Heuristic Considerations On Fast Axopodial Contraction

In The Heliozoa

by

Bruce Eric Davis

April 1994

Institute of Statistics Mirneo Series #2261

BMA Series #38 1

1. INTRODUCTION

The Heliozoa are a polyphyletic assemblage of predaceous freshwater and marine protozoa whose globose cell bodies (diameter"" 8-2600 ]lm) bear many (20-several hundred) radially disposed slender protrusions called axopodia (diameter"" 0.1-10 ]lm; length", 30-500 ]lm). Each axopodium is stiffened by an axoneme consisting of an array of microtubules (numbering 3- '" 500; each ~ 25 nm in outside diameter) cross-linked in a taxonomically diagnostic pattern (Febvre-Chevalier, 1985;

Siemensma, 1991).

Axopodia function in locomotion, cell fusion, cell division, egestion, attachment to substrata, and prey capture. Ciliates, flagellates, and small metazoa that happen accidentally to collide with an axopodium may stick to it owing to a superficial mucous coat discharged by extrusomes. Prey thus ensnared may then be pulled to the cell body by fast contraction of the axopodium (duration of contraction < 100 ms; Suzaki et al., 1980a). Ingestion of prey through phagocytosis ensues and takes place on a time scale of tens of minutes.

Fast axopodial contraction proceeds by the breakage of the axonemal microtubules along their lengths, but the speed of contraction is much greater than rates of microtubule breakdown measured in vitro and in other organisms. Hence, either some component of the mechanism of fast axopodial contraction is faster than a similar component of these processes, or a different mechanism is involved. The most tenable hypothesis supported by experiments to date appears to be that an action potential causes an influx of Ca2+ across the axopodial membrane, which releases some unknown second messenger(s) (e.g., inositol l,4,5-trisphosphate or Ca2+ itself) to cleave the axonemal microtubules by means of a bound second messenger-activated severing factor. A conformational change in a centrin- or spasmin-like calcium-binding protein then drives contraction. (A flowchart summarizing this scheme appears on the next page.) The present paper explores some speculations as to how fast axopodial contraction occurs; a summary of the remaining sections of the report follows below. 2

HYPOTHETICAL CONTRACTION SCHEME

ACTION POTENTIAL

CALCIUM INFLUX

ECOND MESSENGE _

MICROTUBULES

ONTRACTILE ORGANELL 3

SUMMARY OF REMAINING SECTIONS

2. Estimation of length constant

Standard calculations reveal that the length constant is about twice the axopodial length. This fact, together with recordings made by Febvre-Chevalier et al. (1989) of conduction speeds up to 3 mis, imply that the action potential propagation time is a small fraction of the total contraction time.

3. Surface electrostatic charge

Surface electrostatic charge is not expected to impede axopodial motion because the Debye length is very small (of the order of nm).

4. Second messenger diffusion

Calcium diffusion times, bas~d on the classical value of the diffusion coefficient (Hodgkin and Keynes, 1957), are in the 7-50 ms range for large axopodia, and in the 9-120 J-ls range for small ones. If a recently revised value of the diffusion coeficient is used (Allbritton et al., 1992), the diffusion times increase to as much as 660 ms for large axopodia, suggesting that calcium diffusion might be rate-limiting. However, internal calcium stores, possibly activated by the action potential, may appreciably shorten diffusion distances. Diffusion times of 0.5-4.5 ms are expected for large axopodia in this case. For inositol 1,4,5-trisphosphate, diffusion times of 1-88 ms are expected for large axopodia, and 16 J-ls for small ones.

5. Calcium concentration to trigger contraction

For a number of protozoa under in vivo and in vitro conditions, the minimum calcium concentration required for fast contraction is reported. To induce contraction only in the fast mode in a certain marine heliozoan, the threshold calcium concentration in the external medium is 10-6 M.

6. Estimate of force exerted by contractile organelle

Using published values for stresses developed by actomyosin and by spasmonemes, together with cross-sectional areas estimated from electron micrographs, a value for the force produced by the putative contractile organelle in large heliozoa is obtained that is close to that measured experimentally. 4

7. Strain rate

Published data for stalk contraction (isochronous with axopodial contraction) indicate that the time course is exponential (Febvre-Chevalier et al., 1992). A calculation shows that this observation implies that the strain rate is constant.

8. Reynolds number

Reynolds numbers calculated on the basis of the length of the axopodium are certainly < 7 and probably at most of the order of 10-1. This implies that fast axopodial contraction occurs in a viscous flow regime wherein inertial effects can be neglected.

9. Slender body theory

Treating the axopodium as a slender right circular cylinder translating axially normal to an infinite plane wall yields estimates of the force and power of contraction, which are summarized in section 11.

10. Sliding plane theory

Treating the axopodium as an infinite plane sliding at a specified angle into a stationary, infinite plane wall yields estimates of the force and power of contraction, which are also summarized in section 11.

11. Table of estimates of force and power

The caricatures of the two previous sections are applied to a large and a small heliozoan to yield estimates of force and power during contraction that are similar to or below those determined by experiments.

12. Estimates of energy uptake from food

A final consideration is how much energy, in terms of the amount obtained from prey capture, would be spent in the contraction that captures the prey. The slender body theory of section 9 is used in the calculation.

13. Detailed research plan outline

The preceding sections constitute but a preliminary survey of the problem of fast axopodial 5

contraction. This final section sets forth a detailed outline of a theoretical and experimental research plan to elucidate the mechanism. A team of investigators would require several years to carry out this ambitious program. 6

2. ESTIMATION OF THE LENGTH CONSTANT FOR PROPAGATION

OF AN ACTION POTENTIAL IN FAST AXOPODIAL CONTRACTION

According to the idealized passive one-dimensional cable model, the potential across the membrane

decays exponentially:

V(x) = V(O) e -Ixll"0 , (1)

(2)

where V = potential across the membrane (mV), x = position along the length of the cable (cm), 2 "0 =length constant (cm), d =cable diameter (cm), Rm =membrane resistance (0 cm ), and

Ri = resistivity of cytoplasm (0 cm) (Nossal and Lecar, 1991, p. 249). Note that "0 is the distance over which the membrane potential falls to i of its initial value. That is, V("o) = V~O) . Shepherd (1974, Fig. 13, p. 64) gives a graph of "0 vs. d for three values of ~~ appropriate for unmyelinated 1 mammalian nerve processes. In the sequel, I read "0 from the middle curve of Fig. 13.

From Fig. 13 one has that for d = 0.1 JJm, corresponding to axopodia of Ci/iophrys marina and

Heterophrys marina, "0 ~ 100 JJm, which is twice the typical axopodial length (50 JJm) of these heliozoa (Davidson, 1975).

For Echinosphaerium nucleoli/um, d = 5-10 JJm (TUney and Porter, 1965, p. 341), for which

Shepherd's graph indicates "0 ~ 700-1000 JJm. Tilney and Porter (1965, pp. 328, 341) state that the axopodia were as much as 400 JJm long.

The foregoing results are borne out by direct calculation of the length constant from equation (2).

Measurements of am for Heliozoa have not been made, but for ciliates, values corrected to allow for the surface area of the ciliary membranes are 14.3 kO cm2 for Paramecium, and 73 kO cm2 for Stentor 2 (summarized by Wood, 1982, p. 538). [By comparison, for smooth muscle, Rm =40 kO cm (Jack et al., 1983, p. 23).] Typically, Ri = 100 0 cm =0.1 kO cm (Jack et al., 1983, p.23), and for axopodia d = 1 x 10-5_1 x 10-3 cm. Substituting these values in equation (2), we find that for am = 14.3 kO 2 2 3 2 2 3 cm , "0 =2 x 10 -2 x 10 JJm, whereas for Rm =73 kO cm , ~o =4 x 10 _4 x 10 JJm. Observe that 7

these length constants are comparable to those read from Shepherd's graph for unmyelinated

mammalian nerve processes.

GONGL USIONS

If fast axopodial contraction is actually triggered by an action potential in the heliozoa named ~ above, and if values of are close to those for mammalian neural processes, then Ao appears to be I slightly greater than the typical lengths ofaxopodia. Hence, as Dr. Tom Kepler points out

(conversation, 6/23/93), we may be confronted with an ambiguous intermediate case where Ao "" Lo'

~ ~ rather than AO Lo or AO Lo' where Lo is the initial (uncontracted) length of the axopodium.

~ Note that if AO Lo' regeneration would be necessary if an action potential were to serve as a trigger.

For Ao ~ Lo' attenuation of an action potential would be negligible; thus an action potential could easily serve as a trigger in this case.

The values of AO found above are greater than the lengths ofaxopodia. Thus, we may conclude that the action potential undergoes negligible attenuation and the axopodial cytoplasm must be nearly isopotential. As Wood (1989, pp. 361-362) points out, action potentials function chiefly in protozoa to amplify transmembrane potentials and Ca2+ fluxes, a role which has been well-documented in Stentor and Paramecium and which would clearly be of importance in triggering fast axopodial contraction.

In Actinocoryne contractilis, action potentials propagate from head to stalk base at 3 m s-l

(Febvre-Chevalier et al., 1989, p. 241), which implies spread over the maximum axopodiallength (500

Jlm) in 2 x 10-1 ms, much less than the observed latency (7 ± 3 ms; Febvre-Chevalier and Febvre,

1992, p. 587).

Wood (1989, p. 368) has suggested that the propagation of electrical impulses and cell contraction observed in certain ciliates (e.g., Stentor, Vorticella) can be accounted for by the spread of action potentials along the membranes enclosing intracellular Ca2+ stores. Wood notes that these vesicles are commonly in apposition with the plasma membrane, and that their small diameters and the low ionic strength of their contents imply a length constant smaller than that for the surrounding cytoplasm.

(Wood's assumption that the ionic strength of the vesicle contents is low seems doubtful to me.) 8

Hence, the vesicle stores would not be isopotential, and action potentials could serve as signals to bring about the sequential release of Ca2+. It is conceivable that such a scheme might operate in the larger

Heliozoa to couple the passage of an action potential along the axopodial membrane to the discharge of intraaxopodial Ca2+ required to power the contractile organelle. (See, e.g., Matsuoka et al., 1984,

Figs. 3-4 for electron micrographs showing apposition of Ca2+ vesicle and plasma membranes in

Echinosphaerium akamae.) 9

3. ON TIlE ELECTROSTATIC CHARGE AT THE CELL SURFACE

The aim of the present section is to show that electrostatic forces due to surface electrostatic

charge can have only a negligible effect on the motion of a contracting axopodium. This conclusion is

supported by a simple calculation. Owing to lengthy extracts from publications, I enclose my own

parenthetical comments in brackets.

De Loof (1986, p.254) says that "From biochemical data, it is well established that the cell

surface carries a net negative charge (Dolowy, 1984). [Matsuoka et al. (1984a, 1986) and Shigenaka et

al. (1989a) confirm this generalization for Echinosphaerium akamae; see Vommaro et al. (1993) for a

recent determination of negative surface charge in a f1agellat.e.] Fixed surface charges result in the

formation of a diffuse electrical double layer in which the charge at. t.he surface is balanced by charges

of opposite sign in the medium and by the dipole moment of water molecules immediately adjacent to

the surface."

Probstein (1989) discusses diffuse electrical double layers at several points III his text. The following are selected excerpts.

"Electroneutrality or the absence of charge separation holds closely in aqueous electrochemical solutions, although not necessarily organic solutions, everywhere except in thin regions near charged boundaries. These regions are termed double layers or Dcbye sheaths and have thicknesses on the order of 1-10 nm. The double layer is important when we consider very small charged particle interactions and charged surface phenomena, but is generally unimportant with respect to bulk flow characteristics." (Probstein, 1989, p. 43)

Probstein (1989, p. 187, eqn. 6.4.5) gives the Debye shielding dist.ance or Dchye lcngth for an infinite plane double layer as 1 A -( £RT )2 (1) D- 2F2z2c •

l where AD is the Debye length (m), R is the gas constant (8.314 .J K-Imol- ), T is the absolute

4 l temperature (K), F is the Faraday constant (9.648 x 10 C mor ), z is the valence of the ion in

3 question, c is its concentration (mol m- ), and £ is the dielectri<; constant (permittivity) of water. Note 10

that (= (O(r' where (0 is the permittivity of a vacuum ( = 8.851 X 10- 12 C2 J'r1m 2), and (r is the relative permittivity of water. At 25° C( =298 K), (r =78.3. It seems reasonable to use this value in d( the sequel because dT is small (Lide. 1993, 6-148) and 298 K is near t.he t.emperatures at which cert.ain heliozoa have been cultured. In particular, Ando and Shigenaka (1989) grew Echinosphaerium akamae at 20° C (= 293 K), and Davidson (1975) noted generation t.imes for Heterophrys marina and

Ciliophrys marina at ~ 21° C( = 297 K). For a symmetrical electrolyte in aqueous solution at 298 K, the Debye length is given by

nm (2)

(Probstein, 1989, p. 187, eqn. 6.4.6). Probstein observes that "for a univalent electrolyte the Debye

2 3 3 length is thus about 1 nm for a concentration of 10 mol m- and 10 nm for 1 mol m- ."

Now I compute approximately t.he Debye length of a double layer due to Ca2+ on the surface of

H. marina and C. marma. Davidson (1975) states that for the seawater used, c = [Ca2+J = 10

3 mM = 10 mol m- . Taking z =2 (divalent electrolyte) and substituting the foregoing value of c in equation (2) above yields '\0 = 1.5 nm. Since'\o is much less t.han the diameter (l00 nm) and length of an axopodium (50 pm), and the distance between evaginations of the contracted axopodial membrane (Davidson, 1975, Figs. 86-91), it must be that electrost,atic forces offer negligible resistance to contraction.

SO}vIE FURTHER CONSlDERA TJONS

1. Is equation (1) for the Debye length of an infinite plane double layer the same for interface shapes appropriate for heliozoa? Equation (1) also holds for spheres (Russel, 1987, p. 23) and is accurate within a factor of 2 for cylinders (Katchalsky et aI., 1966, pp. 306-307). Thus the conclusion drawn in the present section is unchanged for these more realistic shapes.

2. How does the Debye length calculation turn out for Na+? For Na+, appropriate values are c =460 mM =460 mol m-3 and z = 1. Inserting these data in equation (2) affords '\0 = 0.45 nm.

3. How can one calculate the Debye length for a double layer involving a realistic mixture of ions? For seawater, one may estimate a lower bound on th~ Debye length by taking z = 2 as the 11

maximum valence and letting c = 533 mol m-3 be the total concentration of the principal cations. The latter figure is arrived at by considering K+ (10 mM), Na+ ('160 mM), Ca2+ (10 mM), and Mg2+ (53 mM). We assume that the number of cations and anions in solution is equal. Then equation (2) yields

AD = 0.21 nm as a lower bound.

4. How does contact with anion-exchange resrn beads (Matsuoka et al., 1986; Ando and

Shigenaka, 1989) provoke fast contraction? One may conjecture that anion-exchange resin beads make the potential at the outer face of the axopodial membrrlne more negative without altering the potential at the cytoplasmic face, thereby decreasing the electric field across the membrane and opening voltage­ gated Ca2+ channels (see Hille, 1992, pp. 460-461; Jack et al., 1983, pp. 3'18-3'19).

5. Probstein (1989, pp. 187-188) includes useful additional commentary on the Debye length:

"From equation [1] or [2] it can be seen that AI) decreases inversely as the square root of the concentration. Physically this is a result of the fact that. t.here are more counterions per unit of depth.

The Debye length also decreases with increasing valency because fewer ions are required to equilibrate the surface charge. More importantly, AD increases as the square root of RT. That is, without thermal agitation the double layer would collapse to an indefinitely thin layer."

"From the above considerations we can now define what is meant by electrically neutral solutions. If the dimensions of the system L are much larger than AI), then whenever local charge concentrations arise or external potentials are introduced into t.he solution they are shielded out in a distance short compared with L, leaving the bulk of the solution free of large electric potentials or fields. Based on a Debye length of 1-10 nm, the assumption of electrical neutrality is generally justified for the problems so far considered. However, ... in the case of very small charged microscopic capillaries, such as are characteristic of membranes and finely porous media, the double layer is central to the calculation of the solut.e and ion fluxes."

On p. 188, Probstein derives the Debye-Hiickel approximation and obtains therefrom an alternative expression for the Debye length:

¢J = ¢Jw exp (- ~~ ), (3) where ePw is the wall potential. "The Debye length is thus seen to be the ~ decay distance for the potential and electric field at low potentials. Close to the charged surface where the potential is relatively high and Debye-Hiickel approximation inapplicable, the potential decreases faster than the exponential fall-off indicated."

"The potential ePw can be related to the charge density at the surface by equating the surface charge with the net space charge in the diffuse part of the double layer." 13

4. CALCULATION OF THE MEAN FIRST PASSAGE TI.\IES FOR Ca~H AND IP 3 DIFFUSION FROJ\I THE AXOPODIAL MEMBRANE TO THE AXONEME

The present section estimates the mean first passage times for diffusion of Ca2+ and IP3 (inositol

1,4,5-trisphosphate). These chemical species are typical second messengers in cells, and may be

responsible for triggering fragmentation of the axonemal microtubules by binding to a microtubule-

severing factor (perhaps akin to the microtubule-cleaving factor isolated by Vale (1991) from Xenopus

oocytes or the F-actin-severing protein, gelsolin) postulated here.

I make the following simplifying and idealizing assumptions:

1. There are no interactions among messenger molecules or ions.

2. There is no axial gradient in messenger concentration. That is, the messenger concentration is space-clamped, so ~~ = 0, where z is the axial coordinate.

3. Diffusion occurs in a two-dimensional annulus such that the inner aspect of the axopodial membrane is the source of Ca2+, whereas the axoneme, viewed as a disk, is a sink that binds Ca2+ irreversibly. I assume that there are no other sources or sinks of Ca2+. (See the annexed diagram.)

1---- A ----I

4. Geometrical hindrance due to the filamentous structure of cytoplasm is negligible. This assumption seems reasonable because mesh spacings in the cytoskeleton are ...... 100 nm (Gershon et al.,

1985, p. 5033), but calculations show that diffusion of molecules of radius 0.3 nm would begin to be appreciably impeded at spacings from 10-15 nm (Blum et al., 1989, p. 1000). 14

If a denotes the diameter of the axoneme and A represents the diameter of the axopodium, then the mean first passage time T for diffusion to the axonemal disk is

2 T = 8DA In (A)a , (1)

where D is the diffusion coefficient (Hardt, 1981). For Ca2+, typical published values of D in cytoplasm are 6 x 10-10 m2 s-1 (Hodgkin and Keynes, 1957), 4 x 10-10 m2 s-1 (Kargacin and Fay, 1991), and 3 x 10-10 m2 s-1 (Roberts, 1993, p. 74). However, a recent determination reports

11 2 10 2 DCa = 3.8 X 10- m s-1 (Albritton et al., 1992). Arbitrarily, I shall take 5 x 10- m s-1 as a middling compromise "classical" value of DCa' and I shall call 3.8 x 10-11 m2 s-1 the "revised" value.

2 For IP , Albritton et al. (1992) found D = 2.83 X 10-10 m s-l. Measurements of a and A are seldom 3 1P3 reported in the literature.

Echinosphaerium nucleolilum

Hoffman et al. (1983, p. 375) found that isolated axonemes measured ~ 1.5 pm in diameter, which they maintain is consistent with electron micrographs by Tilney. According to Tilney and

Porter (1965), the axopodia are 5-10 pm in diameter at levels remote from the tip. Using the classical value of DCa' and putting a = 1.5 pm and A =10 pm in equation (1), we obtain

1 T = 5 X 10 ms.

Similarly, for a = 1.5 pm and A = 5 pm,

T = 7 ms.

Hence 15

1 7 ms ~ T ~ 5 X 10 ms (classical).

11 2 Repeating the foregoing calculations with DCa = 3.8 X 10- m s-l yields

2 9 X 101 ms ~ T ~ 6 x 10 ms (revised).

For IP3' equation (1) affords

Echinosphaerium akamae

From an electron micrograph of a cross-section of an axopodium in Matsuoka and Shigenaka

(1984b, Fig. 3), one can readily estimate the diameters of the axoneme and axopodium. Since the magnification factor is x 33,000, 1 cm <-+ 0.30 pm on the published micrograph. The diameters of the images of the axoneme and axopodium are 3 cm and 12 cm, respectively, so a= 0.9 pm and A = 3.6 pm. Thus in the classical case, we obtain for the Ca2+ mean first passage time

T _ (3.6 X 10<6 m)2 In (12) - 8(5 x 10-10 m2 s

T =4.5 ms.

We may repeat the calculation for another electron micrograph of a cross-section probably taken at a different level (Shigenaka et al., 1989b, Fig. 2). Here the a:

T =0.54 ms.

It follows that

0.54 ms ~ T ~ 4.5 ms (classical).

Similar calculations using the revised value of DCa yield the range of times

1 7 ms ~ T ~ 6 X 10 ms (revised).

For IP3' we obtain the time interval

Ciliophrys marina

Davidson (1975) determined that the axoneme is a triad of microtubules in which the links are 15 nm long. Assuming that the microtubules are 25 nm in diameter, it seems reasonable to approximate the axoneme as a disk of diameter 50 nm. Therefore, considering Ca2+ diffusion and taking a = 50 nm and A = 0.1 J.lm,

7 T _ (1 X 10- m)2 In (100) - 8(5 x 10-10 m2 s·l) 50

T = 9 J.lS (classical).

11 2 The same calculation with DCa = 3.8 X 10- m s-1 gives

T = 1.2 x 102 J.ls (revised). 17

For IP3,

TIP = 16 J.ls. 3

GaNGL USIONS

To assess the mean first passage times computed above, we consider the observed latencies for fast axopodial contraction. Davidson (1975, p. 42) remarked that in high-speed films of Heterophrys marina "there is apparently a latent period of 10-20 ms before contraction actually begins, but this is difficult to observe in many cases because the time period between frames is greater than or equal to the latent period". Febvre-Chevalier et al. (1989, p. 239) reported more reliably that the latency for axopodial and stalk contraction in Adinocoryne contractilis following external mechanical and electrical stimulation and intracellular current injection is 7 ± 3 ms (n = 80). Since the axopodia of A. contractilis are from 50-500 J.lm long (Febvre-Chevalier and Febvre, 1992, p. 587), a range which includes the axopodia treated here, we are probably justified in using the latency of this heliozoan as a benchmark. It should be pointed out, however, that no data appear to be available on the diameters of

A. contractilis axopodia.

In the large axopodia of heliozoa of the genus Echinosphaerium, Ca2+ diffusion times based on the classical value of the diffusion coefficient range between 0.54 ms and 50 ms, whereas those founded on the revised value lie between 7 ms and 6 x 102 ms. That the larger estimates exceed the measured latencies noted above would suggest that Ca2+ diffusion might be rate-limiting in large axopodia.

However, internal Ca2+ stores (Matsuoka and Shigenaka, 1984, 1985; Matsuoka et al., 1985), possibly discharged by the action potential, may appreciably shorten diffusion distances. For IP3' time 18

2 discrepancies are harder to reconcile than for Ca + because IP3 derives from the hydrolysis of a plasma membrane-bound precursor, phosphatidylinositol 4,5-bisphosphate (PIP2), and is not released from cytoplasmic stores. In Echinosphaerium, 0.9 ms ~ TI?3 ~ 8 X 101 ms. Observe that the even smaller

10 2 diffusion coefficient of ATP in cytoplasm (DAT? = 1.50 x 10- m s-1; Kushmerick and Podolsky,

1969) implies that the use of cyclic nucleotides (e.g., cAMP) as second messengers probably would not expedite signaling in large axopodia.

In the smallest axopodia, exemplified by those of Ci/iophrys marina, the very brief mean first passage times show that second messenger diffusion occurs amply fast. This conclusion comports with observations of the time course of ciliary beat reversal (cilium diameter ~ 0.2 J.lm): Ca2+ influx from the medium is both necessary and sufficient to induce beat reversal in Paramecium (summarized by

Harold, 1986, pp. 416-419, 501). There is no ultrastructural evidence for intraaxopodial Ca2+ stores in C. marina or H. marina; indeed, apart from the putative contractile element, the axopodia of the latter heliozoan lack mitochondria and other recognizable organelles (Davidson, 1975).

Although the role of Ca2+ as the source of chemical potential for the spasmoneme of vorticellids is well-established, few studies have been made of second messenger signaling in the contractile appendages of protozoa. Evans et al. (1988) used drug treatments to implicate IP3, cyclic nucleotides, and Ca2+ in the control of tentacle contraction in the suctorian He/iophrya erhardi. Suctorian tentacles are comparable in size to axopodia, but their latency (3.7 ± 0.1 s) and contraction time

(18.2 ± 0.2 s) are three orders of magnitude greater (Evans et al., 1988, p. 384; mean ± SE, n = 161).

Moreover, the tentacular membrane is inexcitable and contraction results from the relative sliding of intact microtubules, possibly driven by actomyosin.

It is of interest that in connection with his studies on muscle, A.V. Hill (1948) derived an analytical solution for the problem of inward diffusion of a substance liberated from the wall of a cylinder of radius a. To obtain this result, Hill first solved the initial-boundary value problem for diffusion from an annular source between coaxial cylinders of radii a and b (a>b). The concentration c(r,t) is governed by 19

c(r,t) = cO' b ~ r ~ a

c(r,O) = 0, 0 ~ r ~ b

dcdr -,- 0 r- a

c(O,t) < 00 'v't > o.

The solution is

th where ai > 0, J 1(ai) == 0, and Jk() is the k order Bessel function of the first kind.

Hill then let the thickness of the annular source shrink to zero in the limit as b ... a in equation

(2). Accordingly, conservation of mass implies

(3)

By the mean-value theorem with c E (aibja , ai)'

(4)

But J 1(ai) == O. Thus,

(5)

From Gradshteyn and Ryzhik (1965, Section 8.472, Formula 1, p. 967), Bessel functions of order v obey the recurrence relation 20

z Jz Zv (z) + II Zv (z) = z Zv-l (z).

Putting Zv () = Jv () with v = 1, and evaluating at z =c, we may write

(6)

Since Otib/a < c < Oti ' C -+ Oti as b -+ a, and

Therefore, from (6),

and by substitution in (5),

(7)

as b -+ a.

Insertion of expressions (3) and (7) into equation (2) gives

(8)

Hill (1948, Fig. 2, p. 450) plots c~r,t) vs. D; for values pertinent to diffusion in muscle. 00 a 21

5. [Ca2+J IN RELATION TO CONTRACTILE, OTHER MOTILITY PROCESSES IN PROTOZOA,

INCLUDING THE HELIOZOA

Most of the calcium concentrations reported below pertain to the external medium and are denoted

2+ ~+ by rCa Je' Intracellular free calcium concentrations are denoted by [Ca- Jj . Sources of data are given parenthetically after the scientific names of the organisms.

Heliozoa

A ctinocoryne contraclilis (Febvre-Chevalicr and Febvre, 19S9)

Total inhibition of contraction.

64% slow contraction (6-15 s).

13% fast contraction.

23% no contraction.

27% slow contraction.

67% fast contraction.

6% no contraction.

Only fast contraction.

The authors consider (p. 587) that the threshold for contraction is [Ca2+Je = 10-7 M.

Heterophrys marina and Ciliophrys marina (Davidson, 1975, p. 49) 22

No contraction; axopodia "become very brittle and are

easily broken".

Transformation to fast-swimming flagellate.

Axopodia contract to ::::: one-half their extended length.

Normal contraction.

Echinosphaerium akamae (Matsuoka and Shigenaka. 1984, pp. 426-427; MatslJoka et al., 1985, pp. 69-70)

"Hardly any axopodial contraction." (I have not

2 found a statement of the threshold [Ca +le.)

Axopodia shorten slowlv in the presence of the divalent cation ionophore A23187 (1 J1.g mr1)

and Ca2+. (Time to shorten to 49% of extended length = 10 min.) A much smaller extent of

2 shortening was seen at lower [Ca +le.

Echinosphaerium eichhorni (Schliwa, 1976, 1977)

2 5 1 Slow contraction was observed for [Ca +le as low as 10- M in the presence of 0.5 J1.g mr 2 A23187. The speed of contraction increases with [Ca +le: half-maximal contraction was observed

after 35 min in 1 x 10-4 M Ca2+ and 12 min in 5 x 10-4 1\1 Ca2+. In tile most concentrated

solution tried ([Ca2+]e = 1 x 10-3 M), axopodia shorten to 20% of their extended length within

4 min of exposure.

Is A23187 a carrier or a channel? See Harold (1986, pr. 360-362) for a comparison of 23

transport rates for these classes of ionophores. The roint is that the Ca2+ flux due to A23187

may be much less than that for natural memhrane channels. See also Schliwa's papers for refs. on

A23187.

Vorticellid Spasmonemes

Vorticella (Amos, 1971; Weis-Fogh, 1975, pp. 8"1-85)

Glycerinated stalks remain extended in 10-8 f"Y Ca2+. The threshold concentration for

contraction is 5 x 10-7 M. Stalks remain contrilctcd ill. [Ca2+]e = 10-6 I\L The external work

done in contraction in vivo can be accounted for hy a change of [Ca2+]i from 10-8 to 10-6 M.

See Amos on the rapid cycling of glycerinated spasornoneme on a time scale comparable

to the process in vivo.

Zoothamnium genicu/atllm (Weis-Fogh and Amos, 19(2)

Glycerinated colonies were fully extended in [Ca2+]e = 10-8 1\1, and contracted with stalk

2 flexure in 10-5 M Ca + "a,<; in life but taking several seconds and without suhsequent relaxation"

(p. 303). The isolated spasmoneme shows arpropriate hehavior at the same values of [Ca2+]e'

Ciliate Cell Body Contraction

B/epharisma japonicllm (Matsuoka et aI., 1991)

Results for Triton-extracted models:

The threshold concentration for model elongation is [Ca2+]e = 3 x 10-7 M. The authors

estimate that [Ca2+]. = 10-7 M. Drug treatment implies that elongation is ATP-dependent, 1 • 24

whereas contraction is ATP-independcnt. Elollgated models contract when transferred from

[Ca2+]e = 10-10 M (with ATP) to [Ca2+]e = 10-0 1\1 (without ATP).

Stentor (Huang and Mazia, 1975, pp. 393-391)

Results for EGTA-treated cells:

Cells remain extended in 10-8 1\1 Ca2+.

Cells contract in 10-6 M Ca2+.

Spirostomum (Hawkes and Holberton, 1975; Ishida and Shigenaka, 1988)

Hawkes and Holbert,on found that rnyonemcs do not. evince velocit.y-Ioad relations

appropriate for muscle, but behave instead like an elastomer. (That is, the force developed

is a constant independent of load.) These authors reckoned (p.601) that enough energy for

the work of contraction could be obtained by varying [Ca2+]i from 10-7 M to 10-5 M, thereby

binding 1.93 x 10-14 mol Ca2+. Ishida and Shigenaka's more recent researches on Triton-

extracted cells are consistent with the foregoing range of concentrations.

Ciliary/Flagellar Beat Reversal

Blepharisma japonicum (Matsuoka et aI., 1991)

Results for Triton-extracted models:

Models swim forward for [Ca2+]e < 10-6 ~'l. but swim backward or tumble for [Ca2+]e > 10-6 M.

Chlamydomonas reinhardtii (Melkonian and Rohenek, 1984, p. 244)

Results for isolated flagellar apparatus: 2S

Same as for B/epharis71la, viz., forward swimming for [Ca2+]e < 10-6 ~I, backward swimming for

[Ca2+]e> 10-6 M. [Melkonian et al. (1002, p. 187) assert that "as in other eukaryotic cells, the

the cytosolic free Ca2+ concentration in the nagellate green algae is presumably also kept at a low

and constant level of about 5 x 10-8 l'vl Ca~H".]

Paramecium (literature epitome in Gregson et aI., ] 99:lb, p. (98)

Results for det.ergent-extracted cell models:

Forward swimming for [Ca2+]e:::; 10-7 M, backward swimming for [Ca2+]e ~ 10-6 M.

Contractile Dinophycean Flagella

Peridinium inconspicullm (Hohfeld et aI., ]088, p. 21)

Results on the isolated transverse nagellum:

"Flagella supercoil within a few seconds t.o a highly cont.racted st.at.e" in 5 x 10-5 M Ca2+.

Ceratium tripos (Maruyama, 1985; H6hfeld et. aI., 1988)

Results on reactivated longitudinal nagella:

Contraction requires [Ca2+]e ~ ] x 10-5 l\L

Oxyrrhis marina (Godart and Huitorel. ]902)

Results on longitudinal flagella still attached to permeabilized cells:

[Ca2+]e = 1 x 10-7 M 50% contraction

[Ca2+]e = 3 x 10-7 M 50% autotomy

[9a2+]e = 9 x 10-7 M 00% autot.omy 26

Prymnesiophvcean Haptonemata

Ghrysochromu/ina acantha (l\lelkonian et aI., 1992, p. 207; Gregson et aI., 1993b)

Gregson et a!. (pp. 686, 695) show that haptonemal coiling is Ca2+-dependent.

They establish that "an external concentration of hetween 10-7 1'v1 and 10-6 ]',,1 Ca2+ is the

threshold below which the frequency of coiling on cell death is reduced" in cells treated with

Ca2+jEGTA buffers before fixation. Melkoniall et al. state that it is not known whether centrin

causes coiling of the haptonema.

GONGL USJONS

The axopodia of A. contracli/is and E. n!.:amne can undergo fast contraction in solutions spanning the same range of [Ca2+]e values (typically ...... 10- 7_10- 5 :\1) t.hat. permit.s Ca2+-dependent motility in most other intact and permeabilized protozoa invest.igated to date. By contrast, the axopodia of H. marina and G. marina contract normally only at much higher external Ca2+ concentrations (10-2-10-1

M). It may be that the absence of vesicular Ca2+ stores in these very thin motile appendages means that an especially steep Ca2+ concentration gradient must he set up to supply enough chemical potential energy to drive contraction. In this connection, it is of interest that in Paramecium, in which an influx of Ca2+ across the ciliary m~mbrane serves merely as a signal for heat reversal, not as a source of chemical potential energy for beating, the threshold rCa2+]e for beat reversal is only 10-6

M. 27

6. ESTIMATION OF THE FORCE EXERTED BY THE CONTRACTILE ORGANELLE

According to Bereiter-Hahn (1987, Table 2, p.10), Portzehl (1951) found that the stress exerted by an actomyosin thread prepared from skeletal muscle is 2.5 x 104 N m-2. Interestingly, the tensile stresses exerted by vorticellid spasmonemes in vivo and in vitro are of the same order of magnitude

(Weis-Fogh, 1975, Table I, p.94):

In vivo:

Vorticella 104 N m-2

Garchesium 4 x 104 - 8 x 104 N m-2

In vitro:

Zoothamnium

Now suppose that the contractile organelle of the Heliozoa can develop stresses of the same order of magnitude as the stresses produced by actomyosin and spasmonemes. For simplicity, I shall take this stress to be 5 x 104 N m-2. Suppose further that in Echinosphaerium akamae the X-body is the contractile organelle. Then one JVay approximate the cross-sectional area of the image of the X-body

In a certain particularly clear electron micrograph (Shigenaka et al., 1989b, Fig. 2) as the area 7 enclosed by a rectangle measuring 2 cm x 4 cm. Taking 1 cm +-+ 1.1 x 10- m, it follows that 8 cm2 +-+

1 x 10-13 m2. The force exerted is thus (5 x 104 N m-2)(1 x 10-13 m2) =5 x 10-9 N =5 x 103 pN.

Note that this value is very close to the force of fast axopodial contraction, (4.1 ± 0.9) x 103 pN, measured by Suzaki et al. (1992a) in E. akamae. It is comparable to the estimated drag (8.6 x 103 pN) on the cell body of a contracting Vorticella (Amos, 1975b), but greater than the forces exerted by swimming sperm of various phyla (5-350 pN; Green, 1988).

GONGL USIONS

The foregoing argument demonstrates that the force ofaxopodial contraction could be produced chemomechanically by an actomyosin- or spasmin-like Ca2-f:-binding protein. Although there is 28

ultrastructural and experimental evidence for proteinaceous intraaxopodial contractile organelles (see, e.g., Davidson, 1975; Shigenaka et al., 1982; Matsuoka and Shigenaka, 1985), their chemical composition has not been investigated. Centrin or a centrin-like protein (possibly a spasmin) has been detected by immunofluorescence in the base of the microtubule-containing contractile stalk and near the microtubule organizing centers for the axopodial axonemes of Pseudopedine//a elastica (Koutoulis et al., 1988; Melkonian et aI., 1992, p. 207). It is not known whether centrin or a homologue of centrin also occurs within the axopodia of this heliozoan. 29

7. AN OBSERVATION ON THE TIME COURSE OF FAST AXOPODIAL/STALK CONTRACTION

The exponential shortening curves reported by Febvre-Chevalier (1981) and Febvre-Chevalier and

Febvre (1992) for the stalk and axopodia of Actinocoryne contracti/is imply that

t ~t = -k, L(O) = Lo' k>O (1) or

f = -k, f(O) = 0, k>O. (2)

That is, the strain rate, f, is constant. Note that this result is perfectly consistent with the definition of natural or true strain, fT' appropriate for large deformations (Wainwright et al., 1976, eqn. 2.4, p.9):

(3)

I show this by writing

fT =In (t~ = - kt (4) and differentiating throughout with respect to t:

(5) whence

(6) which is identical with equation (1) above. 0

Now I determine the value of k and the functional form of L = L(t).

The solution of equation (1) is

L(t) =Lo exp( - kt) (7) whence o k =r1ln (LL.) (8) From Febvre-Chevalier and Febvre (1992, p. 588) the mean half-time for stalk contraction in

Actinocoryne contracti/is is 4 ms (n = 20). Hence equation (8) yields

k =! In 2 =0.17 IDs-I. (9) 30

The mean initial stalk length is 185 J.Lm (n = 20). Thus, the time course of shortening, equation (7), becomes

L(t) = 185 exp( - 0.17t). (10)

Now I estimate L = L(t) for the individual time course of stalk contraction shown in Fig. 3, p. 588.

The half-time is 2.2 ms. Hence 1 k = 2~2 In 2 =0.31 ms- (11)

The curve in Fig. 3 begins somewhat above 150 J.Lm. Let x denote the length in excess of 150 J.Lm.

Then taking the correspondence 1.1 em...... 50 J.Lm, we have

140.2 - 50x=> x - 9 . 1 ,.."m. (12)

Since this is but an estimate, I round to 10 J.Lm, whence

Lo = 160 J.Lm. (13)

Therefore

L(t) = 160 exp( - 0.3lt). (14)

I checked this expression for t = 6.5 ms, which gave L == 21 J.Lm, which comports with Fig. 3.

GONGL USIONS

The constancy of strain rate during fast axopodial contraction in A. contraetilis has not been pointed out previously. Time courses of contraction are unknown for other heliozoa, but have been measured for ciliates, including peritrichs (Vorticella: Jones et al., 1970bj Katoh and Naitoh, 1992), heterotrichs (Spirostomum: Hawkes and Holberton, 1974; Stentor: Newman, 1972), and a suctorian

(Discophrya: Hackney and Butler, 1981). The authors cited did not fit curves analytically to these data, and as the published graphs are variously irregular and in some cases both concave and convex, it is doubtful wheter exponentials would be satisfactory.

The observed exponential time course of fast axopodial contraction in A. contraetilis lacks an explanation. An increase in drag as the shortening axopodium moves toward the cell body, length- or speed-dependent chemomechanical properties of the contra.ctile organelle, and an increase in 31

intraaxopodial viscosity as contraction proceeds might singly or jointly account for the exponential time course, but it has not been possible to quantify these effects. In Spirostomum, the force of cell body contraction is independent of speed in solutions of various viscosities (Hawkes and Holberton,

1975, p. 598; Amos et al., 1976, p. 293). It should be kept in mind that further experiments are needed to determine if an exponential time course holds generally for fast axopodial contraction in the

Heliozoa. 32

8. REYNOLDS NUMBERS FOR FAST AXOPODJAL CONTRACTION IN THE HELIOZOA

I compute Re = ~l, where U is the speed of contraction (m s-l) and v is the kinematic viscosity of

water (v ~ 10-6 m 2s- 1). I assume that the charactcristic Icngth I (m) is the distance through which the axopodium contracts. Note that owing to crumpling (huckling) of the axopodial membrane during

contraction, there may be more than one vaillc of Re t.o consider.

Echinosphaerium akamae:

From Suzaki et al. (1992a, p. 432),

3 tJ _ (2.6x 10- m s-I)(J.72x lO- rn) _ 0- 1 R.e - -6 ? -1 - 4 xI. 10 m- s

A ctinocoryne contractilis:

Febvre-Chevalier (1981, p. 340) says that whcn the hase of the head or stalk is stimulated, the axopodia contract to two-thirds of their initial extended length in 2 ms. However, she does not say what the extended length is. Febvre-Chevalicr and Fcbvre (1992, Fig. 1, (d )-(f)) indicate a contraction time == 2.4 ms, and note (p. 587) that the axopodia are 50-500 pm long, Thus, roughly, the speeds of contraction range between ~ (5 x 1O-~3 m) = 2 x 10-2 m s-1 and 2 x 10 m s-I, Accordingly, Re must lie 2 x 10 s between

and

(2 x 10- 1 m s-I)(~ 5 x 10-4 m) _

10-6 m2s- 1 = I. 33

That is,

0.7 < Re < 7.

Heterophrys marina and Ciliophrys marina:

From Davidson (1975), the typical length of an axopodium is 50llm, and an axopodium contracts by ::::: 90% ( ::::: 45 /lm) in 20 ms. Hence,

u =4.5 x 1O-~) m =2.2 x 10-3 Tn s-l. 2.0 x 10-- s

Thus,

Actinophrys sol:

A typical contraction entails a displacement of ::::: 100 11m in < 1 s (Ockleford and Tucker, 1973, pp. 375-376). Hence U> 100 /lm s-l, and

(10-4 m s-I)(10-4 m) _ 10- 2 Re> -G ') -1 -. 10 m- s

The following chart summarizes values of the Reynolds number for fast axopodial contraction in some heliozoa. 34

REYNOLDS NUMBERS FOR FAST AXOPODJAL CONTRACTION IN SOME HELIOZOA

HELIOZOAN U I Re REFERENCE (m s-l) (m)

Echinosphaerium 2.6 x 10-3 1.72 x 10-4 0.4 Suzaki et al. akamae (1992, p. 432)

Actinocoryne '"" (2 x 10-2_ '" (3 x 10-5_ 0.7-7 Febvre-Chevalier contractilis 2 x 10- 1) 3 x 10-4) (1981); Febvre- Chevalier and Febvre (1992)

Heterophrys marina '"" 2.2 x 10-3 4.5 x 10-5 0.1 Davidson (1975) and Ciliophrys marina

Actinophrys sol > 1 x 10-4 1 x 10-4 > 10-2 Ockleford and Tucker (1973)

By way of comparison, Childress (1977, p. 4) gives for Paramecium Re'"" 10-2 for a beating cilium,

and Re '"" 0.2 for swimming of the whole organism, where the Re values are based on the length of the

cilium and cell body, respectively. Childress (1981, Table 1.1, p. 4) classifies Paramecium as a

"Stokesian" swimmer.

Happel and Brenner (1983, p. 3) seem to suggest that the creeping motion (Stokes) equations can e safely be applied for Re < 5. Some published compilations of Re values for various organisms follow. 35

From Vogel (1981, p. 67):

Re A luge wh:z.le swimming ~t 10 m sOl 300.000,000. A run~ swimming ~t the s~e speed 30,000,000

A duck flying :l.t 20 m s"t 300,000 A luge dr~gonfly going 7 m SOl 30,000 A copepod in ~ pulse of 20 em SOl 300 Flight of the sm:z.llest flying insectS 30 An invertebr~te 12.CY~, 0.3 mm long, moving at 1 mm sOl 0.3 A. se~ urchin sperm advancing the species at 0.2 mm sOl 0.03

From Sleigh and Blake (1977, p. 249):

TABLE 1. Dimensions of features of the body and cilia for representative examples of the range of ciliated organisms.

Swimmingt Body velocity Number Cilia length body of body length (L)~ ~S-I Name Group lengths Rb cilia (l)1Jl1I Rc nl Metachronism S-I (n)

Uronema Protozoa 2S 11 SO 46 10-2 200· S 10- 3 103 ? 2 2 3 Tetrahymena Protozoa 70 480 7 10- Soo· 7 10- 3'S X 10 ? Didinium Protozoa 130 12S0 10 10- 1 17S0 12·S 10-2 2·2 x 10· Dexioplectic Paramecium Protozoa 210 1000 5 10-1 SOOO 12 10-% 6 x 10· Dexioplectic 1 Veliger (Aplysia) Mollusc larva ISO ? ? 200 56t - 10- 1-1 X 10· Laeoplectic Blepharisma Protozoa 350 600 2 10-1 7000· 7·5 10-% S x 10· ? 2 2 6 Opalina Protozoa 400 50 0'12 10- lOS IS 10- 1·5 X 10 Symplectic Spirostomum Protozoa 1000 1000 1 1 lOS. 12 10- 2 1·2 x 106 ? Convoluta Flatworm 2000 600 0'3 1 7 x 106 7·S 10-% 5 x 107 Laeoplectic 3 Pleurobrachia Ctenophore 15000 7SOOO 5 10 200(108) SOOt 10 lOS (l010) Antiplectic

·Excludes compound cilia; tindicates compound cilia; tfigures for swimming velocity are "maximal" values, particularly for Pleura- e brachia. R and R are Reynolds numbers based on cell body length 'and cilium length, respectively. b c 36

9. A CRUDE ESTIMATE, ACCORDING TO SLENDER BODY THEORY, OF THE DRAG

AND POWER DISSIPATION IN FAST AXOPODIAL CONTRACTION

I use an asymptotic formula derived by de Mestre and Russel (1975) and corrected by Brennen and

Winet (1977, pp. 351-351) for the tangential force coefficient, Cs' of a slender right circular cylinder translating along its axis normal to an infinite plane wall:

2a II

I h I

L The formula in question is appropriate for a «: hand 2~ ~1:

C_ 2irf.l (1) S- In (Lao)-1.75

(Brennen and Winet, 1977, Table 2, p. 350), where f.l =1 x 10-3 kg m-1 s-l is the dynamic viscosity of water. The drag per unit length is given by

(2)

where U is the speed of contraction. The total drag on an axopodium is thus

Ftot = CsULo, (3) 37

where Lo is the initial (uncontracted) length of the axopodiuTll. The power expended per unit length is

(4) and the total power dissipated is

(5)

The foregoing formulae are used in the sequel.

Ciliophrys marina and Heterophrys marina:

3 Here Lo = 50 /-lm, a = 0.05 Jim, and U = 2.5 x 10- m s-I (Davidson, 1975). (Observe that a =

560h, which likely satisfies the condition a ~ h demanded by equation (I ).) Hence

=3xlO-6 N m- I

= 3 pN /-lm- 1

I Ftot = (3 pN /-lm- )(50 pm) = 150 pN

6 1 3 Ps =(3xlO- N m- )(2.5xIO- m $-1)

= 7.5 x 10-9 W m- 1

= 7.5 fW J.lm- I

1 P tot = (7.5 f\N J.lm- )(50 J.lm)

= 3.8 x 102 fW 38

Echinosphaerium akamae:

3 From Suzaki et al. (1992a, pp. 431-432), I take Lo = 200 11m, U = 2.6 x 10- m s-l, and I estimate that a = 2 pm, noting that the radius of the glass needle is 4 lun. (Observe that a = 510h, which may satisfy the relation a «: h required by equation (1).) Therefore

F_ 271"(1 x 10-3 kg m- I s-I)(2.fi x 10-3 m s-l) s - In 100 - 1.7.5

=6xlO-6 Nm-l

= 6 pN pm-I.

The total force exerted in a maximal contraction of 172 IHn (the greatest displacement of the needle observed - see p. 431 and Table 1, p. 432) is thus

which is of the same order of magnitude as the experimental value, (4.1 ± 0.9) x 103 pN.

6 l 3 1 Ps = (6 x 10- N m- )(2.fi x 10- m 5- )

= 2 x 10-8 W m- I

9 3 1 Ptot = (1 x 10- N)(2.fi x 10- m 5- )

= 1 x 10- 12 W

= 1 x 103 fW 39

CONCLUSTONS

Although it is encouraging that Slender Body Theory yields a value of drag comparable to the measured force of contraction III E. akamae, it should he emphasized that the assumptions of the theory are not closely obeyed. The surface of the axopodillm has a complicated time-dependent form and its slenderness diminishes as contraction proceecls. Moreover, the axopodium always maintains its continuity with the cell body so that there is no fluid gap, and the actual motion is unsteady. It has not been possible to estimate the importance of these discrepancies. 40

10. SLIDING PLANE THEORY

{)=a.

STATIONARY PUl.NE

u {)=o SLIDING PUl.NE

The problem concerns two-dimensional Stokes flow. Introducing the stream function 1/J for mathematical convenience, we solve the biharmonic equation

(1)

We use polar coordinates and let the plane given by 0 = 0 move at constant speed, while the plane at

0= Q remains stationary (see diagram above). Hence the boundary conditions are

(2)

ur =Uo =0 at 0 =0'.

The stream function 1/J satisfies the relations

(3)

whence the boundary condtions (1) become

1 81/J U ~1/Jr = 0 at 0 = 0 r 80 = - , v (4)

1 81/J 0 81/J - 0 0 r 80 = , 8r - at =Q. 41

Taylor (1960) and Moffatt (1964) found that the problem has a similarity solution of the form

1/J(r,B) = l' f(B). (5)

2.1. _ 1 1 Y 'f/ - 1/Jrr + r 1/Jr +"2 1/JBB' (6) l'

Hence, inserting (5) into (6),

2 y27j) = Vf + d ;) 1'\ <10-

whence

which reduces to

(7)

(Note the singularity at l' = 0.)

Consider

(8)

The characteristic equatioll of (8) is

or

Hence m = ± i are roots of multiplicity two, whence 42

Taylor (1960) and Moffat (1964) found that the problem has it similarity solution of the form

ljJ(r,B) = r frO). (5)

(6)

Hence, inserting (5) into (6),

whence

which reduces to

(7)

(Note the singularity at r =0.)

Consider

(8)

The characteristic equation of (8) is

or

? ? (m- + 1)- = O.

Hence m = ± i are roots of multiplicity two, whence 43

Taylor (1960) and Moffatt (1964) found that the problem has a similarity solution of the form

1/;(r,B) = l' f(B). (.5)

(6)

Hence, inserting (5) into (6),

whence

which reduces to

(7)

(Note the singularity at l' = 0.)

Consider

(8)

'1'1\1: c11;lrrt.ct,eristic eClilrltioll of (8) is

4 oJ m +2m- + 1 = 0

or

Hence m = ± i are roots of multiplicity two, whence 44

[(0) = AcosO + BsinO + COcosO + DOsinO, (9)

['(0) = - AsinO + BcosO + C(cosO - OsinO) + D(sinO + OcosO), (10)

where' denotes l(r

Now (4) implies that EN [(0) = &'

(11 ) , 1 EN.. f(O) =r 80°

Thus the boundary conditions (3) may be recast as

f(O) = 0, ['(0) = - U

(12)

[(a) = ['(0) = o.

From (9) and (10) we have

f(O) = 0 :::} A = 0 (13)

['(0) = - U :::} B + C = - U (14) • [(0) = 0 :::} Bsina + Cocoso + Dosino = 0 (15) ['(a) = 0 :::} Bcosa + C(cosa - osino) + D(sina + ocoso) = O. (16)

Eliminating B successively between (14) and equations (15) and (16):

- Bcosa - Ccoso = Ucoso

Bcoso + C(cosa - asino) + D(sino + acoso) = 0

- Cosina + D(sinn+ocoso) = Ucosa (17)

- Bsina - Csino = Usina

Bsina + Cacoso + Dosino = 0

C(acosa - sina) + Dosino = Usino (18) 45

Solving (17) and (18) simultaneously by Cramer's rule:

U cosa sinO' + acosO'

U sina asina

C

- aSIllO' si nO' + O'coSO'

acosa - sina asIIIa

• whence (19)

From (14),

B = - (U + C). (20)

Therefore, inserting (19) into (20) gives

(21)

From (15), we may write D = - (Bsina + Cacosa) asina whence by substitution for Band C, U(a-sina cosO') D= .' (22) 0'2 _ sin 2O' 46

Recalling that t/J = rf(O), and inserting the expressions (13), (19), (21), (22) for A, B, C, D into (9)

yields

2 2 t/J= 2 U~ 2 [-a sinO+sin a(OcosO)+(a-sinacosa)OsinO]. (23) 0' - SIn 0'

In the case of normal incidence, 0' =f. Thus, the streamlines are given by

t/J = --JlL (- ~2 sinO + 0 cosO + f 0 sinO). ~-1 4 (24)

The following figure (slightly modified to conform with my notation) from Batchelor (1967, p. 225)

depicts some typical streamlines. • "'---~-::....-_---,

0' ~ l'. , Figure +8.%. Two-

The tangential stress on the sliding plane is the shear stress TrO evaluated at 0 = 0 (Leal, 1992; p.

143, above eqn. (4-70); p. 154):

TrO 10 = 0 = JJ {r %r C/)+¥ ~~r}o = 0 = o. (25)

From (3) and (23),

2 Uo = - ~t/J = 2 - ~ 2 [- 0'2 sinO + sin 0' (0 cosO) + (0' - sinO' cosa)O sinO] vr 0' - SIn 0' whence

2 r g(:0) = 2 U. 2 [ - 0'2 sinO + sin 0' (0 cosO) + (0' - sinO' cosa)O sinO]. ar\ r(a - SIn 0') 47

Therefore,

(26)

Also from (3) and (23},

w 2 2 ur =}88O· = 2 U. 2 [-a cosO+sin a(cosO-OsinO)+(a-sinacosa)(sinO+OcosO)] a -Sill a whence

ur 2 2 } 88O = 2 U. 2 [a sinO - sin a (2sinO + 0 cosO) + (a - sina cosa)(2cosO - 0 sinO)]. r(a -sm a) Accordingly, 8pr I_ 2U (. ~O 0 - 0 - 2 • 2 a-smacosa). (27) u - r(a -sm a) Thus, by substitution of (26) and (27) in (25),

2~U (. ) T rO I0 - 0 2 . 2 a - sma cosa . (28) - = r(a -sm a)

To apply sliding plane theory to axopodial contraction, I consider the following arrangement: • U=a u STATIONARY PLANES 8=0 o SLIDING PLANE

Hence, from (28), the drag stress on both sides of the moving plane is, by symmetry,

4~U (. ) liD = 2 . 2 a - sma cosa . (29) r(a -sm a)

The criterion for the self-consistency of the assumption of negligible inertia is 48

Re-zr- rU ~ 1, (30)

where Re denotes the Reynolds number, and 1/ is the kinematic viscosity of water (Batchelor, 1967, pp.

225-226). Hence the solution is valid within a neighborhood of the intersection 0 of the stationary and

moving planes given by

r~U·1/ (31)

Now we determine the range of validity of sliding plane theory for parameter values characteristic

of fast axopodial contraction in some heliozoa. We shall take 1/ = 1 x 10-6 m2 s-l.

Heterophrys marina and Gi/iophrys marina

Since U = 2.5 x 10-3 m s-1 (Davidson, 1975), r ~ 4 x 10-4 m = 400 j1.m. Note that the initial

length of the axopodium, 50 j1.m, is k(400 j1.m) •

• Echinosphaeri1l.m akamae Here U = 2.6 x 10-3 m s-1 (Suzaki et al., 1992a). Thus, again, roughly, r ~ 400 j1.m. Observe that

the initial length of the axopodium, :::::: 200 j1.m, is! (400 j1.m).

GONGL USIONS

I suggest that it may be useful to apply Sliding Plane Theory to H. marina and G. manna. E.

akamae seems not to satisfy the criterion for self-consistency.

Values of force and power for H. marina appear in the table comprising Section 11. It can be seen

that Sliding Plane Theory yields values about two orders of magnitude smaller than estimates founded

on Slender Body Theory and Stokes's law. Given that the finite dimensions, time-varying shape, and

unsteady motion ofaxopodia grossly violate the hypotheses of Sliding Plane Theory, it is not surprising

that the results depart drastically from those predicted by Slender Body Theory and Stokes's law. 49

However, I do not know whether these values of force and power are accurate, for they lie in the same

range as has been reported for swimming spermatozoa of various phyla (5-350 pN; Green, 1988; 2-200

fW; Pedley and Kessler, 1992), and single-celled algae (diameter 10 /lm, speed 100 /lm s-l) have been

estimated to expend 2 fW in locomotion (Pedley and Kessler, 1992, p. 318).

• 50

11. ESTIMATES OF FORCE AND POWER IN FAST AXOPODIAL CONTRACTION

HELIOZOAN FORCE POWER SPECIFIC POWER HOW VALUES REMARKS AND REFERENCES (pN) (fW) (W kg-I) OBTAINED

Echino&phaerium 4 X 103 ._------Glass needle deflection Suzaki et al. (1992) akamae

103 103 10° SBT Specific power based on est. mass of entire axopodium; de Mestre and Russel (1975); Brennen and Winet (1977)

103 103 5 X 10°_2 X 102 SBT Specific power based on est. mass of X-body; idem

2 2 Heterophrys 102• 5 X 101_1 X 10 t (1.1-7.5) X 10 • .Cak./measurement by Specific power based on est. mass marina Davidson; tCak. by of entire axopodium; Davidson B.E.D. from Davidson (1975) cine data, Stokes's law e applied to transport of attached carbon sphere

2 4 1.5 X 102 3.8 X 10 1.8 X 10 SBT Specific power based on est. mass ( "'" 10-14 g) of central filament de Mestre and Russel (1975); Brennen and Winet (1977)

~ -

8 X 10° 2 X 101 -_ .._---- SPT, a =f 90% contraction; Taylor (1962), Moffat (1964), Batchelor (1967), Sherman (1990), Leal (1992)

5 X 10° 1 X 101 _.._-----_ ... SPT, a 21r = 3 idem

1 4 X 10° 1 X 10 ----.....---- SPT, a = 3: idem

Slender Body Theory (SBT) and Sliding Plane Theory (SPT) presuppose Stokes flow and constant speed. See the text and references cited for details of additional assumptions. Only external hydrodynamic resistance is considered in the calculations of power and specific power. Sliding Plane Theory is of doubtful applicability at the size scale of E. akamac. 51

12. SOME ROUGH ESTIMATES OF ENERGY UPTAKE FROM FOOD IN HELIOZOA

Laybourn-Parry (1992, pp. 83-84) remarks that measured protozoan digest.ive (= assimilation) efficiencies range between 22 and 76%. As a conservative estimate, I shall take the efficiency to be 20% for the protozoa considered below.

Actinophrys sol (Heliozoa)

Patterson and Hausmann (1981, Fig. 12, p. 44) observe that in A cfin ophrys sol the resorption of the fluid contents of the food vacuole normally is completed t1 h after capture of one Co/pidium colpoda. [From Patterson and Hausmann (1981, p. 40), I gather that the prey is destroyed by lysis typically within 25-35 min after capture.] For simplicity in the ensuing calculation, I shall assume that energy uptake occurs uniformly throughout this 4 h period. J lack the heat of combustion of C. colpoda, but I have a value for the smaller C. campy/um. [According to Blick (1972), C. co/poda is generally somewhat larger than C. campy/urn: C. colpoda length = 100-150 JIm; C. campy/um length = 50-120 JIm.] Laybourn-Parry (1984, p. 140) gives for C. campy/um a heat of combustion of

20 ± 1.5 J mg-1 ash-free dry mass. Note that (Ash-free dry mass) ~ (Dry mass). For the sake of convenience in a rough calculation, and realizing that Colpidium lacks a mineral skeleton, I take the foregoing masses to be equal.

My reading of Fig. 3.7, p. 76 in Laybourn-Parry (1984) indicates an approximate cell volume range of (1-12) x 104 Jlm 3 for C. campy/urn. I shall assume a cell volume of 6 x 104 l.Lm 3. If, following Caron and Goldman (1990, legend of Table 7.4, p. 291), we assume a typical protozoan density of 1.0 g cm-3 and use the empirical relation

Dry protozoan mass = 0.2(Wet protozoan mass), (1) then for one C. campylum cell

Dry mass ~ 0.2(6 x 104 Jlm 3 )(1.0 gjl012 J.Lm 3)

= 1 x 10-8 g 52

= 10 ng.

[By comparison, the dry mass of a well-fed Podophrya fixa rangcs from ~ 4-10 ng (Laybourn, 1976,

Table 1, p. 206), and a large Didinium or small Stentor havc a dry mass of ~ 200 ng (Laybourn-Parry,

1984, p. 99). Thus, I think my estimate is reasonable.]

Therefore, the heat of combustion of one C. campylum is

(20.15 J mg-1)(1 x 10-8 g)(103 mg g-l)

= 2 x 10-4 J.

The energy taken up by A. sol is thus

(A. sol assimilation efficiency)( C. campyillm hCi'lt of combustion) (2)

= 0.20(2 x 10-4 J)

= 4 x 10-5 J.

The average power uptake by A. sol is Energy uptakc (3) Time for assimilation

4 x 10-5 J

6 = 3 X 10 fW.

Owing to the generally larger size of the actual prey, C. colpoda, the probably small difference between dry mass and ash-free dry mass, and the low assimilation efficiency assumed, I consider this figure to be a conservative estimate of power uptake.

According to Patterson and Hausmann (1981, p. 46), an A. sol which has not fed for at least 24 h has a body diameter dB ~ 45 pm. Hence the volume of the cell body is

(4)

= t1r(45 t1m )3

= 4.8 x 104 pm3. 53

A. sol ~ ~ 11m The same typical has 42 axopodia, each of length Lo 100 and basal diameter d A = 1.25

pm. Thus the aggregate volume of the axopodia is

2 VA = 42 X /2il"Lod A (5)

= ~~( 100 Ilm)(1.25)2

3 3 = 1.8 x 10 pm .

The total volume of one A. sol IS

Vtot=VB+VA (6) 4 3 = (4.8 x 10 + 1.8 x 10 ) /lm3

4 =5.0 x 10 11m3.

N6te that this volume is of the same order of magnitude as that of Colpidillm.. [A. sofs gluttony is not unprecedented among heliozoa. Dragesco (1964) descrihes the voracit.y of Echinosphaerillm. eichhorni.

And Capriulo (1990, p. 221) points out that aloricate ciliates (e.g., Didinium.) can ingest prey larger than themselves.] From the cell volumes (pm3) and respiration rates (nl 02 /lm-3 h- 1) given by Caron et al. (1990, Table 8.1, p. 309), I calculate the power dissipated in respiration by Colpidium to be (4.7­

6.2) x 105 fW. Hence, my presumably conservative estimat.e of power uptake by A. sol is 5-6 times the respiratory power expenditure of Colpidium. I conclude t.hat A. sol likely obtains energy from

Colpidium at a rate sufficient to sustain itself, if not to grow and reproduce.

Spongodrymus (Radiolaria) in relation to Echinosphaerillm (Heliozoa)

Anderson et al. (1984) measured the uptake of 14C-labeled algal and crustacean prey by a solitary radiolarian of the genus Spongodrymus. According to Anderson (l983, p. 181), Spongodrymus catches its prey by means of swiftly contracting axopodia. (Like heliozoa, radiolaria are members of the

Actinopoda.) Anderson et al. (1984, p. 206) note that the species they studied had a cell body diameter of ~ 800 pm, comparable to that of the freshwater heliozoan Echinosphaerium eichhorni, which may attain a cell body diameter of over 1000 pm ( = 1 mm; Barrett, 1958, p. 206; Febvre­

Chevalier, 1985, p. 310). Thus, there are grounds for supposing that power uptake rates (derived from 54

measured predation rates) for Spongodrymus may be of the same order of magnitude as those for

Echinosphaerium.

Anderson et al. (1984, p. 205) found that when the dinonagcllate A mphidinium carterae was supplied at a density of 4.2 x 103 cells (ml culture mediurnf 1, one Spongodrymus ingested 983 cells in 2 h. [Anderson et al. (1984, p. 206) caution that the rates ohserved are "ad libitum feeding rates and probably do not represent algal [sic] predation rates in the natural environment where the algal prey density may be lower by several orders of magnitude".] The caloric content of one dinoflagellate cell can be estimated by means of the linear regression formula (Hitchcock, 1982, Eqn. 4, p. 371)

log(ncal ceW 1) = 0.80 log(cell volume in 11m3) + 0.90; r=0.93, n=8. (7)

This formula is founded on data for eight species of dinonagellates, whence n = 8.

From Hitchcock (1982, p. 3(5), the mean volume of A. carterae (based on measurements on a 3 minimum of 25 cells) is 818 j.lm . Hence, regression formula (7) above yields

log (neal cen-1) = 0.80 log (818) + 0.90

= 3.2

Therefore, one cell has a caloric content of

103.2 neal

= 1.7 x 103 neal

= 1.7 x 10-6 cal

= i.l x 1O-6 .J (1 cal = 4.184.1).

Thus the caloric content of 983 cells is

(7.1 x 10-6 J ceW 1)(983 cells)

= 7.0 x 10-3 .J.

Assuming conservatively an assimilation efficiency of 20%, the energy uptake of Spongodrymus is

1.4 x 10-3 .J.

If we suppose that this energy is transduced over a period of 4 h (the same length of time as for A. s00, then the power uptake is 55

= 9.7 x 10-8 J s-l

= 9.7 x 107 fW.

Since the cell body of Spongodrymus is :::: 800 jJm in diameter, its volume is approximately

VB = i7l"(800 jJm)3

-?- _. 7 x 108 jJm.3

I have no information on the number or linear dimensions of the axopodia of Spongodrymus. If we assume that their collective volume is roughly 10% of the cell body volume, then we may take the total volume of one Spongodrymus to be

From Caron et al. (1990, Table 8.1, pp. 309-310), I extract the following data on a heterotrich and a foraminiferan for comparison with my estimates for Spongodrymus. Values for Spongodrymus are in italics.

ORGANISM VOLUME RESPIRATORY POWER/POWER UPTAKE FROM PREY (jJm3) (fW)

7 7 Spiro.ftomum 1.2 x 10 6.7 X 10 amhiguum (heterotrich)

8 8 Roulina 1.25 X 10 3.3 X 10 leei (foraminiferan)

7 Spongodrymu.f 3x 108 9.7X 10 (radiolarian) 56

According to Anderson (1983, pp. 121, 123), Spongodrymus contains symbiotic algae; perhaps

these make a significant contribution to its energy budget.

Now I compare an estimate of the energy dissipated against hydrodynamic drag in axopodial

contraction with the energy acquired in feeding by Spongodrymus and A. sol.

From Suzaki et al. (1992, Table 1, p. 432), the contraction time for E. akamae is 172 p.m 2600 p.m s-l

= 6.5 x 10-2 s.

The total power dissipated is, by Slender Body Theory (cf. Section 9),

3 x 103 fW

= 3 x 10-12 J s-l.

Hence, the total energy expended is

(3 x 10-12 J s-1)(6.5 x 1O-2s)

=2xlO-13 J. 13 Observe that this is only X 10- 100 = 1 x 10-8 % of the energy taken in by Spongodrymus and (2 3J) -13 1.4 x 10- J (2 x 10 5 J) 100 = 5 x 10-7 % of the energy obtained by A. sol in the feeding bouts recounted above. 4 x 10- J The actual percentages are likely larger because I take no account of drag on attached prey, power expended against internal viscous resistance and in breaking down microtubules, etc.

General Comparative Considerations !ill Power Dissipation in Microorganisms

Earlier in this section I have shown that for A. sol, roughly,

For Tetrahymena pyriformis, whose cell volume is of the same order of magnitude, measured respiration rates are generally ,.., 10-5 nl 02 p.m-3 h-1 (Caron et al., 1990, Table 8.1, p. 310). Hence, assuming equality of respiration rates, I estimate that the metabolic power of one A. sol is ,.., 10-6 W.

If we suppose that the power dissipated by A. sol in axopodial contraction is the same as in E. akamae 57

(likely an overestimate because axopodia of A. sol are smaller), then the fraction of metabolic power

( = fraction of metabolic energy) spent to overcome hydrodynamic drag is approximately

12 4 ( 10- W) 100 = 10- %. 10-6 W Note that even if all ::::: 42 axopodia contract at once, the fraction is only about 10-3%. It is useful to compare the foregoing figure with the following estimate due to Bray (1992, p. 6):

Swimming consumes only a small fraction of the cell's energy

The power required to propel a cd! in W'ateris quite small. Itmai-b~ calculated ::s the viscous drag multiplied by the velocity. For a spherical cd! power (61ta11v) X v where 11 viscosity ofwater (lO-2g/ cm sec) a radius ofcell v = velocity ofcell If the cd! h::s a r.o.dius of 1 J.UI1 and is trd.vding at 10 J.UIl!sec, che power con­ sumed is 2 X lOo11 e.rgsIsec (2 X 1O-l8 J/sec). What is this power requirement in tc:III1S ofATP molecules, che princi­ pal currency ofenergy in the cell? Hydrolysis ofone gram mole ofATP releases 1 about 40KJ of useful. energy; hydrolysis ofa single ATP molecule, abOUt 10' ' J. The cd! in the above calculation chert:.:are requires che "hydrolysis ofat le::sc 20 molecules ofATP every second in order to maintain its speed of 10 ~sec. Ifwe ::ssume a low efficiency ofa few' percenc.ge points., chen che ATP hydrolysis due to swimming might be 1000 molecules per second. Since che mecboiic me ofa. ho~, ~g typica.l cd! is around 10' ATP molecules per second, is noe a ma.jor energy case. '.,

Hence, Bray's fictive coccus devotes only 10-2% of its metabolic power to locomotion. 58

An analogous estimate was made by Pedley and Kessler (1902, p. 318):

ENERGY The energy stored within microorganisms is acquired by various means such as photosynthesis. When an algal cell swims it generally uses a very small amount of its stored energy, so that for any given experiment of several hours duration, or for an overnight interval of active swimming by the cell. energy consumption and supply need not be considered. The supply and consumption time scales are not as well separated for bacteria. From the Stokes fonnula one may calculate, for a "typical" algal cell (radius 10 ,Urn. speed 100 ,Urn) a swimming power of2 x 10- 15 J S-I cell-I. The light input can be taken as l W m - 2; using aphotosynthetic conversion efficiency of3%,and the cell's area, the power input is estimated at 10- 11 J s- 1 cell- I. Thus, for algae, swimming requires only a small fraction of the inputenergy. Estimates for spermatozoan swimming power range from 15 2 x 10- to 2 X \0-13 J S-I ceil-I (Bishop 1962); in that case the fluid - medium in which they swim can also supply energy. 1be temperature increase or the fluid due to the swimming power dissipated by a typical cell population is quite negligible. For \07 cells em - J, the temper:uure rises by 10-' °c s - I. If 1 W m-7! light intensity is absorbed and converted to heat by the cells, the temperature rise from that source can be as high -:: -as-iO:~'oc S-I. Illumination can be a source of convection currents in experiments that investigate the ir.teractions of organisms with light.

Hence, Pedley and Kessler's fictive alga devotes only 2 x 10-2% of its power input to propulsion.

Fenchel (1986, p. 83) remarks (referring to Fenchel and Finlay, 1983) that "motility in growing protozoa can at most account for 0.1-0.7% of the energy budget".

Now I adduce data for certain nonheliozoan contractile protozoa.

From Hawkes and Holberton (1975) I have for Spirostomum ambiguum the following facts:

Typical extended length =1.5 x 10-1 cm =1.5 mm (p. 599)

Fast contraction occurs to ~ 43% of extended length (p. 599)

Cell radius =75 pm for extended cell=> cell diameter =150 pm (p.597)

Speed of contraction = 20 cm s-1 (p. 595)

Hawkes and Holberton (1975, p. 601) estimate that the energy dissipated in doing work against the viscous external medium in one body contraction (complete in 10 ms) is 2.13 x 10-3 erg = 2.13 x 10-10

J.

Amos (1971, p.127j 1975b, p. 413) used Stokes's law to estimate the drag on the cell body of 59

Vorticella. Amos treats the cell body as a sphere of radius 20 J.lm, pulled 80 J.lm at an average speed of

23 mm s-l. He finds that the viscous drag is 8.6 x 10-9 N. Hence, the work done to overcome drag in one contraction is

(8.6 x 10-9 N)(8.0 x 10-5 m)

= 6.9 x 10-13 J.

It seems likely that this figure is lower than the true value owing to the steep increase in drag as a sphere closely approaches a plane wall at normal incidence [See Power and Power (1993, Fig. 2, p. 61).]

Also, the calculation omits consideration of drag on the stalk.

The subjoined table epitomizes the foregoing discussion.

ESTIMATED ENERGIES OF FAST CONTRACTION FOR SOME PROTOZOA

ORGANISM TYPICAL DISPLACEMENT TYPICAL SPEED TYPICAL TIME ENERGY (Jlrn) (J.lrn s·l) (rns) (J)

Spiro8tomum 855 2 X 105 -4 2.13 X 10.10 ambiguum

Vorticella 80 2.3 X 104 -3 6.9 X 10-13 conl1allaria

:

13 Echino8phaerium 172 2.6 X 103 65 2 x 10- a.camae 60

13. OUTLINE OF A RESEARCH PLAN TO ELUCIDATE

THE MECHANISM OF FAST AXOPODIAL CONTRACTION

I. INTRODUCTION

A. Synopsis of the Heliozoa

A polyphyletic assemblage of predaceous freshwater and marine protozoa, here considered to embrace pedinellids (Siemensma, 1991).

Globose cell bodies (diameter'" 8-2600 pm) bear many ( '" 20-several hundred; I do not know the max. number yet found) radially disposed slender protrusions called axopodia (diameter '" 0.1-10 pm; length '" 30-500 pm). Each axopodium is stiffened by an axoneme consisting of an array of microtubules (3- '" 500) cross-linked in a taxonomically diagnostic pattern (Febvre­ Chevalier, 1985; Siemensma, 1991).

Axopodia serve several functions, e.g., locomotion, cell fusion, cell division, egestion, attachment to substrate, and prey capture.

B. Fast axopodial contraction

Axopodia can contract rapidly (duration of contraction '" 4-60 ms; absolute speed of contraction '" 2.6-60 mm/s; relative speed of contraction '" 13-300 axopodial lengths/s; extent of contraction = shortening per unit initial axopodial length = 66- '" 100%) in response to various artificial stimuli and contact with prey (Febvre-Chevalier, 1981; Febvre-Chevalier and Febvre, 1992; Suzaki et al., 1992a).

The mechanism of contraction is unknown. Contraction proceeds by breakage of the axonemal microtubules (but see abstract by Suzaki et al., 1992b), requires calcium ions in the external medium, is preceded by an action potential in at least one species, and may be driven by putative contractile elements visible in some electron micrographs.

C. Speed disparity conundrum

I [I.e., fast axopodial contraction entails microtubule (MT) breakdown, yet contraction speeds ~ MT disassembly speeds typically measured in vivo and in vitro.]

Although the speed of fast axopodial contraction is not extraordinary in comparison with that of other forms of protozoan motility (e.g., max. swimming speed of Uronema = 1.15 mm/s =46 cell lengths/s [Sleigh and Blake, 1977]; avg. cell body speed owing to spasmoneme contraction in Vorticella = 23 mm/s [Amos, 1975b, p. 413]; cell body contraction speed in Spirostomum, Stentor '" 200 mm/s '" 100 cell lengths/s [Huang and Mazia, 1975, p. 389]), it is exceptional for a process depending on the disintegration of MTs (Amos and Amos, 1991; Febvre-Chevalier and Febvre, 1992).

Cf. observations by differential interference contrast (DIC) video microscopy of MT shortening in various cell types and in vitro:

In reticulopodia of Reticulomyxa, max. endwise disassembly speed '" 20 pm/s (Chen and 61

Schliwa, 1990). Caplow (1992) remarks that this is the greatest rate of MT depolymerization yet measured in vivo.

By contrast, the mean endwise disassembly rate of MTs showing dynamic instability in CRO fibroblasts is only 32.2 ± 17.7 pm/min (duration of shortening = 4.3 ± 3.3 Sj n=18j Shelden and Wadsworth, 1993). [See Keith and Farmer (1993) for warnings on the use of photobleaching to monitor MT dynamics.]

In vitro endwise disassembly speed for phosphocellulose-purified (hence presumably MAP-free) beef brain MTs in high-calcium ([calcium ion]= 5 mM) solution ..... 1.8 J.lm/s (Gal et al., 1988).

[Aside: Note that Zigmond (1993) reports a milder speed discrepancy for F-actin.]

Cf., as other instances of fast MT breakdown, axopodial contraction in the spumellarian radiolarian Spongodrymus; contraction ofaxopodia and axopodium-like stalk in pedinellids (allied with and possibly ancestral to actinophryid heliozoans); autotomy of flagella In Chlamydomonas, diverse taxa. Find estimates/measurements of the speeds of these processes.

D. Insufficiency of MT breakdown to produce contraction

Cf. experiments of Tilney, Davidson, Schliwa, Salmon, others.

Cf. particle and chromosome movement in vitro by MT depolymerization

Remarks by Oster, Stossel, others on cell movement and filament polymerization/ depolymerization.

Cf. neurite retraction.

E. Buckling, bending conundrum

Why don't axopodia buckle locally or globally or propagate bending waves during normal contraction? [Cf. cilia, non-retractile and retractile flagella, haptonemata. Note that Lee (1989) claims that some haptonemata do not coil.]

II. GOALS

A. Characterize the main physical and biological principles governing fast axopodial contraction.

In particular, I propose scaling relations and compare other protozoan groups, both closely and distantly related. (Comparisons with technological artifacts may be helpful.)

B. Make experimentally testable predictions, suggest further experiments and theory (point out simplifying assumptions) to settle outstanding questions.

[C. Moot a theory/model of how contraction occurs.]

III. WARNINGS

A. Accepting that the Heliozoa are polyphyletic, there may be > 1 basic mechanism of contraction. 62

B. The pitfall of adaptationism

Perhaps the diversity of axonemal MT patterns is not a response to various selection pressures. [Recall Gould and Lewontin (1979).]

Axopodia serve functions other than prey capture by fast contraction. (E.g., slow surface transport, locomotion, cell fusion, egestion.) These functions likely influence mechanical design.

IV. SUMMARY OFAXOPODIAL MORPHOLOGY IN THE EXTENDED AND CONTRACTED STATES

A. Early observations of axoneme birefringence, etc. by Engelmann, MacKinnon, Roskin, Brandt, others.

B. A possible dichotomy: Is it true that centrohelidians have only non-tapering axopodia whereas other heliozoan orders have tapering axopodia? Settle the issue by consulting Smith and Patterson (1986) and marking the table, synopsis in Margulis et al. (1990). Also see Patterson and Hedley (1992, p. 21) for a summary of spp.

Relation between tapering and size?

Do (non-) tapering axopodia always have (non-) tapering axonemes?

How does tapering scale?

Cf. dendrites, axons, reticulopodia, etc. • Electrical implications of tapering? Hydrodynamic implications of tapering?

C. Basal origin ofaxonemes on nuclear plaques or centroplasts

These microtubule organizing centers (MTOCs) would seem to forbid appreciable axial sliding of intact MTs [but see abstract by Suzaki et al. (1992b)]. Also note constraint on axial sliding imposed by cell body diameter.

A telescoping mechanism involving many relatively short segments possibly could provide high speed without axopodial buckling. Do Davidson's (1975) observations of haptocyst motion tell against telescoping in the heliozoans he studied?

D. Miscellaneous Axopodial Organelles

Putative contractile elements seen by electron microscopy (EM) in some genera, not seen in others. Note various conditions of fixation, etc. List all species in which putative contractile elements have been found.

Mitochondria in the axopodia of larger spp. (only?) need not imply an ATP requirement for contraction, may serve other motility, general metabolic processes. Functional implications of dilated, tubular cristae in mitochondria of the cell bodies or axopodia of most heliozoan taxa? See Patterson (1986, Fig. 4.24 , p. 60; p.62).

Extrusomes 63

See Bardele, Grain, Davidson, Hausmann and Patterson, others.

E. The condition of the axopodial membrane, axopodial contents following contraction.

Cf. suctorian tentacles, single myocytes (Krueger et al., 1992).

Does the axopodium thicken appreciably upon contraction?

v. MT PROPERTIES

A. Supramolecular structure 2-D surface crystals and the passage of dislocations, other defects.

B. Polarity [possible implications of Mitchison (1993)?]

C. Flexural rigidity

D. Persistence length

E. Euler buckling, bending as cantilever

F. Low Euler buckling, bending loads imply utility of battens for structural stability

Measurements, observations, theory on other built-up cell structures containing MTs, F-actin.

G. Waugh (1989) on mechanics of the marginal band of newt RBCs

• H. Is an axoneme necessary for mechanical support, other functions?

[Other functions might include service as track for molecular motors, but recall the experiments of Edds (1975a,b).]

Cf. pseudopodia of filose amoebae, axons, reticulopodia, other cell protrusions containing loosely organized bundles of microtubules. N.B. Do pseudopodia of filose amoebae contain MTs or do they derive support from F-actin, other filaments? How large are prey taken by filose amoebae?

Measurements, observations, and theory on built-up cell structures containing MTs, F-actin.

I. Dynamic instability [see data summarized by Erickson and O'Brien (1992)]

J. Disassembly speed increased by calcium ions

K. What is the minimum radius of curvature of a single MT below which breakage occurs?

1. What is the minimum radius of curvature of a cilium, flagellum, or axopodium below which breakage occurs?

M. See Hill and Kirschner's theory (Hill, 1987, pp. 51-56) for the steady-state (Le., constant shortening speed) loss of subunits from linear supramolecular aggregates under compressive force. Also see Hill (1987, pp. 24-28) and Gordon and Bro~land (1987). 64

N. Protofilament numbers in heliozoans See Jones, Tilney, Cachon et al.

O. Biochemical peculiarities of heliozoan tubulins See Luduena.

P. Stable and unstable populations of MTs: Categorize heliozoan axonemal MTs, recall possible function of histones. See Tilney, Schliwa, Matsuoka et al. (1984b), others on colchicine­ sensitivity, etc.

Q. See Pryer et al. (1992) on influence of MAPs on brain MT dynamic instability in vitro. Also see Gelfand and Bershadsky (1991).

VI. THE AXONEME

A. Function as armature (beam-column) for the axopodium

B. Hypothetical design(s) for this purpose

Why aren't axonemes simple circular palisades of MTs?

C. Diversity of actual axoneme patterns, classification of symmetries, comparison with hypothetical design(s) noted in B.

Summary of axoneme structure after Anderson (1988, pp. 242-243).

Actinophryid axoneme structure.

• Compare areal density of MTs with that for neurites and Allogromia reticulopodia (Travis and Bowser, 1991).

D. General principles/concepts of structure, stability, and strength.

Consider tapering, entasis (cf. Math. Intelligencer article) in relation to beam-column theory.

See esp. Ashby (1991), Parkhouse (1987), Wainwright et al. (1976).

Structure of axoneme such as to promote spread of damage?

E. How does the axoneme fail in contraction?

Does the structure of the axoneme promote spread of damage?

Does "intercalary destabilization" in the stalk and axopodial axonemes of Actinocoryne = shear band formation? (Febvre-Chevalier and Febvre, 1992)

Does evidence for stalk oscillation in Actinocoryne indicate possible modes of failure?

Can I make suggestions as to the statistical distribution of lengths of MT fragments immediately after contraction?

What predictions could be derived from a scaling arguIl1;ent comparing the stalk and axopodia of Actinocoryne? What, e.g., would an argument from geometric similarity suggest? Also, 65

consider the possible effects of parallel axonemes on the speed of contraction, etc. Recall that the stalk contains parallel axonemes.

Apart from the simple kinetic arguments adduced earlier, can the endwise disassembly model be debunked by treating it as a Stefan problem?

Vale's (1991) MT-severing factor or a hypothetical gelsolin-like protein (cutting MTs rather than F-actin) would seem to act too slowly. But would such a severing factor have to cut all of the protofilaments in order to be effective?

Is there evidence for or against the existence of periodic/intermittent breakpoints in the axoneme?

Do recent studies of AI/ogromia, axon MT assembly bear on the foregoing question?

Presumably the axoneme must fail reliably when stimulated appropriately. The axoneme may not be perfectly homogeneous; possibly there exists a distribution of breaking strengths along the axoneme or within a given section, whence global failure of the axoneme is governed by both deterministic and stochastic effects. See Gaines and Denny (1993).

Upper bound on speed of breakdown/contraction given by elastic wave speed in the axoneme?

VII. SPEED LIMITATIONS

A. Triggering of contraction, spread of excitation

Role of action potential [see Major's (1993) papers and refs. therein]' implications of space constant, comparison with other protozoans [see Febvre-Chevalier et al. (1989) review].

Electrical implications ofaxopodial tapering?

Electrical phenomena associated with extrusome discharge and contractile element actuation?

Diffusion of calcium ions from calcium channel pores (Barritt, A.V. Hill, many others).

Calcium waves probably are too slow.

B. Scaling relations for contraction

Clues from the stalk of Actinocoryne?

C. Speed limitations inherent in hypothetical breakdown schemes (e.g., series, parallel, etc.)

Upper bound on speed of breakdown/contraction given by elastic wave speed in the axoneme?

VIII. SQME CONTRACTION MECHANISMS THAT QAli BE RULED Q!IT. QR SHOWN I.Q BE IMPROBABLE

A. Actomyosin 66

Are Davidson's (1975) arguments against this valid? (But see Edds, Febvre-Chevalier.)

Cf. speeds of cytoplasmic streaming in characean algae.

B. Dynein, other motor molecules

Cf. speed of MT extrusion from ciliary and flagellar axonemes, etc. Also see Amos and Amos (1991), Peskin et al. (1993) for calculation of max. Brownian ratchet speeds, other data?

Cf. Sheetz et al. (1992) on motor-induced retraction of actin-supported filopodia.

C. Marangoni effect

See Probstein (1986) and Edwards et al. (1992).

D. Membrane tension

See Manton (1964, 1968), Davidson (1975), Peskin et al. (1993), Deiner et al. (1993), others.

[Note that the foregoing schemes in C and D presuppose prior or concomitant disassembly of MTs.]

E. Myonemes

There is no ultrastructural evidence for myonemes of acantharian, peritrich form; however, I should inquire into conditions for satisfactory preservation. Note that the rowing movements of the axopodia of Sticholonche are produced by contraction/relaxation of proximal nonactin filaments.

IX. THE CHEMOMECHANICS OF THE CONTRACTILE ELEMENT

A. Evidence for a centrin/spasmin-like contractile element

Fluorescent anti-centrin antibody binding in the pedinellid Pteridomonas.

B. Evidence against a centrin/spasmin-like contractile element

C. Phase transition in a gel

Cf. papers by Tanaka, Katchalsky, others on gels.

Cf. work of Nanavati (Duke postdoc.) on secretory granule matrix. Note voltage-dependence of swelling.

Cf. research on gels as active materials.

X. EXTERNAL HYDRODYNAMICS

A. Impulsively started flow at low Reynolds number? What is the characteristic time to achieve steady speed?

XI. INTERNAL HYDRODYNAMICS 67

A. Possible relevance of lubrication theory

B. Pressure-driven flow at low Reynolds number

Is the pressure gradient required unreasonably large? See ref. cited in Spero (1982).

Cf. possible piston-like action of euglenid, other protozoan ingestion devices.

C. Transient discharge at low Reynolds number in tubes of small bore

D. Steady low Reynolds number flow in tubes of small bore

XII. COMPARISONS WITH AND CLUES FROM OTHER PROTOZOA, MT-SUPPORTED STRUCTURES, CELL PROTRUSIONS

A. Cf. heliozoan axoneme patterns with putative diffusion-limited aggregation (DLA), other patterns in axonemes of radiolaria and acantharia

Mechanical and morphogenetic implications of tessellations vs. DLA?

See Ashby, Dyson, Lakes, Mandelbrot, Schroeder, others on scaling. Connection between Dyson's scaling relation and scaling relations for strength of honeycombs, other hierarchical structures?

Cf. handedness of actinophryid heliozoan axonemes, radiolarian axonemes

B. Suctorian feeding and prehensile tentacles

C. Piston of Erythropsis

D. Haptonemata

E. Euglenid nemadesms and subpellicular MTs

F. Axostyle MTs

G. Importance of "extracellular matrix" in reinforcing reticulopodia of Astrammina

XIII. MISCELLANEOUS

A. Axopodia as "nanosystems" (sensu Drexler, 1992)

B. Scaling of prey capture in heliozoan "diffusion" feeding

See Fenchel's (1984) analysis of the scaling of prey interception in bacterivorous marine heliozoans.

C. Stirring of medium by axopodia

Verify Suzaki et al. (1980) observations of spontaneous axopodial contraction lD Echinosphaerium akamae.

See Anderson's (1983) observations of spontaneous axopodial contraction in the spumellarian 68

radiolarian Spongodrymus.

See Purcell (1978) for theory on the efficacy of stirring at low Reynolds number.

D. Time course of contraction in Heterophrys marina cannot be deduced from Davidson's (1975) observations of haptocyst motion.

E. A physicist's view of general aspects of fast axopodial contraction

XIV. DISCUSSION

A. Probably only a small fraction of the organism's energy budget is spent on contraction

Give rough calclulations and recall that microorganisms spend relatively little energy on motility.

Energy budgets of protozoa are ably treated by Fenchel (1987, pp. 55-56).

B. Slow reextension compared to other protozoa

Cf. longitudinal flagellum of Ceratium tripos: 1-5 s [Hohfeld et at. (1988) citing Maruyama (1981)]; Stentor cell body reextension: 10-60 s [Huang and Mazia (1975, p. 403)].

Cf. vorticellids, suctorians, other ciliates.

C. Size-dependence, scaling of reextension times

Note that the low speed of reextension probably entails negligible drag.

Consider work to extend the axopodial membrane, diffusive/convective transport of materials, theory of length control devised for rodlike viruses [see Peskin et at. (1993); Nossal and Lecar (1991)].

D. Outstanding questions and decisive experiments for their resolution

E. Concluding summary, prospectus 69

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