A free boundary problem in acrosomal elongation

Math/ Compur. Modelling,Vol. 13, No. I, pp. 49-54, 1990 Printed in Great Britain. All rights reserved

0895-7177/90 $3.00 + 0.00 Copyright 0 1990 Pergamon Press plc

A FREE BOUNDARY PROBLEM IN ACROSOMAL ELONGATION E. Istituto

Matematico

COMPARINI

“U. Dini”, Universitl 50 134 Firenze,

R. Istituto

di Matematica,

Facolta

di Firenze, Italia

viale Morgagni,

67A,

RICCI

di Ingegneria, Universita di Ancona, 60100 Ancona, Italia

Via delle Brecce Bianche,

(Received April 1989; received for publication June 1989) Communicated

by S. A. Levin

Abstract-A free boundary problem, arising from a model for the study of the fertilization of echinoderms, known as acrosomal elongation, is considered. We prove existence and uniqueness of the solution of the mathematical problem, and we study the behaviour of the free boundary.

1.

INTRODUCTION

In this paper we study the free boundary model proposed in Ref. [l] describing a phenomenon observed in the study of the fertilization of echinoderms, known as acrosomal elongation. During fertilization the sperm penetrates the eggs by extending an actin-filled tube, the acrosome. This elongation is caused by actin monomers transported from the base of this tube to the actin filaments at the growing tip, where they are added to the filament by polymerization. A moving boundary diffusion-reaction model for this process was first suggested in Ref. [2]. This model was improved in Ref. [1], allowing the polymerization to be reversible and introducing a convective term into the transport equation. The acrosome can be represented as a laterally insulated cylindrical rod of length s(t), so the process is assumed to be one-dimensional. The monomers, whose concentration will be indicated by u(x, t), move along the acrosome pushed by a fluid filling the acrosome at the speed of elongation. At x = 0 material is entering the rod from a reservoir in which the concentration is assumed constant. At the tip x = s(t) the material arrives by diffusion and is added to the growing filaments, pushing forward the free boundary. The boundary conditions on x = s(t) then follow from mass balance and from a kinetic condition describing the elongation mechanism. The problem obtained is the following: u, - Du, = -Su,,

in D, = {(x, t):

u(O, t) = uo, -Du,(s(t),

t) =jJ(r),

i(f) = ~[k,uMt), f> -M

0 < x < s(t), 0 < t < T},

(1)

O
(2)

O
(3)

O
(4)

where D is the monomer diffusion coefficient, f is the number of actin filament ends per unit cross-section area, I is the length a filament extends when a monomer adds to it and k, and k, are constants related to the reaction between monomers and actin filaments which takes place at the extending tip. Equation (4) represents the polymerization kinetics, while equation (3) is the mass conservation at the free boundary. The quantity f/U(t) gives the rate of absorption of monomers at the tip, due to the polymerization. One would also expect the presence of a convective term 49

50

E. COMPARINIand R. RICCI

u@(t), t);(t) in order to fill the concentration jump across the advancing front. However, this quantity exactly matches the mass transported to the front by the fluid motion, represented by the convection term Su, in equation (1). The initial condition for the concentration u is omitted here. since we are interested in the case in which the free boundary starts from the origin, i.e. s(0) = 0. In order

to have a consistent

problem

we choose Uo>-.

(5) the boundary

value such that

k,

(6)

k,

The ratio k,/k, represents the critical monomer concentration, reached u > k,/k,, then filaments tend to grow: if u < k,/k, , filaments tend to shrink. Remark. condition

Inserting equation (4) in equation (3) we obtain which can be used in place of either equations

2) - hl,

-Du,(s(t), t> =f[k,uMt),

at equilibrium:

if

a different free boundary (3) or (4): O
(7)

No theoretical mathematical results are established in Ref. [l], but the authors give an asymptotic expansion of the free boundary based on singular perturbation techniques. Their computation provides a (quite precise) lower bound for the free boundary, independently of any consideration about “small” parameters. A mathematical proof for the existence of a solution of the problem can be found in Ref. [3]. The authors use a compactness argument to prove existence but no uniqueness result is obtained. In our paper we consider more regular data than those allowed in Ref. [3]. This enables us to prove uniqueness as well as existence by a contractive mapping technique. We restrict ourselves to the case s(O) = 0. When s(O) > 0, we can rely on general results about Stefan-like free boundary problems established in Ref. [4]. In fact the problem can be studied using a transformation similar to that introduced in Ref. [5] to study a problem concerning the swelling of a glassy polymer: s(0

m(x, t) = sX

45, t) dt.

(8)

The function m(x, t) represents the total mass of monomers between the point x and the free boundary, at time t. Obviously, we have m(s(t), t) = 0 and s”(t) = -A[k, m,(s(t), t) + k2] because of equation (4). Moreover, m solves the following parabolic equation in D,:

f ml - Dm,, + r + m, S = 0, ( ) and satisfies the boundary condition m,(O, t) = -u,,. If we consider an initial condition with s(O) > 0, and u(x, 0) = h(x) in 0 < x
2.

AN

AUXILIARY

(9)

the and the the

PROBLEM

To prove existence and uniqueness of problem (1)+4) we make use of a fixed-point argument on the free boundary. So we start solving a moving boundary problem for a given boundary x = r(t), and we define a mapping whose fixed point will be the solution of our problem.

A free boundary

problem

in acrosomal

elongation

51

Let us consider the following moving boundary problem: U, - DU,, = -XJ,,

in D,=((x,t):

U/(0, t) = uo 9 -DUx(r(t),

t) =f[k,

U@(t),

t) -

bl,

O
(10)

O
(11)

O
(12)

Here r(t) is a given C’ function such that r(O)=0

and

r(t)>O,

O
(13)

and O
O
(14)

where K and T are constants to be fixed in the following. The convective term in equation (10) can be easily eliminated using the new variables y =x -r(t) and o(y, t) = U(x, t), so that problem (10)-(12) transforms into the equivalent problem: o( - D& 0(-r(t),

= 0,

in d, = {(y, t): - r(t) < y -c 0,O < t < T},

t) = uo,

- DOJO, t) =f[k,

o(O, t) - k,],

(10’)

O
(11’)

O
(12’)

Notice that now the value of 0 is fixed on the moving boundary. Lemma 1. Problem (10’)-(12’) has a unique classical solution satisfying

k, -
(15)

and ;

[k, - k, u,] < oy < 0.

(16)

Existence and uniqueness of the solution has been proved in Ref. [6]. The proof of inequalities (15) follows from applying the maximum principle in the region 8,. In fact if a minimum is attained on the boundary y = 0 for, say, t =?, then oY(O,$ < 0 (a strict inequality because of the Vyborny-Friedman principle [7]) and, consequently, o(O,$ > k2/k,. Since 0 = u. > k,/k, on the moving boundary, we have 0 > k,/k, everywhere. If there were a maximum on y = 0, then, by the same argument, we would get 0 < k2/k,, contradicting the fact that min 0 > k,/k, . Consequently, the maximum is attained on the moving boundary y = -r(t), and inequalities (15) are satisfied. To prove inequalities (16), we consider the problem solved by z = oY in the region 6,: z, - Dz?, = 0, z,(-r(t), - ;

t) = ;Z(-r(t),

[k ,u,-k,]
in 6,,

(17) t),

(18)

(19)

Again z cannot have negative minima or positive maxima on y = - r (t ) because of equation (18). Then inequalities (16) follow from inequalities (19) and the maximum principle.

E. COMPARINIand R. RICCI

52

3.

EXISTENCE

AND

UNIQUENESS

Now we can prove that the problem has a unique solution for any time. This is done using the Banach theorem on contractive mappings. This gives, as a first step, the existence of a local solution, which means that there exists a time T, such that problem (10)+12) has a (unique) solution up to time T,. Theorem 1. There exists a unique local solution of problem (1)<4).

Set X = {r(t) E C’(0, T): r > 0,O < i: < K}, and, recalling equation (4) define the transformation F by (Y-r)(t) = s(t) =

’A[k, U@(r),

r) -k,]

dr,

s0

(20)

with r(t) E X and U the solution of problem (10)-(12). We prove that 9 is a contractive mapping in the C’ norm, if the maximal time T is small enough. First, the definition of 9 and the estimates (15) imply 0-c;


s > 0.

(21)

Then, choosing K = ;I[k, u, - IQ], we have immediately that Y maps X into itself. To prove the contractive character of 5, consider two functions s, and s2, associated with distinct boundaries r, and rz, and let U, and U, be the corresponding solutions of problem (10)-(12). We have

J,(t) - h(t) = Ak,[VI (r, Oh t) - U*(r2(t),f>l = Ak,Lo,(0, t) - &(O,t)l,

(22)

where the functions oj are defined using the transformation y, = x - r,(t) introduced in Section 2. In the following we indicate by y the space variable in the common domain of y, and y,. Let us define c(t) =inf{r,(t), r,(t)}, F(t) =sup(r,(t), r2(t)} and V = 0, - oz in (0, T) x (-r(t), 0). The function V(y, t) satisfies the heat equation in (0, T) x (-r(t), 0). Moreover,

where D = oi when f = ri, i = 1,2, and D VO, f) = - fk, V,(O, t).

From the above boundary condition it follows that V has neither a negative minimum nor a positive maximum on y = 0, therefore the maximum principle implies that

Using this inequality, for y = 0, one obtains from equation (22):

(23)

A free boundary problem in acrosomal elongation

53

Finally, choosing the time T in the definition of the space X to be strictly less than T,, where D T’=Ak,j-(k,u,-k,)’

the mapping 5 is a contractive mapping from X into X, and there exists a unique fixed point for it, which means that problem (l)-(4) as a unique solution up to time T. 0 The existence of the solution after T is proved in the following way. Theorem 2. The unique solution of problem (1)+4) exists for any time.

The proof of the theorem follows from solving the following problem: in D 7.7.= ((x7 t):

u, - Du,, = - 3u,, uk

Tl = ul(x),

UP, t) = %J, -Du,(s(t),

t) =f[hWt),

t) -

bl,

0
< s(t), T < t < T’},

(24)

o

(25)

T< t < T’,

(26)

T
(27)

-CT’,

s(T) = ~1,

(28)

where U,(x) is the trace of the solution of problem (1)44) at time T and s, is the location of the free boundary at time T. Problem (24)-(28) can be solved locally using a variety of techniques (see the discussion in Section 1). In particular, we can repeat the above proof of existence. Since the time of existence T depends only on the estimates of the maximum of U and U, in the auxiliary problem, it is sufficient to show that these estimates still hold true. This is a trivial consequence of estimates (15) and (16), so that problem (24)<28) has a (unique) solution up to 2T and so on. q

4. CONVEXITY

Here we prove that i(t) is a monotone decreasing function. To do this we start with a lemma. Lemma 2. Let (u, , s, ) and (ZQ,sz) be two solutions of problem (l)-(4) corresponding to data s, (0) = 0 and s*(O) > 0, respectively, with

u,(x, 0) G uo,

0 < x < s*(O).

(29)

Then for any i such that s,(t) < s,(t) in 0 < t < f, we have J,(r) > &(f),

Octci.

(30)

Proof. We again use the transformation y = x - s(t) and u”(y, t) = u(x, t) and we define ti(y, t) = ii,(y, t) - fi,(y, t). Then from inequalities (15) and (29) it follows that 2(-s,(t), t) > 0. Moreover, from equation (7), we have -D&(0, t) =fk,ti(O, t), so li cannot have a negative minimum on y = 0. It follows that

& (i, (t) -

&(t)) = i(O, t) > 0,

o
0

(31)

I

The proof of the convexity of the free boundary can now be deduced from a simple shifting argument. Theorem 3. Let (u, s) be the solution of problem (l)-(4). a(t) < 0.

Then (32)

54

E. COMPARINI and R. RICCI

Let 12be defined as in Lemma 2 and (u’_,, SK,) be defined by fi-,(Y, t) = a(r, t + c),

s-,(t) = s(t + 6).

(33)

Then (u’_,, s_,) is the solution of the same problem but with the time origin shifted to t = -6. That is, s_,(t) = s(t + t) > s(t) for any t > 0 and C_, < u0 for t = 0 so that Lemma 2 applies, and we have i(t) - s(t + 6) = i(t) - s_,(t) > 0. Since t is arbitrary, the function S is non-increasing. Moreover, it can be proved, adapting the regularization argument of Ref. [8] to our problem, that $ exists and so we have X(t) < 0. Finally, suppose that a i exists such that X(?)= 0. Then consider the function w(y, t) = u”,(y, t). Differentiating the condition G( -s(t), t) = uO, and recalling that fiY< 0, we have w( -s(t), t) < 0. Moreover, ~(0, t) = (l/&)?(t) < 0. So the function w attains its maximum at the point (O,?)), and from the Vyborny-Friedman principle it follows that w,.(O,Y)> 0. On the other hand, differentiating equation (5) we have -Dw,(O, t) =fk, ~(0, t). Then the assumption ~(0~7) = 0 would imply w,(O,$ = 0, which is a contradiction. cl REFERENCES 1. A. S. Perelson and E. A. Coutsias, A moving boundary model of acrosomal elongation. J. math. Biol 23, 361-379 (1986). 2. L. G. Tilney and N. Kallenbach, Polymerization of actin. VI. The polarity of the actin in the acrosomal process and how it may be determined. J. cell. Biol. 81, 608423 (1979). 3. P. Colli and A Visintin, A free boundary problem of biological interest. Muthl Meth. uppl. Sci. 11, 79-93 (1989). 4. A. Fasano and M. Primicerio, General free boundary problems for the heat equation, I-III. J. math. Analysis Applic. 57, 696723 (1977); 58, 202-231 (1977); 59, 1-14 (1977). 5. A. Fasano, G. Meyer and M. Primicerio, On a problem in the polymer industry: theoretical and numerical investigation. SIAM JI appl. Math. 17, 945-960 (1986). 6. A. Fasano and M. Primicerio, La diffusione de1 calore in uno strato di spessore variabile in presenza di scambi termici non lineari con l’ambiente, I. Rc. Se&n. mat. Univ. Pudoua L, 269-330 (1973). 7. L. I. Rubinstein, The Srefan Problem. Trans. math. Monographs, Vol. 27. AMS, Providence, R. I. (1971). 8. D. G. Schaeffer, A new proof on the infinite differentiability of the free boundary in the Stefan problem. J. dls Eqns 20, 266269 (1976).