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Mathematical aspects of the cross-bridge mechanism in muscle contraction
N~nlinec~? Analysis. Theo?. Primed in Great Britain.
.Uctho&
MATHEMATICAL
& Appiicmom,
Vol. 7. No. 6. pp. 661-683.
1983.
ASPECTS OF THE CROSS-BRIDGE IN MUSCLE CONTRACTION*
0362-%5&‘83 SO3.00 + .OO @ 1983 Pergamon Press Ltd.
MECHANISM
VALERIANO COMINCIOLI and ALESSMDROTORELLI Istituto Matematico, Universitti di Pavia, Italy (Received 18 November 1981) Key words and phruses: Cross-bridge dynamics, physiological and mathematical model,nonlinearpartial differential equation, existence and uniqueness, asymptotic behaviour, numerical approximation.
1. INTRODUCTION
IN THISpaper we study from the mathematical
point of view a structural model which simulates cross-bridge dynamics. This dynamics is, according to the sliding filament theory of Huxley [3], the basis to explain the contracting mechanism of the muscle. The model studied here is a part of a more comprehensive model simulating the whole process of muscle contraction; it has been recently introduced in the framework of a collaboration between mathematicians and physiologists at the University of Pavia. The aim of the research is both the mathematical analysis of the model and its identification from suitable experimental data. In this paper we will confine ourselves to the mathematical aspects, referring to [5] [6] and [7] and to a paper in preparation [8] for a more extensive description of the physiological model and for the identification results hitherto obtained, together with a convenient literature. From the mathematical point of view we deal with an initial value problem for a nonlinear and nonlocal partial differential equation of first order which seems of new type. The plan of the paper is the following: in Section 2 we give a short description of the physiological background of the problem; in Section 3 we give the mathematical formulation of the problem and we introduce some preliminary considerations; Section 4 contains a local existence and uniqueness result; in Section 5 we prove some a priori estimates for the solutions; in Section 6 we prove a global existence theorem; in Section 7 a global uniqueness theorem is proved; in Section 8 we obtain some results on the asymptotic behaviour of the solution; in Section 9 we consider the numerical approach and in particular we outline the algorithm used in [6] and [7] in the numerical identification process.
2. THE MODEL
The repeating unit of muscle structure is the sarcomere, which consists of an array of interdigitating filaments: the thick (or myosin) and the thin (or actin) filaments. The thick filaments are tied together through their centers and several thin filaments, anchored at the Z-lines, lie adjacent to each thick filament (Fig. 1). Work supported by CNR of Italy through the Istituto di Analisi Numerica of Pavia and G.N.A.F.A. HUSPI Project. l
661
and by the
662
V. CO.MINCIOLI and A. TORELLI
According to Huxley’s theory [3] the generation of muscular force is thought to result from interactions between these filaments. Under the influence of the intracellular calcium concentration [Ca’-] which in turn depends on the time course of the action potential, there is a formation of links or cross-bridges which (once formed) act like springs. An attached cross-bridge is characterized by the distance x between the equilibrium position of the myosin head and reactive site. Let n(x, t) denote the cross-bridge density at time t (number of cross-bridges per unit of cross-bridge length in one half-sarcomere) and N the density of myosin heads (per halfsarcomere) which may form cross-bridges at time t.
attached Cross-bridge actin r
pj$IkzJq
Z-line
-7 Fig.
1.
Schematic
organization
of a sarcomere.
The number N can depend a priori on x, t and also on the length a of the half-sarcomere. If we suppose the cross-bridges to behave as linear elastic bonds with stiffness k, the force developed at time t by all the interacting cross-bridges in the half sarcomere is given by: F,(t) = k
+=n(x, t)x dr. i -2.z
(2.1)
Then to simulate the force generating process we must first describe the dynamics of the cross-bridges population. This dynamics results from the balance of the formation and breakage of bridges; assuming a first order kinetics we have: y
where F and the material cross-bridges made by the
= F(s, x, t)
(N(s, x, t) - n(x, t)) - G(s,x, t) n(x, t),
(2.2)
G are the attachment and resp. the detachment rate functions and d/dt denotes derivative, i.e. the time derivative with respect to a frame moving with the density distribution. As usual in fluid dynamics, conversion to a fixed frame is formula: d _=d++ dt at
where dx “=dt is the velocity of the filaments sliding, which is also equal to the velocity of half-sarcomere
Mathematical aspects of the cross-bridge mechanism in muscle contraction
shortening,
663
that is: dx
ds
(2.3)
v=dt=dr.
It must be emphasized that since the sarcomere shortening is an effect of the formation of cross-bridges, the velocity u is a function of n. In order to specify this dependence we must first incorporate into the model the effects of the elastic properties which are also present in a cardiac muscle. 1
A
A LCEW
i
__
CE
+_
Lk) SE
LSEi L )
1
1
I Fig. 2. Hill’s three component model for cardiac muscle.
To do this we can represent the muscle by a rheological model composed by the following three basic elements (assembled as in Fig. 2 (Hill’s three component model)): the parallel elastic element PE to account for the elasticity of muscle at rest. the series elastic element SE and the contractile element CE, which is freely extensible at rest, but when activated can shorten and develop tension. The contractile element CE is identified with the half sarcomere and then it is described as previously by the sliding filament theory. For simplicity in the model we have neglected the viscous forces which are however present in the cardiac muscle (for a study of such forces in resting myocardium see [2]). The stress-strain characteristics of both SE and PE are expressed by exponential laws. Then, denoting by FPE, FSE the forces supported by the PE and SE, respectively, we can write: &E(t)
=~&xp(~S&SE(~)
-LOSE)
FPE(~) =ap~(exp(bp&(t)
--LOPE)
- 1); - 1);
(2.4) (2.5)
where L(f) represents the muscle length at time t and L OPEthe muscle length corresponding to the zero tension; the &n(t) and LOSEhave an analogous meaning. The muscle tension P(t) is given at any time t by: P(t) = Furthermore
FPE(~) + FsE($
(2.6)
at any instant of time we must have FsE(~) = FcE(~); &E(t)
+ &E(t)
= L(t);
(2.7) (2.8)
V. COMINCIOLI and A. TORELLI
664
where FCE(f) is the force developed by the contracting element (that is the force F,(t) given in (2.1)) and Lc-(t) is the contractile element length (that is the half-sarcomere length s). There are two different important experimental situations: (1) the isometric contraction in which L(f) is kept constant and P(r) is observed; (2) the isotonic contraction in which P(f) is assigned and the muscle length L(r) is computed. In this paper we shall consider the first situation (isometric contraction) into account (2.3) and (2.8), we have:
do
u=x=z=
do -= d&E(t) dt
(0
-- d&E dt
for which taking
(2.9)
.
On the other hand from (2.4) and (2.7) we obtain: dLsE (t) -=&$log dt
( 1+ 3)
=+-$log(l
+L$)),
(2.10)
and then from the definition of FcE(~): u= -Idlog bsE dt
1 +k %E
+xn(x,r)xdx
i -=
. 1
(2.11)
Summarizing, in an isometric situation, the function n(x, t), is the solution for f > 0 and x E R of the following partial differential equation: $
+ u 2 = F(s, x, t) (N(s, x, t) - n(x, r)) - G(s, x, t) n(x, t),
with u given as in (2.11). Assuming initially (i.e. at t = 0) a resting situation, condition:
(2.12)
we can associate to (2.12) the initial
n(x, 0) = 0,
(2.13)
so that to find the cross-bridges distribution n(x, t) we must solve the initial value problem (2.12) (2.13). To conclude it remains to define the functions F, G and N. For what concerns the number N of possible cross-bridges at time t, in this paper we will suppose such number to be a constant, that is independent of x, t and also of s. Obviously it is the independence of s which more simplifies the mathematical problem, but it seems to be allowable at least in the isometric situation (see the identification results in [7]). For the rate functions F, G many different definitions have been proposed in the literature (see for a review e.g. [lo]). In this paper following Huxley [3] ( see also [4]), we will suppose F, G of the form: F = r(t) f(x)
G = g(x)>
(2.14)
where y(t) describes the time course of the intracellular calcium concentration upon stimulation. Such function is of course a physiologic datum, but it can be simulated by means of suitable mathematical models of the intracellular calcium kinetics (e.g. [6], [7]). The qualitative behavior in a twitch following a single stimulus is like this: y(t) rises from zero to a peak soon and then it decays back to zero.
665
Mathematical aspects of the cross-bridge mechanism in muscle contraction
The functions f(x), g(x) in Huxley [3] are of the following very simple form: ifx
f = fix/k g = w/k
O
f = 0, g = gdk
x>h
where h is the largest displacement at which myosin can become attached to actin. Only the three parameters fi, gl, g2 are needed to define f and g and this fact seems the main motivation for the particular choice (2.15). In fact the functions f(x), g(x) must be identified from the experimental findings. A qualitative indication is that f(x) is a nonnegative function with bounded support in the semiaxis x 5 0 and g(x) is a nonnegative function which rapidly increases for x c 0 and x > h. In the present paper the assumptions on f and g will be very general (see Section 3). To simplify the notations and without any loss of generality, in the following we set the various constants present in the previous formulation of the model, that is the constants like k, asE,bsE,Nt.. . , equal to 1. Furthermore we shall use the more “mathematical” notation U(X, t), to indicate the unknown function n(x, t). 3. MATHEMATICAL
MODEL,
PRELIMINARY
CONSIDERATIONS
(a) In order to study the problem of Section 2, we introduce some hypotheses We assume that
on y, f, g.
f, g E Ck(Wv YE Ck([O, +%
(3.1)
the support off is a compact set of W,
(3.2)
where k is a given integer number 2 1. Let us now consider the following precise formulation
of the problem of Section 2:
Problem 3.1. Given T > 0, we look for a function U(X,t) verifying:
x [0,T]),
(3.3)
u(x, 0) = 0, vx E R,
(3.4)
Vt E [0, T], u(x, t) has compact support inx,
(3.5)
u E Ck(R
xu(x, t) dx > -1, Vt E [0,T],
(3-e) (3.7)
= r(t)f(x>11-
u(x, 01 -g(x) 4x7 t>.
(b) Let u(x, t) be a solution of problem 3.1 and z(t) = -log
( I 1 +
)
x4x, t) dx
R
fk
0 = Y(Of(X>
+
g(x);
(3.8)
9
’
P-9)
V. COMINCIOLI and A. TORELLI
666
it follows that the equation (3.7) can be written in the following way: u,(x, r) + z’(t) u,(x, t) = :/(r)f(.r) - H(x, t) u(x, t); this
means that the function u(x. t) verifies the following relation lines)) (Z(I) + y, r), y being a real parameter, ; u(z(r) + Y, 4 = Y#f(Z(~)
+ Y> - H(‘$t)
(3.10)
on the ((characteristic
+ y, f> U(Z(f) + y, t>.
(3.11)
Since (3.11) is an ordinary linear differential equation with respect to the unknown function u(z(t) + y, t), it follows, recalling also (3.4), u+(t) +y,t)
= [Y(J)~(Z(S)
The previous considerations justify the introduction CO([O, Tl), r E [O, 7.1, x, Y E R): V,cV, 0 = [ v(s)f(z(s)
+Y,
+Y> exp(+(z(r)
r) dr) ds.
of the following operators
+ Y) exp( -i’H(z(r)
+ Y, r) dr) ds,
(3.12) (where z E (3.13) (3.14)
V,(x, t) = vz(x - z(t), t). We have also proved the following: LEMMA
3.1. Let z be an arbitrary function belonging to C’([O, T]). Then V,(x, t) is the solution of the problem (3.4) and (3.10). Let also: (3.15) supp,( V,) = support of V,(‘y, t) with respect to y at time t,
(3.16)
supp,( Uz) = support of U, (x, t) with respect to x at time t.
(3.17)
Thanks to (3.2) there exist Nr > 0 and A$> 0 such that: supp(f)
c [-NIT
Nzl.
(3.18)
Using the definitions of V, and U,, it follows easily that: LEMMA 3.2.
For every z E C?([O, T]), we have that: SUPP~(VZ) c suppt(~z)
[-iv1
C [-NI
-
ll4[O.l]~
N2
+
IIZll[o.r]17
(3.19)
21141~0,rJ
(3.20)
- 2//~11[o.r]rN2 +
Let us now consider the following new problem (which later will be proved to be equivalent to problem 3.1); Problem
3.2. Given T > 0, we look for a function z(f) verifying:
z E Ck([O, Tl),
(3.21)
xl/,(x, t) du > -1, Vr E [0, T], IR
(3.22)
Mathematical aspects of the cross-bridge mechanism in muscle contraction
667
(3.23) Remark 3.1. We note that, by lemma 3.2, there exists the integral in (3.22).
The following theorem states that problems 3.1 and 3.2 are equivalent: THEOREM
3.1. Under the hypotheses (3.1) and (3.2) we have that (i) if LI(X, f) is a solution of problem 3.1, then z(r), defined by (3.8), is a solution of problem 3.2; (ii) conversely if z(r) is a solution of problem 3.2, then: (3.24)
u(x, t) = Uz(x, f), is a solution of problem 3.1.
Proof. (i) If u(x, r) is a solution of problem 3.1 and if z(t) is defined by (3.8), then (3.21) holds. By lemma 3.1 and (3.7), it follows that u = U.. Then the relations (3.22) and (3.23) easily can be proved. (ii) Conversely if z(t) is a solution of problem 3.2 and u(x, t) is given by (3.24), then (3.3), (3.4), (3.5) and (3.6) are an easy consequence of lemma 3.1 and the definition of Uz. The relation (3.6) follows from (3.22) and (3.24). The relation (3.7) is a consequence of (3.23) and lemma 3.1. n 4. EXISTENCE
AND
UNIQUENESS
FOR
T SMALL
(a) Let A4 > 0 and Z(T, M) = {z E CO([O,z-1): llz Il[f)r]S M}.
(4.1)
Put also: (Lz)(t) = -log(l
+ //J;(,x,
t) dx),
(4.2) (4.3)
Then we have: LEMMA
4.1. For every M > 0, there exist c > 0 and To > 0 (depending only on M, y,f. g) such
that:
s CT(T), t/zE Z(T, M), VT E [O,z-o]. llLIIl[O,TI Proof
If we put x = z(t) + y, it follows that: xUz(x,t)k= iw
iR
(G)+~)K(y,r)dy,
hence: I~n~~(x,i)dri~~(M+lyl)lVl(y,r)Idy.
(4.4)
V. COMNCIOLI and A. TOEELLI
668
If we put N = max{ Ni, Nz, M},
(4.5)
then it follows (by the definition of V, and by lemma 3.2): xU,(x,t)cLx
SN*~r(T)exp(fT(T)
+g),
where: f= max{]f(x)I,x
E R1,
(4.6)
g = max{]g(x)l, 1x1 G2N).
(4.7)
By the mean value theorem we can conclude the proof of the present lemma.
n
In a similar way we can prove the following: LEMMA4.2. For every M > 0, there exist cl > 0 and TI > 0 (depending such that II LZl
-
~Z211[0,71 c c&T
-
z2]]lo$(T),
only on M, y, f, g),
vri, ~2 E Z(T, M), VT E [O, 7-11
(4.8)
(b) Thanks to lemmas 4.1 and 4.2, it follows that: LEMMA4.3. For every M > 0, there exists T > 0 such that: L(-W,
W)
c W-9 M)
L is a contraction on Z( T, M)
(4.9) (4.10)
and then there exists one and only one z E Z(T, M) such that z = Lz. Lemma 4.3 gives us the following uniqueness result in small: THEOREM4.1. Under the hypotheses (3.1) and (3.2), there exists T > 0 such that problem 3.2 (or equivalently problem 3.1) has at most one solution. Remark 4.1. The existence of a fixed point for the operator L is not yet an existence result for problem 3.2 (or equivalently problem 3.1), since we have proved only that z E C’([O, T]). To this end we will prove the following lemma. LEMMA4.4. If z E C’([O, T]) verifies the relation z = Lz, then there exists T’ E] 0, T] such that z E Ck([O, T’]). Proof. If z = Lz, then we have (taking x = y + z(r)):
that is: exp(-z(t))
- ~(0 /n V,(Y, t) dr - /n~Vz(~, t) - I = 0.
669
Mathematical aspects of the cross-bridge mechanism in muscle contraction
Let us now introduce the following function: G(r, w) = exp(--w)
- w Ix Vdy, 4 dy - _/2yVz(~, f) dy - 1.
Then we have that: Vt E [O, z-1.
G(t, r(t)) = 0, It is easy to verify that the functions iR
Vz Cv, 9 dY, j-R~KC~, f> dy,
belong to Ck([O, T]): this means that also G E Ck. But we have G(0, 0) = 0,
G,(O, 0) = -1.
Since z(f) is implicitly defined by the relation G(r, w) = 0, we can conclude Ck([O, T’]), where T’ > 0 may be smaller than T. n We note that (3.22) is a consequence following local existence result:
THEOREM 4.2.
(or equivalently
of lemma 4.1. We have completely
that z E
proved the
Under the hypotheses (3.1) and (3.2), there exists T > 0 such that problem 3.2 problem 3.1) has at least one solution.
5. SUPPLEMENTARY
PROPERTIES
OF THE SOLUTION
(a) Having in mind to prove a global existence and uniqueness theorem, assume that: f(x) 3 0, g(x) 2 0,
Under the hypotheses arbitrary in C’([O, T]):
(5.1) (5.2)
(3.1), (3.2), (5.1) and (5.2), we have that (z being
0 s U,(x, t) s 1,
Proof. By the positivity of y(t) and f(x) V,(x, t) 3 0. We have also that:
U*(Y + z(r), 9 = vXr7 9 = [P(Y,
vx E R,
Vr E [O, + q.
r(t) 2 0,
PROPOSITION 5.1.
Vr E [0, T].
vx E R,
(5.3)
and using the definition of U,, we obtain that
r) exp (- /MY, I
r) + q(Y, r)l dr) Q,
where: P(Y, 9 = r(t)f(z(t)
from now on we
+y),
4(Y? 4 =dz(t)>
+r)-
(5.4)
V. COMINCIOLIand A. TORELLI
670
Integrating by parts the relation (5.4), it follows that: u,(y + z(r), r) = [ exp (- jhY3
r> +4(Y*
5
-
I j0
dy, 4 exp (-
Recalling (5.1), it follows that uZ(x, t) s 1.
=
61 MY>
d> dr) ds.
(where z E C’([O, T])):
1 exp(-49)
r> + dyt
n
(b) We now introduce the following operator (F,)(t)
4) dr)]k
X[H(x, 0 Ur(x, 0 -
+ IR~~(x, 4 dr i R
~(4 f(x)]
dr.
(5.5)
We remark that, by proposition 5.1, the operator F, is meaningful. We now introduce the following new problem (which will be again equivalent to problem 3.1): Problem 5.1. Given T > 0 we look for a function z(t) verifying:
z E Ck([O, T]),
(5.6)
z(0) = 0,
(5.7)
z’(t) = (F,)(t).
(5.8)
We have also that problem 5.1 is equivalent to problems 3.1 and 3.2, as stated by the following: THEOREM
5.1. Under the hypotheses (3.1), (3.2), (5.1) and (5.2), we have that: (i) if u(x, t) is a solution of problem 3.1, then z(f), defined by (3.8), is a solution of problem 5.1; (ii) conversely if z(t) is a solution of problem 5.1, then u(x, t) = UL(x, t), is a solution of problem 3.1. Proof. (i) The relations (5.6) and (5.7) are obvious. Multiplying (3.7) by x and integrating on Iw, we have that:
1 + J, xu(x, f) dx + JR u(x, t) dx xu,(x, t) dr = ax[:Q) Iw I 1 + J-aXU(X,t) dX
f(x) - H(x, t) u(x,41dr.
But if r(t) is defined by (3.8), it follows that: 1 + Jaxu(x, t) dx = exp( -z(f)) i and then: xu((x, r) cLr = -exp( -z(r)) z’(t). iw By theorem 3.1 we have that u = U, and then: -z’(t)
[ exp(--z(t))
which implies (5.8).
+ L Uz(x, r) d]
= 6.4 y(r>f (4
- H(x, 4 4x,
01 dr,
Mathematical aspects of the cross-bridge mechanism in muscle contraction
(ii) The relations (3.3)-(3-j)
671
are obvious. By lemma 3.1 we have also:
UI(X,t) + z’(t) ux(x, r) = r(t)f(x) We now prove the relation (3.6). By contradiction,
- H(x, r) u(x, r).
(5.9)
assume that:
X=[~E[0,T]:~*u(*,t)dr~-1}c0. Put now T,J = min X. Obviously we have that To E 10, T]. Put now i(t)=-log
( 1+ I,XU(X,f)d+
tE[O,To[.
(5.10)
since lim Z?(t)= + m, r-Toif we prove that f = z, we get a contradiction. Multiplying by x and integrating on W the relation (5.9), we have that: --k’(t) exp(-i(t)) + z'(t)i,u(x,i) dr = jRx[y(r)f(x)- H(x,t)u(x,r)] dx. Recalling that u = iJ, and comparing with (5.8), it follows that: z’(t) exp( -z(t))
= z(t) exp( -z(f)),
Vf E [0, To[
and then j(t) = z(t),
Vt E [O, To[.
So we get a contradiction
and (3.6) is proved. Since we have also proved that f(t) = z(t) in H [0, T], the relations (5.9) and (5.10) imply (3.7). Theorem 5.1 is completely proved.
Remark 5.1. It is easy to verify that the equivalence
stated in theorem 5.1 remains valid, for Tsmall enough, without the hypotheses (5.1) and (5.2) (that is these hypotheses are necessary only in the case in which we want to prove the global equivalence). (c) Let Nr be the number introduced
in (3.18). Let also M and K be two numbers verifying: M>O,
(5.11)
Mz-Nr,
(5.12) where the function r(t) is defined in (4.3). Then we have: 5.1. Under the hypotheses a solution of problem 3.1, then LEMMA
(3.1), (3.2), (5.1) and (5.2) we have that: (i) if u(x, t) is
xu(x, r) dr > -1 + exp(-KM),
Vf E [0, T];
(5.13)
672
V. COMINCIOLI and A. TORELLI
(ii) equivalently
if z(t) is a solution of problem 3.2 (or 5.1) then: z(t) < KM,
'it E[O,Z-1,
(5.14)
where K and M verify (5.11) and (5.12). Proof. Thanks to theorem 3.1 we must prove only the part (i) of the theorem. To this end let u(x, t) be a solution of problem 3.1 and put z(r) = -log(l + JR xu(x, t) du. Suppose, by contradiction, that (5.13) does not hold. Then we can put (k = 0, 1, . . . ,K):
fk= min{t E [0, T]; z(f) 2 /CM}.
(5.15)
Then we have: 0 = f,, < rr < . . . < tK,
x@,
tk)
(5.16)
.?(fk) = /CM,
(5.17)
dx = -1 + exp( -kM).
(5.18)
By theorem 3.1 we have that u = U. and then: xu(x, tk) dr =
j/
[
6’y(s)f(z(s) - Z(fk) + X>exp (- /‘k+) - Z(tk) +x, r) dr)dJ]dx
that is, changing the order of the integrations
and taking
y = z(s) - .?(tk) + x, we have (5.19) where: 11 = /“-’ y(S) kf(Y)(Y
+ z(fk) - Z(S)) exp (- /“H(z(r)
- Z(S) + y, r) dr) dy ds,
(5.20)
+ z(fk) - z(s)) exp (- jRH(z(r)
- z(s) +y, r) dr) dy ds.
(5.21)
s
0
It = 1”
y(s) 1 f(y)(y
0
R
(k-1
Notice that: (5.22)
II 5 0, since ifs E [0, fk_t] it follows that z(&) - z(s) 2 M 3 N1.We have also that:
(5.23)
z, = z,- + IF, where:
G =
1” y(s) j f(y)(y % I&,
+
z(fk)
-
z(s)) exp (- j”H(r(r) I
- z(s) +y, r) dr) dy ds.
673
Mathematical aspects of the cross-bridge mechanism in muscle contraction
It is easy to verify that: 1; so, I; 2 -N,[UQ
(5.24)
- Uhc-,)I
h_f(4
dx.
(5.25)
Thanks to (5.18), (5.19), (5.22), (5.23), (5.24) and (5.25), we have that:
N~[r(h)- r(h-L)]
Ia_
f(x) dx 3 1 - exp(-kM)
and adding with respect to k, it follows: 1 1 - exp( -M)
Comparing (5.12) and (5.26) we get a contradiction. In the special case in which supp(f) c [O, + m[
(5.26)
(5.27)
we can improve the result of lemma 5.1, as stated by: LEMMA 5.2.
Under the same hypotheses of lemma 5.1, if (5.27) holds, then: iw
xu(x, 1) dx 3 0,
Vf E [O, Z-J,
(5.28)
or equivalently: z(r) S 0,
vt E [O, z-1,
(5.29)
where U(X,t) is a solution of problem 3.1 and z(t) is a solution of problem 3.2 (or 5.1). Proof. By theorem 3.1 it is sufficient to prove the relation (5.29). By contradiction that: i = max{z(f), t E [0, T]} > 0.
assume (5.30)
Let also f be such that z(f) = i. Put now u = U,. Then we have (changing the order of the integrations and taking y = z(s) - z(r) + x): /a-WY f) b = l’v(s)
/a (Y + z(f) -
dS))f(y) exp(- ~‘~(44 - 4)
+Y, 4 dr) 4v hj
that is:
Recalling that u = U,, by theorem 3.1 we obtain that z(t) G 0. But this result and (5.30) give us a contradiction. (d) From now on we assume that: 3~ < 4 such that lim g(x) exp( - E]XI) = 0. x-=
(5.31)
V. COMINCIOLI and A. TORELLI
674
Put now (Nt and Nz verifying (3.18) and ,u 2 0): (5.32)
g(u) = max{g(x), x E [ -Nt - 2u, Nz. + 2,u]}, + Ni)[:/+
B(u) =f(N: where T(t) and fare
T(T)g(,u)] + 2f;r(7’)&4
u’,
(5.33)
defined in (4.3) and (4.6) and ‘/ = max{y(t): r E [0, T]}.
(5.34)
W
Then we have: LEMMA5.3. Under the hypotheses
(3.1), (3.2), (5.1) and (5.2), we have that (for every z E
(5.35)
where the symbol /Iz]~~Q~ is defined in (3.15). Proof.
Let I be the integral in (3.35). Then we have: (5.36)
jzj =GI, + I?, where
(5.37) (5.38)
Using proposition
5.1, it follows that: (5.39)
It s $( N; + N:). By the lemma 3.2 and by the definition of U,, we have that: Zt -’
fl(0
(5.40)
(N: + Nt + 211~11fo.rj) g(]lz Il[o.~l)>.
The relations (5.36), (5.39) and (5.40) imply (5.35). Using lemma 5.1 and 5.2, we have that there exists (~2 0 such that: (5.41)
z(t) < Ly,vr E [O, 7-l and then, by the hypothesis (5.31), it follows that there exists p > 0 such that: exp(-u)[B(m)
+
B(U)]
s
p,
Vp e [O, + x[.
a
(5.42)
Then we have: LEMMA 5.4. Under the same hypothesis
of lemma 5.1, if (5.31) holds and if /3 verify (5.42),
then we have that: I2
xu(x, t) dr S -1 + exp(@),
Vf E [0, T],
(5.43)
675
Mathematical aspects of the cross-bridge mechanism in muscle contraction
or equivalently: Vr E [O, T],
z(t) 2 -pr,
(5.44)
where U(X,t) is a solution of problem 3.1 and z(t) is a solution of problem 3.2 (or 5.1). Proofi By theorem 3.1 it is sufficient to prove the relation (5.44). Put now: i(t) = min(z(s), s E [0, t]}.
(5.45)
i E C”“([O, T]), i(0) = 0, i(t) 50, i(r) s Z(f),
(5.46)
It follows that:
i(r) is a nonincreasing function, Ii:(r)] +2’(f)/,
WE [O, z-1,
(5.47) (5.48)
where f:(r) is the right derivative of i evaluated in the point 1. Thanks to (5.41) and (5.46), we have that: II~I~o.~I
hence, by (5.8), proposition
s
mada,
-441,
(5.49)
5.1 and lemma 5.3, Iz’0) I s exd4OP(cr)
+ W-40)1,
(5.50)
and then, by (5.48), I ii(f)
I c exp(z(f) [ B(a) + B( -z(t))].
(5.51)
Now we will prove that: Ii:(r)
Indeed if f(r)
[ d p,
Vf E [O, z-1.
(5.52)
then z:(r) =0 and then (5.52) holds. If i(f) = z(f), then we have (by
IW>I s expHWIP(4
+ W(r) I)1GP.
Since the case i(f) > z(f) is impossible, the relation (5.52) holds. But this implies that i(f) > -,& and then, by (5.46), we obtain the relation (5.44). n
6. GLOBAL Now
we introduce
EXISTENCE
the following fixed point theorem ([9]. see also [l] and [ll]):
6.1. (Browder-Potter). Let X be a normed space, E a convex and closed subset of X, F: [0, l] x E+Xsuch that:
THEOREM
F is continuous, F([O, l]
x E)
is a relatively compact subset of X,
F(A, z) f: z, Vz E aE, VA E [0,11,
(6.1) (6.2) (6.3)
676
V. COM~CIOLI and A. TORELLI
F({O} x L3E)c E.
(6.4)
Then there exists z E E, such that z = F( 1, z). The previous fixed point theorem allows us to prove the following global existence result: 6.2. Under the hypotheses (3.1), (3.2), (5.1), (5.2) and (5.31), for every T> 0 we have that problem 5.1 (or equivalently problems 3.1 and 3.2) has at least one solution.
THEORE,Ci
Proof. With the notations of theorem 6.1, we put: X = C?([O, T]) (with the usual norm), E = {z E X:[/Z[/[O,I~ G max{a, PT + l}}, where cuand /3 have been introduced in (5.41) and (5.42). Let also F:[O, l] x E+ in the following way:
Xbe defined
where (for fixed A) the function z-F_! IS ’ defined in the same way as the function z-+ F, in (5.5), but replacing f by Afand g by & (notice that we must modify in the same way also the definition of H and UJ Let us now verify that F satisfies the conditions (6.1)-(6.4) of theorem 6.1. The function F is obviously continuous and then (6.1) holds. Suitably adapting lemma 5.3, we have that
/lF(A, .z)ll G ATexp(&))B(&), V(A, z) E [0, l] x E,
(6.5)
]]F’(A,z)]] 6 Aexp(&) B(&), V(A, z) E [0, l] x E,
(6.6)
where & = max{Ly,PT + 1) and F’(A, z) is the derivative of the function I--, (F(i, z))(r) (notice that if z E C’([O, T]), then F(A, z) E C’([O, T])). Th e relations (6.5) and (6.6) imply (6.2). Suitably adapting lemmas 5.1 and 5.4, we can verify that if z = F(A, z), with A E [0, 11, then: ]]z]]< maxim, P T + 11,
(6.7)
which means that also condition (6.3) holds. Finally also condition (6.4) is fulfilled because F(0, z) = 0, Vz E C’([O, TJ). Thanks to theorem 6.1, there exists z E E such that F(1, z) = z, or equivalently z’ = F, and z(O) = 0, that is the relations (5.7) and (5.8) are verified. By 11). So we have proved that problem 5.1 the definition (5.5) of F, it follows that z E C kf'([O, has at least one solution. Recalling theorems 3.1 and 5.1, we can complete the proof of the present theorem. n Remark6.1. During the proof of theorem 6.2, we obtained a supplementary that is a solution of problem 5.1 (or 3.2) belongs to Ck”([O, 11).
7. GLOBAL
result of regularity:
UNIQUENESS
In the present section we assume that Zi(i = 1,2) are fixed solutions of problem 5.1. Under the hypotheses (3.1), (3.2), (5.1) and (5.2) we have that:
677
Mathematical aspects of the cross-bridge mechanism in muscle contraction
LE~MMA 7.1. There exists cl > 0, depending only on zl, zz, ‘/, f, g, and T, such that (for all y E R and t E [0, T]): IV&Y, f) - v&Y, t) / s 4Iz1 - ~?11[0.1],
where the notation 11Ijio,t~has been introduced Proof.
(7.1)
in (3.15).
We have that Z = IVJ_Y, 0 - v*,o,, 0 I s
11
+
(7.2)
12,
where Zl=[r(S)exp(
12= i,‘ds)fMs)
--I,’ Z.&G)
+Y> -f(zz(s)
+Y, r> dt ) IfhW
+Y)I exp ( - ~‘H(zdt) +y, 4 dr) -
+YW,
exp ( - jrH(z2(t)
+y, 4 dt)/dr.
Thanks to (3.1) and (3.2) we can put: ft =
max{If’(x)l,xE
(7.3)
R}.
Then recalling that H(x, t) 5 0 see (5.1) and (5.2)), it follows that (by the mean value theorem): (7.4)
- ~2lI[o,rj.
ZI +JV-)llz~ We have also (always because H(x, r) 5 0): zz S I,’ r(s)f(ds)
+ Y) [ jj+d$
+
Y, 3 - f&zz(t) + y, t) ldr] d.s
and then 12
c
121
+
(7.5)
1221
where: I21=
~ks)f(zi(s)
122 = 1’ r($f(zds) 0
+Y> [I,‘ifhW + Y> [ /‘k(ld$ 0
+Y> -f(rdr) + Y> - g(zz(@
+y)lvdr]
b,
+ y) /dr] h.
It is easy to prove that: 121
~ffJ2(T)lh
-
f211[0,+
(7.6)
where f is defined in (4.6). Let now: p = max{llzlllro.
2-b II~~II~o,TI~~
We remark that if y E [ - N1 - p, NZ + p], thenf(z2(s) all y FE[ - N1 - p, N2 + p]. This implies that: 122
+fgJV’)Tllz~
-
(7.7)
+ y) = 0. This implies that 112= 0, for
z2i1[o,r],
(7.8)
V. COMIXIOLI and A. TORELLI
678
where: gr = max{ig’(x)/, -Ni - 2p s x G N2 + 2p}. The relations (7.2), (7.4), (7,5), (7.6) and (7.8) imply the lemma. LEMMA7.2. If h E&!(W),
(7.9) W
then there exists c2 > 0, depending only on ZI, z?, ;J, f, g, T and
h, such that (t E [0, T]):
II a
h(x)[Uz,(x,
t) - uzz(x, 01 &I ~&I
(7.10)
- z&I.~,
Proof. Taking x = zi(t) + y(i = 1,2) and recalling (3.14), we have that:
I = ahb
+ z~(t))v,,O,,
i
t) dy - jRhb
+ zz(~))Vrb’,
(7.11)
t) dy,
where I is the integral in the left side of (7.10). Then: (7.12)
111c 11 + 12, where 11 = R Ih(y + z&)> IIUY, I
r) - V~,(Y, t) I c-b,
12 = R Ih(y + ~20)) - h(y + z,(t)) / V,,(Y, 0 dy. I
Recalling lemmas 3.2 and 7.1 and the position (7.7), we have that: 11 s clL(N, + N2 + 4~) lIzi - ~21110~1,
(7.13)
Z2+I-(W$N1
(7.14)
+N+4p)k
-~2111~1,
where f is defined in (4.6) and: fi=max{lh(x)I The relations (7.11)-(7.14)
+(h’(x)(,xE[-N1-2p,N2+2p]}.
prove the lemma.
LEMMA7.3. There exists c3 > 0, depending only on zl, ZZ, y, IZi(0 - Z?(f) I s c3 IlIz, - z2lh
f, g and
T, such that: (7.15)
r1 dt
Proof. Since z1 and z2 are solutions of problem 5.1, it follows that:
(7.16)
z = Izr(r) - zz(t) 1c II + II, where (recalling also lemma 5.3 and the relation (7.7)): 1 exp(-zl(r))
+ JR Ur,(x, 4 dx
II w
xH(x,
4[ Uz,(x, t) - Liz+,
41 ch
dr,
Mathematical aspects of the cross-bridge mechanism in muscle contraction
679
z2=B(~)lbiexp(-z,(r))+11 )&-exp(-z_(r))+lJ )Jdi. R u (x, l-
,
Z[
R
u qx.r (
the function B(u) being defined in (5.33). Since U:,(x, t) 2 0, it follows that: 11 c c~exp(p) the constant c4 being conveniently 12 6
B W2p
i,’
1 I exp(
--zI(~)
[IIz~
(7.17)
- ~2il[o,d dt,
chosen by lemma 7.2. On the other hand we have that -
exp(--z2(4)
I +
11 [uzI(xT R
4
-
U&,
t>l
do
!I dr,
then: I2 s B(PP~[
e PII~1 - 410,~l +
CSIIZI - .~llp,~] dt,
(7.18)
where c5 is conveniently chosen by lemma 7.2. The relations (7.16)-(7.18) imply the lemma. LEMMA 7.4.
n
We have that: lIZi - ~2ll[O,r]s
(7.19)
c3
where c3 is the constant of lemma 7.3. Proof. It is a consequence
of lemma 7.3 and of the fact that the function Iorki - z211[o,rjdt
is non decreasing. By Gronwall lemma, we can derive the following global uniqueness result: THEOREM 7.1. Under the hypotheses (3.1), (3.2), (5.1) and (5.2), for every T> 0. we have that problem 5.1 (or equivalently problems 3.1 and 3.2) has at most one solution.
8. ASYlMPTOTIC
BEHAVIOUR
In the present section we give only some preliminary results on the asymptotic behaviour of the solution. A forecoming paper will be dedicated to this subject. Let us now introduce the following new hypotheses:
Then we have:
3pi > 0 such that lim v(t) exp(pir) = 0, I--r+=
(8.1)
Zlp2> 0 such that g(x) 2 p2, t/x E W.
(8.2)
680
V. COMLUCIOLIand A. TORELLI
LEIM,MA 8.1. Under the hypotheses (3.1), (3.2), (5.1), (5.2), (5.31) and (8.1), if c(f) is the solution of problem 5.1 (or equivalently 3.2) then there exist c, L 0 (i = 1,2) such that: -cir S z(t) S c2, vr E [O, + p[. Proof. It is an easy consequence
(8.3)
of lemmas 5.1 and 5.4 and of the hypothesis 8.1.
n
Put now: p = min (pi,p:}.
(8.4)
Then we have: LEMMA 8.2. Under the same hypotheses of lemma 8.1, if (8.2) also holds, we have that if U(X,t) is the solution of problem 3.1, then for every p E [O,p[, there exists c > 0 such that: Iu(x, r)j Cc exp( -pt),
V(X, t) E W X [0, + m[.
(8.5)
Proof. If z(t) is the solution of problem 3.2 (or 5.1), then u = U, and so (using the positivity of y and f and the hypotheses (8.1) and (8.2)):
u(x, t)
‘exp( -pis) i0
exp( -p2(t - s)) ds,
which implies (8.5). We conclude this section with the following:
THEOREM 8.1. Under the hypotheses (3.1), (3.2), (5.1), (5.2), (5.31), (8.1) and (8.2), if z(t) is the solution of problem 3.2 (or 5.1) with T = + ~0, then we have that: lim z(t) = 0 I--t+=
(8.6)
Proof. If U(X, r) is the solution of problem 3.1 (with T = + =J), then we have that u = U,.
Put now Z(t) =
ix+,
t) dr =
ixU,(x, t) dx.
By lemmas 3.2, 8.1 and 8.2, we have that: [Z(f) 1s c
exp
(
- ii z t
N2+2qtfc2
/XIdx. )i -Ni-Z(c~r+c?)
This implies that ,lim= I(f) =O. But since z(t) = -log (1 + I(t)), we obtain (8.6).
n
Remark 8.1. More precisely, we could prove that for every p E [O,p[, there exists c > 0, such that [z(t)/ =Sc exp( -pf).
Mathematical aspects of the cross-bridge mechanism in muscle contraction 9. NUMERICAL
681
APPROXIMATION
In order that the mathematical model, studied in the previous sections from the analytical point of view, can be used in practice, we must be able to compute the solution u and then the required muscle tension P(t). To this end there are, indeed, several possibilities available, in correspondence to the different formulations of the problem. We could, for instance, discretize problem 3.1, that is a partial differential equation of first order by using some classical schemes based on finite differences or finite elements methods. A second type of approach could consist in solving numerically problem 5.1, that is an initial value problem for an integrodifferential equation of Volterra type and also this is a classical numerical task. There is another numerical approach which arises directly from the mechanical frame of the model and which, from a mathematical point of view, is connected to problem 3.2 (formulation in terms of characteristic lines). The method, first suggested in [4] (see also [12]) has been used in [6] [7] in order to fit some experimental data and it seems to “work” very well, even if until now the problem of the convergence of the generated solution is essentially open. In this section we limit ourselves to show the underlying idea, referring to [6] [7] and to a future paper for further details and the numerical results. First we introduce a discretization of the time r with a step At; in practice we must also discretize the space variable X; but this is only an implementation detail not essential to understand the method. Since we know the values of u at t = 0, we can suppose to have already computed at the level t, the approximated values ti(x, t) of u for x E R and then we want to compute the approximated values ti(x, t + At) at the successive level t + At. We know (see Section 2) that the cross-bridge population K changes with time because of two simultaneous operating factors: (1) the making and the breaking of bridges and (2) the interfilament motion. The discrete scheme follows essentially from considering these phenomena consecutively. To be more specific we first compute the variation of u from t to r + Af neglecting the interfilament motion. That is, recalling that the balance bridges equation is: (9.1) by discretizing with respect to time this equation we obtain a preliminary result which we shall denote by a*(~, f + At) and which is due only to the association or dissociation of bridges. The final result ri(x, t + At) will be next obtained by only translating ri*, that is by setting: ti(x, t + Ar) = ti*(x + 6, t + At), x E W,
(9.2)
where the shift number 6 is chosen as follows. Bearing in mind the rheological model, the contractile force computed from Li*(x, t + A.t), which we denote by l& (t + Ar), will be no longer in general equal to the series force FE(~), because this latter force is assumed to be in equilibrium with contractile force Fc~(t) correspondent to the function ti(x, t). In order to restore the equilibrium at the time t + A.t we must then search for a shift 6 such that, recalling from Section 2 the mathematical definitions of the contractile and series forces,
682
V. COMLVCIOLIand A. TORELLI
we have:
a*@+ 6, r +
At).&
= exp(Ln(r)
+ 6 - L ~(0)) - 1,
(9.3)
where the series element length L,(t) is known as result of the computation to the time f and the resting length LE(O) is supposed a known constant. Once computed 6, the length LE(~ + At) will be of course given by LE(r) -t 6. Therefore to find the number 6 needed to define ri at the level t + At, we solve the following nonlinear equation: F(6): = exp(Lsn(t)
+ 6 - LsE(O)) - 1 - /_lz li*(x f 6, f + At)xdr = 0.
(9.4)
Under the reasonable assumption that the approximated function ti*, as the function u, is nonnegative with bounded support in x, we have that for 6+ + x the function F(6) becomes positive while it becomes negative for 6-, - 3~. Under further suitable regularity assumptions on fi*, we have also: F’(6) = exp( LE(f) + 6 - LE(O)) - /:zxs(~~ =exp(LE(t)+d-LE(O))+
f 6, t + At)x dr
+xic*(x,r+Ar)~>O. I -z
It follows that there is a unique solution of (9.4) which can be computed for instance by means of Newton’s method. To conclude it is useful to observe that the equation (9.3) is strictly related to the characteristic equation (3.21) in problem 3.2, as it is easily seen passing from the logarithm to the exponential expression. Then we can after all interpret the numerical method, outlined above in an intuitive form, as an application of the idea which is the basis of the classical characteristic methods for a first order hyperbolic equation. Acknowledgements-The authors wish to thank Prof. Magenes for the useful discussions and suggestions and Dr. C. Reggiani and C. Poggesi (Istituto di Fisiologia Umana Universita Pavia) for their valuable help concerning the physiological model. REFERENCES 1. BROWDER F., Probhes non linhires, Lectures Notes University of Montreal (1966). 2. CAPELO A., COMINCIOLI V., MWELLI R., POGGESIC., RECGIASI C. & RICCIARDIL., Study and parameters identification of a rheological model for excised quiescent cardiac muscle, 1. Biomechanics 14. 1-11 (1981). 3. HUXLEYA. F., Muscle structure and theories of contraction, Prog. Biophys. 7, 255-318 (1957). 4. JULIAN F. J., Activation in a skeletal muscle contraction model with a modification for insect fibrillar muscle, Biophys. 1. 9, 547-570 (1969). 5. IMAGESES E., Applicazioni recenti della matematica alla Biologia e alla Medicina, Conferenza Istituto Lombard0 Scienze Lettere (1981). 6. MINELLIR., COMINCIOLIV., CAPELOA., POGGESIC., REGGIAW C., RICCIARDIL. & TERZI R., .Messa a punto di un modelio matematico the simuli il comportamento sia attivo (sistole) the passivo (diastole) dei miocardio di ratto, Progerro HCJSPI 5, 14%173 (1980). 7. MINELLI R., COMINCIOLIV., POGGESIC., REGGIANIC. & RICCIARDI L., Mathematical models for isolated resting and active cardiac
muscle,
Progerto HUSPI 6, 105-130 (1981).
Mathematical aspects of the cross-bridge mechanism in muscle contraction
683
8.POGGESIC., COMINCIOLI V., REGGIAVIC., RICCIARDIL. & MIXLLI R., A model of contracting cardiac muscle Third Meeting of the European Society of Biomechanics University of Nijmegen (21-23 January 1982). 9. POTTER A. J. B., An elementary verston of the Leray-Schauder Theorem, J. London Marh. Sot. 5. llU16 (1972). 10. SCHIERECKP., fnsrunfaneour Elarzicify and Acriue Srure of rhe Left Venrricle, Drukkerij Elinkwijk Buutrecht (1979). 11. SMART D. R., Fixed Poinr Theorem, Cambridge University Press (1974). 12. U'OSG A. Y. K., A model for excitation contraction coupling in frog cardiac muscle 1. Biomechanics 9, 319-332 (1976).
.Uctho&
MATHEMATICAL
& Appiicmom,
Vol. 7. No. 6. pp. 661-683.
1983.
ASPECTS OF THE CROSS-BRIDGE IN MUSCLE CONTRACTION*
0362-%5&‘83 SO3.00 + .OO @ 1983 Pergamon Press Ltd.
MECHANISM
VALERIANO COMINCIOLI and ALESSMDROTORELLI Istituto Matematico, Universitti di Pavia, Italy (Received 18 November 1981) Key words and phruses: Cross-bridge dynamics, physiological and mathematical model,nonlinearpartial differential equation, existence and uniqueness, asymptotic behaviour, numerical approximation.
1. INTRODUCTION
IN THISpaper we study from the mathematical
point of view a structural model which simulates cross-bridge dynamics. This dynamics is, according to the sliding filament theory of Huxley [3], the basis to explain the contracting mechanism of the muscle. The model studied here is a part of a more comprehensive model simulating the whole process of muscle contraction; it has been recently introduced in the framework of a collaboration between mathematicians and physiologists at the University of Pavia. The aim of the research is both the mathematical analysis of the model and its identification from suitable experimental data. In this paper we will confine ourselves to the mathematical aspects, referring to [5] [6] and [7] and to a paper in preparation [8] for a more extensive description of the physiological model and for the identification results hitherto obtained, together with a convenient literature. From the mathematical point of view we deal with an initial value problem for a nonlinear and nonlocal partial differential equation of first order which seems of new type. The plan of the paper is the following: in Section 2 we give a short description of the physiological background of the problem; in Section 3 we give the mathematical formulation of the problem and we introduce some preliminary considerations; Section 4 contains a local existence and uniqueness result; in Section 5 we prove some a priori estimates for the solutions; in Section 6 we prove a global existence theorem; in Section 7 a global uniqueness theorem is proved; in Section 8 we obtain some results on the asymptotic behaviour of the solution; in Section 9 we consider the numerical approach and in particular we outline the algorithm used in [6] and [7] in the numerical identification process.
2. THE MODEL
The repeating unit of muscle structure is the sarcomere, which consists of an array of interdigitating filaments: the thick (or myosin) and the thin (or actin) filaments. The thick filaments are tied together through their centers and several thin filaments, anchored at the Z-lines, lie adjacent to each thick filament (Fig. 1). Work supported by CNR of Italy through the Istituto di Analisi Numerica of Pavia and G.N.A.F.A. HUSPI Project. l
661
and by the
662
V. CO.MINCIOLI and A. TORELLI
According to Huxley’s theory [3] the generation of muscular force is thought to result from interactions between these filaments. Under the influence of the intracellular calcium concentration [Ca’-] which in turn depends on the time course of the action potential, there is a formation of links or cross-bridges which (once formed) act like springs. An attached cross-bridge is characterized by the distance x between the equilibrium position of the myosin head and reactive site. Let n(x, t) denote the cross-bridge density at time t (number of cross-bridges per unit of cross-bridge length in one half-sarcomere) and N the density of myosin heads (per halfsarcomere) which may form cross-bridges at time t.
attached Cross-bridge actin r
pj$IkzJq
Z-line
-7 Fig.
1.
Schematic
organization
of a sarcomere.
The number N can depend a priori on x, t and also on the length a of the half-sarcomere. If we suppose the cross-bridges to behave as linear elastic bonds with stiffness k, the force developed at time t by all the interacting cross-bridges in the half sarcomere is given by: F,(t) = k
+=n(x, t)x dr. i -2.z
(2.1)
Then to simulate the force generating process we must first describe the dynamics of the cross-bridges population. This dynamics results from the balance of the formation and breakage of bridges; assuming a first order kinetics we have: y
where F and the material cross-bridges made by the
= F(s, x, t)
(N(s, x, t) - n(x, t)) - G(s,x, t) n(x, t),
(2.2)
G are the attachment and resp. the detachment rate functions and d/dt denotes derivative, i.e. the time derivative with respect to a frame moving with the density distribution. As usual in fluid dynamics, conversion to a fixed frame is formula: d _=d++ dt at
where dx “=dt is the velocity of the filaments sliding, which is also equal to the velocity of half-sarcomere
Mathematical aspects of the cross-bridge mechanism in muscle contraction
shortening,
663
that is: dx
ds
(2.3)
v=dt=dr.
It must be emphasized that since the sarcomere shortening is an effect of the formation of cross-bridges, the velocity u is a function of n. In order to specify this dependence we must first incorporate into the model the effects of the elastic properties which are also present in a cardiac muscle. 1
A
A LCEW
i
__
CE
+_
Lk) SE
LSEi L )
1
1
I Fig. 2. Hill’s three component model for cardiac muscle.
To do this we can represent the muscle by a rheological model composed by the following three basic elements (assembled as in Fig. 2 (Hill’s three component model)): the parallel elastic element PE to account for the elasticity of muscle at rest. the series elastic element SE and the contractile element CE, which is freely extensible at rest, but when activated can shorten and develop tension. The contractile element CE is identified with the half sarcomere and then it is described as previously by the sliding filament theory. For simplicity in the model we have neglected the viscous forces which are however present in the cardiac muscle (for a study of such forces in resting myocardium see [2]). The stress-strain characteristics of both SE and PE are expressed by exponential laws. Then, denoting by FPE, FSE the forces supported by the PE and SE, respectively, we can write: &E(t)
=~&xp(~S&SE(~)
-LOSE)
FPE(~) =ap~(exp(bp&(t)
--LOPE)
- 1); - 1);
(2.4) (2.5)
where L(f) represents the muscle length at time t and L OPEthe muscle length corresponding to the zero tension; the &n(t) and LOSEhave an analogous meaning. The muscle tension P(t) is given at any time t by: P(t) = Furthermore
FPE(~) + FsE($
(2.6)
at any instant of time we must have FsE(~) = FcE(~); &E(t)
+ &E(t)
= L(t);
(2.7) (2.8)
V. COMINCIOLI and A. TORELLI
664
where FCE(f) is the force developed by the contracting element (that is the force F,(t) given in (2.1)) and Lc-(t) is the contractile element length (that is the half-sarcomere length s). There are two different important experimental situations: (1) the isometric contraction in which L(f) is kept constant and P(r) is observed; (2) the isotonic contraction in which P(f) is assigned and the muscle length L(r) is computed. In this paper we shall consider the first situation (isometric contraction) into account (2.3) and (2.8), we have:
do
u=x=z=
do -= d&E(t) dt
(0
-- d&E dt
for which taking
(2.9)
.
On the other hand from (2.4) and (2.7) we obtain: dLsE (t) -=&$log dt
( 1+ 3)
=+-$log(l
+L$)),
(2.10)
and then from the definition of FcE(~): u= -Idlog bsE dt
1 +k %E
+xn(x,r)xdx
i -=
. 1
(2.11)
Summarizing, in an isometric situation, the function n(x, t), is the solution for f > 0 and x E R of the following partial differential equation: $
+ u 2 = F(s, x, t) (N(s, x, t) - n(x, r)) - G(s, x, t) n(x, t),
with u given as in (2.11). Assuming initially (i.e. at t = 0) a resting situation, condition:
(2.12)
we can associate to (2.12) the initial
n(x, 0) = 0,
(2.13)
so that to find the cross-bridges distribution n(x, t) we must solve the initial value problem (2.12) (2.13). To conclude it remains to define the functions F, G and N. For what concerns the number N of possible cross-bridges at time t, in this paper we will suppose such number to be a constant, that is independent of x, t and also of s. Obviously it is the independence of s which more simplifies the mathematical problem, but it seems to be allowable at least in the isometric situation (see the identification results in [7]). For the rate functions F, G many different definitions have been proposed in the literature (see for a review e.g. [lo]). In this paper following Huxley [3] ( see also [4]), we will suppose F, G of the form: F = r(t) f(x)
G = g(x)>
(2.14)
where y(t) describes the time course of the intracellular calcium concentration upon stimulation. Such function is of course a physiologic datum, but it can be simulated by means of suitable mathematical models of the intracellular calcium kinetics (e.g. [6], [7]). The qualitative behavior in a twitch following a single stimulus is like this: y(t) rises from zero to a peak soon and then it decays back to zero.
665
Mathematical aspects of the cross-bridge mechanism in muscle contraction
The functions f(x), g(x) in Huxley [3] are of the following very simple form: ifx
f = fix/k g = w/k
O
f = 0, g = gdk
x>h
where h is the largest displacement at which myosin can become attached to actin. Only the three parameters fi, gl, g2 are needed to define f and g and this fact seems the main motivation for the particular choice (2.15). In fact the functions f(x), g(x) must be identified from the experimental findings. A qualitative indication is that f(x) is a nonnegative function with bounded support in the semiaxis x 5 0 and g(x) is a nonnegative function which rapidly increases for x c 0 and x > h. In the present paper the assumptions on f and g will be very general (see Section 3). To simplify the notations and without any loss of generality, in the following we set the various constants present in the previous formulation of the model, that is the constants like k, asE,bsE,Nt.. . , equal to 1. Furthermore we shall use the more “mathematical” notation U(X, t), to indicate the unknown function n(x, t). 3. MATHEMATICAL
MODEL,
PRELIMINARY
CONSIDERATIONS
(a) In order to study the problem of Section 2, we introduce some hypotheses We assume that
on y, f, g.
f, g E Ck(Wv YE Ck([O, +%
(3.1)
the support off is a compact set of W,
(3.2)
where k is a given integer number 2 1. Let us now consider the following precise formulation
of the problem of Section 2:
Problem 3.1. Given T > 0, we look for a function U(X,t) verifying:
x [0,T]),
(3.3)
u(x, 0) = 0, vx E R,
(3.4)
Vt E [0, T], u(x, t) has compact support inx,
(3.5)
u E Ck(R
xu(x, t) dx > -1, Vt E [0,T],
(3-e) (3.7)
= r(t)f(x>11-
u(x, 01 -g(x) 4x7 t>.
(b) Let u(x, t) be a solution of problem 3.1 and z(t) = -log
( I 1 +
)
x4x, t) dx
R
fk
0 = Y(Of(X>
+
g(x);
(3.8)
9
’
P-9)
V. COMINCIOLI and A. TORELLI
666
it follows that the equation (3.7) can be written in the following way: u,(x, r) + z’(t) u,(x, t) = :/(r)f(.r) - H(x, t) u(x, t); this
means that the function u(x. t) verifies the following relation lines)) (Z(I) + y, r), y being a real parameter, ; u(z(r) + Y, 4 = Y#f(Z(~)
+ Y> - H(‘$t)
(3.10)
on the ((characteristic
+ y, f> U(Z(f) + y, t>.
(3.11)
Since (3.11) is an ordinary linear differential equation with respect to the unknown function u(z(t) + y, t), it follows, recalling also (3.4), u+(t) +y,t)
= [Y(J)~(Z(S)
The previous considerations justify the introduction CO([O, Tl), r E [O, 7.1, x, Y E R): V,cV, 0 = [ v(s)f(z(s)
+Y,
+Y> exp(+(z(r)
r) dr) ds.
of the following operators
+ Y) exp( -i’H(z(r)
+ Y, r) dr) ds,
(3.12) (where z E (3.13) (3.14)
V,(x, t) = vz(x - z(t), t). We have also proved the following: LEMMA
3.1. Let z be an arbitrary function belonging to C’([O, T]). Then V,(x, t) is the solution of the problem (3.4) and (3.10). Let also: (3.15) supp,( V,) = support of V,(‘y, t) with respect to y at time t,
(3.16)
supp,( Uz) = support of U, (x, t) with respect to x at time t.
(3.17)
Thanks to (3.2) there exist Nr > 0 and A$> 0 such that: supp(f)
c [-NIT
Nzl.
(3.18)
Using the definitions of V, and U,, it follows easily that: LEMMA 3.2.
For every z E C?([O, T]), we have that: SUPP~(VZ) c suppt(~z)
[-iv1
C [-NI
-
ll4[O.l]~
N2
+
IIZll[o.r]17
(3.19)
21141~0,rJ
(3.20)
- 2//~11[o.r]rN2 +
Let us now consider the following new problem (which later will be proved to be equivalent to problem 3.1); Problem
3.2. Given T > 0, we look for a function z(f) verifying:
z E Ck([O, Tl),
(3.21)
xl/,(x, t) du > -1, Vr E [0, T], IR
(3.22)
Mathematical aspects of the cross-bridge mechanism in muscle contraction
667
(3.23) Remark 3.1. We note that, by lemma 3.2, there exists the integral in (3.22).
The following theorem states that problems 3.1 and 3.2 are equivalent: THEOREM
3.1. Under the hypotheses (3.1) and (3.2) we have that (i) if LI(X, f) is a solution of problem 3.1, then z(r), defined by (3.8), is a solution of problem 3.2; (ii) conversely if z(r) is a solution of problem 3.2, then: (3.24)
u(x, t) = Uz(x, f), is a solution of problem 3.1.
Proof. (i) If u(x, r) is a solution of problem 3.1 and if z(t) is defined by (3.8), then (3.21) holds. By lemma 3.1 and (3.7), it follows that u = U.. Then the relations (3.22) and (3.23) easily can be proved. (ii) Conversely if z(t) is a solution of problem 3.2 and u(x, t) is given by (3.24), then (3.3), (3.4), (3.5) and (3.6) are an easy consequence of lemma 3.1 and the definition of Uz. The relation (3.6) follows from (3.22) and (3.24). The relation (3.7) is a consequence of (3.23) and lemma 3.1. n 4. EXISTENCE
AND
UNIQUENESS
FOR
T SMALL
(a) Let A4 > 0 and Z(T, M) = {z E CO([O,z-1): llz Il[f)r]S M}.
(4.1)
Put also: (Lz)(t) = -log(l
+ //J;(,x,
t) dx),
(4.2) (4.3)
Then we have: LEMMA
4.1. For every M > 0, there exist c > 0 and To > 0 (depending only on M, y,f. g) such
that:
s CT(T), t/zE Z(T, M), VT E [O,z-o]. llLIIl[O,TI Proof
If we put x = z(t) + y, it follows that: xUz(x,t)k= iw
iR
(G)+~)K(y,r)dy,
hence: I~n~~(x,i)dri~~(M+lyl)lVl(y,r)Idy.
(4.4)
V. COMNCIOLI and A. TOEELLI
668
If we put N = max{ Ni, Nz, M},
(4.5)
then it follows (by the definition of V, and by lemma 3.2): xU,(x,t)cLx
SN*~r(T)exp(fT(T)
+g),
where: f= max{]f(x)I,x
E R1,
(4.6)
g = max{]g(x)l, 1x1 G2N).
(4.7)
By the mean value theorem we can conclude the proof of the present lemma.
n
In a similar way we can prove the following: LEMMA4.2. For every M > 0, there exist cl > 0 and TI > 0 (depending such that II LZl
-
~Z211[0,71 c c&T
-
z2]]lo$(T),
only on M, y, f, g),
vri, ~2 E Z(T, M), VT E [O, 7-11
(4.8)
(b) Thanks to lemmas 4.1 and 4.2, it follows that: LEMMA4.3. For every M > 0, there exists T > 0 such that: L(-W,
W)
c W-9 M)
L is a contraction on Z( T, M)
(4.9) (4.10)
and then there exists one and only one z E Z(T, M) such that z = Lz. Lemma 4.3 gives us the following uniqueness result in small: THEOREM4.1. Under the hypotheses (3.1) and (3.2), there exists T > 0 such that problem 3.2 (or equivalently problem 3.1) has at most one solution. Remark 4.1. The existence of a fixed point for the operator L is not yet an existence result for problem 3.2 (or equivalently problem 3.1), since we have proved only that z E C’([O, T]). To this end we will prove the following lemma. LEMMA4.4. If z E C’([O, T]) verifies the relation z = Lz, then there exists T’ E] 0, T] such that z E Ck([O, T’]). Proof. If z = Lz, then we have (taking x = y + z(r)):
that is: exp(-z(t))
- ~(0 /n V,(Y, t) dr - /n~Vz(~, t) - I = 0.
669
Mathematical aspects of the cross-bridge mechanism in muscle contraction
Let us now introduce the following function: G(r, w) = exp(--w)
- w Ix Vdy, 4 dy - _/2yVz(~, f) dy - 1.
Then we have that: Vt E [O, z-1.
G(t, r(t)) = 0, It is easy to verify that the functions iR
Vz Cv, 9 dY, j-R~KC~, f> dy,
belong to Ck([O, T]): this means that also G E Ck. But we have G(0, 0) = 0,
G,(O, 0) = -1.
Since z(f) is implicitly defined by the relation G(r, w) = 0, we can conclude Ck([O, T’]), where T’ > 0 may be smaller than T. n We note that (3.22) is a consequence following local existence result:
THEOREM 4.2.
(or equivalently
of lemma 4.1. We have completely
that z E
proved the
Under the hypotheses (3.1) and (3.2), there exists T > 0 such that problem 3.2 problem 3.1) has at least one solution.
5. SUPPLEMENTARY
PROPERTIES
OF THE SOLUTION
(a) Having in mind to prove a global existence and uniqueness theorem, assume that: f(x) 3 0, g(x) 2 0,
Under the hypotheses arbitrary in C’([O, T]):
(5.1) (5.2)
(3.1), (3.2), (5.1) and (5.2), we have that (z being
0 s U,(x, t) s 1,
Proof. By the positivity of y(t) and f(x) V,(x, t) 3 0. We have also that:
U*(Y + z(r), 9 = vXr7 9 = [P(Y,
vx E R,
Vr E [O, + q.
r(t) 2 0,
PROPOSITION 5.1.
Vr E [0, T].
vx E R,
(5.3)
and using the definition of U,, we obtain that
r) exp (- /MY, I
r) + q(Y, r)l dr) Q,
where: P(Y, 9 = r(t)f(z(t)
from now on we
+y),
4(Y? 4 =dz(t)>
+r)-
(5.4)
V. COMINCIOLIand A. TORELLI
670
Integrating by parts the relation (5.4), it follows that: u,(y + z(r), r) = [ exp (- jhY3
r> +4(Y*
5
-
I j0
dy, 4 exp (-
Recalling (5.1), it follows that uZ(x, t) s 1.
=
61 MY>
d> dr) ds.
(where z E C’([O, T])):
1 exp(-49)
r> + dyt
n
(b) We now introduce the following operator (F,)(t)
4) dr)]k
X[H(x, 0 Ur(x, 0 -
+ IR~~(x, 4 dr i R
~(4 f(x)]
dr.
(5.5)
We remark that, by proposition 5.1, the operator F, is meaningful. We now introduce the following new problem (which will be again equivalent to problem 3.1): Problem 5.1. Given T > 0 we look for a function z(t) verifying:
z E Ck([O, T]),
(5.6)
z(0) = 0,
(5.7)
z’(t) = (F,)(t).
(5.8)
We have also that problem 5.1 is equivalent to problems 3.1 and 3.2, as stated by the following: THEOREM
5.1. Under the hypotheses (3.1), (3.2), (5.1) and (5.2), we have that: (i) if u(x, t) is a solution of problem 3.1, then z(f), defined by (3.8), is a solution of problem 5.1; (ii) conversely if z(t) is a solution of problem 5.1, then u(x, t) = UL(x, t), is a solution of problem 3.1. Proof. (i) The relations (5.6) and (5.7) are obvious. Multiplying (3.7) by x and integrating on Iw, we have that:
1 + J, xu(x, f) dx + JR u(x, t) dx xu,(x, t) dr = ax[:Q) Iw I 1 + J-aXU(X,t) dX
f(x) - H(x, t) u(x,41dr.
But if r(t) is defined by (3.8), it follows that: 1 + Jaxu(x, t) dx = exp( -z(f)) i and then: xu((x, r) cLr = -exp( -z(r)) z’(t). iw By theorem 3.1 we have that u = U, and then: -z’(t)
[ exp(--z(t))
which implies (5.8).
+ L Uz(x, r) d]
= 6.4 y(r>f (4
- H(x, 4 4x,
01 dr,
Mathematical aspects of the cross-bridge mechanism in muscle contraction
(ii) The relations (3.3)-(3-j)
671
are obvious. By lemma 3.1 we have also:
UI(X,t) + z’(t) ux(x, r) = r(t)f(x) We now prove the relation (3.6). By contradiction,
- H(x, r) u(x, r).
(5.9)
assume that:
X=[~E[0,T]:~*u(*,t)dr~-1}c0. Put now T,J = min X. Obviously we have that To E 10, T]. Put now i(t)=-log
( 1+ I,XU(X,f)d+
tE[O,To[.
(5.10)
since lim Z?(t)= + m, r-Toif we prove that f = z, we get a contradiction. Multiplying by x and integrating on W the relation (5.9), we have that: --k’(t) exp(-i(t)) + z'(t)i,u(x,i) dr = jRx[y(r)f(x)- H(x,t)u(x,r)] dx. Recalling that u = iJ, and comparing with (5.8), it follows that: z’(t) exp( -z(t))
= z(t) exp( -z(f)),
Vf E [0, To[
and then j(t) = z(t),
Vt E [O, To[.
So we get a contradiction
and (3.6) is proved. Since we have also proved that f(t) = z(t) in H [0, T], the relations (5.9) and (5.10) imply (3.7). Theorem 5.1 is completely proved.
Remark 5.1. It is easy to verify that the equivalence
stated in theorem 5.1 remains valid, for Tsmall enough, without the hypotheses (5.1) and (5.2) (that is these hypotheses are necessary only in the case in which we want to prove the global equivalence). (c) Let Nr be the number introduced
in (3.18). Let also M and K be two numbers verifying: M>O,
(5.11)
Mz-Nr,
(5.12) where the function r(t) is defined in (4.3). Then we have: 5.1. Under the hypotheses a solution of problem 3.1, then LEMMA
(3.1), (3.2), (5.1) and (5.2) we have that: (i) if u(x, t) is
xu(x, r) dr > -1 + exp(-KM),
Vf E [0, T];
(5.13)
672
V. COMINCIOLI and A. TORELLI
(ii) equivalently
if z(t) is a solution of problem 3.2 (or 5.1) then: z(t) < KM,
'it E[O,Z-1,
(5.14)
where K and M verify (5.11) and (5.12). Proof. Thanks to theorem 3.1 we must prove only the part (i) of the theorem. To this end let u(x, t) be a solution of problem 3.1 and put z(r) = -log(l + JR xu(x, t) du. Suppose, by contradiction, that (5.13) does not hold. Then we can put (k = 0, 1, . . . ,K):
fk= min{t E [0, T]; z(f) 2 /CM}.
(5.15)
Then we have: 0 = f,, < rr < . . . < tK,
x@,
tk)
(5.16)
.?(fk) = /CM,
(5.17)
dx = -1 + exp( -kM).
(5.18)
By theorem 3.1 we have that u = U. and then: xu(x, tk) dr =
j/
[
6’y(s)f(z(s) - Z(fk) + X>exp (- /‘k+) - Z(tk) +x, r) dr)dJ]dx
that is, changing the order of the integrations
and taking
y = z(s) - .?(tk) + x, we have (5.19) where: 11 = /“-’ y(S) kf(Y)(Y
+ z(fk) - Z(S)) exp (- /“H(z(r)
- Z(S) + y, r) dr) dy ds,
(5.20)
+ z(fk) - z(s)) exp (- jRH(z(r)
- z(s) +y, r) dr) dy ds.
(5.21)
s
0
It = 1”
y(s) 1 f(y)(y
0
R
(k-1
Notice that: (5.22)
II 5 0, since ifs E [0, fk_t] it follows that z(&) - z(s) 2 M 3 N1.We have also that:
(5.23)
z, = z,- + IF, where:
G =
1” y(s) j f(y)(y % I&,
+
z(fk)
-
z(s)) exp (- j”H(r(r) I
- z(s) +y, r) dr) dy ds.
673
Mathematical aspects of the cross-bridge mechanism in muscle contraction
It is easy to verify that: 1; so, I; 2 -N,[UQ
(5.24)
- Uhc-,)I
h_f(4
dx.
(5.25)
Thanks to (5.18), (5.19), (5.22), (5.23), (5.24) and (5.25), we have that:
N~[r(h)- r(h-L)]
Ia_
f(x) dx 3 1 - exp(-kM)
and adding with respect to k, it follows: 1 1 - exp( -M)
Comparing (5.12) and (5.26) we get a contradiction. In the special case in which supp(f) c [O, + m[
(5.26)
(5.27)
we can improve the result of lemma 5.1, as stated by: LEMMA 5.2.
Under the same hypotheses of lemma 5.1, if (5.27) holds, then: iw
xu(x, 1) dx 3 0,
Vf E [O, Z-J,
(5.28)
or equivalently: z(r) S 0,
vt E [O, z-1,
(5.29)
where U(X,t) is a solution of problem 3.1 and z(t) is a solution of problem 3.2 (or 5.1). Proof. By theorem 3.1 it is sufficient to prove the relation (5.29). By contradiction that: i = max{z(f), t E [0, T]} > 0.
assume (5.30)
Let also f be such that z(f) = i. Put now u = U,. Then we have (changing the order of the integrations and taking y = z(s) - z(r) + x): /a-WY f) b = l’v(s)
/a (Y + z(f) -
dS))f(y) exp(- ~‘~(44 - 4)
+Y, 4 dr) 4v hj
that is:
Recalling that u = U,, by theorem 3.1 we obtain that z(t) G 0. But this result and (5.30) give us a contradiction. (d) From now on we assume that: 3~ < 4 such that lim g(x) exp( - E]XI) = 0. x-=
(5.31)
V. COMINCIOLI and A. TORELLI
674
Put now (Nt and Nz verifying (3.18) and ,u 2 0): (5.32)
g(u) = max{g(x), x E [ -Nt - 2u, Nz. + 2,u]}, + Ni)[:/+
B(u) =f(N: where T(t) and fare
T(T)g(,u)] + 2f;r(7’)&4
u’,
(5.33)
defined in (4.3) and (4.6) and ‘/ = max{y(t): r E [0, T]}.
(5.34)
W
Then we have: LEMMA5.3. Under the hypotheses
(3.1), (3.2), (5.1) and (5.2), we have that (for every z E
(5.35)
where the symbol /Iz]~~Q~ is defined in (3.15). Proof.
Let I be the integral in (3.35). Then we have: (5.36)
jzj =GI, + I?, where
(5.37) (5.38)
Using proposition
5.1, it follows that: (5.39)
It s $( N; + N:). By the lemma 3.2 and by the definition of U,, we have that: Zt -’
fl(0
(5.40)
(N: + Nt + 211~11fo.rj) g(]lz Il[o.~l)>.
The relations (5.36), (5.39) and (5.40) imply (5.35). Using lemma 5.1 and 5.2, we have that there exists (~2 0 such that: (5.41)
z(t) < Ly,vr E [O, 7-l and then, by the hypothesis (5.31), it follows that there exists p > 0 such that: exp(-u)[B(m)
+
B(U)]
s
p,
Vp e [O, + x[.
a
(5.42)
Then we have: LEMMA 5.4. Under the same hypothesis
of lemma 5.1, if (5.31) holds and if /3 verify (5.42),
then we have that: I2
xu(x, t) dr S -1 + exp(@),
Vf E [0, T],
(5.43)
675
Mathematical aspects of the cross-bridge mechanism in muscle contraction
or equivalently: Vr E [O, T],
z(t) 2 -pr,
(5.44)
where U(X,t) is a solution of problem 3.1 and z(t) is a solution of problem 3.2 (or 5.1). Proofi By theorem 3.1 it is sufficient to prove the relation (5.44). Put now: i(t) = min(z(s), s E [0, t]}.
(5.45)
i E C”“([O, T]), i(0) = 0, i(t) 50, i(r) s Z(f),
(5.46)
It follows that:
i(r) is a nonincreasing function, Ii:(r)] +2’(f)/,
WE [O, z-1,
(5.47) (5.48)
where f:(r) is the right derivative of i evaluated in the point 1. Thanks to (5.41) and (5.46), we have that: II~I~o.~I
hence, by (5.8), proposition
s
mada,
-441,
(5.49)
5.1 and lemma 5.3, Iz’0) I s exd4OP(cr)
+ W-40)1,
(5.50)
and then, by (5.48), I ii(f)
I c exp(z(f) [ B(a) + B( -z(t))].
(5.51)
Now we will prove that: Ii:(r)
Indeed if f(r)
[ d p,
Vf E [O, z-1.
(5.52)
then z:(r) =0 and then (5.52) holds. If i(f) = z(f), then we have (by
IW>I s expHWIP(4
+ W(r) I)1GP.
Since the case i(f) > z(f) is impossible, the relation (5.52) holds. But this implies that i(f) > -,& and then, by (5.46), we obtain the relation (5.44). n
6. GLOBAL Now
we introduce
EXISTENCE
the following fixed point theorem ([9]. see also [l] and [ll]):
6.1. (Browder-Potter). Let X be a normed space, E a convex and closed subset of X, F: [0, l] x E+Xsuch that:
THEOREM
F is continuous, F([O, l]
x E)
is a relatively compact subset of X,
F(A, z) f: z, Vz E aE, VA E [0,11,
(6.1) (6.2) (6.3)
676
V. COM~CIOLI and A. TORELLI
F({O} x L3E)c E.
(6.4)
Then there exists z E E, such that z = F( 1, z). The previous fixed point theorem allows us to prove the following global existence result: 6.2. Under the hypotheses (3.1), (3.2), (5.1), (5.2) and (5.31), for every T> 0 we have that problem 5.1 (or equivalently problems 3.1 and 3.2) has at least one solution.
THEORE,Ci
Proof. With the notations of theorem 6.1, we put: X = C?([O, T]) (with the usual norm), E = {z E X:[/Z[/[O,I~ G max{a, PT + l}}, where cuand /3 have been introduced in (5.41) and (5.42). Let also F:[O, l] x E+ in the following way:
Xbe defined
where (for fixed A) the function z-F_! IS ’ defined in the same way as the function z-+ F, in (5.5), but replacing f by Afand g by & (notice that we must modify in the same way also the definition of H and UJ Let us now verify that F satisfies the conditions (6.1)-(6.4) of theorem 6.1. The function F is obviously continuous and then (6.1) holds. Suitably adapting lemma 5.3, we have that
/lF(A, .z)ll G ATexp(&))B(&), V(A, z) E [0, l] x E,
(6.5)
]]F’(A,z)]] 6 Aexp(&) B(&), V(A, z) E [0, l] x E,
(6.6)
where & = max{Ly,PT + 1) and F’(A, z) is the derivative of the function I--, (F(i, z))(r) (notice that if z E C’([O, T]), then F(A, z) E C’([O, T])). Th e relations (6.5) and (6.6) imply (6.2). Suitably adapting lemmas 5.1 and 5.4, we can verify that if z = F(A, z), with A E [0, 11, then: ]]z]]< maxim, P T + 11,
(6.7)
which means that also condition (6.3) holds. Finally also condition (6.4) is fulfilled because F(0, z) = 0, Vz E C’([O, TJ). Thanks to theorem 6.1, there exists z E E such that F(1, z) = z, or equivalently z’ = F, and z(O) = 0, that is the relations (5.7) and (5.8) are verified. By 11). So we have proved that problem 5.1 the definition (5.5) of F, it follows that z E C kf'([O, has at least one solution. Recalling theorems 3.1 and 5.1, we can complete the proof of the present theorem. n Remark6.1. During the proof of theorem 6.2, we obtained a supplementary that is a solution of problem 5.1 (or 3.2) belongs to Ck”([O, 11).
7. GLOBAL
result of regularity:
UNIQUENESS
In the present section we assume that Zi(i = 1,2) are fixed solutions of problem 5.1. Under the hypotheses (3.1), (3.2), (5.1) and (5.2) we have that:
677
Mathematical aspects of the cross-bridge mechanism in muscle contraction
LE~MMA 7.1. There exists cl > 0, depending only on zl, zz, ‘/, f, g, and T, such that (for all y E R and t E [0, T]): IV&Y, f) - v&Y, t) / s 4Iz1 - ~?11[0.1],
where the notation 11Ijio,t~has been introduced Proof.
(7.1)
in (3.15).
We have that Z = IVJ_Y, 0 - v*,o,, 0 I s
11
+
(7.2)
12,
where Zl=[r(S)exp(
12= i,‘ds)fMs)
--I,’ Z.&G)
+Y> -f(zz(s)
+Y, r> dt ) IfhW
+Y)I exp ( - ~‘H(zdt) +y, 4 dr) -
+YW,
exp ( - jrH(z2(t)
+y, 4 dt)/dr.
Thanks to (3.1) and (3.2) we can put: ft =
max{If’(x)l,xE
(7.3)
R}.
Then recalling that H(x, t) 5 0 see (5.1) and (5.2)), it follows that (by the mean value theorem): (7.4)
- ~2lI[o,rj.
ZI +JV-)llz~ We have also (always because H(x, r) 5 0): zz S I,’ r(s)f(ds)
+ Y) [ jj+d$
+
Y, 3 - f&zz(t) + y, t) ldr] d.s
and then 12
c
121
+
(7.5)
1221
where: I21=
~ks)f(zi(s)
122 = 1’ r($f(zds) 0
+Y> [I,‘ifhW + Y> [ /‘k(ld$ 0
+Y> -f(rdr) + Y> - g(zz(@
+y)lvdr]
b,
+ y) /dr] h.
It is easy to prove that: 121
~ffJ2(T)lh
-
f211[0,+
(7.6)
where f is defined in (4.6). Let now: p = max{llzlllro.
2-b II~~II~o,TI~~
We remark that if y E [ - N1 - p, NZ + p], thenf(z2(s) all y FE[ - N1 - p, N2 + p]. This implies that: 122
+fgJV’)Tllz~
-
(7.7)
+ y) = 0. This implies that 112= 0, for
z2i1[o,r],
(7.8)
V. COMIXIOLI and A. TORELLI
678
where: gr = max{ig’(x)/, -Ni - 2p s x G N2 + 2p}. The relations (7.2), (7.4), (7,5), (7.6) and (7.8) imply the lemma. LEMMA7.2. If h E&!(W),
(7.9) W
then there exists c2 > 0, depending only on ZI, z?, ;J, f, g, T and
h, such that (t E [0, T]):
II a
h(x)[Uz,(x,
t) - uzz(x, 01 &I ~&I
(7.10)
- z&I.~,
Proof. Taking x = zi(t) + y(i = 1,2) and recalling (3.14), we have that:
I = ahb
+ z~(t))v,,O,,
i
t) dy - jRhb
+ zz(~))Vrb’,
(7.11)
t) dy,
where I is the integral in the left side of (7.10). Then: (7.12)
111c 11 + 12, where 11 = R Ih(y + z&)> IIUY, I
r) - V~,(Y, t) I c-b,
12 = R Ih(y + ~20)) - h(y + z,(t)) / V,,(Y, 0 dy. I
Recalling lemmas 3.2 and 7.1 and the position (7.7), we have that: 11 s clL(N, + N2 + 4~) lIzi - ~21110~1,
(7.13)
Z2+I-(W$N1
(7.14)
+N+4p)k
-~2111~1,
where f is defined in (4.6) and: fi=max{lh(x)I The relations (7.11)-(7.14)
+(h’(x)(,xE[-N1-2p,N2+2p]}.
prove the lemma.
LEMMA7.3. There exists c3 > 0, depending only on zl, ZZ, y, IZi(0 - Z?(f) I s c3 IlIz, - z2lh
f, g and
T, such that: (7.15)
r1 dt
Proof. Since z1 and z2 are solutions of problem 5.1, it follows that:
(7.16)
z = Izr(r) - zz(t) 1c II + II, where (recalling also lemma 5.3 and the relation (7.7)): 1 exp(-zl(r))
+ JR Ur,(x, 4 dx
II w
xH(x,
4[ Uz,(x, t) - Liz+,
41 ch
dr,
Mathematical aspects of the cross-bridge mechanism in muscle contraction
679
z2=B(~)lbiexp(-z,(r))+11 )&-exp(-z_(r))+lJ )Jdi. R u (x, l-
,
Z[
R
u qx.r (
the function B(u) being defined in (5.33). Since U:,(x, t) 2 0, it follows that: 11 c c~exp(p) the constant c4 being conveniently 12 6
B W2p
i,’
1 I exp(
--zI(~)
[IIz~
(7.17)
- ~2il[o,d dt,
chosen by lemma 7.2. On the other hand we have that -
exp(--z2(4)
I +
11 [uzI(xT R
4
-
U&,
t>l
do
!I dr,
then: I2 s B(PP~[
e PII~1 - 410,~l +
CSIIZI - .~llp,~] dt,
(7.18)
where c5 is conveniently chosen by lemma 7.2. The relations (7.16)-(7.18) imply the lemma. LEMMA 7.4.
n
We have that: lIZi - ~2ll[O,r]s
(7.19)
c3
where c3 is the constant of lemma 7.3. Proof. It is a consequence
of lemma 7.3 and of the fact that the function Iorki - z211[o,rjdt
is non decreasing. By Gronwall lemma, we can derive the following global uniqueness result: THEOREM 7.1. Under the hypotheses (3.1), (3.2), (5.1) and (5.2), for every T> 0. we have that problem 5.1 (or equivalently problems 3.1 and 3.2) has at most one solution.
8. ASYlMPTOTIC
BEHAVIOUR
In the present section we give only some preliminary results on the asymptotic behaviour of the solution. A forecoming paper will be dedicated to this subject. Let us now introduce the following new hypotheses:
Then we have:
3pi > 0 such that lim v(t) exp(pir) = 0, I--r+=
(8.1)
Zlp2> 0 such that g(x) 2 p2, t/x E W.
(8.2)
680
V. COMLUCIOLIand A. TORELLI
LEIM,MA 8.1. Under the hypotheses (3.1), (3.2), (5.1), (5.2), (5.31) and (8.1), if c(f) is the solution of problem 5.1 (or equivalently 3.2) then there exist c, L 0 (i = 1,2) such that: -cir S z(t) S c2, vr E [O, + p[. Proof. It is an easy consequence
(8.3)
of lemmas 5.1 and 5.4 and of the hypothesis 8.1.
n
Put now: p = min (pi,p:}.
(8.4)
Then we have: LEMMA 8.2. Under the same hypotheses of lemma 8.1, if (8.2) also holds, we have that if U(X,t) is the solution of problem 3.1, then for every p E [O,p[, there exists c > 0 such that: Iu(x, r)j Cc exp( -pt),
V(X, t) E W X [0, + m[.
(8.5)
Proof. If z(t) is the solution of problem 3.2 (or 5.1), then u = U, and so (using the positivity of y and f and the hypotheses (8.1) and (8.2)):
u(x, t)
‘exp( -pis) i0
exp( -p2(t - s)) ds,
which implies (8.5). We conclude this section with the following:
THEOREM 8.1. Under the hypotheses (3.1), (3.2), (5.1), (5.2), (5.31), (8.1) and (8.2), if z(t) is the solution of problem 3.2 (or 5.1) with T = + ~0, then we have that: lim z(t) = 0 I--t+=
(8.6)
Proof. If U(X, r) is the solution of problem 3.1 (with T = + =J), then we have that u = U,.
Put now Z(t) =
ix+,
t) dr =
ixU,(x, t) dx.
By lemmas 3.2, 8.1 and 8.2, we have that: [Z(f) 1s c
exp
(
- ii z t
N2+2qtfc2
/XIdx. )i -Ni-Z(c~r+c?)
This implies that ,lim= I(f) =O. But since z(t) = -log (1 + I(t)), we obtain (8.6).
n
Remark 8.1. More precisely, we could prove that for every p E [O,p[, there exists c > 0, such that [z(t)/ =Sc exp( -pf).
Mathematical aspects of the cross-bridge mechanism in muscle contraction 9. NUMERICAL
681
APPROXIMATION
In order that the mathematical model, studied in the previous sections from the analytical point of view, can be used in practice, we must be able to compute the solution u and then the required muscle tension P(t). To this end there are, indeed, several possibilities available, in correspondence to the different formulations of the problem. We could, for instance, discretize problem 3.1, that is a partial differential equation of first order by using some classical schemes based on finite differences or finite elements methods. A second type of approach could consist in solving numerically problem 5.1, that is an initial value problem for an integrodifferential equation of Volterra type and also this is a classical numerical task. There is another numerical approach which arises directly from the mechanical frame of the model and which, from a mathematical point of view, is connected to problem 3.2 (formulation in terms of characteristic lines). The method, first suggested in [4] (see also [12]) has been used in [6] [7] in order to fit some experimental data and it seems to “work” very well, even if until now the problem of the convergence of the generated solution is essentially open. In this section we limit ourselves to show the underlying idea, referring to [6] [7] and to a future paper for further details and the numerical results. First we introduce a discretization of the time r with a step At; in practice we must also discretize the space variable X; but this is only an implementation detail not essential to understand the method. Since we know the values of u at t = 0, we can suppose to have already computed at the level t, the approximated values ti(x, t) of u for x E R and then we want to compute the approximated values ti(x, t + At) at the successive level t + At. We know (see Section 2) that the cross-bridge population K changes with time because of two simultaneous operating factors: (1) the making and the breaking of bridges and (2) the interfilament motion. The discrete scheme follows essentially from considering these phenomena consecutively. To be more specific we first compute the variation of u from t to r + Af neglecting the interfilament motion. That is, recalling that the balance bridges equation is: (9.1) by discretizing with respect to time this equation we obtain a preliminary result which we shall denote by a*(~, f + At) and which is due only to the association or dissociation of bridges. The final result ri(x, t + At) will be next obtained by only translating ri*, that is by setting: ti(x, t + Ar) = ti*(x + 6, t + At), x E W,
(9.2)
where the shift number 6 is chosen as follows. Bearing in mind the rheological model, the contractile force computed from Li*(x, t + A.t), which we denote by l& (t + Ar), will be no longer in general equal to the series force FE(~), because this latter force is assumed to be in equilibrium with contractile force Fc~(t) correspondent to the function ti(x, t). In order to restore the equilibrium at the time t + A.t we must then search for a shift 6 such that, recalling from Section 2 the mathematical definitions of the contractile and series forces,
682
V. COMLVCIOLIand A. TORELLI
we have:
a*@+ 6, r +
At).&
= exp(Ln(r)
+ 6 - L ~(0)) - 1,
(9.3)
where the series element length L,(t) is known as result of the computation to the time f and the resting length LE(O) is supposed a known constant. Once computed 6, the length LE(~ + At) will be of course given by LE(r) -t 6. Therefore to find the number 6 needed to define ri at the level t + At, we solve the following nonlinear equation: F(6): = exp(Lsn(t)
+ 6 - LsE(O)) - 1 - /_lz li*(x f 6, f + At)xdr = 0.
(9.4)
Under the reasonable assumption that the approximated function ti*, as the function u, is nonnegative with bounded support in x, we have that for 6+ + x the function F(6) becomes positive while it becomes negative for 6-, - 3~. Under further suitable regularity assumptions on fi*, we have also: F’(6) = exp( LE(f) + 6 - LE(O)) - /:zxs(~~ =exp(LE(t)+d-LE(O))+
f 6, t + At)x dr
+xic*(x,r+Ar)~>O. I -z
It follows that there is a unique solution of (9.4) which can be computed for instance by means of Newton’s method. To conclude it is useful to observe that the equation (9.3) is strictly related to the characteristic equation (3.21) in problem 3.2, as it is easily seen passing from the logarithm to the exponential expression. Then we can after all interpret the numerical method, outlined above in an intuitive form, as an application of the idea which is the basis of the classical characteristic methods for a first order hyperbolic equation. Acknowledgements-The authors wish to thank Prof. Magenes for the useful discussions and suggestions and Dr. C. Reggiani and C. Poggesi (Istituto di Fisiologia Umana Universita Pavia) for their valuable help concerning the physiological model. REFERENCES 1. BROWDER F., Probhes non linhires, Lectures Notes University of Montreal (1966). 2. CAPELO A., COMINCIOLI V., MWELLI R., POGGESIC., RECGIASI C. & RICCIARDIL., Study and parameters identification of a rheological model for excised quiescent cardiac muscle, 1. Biomechanics 14. 1-11 (1981). 3. HUXLEYA. F., Muscle structure and theories of contraction, Prog. Biophys. 7, 255-318 (1957). 4. JULIAN F. J., Activation in a skeletal muscle contraction model with a modification for insect fibrillar muscle, Biophys. 1. 9, 547-570 (1969). 5. IMAGESES E., Applicazioni recenti della matematica alla Biologia e alla Medicina, Conferenza Istituto Lombard0 Scienze Lettere (1981). 6. MINELLIR., COMINCIOLIV., CAPELOA., POGGESIC., REGGIAW C., RICCIARDIL. & TERZI R., .Messa a punto di un modelio matematico the simuli il comportamento sia attivo (sistole) the passivo (diastole) dei miocardio di ratto, Progerro HCJSPI 5, 14%173 (1980). 7. MINELLI R., COMINCIOLIV., POGGESIC., REGGIANIC. & RICCIARDI L., Mathematical models for isolated resting and active cardiac
muscle,
Progerto HUSPI 6, 105-130 (1981).
Mathematical aspects of the cross-bridge mechanism in muscle contraction
683
8.POGGESIC., COMINCIOLI V., REGGIAVIC., RICCIARDIL. & MIXLLI R., A model of contracting cardiac muscle Third Meeting of the European Society of Biomechanics University of Nijmegen (21-23 January 1982). 9. POTTER A. J. B., An elementary verston of the Leray-Schauder Theorem, J. London Marh. Sot. 5. llU16 (1972). 10. SCHIERECKP., fnsrunfaneour Elarzicify and Acriue Srure of rhe Left Venrricle, Drukkerij Elinkwijk Buutrecht (1979). 11. SMART D. R., Fixed Poinr Theorem, Cambridge University Press (1974). 12. U'OSG A. Y. K., A model for excitation contraction coupling in frog cardiac muscle 1. Biomechanics 9, 319-332 (1976).