(v - - 1 ) . X
N v
=
lx/[l_cos(2rcnz)]
l[1-cos(2rmNz)],
(17)
where we have used (12). Equation (17) reveals the size effect determined by the geometrical structure of the acto-myosin system, namely, some Fourier components of the effective field F were omitted in the limiting case of N - 1 ~ 0 as follows: 1, for nz=integer lira o-,= N~ o0 0, for nz = irrational.
(18)
This gives the exact condition for the coherent sliding motion. That is to say, the effect of oscillating fields (n@0) decreases effectively in the order of 1IN when the number of cross-bridges N increases toinfinity, provided that Z is an irrational number, whereas the uniform contractile force remains invariant. From X-ray diffraction studies on striated muscle, (Huxley and Brown, 1967), it is known that the geometrical ratio k~4/kM(=Z) is about
820
YOJI AIZAWAAND HIROSHI SHIMIZU
1.16, which is not a simple ratio of integers but is surmised to be irrational. In the case that Z is not rational, the contraction speed db~/dt is also constant and the smooth sliding occurs in the limiting case of N - ~~ 0 , i.e., by =flot +by(O),
(19)
where b~(0) is the initial coordinate of vth cross-bridge. As shown in Appendix I, when N is finite as in real muscles, effects of oscillatory fields remain, and consequently the small oscillating behavior appears in the process of contraction. The oscillating term is a quantity of order N -1, as shown from (17). In the case when the effect of the oscillating field is strong enough in comparison with that of uniform field, the motion of the myosin filament will be locked spatially and no sliding motion can occur coherently. Equation (18) shows clearly not only the significant effect of the system size N in the process of muscle contraction but also the indispensable role of the structural parameter Z. (B) Elastic mode of cross-bridges. As mentioned in Section 2, the rigid rod approximation is a simplification of a real muscle system. We consider the small deviation from the rigid rod approximation by the following transformation; b~=b~+ec~. H e r e ~ is a small parameter. The linear variation equations for c~ are obtained from (7) and (15) to give.
dt
= Z {(fln(V)--fln) ei"kAb~+c'c'} n>0
+ • {inkAfln(v)ei.kAb~+c.c.}~c~ n>O v+l
+ ~, A~u(cu-cu-1)~,
(20)
It=v
where c~V(b v - bu)
,
(21)
and c.c. means the complex conjugate of the first term in each bracket { }. Here the higher order terms of o (e2) are neglected. The coefficients A~u defined above are the coefficients representing the rigidity of a myosin filament. Under the approximation that a myosin filament is almost rigid, all the coefficients A~, are assumed to be o(e-1). This is an assumption in the present paper, which should be verified by some experimental data, but is acceptable in a mathematical approach to the elasticity of a myosin
AN EXACT ANALYSIS O F T H E S L I D I N G M O T I O N
821
filament as a first approximation. If the present assumption fails, (20) must be improved by taking into account higher order effects of o(~°). We limit our discussion to the description by (20). The highest order term in the right hand side of (20) is o(e°), consequently the second term of r.h.s., which is of order o(e ~) can be neglected in the following discussion. After the approximation mentioned above, the dominant terms of (20) are obtained by putting e = 1; dc v
d-t- = ~ [(fl"(V)--fl")e*"k"b'+C'C'] +D(cv+l +c~-1 --2cv),
(22)
n>O
where Av, = D6v, u+ 1 (D = const.) are used. Equation (22) is the fundamental equation describing the elastic modes of myosin filament in the process of contraction, Suppose the case of smooth contraction ( N - 1 ~ 0 ) is observed approximately in a real system. In the case where kA/kM is irrational and ~ , ~ 0 (N-I--.0, n:p0), (22) is rewritten as follows: dc~
d r - = ~ [fl"(v)e"k'b'+C'C'] +D(c~+I +c~_, --2cv) n>O
= ~_, [flne"kAa,+C.C.]+O(cv+l +Cv_ 1-2c~).
(23)
n>0
The general solution of (23) is given by
cv =
Z[
F~(t-- t ' )g~,(t' )dt ' ],
Fvu(t)cu(O) +
(24)
3 where cv(0) is initial value of cv, and gu stands for
g.= Z
(25)
n>0
and as is shown in Appendix II:
Fvu(t )-=e -20' E/v+. ,(20 0 - I v .(2Ot)].
(26)
Here Ik is the kth modified Bessel function. If we neglect the effect of initial state, (24) is rewritten as
Fvu(t-- t )[fl, e"kA"~")+C.C.]dt ' ,
CV(o ~--
(27)
/t, n n>O
since each mode except for k = l
decays as e zkt, where 2 5 = - 4 D s i n 2 ( n ( k
822
YOJI AIZAWA A N D HIROSHI SHIMIZU
-1)/2N). The elastic mode is characterized by the Fourier spectrum, which is discussed in Appendix II. Our analysis in this section is based on a lowest order perturbation theory. We do not take into account the feedback effect of elasticity on the sliding motion. This feedback effect will be studied in a forthcoming paper dealing with the generalization of the present model. As mentioned in Section 1, the synchronous coordination of force generation is surmised to originate from the chemical process accompanying cooperative structural changes in the active sites. The structural change of a cross-bridge in rigor gives rise to the behavior of the elastic mode in addition to the dynamical interaction mentioned above. Consequently the power spectrum in real muscle fibers is more complicated than that obtained in Appendix II. The study of the elastic mode may be useful for better understanding of the synchronous mechanism in muscle contraction (Ford et al., 1977). 4. Generalization of Model and Discussion. In the previous section, we developed a dynamical theory of sliding filaments taking the many-body effect of cross-bridge into consideration. In the present section we will study a mathematical framework, in which the nonlinearity appearing in Hill's relation will be understood in terms of the coherent activation of each active site on both actin and myosin filaments. In the case of isotonic muscle contraction, the sliding velocity V is defined by the translational speed of a myosin filament, i.e., 1 ~
=
1 N
da v
ZF--T~fio--T.
(28)
v
Here ~ stands for the value in the case of smooth contraction (N l__,0), and T is the external load. In the derivation of (28), we u s e d the transformed variables in (6). If N is fixed at a finite value, the contractile velocity V has a small oscillation around a steady value depending on T (Appendix I). Equation (28) is the tension-velocity relation obtained by the dynamical model proposed in the previous section, and is quite different from Hill's relation, which is widely accepted in the description of the contracting process: V = b(To - T)(a + T ) - ~i
(29)
where a, b, and To are constants. Hill's relation, which has the same
AN EXACT ANALYSIS
OF THE
SLIDING
MOTION
823
physical background as the equation of motion (28), characterizes the dynamical behaviour of the sliding filaments. Here we would like to emphasize that the monotonically decreasing tendencies derived from (28) and (29) coincide with each other in the region 0 < T < To (or/~0), though details are different in both equations. As we will see in this section, the Hill's relation must be understood by taking into account the many-body effect of cross-bridges as well as the cooperative change of state of a cross-bridge head. Before we go on to the statistical understanding of the Hill's relation, we will briefly discuss the kinetic or phenomenological interpretation of Hill's relation in reference to the Tomita's theory. (Tomita, 1973). His theory was as follows. A head of cross-bridge may have two states, c~-state and fl-state. At the /%state, cross-bridge generates contractile force. After the hydrolysis of ATP, the cross-bridge releases the ADP molecules and returns into e-state. On the other hand, the sliding speed [V] decays due to viscosity as well as to energy dissipation caused by the external tension l,T]. These processes were described by the rate equations d/l-• dr = k ° - kll'ft] - k21-/~]l-V],
(30)
d[V] - dt
(31)
--
(]£3 + k 4 l , T ] ) l ' V ] + ks[/~][V],
where [/~] and [T] are the fl-state fraction of cross-bridges and the external tension, and each ki is a rate constant. F r o m these two equations Hill's relation (29) is derived under the condition of the isotonic case, i.e., d[V]/dt=dl,fl]/dt=O. Parameters in the Hill's relation are obtained as follows, a = k3/1,;4,
b=
kl/k
T o --=( k o k s -- k 1 k 3 ) / k l
2
k3.
(32)
From the dynamical viewpoint, (31) should be regarded as the equation of motion; the differential of velocity is proportional to the force. The first term of r.h.s, of (31) stands for the effective viscosity, and the nonlinear terms l,/~][V] in (30) and (31) represent the nonlinear effect caused by the positive feedback from the sliding motion. The nonlinear term of kgl,T]l,V] in (31) originates from the cooperative effect intermediated by the occurrence of tension in the actomyosin system. In view of the significant roles of tension, this effect is called "cooperative tension effect" in short. We will
824
YOJI AIZAWA .AND H I R O S H I S H I M I Z U
show that Hill's relation can be restored in our model proposed in Section 2 by taking into account this cooperative tension effect. We assume that a cross-bridge takes either the a- or fl-state stochastically in the same way as described in the Tomita's theory. Thus, the basic equation (2) must be rewritten as follows: 7. da ~ ~-=q~F.+
~+1 ~ V(av-a._O-T~,
(33)
/L=V
where T~ is external force acting on the vth cross-bridge arising from the tension, and q~ is the state variable defined by 1, for fl-state 0, for a-state,
q" =
(34)
which describes the attaching state of both filaments. In the case of q~ = 1, the vth cross-bridge creates the contractile force, and the state is called q-active in short. According to the activation of the B-state, the effective viscosity 7* must be different in the two states of the cross-bridge. Here we assume that the coefficient can take on either of two different values; =
+ A*-
(35)
where ~* and A* are constants, and ~ represents the following variable:
~ =
1, 0,
for active state for non-active state.
(36)
Generally speaking, the i-active state is not always equal to the q-active state described by (34). In a special case, however, both active states may be equal. Thus, if these two active states are identical, i.e., ¢ = q (=~0), the active state is considered as the//-state in Tomita's theory. If we define the sliding velocity [V-I by: (37) the velocity-tension relation, obtained from (33), is given by: IV] ~ * = ~q,F v-NIT], v
¥
(38)
AN EXACT ANALYSISOF THE SLIDING MOTION
825
where [T]=I/N~T~. In the derivation of this equation, (13) was used. Using the approximation of smooth sliding, i.e. F~flo, (34)-(38) lead to [-V](~7* + A*({)) = flo(t/) - IT],
(39)
where ( 4 ) and Q1) are mean values of { and tl defined by 1 N
(40) 1
u
The two kinds of variables, {~ and ~7~,are considered to be stochastic since: (1) the activation of actin and myosin molecules depends on the spatially non-uniform and random adsorption of ATP molecules; and (2) the crossbridge has an electrical charge which may oscillate randomly in the process of contraction as was mentioned in Section 3(B). In general, these two causes couple with each other, namely the chemical process of the contractile force generation may be affected by geometrical and mechanical states of actin and myosin molecules as well as by the contracting speed of the actin-myosin filament. In other words, it is reasonable to assume that (~) and (t/) are given functions not only of I T ] but also of IV] as follows: (~) = X([T]. [V]) (41) ( , ) = Y([T]. IV]). where X and Y describe the internal state of a cross-bridge, therefore, we cannot determine the functional forms of (41) in terms of such mechanical variables as {a,} used in (1). In order to know the time variation of (~) and (17) at the state of the smooth sliding contraction, we need adequate information about the process of mechano-chemical energy conversion in the actomyosin system; for example, a kinetic rate equation representing the time variation of (4), etc. For the present it is unavoidable to assume some appropriate functions X and Y. For the sake of convenience, assume ( 3 ) = ( q ) = (/?), and that the time evolution of (fl) is given by (30). The steady state value becomes: (fl)
-
k0 k I H- k2[V ]
( = (4)
= (q)),
(42)
826
YOJI AIZAWA AND HIROSHI
SHIMIZU
corresponding to (41). F r o m (39) and (42), we can derive a relation of Hill's type provided that the term of o([V]Z), which is a higher order small quantity, is neglected. However, it is not the main p u r p o s e of this section to pursue the apparent coincidence between Hill's relation and our theory, but to give a theoretical framework in which Hill's relation can be reasonably formulated by taking into account the equations of motion as well as the internal degrees of freedom of cross-bridges. In this paper we have discussed the dynamical behavior of sliding filaments ,with reference to a statistical effect of many active sites. T w o essentially significant effects were revealed in the present theory concerning the statistical mechanical problems in the contractile system: the first is the effect of system size given by (18) and (AI, 2), and the second is the "cooperative tension effect" represented by (41) and (42).
One of the present authors (Y. A.) thanks Dr. K. Nishiyama (Department of Biology, Faculty of Science, Kyushu University, Japan) and Prof. Y. M a t s u m o t o (Department of Physiology, Emory University School of Medicine, Atlanta, Georgia) for valuable discussion and for giving some important recent informations. This study was supported partly by a grant from the Ministry of Education. APPENDIX I OSCILLATORY CONTRACTION AND SPATIAL LOCKING OF A MYOSIN FILAMENT In order to make clear the physical meaning of (18), an exactly solvable model is analysed here. When fl,=O (except for n=0, _+1) is satisfied, (15) is rewritten as follows: dbv -dt
((AI, 1)
O~-t- O"lfl c o s ( k A b v ~- ~ ) '
where c~, fl, and ~b are appropriate real valued parameters defined from flo, fll and fl_ 1, and the renormalization factor al, is given by, ch -
1 sin(uNz) N sin(uz)
(AI, 2)
Equation (AI, 1) permits two quite different solutions depending on the value of the parameter alfl/a. That is to say, in the case of aiflO
where ctr) = fl" ~ / N ~ ~ o
~1 -2 "e'(k x m=~ / ~ ~ T J "
(AII, 7)
LITERATURE Aiello, E. and M. A. Sleigh. t972. "The Metachronical Wave of Lateral Cilia of Mytilus edulis." J. Cell Biol., 54, 493-506. Aizawa, Y. 1979. "A Mathematical Model of Ciliary Movement." In preparation. Elliott, G. F., J. Lowy and B. M. Millman. 1967. "Low-angle X-ray Diffraction Studies of Living Striated Muscle during Contraction." J. Molec. Biol., 25, 31-45. Ford, L. E., A. F. Huxley and R. M. Simmons. 1977. "Tension Responses to Sudden Length Change in Stimulated Frog Muscle Fibres Near Slack Length." J. Physiol., 269, 441 515. GTansdorff, P. and I. Prigogine. 1971. Thermodynamic Theory of Structure, Stability and Fluctuation. New York: Wiley. Gray, J. 1931. "The Mechanism of Ciliary Movement. Photographic Stroboscopic Analysis of Ciliary Movement." Proc. R. Soc., Lond., B107, 313 332. Huxley, A. F. 1957. "Muscle Structure and Theories of Contraction." Prog. Biophys. Chem., 7, 255 318. . 1971. "The Activation of Striated Muscle and its Mechanical Response." Proc. R. Sot., B178, 1-27. and R. M. Simmons. 1971. "Proposed Mechanism of Force Generation in Striated Muscle." Nature 233, 533-538. Huxley, H. E. and J. Hanson. 1954. "Changes in the Cross-striations of Muscle during Contraction and Stretch and their Structural Interpretation." Nature, 173, 973 976. - and W. Brown. 1967. "The Low-angle X-ray Diagram of Vertebrate Striated Muscle and its Behavior During Contraction and Rigor." J. Molec. Biol., 30, 383-434. Nicolis, G. and I. Prigogine. 1977. "Self-organization in Nonequilibrium System." New York: Wiley. Nishiyama, K. 1977. Private communication. Shimizu, H., T. Yamada, K. Nishiyama and M. Yano. 1976. "Energetic Enzyme Theory of Muscular Contraction. II. Relation Between Hill's Equation and Functions of Two-head Myosin." J. Theor. Biol., 63, 165-189. Tomita, K. 1973. "A Model of Muscle Contraction." In Synergetics (Ed. by H. Haken and B. G. Teubner), pp. 232-240. -
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RECEIVED 3-28-78 REVISED
9-17-78
