I. theor. Biol. (1977) 69, 87-99
Analysis of Bend Initiation in Cilia According to a Sliding Filament Model J. LUBLINER Department of Civil Engineering, University of California, Berkeley, California 94720, U.S.A. AND J. J. BLUM Department of Physiology and Pharmacology, Duke University Medical Center, Durham, North Carolina 27710, U.S.A. (Received 15 November 1976, and in revised form
8 April 1977)
Equations are developed to describe the initiation of bending on a semiinfinite cilium or flagellum in a medium of negligible viscosity when one end of the cilium develops curvature as a result of a forcing function which may cause sufficient local curvature to initiate a propagated wave of bending that then travels at steady wave speed. It is assumed that active sliding is shear-limited, and that the active process begins when a critical curvature is attained, but the magnitude of the critical curvature required for initiation depends on the previous history of the curvature. With these assumptions, it is shown that the general shape of the initiation curves, i.e. position of point of maximum curvature as a function of time, approximately match the experimental data, i.e. the sliding filament model permits a range of initiation times comparable to those observed in nature.
1. Introduction Analysis of bend propagation as a function of viscosity in infinite flagella was initially treated by us in terms of a two-state model for the kinetics of the active contraction process (Lubliner & Blum, 1971a; hereafter referred to as I). This two-state model for the active process was used (Lubliner & Blum, 1971b; hereafter referred to as II) to study the initiation of bending waves in cilia and flagella. Although this treatment adequately described the data for initiation of the bending wave in Codonosiga, where steady wave speed is attained close to the point of insertion, the treatment could not fit data for other cilia and flagella, where initiation occurs more slowly and steady wave
88
J.
LUBLINER
AND
J.
J.
BLUM
speed is not attained until the bend has moved an appreciable distance from the point of insertion. A two-state kinetic model, however, while not necessarily incompatible with a sliding filament model, is more appropriate to a local contraction mechanism. A sliding filament mechanism as the basis of bend generation in cilia has been established by the studies of Summers & Gibbons (1971) on sperm flagella and of Warner & Satir (1974) on gill cilia. Many of the limitations of our original treatment of bend propagation in infinite flagella (I) were removed when, instead of the two-state model for the active process, a model which utilized active sliding was developed (Lubliner & Blum, 1972; hereafter referred to as III). In our original model (I), the contractile process was “time-limited”, i.e. contraction was assumed to begin when a critical radius of curvature was attained and to terminate z ms later. This formulation was based on the notion that the contractile process, once initiated, might have a predetermined kinetic course (for any given temperature, ionic strength, etc.) which would thus in large part automatically determine beat frequency. Since then several lines of evidence indicate that bending is probably shear-limited rather than time-limited. Thus changes in ATP concentration (Brokaw, 1967; Holwill, 1969; Brokaw & Josslin, 1973), temperature (Holwill & Silvester, 1967; Coakely 8z Holwill, 1974) and the number of dynein arms (Gibbons & Gibbons, 1973) alter beat frequency but have little effect on bend angle or wave amplitude, while changes in CO, (Brokaw & Simonick, 1976), attachment of a sperm head to a microscopic slide (Brokaw, 1965), the presence of chemoattractants (Miller & Brokaw, 1970) or the presence of Ca2+ during the extraction procedure (Brokaw, Josslin & Bobrow, 1974) alter bend angle with little effect on frequency. It is reasonable to assume, therefore, that the internal shear-bearing elements responsible for converting active sliding into bending are the elements responsible for limiting the amount of sliding. Recent evidence suggests that the radial link system which had been thought to be the main passive shearbearing elements may instead be active elements which attach at the leading edge of a bend and disengage at the trailing edge. Thus Warner & Satir (1974) have shown that the radial link system is arranged as a vernier opposite a series of projections from the control pair of microtubules. In straight regions, the spokes appear to be functionally detached from the sheath projections, while in bent regions the spokes appear to be attached to these projections. It is this component of shear resistance which, according to Warner & Satir (1974) forms the main part of the mechanism for converting active interdoublet sliding (powered by cyclic cross bridge activity of the dynein arms) into local bending. In the present treatment of bend initiation, we have therefore used a sliding filament model in which bending is shear-limited. This then permits one to specify the kinetics of active sliding very simply In
BEND
INITIATION
IN
CILIA
89
addition to specifying the kinetics of the active sliding process, it was necessary to assume that active sliding was not necessarily initiated simply when a critical radius of curvature, rc*, was attained, but rc* itself might depend on the history of bending just prior to initiation of the active process. This hypothesis was originally introduced (III) to describe the speed of bend propagation in cilia and flagella operating according to a sliding filament mechanism. Support for this hypothesis comes from recent studies of bend initiation in amputated starfish sperm flagella (Okuno & Hiramoto, 1976). These workers found that bending waves of low amplitude which propagated toward the distal end of the cut flagellum were initiated spontaneously in the proximal region in a medium containing MgATP. If, however, the proximal end of an amputated flagellum afExed to the coverslip at its cut end was bent with a microneedle (in the presence of MgATP), the bend was then propagated at a much larger bend angle than in flagella left undisturbed in the presence of ATP. As will be shown below, use of a history-dependent JC*in conjunction with simple kinetics for the active sliding process also enables us to obtain a solution to the problem of bend initiation in cilia and flagella which appears to be compatible with extant experimental data. Although it was a tenable assumption that the initiation of bending at the point of insertion of a cilium or flagellum was partially determined by the basal body or some other specialized mechanism near the transition zone, such as the ciliary necklace (Gilula & Satir, 1972), recent work shows that initiation of bending may occur at any point on a flagellum (Goldstein, Holwill & Silvester, 1970; Brokaw & Gibbons, 1973; Shingyoji, Murakami & Takahashi, 1977). Furthermore, a severed piece of sperm tail may be reactivated to beat asymmetrically as a cilium or symmetrically as a flagellum, depending on whether the cut end is mechanically restrained or free (Lindemann & Rikmenspoel, 1972). It is thus necessary to assume that bend initiation may occur at any point on the flagellum even though initiation is generally confined to the region of the ciliary necklace in situ in most species. In certain trypanosomatids, however, initiation normally occurs at the tip of the flagellum (Holwill & McGregor, 1974). It should be noted that the concentration of free Ca2+ inside the flagellum appears to play a major role in determining the direction of pointing of cilia in paramecium (see, e.g. Machemer, 1976, and references therein), the location of bend initiation and thus the direction of bend propagation in Crithidiu (Holwill & McGregor, 1975, 1976). The present theory does not deal with the issue of where bend initiation will occur nor the direction in which it will propagate. Instead we assume that bend initiation is occurring at some point which we then consider as the origin of a semi-infinite flagellum, and ask whether a sliding filament
90
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I.
BLUM
mechanism, with simple sliding kinetics and a history-dependent critical curvature, can account for the motion of a forming bend from the time of its inception until it moves away from the origin and attains its steady speed.
2. Analysis (A)
MOTION
OF A SEMI-INFINITE
FLAGELLUM
WITH
A SLIDING
FILAMENT
MECHANISM
Let s denote distance along t the time. If the net bending is JC(S,t), and the active shear the flagellum (if all internal (Lubliner, 1973) :
the flagellum axis measured from the base and moment on the cilium is M(s, t), the curvature force is IYI(S,t), then the internal equilibrium of resistances are linearly elastic) is described by
where EZ,is the flexural stiffness of the ith filament, EZOis the flexural stiffness of the flagellum due to the extensibility of the filaments, and K, is the shear stiffness. The external equilibrium is governed by (Machin, 1958): M+M,=O,
(2)
where M, is the bending moment due to viscous resistance. In what follows it will be assumed that the effect of viscosity on bend propagation for cilia and flagella (the terms are used interchangeably throughout the entire section on Analysis) in sea water is negligible, so that M, = 0; hence M = 0. Although high viscosity does affect the wave parameters of flagella (Brokaw, 1966u), the effects are small at viscosities near those of water. Also the effects of increased viscosity on bend angle and beat frequency of reactivated sea urchin sperm flagella decrease with decreasing ATP concentration, becoming almost undetectable at 0.01 mM ATP (Brokaw, 1975). It is therefore unlikely that viscosity plays an important role in determining the course of bend initiation in liquids with the viscosity of water. Furthermore, it can be shown from geometric considerations alone (I) that EZJZ, 4 1, and this quantity will be taken as zero. Lastly, the active shear force m may be replaced by the equivalent shear deformation ys = -m/K, (Lubliner, 1973). Since shear deformation is more likely to be a directly measurable quantity than is shear force (see, e.g. Warner & Satir, 1974) we prefer to use ys. Use of ys also has the advantage of maintaining a consistent notation with our earlier papers. With these assumptions and
BEND
INITIATION
IN
91
CILIA
the definition : a2 = K,/ECli, I
(3)
equation (1) reduces to:
a2K __-
I ol’
as2
JYS K
=
as
(4)
*
An equation of this form, but for a local contraction rather than a sliding filament mechanism, was first derived by Brokaw (19663). It will be noted that, because of the neglect of viscosity, time does not appear explicitly as an independent variable in equation (4) but only as a parameter. The equation may consequently be treated as an ordinary differential equation. The boundary conditions will be taken as: 40, f> = KOW
(54
and K(c0, t) = 0, (5’4 i.e. the basal curvature is assumed given as a function of time and the cilium is regarded as of semi-infinite length. The solution of equation (4) subject to boundary conditions (5) depends on the form of the active function, yS.Brokaw (1971) assumed the equivalent of ys = htc (6) with h a positive constant, i.e. a linear control of active sliding by the curvature. The solution then takes the form: K(S,
t)
=
Kg(t)
(7)
ewp”“,
where p = d 1 + @h/2)’ -ah/2. (8) Clearly this solution does not have the character of a propagating wave, consistent with Brokaw’s (1971) conclusion that wave propagation cannot be sustained on a flagellum with purely elastic internal resistances if the active shear moment, m, is proportional to curvature. (B)
THE
MECHANOCHEMICAL
CONTROL
MODEL
In this paper, we also assume control of the active shear by the curvature, but in accordance with a model we have previously used to analyze flagellar bending (III). At any point s on the flagellum the active process begins at the time t = t*(s) and ends at t = t**(s). The inverses of t*(s) and t**(s) are denoted by s*(t) and s**(t); s = s*(t) describes the location of the activation point and s = s**(t) the location of the deactivation point at time t. If s**(t) does not exist, then the disturbance caused by rcO(f)does not propagate but decays toward the base. If the origin of time is chosen such that
92
J.
LUBLINER
AND
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J.
BLUM
t*(O) = 0, then s*(O) = 0. By definition sqY’(t) < s < s*(t) defines the active zone, and ay$s # 0 in that zone only. The solution of equation (4) satisfying boundary conditions (5) is then : s’(t)
K(S, t) = q)(t) emas-;
-4~-s’l+p(s+s’)] S*L Ce
~YS(S’~ 0_ ds,
ad
(9)
as may be verified by substitution. [Note: d/ds e-‘l’-“l = --c1 sgn (s-s’) * e-als-s’l and d/ds sgn (s-s’) = 26(3-s’). Here sgn (s-s’) = + 1 if s > s’ and - 1 if s < s’, and &s-s’) = co if s = s’ and zero otherwise.] It is further assumed that once activation takes place, ySis independent of curvature, so that equation (9) is an explicit expression if s*(1) and s**(t) are known. Thus, ySis assumed to depend only on the time and distance relative to the activation point, i.e. where t’ E t-t*(s) that:
Y,c%0 = 9xt’, 4 (10) and s’ z s*(t)--s. With these definitions, it is evident
65 --=as
a$ dt*
a4
at’ ds+Z’
(11)
Lubliner & Blum (1972) chose the form i, = a+ bv for the active sliding mechanism (with both a and b constant and v being the constant speed of wave propagation on an infinite flagellum) in an attempt to fit extant data for wave speed and form of flagella as a function of viscosity. It can be seen that their choice corresponds to the choice:
ad --=a a’ b3 , as,= (12) at’ and we shall take this as a first approximation to the properties of 4. This choice insures that when the wave propagates at the constant speed v such that the length of the active region is significantly larger than a-‘, the curvature will be essentially constant [and equal to (u/v)+ b)] over a large portion of the active zone, corresponding to circular-arc-shaped bends. In our previous work (I, II, III), we assumed that the active process was of constant duration 2, i.e. t**(s) = t*(s)+z. (13) We call this the “time-limit” model. In the present paper we shall substitute a “shear-limit” model, i.e. we assume that the active process stops when the active shear yS attains a fixed value ym, so that s**(t) and t**(s) are given implicitly by :
r,[s**(o, tl = YSCS, t**(s)1= Ym9
(14)
BEND
INITIATION
IN
93
CILIA
with the restriction s**(t) > 0. If yS is described by the approximation (10)-(12), and if b = 0, then we recover equation (13) with z = y,,,/a, while ifa=O,wehave: s**(f) = max [s*(t)-?,
01.
Complete specification of the model requires an assumption concerning the nature of the mechanochemical coupling. As before (III) we assume that activation is controlled by the curvature. However, it is not simply the current value of the curvature which is critical; instead the active mechanism possesses a memory extending back to the time t = - T. If the dependence of activation on curvature is linear, then there exists a memory function, M(t), such that activation at the point s occurs when the linear functional: dt’,
-jTM(r-t’)‘~
reaches a critical value, II, which may be thought of as a threshold value of the curvature. As in III, it is convenient to assume that M(t) = 1 + ct. If the above functional is transformed by integration by parts [with the condition that rc(s, -7) = 01, then t = t*(s) or s = s*(t) is given implicitly by: K(S, t)+c
j K(S, t’) dt’ = c.
(16)
-T
It is clear that the value of c is a measure of memory of the active mechanism; c = 0 corresponds to a memory-independent threshold. When equations (12)-(14) are substituted into equation (lo), with s = s*(t), the result is: e -as*(t) = h:. (17) Upon multiplication terms, we obtain:
by eaP@),integration
i rco(t’) dt’+ai
rco(t)+c
-T
+b
{
cash as*(t)-cash
[l $c(t-
by parts, and rearrangement
t’)] [sinh as*(t’)-sinh
of
as**(t’)] dt’+
0
as**(t)+c
i [cash as*(t’)-cash
as**(t’)]
dt’
> = II eas*(t). (18) Equation (18) is a non-linear integral equation for s*(t), the position of the activation point. It is of the same type as encountered in II, i.e. when the 0
94
J.
LUBLINER
integrals in equation (18) rule, a quadratic equation (18) can be carried out, T or, equivalently, s = 0 in
AND
J.
J.
HLUM
are replaced by sums according to the trapezoidal for eas*@)results. Before the solution of equation must be determined. Setting t = 0 in equation (18) equation (16) yields: Ko(o)+c
; -T
Kg(t)
dt’ = K,
(19)
i.e. an expression for the time span, T, of the memory function. If the curvature at the base, /c,(t), is given as a function of time starting at t = -T. e.g. K&)
=
Eu(t+T),
(20)
then equation (19) becomes : u(T)+cju(t’)dt’
= 1.
(21)
0
(C)
COMPUTATION
OF THE
CURVATURE
OF A LONG
FLAGELLUM
Knowledge of the curvature from equation (9) permits computation total bend angle, Q(t), at time t, as: G(t)
= j: K(S, t) ds.
of the
(22)
0
Provided that edaL < 1 so long as s*(t) < L, i.e. the bending point has not yet reached the distal end of the flagellum, the upper limit of the integral in equation (22) may be replaced by a3. The integration with respect to s may by carried out first, yielding: o(t) = fKo(t)-
‘~)(I-e-CE’)a~~~~I)dl.f. s”(f)
Upon substituting equations (1 l)-(12) and, where appropriate, variables, equation (23) becomes, for t < t*(L), o(t) = i Ko(t)+a
j (1 -e-arr(t’)) dt’+b t*[s**(r)l
(23) changing
s*(t)--**(t)k ~~-as**(t)_ e --*CO]}.
(24)
It is also of interest to determine the position at the point of maximum curvature, since this, apparently, is the “position of bend” recorded by Sleigh (1968) in his investigation of bend initiation of cilia from various species. To obtain this analytically, we differentiate equation (9) with respect
BEND
INITIATION
IN
95
CILIA
to s and set the result equal to zero. An explicit expression for the corresponding value of s, which we denote by s,,(t), can be obtained if a = 0, namely : s*(t) = $u In 2 em’@) cash us**(t) - 5; - 1 , (25) { [ 1 1 provided this expression is positive; otherwise am = 0, i.e. the maximum curvature is at the base. (D)
NUMERICAL
SOLUTION
The shear-limit model presents a difficulty in solving equation (18) that was not present with the time-limit model, namely, the lack of an explicit expression for s**(t), except in the two limiting cases a = 0 and b = 0. Since the former case leads to simpler expressions and is a necessary condition for equation (25), and since the purpose of the numerical solution is primarily illustrative, such a solution will be undertaken for this case only. (What this means is a constant active curvature of magnitude b in the active region.) Equation (18) can be put in dimensionless form by introducing the dimensionless time variable i = ct, letting u [introduced in equation (20)] be a function of irather than t, and defining the function, t(i) = KS*(~) and the parameters : B = b/K, (26) D = uy,/b.
(27)
With the auxiliary functions : g(f) = et@)
f(i) = cosht(i)= t [g(i)+&]
equation
1 3g(i) < eD h(i) = cash us**(Z) = 3 g(i) eeD +CL g(t) 1 , g(i) > eD1; ( (18) (with a = 0) becomes : i+T
u(i+T)+
J ~(7’)di’+B 0
I-h(i)+i[f(t’)-h(i’)ldi’ 0
1
= g(f)
(28)
with T = CT given by: II( 7;) + i u(i) di = I.
(29)
96
J.
LUBLINER
AND
J.
3.
BLUM
The numerical solution proceeds as in II, with the integrals replaced by sums according to the trapezoidal rule. The increment in f is l/r, where Y is a large integer. If g, z g(n/r), then, so long as g, < eD at each step we have : (30) where Q, is made up of either given or previously computed quantities. When multiplied by g,,, equation (30) becomes a quadratic equation that can be solved explicitly and that will have real roots (with at least one positive) independently of Q, if: +-B(lf$)]
> 0.
Since I’ is large, equation (31) is equivalent to: O
Analysis of Bend Initiation in Cilia According to a Sliding Filament Model J. LUBLINER Department of Civil Engineering, University of California, Berkeley, California 94720, U.S.A. AND J. J. BLUM Department of Physiology and Pharmacology, Duke University Medical Center, Durham, North Carolina 27710, U.S.A. (Received 15 November 1976, and in revised form
8 April 1977)
Equations are developed to describe the initiation of bending on a semiinfinite cilium or flagellum in a medium of negligible viscosity when one end of the cilium develops curvature as a result of a forcing function which may cause sufficient local curvature to initiate a propagated wave of bending that then travels at steady wave speed. It is assumed that active sliding is shear-limited, and that the active process begins when a critical curvature is attained, but the magnitude of the critical curvature required for initiation depends on the previous history of the curvature. With these assumptions, it is shown that the general shape of the initiation curves, i.e. position of point of maximum curvature as a function of time, approximately match the experimental data, i.e. the sliding filament model permits a range of initiation times comparable to those observed in nature.
1. Introduction Analysis of bend propagation as a function of viscosity in infinite flagella was initially treated by us in terms of a two-state model for the kinetics of the active contraction process (Lubliner & Blum, 1971a; hereafter referred to as I). This two-state model for the active process was used (Lubliner & Blum, 1971b; hereafter referred to as II) to study the initiation of bending waves in cilia and flagella. Although this treatment adequately described the data for initiation of the bending wave in Codonosiga, where steady wave speed is attained close to the point of insertion, the treatment could not fit data for other cilia and flagella, where initiation occurs more slowly and steady wave
88
J.
LUBLINER
AND
J.
J.
BLUM
speed is not attained until the bend has moved an appreciable distance from the point of insertion. A two-state kinetic model, however, while not necessarily incompatible with a sliding filament model, is more appropriate to a local contraction mechanism. A sliding filament mechanism as the basis of bend generation in cilia has been established by the studies of Summers & Gibbons (1971) on sperm flagella and of Warner & Satir (1974) on gill cilia. Many of the limitations of our original treatment of bend propagation in infinite flagella (I) were removed when, instead of the two-state model for the active process, a model which utilized active sliding was developed (Lubliner & Blum, 1972; hereafter referred to as III). In our original model (I), the contractile process was “time-limited”, i.e. contraction was assumed to begin when a critical radius of curvature was attained and to terminate z ms later. This formulation was based on the notion that the contractile process, once initiated, might have a predetermined kinetic course (for any given temperature, ionic strength, etc.) which would thus in large part automatically determine beat frequency. Since then several lines of evidence indicate that bending is probably shear-limited rather than time-limited. Thus changes in ATP concentration (Brokaw, 1967; Holwill, 1969; Brokaw & Josslin, 1973), temperature (Holwill & Silvester, 1967; Coakely 8z Holwill, 1974) and the number of dynein arms (Gibbons & Gibbons, 1973) alter beat frequency but have little effect on bend angle or wave amplitude, while changes in CO, (Brokaw & Simonick, 1976), attachment of a sperm head to a microscopic slide (Brokaw, 1965), the presence of chemoattractants (Miller & Brokaw, 1970) or the presence of Ca2+ during the extraction procedure (Brokaw, Josslin & Bobrow, 1974) alter bend angle with little effect on frequency. It is reasonable to assume, therefore, that the internal shear-bearing elements responsible for converting active sliding into bending are the elements responsible for limiting the amount of sliding. Recent evidence suggests that the radial link system which had been thought to be the main passive shearbearing elements may instead be active elements which attach at the leading edge of a bend and disengage at the trailing edge. Thus Warner & Satir (1974) have shown that the radial link system is arranged as a vernier opposite a series of projections from the control pair of microtubules. In straight regions, the spokes appear to be functionally detached from the sheath projections, while in bent regions the spokes appear to be attached to these projections. It is this component of shear resistance which, according to Warner & Satir (1974) forms the main part of the mechanism for converting active interdoublet sliding (powered by cyclic cross bridge activity of the dynein arms) into local bending. In the present treatment of bend initiation, we have therefore used a sliding filament model in which bending is shear-limited. This then permits one to specify the kinetics of active sliding very simply In
BEND
INITIATION
IN
CILIA
89
addition to specifying the kinetics of the active sliding process, it was necessary to assume that active sliding was not necessarily initiated simply when a critical radius of curvature, rc*, was attained, but rc* itself might depend on the history of bending just prior to initiation of the active process. This hypothesis was originally introduced (III) to describe the speed of bend propagation in cilia and flagella operating according to a sliding filament mechanism. Support for this hypothesis comes from recent studies of bend initiation in amputated starfish sperm flagella (Okuno & Hiramoto, 1976). These workers found that bending waves of low amplitude which propagated toward the distal end of the cut flagellum were initiated spontaneously in the proximal region in a medium containing MgATP. If, however, the proximal end of an amputated flagellum afExed to the coverslip at its cut end was bent with a microneedle (in the presence of MgATP), the bend was then propagated at a much larger bend angle than in flagella left undisturbed in the presence of ATP. As will be shown below, use of a history-dependent JC*in conjunction with simple kinetics for the active sliding process also enables us to obtain a solution to the problem of bend initiation in cilia and flagella which appears to be compatible with extant experimental data. Although it was a tenable assumption that the initiation of bending at the point of insertion of a cilium or flagellum was partially determined by the basal body or some other specialized mechanism near the transition zone, such as the ciliary necklace (Gilula & Satir, 1972), recent work shows that initiation of bending may occur at any point on a flagellum (Goldstein, Holwill & Silvester, 1970; Brokaw & Gibbons, 1973; Shingyoji, Murakami & Takahashi, 1977). Furthermore, a severed piece of sperm tail may be reactivated to beat asymmetrically as a cilium or symmetrically as a flagellum, depending on whether the cut end is mechanically restrained or free (Lindemann & Rikmenspoel, 1972). It is thus necessary to assume that bend initiation may occur at any point on the flagellum even though initiation is generally confined to the region of the ciliary necklace in situ in most species. In certain trypanosomatids, however, initiation normally occurs at the tip of the flagellum (Holwill & McGregor, 1974). It should be noted that the concentration of free Ca2+ inside the flagellum appears to play a major role in determining the direction of pointing of cilia in paramecium (see, e.g. Machemer, 1976, and references therein), the location of bend initiation and thus the direction of bend propagation in Crithidiu (Holwill & McGregor, 1975, 1976). The present theory does not deal with the issue of where bend initiation will occur nor the direction in which it will propagate. Instead we assume that bend initiation is occurring at some point which we then consider as the origin of a semi-infinite flagellum, and ask whether a sliding filament
90
J.
LUBLINER
AND
J.
I.
BLUM
mechanism, with simple sliding kinetics and a history-dependent critical curvature, can account for the motion of a forming bend from the time of its inception until it moves away from the origin and attains its steady speed.
2. Analysis (A)
MOTION
OF A SEMI-INFINITE
FLAGELLUM
WITH
A SLIDING
FILAMENT
MECHANISM
Let s denote distance along t the time. If the net bending is JC(S,t), and the active shear the flagellum (if all internal (Lubliner, 1973) :
the flagellum axis measured from the base and moment on the cilium is M(s, t), the curvature force is IYI(S,t), then the internal equilibrium of resistances are linearly elastic) is described by
where EZ,is the flexural stiffness of the ith filament, EZOis the flexural stiffness of the flagellum due to the extensibility of the filaments, and K, is the shear stiffness. The external equilibrium is governed by (Machin, 1958): M+M,=O,
(2)
where M, is the bending moment due to viscous resistance. In what follows it will be assumed that the effect of viscosity on bend propagation for cilia and flagella (the terms are used interchangeably throughout the entire section on Analysis) in sea water is negligible, so that M, = 0; hence M = 0. Although high viscosity does affect the wave parameters of flagella (Brokaw, 1966u), the effects are small at viscosities near those of water. Also the effects of increased viscosity on bend angle and beat frequency of reactivated sea urchin sperm flagella decrease with decreasing ATP concentration, becoming almost undetectable at 0.01 mM ATP (Brokaw, 1975). It is therefore unlikely that viscosity plays an important role in determining the course of bend initiation in liquids with the viscosity of water. Furthermore, it can be shown from geometric considerations alone (I) that EZJZ, 4 1, and this quantity will be taken as zero. Lastly, the active shear force m may be replaced by the equivalent shear deformation ys = -m/K, (Lubliner, 1973). Since shear deformation is more likely to be a directly measurable quantity than is shear force (see, e.g. Warner & Satir, 1974) we prefer to use ys. Use of ys also has the advantage of maintaining a consistent notation with our earlier papers. With these assumptions and
BEND
INITIATION
IN
91
CILIA
the definition : a2 = K,/ECli, I
(3)
equation (1) reduces to:
a2K __-
I ol’
as2
JYS K
=
as
(4)
*
An equation of this form, but for a local contraction rather than a sliding filament mechanism, was first derived by Brokaw (19663). It will be noted that, because of the neglect of viscosity, time does not appear explicitly as an independent variable in equation (4) but only as a parameter. The equation may consequently be treated as an ordinary differential equation. The boundary conditions will be taken as: 40, f> = KOW
(54
and K(c0, t) = 0, (5’4 i.e. the basal curvature is assumed given as a function of time and the cilium is regarded as of semi-infinite length. The solution of equation (4) subject to boundary conditions (5) depends on the form of the active function, yS.Brokaw (1971) assumed the equivalent of ys = htc (6) with h a positive constant, i.e. a linear control of active sliding by the curvature. The solution then takes the form: K(S,
t)
=
Kg(t)
(7)
ewp”“,
where p = d 1 + @h/2)’ -ah/2. (8) Clearly this solution does not have the character of a propagating wave, consistent with Brokaw’s (1971) conclusion that wave propagation cannot be sustained on a flagellum with purely elastic internal resistances if the active shear moment, m, is proportional to curvature. (B)
THE
MECHANOCHEMICAL
CONTROL
MODEL
In this paper, we also assume control of the active shear by the curvature, but in accordance with a model we have previously used to analyze flagellar bending (III). At any point s on the flagellum the active process begins at the time t = t*(s) and ends at t = t**(s). The inverses of t*(s) and t**(s) are denoted by s*(t) and s**(t); s = s*(t) describes the location of the activation point and s = s**(t) the location of the deactivation point at time t. If s**(t) does not exist, then the disturbance caused by rcO(f)does not propagate but decays toward the base. If the origin of time is chosen such that
92
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AND
J.
J.
BLUM
t*(O) = 0, then s*(O) = 0. By definition sqY’(t) < s < s*(t) defines the active zone, and ay$s # 0 in that zone only. The solution of equation (4) satisfying boundary conditions (5) is then : s’(t)
K(S, t) = q)(t) emas-;
-4~-s’l+p(s+s’)] S*L Ce
~YS(S’~ 0_ ds,
ad
(9)
as may be verified by substitution. [Note: d/ds e-‘l’-“l = --c1 sgn (s-s’) * e-als-s’l and d/ds sgn (s-s’) = 26(3-s’). Here sgn (s-s’) = + 1 if s > s’ and - 1 if s < s’, and &s-s’) = co if s = s’ and zero otherwise.] It is further assumed that once activation takes place, ySis independent of curvature, so that equation (9) is an explicit expression if s*(1) and s**(t) are known. Thus, ySis assumed to depend only on the time and distance relative to the activation point, i.e. where t’ E t-t*(s) that:
Y,c%0 = 9xt’, 4 (10) and s’ z s*(t)--s. With these definitions, it is evident
65 --=as
a$ dt*
a4
at’ ds+Z’
(11)
Lubliner & Blum (1972) chose the form i, = a+ bv for the active sliding mechanism (with both a and b constant and v being the constant speed of wave propagation on an infinite flagellum) in an attempt to fit extant data for wave speed and form of flagella as a function of viscosity. It can be seen that their choice corresponds to the choice:
ad --=a a’ b3 , as,= (12) at’ and we shall take this as a first approximation to the properties of 4. This choice insures that when the wave propagates at the constant speed v such that the length of the active region is significantly larger than a-‘, the curvature will be essentially constant [and equal to (u/v)+ b)] over a large portion of the active zone, corresponding to circular-arc-shaped bends. In our previous work (I, II, III), we assumed that the active process was of constant duration 2, i.e. t**(s) = t*(s)+z. (13) We call this the “time-limit” model. In the present paper we shall substitute a “shear-limit” model, i.e. we assume that the active process stops when the active shear yS attains a fixed value ym, so that s**(t) and t**(s) are given implicitly by :
r,[s**(o, tl = YSCS, t**(s)1= Ym9
(14)
BEND
INITIATION
IN
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CILIA
with the restriction s**(t) > 0. If yS is described by the approximation (10)-(12), and if b = 0, then we recover equation (13) with z = y,,,/a, while ifa=O,wehave: s**(f) = max [s*(t)-?,
01.
Complete specification of the model requires an assumption concerning the nature of the mechanochemical coupling. As before (III) we assume that activation is controlled by the curvature. However, it is not simply the current value of the curvature which is critical; instead the active mechanism possesses a memory extending back to the time t = - T. If the dependence of activation on curvature is linear, then there exists a memory function, M(t), such that activation at the point s occurs when the linear functional: dt’,
-jTM(r-t’)‘~
reaches a critical value, II, which may be thought of as a threshold value of the curvature. As in III, it is convenient to assume that M(t) = 1 + ct. If the above functional is transformed by integration by parts [with the condition that rc(s, -7) = 01, then t = t*(s) or s = s*(t) is given implicitly by: K(S, t)+c
j K(S, t’) dt’ = c.
(16)
-T
It is clear that the value of c is a measure of memory of the active mechanism; c = 0 corresponds to a memory-independent threshold. When equations (12)-(14) are substituted into equation (lo), with s = s*(t), the result is: e -as*(t) = h:. (17) Upon multiplication terms, we obtain:
by eaP@),integration
i rco(t’) dt’+ai
rco(t)+c
-T
+b
{
cash as*(t)-cash
[l $c(t-
by parts, and rearrangement
t’)] [sinh as*(t’)-sinh
of
as**(t’)] dt’+
0
as**(t)+c
i [cash as*(t’)-cash
as**(t’)]
dt’
> = II eas*(t). (18) Equation (18) is a non-linear integral equation for s*(t), the position of the activation point. It is of the same type as encountered in II, i.e. when the 0
94
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integrals in equation (18) rule, a quadratic equation (18) can be carried out, T or, equivalently, s = 0 in
AND
J.
J.
HLUM
are replaced by sums according to the trapezoidal for eas*@)results. Before the solution of equation must be determined. Setting t = 0 in equation (18) equation (16) yields: Ko(o)+c
; -T
Kg(t)
dt’ = K,
(19)
i.e. an expression for the time span, T, of the memory function. If the curvature at the base, /c,(t), is given as a function of time starting at t = -T. e.g. K&)
=
Eu(t+T),
(20)
then equation (19) becomes : u(T)+cju(t’)dt’
= 1.
(21)
0
(C)
COMPUTATION
OF THE
CURVATURE
OF A LONG
FLAGELLUM
Knowledge of the curvature from equation (9) permits computation total bend angle, Q(t), at time t, as: G(t)
= j: K(S, t) ds.
of the
(22)
0
Provided that edaL < 1 so long as s*(t) < L, i.e. the bending point has not yet reached the distal end of the flagellum, the upper limit of the integral in equation (22) may be replaced by a3. The integration with respect to s may by carried out first, yielding: o(t) = fKo(t)-
‘~)(I-e-CE’)a~~~~I)dl.f. s”(f)
Upon substituting equations (1 l)-(12) and, where appropriate, variables, equation (23) becomes, for t < t*(L), o(t) = i Ko(t)+a
j (1 -e-arr(t’)) dt’+b t*[s**(r)l
(23) changing
s*(t)--**(t)k ~~-as**(t)_ e --*CO]}.
(24)
It is also of interest to determine the position at the point of maximum curvature, since this, apparently, is the “position of bend” recorded by Sleigh (1968) in his investigation of bend initiation of cilia from various species. To obtain this analytically, we differentiate equation (9) with respect
BEND
INITIATION
IN
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CILIA
to s and set the result equal to zero. An explicit expression for the corresponding value of s, which we denote by s,,(t), can be obtained if a = 0, namely : s*(t) = $u In 2 em’@) cash us**(t) - 5; - 1 , (25) { [ 1 1 provided this expression is positive; otherwise am = 0, i.e. the maximum curvature is at the base. (D)
NUMERICAL
SOLUTION
The shear-limit model presents a difficulty in solving equation (18) that was not present with the time-limit model, namely, the lack of an explicit expression for s**(t), except in the two limiting cases a = 0 and b = 0. Since the former case leads to simpler expressions and is a necessary condition for equation (25), and since the purpose of the numerical solution is primarily illustrative, such a solution will be undertaken for this case only. (What this means is a constant active curvature of magnitude b in the active region.) Equation (18) can be put in dimensionless form by introducing the dimensionless time variable i = ct, letting u [introduced in equation (20)] be a function of irather than t, and defining the function, t(i) = KS*(~) and the parameters : B = b/K, (26) D = uy,/b.
(27)
With the auxiliary functions : g(f) = et@)
f(i) = cosht(i)= t [g(i)+&]
equation
1 3g(i) < eD h(i) = cash us**(Z) = 3 g(i) eeD +CL g(t) 1 , g(i) > eD1; ( (18) (with a = 0) becomes : i+T
u(i+T)+
J ~(7’)di’+B 0
I-h(i)+i[f(t’)-h(i’)ldi’ 0
1
= g(f)
(28)
with T = CT given by: II( 7;) + i u(i) di = I.
(29)
96
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AND
J.
3.
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The numerical solution proceeds as in II, with the integrals replaced by sums according to the trapezoidal rule. The increment in f is l/r, where Y is a large integer. If g, z g(n/r), then, so long as g, < eD at each step we have : (30) where Q, is made up of either given or previously computed quantities. When multiplied by g,,, equation (30) becomes a quadratic equation that can be solved explicitly and that will have real roots (with at least one positive) independently of Q, if: +-B(lf$)]
> 0.
Since I’ is large, equation (31) is equivalent to: O
