Model for bend propagation in flagella

J. theor. Biol. (1971) 31, l-24

Model for Bend Propagation in Flagella J. LUBLINER~ AND J. J. BLUM$ Polymer Department,

The Weizmann Institute of Science, Rehovot, Israel (Received 6 July 1970)

Previous suggestions that active bending processes in flagella is activated by passively propagated bending have been developed into a theoretical model which describes the effect of viscosity on the shape and propagation velocity of flagellar bends. Our model incorporates a first-order active bending process which is initiated when a critical level of passive bending is reached and a first-order unbending processwhich followsafter a prescribed time interval. The effects of the external viscosity and the internal mechanical properties of the flagellum are included in such a way that it is possible to predict the velocity of bend propagation and the shapes of the bending and unbending transitions for steady waveson an infinite flagellum. These predictions are compared with published data on the effect of viscosity on the velocity of bend propagation along flagella, and values for all the parameters of the flagellar model have heen estimated. 1. Introduction Recent work by Brokaw and his colleagues (Brokaw & Wright, 1963; Brokaw, 1965a) has shown that the planar undulatory bending waves propagating in seemingly sinusoidal fashion along flagella are composed of circular regions of bending separated by straight unbent sections. The implications of these findings led Brokaw (1966a) to develop a theoretical model of propagation which was based on the local passive propagation of bending just ahead of the transition point. As Brokaw observed, such a mechanism is comparable to other examples of wave propagation in non-linear systems, such as the nerve impulse. Brokaw developed his original model in terms of a contractile element which attained its minimum length immediately after activation. Subsequently Brokaw (1966b) extended the theory by introducing a contractile element which shortened at constant speed until the maximum permissible shortening was attained. This modification helped to clarify the interaction t On leave from the Department of Civil Engineering, University of California, Berkeley, California, U.S.A. $ On leave from the Department of Physiology and Pharmacology, Duke University, Durham, North Carolina, U.S.A. ‘T.B. 1 I

2

J.

LUBLINER

AND

J.

J.

BLUM

between the shape of the wave pattern and the velocity of bend propagation in Chaetopterus spermatozoa. An earlier theory by Machin (1958) also considered active bending to be initiated by passively propagated bending, but focused on the passive bending induced by the viscous resistance offered by the fluid to the motion of the flagellum. Brokaw (1966a) presented a qualitative argument indicating that the passive bending induced by viscous drag would generally be small compared to the bending propagated passively by internal shear near the transition region. Both types of passive bending must be considered in any complete model. In order to develop further our understanding of bend propagation in cilia and flagella, we have extended Brokaw’s model in several ways. In our model, active bending will be governed by a first-order rate process, which is initiated when a critical level of passive bending is attained. Unbending will also be governed by a first-order rate process, initiated at a predetermined time interval after the initiation of bending. The kinetics of these processes are developed in section 2(c). Brokaw (1966a) considered a simplified model of a flagellum, containing only two longitudinal elements. In section 2(~), we have considered a more general model, which can be matched, for instance, to the “9 + 2” pattern of flagellar fibrils. This more general model leads to the same equation [equation (9)] obtained by Brokaw (1966a), but the constants in the equation have been defined more generally and more rigorously. Since viscosity is one of the few easily variable external parameters that influence the wave parameters of flagella, we have included the effects of viscosity in the mathematical formulation, as developed in section 2(B). The resulting mathematical model is solved in section 2(D) to obtain the velocity of bend propagation as a function of the various parameters introduced in setting up the model. In section 3(~), these predicted velocities are compared with published data for the effects of viscosity on bend propagation velocity in flagella. In section 2(~), the equations are solved to give the shape of the bent regions as a function of viscosity, and the results are indicated in Fig. 3. Finally, in section 3(~), we have estimated values for the various parameters introduced into the model for flagellar bend propagation. In this paper we restrict ourselves to steady waves (i.e. waves moving without change of speed or shape) in an infinite flagellum. Observations by Sleigh (1968) show that in most cilia and flagella studied, the wave attains the steady state a short distance from the base and is absorbed without reflection at the free end. This restriction to steady waves on an infinite flagellum also allows us to use a sinusoidal approximation for the distribution of viscous bending moment along the flagellum. In reality, both the transient state near

MODEL

FOR

BEND

PROPAGATION

IN

FLAGELLA

3

the base of the flagellum, and end effects on the distribution of viscous forces on the flagellum, will be important influences which must be dealt with in subsequent work. A list of symbols and terms is given in the Appendix. 2. Analysis (A)

KINEMATICS

OF THE FLAGELLUM

MODEL

We consider our flagellum model to consist of n longitudinal contractile elements or fibers and a shear-resistant connecting “web”; the whole flagellum bends in a plane, the local radius of curvature being p - l/p. Let y be the local coordinate in the plane of bending perpendicular to the (bent) longitudinal axis, measured from the centroid of all the contractile elements and positive away from the centre of curvature. Let &i denote the longitudinal strain at the centroid of the ith element and yi its location; then the strain at any point of the element, in accordance with the usual “plane sections remain plane” hypothesis [Popov (1968); assumed for the contractile element, not the whole flagellum] is given by 8 = Ei+/L((y-yi)* (1) If the axial force in the element is Fi and its cross-sectional area is Ai, then, accordingly, the axial stress is 0 = Fi/Ai+Ep(y-yi),

(2)

where E is Young’s modulus for the contractile material. If, lastly, the moment of inertia of the element is Ii, then the resultant bending moment on the whole section is M = ~ Fiyi+ Eel f: Ii. i=l

(3)

i=l

To evaluate ei we shall assume the web to be subject to an average shear y. Neglecting distortion, we then have (see Fig. 1) where y,, gives the location vanishes), and

(4) 6 = v (Yi - Yo>, of the neutral axis (the axis where the strain v = p +

aylas,

s being the coordinate along the (bent) longitudinal axis (see Fig. 1). Suppose, now, that the first m adjacent elements contract by an amount (per unit length) c,, then i= l,...m Ei = - Cg + Fi/EAi, Ei = FilEA,, i=m+l,...n. (5)

J.

LUBLINER

AND

J. J. BLUM

/ Centrold conlrocttle elements

of

FIG. 1. Axial strain in a flagellar model. The left side of the figure shows a cross-section of the flagellum. y! is the distance from a line perpendicular to the y axis and passing through the centroid to the ith contractile element. For further details, see text.

Combining

equations (4), (5) and the equilibrium

condition

$I1 Fi = O

yields VYO= co A,/A where A = $llAi

t e area of all the elements) and A, = f ch i=l

the contracted elements). Hence Fi = EAi [c,(l -AJA)+Vyi], Fi = EAi [ - CO A,IA + Vyi], Furthermore i

i=l

F,yi = E(V i

i=l

Ai (the area of

i = l,...m i = m+l, . . . n.

AiYi2-YcAcCo),

(6)

(7)

where j’, = -l/AC

5 Aiyi. i=l

Now, since CFivi is the moment exerted by the contractile elements on the web, the shear force in the web is S = a/as 2 FiYi. i=l

MODEL

FOR

BEND

PROPAGATION

IN

FLAGELLA

5

The shear y is considered to be made up of an elastic part, S/GA, (where G is the shear modulus of the web and A, its effective cross sectional area), and a part due to active sliding, say yS: Y = S/(W) + us(8) If, now, the viscous bending moment is M,, then the equilibrium condition M+M,=O, together with equations (3), (7) and (8), yields, upon elimination of c Fiyi, I the following equation : 1 a2M, 2a, M,, - 2 + j& -a;i - < P - ;, 9 (9) where a, = E 2 Aiyi’, i=l

a0 = E i

i=l

Ii,

h = i=l5 A;Yi2/AcYcv h2a2 = GA, c = co-h ays/as. Equation (9) is identical with the equation obtained by Brokaw (1966~) on the basis of a two-element model. The symbols ao, aI, a2 and h (r in Brokaw’s work) were introduced by Brokaw, and we retain them for the sake of continuity with his work, although it must be stressed that they are both more generally and more rigorously defined here. Our c corresponds to dL/ds used by Brokaw to denote local contraction per unit length; in the present model, however, c is the effective contraction, which may include local contraction, sliding, or both. It should also be pointed out that this model permits one to include the contributions of the central pair of elements and, when present, the accessory tubules found in many sperm and flagella. (B)

THE VISCOUS

BENDING

MOMENT

We now proceed to derive an explicit expression for M, in terms of the motion of the flagellum. We first note that if U is the longitudinal velocity, I/ the transverse velocity, and 6 the angle with respect to a given reference line at any point on that flagellum, then (see Fig. 2)

ae av at=z

J.

LUBLINER

AND O+xdt

J. J.

BLUM

de

FIG. 2. Longitudinal and transverse velocity components in a moving segment of a flagellum. After time dt the segment ds units long moves so that the points 1 and 2 are translated to points 1’ and 2’, respectively.

Since, however,

we have

a2v -=-

ah as2 at’

(10)

The transverse viscous force per unit length, W, can be represented as w = fcqv,

(11) where 4 is the viscosity of the fluid and ICa geometric factor; the product rcrl is usually designated as the transverse drag coefficient. In general, equation (11) is only an approximation more or less valid for a long slender body (Burgers 1938). rc then has the form 4n (12) K = iqE/d)+cv where L’ is the length and d the diameter of the body, and 6 is a number of order one which depends on the shape of the motion: for a cylinder moving perpendicularly to its axis and for one rotating about a transverse axis, respectively, we have S = 0.50 and 6 = O-80, while for sinusoidal motion (with L’ as the wavelength), 6 = -0.50 (Hancock, 1953). Since a typical flagellum length might be 2Ou and the diameter about 0.25~~ we have ln(2L’/d) N 5, so that the sensitivity to the values taken by 6 is not great, and will be even less for longer flagella.

MODEL

FOR

The equilibrium

BEND

PROPAGATION

IN

FLAGELLA

7

condition is a2M 2 = w, as2 it with equations (10) and (1 l), we obtain

and, on combining

s

ap

ad = xv Iii,

(13)

which we may now use to eliminate M, from equation (9). Before doing this we shall introduce the steady-wave assumption, namely, that p and c, which are function of s and t, depend only on the combination x E s- vt, where v is the (constant) wave speed along the flagellum. We therefore have a/as = d/dx, a/at = - v d/dx, and the resulting ordinary differential equation is $0

(14)

where # = djp/dxj. For a given c(x), equation (14) can be solved by means of a Fourier transform. The solution then depends on the complex roots of a quintic equation and its computation is cumbersome. However, since equation (1 l), and hence equation (13), are approximations, it seems unnecessary to insist on an exact representation of the distribution of the viscous bending moment. Instead we shall make use of Brokaw’s (1966a) notion that the viscous bending moment is distributed sinusoidally with a wave length L, and we shall approximate

Instead of equation (14) we then obtain the simpler equation p-21vp’l/c? /f” = E/c? c, where 2A

=

Icq al (1+2adaJ

a2

[

(L/27rj4+ ;2

(15)

L/2R)2], 2

_- 1+%/al

2a,/h2a2 ’ E = ha2/2a,.

When the viscosity (and hence A) vanishes, equation (15) reduces to the exact equation obtained by Brokaw (1966~). It remains to find an explicit expression for c. Brokaw initially (1966~) assumed an abrupt transition from the relaxed to the fully contracted state and later (Brokaw, 1966b) assumed a transition region with a constant rate of contraction. We shall treat this question in greater generality.

J. LUBLINER (C)

KINETICS

AND

I.

J.

BLUM

OF THE CONTRACTION

We have pointed out that the present model includes both the slidingfilament and local-contraction models, whose relative merits have recently been reviewed by Brokaw (1968). Present experimental evidence does not permit an unequivocal decision to be made about the nature of flagellar contraction, although a sliding-filament system appears most likely (Horridge, 1965; Satir, 1965). We shall assume here that the kinetics of the effective contraction c is governed by a two-state model; that is, that the molecules making up the contractile system may exist in a contracted and an uncontracted state, and that c is proportional to the fraction of molecules in the contracted state. The forward rate of contraction is assumed to be proportional to the fraction of molecules in the uncontracted state (i.e. a first-order process). The rate of relaxation is, similarly, assumed to be proportional to the fraction of molecules in the contracted state. If c equals C, when all the molecules are in the contracted state, then the model is described by dc/dt = k (Z, - c) - k’c,

(16)

where k and k’ are the rate constants. This formalism, however, does not yet incorporate two processes which must occur in a flagellum, namely, initiation of contraction when a critical amount of bending has been achieved (Machin, 1958) and termination of the contractile process to allow for unbending through relaxation. The first of these processes is the mechanochemical coupling, and is necessarily nonlinear-otherwise the governing equation would be parabolic and no wave propagation would be possible (see also Machin, 1963). We shall assume this coupling in the simplest possible form: that k is a step function p-p*, where CL*is a given critical curvature. In a steady wave, then, the point at which p equals p* (the bending transition) travels with a uniform speed, and the equation becomes piecewise linear and thus amenable to classical methods of analysis. As regards the second process, it was pointed out by Brokaw (1968) that maintenance of equal propagation velocities by the bending and unbending points appears to require an internal interaction between bending and unbending, a mechanism for which is not yet apparent. For simplicity, we have chosen to assume that contraction terminates, i.e. k vanishes, a fixed time r after its initiation. The formalism we have adopted for the mechanochemical coupling can then be written as follows: &/dt = kH(p-p*)

[l-H(t-t*+r)]

(7,,,--c)-k’c,

(17)

MODEL

FOR

BEND

PROPAGATION

IN

FLAGELLA

9

where o,z
H(z) = l,z>O and t* (a function of s) is the time at which p equals ,u* (assuming @/at > 0). We assume that c = 0 for t < t*. If 11 > IL* for t* < t < t*+r, then equation (17) may be rewritten as

at/at = (k+k’) CC,-4, aclat = - k2,

t* < f < t*+7, t*+7 < t,

(18)

where

k _ cm= k+k’ cnv The solution to equation (18) is c(r-r*) = 0, -(k+k’)(t-t*) = c,[l-e 19

= c, [1 -e-(k+k’)q e-k’W*-r),

t-t*


O o.

(19)

Note that this solution includes, as a special case, the zero order process assumed by Brokaw (19666); namely, if k = k’ = 0 and c,,, = ol) such that

(k+k’)c, = R, then equation (19) becomes c(t-t*)

= 0,

= R(r-r*), = RT, (D)

THE SPEED

OF BEND

r-t*<0 0 < t-t* c T r-r* > 7.

(20)

PROPAGATION

We have found c as a function r-t*, as given by equation (19). In a steady wave, we may let r* = s/v (i.e. we take the origin of s at the bending transition point at t = 0); hence c is a function of x as previously defined:

x I s--or = -u(t-t*), and we may proceed to the solution of equation (14). For a functionf(x) define its Fourier transform as J(y) = 7 f(x) emirx dx; -00 the inverse is then

f(x) = & -y SW eiyxdy. m

we

10

J.

LUBLINER

AND

J. J. BLUM

Assuming sufficient conditions for the existence of all transforms, immediately find the solution of equation (15) as

we

(21)

where Z(y) = 7 ewiY5c(-t/u) -Co

d&

(22)

c(. . . .) being given by equation (19). Inserting equation (22) in equation (21) and interchanging orders of integration (assuming this to be permissible) we obtain ~(4

=

f

-7

K(x--5)

CC-5/d

d5,

(23)

m

where

K(x) can be broken up into two integrals as follows: (24)

where

.-___ a = a [Jl+(lua)2-Aua], ~~ b = a [Jl + (Ava)2 + ha].

These integrals are in standard form and can readily be evaluated to yield K(x)=~

[H(-x)

e’“+H(x)

eebx],

(25)

where H(x) is the Heaviside step function defined above [see equation (17)]. Equation (25) may be rewritten as follows:

K(x)=2 Jl +a(Aav)2 exp {-aJ1+(AaU)2

~x~-Aua2x}.

(26)

Before we insert this expression into equation (23), we note that c( - t/u) vanishes for c > 0. We may therefore rewrite equation (23) as P(X) = (da’) 7 K(C+x) 0

4514 dt.

(27)

MODEL

FOR

BEND

PROPAGATION

IN

II

FLAGELLA

Inserting, finally, the expression for c given by equation (19), we obtain %I

p(x) = 2&+(I”c?)zj

- {T [l-e-(k+k’)rlv] 0

exp [-aJl+(Aua)2Jx+rJ

-rZuc~~(~+~)]d~+[l-e-(~+~‘)~]~exp[-k’(S/u-r) ---at/~+(lua)~/xf+Lua”(x+S;; dt}.

(28)

We shall evaluate these integrals for general x later. For the present we note that, by definition, ~(0) = II*. Evaluation of the integrals for x = 0 yields the following expression : 1 -exp ( - [ha + J 1 + (AucQ2]a”r} -.1 _-_ 3L”tl+J1+(Iz”c()2 B Jl +(Auc+4 ~-__I-exp { -[(k+k’)/ua+l”cr+Jl +(ilua)2]auz} -____ 1

(k+ k’)/ua+~ua+~l + {I-exp

__~ +(Aua)’

-~___

exp {-[Aua+J1+(Iua)2]am}

[-(k+k’)z]}

k’/ua + Aua + Jl + (AuaJ2

>

,

(29)

where B =

e%

2a2p*

_

%I

2p*h (I+ 2a,/a,)’

We now introduce the following dimensionless variables: 1 = n/z, q = am, m = Aua(= Iq), r = k’z, s = (k+k’)z, p = auz [ha+Jl+(ha)2]

(30)

[= q(m+Jl+m2)].

Equation (29) can then be rewritten as 1 -e-Pes l-e-” 1 -__ 4 -= B &+m2 P+S --1 P

+

(1 -ems)

p+r

e-P

1*

(31)

Equation (29) is a transcendental equation from which the speed of bend propagation, Y, can be obtained as a function of the viscosity q, of the mechanical parameters or,, aI, at, h, L, K, and of the mechanochemical parameters c,,,, k, k’, z, ,u*. Through equation (31) the dimensionless wave speed CJ is a function of the dimensionless viscosity I and the dimensionless parameters B, r, s. (B can be interpreted as a measure of the contractility of the flagellum.)

12

J.

LUBLINER

AND

J. J. BLUM

For vanishing viscosity, equation (31) reduces to 1 -e-qes (1 -ewS) e-4 l-e-¶ - 4+s-. + --4+r. (32) 1 ----4 This equation is, of course, independent of the sinusoidal approximation. When e-‘, e-‘, and, by hypothesis, em4 are all < 1, the limiting value of the dimensionless velocity, q, at vanishing viscosity is q x s(B- 1). The solution of equation (31) can be achieved as follows: replace q by p/(m+&?) and rewrite as follows: 1 B=q

( P+r-p+s l

~s-~~~:-p

1-km2+m~1+mZ=B

l ) e -p-s}

= f(p; B, r, 4.

(33)

Upon squaring both sides and rearranging, we obtain (fb m = J2f(p;

B, r, s)--11

(34)

B, r, s)-1’

from which it follows that f= If(p; B, r, s)--ll/~ (35)

For given values of B, r, s, equation (35) gives a parametric representation q as a function of I, i.e. wave speed vs. viscosity. (E)

THE SHAPE

OF THE BENDING

of

WAVE

We proceed to the evaluation of the integrals in equation (28) and thus the achievement of closed-form expressions for p(x). We shall use the dimensionless parameters defined in equation (30) and, in addition, ____ w = auz (41 +(Iua)2-Iva) = p-2mq T=

222 vz

VT

u = k*. P

Equation (28) takes on different forms in three ranges. I. T < 0 (i.e. s > vt): ahead of the bending wave (no contraction). u = epT.

Here (36)

MODEL

FOR

BEND

PROPAGATION

IN

13

FLAGELLA

II. 0 < T < 1 [i.e. u(t -r) c s < ut]: the region of the bending wave where active contraction is occurring. We obtain e-wT-e-sT

fpe-wT

u=--

&I ____s-w J1+m* { w + ~-e-Pu-T) e-sT-e---P(l-T P + (l-e-“)

P+s e-P(lmT)

(37)

p+r 1’ provided w # s. If w = s, equation (37) becomes

(37a) It will be noted that the wave-speed equation, equation (31), is precisely the condition of continuity of equations (36) and (37) at T = 0. III. T > 1 [i.e. s < u(t--z]: the region behind the unbending transition with only relaxation taking place. Here we have 1 -eew -__-

&f

u=----

1+m*

-

eyw-ems

w

J---K

e-w(T-l)

S-W

-r(T-

e-r(T-l)-e-w(T-l) +(I

-.--

-em”)

+

w-r

e-.m-

1)

9

(38)

p+r

provided r # w # s. (Note that s > r for relaxation to occur). If w = s, we have u=~~{(l~-l)e-~~‘(T-l)+...},

(38~)

while, if w = r, u = $+5

I... + (l-e-‘)

(T-l

+ er:zy)}.

(386)

P+r

At the point T = 1 (the unbending both yield u = g!$

(1~~ W

transition),

equations (37) and (38)

_ ??!t:

+ E).

s-w

p+r

(39)

Equations (36) to (38) give the local curvature as a function of position. The local inclination with respect to a reference direction would be given by

0 (4 = f u (0 d5,

(40)

while the integrals f cos t?(t) d&

f sin O(5) d5

(41)

14

J.

LUBLINER

AND

J.

J. BLUM

would give the coordinates of the point x with respect to some reference system. However, it must be recalled that we are considering here a single wave propagating in an infinite flagellum. In actual cilia and flagella there are many waves, in which segments of approximately constant curvature are followed by more or less straight segments. The present theory, then, can pretend to an adequate description of reality only if this condition can be met, that is, if the exponential decay of the curvature ahead of and behind the wave, as predicted, respectively, by equations (36) and (38), is rapid. Furthermore, the actual shape of the wave is determined by the lengths and orientations of the straight segments [these give the reference points for the integrals (40) and (41)], and these are outside the scope of the present theory. In Fig. 3 we have plotted curves of u vs. T for several values of I, the other parameters being B = 3, r = 5, s = 10 (the reason for this choice will be clear from the next section). It is seen that when I N 0.02, the effect of viscosity on the wave shape is negligible. For I up to about 0.1, the maximum curvature remains virtually the same, but further increases in 1 decrease the maximum curvature and spread out the curved region, making it more sinusoidal. I

I

I

I

I

I

I

I

I

I

I

I

'

--a--_ 0

04

0.8

I.2

I.6

2.0

2.4

2.8

T

FIG. 3. The shape of the bent region as a function of viscosity. The ordinate is the ratio of the reciprocal of the radius of curvature to the reciprocal of the critical radius of curvature. The abscissa is the dimensionless “distance” from the point at which the active bending transition begins, as defined in the text. Curves were computed for several values of I (roughly the viscosity in poises) as follows: curve A, I = 0; curve B, I = 1.79 x 10e2; curve C, I= 5.00x 10e2; curve D, I = 145 x lOTa; curve E, I = 28.1 x lOma. The values of the parameters used for the computation were: B = 3, r = 5, s = 10.

MODEL

FOR

BEND

PROPAGATION

IN

15

FLAGELLA

3. Discussion (A)

DEPENDENCE

OF THE WAVE

SPEED ON VISCOSITY-COMPARISON EXPERIMENT

WITH

From equation (35) it is possible to plot q vs. 1 for assumed values of B, P and s, and examples of such curves are shown in Fig. 4. As expected, an increase in viscosity leads to a decrease in wave speed, with the curves convex down-

3t1og

1

FIG. 4. The effect of viscosity on velocity of bend propagation for infinite cilium. The parameters I and q are proportional to the viscosity and velocity, respectively, as defined by equation (30). They were computed by the use of equation (35). For convenience, the vertical positions of the curves have been adjusted by adding the constant b to log 4. Curve A, s = 10, B = 3, Y = 25. b = 1.2; curve B, s = 20, B = 4, Y = 2.5, b = 0.8; curve C, s = 4, B = 4, r = 2.5, b = 0.9; curve D, s = 3, B = 3, r = 1, b = 1.0; curve E, s = 10, B = 15, r = 2.5, b = 1.0.

ward, down to a limiting point at which dqldl = - co and which occurs at a value of q which is typically between 0.5 and 0.8. Comparison with experimental log ZJvs. log q obtained by Brokaw (19663) shows, in fact, that they are remarkably similar to the log q vs. log I curves predicted by the present theory. We have, therefore, attempted to find values of B, r and s which will lead to a good fit between theory and experiment. It must, however, be emphasized that the parameter values we shall compute below are necessarily rough approximations to the correct values. The reasons for this can be appreciated by considering the parameter r, alone. Since the experimental wave shape shows a rapid transition between the curved and the straight regions, we require that r (and hence s) be significantly greater than one. To limit further the range of choice of r requires use of information on the radius of curvature of the bent region, on wave amplitude and wave-length, all of which are beyond the scope of the present theory, since as mentioned

16

J.

LUBLINER

AND

J. J.

BLUM

earlier, prediction of all the parameters of wave shape demands a theory which utilizes information on beat frequency and accounts for chemical and mechanical interaction between adjacent waves. From preliminary results with such a theoretical treatment (Lubliner & Blum, to be published) it appears that the parameter values computed only on the basis of fitting wave velocity vs. viscosity data are within the order of magnitude (and typically within a factor of 2) of the values obtained when wave shape information is also utilized. We found that the choice B = 3, s = 10, r = 5, yielded curves (see Figs 5 and 6) which satisfactorily fit the data for spermatozoa of Lytechinus and Ciona over the whole range of viscosities studied, and Chaetopterus over a large fraction of the viscosity range, the systematic deviation being in the region of high viscosities (> 30 cP), where the behavior of this sperm was anomalous in other respects (Brokaw, 1966b). The plotting was done by choosing shift factors a and b, such that log,, 4 is equivalent to a+log,, 1 and log u is equivalent to b+Iog,, q. According to the definitions of I and q, we therefore have

and b = -log,,

ar.

It follows from the hypotheses underlying this work that r, aO, al, a2 (and hence a) are intrinsic properties of the flagellum and should be independent of the viscosity of the medium. On the other hand, the results of Brokaw (19666) and Holwill(l965) show that the wavelength decreases with viscosity. Consequently, the fact that the present theory satisfactorily agrees with experiment when a (hence K and L) is assumed independent of viscosity indicates that L is not to be taken as the wavelength of the motion but as a characteristic length of the flagellum model as described by the present theory* It is interesting to compare this finding with the theory of Machin (1958), in which the motion (not merely the distribution of the viscous bending moment) was assumed sinusoidal and in which there also appeared a characteristic length which is neither the wave length nor the length of the flagellum. In Fig. 6 we have also plotted wave speed vs. viscosity for the flagellum of Strigomonas oncopelti which we calculated following the method of Brokaw (1966b) from the data of Holwill(l965). (In calculating the adjusted viscosity we assumed that the methyl cellulose used by Holwill was comparable to the high-molecular-weight methyl cellulose used by Brokaw.) The wave speed of Strigomonas decreases much more sharply with viscosity than that of the spermatozoa previously considered, and the best theoretical fit was obtained

MODEL

FOR

BEND

PROPAGATION

IN

FLAGELLA

17

1.8-

log

qadj

and

D + :OJ 1

FIG. 5. Log-log plots of the propagation velocity, v (in units of pm/set) against the adjusted viscosity, fladl, for spermatozoa of Lyfechinus picfus in sea water (0 -0) and, after glycerination, with 2 mh+ATP (fJ-- - -0) and 0.2 mM-ATP (n-----A). Data taken from Brokaw (19666). The lines are theoretical curves computed from equation (31). For each theoretical curve the parameters s, Band r were 10,3 and, 5 respectively. The constants a and b chosen to obtain a good fit of the theoretical lines to the experimental points were a = 240, & = 1.70 for living sperm, a = 2.55, b = 1.45 for glycerinated sperm in 2 mw ATP, and n = 2.70, b = 1.06 for glycerinated sperm in 0.2 mwATP.

log Q,,,,; ond 2+ log 2 0

I

0

2” 11’1

015

05

I,,!

‘1

I.0 i”

.’

I.0

log qodj and

,*

1

I r,

I.5 1

I

2.0

2-o

a + log 1

FIG. 6. Log-log plots of propagation velocity, u (in units of pm/set) against adjusted vjscosity, qadl, for spermatozoa of Chuetopterus variopedutus (13 -O), Cionu intestinalis and for the flagellum of Strigomonas oncopelti( n ----A). (0 ---Cl), Data taken from Brokaw (1966b) and Holwill(l965). The lines are theoretical curves computed from equation (31). The parameters used to compute the curves and to place them on the graph were: for Chnetopterus, s = 10, B = 3, r = 5, a = 2.50, b = 1.57; for Cio~, s = 10, B = 3, r = 5, a = 240, b = 1.75; for Strigomonus, s = 10, B = 2.0, r = 1.5. The left ordinate and hottom abscissa refer to Chuetopterm and Cionu. The right ordinate and upper abscissa refer to Strigomonus. ? T.R.

18

3.

LUBLINER

AND

J.

J.

BLUM

with B = 2, s = 10, r = 1.5. The theoretical curve fits the experimental data reasonably well in the low viscosity range (up to 15 cP) but overestimates the wave speed in the medium viscosity range (-25 cP), and predicts no wave propagation beyond 38 cP. According to Holwill(1965), wave propagation in these flagella is smooth only up to a viscosity of 30 CP (corresponding to an adjusted viscosity of about 25 cP). At higher viscosities movement becomes erratic and the waves, which normally propagate from tip to base, sometimes die out when they reach the central portion of the flagellum. The transition from smoothwave propagation to erratic waves which maydie out corresponds to a violation of the basic premise of the present theory, i.e. steady wave propagation. Since nothing in the theoretical formulation predicts when the steady wave assumption will no longer apply it is remarkable that the present theory fails precisely in the viscosity range, where, experimentally, its basic assumptions become inapplicable. It should also be pointed out that at high viscosities, where the theoretical curve deviates from the experimental data for Chaetopterus (Fig. 6), these sperm also exhibit anomalies in their wave shape which indicate that a steady wave analysis may be inapplicable. Indeed, Holwill (1966) has already remarked on the similarity in form of beat between the flagellum of Strigowzonas at high viscosities and that of Chaetopterus at high viscosities. It is a common feature of all the theoretical curves used to fit the experimental data that they somewhat overestimate the effects of viscosity at low viscosities; that is, where I w 0.01 (corresponding to r] N I to 2 cP, i.e. water), the observed wave speed lies above the calculated value but below the value corresponding to zero viscosity. However, the discrepancy is small and we can confidently say that in this viscosity range the effects of viscosity, as regards both wave speed and (as was pointed out in the preceding section) wave shape, can be neglected for a single wave on an infinite flagellum. (ES) CALCULATION

OF FLAGELLAR

PARAMETERS

The theory which we have developed as embodied in equations (15) and (17) contains 11 independent parameters: the bending parameters, IZ~,CQ,z2, and h; the mechanochemical parameters p*, c,, r’, k, and k’; and the parameters K and L representing viscous effects. We now proceed to the evaluation of these parameters. The bending parameters, which are defined following equation (9), are obtainable, in principle, from a knowledge of the elastic and geometric properties of a flagellum. It is first necessary, however, to specify which elements offer the primary resistance to bending. This question has been discussed in detail by Holwill (1965, 1966). In view of the facts that motile sperm exist

MODEL

FOR

BEND

PROPAGATION

IN

FLAGELLA

19

without central fibrils and that glycerinated sperm and flagella, largely devoid of outer membranes, are reactivated by ATP, we shall assume that the nine outer doublets are the contractile elements and that they, plus the two central ones, offer elastic resistance to bending. The parameter h, comparable to r in Brokaw’s (1966~) two-element model, can then be evaluated purely from geometrical considerations provided that some assumption is made about which fibrils actively contract during a bending transition. In view of Brokaw’s computations (Brokaw, 1968; Brokaw & Benedict, 1968a,b) that each dynein may participate approximately once during a bending-unbending cycle, it seems likely that about half the doublets may be involved in a bending transition. Such a choice is further indicated by the observation that the fibrils on one-half of a cilium were out of focus while the fibers on the other half were in focus (Lansing & Lamy, 1961). We shall therefore assume that the fibers are distributed around a circle of 0.2 urn diameter, as shown, for example, in Fig. 41 of Sleigh (1962) (see also our Fig. l), with four fibrils (numbered 4, 5, 6 and 7 in Sleigh’s figure) being actively contracted. Direct measurement from such a figure then shows that h = 0.15 urn. To evaluate a0 and ai requires knowledge of Young’s modulus. Rikmenspoel (1966) estimated that E for bull sperm was less than 1 dyne urne2 and more recently he estimated a value of about 0.4 dyne urne2 (Rikmenspoel, 1970). These values, however, are derived from a dynamic treatment of the flagellum as a passive rod driven from one end. Harris (1961) estimated that E for a cilium is about 5 dynes urne2, while Brokaw (1965b) estimated a value of 3 dynes urnb2 from Yoneda’s (1962) data and, also, 6 dynes urnT2 (Brokaw, 1966~). We shall assume that E = 4 dynes ume2. To evaluate the moment of inertia, I, of each of the doublets, we shall take their dimensions as O-037 urn in total diameter and 0.024 urn high (Grimstone & Klug, 1966). No significant error will be made if we assume that the central fibrils are also of this dimension and if we replace each of the 11 elements by rectangles of width u = 0.037 urn and height, b = 0,024 pm. The average moment of inertia of such an element is then given by ub(a2+b2)/24. With these values of a, b and E we compute a,, = 1.7 x 10e6 dyne urn’ and ai = 1.5 x 10e4 dyne urn’. It should be noted that from considerations on the bending of cilia as reported by Yoneda (1962) and Sleigh (1962), Brokaw (19656) estimated that the stiffness of cilia (i.e. ai) wasapproximately 2 x low4 dyne um2. It should also be pointed out that the ratio aO/a, (-O.Ol), is independent of the value of E. To evaluate a2 it is first necessary to evaluate t and a. Because of the abruptness of the bending and unbending transitions, z is essentially the duration of the bent portion of the wave. From Brokaw’s (1965~~) data we compute t = 11.2, 11.6 and 12.1 msec for the sperm of Lytechinus, Ciona and Chaetopterus, respectively. These values, obtained for widely different species

20

J.

LUBLINER

AND

J.

J. BLUM

in sea water, are clearly consistent with our hypothesis that z is a constant of the mechanochemical system. It is also important to establish whether for any given species z depends on viscosity. For viscosities greater than that of sea water, Brokaw (1966b) does not give the angle, (I,,, of the bent region explicitly, but this can be computed from the data by use of his equations (1) to (3) (Brokaw, 196%~). For Lytechinus and Ciona the values of z so computed vary randomly from 12 to 18 milliseconds over the whole viscosity range studied. For Chaetopterus, wave amplitude was not reported. Thus for at least two of the three species of sperm studied, z is essentially independent of the viscosity, i.e. unbending follows bending by a fixed time interval. We shall adopt a value of O-013 seconds for z. If we take log at = - 1.7 as an average value for these sperm, then a- ’ is approximately 0.65 urn. From considerations of the sharpness of the bending and unbending transitions Brokaw (1968) has estimated that a-l, which is a measure of the spatial decay of the bending disturbance, must be significantly greater than 170 8, and significantly less than 3 to 8 urn. It is seen that the value computed on the basis of the present theory (from dynamical measurements of bend propagation velocity as a function of viscosity) conforms to this criterion. Knowing ao, al, h and a, we find that az = 3.4x 10e4 dynes urn-‘. Since A, = h’a,/G, we can compute the effective web area if we know the shear modulus. For most polymeric materials G = E/3. If we assume the stiffness of the web to be the same as that of the contractile elements, we compute that A, = 5 x 10e6 urn’, i.e. somewhat smaller than a square 25 A x 25 A. This is an unreasonably small area of effective shear, being smaller than the crosssection of a dynein molecule (Grimstone & Klug, 1966), and indicates that the web is considerably softer than the contractile elements, as might be expected for a sliding filament type mechanism where the shear is born by cross links between different proteins. The parameter B, defined in equation (29), may be rewritten by using the definitions of E [equation (15)] and of c,,, [equation (1 S)] as follows : B = p-.-f*

kcm

2 (k + k’) h(lwlj

P* = 2p,,;

The maximum radius of curvature for these sperm in sea water was measured by Brokaw (1965~) as 5*05,5*5 and 4.3 urn for Lytechinus, Ciona and Chaetopterus, respectively. For our purposes, we shall take the value of pmax = 5 urn, which, in conjunction with the value of B = 3 used to fit the experimental velocity vs. viscosity data, yields a value of 30 urn for the critical radius of bending, consistent with the failure to observe a transition region between the straight unbent region and the fully bent region of the sperm. [Brokaw (1966b) somewhat arbitrarily assumed p* = 15 urn.]

MODEL

FOR

BEND

PROPAGATION

IN

FLAGELLA

21

Since k/(k +k’) = (s- r)/s, and for the three species of sperm discussed in this paper we have used s = 10, I = 5, we find that k/(k+k’) = 0.5 or k N k’. Using the previously computed values of h (0.15 pm) and of Q/a, (O*Ol), we then obtain c,,,, the maximum change in length (per unit length) of the actively contracting material (due to contraction and/or sliding) as 0.06. Knowledge of the parameters r and s, coupled with our estimates of the experimental values of z now permits the computation of the rate constants for active contraction (k) and for relaxation (k’) of the active elements. For each of these species of sperm, then, k Z k’ Z 400 set- ‘. Since ,the beat frequency for these sperm is about 30 see-l (less at high viscosities) it can be seen that both active contraction and relaxation are fast compared to the duration of the beat cycle It is noteworthy that we have fitted the experimental data for glycerinated Lytechinus, at two concentrations of ATP, with the same values of s, r and B as were used for living sperm. This implies that at concentrations of ATP as low as O-2 mM the rates of contraction and relaxation were the same as in living sperm. Brokaw (1966b) reported that there was no change in the dephosphorylation rate of ATP by Lytechinus in the range from 2.0 to O-2 mM, an observation consistent with the estimate of Raff & Blum (1969) that the Michaelis constant for the ATPase of Tetruhymena cilia was less than 2.5 x 10e4 M and with the finding by Gibbons (1966) that the Michaelis constant for 30 s dynein of Tetrahymena cilia was 1.1 x lo-’ M. Thus the conclusions drawn from the present theory are consistent with existing experimental data. Instead of changing k or k’, therefore, lowering the ATP concentration appeared to increase both T/A’ and C(T(see legend to Fig. 5). At least at low viscosity there was no appreciable change in the radius of curvature, wave amplitude, or wave length (Brokaw, 19663), so that in this range of ATP concentration z probably did not change. The effect of lowering the ATP concentration was, therefore, probably on the physical constants of the flagellum as embedded in the parameters 1’ and a ,but independent methods of measuring these physical constants are required before further interpretation of the effect of reduced ATP on bend propagation can be attempted. Since the viscous parameters, K and L occur together in the expression for 1, we can in principle only evaluate their combination by fitting the theoretical curves to the velocity--viscosity data. Equation (12), however, provides an independent estimate of ICprovided that L’ is related to some experimentally measured parameter of the wave and if a suitable choice of 6 is made. Since, as has already been discussed, the value of ICis not sensitive to the exact choice of L, it is convenient to choose L = L’, i.e. to use that value of L’ in computing x which then leads to computation of the same value of L. We also choose (r = -0.50, corresponding to sinusoidal motion (Hancock, 1953), and

22

J.

LUBLINER

AND

J.

J.

BLUhl

a = 2.5 (see Figs 5 and 6). With these choices, we compute that K - 3 and L = 15 urn. This is just the value to which the observed wavelength of the bending waves (measured along the flagellum) tends in Lytechinus and Ciona

at high viscosity. Since at high viscosity the shape of the observed bending waves tends to become essentially sinusoidal, the wavelength, L, of the sinusoidal approximation to the bending moment might be expected to become identical with the observed wavelength. 4. Conclusion

The present theory provides the first dynamical treatment of wave propagation in cilia as a function of viscosity. Despite its apparent success in accounting for extant data, it must be emphasized that there are as yet insufficient data in the literature to permit uncritical acceptance of the theory. Hopefully, this treatment will provide impetus for the accumulation of more data on the effects of viscosity on wave parameters for sperm, cilia and flagella of many different species and properties. Such data may provide important clues as to which features of the present theory need modification. In particular, the way is now open to try to identify the rate constants k and k’ with what is known about the molecular details of the contractile process. Detailed knowledge of the molecular mechanism of flagellar contractility may require modification of the kinetic formalism adopted in this paper, but since the kinetics enter into the dynamic equations to be solved in a straightforward manner such modifications should be simple, requiring at most the use of computer methods to solve the resulting approximate equation. A more fundamental mathematical problem is the approximation of the viscous bending moment as of sinusoidal form. Although it appears to fit the extant data on the effects of viscosity on bend propagation velocity, this might be an accident of the mathematics and does not necessarily imply that the variation of viscous bending moment is indeed sinusoidal. The value of this approximation, however, is not only that it fits most of the data to remarkably high viscosities, but also that it shows that the velocity of propagation and the shape of both the bending and unbending transitions are essentially unaffected by viscosity in the range 1 to 3 cP. Thus although computations of swimming speed and energy expenditure cannot neglect the viscosity, computations of bend propagation velocity and shape can, at least for viscosities near that of water. Finally, it will be noted that in this as in earlier treatments, it is assumed that active contraction is initiated by bending past a critical radius of curvature. In a filament as thin as a cilium, however, the distortion due to bending should be virtually identical on either side of the centroid. It is not clear, therefore, why active contraction should be confined to one half of the cilium.

MODEL

FOR

BEND

PROPAGATION

IN

23

FLAGELLA

A possible explanation is that the passive process which causes the bending ahead of the bending transition point to reach the critical value places one set of elements under tension and the opposite set under compression and that only in the elements which bend under compression is the active process initiated. One of us (J.L.) gratefully acknowledges support by National Institutes of Health Special Fellowship No. F03-GM43137-01 from the National Institute of General Medical Sciences. The other of us (J.J.B.) gratefully acknowledges the support of Research Career Development Award, 5K3 GM 2341 from the National Institutes of Health and of a Guggenheim Fellowship.

REFERENCES C. J. (1965~).J. exp. Biol. 43, 155. C. J. (19656). Paper presentedat meetings of the Biophysical Society, San Francisco, Feb. 26.

BROKAW, BROKAW,

BROKAW, C. J. (1966~~). Nature, Land. 209, 1. BROKAW, C. J. (19666). J. exp. Biol. 45, 113. BROKAW, C. J. (1966~). Am. Rev. resp. Dis. 93, 32. BROKAW, C. J. (1968). Symp. Sot. exp. Biol. 23,101. BROKAW, C. J. & BENEDICT, B. (1968~). Archs. Biochem. Biophys. 125,770. BROKAW, C. J. & BENEDID, B. (19686).J. gen. Physiol. 52,283. BROKAW, C. J. & WRIGHT, L. (1963). Science, N. Y. 142, 1169. BURGERS, J. M. (1938). Proc. K. ned. Akud. Wet. 16, 113. GIBBONS, J. R. (1966). J. biol. Chem. 241, 5590. GRIMSKXW, A. V. & KLuG, A. (1966). J. Cell. Sci. 1, 35 1. HANCOCK, S. J. (1953). Proc. R. Sot. A 209,447. HARRIS, J. E. (1961). In The Cell und the Orgunism. (J. A. Ramsay & V. B.

Wigglesworth,

eds.) London: Cambridge University Press. HOLWLL, M. E. J. (1965). J. exp. Biol. 42, 125. HOLWILL, M. E. J. (1966). Physiol. Rev. 46,696. HO-P,, G. A. (1965). Proc. R. Sot. B 162, 351. LANSING, A. I. & LAMY, F. (1961). J. Biophys. biochem. Cytol. 9, 799. MACHIN, K. E. (1958). J. exp. Biol. 35,796. MACHIN, K. E. (1963). Proc. R. Sot. B 158,88. Powv, E. P. (1968). Introduction to Mechanics of Solids, p. 180. Englewood Cliffs, N.J.:

Prentice-Hall.

RAFP, E. C. & BLUM, J. J. (1969). J. biol. Chem. 244, 366. RIKMENSPOEL, R. (1966). Biophys. J. 6,471. RIKMENSPOEL, R. (1970). Biophys. Sot. Abstracts 10, 38a. SATIR, P. (1965). J. Cell. Biol. 26, 805. SLEIGH, M. A. (1962). The Biology of Cilia und Flagella. Pergamon Press: London. SLEIGH, M.A.(1968).Symp. Sot. exp. Biol.28,101. YOWA, M. (1962). J. exp. Biol. 39, 307.

24

J. LUBLINER

AND

J. J. BLUM

Appendix For the convenience of readers not interested in mathematical detail, we append the following list of symbols and terms used with references to the equations in the text where they are defined. shearforce in the web [equations(7) and (8)] shearmodulusof the web [equation (S)] sheardue to active sliding [equation (S)] !i Young’s modulusof the contractile material constantsdescribingthe elasticpropertiesof the flagellum[equation (9)] a0, al, a2 c effective contraction; may include local contraction, sliding or both bulk viscosity of medium tl transversedrag coefficient [equation (12)] W L wavelength of the assumedsinusoidaldistribution of viscous bending moment composite constants used to write the equation of motion of the flagellum [equation (1S)] velocity of bend propagation ;, k’ rate constantsfor the contraction and relaxation processes, respectively [equation (16)] local curvature (reciprocal of radius of curvature) IJ critical curvature at which active contraction is initiated P* 7 duration of time interval betweeninitiation and termination of active bending [equation (17)] decreasein length of the contractile system per unit length when cm maximally contracted [equation (16)] B a bending parameter which can be interpreted as a measureof the contractility of the flagellum [equation (29)] viscosity, q as a dimensionless 4 4, m, P, I can be thought of as a dimensionless velocity, s as a dimensionless rate constant of contraction, and r as a r, s, w dimensionless rate constant of relaxation. m, p, and w are other dimensionlessparametersdefinedin equations(30) and (35) angle of inclination of the flagellar segmentlocated betweenpoints x and x-t- Ax along the flagellum,with respectto a referencedirection a ~w,ov--lwzlol b log,ov-logIoq. a and b are shift factors used to fit theoretical curves to the experimentaldata s G